THE INTERFEROMETRY OF REVERSED AND
NON-REVERSED SPECTRA
BY CARL BARUS
Hazard Professor of Physics and Dean of the Graduate Department
in Brown University
PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON
WASHINGTON, 1916
CARNEGIE INSTITUTION OF WASHINGTON
PUBLICATION No, 249
//o
PRINTED BY J. B. LIPPINCOTT COMPANY
AT THE WASHINGTON SQUARE PRESS
PHILADELPHIA, U. S. A.
CONTENTS.
CHAPTER I. — The Interferences of Crossed Spectra.
PAGE
1 . Introductory 7
2. Coincident spectra with one reversed on a given Fraunhofer line. Figs. I, 2, 3 . . 8
3. The same. Further experiments 1 1
4. Coincident spectra with one reversed on a given longitudinal axis. Figs. 4, 5, 6. . 12
5. Interference of the corresponding first-order spectra of the grating, in the absence
of rotation. Figs. 7, 8, 9, 10 14
6. Conclusion 16
CHAPTER II. — Further Study of the Interference of Reversed Spectra.
7. Apparatus with one grating. Figs. 1 1, 12, 13, a, b 19
8. Observations and experiments with a single grating. Fig. 14 22
9. Inferences. Fig. 15, a, b 24
10. Apparatus with two gratings. Figs. 16, 17, 18 26
11. Experiments continued. New interferometer. Figs. 19, 20 30
12. Experiments continued. Homogeneous light 32
13. Experiments continued. Contrast of methods 33
14. Experiments continued. Rotation, etc., of grating. Figs. 21, 22 33
15. Tentative equations. Figs. 23, 24, 25 36
16. Experiments continued. Analogies. Figs. 26, 27 38
17. Subsidiary diffractions. Figs. 28, 29 43
18. Conclusion 45
CHAPTER III. — The Interferences of Non-reversed Spectra of Two Gratings.
19. Introduction. Method. Figs. 30, 31 46
20. White light. Colored fringes. Tables i, 2, 3. Figs. 32, 33, 34, 35 47
21. Homogeneous light. Wide slit. Transverse axes coincident. Tables 4, 5. Fig. 36.. 52
22. Homogeneous light. Fine slit. Transverse axes not coincident. Table 6. Fig. 37. 54
23. Homogeneous light. Slit and collimator removed. Table 7. Fig. 38 55
24. Inferences. Figs. 39, 40 56
25. Rotation of colored fringes. Non-reversed spectra. Figs. 41, 42 58
26. Final treatment of reversed spectra. Hypothetical case. Figs. 43, 44, 45, 46 .... 60
27. Case of reflecting grating. Homogeneous light. Figs. 47, 48 64
28. Non-symmetrical positions. Fore-and-aft motion. Fig. 49 67
CHAPTER IV. — The Distance Between Two Parallel Transparent Plates.
29. Introductory 69
30. Apparatus. Figs. 50, 51 69
31. Equations. Figs. 52, 53 70
32. Method 72
33. Observations and corrections. Preliminary work. Figs. 54, 55 73
CHAPTER V. — Interferometers for Parallel and for Crossed Rays.
34. Introduction. Methods. Figs. 56, 57 78
35. Experiment. Reflecting grating. Parallel rays. Fig. 58 79
36. Experiments. Transmitting grating. Parallel rays 81
37. Experiments. Transmitting grating. Crossed rays. Figs. 59, 60, 61, a, b 82
38. The same. The linear phenomenon. Fig. 62 85
39. The same. Inferences. Figs. 63, 64, 65 87
40. Experiments. Reflecting grating. Crossed rays. Figs. 66, 67 88
41. The same. Compensators 91
42. Miscellaneous experiments. Fringes with mercury light 91
43. Inferences. Figs. 68, 69 92
3
4 CONTENTS.
CHAPTER VI. — Channeled Spectra Occurring in Connection with the Diffraction of
Reflecting Gratings.
44. Introductory 95
45. Apparatus. Fig. 70 95
46. Scattering 95
47. Fringes with white light 96
48. Fringes with sodium light 97
49. Grating on a spectrometer. Fig. 71 98
50. Inferences 100
CHAPTER VII. — Prismatic Methods in Reversed and Non-reversed Spectrum
Inlerferometry.
51. Purpose 102
52. Method and apparatus. Figs. 72, 73 102
53. The same. Crossed rays 103
54. Another method. Fig. 74 104
55. Methods using prismatic dispersion. Fig. 75 105
56. Methods with paired prisms. Fig. 76 106
CHAPTER VIII. — The Linear Type of Displacement Interferometers.
57. Introductory 107
58. Apparatus. Fig. 77 107
59. Film grating. Adjustment. Figs. 78, 79 109
60. Michelson's interferences no
61. Film grating. Another adjustment. Fig. 80 in
62. Equations 1 1 1
CHAPTER IX. — The Use of Compensators Bounded by Curved Surfaces.
63. Introduction 113
64. Lens systems 113
65. Effective thickness of the lenticular compensator. Fig. 81 115
66. Observations largely with weak lenses and short interferometer. Figs. 82, 83 1 16
67. Remarks. Fig. 84 118
68. Observation with lens systems on both sides. Figs. 85, 86 1 19
69. Telescopic interferences. Figs. 87, 88, 89, 90, 91 120
CHAPTER X. — The Dispersion of Air.
70. Introduction. Table 8 124
71. Observations with arc lamp 124
72. Observations with sunlight. Single tube. Table 9 125
73. Two (differential) refraction tubes. Table 10. Fig. 92 127
74. Differential and single refraction tubes. Sunlight. Tables n, 12 129
75. Distortion of glass absent 131
76. Further observations with sunlight. Table 13 131
77. Conclusion I32
CHAPTER XI. — The Refraction of Air with Temperature.
78. Apparatus. Fig. 93. Table 14 133
79. Observations 134
80. Computation 135
81. Final experiments at 100°. Table 15 136
82. Experiments at red heat 137
83. Further experiments at high temperatures. Fig. 94. Table 16 139
84. Flames 140
85. Conclusion I4.1
CHAPTER XII. — Adiabatic Expansion Observed with the Interferometer.
86. Introductory. Table 17 142
87. Experiments with short, bulky air-chambers H3
88. Effect of strained glass 145
89. Equations *4-6
90. Experiments with long tubes. Diameter, I inch. Table 18 148
91. The same. Diameter of tube, 2 inches. Table 19 15°
92. The same. Diameter of tube, 4 inches. Tables 20, 21. Fig. 95 151
CHAPTER XIII. — Miscellaneous Experiments.
93. Effect of ionization on the refraction of a gas 154
94. Mach's interferences. Fig. 96 I55
95. A Rowland spectrometer for transmitting and reflecting gratings, plane or concave.
Figs. 97, 98, 99 156
PREFACE.
The following account of my experiments has been given chronologically.
Although many of the anomalous features, in which the interferences of
superposed coordinated spectra first presented themselves, were largely
removed in the later work, yet the methods used in the several papers, early
and later, are throughout different. It therefore seemed justifiable to record
them, together with the inferences they at first suggested. The pursuit of
the subject as a whole was made both easier and more difficult by the un-
avoidable tremors of the laboratory in which I am working; for it is possibly
easier to detect an elusive phenomenon if it is in motion among other similar
stationary phenomena. But it is certainly difficult, thereafter, to describe
it when found.
It will be convenient to refer to the cases in which one of the two coincident
spectra from the same source is rotated 180° with reference to the other on
a transverse axis (i.e., an axis parallel to the Fraunhofer lines), under the
term reversed spectra; while the term inverted spectra is at hand for those cases
in which one of the paired spectra is turned 180° relative to the other on a
longitudinal axis (i.e., an axis parallel to the r-v length of the spectrum).
In this book the latter are merely touched upon, briefly, in Chapter I, but
they are now being investigated in detail and give promise of many interest-
ing results. The chapter contains a full account of what may be seen with
a single grating — the linear phenomenon, as I have called it, and which, if
it stood alone, would be difficult to interpret.
In Chapter II, therefore, the interferences of reversed spectra are treated
by the aid of two gratings, in virtue of which a multitude of variations are
inevitably introduced. The phenomena are thus exhibited in a way leading
much more smoothly to their identification.
