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Full text of "An Introduction Applied Optics Vol II"

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AN INTRODUCTION TO 

APPLIED OPTICS 




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AN INTRODUCTION TO 

APPLIED OPTICS 



BY 

L. C. MARTIN 

D.Sc , A R CS , D.T C 

ASSISTANT PROFESSOR OF TECHNICAL OPTICS, IMPERIAL 

COLLEGE Or SCIENCE AND II CIINOLOGY, LONDON 
KtALJLK IN TFUINllAL OPTICS IN THh. I'NIVFR-sIIV OP LONDON 




VOLUME II 

THEORY AND CONSTRUCTION 
OF INSTRUMENTS 



LONDON 

SIR ISAAC PITMAN & SONS, LTD. 

1932 



SIR ISAAC PITMAN ONS, LTD. 

PARKFR STRLET, KINGSWAY.l ONDON, W C 2 

IHfc PITMAN PRbSS/BAlH 

FHF KIALTO, COLLINS STREBT, MELBOl KM 
2 WEST 45TH STRLtT,EW YORK 

SIR ISAAC PITMAN & SONS (CANADA), Ln>. 

70 BOND STRttr, TORONTO 



PRINTED IN GREAT BRITAIN 
AT THE PITMAN PRESS, BATH 



PREFACE 

THK author has been encouraged by the kind reception given to 
Volume I of this Introduction to Applied Optics to proceed with the 
preparation of Volume II in which the fundamental optical instru- 
ments are described. There are, of. course, many instruments, of 
utility and interest, other thaji th^^lT) which attention is given in 
those pages, notably vaffrm$j9j^RiTTfic instruments, such as spectro- 
scopes, refractonrujterS, interferometers, and the like, bu^these have 
generally been dealt with quite adequately in other bodks, whereas 
the treatment usually given to the commoner opticalHnstruments, 
oven in optical treatises, is of the slightest description. Even where 
details of construction have been furnished, the material has some- 
times been more appropriate for an optician's catalogue than a 
text-book. It may be that a need will be felt at some future time 
for a book dealing with the optical instruments used in physical 
measurements, but it seemed wiser at present not to re-traverse 
ground which lias been fairly well surveyed by others. 

There are some instruments, like the photographic lens, for which 
no adequate theory can be presented. The refinements of their 
construction can only be understood by those who have been 
successful in the design of such systems. Even so, the result of the 
design is only partially under control. In other cases there are 
many interesting avenues which have been very inadequately 
explored, and into which the writer has ventured some short excur- 
sions, as in the theory of the microscope. Such is the complexity of 
even this limited subject, that one person's experience cannot 
adequately cover the whole range of topics, and previous authority 
has perforce been followed in many respects. In spite of the numer- 
ous shortcomings of the book, it is hoped that the reader may be 
led to take a more critical interest in his instruments and thus 
obtain from them the best performance of which they are capable. 
He is urged especially not to take facile and second-hand opinions 

vii 



viii PREFACE 

about the productions of this maker or that, but to give instruments 
an intelligent trial where selection has to be made. 

As in Volume I, the writer is greatly indebted to the work of 
Professor A. E. Conrady, whose book on Applied Optics and Optical 
Design may be commended to those who would go more deeply 
into the subject. He has also received great help from Dr. W. D. 
Wright, who has been good enough to read the proof sheets of this 
volume. 

One word may be added both as a tribute to optical instrument 
makers and a warning to those who use their products. Treat your 
instruments, and especially their optical surfaces, with the greatest 
respect. Three or four months of intricate calculation may have 
preceded the production of your oil immersion microscope objective. 
Its making involves no less important painstaking and delicate 
work, by operatives of the greatest skill; its surfaces are polished 
true within a fraction of a wave-length of light. A good lens is as a 
jewel to its owner. 

Acknowledgments and thanks are due to the following for the 
loan of blocks and permission to reproduce illustrations : Messrs. C. 
Baker (Figs. 8 and 124), Messrs. R. and J. Beck, Ltd. (Figs. 89 and 
125), the School of Optics, Ltd. (Fig. 103), Messrs. Reichert of 
Vienna (Fig. 129), and Messrs. Carl Zeiss (London), Ltd. (Figs. 64, 
88, 121, and 161). Fig. 196 has been reproduced by the kind per- 
mission of Dr. Felix Jentsch. The frontispiece is due to Sir Howard 
Grubb, Parsons, and Co., and the plate in the chapter on "The 
Microscope" to Messrs. Blackie & Son, Ltd., being reproduced from 
Practical Microscopy, by Martin & Johnson. 

L. C. MARTIN. 

IMPERIAL COLLEGE OF SCIENCE 
AND TECHNOLOGY, 
1932. 



CONTENTS 



CHAP. PAGE 

ONE-METKE REFLECTOR FOR SIMEIS OBSERVATORY, SOUTH 

RUSSIA ...... Frontispiece 

PREFACE ......... Vii 

I. THE MAGNIFICATION PRODUCED BY LENSES THE SIMPLE 

MICROSCOPE I 

II. THE TELESCOPE 1 8 

III. THE MICROSCOPE ....... 75 

IV. BINOCULAR VISION AND BINOCULAR INSTRUMENTS . . 14! 
V. PHOTOGRAPHIC LENSES l66 

VI. THE PHOTOMETRY OF OPTICAL SYSTEMS AND THE PROJEC- 
TION OF IMAGES . . ..... 206 

VII. THE TESTING OF OPTICAL INSTRUMENTS. . . . 24! 



APPENDIX I 

OPTICAL CONVENTIONS AND EQUATIONS . . . .27! 

APPENDIX II 

THEORY OF THE DIFFRACTION GRATING .... 275 

APPENDIX III 

ASTIGMATISM OF A LENS SYSTEM ..... 285 
INDEX 287 

INSET 

MODERN MICROSCOPE STAND BY SWIFT . . facing p. 76 



ix 



AN INTRODUCTION TO 

APPLIED OPTICS 



CHAPTER I 

THE MAGNIFICATION PRODUCED BY LENSES THE 
SIMPLE MICROSCOPE 

IN the first volume of this work, the term magnification has been 
restricted to mean the ratio of a linear dimension of the image to 
the corresponding size of the object ; thus the formulae 

^--L-^-rt 

h~ x~~ f ~ 

express the lateral magnification measured perpendicular to the axis, 
while the equation 

dx' __ A 

dx ~~~ \ h 

gives the longitudinal magnification, or the ratio of image to object 
size measured in the axial direction, close to a particular pair of 

JL / 

points where the lateral magnification is -7-; it is assumed, of 

n 

course, that the image exists in three dimensions. 

The Lagrange relation permits the derivation of an expression 
for magnification in terms of- the distances of object and image 
from any pair of conjugate points for which the magnification is 
known. Thus in Fig. i we might take B and B', P and P' to be 
any two pairs of conjugate points. Let PB = / and P'B' = /'. Let 
BBj = A! and B'B/ = h^ where BB! and B'B/ are a small object 
and image both perpendicular to the axis. We will apply the 
Lagrange relation to the imagery at the points P and P', consider- 
ing the ray BjP and its emergent path P'B/. The ordinary formula 
is 

nhoy = n'h'a*' 

h and h' are now the sizes of a small object and image at P and 
P' respectively, while co and co' are the angles with the axis made 



2 APPLIED OPTICS 

by the rays B X P and P'B/ ; but co = - y and a/ = - --. Hence 

l l 



or 



* 



*'*'/ 



In a particular case of importance, P and P' may represent the 
axial points of the entrance and exit pupils ; the ray passing through 
them then becomes the "principal ray" of the bundle entering the 

instrument. In this case the ratio -p above may be written ,, 

ft // 

where p and />' are the radii of the pupils ; and / and /' can be written 
q and q', using these symbols to denote the distances of object 








2 7 











*B' 




^ 


U) f 


"^H/ 



FIG. i 

and image from entrance and exit pupils respectively. The formula 
then becomes 

V *# 



The great majority of optical instruments arc, however, used as 
direct aids to vision, and the conception of the " magnifying power" 
of an instrument is then 

Magnifying power 

Size of retinal image obtained with the aid of the instrument 
~~ Size of image obtained with the unaided eye 

As explained in Vol. I, it has been found that the accommoda- 
tion of the eye is largely effected by the variation in curvature of 
the lens. In consequence of this the distance of the principal and 
nodal points from the retina varies very little, in fact less than half 
a millimetre in changing the accommodation from distance vision 
to near vision. The " stop " which limits the bundle of rays arriving 
at the retina may sometimes be the pupil of the eye itself, and 
sometimes it may be the exit pupil of some instrument projected 
into the same approximate position. In most cases the action of 



MAGNIFICATION PRODUCED BY LENSES 3 

the accommodation will secure the sharpness of the image, but, if 
not, the image position will be assumed to be defined by the inter- 
section of the principal ray with the retina. Further, since the eye 
entrance pupil and first principal point are only separated by about 
one or two millimetres, the principal ray may be regarded as that 
one passing through the principal points. 

Refer again to Fig. i, and let P and P' now represent the prin- 
cipal points of the eye; the object EB l subtends an angle a> at the 
first principal point P. The ray makes an angle a/ with the axis 
after refraction. The Lagrange relation applied to the principal 
points where the magnification is unity gives 

no) = n'a)' 

if the discussion is limited to paraxial conditions. 

As mentioned above, the size of the image perpendicular to the 
axis may be defined by the intersection of the principal ray with 
the image plane. If this plane is at a distance V from the second 
principal point, we may then write for the reasons given above, 

A/ = -/V = -/'-, 
n 

the size of the image depending mainly on the angle subtended by 
the object at the first principal point, even allowing for variations 
in accommodation. While the angles are small we can deal with 
their angular measure, but when the angles are large, and the 
images are measured on flat screens, we shall have to deal with 
their "tangents." Hence, the above equation for magnifying power 
can be written for small angles, at least 

Magnifying power 

___ Angular subtense of image obtained with the instrument 
~~ Angular subtense of object seen with the unaided eye 

Magnifying Power of an Optical Instrument. It is always neces- 
sary to consider the state of accommodation or "refraction" of the 
eye when dealing with the magnification produced by an optical 
instrument. The condition is conveniently specified by giving the 
position of the accommodation point on which the eye is focused. 
Call this point M (conjugate to the macula M'), and let its distance 
from the first principal point of the eye be k. The condition of 
affairs might be as suggested in Fig. 2, in which "a" represents 
the optical system, and "6" is the system of the eye. 

The object is situated at some point B, and the lens projects an 
image of height h' into the accommodation point of the eye. 



4 APPLIED OPTICS 

The angle on subtended by the image at the first principal point 
of the eye is 

h' 



In order to investigate the dependence of co on the size of the 
object, this equation can be written 

W\i 



-6)1 



)' Vol. 



where x' is the distance F' M. 



a/ 







Cf 


/*;_ 


*""'* I-/ 


B & 






.-- -~~ % f ~""^& 


1 


H 




V 


^ JC 


'--i 






i 


^- I- 





FIG. 2 



Since from Fig. 2, 



CO ~ 



. 

/'a 



In order to see the object distinctly with the unaided eye in the 
same state of accommodation, it will be necessary to place it in 
the point of accommodation, M. This would, of course, be impos- 
sible for a real object with the case shown in Fig. 2, but it would 
be possible in all cases where k is numerically negative. 

When placed at M, the angle subtended at the first principal 
point of the eye by the object is given by 



co = - T = - 



MAGNIFICATION PRODUCED BY LENSES 
Hence magnification 



~ -h/t 



^!-./ a (k + d) ..... (3) 

If /' is the distance of the accommodation point from the -second 
principal point of the lens system, then /' k + d, and the formula 
takes the form in which it is often written 

/' 
Magnification i - j- . . . . . (4) 

In the case of a small lens of high power, it will usually be held 
close to the eye, and the distance of the point of accommodation 
will usually be the "least distance of distinct vision." As has been 
seen from the table in Chapter VIII, Vol. I, this distance varies. 
A mean value for adult vision may be taken as - 250 mm. (or 
about - 10 in.). The magnification then becomes (/' in millimetres) 



or sufficiently nearly, if/' is small, 
Magnification --O-^S./T- -- . . . (5) 

where . / is the power of the lens in dioptres. 

Spectacle Magnification. The user of a pair of spectacles will, 
however, judge "magnification" rather differently in that he will 
compare the apparent size of the more or less indistinct image 
seen with the unaided eye, and the size of the image seen with his 
spectacles, the object remaining fixed. 

From above, the angular subtense of the image will be 



We now calculate the position of the real object corresponding to 
the image position at M, where it must be formed to be seen dis- 
tinctly by the eye. 
The usual formula, 

i ii 

77 T r= 7T 
1 a l o, Jo, 

f V I' 

gives l a fT^-jr ~ - JT "-> omitting the suffix from ./' 

J a~~ I a I ""^ a'S 



6 APPLIED OPTICS 

Let the interval P P' a = t, then the distance of the object from the 
rst principal point of the eye is given by 



If the thickness t be neglected, as it is usually small in spectacles, 
and remembering that l' a = d + k, we have 



And the angular subtense of the object is therefore 
h ( i-(J + ft).; ) 

l*~ n (k + (d-\-k)d.)} 



i.e. -y - i 

angular subtense of image 

Hence magnification - 

angular subtense of object 

- h ] T~7T7' " JL~ /* " ( 

This expression is quite general for any thin lens and any state 
of accommodation, provided that a sharp image is obtained with 
the aid of the spectacle lens. The distance of the lens from the 
eye is not restricted. 

EXAMPLE I. 

Take // = - 4D, . / = 61), and d = i-o (metres) 
Then magnification == i -f- (- 3) 6 



This shows that the lens is forming an inverted image between 
the lens and the eye; and the angular subtense of this inverted 
image is considerably greater than that of the object seen directly. 
The distance of the object from the lens proves to be - 21-4 cm., 
while the distance of the inverted image is 75 cm. from the lens, 

h f 
and -25 cm. from the eye. Thus -r for the lens = -3-5. From 



AN INTRODUCTION TO APPLIED OPTICS. VOL. I 

ERRATA 

Page 35, line 6. Equation to read: J^' = J7, +- .5^. 
Page 47, line 21. Jf k should read J7"* 
Page 49, line i. (Second equation) denominator to be 
i - d 7/. _ _ 

Page 49, line 16. i/ / should be i/ / '. 

Page 54, line 10. Read y l - . n'a^. 

Page 63, paragraph on "Gaussian Constants/' line 3 
of paragraph. Replace C by I//J. 

Page 63, paragraph on "Gaussian Constants," line 4 
of paragraph. Read : h'jh = C and D = - /C. 

T / 

Pagt* 94, line 13. should read . 

/ i 

Pago 102, Inscription to Fig. 64 should be as follows: 

Km. 64. KKHKGV DISTRIBUTION IN THI-: SPECTRA 
OK VARIOUS ILLUMINANTS 

A. Blur sky D. Low sun (Smithsonian) 

1). High siiii (ctita frcitii Smithsonian Instn.) K. Gas fillet 1 tungsten lamp 

(.'. Ives su^^rst^l standard (black 1 ody at 5^00") F. Acetylene flamr- 



Page 114, line 17. '-should read . 

Page 147, line 21. .ft should read . s t . 

Page 184, equation at foot + " should read : 

P.'tgc 230, line 5. ( W A ) should read (n D - u c ). 

305, line- ft. spherical aberration should read 



Page 316, ecjuation (/). Missing letter in denominator 
is '/.' 



(5194) 



MAGNIFICATION PRODUCED BY LENSES 7 

these figures the above value for the magnification is easily checked. 
The student should make a sketch of the arrangement. 

EXAMPLE II. 

Take /'=-4D, and ./^6D (as above), but let d now be 
equal to o-oi metre. Then magnification - 1-06, by the formula. 

The image is now virtual and erect, and the lens is being used 
close to the eye as an ordinary magnifier. The low value for the 
"magnification" is due to the fact that we are comparing the 
angular subtense of the image seen sharply with the lens, and that 
of the diffuse image which is scon without it. These angles are not 
greatly different in magnitude. 

EXAMPLE III. 

Take /' = -\- o5D f ./"= -|- o-2D, d 3-0 (metres). 
In this case the magnification works out to + 2-50. We therefore 
should obtain an erect image of moderate magnification. 

The system is made use of in the so-called "window telescope, 11 
made by some opticians, which consists of a large single lens of 
low power made to hang 011 the window frame. Most persons of 
normal vision can manage to relax the accommodation far enough 
to bring the refraction of the eye to a small positive value, and the 
combination of lens and eye then produces a system having a focal 
length considerably longer than that of the eye alone. An erect, 
magnified image is the result. 

Magnification given by Spectacles giving Distance Correction. 
The above expression for magnification takes a simple form in the 
case where the "glass" employed is suitable for distance correc- 
tion. In that case we have the simple relations 

/' ^ W ' and - ; =~ ;*rV^ ( See Vo1 - r > P- 2f) 5) 

I ./ i-j-a/c 

By a simple substitution the expression for the magnification 
can then be put into the simple forms 

Magnification = - ~ i + d/t .... (7) 

We see at once that a positive correcting lens produces a magnifica- 
tion greater than unity, while a negative lens as used in myopia 
produces a magnification less than unity sometimes called a 
"minification." 

Single Lenses as Magnifiers. The use of single lenses as magni- 
fiers has been known since very early times. The Greeks were 

n * (5494) 



8 APPLIED OPTICS 

familiar with the effects of refraction of light at curved surfaces, 
such as might be studied with solid balls of glass or rock crystal, 
and the perfection of certain antique handwork seems to call for 
the explanation that it was produced with the aid of a n?agnifier. 

Antony van Leeuwenhoek, born at Delft in 1632, made great 
advances in grinding and polishing small lenses for magnifiers, 
and obtained magnifications up to 160. He observed Infusoria and 
Bacteria for the first time. In 1702 J. Wilson produced a pocket 
microscope with which, it is said, magnifications up to 400 could be 
obtained. The convenient mechanical construction of this instru- 
ment ensured its popularity for quite 100 yearsT 

We may distinguish two main functions of single-lens magnifiers. 
First a lens may be used, as in one side of a stereoscope, to obtain 
a general view of a large object or picture at a suitable angle. 
Secondly, a lens may be employed to obtain an enlarged image of 
a very small portion of an object. Naturally the optical arrange- 
ments differ in the two cases. 

Lenses for Viewing Pictures. The first case above brings us 
to considerations very similar to those encountered in the dis- 
cussion of spectacle lenses. The eye turns in its socket in order to 
view the different parts of the object, and the pencils of light 
reaching the retina under different angles have an eccentric passage 
through the lens. The conditions are optically similar to the case 
in which a small stop is situated in the centre of rotation of the eye 
(Fig. 3). The conditions have already been discussed in connec- 
tion with high power spectacle lenses. It was shown in that con- 
nection that one of the chief troubles arising was the astigmatism 
of the oblique pencils. This was shown to be diminished by the 
choice of a lens of suitable figure, generally of meniscus type with 
the concavity towards the eye, but correction could not be given 
beyond a power of about loD and higher without the employment 
of aspherical surfaces. The ellipse of Fig. 188, Vol. I, will provide 
approximate data for the radii required for various powers, but the 
spectacle lens was intended to form images of distant objects, and 
the results do not hold exactly for near objects. Calculations for 
near vision have been made by Whit well. 1 

If it is sought to view a near flat object with such a lens, several 
additional defects in the image are at once noticed. 

i. Roundness of the Field. The exterior parts of the object are 
farther away from the lens than the centre. Even if the lens has 
been freed from "oblique astigmatism/' the object field will not be 
seen in focus as the eye is moved unless it has the curvature of the 
Petzval surface, of radius nf for a single thin lens. With an ordinary 



MAGNIFICATION PRODUCED BY LENSES 



double convex lens, the presence of astigmatism makes matters 
very much worse. If the object is flat we can first bring radial, 
then tangential, lines to a focus in the outer parts of the field by 
bringing the object closer to the lens. 

2. Chromatic Aberration. On tracing a principal ray through the 
system of Fig. 3, it will at once be noticed that considerable lateral 
chromatic aberration must arise through the passage of the pencils 
of light through the outer parts of the single lens. If the object 
consists of a number of small bright points on a dark ground (pin- 
holes in a thin card illuminated from behind) each object point in 



Object 





Position of 
effective 'Stop 1 

b 
r 



FIG. 3. USE OF A SIMPLE MAGNIFIER WITH ROTATING EYE 



the outer part of the field is imaged as a short radial "spectrum," 
the red being innermost. 

In the case of lenses required to give a general view of objects or 
pictures, they are not often required to be of shorter focal length 
than about 5 cm., and, as will be shown below, the spherical aberra- 
tion of the- pencils which are transmitted by the pupil of the eye 
(which will probably not exceed 4 mm. in diameter during the day) 
does not produce any appreciable deterioration of the image. 

3. Distortion, of the pin-cushion variety, is noticeable with a 
single lens. The bending of the rays increases too rapidly in the 
outer parts of the field, and the magnification thus increases with 
the distance from the centre. 

In Vol. I, Chapter I, the question of the proper presentation of 
a perspective projection was discussed, and it was shown that a 
proper judgment of the space values of the picture can only be 
obtained if the latter is seen under the proper angle. 

The majority of pictures obtained by small cameras are made 
with lenses of considerably shorter focal length than the least dis- 
tance of distinct vision ; hence, if the print is to be seen with the 



10 



APPLIED OPTICS 



unaided eye, it must be held at too great a distance to attain the 
proper angle. Matters could be corrected, actually, by placing the 
projection (or print) back in the camera in which the photograph 
was made, illuminating it in some way, and then looking at it 
through the camera lens. 

This, however, would be awkward because the camera lens may 
be provided with a stop through which the eye must look. It 
would be necessary to move the head about/ This type of "keyhole 
observation" is not convenient as we could not get a good view of 
the picture as a whole. Hence, although the perspective conditions 



Centre of 

Rotation 

oFye(B) 




Front Lens: boro-silic.ite crown 
(0*144, J<* na glass catalogue) 

Hack lens : silicate flint 

(0*118, Jena glass catalogue) 

FIG. 4< 



THE "VERANT" LKNS 



require that the viewing lens shall have the same focal length as 
the camera lens, it is desirable that the former shall be corrected 
for use with an (imaginary) stop situated in the centre of rotation 
of the eye when properly positioned with respect to the system. 

The best -known lens of this type is the "Verant," designed by 
M. von Rohr and made by Zeiss. 2 It corrects chromatic aberra- 
tion of the above type, and distortion on the lines laid down by 
Gullstrand, and it is illustrated in Fig. 4. There is some residual 
astigmatism at certain angles of obliquity, but it is not large. The 
curvature of field can be allowed for by variation of the accommoda- 
tion of the eye. 

A lens of somewhat similar character but different construction, 
intended for use with a stereoscope, has been designed by Albada. 3 

The Magnifiers of Wollaston, Brewster, and Coddington. Where 
small lenses were required to obtain a fairly great magnification, 
and yet to give a general view of an object comparable in size with 
the diameter of the lens itself, a device introduced by Wollaston 
in 1812 proved a great improvement on the hitherto prevalent use 
of a small complete sphere of glass; see Fig. 5 (a). Wollaston em- 
ployed two hemispheres of glass mounted together with a small 



MAGNIFICATION PRODUCED BY LENSES 



ii 



stop between them. This was improved by Brewster who cut a 
saddle-shaped groove in a complete sphere, thus obtaining a limita- 
tion of the pencils by a "stop" effectively situated at the centre 
of the sphere. The Coddington lens is similar except in the manner 
of cutting the sphere, which is illustrated in Fig. 5 (c). 

The idea in all cases remains the same. It is evident from Fig. 
b that pencils of light refracted through the centre from any point 
on a spherical object surface? concentric with the sphere must have 
similar optical treatment by the system. In every case the prin- 





(a) Wollaston Magnifier 



(b) Stanhope Lens. 

en 

(c) Coddington Lens. 



cipal ray through the centre of the stop suffers no deviation, and 
the image is therefore free from chromatic aberration of the above 
type, coma, astigmatism, and distortion. 

In cases where the object itself is fairly small, i.e. of the order 
of a few millimetres in diameter, then the use of a magnifier of the 
above type, which should be held close to the eye, would produce a 
retinal image of a part of such a curved object, the part being 
comparable in size to the pupil, but naturally the object itself 
will not, in general, be contained in such a curved surface as shown 
in Fig. 6. This means that cither the object should be moved rela- 
tively to the lens or a variation of accommodation must come into 
play. The "roundness of field 1 ' is naturally extremely marked, and 
severely limits the region of the object which is seen sharply in 
focus at the same time. In the Stanhope lens, Fig. 5 (&) the object 
is intended to be placed on the front curved surface, the central 
point A lying at the principal focus of the back surface. For glass 
of refractive index 1-5, this calls for a radius of one-third of the 
thickness of the lens. The front surface through A is struck about 



12 APPLIED OPTICS 

the same centre C, and complete freedom from chromatic difference 
of magnification, coma, astigmatism, and distortion is therefore 
secured as far as the lens itself is concerned. These lenses were, at 
one time, frequently mounted with very small photographs in pencils 
and the like. The front surface was often made flat, although some 
advantages were lost. The system still has value for examining 
small organisms, etc., which can be placed* in contact with the 
front surface. 

Nature of the Image to be Presented to the Eye. In spite of the 
fact that the human eye is admittedly subject to chromatic aberra- 
tion in the sense of " undercorrection " characteristic of an ordinary 
lens, and also that it suffers from zonal spherical aberration, it 
has been shown in the chapter on the eye, Vol. I, that the acuity 
of normal human vision for the small pupilary diameters charac- 
teristic of day-time is little lower than the limits set by the wave- 
nature of light, even for a perfect optical system. 

The vision of actual objects, either at a great distance, or at the 
near point, manifests no trace of coloured fringes or haziness due 
to spherical aberration. It is, indeed, just conceivable that the 
mental receiving apparatus has some means of automatic compensa- 
tion, which causes the brain to interpret a particular distribution 
of light (in regard to colour) as characteristic of an elementary 
point. However that may be, it has always been judged best 
to design any optical instrument so that the image shall be as per- 
fect as possible, in the physical sense, as it is presented to the eye. 
The defects of vision can then have no worse effect on the appear- 
ance of this image than on the appearance of a real object. 

A few attempts have, indeed, been made to design systems which 
should compensate the chromatic aberration of the eye, and secure 
a greater concentration on the retina. It is possible that systematic 
research on such lines might yield results of interest, but so far 
nothing has been done which has led to any departure from the 
general rule of making the image as physically perfect as possible. 

The Simple Microscope. In contra-distinction from the lens re- 
quired for giving a general view of an object, it is frequently required 
to obtain a greatly enlarged view of a small area small, perhaps, 
in comparison with the diameter of the lens itself. 

It will be shown in the chapter on the microscope that the power 
of the system to yield very sharp images of small objects depends 
upon the angular divergence of the cone of rays derived from the 
object, and brought without appreciable aberration to the retinal 
focus. In Fig. 6, the cone has a small angular diameter w t and this 
could be enlarged by increasing the size of the central stop. On 



MAGNIFICATION PRODUCED BY LENSES 13 

the other hand, this will rapidly increase the spherical aberration 
arising through refraction at each surface, and the effects of chro- 
matic aberration will also become serious. The chromatic aberra- 
tion here referred to is the difference in the focusing position of 
different colours measured along the axis of symmetry of a bundle ; 
thus if the green is focused on the retina, the other spectral com- 
ponents will be represented by blur patches of lessened concentra- 
tion. It is to be noted that this chromatic aberration will be in 




FIG. 



the same sense as that of the eye itself, the shorter wave-lengths 
being focused nearer the lens. 

In order to obtain a perfect image, as projected by the lens, it 
may be required that rays diverging from the object point shall 
all be rendered parallel after leaving the lens, i.e. that the wave- 
fronts shall be plane on entering the eye. The case is then just the 
reverse of that when a lens is to form a sharp image of a very 
distant object. The numerical amounts of the optical path differ- 
ences are therefore the same. 

When we considered the case of primary spherical aberration, the 
maximum residual optical path differences arising at the best focus 
position (midway between marginal and paraxial foci) were shown 
to be one-quarter of the optical path difference between marginal 
and paraxial rays arising at the paraxial focus. The formula (Vol. 
I, Chapter IV) for this latter was 

v 4 



14 APPLIED OPTICS 

so that the residual differences at the best focus of a lens 
exhibiting this primary spherical aberration will be 

v 4 
AA = -A 

The coefficient A was found to be 



In the case where a lens receives parallel light, '/j r- o. Suppose, 
for simplicity, that the lens is plano-convex, with the curved sur- 
face turned towards the incident parallel light. Then '/ 2 = o, and 



Hence A (the sign need not concern us here) , 

-J n Y -2 ' ; L/?!L^ . - 
~'' \n-ij '' 'n-iV -TV +> ' 

* V 2 ^ + 1 ) 
M-I; (n-i) 



As was mentioned above, the path residuals arising in such a 
case will be the same as those which arise when a small object is 
situated at the "best focus" of the lens on the piano side, and 
gives approximately parallel light to a viewing eye. 

Hence, the expression for the optical path residuals becomes 



OPD = 

n-i 



Take the allowable OPD == - - -00013 mm., and v ---- 2 mm. 

4 

(say), allowing for an average pupilary diameter of 4-0 mm. 
Then a quick calculation made with an assumed value of n == 1-5 
gives 

/' = iG-i mm. 

This is the shortest focal length allowable for a good image on the 

250 
above criterion. The magnification = - = 15 approximately. 



MAGNIFICATION PRODUCED BY LENSES 15 

If the lens is reversed so that the curved side is towards the 
object, the conditions are less favourable as far as spherical aberra- 
tion is concerned, and the focal length has then to be increased to 
about double the above value if the aberrations are to be kept 
sufficiently small; the allowable magnification therefore sinks to 
about eight times. 

In practice, however, the above limits may be somewhat ex- 
ceeded before a really marked deterioration of the image begins to 
be manifest. 

In order to test the above conclusions it is interesting to take a 
piano convex lens, of about i cm. radius, and to use it as a magnifier 
held as close as possible to the eye. 

When the plane face is turned towards the eye, the field is of 
fair extent, but the images of bright spots have 
haloes. When the lens is reversed so that the 
plane face is towards the object, the field is 
more restricted, but the contrast at the centre is 
perceptibly improved, the haloes being reduced. 
The test is best made on a number of pinholes 
in a card held up to the light. 

Theoretical or practical tests soon show that 
a lens of 15 mm. focal length will have fairly 
pronounced chromatic aberration on the axis ; 
this results in the presence of coloured haloes in 
the image of a very small source of light, such as 
a pinhole, and the colour is independent as to 
whether the lens is held with the curved or the 
flat side towards the object. 

The " Steinheil " Magnifier. Spherical and 
chromatic aberration can be eliminated by using 
a biconvex crown-glass lens between two menisci of flint. The 
lens is designed after Steinheil's "aplanatic" magnifier or "lupe." 

The symmetrical shape of the lens would tend to give it freedom 
from the aberrations of oblique pencils with regard to principal 
rays passing through the centre, somewhat as in the Coddington 
lens. When, however, the aberrations are reckoned with respect to 
a stop outside the lens represented by the pupil of the eye, or an 
imaginary stop at the centre of rotation of the eye-ball, they may 
not appear in such a good light. The lens is shown in Fig. 7. The 
magnification is about x 6. 

This lens has the great advantage of symmetry, so that if it is 
mounted in a pocket-holder it can be used either way round with 
equal advantage. 




FIG 7 STEINHEIL 

TYPE APLANATIC 

MAGNIFIER 

Th<- triple lens form 
most frequently em- 
ployed in modern 
pot ket magnifiers for 
ma <m nrat ions up to 

v 20* 



16 APPLIED OPTICS 

Symmetry is a very real advantage in a small pocket magnifier ; 
in cases where it is necessary to obtain a compromise between the 
opposing claims of freedom from spherical aberration, and freedom 
from undue astigmatism of the oblique pencils, the double convex 
form of the lens may be frequently selected. 

Small Measuring Magnifiers. It is often very convenient to 
mount a small scale, engraved or photographfed on a thin disc of 
glass, in the focal plane of a simple magnifier. The instrument can 




(r baker, London) 
FIG. 8. DISSECTING STAND 

then often be placed so that the scale is in contact with the object 
to be measured. The illumination is secured by a thin plate of glass, 
placed between the lens and scale, by which light can be reflected 
down on to the object. 

Dissecting Stands. The majority of simple magnifiers are for 
hand use. They are preferably held quite close to the eye in order 
to secure the widest possible field in this " keyhole " type of observa- 
tion. For dissecting purposes, however, "aplanatic" magnifiers of 
somewhat longer focus are mounted in simple mechanical stands, 
one of which is shown in Fig. 8. 

In order to overcome the difficulty of the short working distance 
with the higher powers, Chevalier proposed, in iSjt), to place a con- 
cave achromatic lens above the magnifying glass. A convenient 
form of the arrangement is shown in Fig. 9 (a) ; this is known as the 



MAGNIFICATION PRODUCED BY LENSES 



Briicke lens, but is not often encountered ; it is essentially a com- 
bination of a Galilean telescope with a long focus microscope 
objective. A more modern arrangement is to employ a prism erect- 
ing telescope system, similar to one member of an ordinary field 
glass, in combination with a microscope objective as "front lens 
attachment." Such systems are very convenient for naturalists and 



Galilean 
Telescope 



-Magnifier 




Erecting 
Prismatic 
Telescope 



-Objective 
Front Lens 
Attachment 



(ajBrucke Lens 



(b) Telescope Magnifier Combination 



FIG. 



others. The use of the prism erecting system allows of a larger 
field than is possible with the Briicke system. The theory of the 
telescope systems will be dealt with in the following chapter. 

Such telescopic magnifiers are adaptable for binocular vision, 
sometimes with the aid of suitable achromatic prism systems. 

REFERENCES 

1. A. Whitwell: Reference must be made to a series of papers pub- 
lished in The Optician from about 1916 onwards. See also Kinsley and 
Swaine, Ophthalmic Lenses (Walton l^ess), p. 243. 

2. British Patent 24009 (1903). 

3. Albada: Trans. Opt. Soc., XXV (1923-24), 249. 



CHAPTER II 

THE TELESCOPE 

Historical. Roger Bacon (1216-1294) was* familiar with convex 
lenses, which were about this time beginning to be employed for 
spectacles. He states in his writings that 

" . . . we can give such figures to transparent bodies, and 
dispose them in such order with respect to the eye and the 
objects, that the rays shall be refracted and bent towards any 
place we please, so that we shall see the object near at hand, or 
at any distance under any angle we please. And thus from an 
incredible distance we may read the smallest letters/' 

The above words may be interpreted as a veiled allusion to a 
telescope. Undoubtedly, they contain the germ of the scientific 
idea of the telescope, which is usually an instrument to project 
an image, of a distant object, which can be viewed under a greater 
angle than is possible with the unaided eye. Medieval philoso- 
phers were not in the habit of giving very explicit descriptions of 
what discoveries they made, for the ever present danger of being 
suspected for witchcraft or necromancy was associated with the 
exhibition of any unfamiliar phenomena. 

Somewhat similar hints and allusions appear in the writings of 
Robert Recorde (1551), Battista Porta (1558), and others. The 
earliest circumstantial account of the actual construction of a tele- 
scope dates from 1590, when Zacharias Jansen, the son of a spectacle 
maker, Hans Jansen of Middelburg in Holland, is supposed to have 
invented the instrument. This testimony is due to the son of 
Zacharias. 

There was also, however, another spectacle maker, Hans 
Lippershey, in the same town. It appears certain that he was in 
possession of the invention in the year 1608, A popular story 
ascribes the invention to the children of these two spectacle makers ; 
they mounted a pair of lenses on a piece of wood in play, acci- 
dentally securing the correct distance between the lenses. However 
that may be, Lippershey undoubtedly pushed forward the invention 
and actually constructed binocular telescopes. 

In June, 1609, Galileo heard of the invention without knowing 
any details of the construction. He returned to his laboratory at 

18 



THE TELESCOPE 19 

Padua, and in one day had made his first erecting telescope by 
mounting a plano-convex and a plano-concave lens in a short 
leaden tube. Subsequently, he made a number of telescopes of 
greater length and increased magnification of which examples are 
preserved in the Museo di Fisica, Florence. 

Further historical points will best be given in the development 
of the theory of the instrument. 

Elementary Theory. In its simpler form the telescope consists of 
two lenses mounted coaxially, but the single lenses may be replaced 
by systems of greater complexity for various reasons, as will be 
seen. We will, however, treat the telescope as consisting essentially 
of two main parts: the objective (a), and the eyepiece (b). 

Imagine an object perpendicular to the axis which subtends a 
small angle a at the anterior principal focus F a of the objective. 
Having given this angle we know that, wherever the image formed 
by the objective may be, its height will be given by 

*'.=/. 
where f a is the first focal length of the objective. 

The eyepiece acts as a simple magnifier. Turning to equation (2), 



Objective 



Eyepiece 




OL 



Fir. 



Chapter I, we find that the angle under which the above image 
will be viewed (having given the power ./ b of the magnifier, the 
refraction /' of the eye, and the distance d between the adjacent 
principal points of magnifier and eye) will be 

Since, as before, 

Angular subtense of image 

Magnifying power . - . - , , ?"v- 

b J * l Angular subtense of object 

we may write (if a f) is the angular subtense of the object at the eye) 
Magnifying power of telescope = a - b ' 

This somewhat complex-looking expression becomes greatly simpli- 
fied if it is assumed that : (i) The object is sufficiently distant to 



20 APPLIED OPTICS 

make a and a sensibly equal, as is the case in the great majority 
of applications of the telescope. The expression then becomes 

Magnifying power = f a {.>' ft (i + d/t )-W} (8) 



The distance /' of the image from the second principal point of the 
eyepiece or magnifier is k + d I hence the corresponding distance / of the 
conjugate point for the magnifier system is given by 



k + d 1 ~" '* 

This gives the position of the intermediate image formed by the objective 
and viewed by the eyepiece. We take the case when it coincides with 



F' a . Then the distance 



In the usual case */' b = -/ & , so that 



The above equation gives 

i _ i ,-,, _ i - J^(k + d) 



r ' 

L 7 i -. '/' 



;{ /?-.;;(! t 

But the Gaussian "power" of the instrument is equal to 



Hence from equation (8) above, 

,,, . . . yft Principal point refraction of eye 

Magnifying power = = - ^ *- - j-r - 7 -- 
J r t > Power of the system 

The " Infinity Adjustment." Further, an important case arises 
in which the instrument is adjusted to present a sharp image to 
the normal unaccommodated eye. In this case /' ; = o. Hence 

Magnifying power f a .}' b 

Almost invariably, the objective is in air, so that/' a = -/ . Hence 

f 

Magnifying power = - 7- 

/ & 

The above formula is easily derived in the simplest case from inspec- 
tion of a Gaussian diagram. 



THE TELESCOPE 



21 



Fig. 10 shows the course of the rays in the telescope when a 
virtual image at a finite distance is presented to the eye. If, 
however, the refraction of the viewing eye is to be zero, it must 
receive parallel light. This case is shown in Fig. n, in which F' B 
and F 6 are coincident. A parallel bundle of rays enters the objec- 
tive, is focused in the common focal plane, and diverges to the 
eyepiece from whence it emerges once more parallel. 

Tracing one ray through F a it is seen to pass through F' & ; it is 

Stop to limit Field 
a, 

T U 

Exit 




FIG. ir. ASTRONOMICAL TELESCOPIC 
(Diagrammatic) 

evidently parallel to the axis between objective and eye. The angle 
between the axis and the incident bundle of rays is 

tan a -= a ? (numerically negative in the figure) 

The angle between the axis and the emergent bundle is 

I v O' 

tan a = u , b T vr (numerically positive in the figure) 

1 b 1 ' b 

The ratio of the tangents of the angles becomes equal to the ratio 
of the angles themselves when they are small, and clearly gives an 
equally valid measure of the magnification ; it is as good an approxi- 
mation to relate the impression of "sizes" of the retinal image to 
the tangent of the angle of subtense as to the angular measure; 
neither is strictly accurate in the general sense ; it is ultimately a 
matter of verbal definition. 



in 



usual case. 



tana' P a F a /' a 

== iv i^ " ~ 77~ 
tan a P' 6 K,, J\ 

The ratio of the two focal lengths is therefore a measure of the 
magnifying power. It follows that 

1. To obtain high magnification the focal length of the objective 
must be great in comparison with that of the eyepiece. 

2. If the second focal length or the dioptric "powers" of the 
objective and eyepiece in such a simple instrument are of the same 



22 



APPLIED OPTICS 



sign, the image will be inverted. This is the case in the astro- 
nomical telescope. 

3. If the second focal lengths or the dioptric powers of the eye- 
piece and objective have opposite signs the image will be erect. 
This is the form of telescope invented by Galileo, and also, presum- 
ably, in Holland, though the details of the earliest Dutch instruments 
are not certainly known. 

The Galilean Telescope. The action of the Galilean glass is illus- 
trated in Fig, 12. The eyepiece b is now a negative or diverging 
lens, so that the common focal plane containing F' a and F 6 is shown 
behind the eyepiece. A parallel bundle of rays is focused towards 
the image point B' in the common focal plane, but is intercepted 




FIG. 12. GALILEAN TELESCOPE 
(Diagrammatic) 

by the eyepiece before reaching the focus and rendered parallel once 
again. The image will evidently be erect. If the separation of the 
lenses be diminished, the virtual image seen by the aid of the instru- 
ment can be formed at a finite distance. The formulae for magnifica- 
tion hold good in the present case. 

The Pupils of the Telescope System. The entrance pupil of the 
system is usually considered with reference to an infinitely distant 
object point. Referring back to Fig. ir, the objective "a" is shown 
diagrammatically as limited in diameter. It is usually the case that 
a ray parallel to the axis passing, as shown, through the extremity 
of the objective, is transmitted unhindered by the eyepiece. If, 
therefore, we follow the usual plan for finding the entrance pupil 
by determining the images of the various diaphragms and lens 
rims, etc., formed by the parts of the instrument lying to the left 
of each such diaphragm, it will be found that all these images 
have a greater diameter than the boundary of the objective if the 
above condition regarding the ray is fulfilled. 

The entrance pupil is therefore usually represented by the 
boundary of the objective itself, and the exit pupil is the image of 
this rim formed by all parts of the system lying to the right of 



THE TELESCOPE 23 

it. The axial position of the exit pupil will be found by tracking 
through the system a ray from the centre of the entrance pupil. 
Such a ray is exemplified in the diagram by that through P . It 
cuts the axis in R' beyond the eyepiece. The radius of the exit 
pupil is found by tracing a ray from the boundary of the entrance 
pupil through the system, and finding the axial distance of its 
intersection of the plane through R' perpendicular to the axis. 

Let p and p' be the radii of the entrance and exit pupils respec- 
tively, then inspection of Fig. n shows that (for systems in air) 

t = J^ 

P' A 

in the case of the ordinary adjustment when F' a coincides with 
F. 
But we found above that 

/' 
Magnifying power = - - 

J b 

thus we get also the new result (using diameters now instead of the 
radii of the pupils) 

-.. ., . , . diameter of entrance pupil 

Magnifying power (numerical) = r . - ; ~- 

' diameter of exit pupil 



For purposes of actual measurement, it may be noted that a real 
object of height p l anywhere in the object space, measured perpen- 
dicular to the axis of the telescope, must have an image of height />/, 
where 



which is seen at once by tracing a ray through the top of the object 
parallel to the axis in the object space. Hence the magnifying 

Pi 
power is . In one very useful practical method of measuring the 

magnifying power, the real object may be conveniently repre- 
sented by the points of a pair of dividers opened out to a convenient 
extent fa), and placed immediately in front of the objective. The 
images of the points will be found just within the position of the 
exit pupil, and their separation fa') can be measured with the aid 
of a suitable scale mounted with a magnifier, or with a small travel- 
ling microscope. This is a very reliable and convenient method of 
measuring the magnifying power of any small telescope. 

In another method the whole objective is illuminated by light 
diffused from a sheet of paper which more than covers its aperture, 

3 (5494) 



24 APPLIED OPTICS 

when the ex' pupil can be observed as a uniformly illuminated disc 
behind the eyepiece, and its diameter can be measured (as above) 
with a small scale observed by a magnifier. 

Object at Finite Distance. Tt may be the case that the instrument 
is being used with an object at a finite distance, and that the image is 
formed in the accommodation point at a distance k from the exit pupil 
or eye-ring where the observing eye is situated. This condition is repre- 
sented in Fig. 13. 

Trace a ray from the axial point of the object through the boundary 
of the entrance pupil. It must, therefore, pass through the boundary 
of the exit pupil, and its axial intersection point (real or apparent) 



7.1 (- -, 

* r-\~ 







FIG. 13 

must be in the final image plane at a distance k. Let h and h' be the 
perpendicular dimensions of object and image respectively. 
The Lagrange relation gives 

nha = n'h'a' 

where a and a' are the angles made with the axis by the above incident 
and emergent rays through the axial points of object and image. If p 
and p' are the radii of entrance and exit pupils respectively, and I is 
the distance of the object from the entrance pupil, 

P 
a = -r-(very nearly) 

and a' = j- 

Hence (since n and n', the refractive indices, are almost invariably 
unity, and will now be assumed to be so) 

hp _ h '_ 

/ ~ k 

or t h '' k 

or p' = -kjf 

Angular subtense of image at the eye 

~~ Angular subtense of the object at the entrance pupil of the 

instrument 

We may, therefore, look upon the ratio of the diameters of the entrance 
and exit pupils as representing the "magnifying power" in the above 
sense, even though the telescope may not be in the adjustment for 
infinitely distant object and image. 



THE TELESCOPE 25 

Position of Principal and Focal Points of the System. The for- 
mulae of Chapter II, Vol. I, enable us directly to calculate the 
position of the principal and focal points of a combination of two 
optical systems. 

If a telescope is in the adjustment when F' a coincides with F 6 , 
then the intercept "g" between the adjacent foci is zero. The 
formulae 



PaF 



ff g ' IVP/ g " g 

show that the focal lengths of the system are infinite, and also that 
the principal and focal points are at an infinite distance. The com- 
bination is said to be in afocal adjustment. Hence the general methods 
of discussing the optical performance of the system (position of 
principal planes, focal length, etc.) find very little application. 

Telescope Objectives. The early work * of Chester Moor Hall (1733) 
and Dollond (1758) on the achromatic object glass, and the develop- 
ment of the telescope up to the time of Fraunhofer, can only be ^ 
mentioned here. Fraunhofer's re-discovery of the dark lines of the * 
solar spectrum and their use for exact measurements of refractive 
index first placed the matter of achromatism on a definite basis. 
Fraunhofer achromatized his telescope objectives in the following 
way. He divided the solar spectrum into regions bounded by the 
lines A, B, C, D, E, F, and G, and then measured the ratio of the 
flint dispersion to the crown dispersion for each region. He adopted 
a mean value in which the ratios were weighted according to the 
amount of light in each region ; thus this work involved one of the 
earliest essays in heterochromatic photometry. The mean ratio 
gave a measure of the ratios of the total curvatures ( '^ + ^2) f r 
each component. This condition resulted in lenses which had their 
minimum foci somewhat towards the blue end of the spectrum, 
and Fraunhofer found empirically that a better result was obtained 
by a slight change in the ratio which resulted in a minimum focus 
nearer the red. The condition of bringing together the foci for C 
and F (introduced somewhat later) ensures that the minimum focal 
length of an ordinary objective falls in the apple-green region of 
the spectrum, and produces the maximum bunching together of the 
radiations which are brightest to the eye. Subsequent systematic 
work 2 reveals little to be desired in this provision for visual 
observation. 



26 APPLIED OPTICS 

The type of objective manufactured by Fraunhofer is shown in 
Fig. 14. It consists of a double-convex crown and a nearly plano- 
concave flint. Primary chromatic aberration is corrected in the 
above manner, and spherical aberration is corrected for one zone. 
The lens is practically free from coma, although it must be doubted 
whether Fraunhofer knew of the "sine condition." It is the leading 
type of all small telescope objectives. 

If such lenses are designed with the same radius on the adjacent 
faces, the contact may be cemented. This helps cleanliness and 
avoids internal reflections. Chromatic and spherical aberration can 
still be removed, but freedom from coma can only be secured by the 
choice of suitable glasses. 3 

The absolute elimination of coma is not always considered essen- 
tial by manufacturers in objectives for small theodolites where 



Crown lens D = i'53i 
Flint = 1-616 
*i = + 340, r 2 = - 139 



r 3 = - 136, r 4 = - 620 



Crown 



Flint 



FIG. 14. FRAUNHOFER OBJECTIVE 

good central definition suffices. If the axis of the lens happens, 
however, to be slightly out of its proper alignment, troublesome 
coma may appear in the middle of the field. It is not usual to 
cement lenses with diameters greater than 2 in. 

Variations of the above general design are sometimes used for 
particular purposes. The "Steinheil" form of objective is shown in 
Fig. 15(0). The flint lens faces the object, and somewhat steeper 
curvatures are necessary, but the freedom in removing aberrations 
is much the same as with the Fraunhofer type. 

The removal of spherical aberration for two wave-lengths is pos- 
sible by bending both components of the achromatic combination, 
as suggested by Gauss. The profile of such an objective is shown 
in Fig. 15(6). Such a condition is, generally, secured only by the 
loss of freedom from coma. Such lenses have been used in large 
theodolites. 

Herschel proposed objectives calculated for the removal of 
spherical aberration for two object distances, but this provision is 
not often deemed necessary. 

Large and Small Objectives. The modern manufacture of small 
telescope objectives of apertures up to 2 in. is now largely a matter 



THE TELESCOPE 



27 



of mass production. The data for the system will be computed 
before manufacture begins, and the optical performance will be 
expected to agree with calculation. 

The manufacture of large objectives involves much greater uncer- 
tainty, and figuring by local rubbing is usually necessary to correct 
(a) zonal spherical aberration, (b) errors in the regularity of the 
surfaces, (c) effects of lack of homogeneity in the glass. 

The residual secondary spectrum of an astronomical objective is 
prejudicial in exacting observations. Considerations of cost and the 




Flint 



Crown C 



* 



(b) 



(d) 



Fie;. 15. TYPICAL TELESCOPE OBJECTIVES 

(a) Strmheil obje ti\e 

(b) Gauss ,, 

(c) C<x)kc ,, 

(rf) Astrogr.iphii' objective 
Fessar type (Zeiss) 



in Cookc objective 
i Baryta light flint (O, 54 }) 

2. Borosihrate flint (O, 658) 

3. Crown (O, 374) 
Jena gla^s numbers 



chemical stability of glass usually call for a doublet of crown and 
flint for the largest refractors of aperture loin, or over, and the 
secondary spectrum must perforce be tolerated, but the attempts to 
reduce this in smaller lenses must be noted. 

Reduction of Secondary Spectrum. The attempt to produce 
glasses from which a pair could be selected for the similarity of 
their run of partial dispersions and so eliminate the residual secon- 
dary spectrum of a doublet has already been discussed (Vol. I, 
p. 231). Failing the requisite chemical stability of suitable glasses, 
interest is given to the possibilities involved in separating the crown 
and flint components of a doublet so as to have a space between 
them. The partial dispersions of a typical "hard crown 11 and 
"dense flint" arc 





V 


a 


ft 


y 


Hard crown 


6tV2 


643 


703 


566 


Dense flint 


30-2 


-605 


714 


609 



28 



APPLIED OPTICS 




Notice that the dispersion of the flint is relatively too low in the 
red, but too high in the violet. Separation of the crown and flint 
components of a telescope objective (Fig. 16) would allow of the 
use of a smaller flint lens of shorter focal length. At the same time 
the violet ray is more deviated than the red in passing the crown 
lens, so that the violet meets the "diverging lens" nearer the axis 
fhan the red. This clearly acts in the sense of reduction of the 
secondary spectrum, but the correction so attainable is not suffi- 
cient with a single lens as the rear member without making the 
curvatures of the lenses so great as to negative the advantage 
gained. A proposal by Rogers to use a doublet correcting lens 

which is of zero power for 
mean wave-lengths, and yet 
acts in a "divergent 11 sense 
as between the red and blue, 
solves the problem of the 
secondary spectrum only to 
introduce troublesome sphe- 
rical aberration and chroma- 

,, , A c tic differences of magniiica- 

liG. 16. ACTION OF SEPARATED . _ , . .*> 

COMPONENTS tion. Lenses of similar type 

have been produced by Plossl 

and others under the name "dialyte objectives," but they have 
not come into very common use. 

There are certain advantages attainable by a modest separation 
of the lenses of a doublet objective, in spite of the failure appreci- 
ably to reduce the secondary spectrum. The objective follows the 
changes of external temperature with greater facility, and a suit- 
ably adjusted lens can be freed from spherical aberration for two 
colours. As will be understood from the formula quoted below, the 
shorter focal length of the flint lens tends in the direction of a 
flatter image field. 

The most successful mode of securing the practical elimination 
of the secondary spectrum is by the use of a three-component lens 
in which we may have, for example, two positive components which 
together represent one lens of a glass having a run of partial dis- 
persions similar to that of the negative component. Fig. 15 (c) shows 
the section of a Cooke-Taylor photo-visual objective. 

The relative partial dispersions of the glasses used for the two 
positive components of the Cooke photo-visual objective are given 
in the following table. The mean of their partial dispersions is 
shown in the third line, while the fourth line gives figures for the 
boro-silicate flint used in the negative lens. 



THE TELESCOPE 



29 



Glass 


"D 


V 


Dis- 
persion 


Relative Partial Dispersions 








C-F 


A C 


D-F 


E-F 


F-G' 


F-H 


0-543 


1-504 


50-7 


01115 


3354 


7085 


3309 


5830 


1-1857 


'374 


1-511 


60-8 


00844 


3507 


7026 


3247 


5075 


1-1564 


Mean 








34-20 


7059 


3282 


5763 


1-1730 


0-658 


1-54^ 


50-1 


01090 


3425 


7052 


3278 


5767 


I-I745 



The powers of the two positive components are very nearly equal, 
so that the effect of these lenses combined has a run of partial 
dispersion exceedingly close to that of the flint negative 
component. 

The limits of size of astronomical refractive objectives are set in 
practice by the difficulties attendant on the production of discs of 
optical glass of large diameter. The Ycrkes objective has a diameter 
of 1-02 metres and a focal length of 18-9 metres, and the Lick 
refractor a diameter of 0-91 metres and focal length 17-6 metres ; 
these are large doublets. Generally speaking, the aperture ratio 
(aperture : focal length) decreases with the diameter. With very 
small lenses i : 4 may be reached, but i : 18 is as much as can be 
allowed with large refractors on account of the prominence of the 
secondary spectrum. 

When the triplet lenses are used the necessary curvatures are 
rather large, and it is not easy to obtain a satisfactory performance 
with an aperture ratio better than i : 15. The Gauss condition 
(spherical correction for two wave-lengths) can be secured by separ- 
ating two of the components. Such lenses have been made up to 
I2i in. clear aperture. 

The image given by a triplet apochromatic objective presents 
practically no trace of colour, and such a lens, if used for astro- 
nomical photography, may be focussed visually. 

Astrographic Objectives. If the correction is not to be apochro- 
matic as in the photo- visual objectives mentioned above, the lens 
will be best adapted for astronomical photography if the "F" line 
focus is united with that for H<$ (wave-length = o-4ioi//). This 
ensures a better bunching of the most "actinic" regions of the 
spectrum, using the term to refer to those wave-lengths which most 
affect an ordinary photographic plate. The use of such plates con- 
fines the effective light to a limited spectral band, and tends to 
secure better definition on that account. Modern photographic 
work in astronomy includes, however, work both with ultra-violet 



30 APPLIED OPTICS 

and infra-red radiations, so that the exact requirements for specia- 
lized work may be very varied. The optical design of astrographic 
objectives aims naturally at securing the largest possible flat field 
together with the indispensable high definition required for astro- 
nomical photography, which calls for freedom from spherical 
aberration, coma, and astigmatism. These requirements are met 
by lenses of which the design resembles photographic lenses, except 
in so far as they work at a very much smaller aperture ratio. While 
they imitate the flat field of the photographic lens, they allow of 
much better central definition. Lenses of the type of the Taylor 
triplet, and the Tessar (Carl Zeiss) (Fig. 150) are in use for this 
purpose. It is frequently the case, however, that astrographic work 
is done with doublet lenses very carefully designed, and made to be 
as free as possible from spherical aberration and coma. 

The Image Field and its Curvature. The diagrams so far used 
show the ordinary flat image field of the elementary theory. In 
practice the phenomena of curvature of field and the accompany- 
ing degree of astigmatism are of the greatest significance. Little 
or nothing can be done to remove the astigmatism of an ordinary 
doublet telescope objective, and the curvature of the field and 
astigmatism with those objectives constructed of the usual types 
of glass, such as hard crown and dense flint, are finite and com- 
parable with those of a single lens of the same focal length. 

Thus for thin lenses in air the radius of the Petzval surface is 



*/', 

where n is the refractive index of a lens and /' the focal length. 
E. W. Taylor records an achromatic objective having 

Positive lens: Medium barium crown, n l} = 1-5736, focal length = 3-543 
Negative lens: Light flint, n D 1-6039, focal length 5-391 

The focal length of the combination is 10-06 ; and the radii which 
can be found by calculation and checked by experiment are 

Tangential image field . . . .2-85 
Sagittal image field ..... 6-05 
Petzval surface . . . . .15-6 

In accordance with theory, the tangential surface is three times the 
distance of the sagittal surface from the Petzval.* 

Such a curvature will not be of much significance in the small 
diameter intercepted by an eyepiece, but the curvature of the 
image field for the eyepiece has also to be taken into consideration. 

* Vol. I, p. 136. 



THE TELESCOPE 31 

The cheaper forms (see below) are constructed of two separated 
lenses made out of the same glass. Take, for example, an Huygenian 
form in which the focal length of the field lens may be 3^, say, while 
that of the eye-lens is q. Applying the formula above and assuming 
n = 1*5 approximately 



The focal length for the Huygenian combination (page 43) at a 

3? 
distance of 2q proves to be . This shows that R = 0757, so that 



for Objective 
Tangential- 



Direction of the Light 




Requ!red Image .Field 
for Eyepiece 



FIG. 17 

the Petzval curvature for an eyepiece of focal length i in. would be 
about three-quarters of an inch. 

Object points on a surface of such curvature (assuming astigma- 
tism absent) would give the parallel rays required to emerge from 
the eyepiece. In Fig. 17 is shown in full size the section of the 
image surfaces for the objective discussed above, assuming a focal 
length of 10 cm. The right-hand curve is that on which the image 
should lie if the field of the instrument is all to be "at infinity" 
when viewed with an eyepiece of focal length 1-3 cm. approximately 
(if we may assume the above theory, and also neglect the astigma- 
tism of the eyepiece).* The gap between the two increases very 
rapidly with distance from the axis, but the physical depth of focus 

* Actually the eyepiece will be designed to have some over-corrected 
astigmatism which will flatten the field to some extent. 



32 APPLIED OPTICS 

allows of a certain limited region of the field being apparently 
sharp. If the outer parts of the field are to be viewed in sharp 
focus, the eyepiece may be pushed inwards. The central parts of 
the field may then be viewed clearly by the exertion of accommoda- 
tion. In attempting to observe this with actual telescopes, allow- 
ance will have to be made for the residual astigmatism which will 
be present. In eyepieces like the Huygenian, where the light has 
to pass through a larger field lens and then a smaller eye lens, it is 



'I. 



(b) ( C ) 

FIG. 1 8 

Under-corrected astigmatism 

[b) Correction ; images on Petzval surface 

[c) Over-corrected astigmatism 

N.B. Ihe lens system is imagined to he to the left of these figures. 

possible to modify the amount and direction of the astigmatism by 
suitable curvatures of the lenses, and to produce over-correction. 

Take Fig. i8(a) to represent an ordinary case of curvature of the 
field and "under-corrected" astigmatism (as of the type which 
arises with a simple lens) . If we change the curvatures or separations 
of the component lenses of a complex system without altering their 
focal lengths or refractive indices, we shall sometimes be able to 
alter the amount of the astigmatism, but not the Petzval curvature. 
In Fig. 18(6) the astigmatism is eliminated, and the field has now 
the curvature of the Petzval surface. In Fig. i8(c), the astigmatism 
is over-corrected in such a way as to make the mean curvature of 
tangential and sagittal surfaces zero, so that the field is "flattened" 
in that sense. The tangential field remains at three times the 
distance of the sagittal field from the Petzval surface. The possi- 
bilities mentioned in this paragraph do not, however, apply to 



THE TELESCOPE 33 

ordinary close doublet telescope objectives. The subject will be 
again discussed in connection with photographic lenses. 

Design of a Doublet Telescope Objective. The rough design will 
be worked out for a doublet which is to satisfy (i) the ordinary 
condition of achromatism, (2) freedom from spherical aberration 
for some wave-length of the spectrum. If the lens is to be cemented 
the degrees of freedom will then be exhausted, but if the com- 
ponents may have different radii for the adjacent surfaces, the 
condition of freedom from coma may be added. Choosing for 
trial the glasses Hard Crown and Dense Flint selected for the 
example of Vol. I, p. 229, we had for the powers . )' a and .y^ of the 
components of a doublet 



n v ~ 1-5186 = 1-6041 

V - 60-3 V - 37-8 

The formula for the spherical aberration coefficient of a thin lens 



s 



In the case of the first lens we take f / = - o, and hence 



Calculating the requisite numerical coefficients (by four-figure logs.) 
A a - 165-0 - 55-9I 'i i + 6-21 /,\ 2 

It is more convenient, however, to express the relation in terms of 
$2 where 

,/ K -!)(//! -./?,) 

and >?! = '/ 2 + 5-168 

We then obtain 

A a = 42-0 -|- 8-27 /;, + 6-21 /? 2 2 . . (a) 

For the second lens */\ = J' a = 2-68, and ,>' 6 = - 1-68. The 
equation, when simplified, reduces to 

A 6 = - 11-91 + 9-58^3 - 377^3 2 ( 6 ) 



34 



APPLIED OPTICS 



In order that the lens may be cemented it will be necessary for 
j$ 2 to be equal to _'# 8 . Hence if the sum of the aberrations is to be 
zero 

A + A 6 = o = 30-09 + 17-85 $ + 2-44 /? 2 

a simple quadratic equation in # having roots - 4-67 and - 2-64. 
This determines the forms of two possible lenses for cementing, 
although one (for reasons given below) is much preferable to the 
other. 



First lens : 



0-498 



= 2-52 



// 4 = - 1-885 



= 0-145 



It may be deemed that the contact should be left uncemented 
in order to satisfy the condition for the absence of coma by a 
possible difference of curvature of the adjacent faces. The most 
instructive method to proceed is to plot the values of the aberra- 
tion found from equations (a) and (b) above in a graphical diagram. 
The following table gives points for plotting 



R z or R 3 


-5 


-4 


-3 


-2 





+ 2 


H4 


A fl . 


155-75 


108-22 


73-o8 


50-3 


42-0 


83-38 


I74-38 


A & 


-154-01 


no-55 


-74-58 


-45-^5 


- II-9I 


-6-83 


-33-91 



As is clearly seen from the equations, we obtain two parabolas (Fig. 
19) with vertical axes. It is convenient to plot A b values with 
reversed sign. The parabolas then intersect in the two points with 
abscissae - 4-67 and - 2-64 corresponding to the roots of the equation 
above. 

