DISPLACEMENT INTERFEROMETRY BY THE AID OF THE ACHROMATIC FRINGES PART III BY CARL BARUS Hazard Professor of Physics and Dean of the Graduate Department in Brown University PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON WASHINGTON. 1919 CARNEGIE INSTITUTION OF WASHINGTON PUBLICATION No. 249, PART III is' 77 PRINTED BY J. B. LIPPINCOTT COMPANY AT THE WASHINGTON SQUARE PRESS PHILADELPHIA, U. S. A. CONTENTS. CHAPTER I. — The Displacement Interferometry of Long Distances. PAGE. 1 . Introduction 7 2. Apparatus. Fig. i 7 3. Rigorous equations. Figs. 2,3 8 4. Ocular micrometer. Fig. 4 II 5. Collimator micrometer. Fig. 5 15 6. Half-silvered films 1 6 7. Direct observations 18 8. Indirect observations. Fig. 6 19 9. Ellipses and hyperbolas 21 10. Compensators. Figs. 7, 8 23 1 1 . Number of fringes visible 25 12. Separate adjustable auxiliary mirrors. Figs. 9, 10, n, 12 25 13. Types of achromatic fringes 27 CHAPTER II. — The Interferometry of Small Angles. Methods by Direct and Reversed Superposed Spectra. 14. Introductory 29 15. Method with prism. Figs. 13, 14, 15 29 16. Estimate 30 17. Equations 31 18. Observations. Prism-prism method. Fig. 16 34 19. Interference from rough surfaces. Figs. 17,18 36 20. Reversed rays. Figs. 19, 20, 21 37 2 1 . Second method. Figs. 22, 23 40 22. Equations 41 23. Observations. Figs 24, 25 43 24 . Reversed rays 46 25. Fringes from rough surfaces 48 26. Direct interferences without cleavage prism. Fig. 26 49 CHAPTER III. — The Elastics of Small Bodies. 27. Introductory method. Fig. 27 53 28. Apparatus. Figs. 28, 29, 30 54 29. Preliminary observations. Figs. 31, 32, 33 55 30. Rods in metallic sheath. Figs. 34, 35, 36, 37, 38, 39 59 31. The same. Thinner rods, hard rubber. Figs. 40, 41, 42, 43 59 32. The same. Brass 61 33. The same. Glass. Fig. 44 63 34. The same. Steel 64 35. Modifications of apparatus. Figs. 45, 46 64 36. Observations. Figs. 47, 48, 49 65 37. Apparent yield within the apparatus 66 38. Ocular micrometer. Collimator micrometer 68 39. Summary. Figs. 50, 51 . . 69 CHAPTER IV. — Experiments in Gravitation. I. GRAVITATIONAL ATTRACTION. 40. Introduction 73 41. Equations 73 42. Observations. Floating system. Figs. 52, 53 74 43. Expeditious fringe detection 75 44. Heavy needle in air 76 45. Light needle in air. Figs. 54, 55, 56 76 46. Summer experiments 79 3 4 CONTENTS. II. USE OF THE RECTANGULAR INTERFEROMETER IN CONNECTION WITH HORIZONTAL PENDULUM. PAGE. 47. Introductory 83 48. Apparatus. Figs. 57, 58, 59 83 49. Equations. Fig. 60 84 50. Observations. Fig. 61 86 51. Observations continued. Figs. 62, 63, 64, 65 88 CHAPTER V. — The Interferometry of Vibrating Systems. 52. Introductory 91 53. Telephonic apparatus. Figs. 66, 67 91 54. Observations. Figs. 68, 69 92 55. Binlar systems. Figs. 70. 71 94 56. Further observations. Figs. 72, 73 96 PREFACE. The present report is chiefly devoted to the investigation of methods of research in which displacement interferometry, conducted by the aid of the achromatics discussed in the preceding report, gives promise of fruitful applications. Thus, in Chapter I the method of measuring small angles hitherto suggested is given a practical test. The general theory of the sub- ject in its bearing on the two possible methods is developed at some length and a variety of interferometer devices, with mirror, ocular, and collimator micrometers, are instanced. Unfortunately, it was not till after the end of these experiments that I detected the method of reducing the fringes to the smallest number possible, practically to a single fringe; otherwise the work would have been more satisfactory throughout. As the achromatic fringes can not (in general) be found without first finding the corresponding spectrum fringes and, conversely, since for each type of spectrum fringes (direct or reversed) a corresponding group of achro- matic fringes may be associated, I have devoted Chapter II to spectrum fringes differing in their manner of production. The endeavor here has been to obtain interferences from distant slender luminous objects, without the aid of a slit. Partially at least the work has succeeded, but not as far as I hoped. The experiments are very difficult. The work in the third chapter was undertaken at the request of Prof- W. G. Cady, of Wesleyan University, in the endeavor to obtain the elastic constants of small bodies. The application of the displacement method proved at once to be astonishingly easy in a case where a degree of rough handling is inevitable; but there lurked in the elastic apparatus some dis- crepancy, both of viscosity and hysteresis, the nature of which escaped detection even after many attempts to locate its origin. Chapter IV contains applications of the rectangular interferometer using achromatic fringes to geophysical problems. A method for the determina- tion of the Newtonian constant is worked out. Again, the same interfero- meter is associated with the horizontal pendulum for the detection of small changes in the inclination of the earth's surface. Series of observations extending between January and August are recorded. Finally, in the last chapter, I have investigated corresponding methods for the interferometry of vibrating systems. The luminosity of the achro- matic fringes lends itself easily to this purpose and it was merely necessary to 5 6 PREFACE. design an appropriate vibrating telescope. To test the method, a study is made of the vibration of telephonic apparatus. For the first time I have obtained clear-cut interference vibration curves for two identical telephonic systems joined directly in series, while these forms subsided completely when the telephones were joined differentially. Such a system, from another point of view, is an electric dynamometer capable of appreciating an average alternating current well within a microampere. It could, moreover, be synchronized with an external impulse by aid of the Lissajous curves with the same accuracy as two tuning-forks. CARL BARUS. Providence, August iQi8. CHAPTER I. THE DISPLACEMENT INTERFEROMETRY OF LONG DISTANCES. 1. Introduction. — Methods for the measurement of small angles and of long distances were broached at the end of the last report. It is the purpose of the present chapter to continue the work experimentally, with a view to further development. It will therefore be desirable to collect the useful equations in this place in relation to the form of apparatus to be adopted, as well as to deduce the consequences of these equations in relation to their bearing on displacement interferometry in general. Throughout the chapter the work is done chiefly with the aid of the achromatic fringe groups, as I have called them; but these, as a rule, can be found only by means of the spectrum fringes, wherefore the latter become of coordinate importance. 2. Apparatus. — This is an interferometer of the Jamin-Mach type (fig. i) with four vertical plate mirrors, M,Mr, N,N', in parallel and at 45° to the horizontal beam of impinging white daylight L, from the country beyond. Three of these mirrors are half -silvered, viz, M,N,N', while M' is or may be opaque. The equal distances ab = cd = b constitute the base-line (6) of the appa- $ . ratus and the rectangle abed will be called / ;/ K • ^p ;/ r ^ the ray parallelogram. Its area is 2Kb, /», vicl/:/' : dl/' where ad = bc = 2R. Each of the mir- rors must be provided with thf£e adjust- ment screws for fine motion, so that the mirrors are each slightly revolvable about a vertical and a horizontal axis. In case of the mirrors N, M', these screws must be convenient to manipulate (thumb-screws), for adjustment here will frequently be necessary. The opaque mirror M' is on a Fraunhofer micrometer slide with its screw in the direction of the normal n' to M' and adapted to readings of at least icr4 cm. In the figure S, S' may be regarded as slides of a lathe-bed on which the carriages P and Q carrying the mirrors may move longitudinally. Any distance ab = cd is thus available for a base-line. The telescope is at T and receives both the light from L after two reflec- tions from the paired mirrors and the direct light from K through the half- silvers. It is desirable that the two beams from K and L be of about equal intensity. The pair of mirrors N and Nf is on a vertical axis A, so that it may be rotated as a whole. The amount of rotation may be read off either directly on a divided circle corresponding to the axis A or indirectly by the displacement of M' on the screw at nf in relation to the observed sweep of interference fringes. 7 Vi or" ol 8 DISPLACEMENT INTERFEROMETRY BY Thus there are three objects seen in the telescope: the direct landscape from K, the reflected landscape from L, and the achromatic interference fringes due to the partial beams abc and adc when the apparatus is in adjustment. To find the latter the method pursued in the preceding report suffices. It is first necessary to find the spectrum interferences when L is a beam of intense white light from a collimator and fine slit and the telescope is provided with a direct-vision prism. These are to be centered by moving the micrometer at M' and adjusting the pair of mirrors M'N. The spectro- scope is now removed (swiveled out) and the slit broadened, whereupon the intense achromatic fringes will appear covering the position of the image of the originally fine slit. If not vertical, the fringes may be made so by further slightly rotating M' and N on a horizontal axis, in the absence of compensators. The collimator at L is then removed and the fringes will be found super- posed on the foreground. If they are not bright enough from the light of the landscape, they may be given any intensity by reflecting (by way of adc and abc} a narrow horizontal strip of skylight from white paper into the telescope from L. Intense fringes will be seen transverse to the strip. 3. Rigorous equations. — In addition to the sides of the ray parallelogram b (base) and 2R (R radius of rotation) we shall have to consider the following angles or angular increments: A a the angular rotation of the paired mirrors, A0 the corresponding angular displacement of the fringes, AJV the linear dis- placement of the micrometer mirror in a direction normal to its face, and A
2A(f> cos i
~AN ~ AN /An ~ X
Moreover, if 5 is the angle at the apex of the distance triangle on the base 6,
(4) A s =
(5)
And since the distance d = b/2S = b/2 A a, from (5),
bR F
(6)
2 AN cos *
THE AID OF THE ACHROMATIC FRINGES. 9
so that the sensitiveness is from (6),
f x
(7) w = — »(AAO =
(a) It will next be desirable to deduce the above fundamental equations
more rigorously than has thus far been done. Figure 2 is supplied for this
purpose, and represents the more sensitive case where, in addition to the
mirrors MM,' NN' (all but M being necessarily half-silvers), there is an
auxiliary mirror, mm, capable of rotation (angle a) about a vertical axis a.