This endeavor is given greater promise in Chapter III, which contains a
comparison of the interferences of reversed and non-reversed spectra, the
latter produced in a way quite different from those in my earlier work. Nat-
urally these in their entirety are even more bewilderingly varied, and become
particularly so when, as in Chapter IV, an intermediate reflection of one
spectrum is admitted. But with this I was on more familiar ground, as I
have hitherto, in these publications, given such investigations particular
attention.
The flexibility of the new methods is well shown in Chapter V, where
separated component beams can with equal facility be made to run in parallel,
or across each other at any angle, and perhaps both, with the double result
visible in the field of the telescope. In case of crossed rays a remarkable
phenomenon is shown, in which very small differences in wave-length imply
a remarkably large difference in rotational phase (virtually resolving power)
of the two interesting groups of interference fringes due to each wave-length.
5
6 PREFACE.
Spectra obtained with two, or at times even with one grating, are often
annoyingly furrowed with large transverse fringes. These are investigated in
Chapter VI, and referred to diffractions resulting from residual errors in the
rulings. In Chapter VII, finally, several examples of new methods of investi-
gation are given. They show the important bearing of the diffraction at
the slit of the collimator on all these experiments. The cleavage of a field
of diffracted rays as an essential preliminary is here put in direct evidence.
In Chapters VIII to XIII I have returned to my older experiments with
the displacement interferometer. The subjects adduced, like the dispersion
of air at low and high temperatures, the adiabatic expansion of air, etc., are
pursued less with the object of reaching results of precision than of testing
the limits of the displacement method and developing it.
My thanks are due to Miss Abbie L. Caldwell for very efficient assistance
in preparing the manuscript and drawings for the press.
CARL BARUS.
BROWN UNIVERSITY, Providence, Rhode Island.
CHAPTER I.
THE INTERFERENCES OF CROSSED SPECTRA.
1, Introductory, — If two component spectra from the same source coincide
throughout their extent the elliptic interferences will be spread over the
whole surface, provided, of course, the respective glass and air-path differences
of the two component rays are not so great as to throw the phenomenon
beyond the range of visibility. In the usual method of producing these
interferences, where the corresponding reflections and transmissions of the
two component rays take place at the same points of the same plane surface,
the interference pattern is automatically centered, or nearly so. This is not
the case when, as in the following experiments, the interfering beams are
separated in some other way; and the problem of centering is often one of
the chief difficulties involved; and if the beams are to be treated independ-
ently, it is difficult to obviate this annoyance.
Suppose, now, that one of the spectra is rotated around an axis normal to
both, by a small angle. Will the interferences at once vanish, or is there a
limiting angle below which this is not the case? In other words, how far
can one trench with light-waves upon the case of musical beats, or of inter-
ferences not quite of the same wave-length?
Instead of approaching the question in this form, in which it would be
exceedingly difficult, experimentally, I have divided it into two component
parts. Let one of the spectra be rotated 180° around a longitudinal axis,
parallel to the red-violet length of the spectrum and normal to the Fraun-
hofer lines. In such a case, interference should be possible only along the
infinitely thin longitudinal axis of rotation to which both spectra are sym-
metrical, one being the mirror image of the other. One would not expect
these interferences to be visible. It is rather surprising, however, that this
phenomenon (as I have found) may actually be observed, along a definite
longitudinal band in the spectrum, about twice the angular width of the
distance between the sodium lines and symmetrical with respect to the axis
of rotation. It is independent of the width of the slit, provided this is narrow
enough to show the Fraunhofer lines to best advantage.
Again, let one spectrum be rotated 180° about a given Fraunhofer line
(transverse axis), the nickel or mean D line, for instance. The two coplanar
spectra are now mutually reversed, showing the succession red- violet and
violet-red, respectively. Interference should take place only along the mean
D line and be again inappreciable. Experimentally, I was not at first able
to find any interferences for this case in the manner shown below, but this
may have been due to inadequacies in the experimental means employed,
for the dispersion was insufficient and the reflecting edge of the paired mirrors
too rough. Improving the apparatus, I eventually found the phenomenon,
7
8
THE INTERFEROMETRY OF
but appearing as a single line, vividly colored above the brightness of the
spectrum; or, again, more jet-black than the Fraunhofer lines and located
in the position of the coincident wave-lengths of the two superimposed spectra.
It is possible, however, as will be shown in § 4, to obtain two spectra in
such a way that if their longitudinal axes coincide the Fraunhofer lines
intersect at a small angle, and vice versa. In such a case, for coincident
Fraunhofer lines, interference occurs in a band around these lines and is
absent in the rest of the spectrum; whereas, if the longitudinal axes are
coincident, the interferences are arranged with reference to these axes. These
results seem to bear on the question, but it is difficult to clearly resolve them.
The methods used in this paper consist chiefly in bringing the two first-
order spectra of a grating, or the second-order spectra or their equivalents,
to interfere. In this respect they contain an additional method of inter-
ferometry which may be useful, if for any reason it is necessary that the two
component beams are not to retrace their paths.
2. Coincident spectra with one reversed on a given Fraunhofer line. — In
figure i, L is a narrow vertical sheet (subsequently broadened by the dif-
fraction of the slit) of white sunlight or arc light from a collimator, G the
transparent grating ruled on the side g, from which the first or second order
of spectra gM and gN originate. M and N are opaque mirrors mounted
adjustably on a firm rail, RR, each of them with three adjustment screws
relative to horizontal and vertical axes. M is provided with a slide micrometer
(not shown). From M and N the beams pass to the smaller paired mirrors,
m and n, which should meet in a fine vertical line at a very obtuse angle.
A silvered biprism would here be far preferable, but none having the required
angle was available. From n, m, the beams pass into the telescope T. As
the spectra are each divergent after issuing from g, they can be made to
overlap on leaving n, m, by aid of the adjustment screws on M and N. More-
over, as the spectra are mirror images of each other, as suggested in figure i ,
any spectrum lines (as, for instance, the D} may be put in coincidence on
using one of the adjustment screws specified. It is necessary that the telescope
T be sufficiently near M in order that the micrometer may be manipulated.
REVERSED AND NON-REVERSED SPECTRA.
The D lines placed in coincidence are obviously opposites, each line being
paired with the mate of the other. A fine wire must be drawn across the slit
of the collimator, in order that the vertical coincidence may be tested. One
should expect the interferences to appear between the D lines on gradually
moving the micrometer mirror M, parallel to itself, into the required posi-
tion. As stated above, I did not at first succeed in finding the interferences,
but the experiment is a delicate one. In a repetition with first-order spectra,
it would be advisable to replace the plane mirrors m, n, by slightly concave
mirrors, about 2 meters in focal distance, and to replace the telescope T by
a strong eyepiece. This is the method used in the next paragraph, and it
was more easily successful.
Later I returned to the experiment with the same adjustment, except
that the plane mirrors m, n, were placed beyond the
grating, with the object of using the equivalent of
second-order spectra to get more dispersion. This plan
did not fail, and, having once obtained the interferences,
the reproduction seemed quite easy, as they remained
visible while the micrometer M was moved over about
5 mm. or more, a very important observation. Their
appearance with a small telescope was that of a single
fine line, alternately flaming yellow (very bright on the
yellow background of the surrounding part of the spec-
trum) and jet black as compared with the D lines,
between which the interferential line was situated, and
on an enhanced yellow ground. The flicker is referable
to the tremor of the laboratory, which makes it im-
possible to keep these interferences quiet. Shutting off
the light from either mirror, M or N, naturally quenches
the interferences, but leaves the yellow part of the
spectrum behind.
Obviously, coincidence of the longitudinal axes of the
spectra alone is needed. Therefore, upon moving the
two double D lines apart, by aid of the adjustment screws on the mirror M
and N, symmetrically to the ends of the yellow field in the telescope, the
interferences were isolated and located midway between the D doublets of
each spectrum, i.e., in the center of the field of the telescope. They could
now be observed to better advantage. In the small telescope there is appar-
ently but one dark line. If stationary, its ultimate character, when centered,
would be surmised to be given by the intersection of a vertical diameter with
a series of confocal ellipses, successively bright and dark, as indicated in
figure 2. The light and dark parts alternate or flicker. On moving the
micrometer, the vertical intersector A takes a more and more lateral position
like B, so that the trembling interferences would soon be invisible, as they
rapidly become finer and hair-like (not shown).