Coma. By an extension of the method given in Chapter IV, Vol. 
I, we can investigate the value of the coefficient a 2 in equation (50) 
of that chapter, i.e. the "coma" coefficient. Professor Conrady has 
shown that it is proportional to 

N + i ,,_ 2N + i , N,// 2 

jf^n __________ ' * / 

XT lS'SL\ XT / ' ' 1 -VT 

N * N * N - 1 

for the case of a thin lens. We insert the necessary numerical 
quantities, and find for the above coma coefficients of lenses a and 
6. 



4'445 
- 273 



4-32 



THE TELESCOPE 



35 



These lines are drawn in the diagram again plotting C 6 values 
with reversed sign ; it will thus be seen that the difference between 
C and C & for the abscissa of the lower intersection point of the two 



Parabolic Curve, 
of Spherical 




FIG. 19. GRAPHICAL PRESENTATION OF SPHKKICAL 

ABERRATION AND COMA FOR A DOUBLET 

OBJ ECTIVE 

parabolas is about the same as with that for the higher intersection 
point. The lower intersection point with the shallower curve for 
the contact would, however, be preferable. (We can calculate the 
intersection point of the coma lines as - 3-66.) 

If an uncemented doublet is allowable, we can choose a pair of 
lenses to be free from both spherical aberration and coma. 



36 APPLIED OPTICS 

If C + C & = o, we get by adding the equations 
4-445 4 2 - 273 '// 8 + 6-26 =- o 
/? 3 = 1-63 A* + 2-29 

This relation between ^/ 2 and # 8 will ensure absence of coma. 

Substituting this value for ^ 3 in the expression for A & above, we 
obtain 

A b = - 10-02 /? 2 2 - 12-52 /? 2 - 973 
We had 

A a = 6-21 ',y + 8-27 ''// 2 + 42-0 

if the sum of the aberrations is to be zero, then we get on adding 
o = 3-81 l //- 2 2 + 4-25 /? 2 - 32-27 

This quadratic equation yields two solutions, i.e. '/ a ~3"5 2 or 
+ 2-405. The first of these is the one of practical interest, since the 
second would give a pronounced meniscus form to the lens as a 
whole. The figures therefore become 

^ = i'65 
$* = - 3-52 

#* = - 3-45 
v' 4 = - 0-67 

Suitable thicknesses have now to be assigned, and the system per- 
fected by trigonometrical trials. 

The foregoing account explains the approximate method of de- 
signing a telescope objective which is to be corrected in itself. If 
the objective is to be used for visual observation, it is better to 
correct the spherical aberration for the brightest light in the spec- 
trum (A = o-55ii). 

When an objective is intended for use with a definite eyepiece, 
the axial chromatic and spherical aberration of the latter can be 
found (as the eyepiece design is usually completed first) by tracing 
some rays of an axial parallel beam backward through the system. 
The objective is then designed so as to compensate the axial aberra- 
tions of the eyepiece, and will need to be slightly over-corrected if 
intended for use with an ordinary Huygenian or Ramsden eyepiece. 
The condition for chromatic correction can be adjusted to allow for 
this, even in getting the rough design. 

The approximate solution is turned into a trial formula by find- 
ing the radii, and assigning a suitable thickness and diameter 
necessary to obtain the required aperture ratio. The trigonometric 
formulae are then employed in tracing a group of parallel rays 



THE TELESCOPE 37 

through the objective, a paraxial ray, a marginal ray, and one or 
more rays in mid-zones, preferably using the wave-length for 
brightest light. " Red " and " blue " rays can also be traced through 
the mid-zone. The last radius can be adjusted to produce the 
exact chromatic correction required, and the state of correction as 
regards spherical aberration and coma can be examined. Chapter 
IV of Vol. I contains an explanation of the methods by which the 
phase differences of disturbances arriving in the image can be 
deduced from a knowledge of the geometrical aberrations of ray 
paths. The state of correction as regards coma is examined by the 
so-called "offence against the sine condition." 
The optical sine relation 

nh sin a = rih' sin a' 
and the corresponding paraxial form 

nha = n f h'a f 

give two values of the magnification deduced from the marginal 
and paraxial ray paths, viz. 

n sin a 

r marginal 



varaxial ' u o 

The "sine" condition for freedom from coma (valid in the absence 
of spherical correction) is that m m and m p shall be identical.* The 



corn- 



offence against the sine condition is given by ( - I ). The 

\ m v I 
puting schedule (Vol. I, p. 19) will usually begin with the same 

numerical value for sin a and a n . Hence 



Offence against the sine condition = -- - i = - -- - i 
m v sin a 

The numerical result thus obtained is clearly a measure of the 
coma, taken as the radial distance between the focussing points of 
paraxial and marginal zones divided by the radial distance of the 

* Note that the magnification for a marginal ray is 



n sin a 



n sin a n sin a 



where y^ is the incidence height of the ray and / is the distance of the object 
when this distance is large. Hence when the incident light is parallel to the 

axis, the sine condition may be expressed as the necessary constancy of 



rw 



sin a 



3 8 APPLIED OPTICS 

image point from the centre of the field. It is the general experience 
with ordinary telescope and microscope objectives that it must not 
be allowed to rise above one part in 400, or 0-0025. 

It is not within the scope of the present book to describe in detail 
the systematic trials by which the design of an objective is finally 
completed. The approximate thin lens method saves much time in 
the early stages. Modern advances in design have arisen through 
the ability to interpret the aberrations obtained from the numerical 
work in terms of optical path differences at the focus of the lens, and 
through an exact knowledge of the tolerances allowable. An intro- 
ductory account of the general principles involved is given in Vol. 
I, Chapter IV, but the reader must be referred for fuller details to 
Prof. Conrady's work 4 on Applied Optics and Optical Design. 

Resolving Power of a Telescope. In Chapter IV, Vol. I, it was 
explained that the closest approach of two elementary "star" 
images which still permits of the recognition of the double nature 
of the concentration is approximately such that the centre of one 
Airy disc falls on the first dark ring of the neighbouring image. 

The angle subtended by the radius of the Airy disc at the second 
nodal point of a telescope objective therefore represents the angular 
resolving power. We have the approximate formula for an image 
in air 

p = radius of Airy disc = (see Vol. I, p. 93.) 

The angle w' subtended by this radius at the second nodal point 
is given by 



~~/' a 

where a is the diameter of the objective. 

This is the same angle as that subtended by the objects at the 
first nodal point. If this is put into English units we easily find 

Angular resolving power in seconds = 7 : : r 

& r aperture in inches 

It has been made clear that the theoretical basis for this limit is 
only approximate; the exact figure depends upon the distribution 
of light in the actual "Airy Disc/ 1 and this may be modified by 
spherical aberration and other causes in an actual lens. Dawes's 
Rule derived from experiment gives 

Angular resolving power in seconds = ; ; = 

r aperture in inches 



THE TELESCOPE 39 

In order that it may be possible for the eye to perceive the 
doubling of the image of a close double star, the separation of the 
images must clearly subtend an angle at least as great as the 
minimum separabile for the eye, which Hooke found to be one minute 
of arc. 

As viewed by the eyepiece of focal length f\ , the angle under 
which the radius of the Airy disc is seen is 



A~ A 

== w' M. . . (from above) 

Note that this is, of course, "angular resolving power" multiplied 
by the "magnification." If the angle of view is to be i' of arc 
(= 0-00029 radians), then 



__ 

0-00029 = ---- . M 

It is easy to get an approximate figure if we take A = 0-00058 mm., 
and thus 

a (mm.) , . . , v . 

M = - = 10 x (aperture m inches) . . . approximately. 
2-44 

At such a magnification, however, the resolvable detail, even if just 
visible, would be unendurably small, and even though no fresh detail 
can be rendered it will make for freedom from visual strain to 
increase the magnification to three or four times the above figures. 
Further enlargement serves little or no useful purpose; it is "empty 
magnification." 

The above equations give (for i' visual angle) 



J b 0-00029 \ a 
f' b (mm.) = 2-44 x (aperture ratio number) 

As indicated, however, it will be better to take a focal length 
only one-third or one-quarter of the above value. Thus for an 

astronomical objective with an aperture ratio number = 15, the 

minimum value of f b is 37 mm., but we shall be able to use with 
visual advantage eyepieces with focal lengths as short a^ 12 mm. or 
9mm. 

The equation which gives the minimum magnification for the 
telescope a (mm>) 

Mm. = 

2-44 

4 (5494) 



40 

is useful when we put 
Min. = v 



APPLIED OPTICS 



a (mm.) 



diameter of exit pupil in mm. 



a 
2^44 



It will be seen that in order to do justice to the resolving power of 
the objective the magnification should be at least sufficient to reduce 
the diameter of the exit pupil to 2-4 mm., and we can with visual 
advantage use exit pupils down to 0-6 mm. If a handy test of the 

Exit Pupil 





Exit Pupil 



FIG. 20 



(a) Telescope without field lens 

(b) Telescope with field lens; note enlargement of field and shift of exit pupil 

effective state of magnification is required, the measurement of the 
diameter of the exit pupil gives the information at once. It will be 
understood that the majority of binoculars and instruments with 
large exit pupils come nowhere near making the fullest use of the 
resolving power of their objectives, although the large exit pupil 
is important in maintaining the illumination of the image. 

The Eyepiece. The early astronomical telescope as employed by 
Kepler about 1611 consisted of a convex object glass and a convex 
eye lens. Reference to Fig. 20 (a) shows that all the rays passing 
through the extremities of the image are just intercepted by the 
eye lens, but that if the image were larger, part of the rays would 
fall outside the eye lens. The illumination of the peripheral parts 
of the image would therefore be poor, and the optical performance 
of the instrument would suffer otherwise. It is not known with 
certainty to whom the suggestion of the field lens (Fig. 20(6)) 
should be credited. It will be seen that the use of a convex lens in 
the plane of the image bends the rays towards the axis, although 
it can make no difference to the size of the image. Hence the peri- 



THE TELESCOPE 41 

pheral parts of the image are now viewed by full pencils, the apparent 
field of view being controlled by the margin of the field lens in the 
case shown. 

It is worthy of note that the exit pupil is now moved considerably 
closer to the eye lens, although it remains of the same size. Most 
modern eyepieces, however, do not place the field lens in the focal 
plane, because any specks of dirt on the glass are then seen in 
focus in the field of view. 

The addition of yet a third convex lens, as an erector, trans- 
formed this eyepiece into a terrestrial erecting eyepiece. This will 
be dealt with below, but we must first notice two important forms 
of two-lens eyepiece due to Huygens (1703) and Ramsden (1783). 
Their forms are illustrated in Figs. 21 and 22 respectively. 

In dealing with the simple magnifier, it was mentioned that one 
serious difficulty arises in the variation of magnification with the 
wave-length (or with the colour) of the light. This may be regarded 
as arising from a variation of focal length with wave-length. Con- 
sider, however, the expression for the power of a combination of 
two thin coaxial lenses of the same glass, of refractive index n and 
separated by a distance d. 

<><'= :/ + -;'* - d J' a J' b 
This becomes 

&= (n - i) 4 B + (n - i) '* - d(n - i) 2 4 a f4> 

Let the refractive index of the glass change for some given change 
of wave-length and become n + dn ; then 

. ;; =, ( n + dn - i) st a + (n + dn - i) // 6 - d(n + dn - i) 2 '} a 3* 
Subtracting 



The condition that the focal length may be unchanged, if dn is 
small enough for its square to be neglected, is thus 

o = '/ a + '/ 6 - 2d(n - i) /? a // 






This simple formula furnishes a rough guide to the separation of 
two lenses of the same glass necessary in order to secure "achroma- 
tism of the focal length" of the magnifier, and thus freedom from 
the most objectionable radial colour effects in the field of view. 



APPLIED OPTICS 



The Huygenian Eyepiece. In the Huygenian eyepiece the design 
varies amongst different makers, being adjusted to suit the thick- 
ness of the lenses and the objective distance. In a common form 
the focal length of the "field" lens is about twice that of the "eye" 
lens. The separation called for by the elementary theory is about 



f-2 



f-1 



P' 



0-5 



1-0 



1-5 2-0 




<r-~- * 



FIG. 21. Two FORMS OF HUYGENIAN EYEPIECE 

(a) Illustrates the position of the principal and focal points 
(/)) To illustrate the achromatism 

one and a half times the focal length of the eye lens. Working out 
(by the thin lens theory) the position of the principal and focal 
points of the combination, they are found as in Fig. 21 (a). The 
principal points are so situated that the first is found behind the 
system and the second within it. The focal length of the combina- 
tion is f v times that of the eye lens, and the power is positive, 



THE TELESCOPE 



43 



although it i> clearly impossible to put a real object into the first 
principal focus and obtain an erect magnified image. 

The design usually quoted in textbooks is shown in Fig. 21 (b) ; 
although not of much importance, it will serve to explain some 
features of the arrangement ; in this case the field lens has a focal 
length of 3#, say, an eye lens of focal length q, and a separation of 
29; where q is some suitable unit. The application of the usual 
formulae gives P a P - 3? ; P' & P' = - q ; / = - % q - 9 f = ? . The 

39 Q 

first focal point F is behind the front lens ; the second, F', is - 

2 2 

behind the eye lens. 

In the ordinary use of the Huygcnian eyepiece it will be placed 
so that the image-forming rays from the objective converge towards 



Field Stop 




IMC.. 22. RAMSDKN EYEPIECE 
Showing the position of the principal and focal points 

a point in the first focal plane through F. They reach the field 
lens and form a real image between the two components in the 
plane of the anterior principal focus of the eye lens ft, which is the 
plane of the stop limiting the field of view. The eye lens renders 
the rays parallel after refraction. By use of the plano-convex form 
of each component it is possible to produce "over-corrected" 
astigmatism, and thus to flatten the field to some extent. Distortion 
may also be reduced. A field of 40 may be attained. 

The achromatism of the system arises as follows. A ray 
directed towards the image point B t is shown in the diagram. (It 
may be considered to be the principal ray from the centre of the 
objective.) It is refracted by the field lens, and the dispersion 
causes a greater deviation for the blue than for the red. The 
separation of the field lens and eye lens is, however, such as to make 



44 



APPLIED OPTICS 



the blue ray intersect the eye lens nearer the axis than the red, 
and the greater deviation now produced in the red now renders 
both of them parallel on final emergence. Hence both the blue and 
red images will be seen under the same angle by the eye. 

The Ramsden Eyepiece. Invented by Ramsden for the observa- 
tion of the micrometer webs in reading microscopes, this eyepiece 
has two lenses of equal focal length disposed as shown in Fig. 22. 
The required theoretical separation for achromatism is equal to the 
focal length of either, but with a telescope of any practical length 
this brings the exit pupil too close to the eye lens ; for this reason 
the separation is reduced and the residual chromatic differences 



Field of . 
ViewStop\ 




FIG. 23A. KELLNER 
EYEPIECE 



FIG. 23B. ACHROMATIZED RAMSDKN 
EYEPIECE 



are tolerated. In the case shown, the focal length of each lens is 
q, and the separation is 2^/3; the focal and principal points are 
then disposed symmetrically as shown in the figure, the combined 
focal length being 3^/4. 

The Ramsden eyepiece as actually used does not thus strictly 
secure achromatism of magnification, but is extremely useful in 
"measuring instruments" and "optical sights," because it is pos- 
sible to place cross threads or stadia lines in the common focal 
plane of the eyepiece and objective, and thus to measure directly 
the comparatively undistorted image produced by the latter. The 
angular field represented by the separation of two stadia lines is 
independent of the eyepiece, and we may thus change from one 
magnification to another, if required, by the choice of a new eyepiece. 

Kellner (1849) invented an eyepiece (Fig. 23A) with a double 
convex crown front lens and a cemented eye lens of the ordinary 
"crown and flint" type, which he called "orthoscopic," and for 
which he claimed a greatly improved colour correction in the outer 
regions of the field. In this case, however, the first focal plane lies 
in or very close to the front lens, so that any dust on the surface 
is seen in focus with the field of view. In a later type (Fig. 238), 
introduced by Zeiss the advantages of the Ramsden type were 
retained, and a still better achromatism was effected by the use 



THE TELESCOPE 



45 



of barium silicate crown and silicate flint for the members of the 
cemented eye lens. This type is usually known as the "Achroma- 
tized Ramsden" form. 

In the case of providing for a wide field and sufficient magnifica- 
tion, while maintaining sufficient clearance between the eyepiece 
and the eye ring or exit pupil positions, closely spaced and cemented 
lens combinations are of service. Fig. 24 shows the Abbe ortho- 
scopic eyepiece. 

It is not difficult with such eyepieces to control the chromatic 
difference of magnification fairly exactly, and "compensating eye- 




FIG. 24. ABBE ORTHOSCOPIC 
K YE PIECE 



FIG. 25. TRIPLE 

CEMENTED LENS USED 

AS EYEPIECE 



pieces" may be produced to control the defects of the objective, 
although this is usually only necessary for microscope systems. 
(See page 87.) 

The " Monocentric " eyepieces of Steinheil are not infrequently 
met with. All the surfaces are struck with one centre and therefore 
they have the characteristics discussed in connection with the 
Coddington lens, but they arc not so suitable for use in a case 
where the eye needs to move about a centre which remains steady 
in relation to the eyepiece. For observation where it is necessary 
to have the greatest possible freedom from stray light due to back 
reflections, such eyepieces will be found to give good results up to 
a field of about 20. 

Fig. 25 illustrates a simple astronomical eyepiece made from a 
triple cemented lens. 

Erecting Systems. The Terrestrial Eyepiece. The possibility of 
erecting the image in his simple astronomical telescope by intro- 
ducing an erecting lens between objective and eyepiece was known 
to Kepler. Rheita obtained improved definition by using two inter- 
mediate lenses. The essential form of the four-lens terrestrial eye- 
piece as made by Dollond, Ramsden, Fraunhofer and many others, 
down to the present day, is the outcome of many empirical trials. 
A typical construction is shown in Fig. 26, and the diagram will 



4 6 



APPLIED OPTICS 



be sufficient explanation of the general mode of action ; the erecting 
lens system is followed by a Huygenian eyepiece of usual form. 
The stop in the erector may be made to coincide with an inter- 
mediate image of the entrance pupil, thus removing any stray light 
such as that reflected from the interior of the tubes. A shift of the 



Erector 



Exit Pupil 

I 




FIG. 26. ERECTING, OR TERRESTRIAL EYEPIECE 

erector stop, however, cuts down the aperture of some of the 
oblique pencils, and some opticians use this method of improving 
the definition in the outer parts of the field. Improved erecting 
eyepieces have been devised in which the erecting lens is a cemented 
triplet. In Fig. 27 the field lens is employed to secure the necessary 
convergence of the principal rays. 




KIG. 27 TERRESTRIAL EYEPIECE WITH TRIPLE CEMENTED ERECTING LENS 

It is particularly important with these eyepieces to design the 
objective to suit them ; a certain amount of spherical and chromatic 
over-correction is necessary. 

Pancratic, or Variable-power Telescopes. The possibility of vary- 
ing the magnification of a telescope with erecting lenses was 
realized quite early, and some telescopes were fitted with the erect- 
ing system in an independent draw-tube so that object and image 
distances for the erecting lens could be varied. 

The best known forms of modern variable power telescope arc 
provided with mechanical means whereby the instrument remains 
in focus during the change of magnification. In the "Ross' 1 type, 
the general lay-out of the erector is similar to that shown in Fig. 27 
above. The erecting lens is triple and cemented. When it is desired 
to increase the magnification, the erector is moved nearer the first 
image, and the enlarged second image moves in the direction of 
the eyepiece. The eyepiece system (of the achromatized Ramsden 
type) is mechanically withdrawn in order to preserve the focus. 



THE TELESCOPE 



47 



The eyepiece may also be given an independent movement to adapt 
the focus for the vision of different observers. 

The mechanical means of varying the power consists of three 
tubes fitting one inside the other, the outermost X (Fig. 29) carries 

Objective 
J Low Power 




High Power 

FIG. 28. VARIABLE POWER TELESCOPE (Ross) 

a milled portion which enables it to be turned, carrying with it 
an inner tube Y having two helical slots cut in it. The third tube 
Z is screwed to the main tube of the telescope so that it cannot 
revolve ; it has two straight slots cut in it. Sliding in tube Z are two 



Tube with outer mi lied ring to 
fit over tube Y and rotate it 





r x *. 




Tube with helical slots to Fit 
over tube Z 



- 

r. ' 



i : 

Kr.r=^-,t 



P cs: 



Tube with straight slots carrying 
interior tubes DlE which have pins 
passing through straight slots in 
order to engage in helical slots ofY. 
Tube Z /s screwed to main tube of 
telescope. 

FIG. j(>. TYPICAL SYSTEM FOR MECHANICAL VARIATION OK THE 
SEPARATION OF LENSES CARRIED IN TUBES D AND H 

independent tubes in which the erecting lens and the eyepiece 
respectively are mounted. These tubes each carry a pin which pro- 
jects through a straight slot of the fixed tube Z into the corre- 
sponding helical slot of tube Y. Thus any rotation of the latter 
causes translatory movements of the erector and eyepiece. 

Assuming the usual notation, the distance k between object and 
image for the erector system is given by 



48 APPLIED OPTICS 

where d is the distance between the principal points of the erector 
system. 



dk P + 2lf 
Differentiating, ^- = - ^777^2 

This is clearly zero (and the distance between object and image is 
a minimum) when I = - 2f, the condition when the magnification 
due to the erector only is - i. The differential coefficient gives the 
ratio of the pitches of the helical slots controlling the movement of 
the eyepiece and erector respectively. The diagram shows that the 

Objective 



FIG. 30. VARIABLE POWER TELESCOPE (OTTWAY) 

pitch of the right-hand (eyepiece) slot is zero at one point, and the 
eyepiece thus moves very little for comparatively large changes of 
magnification when it is nearest the objective. 

In the Ottway type of variable power telescope, the eyepiece 
remains stationary, the effective focal length and position of the 
erector system both being varied. This is accomplished by using 
a two-lens erector, as shown in Fig. 30, one lens being a cemented 
doublet. The power of the erecting system is varied by the varia- 
tion of the distance between the components, each member being 
given the displacement required to keep the image stationary. 

Alternative Powers. An alternative arrangement to the provision 
of continuous variable power in a telescope is to give two or three 
magnifications by separate eyepieces mounted on a swivel, so that 
the change from one to another can be made very quickly. The 
apparent angular field of view in various powers dobs not vary 
greatly, so that the real field is smallest with high powers and 
greatest with low. The great advantage of a variable power is that 
an object may be "picked up" easily with a large field, and then 
observed in detail by the use of a higher power. The magnifying 
powers usually obtainable with variable power systems range from 
about five to twenty. 

The Field of View of a Telescope. In all telescopes we can 
distinguish a diaphragm which limits the angular divergence of 



THE TELESCOPE 



49 



the rays passing through the centre of the exit pupil. Refer- 
ring to Fig. n, the stop limiting the field is seen in the common 
focal plane. If it reaches the size when the full bundle of rays 
passing through a point in the margin cannot be transmitted by the 
eyepiece, the illumination of the boundary of the field will suffer. 
In most eyepieces intended for ordinary observation, the diameter 
and position of the stop is specified in the complete design. Take 
the case of the* Huygenian eyepiece shown in Fig. 31, for which 

ou i b 




FIG. 31 

/' a = 2/' b and d = * --. The stop is situated in the first focal plane 
of lens 6. If its radius is v, the apparent angular field is clearly. 

2\ tan' 1 77- ). The entrance pupil of the eyepiece is evidently formed 
\ / &/ 

at a distance / from the field lens for which the conjugate V = and 

ill 

if 

. The radius of the effective entrance pupil is 

yl 4v 



and thus / = 
therefore 



Hence the real field of view will be 2 tan 



-i/4\ 

Wv f 



where f' is the 



focal length of the objective. If f' c is the focal length of the 
eyepiece as a whole, i.e. 4/' b /3, the apparent angular field is 

2 tan" 1 ( j f - 1. If the angles are small, tjae ratio of apparent field to 

/' 
real field will therefore reduce to ^r, the ratio of the focal lengths 

/ e 
of objective and eyepiece which is the angular magnification of the 



50 APPLIED OPTICS 

telescope. More generally, we calculate the radius y 9 of the effec- 
tive stop situated in the first focal plane of the eyepiece system, 

and find the real field as 2eo = 2 tan- 1 ( - ), and the apparent field 

f y '\ \J oJ 

as 2o/ = 2 tan- 1 1 ~ 1. The distance of the exit pupil from the eye 

lens can be calculated. Let it be c, say. The radius p' of the exit 
pupil can also be calculated from where p is the radius of the 

entrance pupil and m the magnification of the telescope. Then the 

c .y' 
required radius of the eye lens will be p' + c tan co' = p' + -jr~ 

this will allow the full oblique pencil to be transmitted. * e 




FIG. 32. EYE PUPIL FALLS IN 
A LARGER EXIT PUPIL 



FIG. 33. EXIT PUPIL FORMED IN 

THE CENTRK OF ROTATION OF 

THE EYE 



Observation with Rolling Eye. In the case when the pupil of the 
eye is brought into coincidence with the exit pupil of the instru- 
ment, and the latter is smaller than the eye pupil, then the whole 
field of the instrument will be projected on the retina. In using hand 
instruments, small telescopes or binoculars, the head and instrument 
can easily be moved together in order to bring different images 
into the central region of distinct vision. If, on the other hand, the 
instrument is held on a stand and not easily moved, the eye can 
observe different parts of the field by turning the head so as to 
observe as it were through the small window of the exit pupil, or 
by holding the head stationary and moving the eye in its socket. 

The eye, observing through the eye lens, sees the image of the 
stop (more or less well defined) which limits the field. This image 
forms the "exit window," and in the case of telescopes with positive 
eyepieces it is usually seen "at infinity" and sharply defined. 



THE TELESCOPE 5* 

If the exit pupil coincides with the eye pupil, let the point of 
rotation be situated at a distance p behind the pupil, and let p e be 
:he radius of the pupil of the eye. When the eye is turned so as to 
bring the margin of the field to the fovea, the required radius of the 
?xit pupil is seen from Fig. 32 to be 

p' = p tan co f + p e sec co' 

The eye will then lose no light on rotation. 

It may be possible to design the instrument so that the exit 
pupil falls into the centre of rotation of the eye in ordinary condi- 
tions of vision. This case is illustrated in Fig. 33, and it will be seen 
that the radius of the actual exit pupil need only be p e sec to', in 
order that the eye pupil may be kept completely filled with light. 

On the other hand, it will often be the case that the exit pupil 
of the instrument is smaller than the eye pupil, and the limit to 




FIG. 34. KYE PUPIL FALLS 

TOGETHER WITH A SMALLER 

Kxrr PUPIL 



FIG. 35 OBSERVATION WITH 
STATIONARY EYE 



which the eye can move without experiencing some curtailment of 
the light will be restricted. In Fig. 34, PQ represents the exit pupil 
of the instrument, and RS the pupil of the eye. If the eye rotates, 
the iris will begin to cut off some light from the image when S 
reaches Q. No illumination of the image will be possible when S 
reaches P. Hence 

Total rotation possible while retaining full illumination 

JL i t 

=3 ----- (approx. angular measure) 
P 

Total rotation possible while retaining partial illumination 
P.+P' 



52 APPLIED OPTICS 

When the instrument exit pupil coincides with the centre of 
rotation of the eye, no light will be cut off in this way for observa- 
tion with the rolling eye until the margin of the field falls on the 
fovea, provided that the optical system gives full parallel bundles 
through the exit pupil up to the limit of the angular field. On the 
other hand, it will be noticed that the stationary eye at rest in the 
symmetrical position (Fig. 35) will not secure fulf illumination of 
the whole field simultaneously unless the radius of the eye pupil 
is at least 

p e = p tan <o' + p' 

where p is the distance of the exit pupil behind the eye pupil. 

If it should be smaller than the required amount, we can see that 
the limit of the angular field for full illumination will be 



while the limit of the angular field for partial illumination will be 

tair 1 



Visible Field in Relation to the position of the eye. In observ- 
ing from the axial point of the exit pupil of an instrument, there 




FIG. 36. EXIT PUPIL AND EXIT WINDOW 

will be some lens mount or diaphragm which limits the angular 
extent of the field of view ; the image thus seen will be called the 
exit "window." 

Let WW' (Fig. 36) be the exit window and PP' the exit pupil 
of a telescope. If the eye is placed anywhere within the quadri- 
lateral APBP' it will be clear that the limits of the visible field of 
view will be controlled by the margin of the "window" WW'. 
On the other hand, if the eye is placed outside the quadrilateral 
beyond B, as at E 2 , the angular subtense of the visible field will be 



THE TELESCOPE 53 

controlled by the exit pupil PP' which now takes on the role of the 
exit window. The term "position of the eye" used above may 
refer either to the nodal point of a fixed eye, or to the centre of 
rotation of a rolling eye. 

The case just mentioned, when the exit pupil limits the angular 
field, is typical of the Galilean telescope, and it is important enough 
to merit a closer discussion. 

The Field of View of the Galilean Telescope. The "entrance 
pupil" of the Galilean telescope will be assumed to be the diaphragm 
which limits the objective, of which the radius is p. The radius of 

the exit pupil is where m is the magnification of the telescope. 

The distance of the entrance pupil from the first principal point 
of the eyepiece is 

' = -(/".+ A) 

Hence /', the distance of the exit pupil from the second principal 
point, is given by 



Tl * * 

Then = 

1 J b 



A (/'+/') 



Let the distance of the nodal point of the eye from the second 
principal point of the eyepiece be d, then its distance p from the 
exit pupil is given by 



- 37(0) shows Ihe formation of the exit pupil PP', and a bundle of 
parallel rays originally derived from an infinitely distant image 
point. Let the pupil of the eye be situated at EE' (shown separ- 
ately in lower figure). A bundle of parallel rays from the exit pupil 
must completely fill the eye pupil, provided that the inclination 
of the rays to the axis is not greater than that of the line (a) in the 
diagram ; the angle with the axis is thus given by 



54 



APPLIED OPTICS 



Maximum angle of full illumination 
m~*' 



w i / P - w # \ 
= tan' 1 - t r = tan" 1 I = ^7-7 : J 

<- A (.-i) ^-/W 




FIG. 37. To EXPLAIN THE FIELD OF VIEW OF A GALILEAN TELESCOPE 

When the inclination of the bundle has increased to that of the ray 
(6) in the figure, no light can reach the retina. 

Maximum angle of partial illumination 



= tan- 1 



A- 



^ ( 

\ 



md-f b (m-i) 



The visible field is therefore bounded by a ring of decreasing illu- 
mination which may be looked upon as the out-of-focus image of the 
exit pupil. We may clearly enlarge the field of view by increasing 
p, and diminishing d as far as possible. The quantity /' 6 will be 
numerically negative so that by decreasing it (other things being 
equal) we shall increase the field of view, but this would call for a 
shorter focal length for the object glass to preserve the same mag- 
. nification. The limit to progress in this connection is the difficulty 
of correcting the aberration* of the negative eye lens when the focal 
length is reduced below i in. or thereabout. The objective radius 



THE TELESCOPE 



55 



p may be increased in a binocular until the mounts of the two 
objectives practically touch each other (if the interocular distance 
is to be adjustable, this condition must be reached at the lowest 
separation), and d is decreased as far .as possible by bringing the 
eyes as close as possible to the eye lens. 
The so-called "mean field" is given by 

P 



tan- 1 



md -f' b (m - i) 



and it will be seen that the effect of increasing m while other things 
remain the same will be to decrease the field. (The magnification 
would be increased by a longer focal length of the objective.) 

Reflecting Erecting Systems* Perhaps the most important means 
of erecting the telescope image is now by the use of erecting prisms ; 



/7\ 





(b) (c) 

FIG. 38. MIRROR REFLECTIONS 




(d) 



the suggestion was originally made by Porro, and put into practical 
use by Abbe. 

The image of an object formed by reflection in a horizontal mirror 
is seen inverted (Fig. 38 (a)) ; the image formed by a vertical mirror 
is reversed, left to right (Fig. 38 (6)). The reversed image of the ver- 
tical mirror may be viewed in the horizontal mirror (Fig. 38 (c)). The 
result is a complete reversion and inversion such as is found in the 
real image formed by a telescope objective. If, therefore, the light 
from an objective suffers two such reflections between the objec- 
tive and the image plane, it will be seen erect and correctly disposed 
when examined by the eyepiece. The light must not fall on the 
mirrors at too great an angle of obliquity, so that in general the 
direction of the principal ray will be more or less deviated after 
the two reflections. 

If the two mirrors are placed together so that their planes inter- 
sect at right angles, reflections may take place at either one first, 
and in addition to the two images formed by one reflection in 
mirror, there will be an inverted and reversed image t 
reflection in both mirrors has contributed (Fig. 38 (d)). 

5 (5404) 



56 APPLIED OPTICS 

In Fig. 39 (a), a ray shown in broken lines suffers reflection at two 
mirrors shown by full lines. It is easy to show that if the angle a 
between the mirrors is reckoned as shown in the diagram, then the 
deviation of the ray is 2a. Fig. 39 (b) is a " polar diagram " in which 





FIG. 39 

Reflection at two inclined mirrors (a) and the corresponding polar diagram (b). (In (b) the 
mirror direction is such that the reflecting side is met when going clockwise round the circle) 




FIG. 40 

Reflection at two inclined mirrors 
Polar diagram 

the directions and senses of the mirrors A and B are shown by the 
outward directions of the corresponding radii. The convention is 
that the reflecting surface represented in the polar diagram shall 
be met by going clockwise round the circle. In these and subsequent 
polar figures the ray directions are also shown by the outward 
directions of the corresponding radii. In Fig. 40 (a) the reflecting 



THE TELESCOPE 57 

surfaces of the mirrors include an acute angle, and Fig. 40 (b) is the 
corresponding polar diagram. Let ray I pass through an object- 
point O between the reflecting surfaces; the path after reflection 
in mirror A is in a line passing through P, the mirror image of O. 
After reflection in B, the ray 3 appears to diverge from the point Q ; 
it is easily proved that the angle subtended at the centre by the 
arc OBQ is equal to 2ft, where ft is the angle included between the 
reflecting surfaces of the mirrors. We might, however, draw a ray 
through O which strikes first the mirror B, then A. This ray would, 
after the double reflection, appear to originate from the point R, 
where the arc OAR also subtends an angle 2ft at the centre. The 
points Q and R will evidently coincide only when ft = 90, and a 




FIG. 41. EQUIVALENCE OF AN ERECTING PRISM AND A 
PARALLEL PLATE OF GLASS 

single image will be seen. If ft is not 90, the angular separation of 
Q and R will be 360 - 40. 

A combination of two reflecting surfaces mounted so that reflec- 
tion can first take place at either of them forms a "roof" reflector. 
If used in an instrument, the doubling of the image produced by 
any slight difference of ft from 90 must not be perceptible to 
the eye. The tolerance will depend on the optical arrangements, 
but an accuracy within one second of arc may sometimes be called 
for. The eye would scarcely detect the doubling of the image if 
the separation of the components subtends an angle of less than 
30 sec. in the visual field of view, but this tolerance is still somewhat 
large in regard to the full contour acuity of the normal eye. (Vol. 
I, Chapter V.) 

Prism Faces as Reflectors. The equilateral inverting prism ABC 
(Fig. 41) partly overcomes the difficulty, illustrated in Fig. 38 (a), 
that the observer has to look into a new direction in order to see 



58 APPLIED OPTICS 

the image. The optical effect is conveniently studied by "develop- 
ing" the prism and drawing the mirror image of the refracted ray 
paths, and we therefore find the effect as that of a thick inclined 
plate of glass in the path of the light. The effect of dispersion 
clearly separates a red and blue ray, but they emerge parallel. 
The lateral displacement has no effect if the object or image is at 
a distance so great that the confusion of the images is not percep- 
tible, but it will be serious if the image is close to the prism. Parallel 
rays from a distant object will all be subjected to the same action 
by a parallel plate, so that no aberration can arise, but if the image 
is close to the prism surface, so that rays passing through any 

A 

b 




b 

FIG. 42. EFFECT OF ANGLE ERROR 

point of it to the prism surface are incident at various angles, 
then serious aberrations arise by refraction at the plane surfaces, 
which will clearly be especially objectionable if the axial ray to the 
central point of the field is obliquely incident. Hence, if any refract- 
ing prism surface must be placed in the path of a divergent or 
convergent beam, the surface should be normal to the axial ray in 
order that the central part of the field, where the corresponding 
angles of incidence have small obliquity, may be subject to the 
least possible disturbance. Fig. 42 clearly shows that if the base 
angles are not equal, the inverting prism will have a residual dis- 
persive action which would (for example) draw the image of a star 
into a short spectrum. 

Such a prism as that of Fig. 41 can, however, only produce inver- 
sion in the "up and down" sense while its base is horizontal. If it 
is placed in front of the objective of an inverting telescope the. 
image will be erect, but reversed "left to right." 

We can, however, obtain a prism which also gives reversion, by 
substituting the two faces of a roof reflector for the horizontal 
reflecting face ; each of the roof faces will be inclined at 45 to the 



THE TELESCOPE 



59 



horizontal plane, and the roof edge will be a horizontal line. Fig. 43 
shows the end view and side elevation of such a prism. There is no 
need to retain the vertical sides of the simple reflecting prism (indi- 
cated by the dotted lines in the end view). The top faces are, there- 




Side 

FIG. 43. SIMPLE "ROOF" PRISM 



fore, also cut at an angle of 45, but only the end faces and the two 
roof faces are polished ; the top faces are left rough-ground. 

We can follow the action of such a prism by the aid of a three- 
dimensioned polar diagram as suggested by Instructor Captain T. Y. 
Baker. 5 It must first be understood that when reflection takes 




(3') 
fa) (b) 

FIG. 44. ANGULAR SWING OF THE IMAGE 

place at two mirrors in turn (Fig. 44), the image may be geometrically 
regarded as the result of swinging the object around the axis marked 
by the intersection of the reflecting surfaces. If R is a real object 
lying in the plane of the diagram, images 2 and 2' are formed after one 
reflection, 3 and 3' after two reflections. The angle of swing is double 
the angle included between the reflecting surfaces. Fig. 44 (6) is a 



6o 



APPLIED OPTICS 



two-dimensional polar diagram of the same case, having significance 
only with regard to the orientation of the images and ray direc- 
tions. The letter R in positions i, 2, and 3 is parallel to those in 
the left-hand diagram. The final image orientation at 3 is the result 
of a swing 2a round the axis, and is, of course, independent of the 
exact position of the reflecting surfaces, provided their line of inter- 
section remains the same and that they include an angle a. 

We are now ready for the three-dimensional diagram. Fig. 45 
represents a sphere with centre O, and the radius vector Oi to the 

point i on the circumference 
represents a downward di- 
rected ray corresponding to 
an inverted image suggested 
by the letter R. The point 
i is in the middle of the 
vertical stroke. Point 2 
gives by the outward radial 
direction the required direc- 
tion corresponding to an 
erect image. If both vertical 
strokes of the letters R are 
in a vertical great circle, it 
will be seen that rotation 
through 180 about a radius 
of the sphere, marked OP 
in the diagram, will be suffi- 
cient to effect the required change. This line OP, therefore, is 
the required direction of the intersection of the two reflecting sur- 
faces required, and the angle between them must be 90. Their 
directions can be shown (in the sense of the polar diagrams) by a 
pair of lines drawn on the surface at 90 through P. This repre- 
sents the simple case discussed above when we use a roof prism 
with its roof edge horizontal. The roof prism is allowable since the 
rotation is to be exactly 180. If some other angle, say 179, of rota- 
tion were required, it would not be permissible to use a roof prism, 
because reflection must now take place first at one surface and then 
at the other for all rays if overlapping images are to be avoided. 

In general it can be shown that the displacement of a rigid body 
from one position to any other can be effected by (i) a movement 
of translation along a certain axis, and (2) a movement of rotation 
about that axis. 

An image can be given a shift of pure translation in any direc- 
tion and of any amount by reflection in two suitably-disposed 




FIG. 45. POLAR DIAGRAM TO ILLUSTRATE 
ACTION OF "ROOF" REFLECTOR 



THE TELESCOPE 



61 



parallel mirrors (one reflection produces a reversal and a second is 
necessary to annul the reversal). Any required movement of rota- 
tion can be produced by successive reflection in two surfaces for 
which the roof edge represents the rotation axis. It follows that 
reflection from four surfaces will suffice to effect any required change 
in the position and disposition of an image, so as (for example) to 
bring it into the field of view of an eyepiece placed in any particu- 
lar position. The vast majority of prism erecting systems employ 
four reflections. 

The polar diagrams are helpful because they permit attention to 
be given in the first place to the required change of direction of the 
ray and orientation of the image. The 
lateral shift can be dealt with after- 
wards. General considerations are 
illustrated by Fig. 46. The outward 
radius to A represents the direction 
and sense of the axial ray before 
entering a prism system, while the 
point X represents the axial ray on 
leaving the system. The orientation 
of the object-space field is represented 
by the short perpendicular lines AB 
and AC which might represent the 
12 o'clock and 3 o'clock directions 

respectively. The corresponding lines XY and XZ represent these 
respective directions in the image field. 

The pole of the great circle containing AB is the point c, while 
the point z is the pole of the great circle containing XY. If we are 
to find some axis of swing, rotation about which will bring AB into 
the direction XY, this swing will bring the pole c to the point z. 
If we bisect AX at right angles by the line PR, the axis must lie 
on this line ; equally, it must lie on the line QR which bisects cz at 
right angles. These bisectors intersect in R, the radius to which 
point marks the axis of swing. The angle of swing is clearly given 

by ARX. The necessary angle between the reflecting surfaces, con- 
sidered in the sense of the polar diagrams, will be half ARX. Cap- 
tain Baker's original paper shows how to use the constructions in 
the graphical design of prism systems. 

" Prism Binocular " System. In prism binoculars it is usually 
required for the final ray to be parallel to the direction of the 
incident ray. Fig. 47 (a) illustrates the case in which the incident 
ray travels "south east north vertical south." This is 




FIG. 46. GENERAL CASE 



62 



APPLIED OPTICS 



accomplished by the aid of (i) a pair of reflectors at 90 with roof 
edge vertical, (2) a second pair with roof horizontal (see Fig. 48). Re- 
ferring to the polar diagram, in Fig. 47 (6), we see that the effect of 
(i) will be to swing the a round the vertical axis through 180. The 

,0 




FIG. 47 (a). PATH OF RAY IN PORRO PRISM SYSTEM 




Roofedgeti) 
(I) 6 
FIG. 47 (b). POLAR DIAGRAM FOR PORRO PRISM SYSTEM 

effect of (2) swings the image through 180 round the horizontal 
roof edge. The final result is R distinguished with a ring in the 
diagram. Figs. 48 and 49 illustrate the arrangements of the prisms 
in practice. It will, of course, be clear that the system could be 
rotated as a whole round the incident ray (O) as an axis ; the "erect- 
ing" effect would remain precisely the same, although the emergent 
ray would have moved with the system while remaining parallel 
to its original direction. The reflecting faces can be grouped in 
pairs for the purpose of the above theory irrespective of the order 
in which they are encountered by the light. 

Such a system may, if necessary, be divided as shown in Fig. 49 (a) , 
where part of the farther prism has been lifted vertically. In cer- 
tain forms of periscopic telescope, the upper part of the system may 



THE TELESCOPE 63 

be placed just over the objective, while the lower part of the system 
is immediately followed by the eyepiece. Fig. 49 (6) shows a different 
arrangement; the reflections take place in a different order. All 




FIG. 48. PRISM BINOCULAR SYSTEM 





FIG. 49. VARIATIONS OF AN ERECTING SYSTEM 

these cases involve a lateral displacement between the incident and 
emergent principal rays ; reflection takes place at the various sur- 
faces consecutively, and the extreme accuracy required for roof edge 
angles can be avoided. The 90 angles must be accurate to 3' 
of arc, and the prisms should also be free from pyramidal error, 
i.e. non-parallelism of the 90 edge to the opposite face. 



6 4 



APPLIED OPTICS 



" Direct Vision Erecting Prisms." In many cases the loss of align- 
ment between the axes of objective and eyepiece is undesirable, 
and the Abbe prism shown in Fig. 50 (a) illustrates an arrangement 




(e) 



FIG. 50. A GROUP OF KRBCTING PRISMS 



(a) Abbe direct vision erecting prism 

(b) Leman prism 

(c) Modified Leman prism 



(d) K6nig prism 
(f) Daubresse prism 



(plan) 



by which this is avoided. The disadvantage of this arrangement 
is the somewhat awkward size of the prism, which calls for a 
considerable enlargement of the telescope tube. An arrangement 



THE TELESCOPE 65 

due to A. KSnig, Fig. 50 (d), is somewhat similar, but allows of the 
use of a smaller prism ; it will be seen that the division of the prism 
into two parts brings an air film into use, at which total reflection 
of the incident ray can take place at any point. When the ray 
returns from the roof edge it is incident at a very small angle, and 
passes through the air film with but slight loss. 

A "prism" due to Leman is shown in Fig. 50 (6), and a modified 
form in Fig. 50 (c) ; the latter was widely used during the War in an 
optical machine-gun sight in which the emergent ray is lowered by 
about 2 in. in order to secure useful protection to the eye of the 



Axis of 1st. Swing 
^ (120*) 6 




1st. Orientation of Image 
in Incident Light 



is of 2nd. Swing (180*) 
FIG. 51. ACTION OF MODIFIED LEMAN PRISM 

gunner. The optical action of the system of Fig. 50 (c) is illustrated 
by the polar diagram of Fig. 51. The first axis of swing is horizontal, 
and the angle of swing is 120. The next swing is one of 180 about 
an axis inclined at 60 to the horizontal (the lower roof edge). These 
two swings are clearly sufficient to erect the image. 

The prism of Fig. 50 (e) is due to Daubresse, and can, perhaps, be 
understood more readily from the plan view (Fig. 50 (/)). It consists 
essentially of a right-angled prism ABC, together with one of the 
familiar prisms (the so-called "pentag") used in rangefinders 
(CDEFB). The "pentag" alone produces a deviation of 90 in 
the path of a beam, when it is important that the deviation shall not 
alter with any angular shift of the prism itself. Reflection in the 
"pentag" takes place at silvered surfaces corresponding to the lines 
DE and FB ; the directions of these lines are at an angle of 45 
with each other. Thus the "pentagonal prism" system consists 
essentially of two inclined mirrors plus a plane parallel block of 



66 



APPLIED OPTICS 



glass. Rotation of such a system produces no angular deviation 
of a parallel beam doubly reflected from the mirrors. 

In the Daubresse prism, the face FB is replaced by two roof 
faces which meet at 90 in a horizontal line. The action can now be 
understood from the polar diagram (Fig. 52). The inverted d at 
(i) marks the direction of the incident beam. The first axis of swing 
is vertical, and the swing is 225 produced by surfaces AC and ED 
inclined (in the sense of the polar diagram) at 112-5. This brings 
the image to the position (2). The next swing must naturally be 



Axis of. 1st. Swing (225*) 



Ang/e between Mirrors 
(conventional sensc)~m-5* 




Axis of 2nd. Swing 
O80 9 ) 



FIG. 52. ACTION OF DAUBRESSE PRISM 

one of 180 about an axis symmetrically inclined to both (i) and 
(2), which makes an angle of 112-5 w ^h the final direction i. 
This is the direction of the roof edge HG in Fig. 50 (/). It will be 
clear that the prism ABC could be separated from the remainder 
of the system if it were desired to obtain a considerable separation 
of the incident and emergent rays, as, for example, in a periscope. 

Reflecting Telescopes. In 1639 Mersenne proposed to make a 
telescope from spherical reflecting surfaces* by substituting a con- 
cave reflector for the objective, and a convex reflector for the nega- 
tive eyepiece of a Galilean telescope. While it is true that reflecting 
astronomical telescopes are of great importance, the eyepiece in 
modern instruments is always of the refracting form. 

The "parabolic" mirror (of which the surface is a paraboloid 
of revolution) brings a bundle of rays parallel to the axis to a focus 



* Reflecting systems are sometimes spoken of as "catoptric" systems, i 
distinction from "dioptric" systems of lenses. Combinations of mirrors an 



lenses may be called "catadioptric" systems. 



in 
and 



THE TELESCOPE 



67 



without spherical aberration. In the first method of using the 
mirror (Fig. 53), a photographic plate is mounted in the focal plane 
to register the image, which is not observed visually. The plate 
obscures a certain proportion of the mirror aperture ; the concentra- 
tion of light in the image "star discs" is slightly diminished; a 
circular mount will be desirable for the plate in order to avoid radial 
irregularities in the concentration. 

Aberrations of the Image. Consider the image formation at the 
focus of a parabola AP (Fig. 54(0)) ; while it is free from spherical 
aberration, it is afflicted with coma. 
It was shown above that the opti- 
cal sine condition for freedom from 



coma is 



-?, = const, 
sin a 



In our case 



sin a 



is represented 





FIG. 53. PLATE IN Focus OF 
ASTRONOMICAL REFLECTOR 



(a) 



FIG. 54(a). REFLECTION AT A 
PARABOLOIDAL SURFACE 



by the length FP for each zone, and is clearly not a constant ; let 
P be the point x,y, A being the origin of co-ordinates, and AF the 
axis of x. Let AF = /, the " focal length." 

FP* = FQ 2 + PQ 2 

= (/-*) z + :y 2 

The equation to the parabola isjy 2 = 4#, and a =f, so that 
FP = ( f- y . 



i.e. 



In the absence of spherical aberration, the magnification, m m , for a 
marginal zone may be written, using the sine theorem and putting 

V 
sin a =-j, where y is the incidence height on the mirror and / is 



FP = {/+^-, 



68 



APPLIED OPTICS 



the object distance (assumed to be very great), 

A _ (*\ L? !L -?-- 

h) m ~~ \n ' J sin a' ~~ n' /sin a! 
k / \ / 

For the paraxial zone the magnification is 

_ w AF 
m " ~ n' "/ 



FP 



Petzval surface. 




FIG. 54(6). IMAGE FIELDS WHEN OBJECT FIELD is AT INFINITY 





FIG. 54 (c). REFLECTION AT A 
HYPERBOLOIDAL SURFACE 



v y 



(d) 

FIG. 54(<f). REFLECTION AT AN 
ELLIPSOIDAL SURKACE 



Hence the offence against the sine condition is 



AF 



THE TELESCOPE 69 

v i 

For the common value 7- = for a mirror, the above amount comes 

/ 20 

to -v . We mentioned above that general experience shows that 

for ordinary aperture ratios, the offence against the sine condition 
should not be allowed to exceed one part in 400; hence for para- 
bolic mirrors of ordinary aperture, the coma is well within the 
tolerance and compares well with that characteristic of well-designed 
refractors; the amount speedily grows, however, when relatively 
large aperture ratios are used. 

The Astigmatism of the Oblique pencils will be encountered in 
precisely the same way and for the same cause as that in a refract- 
ing telescope objective passed centrally by the principal ray, i.e. 
the fore-shortening of the aperture with regard to the tangential 
rays ; the amount will be approximately the same as that calculated 
on page 147 of Vol. I. 

The curvature of the field may be calculated from the ordinary 
expression for that characteristic of a refracting surface (Vol. I, 
page 139). We had 



i _ i In' -n\ 

n'R\. " "ft'Ri ~" ~ V VV/ 



Putting n' r= - n, the field curvature for an infinite radius in the 

2 
object field comes out to -; the radius of the Petzval surface is 

equal to the focal length. Hence the result is better than with a 
refracting telescope objective. 

Tangential and Sagittal Fields for a Concave Mirror. When the 
ordinary equations for the conjugate distances in narrow tangential 
and sagittal fans refracted at a spherical surface (Vol. I, page 296) 
are adapted to the case of reflection by putting n' = - n t we obtain 

II 
Formula for tangential fan : -, + - = 

* t 7 



COS. I 



. * , I I 2 COS. I 

Formula for sagittal fan : - , + - = 

b s s r 



Referring to Fig. 54(6) we see that the image distance when t is 
infinite is given by /' = ( 1 cos. i , and is therefore found by drawing 

a perpendicular FT from the focus F to the principal ray AP. The 
tangential field near the axis is therefore spherical and has a radius 



70 APPLIED OPTICS 

, since from elementary geometry: "the angle in a semicircle is 

4 

a right angle." 

Again, s' = -. .*. The sagittal field is evidently flat. If FP 

(2 COS. if) 

is the Petzval surface, 

h 2 
the interval TS = . . , approx., where h = FS, 

2 (?) 

and SP = 



so that TS = 2 SP, and, as we expect from theory, the tangential 
focal line is three times as far as the sagittal line from the Petzval 
surface. The above results apply strictly, of course, to very narrow 
apertures, but the system compares favourably with a refracting 
telescope. 

The most important advantage of the reflector is, however, that 
the chromatic aberrations, both axial and radial, are absent. 

The main difficulty in securing a good performance with a reflec- 
tor is due to the necessary extreme accuracy of figure of the surface. 
In order that the disturbance from the object point may meet in 
the image within the usual tolerance of JA path difference, the 
figure of the surface must be accurate to within JA. For methods 
of testing see page 249. 

Cassegrain Telescope. In order to appreciate the action of the 
Cassegrain and Gregorian telescopes, we may introduce a simple 
theorem in analytical geometry. If AP, AT' are two branches of 
a hyperbola (Fig. 54 (c)), 



b* 
the normal at the point P (x'y f ) is 

x - x' y -y' 

which intersects the x axis at the point T given by putting y = o, 
and obtaining 



THE TELESCOPE 71 

But if e is the eccentricity of the hyperbola, 

fe 2 

= - (1 - **) 

a 2 v ' 
so that x = #'d 2 

and the distances of the intersection point from the two foci are 



ST = CT + CS' = x'e* + ae 
Now we have 

(ST) 2 = (ae + x'}* + / 2 
and y* = (a 2 -* 2 )(i -* 2 ) 

so that (ST) 2 =-- (xe + a) 2 

and similarly (SP) 2 = (xe - a) 2 

SP xe-a ST 
Hence ^p = '+~ a = gf ( from above ) 

By Euclid, VI, A : " If, in any triangle the segments of the base 
produced have to one another the same ratio as the remaining sides 
of the triangle, the straight line drawn from the vertex to the point 
of section bisects the external angle." 

Hence a ray OP proceeding to one focus S of the hyperbola 
would, if reflected at P, proceed towards the other focus S'. The 
action is exactly analogous to that of the conjugate foci of an ellip- 
tical reflector (Fig. 54(d)). 

In the Cassegrain reflector (Fig. 55(*))* the main mirror is pierced 
by a central aperture. Instead of the Newtonian plain mirror, a 
convex mirror of hyperbolic form is placed so that its two foci are 
situated at S' and S (the main mirror focus and the region of the 
aperture respectively). In this case an image free from spherical 
aberration will be formed at S. Sampson recommends an eccen- 
tricity of about 3. This determines the position of the auxiliary 
reflector. 

It will be noticed that the use of such a reflector is like that of a 
negative lens in a telephoto system ; it increases the magnification, 
approximately in the ratio of the distances of the surface from the 
two foci of the hyperbola. The diameter of the small mirror and 
that of the aperture in the large one are about equal in the usual 

case. If the diameter of the small mirror is - that of the large one, 

P 

then the resultant focal length can easily be shown to be (p - i)/' 
approximately, where/' is that of the main mirror. 

6 (3494) 



72 APPLIED OPTICS 

The Gregorian Reflector. In this case the auxiliary reflector is 
placed beyond the focus of the main mirror, and should therefore 
be given an elliptical section. The Gregorian form is less common 
than the Cassegrain. 

HerschePs Telescope. Herschel's method of using the reflector is 
not widely used now; it is shown diagrammatically in Fig. 55(6). 
While it possesses some advantages in simplicity, and consequent 
conservation of light, over the arrangements described below, it is 

V V 
clear that the minimum angular tilt will be about or -. Herschel's 

I VI 

mirrors had an approximate aperture ratio of , i.e. ;.= . 
rr v 10 2/ 40 

Consequently, the inclination of the principal ray to the axis will 
be about 1-5. Therefore the difference of the sagittal and tangen- 
tial focussing distances will be /tan 2 (1-5). As K6nig 8 points out, 
for a mirror of 1-22 metres diameter and 12-2 metres focal length 
the intercept between the astigmatic foci is therefore 4-2 mm., and 
if an eyepiece is used which will make the telescope magnification 
400, the intercept requires a 4-dioptre difference of accommodation 
for the two foci. The removal of this defect calls for special figuring 
of the mirror. In addition to the astigmatic defect, the effects of 
coma also become objectionable at such a comparatively large 
distance from the axis. 

Newton's Telescope. It will be remembered that Newton saw no 
hope of making achromatic lenses, and therefore turned his atten- 
tion to reflecting telescopes (1672). In the Newtonian form (Fig. 
55 (c)), a small auxiliary mirror, at 45 to the principal ray, deflects 
the convergent beam through 90, so that the focus is formed 
approximately in the locus of the surface of the telescope tube 
where it may be conveniently examined by an eyepiece. If D is 
the diameter of the objective, the distance of the mirror from the 

D 

focus is -. The shape of the mirror must clearly be an elliptical 

. D 2 

conic section, the minor axis being , and the major axis approxi- 
mately -r--r . By this arrangement, the centre of the field lies on the 

/V 2 
axis, and unsymmetrical aberration is avoided. 

General Remarks on Reflectors. It is sometimes the practice to 
equip a reflecting telescope with alternative methods of securing the 
image. Thus in the Mount Wilson photographic reflectors, the image 
can be secured either with a Newtonian reflector, or with a com- 
bination of Cassegrain reflector and auxiliary Newtonian reflector, 



THE TELESCOPE 



73 



the latter being placed close to the mirror. Practically all modern 
mirrors are silvered on glass ; the use of speculum metal has been 
very largely discontinued. Many difficulties are encountered when 
very large mirrors are required, since thick glass discs are neces- 





FIG. 55(0). CASSEGRAIN REFLECTING TELESCOPE 




Fir.. 55(6). HRRSCHEL'S REFLECTING TELESCOPE 



j . 



(C) 



FIG. .5.5 (r). NEWTON'S REFLECTING TELESCOPE 

sary which are difficult to keep free from the disturbing effects of 
internal strain and temperature variations ; these may distort the 
surfaces. For this reason it has been proposed to grind the discs 
in fused quartz, which has a much lower coefficient of expansion 
than glass. 

REFERENCES 

1. An account of early history will be found in a paper by Court and 
von Rohr, Trans. Opt. Soc. t XXX (1928-29), 207. 

2. R. Kingslake: Trans. Opt. Soc. t XXVIII (1926-27), No. 4. 



74 APPLIED OPTICS 

3. Smith and Cheshire: Constructional Data of Small Telescope Ob- 
jectives (Harrison, London, 1915)- 

4. Confady: Applied Optics and Optical Design (Oxford University 
Press). 

5. Baker: Trans. Opt. Soc., XXIX (1927-28), 49. 

6. Konig: Die Fernrohre und Entfernungsmesser (Berlin. 1923). P- 7- 



CHAPTER III 

THE MICROSCOPE 

IT is possible that compound microscopes were constructed by 
Giambattista della Porta, but the actual introduction of the instru- 
ment must be credited to Zacharias Jansen, and Lippershey, of 
Middelburg, about the year 1610. The principle of the telescope 
would naturally suggest that of the microscope, and it is known 
that Galileo also developed both these instruments. 

The practice of microscopy was developed by Hooke, whose 
Micrographia was published in 1665. The researches of Leeuwenhoek 
(born 1632) with the simple microscope, and Bonnani with his com- 
pound instrument (1697), helped on the development, but the early 
objectives were necessarily made of very small aperture in order to 
avoid overwhelming chromatic aberration. The discovery (1733- 
1758) of the achromatic lens finally enabled a great advance to be 
made in this instrument, but progress was slow until the beginning 
of the nineteenth century when Marzoli (1808-1811) constructed 
plano-convex cemented objective lenses, used with the plane side 
turned towards the object. He was followed in this step by 
Chevalier (1825). About this time J. J. Lister, in England, and G. 
B. Amici, in Italy, began their work. In 1830 Lister published his 
discovery of the two pairs of spherical-aberration-free conjugate 
points for these plano-convex doublets. Principles of compensation 
were used in Amici's objectives of 1827. Lister was one of the 
earliest to realize the value of a wide aperture in the object glass. 

The theory of the "Lister" points will be understood from Fig. 
56(0). The "thin lens" approximate discussion of Chapter II indi- 
cates a shape somewhat similar to that of the diagram for the 
aplanatic crown-flint achromatic combination used for a telescope 
object glass. If the object point B is brought nearer the lens it is 
still possible to find a lens corrected for spherical aberration, and 
approximately for coma, which has a form not greatly different. 
Reference to the spherical aberration equation shows, however, 
that having once calculated such a form we could derive an equa- 
tion giving the aberration of the crown lens in terms of f ( / lt the 

vergence (y J to the object. The flint lens would furnish another 
equation in J/ A9 which would be transformed by putting 



75 



7 6 



APPLIED OPTICS 



We should therefore obtain a quadratic equation in //i, for the 
spherical aberration of the complete lens, and the solution would 
furnish two roots which represent two pairs of conjugate points free 
from spherical aberration for small apertures of the lens. It is found 
that the residual coma has an opposite sign in the two cases so 
represented. 

Two such pairs are shown in Fig. 56(0). One pair, B and B', are 
real object and image points, but of the other points, C and C', 
the point C represents a virtual object. 

Fig. 56(6) shows an aplanatic combination of two cemented doub- 
lets. The first lens (a) works on the conjugate pair represented by 




FIG. 56(a). THE APLANATIC POINTS OF A DOUBLET LENS 



B' 




FIG. 56(6). LISTER OBJECTIVE 

B and B 7 in Fig. 56(0) ; but lens (b) works on the conjugate pair C 
and C'. Reverse the direction of the light, treating C' as the object 
point, and we have an aplanatic objective of the Lister type. The 
advantage thus gained is the production of a system of short focal 
length in which the aberrations are not excessive. If it is sought to 
attain the same short focal length by using one doublet lens and 
increasing the curvatures of both components, then the aberrations 
of the system become much more troublesome, for even if spherical 
aberration is corrected for one zone, there are very serious amounts 
of spherical aberration for other zones (zonal aberration), for the 
restriction of which it is necessary to diminish the aperture of the 
lens very severely. 

The importance and exact significance of the "aperture" of the 
lenses will be dealt with before discussing the further development 
of the miscroscope objective. 

Magnification. The microscope consists of two positive systems, 
"a" and "b" (Fig. 57), of which the adjacent principal foci are 



THE MICROSCOPE 



77 



separated by a distance g ; the systems are the objective and eye- 
piece respectively. 

The object of height h has an image of height h' which is pro- 
jected by the objective into the first principal focus of the eyepiece. 

*' = -*_ 

The image is viewed under the angle >' where 

/ A' hg 

(U -JJ- = rt fT 
J b J aJ b 



Af *~~ 




- 3 ^ 




/?' f^". 


x > 




PL' CT-*~~^ JT' 
a, /^ ^-*.^^ ,/t 




yZ'^-''' 


Ob 


//ft 


?/?"yi/ 




"'' 



Eye-piece 

FIG. 57. A CVAUSSIAN DIAGRAM OF THE MICROSCOPE 

An object viewed by an unaided eye will naturally be placed in 
the nearest point of accommodation, which is at a distance ft, say, 
from the first principal point of the eye. Hence, 

Angular subtense of object to unaided eye = - -3 

Hence, 

Angular subtense of image to eye ___ 

^ Angular subtense of object when at near point 



_ 

faf'b 

The expression can be written ( jr ) ( j,\ and it will be seen from 

above that the first bracket represents the ''first magnification" of 
the objective, i.e. ratio of linear size of image to object measured 

/ ft \ 
transversely to the axis. The second bracket ( /r) is clearly the 

"magnification" of the eyepiece used as a simple magnifier. 