The mirrors, M ---- N', in their original position, are
conveniently at 45° to the rays of light, while mm is 2 (sin 0o+0 cos 00)2+e2a2/v = e*Kp?
where K = 4 (/x2 — sin2 i] .
This is an ellipse if n at the center corresponds to a maximum value, in
terms of the variables 0 and a, so long as K and /* are considered constant.
But as n and therefore K vary with X, though slowly, it is true the equation
is more complicated.
When the center of ellipses is not in the field, but passes through the ver-
tical plane corresponding to the center of the field of view, the ellipses may
soon become appreciably straight lines in their visible contours, and the
fringes must rotate in one direction or the other, according as the center is
above or below the field. Rotation will be rapid when the vertical axis of
the ellipses is relatively long. To bring the center into the field (for a proper
value of N), the angle a must be zero, i. e., the two corresponding opaque
mirrors which reflect the interfering beams must be rotated on a horizontal
axis towards each other, or from each other, until a = o, or the horizontal
plane through the field is a plane of symmetry.
Furthermore, since the fringes necessarily move toward or from the center
of ellipses with change of N, the motion of fringes will necessarily be oblique
if the center of ellipses is obliquely outside the field of view. In the limit, if
the center is in the vertical plane specified, horizontal fringes will rise or fall.
Finally, if n passes through a minimum instead of a maximum, the fringes
will be roughly of the hyperbolic type.
At the center of ellipses in case of spectrum fringes, n is therefore a maxi-
mum relatively to points of the spectrum in the same vertical or transverse
line of homogeneous color. This maximum is due to obliquity, the hori-
zontal one to change of X. In the case of fringes produced with white
light (without dispersion), like the colors of thin plates generally or the achro-
matic fringes discussed elsewhere, the center of ellipses (which are now cir-
cles) is an absolute maximum, horizontally or vertically, i. e., relative to
points in all directions from the center and for each color. The center of
THE AID OF THE ACHROMATIC FRINGES. 23
spectrum ellipses, therefore, has no direct relation to the center of white light
fringes; for the latter occur only when the rays pass the plate normally.
On the other hand, when the white-light fringes are straight lines correspond-
ing to very oblique incidence of interfering rays, the spectrum fringes are
none the less perfect ellipses.
It is finally necessary to account for the coincidence in adjustment of the
center of spectrum fringes and the achromatic fringes, as the latter overlie
the coincident white slit-images from which the superposed spectra are
produced by the grating. This is easily seen to be referable to the fact that
interferences with white light can only be visible if the light in the region
of interference, when analyzed spectroscopically, contains but few dark
bands. Since the number of bands in the spectrum is least near the center
of ellipses, and is further reduced on making them as large as possible, the
relation is obvious. In the case of strong, large achromatic fringes, a single
fringe virtually occupies the whole spectrum. The light is either white or black.
The displacement of the center of ellipses with the angle of incidence for
a given adjustment may be computed from the original equation for centers
N =e (cos r-\-2 J?/X2 cos r), where r is the angle of refraction for the incidence
i, e the thickness of plate, and B the dispersion constant. When N and e
do not vary it may be shown that (since n = A +.B/X2) ,
d\_ X sinr cos i (2.B/X2 — ^ cos2 r)
di 2B/\2 fj. cos2 r (2+cos2 r)
To obtain an estimate 2B/\2 = 0.026 may be neglected as compared with
H cos2 r and the equation given the approximate form (^ = 1.5)
d\ X sin * cos i
— =- - = 10 X sin i cos *, nearly.
di 0.026 2.5 jj.
Thus, if t' = 45°, d\/di = $\ or 0.09 X per degree of i, which is about 100 times
the distance of the sodium lines. If i = o° or 90°, the shift vanishes.
10. Compensators. — When the fringes are found they may be erected as
stated by rotating either pair of the diagonal mirrors of the ray parallelogram
towards or from each other, usually on a horizontal axis. The fringes may be
enlarged by rotating the paired mirrors on either end of the ray parallelo-
gram and restoring the fringes after each small step of rotation by displace-
ment AAT at the micrometer. But these processes are tedious and must be
very cautiously performed or the fringes are liable to be lost. The same re-
sult may be accomplished by the aid of plate-glass compensators, about
0.5 to i cm. thick, placed normally in each of the two interfering beams
and originally parallel and vertical. (See fig. 7, C and C'.) In addition to
rotation and enlargement, these compensators serve with further advantage
in equalizing the two beams in intensity. For this purpose it is merely
necessary to half-silver lightly the compensator in the stronger beam of
the ray parallelogram. If the fringes are more nearly vertical (between
24 DISPLACEMENT INTERFEROMETRY BY
45° and the vertical) they may be erected by rotating either compensator
or both around a horizontal axis and enlarged by rotating around a vertical
axis. The two compensators should be actuated together in opposite direc-
tions if there is danger of the fringes leaving the field; i. e., if the adjust-
ment is considerable. Similarly, if the fringes are more nearly horizontal,
and particularly when horizontal fringes are wanted, the fringes may be
leveled by rotating the compensators in opposite directions around a vertical
axis and enlarged by rotating around a horizontal axis. Horizontal fringes,
which climb up and down the broad white slit-image and may be made
quite large, are often very advantageous. The illuminated field is much
more extensive in the vertical direction.
For slight adjustments it is convenient to have the compensators nearest
at hand rotate in the same direction as the fringes. This may be done by
working either on one side or the other of the center of fringes. The com-
pensators may easily be ma-
nipulated by hand (without ,-' T~ "\
tangent screws) and they are
most efficient when nearly
normal to the respective
beams. To pass from vertical
to horizontal fringes one would
first rotate the compensators in opposite directions around a horizontal axis
until the fringes are inclined about 45°, after which the further rotations
would be made in opposite directions around a vertical axis. The motion of
fringes indicates the proper direction of the rotation of the plates.
To account for these apparently complicated effects, it is sufficient to re-
call that the compensators displace the center of fringes, usually enormously
distant outside of the field of view, and besides that invisible with white light ;
for fringes are visible only in the narrow strip for which the spectrum fringes
are very large and centered. Hence the result of rotation around a hori-
zontal axis is to change fringes (fig. 8) of the type a, through c (vertical)
into b, while the center moves downward, and vice versa. Again, rotation
around a vertical axis changes fringes of the type a, through c' (horizontal)
into d, as the center moves from left to right, and vice versa. The effects are
necessarily opposite for the two beams. If the fringes are made vertical
as at c, rotation of a compensator around the vertical axis can have no effect
of rotation of fringes ; for the center moves in the line of symmetry ; but the
effective or differential thickness of plates (e) is changed and hence the
fringes are increased or decreased in size.
In view of the presence of compensators, C and C', figure 7, the original ad-
justment is much simplified ; for it is necessary merely that the spots of sun-
light on the mirrors at a and b, figure 7, and at a' and br, one or more meters
off, be at the same level and at the same distance apart, nearly. The accu-
rate adjustment at c and d for coincidence horizontal and vertical is then
made with the telescope at T. When the distances are approximately equal,
THE AID OF THE ACHROMATIC FRINGES. 25
fine spectrum fringes will nearly always appear. These are enlarged and
centered as specified. A broad slit with the spectroscope removed will then
show the achromatic fringes, which may in turn be enlarged and rectified.
Horizontal fringes, though often convenient, are at fault, inasmuch as they
must rotate with the displacement of the micrometer AN. This appears at
once from figure 8, for the center of ellipses is shifted. Vertical fringes alone
are free from rotation in relation to AN.
11. Number of fringes visible, etc. — To get the most promising conditions
for observing coincidence in case of range finding, the direct and reflected
images should be about equally intense. Hence, if a is the coefficient of
reflection, the two equal intensities are
(i —a2) = 2(1 — a) a2, or a= 1/2
It is best, however, to have a less than this and to darken the direct beam
if necessary by a thin half-silver plate interposed in the beam. If a = 1/2 the
images are too dark and require higher illumination of the foreground than
is usually present. As for the achromatic fringes themselves, they may be
obtained with clear plate and opaque mirrors almost as well as with half-
silvered plate, if the supernumerary images are partially screened off. With
optic plate glass they would not appear. The surprising appearance of
satellites — i. e., repetitions of the group with increasing faintness — is also
common with clear plate.
A series of experiments were made by replacing the half-silver plates with
grid-like opaque mirrors. These are easily made by removing the silver along
parallel lines (using a T-square) with a sharp wet stick. The slit images
were then also gridlike in appearance and the achromatic fringes occurred
only on the dark bars. For clearly the superposition of beams takes place
on reflection from glass only. In this way the fringes on the supernumerary
slit images were identified. These occurred on the bright bars. The two
phenomena are therefore complementary.
The reason for different numbers of visible fringes is less easily understood.
In the original experiments two achromatic fringes (black or white) , with about
three green-reddish fringes on either side and rapidly fading out, were alone
visible. This narrow grid is very advantageous if displacement interfero-
metry is in question, for the achromatic fringes are easily recognized. Subse-
quently, however, large numbers of less distinctive fringes (20 or more) were
usually obtained. As a small number of fringes is as frequently obtained
with clear glass as with half-silvered plate, the occurrence is not attributable
to the silver. (§ 13.) A variety of experiments were made with lenticular
compensators, convex or concave, in each beam. The fringes, though
obtained without difficulty, were usually rounded and irregular and the
results without interest.