'On using higher magnification (larger telescope), two black lines bordering
3)
10
THE INTERFEROMETRY OF
a bright line, or a black line between two bright lines, seemed to be visible;
but the interferences would have to be stationary to be definitively described,
since the width of the pattern is not more than one-third to one-half of the
distance between the sodium lines.1 The interferences, moreover, did not
now readily conform to the design B, figure 2, anticipated, but were more of
the type C, with long, dark lines slightly oblique to the vertical, and vibrating
within a vividly yellow band. Sometimes these were heavier, with two or
three faint lines on one side.
Further experiment showed that the phenomenon is not influenced by
the width of the slit, except that it is clearest and sharpest with the narrowest
slit possible and vanishes when the slit is made so wide that the Fraunhofer
lines disappear. It may easily be produced by the modified method following,
in any wave-length red, yellow, green, etc., with no essential difference except
in size. It is present, moreover, in all focal planes, i.e., the ocular of the
telescope may be inserted or pulled out to any distance, yet the same phe-
nomena persist on the vague, colored background. A number of observations
were made to detect the change
of the pattern of the interference,
between its entrance into the field
and its eventual evanescence, in
case of the continuous displace-
ment of the mirror M over 5 mm.
In figure 2 this would be equiva-
lent to a passage of B into B'
through A, and the fringes for a
distant center should therefore
rotate, as they actually do in
the experiments of the next para-
graph. But in the present case the type C persists; the lines may become
longer or all but coalesce and their inclination may change somewhat.
They nevertheless remain fine and nearly vertical, until they vanish completely
and there is no rotation. Nor could the phenomenon be found again within
the length of the given micrometer screw. Hence it is improbable that these
interferences conform at once to the ordinary elliptic type for which figure 2
applies, even if the ellipse is considered exceptionally eccentric. The use of
two slits, one following the other, does not change the pattern.
The modified method of experiment was one of double diffraction. In
figure 3, L is the blade of light from the collimator, which passing under the
plane mirror, m, penetrates the grating G, whence the diffracted first-order
beams reach the opaque mirrors M and N. These return the beams, nearly
normally but with an upward slant, so that the color selected intersects the
1 The use of the D\ D2 distance of the sodium lines for the measurement of the breadth
of the interference phenomenon is a mere matter of convenience in describing it. It will
be shown in the next report that the breadth of the strip carrying interference fringes is
quite independent of the dispersion of the optic system.
REVERSED AND NON-REVERSED SPECTRA. 11
grating at a higher level than L. A second diffraction takes place at about
the same angle, 6, to the direct ray t, and the coincident rays now impinge
on the mirror m. They are thence reflected into the telescope at T. This
method admits of easier adjustment, as everything is controlled by the adjust-
ment screws on M and N. Plane mirrors M, N, and m only are needed, the
latter being on a horizontal axis to accommodate T. The direct (white)
beam is screened off after transmission through the grating, if necessary.
But it rarely enters the telescope.
3. The same. Further experiments. — In place of the plane mirror, m, a
slightly concave mirror (2 meters in focal distance, say) may be used with
advantage and the telescope T replaced by a strong eyepiece. In this way
I obtained the best results.
It is to be noticed that the apparatus (fig. 3) may serve as a spectrometer,
provided the wave-length X of one line and the grating space D are known,
and the mirror, M, is measurably revolvable about a vertical axis. In this
case any unknown wave-length, X', is obtained by rotating M until X' is in
coincidence with X. Supposing the X's ofthe two spectra to have been origi-
nally in coincidence and that 6 is the angle of M which now puts X' in coin-
cidence with X, it is easily shown that
X'-X = X (2 sin2 0/2 + \£2A2-isin 6)
Angles must in such a case be accurately measurable, i.e., to about o.i minute
of arc per Angstrom unit, if the grating space jD = 35iXicr6, as above.
Counter-rotation of the mirror N till the X's coincide would double the accu-
racy. The usual grating, however, has greater dispersion and would require
less precision in 9.
Finally, a still simpler and probably more efficient device consists in com-
bining the mirror m and the plane grating G, or of proceeding, in other words,
on the plan of Rowland's method for concave reflecting gratings. In such
a case the light would enter in the direction TG, figure 3, be reflected along
GM, back along MG, and then return along GT at a slightly higher or lower
level than on entering. The equation just given would still apply, and many
interesting modifications are suggested. Experiments of this kind are to be
tested. Moreover, in case of the plane-transmitting grating and plane
mirror, as above shown, the same simplification is possible if the lens is
replaced by the telescope at T. But in this case the spectra are intersected
by strong, stationary interferences due to reflections from front and rear
faces and consequently not conveniently available. A reflecting grating
and telescope would not encounter this annoyance. In general, however,
as in the disposition adopted in figure 3 , the light enters opposite the observer,
and, as the light directly transmitted can be screened off, this is a practical
convenience in favor of the transparent grating. The reflected spectra used
may be placed at any level by rotating the mirror m on a horizontal axis.
On further repeating the work by the use of the concave mirror m, a strong
eyepiece at T, figure 3, and using a compensator, I eventually succeeded in
12 THE INTERFEROMETRY OF
erecting the interference design C, figure 2. It then took the form given at D,
and this seems to furnish the final clue to the subject. In other words, the
design consists of a new type of extremely eccentric ellipses, with their long
axes parallel to the Fraunhofer lines, each end having the outline of a needle-
point, possibly even concave outward. Only one end of the long, closed
curves is obtainable. These jet-black lines dance on the highly colored back-
ground of less than half the width between the two sodium lines. The inter-
ference design would, therefore, be the same (apart from color) as that which
would be obtained if the spectrum containing ordinary elliptic interferences
were to shrink longitudinally from red to violet, till it occupied less than half
the space between the two D lines. In fact, I have at other times obtained
just such patterns, with all the colors present, but not in the pure yellow, as in
the present case. Vertically, the path-difference is always due to more or less
obliquity of the rays passing through the plate of the grating. Horizontally,
however, the equivalent path-difference is complicated, in the present case, by
the fact that one wave-length of a pair has increased, whereas the other has
diminished, while both may pass through the same thickness of glass and air.
4. Coincident spectra with one reversed on a given longitudinal axis. — For
this experiment it is necessary to reflect the first-order spectra issuing at
the grating G, figure 4, from the ruled face g (a narrow, preferably horizontal,
blade of white light is here furnished by the collimator L with a horizontal
slit, and the rulings of the grating are also horizontal and parallel to it), twice
in succession and preferably from mirrors M and N and in and n, reflecting
normally to each other and inclined at an angle of, roughly, 45°. Each of
the mirrors M and N must be revolvable about a horizontal axis parallel to
the slit and furnished with three adjustment screws relatively to axes normal
to each other, one of which is horizontal. The mirrors m, n are the silvered
faces of a prism right-angled at the edge. It is, moreover, to be placed on
the slide of a Fraunhofer micrometer so that the prism may be moved, grad-
ually up and down, for the adjustment of distances.
On leaving the mirror m, n, the two spectra are carried by nearly horizontal
and parallel sheets of divergent rays, which pass outward from the diagram.
But it will be seen that one of the two spectra reaching the observer is reversed
on the longitudinal axis relatively to the other; i.e., if one is in the position
, f top 1 , ( bottom ) . ,
red ] , I violet, the other will be red ] > violet.
( bottom j ( top J
The subsequent passage of the rays is shown in figure 5, which is the side
elevation and therefore at right angles to the preceding figure. The rays
from m and n impinge on a distant, slightly concave mirror K (about 1.74
meters in focal distance) , placed somewhat obliquely, so that when the rays
come to a focus at F near the micrometer they may just avoid it. The partially
overlapping spectra at F are viewed by a strong eyepiece, E. The observer
at E can then control the Fraunhofer micrometer by which m, n is raised and
lowered, and the three adjustment screws of M.
REVERSED AND NON-REVERSED SPECTRA.
13
The adjustment consists in first roughly placing all parts in symmetry with
sunlight, until the two spectra appear at E. The lens may be removed.
There should be a bright, narrow spectrum band on each side of and near the
edge of the prism mn; for it is clear that after passing the lens E, correspond-
ing rays from M and AT must both enter the pupil of the eye to be seen together.
To make the spectrum parallel, the mirror mn is rotated, as a whole, around
a vertical axis. The three screws on the mirrors M and N then assist in
completing the adjustment; the rotation around the horizontal axis brings
the sodium lines in coincidence (both must be clearly seen and sharp and at
an appreciable distance apart); that around the oblique axis gives rise to
more or less overlapping, as required. The need of a sharp coincidence of
the sodium lines is very essential in all these experiments.