Modern models of the microscope have a length such that the 
normal value of g, the "optical tube length" is iSomm. The 
older English tube had an optical tube length of 10 in. 



78 APPLIED OPTICS 

Amongst English makers it is usual to specify the eyepieces by 
the magnification, i.e. (approximately) 

loin. 

focal length of eyepiece in inches 
250 



or 



focal length of eyepiece in millimetres 

taking 10 in. or 250 mm. as measures of the least distance of dis- 
tinct vision. In the Continental catalogues, however, the objective 
magnification is listed as 

Objective magnification (Continental) 

250 

focal length of objective in millimetres 
and 
Eyepiece magnification (Continental) 

180 

focal length of eyepiece in millimetres 
This is equivalent to writing the microscope magnification as 



Magnification of microscope 



~(yv (A) 



which arose from Abbe's special method 1 of deriving the magnifica- 
tion formula, which need not be reproduced here. We may, how- 
ever, note that when the image formed by an instrument is projected 
"to infinity/' then 

Q 

Magnification = - 77 

where/' is the focal length of the combined system, given by 

/' __ _f'*f' b 

Bg 

Thus Magnification = 7,^-57- 

J at * 

Resolving Power of the Microscope System. It will be assumed 
in the first instance that the object field consists of an assembly 
of discrete self-luminous points, although such a condition rarely 
obtains in the actual use of the microscope, and it will be necessary 
later to review the conclusions reached by such an assumption. 

In Fig. 58 (a) let R and R' be the entrance and exit pupils respec- 
tively of the whole image-forming system, including the eye if 



THE MICROSCOPE 79 

visual observation takes place. Let a and a' be the extreme angles 
with the axis, made by rays passing through the axial points of 
object and image respectively. Then 

6iA' 

Radius of Airy disc in the image = h f = - . , 

J & sin a 

where A' is the wave-length of light in the image space. 

In order to find the dimension h in the object plane, which has 
an image of size h' (the radius of the Airy disc) in the image field, 
(assuming the system to fulfil the " sine condition") we use the sine 
relation 

nh sin a = n'h' sin a' 
whence 

n'h' sin a' _ 0-61 n' X __ 0-61 A 
~ n sin a n sin a n sin a 

where A is the wave-length of light in the air. 




FIG. 58(0) 




k- W- 



FIG. s$(b) 

It was explained in Vol. I, page 108, that the closest possible 
approach for two elementary star images which allows of recog- 
nizable resolution is approximately this radius of the Airy disc. 
Thus the distance of closest approach for two object points is in- 
versely proportional to the product of the refractive index of the 
object space, and the sine of the semi-aperture of the cone of rays 
entering the entrance pupil of the system from the object point. 
Abbe called this product the "Numerical Aperture" of the system, 
and we shall write it NA, thus 

_ Q-6iA 

* ~~ NA 



8o APPLIED OPTICS 

It is remarkable that the only necessary assumption about the con- 
struction of the optical system is its ability to produce images free 
from aberrations of optical path. 

Visual Resolutions. As in the case of the telescope, the visual 
resolution of the images requires that they shall be presented to 
the eye under a sufficient angular magnitude. If co is the angle 
subtended by the image at the eye, then 

co = (Angle subtended by the object at the near point) 
X (magnification) 



= ( lw a , numerically, where m a is the magnification, 
o-6iA m a 



if h is the separation necessary to secure that the images are just 
resolved physically. In order to present the image separation under 
the very minimum angle for visual resolution (say one minute of 
arc, which is -00029 in angular measure), we get from the formula 



by taking A = -00058 mm. and ft = 250 mm. 

m a (the minimum essential magnification) = 200 NA approx. 

It will be found, however, that this critical image distance should, 
in actual practice, be presented under a much larger angle than one 
minute. Four or five minutes will not be too large. The formula 
required is 

m a = 200 NA (to in minutes) 

so that if we are working with NA = 1-2, and we desire to make 
a) = 3' in order to have comfortable observation, a magnification 
of 720 will be called for. The required eyepiece can then easily be 
calculated from the formula 

_ 250 x 160 
J * ~ fa X m a 
Take, for example, a case in which m a = 750 and/' = 2 mm. 



TM_ , 2 5 x 

Then / b = = 27 mm. approx. 

In the English system this represents a magnifying power for 

250 
the eyepiece of = xg approx. The "Continental magnifying 



THE MICROSCOPE 81 

180 
power" is = x 7 approx. It is quite usual to work with magni- 

fying powers considerably higher than these when employing a 
2 mm. immersion lens. In fact, the ordinary microscope system 
employs a great deal of "empty" magnification in contrast to the 
case of small telescopes and prism binoculars, where the magnifica- 
tion is usually not nearly high enough to use the full resolving power 
of the system. 

Size of Exit Pupil and Measurement of Magnification. Let R and 
R' (Fig. 58(6)) be the axial points of the entrance and exit pupils 
of the optical system of a microscope, while B and B' are the axial 
points of object and image respectively. The extreme ray from B 
passes through the margin of the exit pupil after refraction by the 
system, since entrance and exit pupils are conjugates, and it must 
be directed from B'. If h and h' are the sizes of object and image, 
then 

nh sin a = n'h' sin a' 

P' 
But n sin a = NA , and sin a' = p>v>> within allowable approxima- 

tion if R'B' is large. The sine relation becomes 



= (n'p') X (angular subtense of image 
taken at centre of exit pupil)* 



h fn'p'\ 

Hence, 3 . NA = ( -A 1 X (angular subtense of image) 

But -3 is the numerical value of the angular subtense of the object 

held at the near point. Hence since n' = i, 

NA . ft Angular subtense of image 

7 = - -- ------ -- r~TT* . = Magnification. 

p Angular subtense of object 

This gives a useful method of finding the magnification of the 
microscope. 

Since we found above a minimum value for the necessary mag- 
nification 

m a = 200 NA 



" 200 N A 

* We are only concerned with numerical relations, and shall not need to 
consider the signs of the angles in this section. 



APPLIED OPTICS 



The conventional value for ft is 250 mm., so that p 9 should then 
be 1-25 mm. The diameter of the exit pupil of the microscope 
ought to be at most 2-5 mm. if all the resolvable detail is to be 
visible to the eye. It will be an advantage, as shown above, to 
employ magnification four or five times as great, so that the exit 
pupil diameter may well be reduced to o5 mm. 

Depth of Focus in the Object Space. Referring to Fig. 5 8 (a), let 
B and B' represent conjugate points for which paraxial and mar- 
ginal optical paths are equal. We may displace the object point 
to B! along the axis, so as to cause a difference of paraxial and mar- 
ginal optical paths of -before the deterioration of the image becomes 

4 

apparent through the phase differences of the disturbances arriving 
in the image. See Vol. I, pages 108-112 and page 141, Join 
and draw BC perpendicular to B X P. The difference 

Increase of axial path - increase of marginal path 

= B X B - BjC 
within a small quantity of the second order, giving 

Optical path difference = n . BjB (i - cos a) 
and writing B^B = dx, we obtain 

Optical path difference = 2n . dx . sin 2 - 

If this may amount to -, 

4 



8n sin 2 a/2 

where A is the wave-length of light in air. Note that a is 

(NA\ 
) where NA is the numerical aperture of the microscope 

objective, and n is the refractive index of the object space. 

Since the step is allowed on either side of the focus, the total range 
is double the above, and the following approximate table may be 
calculated for blue-green light. 

TOTAL DEPTH OF Focus 



Numerical Aperture 
of Objective 


Depth of Focus 
in Air 


Depth of Focus in 
Medium of Refractive Index 

i'5 


25 
50 
75 

1*00 

1-25 


0079 mm. 
OOIQ 
0008 ff 


0122 mm. 
0030 
0013 
0007 
0004 



THE MICROSCOPE 83 

The depth of focus in the image depends on the aperture ratio 
of the convergent beam. With average objectives it may be of the 
order of one or two millimetres. When a projection eyepiece is 
used to form the final image for photomicrography the depth of 




JFiG. 5 9(a) 



Concentric 
Surface 



Aplanatic 
Refraction 




FIG. 59(6) 

focus in the final image will be very considerably greater still, 
owing to the very small angular apertures of the convergent pencils ; 
it may be several inches or more. Note that this does not refer to 
focus changes produced in other ways than by moving the final 
receiving screen. 

High Power microscope Objectives. It will now be appreciated 
that the resolving power of the objective is dependent on the 
Numerical Aperture which can be attained, and, hence, the means 
for increasing the numerical aperture to the widest possible limits 
must receive attention. 

In Vol. I, page 20, an explanation was given of the aplanatic 



8 4 



APPLIED OPTICS 



surfaces of a sphere. Referring to Fig. SQA, the sphere S of radius 
r has a refractive index n' t and is immersed in a medium of refrac- 
tive index n. An object point B within the sphere lies on a con- 



rn 
centric circle drawn with radius ; all rays from this point suffer 

aplanatic refraction at the surface of the sphere. The corresponding 

rn' 
virtual image B' is situated on a corresponding circle of radius , 

and is free from spherical aberration and coma. Thus the wide 
divergence of the fan of rays from B is very considerably reduced. 








FIG. 6o(a). SECTION OF 
IMMERSION ACHROMAT 

2mm., t /K'.&/- x '*5 



FIG. 60(6). SECTION 

OF TYPICAL 

APOCHROMATIC 

OBJECTIVE 



FIG. 6o(c). SECTION OF 
TYPICAL DRY LENS 



g 



0-65 



2 mm., 



* 3 x '4 



Such a fan, with reduced divergence, may now encounter a 
spherical refracting surface which has its centre coincident with the 
common divergence point B' of all the rays, which thus suffer no 
deviation and no aberration. This condition is illustrated in Fig. 
59(6), where the rays enter the first surface of the meniscus lens. It 
is clear, however, that the back surface of this lens can also be made 
to produce an aplanatic refraction if B' is one of the aplanatic 
points, and the divergence of the rays can be still further reduced. 
In this stage they may be made to encounter a "Lister" combina- 
tion or other arrangement by which the divergence from a virtual 



THE MICROSCOPE 85 

image is changed into convergence towards a real image, as sug- 
gested in Fig. 60 (a). 

The object point cannot be placed, of course, within a spherical 
lens ; hence this front lens is cut off by a plane face, and the con- 
tinuity of refractive index of the media is ensured by placing a 
film of liquid, usually cedar oil of refractive index 1-517, between 
the lens and the cover glass (n = 1-515, about) which is practically 
in contact with the object. There is then little change of refractive 
index between the front lens and the farther side of the cover glass, 
as the front lens is usually of glass of fairly low refractive index, 
not above 1-54. The above principles represent the main basis of 
the control of spherical aberration and coma in high power immer- 
sion lenses. 

On the other hand, the two aplanatic refractions involve a large 
amount of chromatic under-correction which, therefore, has to be 
compensated by the back lens or lenses of the system, and which 
is usually only attained by the use of fairly deep curves in the con- 
tact faces of these lenses. The result is that although spherical 
aberration can be compensated for one zone without very great 
difficulty, it is somewhat difficult to obtain it for all zones, i.e. zonal 
aberration is likely to be present. The successful balancing of the 
aberrations calls for considerable skill and resource on the part of 
the designer. 

Apochromatic Objectives. Efforts to reduce the secondary 
spectrum characteristic of the "achromatic" systems made of 
glasses of the older types were made by Abbe when the new Jena 
glasses were introduced, but the success of his "apochromatic" 
lenses was largely due to the use of "fluorite" as a component of 
some of the lenses. The "optical constants" of fluorite as com- 
pared with two other glasses are (compare Vol. I, page 233) 





*D 


H f - W C 


V 


n f - n D 


G1 - H V 


P 


y 


Fluorite 


1-4338 


00454 


95'5 


00321 


00256 


707 


563 


Boro-silicate crown 


1-5160 


OOSOC) 


68-3 


00567 


00454 


-701 


501 


Telescope flint 


i\5237 


01003 


5^-2 


00708 


00577 


706 


575 


Dense flint . 


1-6225 


OI72<) 


36-0 


01237 


01052 


715 


608 



It will be recalled in the discussion of achromatism (Vol. I, 
Chapter VII) that the "powers" of opposing lenses in achromatic 
combinations are lowest when there is a large difference of V values ; 
when the powers of the lenses are relatively low, the curves are rela- 
tively shallow, and zonal aberration is less likely to arise. It was 



86 APPLIED OPTICS 

also shown that the residual secondary spectrum was reduced by 
a large difference of V values ; on the other hand, the relative par- 
tial dispersions of fluorite are quite comparable with those of 
ordinary crown glasses, and also come fairly close to those of an 
extra light flint such as "telescope flint." Hence with combina- 
tions using fluorite in place of the "crown" lens, and "extra light 
flint" in place of the ordinary flint lens of an achromatic combina- 
tion, it is possible not only to reduce the secondary spectrum, but 
also largely to reduce the zonal aberrations. Fig. 60(6) shows an 
" apochromat." Low powers (dry) are also made. 

Apochromatic objectives are found usually to possess noticeable 
chromatic difference of magnification, which is best compensated 
by the use of special eyepieces. (See page 87.) 

High Power " Dry " Lenses. Such lenses are made with numer- 
ical apertures up to 0-95, and, therefore, the extreme rays in the 
object space may have to make an angle of shr 1 0-95 = 72 with 
the axis. The arrangement of the majority of dry lenses of inter- 
mediate power is similar to that shown in Fig. 6o(c), which might be 
regarded as a Lister pair combined with a hemispherical front lens. 
In this case, however, considerable spherical and chromatic aberra- 
tion arises at the front face of the "hemispherical" lens, and this 
has to be corrected by the rear components ; not only this, but the 
aberration arising at the cover glass surface must be considered. 
If the front lens and the cover glass are of equal refractive index, 
the cover glass may be considered as an addition to the thickness 
of the front lens in the course of design, but this means that the 
system will usually be very sensitive to the thickness of cover glass 
with which it is used. Referring to Vol. I, page 14, it will be found 
that refraction at a plane surface transforms a spherical wave-front 
into an elliptical wave-front, and, if a decrease of refractive index is 
involved, the major axis of the ellipse is the normal to the surface, 
so that the disturbances travelling along oblique paths suffer rela- 
tive retardation. Hence, since an infinitely thin cover glass could 
produce no effect, it will be clear that increasing thickness of cover 
glass produces more "over-correction" in the spherical aberration 
sense. 

In practice, slight over-correction arising through the use of a 
too thick cover glass can be compensated by using the objective 
with a shorter tube length and, consequently, a greater working 
distance between the front lens and the object point ; this produces 
an opposing tendency to under-correction, mainly in the front lens. 

Eyepieces. Little need be added to the discussion of eyepieces 
given hi the preceding chapter. The Huygenian type is perhaps the 



THE MICROSCOPE 87 

most widely employed, the design varying with the required magni- 
fication ; the focal length of the field lens may not exceed 1-5 times 
that of the eye-lens at the lowest magnifications. Rings are fitted 
to a set of eyepieces so that they are parfocal, and can be inter- 
changed without large changes of focus ; the optical tube-length is 
thus kept constant. Various eyepieces of the " orthoscopic " and 
other types are in favour with some microscopists who prefer a very 
wide angular field. 

Ordinary achromatic objectives have little chromatic difference 
of magnification, and are best used in conjunction with ordinary 
Huygenian eyepieces. With the jV in. oil immersion objectives, and 
with the apochromatic objectives, there is a slightly greater mag- 
nification for the "blue" or shorter wave-lengths, as compared with 
the red, and this is best overcome by the use of so-called "com- 
pensating" eyepieces which are designed to give a correspondingly 
greater magnification to the red than to the blue. If held to the 
light the border of the field stop 'appears tinged with orange in a 
compensating eyepiece, and blue in an ordinary Huygenian. Abbe 
designed a series of apochromatic lenses for the firm of Zeiss in which 
all had the same chromatic difference of magnification, and thus 
could be used to advantage with the same compensating eyepieces, 
and various other makers have special arrangements in this 
connection. 

The "projection eyepiece" usually has a single field lens with a 
triple projecting lens. The exit pupil is limited by a small stop in 
order to cut out stray light. 

As mentioned above, the ordinary forms of eyepiece have a 
negative curvature of field according to the usual formula 

f - i > 



where n is the refractive index and/' the focal length of a constituent 
lens of the system reckoned as for an infinitely thin lens with faces 
of the same curvature. In 1918, Conrady proposed to introduce as 
the front lens of an eyepiece system, a carefully computed achromatic 
lens of negative power which would be sufficient to give a positive 
curvature to the field of a telescope objective. This might then be 
followed by a suitable eyepiece. 

An anastigmatic flat field telescope on these lines was later 
independently designed and produced by H. Dennis Taylor. 

In microscope objectives, the curvature of field of the primary 
image is so much greater that an achromatic negative combination 
can hardly be expected to do more than flatten the field of the 

7^(5494) 



88 APPLIED OPTICS 

primary image without compensating the curvature due to an 
observing eyepiece. 

In 1922, H. Boegehold and A. Kohler, of Messrs. Carl Zeiss, 
brought out the "Hoir^l," a projection lens of negative power, 
which flattens the field of the microscope objective for which it is 
designed. As in the case of the Galilean telescope, the exit pupil 
lies within the lens ; the field of view to an eye placed behind it is 
very small indeed, and the system is useless for visual" work, but 
the system is quite satisfactory for photography. 

Numerical Aperture: General Remarks. Since "numerical aper- 
ture" is the product of refractive index and the sine of the angle of 
obliquity of the extreme ray, a medium of refractive index n can- 
not transmit a ray of numerical aperture greater than n. Hence, if 
an object is in air, the numerical aperture of the extreme rays 
cannot exceed unity. The law of refraction, n sin i = n' sin t', 
shows that the "numerical aperture" characteristic of a ray is 
independent of any refractions at plane surfaces normal to the axis, 
such as those of the cover glass, etc.; therefore, if the original 
object is in air, the objective cannot work at a numerical aperture 
greater than unity. If the medium is water (refractive index 1-333), 
the limit of possible numerical aperture is equal to or under 1-333, 
provided that all the media have at least this refractive index. To 
take full advantage of an objective having any high NA such as 
1-6 (such could be constructed), it would be necessary to have the 
mounting medium, cover glass, and immersion medium of this 
refractive index at least. The action of dark-ground illuminating 
systems has to be considered with this point in mind. (See below, 
page 119.) 

Historical Note. As remarked above, Lister's paper on the 
aplanatic points of an achromatic doublet was published by the 
Royal Society in 1830. About the same time, G. B. Amici, in Italy, 
was working somewhat on the same lines, but attempting to find 
how the spherical aberration of higher orders and the coma could 
be eliminated by the mutual action of doublet lenses, none of which 
were necessarily working strictly aplanatically. In the "fifties," 
Amici introduced the strongly curved plano-convex lens as a front 
lens, which later developed into the hemispherical and hyper- 
hemispherical form of modern high power objectives. This allowed 
of a great increase of numerical aperture, the importance of which 
in resolution of fine structures had been realized by various workers, 
and also noticed by Lister in 1830. 

Immersion systems were first introduced as a means of over- 
coming the effects of the thickness of the cover glass, and also its 



THE MICROSCOPE 



possible variations and irregularities; the liquid employed was 
almost invariably water. Such systems were employed by Amici 
and others. The great advantages of immersion systems in high 
power work were only slowly realized, although advances had been 
made by Tolles in America, who used glycerine and balsam immer- 
sion lenses. Abbe published his diffraction theory in 1877, and 
explained on this basis the theory of the effect of homogenous 
immersion in increasing numerical aperture and securing increased 
resolving power. He also introduced the apochromatic lenses. 

The Illumination of the Object. The assumption made above, 
that the object in the microscope consists of an assembly of dis- 
crete self-luminous points, is seldom realized in practice. The 




Source 
of light 



Condenser 
Diaphragm 



J 



Diaphragm 



FIG. oi. THE USE OF THE SUB-STAGE CONDENSER 
( Diagrammatic) 

majority of biological specimens are viewed by transmitted light, 
while opaque objects, such as metallurgical specimens, are illu- 
minated by light which they reflect into the instrument. 

Material objects which are non-self-luminous may obstruct light 
completely or partially. They may also reflect, refract, diffract, or 
scatter light. By these actions they become "visible/ 1 and the 
interpretation of the "image" formed on the retina usually leads 
us to a more or less correct knowledge of the physical character of 
the corresponding object, but often mainly through collected experi- 
ence. The simplest case is that of a "silhouette" pattern in which 
the geometrical outline pattern of a flat object is seen by the 
obstruction of light, partial or complete. If the object exercises 
selective absorption of light, the geometrical boundaries may be 
indicated by the fields of various colours. 

If such an object is to be viewed away from a microscope, we may 
hold it in front of a uniformly luminous white surface to obtain the 
best view, and the "true" colours. In the microscope we may 



go APPLIED OPTICS 

project the image of a uniformly illuminated or luminous white 
surface approximately into the plane of the object by the aid of the 
substage "condenser" system (Fig. 61), and this affords a close 
approximation to the best condition for obtaining an image pat- 
tern similar to the object itself in a geometrical sense. 

The above conditions may not, however, be at all suitable for 
yielding an image from which the physical characteristics of certain 
other classes of object can be inferred. As an experiment, take a 
piece of glass with a pattern moulded in its surface, such as is em- 
ployed for doorways and the like. Where such a glass is held close 
to a broad, evenly illuminated source of light, the ''pattern" prac- 
tically disappears. In order to obtain a strongly marked, easily 
visible appearance of some kind, the glass is held between the eye 
and a small source of light which is thus giving a more or less 
directed beam. The "pattern" stands out strongly because cer- 
tain parts of the glass refract or reflect light into the eye, and other 
parts do not. The effects of cumulative experience again largely 
enter into the interpretation, which may or may not be true. Hence, 
we need to guard against the easy assumption that there is always 
a possibility of obtaining a "true" picture of the object. The 
successful microscopist will adapt his illumination to the end in 
view. He will regard any image pattern not so much as a picture 
of the object, as a piece of physical evidence, from which the physical 
characteristics of the object can possibly be established. The differ- 
ent methods of illumination possible with the modern substages 
should be regarded as additional weapons in the armoury. 

Substage Condensers. Theoretical considerations (to be developed 
more fully below) show that in order to obtain the optimum resolv- 
ing power, i.e. the distinction of the finest possible detail in many 
classes of object, (but not necessarily every case), it should be pos- 
sible, if required, to pass light through the object in such a way 
that the rays diverging from any one point of it spread out and 
fill, uniformly, the whole aperture of the objective. The case is 
illustrated in Fig. 61. If the refractive indices of the media on the 
two sides of the object slide are n^ and n 2 , and the angular diver- 
gence of the extreme ray which can enter the objective from an 
axial point of the object plane is <%, then we must have 

n^ sin (*! < n 2 sin a a 

i.e. the numerical aperture of the condenser system must at least 
be equal to that of the objective if the above condition is to be 
fulfilled. 

In the simplest case, there is no lens system in the substage. 



THE MICROSCOPE 91 

An image of a fairly broad source of light, i.e. a lamp flame or an 
opal glass electric lamp, is thrown approximately into the object 
plane by a concave mirror. The divergence of the light from the 
object plane is sufficient to fill an objective of small numerical aper- 
ture up to about 0-25. It is then advisable to use a small stop 
immediately below the object to limit the illumination to the part 
of the object required. Sometimes when a broad source of light is 
available, the plane mirror may be used to reflect the light through 




FIG. 6*. ABBE TWO-LENS CONDENSER 
. -I Av' 1-2 

the object. The angular divergence of the extreme rays passing 
through any point of the object plane is then determined by the 
angular subtense of the source of light. 

For larger apertures a lens system in the substage is employed. 
Fig. 62 shows a type frequently employed, due to Abbe, with which 
a numerical aperture of 1-2 can be obtained. There is, however, 
considerable spherical aberration, so that the rays do not all pass 
through one point, and there is no correction for chromatic aberra- 
tion. The disadvantage of this aberration will be understood from 
Fig. 63, in which the condenser S brings the marginal parallel rays 
from a small distant source to an axial focus at C intermediate 
rays to B, and paraxial rays to A. If an object is placed in the plane 
of the point C, any point of it can only be illuminated by marginal 
and paraxial rays. A pinhole at C would block the rays from an 
intermediate zone. In practice, the difficulty is usually overcome 
by using a broad source of light, and rays from another part of the 



92 APPLIED OPTICS 

source may then pass through the intermediate zone and through 
the axial point of the object. This could be studied by drawing a 
figure similar to Fig. 63, and bringing a bundle of rays into the 
condenser at a suitable angle with the axis. The use of a broad 
source of light is, however, likely to be accompanied by difficulty 
in controlling the light and restricting the presence of /'stray" and 
unwanted light. 

This uniform distribution of light into the objective is most im- 
portant in practice. It should always be checked by removing the 
eyepiece of the focused microscope, and observing the image 




FIG. 63. EFFECT OF SPHERICAL ABERRATION IN A CONDENSER 

(formed by the objective) of the diaphragm below the condenser, 
or of whatever stop limits the numerical aperture of the extreme 
rays passing through the system. If light diverges uniformly to 
the objective from any point of the object, this image (which is 
seen just above the objective) will appear as a uniform disc of light. 
If a small source is employed, and the condenser is subject to 
spherical aberration, then the disc at the back of the objective will 
probably be illuminated either at the margin and the centre only, 
or in an intermediate zone. This must be remedied by the use of 
a larger source of light, or by the use of an auxiliary condensing 
lens s described below. 

If three or four lenses are employed in the construction of the 
substage condenser, as shown in Fig. 64, it may be made free from 
spherical aberration and given an NA up to 1-40. It is sometimes 
useful thus to be able to use quite a small source, such as a glowing 
thorium pastille heated by a small gas flame, or the glowing tungsten 
ball of a small Point'olite lamp*, as the illumination of the object 

* Manufactured by Messrs. Ediswan Electric Co., Ltd. If a Point'olite 
lamp is used for visual work the intensity of light will have to be reduced by 
a suitable light filter or screen. 



THE MICROSCOPE 93 

can then be limited to one small part under examination. This is 
most helpful in avoiding strong light. In order to make perfectly 
sure of the above test, it is advisable to make the observation 
through a small pinhole in a card placed at the end of the tube. 
This will save any mistake due to glare. 

Chromatic aberration is a little troublesome in cases where it is 
desired to use such small sources, and the best substage condensers 
are of the achromatic type. They are made on practically the same 
lines as objectives of equivalent numerical aperture, but the degree 
of precision required in their manufacture is naturally less, as the 
sole requirement is that they shall secure this uniform distribution 





Three-lens "Aplanatic" Achromatic Condenser 

Condenser NA = 1-40 NA i o 

FIG. 64. TYPICAL CONDENSERS (Zeiss) 

of light into the objective, and any small deficiencies are not noticed 
because a source of light of finite size is invariably employed. The 
main advantage they confer is the possibility of obtaining the proper 
conditions with the minimum of trouble when it is desired to limit 
the illuminated area of the object. A diagram of such an achromatic 
condenser is shown in Fig. 64. In certain cases it may be useful 
to use an objective (reversed) as a condenser, especially when using 
immersion systems of high numerical aperture ; in such a case the 
object will be mounted between two cover glasses. 

The diaphragm which limits the aperture of the condenser is an 
important adjunct. It is usually of the "iris" variety, so that its 
aperture can be reduced from the full lens diameter down to a few 
millimetres, and it is preferably mounted so that it can be displaced 
perpendicular to the axis of the system in order that a narrow cone 
of very oblique light may be projected through the object. This is 
useful in cases where evidence of the very finest structures is sought 
for. 

Condensers required to give a greater numerical aperture than 
i-o must be used as immersion systems, a spot of cedar-wood oil 
being placed between the surface of the objective and the slide, 
otherwise n sin a can never rise above unity, even if the extreme 
rays emerge from the condenser into the air practically at right 
angles with the axis. Even if the condenser is not designed for an 



94 APPLIED OPTICS 

NA greater than i-o, the use of oil immersion saves much loss of 
light by reflection at very oblique angles. If the objective in use 
has an NA of 1-2 or more, it is only very seldom, however, that the 
allowable NA of the condenser (as actually stopped down for use 
in observation) may exceed i-o, and some observers, if pressed for 
time, will omit the use of the oil for the condenser on this account. 
Naturally, an immersion objective must always be oiled on. 

The working distance of the condenser should be so arranged 
that if the source of light is placed at about 10 to 12 inches 
from the mirror, an image of the source will be projected into the 
plane of the object as mounted on a slide of normal thickness, say, 

^ Working Distance of Microscope Condenser^ 

L 2 





Source Auxiliaru Microscope 

Condenser Condenser 

FIG. 65. USE OF AUXILIARY CONDENSER * 

i-o mm. The distance between the front surface of the condenser 
and the slide should then be about 0-2 mm., so that a drop of oil 
may be squeezed out between the surfaces. The condenser will 
always be held in a focusing mount, and a good rack and pinion 
gives ample accuracy for the adjustment. It saves anxiety if a 
suitable stop prevents the surface of the condenser from actually 
rising above the plane of the surface of the stage on which the 
slide has to rest. There can then never be any danger of breaking 
the slide in that way. 

In practice, a number of difficulties may be encountered in secur- 
ing correct illumination which is evenly distributed over the required 
region of the object, amongst which are the following 

I. The Image of the Source is Too Small to Illuminate the Required 
Area of the Object. In this case an auxiliary condenser must be used, 
as shown in Fig. 65. The lens L a projects an image of the source S 
into the lens 1^, which represents the microscope condenser. If Lj 
is placed at the working distance for 1^, and further, if L x projects 
the image without spherical aberration, it (L x ) will act as a uniform 
source, projecting light into the image of S. Clearly, this image 



THE MICROSCOPE 95 

should be large enough to fill the aperture of L 2 sufficiently to attain 
the required AM of the illuminating beam. 

Special "aplanatic" auxiliary condensers can be obtained for the 
above purpose. If only a plano-convex bulls-eye condenser is avail- 
able, it should be employed with the plane side towards the shorter 
conjugate distance to minimize the spherical aberration as far as 
possible. 

If the above arrangement proves impossible for any reason, 
owing, perhaps, to the difficulty of filling the microscope condenser 
aperture, use may be made of the double condenser system shown 
in Fig. 66. The first auxiliarly lens L! projects an image S' of the 
source S into the lens 1^. The lens Lg now projects an image of 





I, 


Iz 


S 

^ 


I 


^^5t>r: 


L, 




~^2* 

** ** 




\ 

... ^3 0/ec 
Microscope Plane 
Condenser 

Vic,. 06. DOUBLE CONDENSER SYSTEM 

L! into the aperture of the microscope condenser, and if L x is reason- 
ably free from spherical aberration, the illumination of the condenser 
will be uniform. The microscope condenser projects an image of 
Lg into the plane of the object, so that the original source of light 
is finally imaged there also. 

If the lenses L! and 1^ are fitted with iris diaphragms, shown in 
the diagram by I x and I 2 , the use of I I clearly controls the aperture 
of the microscope condenser which is illuminated and, hence, the 
numerical aperture of the illumination, while I 2 clearly controls 
the area of the object plane which is illuminated. 

2. The Working Distance of the Condenser is Too Small or Too 
Great. If a very thick slide is used, it may be found that the con- 
denser will not give a satisfactory image of the source or effective 
source when used at the ordinary working distance, even when the 
condenser is racked up as far as possible. If this results in uneven 
distribution of the light into the objective, the source must be 
brought much nearer to the condenser, or an image of it projected 
into such a position as may be required. 

If, on the other hand, the working distance is too great, the 



9 6 



APPLIED OPTICS 



condenser may, of course, be racked down to obtain the required focus, 
but difficulty may then be found in retaining a film of immersion 
oil between the condenser and the slide. The space may, however, 
be partly filled by one or two cover glasses, oiled on both sides, and 
this usually overcomes the difficulty. 

3. The Source of Light may be Unsuitable. If an electric lamp is 
used it should have an opal glass bulb inside a suitable housing 
with one or two windows, the size of the aperture being regulated 
by iris or other diaphragms (see Fig. 67). 

If the electric lamp has visible filaments, the light should be 
diffused by a screen of "ground" or "opal" glass, or even by tissue 




"olour Filter 
Holder 



FIG. 67. SIMPLE MICROSCOPE LAMP 

paper in emergency. In any case, some scheme should be adopted 
for visual observation to obtain a small, bright, uniform patch of 
light about I in. diameter; it is helpful to have an iris or other 
means of reducing this diameter if required. The Point'olite tung- 
sten arc may be adapted for visual observation by one or more of 
the means described above, but some "light filter" preferably with 
variable density will usually be required. Suitable "wedges" of 
neutral glass may be mounted to be moved oppositely so as to 
secure more or less dimming ; an alternative is to vary the thickness 
of a layer of a suitable absorbing liquid. 

Setting up the Instrument for Observation. Perhaps the best 
approach to the theory of the microscope is by an experiment 
which can be followed either practically, or from the descrip- 
tion below. This description gives, in fact, an outline of the pro- 
cedure usually adopted in setting up a microscope for careful 



THE MICROSCOPE 



97 



observations on any object. A microscope is chosen with a well- 
corrected achromatic condenser fitted with an iris diaphragm. The 
source of light is a lamp, such as is shown in Fig. 67, placed at, say, 
10 in. from the mirror. The objects for the experiment are two 
diffraction rulings mounted on an ordinary 3 in. x i in. microscope 
slide and covered by a suitable cover glass. The gratings should have 
bands of 30,000 and 15,000 lines to the inch respectively, and are 
to be used with a 16 mm. apochromatic objective of first class 
quality, which has a numerical aperture of 0-30. The eyepiece 
may be a " x 15 compensating." (Such rulings as mentioned were 
obtainable, commercially, under the name of Grayson's rulings,* 
but there is at the time of writing some difficulty in procuring 
specimens.) 

In setting up the instrument, a i in. or 2 in. objective is first 
used in the microscope with a low power eyepiece. The plane side 
of the mirror is used to reflect the maximum light through the 
system, and the diaphragm of the condenser is closed to its smallest 
diameter. By racking the objective up and down, the image of the 
diaphragm may be found in the field, and it is made central by the 
use of the centring screws of the condenser. The object slide is 
clipped on the stage and the image of the object plane is then 
focused. The condenser is racked up till the image of the diaphragm 
on the lamp is brought into the object plane. Hence the lamp iris 
permits of the control of the illuminated area in the object plane. 

This illuminated area is reduced to a small portion in the centre 
of the field. Then the 16 mm. apochromatic lens and the compen- 
sating eyepiece are put in position and the object focused. In order 
to see the rulings it will probably be necessary to close down the 
aperture of the condenser, and this will easily enable the "15,000" 
band to be focused. 

Now remove the eyepiece and look into the back of the objective. 
A symmetrically placed circular patch of light should be seen, 
apparently filling a part of the aperture of the back lens. If the 
iris of the substage condenser is opened out the patch expands, and 
vice versa. It should appear perfectly uniformly illuminated. If 
this is not the case, the illumination system must be put into better 
adjustment, until uniformity is obtained even for the largest sub- 
stage apertures. Disregard for the moment any dimly coloured 
lateral patches of light which may be seen. 

Having made sure of this point, replace the eyepiece, and, having 
focused the plane of the object as carefully as possible, open out 

* A typical Grayson's ruling may have bands of 5,000, 10,000, 15,000 up 
to 60,000 lines to the inch in the same slide. 



g8 APPLIED OPTICS 

the condenser aperture slowly, watching for the appearance of the 
30,000 band. (If it fails to appear, a blue filter placed in front of 
the source of light may aid matters. If it is desired to use mono- 
chromatic blue light, the opal bulb lamp may have to be replaced 
by a Point'olite with auxiliary condenser in order to obtain sufficient 
intensity.) Provided the system is adequately corrected, the 30,000 
band will probably appear when the condenser has a wide aperture 
nearly "filling" the objective; but diminution of the condenser 
aperture will cause the bands to vanish. Returning for a moment 
to the theory already developed, it can be ascertained from the 

formula, i.e. h = , r . ', that the closest limit of approach for two 



self-luminous objects which are to be still resolved in the image 
given by a lens of NA 0-3 is 

o-6iA 
h = -- 
0-3 

Taking A = 0-45 X io* 3 mm., and remembering that i mm. = ---- 

2 5 '4 
of an inch, we find that we should not expect to resolve a pair of 

elementary objects spaced much more closely than - of an inch 
J J * J 27,000 

apart if they were self-luminous with blue- violet light. Actually, 
then, if we are resolving lines in a 30,000 band, the performance is 
rather better than the "self-luminous" theory would seem at first 
to indicate, but the formula, it will be remembered, is not of a hard 
and fast character*, so that great significance must not be attached 
to the lack of correspondence. The possibility of the resolution of 
the 30,000 band under the above conditions may, however, be estab- 
lished as an experimental fact. 

Now remove the eyepiece and observe that almost the whole of 
the back of the objective is filled with the direct light from the 
condenser, i.e. the objective is working at practically full aperture 
for the direct light. What will occur if the condenser is stopped 
down, so that the direct light only fills a small fraction of the aper- 
ture of the objective? If the resolving power is dependent on the 
aperture thus filled with the direct light, we may expect that on 
stopping down the condenser to give, say, one-eighth of the full 
aperture the distance between the lines would have to be eight 

* The distribution of intensity in the neighbourhood of the image of a self- 
luminous line object has been investigated. The correspondence in the in- 
tensity distribution perpendicular to the elementary line image and across 
the diameter of the elementary "Airy disc" image is close enough to justify 
the tentative use of the same formula. 



THE MICROSCOPE 



99 



times as great to permit of resolution. A little care will, however, 
show that the 15,000 band can be resolved with quite small con- 
denser apertures ; the resolution is therefore diminished to one-half 
instead of to one-eighth. It is evident that the above theory is inadequate 
to deal with this case. 

While the condenser aperture is small, the eyepiece may be 
removed and the back of the objective inspected. The eye is placed 
as close as possible to the plane of the image formed by the objec- 
tive. (A suitable spectacle lens will help to bring the back of the 
objective into focus.) In this way the direct beam will be seen to 
be flanked by coloured lateral patches of light which have been 





FIG. 68 

(a) Appearance at back of objective with "15,000 lines band" just resolved, 

using small aperture condenser 

(6) Appearance at back of objective with " 5,000 lines band" ; same condenser 
and objective 

diffracted by the grating, their size and shape corresponding to that 
of the circular patch of the direct beam. The diffraction colours 
naturally show violet and blue nearer the centre followed by green 
and red, but they will not be well seen with the 15,000 band and the 
0-3 lens, since only the blue parts of the diffracted images will prob- 
ably be visible on each side. The succession will be better seen by 
using a coarser ruling, say, 5,000 to the inch, when not only the 
first order, but also the second order, diffraction patches will be 
seen. When using the 30,000 band, the diffracted light cannot enter 
the objective till the condenser aperture approaches its maximum. 
Such experiments as the foregoing form an instructive introduc- 
tion to the theory of the microscope. Abbe carried out experiments 
in which special gratings were used as objects; these were so ruled 
that the elements had definite angular shapes and the paths of all 
refracted rays could be calculated. It was found, however, that 



ioo APPLIED OPTICS 

although in certain cases the rulings could not be resolved when the 
aperture of the objective of the observing microscope was just 
wide enough to admit all the directly refracted rays, the resolution 
of the images could be secured by a still greater increase of aper- 
ture. This led Abbe to the realization of the part played by the 
diffracted light, and to the formulation of the famous "Abbe 
Theory." 

The Theory of the Microscope. It was mentioned above that a 
very suitable illumination for a microscopic object consisting of a 
plane containing opaque and transparent areas would be the pro- 
vision of a self-luminous surface immediately behind it. The trans- 
parent areas would then function as self-luminous sources, and the 
theory of the formation of the image would be similar to the simple 
case of the telescope. Subject to the absence of aberrations and 
stray light, the resolving power so obtained would depend only on 
the numerical aperture of the objective. 

In practice, lenses suffer from certain imperfections, and the 
object has to be illuminated with the aid of a condenser. It is an 
experimental fact that certain very delicate structures cannot be 
detected unless the illuminating cone is of somewhat smaller angu- 
lar aperture than is required to fill the whole aperture of the objec- 
tive. In such cases it is easily shown experimentally that the light 
entering the objective from the object is partly diffracted light, and 
any theory of image formation must take account of the effects of 
such diffracted light. 

Now the theoretical investigation of the diffraction of light by 
material objects is generally a matter of the most difficult nature. 
It only becomes simple to calculate the main effects when the 
"object" is of a very simple character. A straight edge, a circular 
hole, a rectangular slit, or a row of similar apertures or structures 
are among the cases where simple methods are possible. This 
partly accounts for the fact that the question of the formation of 
the image of a "grating" (or row of slits or similar small structures) 
is of such theoretical importance. It is true, of course, that very 
many natural objects show a regular structure, and this makes the 
corresponding theory of particular interest. 

The thorough analysis of the mode of image formation in any 
condition of illumination is a very difficult and intractable problem, 
but light may be gained in considering specially simple cases, especi- 
ally that of the grating. 

i. The theory developed by Abbe and others 2 supposes that the 
light derived from the condenser and transmitted by the object 
may be resolved into systems of plane waves. The basis for this 



THE MICROSCOPE 



101 



assumption has been examined by Stoney.* Thus in Fig. 69, if we 
concentrate attention on one such system represented in the 
diagram by a "parallel beam" travelling in a particular direction, 
we shall have diffraction occurring in the grating in the object 
plane; the objective will receive the direct and such diffracted 
beams of which the angular directions fall within its limits of aper- 
ture. Such beams will be concentrated by the objective to form 
sharp maxima near the back focal surface, and these maxima 
will have related phases, so that their effect in any other surface 
can be calculated. They will give an interference system, and it 



Gratinf 



.Spherical reference 
( surface, centred in B' 



Image 
ne 




Objective 



FIG. 69. DIFFRACTION IN THE MICROSCOPE 

( Abbe pniu iple) 

can be shown that the frequency of the maxima in the image plane 
conjugate to the object plane for the objective corresponds to the 
enlarged dioptric grating image. In order to deal with the prac- 
tical problem, however, the superposed effects of all the various 
beams passing in different directions through the object plane have 
to be worked out. 

2. If it is assumed that the source of light is accurately focused 
in the plane of the object, then it is possible to proceed in the first 
place by dealing with one elementary point of the source at a time. 
The distribution of intensity in the plane of the object becomes 
known, and thus also the amount of light diffracted into different 
directions. In this way the distribution of light in the final image 
plane can be arrived at, but even in this case it is convenient first 
to consider the distribution of intensity near the back focal sur- 
face of the objective. This procedure has been worked out by the 
present writer 3 , but only for the simple case of a grating, and assum- 
ing an optical system with rectangular apertures. Once more it is 



* Phil. Mag., Oct., 1896, p. 335. 



102 



APPLIED OPTICS 



necessary to extend the results by considering the superposition of 
the effects of all points in the source. 

The Abbe Principle. On the basis of the experiments which he 
made with specially ruled gratings, Abbe reached the conclusion 
that the necessary condition for the resolution of a regular struc- 
ture was that the directly refracted light, and at least one of the 
diffracted beams should enter the objective. 

In Fig. 70 we have pictured a parallel beam incident on a grating 
at an angle 6 in a medium of refractive index n . The angle of 
transmission is 0^ in a medium of refractive index n v The general 





FIG. 70. DIFFRACTION OF AN OBLIQUE BEAM 

formula giving the position of a diffraction maximum in the second 
medium is, writing x for the width of the grating element, 

x(n sin Q - n sin ) = pk (see page 283) 

where p is an integer and A is the wave-length of light in air. If 
p = o, this reduces to the law of refraction. 

Now, if the incident parallel light is incident normally, as in 
Fig. 69, the angle of the first order diffracted beam is 

sin O l = A 



or 



x = 



sin 6 l 



If, therefore, the numerical aperture of the objective enables it to 
receive rays at a maximum obliquity of lf this equation would give, 
for the minimum value of x enabling the first order diffraction 
maximum on each side of the axis to enter the objective, 



On the other hand, the light may be incident obliquely as in Fig. 
70, and it is clear that the most favourable conditions under which 
the refracted beam and first diffracted beam can enter the objective 
is that they should lie symmetrically on each side of the normal. 



THE MICROSCOPE 103 

In this case if the angle of the diffracted light is 9^ as shown in the 
figure, the above equation becomes 

x (n sin ^ + n sin ) = A 

Note the "plus" because <p is measured on the other side of the 
normal. But if O l is the angle of refraction, 

M! sin O l n sin 
so that the diffraction equation is 

x (n sin q^ + HI sin 0^ = A 
and thus if t =-= 9?! 

A 

# = - ; -- 

2ttjL sin O x 
And if the objective can receive rays as oblique as O l 

*m*n = (for oblique light)* 



Now in the case of a full cone of illumination there will be some 
components of the light incident obliquely, and, if these secure 
resolution, the resolving power of the microscope will be given by 
the last formula, which closely agrees with that determined for a 
self-luminous object structure. On the other hand, if the illumin- 
ating beam is cut down to an extremely narrow angle, then, accord- 

ing to Abbe's principle, the resolving limit would be x = ^-7, which 
Agrees with experiment. 

Investigation of the Intensity Distribution in the Image. We must 
now go more thoroughly into this mode of analysis of the image 
formation. Referring to Fig. 69, we have the case of a normally 
incident beam which is diffracted at an angle within the limits of 
numerical aperture of the objective which thus forms three maxima, 
one (D) due to the directly transmitted light, and one on each side 
of the centre (A and C) due to the first order diffraction. The grat- 
ing is symmetrical with regard to the point B. 

In Vol. I it was shown how to find the resultant of two vibra- 
tions of equal amplitude, but different phases. If one phase angle 
is + a and the other is - a, the phase angle of the resultant will be 
zerp. Arguing on this simple basis we can see that if the grating is 
symmetrical with regard to the axial point B, and symmetrically 
illuminated, the phases of the disturbances sent in any chosen 

* If the numerical aperture of the condenser beam cannot exceed a value 
NA e , which is less than NA the numerical aperture of the objective, the 

optimum condition then is x min 
8 (5494) 



104 APPLIED OPTICS 

direction from grating elements on one side of the axis will have 
positive phase angles, while those from the other side will be 
numerically equal but negative. Hence, the phase of the resultant 
disturbance in any direction will agree with that from B. 

Further, we know that if the objective is free from spherical 
aberrations, any disturbances originating from B will meet without 
phase differences in the conjugate image point B', and they will, 
therefore, be in the same phase in a spherical surface ADC struck 
with B' as centre. Therefore the diffraction maxima must have the 
same phase in such a spherical reference surface.* 

Let AD = DC = y, and let the distance from the point D to the 
image point B' be /'. Let P be a point (near the axis) in a plane 
through B' perpendicular to the axis, and let PB' h'. Then if 
AC is small in comparison with /', the disturbance from A arriving 
at the point P will lead in phase by 

2?r fyh' 
T \T 

while that from C will lag by an equal amount. Let us first suppose 
that the grating apertures are all indefinitely small ; then the direct 
and diffracted maxima all tend to have the same amplitude which 
we will write, a. 

Hence the resultant of A and C in the image plane will bo an 
amplitude of 

'27ryA 

2(1 



(a) 



and adding the effect of D (same phase) we get the result 

/27ryft 
-- 



7ryft'\ ) 

-jj, ) f . 



This expression represents a curve (see Fig. 71) with positive 

/Try/A 
maxima of value 30 for the values of ( -.7,- 1 equal to o, TT, 277-, 

77 7T 

etc., and negative maxima = a for, 3-, etc. The intensities will, 

2 2 

therefore, be proportional to qa* and a 2 respectively. 

* Although their amplitudes may have positive or negative signs. In the 
case where there is a grating aperture on the axis of symmetry, and the 
apertures are all very small, the amplitudes of the maxima will have the same 
sign; if the grating is moved so that a "bar" is on the axis the first lateral 
maximum will be opposite in sign to the central one. 



THE MICROSCOPE 



105 



It can easily be shown that the separation of the main maxima 
given by 



"AT" 77 






corresponds to the separation of the dioptric images of the grating 
apertures which would be formed if the latter were self-luminous. 
The angle of diffraction for the first order diffracted beam is given 



by 



n^x sin 0^ = A 



8 

7 
6 
S 
4 
3 
2 
1 

-1 

mt 


1 


I 






: 


\ 


1 


t 






I 


\ 


I 
1 


\ 






i 


\ 
\ 


j 


1 
1 






f 


\ 


1 

t 








t 

t 


i 


1 . 


t 






t 
t 


\ 


\/ 


"Ny 






\f 


\\ 


'/ 


^ 






y 


"x 






Vx 


*^Jt< 










N^ 


^/ 



















/) its 7T 

^r- % 

FIG. 71. AMPLITUDE AND INTENSITY 

Dotted curve: Relative intensity 
1'ull curve : i j- 2 cos (2TC vh'lfo') i e Amplitude 

where x = the grating element separation and 6 l is the angle of 
diffraction. Now referring to Fig. 69 we see that the angle made 
with the axis by the extreme ray reaching the axial point in the 

image plane will be ( 77 ) Hence, if h' x is the image element corre- 
sponding to the dimension x in the object space, the optical sine 
relation gives 

n^x sin O l = ~ , 

the image being in air. And hence y = -,7-, which gives h' = h' x . 

n x 

It will be noticed that if further maxima are taken up by the 
objective, the resultant becomes of the type 

S = a (i + 2 cos 9? + 2 cos 2^ + 2 cos 39? + etc.) . (/?) 



io6 



APPLIED OPTICS 



the number of terms depending on the number of maxima. This 
series approximates the more closely to a discontinuous succession 
of equal isolated very narrow maxima as the number of terms 
increases. The object was a grating of indefinitely narrow spaces. 
Hence we realize the importance of a wide angular aperture for the 
objective if the "image" is to bear the closest possible resemblance 
to the object. Fig. 72 shows the resultant of the first three terms 




FIG. 72. FIRST THREE TERMS OF SERIES 

cos q) + cos 2<p + cos $(p + etc. 

Resultant : dotted curve 



of a cosine series which already has large values for 99 = o, 2?r, 477, 
etc., and much smaller values elsewhere; the addition of further 
terms will add to the effect. 

The above discussion relates to a mere succession of indefinitely 
small apertures for the grating in the object plane, but if the spaces 
are widened it is found that the phases and relative intensities of 
the lateral maxima may change, and thus produce a distribution of 
light more closely representing the relative widths of dark and 
light spaces as the numerical aperture of the objective is increased 
and more diffraction maxima are allowed to contribute their part 
to the image. This action has been discussed by Conrady. 4 



THE MICROSCOPE 107 

Notice also that the blocking out of the central maximum in 
the above case (first order only admitted) would have brought the 
expression into the form 

S = za cos . 

in which the maxima are all of the same brightness, and have twice 
the frequency of the main maxima in the case of equation (a) 
above. Therefore if one observes the image plane while the cen- 
tral maximum is screened, the spacing of the bright maxima 
appears to be halved. A similar effect would be noticed if the first 
and second order images were acting at first, and then the first 
order is screened off. 

Abbe himself performed many experiments of this type, and 
thereby somewhat alarmed microscopists, who felt that the condi- 
tions employed by him, viz. the use of a very narrow-angled illu- 
minating cone derived from a small aperture in the condenser 
diaphragm might easily lead to artificial and erroneous results. 

While there is nothing invalid in the theory so far as it goes, 
the real difficulty in applying it to practical cases is in integrating 
the effects for the beams at varying obliquities which must be 
assumed present in illumination by condensers of finite aperture. 
Closer discussion shows that while interferences from one set of 
related maxima could be observed in any plane where the beams 
overlap, the interferences from various sets will only agree in the 
focal plane. Thus the use of illumination of increasing angular 
aperture gives increasingly the characteristics of an ordinary image 
formation. 

New Method of Analysis. We may now consider the second 
mode of analysis in which the source of light is focused in the grat- 
ing, assumed to be a row of indefinitely narrow apertures as before. 

Fig. 73 gives a diagrammatic representation of the conditions. 
The source of light is considered to be a point source in the first 
place ; it will ultimately be necessary to integrate the results to find 
the effect of a continuous source. Assuming the condenser to be a 
well corrected optical system of not too great aperture, it may be 
assumed that the distribution of light in the focus of the condenser 
is of the "Airy Disc" type in the case of a circular aperture; the 
case of a circular aperture is, however, too difficult mathematically, 
and we are forced to take the simpler case of a rectangular aperture, 
in which the distribution of the light in the image follows a simpler 
law. By assuming this aperture very small in one direction, the 
discussion can be limited, in a well-understood manner, to two 



io8 



APPLIED OPTICS 



dimensions, neglecting the spreading in three dimensions which 
must occur in practice. In this case, the illumination in the object 
plane due to the condenser will be represented by an expression of 
the form (see Fig. 74 and Appendix, page 278) 

. sin U 

Amplitude = T 




Condenser Objective 



FIG. 73. ILLUMINATION BY A SOURCE OF LIGHT FOCUSED IN 
THE OBJECT PLANE 

It must be carefully remembered that these vibrations in the 
object plane G are practically in the same phase, 5 and that although 
the amplitudes in the successive maxima alternate in sign, there 




FIG. 74 

Full line: amplitude *^ 
Dotted line : intensity 



relative 
relative 



is no continuous change of the phase from that of one maximum 
to the next. It is shown in the Appendix that the phase of the 
resultant vibration in the diffraction disc must be identical with 
that of the disturbance derived from the mid-element of the aper- 
ture, and therefore the actual phase change in the plane of G is 
negligible near the axis, in fact, over the range of many maxima 
and minima of the diffraction pattern. 

Note that the first lateral zero value of the amplitude occurs 
.? ?, di ? erence of P ha *e between the disturbances arriving 
in the illumination image" from the extremities of the condenser 



THE MICROSCOPE 



109 



aperture is equal to 277, and the linear distance of this from the 
central maximum will be given by 

h = ^ 

NA. 

where NA C is the numerical aperture of the condenser. 

Let the row of apertures forming the "object," and situated in 
the plane G, be situated symmetrically about the central maximum 
of illumination. The relative amplitudes of the vibrations occurring 
in successive apertures may then be given by such terms as 
sin u sin 311 sin 5 

_ _ ___ _ _ air* 

, , . V.LL. 

u yi 511 

The final resultant of the light diffracted into any given direction 
will be found by 

A 2 = [Z(a sin d + a sin (- d) J ] 2 + [Z {a cos d + a cos (- d)} ] 2 

since there will be equal lag and lead in the phases of elements on 
each side of the centre. Hence 

A = Z 1 2a cos d 

Let x be the grating interval, then the first pair of apertures 
contribute an amplitude, in the direction making an angle 
with the axis, given by 

/sin u\ (TT . \ 
o = 2 I ---- J cos I j- % sin \ 

The second pair give 



cos 



(sin 3/ 
^r 



The whole effect of the grating is thus 

4 i- A j (sin i (TT . \ sin 3^ (3* . \ 

Amplitude 21 ---- cos I T x sin ] H ------ cos I -r- x sin 1 

F ( u \l ] 3^ \* J 

+ etc.j 

We have implicitly adopted the conception usual in the discussion 
of Fraunhofer diffraction phenomena, i.e. that the effects are 
realized at infinity, or in the focal plane of some lens which brings 
the disturbances together without further relative changes of phase. 
Referring again to Fig. 73, notice that the phase of the resultant 
of each pair of apertures, symmetrically situated about the centre, 
will be identical with that of an imaginary disturbance starting 
from the centre; all vibrations being in the same phase in the 
object plane. 



no 



APPLIED OPTICS 



But the phases of disturbances starting from the centre point B 
will be identical at B', and therefore in any surface concentric with 
B'. Hence the phases of the resultants will be identical in such a 
surface as W of Fig. 73, and the contributions of the various pairs 
may be directly added together. Note that the reference surface 
must lie close to the principal focal surface of the lens O, in order 
that the above summation may be considered to be .effective. 




,1 








*SJ *" 


-/& 


*5 





FIG. 75(a). SHOWS SUPERPOSITION OF 
FIRST Two TERMS IN THE SERIES 

sin p -f $ sin 3^ + A sin 5/> -\- etc. 



FIG. 75(6). FOURIER SINE SERIES 
RESULTANT 



The number of terms to be included clearly depends only on the 
number of apertures in the grating free to transmit light. 

Writing v = I j \x sin 6 we obtain the series in the form, 
Amplitude 



= - ) sin u 
u( 

= 1 j(sini* 



cos v + J sin 3 cos $v + J sin $u cos 51; + etc. 



etc.) + (sin u - v -f J ) 
.) ) 



sin 3 ft - v + etc.) 

These series represent well-known "Fourier expansions." If we 
plot graphically a series of terms such as 

sin^> + J sin $p + ? sin 5^ + 1 sin ip + etc. to infinity 
we obtain a result suggested in Fig. 75 (a) and Fig. 75(6). It will be 
seen that the tendency of the addition of successive terms is to 

give a value which is stationary ( actually it is ) from p = o to 

\ 4/ 

p = TT, and then again stationary with negative value from p = TT 
to p = 27r, and so on. In Fig. 75 (a) the effect of the addition of the 
first two terms is seen; the student should draw the figure himself, 
and investigate the addition of the term sin 5/>. 



THE MICROSCOPE 



in 



The effect of the two series in the equation above may then be 
easily found graphically in particular cases. Take, for example, the 

case when u = . We then have the two series as represented in 

Fig. 76. The sum of the two is seen to represent regions of amplitude 
2, alternatively positive and negative, separated by regions of no 
disturbance. 

The central band of amplitude 2 extends from v = to -\ 

/TT\ 4 4 

but v = ( -j \x sin where A is the wave-length of light in the object 

space. The equation becomes 

more general by introducing n, o-+2 -* 

the refractive index of the object 

space, and A , the wave-length of 

light in air. The equation thus 

modified is 

nx n sin TT 



/r 



-/ 



7T 



where (NA)o is the "numerical 
aperture" of the ray direction 
given by 0. Now it is easily 
shown that having given a con- 
denser of numerical aperture NA C , 
the difference of optical path with 
which disturbances from the ex- 
tremes of the condenser meet in 
the focus of the condenser system 
at a lateral distance h from the central maximum is 

27T 

-j- . 2h.NA c = 2U 

Hence using A for the wave-length of light in air from this point 

l.II 
onwards, h = - 



FIG. 76. EFFECT OF Two SERIES 

(< - 7T/4) 



When the value of h is L , the distance of the innermost grating 

2 

aperture from the central point, the corresponding value of U is 
w (say). Hence x 



or 




APPLIED OPTICS 

Hence the above expression for v becomes 

(NA) e 






We saw that when u = -, we found v for the central maximum to 

4 

extend to -, showing then that (NA)o = NA C> and that in 

4 

this case, at any rate, the central maximum corresponds to the 
aperture of the objective which is "filled" by the condenser. This 
can be justified in the general case; it can also be shown that a 
similar distribution of the maxima arises, no matter what the rela- 
tive positions of the grating apertures in regard to the illumination, 
and also even if the grating has finite apertures. 

The calculation of the resultant effects in the final image plane 
of the microscope involves now, first, the summation of the effects 
of the central maximum and all the lateral diffraction maxima con- 
sidered as derived from one elementary point of the source of light, 
and, second, the addition of such effects for all the elementary 
points in the source of light. 

It will not be possible in the limited treatment necessary here to 
do more than indicate the procedure. In the case above, the inten- 
sity distribution in the reference surface showed a central maxi- 
mum of amplitude -f 2, and lateral maxima with amplitudes alter- 
natively - 2 and + 2. Let us suppose that the aperture of the 
objective is only wide enough to focus the central and two lateral 
maxima. We can then find the resultant effect in our case (which 
case is limited to the elementary axial source of light). 

The central maximum forms its "iso-phasal" image in a plane 
perpendicular to the axis, and the distribution of amplitude is 
represented by sin W 

"vT~ 

where 2W is the difference of phase of the disturbances arriving, at 
the point in the image plane, from the extremities of the central 
maximum. If Y is the diameter of the central maximum, f is the 
focusing distance, and h' the distance of the point in the 'mage 
plane from the axis, then the path difference d corresponding to 2\V 
is , YA' 

d== T 

and the phase difference 2W is 



, ir YA' 

W = 



THE MICROSCOPE 113 

Consider now one of the lateral maxima in the above case 

/ 11 = - j. Its centre lies at a distance from the axis equal to 2 Y, 

where Y is the diameter of the central maximum. It may be 
regarded as producing a distribution of the above type in a plane 

2Y 

whose normal makes an angle y = -,- with the axis (see Fig. 77). 

/i/ . ^ Objective 
Object System 
Plane 



Condenser* 
Beam 





FIG. 77 

If the aperture is small there will be considerable depth of focus, 
so that we may consider the lateral maxima to produce a - - 

amplitude distribution in the normal plane, but with a progressive 
relative phase change. This phase change 9? at a distance h r from 
the axis will be 



27T, 2Y 
A . 



Taking account of the first order maxima on the two sides 
central one, and calling the amplitude at any chosen point 
the central maximum ci 09 while the amplitude due to the 
maximum is <f lf the resultant will be 

A 2 = -- - 2 



of the 
due to 
lateral 



sn 



cos - 



f sin (-y)l* -h {<*<> i- a i cos m 

i.e. A a -}- za l cos <p 

Hence we find the expression for the amplitude to be 
Amplitude 

^sinW_ siiny sinWr /27r .. 2Y 

W" 2 W 



cos q> = --V.T"" 



114 APPLIED OPTICS 

Putting in the value of h', i.e. 



h' = 



WA/' 

TrY 



the amplitude is found to be 

sin W 



(i - 2 cos 4\V) 



This function is plotted in Fig. 78, the dotted curve shows the 

sin W 

variation of llr - . It will be seen that the separation of the main 
W w 

intensity maxima is given by a difference of W values of 90 or - . 
The corresponding h' value is therefore 



30 60*\90*i 120* ISO 




- Relative Values of sin tyfa 

> . 1-2 cos 4W 

n [lsinW)U-2cos 4 W)/W] 2 ~ Intensity 

FIG. 78 

We shall see that this corresponds to the separation of the spaces 
in the object plane, for we had the separation of the apertures % 
given by /^x 

i ^IT) i 
AJ A 



* ~ 7rNA c 



n our case. 



THE MICROSCOPE 115 

Now we may use the optical sine relation to derive the mag- 
nification. The "numerical aperture" of the extreme ray from the 
condenser, NA e , corresponds to that of the corresponding ray 
imagined to pass from the boundary of the central diffraction 
maximum in the reference surface to the focal point. The sine of 

Y 

the angular inclination of this latter ray is f - t where Y is the width 

of the maximum (see Fig. 77). Hence the sine relation gives 

xNA.= ^ 

Since from above ^ 

e = - 

A/' 
we get h' = y 

Hence the main intensity maxima in the figure represent the 
"images" of the apertures in the grating so far as they can be 
rendered by a very small source of light. If, now, the aperture of 
the objective could be extended we should obtain the effects of 
higher order diffraction "spectra/ 1 and, provided that the above 
treatment can be taken as more or less valid, the amplitude would 

become sin XV , WT ^ T 

- (i - 2 cos 4 XV + 2 cos 8XX r - etc.) 

the number of terms in the bracket depending on the number of 
spectra admitted by the objective. The above series can be written 



t- 2 cos 12 f XV + - J + etc. 