12. Separate adjustable auxiliary mirrors. — To obtain strong interfer-
ences the two component rays eadT and bfcT of the rectangular inter-
26
DISPLACEMENT INTERFEROMETRY BY
ferometer must not only be parallel on entering the telescope T (fig. 9), but
they must be locally coincident at the mirror M; i. e., the two pencils from
the collimator L (M',N,Nf being half-silvers) finally entering T, must very
nearly coincide. Otherwise, even when the path-difference is annulled
and there is perfect coincidence of the slit image, no fringes may be obtained.
This is often a great annoyance when the mirror M' is on a micrometer screw
(ri) normal to the face of the mirror; for on continuously displacing M' the
y
xe*
N
Tft e — • >
,m', ^
a*
/eA"'
\
d
1 /"- * 7 c-"
9
ir
_v lo
T7l
-*fi
\ a- /
^
^ ^' /
:*' ^"
•-U
rays cMT and dM'T separate more and more fully and the fringes soon
vanish, unless a fresh adjustment for local coincidence is made. It is for
this reason that the fringes are often so hard to find. The achromatics are
much less sensitive to this disadjustment than the spectrum fringes; but the
former are so mobile and easily lost that they have to be found as a rule by
the aid of the latter.
To meet this difficulty there must be one mirror available which reflects
the component beam normally and which may be displaced parallel to itself ;
i. e., whose micrometer screw is parallel to the incident and reflected ray.
This condition is most easily secured by separating the auxiliary mirror into
the parts m and m', each normal to its respective ray, while m' only is on a
micrometer screw n1 ' . Under these circumstances there is no difficulty in
finding the fringes after the adjustment for parallel rays and local coincidence
at M' has been made once for all by actuating the mirror m' in one direction
or another. Moreover, it makes no difference, within limits, how the paral-
lelism and local coincidence are secured by moving any of the mirrors M,M',
N,N', m,m', all of which must be on three leveling screws. Finally, if the
mirror m and m' rotate as a rigid system about a common axis, it is still
possible to use mm' for the measurement of small angles.
If the rays a and c, which may be of any length, are very long, the adjust-
ment shown in figure 10 is preferable, as the observer at T is near the
micrometer screw n'. Here M, N, N' are half-silvers.
THE AID OF THE ACHROMATIC FRINGES. 27
For the case, however, in which the body whose small angles of rotation are
to be measured is part of another apparatus which does not admit of manipu-
lation, the method may be modified as in figure n. Here all the plate
mirrors M,M',N,N' are half-silvers and the rays from the collimator L form
the interfering pencils LemeagdT and Lbfm'fct. The mirror m and m'
meet their rays normally and m' is in the micrometer screw at n' parallel
to /. The mirror m", on the axis A normal to the paper, rotates, and its
small angles of displacement are to be measured. Considerable light is lost
in the three penetrations of half-silver films by each ray; but in case of the
achromatic fringes the light is usually in excess, so that the diminution
of light is an advantage. It is more difficult, however, to find the spectrum
fringes, as these require a slit.
The plan of figure n is carried out more simply in figure 12, where both
reflections take place at the same mirrors M and M', respectively, the compo-
nent rays being Lbm"bach T and Ldem"efgi T. It is necessary to incline the
parallel mirrors m' and m" on a vertical axis, in order to avoid the entrance
ray L' into the telescope at T. But this separates the component rays h and i
locally, so that means must be employed (compensators, rotation of the other
mirrors m", N, or M') to obviate this as far as necessary. In both cases of
figures ii and 12 it is therefore not easy to find the fringes, and I did not
persevere in the quest because of an eye affliction contracted at the time.
Similarly the system M, M',m" of figure 12 might be used if a half -silver
is placed at r and the telescope at T' to the right of it. In this case the mirror
m" must be in two parts, with adequate air-space inserted into the shorter
ray Lb.
13. Types of achromatic fringes. — The difficulty of obtaining fringes of
the strictly achromatic type (i. e., two strong fringes with a black line be-
tween and the remaining fringes green-reddish and faint) in the rectangular
or other interferometer, has been frequently referred to in the text above.
As a rule the fringes found are more or less diffuse, non-symmetric, with
large numbers (10 or more) about equally strong. Such fringes are, of course,
useless in displacement interferometry. When the sharp fringes needed are
obtained, their definition is independent of the particular part of any of the
glass plates used, and any plate may be rotated 180° in its own plane without
spoiling the sharpness of the fringes. Hence such slight curvatures or wedge-
shapes as the plates may possess are without influence on the phenomenon.
To further test this I devised a screw-press adapted to push the vertical
edges of a plate to the rear and the middle forward, so as to give the plate
marked cylindricity. Quarter and eighth inch plates were operated on, in
the latter case sufficiently to give the two superposed slit-images quite
unequal width; but no essential or useful improvement of the fringes was
observed. The type of the fringe was not altered. Again, the symmetrical
fringes may be obtained from plates thickly or thinly silvered, without essen-
tial difference.
28 DISPLACEMENT INTERFEROMETRY.
It follows, therefore, that the relative thickness of the glass paths traversed
by the interfering beams can alone be of influence in shaping the fringe pat-
tern in the manner in question. This is in consonance with the general
theory of achromatic fringes, the result being a superposition of the color
phenomena due to the dispersive refraction of the glass and the colors re-
sulting from the wave-lengths of the interfering rays. To test this the ap-
paratus, figure 10, is particularly convenient, as the fringes are easily found.
Moreover, both rays, a and c, from the collimator at L, eventually pass
through the plate N' before reaching the telescope at T. It is thus merely
the thickness of the half -silvers M and N, both at 45°, that is here in ques-
tion. If this thickness is the same, the sharp symmetrical design of but
two strong fringes appears. If the difference of thickness is but little over
0.5 mm., many fringes, non-symmetric in distribution, are the rule. If the
differential thickness is several millimeters there may be hundreds of fringes.
If these are small they may be enlarged at pleasure; but they are always
faint and useless for measurement.
CHAPTER II.
THE INTERFEROMETRY OF SMALL ANGLES, ETC.— METHODS BY DIRECT
AND REVERSED SUPERPOSED SPECTRA.
14. Introductory. — It occurred to me that a number of the methods treated
in my papers on direct and reversed spectrum interferometry might be used
directly for the measurement of small angles and possibly of the distance
of the source of light. Such a procedure would have an apparent advantage,
at least theoretically, of not calling for the preliminary superposition of two
images of distant objects, as the superposition is inherent in the method
itself. But there are large constants involved, which make the result very
problematical unless these constants can be removed by a compensator. In-
deed, it is also very questionable whether such interference can at all occur.
A further difficulty which hampers the method is the decrease of size of
objects as their distance increases. Nevertheless a progressive investigation
with the object of ascertaining to what degree the experiment is feasible is
worth while, and as it will be convenient to develop the methods without
reference to the ulterior conditions which limit the interferences, this method
has been pursued.
15. Method with prism.— Figure 13 is a sketch of one of the methods in
which 5 is the distant source of light, from which rays d and d' strike the
mirrors m and n, and are thence reflected to
the silvered sides of the right-angled prism P.
After leaving it the rays enter the spectroscope
at T in parallel. If the proper angles are se-
lected the prism P may be replaced by one of
any angle or by a reflecting grating.
Suppose now the system mPn is securely at-
tached to a rigid metallic beam or rail capable
of rotating around a vertical axis at its center
(P). This is indicated in figure 14, where the
direction of rays and the normals of mirrors
have been drawn and where the angle of rota-
tion a has placed mPn into the position m'Pn' .
The result is that a part y of the ray d is cut
off on the left side and a part x added to the
ray d' on the right side, so that the path-difference, which may be assumed
to have been zero originally, is now appreciably incremented, but not sym-
metrically for both sides.
It may be shown, however, that the rays n'P2Tz and m'P\T\ still enter
the telescope in parallel and that therefore the conditions of interference have
29
30 DISPLACEMENT INTERFEROMETRY BY
not been disturbed. This is the interesting feature of the method, for the
angle a between the two positions of the rigid beam will also be the angle
between all corresponding normals of the mirrors, as indicated in the diagram.
If we take the case on the left, the angle between incident and reflected ray
at P will therefore be ^2+40: for the original mirror at P and TT/ 2 -\-4a- 2
a = 7r/2 + 2Q! for the new position at PI. But the angle between the rays re-
flected at m and m' respectively as 2 a. Hence if T\P\ is prolonged backwards
it must intersect the line mn at the original angle \ and thus P\Ti is paral-
lel to PT, Similar reasoning applies on the other side for PiTi and will still
hold if the direction of the ray Sn prolonged is reversed. Finally, ir/2 may
be any reasonable angle.
It will contribute to a more adaptable design of the apparatus for general
interferometry if the ray Sn' may also be reversed by reflection (fig. 15,
mirror n") in parallel to itself, allowing a small lateral offset, similar on both
sides for clearance of the mirrors. Reflection between fixed parallel mirrors
on the left in d and between mirrors set at a reentrant right angle on the
right, say at n" ', would accomplish this at corresponding distances for the
transverse rays. Again, half -silvers may be used at m and n for reflection,
which method is probably best. These details will here be disregarded.
If small angles are to be measured the direct method is enormously more
sensitive.
16. Estimate. — The full expression for the path-difference corresponding
to the rotation of rail a will be complicated and of no interest here. It is not
sufficient to regard the intercepts y and x as solely contributing to the path-
difference, which would therefore be x-\-y for the direct case and x — y for
the case when the ray d' is reversed somewhere at n" (fig. 15) and returned
parallel to itself. It may be shown that for small angles a, if /3 is the angle
between incident and reflected rays originally at m and n and b the distance
mP = Pn, d the distance Sm = Sn,
are sufficiently approximate equations up to the squares of small quantities
to meet the interference for the direct and the reversed cases respectively.