After proper vertical position of mn has been found by slowly moving the
micrometer screw up and down, the fringes appear. They are usually very
fine lines, possibly indicating distant centers of the ellipses to which they
belong. The appearance is roughly suggested in figure 6 . They are thus totally
different from the preceding set, § 3. They pass from the type a through b
(contraction toward the violet end was not noticed) into the type c, when
the mirrors mn move in a given direction. The center of the ellipses is in
the vertical through the field of view for the adjustment b, in which case the
lines pass from end to end of the spectrum as a narrow band near the longi-
tudinal axis of actual coincidence of spectra, symmetrically.
The height or breadth of the longitudinal interference band, d in figure 6,
is not greater than 1.5 to 2 times the distance apart of the sodium lines at
right angles to the band. From this the angular divergence of the breadth
of the band may be found, since X = D sin 6, where X is the wave-length of
light, D the grating space, and 6 the angle of diffraction. Hence for the two
sodium lines
= AX/£>cos 6
14
THE INTERFEROMETRY OF
Since Z) = 35iXicr6, cos # = .986, and AX = 6Xicr8; therefore &9=i.'jXio-4
radians. Since the width of the band is about twice this, it will be 68 seconds
of arc, or, roughly, about a minute in breadth. Within the strip, when the
fringes are horizontal, I counted about five of them, so that their distance
apart would be about 14 seconds of arc.
It appears, therefore, that rays of a given color, say of the wave-length at
D, which leave the grating at a given point and at an angle of about one
minute in the plane of the D line, are still in a condition to interfere ; whereas
one would anticipate that only those rays which lie in the common longitudinal
axis of rotation of the two coincident spectra, symmetrical to this, should be
in this condition. Such interference should not be appreciable, since the
white rays are independent and apparently come from two different points
of the slit. If we consider the angular deviation of pencils of parallel rays
crossing the grating to be equivalent to the divergence of their respective
optical axes at the collimating lens (about 45 cm. in focal distance), the dis-
tance apart of two points of the slit, the rays of which are still able to produce
interference, is
45X i. 7X10-^= 7.6Xio-3 cm.
or nearly o.i mm. Hence points of white light in the slit about o.i mm. apart
along its length are included in the band of interferences in question, extending
in colored light from red to violet. This seemingly anomalous result will be
fully interpreted at another opportunity.
5. Interference of the corresponding first=order spectra of the grating, in
the absence of rotation. — This apparatus seemed to be of special interest,
since the rays used do not retrace their path and are thus available for experi-
ments in which rays traveling in one direction only, are needed.* I have
tried both the adjustments given in figures 7 and 8, the latter, since the rays
are more nearly normally reflected at the mirrors M and AT, having some
advantages ; but the other succeeds nearly as well. The difficulty encountered
is a curious one of adjustment, which was not anticipated. In other words,
if the longitudinal axes of two identical spectra are in coincidence, the Fraun-
hofer lines are likely to be at a small angle to each other and complete inter-
* Cf. Am. Journal of Sci., xxxiv, p. 101, 1912, on the interferometry of an air column
carrying electrical current.
REVERSED AND NON-REVERSED SPECTRA. 15
ference is therefore impossible. Again, if the spectrum lines are in coincidence,
the longitudinal axes usually diverge by a small angle. Furthermore, the
interferences are almost always eccentric and the lines hair-like, indicating
distant centers. I have not succeeded in making a perfect adjustment,
systematically; but the discrepancies indicated are themselves interesting in
their bearing on the subject of this paper.
In figures 7 and 8, L is a vertical blade of white light from a collimator
with fine slit, and G is the grating. The two first-order spectra leaving the
ruled face at the line g strike the opaque mirrors M and N, the former on a
micrometer moving the mirror parallel to itself. From M and N the rays
reach the half -silvered plate of glass HS, where one is transmitted and the
other reflected into the telescope T. The coincident rays R are superfluous.
After placing the parts and roughly adjusting them for symmetry with
sunlight, the finer adjustment may be undertaken. It may be noticed that
the two systems M and N, and G as well as HS, can be used for further
adjustment separately. All are provided with adjustment screws relatively
to rectangular axes. To put the mirrors M and N in parallel and in the
vertical plane with the grating G, the half-silvered plate should be removed
/
10
and replaced by a small white vertical screen of cardboard, placed at right
angles to the direction of HS in figure 7 and receiving both spectra. A fine
wire is drawn across the slit to locate the longitudinal axis, and an extra
lens may be added to the collimator and properly spaced until the doublet
insures sharp focussing. Both mirrors, M and N, are now rotated on hori-
zontal axes, until the longitudinal black lines in their spectra cease to diverge
and coincide accurately. G, M, N, may now be considered in adjustment.
On returning the half-silvered plate, HS, it in turn is to be carefully rotated
around horizontal and vertical axes, until the horizontal black line in the
spectrum and the sodium line (always incidentally present in the arc lamp)
both coincide. But, as a rule, it will be found that if the longitudinal axes,
ww, figure 9, coincide, the D lines cross each other at a small angle, exagger-
ated in the figure. The interferences, when found by moving the micrometer
at M, are usually coarse, irregular lines, indicating a center not very distant
and located on the level of a band where the D lines cross.
On the other hand, if the D lines are brought to coincidence by moving
the adjustment screws on M and N (which throws them out of parallel), the
longitudinal axes ww, w'w', figure 10, diverge at a small angle and the inter-
ferences are found in a vertical band where the lines ww and w'w' cross. This
band is relatively wide, however, as compared with the cases in paragraphs 2
and 3. Nevertheless, I have looked upon these results as additional proof
16 THE INTERFEROMETRY OF
of the possibility of interference; for in neither case ought they to occur if
the spectra are not quite coincident horizontally and vertically. If they do
occur, it would at first sight seem that a certain small latitude of wave-length
adjustment is permitted even with light-waves.
I was at first inclined to refer the cause of this lack of simultaneous parallel-
ism to the grating itself, as it occurred with an Ames grating ruled on glass,
with a Michelson reflecting grating, and with a film grating, in about the
same measure. But subsequently, on adopting the method of figure 8, the
divergence was largely removed and the interferences were now visible
throughout the whole of the spectrum. The discrepancy is probably due to
insufficient normality of the plate of the grating to the incident white ray,
since one of the rays is twice reflected. In any case the adjustment of the
coincident sodium lines must be very accurate if the fringes are to be sharp ;
certainly as little as half their distance apart will obscure the phenomenon.
Though the spectra are bright, the interferences are not as good as with
the usual method (paragraph i); i.e., the dark lines are not black. Neither
have I found an available or systematic method for centering the fringes, so
that the lines obtained are usually delicate. Again,. the position of the colli-
mator, both as regards slit and lens, is here of very serious importance. Any
micrometeric horizontal motion of either, in its own plane, will throw the
fringes out. Finally, the whole spectrum travels with the motion of the
micrometer mirror M. The apparatus is thus too difficult to adjust for use,
to be of practical interest when simpler methods are at hand. The effect of
tremors acting prejudicially on so many parts is exaggerated.
6. Conclusion. — The phenomena of paragraphs 2,3, and 4, showing definite
and characteristic interference in case of two coincident spectra crossed either
on a longitudinal or transverse axis, represent the chief import of the present
chapter. These results can not be directly due to the diffraction of a slit
(regarding the line of coincidence as such), owing to their relatively small
magnitudes and their independence of the breadth of the slit. Since there
is in each case but a single line of points or axis, the disturbance of which
comes from identical sources, we might regard the image of this line in the
telescope to be modified by the diffraction of its objective. But if the inter-
ferences originated in this way, the Fraunhofer lines of the spectrum should
show similar characteristics and the diffraction pattern should differ from
those observed. Thus the conclusion is apparently justified that distinct and
independent points of the narrow slit whose distance apart on its length is
not greater than o.i mm. contribute rays to the field of interference in each
of the colors of the spectrum (longitudinal axes coinciding).
The phenomenon of inversion is virtually one of homogeneous light, the
same type of interference occurring in each color from red to violet. When
the fringes are horizontal, homogeneous light and a correspondingly broad
slit would replace the spectrum. They belong, moreover, to the elliptic
category, being of the same nature, apart from their limitations, as those
REVERSED AND NON-REVERSED SPECTRA. 17
used in displacement interferometry. With the exception of the points lying
on the longitudinal axis of rotation or of coincidence, all the pairs of points
of the two coincident spectra owe the major part of their light to different
sources; i.e., the points of the superposed spectra are not colored images of
one and the same point in the slit.