It was pointed out above that if we take a series of the general form 
cos -|- cos 26 -|- cos 30 + etc. 

and plot the cosine curves and add the amplitudes, we find that 
by taking a great number of terms the ordinates of the resulting 
curve get very large at the values of o, 27r, 477, etc., for 0. Hence, in 
the above series, we shall get finite values for 

= o, 27T, 4?r, etc., 



n6 APPLIED OPTICS 

Hence the maxima will represent the grating apertures both in 
spacing and relative position, but the maxima will only be illu- 
minated in proportion to their illumination by the "illumination 
image" formed in the object plane from the elementary point 
source of light. 

Such a discussion has to be completed by extending the argument 
to take account of the case of an assembly of such point sources 
of illumination; this has been done in a paper 3 by the present 
writer. It appears that the effective result will be to illuminate 
more and more of the maxima in the final image, but the intensity 
maxima in the reference surface will retain the same locations and 
relative intensities. 

Summary and Conclusions. In the Abbe method, the final image 
is regarded as due to the superposition of an indefinite number of 
"secondary" interference systems derived from sets of related 
maxima. No account is taken of any phase relations between such 
sets, which are found at the so-called "homologous points' 1 of the 
observed diffraction maxima of finite extent. 

At first sight this view appears to contradict the other discussion 
in which the broad maxima appear as regions of uniform phase, 
but this, it will be noted, was considering only one elementary 
point in the source of illumination. These apparent differences arise 
only in the method of analysis, and have nothing to do with the 
physical nature of the action. The great mistake very many 
theorists have made was in attributing an undue physical significance 
to the details of the theory. Properly understood, however, each 
method of analysis has something to teach us, and we may proceed 
to draw a few conclusions. 

1. The objective must be well corrected. The necessity of equal 
optical paths for various routes between corresponding object and 
image points is clearly brought out by the second mode of analysis. 
If the paths are unequal, the phases of the maxima in the spherical 
reference surface for any image point will vary, and the concentra- 
tion of light in the image will deteriorate. If, however, a poorly 
corrected lens is used, the "resolution" obtained with narrow 
aperture pencils may not seem markedly inferior to that of a good 
lens ; but such resolution largely depends on secondary interference 
phenomena, such as were discussed in connection with the Abbe 
principle. 

2. Where the object consists mainly of opaque and transparent por- 
tions the fullest aperture illuminating cones must be employed. The 
point to point correspondence of the illumination of object and 
image plane depends largely on the concentration represented by 



THE MICROSCOPE 



117 



sin \V 

the TST~- term * n t ^ ie expression above (page 114). This concentra- 
tion is improved by having the largest possible aperture both for 
objective and illumination. The limit is set by considerations of 
the increase of aberration with aperture and the effects of "glare." 
Unless the objective is of extremely good quality, it will usually be 
found that the optimum results are secured with the condenser 
aperture about two-thirds to three-quarters that of the objective. 

3. Where evidence of the finest detectable regular structures in the 
object is sought for, it is advisable to use an oblique beam of narrow 
aperture to illuminate the object. 

Under such conditions the appearance in the image plane is 
largely of the nature of an interference phenomenon, and the one 
to one correspondence of object and image is spoilt in great measure, 
only the coarser features of the former being rendered by the 
limited angular aperture of the illuminating beam. If, however, 
there are regular structures present in the object, even periodic 
variations of refractive index, these latter have their best chance 
to produce an intensity variation of proportionate frequency in the 
image plane. The location of the interferences in the "picture" 
will roughly correspond to that of the corresponding structure in 
the object, but we must regard any element of the interference 
pattern as due to the whole of that structure, and abandon the idea 
of point to point correspondence. 

Abbe pointed out that under the conditions of oblique lighting 
by narrow pencils, used with an object consisting of a layer contain- 
ing media of varying refractive index, the optical path differences 
arising between disturbances traversing any neighbouring parts of 
the structure may vary with the inclination of the light. Hence 
the relative phases of the various diffraction maxima would be 
likely to show a corresponding variation, and this would be likely 
to result in a weakening of the interference phenomenon if wide- 
angled cones were used for illumination. It is a perfectly legitimate 
conclusion that we are more likely to find evidence of delicate and 
fine structures when using narrow cones of illumination, but the 
appearances in such a case are mainly secondary interferences and 
are not to be interpreted in any other way. 

Let us, for example, consider the theoretical aspect of the observa- 
tion of a typical microscopic object consisting of a regular structure 
so fine that, when the condenser aperture is very small, the first 
order diffraction maxima lie entirely outside the limits of the objec- 
tive aperture. Fig. 79 suggests the appearance within the ring of 
the objective O. D indicates a direct beam, and D x a first order 



n8 APPLIED OPTICS 

diffracted beam ; as we know, the angular apertures of these corre- 
spond to the aperture of the condenser. As this aperture is increased, 
the first order maxima begin to appear, and the homologous points 
of the Abbe principle may be imagined now to begin to set up their 
interferences. The conditions should, at first sight, improve till the 
condenser aperture is equal to that of the objective, but this is 
dependent on the avoidance of relative phase changes between such 
pairs of homologous points, a matter clearly dependent on the 
nature of the object. In the majority of cases the resolution begins 
again to be lost as the aperture of illumination is yet further in- 
creased ; this is partly due to glare, and partly due to the loss of the 
secondary interferences owing to the above reason. 

The above case is well illustrated by the familiar diatom, amphi- 
pleure pelhicida, which is "resolved" by a first-class microscope 




FIG. 79 

objective of NA at least 1-2 when working with a condenser giving 
a symmetrical cone of illumination of NA 0-9 or thereabouts, but 
not with a smaller aperture of illumination. This resolves the 
structure of 10,000 lines to the inch, but the (only slightly finer) 
lines perpendicular to the first can only be "resolved" by the use 
of oblique light, presumably because the variations of optical path 
involved in the second structure are much smaller, and the diffrac- 
tion maxima are much weaker relatively to the direct light. Al- 
though the frequency of the structures is thus known, the ultimate 
form of the siliceous frustules of the diatom is quite unknown. 

It seems fair to conclude that whereas the majority of fine-resolu- 
tion effects with the microscope are obtained with apertures of the 
illumination considerably smaller than the objective, the finer detail 
of the image of regular structures must be usually ascribed to " secon- 
dary" interference effects. When, however, the conditions are such 
that the aperture of the condenser can be increased to equality 
with the objective, and when we are dealing with an object con- 
sisting merely of variations of opacity in one plane rather than of 



THE MICROSCOPE 119 

refractive index in a thin layer, then we may regard the image 
formation as equivalent to that of a self-luminous object. 

4. There are thus two ways of regarding and using the microscope. 
Firstly as a camera, secondly as a kind of interferometer. In the first 
case the aim is to obtain a "picture" of the object in the conven- 
tional sense, with a "one to one" correspondence between details. 
A precise meaning can only be attached to the picture when the 
object consists of a plane containing transparent and, more or less, 
opaque layers with definite geometrical distribution. When the 
object is three-dimensional or consists of variations of refractive 
index between two plane boundaries, the one to one correspondence 
loses much of its meaning. The effort to interpret the picture in a 
conventional sense becomes increasingly difficult, and the "image" 
is best regarded as giving "evidence" of structure (and that only 
when the illumination has been taken into account) rather than 
as a picture at all. 

The use of these very narrow aperture bundles with a microscope 
objective suffering from spherical aberration illustrates some of the 
views expressed above. The presence of a moderate amount of 
spherical aberration does not preclude the formation of the diffrac- 
tion maxima and the interference phenomena, but the latter may 
often be best seen in a different focus from that in which the general 
features of the object seem to be best revealed. 

Dark-ground Illumination. Given a condenser having a greater 
numerical aperture than the objective of a microscope, a stop may 
be placed in the condenser diaphragm (Fig. 80), so that the illu- 
minating beam is highly oblique and no "direct" light enters the 
objective after passing through the object plane, provided that we 
only have homogeneous media present in the slide, object layer, 
and cover glass, etc. If, however, the object layer contains bodies 
which are capable of refracting, scattering, or diffracting the light, 
a proportion of this deflected light may enter the objective. 

The calculation of the amount so deflected is usually very difficult, 
and the problem only becomes solvable in special cases such as 

(a) Regular structures, or apertures, gratings, etc. The relative in- 
tensities of light diffracted in various directions may be estimated 
in simple cases. 

(b) Spherical globules or other bodies of regular shape present in 
the object layer. The laws of reflection and refraction allow the dis- 
tribution of the emergent light to be calculated if the bodies are 
large enough. 

(c) Very small particles or filaments of material having a refrac- 
tive index differing from the surrounding medium. 

9 (5494) 



120 



APPLIED OPTICS 



In this case the relative amount of light scattered into different 
directions by particles of various sizes can be calculated. A few 
notes may be helpful in this connection. Referring to Fig. 81, L 
represents a beam of light, and P a scattering particle. In the theory 
due to the late Lord Rayleigh, 6 the particle "loads the ether" and 




FIG. 80. SIMPLEST FORM OF DARK-GROUND ILLUMINATOR 

becomes the seat of disturbances propagated in all surrounding 
directions. Fig. 81 is a perspective view. The electric forces in the 
wave-front are perpendicular to the direction of propagation, and 
may be resolved into horizontal and vertical components H and V. 




FIG. 8 1 

Both H and V produce equal effects at C in the line of propagation 
or at D (backwards), but only V can act at A or B in the direction 
perpendicular thereto. 

Rayleigh's formula for the intensity of the scattered light in a 
direction making an angle with the incident ray is, if the incident 
light is unpolarized, 

4 .(D'-D) . _ T 



where r is the distance of the particle from the point of observa- 
tion, A is the wave-length, T is the volume of one particle, m is the 
number of particles, D' and D are the densities of the material of 



THE MICROSCOPE 



121 



the particles and the medium respectively, and A 2 is the intensity 
of the incident light. Thus the intensity in the forward or back- 
ward direction will be double that at right angles to the incident 
light. 

It is easy to understand that the light scattered in a direction 
perpendicular to the incident beam is plane polarized. 

With very small particles, the amount of light scattered is inversely 
proportional to the fourth power of the wave-length, accounting for 

Object 
Plane 




FlG. 82. NACHET'sJPRISM 

the "blue" of fresh smoke, etc. With coarser particles the light 
becomes white to ordinary observation. The conditions of transi- 
tion between the pure scattering effects and the refraction effects 
characteristic of larger bodies have not been fully explored, and 
this lack of knowledge causes much difficulty in the interpretation 
of phenomena observed when using dark-ground illumination. 

Optical Systems. One of the earliest inventions for one-sided, 
dark-ground illumination was Nachet's prism 7 ; Fig. 82 will be self- 
explanatory. Referring now to Fig. 83, it will be seen how the use 
of a central dark stop in the diaphragm of the condenser produces 
annular illumination, with a great gain in intensity as compared 
with the arrangement of Fig. 80. This method is often used with 
low power objectives. 

Another device was, however, described by Wenham 8 in 1850, 
which was the forerunner of modern reflecting dark-ground illu- 
minators. Fig. 84 shows a section of the paraboloidal reflector; the 



122 



APPLIED OPTICS 



top of the parabola is absent, so that the rays concentrated through 
the focus can proceed onwards. Wenham compensated for the 
spherical aberration arising in the microscope slide by the provision 





FIG. 83. USE OF ANNULAR STOP 

of a meniscus lens as shown in the diagram. In later reflectors a 
solid truncated paraboloid was made in glass, and the top is oiled 
on to the slide, thus escaping the spherical aberration* arising by 




FIG. 84. WENHAM'S PARABOLOID ILLUMINATOR 

oblique transmission at a refracting surface, T, Fig. 85. Such para- 
boloids are still made and used. 

The difficulty of manufacturing paraboloidal surfaces of sufficient 
accuracy led to the invention by Ignatowsky and Siedentopf lo of 
double mirror reflecting condensers. 

Theory of Cardioid Condenser. To understand their action, we re- 
call the formula expressing the relation between the length of the 

* Vol. I of this book, p, 13. 



THE MICROSCOPE 



123 



radius vector at any point on a polar curve, and the angle between the 
radius vector and the normal. 




FIG. 85. MODERN PARABOLOID 





V ( T U W 



FIG. 87. THE CARDIOID 

The equation to the curve is /(K,0) =- o 
and the angle is i. 

Joining the origin O to two indefinitely close points PjP 8 on the 
curve (Fig. 86) and dropping the perpendicular P^ to OP 2l we find 

DP a <SR 
tan * = V-^FT = 



124 APPLIED OPTICS 

Hence in the limit, 



For the cardioid OPS (Fig. 87), whose equation is 

R == r (i + cos0), 

OS 
where r = , we obtain 

d ~ - - ' 

dO " " r sm 

_ , . - r sin j / N 

and tan i = : HT = - 1 an v - ; 

r (i + cos0) \2/ 

Hence if a ray started from O and were reflec'ed at the cardioid at P, 
it would travel towards a point U on the line OS, for which OUP is an 

isosceles triangle. Take a point T at a distance - from O and draw 

TV cutting PU in V, so that VTU = 0. Then with T as centre and 
TV as radius, draw the circular arc VW ; it is evident that if the ray 
PV were reflected at this arc, the final direction must be VY parallel 
to the line OS, since the incident and reflected rays then make equal 
angles with the normal to the surface. To show that the same spherical 
locus produces the same final direction, parallel to OS, for all rays, it 
must be shown that TV is independent of 0. 

Now TV = 2TU cos 0, and TU -= -- ^1L _ ! = -- ( --l co j* * - - 

2 COS 02 2 COS 2 

r 



2 COS 
TT -rw 2r COS 

Hence IV = ^ y 

2 cos 

If k = distance of ray VY from the axis OS, then evidently 

h 

~ = sln 

h 

or -. 7: = const. 

sin 

It, therefore, appears that if we send a parallel beam (from right to 
left in the diagram) so that reflection takes place first at the sphere 
and then at the cardioid, we shall obtain an aplanatic refraction free 
from spherical aberration and coma, so that a distant source of light 
of reasonably small angular size will be sharply imaged in the plane, 
perpendicular to the axis, through O. 

Fig. 88 shows the course of the rays through a cardioid con- 
denser as made by Zeiss, and a plate of uranium glass placed above 
it. The ray-; meet the "spherical" surface first. In practice the 
second cardioid surface, being only a comparatively narrow one, 



THE MICROSCOPE 



125 






can be represented sufficiently well in practice by a ring of the 
approximating spherical surface. The range of numerical aper- 
ture in the illuminating cone is about 1-2 to 1-33. If no direct light 
is to enter the objective the NA of the latter must be below about 
1-05. Objectives of higher NA have to be stopped down. 

The NA of a ray perpendicular to the axis is equal to n sin 90, 
i.e. to n, the refractive index of the medium. Hence if a ray in 
.glass has an NA of, say, 1-4, 
it cannot enter a layer of water 
perpendicular to the axis, but 
will be totally reflected at the 
boundary since the refractive 
index of water is 1-333. Hence 
the NA of 1-333 is the highest 
which can be used to illuminate 
an object in a watery medium. 
If the object is immersed in 
a medium of higher refractive 
index the NA of the illumi- 
nating beam can be increased, 
thus allowing objectives of 
higher NA to be used with- 
out stopping down. 

When light is totally re- 
flected at a glass-air surface, 

say, it is well known that a certain amount of light energy does 
pass into a very narrow layer (of thickness comparable with the 
wave-length of light) of the air at the surface. Advantage is taken 

of this in illuminating so-called 
"smear" preparations of bacilli 
with such dark-ground illumin- 
ators, the organisms adhering 
| to the totally reflecting sur- 
face being thus brilliantly 
I illuminated. 

A number of other types of 
| illuminator have been pro- 
duced. Fig. 89 shows the 
Ionising dark-ground illumin- 
,itor of Messrs. R. and J. Beck, 
Ltd., which allows control of 

v Q T , <* * ' '** M) the focusing point. With other 

FIG. 89. FOCUSING DARK-GROUND , .:. ,<.__ :* ;,. 

ILLUMINATOR types of illuminator it is 






(Carl Zeiss, Jena) 

FIG. 88. COURSE OF THE RAYS 
THROUGH A CARDIOID CONDENSER 




126 



APPLIED OPTICS 



necessary to choose an exact thickness for the microscope slide 
(usually about 1-2 mm.). The focusing type can be used with 
other thicknesses within reasonable limits. 

The 'Slit Ultra-microscope. This arrangement, introduced by 
Siedentopf and Szigsmondy, 11 has been employed chiefly in the 
study of colloid particles in various media. The illumination of the 
medium is concentrated into the region of a sharply limited image 
of a bright slit, and only such particles as lie within such a region 
are visible. Fig. 90 illustrates the optical system. The image of 
the crater of the arc is projected by the lens / on the micrometer 
slit g. The second lens h then projects an image of g into a point 
at the proper working distance from the objective i. Thus we 



i. M 




FIG. 90. THE UI/TRA-MICROSCOPE 
(Diagrammatic) 

obtain in the object medium a doubly reduced image of g. If the 
particles to be studied are disposed in a liquid, this is usually con- 
tained in a tube of square section /. The observing objective O is 
usually corrected for water immersion. The particles are seen when 
the region of illumination is brought by adjustment of i into the 
proper position with regard to O. A polarizing prism may be intro- 
duced into the illuminating beam for special purposes. 

General Remarks on the " Cardioid " and " Slit " Ultra-micro- 
scopes. In these ways it is possible to make visible any particles, 
filaments, or differentiated structures capable of scattering sufficient 
light into the objective. Small objects thus seen appear bright on 
a dark background. If the dimensions of such objects are small in 
comparison with the resolving limit of the objective, the size of the 
patch of light representing the image will be determined by the 
considerations of diffraction. 

The method is employed for the study of colloidal particles of 
diameters down to 4 x io" 6 mm. (Siedentopf), and light has been 
thrown on the phenomena of pedesis (Brownian movements) and 



THE MICROSCOPE 



127 



the absorption of light by colloids in liquids and glasses, etc., more 
especially by using the one-sided illumination of Siedentopf and 
Szigsmondy. The concentric dark-ground illuminator is capable of 
revealing fine structures, such as flagellae of micro-organisms, 
delicate spines, etc., which cannot be readily observed in other ways. 
The inner structure of bacteria is also shown up by dark-ground 
illumination where it is not detectable by ordinary transmitted light. 

While this method has a great advantage in the extreme contrast 
produced in the image, it suffers from disadvantages in a loss of 
resolving power under certain con- 
ditions. 

Resolving Power for Dark-ground 
Illumination. The first investigation 
of this subject is due to M. J. Cross. 12 
He applies the Abbe principle. Con- 
sider the illumination of a grating by 
an oblique beam of narrow angular 
aperture from the illuminator of a 
microscope. Let it be assumed then 
that in dark-ground illumination none 
of the "direct light" enters the objec- 
tive, but that at least two diffraction 
maxima, say the first order and second 
order, must be formed at angles with- 
in the numerical aperture of the objective if the grating is to be 
"resolved." If the distribution of the diffracted beams is as repre- 
sented in Fig. 91, the first order and second order being at angles 
y^ and <F 2 , the equation for <f 2 is 




nx sn 



+ nx sln = 



where x is the grating space, is the angle of the incident beam, and 
n and n' arc the refractive indices on the two sides of the grating. 
In order to understand the optimum result we may imagine a grat- 
ing of variable spacing. By broadening the spaces the second order 
beam will swing towards the direct beam D, and we could alter the 
angle between beams i and 2. By altering the angle of illumination 
we could then place I and 2 at equal angles with both sides of the 
normal, as shown in the figure, and thus it could be arranged that 
the first and second order maxima fall just within the objective 
aperture. This represents the optimum conditions for the objective 
with dark-ground illumination. Note that if we increase or diminish 
the illumination aperture, the maxima may not be focused by the 
objective. 



128 APPLIED OPTICS 

If a i is the numerical aperture of the illuminating beam, and 
a is the numerical aperture of the objective, the above condition is 



If the first order maximum falls just within the objective aperture 
on the other side, the condition is 



Subtracting these equations we obtain 



or x d = - 

so that if the optimum conditions are obtainable, the resolving 
power may become equal to that for bright-ground illumination; 
but on adding the equations we find that 



so that in order to get these optimum conditions we must have 



Since with the majority of cases the aperture of the illuminating 
beam will not greatly exceed that of the objective, and there will 
be few cases in which the absolute optimum is realized, the theory 
indicates that x d must be about double the spacing resolvable with 
bright ground. This indicates that the resolving power may be 
halved. 

Experiments by Siedentopf 13 have shown that this expected 
result is realized in the case of gratings, but Beck 14 and various other 
observers working with objects such as diatoms have maintained 
that the resolution is not impaired. The reasons for the discrepancy 
in the observations is not yet adequately explained. It may be 
pointed out that the illumination of special objects is difficult to 
define very precisely. A considerable amount of light may be re- 
flected back to the top surface of the object from a " total reflection " 
effect at the top of the cover glass. It should also be remembered 
that the elementary theory takes no account of the state of polariza- 
tion of the light, which may be very significant in the diffraction 
effects. 

Owing to the drastic variation of diffraction angles with colour, 
it is usually the case that " dark-ground " images show brilliant 
chromatic effects when a lamp giving white light is employed as 
the source of illumination. 



THE MICROSCOPE 



129 



Diffraction Images of Various Objects with Unilateral Dark-ground 
Illumination. When polarized light is used for unilateral dark- 
ground illumination, we may consider the intensity of light scattered 
into different directions by a small particle. The main necessary 
facts were considered above and are put into graphical form in 
Fig. 81. If the vibrations in the incident ray are perpendicular to 

Vibration 




FIG. gz 

the plane containing the ray and the microscope axis, the case will 
be represented by the " plan " diagram of Fig. 92 (a). The circle repre- 
sents the aperture of the objective, and the amplitudes of the 
vibrations are greatest along one diameter. They are uniform along 
this line, but vary in any perpendicular direction as suggested by 
the length of the arrows in the figure. 

When the vibrations of the polarized light are parallel to the axis 
of the microscope, the intensity is zero in the centre of the aperture 
and rises towards the margin (Fig. 92 (&)). When the vibrations are 
inclined, say, at 25 to the plane containing the ray and the micro- 
scope axis, the point of zero illumination is clearly displaced away 
from the centre of the aperture, and the intensity has an unsym- 
metrical distribution as suggested in Fig. 92 (c). 

The resulting image disc in these cases will differ somewhat from 
the Airy disc. In case (a) the disc will be elongated somewhat in 
the direction of the vibrations. In (6) the surrounding rings will be 



130 



APPLIED OPTICS 



of considerably greater intensity than in the normal Airy disc, and 
the image distribution has a black centre. In (c) there may be an 
unsymmetrical patch of light with a one-sided dark spot. 

When unpolarized light is used for illumination, however, the 
resultant effect differs little from the usual Airy disc, merely show- 
ing a slight "astigmatism" due to the unilateral direction of the 
illumination. Even this disappears when symmetrical dark-ground 
illumination is used. 

The above particular effects are entirely due to the wave-nature 



Diffracted Conical Wave-front 




Diffract}^ or 
Scattering 
Filament I 

Incident / 
Wave-frp'nt 



FIG. 93. DIFFRACTION BY A FILAMENT 

of light, and must not be interpreted as indicating any structure 
in the ultra-microscopical particles. 

Effect of a Linear Filament. Referring to Fig. 93, let WW be a 
plane wave-front of an incident parallel bundle of rays falling on 
a straight filament ABCD capable of scattering light. It is assumed 
that the elements of the filament act as scattering centres. By the 
time the wave reaches D, the secondary disturbances scattered from 
C, B, and A will have spread to proportionate distances. The 
envelope of the spherical surfaces, which we regard as the diffracted 
wave-fronj;, is a cone of which the axis is the line DA. The direc- 
tions of the diffracted rays will also be represented by a cone sym- 
metrical round the filament ; this cone contains the incident ray, and 
also that "reflected" ray lying in the plane of the incident ray and 
the filament. We may look on the incident ray as a generating line. 
If the filament is perpendicular to the incident ray there will be a 
cylindrical wave resulting from the diffraction. 

Such diffracted wave-fronts will evidently tend to give extremely 



THE MICROSCOPE 

astigmatic images in the microscope ; although the concentration of 
the image will tend to represent a narrow thread, there can be no 
genuine point to point representation. 

Now imagine a straight filament in the "ultra-microscope" in- 
clined at an angle ft to the incident ray (see Fig. 94). The diffracted 




FIG. 94 



light is limited in angular direction to the cone of which the inci- 
dent ray is a generating line. In the case shown, no light will enter 
the objective of semi-angular aperture a unless 2ft < (90 + a) and 



\ 


/ 


\ 


/ 


\ 


/ 


\ 


/ 


\ 


/ ^ 





FIG. 95 

20 > (90 - a.) The filament is assumed to be at the proper working 
distance for the objective and its size is greatly exaggerated in the 
figure. If we imagine the illuminating beam rotates round the axis 
of the microscope, assumed vertical, we may consider the case of a 
filament inclined at a small angle ft to the horizontal. In Fig. 95 (a) 
the cone does not enter the cone of ray directions representing the 
aperture of the objective, but when the direction of the illuminating 



132 APPLIED OPTICS 

ray has been rotated round the vertical Fig. 95 (6) (or the filament 
rotated round a vertical axis) the diffraction cone may enter the 
objective. Thus there are ranges of filament directions in which no 
light can reach the objective. 

Sometimes the object seen by dark-ground illumination consists 
of very delicate needle-like crystals which are in motion in the 
medium, flashing into view as the angles vary. Sometimes the object 
consists of a spiral thread; only those portions lying within a cer- 
tain range of angular directions are visible, and the object may 
appear as a series of detached sections. 

Microscopy with Ultra-violet Radiation. The expression for the 
resolving limit of a microscope objective 



NA 

shows that when optical design has increased the angular aperture 
of the objective to the practicable limit, and has employed the 
practicable immersion media of the highest refractive indices, the 
only remaining way of increasing resolving power (diminishing A) 
is to employ radiation of shorter wave-length. The limit of trans- 
mission for stable glasses in the ultra-violet region is approximately 
0-3^ (Vol. I, page 245). Optical media available for regions of 
shorter wave-length are practically limited to quartz (crystalline 
and fused), and fluorite. Of these, fused quartz and fluorite are 
both difficult to obtain in a perfectly homogeneous condition. 
Crystalline quartz has only limited uses in lens systems owing to 
its double refraction ; it can be employed in eyepiece lenses, or even 
in the back components of microscope objectives, but not in lenses 
where a great divergence of angular directions must be allowed for 
the rays. Both fused quartz and fluorite have rather low refractive 
indices, and any attempt to make high aperture achromatic 
combinations from these materials would meet with the greatest 
difficulties. Present practice is mostly limited to the use of so-called 
"monochromat" objectives constructed entirely of fused quartz, 
and using radiation corresponding to a single apparent "line 11 of 
the spark spectrum of cadmium, generally A = 0-2749^. The con- 
struction of such a lens is shown in Fig. 96 on a greatly enlarged 
scale. It can be freed from spherical aberration and coma in a 
very satisfactory way. Projection eyepieces of crystalline quartz 
are employed, and the image is registered by photography. 

Cover glasses are worked in fused quartz; slides usually are of 
crystalline quartz, but might be made of one of the ultra-violet 
transmitting glasses. 



THE MICROSCOPE 



133 



Homogeneous immersion can be secured by the use of solutions of 
glycerine or cane sugar which are transparent to this radiation. 



Immersion 
Front 




FlG. 96. MONOCHROMAT OBJECTIVE 

Suitably adjusted mixtures can be freed from hygroscopic variations 
of refractive index. 

Calculation of the depth of focus of an immersion monochromat 
of NA 1-25 shows it to be about o-2/*, or eight millionths of an inch. 
The fine adjustment focusing of the objective must, therefore, be 




Objective 



P (Plane Surface) 




(a) Sitle virw 



FIG. 97 



(b) Front view 



such that this interval corresponds to a measurable movement of 
the control. Satisfactory results arc obtained from well-made 
micrometer screws with tangent screw, or by well-regulated elastic 
displacements. 

The image is invisible to the naked eye. A fluorescent finder, 
which may be likened to an "eye" with a quartz lens, and a 
"retina" of fluorescing uranium glass viewed from behind by a 



134 APPLIED OPTICS 

magnifier, permits the finding and focusing of bright images when 
the objects have well-marked contrast, but not of the smaller and 
delicate images unless some opaque material is introduced into the 
slide. Some workers have used a "carbon pencil" line drawn on 
the under side of the cover glass. In such cases the method used 
by Barnard is to employ two objectives in special mounts (Fig. 97), 
which can thus be interchanged without disturbing either the object 
or the remainder of the microscope. One is a visual objective and 
the other a monochromat, and they are made as nearly as possible 



Collimatinf Lens 
artzf7 

lass Lens 




FIG. 98. ILLUMINATING SYSTEM FOR ULTRA-VIOLET MICROSCOPE 

"parfocal," so that, when the image has been found and focused by 
the visual lens, it requires only a small measured motion of the fine 
adjustment to obtain the ultra-violet focus for the monochromat 
when the latter is placed in position. 

Yet another alternative which is successful in some cases is to 
use the monochromat (a) with "homogeneous" visible light such 
as that obtained from the green line of the mercury spectrum, (b) 
with the ultra-violet. Although the definition of the visual image 
obtained in the first place is poor, owing to the spherical aberration 
of the lens for that wave-length, the difference of the focus in the 
two cases can be measured with sufficient accuracy to come very 
near the ultra-violet focus from the visual setting. In all methods, 
one or two trial "shots" with slight adjustment of the focusing 
are usually necessary to obtain the sharp image. 

The general arrangement of the apparatus for making photo- 
micrographs of transparent objects is shown in Fig. 98, which is 
largely self-explanatory. The lens 1^ projects the images of the 
spark gap, i.e. the spectrum, into the plane of the duplex condenser 



THE MICROSCOPE 



135 



D ; one " line" must be broad enough to fill the aperture of the con- 
denser, of which only the centre part is constructed of quartz. The 
outer part consists of a dark-ground illuminator of the concentric 
type as made by Messrs. R. and J. Beck, Ltd. (Fig. 99). The advan- 
tage of the arrangement is that objects in a slide can be found very 
easily with the aid of this illuminator and the visual objectives; 




Fir,. 90. CONCENTRIC TYPE CONDENSER ANU ILLUMINATOR 



A 
K 

C ) 
D [ 
E \ 



Movable component of dark ground illuminator (las s) 
: Front component of dark ground illuminator (quartz) 

- Condenser components in quartz 
Slide (i]u.irt/) 



they can then be photographed by the ultra-violet radiation without 
disturbing the slide in any way. 

It is convenient to produce the discharge from a small trans- 
former working from an A.C. supply ; the brightness is found to be 
independent of the frequency of the cycles of the current within 
ordinary limits. The effective brightness of the spark is dependent 
mainly upon the total energy consumed at the gap. The spark gap 
with the arrangement of Fig. q8 is about 3 mm., the secondary 
volts are about 5,000, and there is a condenser of capacity of about 
01 /iF in parallel with the gap. The circuit consumes about half a 
kilowatt. 

The earliest steps in ultra-violet microscopy were made by 
Kohler 16 and von Rohr of the firm of Carl Zeiss prior to 1904, but 

io (5494) 



136 



APPLIED OPTICS 



the method found few users till after the War, when Barnard 16 
introduced many alterations in the apparatus and technique in con- 
junction with the firm of Messrs. R. and J. Beck, Ltd., and used the 
method with great success in biological work. The experimental 
methods, particularly in regard to the stabilization ot the immer- 
sion media, and the use of elastic displacement fine adjustments, 
have also been explored by Martin and Johnson. 17 18ils Johnson 
has given an experimental demonstration that the expectation of 
increased resolving power has been almost realized. 

Polarization Tests the Optical Indi- 
catrix. Microscopic tests, carried out 
with the aid of polarized light, are of 
great importance in the identification 
of minerals in small crystals, or in thin 
rock sections. Similar tests are now 
of increasing importance in biology 
and in colloid physics. 

The section on crystal optics in Vol. 
I, Chapter VI, may be consulted for 
introductory study of the phenomena 
of polarized light. We may add here 
that in a doubly refracting medium 
there are in general three chief vibra- 
tion axes ; let them be o, 6, and c. 

Light travels fastest when its vibra- 
tions are parallel to the direction a, 

and most slowly when parallel to c. The other direction 6 is a 
direction of intermediate case. 

The corresponding indices of refraction for rays having vibrations 
parallel to these directions a, 6, c, are a, /?, and y respectively, the 

velocities being proportional to -, -r, and -. Thus it is possible to 

represent the refractive index for a vibration in any direction by 
a three-dimensional figure such as shown in Fig. 100. This is the 
so-called "optical indicatrix," an ellipsoid with semi-axes a, /?, and 
y. Any section of such an ellipsoid by a plane passing through the 
centre is, in general, an ellipse with different axes. From the 
ellipsoid we see, for example, that a disturbance propagated in 
the direction OB and vibrating parallel to OA has a velocity pro- 
portional to -, while one vibrating parallel to OC has a velocity 
proportional to -. 




FIG. ioo. THE OPTICAL 
INDICATRIX (FLETCHER) 



THE MICROSCOPE 137 

Consider a ray propagated parallel to the direction OC. The 
vibration, perpendicular to OC, can be resolved into components 
vibrating parallel to OA and OB respectively, for which the refrac- 
tive indices are a and ft. 

Now since a < ft < y t there will be two directions OD, OD' 
between OA and OC and between OA ; and OC, for which the semi- 
diameter of the ellipsoid will be equal to ft. The "ellipsoids" 
through B and D and through B and D' will, therefore, become 
circles, and the directions normal to these planes will be ray direc- 
tions for which there is no difference of velocity for vibrations 
taking place in various azimuths. These directions of single ray 
velocity are the "optic axes" of the doubly refracting medium. 

In any thin crystal section, or section of other doubly refracting 
medium, there will be fast and slow vibration directions corre- 
sponding to least and highest refractive indices. These may be 
represented by the "index ellipse" for the section. 

The effects of double refraction associated with increasing paths 
through a crystal plate, the colour effects using white light, and the 
estimation of the double refraction with the aid of a wedge com- 
pensator have been described in Vol. I. 

The Polarization Kicroscope. The polarization microscope is de- 
signed in the first place to permit of the application of such tests 
to very small and thin specimens. It is fitted (Fig. 101) with a sub- 
stage polarizer (Nicol, Glan-Thompson, etc.), which can be easily 
swung in and out as desired. This is followed by a diaphragm and 
by a sub-stage condenser, usually of a two-component type, of 
which the upper highly converging system can be removed from 
the path of the light if it is required that the object shall be traversed 
by comparatively "parallel" pencils of rays. It is thus possible 
to illuminate the object with plane-polarized light. The objectives 
and eyepieces are of normal types, but it is considered a great 
advantage to have the widest possible field of view. An analyser 
may be inserted in the tube below the eyepiece. It is preferably 
of the square-ended variety, so that no displacement of the image 
is produced by its rotation. An auxiliary analyser with a divided 
circle to measure rotation ("cap analyser") may be fitted over the 
eyepiece with a swing-out movement. 

Slots are usually provided above the objective and sometimes 
under the eyepiece, so that various compensators, such as a mica- 
quarter wave plate, quartz wedge, or Babinet compensator, can be 
inserted into the path of the light. 

Petrological tests with the polarization microscope usually aim 
at the identification of various constituents of a specimen of rock, 



138 



APPLIED OPTICS 



and for this purpose a thin section is made by grinding, and is 
cemented between a slip and cover-glass. It is desirable to know 
the thickness of the specimen, which is usually of the order of one- 
thousandth of an inch. 

It is not within the scope of the present book to deal fully with 
all the numerous tests and criteria which can be applied for the 
identification of the various materials ; we can only mention those 
which depend on the polarization of light. The doubly refracting 
materials are immediately recognized by the restoration of light 
between crossed Nicols. Suppose that an unknown doubly refract- 
ing substance is present, the crystallographical axes are likely to 
have all kinds of inclinations relatively to the section in different 
parts. Any element traversed by the light has its maximum and 
minimum refractive indices, represented by the major and minor 

5 






FIG. loi. OPTICAL SYSTEM OF POLARIZING MICROSCOPE FOR ORDINARY 

OBSERVATION 

P = Polarizer 

C = Condenser 

O = Objective 

W = Wedge or retardation plate sometimes introduced 
AI ^ Analyser used when compensator B is not in use 

a = Babinet compensator. (Other compensators may be introduced here) 

R *-= Ramsden eyepiece 
A. 2 --= Analyser used when compensator is in use 

axes of the ellipse representing the section of the indicatrix ellip- 
soid taken normally to the direction of the light. If the major or 
minor axis of the ellipse is parallel to the vibration direction of the 
polarizer, the light is not restored between the Nicols, but if the 
specimen is rotated through 45 the components *of the vibration 
separated at the crystal surface are equal, and the colour is at 
maximum intensity. The highest order colour will be found in 
sections containing OA and OC where the maximum difference of 
refractive index is found; sections normal to OD or OD' (the 
axes) would not show double refraction at all. As mentioned 
in Vol. I, page 211, the Chart of Michel Levy shows the colours 
characteristic of different thicknesses of crystal plates of varying 
bi-refringences, and in this way the maximum or sometimes the 
average bi-refringence of the mineral can be estimated which, 
together with other signs, is usually sufficient for identification. 



THE MICROSCOPE 



139 



Alternatively, the bi-refringence may be compensated, and thus 
measured with the aid of the quartz wedge or more readily with 
the Babinet compensator. 

Extinction Directions. Another important test may concern the 
angle between a principal extinction direction for the section, and 
either the cleavage marks or crystal edges which are often recog- 
nizable. It is necessary in such a case to have a cross-wire in the 
field, marking the extinction direction of the analyser, and a means 
of measuring the rotation of the stage. In order to facilitate 
the extinction direction setting, various auxiliary devices can be 
employed. 

Examination in Convergent Light. A system for examination of 
mineral sections in convergent light (the so-called konoscopic 




FIG. 102. SYSTEM FOR OBSERVATION OF CRYSTAL SECTIONS IN 
CONVERGENT LIGHT 



P = Polarizer 
C = Condenser 
O = Objective 



B = Bertrand lens 
E - - Eyepiece 
A - Analyser 



observation) is obtained in a simple way with the microscope, by 
using a condenser of power equivalent to that of the objective and 
removing the eyepiece, so as to study the appearances in the upper 
focal surface of the objective, where the so-called " stauroscopic " 
figures are found. A short discussion of the "ring and brush" 
appearances was given in Vol. I, pages 218-220. The part of the 
slide illuminated by the condenser should be confined to the area 
of the specimen under test ; this is easily arranged with the eyepiece 
in position. 

The use of a "Bertrand Lens" between the objective and eye- 
piece facilitates the observation of the upper focal surface of the 
objective, as shown in Fig. 102, from which it will be seen that the 
combination of Bertrand Lens and eyepiece forms a low power 
microscope projecting the image of this focal surface into the focal 
plane of the eyepiece. In the usual arrangement, this auxiliary 
lens is so fitted to the microscope that it can be pushed into posi- 
tion or withdrawn easily as required. 

The brief mention given to petrological tests in this book by no 
means represents their great development. Owing to their com- 
mercial importance they have received a very great amount of 



140 APPLIED OPTICS 

study since Sorby, in 1858, took the initial steps in this branch of 
microscopy. 

REFERENCES 

1. Czapski: Eppenstein, Theorieder Optischen Instrumente (Barth, 
Leipzig), p. 477. 

2. O. Lummer and F. Reiche: Die Lehre von der Bildentstehung im 
Mikroskop von Ernst Abbe (Braunschweig, 1910). 

3. Martin: Proc. Phys. Soc., Vol. XLIII, Part 2 (1931), p. 186. 

4. Conrady: Jour. Roy. Microscopical Soc., Dec., 1904; Oct., 1905. 

5. Rayleigh: Phil. Mag., August, 1896, p. 167. 

6. Rayleigh: Phil. Mag., XLI (1871), 107-120. 

7. See Trans. Microscopical Society, III (1852), 74. (Paper read by 
Shadbolt in 1850.) 

8. Wenham: Trans. Microscopical Society, 111 (1852), 83. (Paper 
read in 1850.) 

9. Zeit.f. Wiss. Mikros., XXV (1908), 64. 

10. Zeit.f. Wiss. Mikros., XXVI (1909), 391. 

11. Ann. d. Phys., 1903 (4), 10, 1-139. 

12. Knowledge, 1912, p. 37. 

13. Zeit.f. Wiss. Mikros., XXXII (1915), 16-33 

14. Beck, The Microscope, Vol. II, p. 125 (London, 1925). 

15. Kohlerand von Rohr: Zeit. f. Inst., XXIV (1904), 341 

16. Barnard: The Lancet, i8th July (1925), p. 109. 

17. Martin and Johnson: Jour. Sci. Inst., V (1928), pp. 337 and 380; 
VII (1930), Jan. 

18. Johnson: Jour. Roy. Mic. Soc., XLVIII (1928), 144. 

19. Johnson : Phys. Soc. Proc., XLII (1929), 16 ; XLIII (1931), Part i. 



CHAPTER IV 

BINOCULAR VISION AND BINOCULAR INSTRUMENTS 

Physiology of Binocular Vision. Studies of binocular instruments 1 
and binocular vision 2 - 3 have formed the subjects of complete 
treatises, and it will not be possible in the limits of this chapter to 
do more than give the barest introduction, and to indicate the lines 
along which the study may be developed. 

The optic nerve in each eye leaves the eyeball in the region of the 
optic disc, and passes through the optic foramen, an opening in the 
end of the orbit (the conical cavity in the bone which contains the 
eyeball). The nerve then continues to the chiasma. or crossing 
point of the nerves from each eyeball. It is here that a separation 
or decussation takes place, the nerve fibres belonging to the nasal 
parts of each fundus passing to the opposite hemisphere of the 
brain, and those belonging to the temporal parts passing to the 
corresponding hemisphere of the brain; thus impressions on the 
right of each fundus are conveyed to the right hemisphere of the 
brain and vice versa. (See Fig. 103.) 

In normal health of the visual system, both eyes are fixated on 
any object under examination, and a single visual fused impression 
results. The fixation is maintained by the motor muscles of the 
two eyes which are largely yoked in their action, so that the fusion 
of the images is maintained without conscious effort. 

There is also an important inherent connection between the rela- 
tive convergence of the visual axes of the two eyes and the accommoda- 
tion of each eye, so that when a healthy pair of eyes views a near 
object, they converge to exactly the right amount to bring the 
fixated image to the fovea of each eye. This relation, however, 
is not of an inseparable character, and the convergence of the visual 
axes can be varied independently of the accommodation by special 
means, such as placing a weak prism in front of one eye, or altering 
the relative positions of pictures in a stereoscope. This, however, is 
apt to lead to strain and discomfort. 

The normal condition of muscular balance in which the visual 
axes are parallel when the muscles of the eyes are at rest is known 
as orthophoria ; the condition of muscular imbalance, in which there 
is a lack of parallelism with the muscles at rest is known as hetero- 
phoria. The lack of balance may be in the horizontal or vertical 

141 



142 



APPLIED OPTICS 



directions ; there may be excessive or defective adduction or con- 
vergence, or it may be that there is an elevation of one visual axis 
relatively to that of the other. In such cases, fusion is only main- 
tained by increased innervation of the weaker muscles, and when 
fusion is prevented by any means, as by the influence of drugs, 
diplopia or double vision will result. 

In the case of strabismus, or squint, the muscular; imbalance is 
so marked that fusion is impossible, and the visual axes of the two 
eyes cannot both be directed to one and the same object. 

In testing for Heterophoria, use is made of the Maddox Groove, 



A External rectus(musrle) 
li Superior oblique 
(' - Internal rectus 
/) Superior rcrtus 
E -- Inferior rectus 
F =-= Inferior oblique 
G ~ Ciliary body 
// = Ins 
/ = Chiasma 
J = Uncrossed hbres 

of optic tracts 
A' Crossed fibres 
L Optic tracts 
M = Superior colliculus 
N Pulvmar 
lateral gemrulata 
P ~ Medial geninilate 
Q Nucleus of third riervr 
R --- ,, of fourth nerve 
.S - ,, of sixth nerve 
T - Occipital lobes 




FIG. 103. DIAGRAM OF VISUAL TRACTS AND THE CONNECTIONS 
OF THE THIRD, FOURTH, AND SIXTH NERVES 

consisting of a cylindrical groove ground in a piece of red glass and 
held before one eye; the glass is blackened with the exception of 
the groove. This eye then sees a lamp as a bright red streak per- 
pendicular to the axis of the cylinder. The test chart consists essen- 
tially of a scale with a small aperture brightly illuminated in the 
middle, and the test is illustrated in Fig. 104; it is seen that the 
Maddox groove is held horizontally before the left eye, which then 
perceives nothing but a vertical streak which cannot possibly be 
fused with any part of the scale seen by the right eye. There is, 
therefore, nothing to bring the muscles out of the condition of rest, 
and the apparent position of the streak on the scale gives an indica- 
tion of the Esophoria (eyes tending to turn inward) or Exophoria 
(eyes tending to turn outwards) which may be present. The ver- 
tical test may be made by turning both scale and groove into the 
vertical direction. In Hyperphoria one eye tends to turn upwards ; 



BINOCULAR VISION AND INSTRUMENTS 143 

in Hypophoria one eye turns downwards. Combinations of these 
conditions may be found. The amount is measured in prism 
diopters. (Vol. I, page 306.) 

Diplopia Inhibition. It was mentioned above that when both 
eyes fixate the same object a single fused image is perceived. 
Conditions frequently occur in which different images are presented 
to the two eyes even in normal vision, while this is always the case 
in strabismus. 

When both eyes are equally good and diplopia is produced, say by 
slight pressure with the finger on the eyeball at the rim of the 
orbit, both images are equally strong at first. By slightly increasing 
the pressure on the eyeball, however, the displaced image is weakened, 




MaddoxRod 

FIG. 104. TEST FOR HETEROPHORIA 



and may then practically disappear from consciousness. Sometimes, 
by mental concentration on one of two almost equally strong images 
in diplopia, the other can be caused to be hardly noticeable. This 
effect can well be illustrated by putting two "infusible" pat- 
terns, say a circle and a cross respectively, into the two fields of a 
stereoscope. 

Again, if different colours be presented to the two eyes, it is 
usually the case that there is a "struggle" between the two colours. 
Either one or the other is perceived almost at will. It is, however, 
sometimes found that if the colours are distributed in easily fusible 
patterns, then a binocular colour mixture in the additive sense may 
occur. Some observers, however, cannot see this effect. 

Suppressions of this kind are related to well-known phenomena 
in the nervous system by which nerve pulses are inhibited, especi- 
ally at the crossing points of nerve junctions; a strong pulse may, 
as it were, take up the whole road to the exclusion of any other. 

It seems, in fact, as though one eye of the two is generally the 
one for which the pulse is the master. Let any one try this simple 
experiment. Make a ring with the thumti and forefinger; then with 
both eyes open raise the hand quickly so as to view, through the 



144 



APPLIED OPTICS 



ring, some object on the other side of the room. It will usually 

occur that the ring comes into the line of sight of the " master eye." 

Stereoscopy. Let the two eyes fixate a point represented in Fig. 

105 by the point P. This can be represented in practice by a pencil. 




FIG. 105 

If another pencil is held at any point Q within the angle formed 
by the crossing of the visual axes, and either nearer or farther 
away than P, as at Q', then the image of Q or Q' is seen doubled. 

But if the second pencil is transferred to 
R, outside this region of diplopia, it no 
longer appears doubled in consciousness, 
but shows the stereoscopic effect of a 
single object situated in space in a recog- 
nizable position in respect to P. This 
effect persists for a certain range of dis- 
tances of R, both smaller and greater than 
that of P, but outside such a range the 
stereoscopic impression breaks down, and 
a doubled image with no stereoscopic 
effect is manifest. 

The Presentation of Space to Conscious- 
ness. The visual image presented to con- 
sciousness when using either eye is that 
of a perspective presentation taken with 
the nodal point of the eye as centre. 
How this is effected by the retino-cerebral 
system is unknown. In common with a 
chick who can start to peck at food on 
emerging from the egg, we are endowed 
with this intuitive faculty of recognizing 
the geometrical relations of the positions 
of objects. 

Let us now consider the binocular vision of two points P and Q 
at different distances from the observer, whose eyes are A and B 
(Fig. 106). Now the presentation of these points to eye A is effected 
through the image points p a q a on the retina ; similarly, the image 




FIG. 1 06 



BINOCULAR VISION AND INSTRUMENTS 145 

points p b q b determine the presentation to eye B. If both eyes 
fixate the point P, then p a and p b fall on the fovea in each eye 
respectively, while q a q b fall at different distances in the two eyes. 
Let N a , N ft be the nodal points of the two eyes. It is sufficient to 
consider one nodal point for each eye as in Vol. I, page 156. 




FIG. 107. WHEATSTONE STEREOSCOPE 

Let/ be the anterior focal length of each eye; then the difference 
of p b q b and p a q a is evidently (remember that the distance from the 
second nodal point to the retina is equal to the anterior focal length) 



provided the angles are not great. The difference of the two angles 
is the binocular parallax. 

The Stereoscope. We therefore ascribe a stereoscopic perception 
of relative distance to the difference in the perspective images pre- 
sented to the two eyes. Acting 
on this reasoning, Wheatstonc* 
argued that if a suitable perspec- 
tive picture were presented to 
each eye simultaneously, then a 
sensation of a solid object should 
result ; this is the principle of the 
stereoscope. Wheatstone's mirror 
stereoscope is shown in plan in 
Fig. 107 ; M! and M 2 arc mirrors, 
reflecting the images of I\ and 
P 2 , the two perspective pictures. 

The modern stereoscope (Fig. 
108) is generally adapted for 
stereoscopic perspective pairs 
P t P 2 mounted at the interocular 





FIG. 1 08. THE STEREOSCOPE 
(DIAGRAMMATIC) 



distance, and held in the focal plane of viewing lenses L x and Lg. 
If any pair of corresponding points in the pictures are "fused" 



146 



APPLIED OPTICS 



when the visual axes are parallel, the accommodation which is at 
"infinity" corresponds perfectly to the convergence for these 
points, but there will be some deviation from the normal relation 
when points at different apparent distances are fixated. Wheatstone 
also used stereoscopes of this type. For the types of lenses used in 

R A A_ _ 





FIG. 109 

modern stereoscopes see the Verant, page 
io/also Albada's lenses, page 17. 

Brewster's stereoscope (Fig. 109), ex- 
amples of which are still in fairly wide use, 
was adopted for viewing larger pictures 
of which the centres were mounted at a 
greater distance than the ordinary inter- 
ocular distance; this is effected by the 
lens-prisms. When such pictures are 
viewed by a modern stereoscope the inner 
parts of the viewing lenses are employed, 
so that the prismatic effect can again be 
obtained. 

Stereoscopic Perspective Pairs. Let G x 
and G 2 (Fig. no) be two perspective 
centres situated at the ordinary inter- 
ocular distance; think of them, for the 
moment, as the two eyes. We can imagine 
a projection plane R of glass on which, closing one eye at a time, 
we mark out the outline of a solid object ABCD ; two perspectives 
are thus obtained of the kind suggested in Fig. in. Another way of 
obtaining these perspectives would be to employ two cameras with 
lenses at G x and G 2 having the focusing distance of each lens from 
the back nodal point to the picture plane exactly equal to g, the 
distance from perspective centre to projection plane; the lenses 
are placed more particularly with their front nodal points at G l and 




bdccu 



FIG. no. PERSPECTIVE 
PROJECTION 



BINOCULAR VISION AND INSTRUMENTS 147 

G 2 . The photographs thus obtained when mounted (duly reversed) 
in the plane of R will correspond geometrically to the drawings just 
imagined. 

If two such projections are held in the original positions before 
the eyes, provided that g is great enough, most persons can then 
" fuse " the pictures without optical aid and obtain a visual sensation 
of a three-dimensional structure. 

Difficulty is, however, experienced by many persons in securing 
this fusion of the two projections, because, when one of them is 
viewed, the relation between accommodation and convergence 
comes into play and causes the visual axes to converge in the pro- 
jection plane rather than behind it. Success may sometimes be 
attained by looking downwards into a mirror at the image of a 





FIG. in. PERSPECTIVE PAIRS 



distant object; the visual axes then become parallel; if now the 
"stereoscopic picture" is quickly brought between the eyes and the 
mirror (not too close to the eyes, say about 24 in.) it may be possible 
to secure the fusion. Another method is to look at the pictures 
through two pinholes, one before each eye ; the accommodation is 
less definite and a fusion can be effected. 

The function of the lenses in the stereoscope will now be more 
easily understood ; the fusion of a pair of corresponding points can 
be effected without straining the relation between convergence and 
accommodation. We also secure the important advantage that by 
using short focus .lenses we can deal with wide angle views ; with 
the ordinary parallel-axis stereoscope the pictures have to be placed 
in the projection plane with their centres, say, 65 mm. apart, and 
therefore, their effective breadth cannot exceed this amount. It is 
clear that the correct angular projection of the perspective views 
cannot be obtained unless the focal lengths of the viewing lenses 
agree with those of the taking lenses. The importance of viewing 
a perspective presentation under the correct angle was explained 
in Vol. I, page 4. 



148 



APPLIED OPTICS 




Stereoscopy and Convergence. The following experiment will 
illustrate some important conclusions. A stereoscopic reconstruc- 
tion is obtained by fusing two pictures in a stereoscope; the pic- 
tures are two projections made in any suitable way, and are mounted 
separately so that they can be moved independently of one another. 
When fusion has been obtained they are moved slightly apart. It 
is found that in the duration of the movement the whole field 
appears to retreat; if they are made to approach each other the 
field appears to draw neater; but such apparent movement from 
or towards the observer ceases immediately the pictures are still, and 
the whole field looks to be at the same absolute distance as before. 
We therefore conclude that the sensation of relative distance in 



Angular 
Separation 

of 
AandB 



1* 2* 3 C 

Angular Separation 
ofCandD 

FIG. 112. LIMITS OF BINOCULAR PARALLAX FOR STEREOSCOPY 

stereoscopy is entirely independent of the absolute state of con- 
vergence or divergence of the visual axes. These axes may, in fact, 
be made divergent while a perfectly natural-looking stereoscopic 
view is obtained. If the separation of the picture is increased too 
far, however, the muscular control breaks down and diplopia results. 
The only real guide to the apparent absolute distance of the 
"objects" in a stereoscopic field is their angular size; the stereo- 
scopic sense gives an impression of their relative distances, and 
this results through the variation of the binocular parallax, but, as 
mentioned above, this is limited in amount. 

According to J. W. French, 6 if we are fusing the binocular images 
of two points subtending an angle of one degree to one eye, then 
the limiting binocular parallax is about half a degree, i.e. if two 
points A and B subtend an angle of i to the left eye, and two 
other points C and D in the same horizontal line as A and B are 
viewed by the right eye, and if then the images of A and C are 
fused, it will be impossible to fuse B and D into a stereoscopic 
image unless the angular separation of C and D to the right eye 



BINOCULAR VISION AND INSTRUMENTS 149 

lies between about 0-5 and 1-5 (French gives 0-6 and 1-6 more 
exactly) ; for smaller or greater separations of C and D stereoscopic 
vision fails, and the images C and D are not fused. The limiting 
separations depend, according to the same writer, on the angular 
separation of A and B as shown in Fig. 112. In connection with these 
results the limited visual acuity even a short distance from the 
fovea has to be remembered. 

Production of Stereoscopic Effects by Projection and Otherwise. 
The essence of the stereoscopic sensation is the binocular parallax 




FIG. 113. (iRiD ARRANGEMENT FOR STEREOSCOPY 

arising in the different perspectives presented to the two eyes; it 
cannot arise when both eyes view one and the same picture; but 
pictures in different colours may be projected on the same screen, 
as in red and blue-green. If then the eyes are furnished with colour 
filters transmitting only these regions of the spectrum, the two 
pictures (which may be two suitable perspectives) may be combined 
visually to produce stereoscopic results. 

Alternatively, the two pictures may be printed in inks of red and 
blue-green to be viewed in a similar way. 

If a suitable grid is held before a screen, the grid consisting of 
close equal bars and spaces of equal width, it is possible to arrange 
matters so that the two eyes view different strips of the same 
screen (Fig. 113). Thus two perspective projections can be made; 
one photographed in one set of strips; the other photographed in 
another set of strips; in this way a stereoscopic effect can be ob- 
tained. There seems, however, to be no escape from the necessity 
of a fairly close relation between the distances of the grid and the 
observer's head, so that the effect cannot be shown to many people 
at once. 



I5o APPLIED OPTICS 

Yet another possibility is to project the pictures alternatively on 
the same screen, providing the observer with a rotating sector 
disc to obscure the eyes alternately, so that one eye views one set 
of pictures only. Perfectly satisfactory stereoscopic effects can be 
obtained in this way, but the necessity of the sector disc for each 
person has prevented the system from being used for entertainment 
purposes. 

Interocular Distance. Binocular instruments must necessarily 
take account of the variations of interocular distance between 
different individuals, which ranges in adult men from about 56 mm. 
up to 72 mm. with a mean of 63 mm. ; the mean for women is some- 
what smaller, being about 61 mm. Particulars of measurements 
may be found in a paper by J. W. French. 6 Prismatic binoculars 
are usually made adjustable between the limits of 57-70 mm., but 
owing to the much larger eye lenses and exit pupils of Galilean 
binoculars there is not the same need for the provision of the ad- 
justment, although the necessity of making the glasses usable by 
a person of small interocular distance limits the diameter of the 
objective and, therefore, the field of view. 

Effect of Duration of Illumination. Dove 7 performed an experi- 
ment many years ago (1841) which demonstrated the occurrence of 
the perception of stereoscopic relief under the very brief illumina- 
tion given by an electric spark, thus demonstrating that ocular 
movements or changes of convergence of the visual axes are not 
an essential element of stereoscopic sensation, for although the 
sensation lasts much longer than the spark itself, any movements 
of the eyes would not move the location of the retinal image in 
either eye. Much more recently, Langlands* has shown that an 
improvement in stereoscopic acuity occurs when the duration of 
the illumination rises above o-i sec. ; this suggests that ocular 
movements, when possible, do definitely increase the stereoscopic 
sense. 

Stereoscopic Acuity. Many experiments have been made to deter- 
mine the lower limit of binocular parallax which is capable of 
producing the sensation of stereoscopic relief. Under favourable cir- 
cumstances a parallax of under 5 sec. of arc may be appre- 
ciated under conditions of steady observation ; but 10 sec. of arc 
is usually required for instantaneous observations as in the experi- 
ments of Dove and Langlands. Such figures are, however, only 
valid for persons with excellent form vision and adequate training 
in stereoscopic observations. Some observers may find it difficult to 
detect parallaxes of whole minutes of arc without experience, and 
some appear to be lacking in the stereoscopic sense altogether. 



BINOCULAR VISION AND INSTRUMENTS 151 

Pulfrich 9 has devised interesting test charts for use in a stereoscope ; 
the elements of the chart embody various parallactic displacements, 
and they afford a ready means of testing the capabilities of various 
observers. 

Pseudoscopic Effects. If the left-hand perspective projection is 
presented to the right eye, and vice versa, the "stereoscopic depth" 
sensation is reversed and opposes the ordinary interpretation of the 




FIG. 114. STRATTON'S PSEUDOSCOPE 

perspective. The effect may be seen by cutting an ordinary stereo- 
scopic pair and putting the left-hand picture on the right of the 
stereoscope, the right-hand picture on the left. 

A similar effect can be produced by an arrangement of mirrors, 
as in the so-called " pseudoscope " of Stratton (Fig. 114). Two mir- 
rors M! and M 2 are mounted together on a board at 45 to the direc- 
tion of view of the two eyes L and R, so that rays from any object 
reach L directly, but must be reflected from M l and M 2 before 
entering R. 

The perspective centre for the view presented to the right eye R 
is evidently at R x , the image of the nodal point of R formed by the 
two mirrors. Let us draw the projected image of the right eye; 
then we may remind ourselves which is the nasal side by the con- 
vention in the diagram as suggested by von Rohr. 

xx (5494) 



152 APPLIED OPTICS 

The experiment itself is a very easy and interesting one to make, 
and should be carried out by the student. 

The Telestereoscope. Referring to Fig. 115, let A and B be two 
visual perspective centres, and let the separation be b. If the points 
P and Q are two points in the field situated near the line bisecting 
AB at right angles, and if their distances are r v and r q from the 
mid-point of AB, then we shall have, sufficiently nearly if 6 is rela- 
tively small 

APB = and AQB = - 
It was shown above that the parallax for the points P and Q 

x*v. <x"x. 

depends on the difference of PAQ and PBQ. 

i 
i 
i 
l 









FIG. 116 



Hence parallax = PBQ - PAQ - APB - AQB 



* 



If the difference of distances is small so that we may write it dr, 
and the product of r p r q as r 2 , then 

bdr 
parallax = 



BINOCULAR VISION AND INSTRUMENTS 153 

We have made it clear that the stereoscopic phenomenon depends 
solely on the binocular parallax, and it is in a sense immaterial 
how the perspectives are presented to the two eyes provided that 
corresponding points can be fused. By means of optical arrange- 
ments such as the pseudoscope above, we may present to the two 
eyes the perspective obtained from any desired centres, and the 
binocular parallax valid in vision is now to be calculated from the 
perspective centres actually in use. Thus in the telestereoscope 
(Fig. 116), the actual perspective centres are the mirror images of 
the nodal points of the eyes n' a , n\ formed by the double reflec- 
tion. If the separation is now B the parallax for any pair of points 

Bdr 

IS jj 

The effects of such an arrangement were discussed by Helmholt/. 

Stereo-telescopes. If O is at an infinite distance, and P is a point 
at a distance r which can just be recognized stereoscopically with 
the ordinary intcrocular base ft, as a nearer point, then the parallax 
must be the lower limiting value p ot say, which is recognizable. 

Hence since parallax 

I) 
we have p 

The distance r is the " stereoscopic radius." 
If the base length is increased to B, the stereoscopic radius R 
will now be given by T> 



since the stereoscopic radius is increased proportionally to the 
effective base length. 

In the stereo-telescopes we not only have an arrangement of 
reflectors to increase the base length, but an erecting telescope 
system is also held in front of each eye. The effect is, therefore, to 
magnify all visual angles by a factor m, ^ay, which is the magni- 
fying power of the telescope. 

The binocular parallax valid at the actual perspective centres 

will now be presented to the eyes magnified in times ; it will, there- 
fore, be p and thus recognizable. The equation for the stereoscopic 
radius R m is now 



m R 



m 



154 APPLIED OPTICS 

The parallax for any given pair of points at an approximate 
distance r, and difference of distance dr t is now 



and if this is equal to or greater than the lower limit of discernible 
parallax p ot the stereoscopic separation of the two points will be 
recognizable ; then the limiting condition is 

mBdr 
P = 3~- 



so that dr=~ 

mB 



Binocular Telescopes. In considering the perspectives presented 
to the eyes, it is frequently helpful to imagine the image of the eye 





FIG. 117 FIG. 118 

centre projected by the optical system into the object space. Fig. 
117 is intended to represent a non-erecting binocular telescope. 
The eyes are diagrammatically represented in a convention suggested 
by von Rohr, the nasal side being indicated. Since the eye pupil 
is placed in the exit pupil of the instrument on each side, we con- 
sider an inverted image of each eye to be projected by the eyepiece 
system into the corresponding entrance pupil. It is quite clear 
that the effective projections will be reversed as well as inverted, 
and that the space image will be pseudoscopic. The erecting system 
in ordinary binoculars is therefore necessary for this reason also. 
We may note that if both eyes use the same optical system (Fig. 
118), and if we consider the effective centres, we find that the 



BINOCULAR VISION AND INSTRUMENTS 155 

relation of the two eyes is the same in the projected image in the 
entrance pupil, and the effect cannot therefore be pseudoscopic, 
though there will still be inversion and reversion. 