Hence, if for instance a=i° =0.0175; b=i meter = io2cm.; d= i kilom. = io5
cm; X = 6X io~6 cm., the number of fringes corresponding to each of the terms
may be computed as
(Direct) w = 6Xio4— io3+6o (Reversed) w = io3 — 60
In the first case over 61,000 fringes pass per degree of rotation, a= i°. This
makes about 2. 9 Xio~7 or about 0.06 second of arc per fringe. But the
method is insensitive as regards distances d, unless the first two terms can be
compensated. In the second or reversed-ray case, the method would be
relatively much more sensitive as regards d if the first term 2baz could be
compensated. The difficulty lies in the occurrence of «2 in the term,
whereas most compensators would act as the first power of a.
THE AID OF THE ACHROMATIC FRINGES. 31
Furthermore, if the angle a is small and S is displaced over an angle a or
a distance r = da to the right, the original triangle may be regarded as
restored. Hence the same number of fringes roughly should pass back
again. In the second case, supposing 2bo? can be removed by compensation,
r = da and a = \d/2b2, nearly, or
or the object should be located to 30 cm. at a kilometer for each fringe pass-
ing. In this case d need not be known, since
n\
and n fringes are observed to pass for the angle of rotation a in the compen-
sated apparatus. The direct rays without compensation would of course
give indefinitely better results if d is known; for the angle per fringe has been
found as a = 2.9Xio~7 when r = da = 0.03 cm. per fringe if d is i kilometer.
Unfortunately, however, the method of figure 14 can not be rigorously
carried out experimentally. For in any practical apparatus the mirrors
M and N would have to rotate at a fixed distance from each other, apart
from the micrometers; i. e., the two mirrors rotate on a rigid radius or rail
and are therefore both rotated and displaced. It is this displacement which
is relatively of much importance and by it all terms involving the first order
of distance d are wiped out, so that terms of the second order in b/d only
remain.
17. Equations. — To derive these equations certain intercepts of the rays,
figure 14, in addition to x and y, b and b' may be defined. PI PZ is the trace
of the vertical plane of symmetry of the right-angled prism, if rotated at
an angle a to the right. In this case the reflected ray n'P^q on the right cor-
responds to the reflected ray m'P\ on the left, both terminating in the common
wave-front P\qs before entering the telescope.
Let n'P2 = c' = b sin /3/cos a sin (f} — a)=zl/sin a cos a
m'Pi = c =b sin /3/cos a sin (/3+a) =z/sin a cos a
P2t = z' = b sin a sin /3/sin (/3 — a)
tq = z =6 sin a sin /3/sin (/3+a)
and nn' = x = b sin a/sin (/3 — a) = z'/sin /3
mm' = y =b sin a/sin (/3+a) =2/sin /3
since the original angles at the ends of the base are /3 and the rotation a.
The angles between incident and reflected rays are respectively /3 — 2a at n',
/3+2a at m', 90° — 20. atP2, and go° + 2a at P\. Most of the angles are indicated
in the figure. The new radii m'P = b' = b sin /3/sin (0 — a); n'P = b" = b sin
/3/sin 03+ a).
The rays, however, do not reach the planes of symmetry, but are reflected
by the faces of the right-angled prism, and this may be sketched in, in the
rotated position (angle a) at Pipp'. The path of the reflected rays from n'
32 DISPLACEMENT INTERFEROMETRY BY
is now n'rs and from m', m'P\ before they meet in the common wave-front
P\qs. Hence the intercepts
rs = v = (2+2') (cos a — sin a)/(sin a-f cos a)
rP2 = w = (0-fV)/cos a (sin «+cos a)
will enter in treating the path-differences. On the left the rays have not
been distributed.
If we take the direct case first the original path-difference SnP and SmP
may be regarded zero or n and m in the same phase. On rotation, therefore
(angle a), the path-difference is increased on the right by x-\-cf — w-\-v and
increased on the left by —y+c, so that the total path-difference is equiva-
lent to the equation
riK = c' — c—(w — v)-\-x+y
If the above equivalents are inserted, this equation may be reduced to
sin /3 cos /3/cos a — sin2 /3 sin a + sin /3 cos a
n\ = 2b sin a-
sm (p-\-a) sin (p—a)
in which the three terms in the numerator correspond to the respective
intercepts c' — c, w — v, x-\-y.
Since a and /3 are small angles, we may write sin a = a, cos a=i— a2/2.
and cos 0 = b/d. Therefore the equation would, for practical purposes,
become
n\ = 2ba — 2baz-\-2bza/d
the three terms corresponding to the xy, wv, and cc' effect.
In the case of reversed ray (fig. 2) we may consider the points m' and
n' in the same phase. Hence the original path-difference (« = o) is x — y.
The path-difference after rotation c' — w-\-v — c. The total change of path-
difference due to rotation is thus given by
n\ = c' — c— (w — v) —x-\-y
This differs from the preceding by the deduction of 2*. The rays again ter-
minate in the common wave-front P\qs to enter the telescope. Hence after
reduction
sin /3 cos /3/cos a — sin2 8 sin a — sin a cos /3
n\ = 20 sin cc -
sin (j3-f a) sin (/3 — a)
the terms showing the cc', wv, and xy effects. The approximate form of this
equation is thus practically
The wv effect predominates, the cc' effect is intermediate, and the xy effect
very small if d is large, as already instanced.
The preceding equations may also be obtained geometrically by letting
fall the normal from n (fig. 14) to the prism-mirror and prolonging the ray
at s backward. In the isosceles triangle so formed the angle at the base is
45° — a. Hence in the above notation the path-difference takes the form
x+2 (c'—w)cos2 (45° — a) — (2' — 2) — (c— y)
THE AID OF THE ACHROMATIC FRINGES. 33
On inserting the values of the quantities as given above and reducing, an
equation identical with the above appears, which for small a. is
n\ = 2ba (cosec /3 — a+cot 0)
If the prism has its nose at P (nearly), or in the axis of rotation, a small
correction is to be added to the preceding expression. The path-difference
on the right is increased by
, i + cos (90° — 20) t 2 sin a
cos a (cos a+sin a) cos a — sin a
and increased on the left by
z 'I (cos a — sin a) cos a
Hence the correction is on reduction
2b sin2 a sin 0/sin 5 = 2&a2, nearly.
This merely wipes out the small middle term, — a, of the above equation,
leaving n\ = (2ba/sin 0) (i +cos 0)
When the prism is reduced to reflectors in its plane of symmetry, as treated
at the beginning of this paragraph, the equation loses the terms w — v and
reduces to
n\ = xJt-y-\-z-\-z'-\-c' — c, or to n\ = 2ba (i/sin /3-f-i)
In the practical apparatus the mirrors m and n rotate on a fixed radius
b, whereas b in the diagram elongates on the right and contracts on the left
respectively to
b" = b sin 0f sin 8 b' = b sin j3/sin a
Hence the mirrors in the apparatus are displaced normally on the right and
left by
e—(b" — b) cos j8/2, inward, and e' = (b — b') cos 0/2, outward.
The path-difference thus introduced is the sum of the decrease on the right
and increase on the left and its value is 2e cos i, when i is the angle of inci-
dence in question. Thus the correction is (after reduction)
(b" -b} (cos a+cos (0-a)) + (6-b') (cos a+cos (0-f-a))
The expression may be further reduced to
260: cos 0 sin 0 .
-(cos 0+cos a)
sin a sin 5
when a is a small angle, or to
260: (i/sin 0 — sin /3-f-cot 0)
If this quantity is deducted from the above equation for path-difference and
direct rays there remains simply n\ = zba sin 0. The latter, therefore, is
the equation to be used in interpreting the observation. So that generally
when i = 0/2 for the micrometer at n
2 cos i A./V/Aa: = 26 sin 0
In the case of reversed rays the conditions on the left remain the same
as before. But on the right the mirror n is set at an angle 0/2 to the rail
3
34 DISPLACEMENT INTERFEROMETRY BY
and at right angles to its former position. Hence the normal displacement is
e = (b" — b) sin 0/2 . The angle of incidence is i = 90° - (13/2 - a) . Thus the
path-difference here to be deducted is 2 e cos i or
2 (6" -b) sin 0/2 . sin (0 — «) = (&" — 6) (cos a— cos(0— a))
and the total deduction from both sides is therefore
(&" — &) (cos a — cos 5) + 2 (6 — 6') (cos a+cos a}
This expression when reduced gives for small a
zba (cot 0 — a cot 0/sin 0)
or more simply
2ba cot 0
It is practically as large as the total path-difference for reversed rays found
above. If, therefore, the two effects are opposite in sign, the path-difference
introduced by rotation would be zero, apart from the change of glass paths
and second-order effects which are relatively small. In fact, the experiments
show that the rotational effect, Aa, in case of reversed rays, is relatively
negligible as compared with the effect in case of rays not reversed. In other
words, if from the equation for direct rays
ri\ = 2ba (i/sin 0+cot 0)
we deduct
. . , . , / / • 0 \ a\
2X-\-2e cos i-\-2e cos ^ = 2ba (i/sin 0-)-cot 0)
the right-hand member vanishes to the second order of small quantities.
