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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'NIVFRsIIV 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
textbook. 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 retraverse
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 secondhand 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 wavelength 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
ONEMETKE 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 = io (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  214 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 = 35. 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 ooi metre. Then magnification  106, 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 ./"=  o2D, d 30 (metres).
In this case the magnification works out to + 250. We therefore
should obtain an erect image of moderate magnification.
The system is made use of in the socalled "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 ./ ija/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 singlelens 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 pincushion 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: borosilic.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
saddleshaped 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 15, this calls for a radius of onethird 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 daytime 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 contradistinction 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 wavelengths
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 onequarter 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 planoconvex, 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^ . 
~'' \nij '' 'niV TV +> '
* V 2 ^ + 1 )
MI; (ni)
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 =
ni
Take the allowable OPD ==   00013 mm., and v  2 mm.
4
(say), allowing for an average pupilary diameter of 40 mm.
Then a quick calculation made with an assumed value of n == 15
gives
/' = iGi 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 crownglass 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 eyeball, 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 pocketholder 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 (192324), 249.
CHAPTER II
THE TELESCOPE
Historical. Roger Bacon (12161294) 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 planoconvex and a planoconcave 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 complexlooking 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 = =  ^ *  jr  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 eyering 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 rediscovery 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 applegreen 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 doubleconvex 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 = 1616
*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 wavelengths 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
302
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 wavelengths, 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 threecomponent 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 CookeTaylor photovisual objective.
The relative partial dispersions of the glasses used for the two
positive components of the Cooke photovisual 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
borosilicate flint used in the negative lens.
THE TELESCOPE
29
Glass
"D
V
Dis
persion
Relative Partial Dispersions
CF
A C
DF
EF
FG'
FH
0543
1504
507
01115
3354
7085
3309
5830
11857
'374
1511
608
00844
3507
7026
3247
5075
11564
Mean
3420
7059
3282
5763
11730
0658
154^
501
01090
3425
7052
3278
5767
II745
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 102 metres and a focal length of 189 metres, and the Lick
refractor a diameter of 091 metres and focal length 176 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 wavelengths) 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<$ (wavelength = o4ioi//). This
ensures a better bunching of the most "actinic" regions of the
spectrum, using the term to refer to those wavelengths 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 ultraviolet
30 APPLIED OPTICS
and infrared 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} = 15736, focal length = 3543
Negative lens: Light flint, n D 16039, focal length 5391
The focal length of the combination is 1006 ; and the radii which
can be found by calculation and checked by experiment are
Tangential image field . . . .285
Sagittal image field ..... 605
Petzval surface . . . . .156
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 eyelens 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 threequarters 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 righthand 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 13 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 overcorrected
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
Undercorrected astigmatism
[b) Correction ; images on Petzval surface
[c) Overcorrected 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 overcorrection.
Take Fig. i8(a) to represent an ordinary case of curvature of the
field and "undercorrected" 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 overcorrected 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 wavelength 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 ~ 15186 = 16041
V  603 V  378
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 fourfigure logs.)
A a  1650  559I 'i i + 621 /,\ 2
It is more convenient, however, to express the relation in terms of
$2 where
,/ K !)(//! ./?,)
and >?! = '/ 2 + 5168
We then obtain
A a = 420  827 /;, + 621 /? 2 2 . . (a)
For the second lens */\ = J' a = 268, and ,>' 6 =  168. The
equation, when simplified, reduces to
A 6 =  1191 + 958^3  377^3 2 ( 6 )
34
APPLIED OPTICS
In order that the lens may be cemented it will be necessary for
j$ 2 to be equal to _'# 8 . Hence if the sum of the aberrations is to be
zero
A + A 6 = o = 3009 + 1785 $ + 244 /? 2
a simple quadratic equation in # having roots  467 and  264.
This determines the forms of two possible lenses for cementing,
although one (for reasons given below) is much preferable to the
other.
First lens :
0498
= 252
// 4 =  1885
= 0145
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 .
15575
10822
73o8
503
420
8338
I7438
A &
15401
no55
7458
45^5
 II9I
683
3391
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  467 and  264 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
432
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  366.)
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
4445 4 2  273 '// 8 + 626 = o
/? 3 = 163 A* + 229
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 =  1002 /? 2 2  1252 /? 2  973
We had
A a = 621 ',y + 827 ''// 2 + 420
if the sum of the aberrations is to be zero, then we get on adding
o = 381 l // 2 2 + 425 /? 2  3227
This quadratic equation yields two solutions, i.e. '/ a ~3"5 2 or
+ 2405. 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
$* =  352
#* =  345
v' 4 =  067
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 = o55ii).