FIG. 119. STEREO TELESCOPE SYSTEM 

The chapter on the telescope includes a number of diagrams of 
prism erecting systems for telescopes in which two parts can be 
separated ; if the ray direction between these parts is at right angles 




FIG. i2o MILITARY STEREO-TELESCOPE 



156 APPLIED OPTICS 

to the initial and final directions, such a system will be useful in 
obtaining a binocular system with a long base and enhanced stereo- 
scopic effect. Fig. 119 is a diagram of the arrangement of the optical 
parts in a stereo-telescope, and Fig. 120 shows a military instru- 
ment. The two arms of the apparatus can rotate about an axis 
parallel to the final directions of the optical axes of the eyepieces, 
and, as will be understood from Fig. 120, this enables a given inter- 




(Carl Zeis^y Jena) 
FIG. 121. TRACE OF KAYS THROUGH A FIELD GLASS 

ocular distance for the axes to be obtained either for the closed 
or open positions of the arms. An ordinary "binocular" system is 
shown in Fig. 121. 

Adjustment of the Axes. As mentioned above, the eyes are sub- 
ject to the habit of a fairly close relation between accommodation 
and convergence. If in using a binocular telescope the accommoda- 
tion must be for "infinity/ 1 the corresponding normal direction for 
the visual axes would be parallel. It is not easy to produce or 
tolerate any considerable divergence of the visual axes; seven to 
eight minutes of arc may be managed without discomfort. Much 
greater amounts of convergence can be tolerated with the accom- 
modation still at oc; 20 to 25 min. causes little or no discomfort. 
Very little allowance is possible in the variation of the vertical 
directions of the eyes ; the enforced Hyperphoria of one eye should 
not exceed seven to eight minutes of arc. These limits set a severe 
demand on the mechanical construction of binocular instruments, 
which therefore have to be mounted with their axes very closely 
parallel. 

Let l\ and p l9 P 2 and/> 2 (Fig. 122) be the entrance and exit pupils 



BINOCULAR VISION AND INSTRUMENTS 157 

of the two members of a binocular system mounted with the optical 
axes at an angle a. The systems are presumed to be erecting systems, 
so that it is clear that if parallel rays enter the centres of the two 
entrance pupils P l and P 2 , these rays will emerge from the centres 
of the exit pupils at a mutual angle of (M - i)a, which must not 
exceed the limits laid down above if comfort is to be preserved in 
use. 

Thus with a pair of binoculars of magnifying power x6, the 
limit of divergence of the optical axes towards the observer 
(requiring divergence of the visual axes) is, say, 

7! min. 



6-1 



= it min. 




FIG. 122 



The same applies to the vertical variation. The limit of con- 
vergence of the optical axes is about 

22 i min. 



6-1 



mm 



These are the tolerances generally treated as standard ones. 

Correspondence of Magnification. Another important matter in 
connection with binocular instruments is the necessity of the exact 
correspondence of the magnification in each part. Any inequality 
in this respect will produce very unpleasant effects to vision, and 
will upset the stereoscopic presentation of the binocular space 
image. Another defect which cannot be tolerated is that of imper- 
fect erection of the images, which will also upset the stereoscopic 
effect, if present, and prevent the effective fusion of the two parts 
of the field. Particulars of methods of testing binocular telescopes 
will be found in a paper by Lt.-Col. Williams. 10 

Space Presentation with Binocular Instruments. As mentioned 
above, a ready way of obtaining the position of the effective centre 
when using any instrument before one eye is to find the image of 
the nodal point of the eye as projected by the instrument. Take, 
for example, the simple cases represented in Figs. 123 (a) and 123 (6). 
A rectangular framework is being viewed with the aid of a lens; 



158 



APPLIED OPTICS 



N is the nodal point of the eye, and N' its image projected by the 
lens. In the first case the eye is close to the lens (the case of the 
spectacle lens), and the positions of N and N' are not far removed 
from each other; the framework is evidently seen under natural 
perspective, since the side nearer to the eye is seen under the greater 
angle. In the second case, the perspective centre is projected 



Object 




FIG. 123. PERSPECTIVE EFFECTS WITH MAGNIFIER SYSTEM 

(a) Effective viewpoint on same side. Natural perspective 
',(6) Effective viewpoint on far side. Unnatural perspective 

beyond the lens and framework ; the latter is seen under unnatural 
perspective, since the side farther from the eye is seen under the 
greater angle. The student should draw for himself the diagram of 
the case when the perspective centre is projected to an infinite 
distance; objects of equal height then subtend equal angles at the 
eye. Prof, von Rohr calls these three types of perspective by the 
names "Entocentric," "Hypercentric," and " Telecentric " ; they 
are all possible with monocular optical systems. 

When we consider binocular systems, we may have two general 
cases of space presentation through the stereoscopic sense, i.e. 
orthoscopic, in which the two perspective centres have their natural 
relation; pseudoscopic, in which the centres are reversed as ex- 
plained above; and sometimes the system may project the two 
perspective centres into one point, when the effect is " synopic," 
and the stereoscopic sense cannot exist' since binocular parallax 
must be absent. Theoretically, as von Rohr points out, the combina- 
tion of the three kinds of perspective with the three binocular 
possibilities makes nine modes of space presentation possible with 
binocular instruments. 



BINOCULAR VISION AND INSTRUMENTS 159 

Most of the traps in this respect are avoided in the familiar 
instruments in which Entocentric perspective unites with ortho- 
scopic presentation, but designers of new systems have to bear in 
mind the possibilities of error. 

Binocular Systems for Microscopes. Owing to the necessary prox- 
imity of the objective and the object, two independent optical 
systems are only possible with the lowest powers, as in the Green- 
hough microscope, Fig. 124. 

If the argument used above in the case of the binocular telescope 
is followed, it will be seen that each eye pupil will be projected by 





(C. 

MICROSCOPE 



(R. 6- /. Beck, Ltd.) 

FIG. 125. BINOCULAR 
MICROSCOPE SYSTEM 



the system roughly into the position of the objective, and that the 
natural disposition of the nasal and temporal sides will be upset 
unless each system is erecting. It follows that an erecting device 
must be included in such a double-barrelled instrument. Use is 
made of the eccentricity of the eyepiece (when a prism erecting 
system is used) to obtain interocular distance adjustment, each 
member being rotatable about the axis of its objective. 

When higher powers are to be employed, only a single objective 
is possible, and the beam from the objective must therefore be 
divided if a binocular system is used. Tf the whole aperture of the 
objective is to remain in net ion lor each image so as to retain the 
utmost resolving po\\er, then (he dixision must be effected by means 
of a partly transmitting and partly reflect ing surface. 



i6o 



APPLIED OPTICS 



Fig. 125 represents a modern prism system due to Messrs. R. and 
J. Beck, Ltd., by which this is done with the aid of a half-silvered 
surface. The final direction of the two principal rays is such as to 
call for a convergence of the eyes to a point approximately at the 




FIG. 126. ALTERATION OF Focus POSITION DUE TO BLOCK OF GLASS 

near point of the average eye, say 250 mm. distant. This will secure 
the usual relation between convergence and accommodation if the 
image is formed at the near point. 

The necessity of introducing the double reflection in the prism 




FIG. 127. ABBE STEREOSCOPIC EYEPIECE SYSTEM 

elongates the path in the corresponding beam, and this would tend 
to make it come to a focus at a point unduly below the other eye- 
piece. This difficulty is countered by adding an additional cubical 
member to the prism. Fig. 126 will illustrate how such an arrange- 
ment may extend the focusing distance. 

In this arrangement the interocular distance is varied by the 



BINOCULAR VISION AND INSTRUMENTS 161 



alteration of the tube length in each member. If abnormal adjust- 
ment in this respect should be necessary, some deterioration of the 
image might result, and it would be better to withdraw the prism 
and make use of the instrument as a monocular. 

The division of the field need not be effected immediately above 
the objective ; various systems are now in use which can be adapted 
to the draw-tube of the instrument as binocular eyepieces. The 
Abbe "stereoscopic" eyepiece is of this kind, and is illustrated in 
Fig. 127 ; the separating surface* is a thin air film. In order to bring 
the two exit pupils to the same level, two special eyepieces of differ- 
ing construction arc used. One works as a "Ramsden," the other 



\ 




; 
A 




Fir.. 128. HINOCULAR KYEPIECE SYSTEM 

as a Huygcnian eyepiece. In this arrangement the brightness of 
one image is twice that of the other. Modern arrangements have a 
more symmetrical design. 

Fig. 128 shows the principle of an arrangement used by Leitz and 
other makers in which the glass path in each arm is equal. A 
specially built body is usually required for systems of this kind. 
Other makers have brought out similar systems in which the eye- 
pieces are given a backward inclination, so that a comfortable 
position for the head may be maintained even when the axis of the 
objective is vertical. 

Fig. 129 shows a construction by Reichert arranged so that 
monocular or binocular observation can be obtained by turning a 
knob. 

Messrs. R. and J. Beck, Ltd., have recently produced a binocular 
eyepiece of this kind with converging tubes. Many observers agree 



162 



APPLIED OPTICS 



that Convergent vision is the more natural with the "downward 
look " position of the head usual when using the microscope, or in 
viewing objects on the bench. 




FIG. 129. MODERN BINOCULAR MICROSCOPE (REICHERT) 



Production of Stereoscopic Effects. No stereoscopic effect can be 
obtamed while the whole aperture of the objective is in use fcr 
each eye. Let us consider the effective perspective centres obtained 
say, with the Abbe eyepiece. The essential optical conditions are 
represented in Fig. 130 (), which shows (conventionally) the two 
eyes of the observer, R and L. No erecting systems arc included 
(the roughly parallel mirrors in the one path produce no erection) 
so that the pupils are projected reversed and superposed into the 
entrance pupil. v 

Now our discussions so far have not dealt properly with lenses 
of finite aperture, and we must consider such a case before going 
further In l-ig. 131, the lens L projects an image of the object 
points A and B into the plane P. Now the image of A is in focus, 
while that of B is not in focus, and, consequently, the position of 



BINOCULAR VISION AND INSTRUMENTS 163 




ft i 

/ \! 



,i 



f Object 



/Vw 



Object 



(a) (b) 

FIG. 130. THKORV OF STEREO KYEPIECE SYSTEM 




of apparent 
r image ofB. while 
^ using top ha/ f of 
<** */*/& 



Centre of apparent 
image ofB, while 
using lower half of 
Jens 

FIG. 131 



164 APPLIED OPTICS 

the centre of the image patch on the local plane P will differ accord- 
ingly as the top or bottom halves of the lens L are used. The two 
pictures are much as we might obtain from two different lenses 
with centres at C t and C 2> which are really the perspective centres. 
We can look on the top or bottom half of the lens as two thin lenses 
plus a prism of small angle; thus it happens that these perspectives 
are superposed. 

Let us now go back to the diagram of the optical arrangements 
in the microscope; let screens be introduced to cover the nasal 
halves of each pupil; they are shown imaged in front of the nasal 

D DO 
O D O 
O DO 

Stereojftopic Pseudoscopic 

FIG. 132 

parts of projected pupils R' and I/. Consequently, the perspective 
centre will be on the C t side for R', and on the f 2 side for I/, where 
Cj and C 2 are towards the temporal sides of the projected pupils. 
A true stereoscopic effect will result. 

On the other hand, if the screens hac^ covered the temporal sides 
of each real pupil we should have the/conditions suggested in the 
smaller figure (Fig. 130 (fe x ) ; speaking of the projected pupils we now 
see that the centre for R' is towards C 2 , and that for L' is towards 
C x ; it is clear that the effective viewpoints arc now interchanged, 
left for right, and a pseudoscopic effect must result. 

Abbe pointed out that it is unnecessary to consider the particular 
means by which the separation of the centres is made, i.e. whether 
by a prism dividing the beam from the objective, or whether by 
screens in the pupils. Nor need the differentiation of the centres 
be effected except by a screen in one pupil, unless the fullest degree 
of relief is required. If the pupils appear as on the left of Fig. 132, 
the effect will be stereoscopic ; if as on the right, a pseudoscopic effect 
will be seen. 

R INFERENCES 

1. von Rohr : Die Binokularen Instrumente (Berlin, J. Springer, 1920). 

2. Hoffmann: Die Lehre vom Raumsinn des Auges (Berlin, J. 
Springer, 1920). 



BINOCULAR VISION AND INSTRUMENTS 165 

3. Helmholtz: Physiological Optics. Section on binocular vision. 
(English translation, Optical Society of America, 1925.) 

4. Wheatstone: Phil. Trans. Rov. Soc. (1838), 371-394. 

5. J. W. French: Trans. Opt. So'c., XXIV (1922-23), 226. 

6. J. W. French: Trans. Opt. Soc., XXTII (1921-22). 

7. Dove: Benchte der Berliner Akademie, 1841. 

8. Langlands: Medical Research Council. Special Report Series, 
No. 133 (H.M. Stationery Office), p 64. 

9. Pulfrich: Zeit f. lnstr. t XXI (10.01), 249. 

10. Williams: Trans. Opt. Soc., XX (1918-19), 97. 



CHAPTER V 

PHOTOGRAPHIC LENSES 

PHOTOGRAPHIC lenses are usually required to project an image on 
a flat plate or film held perpendicular to the optical axis. As far 
as requirements are concerned, they may be divided into a number 
of groups of which the first comprises snapshot cameras. These 
are required to allow "snapshot" exposures of, say, L / : ,th sec. in 
medium sunlight, or even in the absence of direct sunlight on a 
bright day. This requires a minimum aperture or stop number of 
"//n," with the photographic materials available at the present 
time. Such lenses are usually of the "landscape" type, consisting 
of a single or sometimes an achromatic lens with a suitable stop. 

The stop number is the quotient of the focal length of the lens divided 
by the diameter of the effective stop or diaphragm. Soe below. The 
focus is usually non-adjustable, and great depth of focus in the 
object field is required. 

The pictures produced by such landscape lenses are expected to 
be reasonably sharp to unaided vision, but not necessarily to l>ear 
any great enlargement. They are seldom used for architectural sub- 
jects, and some distortion in the image can be permitted. 

The second group includes lenses which are fitted to more ambi- 
tious cameras for amateur work. The main requirements are im- 
proved definition, allowing of enlargement ; reasonable absence of 
distortion in the image ; focusing by visual setting ; maximum rela- 
tive aperture possible without undue cost. In this group we find 
lenses of many optical types and relative apertures, ranging from 
//8 to about 7/4. 

In the third group which comprises lenses for Press and com- 
mercial work, the above requirements have to be considered together 
with the necessity of obtaining the utmost speed (and therefore 
the greatest relative aperture), more or less regardless of cost. Here 
we find relative apertures from //4 to 7/2-5 or even f/2. Snapshot 
exposures can be obtained with such lenses in very poor light, and 
when sunshine is available fast moving objects can be photographed 
with exposures as short as TT nny sec. or even less. 

Kinematograph lenses may be included in this third group. They 
are required to give the maximum illumination of the image, but, 
of course, the scale of the picture is in this dase a very small 

166 



PHOTOGRAPHIC LENSES 167 

one. The definition must be sharp enough to bear very great 
enlargement. 

Lenses are also made to fulfil special requirements such as sur- 
vey work, where a special freedom from distortion is required; 
wide angle photography, in which the image must be flat and 
well defined over an exceptionally wide field ; process and copying 
work, where again special attention has to be paid to freedom from 
distortion, and (especially in three-colour printing) to the freedom 
from differences of magnification for different colours; in these 
cases, however, a large relative aperture may not be called for. 

Projection lenses have optical principles similar to those of photo- 
graphic lenses, although the path of the light is reversed and the 
object is not self-luminous. 

Lastly, the class of telephoto lenses have the function of project- 
ing an image, of a distant object, larger than can be obtained from an 
ordinary lens with the same working distance. This is done by 
employing a construction giving an effective optical focal length 
much greater than that of the ordinary camera extension. 

The Illumination of the Image. Reference may be made to Chapter 
VI for the fundamental photometric concepts ; it will be convenient 
to deal with the photometric aspects of photographic lens theory 
immediately. In photographic lenses the diaphragm is within or 
close to the lens system ; the entrance pupil is usually the image of 
this diaphragm seen from the front, and the exit pupil is the cor- 
responding image seen from the rear. Let us consider a small ele- 
mentary object of area a, and of brightness B perpendicular to the 
axis in the object space. Fig. 133 will be of help. The corresponding 
image is of area a'. If m is the linear magnification, then 

a' m 2 a 

Let p be the radius of the entrance pupil, then the amount of light 
entering the lens from the elementary object area is (if q is the 
distance of the object from the entrance pupil) 



This light suffers partial absorption by the lens, and a fraction k 
reaches the image, where the amount of light falling on unit area 
(the illumination) is 



Til .- r.t. - /&rB7T/> 2 \ 

Illumination of the image = ( -- j j 1 -f- m z a 



"-(5494) 



i68 



APPLIED OPTICS 



In accordance with the usual notation, the symbol x denotes the 
distance of the object from the first focal point of the lens. Then 
we have 

? = (/+/+*). say, 
where vf is the small distance of the first principal point from the 

h f I f\ 
entrance pupil; but since m - j = -( L % J, (see Vol. I, page 40), we 

get ' V' 



and 



mq = 



Diaphragm 

\ f- Entrance Pupil 




<- 



(T* 



^Position of Principal Plane 
FIG. 133 



The expression for illumination is then 

ftBir 



Illumination ~ -, - 



The fraction 



-|- v) m - i} 

( - 1 is half the reciprocal of the stop number, since 

the stop number was defined above as the quotient of the focal 
length divided by the diameter of the effective stop or diaphragm. 
Note that we measure the entrance pupil,* i.e. the image of the 
diaphragm seen from the front of the lens, and not the diaphragm 
seen directly. It is also to be noted that m will be numerically nega- 
tive for an inverted image, and that when the object is infinitely 
distant its value will be o. The value of v will be negligible in many 
practical cases. 

* It may be measured in practice with the help of a reading microscope 
with a fairly long focus objective. 



PHOTOGRAPHIC LENSES 



169 



Illumination of the Image Away from the Axis. Consider the 
case when the lens projects the image of an extended uniformly 
diffusing surface of brightness B. A small area p (Fig. 134) sends 
light to the entrance pupil so that the rays make an angle a with 
the axis of the lens, and with the normal to the surface at p. The 
amount of light dV radiated by the element into a solid angle dco 
in this direction is (page 207) 

tiV pH cos a d( 



*-Image 
Plane 




IMC;. 134 
and the solid angle subtended by the pupil is 

7T/> 2 COS a 

/t t t \ _ 

Ml') - . xo 

(q sec a)* 

where q is the distance of the entrance pupil from the object plane, 
and p is the radius of the pupil. 

Hence dV 7rBp/>* cos 4 a 

In cases where there is no distort 'on, the area of the corresponding 
image patch will be uniform and independent of a. Hence the 
illumination of the image plane will vary as cos 4 a, even if vignetting 
by the diaphragm does not occur (see below). It is easily calculated 
that cos 4 a becomes 0-56 at 30, and 0-25 at 45. 

The effects of vignetting were referred to on page 107, Vol. I. 
It may be illustrated by reference to the symmetrical system shown 
in Fig. 135, where two similar lenses have a stop between them. If 



170 



APPLIED OPTICS 



the stop is opened out to full aperture so that an incident bundle 
of rays parallel to the axis is completely transmitted, we see that 
the actual separation produces such vignetting that, when a bundle 
of the same diameter is incident at about 30 with the axis, the 
part of the back lens transmitting light is limited to the lune-shaped 
area shown in the side projection. The effect of the vignetting in 
reducing the relative illumination of the outer parts of the image 
can be avoided by using a sufficiently small stop. Wide-angle 
lenses are used with small stops for this reason. 

Ordinary modern lenses will rarely transmit any rays at all at 
greater angles with the axis than 45 or 50. The diagonal of the 



'ffect with 
"large stop 

fffect with 
small stop 




135 



plate may be expected to subtend an angle of 40 to 50, so that 
the most oblique rays will not make angles with the axis much 
over 25. The loss of light, as well as the optical aberration, sets 
a limit to the area which can be usefully covered by the lens. 

Requirements for the Formation of Images on a Plane. Fig. 136 
shows an optical system which is forming an axial image B' of an 
axial object B. The system is free from spherical aberration for 
these conjugate points; hence all rays from B pass through B'. 

If now, in addition, we have another object point B lf situated in 
the plane perpendicular to the axis through B, and near to the axis, 
the condition that the corresponding image B/ may be sharply 
defined was worked out in Vol. I, page no, and is known as the 
sine condition. Let co, co'; co lt 0V, be the angular inclinations of 
corresponding rays passing through B and B' ; then the sine condi- 
tion states that 



sn a 

_- - 

sin co 



sn 



sin 



-_ constant. 



This relation must, therefore, be fulfilled if we are to obtain good 
definition of the images of points surrounding B, wherever these 
images are situated. 

The sine condition is the necessary criterion for freedom from 
"coma," or differences of magnification for different zones of the 



PHOTOGRAPHIC LENSES 



171 



lens, but its fulfilment does not, as shown in Vol. I, page 135, secure 
freedom from oblique astigmatism, which often tends to arise on 
account of the differential limitation of the perpendicular width of 
a bundle of rays passing obliquely through a round stop. The fore- 
shortened width in the "tangential" direction is d cos a (where d 
is the diameter of the stop, and a is the inclination), as against a 
width of d in the sagittal direction. If a pencil goes centrally through 
a thin lens, the same approximate relative retardation is impressed 
on the central parts, as compared with the marginal parts of each 
section; since the "tangential" section is the narrower, we can 
visualize a refracted wave of greater curvature and shorter focal 
distance than for the sagittal section. In Chapter VIII of Vol. I, 




FIG. 136 

pages 300 to 305, it was shown, however, that the effects of astig- 
matism, at least for a very narrow bundle, could be removed by the 
use of an axial stop at a finite distance from the thin lens when the 
latter was bent into a suitable meniscus form concave to the stop. 
We may, therefore, understand that there are similar possibilities 
in connection with more complex lens systems if a suitably disposed 
stop or diaphragm is allowed.* 

Provided that astigmatism is eliminated, the sharp images are 
found in cases of simple lens systems to lie on the so-called Petzval 
surface. From the equation (52) (Vol. I, page 139) it follows that 
the radius R of this surface (assuming a flat object plane as in our 
case) is given by the equation 



i 
R 



- etc. 



where n lt 14, are the refractive indices of the glasses of which the 
successive lenses are composed, and f l9 f 2 , are the focal lengths 
which would be found for the lenses if calculated from their radii 
and refractive indices while neglecting the thickness. According to 

* Appendix III gives a short discussion of the curvatures of the tangential 
and sagittal image fields by the simpler "third order" theory. The results 
are, however, only reliable for rays making small angles with the axis. 



172 



APPLIED OPTICS 



the simple first order theory, the sharp image should be flattened, 
then, by choosing the glasses and radii to make R infinite. In prac- 
tice, the effect of finite thicknesses in the lenses causes some departure 
from this provision of the simple theory. 

Distortion. Provided then that we satisfy all the above require- 
ments, we may expect to find a reasonably sharp image in the 
neighbourhood of the axis, which would fall into focus on a flat 
plate; the definition might be expected, however, to deteriorate 
when we exceed a certain distance from the axis, where the angular 
inclination of the rays with the axis invalidates the simpler theory. 

We have, however, still to inquire whether the image in the plane 
S' will be geometrically similar to that in S. We know that the scale 




FIG. 137 

of the magnification in the plane of B' is given by ~ - at least 

for very small objects, so that there should be geometrical similarity 
to the object in the close neighbourhood of the axis. Our theory 
does not allow us to investigate the presence or absence of distor- 
tion far from the axial region in these planes S and S', but as we 
know that photographic lenses are often required to work at all 
kinds of relative conjugate distances, it will be of interest to con- 
sider a second pair of conjugate planes through T and T', which we 
may first assume to fulfil all the requirements of the foregoing para- 
graphs. These are shown in Fig. 137. Now a ray through B in the 
object space passes through B' in the image space, and it passes 
through the planes through T and T' in two points L and L' at 
distances h and h' from the axis; we will suppose that L' represents 
a reasonably sharp image point. Then, the ratio h'/h represents 
the magnification and is given by 



*' 

h 



q' tan a>' 
q tan w 



where q and q' are the distances from S and S' to T and T' respec- 
tively. In order that the magnification ratio may be constant for 



PHOTOGRAPHIC LENSES 173 

all sizes of object, we must have relations such as 

a' tan c' 

_. constant for any values of oj and co . 

q tan oj J 

But we found above that 

sin ft/ 

==- constant 

sin ( 

was the necessary condition for sharp definition in the planes 
through S and S'. Since the above conditions are incompatible, 
we find at once that even if the distortion is completely corrected 
for pianos containing a pair of truly aplanatic points, there must 
be definite distortion in anv other pair of planes. This conclusion 




FIG. 138 



is justified in practice, but in cases where the extreme angles o/ 
and o) do not exceed about 4 or 5, the difference between the 
sines and tangents will be less than 0-3 in 100, so that the distortion 
would not necessarily be serious for small aperture systems. 

In practice, modern photographic lenses are not completely cor- 
rected for spherical aberration ; this aberration has to be kept 
below certain limits, but these limits are wide enough to make it 
much easier to achieve freedom from distortion and other aberra- 
tions in a way which is impossible if strict aplanatism is called for. 

If, then, spherical aberration is to some extent permissible, the 
position of an image "point" will be fixed by the principal ray 
through the centre of the diaphragm. We will, therefore, consider 
a case shown in Fig. 138, where the centre of the diaphragm is 
situated at the point D. Let us trace a ray through this point in 
both directions to its intersection with the planes S and S', which 
represent the object surface and the photographic plate respectively. 
The resulting intersection points V and V are now object and 
"image." If the image is subject to spherical aberration, its 



174 APPLIED OPTICS 

position is still marked by the intersection of the principal ray with 
the plane. S'. 

Looking into the front of the lens, the axial point of the entrance 
pupil P would be seen as the image of D. From the rear, the exit 
pupil would be seen at P'. (Vol. I, page 106.) 

Now it is likely that the axial positions of the entrance and exit 
pupils will be to some extent dependent on the inclination of the 
principal ray through D; these images may be subject to the effects 
of spherical aberration. Let the Gaussian positions of P and P', 
calculated for very small inclinations of the principal ray, be denoted 
by P and P' . Let CD and co' be the axial inclinations of the prin- 
cipal ray in the object and image spaces, while x and x' are the 
intervals P B and P' B' from the Gaussian pupils to the conjugate 
planes respectively; also let PP = d, and P'P' ^ d'. Thus PB 
= x + d and P'B' = x' Q + d'. Then object and image heights are 
related by the equation 

B * *' 1-3' tan a/ 



~~ A ~" (x + <5)tanft 

In the case of an indefinitely small object and image let 

limit h' _ 
(h = o) h - m 

Then a convenient specification of the amount of the distortion D is 



p __ i ((*', + ($') tan ft/) 
>o I (*o + <5) tan o) ) 



I 



This specification is used in Wandersleb's important papers 1 
(1907) on distortion, in which the distortion of most of the important 
photographic lenses of that period is given graphically in terms of 
D for given angles with the axis. 

When the object is at an infinite distance we need to transform 
the above equation. Remembering from Vol. I, page 40, 

/ 
h x 

f 
= , for a lens in air, sufficiently nearly 

x o 

so that m (x + d) = /' + m d 

= /' 
in the limit since m is indefinitely small, and d is also small. Thus 



m = v = - - 



((*' + ?') tan a/) 
( f tan to ) 



PHOTOGRAPHIC LENSES 175 

Wandersleb uses a value N in his discussion which is the reciprocal 
of the magnification m. 

If the discussion of Chapter IV, Vol. I, is carefully re-read it will be 
seen that the expression in equation 50 represents the optical path 
difference between distances derived from the centre, and from a point 
in the effective aperture, of the exit pupil when they arrive at the point 
given by the intersection of the auxiliary optical axis with the Petzval surface. 
Even if the coefficients a,, 2 , a s are zero an optical path difference 
remains; and thus when the transverse aberration T'y is calculated, a 
term remains which is proportional to the cube of the image "height," 
indicating a displacement of the image in a radial direction which does 
not depend on S and therefore affects the principal ray itself. 

Analytical calculations of distortion may be made by the help of 
formulae quoted in books such as Conrady's Applied Optics and Optical 
Design, but they are not of a simple character, and fail to give useful 
results when thick lenses and very oblique rays are dealt with, so that 
they have to be supplemented by empirical calculations. The type of 
distortion arising in simple cases can often be understood from first 
principles. No distortion results from thin lenses passed centrally by 
oblique pencils. If a stop is used with a thin convex lens behind it, the 
deviation towards the axis of the oblique principal rays tends (for usual 
forms of lens) to be over-great, and "barrel" distortion, resulting from 
the lower magnification for the greater image height, results. This is 
the most common form. A similar thin convex lens in front of a stop 
produces pin-cushion distortion. The "symmetrical" combination is 
more likely to be distortion-free, and it is important to retain a sym- 
metrical design in any case where the utmost freedom from distortion 
is essential, as will be understood when symmetrical systems are dis- 
cussed below. 

Tolerance for Distortion. Distortion may be expressed as above 
by the percentage 

fm \ 

100 D = 100 I i ) --= percentage distortion. 

\m J 

In the best symmetrical form lenses the distortion may be kept 
within o-i per cent up to 30 or more with the axis. The value is 
sometimes greatest for an intermediate inclination of the pencils. 
With the "Hypergon" (see below) it is only 0-3 per cent at 54 with 
the axis. It is more difficult to control with the unsymmetrical 
forms. The curves for an early Cooke lens are given below, but much 
smaller amounts are found with later lenses. Thus with one form 
Wandersleb finds the distortion to be zero at 27 with the axis, and 
reaching a maximum of 0-6 per cent. The tolerance for distortion 
varies enormously with the object of photography. Several units 
per cent can be tolerated for landscapes, but the requirements for 
architectural subjects are much more stringent, say, 0-5 per cent to 



176 



APPLIED OPTICS 



i-o per cent. Photograptdc mapping requires distortion within 
o-i per cent if possible. 

The Landscape Lens. The lenses fitted to inexpensive hand 
cameras are often of simple meniscus form; they are placed at a 
suitable distance behind a circular stop. There is no chromatic 
correction ; the focus is fixed once for all, reliance being placed on the 
fact that the spectral region of actinic activity for ordinary photo- 
graphic emulsions is fairly well defined in the spectrum. 




FIG. 139. MENISCUS LANDSCAPE LENS 

Fig. 139 shows a diagrammatic section of such a lens, traversed 
by an oblique bundle of parallel rays. Such a lens suffers in the first 
place from spherical aberration of the ordinary "under-corrected" 
type which causes the outer rays to reach the focus closer to the 
lens than the inner ones. In this case, however, coma is also present , 
the upper part of the lens is traversed more or less symmetrically 
by the rays in the diagram; the deviation produced by a prism, 
it will be remembered, is a minimum where there is such a sym- 
metrical disposition. This then tends to lengthen the focus in the 
upper part, but the rays in the lower part traverse the lens very 
unsymmetrically, so that there is relatively great deviation in the 
sense suggested by the figure. Hence, the coma opposes the spherical 
aberration in the upper part and supports it in the lower part of 
the lens under the conditions drawn. 

For this reason a stop is placed in front of the lens which limits 
the oblique pencils, as indicated in the diagram; the spherical 
aberration and coma thus tend to balance their effects in the 
oblique pencils. Since the spherical aberration must be tolerated, 



PHOTOGRAPHIC LENSES 177 

the allowable amount sets a limit to the useful diameter of the stop. 
f/n can be attained. 

Nor is this all ; the theory of Chapter VIII, Vol. I, indicated such 
a disposition of lens and stop to overcome the effects of astigmatism ; 
the problem in designing such a lens consists of striking a suitable 
balance between the aberrations. It can be shown that a favour- 
able over-correction for astigmatism is obtained when coma and 
spherical aberration are balanced in the above way, by the disposi- 
tion of the stop. 

Astigmatic Corrections. We will suppose that conditions allow us 
to abolish astigmatism, or even to over-correct it in a simple lens 
system. 

Fig. 140 shows the cases of under-corrected, corrected, and over- 
corrected astigmatism. In the first case, t'lc PeUval surface has a 




(c) 

'' i 

Under -corrected Corrected Over- corrected 

astigmatism 



negative radius of curvature, and the tangential and sagittal 
image surfaces are similarly disposed ; it will be remembered that 
the distance between the tangential and Pct7,val surfaces is always 
three times that between the sagittal and Pctzval. 

In the second case the astigmatism is corrected, and the sharp 
image lies on the curved Petzval surface. 

In the third case the astigmatism is so far over-corrected that 
the tangential field is flat while the sagittal one is slightly round; 
this condition represents a very usual compromise in the direction 
of getting a flat field for the image. It has been tried in some cases 
to go still further, so that the tangential and sagittal fields are 
symmetrically disposed on each side of a plane surface, but this is 
only at the disadvantage of a greater amount of astigmatism which 
itself causes a severe loss of definition. The apparently obvious 
method of improving matters, when more complex lenses are used, 
is to choose glasses, such that the Petxval curvature will be zero 
or approximately so: it will be found in practice, however, that 
the finite thicknesses of the lenses and the effects of the oblique 



I 7 8 



APPLIED OPTICS 



aberrations away from the axis are such as to make the theorem 
only a very rough guide. 

Achromatic Landscape Lenses. In seeking to improve on the 
single meniscus lens it is natural to seek to produce an achromatic 
combination which will allow the image to be focused visually. 
In this case we shall equalize the focus for the lines G' and D, so 
that the V values used for the formulae will need to be recalculated. 
Our new V will be found from 

v = tozl) 



This brings together the brightest visual region of the 
spectrum and the region which has the greatest 
action on ordinary photographic emulsions. The 
formulae now used are of the same form as those of 
Vol. I, page 228. In practical designs the lens has 
still an outer meniscus form, with the concavity to- 
ward the stop (Fig. 141). 

Old and New Achromats. When the glasses used 
in achromatic combinations have a large difference 
of V values, it is possible to keep fairly shallow 
curves for the lenses, and thus to avoid zonal spheri- 
cal aberration. This is a very important matter in 
telescope objectives, and keeps the "old achromat 11 
crown and flint combination, with its large difference 
of V values, of outstanding importance. 
Some of the more recently produced glasses have different optical 
properties. Let us compare the following combinations 



LI/ 



FIG. 141. 

ACHROMATIC 

LANDSCAPE 

LENS 



Glass 


"D 


V 


a 


ft 


y 


< Hard crown 
< Dense flint 


1-5186 
1-6041 


60-3 
37-8 


295 
286 


705 
714 


569 
606 


< Dense barium crown . 
2 (Light flint. 


1-6089 
1-5472 


57'4 
45-8 


294 
291 


.706 
709 


572 
591 



We notice that in the second pair, the glass with the higher refrac- 
tive index has the higher V value (or lower relative dispersion). Cal- 
culating an ordinary achromatic combination from each, we find 
for unit focal length (a and b standing for crown and flint respec- 
tively) 

Combination i j ^ ^^ Combination 2 j = 

(Old achrornat) (New achromat) 



- 0-2533 



PHOTOGRAPHIC LENSES 



179 



We evidently have much shorter focal lengths and heavier curves 
in the units of combination 2, a type of "new achromat." On the 
other hand, the much closer correspondence between the partial 
dispersions of the glass used in the new achromat would ensure a 



Light Barium 
Flint* ICrown 




Ordinary 



Converging 
Face 






^Diverging 
Face 

B 



Crown of Ordinary 
Jfigh Index Crown 

FIG. 142 "NEW" \ND "OLD" ACHROMATS 

A Achromat of new glassp-s Contat t surface has com erring tnd*ncy. (Diagrammatic only) 

Sut h a lands* ape lens will rover a 90 neltl at //ifi) 
/?. Arhmmat of old glasses. Contact surface has di\ergmi/ tendency. (Diagrammatic only) 

much less pronounced secondary spectrum. Calculating the radius 
of the Petxval surface in each case we find by the usual formula, 



R = - 1-38 for the old achromat, and R = - 1-94 for the new. 
Hence the image field, if corrected for astigmatism, will be much 
flatter with the new achromat than with the old. A diagram (Fig. 
142) of possible arrangements of the old and new achromats will help 
to show that the contact surface in the new has a converging 
tendency, while in the old the contact has a diverging tendency. 
On account of the flatter Petzval surface, and the different action of 
the contact face, it proves to be easier to give satisfactory "photo- 
graphic" corrections to a new achromat than to one of the older 
type. A well-designed landscape lens will cover a 90 field at an 
aperture //i6, but the image will be subject to distortion. 
Anastigmatic Correction. In Vol. i, page 302, we obtained the 



i8c APPLIED OPTICS 

following expression for the astigmatism arising by refraction at a 
surface 



(Qi-Q<) 2 \'t' 

It is clear that the squared terms will always be positive, and 
therefore the sign of the astigmatic contribution will be given by the 
second bracket term 



\n't' nt) 

The symbols t' and t may be taken as the conjugate image and 
object distances respectively for the surface in question. Clearly, 
in the case where we have a converging surface with the object 
distance negative and the image distance positive, both terms in the 
equation would be positive. 

For small angles of incidence, the tendency of a convergent face 
is to give a positive sign to the bracket, while a negative or diverging 
face would give a negative sign, although contributions given may 
vary in sign if the incident light is very convergent or divergent. 
Students should work out typical cases. 

Now both with the "old achromat" and the "new achromat" 
type of landscape lenses it is possible to "flatten the field" in the 
sense that the usual astigmatism may be over-corrected so that the 
tangential field is flat or even "hollow" (i.e. turning its convexity 
towards the lens) but with the ordinary constructions the astigmatic 
error constantly increases in the same sense towards the margin of 
the field, as does also the spherical aberration. 

If the Petzval surface is "round" as seen above, the flattening of 
the tangential field can only be achieved if a considerable amount 
of actual astigmatism is tolerated. 

The "concentric" lens designed by Schroder about 1887 and made 
by the firm of Ross, achieved fairly flat image fields by the choice of 
suitable glasses, but only at the expense of a large amount of under- 
corrected spherical aberration. 

In seeking to improve on these conditions Steinheil, and also 
P. Rudolph, evolved (about 1881 and onwards) more or less sym- 
metrical combinations in which thick doublet lenses of outward 
meniscus shape were mounted on each side of a stop, the concavity 
of each meniscus being turned towards the stop. With the aid of 
this construction it was found possible to correct spherical aberra- 
tion for one zone of the system, chiefly through the opposing action 
of the inner contact faces, although there was considerable residual 
zonal aberration. Under these conditions the astigmatism, even if 



PHOTOGRAPHIC LENSES 



181 



it were of the usual under-corrected type for small angles of the 
field, and showed at first a tendency to give "round" fields, was 
now found to be subject to a correcting influence at higher angles s<f 
that the fields tended to become hollow. Even with the old glasses 
it is possible with special constructions to make the tangential and 
sagittal fields intersect towards the edge of the field, although the 
residual bending of the image surfaces is large in other regions. 

Great improvements were possible by the employment of the new 
glasses. An "old achromat" used in front of the stop, with a "new 
achromat " behind, gave the possibility of a flatter Pctzval surface, 



Diverging 
Face 



Converging 
Face 




Crown of 
High Index 




Ordinary 
Crown 



FIG. 143. "TRIPLE CEMENTED TYPE" 

and less difficulty in the correction. Lenses of this type were made 
by the firm of Zeiss, and in such cases it may be considered that the 
correction is largely secured by the opposing action of the two 
contact faces, the negative astigmatism of the first with its diverging 
action acting against the positive astigmatism of the second (con- 
verging) contact. 

The next step taken by Rudolph, about 1891, was to combine in 
a triplet lens, three glasses as shown in Fig. 143. The refractive 
index shows a step upwards at both contact faces, one being diverg- 
ing and the other converging. It is thus possible to secure anastig- 
matic correction by the opposing action at these contact faces. 
Rudolph also showed how it was possible to get a similar effect with 
other constructions; the middle lens of the three might be "double 
convex," while the refractive indices are still progressive. 

The "Amatar" lenses of Messrs. Zeiss are of the type shown in 

Fig- 143- 



182 



APPLIED OPTICS 



The use of lenses of this type in a symmetrical system was worked 
out by von Hoegh independently, and patented by the firm of 
Goerz in 1892. (Goerz double anastigmat.) 

Still further improvement was found possible by Rudolph in 1894 
when the combination now used in the " Protar " lenses of Zeiss was 
patented. From Fig. 144 it will be seen that a modified "new 
achromat" is combined with an "old achromat" combination. 
Although the general construction may be so described, the pairs 
are not now achromatic in themselves, but achromatism is estab- 
lished for the lens as a whole. There are now three contact faces, 
and it is possible to secure a very complete anastigmatic correction 



Modified 
Old 



Ft Cr. 




FIG. 144. COMBINATION OF MODIFIED 

NEW AND OLD ACHROMAT IN THE 

PROTAR LENS (ZEISS) 

(Designed by Rudolph) 




Crown of 
High Index 



Ordinary 
Crown 



FIG. 145. ANASTIGMATICALLY 
CORRECTED, SEPARATED DOUBLET 



over a field of 30 on each side of the axis at //I2\5. The lens will 
cover a 90 field if used with a smaller stop, and since the stop is 
very close to the system the distortion is smali. 

Yet another method of securing anastigmat correction is to 
separate the components of a landscape lens as shown in Fig. 145. 
The rear faces now have a converging tendency and the front faces 
a diverging effect, so that opposing spherical and astigmatic con- 
tributions may be obtained by the choice of suitable bendings. 

Symmetrical Lenses. Important advantages are obtainable from 
the employment of systems which are built symmetrically with a 
central stop. In Fig. 146, let X and Y be two lenses of such a 
system ; the central stop is at R. Consider two parallel rays, AB 
and CD, between the lenses, symmetrically situated with regard to 
R. If these are traced through the lenses their crossing points B l 



PHOTOGRAPHIC LENSES 



183 



and B/ in the object and image region must also be symmetrical 
with regard to the system. Also, there must be some ray through 
the centre of the stop, which will pass through both B t and B/, 
Hence the system must be free from coma for one zone at least, 
when the image size is exactly equal to the object size as must now 
be the case ; further, the exact equality of object and image dimen- 
sions must result in the complete elimination of distortion. 

Also, since the above reasoning would apply to rays of differing 




FIG. 146. SYMMETRICAL SYSTEM 

wave-length, the image must now be free from chromatic difference 
of magnification. 

The symmetry of the system does nothing to remove the effects 
of axial chromatic aberration, spherical aberration, astigmatism or 
curvatun^of the field, but the freedom from coma, distortion, and 
chromatic difference of magnification is a very important advantage. 

Let us now consider the conditions when the object and image 
are situated at different distances from the system. In the first 
place the parts of the principal ray outside the lenses will be parallel 
to each other, so that <o co' (see equation above). The absence of 
distortion then requires d = d', or a constancy of the positions of 
the images of the centre point of the stop for all inclinations of the 
principal ray. A similar argument applies to differences of magnifi- 
cation due to colour; in order to avoid these, the position of the 
entrance and exit pupil should be independent of wave-length. 
Further, the argument which established the freedom from coma is 
o longer valid. 

In all these cases, however, the symmetry of the system is a very 
great advantage. If the separate systems are moderately well cor- 
rected in themselves they can be combined in the symmetrical 
manner without fear cf introducing serious coma, and with the 

13~(5494) 



184 



APPLIED OPTICS 



reasonable assurance that the distortion and any chromatic differ- 
ence of magnification will be made practically negligible. Moreover, 
the shortening of the focal length due to the combination produces a 
lens working satisfactorily at double the relative aperture, or more. 
The Hypergon lens of Goerz (Fig. 147) consists of two deep menisci, 
made of one glass, of which the outer surfaces very nearly form a 
sphere. Owing to the absence of chromatic correctipn it must be 

used with a small stop, but the sym- 
metry of the system removes the 
coma, while the astigmatism, as 
discussed above, page 177, can be 
removed in such a system by a suit- 
able position of the stop and bending 
of the lenses. The smallness of the 
stop produces a sufficient depth of 
focus to allow satisfactory definition 
on a flat plate, and the lens has a 
total field of 135 free from coma, 
astigmatism, and distortion. It is, 
therefore, suitable for photographic 
surveying. 




FIG. 147. THE HYPERGON 
(DIAGRAMMATIC ONLY) 



A "Rapid Rectilinear" lens usually consists of a pair of lenses 
of the cemented landscape type mounted symmetrically with a 
stop between them. The system is, however, not so important as 





FIG. 148. THE GOERZ 
DOUBLE ANASTIGMAT 



FIG. 149. THE Ross 

HOMOCENTRIC LENS 



it used to be, owing to the round field and lack of the anastigmatic 
correction which can be attained by lenses of simpler construction 
as will be seen below. 

Symmetrical anastigmats are still of importance. The Goerz 
Double Anastigmat (Fig. 148) will be recognized as the combination 
of two lenses similar to that of Fig. 144. The Ross Homocentric 
(Fig. 149) is a combination of two pairs of separated doublets. The 



PHOTOGRAPHIC LENSES 




Crown. n a 
Flint tf 



THE PETZVAL PORTRAIT 
OBJECTIVE 

-- 1-5181 c Flint 1-5783 

1-5783 d Crown 1-5152 



Double Protar of Zeiss is (in one form) a symmetrical combination 
of two Protar lenses of the type shown in Fig. 143. The components 
of such systems may be used separately as long focus lenses ; thus 
the Double Protar (//63) has a back component which can be used 
as a long focus lens at an aperture of //I2-5. This availability of the 
back component is of very considerable advantage to a photo- 
grapher. See also reference to the Taylor Hobson anastigmat below. 

Modern anastigmats usually give good definition at full aperture 
over a plate of which the diagonal is equal to the focal length of the 
lens; i.e. a field of about 50. 
With a small stop the field 
may be often enlarged to 70 
to 80. 

Henri-symmetrical Systems. 
Somewhat better results may 
in some cases be obtained by 
altering the scale of the two 
components, while their FIG. 150. 
separation from the stop is 
adjusted proportionally. The 
two lenses are still indepen- 
dently corrected; thus the user of such a system has the choice 
of three focal lengths, i.e. the combination and either the front or 
the back component. 

Asymmetrical Lenses. We can only describe a few lenses of the 
principal types. The Petzval portrait objective (Fig. 150), designed 
by Petzval as long ago as 1840, secures a large relative aperture, 
//3, and satisfactory correction of chromatic and spherical aberra- 
tions, as well as chromatic difference of magnification. The coma 
also is small. With an aperture of this magnitude, great pains had 
to be taken to secure freedom from zonal spherical aberration. 

Since the axial region of the image is well corrected, while the 
marginal points suffer from astigmatism and curvature of field, the 
lens is mainly suitable for portraiture where good definition is 
usually only necessary over a limited region to include the features 
of the sitter, and is even objectionable elsewhere. 

Owing to the large separation of the components, there is con- 
siderable loss of light away from the axis, due to the restriction of 
the effective aperture for oblique pencils. 

The lens is still in wide use, and is often employed as a projection 
lens in projection lanterns, and for enlarging. 

The Cooke Lens. Perhaps the most famous type of unsym- 
metrical anastigmats includes the series of "Cooke" lenses, 



i86 



APPLIED OPTICS 



originally designed by Mr. H. Dennis Taylor, who was then optical 
designer to the firm of T. Cooke and Sons, of York. (The Cooke 
lenses are now manufactured by Messrs. Taylor, Taylor, and 
Hobson, Ltd., of Leicester.) The general arrangement of these 




FIG. 151. THE COOKE LENS 
* j = Dense barium crown b = Light silicate flint 

lenses is shown in Fig. 151. The positive outer lenses, in one type 
of the system, are made of dense barium crown glass, while the 
negative inner lens is of light silicate flint. It is not within the scope 
of the present book to give a theoretical account of the optical 





FIG. 152(0). THE ZEISS 
TESSAR (7/4-5) 



FIG. 152(6). THE ALDIS 
ANASTIGMAT 



principles. The system has been modified in many ways, while 
retaining the same general principle of construction. 

In the Zeiss "Tessar," Fig. 152(0), the back positive component 
of a similar system is made into a cemented doublet ; in the Taylor 
Hobson//2-5 anastigmat, Fig. 153(0), the back component is separated 
into two lenses which diminish the spherical aberration component 
due to the last lens, and allow of a bigger aperture ratio than with 
the ordinary three-lens system. Compare this with the symmetri- 
cal/^ anastigmat of the same firm, Fig. 153(6) . The Aldis anastigmat , 
Fig. 152(6), is a much modified case of the "Cooke" principle, the 
front components being cemented together, and the corrections 
being secured by the last lens of the system. 



PHOTOGRAPHIC LENSES 



187 



Graphical Representation of the Aberrations of Photographic 
Lenses. The multiplicity of types of photographic lenses is some- 
what confusing to the would-be user, and comparatively few details 
of the performance of many modern lenses are available. Von Rohr's 
treatise 2 , Der Theorie und Geschichte des Photographischen Objectivs, 
gave, however, details of the performance of many of the types 




* 



FIG. i53(a). THE TAYLOR-! IOBSON (7/2-5) ANASTIGMAT 

** v n d v 

1. ?6i3 56-5 3- I-6I3 58-3 

2. 1-651 337 4- I'6i3 5$'5 




FIG. 153(6). THE TAYLOR-HOBSON (//2) ANASTIGMAT 

extant up to 1899, and since his method of presenting the facts has 
frequently been used since that time we may give examples here, 
viz. for a French landscape lens consisting of an achromatic menis- 
cus behind a stop, next for a Cooke portrait lens, and next for an 
anastigmat (single) of the Protar type. See Figs. 154, 155, 156- 
The curves must now be explained. 

In order to make the lenses comparable, the results are given in 
each case for a lens made on a scale which would give it a fpcal 
length of 100 mm. for the D line. Then the ordinates of the spherical 
aberration curves represent the incidence height in millimetres of 
the incident ray (parallel to the axis) ; the abscissae of the broken 
line curves are the longitudinal aberrations of the focal length, and 
those of the full line curves represent the aberrations of the axial 
intersection distances of the rays. 



i88 



APPLIED OPTICS 



In the curves showing the astigmatism, the ordinates represent 
"Grades"* of the- semi-angle of the field, i.e. the angle between the 
axis and the principal ray in the image space. The abscissae of the 
full line curves give the distances by which the focusing screen 
must be removed from the axial focus position, to bring the sagittal 
bundles into focus; the broken line represents the corresponding 
distance for the tangential bundles. 

The coma is not plotted directly in these diagrams, but it will 
now be shown that a measure of the coma is obtained by finding 



25 



o ( 

-3*9 


7 1Q 20 SP 


yv-oT j 

Distortion 





-2 -7 -'5 --5 

Spherical Aberration Astigmatism 

FIG. 154. FRENCH LANDSCAPE LENS (7/15) 



Vo 



the horizontal distance between the two curves, dotted and full 
line, in the spherical aberration diagrams. From the relatively small 
intercepts in the curves, it will appear that the elimination of coma 
is regarded as one of the most important conditions to be secured 
in the design of such lenses. 

The Sine Relation and Coma. The investigation of the "optical 
sine relation," Vol. I, page no, showed that the dimensions of 
object and image are governed by the relation 

nh sin a = n'h' sin a'. 

The formula strictly means that if we have an object point at a 
distance h from the axis, and we consider the action of a certain 
zone of the lens system, the corresponding "image" point in which 
the disturbances from the object point come together in the same 
phase will be situated at a distance h' from the axis given by the 
above formula ; the angles a and a' are the angles made with the 

* The Grade divides the right angle into 100 parts. At one time it was 
expected that this unit of angular measure would supersede the degree, but 
it has not come into prominence so far. 





(A 

U 

,a 



-s s 

^ I? 

04 

w 
us 
o 
o 

u 




* 



igo 



APPLIED OPTICS 



axis by rays starting from the corresponding axial point of the object 
plane, traversing the above particular zone of the lens system, and 
coming to a focus at the axial point of the corresponding image 
surface. In other words, the sine relation determines the physical 
image point for a particular zone. 

y 
Note that if the object is at a great distance, then sin a = j 

^ V 

sufficiently nearly, where y = the incidence height of a ray in the 



-5 



V 



3 



Distortion 




-0-B 



+2-0 



w 



-/ +; 
Spherical Aberration Astigmatism 

FIG. 156. RUDOLPH ANASTIGMAT, ZEISS PROTAR TYPE (7/12-5) 

first principal plane,* and / = distance of the object from this 
plane. Hence for a very distant object, 



_ h fn\ ^y_ 
~/U'/sina' 



where o> is the angular distance of the object point from the axis. 
This shows that for a given value of A, the different zones will 

* Which we will here define as the plane perpendicular to the axis, passing 
through the first principal point. We are not supposing the existence of true 
"principal planes/' as will be seen. 



PHOTOGRAPHIC LENSES 



191 



y 

produce an image of the same size if . , is constant. Since by 



sin a 



our ordinary conceptions of "focal length" we have 
h' = /tan co (Vol. I, page 47) 



tana, 



we may therefore interpret the quantity ( . , 1 as the equivalent 

focal length of the zone under consideration, and the condition 
that images formed by successive zones of the system shall be of 

the same size is that I .- - t } be constant. Provided the system is 

\sm a J J 



^ 


I t \ 




r-*.! 

L / S 




i "'-V . 
j i 








i 1 
i 1 

\/ 




F' 



FIG. 157. THE "PRINCIPAL SURFACE" OF A COMA-FREE LENS 

free from spherical aberration, this condition is enough to secure 
the absence of coma in the regions near the axis. 

If coma is present, let h' m be the "height" of an image for a 
marginal zone, and let //' be the corresponding value for a paraxiai 
zone ; then the amount of the coma is reckoned by the fraction 

*' - h ' 
Measure of coma -= 



, 
n 



- 

/' 



. v ' sinct ' 

.y sin a' m 



the vSuffix m being used to denote "marginal zone" values. This 
measure of coma is also a measure of the "offence against the sine 
condition." 

Still dealing with very distant object points, and taking a number 
of incident parallel rays into a system at different distances from 

y 

the axis (Fig. 157), we see that the constancy of -T , will require 

sin ct 



192 



APPLIED OPTICS 



that the incident and emergent rays shall intersect each other in a 
spherical surface. The second "principal surface" of the system is, 
therefore, a sphere centred in the principal focal point. 

Meaning of the Sine Relation in the Presence of Spherical Aberra- 
tion. In Fig. 158 we represent a lens system with the apex A of the 
last surface, the stop, or exit pupil, with centre R, and the marginal 
and paraxial image points on the axis bv B' m and B'. respectively. 
The focal surfaces for a marginal and paraxial zone of the exit pupil 
will be (sufficiently nearly) planes in the neighbourhood of the axis, 
and we may represent the corresponding extra-axial image points 




FIG. 158 

formed by these marginal and paraxial zones by C' m and C' respec- 
tively, where B'C' = h' and B' w C' m = h' m . 

Now it is clear the amount of the coma under the present condi- 
tion is the vertical intercept c = QC' m between the line RC and 
the point C' w . If it is zero, the lateral dis-symmetry of the image 
patch in any plane will disappear. 

Hence in the presence of spherical aberration 

c 
Measure of coma = p 

But c = h' m - B'^Q 

and from the similar triangles RB' m Q and RB'C', we find 

Bi r\ r*rr 
W U = J3 L 



Let AB' be denoted by l' t and AR (the distance of the exit pupil 
from the apex) by /'; also let the distance AB' m to the marginal 
focus be L' ; then 



PHOTOGRAPHIC LENSES 193 

Thus, 

Measure of Coma = 77 = ,7" - -r, jr^ 
n n L I j) 

Writing I/ - /' - (/' - /',) - (/' - L') 

we obtain 

_ _ = T _ l __ 



- 
"V 



1 /' 



f V'-i' 






Now if the stop is close to the second principal surface of the lens, 
then/' will not be very different from (/' -l' p ), when the object is 
at infinity. Hence very nearly. 

f I ' I '} ( f - f } 
Measure of Coma = V ----- - 

_ (axial spherical aberration) - (difference of focal lengths) 

focal length 

Thus the measure of the coma is found as mentioned above, by 
the horizontal intercepts between the "spherical aberration 11 curves 
of von Rohr's diagrams, but only under the limitations stated. 

Distortion. Particulars of the distortion of many photographic 
lenses were given graphically by Wandersleb 

Put in a simple way, the distortion is given by the ratio between 
the image dimensions, h', i.e. the actual "image height," and V, 
i.e. the corresponding size it would have if free from distortion. In 
order to get the value of h' we should multiply h by the magnifica- 
tion ratio found for very small objects and images. Then 

h' 

Distortion = 77 - 1 
/i 



and the percentage value, i.e. looj (J-ifis plotted in the curves. 

The distortion curves are also given for different values of N, the 
reciprocal of the magnification. Thus N = QO for an infinitely disr 
tant object, and = I for the case when object and image are of the 
same size. With the old form of the Cooke portrait lens, note how 
the distortion changes very rapidly with change of magnification. 

It is much to be desired that opticians should give specifications 



I 94 APPLIED OPTICS 

of the performances of their lenses on these lines, or by some other 
simple method. Such information would be of the greatest assis- 
tance in determining the suitability of the lenses for various 

purposes. 

Depth of Focus; Photographic Definition. The "depth of focus" 
in the image may be defined as that total displacement of the plate 
on each side of the true focus which is possible without producing 
an appreciable spreading or loss of definition in the image. In 
Vol. I, page 141, the matter was discussed in terms of optical path 
differences between marginal and paraxial disturbances, and the 
equation for the shift on one side of the focus is 



where dp is the allowable difference of optical path. Remembering 
that the total range will be approximately double the above, and 

that the stop number is ( ^- j, we get for a medium where n' = I 

depth of focus = 16 (allowable path difference) (stop number)* 
The allowable path difference is, however, not easy to specify very 
exactly; the requirements for photographic recording are usually 
less severe than for direct visual observation of optical images 

where the Rayleigh limit of - may be necessary. If in photography 

A ^ 
the limit were taken as -, say, then with a lens working at //8 and 

wave-length = 0-5/4, the depth of focus would be 256^, or about a 
quarter of a millimetre. 

The tolerance for the loss of definition of the image is usually 
such that a photograph held at the least distance of distinct vision 
should appear reasonably sharp to the eye, and it is usually esti- 
mated that the photographic disc or patch representing an image 
point may subtend an angle of, perhaps, two minutes of arc. 

If d is the diameter of the patch in millimetres, the limit would 
thus be given by the equation 

d , 

= 2 mm. in angular measure 
250 6 

d= 350 
1710 

so that d will not exceed about one-seventh of a millimetre, or about 
T | F in. This limit is on the severe side, and a patch of pf^ in., or 
0-25 mm., may be tolerated in some cases. 



PHOTOGRAPHIC LENSES 



195 



If the diameter d of the patch were determined simply by the 
diameter of a cone of rays passing through a point in the focus, 
taken at a distance df therefrom, we should have 

df 

-r = stop number 

and if d were 0-25 mm., then df = 2-0 mm. for //8, and the total 
depth of focus would be of the order of 4-0 mm. It is clear that the 
usual criteria of optical path are far too severe for this case; on 




2 4 



6 8 10 12 14 16 18 
Aperture (mms. ) 



20 



FIG. 159. FOCAL RANGE FOR AN ANASTIGMAT LANDSCAPE LENS 

the other hand, the distribution of light is not well represented by 
the diameter of the supposed "cone of rays, M which gives very mis- 
leading results near the focus. 

It is thus difficult to give a satisfactory theoretical discussion on 
any simple lines. Considerable light is thrown on the matter by 
the experimental work of Miss H. G. Conrady, 3 who used an anastig- 
mat landscape lens (flj approx.) suffering from a residual spherical 
aberration of known amount, and investigated the position of the 
best focus, and the focal range for various apertures. The results 
are shown in Fig. 159, in which curve A represents the focal range 
giving fairly good definition for practical purposes ; curve B shows 
the range over which no loss of definition is at all perceptible; 
curve C shows the range predicted by physical theory for a perfect 

lens, using the "Rayleigh limit" of - for allowable path differences. 

4 
It appears that the range is greatly increased by the presence of 



xg6 



APPLIED OPTICS 



slight spherical aberration, and that it may be anything from one 
to eight times the "Rayleigh limit" range, depending on the 
amount of spherical aberration, which is naturally increasing in the 
diagram with increasing aperture. On the whole, it is clear that 
neither the physical theory nor the geometrical discussion of discs 
of confusion have a precise significance in this problem. Failing 
more exact knowledge, however, we usually find that it is very 
rarely with these photographic lenses that the diameter of the 
effective patch of light exceeds the limits of the calculated geo- 
metrical disc of confusion of the rays, and, consequently, it is the 
general experience that useful tolerances can be obtained by 
geometrical theory when discussing focal depth in the object space. 




FIG. 160. THEORY OF FOCAL DEPTH IN THE OBJECT SPACE 

Focal Depth in the Object Space. With the above limitations in 
mind, let us consider with the help of Fig. 160 an optical system, 
having an entrance pupil of diameter a, which is forming, in the 
plane B', a sharp image of all points in the plane B. Consider now 
a point P situated in the plane C. Rays from a limited area (dia- 
meter c) of the plane B can reach the entrance pupil by passing 
through P. Hence we see that the rays from the point P will inter- 
sect the image plane B' in all parts of a disc corresponding to the 
image of the disc c. If the magnification is m, the diameter of this 
image = me. 

Similarly, the image of a point Q in the plane A will correspond 
to that of a finite disc in the plane B. 

First Criterion. If the criterion for sharpness of definition is 
merely such that the dimension of me must not exceed a certain 
limit, the distances of the planes A and C from B will be limited 
also. Taking the dimensions shown in the diagram, we obtain from 
the similar triangles with a common apex in Q the relation 



a 



PHOTOGRAPHIC LENSES 197 

At l 7/ 1 x \ k 
giving o/i = - / I --- I =- -- 

*> * * a/ \c a) a-c 

and from the triangles with their apex in P, 

(Mg _ I - 61 2 

c a 

., I //i i\ fc 

giving 0/2= -/i-H I -- 

& to * a/ \c aj a + c 

These equations give the focal depths on each side of the exact 
focus. The total focal depth is given by adding the equations. 

Thus Bf c , /I I \ , 2a 

dl + <$/ = lc - -I ------ .\ = ic - 

1 4 \a-c a + c) at-c* 

2 ale 

Total focal depth = 2 - - 4 

r a 2 - c 2 

Second Criterion. On the other hand, it may not be intended that 
the picture shall be viewed simply at the distance of distinct vision, 
but, perhaps, under the proper angular magnitude, possibly with 
the aid of a suitable lens of focal length equal to that of the camera 
lens. Under these circumstances a disc of confusion will subtend 
an angle at the eye equal to that subtended by the corresponding 
disc of confusion in the plane B at the entrance pupil of the camera 
lens. 