Again, in case of rotation of one of the coincident spectra around a trans-
verse axis (Fraunhofer line) , colors which differ in wave-length by about half
the distance apart of the two sodium lines seem also to admit of interference.
This permissible difference of wave-length is thus relatively about
AX_.5X6Xio-8
\ 59X10-0
or less than o.i per cent. The character of these interferences is distinctive.
They are not of the regular elliptic type, but arise and vanish in a succession
of nearly vertical (parallel to slit), regularly broken lines. Later observation,
however, revealed as their true form a succession of long spindles or needle-
shaped designs. The chief peculiarity observed is their almost scintillating
mobility, which in the above text has been referred to the inevitable tremors
of the laboratory. It is, however, interesting to inquire into the conditions
of the possibility of observable beating light-waves. For two waves, very
close together, of frequency n and n' and wave-lengths X and X', if V is the
velocity of light, the number of beats per second would be
AX
Therefore in case of the two sodium lines, for instance,
w'-n = 3 X io'°X6X io-Y348oX io-12= 5 X xo11
i.e., about sX io10 beats per o.i second, the physiological interval of nickering.
Naturally this seems to be out of all question, even if one is confronting a
source which is an approach to a mathematical line. The endeavor will have to
be made to produce these interferences under absolutely quiet surroundings.
Their appearance is altogether singular and not like the case of paragraph 4,
where there is also perceptible tremor, or with the general case of trembling
interference patches, with which I am, unfortunately, all too familiar.
In this place, however, it is my sole purpose to present, at its face value,
an observation which is spatial, independent of time consideration; and the
laterally cramped character of the new interference, with its long, hair-like
lines thrust into a strip less than half the distance apart of the sodium lines,
is the only evidence submitted. If the coincident path of two rays of slightly
different wave-lengths, X and X', which interfere, is %, then there are x/\ and
x/\', complete waves in the given path, and, in case of original identity in
phase, instantaneous reinforcement will occur when
x(i/X-i/X') = i,2,3, ...... n
In other words, at the wth reenforcement
18 REVERSED AND NON-REVERSED SPECTRA.
Hence, since X2 is very small and x relatively very large, the small value of
AX (i.e., the very thin strip of spectrum within which the phenomenon occurs)
is apparent. In the above experiments the estimates, in round numbers,
were AX = 2.4Xicr8, X2 = 36Xio~10. Hence if n = i,
# = 36Xicr10/2.4Xicr8 = .i5 cm.
so that one reenforcement would have to occur about at each 1.5 mm. along
the rays. Nevertheless, the formidable difficulty remains to be investigated,
viz, why these nominally beating wave-trains, with an infinitesimal group
period (icr11 sec.), could be recognized at all.
The characteristic feature of the new phenomenon is this, that apart from
intensity it persists, without variation, through a path-difference of over 5 milli-
meters; i.e., through 15,000 or 20,000 wave-lengths. It follows, since the
optical paths grating-mirror-grating are alone significant, that two individual
light -waves of the same ray over 15,000 wave-lengths apart are still appre-
ciably identical. Beyond that the waves under consideration no longer
correspond in orientation and can not interfere in a way to produce alterna-
tions of accentuated brightness and darkness.
CHAPTER II.
FURTHER STUDY OF THE INTERFERENCE OF REVERSED SPECTRA.
7. Apparatus with one grating. — The different methods suggested in para-
graph 3 were each tried in succession, but none of them were found equally
convenient or efficient in comparison with the method finally used in the
preceding paper. To begin with the annoyances encountered in the use of a
reflecting grating, it was found that the impinging light from the collimator
and the reflected doubly diffracted beam from the grating lie too close together,
even if all precautions are taken, to make this method of practical value. The
use of Rowland's concave grating without a collimator is out of the question,
since the spectra formed on the circular locus of condensation, if reflected
back, will again converge into a white image of the slit, colored if part of the
spectrum is reflected. The plane-reflecting grating, though not subject to
this law, requires a collimator, and, since marked obliquity of rays is excluded,
it will hardly be probable that the elusive phenomena can be obtained in this
way. A compromise method, in which both the reflecting and the transmitting
grating are used, will be described in paragraph 10. Though apparently the
best adapted of all the methods used, it has only after difficult and prolonged
research led to results. These, however, proved very fruitful in their bearing
on the phenomena.
For first-order spectra, where there is abundance of light (it is often difficult
to exclude all the whitish glare in the field of the telescope completely), the
method of figure n, which shows normal rays only, is still preferable. Here
the impinging collimated beam L passes below the opaque mirror m and
through the lower half of the grating G. The diffracted pencil is reflected
nearly normally but slightly upward, by the mirrors M and N (the former
carried on a micrometer slide) , to be again diffracted at the grating and there-
fore to impinge as definitely colored light on the lower edge of the concave
mirror m (about 1.5 to 2 meters in focal distance), whence it is brought to a
focus at F and viewed by the strong eyepiece E. Considerable dispersion
and magnification is obtained in this way; indeed, the two D lines stand far
apart and the nickel line is distinctly visible between them. There must be
a fine hair wire across the slit so that the longitudinal axes of the spectra may
be accurately adjusted. The mirror m above the impinging beam must be
capable of rotation about a vertical and a horizontal axis in order that the
focus F may be appropriately placed between M and N. With G at i meter
and m at 2 meters from F, the disposition is good. The micrometer M is
easily at hand. Though the direct beam may be screened off, the glare
reflected back from the grating and the glare from the objective of the colli-
mator are not excluded, as stated. In fact, it was eventually found necessary
19
20 THE INTERFEROMETRY OF
to carry this pencil in an opaque tube reaching from the objective of the
collimator, as far as the grating.
With first-order spectra this method always succeeded satisfactorily, and
in case of a ruled grating the phenomenon is exhibited brilliantly, if the paths
GM and GN are optically nearly equal. After some experience it is fairly
easy to find it. I have not, however, been able to obtain it with a film grating,
even after using a variety of excellent samples. This is not remarkable, for
the film grating is hardly sufficiently plane to produce clear regular reflection,
and the corresponding paths GM and GN would not, therefore, be definite.
Second-order spectra are too faint and can not be seen, unless the glare is
excluded in the manner stated. All modifications of the method seemed with-
out avail, until finally the light was led from the collimator objective C,
figure n, to the grating G, in a cylindrical tube, whereupon both the glare
from the objective and the rearward reflection from the grating were effec-
tively screened off. This tube must, of course, lie below the returning pencil,
•i.e., it must not (in section) cover more than the lower half of the grating.
In this case the second-order spectra, though faint, were seen clearly; but
the scintillating interferences could not be observed until the very weak
eyepiece, E, was used with the concave mirror m; or a weak telescope with
a plane mirror. It was then detected, but showed no essential difference from
the case of first-order spectra. The larger dispersion, in other words, was
unavailable. The phenomenon was seen most distinctly by drawing out the
eyepiece of the telescope, as the light is thereby concentrated, although the
Fraunhofer lines vanish. Second-order spectra are therefore not necessarily
advantageous. The phenomenon is very hard to find, and the experiments
were persisted in only to obtain the result under different conditions.
The tube-like light conductor referred to above is, of course, advantageous
in case of first-order spectra. If the concave mirror is used, the phenomenon
may even be seen brilliantly with the naked eye.
An alternative method of half-silvering the ruled face of the grating and
then using it as a reflector was tried with success. The beam of parallel
rays from the collimator L, figure 12, is transmitted by the grating (ruled,
half-silvered face, g toward the mirrors M and AT) and the two diffracted
beams then returned by the opaque mirrors M and N, to be in turn diffracted
by reflection into the telescope T. In fact, this method succeeds with the
unsilvered grating; for the rays diffracted, by reflection, from the ruled face
(toward the telescope), but not very well. The reflection from the rear face
REVERSED AND NON-REVERSED SPECTRA. 21
of the grating is so cut up by the strong, stationary interferences that it is
unavailable. The grating plate must, of course, be slightly wedge-shaped,
otherwise all the spectra would be superposed. In case the ruled face is
half-silvered, however, the stationary interferences are practically absent,
while two strong spectra are reflected from the silvered side. The phenome-
non may then be produced at all distances of G from M and N (2 meters and
less), but best at distances within i meter. It is, however, frequently
hard to find unless different distances apart of the mean D lines are tested.