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 overcorrected 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 midzones, preferably using the wavelength for
brightest light. " Red " and " blue " rays can also be traced through
the midzone. 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
socalled "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 00025.
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
(= 000029 radians), then
__
000029 =  . M
It is easy to get an approximate figure if we take A = 000058 mm.,
and thus
a (mm.) , . . , v .
M =  = 10 x (aperture m inches) . . . approximately.
244
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 000029 \ a
f' b (mm.) = 244 x (aperture ratio number)
As indicated, however, it will be better to take a focal length
only onethird or onequarter 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. =
244
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 24 mm., and we can with visual
advantage use exit pupils down to 06 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 twolens 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
wavelength (or with the colour) of the light. This may be regarded
as arising from a variation of focal length with wavelength. 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 wavelength 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
f2
f1
P'
05
10
15 20
<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 imageforming 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 planoconvex form
of each component it is possible to produce "overcorrected"
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 fourlens 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
overcorrection is necessary.
Pancratic, or Variablepower 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 drawtube 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 layout 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 righthand (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 twolens 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 = ^77 : 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
^ (
\
mdf b (mi)
The visible field is therefore bounded by a ring of decreasing illu
mination which may be looked upon as the outoffocus 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 socalled "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 roughground.
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
twodimensional 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 lefthand 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 threedimensional 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 suitablydisposed
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 objectspace 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. nonparallelism 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 machinegun 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 rightangled prism ABC, together with one of the
familiar prisms (the socalled "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 1125. 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)~m5*
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 1125 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 coordinates, 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 welldesigned
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 foreshortening 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 xea 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 15. Therefore the difference of the sagittal and tangen
tial focussing distances will be /tan 2 (15). As K6nig 8 points out,
for a mirror of 122 metres diameter and 122 metres focal length
the intercept between the astigmatic foci is therefore 42 mm., and
if an eyepiece is used which will make the telescope magnification
400, the intercept requires a 4dioptre 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 rr . 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 (192829), 207.
2. R. Kingslake: Trans. Opt. Soc. t XXVIII (192627), 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 (192728), 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 (18081811) constructed
planoconvex 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 sphericalaberrationfree conjugate
points for these planoconvex 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 crownflint 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/
"''
Eyepiece
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 selfluminous 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 imageforming 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 wavelength 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' _ 061 n' X __ 061 A
~ n sin a n sin a n sin a
where A is the wavelength 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 semiaperture 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
_ Q6iA
* ~~ 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,
o6iA 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 = 12, 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 125 mm. The diameter of the exit pupil of the microscope
ought to be at most 25 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 108112 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 wavelength 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 bluegreen 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
125
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
065
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 1517, between
the lens and the cover glass (n = 1515, 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 154. 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 undercorrection 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
14338
00454
95'5
00321
00256
707
563
Borosilicate crown
15160
OOSOC)
683
00567
00454
701
501
Telescope flint
i\5237
01003
5^2
00708
00577
706
575
Dense flint .
16225
OI72<)
360
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 095, and, therefore, the extreme rays in the
object space may have to make an angle of shr 1 095 = 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 wavefront
into an elliptical wavefront, 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 "overcorrection" in the spherical aberration
sense.
In practice, slight overcorrection 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 undercorrection, 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 15 times
that of the eyelens 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 tubelength 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 wavelengths, as compared with
the red, and this is best overcome by the use of socalled "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 1333),
the limit of possible numerical aperture is equal to or under 1333,
provided that all the media have at least this refractive index. To
take full advantage of an objective having any high NA such as
16 (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 darkground 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 planoconvex 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 selfluminous points, is seldom realized in practice. The
Source
of light
Condenser
Diaphragm
J
Diaphragm
FIG. oi. THE USE OF THE SUBSTAGE 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 nonselfluminous 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 025. 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 TWOLENS CONDENSER
. I Av' 12
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 12 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 140. 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
Threelens "Aplanatic" Achromatic Condenser
Condenser NA = 140 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
io must be used as immersion systems, a spot of cedarwood 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 io, 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 12 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 io, 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 *
io mm. The distance between the front surface of the condenser
and the slide should then be about 02 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 planoconvex bullseye 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 030. 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
selfluminous objects which are to be still resolved in the image
given by a lens of NA 03 is
o6iA
h = 
03
Taking A = 045 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 selfluminous with blue violet light. Actually,
then, if we are resolving lines in a 30,000 band, the performance is
rather better than the "selfluminous" 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, oneeighth 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 onehalf
instead of to oneeighth. 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
03 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 selfluminous surface immediately behind it. The trans
parent areas would then function as selfluminous 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 wavelength 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
selfluminous 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 selfluminous.