18. Observations. Prism=prism method. — In this case (fig. 19 below) a
sharp-angled prism at S, with its knife-edge vertical, cleaves the beam
of white light issuing from a collimator, reflecting the beams d and d' as
described in my earlier papers. The system should be leveled so that all
corresponding rays lie in a horizontal plane. By making the strips of light
on both mirrors m and n (figs. 13, 14) coincide horizontally and vertically
(using an auxiliary lens, if necessary) and then placing the prism P so that
the rays mP and nP all but escape at its edge, the adjustment may be com-
pleted by aid of the telescope at T. The two slit-images, which should be
equally bright, are made to coincide horizontally and vertically by the ad-
justment screws on m and n. If now the direct-vision spectroscope (prism
grating) is swiveled in front of the objective of T, fringes will usually appear
when the path-difference is annulled. For this purpose the prism P is placed
on a Fraunhofer micrometer with the screw in the direction mn. The spec-
trum fringes are as a rule easily found and are quite strong, but they can not
be centered in the field of view, for the occurrence of ellipses presupposes
the rigorous superposition of the two strips of light on the edge of the prism P,
which is not possible. The fringes, if too oblique, may be erected by a plate
compensator with a horizontal axis, or the prism P may be rotated on a
horizontal axis. Vertical spectrum fringes are not usually wanted in these
experiments, for they are to serve only as an essential aid to finding the
achromatic fringes.
THE AID OF THE ACHROMATIC FRINGES.
35
16
It is a curious fact that although the ellipses can not be produced nor the
slit much widened, apparently achromatic fringes do occur in white light for
a micrometer placement at P such as should produce centered ellipses.
Moreover, as the white slit-image is linear, the achromatic fringes are pref-
erably made to run transverse to it. They are then exceedingly brilliant,
extending much beyond the slit-image, and they travel up and down it with
the motion of either micrometer at P or at n, (A./V) , or with the rotation of
the rail (Aa) . As there are but four or six fringes with but one or two strong
and brilliant, they make an exceedingly sensitive index for measurement.
The occurrence of achromatic fringes may also be detected in the solar spec-
trum, as all the Fraunhofer lines (homogeneous light) become helical and
broad from the cross-hatching due to the fringes. Here with homogen-
eous light the fringes are indefinite in number and follow each other contin-
uously, whereas with white light but one or two intense black-white fringes
appear. Though the achromatic fringes are by far the most
brilliant part of the phenomenon, they rarely occur without
streamers. The general appearance is roughly suggested in
figure 1 6, where ss is the white slit-image in the telescope and
a the achromatic fringes moving up and down 55 when A./V or
A« change. In the lateral glare of the field, however, fan-
shaped or radiating coarse fringes bb are seen, intersected with
very fine hairlike fringes cc. Probably there is also an intermediate group.
These streamers are very useful to register the approach of the achromatic
fringes, which move so rapidly that they are easily lost.
A few measurements or rather estimates were made to coordinate the values
of A./V of the micrometer displacement at n and the corresponding rotation
Aa of the rail necessary to annul this displacement. To do this the achro-
matic fringes were placed on the cross-hair, or better, on the image of the
cross-hair at the slit of the telescope, and both readings were taken. They
were then displaced by rotating the rail and restored by moving the microm-
eter. To measure the rotation an index was placed at the end of the rail
(radius 27 cm.) moving over a millimeter scale observed with a lens. The
constants of the triangle, figure 13, were
6 = 20 cm. d = 62 cm. /3 = 7i.3° * = 35-6°
Corresponding readings were found as follows in two separate adjustments:
10* AN
I04Aa
10* AN
io4Aa
0
O
0.0
o
204
9
26.9
7
493
19
76.0
30
Mean A7V/Aa = 26.5 cm. /radian
79.8
132.5
33
56
178.3
74
1/9-3
74
204.0
85
Mean AN/Aa = 2^ cm. /radian
36 DISPLACEMENT INTERFEROMETRY BY
Since AAr/Aa = - ^ = 23.3 the observed data are above the computed
2 cos i
values, but not more so than the difficulties of these measurements on an
improvised apparatus imply. A much more refined method for finding Ac*
is, of course, essential.
19. Interference from rough surfaces. — The question now at issue is
whether the interferences can be retained when the collimator is removed and
the light comes directly from a ground-glass surface or a Nernst filament. The
spectrum fringes go at once when the slit is widened ; not so the achromatic
sets. Having produced them clearly with sunlight, I found that a ground-
glass screen or a scratched mica film could be placed at c or b or a, figure 17,
whereas 5 is the slit and L the coliimating lens.
The fringes should be transverse, as in figure 16,
as vertical fringes are too easily confounded with
the white slit-image. The slit was now broadened
or quite removed; but the fringes, though less
prominent from excess of non-interfering light, remained in place distinctly
and without other change. On removing the lens, however, the fringes invariably
vanished .
I now replaced the sunlight by the light of a Nernst filament, under the
impression that ground glass might to a small degree still behave like plate
glass. The same experiments were made, the filaments at e (fig. 17) replacing
the ground glass. In this case, however, I first removed the lens L and it
was then seen that the two washed slit-images were not superposed, as is
otherwise obvious; but it accounts for the failures of the experiments with
sunlight. Superposing the two vague images both out of focus, a position
was soon found in which the achromatic fringes appeared brilliantly. The
slit could now be widened or removed at pleasure, yet the fringes persisted
strongly, but with loss of brilliancy.
It is thus possible to obtain these achromatic fringes directly from the
Nernst filament and without a collimator; but they are so mobile, with
change of Aa and AJV, that to find them it is necessary first to produce the
spectrum fringes with collimator and spectro-telescope ; then to find the
achromatic fringes on removing the spectroscope; next to remove the lens
of the collimator and adjust for superposed images; and finally to remove
the slit. These non-collimated achromatic fringes are best seen in a par-
ticular focal plane of the telescope and they change their focal plane with
displacement (Aa, A/V). They practically cover the whole width of the
washed slit-image. They usually measure about 0.5° in width, but the
streamers may extend laterally five times further, depending on the adjust-
ment. When pronounced, the slit -images may even be separated as in figure
1 8, while each alone retains the achromatic fringes. This puzzling phenom-
enon, which I had previously obtained, is probably due to the intersection and
interference of rays in a region in advance of the plane of vision. Finally, as
THE AID OF THE ACHROMATIC FRINGES.
37
the transverse arrow in figure 1 9 (indicating the right and left side of the slit-
images) show, after the reflections at p m n P, although the slit-images are not
reversed, the superposed rays are reversed; for these constitute the right-
hand and left-hand radiation from the filament,
separated by the prism p. If the slit is widened or
removed, there is only one vertical line of rays (co-
inciding with the position of the slit before removal)
which can interfere. The remaining light does not
interfere and its admixture robs the phenomenon
proper of its brilliancy.
A few experiments made on the nature of the achromatic phenomenon here
obtained showed that the fringes are probably Fresnellian interferences. To
test this the objective of the collimator was removed and strong fringes were
obtained by passing the two washed images of the slit over each other laterally,
by moving the corresponding adjustment screw on the mirror n. It was
found that the fringes passed from horizontal maxima in size, gradually to
vertical hair-lines, as the images slid horizontally from contact of their nearer
edges to the contact of the further edges. The coarse fringes were even
strongly present in the narrow gap between slit-images before contact. The
telescope was now focussed on the slit, so that sharp linear images appeared.
The fringes vanished; but it appeared that the coarse fringes corresponded
to coincident sharp slit-images when observed out of focus, and the fine
fringes to sharp slit-images far apart. The whole phenomenon thus depends
on the distance apart of two lines of light and the interferences are observ-
able before or behind their plane.
20. Reversed rays. — The apparatus was now adjusted for the reversal of
the rays d' by adjusting a mirror at some place n" (fig. 15) on a fixed microm-
eter and in such a way that the rays on reflection retraced their path. The
mirror at n being a half-silvered plate, in turn reflected the
rays toward the prism P. This modification of apparatus in-
troduces very considerable path-difference, 2 nn", on the right,
which must therefore be compensated on the left. It is diffi-
cult to accommodate the micrometer and leveling devices at
n and n" without an allowance of 5 to 10 cm. of path-excess.
In my first experiments, which were merely tentative, the com-
pensation on the left was secured by inserting a glass column
about 15 cm. long. With this and the right-and-left microm-
eter displacement of the prism, or the to-and-fro motion of
the mirror n", path-difference was easily annulled and the
ellipses found in the spectrum. They are centered as usual by rotating the
glass compensator on a horizontal and a vertical axis, till with the occurrence
of parallel rays at T the illuminated strips on the prism coincide, locally, to
the eye.
In view of the long glass path and therefore of considerable dispersive effect,
38 DISPLACEMENT INTERFEROMETRY BY
the ellipses are small and the spectrum is filled with innumerable lines. More-
over, in view of the prism separation (at p, fig. 19) the ellipses are throughout
half -ellipses, all terminating in the vertical axis. For the two areas or strips
of light (ab, fig. 20) seen on the face of the grating and entering the spectro-
telescope are each single, being one-half of the full area of light rays capable
of interference obtained at the collimator. This results in the half -ellipses e.
If the prism is replaced by a half-silver plate as in the next paragraph,
the strips ab1 and a'b are both double, the full areas being superposed;
thus the areas ab and a'b' give rise to the full ellipses ee' . Hence, also, the
vertical axis in e, being at the edge of the prism P, is not quite clear. Hori-
zontal lines do not occur. These half-ellipses move with displacement of
the micrometer at n', or at P, or on rotation of the rail mPn, as a body. It
is difficult, however, to use them for measurement, as their vertical terminus
is not sharp enough. If AA/" is the micrometer displacement corresponding
to the rotation Act, we may write
aAAT/Aa = o
I did not succeed, however, in obtaining trustworthy results with the half-
ellipses.
The achromatic phenomenon can not occur when glass columns are used
for compensation from the great number of lines in the spectrum. To obtain
large ellipses the dispersion effect B/\2 must practically vanish. Hence an
air-path compensator is to replace the glass column. This is conveniently
made (as shown in fig. 21) of two pairs of parallel opaque mirrors ab and cd.
The pair ab are clamped between short lengths of square brass tubing and
cd similarly and at right angles (nearly) to the pair ab. Both are mounted
normally to a horizontal brass table t, provided with three leveling-screws,
capable of being raised and lowered and of rotating around a vertical axis.