This may be due to the fact that the silver film is not quite equally thick.
Besides the symmetrical position, gT, figure 12, the two corresponding
unsymmetrical positions g'T' were tested with success; and it appeared that
while in the case gT the phenomenon is virtually linear, dark or bright, like
a Fraunhofer line, a succession of dark lines inclined to the vertical may
appear for the unsymmetrical position g' T . Dark lines are apt to be broadened.
Questions relative to the effect of oblique incidence were also tested by
aid of the concave-mirror method shown in figure 1 1 , the white light from C
to G being conducted in an inch tube of pasteboard, immediately under the
concave mirror, m. Figure 13,0, shows the general disposition of apparatus.
The angle of incidence i is gradually increased, until the return rays from N
meet the grating at nearly grazing incidence. No essential difference in the
phenomenon was observed, however, except that it was apt to be broader
in the non-symmetrical positions and to suggest fine new lines in parallel
with the old. In a return to the symmetrical position, sharp lines were
especially distinct, usually showing one dark and two bright lines, while two
dark and one bright occurred less frequently. It could be seen quite vividly
with the naked eye. When the telescope was used and the ocular drawn
far forward, the multilinear form was often suggested. On broadening the
slit the black lines vanish first and a flickering band remains after the Fraun-
hofer lines are gone. Finally, the phenomenon could be seen even when the
longitudinal axes of the spectra were not quite coincident, but it rapidly
became fainter in intensity.
Figure 13, &, suggests a method of using a reflecting grating, either plane
or (possibly, if the incident light is parallel) concave, for the production of
the phenomenon. G is the grating, receiving the collimated white light, L,
which is diffracted toward M and N, thence reflected (at a different elevation)
back to G, to be again diffracted towards T, above or below the direct beam,
where it is observed. I have not, however, been able to obtain results with
these methods owing to subsidiary difficulties.
22 THE INTERFEROMETRY OF
8. Observations and experiments with a single grating. — On considering
figure n, it will be seen that the doubly reflected, doubly diffracted rays are
also in a condition to interfere. Thus the rays GMGNG and GNGMG have
identical path-length, or at least path-difference; but it is improbable that
superimposed on the strong spectra this effect could be seen, for the reflec-
tion from the ruled face of the grating is very slight and the divergent
spectra have weakened seriously. The scintillating interferences, on the
other hand, are much brighter than the superposed spectra. Such interfer-
ences, also, should be independent of the play of the micrometer M , since
the path-difference of these beams is not changed thereby, each being identi-
cally lengthened or shortened. Furthermore, the interposition of a thick
plate-glass compensator in CM should have no effect. Neither of these infer-
ences applies for the phenomenon in question, which persists for a definite
displacement of M, only, and the introduction of a compensator requires the
usual equivalent displacement of M, within the range of the phenomenon.
Finally, the interferences relatively to a phenomenon produced by double
diffraction would not be modified.
Many experiments were made to ascertain the path-difference within which
the phenomenon is visible. This can not be accurately determined, since it
is a question of stating when an observation, which is becoming rapidly less
distinct, has actually vanished. Moreover, any imperfection of the microm-
eter throws out the coincidence of longitudinal spectrum axes, while a read-
justment breaks the continuity of the micrometer displacement, or reading.
Results were obtained as follows, for example, AAT being the displacement of
the mirror M:
With telescope AA7 = 0.34, 0.45, 0.41 cm.
With concave mirror and lens 0.45, 0.35, 0.41 cm.
With concave mirror and adjustment 0.50 to 0.60 cm.
The low readings are due to the micrometric wabbling of the micrometer
slide. Since A./V is the double path-difference, the number of wave-lengths
in question may be put
6oXio-6
i.e., the distances along the ray are 15,000 to 20,000 wave-lengths apart,
about as estimated in the above paper. This is the characteristic feature of
the phenomenon.
Between its extreme ranges of visibility the appearance of the phenomenon
scarcely changes. It ceases to be visible rather suddenly; and this is to be
expected, since we are dealing directly with two wave-trains displaced rela-
tively to each other. It is visible for a wide slit even after the Fraunhofer
lines vanish. It disappears by decreasing in width, when the slit is closed.
If the ocular of the telescope is drawn out, the phenomenon may even be
observed after the Fraunhofer lines have vanished, in the dark, stringy spec-
trum of an extremely fine slit. When the longitudinal axis of the spectrum
is indicated by a fine wire across the slit, the adjustment consists in bringing
REVERSED AND NON-REVERSED SPECTRA. 23
the black longitudinal lines of the two spectra together. The question thus
arises how close this coincidence is to be. When the phenomenon is sharp,
it has been found possible to displace the two black lines so that a fine,
bright strip of spectrum may just be seen between, without quite destroying
the interferences. Naturally they are then much weaker. This result is in
harmony with the observations made on rotating one spectrum, on a longi-
tudinal axis, 1 80° with reference to the other.
Since the phenomenon was originally produced with sunlight, it might be
supposed that the edges of the Fraunhofer line, under conditions of tremor,
would interfere with each other as indicated in figure 14, where A is one and
B the other of the two superposed spectra. The change of wave-length is
suggested by the slant of lines on the
diagram. In such a case, whereas the
conditions a and c would show bright
overlapping spectra, the dark line would
appear under condition b. But even
in this case, lines of slightly different wave-length would have to interfere
with each other. The crucial test was made by using an arc-lamp spectrum,
and it was then found that the phenomenon appeared as well as with sunlight.
A further question at issue is the breadth of spectrum needed to produce
the phenomenon; for the observed breadth would be influenced by the quiver
of the apparatus. With this end in view, different lines of the spectrum were
placed in full coincidence, and it was found that for none of the secondary
lines in the orange-yellow spectrum was it extinguished or even modified.
If, however, the corresponding D lines of the spectra (D\D\; D2 D2') were
superposed, the phenomenon in these experiments played like a wavy strip at
their edges only. Sometimes a bright line flashed through the middle of the
coincident lines. One would conclude, therefore, that the part of the spectrum
used in producing these interferences is not much broader than either the
DI or Dz lines, while the other marked lines in the orange-yellow are too
narrow to appreciably influence it. These results will be greatly amplified
in the work done with two gratings below.
A corresponding experiment was now made with sodium light. To obtain
a sufficiently intense source, solid caustic soda was volatilized between the
carbons of the electric arc, A and B, figure 12, or the corresponding case in
figure 1 1 . On drawing the carbons apart, strong D lines were seen, in the entire
absence of an arc spectrum, at first so broad as to be self -reversing. Gradu-
ally they became finer and eventually reached the normal appearance of the
DI, D2 lines. In order to facilitate adjustment and with the object of obtain-
ing cases correlative with the results for the dark-line spectrum, a beam of
sunlight (as at L, figure 12) was introduced between the carbons and the phe-
nomenon established faultlessly in the usual way. The pencil of sunlight was
then screened off and the arc light substituted, or the two were used together.
These observations seemed to show that when the normal DI or D2 lines
were placed in coincidence, the thread-like phenomenon fails to appear with
24 THE INTERFEROMETRY OF
all the characteristics visible in the case of sunlight. When the slit is broad-
ened an alternation of brightness, or nicker of light, may be detected vaguely.
With a slit of proper width to show the Fraunhofer lines all this seemed to
vanish. The actual phenomenon was therefore apparently not reproduced
or improved either by homogeneous light or by widening the slit. Such exper-
iments alternating with sunlight were made at considerable length, but the
adaptation of methods for two gratings discussed in paragraph 10 will never-
theless throw out this conclusion.
If the narrow sodium line is broadened by adding fresh sodium at the car-
bon, so that the yellow spectrum is again self -re versed, the phenomenon plays
with extreme vividness around either of the reversed and coincident D\ or
DZ lines, or even within the black line in question, if narrow. But here the
light is no longer homogeneous. Sometimes when the solar spectrum is used,
a black line preponderates; in other adjustments a flashing bright line is in
place; but the reason for this can not be detected by the present method.