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 narrowangled 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 wellunderstood 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 midelement 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 wellknown "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 wavelength 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 wavelength 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 wavelength 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 "isophasal" 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
> . 12 cos 4W
n [lsinW)U2cos 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 socalled "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 twothirds to threequarters 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 firstclass microscope
FIG. 79
objective of NA at least 12 when working with a condenser giving
a symmetrical cone of illumination of NA 09 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 fineresolu
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 selfluminous 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 threedimensional 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.
Darkground 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 DARKGROUND ILLUMINATOR
becomes the seat of disturbances propagated in all surrounding
directions. Fig. 81 is a perspective view. The electric forces in the
wavefront 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 wavelength, 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 wavelength, 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 darkground illumination.
Optical Systems. One of the earliest inventions for onesided,
darkground 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 darkground 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 12 to 133. If no direct light
is to enter the objective the NA of the latter must be below about
105. 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, 14,
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 1333. Hence
the NA of 1333 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 glassair 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
wavelength of light) of the air at the surface. Advantage is taken
of this in illuminating socalled
"smear" preparations of bacilli
with such darkground 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 darkground 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 DARKGROUND , .:. ,<.__ :* ;,.
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 12 mm.). The focusing type can be used with
other thicknesses within reasonable limits.
The 'Slit Ultramicroscope. 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/TRAMICROSCOPE
(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 " Ultramicro
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 onesided illumination of Siedentopf and
Szigsmondy. The concentric darkground illuminator is capable of
revealing fine structures, such as flagellae of microorganisms,
delicate spines, etc., which cannot be readily observed in other ways.
The inner structure of bacteria is also shown up by darkground
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 Darkground
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 darkground 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 darkground 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 brightground 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 " darkground " 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 Darkground
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 onesided 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 darkground
illumination is used.
The above particular effects are entirely due to the wavenature
Diffracted Conical Wavefront
Diffract}^ or
Scattering
Filament I
Incident /
Wavefrp'nt
FIG. 93. DIFFRACTION BY A FILAMENT
of light, and must not be interpreted as indicating any structure
in the ultramicroscopical particles.
Effect of a Linear Filament. Referring to Fig. 93, let WW be a
plane wavefront 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
wavefronj;, 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 wavefronts 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 "ultramicroscope" 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 semiangular 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 darkground illumination consists
of very delicate needlelike 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 Ultraviolet 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 wavelength. The limit of trans
mission for stable glasses in the ultraviolet region is approximately
03^ (Vol. I, page 245). Optical media available for regions of
shorter wavelength 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 socalled
"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 = 02749^. 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 ultraviolet
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 125 shows it to be about o2/*, 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 wellmade
micrometer screws with tangent screw, or by wellregulated 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 wellmarked 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 ULTRAVIOLET 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 ultraviolet 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 ultraviolet. Although the definition of the visual image
obtained in the first place is poor, owing to the spherical aberration
of the lens for that wavelength, the difference of the focus in the
two cases can be measured with sufficient accuracy to come very
near the ultraviolet 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 selfexplanatory. 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 darkground 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 ultraviolet 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 ultraviolet 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 threedimensional figure such as shown in Fig. 100. This is the
socalled "optical indicatrix," an ellipsoid with semiaxes 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, GlanThompson, etc.), which can be easily
swung in and out as desired. This is followed by a diaphragm and
by a substage condenser, usually of a twocomponent 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 planepolarized 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 squareended 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 swingout 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 coverglass. 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
birefringences, and in this way the maximum or sometimes the
average birefringence of the mineral can be estimated which,
together with other signs, is usually sufficient for identification.
THE MICROSCOPE
139
Alternatively, the birefringence 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 crosswire 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 socalled 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 socalled " stauroscopic "
figures are found. A short discussion of the "ring and brush"
appearances was given in Vol. I, pages 218220. 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), 107120.
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, 1139.
12. Knowledge, 1912, p. 37.
13. Zeit.f. Wiss. Mikros., XXXII (1915), 1633
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 wellknown 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 retinocerebral
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
lensprisms. 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 threedimensional 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 parallelaxis 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 naturallooking 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 05 and 15 (French gives 06 and 16 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 bluegreen. 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
bluegreen 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 5770 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 oi 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 lefthand 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 lefthand picture on the right of the
stereoscope, the righthand picture on the left.