The path-excess introduced is thus equivalent to ab and cd and the ray dm
is collinear with ra. This compensator not only introduces path-differ-
ence, but since the mirrors are capable of rotating as a whole both around
a vertical and a horizontal axis (leveling-screws), the beam dm may be
moved right and left or up and down without ceasing to be parallel to ra.
If, therefore, the ray entering T (fig. 19), were first made parallel, the ray
d may be adjusted by the compensator until the strips of light on P practi-
cally coincide at its edge.
With the use of this air compensator or offset, the fringes were found with-
out much difficulty and enlarged as specified. In view of the reflection at
P, only half fields are returned; full ellipses or horizontal lines are not ob-
tainable, as explained. But on removing the spectroscope and cautiously
advancing the micrometer at N, the achromatic fringes eventually appear.
In the present experiments these fringes did not take the usual and desirable
form, consisting of but few fringes with the middle member in black and white.
Probably because of the many reflections at mirrors (fig. 21), none of which
was perfect, the fringes were now colored and present in large number
THE AID OF THE ACHROMATIC FRINGES.
39
without much distinction between fringes. On being made transverse to
the white image of the fine slit, they cross-hatched it from top to bottom.
Nevertheless, their rapidity of motion is such that they serve quite well for
measurement, the datum being more accurate than the measurement of
Aa. The comparison was carried out in the same manner as before, the pres-
ence of the achromatics being successively destroyed by rotation (Aa) and
restored by the normal displacement (A7V) of the mirror at n. In this way the
following data among many others were obtained. It is necessary to dis-
place the mirrors very carefully; for if the fringes are lost they are extremely
difficult to find without beginning with the spectrum fringes all over again.
AaXlo3
AiVXio8
AaXio8
AA7Xio«
O.
o.o cm.
XH-4
16.0 cm.
2.6
2.1
18.2
19.0
6.7
7-o
21.8
21.4
8.2
9-4
25-5
25-9
"•5
12.3
The range of Aa is much increased by removing the objective lens of the col-
limator, and this is done after the observation marked x in the table. The
fringes are perhaps even more distinct when present in the absence of the
lens. The constants of the apparatus were:
6 = 21 cm.; 18 = 70.7° ; ^ = 64 cm.
From this the rate AAf/Aa= 1.05 was found graphically. In the other series
the rates were above 0.9. Approximate estimates of the same value were ob-
tained with the spectrum ellipses and the glass column. This result again
differs from the computed value, AN = o. The reason may lie in the fact that
the plane of symmetry of the prism P (fig. 14) did not pass through the axis
of rotation, or was not originally midway between the mirrors m and n. To
test this inference (which will again be treated in the next section) the fol-
lowing experiments were made:
The prism P was as carefully as possible centered by the eye, so that its
plane of symmetry passed through the axis of rotation. In this case the
relative measurements
icfi AN = 0.0
10'Aa =O.O
1-5
2.2
4-5
4.8
6.6
7-4
8.1 cm.
9-3
showed a mean coefficient of AN/Aa = o.8j. Finally the prism was moved
to the right, i. e., with its plane of symmetry on the other side of the axis.
The results were
io'AN=o.o 3.0 4.4 7.0 cm.
io3Aa =0.0 3.0 4.4 6.7
giving a mean rate AA/"/Aa = 1.05. Thus the shifting of the prism right and
left has made but little difference and can not account for the discrepancy.
It is probable that the coefficients found are largely due to the half-silver
glass mirror n (fig. 15), which rotates with the rail mn. To test this a com-
40 DISPLACEMENT INTERFEROMETRY BY
pensator c of the same thickness and g]ass may be placed in the beam mP
on the left. If c and n are parallel, both originally at an angle /3/2 to the
beams traversing them, it is obvious that the compensation will not be de-
stroyed by the rotation of the rail, provided c is fixed while n rotates. If c
is effectively thicker than «, the part of the coefficient due to the compensator
may become negative. This is apparently the case in the following experi-
ments, in which a compensator was installed as in figure 15 at c:
AATXio3= —o.o —.4 —1.4 — 2.8 —4.0 —5.2 cm.
AaXio3= o.o 2.6 5.9 9.3 12.2 16.0
The rate here is AN/Aa = —0.31, so that the zero value is exceeded. How-
ever, the path-difference in the compensator of thickness e at an angle of
incidence i and refraction r, viz, e (cos i — sin i tan r) a, where 2 = 90° —
jS/2, here becomes 0.77 (0.816 — 0.578X0. 6i8)a = o. 3530:, so that the whole
difficulty is not explained away.
Finally, a few experiments were made to compare the effect of displacing
(A7V) the micrometer at n" (fig. 15) as compared with the effect of a microm-
eter (AN') which displaces P in the direction of its plane of symmetry.
The latter (AN1) is zero when a = o. Generally if e is the normal displacement
of the prismatic faces, the path-difference is
ze (cos (45° —a) — cos (45° -fa)) = 2AN sin a.
since e = AN sin 45°. Hence, as AAf is a normal displacement
AN /AN' sin a
The results obtained were
io3AiV' = o.o 50 100 150 200 225 cm.
io3&N =0.0 1.5 3.4 5.4 7.2 9.2 cm.
Thus the mean rate is a = AN /AN' = 0.036 or a is a little over 2°. To obtain
these data the achromatic fringes were used as above. When the slit-images
seen in the telescope are not quite parallel, they may be made so by slightly
rotating the slit on a horizontal axis normal to its plane. The images
rotate in opposite directions. A slight angle between the images is, however,
of no consequence.
21. Second method. — In view of certain difficulties encountered in the use
of reflecting prisms, in particular the loss of rays at the edge, the limitation
to half -ellipses, etc., the method of figure 22 enlarged in figure 23 was devised.
In this the prism is replaced by a half -silvered plate PP'. Hence the rays
issuing at 5 and reflected by the opaque mirrors at m and n are thereafter
respectively transmitted and reflected by the half -silvered plate at p, and then
reach the spectro-telescope at T together. When the path-differences are
sufficiently equal, elliptic interference fringes will be seen in the spectrum.
When first found they are usually very fine straight lines; but they may be
rectified by plate compensators in the beams d and d' or mp and np, though
THE AID OF THE ACHROMATIC FRINGES.
41
the operation is not easy. Leaving these details for further consideration,
the procedure for angular measurement may advantageously be treated here.
For this purpose the half -silver P and one opaque mirror, n, for instance, are
mounted on a rigid bar with an axis at P. The other mirror m is to remain
fixed. If the bar is now rotated over a small angle
a, figure 23, the mirror at n is displaced to n' and
the ray Sn prolonged (intercept %) is now reflected
from n' to q and thence along T into the spectro-
telescope, parallel to its original direction or to
the other ray mp. Hence the interferences remain
intact, but many fringes pass during the transfer.
The persistence of parallelism is easily seen, be-
cause the angle between the incident and reflected
ray at n is decreased by 20. when n passes to n',
but is again increased by 2 a owing to the rotation
of PP\ to PP' over the angle a.
To control the fringes either the mirror at n (or at m} may be displaced
on a micrometer screw normal to itself, or the half-silvered plate at P may
be displaced parallel to itself. If the angle of incidence at n is i and the nor-
mal displacement of n is e, the path-difference introduced will be 2e cos i.
Similarly if the normal displacement of the plate P is e' and the angle of
incidence i1 ', the path-difference will be ze' cos i'.
As in the preceding experiment, the mirror at n may be a half -silver, so
that the ray d' passes through it and may then be returned in its own path
by a mirror at n" on a fixed standard. The displacement of this mirror over
a distance e, parallel to itself, introduces the path-difference ze, so that the
cosines are avoided. But a much more important result is the fact that the
rays np or n'g now are stationary. The strips of light originally at p do not
therefore travel over each other while one passes from p to q and the inter-
ferences are kept at full intensity throughout. This is a great advantage.
Moreover, the half -silver plate at n compensates the half -silver Pp, which
is a further advantage, since both paths within are glass paths with high dis-
persion coefficients. It is obvious that the path-excess nn" on the d' side,
must be separately compensated on the d side. The method of doing this
by an air compensator (fig. 21) will presently be considered, as a long glass
compensator would not in general be desirable because of the sluggish motion
of the small ellipses thus produced.
22. Equations. — The equations for this case are apparently very compli-
cated. If in figure 23, m and n are in the same phase and Pp is symmetrical,
there will be no path-difference at p. When Pn is rotated over an angle a. into
Pn', the path on the right becomes nn'+n'q+qs while (ps wave-front) the
path on the left remains mp as before. The path-difference is thus the differ-
ence of these quantities, to which, however, the increased glass path at PP'
would have to be deducted, and the surface PP'.must pass through the axis P,
42 DISPLACEMENT INTERFEROMETRY BY
If the angle SnP is /3 and Pnp is 7, the values of the branch paths may be
found to be (since nP = mP = b), if /3 — a = d and 7 — a = r,
wp = w£ = 6/cos 7
wn' = b sin a/sin 5
nq =b sin /3/ sin 5 sin T
_ ^ f sin a sin /3 sin r+sin2 7 sin 6 cos r 1
sin o cos T cos 7 [ — sin /3 sin 7 sin r cos r j
Hence the path-difference is equivalent (after some reduction) to the equation
% _ & [sin a (cos 7 cos r + sin fi sin r ) + sin /3 cos 7
sin 5 cos T cos 7 j — cos2 7 cos r sin 5 — sin /3 sin 7 sin r cos rj
If a = o, then /3 = 5, 7 = ^, and the right-hand member becomes zero, as it
should. I have not succeeded in putting this equation in a much more con-
venient form.