9. Inferences. — If the wave-length of the two spectra is laid off in terms
of the angle of diffraction, 6, measured in the same direction in both cases,
the graph will show two loci as in figure 15, a, intersecting in the single point
of coincident wave-lengths X0. It appears, however, as if the wave-lengths at
<pi and tps, (pi and <p4, are still in a condition to interfere. The phases <pi and
<P2, (f>3 and (pi, differ because of path-difference introduced for instance at the
micrometer, the phases <p\ <ps, ipz y\ differ because of color differences, having
passed through refracting media of glass and air. Probably the phase-differ-
ence (f>\ — (p3 = <pz — <f>4, these having the same color-difference; and if>i — (pz =
<P3 — <P4, having the same path-difference. At X0, do, the two phases <p0 are
due to path-difference only.
To allude again to the question of beats: if ten beats per second are dis-
cernible, the beating wave-trains in the case of the given grating would be
only 6Xio"10 second of arc apart in the spectrum. If the phenomenon has
a breadth of 3Xio~8 cm. in wave-length, as observed, then the number of
beats in question will be 2.5 X iou per second. All this is out of the question,
so far as the phenomenon appreciable to the eye is concerned. If beats were
due to a difference of velocity resulting from the dispersion of air, and if T
is the period of the beats, X the mean wave-length, d the difference of the
reciprocal indices of refraction, we may write
T-
If, furthermore, » = A-B/\2, where B= 1.34 Xio-14, 5X = 2.4Xicr8,
X4 1.3 X i o-17
_ ¥. - = 7 N/ T r\-o cpp
2vBd\ 2X3Xio1°Xi.34Xio-14X2.4Xio-8
Ni= i.4Xio6 beats per sec.
which would also be inappreciable.
REVERSED AND NON-REVERSED SPECTRA. 25
If both the difference of wave-length and wave-velocity are considered,
we should have for the first spectrum v and n, and for the second spectrum
v and n'. The conditions would be left unchanged, if the second velocity is
taken equal to the first and the frequency n'(v'/v) replaced by n'. From this
it follows that the number of beats N is nearly
If 5X is considered negative, if ^ = A— B/\2 and the multipliers ju and ju2 be
neglected,
which is the difference of the two cases above computed. As the first is very
large compared with the second, the visibility of the phenomenon is not
changed.
The theory of group waves usually introduces a factor 2. Thus if Xi, Vi, HI,
be the group wave-length, velocity, and frequency,
or,
or with the above data
results otherwise like the above and without bearing here. There is a possible
question whether differences of wave-length due to velocity and not to period
can be treated as dispersion.
The occurrence of forced vibrations has also been looked to as an explana-
tion. Though here again, even if the spectra are almost always of unequal
intensity, the reason for the preponderance of one would have to be stated.
True, equal mean strength is not equivalent to equal instantaneous strength.
In the case of forced vibrations, however, if the harmonic forces of one spec-
trum are F = A cospt (forced, T = 2Tr/p), of the other F = A'cosqt, (free,
T = 2ir/q) and there is no friction, the resulting harmonic motion will be
given by
r-t
Now if we regard the case of figure 15, on one side of the line of coincidence
Xo, q2>p2', on the other side, p2>q2. Hence, whenever a brilliant line flashes
out due to coincident phases, there should also be a black line due to opposi-
tion; and, in fact, when the phenomenon is produced under conditions of
perfect symmetry of the component beams, this seems to be its character;
i.e., the enhanced line cuts vertically across the breadth of the spectrum.
The case q2 = p2, being of infinitely small breadth, would not be visible. It is
not to be overlooked, however, that in certain adjustments, particularly in
26 THE INTERFEROMETRY OF
the non-symmetrical case of figure 13, more than two black lines frequently
occur. (Cf . §15.) These accessor}' lines are ordinarily very thin and crowded
on one side of the phenomenon only. It is thus merely the prevalent occur-
rence of paired dark and bright lines that are here brought to mind. Again,
the suggestion of many oblique lines has occurred in some of the observations.
These would be quite unaccounted for.
Finally, many attempts were made to find whether the phenomenon would
occur again beyond its normal range of about 2X0.5 cm. of displacement.
But, though the micrometer screw actuating the mirror M was effectively
2X3 cm. long, no recurrence could be found. At the ends of its range the
phenomenon drops off rather abruptly.
None of the inferences put forward adequately account for the phenome-
non as seen with a single grating, as a whole. In this dilemma I even went
so far as to suppose that a new property of light might be in evidence. One
feature, it is true, has been left without comment, and that is the width of
the slit-image. If ab, figure 15 b, is the angular width (d&) of this image,
the case of figure 150 should be additionally treated in terms of figure 156.
But within the limits of the present method
of experiment, with but one grating, this
circumstance seems to offer no clue. If, for
instance, the spectra actually coincide in
color throughout their extent, as in ordi-
nary interferences, the interference patterns ~8
should be enormous, for the path-difference
may be zero. The invariability of the present phenomena as to size within its
long range of presence, the occurrence of intensely sharp and bright or dark
single lines, with a distance (dff) much less than the distance apart of the DI, Z)2
lines, is in no way suggested by the width of slit-image. Moreover, in spite
of its persistence, the interference phenomenon of reversed spectra has the
sensitiveness of all interferences. Slight tapping on the massive table throws
it out altogether. Clearly, therefore, a modification of method is essential if
new light is to be thrown on the phenomenon, and from this viewpoint a
separation of the two diffractions seems most promising.
10. Apparatus with two gratings. — All the varied experiments described in
the preceding paragraph failed to show any essential modification of the linear
interference pattern obtained. In a measure this was to be anticipated, inas-
much as both diffractions take place at the same grating. It therefore seemed
promising to modify this limitation of the experiments, although the difficulty
of finding the phenomena would obviously be greatly increased. The separa-
tion of the two diffractions, however, seemed to be alone capable of resolving
the phenomenon into intelligible parts.
In the present method the glass grating G, figure 16, receives the white
beam L from the collimator, which is then diffracted to the opaque mirror M
REVERSED AND NON-REVERSED SPECTRA.
27
(on a micrometer slide) and N, thence to be reflected to the reflecting grating
G', plane or curved. Here the two beams of the identically colored light
selected are again diffracted to the telescope or lens at T. Since the gratings
G, G', rarely have the same grating constant, their proper position must be
found by computation and trial. In my work the distances to the line of
mirrors NM were 165 cm. for G and 90 cm. for G'. This method automati-
cally excludes the direct beam a and all glare, and gives excellent spectra both
in the first and second orders. The use of two gratings, however, introduces
the difficulties of adjustment specified, as the two D doublets corresponding
to N and M will not, as a rule, be parallel and normal to the longitudinal
axes of the spectrum, unless all cardinal features, like the rulings and their
planes, are quite parallel. If the grating is not normal to the impinging
beam, the axis of the corresponding spectrum is a curved line. The spectra
are, moreover, likely to be unequally intense, a condition not infrequent
even in the preceding method. It is possible that this may be due to the
grating itself, but probably unequal parts of the corresponding beams are
used in the two cases, or the mirrors are unequally good. As a result, in my
earlier work I was not able to produce the phenomena with two gratings,
after many trials, in spite of the clearness of the overlapping spectra; but
the same serious difficulties are encountered whenever interferences are
produced from two independent surfaces.
Later, having added a number of improvements to facilitate adjustments,
I returned to the search again and eventually succeeded. There are essen-
tially four operations here in question, supposing the grating G approximately
in adjustment. By aid of the three adjustment screws on each of the mirrors
M and N, figure 16, the fine wire drawn across the slit may be focussed on the
grating, if an extra lens is added to the collimator and the black horizontal
shadows of that wire, across the corresponding spectra, placed in coincidence.
The grating G' is then to be moved slowly fore and aft, normal to itself, on
the slide, so that the position in which the sodium lines are nearly in coinci-
dence to an eye placed at the telescope, T, may be found. The grating G'
is next to be slowly rotated on a line (parallel to LT) normal to its surface,
to the effect that the black axes of both spectra (i.e., the spectra as a whole)
may coincide. This must be done accurately, and the last small adjustments
may be made at the screws controlling M and N. Finally, the micrometer
28
THE INTERFEROMETRY OF
a
slide carrying M is to be moved fore and aft until the interferences appear.
These operations are difficult even to an experienced observer. The fringes
are very susceptible to tremors, and only under quiet surroundings do they
appear sharply. At other times they move, as a whole, up and down and
intermittently vanish.