But the condition for sharp images is that the angle subtended 
by the disc of confusion at the entrance pupil shall not exceed a 
definite limit, say a, i.e. 

At the limit, c -- la 

I . la la 
Hence d/ 



a - la 



and <5/ 2 = ---- 

a 



T-" 



In the position when a -r, the focal depth will extend outwards 
to infinity. This gives 



a 

and / - 6L = 

* 2a 



198 APPLIED OPTICS 

so that the focal depth of sharp focus will extend from a distance 

u 

of to infinity. 
2a J 

We may now note that this depth of focus in the object space only 
depends on the diameter of the aperture of the entrance pupil. 
Hence, small snapshot cameras using lenses with big aperture ratios 
have a greater depth of focus in the object space as compared with 
larger lenses, and have the advantage that the smaller lenses are 
much cheaper. 

The image diameter corresponding to the angular subtense a 
above will be fa where/' is the focal length of the camera lens. If 




(Carl Zeiss, Jena) 



FIG. 161. FOCAL DEPTH FOR A 6 x 4^ CM. CAMERA USING VARIOUS 

LENSES 



the photograph is viewed under the correct conditions it will be 
held at a distance/' (Vol. I, page 2), so that the value of a must be 
determined by the resolving power of the eye. A practical value is 
2 min. , or 0-00058 radians. If, however, /' is shorter than the distance 
of distinct vision, the picture should be viewed with the aid of a 
magnifying lens as in a stereoscope (or with such a lens as the 
Verant), having a focal length equal to that of the camera lens. 
Alternatively, the picture may be enlarged photographically, and 



PHOTOGRAPHIC LENSES 199 

the scale of the enlargement should be 

Distance of viewing of picture 
Focal length of camera lens 

if the picture is to be seen under correct conditions of perspective. 
Take, for example, a very small camera with a lens of focal length 
5 cm. If the resulting pictures are to be viewed at a distance of, 
say, 30 cm., the best scale for enlargement would be 6 diameters. 

An interesting presentation of the depth of focus effects when 
using lenses of about 8 cm. focal length covering a plate of about 
8 cm. diameter (about 2j in. by 2j in. plate) is given by Messrs. 
Carl Zeiss (Fig. 161). The criterion of focal depth is that the disc of 
confusion must not subtend an angle greater than 2\ min. The 
depth of focus is shown for the series of aperture ratios: 

f f f f 

~7~' ~ f "7" anc * S.~~* Since the focal lengths are the same, the 

/OJT 1 *) J 

actual entrance pupils differ in size. 

The first has a focal length of 8 cm., the three latter have focal 
lengths of 7-5 cm., with a slightly wider angular field. The prin- 
cipal distance, i.e. the distance for best focus, is 3-0 metres; and the 
diagram is interesting as showing the kind of depth of focus obtain- 
able for "personal" photographs with small pocket cameras. 

The Telephoto Lens. Consider a combination of two lenses of 
positive and negative powers numerically equal. Referring to page 
50 of Vol. I, we find the power of a combination given by 

. *> | X J X / 

y = ./+>'- d. ' a /& 

and the distance of the second principal plane from the second lens 
is 



Take, for example, the case where ./^ = loD, .% = - loD, and 
the separation d = 4 cm. (= -04 metres), then 

./'= 10 - 10 - -04 (io)(- 10) 
= 4-0 

The focal length is therefore 25 cm. 

- 10 (-04) i 

The equation for P ft P' gives this length as - = -- metre 

4 10 

= - 10 cm. ^ 

Hence the second principal surface lies - 10 cm. in front of the 
second lens. 
The effect of the combination is, therefore, that of a lens of 

*4 (5494) 



2OO 



APPLIED OPTICS 



power 4'oD placed 6 cm. in front of the first component as sug- 
gested in Fig. 162. The back focusing distance of the combination 
is clearly 15 cm., whereas the actual focal length is 25 cm. Hence 

25 5 
the images of distant objects will be larger in the proportion = - 

than those of an ordinary lens with the same back focal distance. 




Position of equivalent 
thin lens 

FIG. 162. TELEPHOTO COMBINATION 

Modern telephoto lenses employ two separated systems of doublet 
or triplet type ; the first having a positive power, and the second a 
negative power. Each component must be separately corrected for 
chromatic aberration owing to the large distance between them, 

A 




nff 




FIG. 163. CONDITION OF ACHROMATISM 

for suppose that we consider rays i and 2 in Fig. 163; we may 
imagine that ray i is subject to dispersion by lens A, and that the 
under-correction, by which the blue ray is deviated more than 
the red ray, is corrected by lens B, the blue now suffering more 
deviation towards the margin of the negative lens than the red. 
If correction were thus given for image points very near the axis, 
consider ray 2 which suffers the same type of aberration in lens A, 
but meets lens B on the other side of the axis. The aberration is 
evidently exaggerated. Hence each component must be separately 
achromatized. 

This can also be argued from the expressions for the magnification 
of thin lenses. Let the two thin lenses be a and b ; assuming homo- 
geneous light, let the distances of object and intermediate image from 



PHOTOGRAPHIC LENSES 201 

lens a be l a and l' a ; the distances of intermediate image and final image 
from b are l b and /' 6 . Let h a , h' a , and h' b be the sizes of object, inter- 
mediate image, and final image. Then 

k ' V ,H *' *' 
r- = y-, and -r-, -= 7- 

"a *a "a 'ft 



so that $ = -* 

h a l a l b 

If now the final magnification is to be independent of the wave-length 
of the light, and the position of the image is to be constant also, then 

1 

l' b , and / rt are all constants. Therefore since jr 

*(*'./*) _ n 
~ 



(J-/\ 
~J t 



and (dl' a }l*-l 

But the separation of the lenses, i.e. /' - l b (remember our sign con- 
veations), is constant, since the separation is prescribed by the equation 
on page 45, for the achromatism of the focal length. Therefore 

<//' a ^ dl b 
and this, substituted in the previous equation, gives 

<'(/> -/') -o 

But the bracket is equal numerically to the lens separation and cannot 
be zero. Hence dV a must be zero and dl b = o also by a similar argu- 
ment. The position of the intermediate image is independent of the 
wave-length. Hence each lens must be separately achromatized. 

This discussion deals with two separated thin lenses only, and obvi- 
ously neglects the possible/chromatic variation of the principal points 
with wave-length which may be expected in thick lenses. It often hap- 
pens in complex lens systems (like the Cooke or Aldis lenses) that we 
have single uncorrected lenses used, so the meaning of the above dis- 
cussion must not be pushed too far. It is, however, to be remembered 
when the use of widely separated and more or less thin lenses is con- 
templated. 

Supposing for a moment that we restrict the further demands to 
freedom from spherical aberration and coma. Lens A might be a 
cemented doublet of suitable glass of the telescope objective type. 
Lens B might also be a cemented doublet with negative lens of crown 
glass and positive lens of flint. 

In the section on microscope objectives, it was explained that a 
positive cemented doublet has two pairs of conjugate points free from 
spherical aberration, and we should expect to find similar pairs of 
points for a negative combination with a strong negative crown lens 
and a weaker flint lens cemented together. Although it is possible to 
use more or less aplanatic components, this is only at the expense of 
marked pincushion distortion, and modern telephoto lenses usually 
have components which are not separately corrected in themselves, as 
will be seen below. 

The early telephoto lenses mainly used a negative system behind 
a photographic lens system of some ordinary type; the power 



2O2 



APPLIED OPTICS 



could be varied by varying the separation. The " magnifying effect " 
m of the telephoto attachment is given by 

Dimensions of image with telephoto attachment 



m = 



Dimensions of image with positive lens only 

But for an infinitely distant object subtending an angle a, the 
dimension of the image is / tan a ; also for a combination of two 
lens systems placed with their focal points so that F' a F 6 = g, we 
had 



Therefore 



m = 



/tana 



f a tan a ~~ g 
ray for transmission 



Extreme ray for 
full illumination 




FIG. 164 

When opticians supply a negative telephoto attachment with a 
variable separation, a scale on the mount is usually made to indi- 
cate values of m or g. The use of such attachments has, however, 
largely been abandoned in favour of the modern fixed-focus tele- 
photo lens. 

As distinct from the magnifying effect, the expression "telephoto 
effect" is sometimes used: 

T-ii.,. Focal length of combination 

Telephoto effect = * n -~- 

Back focal length 

the "back focal length" being measured from the last surface of 
the lens system, and representing the approximate focal length of 
an ordinary lens used at the same camera extension. 
It is easily shown that this is equivalent to 

/' 
Telephoto effect = J * 



where d is the separation of the lenses. 



(/'-<*) 



PHOTOGRAPHIC LENSES 203 

Effective Aperture. Consider a lens system A used first alone, 
then with a negative attachment B. The entrance pupil will be 
assumed to be the same as that of the system A ; the diameter of 
B is sufficient to allow of this. Then 

Focal length 

Stop number = =r; - 7 - ^ 7 - r. 
r Diameter of entrance pupil 

Stop number with telephoto lens / 
' Stop number without telephoto f a ~~ 

The exposure must be proportional to m 2 . 

Field of View. In Fig. 164, let A and B be the positive and nega- 
tive lenses of a thin lens telephoto system, and let y a and y b be the 
radii of the diaphragms which limit them. The tangent of the angle 
with the axis made by the most oblique ray which can pass between 
the lenses is clearly 



If /' a is the focal length of the thin lens A, the deviation in a ray 
produced by a transmission through it at a distance v a from the 

v 
axis is approximately -. Hence, tracking this most oblique ray 

J a 
backwards through A, we find the inclination to the axis for the 

most oblique ray which can enter lens A and be transmitted by the 
system. Provided that we are dealing with angles small enough to 
take the tangent of an angle as its numerical value, the field of view 
is given by 



d fa 

This represents the extreme limit of the field. It will be seen that 
the limit of the fully illuminated field is 



These equations are, however, not strictly accurate with the thick 
lenses encountered in practice. 

Modern Telephoto Lenses. The modern telephoto lenses are 
mostly fixed focus combinations with anastigmatic correction. The 
telephoto effect is low, being only two to three, but this suffices 
for a great number of purposes, more especially as several advan- 
tages are obtained, viz. high relative aperture (the aperture of some 



204 APPLIED OPTICS 

telephoto lenses has been increased to ) anastigmatic correc- 

j 3 

tion giving sharp images capable of enlargement, and reasonable 
economy in size. Fig. 165 shows the Dallon (of Messrs. Dallmeyer, 





FIG. 165. DALLON LENS (//5-6) 

Ltd.) designed by Mr. L. B. Booth, who was a pioneer in the con- 
struction of such systems. Typical glasses in lenses of this kind 
are 

1. Dense barium crown. 

2. Dense flint. 

3. Light flint. 

4. Medium barium crown. 

A general review of the development of modern telephoto objec- 
tives has been given by Lee, 4 who was successful in producing a 





FIG. 166. //5. DISTORTION-FREE TELEPHOTO LENS DESIGNED BY LEE 

(Messrs. Taylor, Taylor & Hobson) 

distortion-free telephoto lens. He says: "If we consider the con- 
struction of the telephoto, a positive lens placed in front of the dia- 
phragm, which will possess pin-cushion distortion, and a negative 
lens behind the diaphragm which tends to produce the same kind of 
distortion, it is not surprising that telephotos are afflicted with much 
pin-cushion distortion, and some designers have considered it 
inevitable. ... By separating the components of the negative 
lens, it was possible to utilize the astigmatism in these surfaces, 



PHOTOGRAPHIC LENSES 205 

which is fairly large, to correct the pin-cushion distortion. . . . 
We are then left with a residuum of under-corrected astigmatism, 
which is neutralized in the front positive lens by dividing it up into 
two menisci, one of which is a doublet, for the purpose of achroma- 
tism." These words give a brief picture of some of the main stages 
in the design of a new system. The distortion free lens, B.P. 222, 
709, is shown in Fig. 166. 

REFERENCES 

1. Wandersleb: Zeit. f. Inst. t XXVII (1907), 33 and 75. 

2. von Rohr: Der Theorie und Geschichte des Photographischen 
Objectivs. 

3. H. G. Conrady: Jour. Roy. Phot. Soc., LXV1 (1926), 22-25. 

4. Lee: Proc. Opt. Convention, 1926, p. 869. 



CHAPTER VI 

THE PHOTOMETRY OF OPTICAL SYSTEMS AND THE PROJECTION 

OF IMAGES 

IN the foregoing discussion of the principles of the telescope and 
microscope, no attention has been given to the question of the 
brightness of the image ; this aspect of the subject is, however, of 
the first importance, and must now be considered. A discussion of 
the sensitiveness of the human eye to light has been given in Vol. 
I, Chapter V. 

For the purposes of elementary discussion, it is assumed that 
radiant energy would spread out from an elementary source of 




infinitesimal size along the paths represented by the "rays. 11 The 
amount of energy per unit time passing any cross-section of a tube 
whose walls were made up of such rays would, therefore, be constant. 
Rays are straight in a homogeneous medium; hence if we con- 
sider a conical tube representing a very small solid angle do* with its 
apex in the elementary source C (Fig. 167), the normal cross-sec- 
tional areas at distances r^ and r 2 would be r^do) and r 2 2 do). Let 
the amount of energy passing in unit time be dV ; then the energy 
per unit area at these sections will be 



and the quantities must be equal in the absence of absorption, i.e. 
(putting the result into words) the amount of energy per unit area 
falling on an elementary area held normal to the incident light is 
inversely proportional to the square of its distance from the source. 
This is the "inverse square" law; but it only has an exact meaning 
in regard, to an imaginary source of infinitesimal size, and, therefore, 
may only hold approximately in practical cases. 
When the energy is evaluated according to the luminous sensation 

206 



PHOTOMETRY OF OPTICAL SYSTEMS 207 

rfF 

produced, the symbol F represents an amount of "light," and j- 

represents, for some particular direction, "the amount of light per 
unit solid angle," which is the "candle-power" (intensite lumineuse) 
of the source for that direction. The candle-power is usually denoted 
by J. Thus 

rfF 

Candle-power J = - 

do) 

The amount of light falling per unit area of a surface represents the 
"illumination" (usually denoted by E), so that 

dF 

Illumination E =-- --.- 

ds 

where ds represents an elementary area of the surface. 

Lastly, the "brightness" (B) of a surface is defined as the candle- 
power per unit projected area in the direction under consideration, 
so that 

d] 

Brightness B = -~ 

as 

The Cosine Law. Practical observation of self-luminous surfaces 
(a red-hot poker is a good example), and of many types of diffusely 




reflecting surface (such as blotting-paper) shows that their apparent 
brightness under visual observation is very nearly independent of 
the direction of observation. It will be shown that this can only 
be explained by the assumption that the "candle-power" of a 
small element of such a surface (regarding it for the moment as a 
source of light) in any direction is proportional to the cosine of the 
angle between that direction and the normal to the surface. 

This relation enables us to calculate the amount of light radiated 
from one small elementary surface to another, when these surfaces 
are inclined at any angle to the straight line drawn between them. 
In Fig. 168 let ds L and ds% represent the elementary areas, and let 
the normals to these areas make angles 6 lt 6 2 with the line joining 
them. Let the normal brightness of ds l be B; then the normal 
candle-power will be 



208 APPLIED OPTICS 

The solid angle subtended by ds 2 at the distance r is given by 

ds 2 cos 6 2 
d < = ^ 

Assuming the above cosine law of radiation to hold, the brightness 
of ds, in a direction inclined at an angle O l to the normal is B cos Q lt 
and therefore the candle-power of the elementary surface in this 
direction is Bds l cosO l . This is the light radiated per unit solid 
angle in this direction, so that the amount radiated into the solid 
angle dco subtended by ds 2 is 

B ds* ds 2 cos QI cos 2 
B . ds l . do} . cos 0, = * 2 y2 

i.e. we multiply the brightness of the source, the area of either one 
of the elements, and the cosine of its angle, with the solid angle 
subtended by the other element. 

The illumination of the element ds 2 , the amount of light per 
unit area, is obtained by dividing the above expression by ds 2t i.e. 
it is 

r 2 Bds l cos O l cos 2 

and is clearly proportional to cos 2 ; if we hold a small surface in 
a beam of light, the illumination is proportional to the cosine of 
the angle turned from the position of normal incidence. This rela- 
tion is one of a purely geometrical character, it does not depend, as 
does the "cosine law" of radiation mentioned above, on the physical 
properties of the surface. 

Apparent Brightness of a Radiating Surface. The apparent 
brightness of a surface is usually directly dependent on the illu- 
mination of the retinal image. Let the elementary area ds 2 (Fig. 168) 
represent the entrance pupil of an eye, and let 2 be zero, so that 
the eye "looks" at ds v In Vol. I, page 47, it was shown that the 
size of the image of an object subtending a plane angle co p will be 

h' = ftanco p 

Correspondingly, the area of a small retinal image of an object 
subtending a small solid angle co 8 will be found by squaring the 
simplified form of the last equation when it has been written 

*'=/, 
thus obtaining 

h'*=f* p 9 

or ds' =/ 2 co a 

This area ds' is taken to be uniformly illuminated by light reaching 



PHOTOMETRY OF OPTICAL SYSTEMS 209 

the entrance pupil of the eye from the object. The quantity of light 
dF reaching the pupil is (from above) 

,^ Bds<i dsn cos 6\ 
dF = + 

Of this, however, only a fraction kdF is transmitted by the media 
of the eye. But the solid angle subtended by the object at the eye 

(d$i cos 0) 
i s _ 9 an( j if the distance from the object to the front focal 

point is large in comparison with the short distance from the focal 
point to the pupil, we may rewrite the above expression 



Hence the illumination of the retinal image is 



ds' ~~ f*a> 8 ~ 2 

It is clearly independent of the angular position and distance of 
the object, provided that the accommodation of the eye is unchanged, 
and that the normal brightness of the surface is constant. Hence, 
if a surface radiates or reflects in accordance with the cosine law, 
it will appear to have the same apparent brightness no matter from 
what distance or under what angle it is seen. This statement is 
subject to several limitations. In the first place it does not apply 
when the geometrical image of the object is of a size comparable 
with, or small in comparison to, the physical concentration of the 
Airy disc elementary image. Under such circumstances the area 
of the image is independent of the distance or angular position of 
the source, and it is too small to give the sensation of any finite 
extension. The relative apparent "brightness" is, therefore, depen- 
dent only on, and directly proportional to, the total amount of 
light received by the eye, i.e. 

B ds l ds% cos O l 

uV == 

r 2 

In order that the light may be perceived by the eye, the quantity 
must exceed the "threshold value" for the retina. It is found in 
practice that in very weak illumination, the retina has the power of 
integrating the light received over a small area subtending about 
one degree of arc in the visual field, so that if the total light radiated 
on such an area exceeds a certain amount, the sensation of light 
will result. 

Total Light Radiated by a Self-luminous Surface. Imagine a 
small element, of area ds, of self-luminous surface at O, Fig. 169; 



2io APPLIED OPTICS 

assuming that radiation takes place in accordance with the cosine 
law, it is possible to calculate the total light radiated by the element 
into the space above it. Imagining a hemisphere described above 
the element, we may calculate the light radiated to a circular strip 
limited by the angles 6 and + dO between the radii drawn from 
O and the normal OB. If r is the radius of the hemisphere, the 
area of the strip is clearly rdO . 2irr sin 0, and the solid angle do> 
subtended at O is therefore 

dfo = 2?r sin OdO 

If B is the normal brightness, the candle-power of the element in 
the direction is 

Bds cos 

Hence the light radiated to the annular strip is 
dF = Eds . 2n sin cos 6 dO 

To find the total light radiated 
into a symmetrical cone of angle B 
we integrate thus 

F = / Eds . 27r sin cos d& 

= nBds sin 2 
When the angle is -, so that we 

consider the whole of the space above the element, the result is 
TrBds. 

Brightness of Optical Images. The first case considered is that 
in which the image is presented directly to the eye, without being 
projected on a diffusing screen. The second case will be that of 
the photographic image or projection lantern image where the light 
falls on a projection plate or screen. 

The simplest case to be considered is that of a symmetrical instru- 
ment, Fig. 170, of which the centres of the relevant entrance and 
exit pupils are at R and R'. Let ds be a small element of self- 
luminous surface with the optical axis as its normal, and let ds' 
be the corresponding image. The limiting angular divergence of the 
rays entering the entrance pupil is a, and the corresponding angle 
made by the extreme rays with the axis in the image is a'. 

Let the normal brightness of the object be B ; then the amount 
of light F entering the instrument is 

F = n&d$ sin 2 a 




PHOTOMETRY OF OPTICAL SYSTEMS 



211 



The luminous energy flows through the system by various paths 
which are represented by the ray tracks. If the system fulfils the 
"sine condition," then when we write the optical sine relation 

nh sin a = n'h' sin a' 

we know that the ratio of sin a to sin a will be the same for all 
zones of the system. The energy radiated from the object element 
into the entrance pupil of the instrument will be subject to some 





FIG. 170 

losses by reflection and absorption. If the transmission factor is k, 
the amount of light reaching the image will be 

kV - knBds . sin 2 a 

Assuming now that the corresponding image area, ds', has a normal 
brightness B', and that it radiates in accordance with the cosine 
law, the amount of light passing through the aerial image will also 
be expressed as 

A>F - nB'ds' sin 2 a' 
i.e. 

kirBds . sin 2 a -- nWds' . sin 2 a' . . (a) 

By squaring the sine relation we obtain 

u*ds . sin 2 a = w' 2 rfs' . sin 2 a' 
and, dividing into the previous equation 

kit __ B' 
ii* " n' 2 



or 



B' - 



Returning to equation (a), let us differentiate it with regard to 
a and ', obtaining 

zkir&ds sin a cos a da = 2rrB'rfs' sin a' cos a' da' 

Referring back to the investigation above, page 210, we see that 
the assumption that the image radiates in accordance with the cosine 
law is founded on the supposition that the light radiated by the 
object into an annular cone of angles a and a + da passes through 
the image in a corresponding annular cone of angles a! and a' + da' 



212 APPLIED OPTICS 

which can be calculated by the assumption of the sine condition. 
Differentiating the squared form of the sine relation we obtain 

zntds sin a cos a da = 2,n'*ds' sin a' cos a' da! 

which, dividing into the above equation, gives the same result for 
B'. The investigation is, therefore, in strict accord with the funda- 
mental assumption that the quantity of light remains constant for 
all cross-sections of a cone whose walls are made up of rays of light. 
The condition that the image may radiate in accordance with the 
cosine law is that the transmission factor of the instrument may be 
the same for all zones. In practice, this is not likely to hold exactly, 
owing to the different obliquities of the rays at the different sur- 
faces which become much greater towards the margins, thereby 
greatly increasing the reflection losses. 

Before going further, it will be well to re-state the result and draw 
certain inferences. Few optical theorems have been the subject of 
so much misunderstanding as this. 

Firstly, although the image ds' is radiating in a homogeneous 
medium, it is unlike the object insomuch as it can only radiate 
within definite angular limits; the extreme rays cannot exceed an 
angle a' with the axis. 

Provided the eye is so placed that the pupil lies entirely within 
this angular limit, the image when observed will have the apparent 

(ri*\ 
brightness B' = kB ( ~y 1 ; but if the eye is outside the cone of 

radiation, the image will not be seen. Now, if the image space 
medium is air, as is usually the case in visual observation, then 

(n'*\ 

( ^ 1 is unity, and the subjective brightness of the image is that of 

the object multiplied by a transmission factor k which is always 
less than unity. The formal proof has only been given for the most 
simple case, but it may be taken as a general rule that the apparent 
brightness of the image of a luminous surface formed by an optical 
system cannot exceed the apparent brightness of the object surface 
observed directly, provided that the apparent size of both object and 
image is not very small. 

As will be seen below, cases arise when the image projected by 
a system is viewed so that the pupil of the eye is not wholly illu- 
minated, as, for example, when using a microscope. The apparent 
brightness of the image will be further reduced in the proportion 

fPY 

( 1 , where p and p are the restricted and full radii of the pupil 

respectively. This assumes, however, that the restricted pupil is 



PHOTOMETRY OF OPTICAL SYSTEMS 213 

uniformly illuminated. Owing to the reflection and absorption 
losses in instruments, losses which are usually larger for the more 
oblique pencils, the assumption is rarely strictly true; it may be 
used to obtain a first approximation to the answer of several prob- 
lems, as will be seen. 

Transmission Factor of Photographic Lenses and other Instru- 
ments. The above theorem indicates a ready means of measuring 
the transmission factor of an optical instrument such as a photo- 
graphic lens or a telescope. There are instruments, known as 
brightness photometers, 1 which allow of the direct measurement of 
the apparent brightness of a luminous surface. We have, therefore, 
only to measure (i) the apparent brightness of the surface observed 
directly; (2) the apparent brightness of the image of this surface 
as projected by the photographic lens or other system. Then the 
ratio of image to object brightness gives the transmission factor. 
It is to be noted that the relative positions and distances of the 
luminous surfaces are immaterial, provided that a clear view of them 
can be obtained with the photometer, since the subjective bright- 
ness of a surface is, as shown above, independent of its distance 
provided that the angular subtense does not decrease below a 
certain value. 

A suitable luminous surface is obtained by illuminating a piece 
of opal glass from behind. A photographic lens may be supported 
near the surface, and the brightness of the image of the surface 
is observed, holding the photometer aperture close to the lens. 
The latter is then removed and the brightness of the surface deter- 
mined directly. For measurements on telescopes, the surface is held 
in the exit pupil of the instrument ; its image is then observed in 
the entrance pupil, and the brightness can be determined with the 
photometer. These methods are much quicker and more accurate 
than the older methods using the optical bench. 

Brightness of the Image in a Telescope. The above theorem 
regarding the "brightness" of the image projected by an optical 
system dealt with the brightness as a physical quantity. Given two 
surfaces of the same physical brightness, and radiating in accordance 
with the cosine law, they will only appear of the same brightness if 
the eye observes them under the same conditions ; for example, with 
the same diameter of the eye pupil for each observation. 

Consider first the case of the telescope; suppose it is used to 
view a distant luminous surface such as a region of the moon over 
which the brightness B is constant. 

Let the unaided eye regard this surface ; it receives an impression 
of brightness B, using the pupillary area <fe 8 . Now suppose that the 



214 APPLIED OPTICS 

telescope of magnifying power M is held before the eye. Let the 
area of the object glass be A, and let the transmission factor of the 
instrument be k. The "brightness" (in the limited physical sense) 

of the image projected by the telescope is now kB, and the area of 

^ 
the exit pupil of the instrument is . If this area is larger than 

that of the eye pupil the impression of the brightness of the tele- 
scope image will be kB 9 since the eye pupil is wholly illuminated ; 
but if the exit pupil is smaller than the eye pupil, the impression 
of brightness will now be proportional to the available aperture of 
the eye pupil, i.e. 

Apparent brightness with telescope __ k / A \ 
Apparent brightness without ds 2 \M 2 / 

We may calculate this result in another way which will recall 
first principles. Let the distance from the surface to the eye, or 
to the observing telescope, be r\ then using the above symbols, 
the amount of light, from an area ds l of the object, radiated to the 
eye pupil in the case of direct observation must be 



Let the area of the retinal image be <fe 3> and let the transmission 
factor of the eye media be t, then the illumination of the retinal 

image will be ^-, , . 

** Bds l ds 2 t 



When using the telescope, provided that the full pencils of light 
can pass unobstructed into the eye, all the light entering the object 
glass (less absorption losses, etc.) passes to the retina. The light 
received by the object glass is 



But the retinal image now has an area M 2 .rfs 3 owing to the telescope 
magnification. Hence the illumination of the retinal image is 

Sj A t 



JT Illumination of retinal image with telescope ___ k I A \ 

Illumination of retinal image without ~~ d% \M*/ 



/Area of exit pupil\ 
' \Area of eye pupil/ 



PHOTOMETRY OF OPTICAL SYSTEMS 215 

It is easy to show that when there is obstruction of the full 
pencils owing to the exit pupil being larger than the eye pupil, 
then the effective entrance pupil of the telescope is of area M 2 rfs 2 , so 
that the ratio of the illumination in the two cases is simply deter- 
mined by the transmission factor of the telescope. 

Case of Star Images. The above considerations do not apply 
unless the distant luminous surface is of appreciable angular magni- 
tude. In the case of such an object as a star, the area of the retinal 
image, in the case both of direct and aided observation, will usually 
be so small that its magnitude will be determined mainly from the 
optical imperfections of the eye system and the physical spreading 
of the image. 

At any rate, there will not be a great difference in the 
retinal area illuminated in aided and unaided observation; but 
whereas in unaided observation we merely have the small area of 
the pupil to receive light, we can with a telescope capture the light 
received by the much greater area of a large object glass, and con- 
centrate most of it into a retinal image of much the same extent 
as that found in unaided observation of a star. Given a large aper- 
ture telescope of, say, i metre diameter, the area is of the order of 
fifteen thousand times the eye pupil (8-0 mm.) at night, and the 
telescope, therefore, has a greater range in detecting distant stars. 
But since the light received from a star is inversely proportional to 
the square of the distance, the range is only increased by the ratio 
of the diameter of the object glass to that of the eye pupil, or about 
125 to i in the above case. 

Again, if there is any general luminosity in the "background" of 
the sky, this will be diminished in the telescope image as calculated 
above, so that this effect tends to increase the contrast between the 
brightness of the star and the background. 

Night Glasses. It was mentioned above that the eye has the power 
of integrating the light in feeble stimuli spread over a retinal area 
corresponding to an angular diameter of about one degree in the 
field of vision. The "threshold" of perceptible "brightness"* is, 
therefore, inversely proportional to the square of the angular 1 sub- 
tense of the stimulus. The smaller the "threshold," the easier the 
vision in faint light ! When this angle rises above one degree, the 
threshold is then inversely proportional to the angular subtense 
itself, and not to the square ; this is true up to a subtense of about 
four or five degrees, after which the threshold tends to become 
independent of the angular size of the stimulus. We may put these 

* Brightness is defined as the candle-power per unit area, in the usual 
photometric sense. 

15 (5494) 



216 APPLIED OPTICS 

results into symbols; let / be the threshold brightness, and a the 
angular subtense, then 

t oc . . up to one degree 
cr 

t oc - . . from one to four degrees 
a 

When using a telescope in faint light we have two cases to con- 
sider. In the first the exit pupil is larger than the pupil of the 
eye; the brightness of the image is kB and independent of the 
magnification. While this is true, we shall obtain continuous advan- 
tage by increasing the magnification since the angular size of the 
image will be proportional to m ; this will hold good till the image 
we wish to observe subtends more than four to five degrees. 

In the second case, the magnification has been so far increased 
that the exit pupil is now smaller than the pupil of the eye. The 
apparent brightness of the image is now inversely proportional to 
the square of the magnification, so that to maintain a given apparent 
brightness of the field the external object brightness B must be 
proportional to w 2 ; but if the stimulus is a small one, so that its 
magnified image subtends less than one degree in the visual field 

of the eye, the allowable "threshold" varies as - 9 - by reason of 
J m 2 a 2 J 

the physiological effect (where a is the angular subtense of the stimu- 
lus to the unaided eye). Hence 

m 2 

t oc -^~ 9 
m 2 a 2 

i.e. / oc - 5 

a 2 

so that the threshold for small stimuli is almost independent of 
the magnification of the telescope. For larger stimuli with an 
apparent angle, in the instrument, of over one degree, the increase 
of magnification will obviously cause a disadvantage. 

Experiments by the writer 2 seemed to show, however, that with 
small stimuli the visual threshold for the unaided eye was inversely 
proportional to a slightly higher power of the angular subtense 
than the second power up to about one degree of subtense. If 
this can be accepted then, when using the telescope, 



oc 



m 2 



(ma) 2 + * 
where x is a small positive quantity. Confirmatory experiments 



PHOTOMETRY OF OPTICAL SYSTEMS 217 

seemed to indicate that the expected slight advantage with increas- 
ing magnification was realized until the stimulus reached an angle 
of about one degree. 

The majority of night glasses are binoculars for hand use, and 
the magnifying power should then not be greater than about six 
times, while the greatest efforts are made to secure the largest 
possible exit pupils and the least possible internal losses of light. 

Brightness of the Image in the Microscope. It is seldom that 
the object is self-luminous; in the majority of cases the condenser 
projects the image of some luminous surface into the object plane. 
Let k be the transmission factor of the microscope (supposed uni- 
form for various ray paths), and ft the transmission factor of an 
object element ; then, if the eye pupil were wholly filled with light 
when observing the image, the apparent brightness of the image 
element would be Bkb. But in the majority of cases the eye pupil 
is only partially illuminated. It was shown above that the radius 
/>' of the exit pupil of the microscope is 



., 
P - 



where . !'/ /is the numerical aperture, ft is the distance of distinct 
vision, and m is the visual magnification. Hence, if p is the radius 
of the eye pupil when observing the source directly, we have, to a 
first approximation, 



Apparent brightness of microscope image 
Apparent brightness of source 



/ />'\ 2 
\Po/ 



Hence, other things being equal, the apparent brightness of the 
image will be directly proportional to the square of the numeri- 
cal aperture, and inversely proportional to the square of the 
magnification. 

Note that if the aperture of the objective is not filled by the con- 
denser, then the illuminated area of the exit pupil will be smaller ; 
the "numerical aperture 11 to be used in the above equation is, 
therefore, that numerical aperture of the objective which is effec- 
tively filled by the condenser. 

Exposure in Photomicrography. In photomicrographic work we 
shall be concerned with the illumination of the image projected on 
to the screen. The condenser, we will suppose, projects an image of 
the source or effective source into the object plane, which may, as 
a first approximation, be supposed to radiate as a perfectly diffusing 



218 APPLIED OPTICS 

source into that aperture of the objective which is filled by the 
condenser. Let the "brightness" be B', then the light received by 
the objective from a small area ds will be 



, f . . 
nB'ds sm 2 a = 



n 



where a is the angular divergence of the extreme rays from the 
condenser, . I .' / is the numerical aperture, and n' is the refractive 
index of the medium. But if B is the original brightness of the 
source (in air, say), and k c is the transmission of the condenser, 
then the light entering the objective will be 

7Tbk e Bds (. I / /)* 

where b is the transmission of the object element as before so that 

B' = k c b&n f * 

The transmission of the microscope is k m say, and the size of the 
corresponding image patch will be M 2 rfs where M is the linear 
magnification. Hence the illumination of the image will be 

. ,. 7Tbk r k m Bds(. |/ /) 2 

Illumination -= 



M 2 ds 
= Trbk 




In practice the useful part of this result will be that the illumina- 
tion is proportional directly to the square of the . I '/ /, and inversely 
to the square of the magnification ; a relation which is of service 
in making photomicrographs of the same object (say) with different 
objectives. Assuming the reciprocity relation, the exposure will be 
proportional directly to M 2 , and inversely to (.yKV) 2 . 

The Projection Lantern. The most widely used types of projection 
apparatus are those for the projection of transparencies, "lantern 
slides" and kinematograph film images. The "magic lantern" is 
said to have been invented by Roger Bacon, but detailed informa- 
tion on the lantern was first given by della Porta (1538-1615). 
One of the chief early difficulties was the lack of suitable light 
sources, as the early oil lamps without chimneys were not super- 
seded by the Argand burner till the end of the eighteenth century. 

It might appear that the only requirements for the projection of 
a transparency would be the provision of an illuminant and a pro- 
jection lens ; but this is only the case under very restricted condi- 
tions. Using an opal bulb lamp O, and a projection lens L, Fig. 
171, it is possible to project an image of a small transparency such 
as might be made from a vest pocket camera picture, but the 



PHOTOMETRY OF OPTICAL SYSTEMS 



219 



illumination will not be very strong, and its uniformity will depend 
on the uniformity of the opal glass. Starting from the boundary 
of the objective and drawing "rays" intersecting the various 
points of the object plane, we find that t l and t 2 are the extreme 
points which would be seen projected against the bright background 
of the bulb if viewed from all parts of the lens L. A point such 
as t is therefore only partially illuminated, and the projected 
image will have a boundary in which the illumination fades away. 
Evidently the region of fading will be the smaller the closer the 
transparency to the source of light. 

In the majority of cases the transparency to be projected will be 
larger than the source of light. British lantern slides measure 3 J in. 








FIG. 171 



X 3^ ill. Continental sizes vary, but a common size is 83 mm. x 
100 mm., while larger slides at 12 cm. x 9 cm. are sometimes used. 
The American size is 3^ in. x 4 in. (83 mm. x 102 mm.). Then in 
photographic enlargers provision may be required to deal with 
negative transparencies up to post card size. The most important 
case is, however, that of the standard kincmatograph film picture 
which is i in. wide x f in. high in the direction of the length of the 
film. 

On the other hand, the practical sources of light are mainly of 
small size. The most important and widely used source is the elec- 
tric arc. In the common form the crater of the arc is the actual 
radiant, and this presents a white disc of size varying with the 
current. Baby arcs taking about 5 amp. give a disc of 3 mm. to 
4 mm. diameter, but with 20 to 100 amp. a radiant of much greater 
diameter can be obtained. According to S. Harcombe, in the Proc. 
Opt. Conv. t 1926, the results of various measurements on the arc 
crater show that in low current density arcs, where the current 
density is about 0-15 amp./mm. 2 of crater area, the intrinsic 
brilliancy of the crater is about 135 c.p./mm. 2 In medium cur- 
rent density arcs (7 amp./mm. 2 ) the intrinsic brilliancy is about 
200 c.p./mm. 2 In high current density arcs, where the current 



220 APPLIED OPTICS 

density is i-o amp. per sq. mm. of crater, the intrinsic brilliancy is 
750 c.p./mm. 2 If d is the diameter of the crater in millimetres, and 
D that of the positive carbon in millimetres, and I the current 
in amperes, then d = K\/DI where K = 0-344 for ordinary arcs, 
and 0-285 for high current density arcs. 

Increasing use is being made of these high intensity "searchlight 
arcs," in which cerium and other salts are introduced into the cores 
of the carbons ; in this way it is possible to maintain stable arcs at 
much greater current densities, and to reach a still greater intrinsic 
luminosity in the radiant, since the gases from the core reach a 
much higher temperature than the carbon itself. With cored car- 
bons taking up to 150 amp., craters up to J in. diameter can be 
obtained, but lower currents are usual in kinema projection, 60 to 
70 amp., when the crater will be rather smaller. 

The electric arc, although the most satisfactory source of light 
when high brightness of the picture is required, is somewhat trouble- 
some in operation ; even though more or less satisfactory automatic 
feeding mechanisms are in use, they require a certain amount of 
skilled attention. Direct current must be employed almost always, 
especially in kinematograph projection. For small range projection 
or for photographic enlarging, gasfilled tungsten lamps with coiled 
filaments are now very widely used. The filament coils are grouped 
fairly closely together (in a small lamp taking 5 amp. at 100 volts 
they may be mounted within a projected area i cm. square as seen 
from the centre of the projector lens), but the possibility of short 
circuiting through thermionic effects sets a limit to the practical 
closeness of grouping. The higher voltage lamps necessarily have 
larger filament coils and are less satisfactorily concentrated. 

Other lamps of value are the Point 'olite tungsten arcs of the 
Ediswan Electric Company. In the smaller 100 c.p. lamps the 
chief source of light is a tungsten ball about 2-5 mm. in diameter, 
but in the larger 500 c.p. lamps the light is mostly derived from 
a glowing plate 5 mm. square. 

The wide-spread availability of electricity is making the older 
sources of less importance, but the oxy-hydrogen flame used to 
heat a "lime" cylinder, or (more recently) a thorium pastille, is 
capable of producing a source of very high intrinsic luminosity 
distributed in a fairly uniform patch. 

Arrangement of Optical Systems for Projection; Screen Bright- 
ness. First Arrangement. In a very usual optical arrangement, 
where the source of light is small in comparison with the trans- 
parency, it is arranged that the condenser shall project an image 
of the source into the entrance pupil of the projection lens; the 



PHOTOMETRY OF OPTICAL SYSTEMS 



221 



transparency is placed immediately after the condenser. The 
arrangement is shown diagrammatically in Fig. 172. 

Assuming the angular divergence of the beams to be small, we 
may calculate an approximate expression for the illumination of 
the screen. Let ds l be the area of the source and B its average 
brightness, and let ds^ be the size of the corresponding image in 
the entrance pupil of the condenser projection lens L. Let a 9 be 
the angular convergence (relatively to the axis) of the extreme rays 
from the condenser C to the lens L. If k is the transmission factor 



Condenser Plane of slide 

T>< * *^ 



Screen 




FIG. 172. FIRST ARRANGEMENT FOR PROJECTION (DIAGRAMMATIC ONLY) 

The image of the source is formed in the entrance pupil of the projector lens 

for the condenser, then the average brightness of the image is fcB, 
and the total amount of light F passing into the projection lens is 
therefore given by 

F = rrkBdSi sin 2 a' 

The entrance pupil will not lie far from the nodal point ; assuming 
the angular divergence of the rays after passing through the projec- 
tion lens to be a', the area of the screen illuminated at a distance V 
will be (very nearly) TT (/' sin a') 2 ; hence the illumination of the 
screen is 

Tr&AjB dsi sin 2 a' ___ k^B ds^ 
r^iSV ^ T* 

where k is the transmission factor of the projection lens. 

This illumination is evidently proportional to the brightness and 
area of the source, provided that the projection lens is sufficiently 
large to transmit the whole of the light. 

Further, it is easy to see that our equations imply that the 
illumination of the transparency is equivalent to illumination by 
an extended source having the same average brightness as the 
actual one and placed just behind the slide, but allowing, of course, 
for the losses in the condenser lens. 

Consider a small circular area ds 2 in the plane of the transparency. 
If B x is the effective brightness of this plane, the light sent from 



222 APPLIED OPTICS 

this area to the area ds^ (the image of the source) in the entrance 
pupil of the projector is 

B l ds 2 dsS 
I 2 

where / is the distance from transparency to entrance pupil. 
Assuming that ds% is a small circular area of radius a we obtain 

, a 2 ^ , , . , 
7rB l ds^ -- = TrBj rfs/ sin 2 a' 



l -- 
/ * 

where the angle of convergence of the rays between the axis and 
the most oblique ray from ds% passing through the axial point of the 
image ds^ is a' . 

But we know that the effective brightness of the source image 
ds must be kB, where A' is the transmission of the condenser and 
B the brightness of the source. Hence the light passing through 
this image and derived from ds% is 

nkB d Sl ' sin 2 ' 

Equating the two values for this light from ds 2 we get 

TrBj d$i sin 2 a' = rrkB ds^ sin 2 u' 
so that B x = kB 

The effective brightness of the plane of the ''object " is the bright- 
ness of the source multiplied by the transmission factor A. 

Provided that all the light goes unhindered through the projec- 
tion lens, there is no need for the strict condition that the image 
of the source shall be formed in the entrance pupil, but we can 
easily see that if the condenser is free from spherical aberration, so 
that a sharp image of the source is formed, then the above arrange- 
ment will allow us to use the minimum aperture of the projection 
lens while obtaining the maximum possible light with the given 
source and condenser, and at the same time illuminating as large 
a transparency as is possible. If under these conditions the entrance 
pupil of the projection lens is not completely filled with light, 
advantage may be obtained by using a larger source, or by pushing 
the source nearer to the condenser (see below). 

Again, there is no need to place the transparency immediately 
behind the condenser. Provided it is wholly illuminated it may be 
placed anywhere between the condenser and the projected image 
of the source, provided that the projector lens can still receive all 
the light, and that the required projection can be effected. 

Second Arrangement (Kinematograph Projector). We noticed 



PHOTOMETRY OF OPTICAL SYSTEMS 



223 



above that the illumination of the screen is dependent (granted a 
sufficient aperture of the projection lens) on the area of the pro- 
jected image of the source of light. If this source is practically 
uniform its image may be projected into the plane of the trans- 
parency, provided that the picture can be completely covered. In 
practice, the arrangement is mainly of interest in kinematograph 
projection, since the small area of the film picture can be covered 
by the projected image of the arc crater. The " condenser " is usually 
represented in practice by a mirror, but the theory can be illus- 
trated simply by Fig. 173, in which the condenser is shown dia- 
grammatically by a lens. The symbols ds, ds' t and ds" represent the 
areas of the source, intermediate image, and final image respectively. 

Plane of film 




FIG. 173 SECOND ARRANGEMENT FOR PROJECTION APLANATIC CONDENSER 
UNIFORM SOURCE PROJECTED INTO PLANE OK TRANSPARENCY TO BE 

PROJECTED 

It will be clear that the illumination of the final image will be 
dependent on the angular aperture of the beam which is transmitted. 

If the angles a and a' represent the angles with the axis made 
by the extreme rays diverging to the projector lens and converging 
to the final image, we can write, using the symbols in the same sense 
as above, 

Total light received by the projector lens = - rrkBds' sin 2 a 

i Brfs' sin 2 a 



Hence the illumination of the screen 

But the optical sine relation gives 

sin 2 a sf'ii 2 a' 

ds" ~~ ds' 
so that the illumination of the screen == 



ds" 



sin 2 a 

MjB (utilized area of the stop of the projection lens) 
Square of distance from lens to screen 

With such an arrangement, a projection lens of large aperture 
(low stop number) can be usefully employed if the condenser gives 



224 



APPLIED OPTICS 



a beam of equally large angular aperture. The actual illumination 
on a screen, therefore, depends in each case, first on the illuminated 
aperture of the projection lens which would be seen to be illu- 
minated when looking through a pinhole in the screen ; secondly, on 
the intrinsic brightness of the source ; thirdly, on the transmission 
of the optical system. With a condenser of large aperture we shall 
be able to utilize a given area of the stop with a projector of shorter 
focal length than with a condenser of smaller aperture, and hence 
to secure a proportionately larger picture of the same brightness. 
The advantage of the second arrangement is, however, that a 
mirror can be used as condenser as shown in Fig. 174. The plane 
of the image of the crater is uniformly illuminated, but other sec- 



Cone of shadow from 
carbon supports 




Plane of 
film 



. f . 
Ellipsoidal mirror 

magnifying about 
6 times 



Negative lens 
magnification 
about 2 times 




Image of arc 
crater magnified 
about 12 times 



FIG. 174. USE OF MIRROR CONDENSER 



tions of the beam will show more or less shadow due to the carbons. 
Also, when high intensity arcs are employed, the flame of the arc 
contributes a proportion of the light much greater than in the case 
of the ordinary arc lamp, in which practically all the light is derived 
from the crater. The result is that the projection of the image of 
the source exactly into the plane of the transparency must be very 
precise, and it is advisable to use some special device to ensure 
that the negative carbon crater maintains its position very exactly, 
and that the length of the arc also remains constant. 

Selection of Projection Lens. The focal length of the projection 
lens required under given conditions can be found from the approxi- 
mate relation 

Focal length in inches 

( * hrow from les to screen in feet) A (diameter of transparency in inches) 
(Corresponding diameter of image in feet) 



PHOTOMETRY OF OPTICAL SYSTEMS 



225 



For example, take a standard size slide with a picture 3 in. sq., 
say, and suppose it is desired to project a picture 8 ft. sq. at a 
distance of 40 ft. Then 40 x 3 

Focal length - =- 15 in. 

o 

In the case of the kinema projector, the width of the "gate" 
aperture exposing the film may be about 0-91 in. ; to find the focal 
length of a lens to give a 13 ft. picture at a 50 ft. throw we have 



,, . . .. 50 x o-oi 
tocal length = --- - - 

o 



. 
in. 



Screen 



Plane of 
/slide 




Paraxial image of 
source of light 




Project/on lens 
(diagrammatic) 



Ki<;. 175. PROJECTION WITH CONDENSER LENS SUFFERING FROM 
SPHERICAL ABBERRATION (DIAGRAMMATIC) 

If piojei tion lens is moved too far to the right the outer p.irl of held (rays 4) will l>e 
rut off , if i MOV i-d too f.ir to the left an annular zone of light (ravs 2 and ^) will be < ut 
off In the < ase shown above, the source of light might, if small, be pushed closer to 
the condenser to obtain better illumination ; but the aboxe jiosition allows for a finite 
si/e of the image of the soun e 

Lantern Slide Projection Systems. It commonly occurs that the 
projection systems in ordinary use employ condensers having large 
residuals of spherical aberration. A common type of condenser has 
two "bulls-eye" lenses, each plano-convex, the convexities being 
turned together. The powers of the components are arranged so 
that roughly parallel light passes between the two members. The 
plano-convex lens is cheap to manufacture, and represents an approx- 
imation to the crossed lens giving minimum spherical aberration. 

The size of the best concentration of light in the beam focused 
by the condenser is dependent partly on the spherical aberration 
of the condenser, partly on the chromatic aberration but only to 
a very small extent, and partly on the si/e of the source; the latter 
more especially when the source is of considerable magnitude, as 
in the case of a gasfilled lamp. The dimensions of the image of the 



226 



APPLIED OPTICS 



source (neglecting spherical aberration for the moment) will be the 
greater, the shorter the distance from the source to the condenser; 
but the total amount of light taken up by the condenser will be 
the greater if the source is pushed closer to it. It is, therefore, 
advisable to use a projection lens having a fairly large diameter 
of the back lens in relation to the focal length ; then the source may 
be pushed in towards the condenser till the cones of light begin to 
be cut off by the mounts of the projection lens. The best condition 
will usually be such that the image of the source formed by the 
paraxial regions of the condenser lies beyond the projector, and 
since the spreading of the beam will be less with smaller sources, 




FIG. 176. TRIPLE LENS CONDENSER (WITH DIMINISHED SPHERICAL 
ABERRATION) INCORPORATING WATER CELL FOR HEAT ABSORPTION 

they may be brought closer to the condenser than when using 
larger ones without loss of light. The condition is illustrated in 

Fig. 175- 

If specially figured or corrected lenses are employed in the con- 
denser system, then the beam will be free from any spreading due 
to spherical aberration, and this again will allow of the maximum 
amount of light being taken up by the system consistent with given 
dimensions of the source. Fig. 176 shows a triple lens condenser 
which gives considerably diminished aberration. It has a trough 
containing water for cooling purposes. 

The lens type chosen for projection is usually on the lines of the 
Petzval portrait objective, and may work at a high aperture, 

to - or thereabouts. The Petzval type has then the advan- 
2 "5 4 

tage of the large diameter of its lenses for a given focal length, and 
consequent economy of light. These lenses are usually well cor- 
rected for spherical aberration and coma, so that the centre of the 
field is well defined. Owing to the long throw which is usually 
required, the angular field to be covered is not usually very large, 
and there is then no need for a more complex and expensive anastig- 
matic system, although cases do arise in which a high magnification 
is required for a comparatively short throw, and another type of 
lens giving a wider field may then be advisable; an anastigmat 



PHOTOMETRY OF OPTICAL SYSTEMS 



227 



may then he chosen. The condenser must naturally be capable of 
giving the angular field required. 

Loss o! Light in Projection Systems. The average efficiency of 
ordinary projector systems is very low, only about 5 per cent of the 
total light from the source reaching the screen in many cases. In 
kinematography a further loss occurs through the cutting off of 
light by the revolving shutter. 

Even though a mirror may be used behind the source of light, 
it is difficult to get much more than 10 per cent of the light through 
the slide carrier with ordinary systems, and reflections at the sur- 
faces of the slide and projection lens cause further serious 
weakening. Numerical details of the kinematograph projector may 
be of some interest. It may be reckoned that at least 4 per cent 
of light is lost by reflection at a glass-air surface, and an absorption 
of 5 per cent per centimetre thickness of glass is not uncommon 
in the inferior glass often employed in condenser lenses. A two-lens 
condenser may therefore transmit only 70 per cent of the incident 
light, and the loss in a triple condenser would be considerably 
greater. 

The gate at which the film appears is situated either at the focus 
of the condenser system, or at the "waist" of the converging cone 
if the system is subject to spherical aberration, and the loss here 
varies greatly with the source and the optical system employed. 
It is evident that the area of uniform illumination must overlap 
the gate widely in order to allow for slight variation in the position 
of the source of light due to wandering of the arc, or mal-adjust- 
ment of the carbons in the focus. Even in favourable circumstances 
a loss of about half the light seems difficult to avoid if a safe overlap 
is to be given. 

The film itself, even in the most transparent part, removes about 
20 to 25 per cent of the direct light, and the objective, usually with 
six "air-glass 11 surfaces, may remove a further 40 per cent, even if 
the absorption in the glass is negligible. We then have to consider 
the loss due to the kinematograph shutter which masks the image 
while the film is in motion from one picture to the next. This 
inevitably cuts off 50 per cent. Hence the transmissions of the 
various parts are likely to yield the following approximate table 
APPROXIMATE PERCENTAGE TRANSMISSIONS 



Condenser 


Gate 


Film 


Projector 


Shutter 


Combined 


70 


5<> 


75 


60 


50 


7-8 



228 



APPLIED OPTICS 



The arc lamp employed in kinema work may have a candle- 
power of 12,000, according to the current taken. Assuming that 
the condenser intercepts a cone of radiation of unit solid angle, 
which is an approximation to the truth, the total light entering the 
system will be approximately 12,000 lumens, of which 7-8 per cent 
reach the screen, i.e. 936. If the screen area is, say, 250 sq. ft., 




FIG. i77(). SIMPLE EPISCOPE 



FIG. 177(6). SPHERE 
KPISCOPK 



this means an illumination of 3-7 lumens per sq. ft., which is fairly 
satisfactory in practice. 

The condenser losses are considerably reduced with the mirror 
and negative lens combination. The glass surface, instead of reflect- 
ing the light back, in this case sends it onward in the required 
direction, and the only serious losses are those due to any imper- 
fection of the mirror reflection and reflection at the negative lens, 
which should not amount to more than 15 or 16 per cent when the 
surfaces are in good condition. 

Episcopes. Episcopes are instruments designed for the projection 
of images of opaque objects. Such objects are illuminated very 
intensely with the aid of powerful lamps and reflectors, and their 
images are projected by anastigmat lenses of very large aperture. 
Let us assume that an element of perfectly reflecting and diffusing 
surface, of area ds, is illuminated in the first place by a powerful 
electric lamp of 2,000 candle-power, say, at 6 in. from the sur- 
face, as in Fig. 177 (a). The lamp may be furnished with a reflector 
to enhance the illumination, and although the incidence of the light 
must be oblique, it may be possible to attain an illumination of 
8,000 ft. candles on the surface, i.e. 8,000 lumens per sq. ft. must 
be re-radiated; but the total radiation =--- trB lumens per sq. ft. 



PHOTOMETRY OF OPTICAL SYSTEMS 229 

Sooo 
Hence, B = . The candle-power of the elementary area ds, is 

77 

therefore (8000 dsj 



7T 



and the lumens radiated into a cone of semi-apical angle will be 

. o/>j /8ooo\ 
IT sin 2 Ods l [ 1 

\ 7T / 

If the lens is working at -, say (the ratio of diameter of entrance 

pupil to focusing distance from object to pupil is i, so that tan -1 
= J, and we find ----- 14), sin 2 = -^ approximately, so that the 
amount of light is 500^ lumens. This light, or a proportion of 
it depending on the transmission of the lens, is to be distributed 
over the magnified image of ds^ If the magnification would be 
sufficient to enlarge a 4 in. picture up to 6 ft., the linear magnifica- 
tion will be itS times, and the area magnification = i8 2 = 324. 
Hence, if the transmission of the lens is 80 per cent, we shall have 
400 ds l lumens distributed over an area 324 ds, so that the lumens 
per unit area ,{!J'/ -- 1-23. This is the illumination of the screen 
in foot-candles, but allowance must further be made for the reflec- 
tion factor of any actual object surface. 

The principle of the Ulbricht integrating sphere has been applied 
by Bechstcin to the problem of Episcopic projection. Fig. 177(6) 
illustrates the principle. The sphere S has two, main openings, one 
of which is filled with the projection lens L, the other is closed 
by the picture or surface, the image of which is to be projected. 
The interior of the spherical surface is coated with a matt white 
paint. Suitable paints can be secured with very high total reflec- 
tion coefficients, i.e. up to 98 per cent. 

Suppose for a moment that the paint was totally reflecting and 
absorbed no energy, also that the surface of the diagram P absorbed 
a negligible amount of energy, then if we introduce a source of 
light E into the sphere, the aperture of the lens L will be the only 
path by which the light can escape (perhaps after many internal 
reflections), and thus clearly all the light radiated by E would have 
to pass through L. It can be shown that the proportion of the light 
derived from the circular area P to the total light will be represented 
by sin 2 0: i, where is the semi-angular subtense of P (provided 
that we neglect the light directly escaping from E through L with- 
out internal reflection, or assume that a small diffusely reflecting 
screen is placed to stop such direct escape). 



230 APPLIED OPTICS 

The above performance would be an extremely efficient example 
of projection, even as compared with the case of transparent 
objects; but in practice no such efficiency is attainable since the 
object for projection will absorb much light, the lens L will have a 
very appreciable aperture and may allow some direct light to 
escape, and a considerable absorption of light energy will take 
place at the walls of the sphere. In spite of these drawbacks, a 
good performance can be obtained. 

Epidiascopes. In recent years a number of makers have intro- 
duced systems combining an episcope with an ordinary projection 
lantern; a mirror and condenser can be brought into action when 
required in order to change from one system to the other; such 
instruments arc known as "epidiascopes." Lantern slides for pro- 
jection are often called " diapositives " in Continental literature. 

The Projection of Light. Apparent Brightness of Image-forming 
System (Maxwellian view). In a previous section we have calculated 
the apparent brightness of the image projected by an optical 
system, assumed to be free from spherical aberration and to fulfil 
the optical sine condition. Referring back to Fig. 170 and the accom- 
panying discussion, the illumination of the area covered by the 
image in the image plane (light per unit area) is 

rrBk sin 2 a 

V 

if a! is small, sin a' = "j- f where v is the radius of the exit pupil of 

I 

the image-forming system, so that 
Illumination = 



__ Bk (area of exit pupil) 

We therefore find that the whole exit pupil of the image-forming 
system is now radiating as a source of brightness Bk, i.e. the bright- 
ness of the source multiplied by the transmission factor of the optical 
system. The "candle-power" of the radiant area is thus Bk (area 
of exit pupil). 

It is not very easy to give an entirely satisfactory general proof 
that the same thing holds good whenever the eye views an optical 
system projecting the image of a uniform source in such a way that 
any ray from the eye traced back through the system intersects 
the source; the following treatment, however, may indicate that 
the principle is wider than might be inferred from the special case 
above. 



PHOTOMETRY OF OPTICAL SYSTEMS 



231 



Let A be a small area of an object of brightness B, and let RT (Fig. 
178) be the trace of a refracting surface having its normal in the plane 
of the diagram ; let the small area be rectangular with one diameter 
in the plane of refraction and one diameter perpendicular thereto. Let 
* and *' be the angles of incidence and refraction, then the angular 
width of the fan of rays (in the plane of the diagram) from the element 
to a point R of the surface is di, and the corresponding width when 
refracted is di'. The breadths perpendicular thereto are proportional 
to sin * and sin i' respectively. In order to see this, imagine per- 
pendiculars dropped from each fan to the normal and that the 



.(Area of element)* cosy ~ 




IMG. 178. DIAGRAM 

diagram is rotated through a small angle dO about the normal. The 
locus of the surface will be constant in the neighbourhood of the point 
R, and the arcs described by corresponding object space and image 
space elements at distances / and /' from R will be / sin idO and /'sin t 'dO. 
Let r^ and o> 2 be the solid angles of these bundles, then 

<o, sin i di 



But 
and 

Hence 
so that 



co a sin i' di' 
n sin i = n f sin i' 
n cos i di = n' cos i'di' 
sin i di n' 2 cos t' 
sin V di' ~ n* cos i 
>! __ n'* cos i' 
o> 2 n 2 cos i 



Let the projected area of the element A in the direction of R be <r, 
then the light sent from the radiating element to a small area ds on 
the.surface is (B* ds cos <)// = B cos i ds Wl 

and the amount of light it radiates is 

B! cos r ds co 2 

where B 4 is the effective "normal brightness" of the element R, which 
will agree with its apparent brightness to an observing eye of which 

16 (5494) 



232 APPLIED OPTICS 

the pupil is filled with the light. In order that we may have the amount 
radiated equal to k times the amount received (k being the transmission 
factor), we must have 

B x cos i' ds eo 2 = kB cos i ds o^ 

kBw l cos i 
sothat Bl== to, cos*' 

- n -kB 

Hence the apparent brightness of the optical surface would be equiva- 
lent to that of the source if there were no reflection losses ; and if the 
initial and final media were the same, since the equation could be 
applied to any number of refractions. Thus 



Reflection can be looked upon as a particular case of refraction. 

This theorem is a very important one ; it indicates, let us repeat, 
that when the eye pupil is filled with the beam from a projector 
of any kind, and when all rays which could be traced from the eye 
back through the projector intersect the source, then the whole of 
the projector system has the apparent brightness of the source. 

The appliances to be dealt with under this heading comprise 
searchlights, motor-car headlights, light-house projection systems, 
signalling lamps, and the like. It will not be possible to do more 
than to give the briefest outline of the theory and practice, since 
a very large technical literature exists in regard to all of these. 

We showed above that the apparent brightness of the radiating 
aperture forming an image of a surface of brightness B is kB, where 
k is the transmission of the system, and this agrees with the apparent 
brightness of the image when it is observed by the eye. If the eye 
moves to various distances, the apparently illuminated part of the 
radiating aperture of the system will always have the same apparent 
brightness KB, provided that the pupil of the eye is filled with 
light, or lies within the cone of radiation from the optical system ; 
if, however, the pupil is not completely filled with light, the radiating 
aperture will appear less bright ; or if there are possible ray paths 
between the eye and the optical system which, on being traced 
backwards, fail to intersect the source of light, then the corresponding 
parts of the optical system will appear dark. 

To make these principles more definite we will refer to Fig. 179. 
The eye is withdrawn behind the image ds' of a radiating element 
ds; when the eye pupil was coincident with ds' the whole back 



PHOTOMETRY OF OPTICAL SYSTEMS 



233 



surface of the lens appeared to be of brightness kB, since ds' was, 
we will say, slightly larger than the eye pupil. When the eye is 
withdrawn behind the image ds' t the area of the latter acts like a 
circular stop, and only a limited area in the centre of the lens can 




FIG. 179 

send light to the whole of the eye pupil ; this part, therefore, appears 
of the full brightness kB, and is surrounded by a penumbral shadow. 
Again, consider the case of Fig. 180, in which the image of a very 
small source is projected near the eye by a lens exhibiting strong 
spherical aberration ; the centre of the lens appears filled with light, 




FIG. 180 

but the marginal zones are in shadow, since the marginal rays pass 
outside the pupil. We could infer the same thing if we trace a 
ray (dotted) from the pupil centre backwards through the margin 
of the lens system, and find that it passes outside the source; if, 
again, we take a ray through the marginal point P from the bottom 
point of the pupil and find that it intersects the source, while a ray 
from the top of the eye pupil fails to intersect it, we shall infer a 
partial illumination of the surface of the system near P. 

If the aperture appears wholly illuminated to the observing eye, 
technical parlance speaks of a "complete flash" ; otherwise we may 
have a "partial flash" if the apparent illumination is incomplete. 