A similar effect can be produced by an arrangement of mirrors,
as in the socalled " 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
midpoint 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/.
Stereotelescopes. 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 stereotelescopes 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 nonerecting 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 STEREOTELESCOPE
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 stereotelescope, 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.
61
= 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.
61
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 doublebarrelled 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 halfsilvered
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 drawtube 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 lig. 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), 371394.
5. J. W. French: Trans. Opt. So'c., XXIV (192223), 226.
6. J. W. French: Trans. Opt. Soc., XXTII (192122).
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 (191819), 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 nonadjustable, 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 7/25 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 threecolour 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 selfluminous.
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 056 at 30, and 025 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 luneshaped
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. Wideangle
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 socalled 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 03 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 * *' 13' 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 reread 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 overgreat, 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 pincushion distortion. The "symmetrical" combination is
more likely to be distortionfree, 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 oi 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 03 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 06 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, 05 per cent to
176
APPLIED OPTICS
io per cent. Photograptdc mapping requires distortion within
oi 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 "undercorrected"
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 overcorrection 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 overcorrect it in a simple lens
system.
Fig. 140 shows the cases of undercorrected, 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 overcorrected 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
15186
16041
603
378
295
286
705
714
569
606
< Dense barium crown .
2 (Light flint.
16089
15472
57'4
458
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)
 02533
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 glassps 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 =  138 for the old achromat, and R =  194 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 welldesigned 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
(QiQ<) 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 overcorrected 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 undercorrected 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
wavelength, 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 wavelength.
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
 15181 c Flint 15783
15783 d Crown 15152
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 //I25. 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.
Henrisymmetrical 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 (7/45)
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//25 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 threelens 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 wouldbe 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 (7/25) ANASTIGMAT
** v n d v
1. ?6i3 565 3 I6I3 583
2. 1651 337 4 I'6i3 5$'5
FIG. 153(6). THE TAYLORHOBSON (//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 semiangle 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
yvoT 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
0B
+20
w
/ +;
Spherical Aberration Astigmatism
FIG. 156. RUDOLPH ANASTIGMAT, ZEISS PROTAR TYPE (7/125)
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 COMAFREE 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 extraaxial 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 dissymmetry 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 (Jifis 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
wavelength = 05/4, 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 oneseventh of a millimetre, or about
T  F in. This limit is on the severe side, and a patch of pf^ in., or
025 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 025 mm., then df = 20 mm. for //8, and the total
depth of focus would be of the order of 40 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) ac
and from the triangles with their apex in P,
(Mg _ I  61 2
c a
., I //i i\ fc
giving 0/2= /iH 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 \ac a + c) atc*
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 000058 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 75 cm., with a slightly wider angular field. The prin
cipal distance, i.e. the distance for best focus, is 30 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)
= 40
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
undercorrection, 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 wavelength
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
wavelength. 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 wavelength 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 fixedfocus tele
photo lens.
As distinct from the magnifying effect, the expression "telephoto
effect" is sometimes used:
Tii.,. 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 (//56)
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. DISTORTIONFREE TELEPHOTO LENS DESIGNED BY LEE
(Messrs. Taylor, Taylor & Hobson)
distortionfree 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 pincushion 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
pincushion 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 pincushion distortion. . . .
We are then left with a residuum of undercorrected 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), 2225.
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 crosssection 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 crosssec
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 "candlepower" (intensite lumineuse)
of the source for that direction. The candlepower is usually denoted
by J. Thus
rfF
Candlepower 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 selfluminous surfaces
(a redhot poker is a good example), and of many types of diffusely
reflecting surface (such as blottingpaper) 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 "candlepower" 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
candlepower 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 candlepower 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 Selfluminous Surface. Imagine a
small element, of area ds, of selfluminous 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 candlepower 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 crosssections 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 restate 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 (80 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 candlepower 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 selfluminous; 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 (15381615).
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 015 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 io 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 = 0344 for ordinary arcs,
and 0285 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 25 mm. in diameter,
but in the larger 500 c.p. lamps the light is mostly derived from
a glowing plate 5 mm. square.