If a is very small, so that differential expressions may be introduced, the
rigorous equation, to an approximation of the second order in a, may be
reduced to / fr,\\ to \
cos 7(i-f-cos (p — 7); i+cos (p — y)
n\ = ba — - = ba—
sin 5 cos r sin j8
If /3 is nearly 90° and if the distance Pp is p, then
n\ = ba (i-j-sin 7) =ba-\-pa cos 7
The same expression may be obtained geometrically by prolonging n'p and
T'q to m'. The triangle n'qm' is isosceles. Hence if we draw the wave-
front Pq', normal to pT, we may deduct the common length pq' from both
rays, or add it to both. Hence the path on the right will be x+b' cos (y — a)
-\-p sin 7, where bf is the line Pn' and on the left as before, 6/cos 7. Hence
(if b = p/tan 7) the path-difference is
/sin a b' i \
b I — ; h~cos (7 — a)+tan 7 sin 7 I
\ sin 5 b cos 7/
which, as above, also reduces to
ba. (i+cos (18-7)) / sin /3
There is a source of discrepancy which enters when the face PP' does not
pass through the axis P, but is eccentric. In such a case, if e is the distance
of the plate from the axis, a correction equivalent to e (i— cos a) cos 7 =
e«27, nearly, will have to be supplied. Again, if there is lack of symmetry
from such a cause, the base-lines will be b and b' and the angles 7 and 7',
so that a modified equation is suggested. Finally, from all these expressions
the changes of glass path on the left with a, if not compensated, must be
deducted. As the method admits of a good achromatic phenomenon of
reversed slit-images, it is theoretically interesting and I have given it some
study.
For the case where the ray Sn prolonged returns on itself, as from n" in fig-
THE AID OF THE ACHROMATIC FRINGES. 43
ure 23, the mirror n being a half -silvered plate, the quantity nw" = 2&a/sin .u== l-55> and 7 = 27.1°.
A further discrepancy may be sought for in the fact that if the surface
Pp (fig. 23) does not pass through the axis of rotation, this plate is both
rotated and displaced. The error so introduced may be either positive or
negative. If the displacement of plate is e' = e(i— cos a), the equation
should read
cos 7
2 AN COS
= b (sin /3 + sin 7)
where e is the distance of the plate from the axis. The following experi-
ments were therefore made with some consideration to greater symmetry of
apparatus. The constants were: /3 = 7o.5°, 7 = 27.1° and therefore i =
48.8°, or 2 cos 1 = 1.32. The data found were in different adjustments;
AATXio3
AaXlO3
AA7/Aa
2 AN cos i/ Act
110.9
5-2)
203.7
9.6
21.2
28.0
246.5
II. 6J
102.9
208.4
4-8 \
10. 0 /
20.5
27.1
47.6
2.6'
80. i
4 I
170.4
8.5
20.7
27-3
223.6
10.7 ]
257.6
12.7 J
46 DISPLACEMENT INTERFEROMETRY BY
These results agree much better with the theoretical equation than the
former, and may be considered as coinciding with it.
In the present case the attempt to get interference from rough surfaces
was not at first successful. The slit-images are reversed, as indicated by
the transverse arrows in figure 2 2 . Hence if the white slit-images are wide
there can be coincidence only in a single vertical line. Fringes with white
light will occur as a case of the interference of fine slit-images. To produce
them it is first necessary to obtain the spectrum fringes with the ellipses,
or else with horizontal fringes in the field. If now the spectroscope is removed
and the white slit-images put out of focus, the phenomenon indicated in
figure 24, where 5 is the superposed, washed slit-images wiJl usually appear, or
may be found on cautiously moving the micrometer screw. Within the slit-
image the fringes are coarse and colored, but they send out fine oblique stream-
ers into the field of diffuse light or glare, on both sides of s.
When the slit is widened these fringes are liable to vanish just 24 j//// £ ->
as the spectrum fringes vanish, except perhaps at the edges
of the images. These achromatic fringes climb up and down
the slit-image with motion of the micrometers (A/V, Aa) with
extreme rapidity and are easily lost, as there are not usually
more than 10 or 20 of them. If the spectrum ellipses are huge, the white
fringes are almost too coarse to be seen and too mobile to be controlled.
I next removed the objective of the collimator. The fringes, though much
changed in appearance, practically black and white, were not destroyed.
In such a case the slit-image shrinks vertically. To obtain a long strip a
highly illuminated ground-glass screen (sunlight and weak condenser)
should be placed in front of the slit as a source of very diffuse light. In
such a case this long white post (as it were) is covered from top to bottom
with sharp blackish and usually oblique lines, which vanish at once, up
or down, on moving the micrometer. No fringes are seen if the slit is in
focus. When considerably out of focus (as in case of the diffraction patch
in fig. 25,) strong, sharp-colored cross-markings are present, which would
be quite available for measurement. However, in this experiment, when the
slit was widened or removed, the fringes apparently vanished. The phe-
nomena as a whole seem to me to be fringes of the two white slit-images,
and seen either behind or in front of their focal plane, like the complemen-
tary fringes described elsewhere. This is confirmed by experiments pres-
ently to be described.
24. Reversed rays. — The apparatus was now adjusted for a reversal of
rays by putting a half -silver plate at n and an opaque mirror on a micrometer
(with the screw normal to its face) at some position n" (fig. 23) fixed inde-
pendently of the rotation. In this case, therefore, the intercept nn' changes
sign. Moreover, the angle of incidence at n" is o°. Hence the equation
should be
a = b (sin 0 — sin 7)
THE AID OF THE ACHROMATIC FRINGES.
47
The constants of the apparatus were
6 = 20 cm. ,8=71. 3°
Thus
7 = 34-9
— 0.57 2) = 7. 5
To obtain the ellipses a thick plate-glass compensator may be placed in
the d ray to provide for the elongation 2% in d' '. About 14 cm. of glass col-
umn were necessary. This makes it very easy to center the ellipses and to
obtain them intensely black on a colored ground by rotating the compen-
sator on a horizontal and vertical axis until the two strips of illumination at
p quite coincide when the rays T, T' are parallel. But on the other hand,
because of the thickness of glass used, the small ellipses obtained move rel-
atively sluggishly with displacement of the micrometers. The sensitiveness
decreases proportionately to the thickness of glass path.
Experiments, of which the following data are examples, were made by
alternately restoring the center of ellipses to the D lines of the solar spectrum
first by the micrometer (A/V) and thereafter by the rotation of rail (Aa).
The adjustments were very different.
ANXio*
AaXlo3
ANXio3
AaXlO8
o.ocm.
o.orad.
o.ocm.
o.orad.
53-2
13-7
24-3
1-9
66.1
17.0
33-3
4.4
58.1
14.8
44.1
"•5
43-3
7-4
21. 1
5-2
57-4
10.7
68.8
14.0
Mean 2AA7'/A« = 7.6
Mean 2A]V/Aa = 6.8
The second series changed its rate enormously, almost one-half, owing
to necessary intermediate adjustments (inserting a new zero). Otherwise
the observations are as good as the apparatus permitted; but the computed
2 AA/YAa = 7.5 is above the observed value, usually, possibly owing to an ec-
centric position of the plate Pp relative to the axis of rotation. To test this
point of view the plate Pp was displaced eccentrically toward the left of the
axis. The result should be a modified coefficient, but the following data
obtained in the same way as before fail to bear this out, however:
AAfXio3 = o.
19.9
5.9
40.2
12.2
56.8 cm.
16.7 rad.
The rate, 2AA/"/Ao: = 6.6, does not differ essentially from the above. With
these small ellipses there can not, of course, be an achromatic phenomenon.
To obtain large ellipses the glass-path difference (i. e., the dispersion) must be
abolished on both sides and an air-path difference introduced, preferably
in a way which has been shown above in figure 21. As such experiments
are so mach more trustworthy and sensitive, I did not pursue the glass-
column work further.
48 DISPLACEMENT INTERFEROMETRY BY
25. Fringes from rough surfaces. — Experiments were now made with the
use of the air-path compensator (fig. 21,) placed in the d rays when these
rays were reversed. A magnificent set of achromatic fringes were here
found, only about five in number, with the central members in black and
white. Tests similar to the above showed that they are Fresnellian inter-
ferences. To prove this the objective of the collimator was removed and
even more brilliant fringes were found on placing the washed slit-images in
contact. If these patches of light were slid over each other horizontally,
by moving the adjustment screw for rotating the micrometer mirror on a
vertical axis, the fringes rotated nearly 180°, passing from vertical hair-lines,
through a maximum of coarseness for the horizontal fringes, back to hair-
lines again. On focussing the telescope on the slit it was then found that the
large horizontal fringes corresponded to coincident slit-images in focus,
whereas for the very fine fringes the focussed slit-images are far apart. No
fringes appear on the slit-images in focus, in any case. They lie in front
of and behind the image plane. This is exactly the case found above, ex-
cept that here the edges of the washed slit-images are exchanged.
The endeavor to obtain the fringes without the slit was next tried. For
this purpose a ground-glass screen illuminated by sunlight (a, fig. 17) was
first placed in front of the slit 5 in the absence of the objective (L) of the col-
limator. The fringes were still very prominent, though the light was darker.
The slit 5 was now also removed. The fringes could then no longer be seen;
but on narrowing down the illuminated ground-glass screen a to a vertical
strip of light i to 2 mm. broad, they were unquestionably present. In such
experiments, therefore, the chief function of the slit 5 is to cut off the light
which does not interfere, so that the fringes are lost in the glare. In the
absence of such excess of light the fringes are quite visible and therefore
certainly always present. By aid of the offset air compensator huge achro-
matic fringes may be easily produced ; but they are so sensitive as not to be
manageable in an improvised apparatus.