Effect of non-fulfilment ot the Sine Condition. We have seen 
that in a simple case the apparent brightness of the flash is kB K 
and this was independent of any fulfilment of the optical sine 
condition, i.e. the constancy of magnification for the different zones 
of the system, although the optical sine relation expressing the 



234 



APPLIED OPTICS 



magnification for a zone was used in the simple discussion. If the 
system does not fulfil the sine condition, the images of the source 
formed by different zones will have different sizes ; hence the pro- 
jected image will have a diffuse boundary. If the outer zones give 
the larger image, then an eye observing the lens from a point in 
the outer region of the image will see the margin of the lens bright 
while the centre parts are dark, and vice versa. Evidently, in order 
to obtain a sharply-bounded image, the fulfilment of the sine condi- 
tion will be of importance. 

Searchlights and Headlights. Searchlights almost Invariably em- 
ploy mirror reflectors rather than condensing lenses, since mirrors 




Kic; 181 

give freedom from chromatic aberration, and, moreover, involve 
less loss of light. The geometrical form of the paraboloid of rotation 
renders rays from the focus strictly parallel on reflection, although 
the system does not fulfil the optical sine condition. Hence, rays 
diverging from a point (in the focal plane) away from the axis are 
not strictly parallel. 

The divergence from the sine condition for a zone of diameter 
D of a parabolic mirror of focal length /is shown by the relation 



If the sine condition were fulfilled (page 37), we should have 

y 

^ , as a constant. The defects are quite easy to realize geometric- 
ally (see Fig. 181); - m y t is the distance PF for any zone; the 



sin a 



above equation can easily be calculated from the equation to the 
parabola (see page 68). The rays FP and FQ are rendered parallel 



PHOTOMETRY OF OPTICAL SYSTEMS 235 

X"X 

on reflection, but we can see that the angle FPB is greater than the 

angle FQB if BP and BQ are rays from the extra-axial point B ; 
the paths of these rays after reflection are PC and QD respectively, 

and hence the angle RFC is greater than SQD, so that PC and QD 
are clearly convergent. Therefore the searchlight beam departs 
from parallelism, with a finite size of the source, on account of the 
geometrical properties of the image formation. 

In practice, a single reflecting surface is too liable to tarnishing 
and damage. Hence, reflectors of silvered glass are generally em- 
ployed, the silver backing being suitably protected by coppering 
and painting. 

One plan is to grind a glass reflector of which both surfaces are 
congruent paraboloids of rotation; 4 thus the reflected component 
from the front glass surface should be "parallelized*' in addition 
to that from the back. But, in practice, the finite thickness of the 
glass introduces a certain amount of spherical aberration into the 
beam reflected from the back, so that it is advisable to modify 5 
the shape of the back surface in order to avoid this defect, and it is 
no longer a true paraboloid. 

In both the above cases, then, the reflected components from both 
front and back surfaces are more or less parallelized, and this is of 
importance in long distance projection where the utmost economy 
of light is required. 

In the case of motor headlights, however, the lateral spreading 
of a certain amount of the light is highly desirable, and it is then 
possible to use reflectors (as in the Zeiss systems) in which the back 
reflecting surface is truly spherical, and the front surface is figured 
to a suitable non-spherical curve, refraction at which corrects the 
spherical aberration arising from reflection at the back surface. 

The Mangin mirror (Fig. 182) (described by A. Mangin in 1876) 
consists of a glass reflector, silvered on the back, each surface of 
which is a true sphere. The curvatures may be so chosen that the 
system is freed from spherical aberration, and also from coma ; thus 
it fulfils the optical sine condition. In consequence of this the pro- 
jected image of a small source placed at the focus is fairly sharply 
defined, and the lateral spreading of the light is greatly restricted as 
compared with the effect of a parabolic reflector. The mirror is, 
consequently, very useful for signalling lamps where only a very 
limited region near a receiving station may receive the light. 

The form of the glass reflector itself is that of a diverging meniscus 
lens, and the centre of curvature of the hollow side may nearly 



236 APPLIED OPTICS 

coincide with the focus. Since the thickness of the glass rapidly 
increases towards the margin, this sets a limit to the aperture ratio 
which can be effectively used ; but the mirror can be made to sub- 
tend 130 to 140 at the focus. 

Though the Mangin mirror is effectively employed for the smaller 
searchlights up to about 20 in. in diameter, the parabolic reflectors 

Spherical 

Surface 

Silvered 




FIG. 182. THE MANGIN MIRROR 

(which avoid too great a marginal thickness of glass) are employed 
for the larger sizes. 

Illumination due to Searchlight. Consider a small area in the 
field of illumination taken near the axis in a plane perpendicular 
to the axis. It is to be at such a distance from the projector that 
any ray taken from this area to the exit pupil of the system and 
traced onwards, will intersect the source of light. Under these 
conditions an eye placed at the position of our area would observe 
a "complete flash" in the projector. 

According to the theory the illumination will then be 

Bk (area of projector) 
H 

so that the effective "candle-power" of the projector in this area is, 
as before, B& x (area of exit pupil). The coefficient k must, how- 
ever, include the possible effects of atmospheric absorption. 

Large searchlights are made with mirrors up to 5 ft. in diameter, 
but the difficulty of producing an accurate figure of the surfaces, 
as compared with the smaller 3 ft. mirrors, causes the results 
obtained from the larger mirror to fall short of expectations. Assum- 
ing a 3 ft. mirror and a carbon arc crater of an intrinsic brightness 



PHOTOMETRY OF OPTICAL SYSTEMS 237 

of io 5 candles per square inch, a complete flash would give a gross 
candle-power of 

TT . i8 2 . io 5 = io 8 candles (approx.) 

But this figure will, in practice, be reduced very considerably by 
the obstruction of the negative carbon of the arc, the reflection 
losses in the mirror and in the front "window" which may be fitted 
to protect the arc from wind ; so that the net result is not likely to 
exceed 60 per cent of the above figure in a clear atmosphere, even 
with a fairly perfect reflector. The majority of reflectors fail, how- 
ever, to give a really complete flash owing to optical imperfections. 

Effect of Various Sources. In the simple theory above, the 
source was assumed to be an elementary disc perpendicular to the 



Pfa&: 




FIG. 183. OPTICAL PROPERTIES OF PARABOLOIDAL REFLECTOR 

axis. The nearest approximation to this in practice is the crater of 
the arc in a searchlight, but smaller projectors may employ other 
sources, such as filament lamps or acetylene flames. We can use- 
fully consider the illuminated area and the "flashing" of the 
radiant area under other conditions. 

A typical example, Fig. 183, concerns a spherical source of radius 
FK = r, situated so that its centre falls into the focus. A ray 
following the path FP would be reflected parallel to the axis, but 
one derived from the extremity of the source as seen from P (i.e. 
one following the tangential direction KP) will be reflected through 

B, where FPK = QPB == 0. Since PQ and AB are parallel, the 
angle ABP is also 0, and we have 

sin e = - ra = (approx - ) 



238 APPLIED OPTICS 

if AB is very large in comparison toy, the radius of the zone marked 
by P. We have, by the geometry of the parabola, 



-^5 very nearly, if r is not large 
A 15 



and AB = - I / | - f } very nearly. 

r \ 4// 

It is clear that a "complete flash" will be obtained at any axial 
point beyond this distance, and that the illumination at such 
points will vary (neglecting absorption effects) according to the 
"inverse square" law. A greater distance is required as the aper- 
ture of the mirror is increased. 

A ray from the apparent extremity of the source L will be reflected 

so as to make an angle CPQ with the line PQ. If ECB marks a 
plane through B perpendicular to the axis, the point P of the mirror 
will be apparently illuminated to an eye placed anywhere between 
C and B ; on the oth^r hand, a point R on the other extremity of 
the mirror diameter will be dark, and a complete flash will not be 
obtained. The flashing of the whole mirror will therefore only be 
obtained from points within the cone, represented by the shaded 
area in the figure, the generating line of which is the prolongation 
of the line PB or RB. At a distance D from the mirror the diameter 
of the fully illuminated area taken perpendicular to the axis is, 
writing d for AB, 

Diameter of fully illuminated area = 2 (D-d) tan 

2(D-d)r 
very nearly. 



The "inverse square" law can only be supposed to hold along 
those parts of straight lines from the point A which lie within the 
shaded area. 

Lantern and Lighthouse Projection Systems. Lenticular or "di- 
optric" condensers for the projection of light are of considerable 
importance in connection with lanterns and lighthouses. In a 
ship's lantern, concentration is only required in the sense that light 
should not be wasted in going much above or below the horizontal, 



PHOTOMETRY OF OPTICAL SYSTEMS 



239 



but must spread freely in azimuth. The lighthouse beam must often 
be restricted in both directions. In the lighthouse the "lens" can 
be made of much larger aperture than would be practicable for any 
single parabolic reflector of such size, and several projectors can be 
grouped around a single source. Fig. 184 shows the section of a 
typical lighthouse "lens" ; the whole would be realized by rotating 
this section about the horizontal axis through the source; a "ship's 
lantern " system would be obtained by rotating the central elements 



Catadioptric ( ^&1 I" " 
(Refoctors)\ ' A ' 
Back Pri s/ns 




*~ Mean Ray 

> Catadioptric 
(Reflectors) 

Angle of 
Vertical 
Divergence 
M5^> Mean Ray 

Dioptric 
(Refractors 



Catadioptnic 
(.Reflectors) 



Ray 



FIG. 184. LIGHTHOUSE PROJECTION SYSTEM 



of the section about a vertical axis through the source. These 
stepped lenses were invented by Fresnel in 1822. 

The construction of the inner elements is a method of overcoming 
the great thickness of the lens which would be necessary if the front 
face were continuous; the discontinuous elements allow a great 
saving of weight and of absorption of light in the glass. For the 
outer zones, the elements are reflecting prisms, but one of the refract- 
ing surfaces of these, like the surface of the inner prism elements, 
is not of perfectly straight section; they are given a curvature 
which should make a ray from a point source at the focus emerge 
in a horizontal direction. The refraction at the inner prism ele- 
ments is subject to chromatic aberration, but extreme spherical 



240 APPLIED OPTICS 

aberration of the outer refracting zones, which would be charac- 
teristic of a single lens with spherical surface, can be avoided by 
giving the correct form to the outer ring elements. 

A theoretical account of the distribution of light in the beams 
has been given by W. M. Hampton. 6 

REFERENCES 

1. Walsh: Photometry (Constable), p. 346. 

2. Martin : Dept. of Scientific and Industrial Research, Bulletin No. 3. 

3. German Patents 250314 of 1911, 252920 of 1911, 31.6050 of 1919. 

4. Munker and Schuckert: German Patent 35477 of i8th Aug., 1885. 

5. Straubel: U.S.A. Patent 1151975 of 3ist August, 1915. 

6. W. M. Hampton: Trans. Opt. Soc., XXX (1928-29), 185. 



CHAPTER VII 

THE TESTING OF OPTICAL INSTRUMENTS 

NEEDLESS to say, the fundamental test of any instrument or appli- 
ance is that of satisfactory performance under ordinary conditions 
of usage. With optical instruments for visual observation, how- 
ever, it is not always easy to tell whether or not the performance 
is really satisfactory without spending much time and trouble. 
Thus the purchaser of a microscope may obtain what seems a 
satisfactory image of some object ; it appears well defined and free 
from obvious defects; but a person of keener eyesight, and using a 
more critical method, may discover deficiencies in the performance. 
The same is true of a telescope or of projection apparatus. It is, 
therefore, advisable to adopt methods of testing which are as free 
as possible from likely errors due to any defective eyesight or 
prejudices of the observer. 

The inspection department of an optical factory will employ 
sensitive qualitative tests for instruments and their components, 
as a matter of routine. Occasionally, when a new system is 
being perfected, the physical measurement of any defects has to 
be undertaken. Similar physical measurements of aberrations are 
of importance to many who have to use optical instruments for 
very exact measurements. Thus, astronomers commonly deter- 
mine numerically the aberrations of their reflectors and refractors, 
and surveyors using photographic methods may require to measure 
very accurately the distortion of their lenses. Since there does not 
seem to be a concise account in English of the chief optical methods, 
the details being mainly found in scattered handbooks, it is hoped 
that the following short summary may be of interest. 

The ** Star " Test. (Telescope and Microscope Objectives.) The 
simplest possible theoretical object is a "point source 11 of light; 
since this only exists in imagination, we remember that the image 
of a very small source of light never decreases in dimensions beyond 
the theoretical limits set by the aperture ratio of the system and 
the wave-length of the light in the image region. The reduction of 
the dimensions of the source beyond certain limits in which its 
geometrical image becomes small in comparison with the diameter 
of the "Airy disc," produces no further appreciable change in the 
diffraction image except a diminution of its brightness. 

For telescopes, real stars are ideal test objects ; and manufacturers 

241 



242 



APPLIED OPTICS 



of astronomical telescopes of all sizes naturally consider final tests 
on real stars as indispensable, but during the course of manufacture 
"artificial stars" can be of great use. 

A small spherical mercury thermometer bulb may form a very 
small image of the sun, or a nearby source such as a small electric 




FIG. 185. VARIATIONS OF AN ARTIFICIAL STAR FOR TELESCOPE 

TESTING 

a = Mercury bulb. 

b = Source behind pinhole. (This may give narrow beam) 

c = Arrangement for wider-angled beam 

d = Arrangement where smallest possible "star" is required for closer work 

lamp, etc. ; when viewed from a distance such a " star " may be very 
satisfactory. Alternatively, a small pinhole in a sheet of tinfoil may 
be illuminated from behind by an arc lamp (a carbon arc or some- 
times a mercury arc), or perhaps by a condensed electric spark. 
If the "star" is to radiate light over a wide angle, then the image 
of the source may be focused on the pinhole by a suitable condenser 
lens. The pinhole may be too large ; it can be used at the proper 
working distance from a reversed microscope objective, which then 
projects a diminished image of the pinhole into the plane which 
would be occupied by the microscopic object in the ordinary use 
of the lens. 

Telescopes intended for use in daylight should be tested by the 
"sun and thermometer bulb" arrangement, if possible, as the effects 
of chromatic aberration are more easily recognizable than when 
artificial light is employed. If it is necessary to use artificial light 
an "artificial daylight " colour filter can, however, be used. Fig. 185 
shows some variations of an artificial " star " for telescope testing. 



TESTING OF OPTICAL INSTRUMENTS 243 

Artificial "stars" for the microscope may be obtained by silver- 
ing the undersides of a number of cover-glasses of varying thickness, 
without taking too great pains to remove dust and specks from the 
surface. It will usually be found that a number of small holes 
result in the film, which are well below the resolution limit of ordinary 
objectives. The glasses are cemented down on 3 in. x i in. slides, 
silvered faces down, and the "stars" are then illuminated by the 
microscope condenser, which forms the image of an arc or Point 'olite 
lamp, etc., in the plane of the film. 

Another device useful with low powers is to produce, by holding 
a slide above boiling mercury, a deposit of extremely minute mer- 
cury globules which form correspondingly small images of a source 
of light suitably disposed. The light may be thrown on the globules 
by the help of a small mirror, or even by a vertical illuminator 
system in the microscope. 

Having secured a suitable "star" source, and made sure that it 
can send unrestricted light into the whole aperture of the objec- 
tive, the appearance at the focus is then examined with the eye- 
piece or eyepieces supplied with the instrument. In many cases, 
nowadays, the objective and eyepiece systems are designed together 
and should, therefore, be tested together; but if odd objectives are 
to be tested, a good idea of their individual performances can still 
be obtained with the aid of a good set of Huygenian eyepieces of 
focal lengths ranging from, say, 25 mm. to 7 mm., if objectives of 
known performance are available for comparison. When using the 
highest power the Airy disc will be clearly seen in appreciable 
extension at the best focus with its surrounding rings, that is, with 
telescope and microscope objectives of usual types. 

A complete discussion of the various points in the "star" tests 
would take too long ; space can only be found for brief notes. Fuller 
details will be found in a booklet by Mr. H. D. Taylor 1 (The Adjust- 
ment and Testing of Telescope Objectives). 

Axial Images; Centring. The first test is for centring of the 
lenses. The star image is brought into the centre of the field and 
the appearances arc observed, both at the best focus and also when 
the eyepiece has been moved slightly within and without the best 
position. Complete symmetry of the distribution of light and colour 
around the axis indicates the correct centring and mounting of 
the lenses, but if one component lens has its optical axis displaced 
from the general axis of the system, unsymmetrical colour effects 
will appear. The eyepiece is assumed to be sufficiently well centred 
by the tube of the instrument. 

Squaring-on. Slight asymmetry of the appearance of the Airy 



244 APPLIED OPTICS 

disc rings near the best focus may indicate imperfect " squaring-on " 
of the objective ; the optic axis of the objective does not pass through 
the centre of the eyepiece. If this is suspected, tests should be 
made with a special "squaring-on" eyepiece. 2 

Colour Correction. An uncorrected objective, such as may be 
encountered in old telescopes or microscopes, shows brilliant colour 
effects with the artificial " star," especially on each side of the best 
focus. Inside the focus, the disc is fringed with bright red, passing 
through yellow to blue in the centre ; outside, the disc is fringed with 
blue-violet passing through green and yellow to red in the centre. 

An achromatic objective, if corrected for visual observation (Vol. 
I, page 231), has its minimum focus for the apple-green region of 
the spectrum. Practically no colour is seen at the best focus, but 
inside the focus a yellowish disc (greenish at the centre) is fringed 
with an orange-red border, but the colour is very much less marked 
than with a non-corrected lens; outside the focus there appears an 
outer fringe of apple-green surrounding a yellowish disc with a 
faint reddish-violet centre. 

The beginner should be warned that some of the colour effects, 
especially the red fringe inside the focus, must be partly attributed 
to the eyepiece and the eye, which are not chromatically corrected, 
and that the amount of colour may vary with different eyepieces, 
being less marked with the higher powers. It is advisable for him 
to make a start by observing with lenses known to be well corrected, 
and to work with different eyepieces so that the eyepiece effect may 
be allowed for; so, until much experience has been obtained, any 
unknown lenses should be tested against similar ones of proved 
performance. 

Apochromatic objectives should exhibit practically negligible 
axial colour (as also will, of course, reflector objectives) : any con- 
siderable colour effects must be ascribed to the residual errors of 
eyepiece and eye, unless they are marked enough to be ascribed 
to faulty construction. Experience is necessary. 

Note that the Airy disc with white light is formed of overlapping 
discs in various colours, and that these have different radii. No 
correction of the lens can overcome this effect which may be noticed 
if very careful observation is made. 

" Photographic " Colour Correction. It will be remembered that 
lenses designed for photography are usually achromatized by 
uniting the foci for the G' and D lines of the spectrum, thus bringing 
the minimum focusing distance for the lens into the F line region 
(blue green). The result of this will naturally be to enhance the red 
fringe inside the focus, and to make a blue-green instead of an 



TESTING OF OPTICAL INSTRUMENTS 245 

apple-green appearance in the outer fringe outside the focus. Here, 
again, the performance of any lens should be compared with one of 
known colour correction or the actual chromatic variation of the 
focus can be measured. (See below.) 

Spherical Aberration. This defect may affect axial images from 
telescope and microscope objectives, and may be due to various 
causes such as incorrect figure, wrong working distance, etc. The 




KIG. i BO. TYPICAL STAR FIGURES FOR CORRECTED LENS AND FOR 
SPHERICAL ABERRATION 

Row i. Spherical under-correction . OuUide foe s; at focus; inside focus 
,, 2. Corrected lens: Outside focus; at focus; inside focus 
,, 3. Mid zone with short focus : Outside focus ; at focus ; inside focus 
(Zonal spherical aberration) 

simple theory was discussed in Vol. I, Chapter IV. In "under-cor- 
rection," the type of aberration due to a double convex lens, the 
marginal zones have the shortest focusing distance, and the focal 
point for any zone is farther from the lens the smaller the 
radius of the zone. The reverse is the case in "over-correction." 
In "zonal" spherical aberration, however, the rays from an inter- 
mediate zone of the lens may have too small or too great a focusing 
distance in the cases of zonal under-correction and over-correction 
respectively. The various typical results must be left to the accom- 
panying diagrams and pictures, which will explain them better than 
a great deal of verbal description. (See Figs. 186 and 187.) 

Defects in the Lens, Striae, Strain, etc. Striae in the glass of the 
lens may produce a marked "fuzziness" at the best focus and 



246 



APPLIED OPTICS 



irregularities in the extra-focal appearances. Strain in the glass 
may be due to bad annealing, or undue pressure by the mount. 
The resulting distortion of the surfaces is also manifested clearly 




(1) Perfect Lens. Symmetrical Appearances 
on Each Side of the Focus 



Bright Outer Ring, Weak Cent re 




Bright Centre, Weak 
Surround in f Patch 
(Z) Spherical Under- correction 



Bright Centre, Weak Surround 




Outer ffing, 
r* Centre 



(3) Spherical Over- correct ion 



Bright Intermediate 




~ZZDark Intermediate 
King 



(4) Zonal Under-correction 

FIG. 187. RAY DIAGRAMS FOR STAR TESTS 

in the distortion of the extra-focal rings. Striae and strain are to 
be tested for independently by the Foucault test and the polariscope 
respectively. (See Vol. I, pages 257, 208 and below.) 



TESTING OF OPTICAL INSTRUMENTS 247 

General Note. Unless there is a very marked physical defect in 
the glasses or surfaces, there will usually be one stage of the focus 
in which a concentration very closely approximating to the Airy 
disc appears, but when aberration is present light leaves the centre 
disc and appears in the surrounding rings. 

Extra-axial Aberrations. If chromatic or spherical aberration are 
present on the axis they will persist over the whole field, but if 




FIG. i 88 

a Coma, extra-axial and focal 
6 Astigmatism, extra-axial and focal 
c Heavy coma and astigmatism together 
The centre image represents the best focus appearance in both (a) and (fr) 

they are absent on the axis, the aberrations which may afflict the 
image definition in the outer part of the field are chromatic difference 
of magnification, coma, and astigmatism. The general nature of these 
defects have already been explained in Vol. I, Chapter IV. Again, 
it is important to test an objective with the proper eyepiece, for 
we cannot easily dissociate the effects of the two. Chromatic differ- 
ence of magnification causes an extra-axial "star" to appear as a 
short radial spectrum (with the colours crowded together and mostly 
overlapping). Coma gives an unsymmetrical side distribution of 
light (see Fig. 188) ; while astigmatism gives a radial or tangential 
line. All ordinary telescope objectives show astigmatism, and if 

17 (5494) 



248 APPLIED OPTICS 

more or less coma happens to be present also, the appearances are 
very complex ; Fig. i88(c) shows a typical appearance. This inherent 
astigmatism does not appear on the axis. If the lens suffers slightly 
from coma, then the smallest error of squaring-on will cause the 
central image to show a coma effect. 

Other Test Objects. Numerous test objects are employed by 
practical workers in addition to, or even to the exclusion of, the 
"star" test, especially in microscopy ; reference should be made to the 
various textbooks. The " star " test for experienced workers yields 
very satisfactory results, but should be supplemented by a test of an- 
other type. The fact that at one focus the " star " image suffering 
from spherical aberration approximates very closely in size and 
general appearance to the Airy disc means that any image, even in 
the presence of this aberration, will show more or less sharp detail. 
Hence, mere sharpness of detail is not a sufficient indication of a 
well-corrected system. The supplementary test is one for the con- 
trast of the image, and the object should be one with sharp demarca- 
tion between black and white. For the microscope this is repre- 
sented by Abbe's test plate, consisting of transparent rulings in 
an opaque silver film on the underside of a cover glass. 

It is easily possible to interpret the extra-focal appearances on the 
image so as to inter the defects of the objective, but less simply 
than with the " star " test. 

For the telescope, contrast objects are numerous; the twigs of 
trees ; towers against the sky, and so on, arc easily found. For other 
test objects see Johnson's Practical Optics? page 129. 

The Measurement of Aberrations. Aberrations may be expressed 
in terms of the geometrical ray paths, or in terms of phase rela- 
tions of vibratory disturbances in the image ; it is usually possible 
to calculate the one expression from the other. Likewise, the 
methods of measurement can be divided into two general classes ; 
one which aims at tracking ray paths by various means, the other 
which gives a direct indication of the optical phase aberrations in 
an interferometer pattern. Broadly speaking, the first class involves 
little apparatus, but much time and trouble; the second involves 
expensive apparatus, but comparatively little time. 

Visual Ray-path Method. The "ray" is a mathematical concep- 
tion and cannot be physically realized. If we place a diaphragm 
containing a small aperture of finite size in the path of a convergent 
wave-front, the maximum concentration of energy will be for prac- 
tical purposes in the centre of curvature of the element thus exposed. 
If the aperture has symmetry with respect to some point in its 
plane, the distribution of light in the convergent beam will show 



TESTING OF OPTICAL INSTRUMENTS 249 

a corresponding symmetry with respect to the line joining this 
point to the centre of curvature; the centre of symmetry of the 
diffraction pattern in the beam may thus be conceived as a ray 
track. 

The most direct method of procedure is to arrange a diaphragm 
behind the lens which forms the image of an artificial " star." The 
distribution of light in any plane can then be examined with the 
aid of an eyepiece. (Fig. 189 (a) and 189 (ft).) 

A useful method of finding the foci of various zones is suggested 
by C. Beck, who uses a pair of apertures disposed in the diaphragm, 
as shown in Fig. 189 (a). The position of the focus of the corre- 
sponding zone is recognizable by the cruciform symmetry of the 




w 

FIG. 189. DIRECT OBSERVATION OF ZONAL Focus 

double diffraction pattern. With the aid of a series of diaphragms, 
it is thus possible to find the focusing points of a series of zones, 
or for a series of different wave-lengths of light, and hence to investi- 
gate the spherical and chromatic aberrations of the system, although 
it would be necessary in investigating an objective alone carefully 
to allow for the corresponding aberrations of the system of the eye- 
piece and eye, by direct observations on very small "mercury 
globule" stars supported in the focal plane of the eyepiece. 

The difficulty attaching to this and allied methods is, however, 
very considerable in practice, and seems to arise through the 
irregularities of the refracting media of the eye. The cornea is 
traversed, in this test, by two very narrow beams which can easily 
suffer considerable deflection by a minor irregularity which would 
make little or no difference to a broad beam. For this reason it is 
usually difficult to estimate the correct focus of a particular zone ; 
probably some persons would find the experiment much easier than 
others would. 

The Foucauli Test. The Foucault test 4 was described in Vol. I, 
page 257, but some notes will be included here to make the chapter 
complete; it has the merit that it avoids the criterion of a judgment 



250 



APPLIED OPTICS 



of position, using rather an estimation of equality of intensity of 
light ; the setting is photometric. Originally applied to test astro- 
nomical mirrors, and still mainly employed for that purpose, it can 
also be applied to "refractors" with useful results. 5 

A small source of light s, conveniently a pinhole backed by a 
flame or opal bulb lamp, sends light to the mirror (Fig. 190), which 
forms a corresponding image s'; the eye is placed immediately 
behind this image, and sees the mirror under the condition of the 
Maxwellian view in which it is filled with light, provided that all 
rays from the mirror enter the pupil unobstructed* 

If a knife-edge k^ is brought upwards into the focus, the whole 
mirror darkens uniformly if the light from all zones is focused in 
the same point ; but if spherical aberration is present, then charac- 
teristic distributions of light and shade appear on the surface of the 




(6) 
FIG. 190. ZONAL DIAPHRAGM FOR FOUCAULT TEST 

mirror. The shadows naturally occur on those regions the rays 
from which are cut off by the knife-edge. The necessity of conducting 
the mirror test with image and object point away from the axis 
makes the characteristic shadow figures unsymmetrical with the 
reflector, but symmetrical figures are obtained when testing refrac- 
tors forming really axial images. 

Experienced workers recognize the figures characteristic of para- 
bolic, elliptical, hyperbolic, and other forms of reflecting surface, 
but the test is made quantitative by measuring the focus for suc- 
cessive zones by the use of diaphragms, such as shown in Fig. 190. 
It is clear from the figure that when the knife-edge intersects the 
beams either nearer or farther than the focus, the two patches 
exposed by the diaphragm will not darken simultaneously. Con- 
siderable precision is attained by this method if care is taken with 
the precautions usual in photometric work. 

If the pinhole and the knife-edge are moved on the same carrier, 
the pointer attached thereto may be made to read the relative 
positions of the centres of curvature of successive zones ; but if the 
knife-edge alone is moved while the pinhole remains stationary, 
the separation of the images will be clearly double that of the 



TESTING OF OPTICAL INSTRUMENTS 251 

corresponding zonal centres of curvature. Thus, if the radius of the 
zone isy, and the approximate radius of curvature is r t the centres of 

y* 

curvature of the paraxial and zonal foci are separated by for a 
parabolic mirror. 

The Foucault test is not only used for testing the main mirrors, 
but also the secondary mirrors of Cassegranian and Gregorian 
reflecting telescopes. According to Hindle (Mon. Notices, R.A.S, 
XCI (1931) 592), it is best carried out for the secondary mirrors 
independently with the aid of an auxiliary spherical mirror which 
has itself been tested alone. In this method one focus of the hyper- 
bolic or elliptical mirror will lie at the centre of the spherical mirror, 
the other focus being at the point of observation. 

Of the other tests adapted for visual use, those of the shadow 
fringe methods developed by Ronchi and Jentsch (see below) may 
be mentioned, but if actual measurements are desired they may 
best be made with the help of photographic recording. The same 
applied to the general methods suggested by Chalmers, which have 
been revived in a more practical guise by Gardner and Bennett. 
We shall now proceed to describe several methods involving photo- 
graphic recording. 

The Hartmann Test. 6 The objective under test is directed to a 
suitable star or artificial star, and a diaphragm placed behind it 
has circular apertures, A 1 // 2 , representing a particular zone of 
the lens. A photographic plate-holder is arranged so that photo- 
graphic plates can be exposed when held perpendicular to the axis 
in two positions, within and without the focus, at a measured distance 
apart. The light from each aperture forms a small diffraction patch 
on the plate, the centre of which may be regarded as representing 
the track of the ray from the centre of the aperture. A suitable 
diameter for the holes is about ^ooth of the focal length. 

If A x and A 2 are the plate-holder scale readings corresponding to 
two plate positions, and l lt 1% arc the distance of the centres of the 
two dots on each plate, then the scale reading A corresponding to 
the focus of the zone is (see Fig. 191) 



The possibility of cylindrical errors of the surfaces has to be 
remembered however. It follows from the theory of Vol. I, page 
280, the astigmatic difference of focus between the diameter of 
minimum focal length and the diameter at an angle of 9 with the 
first will be given by 

A0 = A m , n , mum + a sin 2 



252 APPLIED OPTICS 

where a is the " astigmatism." Then 



and the mean 

Afl + A0+ 9 o 

- 



We may, therefore, eliminate irregularities due to astigmatism, 
when taking results from apertures in different diameters, by always 
taking observations for one zone in two perpendicular diameters. 




FIG. 191. THE HARTMANN TEST 

There is thus no reason to limit the diaphragm to two holes. A 
specification for the arrangement of apertures in the diaphragm for 
a typical case is as follows 



Objective 80 mm. Diameter 
Angles (0) 



i metre Focal Length 
Radii of Zones containing 
Apertures (in mm.) 






90 


10, 18, 26, 


38 


22-5 


112-5 


30 




45 


135 


6, 14,22, 


34 


67-5 


157-5 


30 





The diameter of the holes are 4 mm. in the outer, and 3 mm. in the 
inner zones. 

It is well to add a single hole in the diaphragm, so that the proper 
orientation of the plates may be recognized after development. 
The numerical part of the work lies in the systematic measurement 
of the values of /j and 1 2 for the various zones and the calculations 
of the corresponding focusing points. 

If the diaphragm itself is not too large it can be measured up on 
the measuring microscope, and one photograph may thus be dis- 
pensed with if the distance from diaphragm to plate is known, and 
also whether the plate is taken inside or outside the focus. 



TESTING OF OPTICAL INSTRUMENTS 253 

The main difficulty experienced in this test is the distortion of 
the dots arising from interference effects between the light from 
successive dots. This effect sets a limit to the reduction of the dis- 
tances between the adjacent holes in the diaphragm, and also makes 
it imperative to take the negatives at a distance from the focus 
sufficient to avoid marked interferences. 

With ordinary aperture ratios it is not difficult in this way to 
ascertain the relative differences between the foci of successive 
zones to within a few tenths of a millimetre. The way in which the 
geometrical aberrations so found can be interpreted in terms of 
phase differences of disturbances at the focus was shortly explained 
in Vol. I, pages 119 and 120. A full discussion of the application of 
the test to a microscope objective will be found in a paper by the 
writer. 7 Kingslake 8 has pointed out that the assumption made in 
the above theory that all rays cross the optical axis is not justified, 
and he prefers to obtain a three-dimensional trace of the rays from 
a large refractor. The co-ordinates for any ray in any chosen focal 
plane can thence be calculated. 

Hartmann has suggested a criterion for the comparison of astro- 
nomical objectives; it is a magnitude expressing the "weighted 
mean diameter " (in hundred thousandths of the focal length) of 
the cones of light from the various zones in the plane where the 
circle of light containing all of the converging pencils is the smallest. 
Weights are given according to the light gathering power of the 
zones. 

Since the area of a zone is 27irdr, the weight is proportional to 
the radius and the criterion (denoted by T) is 

io 5 Zrd 
T = T ZV 

where d is the diameter of the circle of light from a zone of radius r. 

Hartmann's original papers gave particulars of similar methods 
for the measurement of oblique aberrations. The subject has also 
been developed by Kingslake, 9 who bases the method of the deter- 
mination of the foci for pairs of rays derived from the extremities 
of chords of a zone drawn parallel to the axis of tilt of the lens, 
using formulae developed by Conrady. 10 The method has proved 
very convenient in practice; reference should be made to the 
original paper for particulars. 

Interference Methods Fizeau's Experiment. 11 Fizeau placed two 
parallel slits behind an object glass forming the image of a star. 
If the distance of the slits is d, Fig. 192, and the distance from slits 
to focus is/', the appearance found in the focal plane is a series of 



254 APPLIED OPTICS 

interference fringes of which the successive maxima are separated 
by the distance -^-. The angular subtense, at the back nodal point 
of the lens, of the distance from a maximum to the neighbouring 
minimum is therefore If the telescope is directed towards a 

double star with components at this angular separation, the maxi- 
mum of one fringe system will coincide with the minimum of the 




FIG. 192. FIZEAUS' EXPERIMENT 

other system, and the pattern will become a uniform band if the 
two systems are of equal brightness. This is the basis on which was 
developed Michelson's method of measuring the angular diameters 
of stars, and a number of methods of measuring the diameters of 
ultra microscopic particles developed by Gerhardt and others. 12 



D 
h 





6 



FIG. 193. THEORY OF CHALMERS' TEST 

Chalmers 9 Tests. If the slits in Fizeau's experiment are no longer 
symmetrically disposed with respect to the axis, the central fringe 
of the interference system will still fall in the axial focal point if 
the lens is free from aberration, because the disturbances from all 
parts of the wave-front will reach this point in the same phase. 
But if spherical aberration is present, the central fringe will be, in 
general, displaced away from the axis to some position where the 



TESTING OF OPTICAL INSTRUMENTS 255 

phase agrees. In Fig. 193 let A and B be the two apertures ; let C 
be the axial focal point, and let D be the position of the central 
"maximum" of zero path difference. 

In the actual use of the lens, the image will be formed at the 
point C, and we are, therefore, concerned to write down the phase 
difference with which the disturbances from A and B arrive in the 
point C. 

From the point D drop a perpendicular DE on the line AC. The 

angles ACG and EDC are equal, and will be written a v Now 

AC = AE + EC 
- AD + EC 

within a small quantity of the second order if AG and CD are small 
in comparison with GC. 
Writing h = CD 

we thus obtain AC, - AD = h sin a^ 

and, similarly, BC - BD = h sin a 2 

Now (AC - AD) and (BC - BD) represent the net changes of 
optical path of the disturbances from A and B when meeting at 
C, rather than at D where we know that they have the same phase. 
Hence the path difference at C is 

Optical path difference 

h (sin aj - sin Oj) j- f (\\ - v 2 ), very nearly. 

By starting with one aperture on the axis and one at a distance 
y l9 we may, therefore, find the optical path difference for the first 
zone with respect to the paraxial zone. Now moving one step 
outwards, so that the inner aperture falls into the position previ- 
ously occupied by the first one, and so on, we may find by addition 
the optical path differences for a number of successive zones with 
respect to the paraxial zone. Delicate measurements with an eye- 
piece micrometer are involved in the practical application of this 
test, which is not particularly easy to carry out quantitatively, 
although it makes a ready means of testing an objective by visual 
estimation ; for this purpose the slits may be made in a diaphragm 
held in front of the objective, while the appearances in the focal 
plane are watched with a high power eyepiece. 

Gardner and Bennett's Method. 13 This is essentially an exten- 
sion of Chalmers' method in which a photographic record of the 
fringe positions is obtained for a number of zones simultaneously ; 
it is taken away from the focal plane, thus resembling the Hartmann 



256 



APPLIED OPTICS 



test ; in this case, however, the interference effects which spoil the 
accuracy of the Hartmann record are turned to good use. 

In Fig. 194, let A and B be two apertures ; C is the focus of the 
lens for the central zones, and D the displaced position of the cen- 
tral fringe from A and B. If we take a record inside the focal 
plane, the fringe will be at E' on the straight line joining D to the 
mid-point between A and B, whereas it would have been at E on 




FIG. 194. THEORY OF GARDNER-BENNETT TEST 

the corresponding line to C if the aberration were zero. If EE' is 
h and the distance from diaphragm to the recording plate is /', 

then = , 

If we measure the distance co between successive maxima in the 
fringe system due to A and B, the interval will be 



0) 



But in the theory of Chalmers' method we found that the optical 
path difference (O.P.D.) for disturbances arriving at C is 



or, 



path difference in wave-lengths = 



hi 

CO 



In practice, a diaphragm is prepared with a number of equidistant 
holes arranged symmetrically with the axis, and the photograph is 
taken sufficiently far inside or outside the focus to obtain satis- 
factory interference fringes in the pattern. The width of the fringes 



TESTING OF OPTICAL INSTRUMENTS 257 

is measured, and also (for each pair of holes) the distance of the 
centre fringe from the axis, i.e. E'H in the diagram. Then 

FF' F'H (/*- 
EE =EH 



So that the optical path differences can be obtained for any assumed 
focus C. 

It is to be noticed that the "focus" is not an exactly ascertain- 
able point. We can select a likely point, and then find the optical 
path differences of the disturbances meeting there ; a good criterion 
for the performance of a well-corrected system is the residual path 
difference between any two zones, at the focus where these residuals 
are numerically least. These remarks apply also to the ordinary 
form of the Hartmann test described above. 

The Shadow-fringe Methods. These methods have been developed 
by Ronchi 14 , 15 and Jentsch, 16 who employ gratings of various 
descriptions, chiefly straight line gratings. Ronchi chiefly recom- 
mends rulings with 10-20 lines per mm., while Jentsch uses com- 
mercial process screens with 4-8 lines per mm. The gratings are 
placed in the convergent beam from the test lens which forms an 
image of an artificial star. If the focus is a perfect one, the projec- 
tion pattern of the grating has one point, i.e. the focus, as perspec- 
tive centre, and the shadow pattern in any plane is an undistorted 
series of dark and light bars ; but if the lens suffers from spherical 
aberration, then the more oblique rays have a different centre, and* 
distortion results in the shadow pattern. Ronchi develops the 
theory on the basis of the interference of light, which is a more 
suitable method for dealing with the effects under the conditions 
he uses. A brief geometrical treatment due to Jentsch may be 
reproduced here. 

Refer to Fig. 195. A ray diverges from the point B at an angle 
u with the Z axis of co-ordinates, and projects the image of the 
point P in the grating plane as P' in the projection plane. The 
origin of co-ordinates O is the paraxial focus, and the intercept 
OB = s is the axial spherical aberration associated with the angle 
u. The co-ordinates of the points P and P' are x, y, g, and f , 17, p 9 
respectively. 

From the geometry of the figure, 

i3_if 

x y g-s 

A 4. V^T? 

and tan u = - - - 

p-s 



2 5 8 



APPLIED OPTICS 



In the simplest type of spherical aberration 

s = A tan 2 u . . . , say, sufficiently nearly. 
By elimination of s and tan u in the above equations we obtain 

A (| - *) 8 (! 2 + if) = fe* - P*)(P - g)*P 
If the aberration is zero, A = o, and the equation reduces to 



So that if the grating is rectilinear with parallel bars, then the 
projection will show a series of parallel bars. 



Plane of 
Projection 




FIG. 195. THEORY OF JENTSCH'S TEST 

Again, when A is finite we may write x = N<5, where 6 is the 
grating interval, and N is a constant which may be assigned a 
succession of positive and negative integral values ; thus giving the 
curves of the various shadows in the projection plane. 

For a central bar, x = o, and the equation splits into two parts. 

i.e. | = o (a straight line), and 

2 I 2 o it \o 



this latter curve is a circle, but it can only be real if g and A have 
the same sign, i.e. it will only appear on one side of the focus. 
Fig. 196 shows typical shadow fringes inside and outside the focus 
of a lens suffering from aberration. 

For more exact quantitative measurements, Jentsch advises 
photographic recording of the shadow fringes and measurement of 
the plates along the X direction; the calculation will be carried 
through in ways similar to those already described. 




TESTING OF OPTICAL INSTRUMENTS 259 

Interferometer Tests on Optical Apparatus. The interference of 
two beams reflected from the surfaces of a thin film was discussed 
in Vol. I, page 184, et seq. It was shown that if the faces of the 
film are parallel, and the angle of reflection inside the film is *', 
then the relative optical retardation of the second disturbance is 
2rit cos i', where n f is the refractive index of the medium of the 
film. Remembering the phase change of TT consequent on reflec- 
tion at one face (that which would involve 
a step from a lower to a higher refractive 
index for the transmitted light), the con- 
dition for destructive interference is 

2n't cos i' = mh 

where m is o or an integer. 

When the faces of the "film" are truly 
parallel, interference effects can be seen at 
separations much greater than those char- 
acteristic of the soap bubble, in fact for 
quite large separations if light of good 
spectral homogeneity is employed. The 
limits of optical path difference up to which FlG - 196. JENTSCH'S 
interference effects can be observed are not GRID METHOD 

certain, but Michelson's experiments showed ' "^'cSST frmges uts ' de 
that they are not less than half a metre. 3 an t d he 4 cS w fringcs msuie 

One very useful method of testing The g^SHto?^^^ the 
employs the so-called Haidinger's fringes. 

Referring to Fig. 197, we see that the thin glass plate reflects 
the green light from a mercury vapour lamp (preferably a low 
pressure glass lamp) downwards on a piece of glass G with 
plane parallel faces. The parallel components of the reflected light 
are brought to a focus on the retina of the eye accommodated for 
infinity, and the condition for destructive interference is still given 
by the above equation. The optical path difference for the distur- 
bances following the exactly normal path is 2n't, and it is clear 
that cos i' will be constant for any circle surrounding the normal. 
The fringes therefore appear as circles with the normal as centre. 

If the plate G is moved about and the thickness is slightly irregu- 
lar, we may find a region in which t increases. Now one definite 
fringe corresponds to a fixed amount of retardation. It will, there- 
fore, move to a place where cos i 9 diminishes in order that 2n't cos i' 
may remain constant. This means that i' must increase, and hence 
the fringes move outwards from the centre, fresh fringes developing 
at the central point and expanding outward; the development of 



26o 



APPLIED OPTICS 



one fringe corresponds to a variation of one wave-length in zn't. 
The reverse effect takes place if the thickness diminishes. Hence 
by systematic movements of tjhe plate it is possible to examine the 
variation of thickness over the whole area. 

It may be mentioned that the interferences arising in a very 
similar way at two parallel surfaces are found and examined in the 
Fabry-Perot interferometer, but in this case two half-silvered glass 
surfaces held exactly parallel separate a film of air, and the circular 
fringes are viewed by transmitted light. This system has been used 



Eye 




Mercury 
Lamp 



FIG. 197. PRODUCTION OF HAIDINGER'S FRINGES 

by Barnard to test the accuracy of the slides of a microscope. One 
plate is mounted on the stage of the instrument, the other in place 
of the objective; the fringes remain central while the plates are 
truly parallel. 

Fizeau's Apparatus. Fizeau used a device by which the angle 
of incidence of the light is constant over the reflecting surfaces, but 
a lens is then necessary in order to bring the light from the different 
parts into the eye. The optical parts of an arrangement similar to 
Fizeau's are shown in Fig. 198, which is adapted for sensitive inter- 
ferometric tests of an optical surface. ^ 

The glass plate S has a truly plane lower surface and is supported 
by a stand with levelling screws, so that this surface can be brought 
closely parallel to the upper surface of plate T, which is to be tested 
for its "figure." The mercury lamp illuminates the pinhole H, 
which gives a diverging beam reflected downwards by the plate P. 
The light is then rendered parallel by the lens L. The components 
reflected from the adjacent glass surfaces of S and T show inter- 
ferences, the geometrical path difference at any point being twice 



TESTING OF OPTICAL INSTRUMENTS 



261 



the thickness of the air film, since the incidence is everywhere 
normal. The light on its return journey is brought to a focus in 
the eye, so that a Maxwellian view of the interference field is 
obtained showing a distribution of fringes which represent, in fact, 
a contour map of optical thickness. 

If the plate S is slightly prismatic, reflected light from its upper 
face may be avoided; the lower surface of T may be blacked or 
greased. 

In cruder applications of the test the surfaces S and T may be 




FIG. 198. FIZEAU TYPE INTERFEROSCOPE 

carefully cleaned and then slid or wrung together in contact. Any 
irregularities of the test surface are then again revealed by the 
variation in thickness of the separating film. The disadvantage in 
this method lies in the possibility of the deformation of one or both 
plates. 

The Michelson Interferometer. The optical system of the 
Michelson interferometer is analogous to the arrangement for the 
production of Haidinger's fringes. It is illustrated in Fig. 199. 
Light from an extended source meets the mirror M 8 , and is half 
transmitted and half reflected by the lightly silvered back surface. 
The components proceed to the mirrors M 2 and M x respectively, 
from which they retrace their path, combining again at the surface 
of M 3 . The eye placed in the position E can view the interferences 
with or without the aid of a telescope ; the formation of two inter- 
fering components in an oblique direction is very closely analogous 
to that of Fig. 197 (Haidinger's Fringes), but there is the immense 
advantage that one surface does not "get in the way" of the other. 



262 



APPLIED OPTICS 



The system is equivalent to a plate formed by M lf and the image 
of M 2 in M 3 . 

The interferences are ascribed to the variation of relative retarda- 
tion with the obliquity of the light, and the fringes are grouped 
around the normal as before. The component reflected from M x 
has a double transmission through the plate M 3 ; this is compensated 
for the M 2 component by the insertion of the plate P equal in thick- 
ness to M 3 ; the fringes are then circular. 

Interferences are only visible with white light in very thin films ; 



Extended ^ 
Light Source 





Interference 
fringes round 
in Focal Plane 



FIG. 199. MICHELSON INTERFEROMETER 

the colours in Newton's rings soon fade out as the thickness of the 
film increases, owing to the superposition of the systems of different 
sizes in different wave-lengths. Equally, the fringes are only found 
with white light in the Michelson interferometer when the "equiva- 
lent plate " is very thin ; i.e. when the effective lengths of the " arms " 
of the apparatus are very nearly equal, but with spectrally homo- 
geneous light, interference takes place up to very large path 
differences. 

The Prism Testing Interferometer. As the Michelson interfer- 
ometer is analogous to the Haidinger fringe apparatus, so the lens 
and prism testing interferometer, the earliest form of which was 
patented by Twyman and Green in igiG, 17 is analogous to the Fizeau 
arrangement. A simple form of the apparatus is shown in Fig. 200. 
As in the Fizeau apparatus, the beams are brought to a focus by a 
lens which gives a Maxwellian view of the interference field; this 



TESTING OF OPTICAL INSTRUMENTS 



263 



is in effect a contour map of the optical thickness of the " equivalent 
plate" of the system. 

We may obtain a simple view of the action by considering a 
point source and a perfect lens giving two wave-trains of plane 
waves which unite after passing through the system (Fig. 201). 
If their directions are inclined at a small angle a and the wave-length 
is A, it can easily be seen that a steady interference system of 
maxima and minima distributed in planes equally inclined to each 



Pin-hole Light 
Source 






Objective ^ 
Interferences 
in this Region 



FIG. 200. TWYMAN INTERFEROMETER 

group of waves will result. The distance between two "maximum" 
planes is strictly A cosec a cos ( - ), but for all practically interesting 

cases this becomes - since a is always very small. The objective 
a 

existence of these maxima and minima in the region of super- 
position can easily be demonstrated with the aid of a piece of ground 
glass on which the fringes can be found, or by exposing a photo- 
graphic plate shielded from extraneous light. When the value of 
a is zero, a single dark or light fringe broadens so as to fill the 
whole field. 

Imagine now a very slightly wedge-shaped piece of glass to be 
placed in the path of the M a beam, the thicknesses at the two ends 
of the plate being ^ and t 2 respectively. The increase of optical 
path owing to the introduction of a thickness t of a medium of 
refractive index n into a space previously filled with air is (n - i)t. 

18 (5494) 



264 APPLIED OPTICS 

Hence the relative path difference between disturbances passing 
through the extremities of the plate is (n-i)(t l -t 2 ). The double 
transmission in the interferometer will make the actual optical path 
difference 2 (n - 1)(^ - ^), and the number of resulting fringes will 

be 2 ( n ~ I )(*i~ f 2) t Note now that the same plate tested by the 

2ffc It 1^\ 

Fizeau method would show ~ fringes, so that if n = 1-5 

U^ A * 

we shall only find one-third the number of Fizeau fringes when 



(1) Combined Crests with Crests (maximum) 
&) Crests with Troughs (minimum) 




Y~ ^Minimum 
\/ 
J (f) ] Maximum 



The slanting full lines are "Crests" 
i> broken n "Troughs 9 ' 

FIG. 201. THE SUPERPOSITION OF PLANE WAVE-TRAINS 

using the interferometer; though it is less sensitive for this par- 
ticular purpose, there are other advantages which more than 
compensate. 

If, for example, a prism of considerable angle is placed in one 
arm, the mirror can be rotated so that it is normal to the incident 
light, which therefore retraces its path. The optical path differences 
along the various ray routes will now be zero if the surfaces of the 
prism are uniform and the glass is homogeneous ; but local irregu- 
larities in the surfaces, or variations of refractive index of the speci- 
men, will cause variations which are seen in the contour map of 
optical thickness. 

Slight pressure on the iron bed of the instrument on which the 
mirrors are mounted can be applied so as to elongate or diminish 
the path in one arm. The fringes will therefore (moving so as to 
retain the same retardation) proceed to places of lesser or greater 
path in the specimen, thus giving a criterion of the sign of the 
irregularity. 

Suppose, for example, that a prism for use in a spectroscope has 



TESTING OF OPTICAL INSTRUMENTS 265 

a local increase of refractive index in one region ; this part may be 
marked out on the surface of the prism by a brush or grease pencil, 
and subjected to local rubbing by a chamois leather pad charged 
with rouge. In this way sufficient glass may be removed to com- 
pensate for the local variation of index and perfect the interference 
pattern. In such ways as this it is possible greatly to improve the 
action of prisms and optical parts, although the compensation will 
only be exact for one direction of the light, and nothing can wholly 
compensate for any lack of homogeneity in the glass. 

The method of testing lenses for axial aberrations will be under- 
stood from Fig. 202. The convex mirror, which must be of reason- 
ably perfect figure (correct to within A/8), may be so disposed that its 
centre of curvature coincides with the focus of the lens. This system 
now replaces the plane mirror M 2 of Fig. 200. With a perfect lens 




i 




Mirror 



F' 



FIG. 202 

the "rays" will all fall normally on the surface of the mirror, and 
will return along their own paths; consequently, no difference of 
direction can arise for any part of the wave-front, and a uniform 
interference pattern will be obtained. 

Aberration Path Differences with Lens and Mirror System. Refer- 
ring still to Fig. 202, the lens is shown with principal focus F', 
and the mirrors with centre of curvature C. These will not in 
general coincide at first, although the apparatus will be so con- 
structed that C falls on the axis of the lens, and in cases of practical 
interest C will be very close to F', so that rays will return very 
nearly along their own paths. The wave-front has the radius 
(f'-g) on reaching the apex of the mirror, f being the vertex focal 
length of the lens, and g the separation of the adjacent apices of 
the lens and mirror. 

Assuming as the usual basis of Fermat's theorem (Vol. I, Chapter 
IV) that we can calculate optical paths along known ray routes, 
the extra geometrical path for the marginal over the paraxial ray 
will be represented by double the marginal gap between the circles 



266 APPLIED OPTICS 

centred in C and in F respectively ; that is, if the lens is free from 
aberration. 



Marginal path - axial path 



= *y* | - 



The optical path difference will be proportional to y*, and the 
frequency of the circular fringes will increase from the centre out- 
ward, but the central fringe will broaden out to fill the whole field 
when r = (f -g), i.e. when the mirror is drawn back so that C 
and F coincide. 

Measurement of Aberration. If the lens suffers from spherical 
aberration, the optical path difference for marginal and paraxial 
rays at the paraxial focus can be represented by a series (Vol. 
I, page 115). 

Marginal path - axial path = c 2 y* + c$y* + etc. 

Hence, in general, the optical path difference with lack of coinci- 
dence of the centre of the mirror with the paraxial focus, and in the 
presence of spherical aberration, will be represented by 

w 2 + c *y* + w* + etc - 

This equation may be discussed in the manner used in Vol. I, 
page 121, but it will suffice here to point out that if the focus for 
any particular local zone of the wave surface coincides with the 
centre of curvature of the mirror, there must necessarily be a local 
zonal maximum or minimum of optical path. Therefore, on apply- 
ing the test above (slightly lengthening one branch) the fringes 
will gather towards or spread from this zone in both radial direc- 
tions ; a fringe at such a zone will be evanescent if the other fringes 
crowd towards it. 18 This criterion affords a ready means of setting 
the radius of curvature of the mirror into coincidence with the foci 
of successive zones of a lens suffering from spherical aberration, 
and hence allows the axial aberration to be measured if the mirror 
is furnished with a suitable micrometer screw. At the same time, 
the optical fringe system is a "contour map" showing the optical 
path differences with which the disturbances from particular zones 
meet in the focus corresponding to the centre of curvature of the 
mirror. 

Sensitiveness. The Rayleigh limit of - for the allowable differ- 

4 

ences of path at the best focus for a telescope objective would, if 
present in a lens tested on the interferometer, give optical path 

differences of - owing to the double transmission. Taking a case 



TESTING OF OPTICAL INSTRUMENTS 267 

when this arises through "first order" spherical aberration, the 
fringe system might then be (say) at its brightest for a mid-zone 
where the phases of the re-united waves agree, and shade off to the 
minimum both at centre and margin where the path differences 
would reach half a wave-length. In practice, the difficulties in 
making the surfaces, especially the convex surface of the mirror, 

accurate within say ^ are considerable, and the above would repre- 
sent something like the lower limit of aberration which it is possible 
to test with any accuracy. The most serious defects of lenses, when 

the theoretical aberrations are reduced below about - or so, are 

generally faults of regularity, centring, etc., which can readily be 
detected on the interferometer. For this reason it is not usually 
possible to "measure" the axial aberration in the way suggested 
above when the error is small. 

Chromatic Aberration. When the centre of curvature of the mirror 
coincides with the principal focus of the lens for any zone, the 
combination of lens and mirror is clearly equivalent to a plane 
reflecting surface, at least in the zone considered. This consideration 
is the basis of the method of measuring the chromatic aberration 
on the interferometer; it is, however, advisable to employ a chro- 
matically corrected lens to collimate the light. With the aid of a 
hydrogen tube and a mercury lamp with a monochromator, or with 
suitable colour filters of the Wratten series, it is possible to set the 
mirror to the focus of the lens for a series of wave-lengths through- 
out the spectrum. 19 With a photographic lens it is possible to study 
the chromatic variations for one particular zone in observing the 
evanescence of the fringes, but with more highly corrected* lenses 
it will be necessary to study the pattern with the minimum number 
of fringes for the lens as a whole. 

Measurement of Oblique Aberrations. The earlier simple forms 
of the lens testing interferometer used for testing telescope objec- 
tives have been followed by other more complex arrangements by 
the aid of which the aberrations of other lenses, such as camera 
lenses and microscope objectives, may be ascertained. The camera 
lens interferometer 20 allows of numerical tests of the oblique aberra- 
tions, and Kingslake 21 has shown how the numerical coefficients of 
the Seidel aberrations may be deduced from an interferogram. 

Microscope Interferometer. Fig. 203 illustrates the principle of 
one form of the microscope interferometer. It will be seen that the 
plane mirror in one arm is replaced by a negative lens followed by 



268 



APPLIED OPTICS 



the microscope objective. The virtual image of the pinhole source 
formed by the negative lens must lie at the proper working point 
fixed by the appropriate tube length of the objective. The spherical 
mirror in this case is sometimes represented by a globule of mercury. 
These globules, when sufficiently small, become sufficiently spherical 





FIG. 203. INTERFEROMETER TEST ON A MICROSCOPE OBJECTIVE 

owing to the great surface tension pressure. Details will be found 
in a paper by Twyman. 22 

General Aspects of the Interferometer Tests. Although they re- 
quire expensive apparatus, the interferometer tests yield direct 
results with the least expenditure of time. General points in the 
theory of the instruments have been discussed by T. Smith. 28 In 
order to be able definitely to associate the faults of a system with 
their apparent places in the interference field, we should place 
the system and mirror as close together as possible, and view or 
photograph the field with a lens focused on the mirror. 

While the interferometer is at the service of the professional 
optician, the older methods will still be fruitful in the hands of 



TESTING OF OPTICAL INSTRUMENTS 269 

those who may only have occasional necessity for the accurate 
measurement of aberrations. 

REFERENCES 

1. H. D. Taylor: The Adjustment and Testing of Telescope Objectives 
(York: Messrs. Cooke, Troughton, and Simms). 

2. F. J. Cheshire: Trans. Opt. Soc., XXII (1920-21), 235. 

3. B. K. Johnson: Practical Optics (London: Benn Brothers). 

4. Foucault: Ann. de l'Obs. de Paris, V (1859), 197. 

5. H. G. Conrady: Trans. Opt. Soc., XXV (1924), 219. 

6. Hartmann: Zeit.f. Inst., XXIV (1904), i; and subsequent papers 
in 1904. 

7. Martin: Trans. Opt. Soc., XXIII (1921-22), 28. 

8. Kingslake: Trans. Opt. Soc. t XXIX (1927-28), 133. 

9. Kingslake: Trans. Opt. Soc. t XXVII (1925-26), 221. 

10. Conrady: Applied Optics and Optical Design (Oxford University 
Press). 

11. Fizeau: Compt. Rend., LXVI (1868), 934. 

12. A general review is given by Kiihne, Ann. d. Phys. 5 Folge, IV 
(1930), 215. 

13. Gardner and Bennett: Jour. Opt. Soc. America, XI (1925), 441. 

14. Ronchi: Ann. d. R. Scuola Normale Superiore di Pisa, Vol. XV 

(1923). 

15. Ronchi: Revue d'Optique, VII (1928), 49. 

16. Jentsch: Physikal. Zeitschr, XXIX (1928), 66. 

17. Twyman and Green: British Patent 103832 (1916). 

18. Perry: Trans. Opt. Soc., XXV (1923-24), 97. 

19. Martin and Kingslake: Trans. Opt. Soc. t XXV (1923-24), 213. 

20. British Patent 130224 (1918). 

21. Kingslake: Trans. Opt. Soc., XXVIII (1926-27), i. 

22. Twyman: Trans. Opt. Soc., XXIV (1922-23), 189. 

23. T. Smith: Trans. Opt. Soc., XXVIII (1926-27). 104. 



APPENDIX I 

OPTICAL CONVENTIONS AND EQUATIONS 

IT may be a convenience for some readers to have a concise state- 
ment of the sign conventions used in the present book, together 
with some of the elementary equations deduced in the early chapters 
of Vol. I. 

Symbols. Symbols relating to the image space are distinguished 
from those of the object space by the addition of a dash or accent, 
thus: /'. 

Refractive indices (object and image spaces) . . n t ri. 

Conjugate distances of object and image . . . Z, /' 
(measured from the principal points) 

Conjugate distances of object and image . . . x t x' 
(measured from focal points) 

Perpendicular heights of object and image . . h, h' 

Angles between a ray and the axis (object and image 
spaces) a, a' 

The reciprocals of distances are denoted by the cursive 

form of capitals, thus i/l and i/Z' . . . . c / and X 

The power of an optical system riff . . . ,/ 

Sign Convention. Distances 
measured to the right along the ruri 
axis are counted positive; those *2iJ 
to the left are negative. Similarly, ^> 

those measured upwards, perpen- - ^^ 
dicular to the axis, are positive, ^ 
and those measured downwards 
are negative. 

The angles at which the ray 
directions meet the axis are 
counted positive if a clockwise FlG> Al 

turn will bring a line from the axis direction to the ray direction 
by the lesser angular movement. 

The refractive indices are numerically positive when the direction 
of the light is from left to right, and numerically negative when the 
direction of the light is from right to left. The form of the equation 
remains the same for either direction. 

271 



272 APPLIED OPTICS 

Equations. 

(Paraxial equations.) 

The law of reflection 

*" = -* 
The law of refraction 

n' sin i' = n sin i 
Conjugate distance relation (single refracting surface) 



-- 7 = - -;or </'- c /=.^. 

The above can be converted to the relation for a single reflecting 
surface by putting fi' = - ^ and simplifying 
Conjugate distance relation (single reflecting surface) 



Magnification relation (single refracting surface) 



*' ~" A 
Smith-Helmholtz-Lagrange relation (for an optical system) 

#' # a' = wAa 
Conjugate distance relation (thin lens in air) 



Magnification relations (optical system) 

A j A f 
The "Newtonian equation" 
xx'=ff' 
Longitudinal magnification 

= (- Y - 

rf# "" \A/ n' 

Focal lengths of an optical system 

-7 = 77 = ^ss Power. 

Power of thick lens; surfaces of power J and J 
(thickness) + (refractive index) = 5 



APPENDIX I 273 

Power of a system of two thin lenses ; separation d \ and correspond- 
ing focal length equation 

i i i d 

/? * tf*S i '"V J ** * * i 

./'== ,/ a -f- ./fc-a./' ./fc; ft=-fr +7;" "~ 77F7- 

/ /a / 6 /a/ft 

Distance from first lens (of system of two thin lenses) to first 
principal point of system 



and corresponding distance from second lens to second principal 
point of system 

P 6 F = -fd/f a 



APPENDIX II 

THEORY OF THE DIFFRACTION GRATING 

THE simplest approach to the general theory is to consider the 
grating to consist of a series of parallel rectangular apertures. The 
investigation first deals then with the diffraction of a plane wave 
by a single rectangular aperture ; the discussion is then extended to 
the case of a series of apertures. 

1. Diffraction of a Plane Wave by a Rectangular Aperture. We 
may imagine a plane screen of indefinite extent with a rectangular 




FIG. A2 

aperture ABCD (Fig. As). A wave-train with its wave-fronts 
parallel to the screen produces vibrations in the plane of the aperture 
which are all in the same phase. It is now desired to find the relative 
illumination, due to the aperture, at various points in a second screen 
at an indefinitely great distance, or a distance so great that if N is 
a point on the normal through the centre O of the aperture, the 
difference between the distances AN and ON, or CN and ON, is not 
more than a negligible fraction of the wave-length of light. In this 
way the disturbances spreading from all the parts of the aperture 
and reaching the point N will arrive in sensibly the same phase, 

275 



276 



APPLIED OPTICS 



and their amplitudes will simply be added together to find the 
resultant. 