The widespread availability of electricity is making the older
sources of less importance, but the oxyhydrogen 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 091 in. ; to find the focal
length of a lens to give a 13 ft. picture at a 50 ft. throw we have
,, . . .. 50 x ooi
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 id 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 "bullseye" lenses, each planoconvex, the convexities being
turned together. The powers of the components are arranged so
that roughly parallel light passes between the two members. The
planoconvex 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 glassair 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 twolens
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 maladjust
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 "airglass 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
78
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 78 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 37 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 candlepower, 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 reradiated; but the total radiation = trB lumens per sq. ft.
PHOTOMETRY OF OPTICAL SYSTEMS 229
Sooo
Hence, B = . The candlepower of the elementary area ds, is
77
therefore (8000 dsj
7T
and the lumens radiated into a cone of semiapical 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'/  123. This is the illumination of the screen
in footcandles, 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 semiangular 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 Imageforming
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 imageforming system, so that
Illumination =
__ Bk (area of exit pupil)
We therefore find that the whole exit pupil of the imageforming
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 "candlepower" 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, motorcar headlights, lighthouse 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 nonfulfilment 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 sharplybounded 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 extraaxial 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 nonspherical 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 "candlepower" 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
candlepower 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 (Dd) tan
2(Dd)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 (192829), 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 wavelength 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 widerangled 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 coverglasses 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.
Squaringon. Slight asymmetry of the appearance of the Airy
244 APPLIED OPTICS
disc rings near the best focus may indicate imperfect " squaringon "
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 "squaringon" 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
blueviolet 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 applegreen 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 orangered border, but the colour is very much less marked
than with a noncorrected lens; outside the focus there appears an
outer fringe of applegreen surrounding a yellowish disc with a
faint reddishviolet 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 bluegreen instead of an
TESTING OF OPTICAL INSTRUMENTS 245
applegreen 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 undercorrection . 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 "undercor
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 "overcorrection."
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 undercorrection and overcorrection
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 extrafocal 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 Undercorrection
FIG. 187. RAY DIAGRAMS FOR STAR TESTS
in the distortion of the extrafocal 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.
Extraaxial 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, extraaxial and focal
6 Astigmatism, extraaxial 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 extraaxial "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 squaringon 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
wellcorrected 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 extrafocal 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 Raypath 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
wavefront, 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 wavelengths 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 knifeedge 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 knifeedge. 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 knifeedge 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 knifeedge 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
knifeedge 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 plateholder 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 plateholder 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
225
1125
30
45
135
6, 14,22,
34
675
1575
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 threedimensional trace of the rays from
a large refractor. The coordinates 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 wavefront 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
midpoint between A and B, whereas it would have been at E on
FIG. 194. THEORY OF GARDNERBENNETT 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 wavelengths =
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 wellcorrected 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 Shadowfringe 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 1020 lines per mm., while Jentsch uses com
mercial process screens with 48 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 coordinates, and projects the image of the
point P in the grating plane as P' in the projection plane. The
origin of coordinates O is the paraxial focus, and the intercept
OB = s is the axial spherical aberration associated with the angle
u. The coordinates 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 gs
A 4. V^T?
and tan u =   
ps
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 socalled 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 wavelength 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
FabryPerot interferometer, but in this case two halfsilvered 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 wavelengths. 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 wavetrains of plane
waves which unite after passing through the system (Fig. 201).
If their directions are inclined at a small angle a and the wavelength
is A, it can easily be seen that a steady interference system of
maxima and minima distributed in planes equally inclined to each
Pinhole 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 wedgeshaped 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 (ni)(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 = 15
U^ A *
we shall only find onethird 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 WAVETRAINS
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 wavefront, 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 wavefront 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 midzone
where the phases of the reunited waves agree, and shade off to the
minimum both at centre and margin where the path differences
would reach half a wavelength. 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 wavelengths 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 (192021), 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 (192122), 28.
8. Kingslake: Trans. Opt. Soc. t XXIX (192728), 133.
9. Kingslake: Trans. Opt. Soc. t XXVII (192526), 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 (192324), 97.
19. Martin and Kingslake: Trans. Opt. Soc. t XXV (192324), 213.
20. British Patent 130224 (1918).
21. Kingslake: Trans. Opt. Soc., XXVIII (192627), i.
22. Twyman: Trans. Opt. Soc., XXIV (192223), 189.
23. T. Smith: Trans. Opt. Soc., XXVIII (192627). 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
SmithHelmholtzLagrange 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 ./fca./' ./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 wavetrain with its wavefronts
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 wavelength 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 twodimensional
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 {(2rrm 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 wavetrain 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 midpoint.