A number of measurements were now made with the achromatic fringes
set at convenient (small) size by the air-compensator. In this work the
plate PP' (fig. 23) was moved in three steps over about i cm. For each
step a set of data was investigated. The results were in succession
f N f AArXio3 = o.o 1.9 6.1 12.4 17.8 21. 9 cm. \ 2&N
(*) 1 A V 1 J-
(A«Xio3 = o.o .4 1.2 2.3 3.3 3. 9 radian]
, . j A/VXio3 = o.o 7.1 12.2 19.2 26.6 32.6cm. 1 2A^
(2) 1 , . — = Q.6
[ Aa X ioj — .0.0 i.i 2.2 3.7 5.2 6. 3 radian J &a
, . f A/V Xio3 = o.o 9.3 19.2 27.7 37. 7 cm. 1 2AA/"
[ Aa X io3 = o.o 1.8 3.7 5.9 7.4radian j ±a
The coefficients so obtained are practically identical ; and they agree as nearly
THE AID OF THE ACHROMATIC FRINGES. 49
as may be expected with the equation, since the angle B and T are not
easily specified with accuracy. They were
18 = 71.3° 7 = 28.4° 6 = 21 cm.
so that theoretically
o! = 21(0.947— 0.476) =9-9
26. Direct interferences without cleavage prism. — The next step in ad-
vance was made by dispensing with the sharp prisms heretofore used for
cleaving the rays issuing from a collimator (or the slit simply) in the endeavor
to obtain two rays capable of interference. The assembly of apparatus is
shown in figure 26, where S is the slit (to be replaced by a Nernst filament
or a tungsten filament), m and n the opaque mirrors, pp' the half-silvered
plate. The rays dd', diffracted at 5, pass after reflection into c and c' and
may be observed by spectre-telescopes placed either at T or T1 '. In the first
experiments the distance Spr was about 4 meters and the distance mn 10 cm.
The mirrors « and pp' were on micrometers with the screws normal to their
respective faces. The distance mn must be within the limits of the wedge
of light from the slit and is therefore small, unless d is very large. Both
pp' and n are on the rotating rail (as above), whereas m is fixed. The appa-
ratus was also adjustable for reversed rays by attaching an auxiliary mirror,
normal to the rays d' prolonged through n. S being distant, this slit must be
long, as otherwise the spectrum band will be a mere horizontal line and
the fringes difficult to detect. A doublet of lenses, each about 10 cm. in
diameter and of the same focal power (1.60 cm.), but respectively convex
and concave and having a combined focal distance of about 5 or 6 meters,
is of advantage for focussing a large solar image, i to 2 inches in diameter,
on the slit. The Nernst or tungsten filament gives the same advantages at
once; but the former is too thick, at least for the initial experiments at
shorter distances.
The fringes are exceedingly difficult to find in spite of the brilliant spectra.
It was not until after about three days of searching, in which (besides sunlight)
the filaments as well as the methods of direct and of reversed rays were used,
that the experiment ultimately succeeded with sunlight. The filaments are
much less gracious. To obtain the fringes calls not only for very accurate
adjustment for horizontal and vertical spectrum coincidence, but the fringes
lie quite sharply in a definite focal plane, usually between that of the slit-
image and the principal focal plane; the rays must interpenetrate at the
plate and finally path-difference must be nearly annulled. And there are
other conditions presently to be stated. After being found they are quite
strong elliptic spectrum fringes, but when lost nevertheless difficult to
rediscover. The slit may be broadened till the spectrum lines vanish,
perhaps to more than a millimeter, before they disappear in a uniform
spectrum band.
The achromatics which coincide in adjustment with horizontal spectrum
4
50 DISPLACEMENT INTERFEROMETRY BY
fringes and are seen with the slit-image out of focus are also difficult to
find because of the short length of the slit-image. As first obtained they
lacked brilliancy and were not easily observed. Similarly, experiments
with filaments failed to show the fringes, although made in parallel with the
successful result with sunlight.
A considerable assistance in finding the fringes is an opaque screen with
a vertical slit 2 mm. to 4 mm. wide placed just in front of the objective of
the spectro-telescope, in the best position as to symmetry. This screen cuts
out rays which do not interfere and makes the fringes stronger, even though the
background is darker. Fringes are frequently found in this way when they
are all but invisible in the full spectrum.
In addition to the regular fringes, a much larger vague set seems to be
present in another focal plane. They also rotate, etc., like the regular
fringes, but the experiment led to no decision with regard to them, as
they appeared in the field erratically and could not be produced at will.
They may be shadow interferences of the principal set.
An attempt was made to register slight lateral displacements of the slit
in terms of the displacement of fringes, but as the slit-images are thrown out
of coincidence when the slit moves, trustworthy numerical data can not be
obtained. One may estimate that as a first approximation
d^
x = —\
c
if x is the lateral displacement of the slit and c the distance between mirrors
m and n. Hence
400
x0 = - 6Xio~5 = o.oo24 cm.
10
should have been equivalent to the passage of one fringe in the given appa-
ratus, or generally
2AAT cos (/3-f 7)/2 =cx/d
If (j3+7)/2 = 70°, A7V=io~4 cm., and d/c = 4o,
then
. = 2.Xio~3 cm.
could have been registered. Incidentally it appears that two vertical lines
of the slit, xo/2 =0.0014 cm. apart, would wipe out each other's interferences;
but this is not the case, as much greater slit-widths are admissible. To the
right and left of the line of the slit capable of producing interferences the
parallel lines either cease to produce parallel rays, or parallel rays come
from symmetrical but different lines.
After completing these experiments, the distance between slit S and the
mirrors m and « was increased to about 9 meters. The same lens doublet,
focussing a large solar image on the slit, was used as before. With the aid of
the slotted screen in front of the telescope and the micrometer distances
from the preceding experiment, the fringes were found without difficulty.
THE AID OF THE ACHROMATIC FRINGES. 51
In fact, in view of the longer distance d, the slit could be opened to over
a millimeter of breadth before the fringes quite vanished from the spectrum;
but on using a somewhat stronger condensing system (concave lens of doublet
preceding the convex lens) and consequently more oblique rays, a very fine
slit was needed to show the fringes. They are thus more easily found when
the rays are more nearly parallel.
With artificial light, again, I obtained no results, even after long searching.
Operating with two successive slits at about 9 meters from the interfer-
ometer, one of which received the light through the other, I found that two
independent sets of fringes very different in size and inclination could be put
in the field together. The further investigation eventually showed that the
size and inclination of the fringes is essentially dependent on the degree of
parallelism of the two slit-images. When the images are parallel, the fringes
are of maximum size and vertical. When the images are not quite parallel
(they incline in opposite directions when the slit is slightly rotated in its own
plane from the vertical), the fringes rapidly grow smaller and rotate. With
parallel slit-images the spectrum ellipses are centered in the field; otherwise
they are very far out of center. The adjustment for actual (not X-like,
coincidence must therefore be made with precision if large fringes are wanted.
Further work was also done with sunlight to obtain more pronounced
achromatics. For this purpose a compensator was inserted to equalize the
glass path in the half -silvered plate. Huge spectrum ellipses were obtained
in this way and their centers were placed above the telescopic field, so that
the fringes seen were large horizontal bars. On removing the spectroscope
and placing the slit-images out of focus, brilliant achromatics were in fact
obtained, of the concentric hyperbolic type, vividly colored and broad be-
tween the apices, and diminishing to hair-lines laterally. With these it
was possible to enlarge the slit to at least 3 mm., without destroying the
fringes, though they became more vague. It is necessary that the slit-images,
when in focus, should be quite parallel, otherwise any broadening of the slit
will wipe out the achromatics. It was possible to place a plate of ground
glass on the far side of the slit without destroying the fringes, but not on the
side towards the interferometer. In other respects the behavior was as
described in the case of achromatics in the earlier experiments with a cleav-
age prism.
Finally, the spectrum fringes and the corresponding achromatics were
obtained with the light of a Nernst filament, at first by focussing an image
of it with a strong condenser lens on the slit. The experiments, however,
are very difficult. The spectrum fringes are often weak, out of focus, and
extremely sensitive to small disadjustments in the horizontal and vertical
coincidence of the slit-images. They require a fine slit. When well produced
the achromatics are also obtainable on removing the spectroscope when the
spectrum fringes are horizontal bars. The achromatics may also be obtained
brilliantly without the condenser lens, but the adjustment must in such a
case be made first with sunlight, as the spectrum from the Nernst filament
52
DISPLACEMENT INTERFEROMETRY.
is too feeble for detecting fringes so elusive as the present. The achromatics,
however, are strong and brilliant even here (Nernst filament) .
An interesting result is obtained in case of the achromatic fringes by nar-
rowing one of the beams, for instance that coming from the mirror m (fig.
26), by a screen with a vertical slit about 2 mm. wide.
In such a case the slit-image (out of focus) is correspond-
ingly narrowed. It may be passed from side to side of
the broad washed slit-image coming from the mirror n,
by moving its adjustment screws (vertical axis). The
fringes then appear only in a particular position of the
narrow image in the field of the broader ; but when they
do appear they spread far beyond the margins of the nar-
row image on both sides. Interference thus apparently
occurs where but one beam is present. The phenom-
enon is like those instances above (figs. 18, 24, Chapter
II) and means, as I understand it, that the beams have
met in some other focal plane, though one is tempted to
conclude that interference is stimulated by resonance,
in particular as it is often impossible to find a plane in
which they have met. The achromatics may sometimes be seen before and
behind the principal focal plane, but more frequently either in the one or in
the other region only.
71'
CHAPTER III.
THE ELASTICS OF SMALL BODIES.
27. Introductory method. — At the request of Professor W. G. Cady, who
was in need of Young's modulus in case of certain crystals used in experiments
in which he is interested, the endeavor was made to adapt the above: inter-
ferometer for measuring small angles with an auxiliary mirror for this pur-
pose. The project seems feasible and apparently simple in execution when
the method of end-thrust indicated in figure 27 is used.
Here F is a rigid metallic bar subjected to the force couple
P,P', carrying the coplanar mirrors m,m' and capable of ro-