Effects of this class were discussed by Fraunhofer, and are usually 
known as Fraunhofer diffraction phenomena. As will be explained 
below, the effects on a screen at an "infinite distance" are similar 
to those found in the focal surface of a lens placed behind the 
aperture and focusing the light; hence the discussion has more 
than a theoretical interest. 

The second plane is also normal to the line ON ; let us consider a 

point P in this plane such that NOP = 0, say, and NP is parallel to 
GH, a diameter of the rectangle ABCD. It will be clear that the 




FIG 



distances from P of all points on a line, such as AD or BC, per- 
pendicular to GH, will still be sensibly equal, but the distances 
from P of various points on the line GH will differ considerably if 6 
is finite. The calculation of the effect due to the whole aperture may 
therefore be effected by dividing this aperture into a series of strips, 
such as EF, and imagining the whole effect of the strip as equivalent 
to a proportionately intense source of disturbance situated at the 
central point K of the strip. 

The problem can now be discussed with a two-dimensional 
diagram (Fig. A3), as if the rectangular aperture were a line source 
of light, giving rise to disturbances starting with the same phase. 

Let the sides of the rectangle be m = AB, and n^ = AD respec- 
tively ; the area is therefore m^n^ let the amplitude at N be written 
km^ since all disturbances arrive at N in the same phase. In seek- 
ing the effects at a point P l the relative phases of the elementary 
disturbances arriving there must be considered. The dotted line 
MON in Fig. AS, is perpendicular to OP. All optical paths from 
points on MON to the point P are considered equal ; hence it will 



APPENDIX II 277 

be clear that the disturbance derived from the point R at a distance 
m from the point O will have an extra path to travel as compared 
with the disturbance derived from O. This extra path is m sin 
(where OR = m) and the corresponding lag of phase is (2irm sin 0)/A. 
In Vol. I, page 81, it was shown that the amplitude A resultant 
from a number of successive contributions with various phases is 
given by 

A 2 = [Za sin <5} 2 + {Za cos <S} 2 

where a is the amplitude and 8 the phase angle of a single contribu- 
tion. In our case the area will be divided up into equal strips of 
width dm so that the contributions of successive elements will have 
equal amplitudes but differing phases. The area of a single strip is 
n^dm, so that the amplitude at P produced by it will be kn^dm where 
k has the same meaning as above. 
The summation will therefore be 

m -. 2S 2 

A 2 = Z (kn^dm) sin {(27rw sin 0)/A} 
LWI J 



[m i -.2 

Z 2 (kn^dm) cos {(2-rrm sin 0)/A} 
. M m i - 

/n = 

2 

Consider, however, the summation Za sin d characteristic of the first 
main term above. For one strip the component will be 

kn^dm sin {(zirm sin 0)/A}, 

but there will be another strip on the other side of O, at a distance 
- m from O which will have an equal phase angle, but negative, so 
that the two terms will cancel each other. On the other hand, such 
terms will add numerically in the cosine summation, since the cosine 
of a negative angle is equal to that of the equal positive angle. We 
thus see that the first bracket above reduces to zero, and we get 



A = 2 (kn^dm) cos {(mm sin 



m = * 

2 



/-" 
^ I 



dm cos (zirm sin 0)/A } 



278 APPLIED OPTICS 

r' 



sn {ZTrm sn 



m -- * 

2 



277 sin 



(277 sin 0)/A 

t. /277 W x \ . /277 Wj . \"| 

sm ( -T- . sin ] + sm I -T- . --^sin 
\A2 / \A2 /J 



knyh . /TTWi sin 0\ 

A == : 2 sm I i ) 

77 sm \ A / 
= ^w^ 



. 

/77W! sm 

v r~ 



sinU 



where K is the central intensity, and U = (77m x sin 6)/L A similar 
law must also hold for a direction from N taken parallel to the other 
side of the rectangle ; in the case of U = (TTW I sin 0)/A. 

As regards the phase of the resultant vibration, we can see by the 
ordinary graphical construction, Vol. I, page 80, that if we have. 
two equal components with phase angles of opposite sign but 
numerically equal, the resultant phase angle must be zero. In the 
above case, the resultant disturbance from the whole aperture has 
the phase of the component derived from the central point O. 

Effect of Two Parallel Rectangular Apertures. The result of the 
investigation for a single aperture must now be extended to the 
case of a number of apertures. Let us first of all consider the case 
of two equal parallel rectangular apertures in a screen on which a 
plane wave-train is incident. The width of each aperture is m l as 
before, and its height n v In a very similar manner, we may reduce 
the effect of each aperture to that of a line source in the plane of 
the diagram. We found above that the resultant of a single aperture 
has the phase of a disturbance starting from the mid-point. 

Let the distance between the central points of each aperture be x 
and let 6 be the angle of diffraction considered. In Fig. A4, the 
two apertures are A and B and the mid-point is C. Drawing a 
dotted line ECD through C perpendicular to the direction of diffrac- 
tion, it will be clear that the disturbance from B would arrive with 
a lag of phase (<$ say), while that from A would arrive with a numeri- 
cally equal lead of phase, as compared with an (imaginary) distance 
derived from C. If Aj is the amplitude due to a single aperture in 



APPENDIX II 



279 



the direction 0, then the resultant amplitude due to each of them 
will be 

A 2 = {A x sin 6 + A t sin (- d) } + {Aj cos d + A x cos (- <5) } 2 

= 4A 2 ! cos 2 6 
Now 5 is evidently given by 




FIG. A4 



so that. the resultant amplitude is given by 

/TT* sin 6\ 
I j 1 



A = 



cos 



But from above we had that 

A! = K(sin U)/U 

where U = (7rm l sin 0)/A, so that the complete expression for the 
amplitude is 

2K sin U fnx sin 

T 



A = 



U 



cos I - 



Hence the distribution due to a pair of very thin slits, which would 
be a set of interference fringes in which the amplitude follows the 
law A cos (nx sin 0/A), and the intensity follows the corresponding 
"cos 2 " law, will, in practice, when using slits of finite aperture, be 
modified by the (sin U)/U term. Students who are interested in 

19^(5494) I* PP- 



28o 



APPLIED OPTICS 



the study of diffraction should make the experiment and plot the 
curves for particular cases, as they are most instructive. The subject 
is treated more fully in textbooks of Physical Optics. 

Effect of a Series of Parallel Apertures. Let the width of each 
aperture be m l and the length x as above. Exactly as before, we 
can reduce the effect of each aperture to that of a line source in its 
small diameter shown in the plane of the diagram (Fig. AS). Let 
the common distance between the centres of the apertures be x, 





I FIG. AS 

then also, as before, the disturbances derived from each aperture 
will have the phase of one derived from the mid-point. 

Let C be the central aperture and A and B the first apertures on 
each side. It will be clear that the disturbance from B will have a 
lag, and that from A an equal lead in phase (6 say) considering the 
diffraction angle 6. For the p th aperture away from the centre the 
phase difference will be pd. Let A l be the amplitude due to a single 
aperture in the resultant taken for the angle 0, then the effect of all 
the apertures will be given by 



A 2 = Z {^sinpd + A! sin (-pd) } 



r 

A x 

L 



p = (N - 1)/2 

Z {A 1 cos/>(5 



cos - 



-.2 

} 

J 



for a total number of apertures N including the central one. It 
will be clear that the sine terms will mutually cancel each other 
when N is odd, leaving 



A = A! + 



cos pd 



APPENDIX II 281 

or putting in the value of d, i.e. (2ir - x sin 0)/A 

A = A! + 2Z A! cos{(27r/># sin 0)/A} 

The sum of the series may be found as in Vol. I, page 83; it 
proves to be given by 

sin{(7rN*sin0)/A} 
~~ l sin {TTX sin 0)/A } 

We can simplify the expression for the case when is small in 
the neighbourhood of the central maximum. It then becomes 

sin{(7rN#sin0)/A} 

A = A, 



(TTX sin 

sin{(7rN*sin0)/A} 
(TrNx sin 0)/A 

= (effect due to whole of apertures) {sin W)/W } 
where W = (irNx sin i 



This last expression will, however, become inaccurate, as be- 
comes greater. It is not so easy to see the way in which the ampli- 
tude varies from this expression as from the one above, i.e. 

/>-(N-l)/2 

A = A! 4 2% Aj cos pd 

/=! 

where 6 == (2irx sin 0/A). 

If we plot a succession of curves y l = cos d, y^ = cos 26, 
y 3 = cos 3<5, etc., and add the ordinates, we shall find that the sum 
of the ordinates keeps on increasing at d = o, d = 27r, d = 477, etc., 
but that at intermediate points the contributions of successive 
terms vary in sign and tend to cancel each other. With two terms 

A l + 2A l cos d 

(which represents three grating apertures) there will be a result 
shown in Fig. 71 of this volume; the main maxima are separated 
by one intermediate maximum. With three terms (representing 
five grating apertures) there will be three intermediate maxima. 

The student should draw the amplitude curve, and then the curve 
given by squaring the amplitudes, thus obtaining an intensity 
curve with all positive ordinates. 



282 



APPLIED OPTICS 



We find that when the number of terms is indefinitely large (i.e. 
when there are a large number of diffracting apertures, the sum of 
the amplitudes only has a finite value when d in the above expres- 
sion is zero or some positive or negative multiple of 27r, i.e. when 

x sin = r%, 
where r is an integer or zero. 



Tt, 




FIG. A6 

We may note that in Fig. A4, the amount of x sin would be 
EA + BD, or simply the path difference between the disturbances 
derived from corresponding points of two adjacent apertures. If 
this is zero, or corresponds to one or more whole wave-lengths, we 
get a diffraction maximum. The more exact expression for the 
sum at the foot of page 280 gives 

A = A! (sin (N(5/2}/sin (5/2) 

We found that A lf the effect of a single aperture, was given by 
(K sin U)/U so that the complete expression for the amplitude is 



sm 



A = constant x 



/TTWj sin 

\ r~ 

irm l sin 6 
I 



\ . /wN*sin0\ 

j -(IT- ) 

(irx sin 0\ " 
-T-) 



APPENDIX II 



283 



Wave Incident at Another Angle. We can easily see how the 
above expressions will be modified to deal with the case in which 
the plane wave-fronts incident at the apertures are not parallel to 
the screen, but are inclined with the normal in the plane of the 
diagram (which is supposed to be perpendicular to the screen) and 
when there is a change of refractive index at the plane of the screen. 

We can show, as before, that the effect of the whole aperture will 
be equivalent to that of a line source at the middle of the aperture. 
This centre line can be shown in the diagram (Fig, A6). Let C be 
its central point, and let F be a point at a distance m from the centre. 



W 



W 




FIG. A- 



The plane waves pass up to the aperture at an angle 6 with the 
screen ; we desire to find the diffraction effect in a direction making 
an angle 6 L with the normal beyond the aperture. Draw the per- 
pendicular CE and FD to the "incident" and "diffracted" paths. 

The disturbances passing through F have a shorter optical path 
than those through C, the difference being 

ri CD - n EF = n'm sin O l - nm sin , 

but for those passing through F', the symmetrical point on the other 
side of C, there will be an equal and opposite path difference. 
Hence the term sin in the above expressions will be replaced by 
n' sin QI - n sin 6 . In the case of the diffraction grating, for example, 
the condition for a bright maximum will be 

x(n' sin 6 - n sin O n ) = ph 

where p is zero or an integer. 

Actual Gratings. Actual gratings consist in practice of rulings 
made by a diamond point in the reflecting or transmitting surface. 

I9A (5494) 



284 APPLIED OPTICS 

"Gratings" or test rulings made as objects for testing the resolving 
power of a microscope are usually on a transparent surface. Diffrac- 
tion gratings with high dispersion are usually ruled on a reflecting 
surface, but a cast of the grating can be taken in a thin film of 
celluloid and mounted on a plate of glass. The general subject may 
be studied from the accounts given in Baly's Spectroscopy, Wood's 
Physical Optics, and other articles. 

It must suffice to say here that, although the theory of the grating 
given above concerns only a series of apertures in a thin opaque 
screen, the effect of any regular structure producing changes in the 
optical path of some disturbances comparable with the wave-length 
of light must be of a similar character. Fig. Ay represents a plane 
wave-front passing into such a thin film with a regular structure 
shown in section. The grating element has a spacing x. 

In any direction 6 with the normal to the face of the grating, any 
single element will produce a resultant amplitude a lt say, and 
having the phase of an elementary disturbance propagated from 
some point A, say, which will be exactly similarly placed for every 
grating element. The effect of the grating will therefore be that of 
a series of sources at A, A lf A 2 , etc., with a constant step of phase 
if the incident wave is oblique, or co-phasal if the wave-front is 
incident normally. 



APPENDIX III 



ASTIGMATISM OF A LENS SYSTEM 



IN Vol. I, page 302, we obtained the equation for the astigmatism 
produced by a lens system, in the form 



where I t and I, are the tangential and sagittal image points respec- 
tively, both situated on the principal ray through the centre of the 
stop, and O t O 8 are the corresponding points in the object. 




FIG. A8 

With the aid of this equation we can obtain a simple expression 
for the radii of curvature of the tangential and sagittal image fields, 
supposing them to be represented, near the axis of the system, 
sufficiently nearly by spherical surfaces. We must also refer to the 
general discussion of aberrations given on pages 131-139 of Vol. L 
The expressions on page 132 give the transverse displacements of a 
ray from Q in they and z directions, and it was shown on page 135 
that the tangential focus lies at a distance from the Petzval surface 
which is three times the distance of the sagittal focus from the same 
surface. In Fig. A8 let 1^,1,, and &'p represent points on the tan- 
gential, sagittal, and Petzval surf aces respectively, of radii R', R' f , 
and RV For small angles of slope we may calculate the intercepts 
by the spherometer formula. 

285 



286 APPLIED OPTICS 

Thus 1,1. = 2l 8 B' p = 2 (~ r ~ 1 

\2IV $ Z. 

_ / _i i_> 

" IT?' 1?' , 

\K* K s> 

Similarly O t O, = A 2 ( ^ - ^- ) 

\ K t K 8/ 



The above equation therefore takes the form 

i / i : 



Then 



'R', nR, 



from the equation 52 on page 139, Vol. I. This gives the radius of 
the sagittal image field near the axis. 

/A' 2 A' 2 \ 
Since also I t l s = $(I,B',) = ^ - k/ -J. 

we can obtain in a similar way 



(Q, - 

giving the radius of the tangential image field. The definitions oi 
Qt and Q t were 



where / and /' refer to the distance of the intermediate entrance and 
exit pupil for the surface. 



where / and t' refer to the distance of the intermediate object and 
image for the surface. 

The equations above are not satisfactory for very oblique rays 
with large values of astigmatic differences of focus, and the curva- 
tures so found only apply to the fields near the axis. It will be well 
for the student to revise the theory of a thin lens used with a stop, 
given in Vol. I, pages 302-304. It is clear from the above how the 
tangential image surface lies at three times the distance of the 
sagittal surface from the Petzval surface, the radius of the latter 
being determined by the first sum on the right of each equation. 



INDEX 



Abbe, 55, 87, 89, 99 

condenser, 91 

orthoscopic eyepiece, 45 

prism, 64 

stereo eyepiece, 160 

theory, 100, 102 

Albada, 17 

Aldis anastigmat, 186 
Amatar lenses, 181 
Amid, 75, 88, 89 
Amphipleura pellucida, 118 
Anastigmatic correction, 179 
Apochromatic objectives (micro- 
scope), 85 

(telescope), 27 

Arc lamps, 219 
Artificial star, 242 
Astigmatism, 285 

Astrographic telescope objective, 27, 
29 

Bacon, Roger, 18, 218 
Baker, Messrs. C., 159 

, T. y., 59, 74 

Barnard, 134, 136, 260 
Bechstein, 229 
Beck, C., 140, 249 

, Messrs. R. and]., 125, 135, 

136, 159, *6o, 161 
Bennett (and Gardner), 251, 255 
Bertrand lens, 139 
Binocular microscope, 150 

telescope, 154 

vision, 141 

Boegchold, 88 

Bonnani, 75 

Booth, 204 

Brewster, 10 

Brewster's stereoscope, 146 

Brightness, 207 

Briiche lens, 17 

Bull's-eye condenser, 95 

CANDLE power, 207 
Cardioid condenser, 122 
Cassegrain telescope, 70, 251 
Centring, 243 
Chalmers, 251, 254 
Chevalier, 16, 75 
Cheshire, F. >., 269 

, /?. W, 74 

Chiasma, 141 
Cinematograph projector, see 

Kinematograph 
Coddington, 10 



Colour correction tests, 244 
Compensating eyepieces, 87 
Complete flash, 233 
Concentric lens, 180 
Condenser (lantern), 226 
Condensers (substage), 90 
Conrady, A . E., 38, 74, 87, 106, 175, 

253 

, H. G., 195, 269 

Conventions and signs, 271 
Cooke photographic lenses, 185. 187 
telescope objective, 27, 28 



Cosine law, 207 
Court, 73 
Cross, 127 
C zap ski, 140 

DALLON lens, 204 

Dallmeyer, Messrs., 204 

Dark-ground illumination, 1 19 

Daubresse prism, 65 

Dawcs' rule, 38 

Depth of focus (microscope), 82 

(photographic lens), 194 

Dispositives, 230 

Diffraction grating theory. 275 

Diplopia , 142 

Dissecting stand, 16 

Distortion, 172 

Dollond, 25, 45 

Dove, 150 

Ediswan Electric Co., 220 
Emsley (and Swaine). 17 
Epidiascope, 230 
Episcope, 228 
Erecting eyepiece, 41 

systems, 55 

Eyepieces, telescope, 40 

, microscope, 86 

Esophoria, 142 
Exophoria, 142 

Fabry-Perot interferometer, 260 
Field-glass, 156 
Field of view (telescope), 48 
Fizeau's experiment, 253 

apparatus for testing surfaces, 

260 

Foucault test, 246, 249 
Fraunhofer, 25, 45 
French, 148, 150 
Fresnel, 239 



Galileo, 18, 75 



287 



288 



APPLIED OPTICS 



Galilean telescope, 22, 53 
Gardner and Bennett, 251, 255 
Gauss' telescope objective, 26 
Gerhardt, 254 
Goerz double anastigmat, 182, 184 

hypergon, 184 

Green, 262 

Greenhough microscope, 159 
Gray son's rulings, 97 
Greeks, 7 

Gregory's telescope, 70, 7?, 251 
Gullstrand, 10 

Haidinger's fringes, 259 

Hall, Chester Moor, 25 

Hampton, 240 

Har combe, 219 

Hartmann test, 251 

Headlights, 234 

Helmholtz, 165 

Herschel telescope objective, 26 

mirror telescope, 72 

Heterophoria, 141 
HindU, 251 

Hoegh, von, 182 

Hoffmann, 164 

Homal, 88 

Hooke, 39, 75 

Huygenian eyepiece, 41, 42 

(curvature of field), 31 

Hypergon, 184 
Hyperphoria, 142 
Hypophoria, 143 

Ignatowsky, 122 
Illumination, 207 
Immersion systems, 85, 89 
Interferometer, camera lens, 267 
, microscope, 268 

tests, 259, 262 

(Twyman and Green), 262 

Interocular distance, 150 

Jansen, 18, 75 
Jentsch, 251, 257 
Johnson, 136, 248 

Kellner eyepiece, 44 
Kepler, 40, 45 

Kinematograph projector, 222 
Kingslake, 73, 253, 267 
Kdhler, 88, 135 
Konig, 72, 74 
Kdnig prism, 65 
Kuhne, 269 

Lagrange relation, i 
Landscape lens, 176, 187 
Langlands, 150 
Lantern slide sizes, 219 



Lee, 204 

Leeuwenhoek, S, 75 

Leitz, Messrs., 161 

Leman prism, 65 

Levy, M., 138 

Lick objective, 29 

Lighthouse systems, 238 

Lippershey, 18, 75 

Lister, 75, 88 

Lummer and F. Reiche, 140 

Maddox groove, 142 
Magnification, i . 
Magnifiers, 7 
Magnifying power, 3 
Mangin, 235 
Martin, 136, 140, 240 

and Kingslake, 269 

Marzoli, 75 
Mersenne, 66 

Mic kelson, 254 

interferometer, 261 

Microscope, simple, i 

, compound, 75 

Monocentric eyepiece, 45 
Mount Wilson reflector, 72 
]\ Junker and Schuckert, 240 

Nachet's prism, 121 
Newton's telescope, 72 
Night glasses, 215 
Numerical aperture, 79 

ORTHOPHORIA, 141 
Ottway telescope, 48 

PANCRATIC telescopes, 46 
Perry, 269 
Perspective, 158 
Petrological microscope, 137 
Petzval objective, 185 
Photographic lenses, 166 
Photometry, 206 

Photomicrography, exposure, 217 
Plossl, 28 

Point'olite, 92, 96, 220 
Polarization microscope, 137 
Porro, 55 

(prisms), 62 

Porta, Battista, 18, 75, 218 
Prism binoculars, 61 
Projection eyepiece. 87 

lantern, 218, 225, 227 

lens, 224 

Protar lens, 182, 185, 187 
Pseudoscopic vision ,151 
Pulfrich, 151 

Ramsden, 45 

eyepiece, 41, 44 

(achromatic), 45 



INDEX 



289 



Rayleigh, 120, 140 
Records, Robert t 18 
Rectilinear lens, 184 
Reflecting telescopes, 66 
Reichert, Messrs., 161 
Resolving power, of telescope, 38 

, of microscope, 80 

Rogers, 28 

Rohr, M. von, 10, 73, 135, 158, 187 

Ronchi, 251, 257 

Ross Homocentric lens, 184 

, Messrs., 180 

telescope, 46 

Rudolph, 1 80. 181 

Schroder, 180 
Schuckert, 240 
Searchlights, 234 
Secondary spectrum, 27 
Shadbolt, 140 
Ship's lanterns, 238 
Siedentopf, 122, 126 
Signs and conventions, 271 
Simple microscope, 1 2 
Smith, T., 74, 268 
Spectacle magnification, 5 
Squanng-on, 243 
Stanhope lens, 1 1 
Star test, 241 
Steinheil eyepiece, 45 

magnifier, 15 

telescope objective, 26 

Stereoscope, 145 
Stereo-microscope, 162 
Stereo-telescope, 153 
Stoney, 101 
Stop-number, 166 
Strabismus, 142 

Strain, 245 

Stratton pseudoscope, 1 5 1 

Straubel, 240 

Striae, 245 

Swaine (Emsley and), 17 



Szigsmondy, 126 

Taylor, E. W. t 30 
Taylor, H. D., 87, 186, 243 
Taylor Hobson anastigmat, 186 
Taylor, Taylor, and Hobson, Ltd., 186 

204 
triplet telescope lens, 27, 30 



Telephoto lenses, 199 
Telescope, 18 

, astronomical, 21 

, Galilean, 22 

objectives, 25 

, design of, 33 



Telestereoscope, 152 
Terrestrial eyepiece, 45 
Tessar astrographic objective, 27, 30 
photographic lens, 186 



Testing methods, 241 

Tolles, 89 

Twyman, F. t 262, 269 

Ulbricht integrating sphere, 229 
Ultra -microscope, 126 
violet microscopy, 132 

VARiABLE-power telescopes, 46 
Verant, 10 

Walsh, 240 
Wander sleb, 174, 193 
Wenham, 121 
Wheatstone, 145 
Whitwell, 8 
Williams, 165 
Wilson, J , 8 
Wollaston, 10 

Yerkes objective, 29 

Zeiss, 10, 30, 87, 88, 124, 181, 199, 
235 



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REINFORCED CONCRETE, DETAIL DESIGN IN. By Ewart S. 

Andrews, B.Sc. (Eng.) 60 

REINFORCED CONCRETE. By W. Noble Twelvetrees, M.I.M.E., 

A.M.I.E.E 21 

REINFORCED CONCRETE MEMBERS, SIMPLIFIED METHODS OF 

CALCULATING. By W. Noble Twelvetrees. Second Edition. 5 
SPECIFICATIONS FOR BUILDING WORKS. By W. L. Evershed, F.S.I. 5 
STRUCTURES, THE THEORY OF. By H. W. Coultas, M.Sc., 

A.M.I.Struct.E., A.I.Mech.E 15 

CIVIL ENGINEERING, BUILDING, ETC. 

AUDEL'S MASONS' AND BUILDERS' GUIDES. In four volumes 

Each 7 6 

1. BRICKWORK, BRICK-LAYING, BONDING, DESIGNS 

2. BRICK FOUNDATIONS, ARCHES, TILE SETTING, ESTIMATING 

3. CONCRETE MIXING, PLACING FORMS, REINFORCED 

STUCCO 

4. PLASTERING, STONE MASONRY, STEEL CONSTRUCTION, 

BLUE PRINTS 
AUDEL'S PLUMBERS' AND STEAM FITTERS' GUIDES. Practical 

Handbooks in four volumes ..... Each 7 6 

1. MATHEMATICS, PHYSICS, MATERIALS, TOOLS, LEAD- 

WORK 

2. WATER SUPPLY, DRAINAGE, ROUGH WORK, TESTS 

3. PIPE FITTING, HEATING, VENTILATION, GAS, STEAM 

4. SHEET METAL WORK, SMITHING, BRAZING, MOTORS 
BRICKWORK, CONCRETE, AND MASONRY. Edited by T. Corkhill, 

M.I.Struct.E. In eight volumes .... Each 6 
" THE BUILDER " SERIES 

ARCHITECTURAL HYGIENE; OR, SANITARY SCIENCE AS 
APPLIED TO BUILDINGS. By Sir Banister Fletcher, 
F.R.I.B.A., F.S.I., and H. Phillips Fletcher, F.R.I.B.A., 

F.S.I. Fifth Edition, Revised 10 6 

CARPENTRY AND JOINERY. By Sir Banister Fletcher, 
F.R.I.B.A., F.S.I., etc., and H. Phillips Fletcher, 
F.R.I.B.A., F.S.I., etc. Fifth Edition, Revised . . 10 6 
QUANTITIES AND QUANTITY TAKING. By W. E. Davis. 

Seventh Edition, Revised by P. T.Walters, F.S.I., F.I.Arb. 6 
BUILDING, DEFINITIONS AND FORMULAE FOR STUDENTS. By T. 

Corkhill, F.B.I.C.C., M.I.Struct.E - 6 

BUILDING EDUCATOR, PITMAN'S. Edited by R. Greenhalgh, 

A.I.Struct.E. In three volumes . . . . 63 
BUILDING ENCYCLOPAEDIA, A CONCISE. Compiled by T. 

Corkhill, M.I.Struct.E 76 

ENGINEERING EQUIPMENT OF BUILDINGS. By A. C. Pallot, 

B.Sc. (Eng.) 15 



8 PITMAN'S TECHNICAL BOOKS 

Civil Engineering, Building, etc. contd. St </. 

HYDRAULICS. By E. H. Lewitt, B.Sc. (Lond.), M.I.Ae.E., 

A.M.I.M.E. Fourth Edition 10 6 

JOINERY & CARPENTRY. Edited by R. Greenhalgh,A.I.Struct.E. 

In six volumes ....... Each 6 

MECHANICS OF BUILDING. By Arthur D. Turner, A.C.G.I., 

A.M.I.C.E 50 

PAINTING AND DECORATING. Edited by C. H. Eaton, F.I.B.D. 

In six volumes Each 7 6 

PLUMBING AND GASFITTING. Edited by Percy Manser, R.P., 

A.R.San.I. In seven volumes .... Each 6 

SURVEYING, TUTORIAL LAND AND MINE. By Thomas 

Bryson 10 6 

WATER MAINS, LAY-OUT OF SMALL. By H. H. Hellins, 

M.Inst.C.E 76 

WATERWORKS FOR URBAN AND RURAL DISTRICTS. By H. C. 

Adams, M.Inst.C.E., M.I.M.E., F.S.I. Second Edition. . 15 



MECHANICAL ENGINEERING 

AUDEL'S ENGINEERS' AND MECHANICS' GUIDES. In eight 

volumes. Vols. 1-7 Each 7 6 

Vol. 8 15 

CONDENSING PLANT. By R. J. Kaula, M.I.E.E., and I. V. 

Robinson, Wh.Sc., A.M.Inst.C.E 30 

DEFINITIONS AND FORMULAE FOR STUDENTS APPLIED ME- 
CHANICS. By E. H. Lewitt, B.Sc., A.M.I.Mech.E.. . - 6 

DEFINITIONS AND FORMULAE FOR STUDENTS HEAT ENGINES. 

By A. Rimmer, B.Eng. Second Edition. . . - 6 

ENGINEERING EDUCATOR, PITMAN'S. Edited by W. J. 
Kearton, M.Eng., A.M.LMech.E., A.M!nst.N,A. In three 
volumes . . . . . . . . . 63 

ESTIMATING FOR MECHANICAL ENGINEERS. By L. E. Bunnett, 

A.M.I.P.E 10 6 

EXPERIMENTAL ENGINEERING SCIENCE. By Nelson Harwood, 

B.Sc 76 

FRICTION CLUTCHES. By R. Waring-Brown, A.M.I. A.E., 

F.R.S.A., M.I.P.E 50 

FUEL ECONOMY IN STEAM PLANTS. By A. Grounds, B.Sc., 

F.I.C., F.Inst.P 50 

FUEL OILS AND THEIR APPLICATIONS. By H. V. Mitchell, 

F.C.S. Second Edition, Revised by A. Grounds, B.Sc., A.I.C. 5 

MECHANICAL ENGINEERING DETAIL TABLES. By P. Ross . 7 6 

MECHANICAL ENGINEER'S POCKET BOOK, WHITTAKER'S. Third 
Edition, entirely rewritten and edited by W. E. Dommett, 
A.F.Ae.S., A.M.I.A.E 12 6 

MECHANICS' AND DRAUGHTSMEN'S POCKET BOOK. By W. E. 

Dommett, Wh.Ex., A.M.I.A.E 26 

MECHANICS FOR ENGINEERING STUDENTS. By G. W. Bird, 

B.Sc., A.M.I.Mech.E., A.M.I.E.E. Second Edition . .50 



AVIATION 9 



Mechanical Engineering contd. 5. d. 

MECHANICS OF MATERIALS, EXPERIMENTAL. By H. Carrington, 

M.Sc. (Tech.), D.Sc., M.Inst.Met., A.M.I.Mech.E.,A.F.R.^E.S. 3 6 
MOLLIER STEAM TABLES AND DIAGRAMS, THE. Extended to the 

Critical Pressure. English Edition adapted and amplified 

from the Third German Edition by H. Moss, D.Sc., A.R.C.S., 

D.I.C 76 

MOLLIER STEAM DIAGRAMS. Separately in envelope . .20 
MOTIVE POWER ENGINEERING. By Henry C. Harris, B.Sc. . 10 6 
PULVERIZED FUEL FIRING. By J. Foden . . . .76 
STEAM CONDENSING PLANT. By John Evans, M.Eng., 

A.M.I.Mech.E 76 

STEAM PLANT, THE CARE AND MAINTENANCE OF. By J. E. 

Braham, B.Sc., A.C.G.1 50 

STEAM TURBINE OPERATION. By W. J. Kearton, M.Eng., 

AJVLI.Mech.E., A.M.Inst.N.A 12 6 

STEAM TURBINE THEORY AND PRACTICE. By W. J. Kearton, 

A.M.I.M.E. Third Edition 15 

STRENGTH OF MATERIALS. By F. V. Warnock, Ph.D., B.Sc. 

(Lond.), F.R.C.Sc!., A.M.I.Mech.E 12 6 

THEORY OF MACHINES. By Louis Toft, M.Sc.Tech., and A. T. J. 

Kersey, B.Sc. Second Edition 12 6 

THERMODYNAMICS, APPLIED. By Prof. W. Robinson, M.E., 

M.Inst.C.E. ... 18 

TURBO-BLOWERS AND COMPRESSORS. By W. J. Kearton, 

A.M.I.M.E 21 

UNIFLOW BACK-PRESSURE AND STEAM EXTRACTION ENGINES. 

By Eng. Lieut.-Com. T. Allen, R.N.(S.R.), M.Eng., 

M.I.Mech.E 42 0. 

WORKSHOP PRACTICE. Edited by E. A. Atkins, M.I.Mech.E., 

M.I.W.E. In eight volumes Each 6 



AVIATION 

AERO ENGINES, LIGHT. By C. F. Caunter . . . . 12 6 
AEROBATICS. By Major O. Stewart, M.C., A.F.C. . . .50 
AERONAUTICS, DEFINITIONS AND FORMULAE FOR STUDENTS. 

By J. D. Frier, A.R.C.Sc., D.I.C., F.R.Ae.S. . . - 6 
AEROPLANE STRUCTURAL DESIGN. By T. H. Jones, B.Sc., 

A.M.I.M.E., and J. D. Frier, A.R.C.Sc., D.I.C. . . . 21 
AIR AND AVIATION LAW. By W. Marshall Freeman, Barrister- 

at-Law 76 

AIR NAVIGATION FOR THE PRIVATE OWNER. By Frank A. 

Swoffer, M.B.E 76 

AIRMANSHIP. By John McDonough 76 

AIRSHIP, THE RIGID. By E. H. Lewitt, B.Sc., M.I.Ae.E.. . 30 
AUTOGIRO, C. 19, BOOK OF THE. By C. J. Sanders and A. H. 

Rawson ......... 5 

AVIATION FROM THE GROUND UP. By Lieut. G. B. Manly . 15 



10 PITMAN'S TECHNICAL BOOKS 



Aviation, etc. contd. 5 . <f. 

CIVILIAN AIRCRAFT, REGISTER OF. By W. O. Manning and 

R. L. Preston 36 

FLYING AS A CAREER. By Major Oliver Stewart, M.C., A.F.C. 3 6 
GLIDING AND MOTORLESS FLIGHT. By C. F. Carr and L. 

Howard-Flanders, A.F.R.^E.S. Second Edition . .76 
LEARNING TO FLY. By F. A. Swoffer, M.B.E. With a Foreword 

by the late Sir Sefton Brancker, K.C.B., A.F.C. 2nd Ed. . 76 
PARACHUTES FOR AIRMEN. By Charles Dixon . . .76 
PILOT'S " A " LICENCE Compiled by John F. Learning, Royal 

Aero Club Observer for Pilots Certificates. Fourth Edition . 3 6 

MARINE ENGINEERING 

MARINE ENGINEERING, DEFINITIONS AND FORMULAE FOR 

STUDENTS. By E. Wood, B.Sc - 6 

MARINE SCREW PROPELLERS, DETAIL DESIGN OF. By Douglas 

H. Jackson, M.I.Mar.E., A.M.I.N.A 60 

OPTICS AND PHOTOGRAPHY 

AMATEUR CINEMATOGRAPHY. By Capt. O. Wheeler, F.R.P.S. . 6 
APPLIED OPTICS, AN INTRODUCTION TO. Volume I. By L. C. 

Martin, D.Sc., D.I.C., A.R.C.S 21 

BROMOIL AND TRANSFER. By L. G. Gabriel . . . .76 

CAMERA LENSES. By A. W. Lockett 26 

COLOUR PHOTOGRAPHY. By Capt. O. Wheeler, F.R.P.S.. . 12 6 
COMMERCIAL PHOTOGRAPHY. By D. Charles . . . .50 

COMPLETE PRESS PHOTOGRAPHER, THE. By Bell R. Bell. . 6 
LENS WORK FOR AMATEURS. By H. Orford. Fifth Edition, 

Revised by A. Lockett . . . . . . .36 

PHOTOGRAPHIC CHEMICALS AND CHEMISTRY. By J. South- 
worth and T. L. J. Bentley 36 

PHOTOGRAPHIC PRINTING. By R. R. Rawkins . . .36 
PHOTOGRAPHY AS A BUSINESS. By A. G. Willis . . ,50 
PHOTOGRAPHY THEORY AND PRACTICE. By E. P. Clerc. Edited 

by G. E. Brown 35 

RETOUCHING AND FINISHING FOR PHOTOGRAPHERS. By J. S. 

Adamson. Third Edition 40 

STUDIO PORTRAIT LIGHTING. By H. Lambert, F.R.P.S. . . 15 

ASTRONOMY 

ASTRONOMY, PICTORIAL. By G. F. Chambers, F.R.A.S.. . 2 6 
ASTRONOMY FOR EVERYBODY. By Professor Simon Newcomb, 

LL.D. With an Introduction by Sir Robert Ball . .50 
GREAT ASTRONOMERS. By Sir Robert Ball, D.Sc., LL.D., F.R.S. 5 
HIGH HEAVENS, IN THE. By Sir Robert Ball . . .50 
STARRY REALMS, IN. By Sir Robert Ball, D.Sc., LL.D., F.R.S, 5 



MOTOR ENGINEERING 11 

MOTOR ENGINEERING s. d. 
AUTOMOBILE AND AIRCRAFT ENGINES By A. W. Judge, 

A.R.C.S., A.M.I.A.E. Second Edition . . . . 42 

CARBURETTOR HANDBOOK, THE. By E. W. Knott, A.M.I.A.E.. 10 6 

GAS AND OIL ENGINE OPERATION. By J. Okill, M.I.A.E.. . 5 

GAS, OIL, AND PETROL ENGINES. By A. Garrard, Wh.Ex. . 6 
MAGNETO AND ELECTRIC IGNITION. By W. Hibbert, A.M.I.E.E. 

Third Edition 36 

MOTOR BODY BUILDING, PRIVATE AND COMMERCIAL. By H. J. 

Butler 10 6 

MOTOR-CYCLIST'S LIBRARY, THE. Each volume in this series 
deals with a particular type of motor-cycle from the point 
of view of the owner-driver ..... Each 2 

A.J.S., THE BOOK OF THE. By W. C. Haycraft. 

ARIEL, THE BOOK OF THE. By G. S. Davison. 

B.S.A., THE BOOK OF THE. By " Waysider." 

DOUGLAS, THE BOOK OF THE. By E. W. Knott. 

IMPERIAL, BOOK OF THE NEW. By F. J. Camm. 

MATCHLESS, THE BOOK OF THE. By W. C. Haycraft. 

NORTON, THE BOOK OF THE. By W. C. Haycraft 

P. AND M., THE BOOK OF THE. By W. C. Haycraft. 

RALEIGH HANDBOOK, THE. By " Mentor." 

ROYAL ENFIELD, THE BOOK OF THE. By R. E. Ryder. 

RUDGE, THE BOOK OF THE. By L. H. Cade. 

TRIUMPH, THE BOOK OF THE. By E. T. Brown. 

VILLIERS ENGINE, BOOK OF THE. By C. Grange. 
MOTORISTS' LIBRARY, THE. Each volume in this series deals 
with a particular make of motor-car from the point of view 
of the owner-driver. The functions of the various parts of 
the car are described in non-technical language, and driving 
repairs, legal aspects, insurance, touring, equipment, etc., all 
receive attention. 

AUSTIN, THE BOOK OF THE. By B. Garbutt. Third 

Edition, Revised by E. H. Row . . . .36 

MORGAN, THE BOOK OF THE. By G. T. Walton . .26 

SINGER JUNIOR, BOOK OF THE. By G. S. Davison. . 2 6 

MOTORIST'S ELECTRICAL GUIDE, THE. By A. H. 

Avery, A.M.I.E.E 36 

CARAVANNING AND CAMPING. By A. H. M. Ward, M.A. 2 6 

ELECTRICAL ENGINEERING, ETC. 

ACOUSTICAL ENGINEERING. By W. West, B.A. (Oxon), 

A.M.I.E.E 15 

ACCUMULATOR CHARGING, MAINTENANCE, AND REPAIR. By 

W. S. Ibbetson. Second Edition 36 

ALTERNATING CURRENT BRIDGE METHODS. By B. Hague, 

D.Sc. Second Edition 15 



12 PITMAN'S TECHNICAL BOOKS 

Electrical Engineering, etc. contd. 5. d. 

ALTERNATING CURRENT CIRCUIT. By Philip Kemp, M.I.E.E.. 2 6 
ALTERNATING CURRENT MACHINERY, PAPERS ON THE DESIGN 

OF. By C. C. Hawkins, M.A., M.I.E.E., S. P. Smith, D.Sc., 

M.I.E.E., and S. Neville, B.Sc 21 

ALTERNATING CURRENT POWER MEASUREMENT. By G. F. 

Tagg, B.Sc 36 

ALTERNATING CURRENT WORK. By W. Perren Maycock, 

M.I.E.E. Second Edition 10 6 

ALTERNATING CURRENTS, THE THEORY AND PRACTICE OF. By 

A. T. Dover, M.I.E.E. Second Edition . . . . 18 
ARMATURE WINDING, PRACTICAL DIRECT CURRENT. By L. 

Wollison 76 

CABLES, HIGH VOLTAGE. By P. Dunsheath, O.B.E., M.A., B.Sc., 

M.I.E.E 10 6 

CONTINUOUS CURRENT DYNAMO DESIGN, ELEMENTARY 

PRINCIPLES OF. By H. M. Hobart, M.I.C.E., M.I.M.E., 

M.A.I.E.E. 10 6 

CONTINUOUS CURRENT MOTORS AND CONTROL APPARATUS. By 

W. Perren Maycock, M.I.E.E 76 

DEFINITIONS AND FORMULAE FOR STUDENTS ELECTRICAL. By 

P. Kemp, M.Sc., M.I.E.E - 6 

DEFINITIONS AND FORMULAE FOR STUDENTS ELECTRICAL 

INSTALLATION WORK. By F. Peake Sexton, A.R.C.S., 

A.M.I.E.E - 6 

DIRECT CURRENT ELECTRICAL ENGINEERING, ELEMENTS OF. 

By H. F. Trewman, M.A., and C. E. Condliffe, B.Sc.. . 5 
DIRECT CURRENT ELECTRICAL ENGINEERING, PRINCIPLES OF. 

By James R. Barr, A.M.I.E.E 15 

DIRECT CURRENT DYNAMO AND MOTOR FAULTS . By R. M. Archer 7 6 
DIRECT CURRENT MACHINES, PERFORMANCE AND DESIGN OF. 

By A. E. Clayton, D.Sc., M.I.E.E 16 

DYNAMO, THE: ITS THEORY, DESIGN, AND MANUFACTURE. By 

C. C. Hawkins, M.A., M.I.E.E. In three volumes. Sixth 

Edition 

Volume I 21 

II 15 

III 30 

DYNAMO, How TO MANAGE THE. By A. E. Bottone. Sixth 

Edition, Revised and Enlarged 20 

ELECTRIC AND MAGNETIC CIRCUITS, THE ALTERNATING AND 

DIRECT CURRENT. By E. N. Pink B.Sc., A.M.I.E.E. . 3 6 

ELECTRIC BELLS AND ALL ABOUT THEM. By S. R. Bottone. 
Eighth Edition, thoroughly revised by C. Sylvester, 
A.M.I.E.E 36 



ELECTRICAL ENGINEERING 13 

Electrical Engineering, etc. contd. s. d. 

ELECTRIC CIRCUIT THEORY AND CALCULATIONS. By W. Perren 
Maycock, M.I.E.E. Third Edition, Revised by Philip Kemp, 
M.Sc., M.I.E.E.. A.A.I.E.E 10 6 

ELECTRIC LIGHT FITTING, PRACTICAL. By F. C. Allsop. Tenth 

Edition, Revised and Enlarged 76 

ELECTRIC LIGHTING AND POWER DISTRIBUTION. By W. Perren 
Maycock, M.I.E.E. Ninth Edition, thoroughly Revised by 
C. H. Yeaman In two volumes .... Each 10 6 

ELECTRIC MACHINES, THEORY AND DESIGN OF. By F. Greedy, 

M.A.I.E.E., A.C.G.I . 30 

ELECTRIC MOTORS AND CONTROL SYSTEMS. By A. T. Dover, 

M.I.E.E.. A.Amer.I.E.E 15 

ELECTRIC MOTORS (DIRECT CURRENT): THEIR THEORY AND 
CONSTRUCTION. By H. M. Hobart, M.I.E.E., M.Inst.C.E., 
M.Amer.I.E.E. Third Edition, thoroughly Revised . . 15 

ELECTRIC MOTORS FOR CONTINUOUS AND ALTERNATING CUR- 
RENTS, A SMALL BOOK ON. By W. Perren Maycock, M.I.E.E. 6 

ELECTRIC TRACTION. By A. T. Dover, M.I.E.E., Assoc.Amer. 

I.E.E, Second Edition 25 

ELECTRIC TRAIN-LIGHTING. By C. Coppock . . . .76 

ELECTRIC TROLLEY Bus. By R. A. Bishop . . . . 12 6 

ELECTRIC WIRING DIAGRAMS. By W. Perren Maycock, 

M.I.E.E 50 

ELECTRIC WIRING, FITTINGS, SWITCHES, AND LAMPS. By W. 
Perren Maycock, M.I.E.E. Sixth Edition. Revised by 
Philip Kemp, M.Sc., M.I.E.E 10 6 

ELECTRIC WIRING OF BUILDINGS. By F. C. Raphael, M.I.E.E. 10 6 

ELECTRIC WIRING TABLES. By W. Perren Maycock, M.I.E.E. 

Revised by F. C. Raphael, M.I.E.E. Sixth Edition . .36 

ELECTRICAL CONDENSERS. By Philip R. Coursey, B.Sc., 

F.Inst.P., A.M.I.E.E 37 6 

ELECTRICAL EDUCATOR. By Sir Ambrose Fleming, M.A., 

D.Sc., F.R S. In three volumes. Second Edition . 72 

ELECTRICAL ENGINEERING, CLASSIFIED EXAMPLES IN. By S. 
Gordon Monk, B.Sc. (Eng.), A.M.I.E.E. In two parts- 
Volume I. DIRECT CURRENT. Second Edition. . .26 
II. ALTERNATING CURRENT. Second Edition . 3 6 

ELECTRICAL ENGINEERING, ELEMENTARY. By O. R. Randall, 

Ph.D., B.Sc., Wh.Ex 50 

ELECTRICAL ENGINEERING, EXPERIMENTAL. By E. T. A. 

Rapson, A.C.G.I., D.I.C., A.M.I.E.E 36 

ELECTRICAL ENGINEER'S POCKET BOOK, WHITTAKER'S. Origi- 
nated by Kenelm Edgcumbe, M.I.E.E., A.M.I.C.E. Sixth 
Edition. Edited by R. E. Neale, B.Sc. (Hons.) . . 10 6 

ELECTRICAL INSTRUMENT MAKING FOR AMATEURS. By S. R. 

Bottone. Ninth Edition 60 

ELECTRICAL INSULATING MATERIALS. By A. Monkhouse, Junr., 

M.I.E.E., A.M.I.Mech.E 21 



14 PITMAN'S TECHNICAL BOOKS 

Electrical Engineering, etc. contd. s. d. 

ELECTRICAL GUIDES, HAWKINS'. Each book in pocket size . 5 

1. ELECTRICITY, MAGNETISM, INDUCTION, EXPERIMENTS, 

DYNAMOS, ARMATURES, WINDINGS 

2. MANAGEMENT OF DYNAMOS, MOTORS, INSTRUMENTS, 

TESTING 

3. WIRING AND DISTRIBUTION SYSTEMS, STORAGE BATTERIES 

4. ALTERNATING CURRENTS AND ALTERNATORS 

5. A.C. MOTORS, TRANSFORMERS, CONVERTERS, RECTIFIERS 

6. A.C. SYSTEMS, CIRCUIT BREAKERS, MEASURING INSTRU- 

MENTS 

7. A.C. WIRING, POWER STATIONS, TELEPHONE WORK 

8. TELEGRAPH, WIRELESS, BELLS, LIGHTING 

9. RAILWAYS, MOTION PICTURES, AUTOMOBILES, IGNI- 

TION 
10. MODERN APPLICATIONS OF ELECTRICITY. REFERENCE 

INDEX 
ELECTRICAL MACHINERY AND APPARATUS MANUFACTURE. 

Edited by Philip Kemp, M.Sc., M.I.E.E., Assoc.A.I.E.E. 

In seven volumes ....... Each 6 

ELECTRICAL MACHINES, PRACTICAL TESTING OF. By L. Oulton, 

A.M.I.E.E., and N. J. Wilson, M.I.E.E. Second Edition . 6 
ELECTRICAL MEASURING INSTRUMENTS, COMMERCIAL. By R. M. 

ARCHER, B.Sc. (Lond.), A.R.C.Sc., M.I.E.E. . . . 10 6 
ELECTRICAL POWER TRANSMISSION AND INTERCONNECTION. 

By C. Dannatt, B.Sc., and J. W. Dalgleish, B.Sc. . . 30 
ELECTRICAL TECHNOLOGY. By H. Cotton, M.B.E., D.Sc. . 12 6 
ELECTRICAL TERMS, A DICTIONARY OF. By S. R. Roget, M.A., 

A.M.Inst.C.E., A.M.I.E.E. Second Edition . . .76 
ELECTRICAL TRANSMISSION AND DISTRIBUTION. Edited by 

R. O. Kapp, B.Sc. In eight volumes. Vols. I to VII, Each 6 

Vol. VIII 30 

ELECTRICAL WIRING AND CONTRACTING. Edited by H. 

Marryat, M.I.E E. f M.I.Mech E. In seven volumes . Each 6 

ELECTRO-TECHNICS, ELEMENTS OF. By A. P. Young, O.B.E., 

M.I.E.E 50 

FRACTIONAL HORSE-POWER MOTORS. By A. H. Avery, 

A.M.I.E.E 76 

INDUCTION COIL, THEORY AND APPLICATIONS. By E. Taylor- 
Jones, D.Sc 12 6 

INDUCTION MOTOR, THE. By H. Vickers, Ph.D., M.Eng. . 21 

KlNBMATOGRAPHY PROJECTION! A GUIDE TO. By Colin H. 

Bennett, F.C S., F.R.P.S 10 6 

MERCURY- ARC RECTIFIERS AND MERCURY-VAPOUR LAMPS. By 

Sir Ambrose Fleming, M.A., D.Sc., F.R.S. . . .60 

METER ENGINEERING. By J. L. Ferns, B.Sc. (Rons.), A.M.C.T. 10 6 



TELEGRAPHY, TELEPHONY, AND WIRELESS 15 
Electrical Engineering, etc. contd. s. d. 

OSCILLOGRAPHS. By J. T. Irwin, A.M.I.E.E. . . .76 
POWER DISTRIBUTION AND ELECTRIC TRACTION, EXAMPLES IN. 

By A. T. Dover, M.I.E.E., A.A.I.E.E. 3 6 

POWER STATION EFFICIENCY CONTROL. By John Bruce, 

A.M.I.E.E 12 6 

POWER WIRING DIAGRAMS. By A. T. Dover, M.I.E.E., A.Amer. 

I.E.E. Second Edition, Revised 60 

PRACTICAL PRIMARY CELLS. By A. Mortimer Codd, F.Ph.S. . 5 
RAILWAY ELECTRIFICATION. By H. F. Trewman, A.M.I.E.E. 21 
SAGS AND TENSIONS IN OVERHEAD LINES. By C. G. Watson, 

M.I.E.E 12 6 

STEAM TURBO -ALTERNATOR, THE. By L. C. Grant, A.M.I.E.E. 15 
STORAGE BATTERIES: THEORY, MANUFACTURE, CARE, AND 

APPLICATION. ByM. Arendt, E.E 18 

STORAGE BATTERY PRACTICE. By R. Rankin, B.Sc., M.I.E.E.. 7 6 
TRANSFORMERS FOR SINGLE AND MULTIPHASE CURRENTS. By 

Dr. Gisbert Kapp, M.Inst.C.E., M.I.E.E. Third Edition, 

Revised by R. O. Kapp, B.Sc 15 

TELEGRAPHY, TELEPHONY, AND WIRELESS 

AUTOMATIC BRANCH EXCHANGES, PRIVATE. By R. T. A. 

Dennison .... ..... 12 6 

AUTOMATIC TELEPHONY, RELAYS IN. By R. W. Palmer, 

A.M.I.E.E 10 6 

BAUD&T PRINTING TELEGRAPH SYSTEM. By H. W. Pendry. 

Second Edition 60 

CABLE AND WIRELESS COMMUNICATIONS OF THE WORLD, THE. 

By F. J. Brown, C.B., C.B.E., M.A., B.Sc. (Lond.). Second 

Edition ......... 7 6 

CRYSTAL AND ONE-VALVE CIRCUITS, SUCCESSFUL. By J. H. 

Watkins 36 

RADIO COMMUNICATION, MODERN. By J. H. Reyner, B.Sc. 

(Hons.), A.C.G.I., D.I.C. Third Edition . . . .50 
SUBMARINE TELEGRAPHY. By Ing. Italo de Giuli. Translated 

by J. J. McKichan, O.B.E., A.M.I.E.E 18 

TELEGRAPHY. By T. E. Herbert, M.I.E.E. Fifth Edition . 20 
TELEGRAPHY, ELEMENTARY. By H. W. Pendry. Second 

Edition, Revised 76 

TELEPHONE HANDBOOK AND GUIDE TO THE TELEPHONIC 

EXCHANGE, PRACTICAL. By Joseph Poole, A.M.I.E.E. 

(Wh.Sc.). Seventh Edition 18 

TELEPHONY. By T. E. Herbert, M.I.E.E 18 

TELEPHONY SIMPLIFIED, AUTOMATIC. By C. W. Brown, 

A.M.I.E.E., Engineer-in-Chiefs Department, G.P.O., London 6 
TELEPHONY, THE CALL INDICATOR SYSTEM IN AUTOMATIC. By 

A. G. Freestone, of the Automatic Training School, G.P.O., 

London 60 

TELEPHONY, THE DIRECTOR SYSTEM OF AUTOMATIC. By W. E. 

Hudson, B.Sc. Hons. (London), Whit.Sch., A.C.G.I. . .50 



16 PITMAN'S TECHNICAL BOOKS 

Telegraphy, Telephony, and Wireless contd. s d 

TELEVISION : TO-DAY AND TO-MORROW. By Sydney A. Moseley 
and H. J. Barton Chappie, Wh.Sc., B.Sc. (Hons.), A.C.G.I., 
D.I.C., A.M.I.E.E. Second Edition . . . .76 

PHOTOELECTRIC CELLS. By Dr. N. I. Campbell and Dorothy 

Ritchie. Second Edition 15 

WIRELESS MANUAL, THE. By Capt. J. Frost, I.A. (Retired), Re- 
vised by H. V. Gibbons. Third Edition . . . .50 

WIRELESS TELEGRAPHY AND TELEPHONY, INTRODUCTION TO. 

By Sir Ambrose Fleming, M.A., D.Sc., F.R.S. , . .36 

MATHEMATICS AND CALCULATIONS 
FOR ENGINEERS 

ALTERNATING CURRENTS, ARITHMETIC OF. By E. H. Crapper, 

D.SC.M.I.E.E 46 

CALCULUS FOR ENGINEERING STUDENTS. By John Stoney, 

B.Sc., A.M.I.Min.E 36 

DEFINITIONS AND FORMULAE FOR STUDENTS PRACTICAL 

MATHEMATICS. By L. Toft, M.Sc - 6 

ELECTRICAL ENGINEERING, WHITTAKER'S ARITHMETIC OF. 

Third Edition, Revised and Enlarged . . . .36 
EXPONENTIAL AND HYBERBOLIC FUNCTIONS. By A. H. Bell, 

B.Sc 36 

GEOMETRY, BUILDING. By Richard Greenhalgh, A.I.Struct.E. 4 6 
GEOMETRY, CONTOUR. By A. H. Jameson, M.Sc., M.Inst.C.E. . 7 6 
GEOMETRY, EXERCISES IN BUILDING. By Wilfred Chew . 1 6 
GRAPHIC STATICS, ELEMENTARY. By J . T. Wight, A.M.I. Mech.E. 5 
KILOGRAMS INTO AVOIRDUPOIS, TABLE FOR THE CONVERSION OF. 

Compiled by Redvers Elder. On paper . . . .10 
LOGARITHMS FOR BEGINNERS. By C. N. Pickworth, Wh.Sc. 

Eighth Edition 16 

LOGARITHMS, FIVE FIGURE, AND TRIGONOMETRICAL FUNCTIONS. 

By W. E. Dommett, A.M.I.A.E., and H. C. Hird, A.F.Ae.S. 1 
LOGARITHMS SIMPLIFIED. By Ernest Card, B.Sc., and A. C 

PARKINSON, A.C.P. Second Edition . . . .20 
MATHEMATICS AND DRAWING, PRACTICAL. By Dalton Grange. 2 

With Answers . . . . . . . .26 

MATHEMATICS, ENGINEERING, APPLICATION OF. By W. C. 

Bickley, M.Sc 50 

MATHEMATICS, EXPERIMENTAL. By G. R. Vine, B.Sc. 

Book I, with Answers . . . . . . .14 

II, with Answers .14 

MATHEMATICS FOR ENGINEERS, PRELIMINARY. By W. S. 

Ibbetson, B.Sc., A.M.I.E.E., M.I.Mar.E 36 

MATHEMATICS, PRACTICAL. By Louis Toft, M.Sc. (Tech.), and 

A. D. D. McKay, M.A 16 

MATHEMATICS FOR TECHNICAL STUDENTS. By G. E Hall, B.Sc. 5 
MATHEMATICS, INDUSTRIAL (PRELIMINARY), By G. W. String- 
fellow 20 

With Answers 26 



MISCELLANEOUS TECHNICAL BOORS 17 

. Mathematics for Engineers contd. s. d. 

MEASURING AND MANURING LAND, AND THATCHER'S WORK, 

TABLES FOR. By J. Cullyer. Twentieth Impression . .30 

MECHANICAL TABLES. By J. Foden 20 

MECHANICAL ENGINEERING DETAIL TABLES. By John P. Ross 7 6 

METALWORKER'S PRACTICAL CALCULATOR, THE. By J . Matheson 2 

METRIC CONVERSION TABLES. By W. E. Dommett, A.M.I.A.E. 1 

METRIC LENGTHS TO FEET AND INCHES, TABLE FOR THE CON- 
VERSION OF. Compiled by Redvers Elder. On paper. . 1 

MINING MATHEMATICS (PRELIMINARY). By George W. String- 
fellow 16 

With Answers 20 

NOMOGRAM, THE. By H. J. Allcock, B.Sc. f A.M.I.E.E., 

A.M.I.Mech.E., and J. R. Jones, M.A., F.G.S. . . . 10 6 

QUANTITIES AND QUANTITY TAKING. By W. E. Davis. Seventh 

Edition, Revised by P. T. Walters, F.S.I., F.I.Arb. . .60 

SCIENCE AND MATHEMATICAL TABLES. By W. F. F. Shearcroft, 

B.Sc., A.I.C., and Denham Larrett, M.A. . . .10 

SLIDE RULE, THE. By C. N. Pickworth, Wh.Sc. Seventeenth 

Edition, Revised 36 

SLIDE RULE: ITS OPERATIONS ; AND DIGIT RULES, THE. By A. 

Lovat Higgins, A.M.Inst.C.E. - 6 

STEEL'S TABLES. Compiled by Joseph Steel . . . .36 

TELEGRAPHY AND TELEPHONY, ARITHMETIC OF. By T. E. 

Herbert, M.I.E.E.. and R. G. de Wardt . . .50 

TEXTILE CALCULATIONS. By J. H. Whitwam, B.Sc. . . 25 

TRIGONOMETRY FOR ENGINEERS, A PRIMER OF. By W. G. 

Dunkley, B.Sc. (Rons.) 50 

TRIGONOMETRY FOR NAVIGATING OFFICERS. By W. Percy 

Winter, B.Sc. (Hons.), Lond 10 3 

TRIGONOMETRY, PRACTICAL. By Henry Adams, M.I.C.E., 

M.I.M.E., F.S.I. Third Edition, Revised and Enlarged . 5 

VENTILATION, PUMPING, AND HAULAGE, MATHEMATICS OF. By 

F. Birks 50 

WORKSHOP ARITHMETIC, FIRST STEPS IN. By H. P. Green . 1 

MISCELLANEOUS TECHNICAL BOOKS 

BOOT AND SHOE MANUFACTURE. By F. Plucknett . . . 35 
BREWING AND MALTING. By J. Ross Mackenzie, F.C.S., F.R.M.S. 

Second Edition . . . . . . . .86 

BUILDER'S BUSINESS MANAGEMENT. By J. H. Bennetts 



A.I.O.B. 

CERAMIC INDUSTRIES POCKET BOOK. By A. B. Searle 
CINEMA ORGAN, THE. By Reginald Foort, F.R.C.O. 
COST ACCOUNTS IN RUBBER AND PLASTIC TRADES. By T. W 

Fazakerley ...... 

ELECTRICAL HOUSECRAFT. By R. W. Kennedy 
ENGINEERING ECONOMICS. By T. H. Burnham, B.Sc. (Hons.) 

B.Com., A.M.I.Mech.E. Second Edition 



10 6 

8 6 

2 6 

5 

2 6 

10 6 



18 PITMAN'S TECHNICAL BOOKS 

Miscellaneous Technical Books contd. s. d. 

ENGINEERING INQUIRIES, DATA FOR. By J. C. Connan, B.Sc., 

A.M.I.E.E., O.B.E 12 6 

FARADAY, MICHAEL, AND SOME OF His CONTEMPORARIES. By 

Prof. William Cramp, D.Sc., M.I.E.E 26 

FURNITURE STYLES. By. H. E. Binstead. Second Edition . 10 8 

GLUE AND GELATINE. By P. I, Smith 86 

GRAMOPHONE HANDBOOK. By W. S. Rogers . . .26 

HAIRDRESSING, THE ART AND CRAFT OF. Edited by G. A. Foan. 60 
HIKER AND CAMPER, THE COMPLETE. By C. F. Carr . .26 
HOUSE DECORATIONS AND REPAIRS. By W. Prebble. Second 

Edition 10 

MOTOR BOATING. By F. H. Snoxell 26 

PAPER TESTING AND CHEMISTRY FOR PRINTERS. By Gordon A. 

Jahans, B.A 12 6 

PETROLEUM. By Albert Lidgett. Third Edition . .50 

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REFRACTORIES FOR FURNACES , CRUCIBLES, ETC. By A. B. Searle 5 
REFRIGERATION, MECHANICAL. By Hal Williams, M.I.Mech.E., 

M.I.E.E., M.I. Struct.E. Third Edition . . . . 20 

SEED TESTING. By J. Stewart Remington . . . . 10 6 
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seven volumes. Vols. 4-7 (In prepai'ation) . . Each 3 6 
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stein. Second Edition 76 

TALKING PICTURES. By Bernard Brown, B.Sc. (Eng.). Second 

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TEACHING METHODS FOR TECHNICAL TEACHERS. By J. H. 

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BELTS FOR POWER TRANSMISSION. By W. G. Dunkley, B.Sc. 
BOILER INSPECTION AND MAINTENANCE. By R. Clayton. 
CAPSTAN AND AUTOMATIC LATHES. By Philip Gates. 
CENTRAL STATIONS, MODERN. By C. W. Marshall, B.Sc., 

A.M.I.E.E. 
COAL CUTTING MACHINERY, LONGWALL. By G. F. F. Eagar, 

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A.C.G.I., A.Amer.I.E.E. 
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COTTON SPINNING MACHINERY AND ITS USES. By Wm. Scott 

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DIESEL ENGINE, THE. By A. Orton, A.M.I.Mech.E. 
DROP FORGING AND DROP STAMPING. By H. Hayes. 
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ELECTRICITY IN STEEL WORKS. By Wm. McFarlane, B.Sc. 
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