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 midpoint 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 midpoint.
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.
/>(Nl)/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 wavelengths, 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 wavefronts 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 wavelength
of light must be of a similar character. Fig. Ay represents a plane
wavefront 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 cophasal if the wavefront 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 131139 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 302304. 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'seye 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
Darkground 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
FabryPerot interferometer, 260
Fieldglass, 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
Squanngon, 243
Stanhope lens, 1 1
Star test, 241
Steinheil eyepiece, 45
magnifier, 15
telescope objective, 26
Stereoscope, 145
Stereomicroscope, 162
Stereotelescope, 153
Stoney, 101
Stopnumber, 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
VARiABLEpower 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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deals with a particular type of motorcycle from the point
of view of the ownerdriver ..... 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 motorcar from the point of view
of the ownerdriver. The functions of the various parts of
the car are described in nontechnical language, and driving
repairs, legal aspects, insurance, touring, equipment, etc., all
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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
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ALTERNATING CURRENT BRIDGE METHODS. By B. Hague,
D.Sc. Second Edition 15
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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
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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
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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.
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DYNAMO, THE: ITS THEORY, DESIGN, AND MANUFACTURE. By
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Edition
Volume I 21
II 15
III 30
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Edition, Revised and Enlarged 20
ELECTRIC AND MAGNETIC CIRCUITS, THE ALTERNATING AND
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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
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ELECTRIC CIRCUIT THEORY AND CALCULATIONS. By W. Perren
Maycock, M.I.E.E. Third Edition, Revised by Philip Kemp,
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ELECTRIC LIGHT FITTING, PRACTICAL. By F. C. Allsop. Tenth
Edition, Revised and Enlarged 76
ELECTRIC LIGHTING AND POWER DISTRIBUTION. By W. Perren
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C. H. Yeaman In two volumes .... Each 10 6
ELECTRIC MACHINES, THEORY AND DESIGN OF. By F. Greedy,
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ELECTRIC MOTORS AND CONTROL SYSTEMS. By A. T. Dover,
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ELECTRIC MOTORS (DIRECT CURRENT): THEIR THEORY AND
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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 TRAINLIGHTING. 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.,
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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.
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ELECTRICAL INSULATING MATERIALS. By A. Monkhouse, Junr.,
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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
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Vol. VIII 30
ELECTRICAL WIRING AND CONTRACTING. Edited by H.
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ELECTROTECHNICS, ELEMENTS OF. By A. P. Young, O.B.E.,
M.I.E.E 50
FRACTIONAL HORSEPOWER 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 MERCURYVAPOUR 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
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OSCILLOGRAPHS. By J. T. Irwin, A.M.I.E.E. . . .76
POWER DISTRIBUTION AND ELECTRIC TRACTION, EXAMPLES IN.
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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 ONEVALVE 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., EngineerinChiefs 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
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TELEVISION : TODAY AND TOMORROW. By Sydney A. Moseley
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D.I.C., A.M.I.E.E. Second Edition . . . .76
PHOTOELECTRIC CELLS. By Dr. N. I. Campbell and Dorothy
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WIRELESS MANUAL, THE. By Capt. J. Frost, I.A. (Retired), Re
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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,
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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.
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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
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. Mathematics for Engineers contd. s. d.
MEASURING AND MANURING LAND, AND THATCHER'S WORK,
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METALWORKER'S PRACTICAL CALCULATOR, THE. By J . Matheson 2
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METRIC LENGTHS TO FEET AND INCHES, TABLE FOR THE CON
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SLIDE RULE: ITS OPERATIONS ; AND DIGIT RULES, THE. By A.
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STEEL'S TABLES. Compiled by Joseph Steel . . . .36
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BUILDER'S BUSINESS MANAGEMENT. By J. H. Bennetts
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COST ACCOUNTS IN RUBBER AND PLASTIC TRADES. By T. W
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5
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GRAMOPHONE HANDBOOK. By W. S. Rogers . . .26
HAIRDRESSING, THE ART AND CRAFT OF. Edited by G. A. Foan. 60
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HOUSE DECORATIONS AND REPAIRS. By W. Prebble. Second
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MOTOR BOATING. By F. H. Snoxell 26
PAPER TESTING AND CHEMISTRY FOR PRINTERS. By Gordon A.
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ELECTRIC MOTORS, SMALL. By E. T. Painton, B.Sc., A.M.I.E.E.
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ELECTRICITY IN STEEL WORKS. By Wm. McFarlane, B.Sc.
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