The fringes so obtained, figure 17, were totally different from the preceding
and consisted of short, black, equidistant, nearly horizontal lines across the
active yellow strip of spectrum, at the axis of coincidence. The strip was
about of the same width as above. Thus the pattern presented the general
appearance of a barber's pole in black and yellow, the width being less than
the sodium interval, D\, D2, and the distance apart of fringes usually smaller.
They were visually in motion up and down, rarely quiet, no doubt owing to
tremor. Since the fringes were nearly horizonta1 or less than 30 degrees in
inclination, it was possible to enlarge the width of the slit without destroying
them, as in case of the hair-like vertical fringes in paragraph 2 above. In
this way a breadth of strip greater than the distance Di, D2, could be obtained
with sunlight or arc light, though a moderately fine slit was still desirable.
34
b
d
17
e
f
In general, the characteristics noted above were again observed. Thus on
moving the micrometer screw controlling M, the interferences appeared rather
abruptly. They vanished in a similar manner, after about 0.4 cm. or more
of the micrometer screw had been passed over. In other words, the fringes
remain identical for a path-difference of about 2X0.4 cm., or nearly 15,000
wave-lengths.
If we call the four D lines available in the two solar spectra DI, D2, D\, D'2,
respectively, a number of curious results were obtained on placing them
variously in approximate coincidence. Thus figure 17 a, when each D line
of one spectrum coincides with the mate of the other (Di, D'$\ D'i, D2), equi-
distant dots, surrounded apparently by yellow luminous circles, appeared
between the two doublets. On widening the slit the dots changed to a grating
of nearly horizontal lines covering the strip DI, D2, figure 176. The lines in
one part of the slit seemed to slope upward and in another to slope downward.
With a large telescope the phenomenon was more dim and quiet, apparently.
The fringes often lie in more definite focal planes and cease to be visible when
the ocular of the telescope is far outward, differing from the case above.
The phenomenon of chief interest, however, was observed (figure 1 7 c] in
placing two identical D lines in coincidence (D\; D2 D'2; D\}. The fringes
REVERSED AND NON-REVERSED SPECTRA. 29
were then seen across the coincident lines, now no longer visible, quite inde-
pendent of the absence of light. This would seem to mean that the otherwise
quiet ether within the black line is stimulated into vibration by the identical
harmonic motions of the bright fields at and beyond the edges of the line
(diffraction). The question will presently be broached again in a different
way. Here I may note that in the above cases of transverse lines (§ 8) it is
often possible to observe a very fine parallel yellow line within the coincident
D2, D'2, or DI, D\, doublets, excited, therefore, in the dark space and splitting
the line.
The experiments were now repeated with the sodium arc, and these also
gave some striking results. Thus in the case of figure 17 d the lines were
separated, but the yellow striations seemed to show across the dark space
between D2 and D'2. When the yellow light was too weak, cross-hatchings
were seen only across D'2, as in figure 17 e. Frequently the phenomenon
figure 17 / occurred on broadening the slit, in which D2 and D'2 interfered,
but only D'2 was marked. Screening off D2 (left mirror) at once removed the
fringes. I have interpreted this observation as the result of parallax, due to
the fact that the lines and the interferences are seen in different focal planes.
A v,x It It If
*' - g. ~
,8
On the basis of these results one might with some plausibility adduce the
following remarks in explanation of the phenomenon: In figure 18 a, let Si
and Sz be the overlapping reversed spectra and let the line of symmetry be
at Xi, X2. Then if identical ether vibrations can react on each other across a
narrow ether gap, rays as far as X'i, X'2 and X"i, X"2 being of identical source
and wave-length, respectively, are still in a condition to interfere. There
would then be three groups of interferences, Xi X2, X'i X'2) X"i X"2. If, figure
1 8 b, all are in phase, we should have a brilliant line; if all are in opposite
phases, a dark line on the principle of figure 18 c. Naturally, if wave-trains
react on each other across an ether gap, small as compared with the DI, D<t
interval, the assumption made above relative to interference of different
wave-lengths is superfluous. My misgiving in the matter arises from the
misfortune of having taken down the original apparatus, for modification,
and having since been unable to reproduce them with anything like the
decisiveness with which they were at first apparently observed. I can not
now be certain whether what occurred was actually what I seemed to see,
or whether the broad illumination of the sodium flash (broad individual
lines, DI to D2, virtually a continuous spectrum) may not have misled me.
The experiments were continued, as follows.
30
THE INTERFEROMETRY OF
11. Experiments continued. New interferometer. — At the outset it was
necessary to ascertain the reason for the difference of the phenomena, as
obtained with one grating in paragraph 8 and with two gratings in para-
graph 10. As the probable cause is a lack of parallelism of the rulings in the
latter case, it was necessary to remount the second grating G' in the manner
shown in figure 19. Here A A is a baseboard, capable of sliding right or left
and of rotating on a horizontal axis parallel to the grating. The latter (in a
suitable frame) is held at the bottom by the axle, e, normal to the grating
and by the two set-screws a and b carried by the standards c and d. Thus
the grating could be rotated around an axis normal to its plane. At first a
Michelson plane-reflecting grating G' and a telescope were used, as in figure
16; but it was found preferable (fig. 20) to use a Rowland concave reflecting
grating G', with the strong lens at T, the grating receiving a beam of parallel
rays of light for each color from the collimator and first grating G. In this
case, with sufficiently high dispersion, a large, strong field was obtained, in
which even the very fine lines of the solar spectrum were quite sharp. Rotating
grating G' around a parallel horizontal axis, like AA, figure 19, made little
a
difference, relatively speaking; but rotation around the axis e, normal to its
plane, carried out by actuating a and b in opposite directions, made funda-
mental differences in the appearance of the phenomenon and eventually
suggested a new interferometer for homogeneous light.
The adjustments are the same as in case of figure 1 6, G being the transparent
grating, except that G' is now a concave grating and T a strong eyepiece.
The distances G'T and GT were of the order of i and 2 meters.
On rotating the grating G' on an axis normal to its face, from a position of
slight inclination of the rulings toward the left, through the vertical position,
to slight inclination to the right, the fringes passed through a great variety
o; forms, to be described in detail in § 13 below. Difference of focal planes
between the Fraunhofer lines and the interferences were common, so that
effects of parallax were apt to occur. Thus when D* and D'z coincide, the
ladder-like phenomenon may lie between D'% and D'\\ or the ladder may pass
obliquely between the Dz D\ and D\ D'z doublets. The first experiment
with the new and powerful apparatus (plane transparent grating G, grating
space 35iXio~6 cm., and the concave reflecting grating G', grating space
173 Xicr6 cm., fig. 20) was made with the object of verifying, if possible, the
REVERSED AND NON-REVERSED SPECTRA. 31
reaction of parallel ether wave-trains on each other across a very narrow
ether gap. The sodium arc lamp was used as a source of light. The results
as a whole were negative, or at least conflicting. Usually when strong inter-
ferences were observed for coincident positions of DZ, D'2, for instance, there
was no passage of fringes across the dark space when DZ and D'% were slightly
separated. At the beginning of the work (possibly as the result of lines
broadened by a flash of sodium light) the stretch of interference fringes across
the dark space was certain; but such evidence is not quite trustworthy, for
a continuous spectrum (i.e., lines broadened by the flash) would necessarily
produce the striations. With a very fine slit the coincident DI, D\ or D2, D'2
was frequently much broadened by a sort of burr of fringed interferences.
When the lines are self -re versed, superposition of DI, D'\, etc., frequently
showed vivid interferences across the intensely black middle line. This and
the passage of the bright and dark lines across the superposed DI, D\ lines
of the solar spectrum are thus the only evidence of the reaction of separated
light-rays on each other across an ether gap observed in the new experiments,
and the above results could not be repeated.
On introducing a refined mechanism to establish the sharpest possible
coincidence of the DI, D\ or D2, D'2 lines, it seemed as if these lines could
at times be brought to overlap with precision, without the simultaneous
appearance of the interferences around then ; but on drawing out the ocular
of the telescope or the lens the cross-hatching invariably appears. If the
coincidence is not quite sharp, the phenomenon is usually very strong in the
isolated bright strip. Horizontal fringes are best for the test.
An additional series of experiments was made some time later by screening
off parts of the concave grating G', in order to locate the seat of the phenom-
enon at the grating. Screening the transmitting grating G was without con-
sequence; but on reducing the area G' to all but the middle vertical strip
about 5 mm. wide, a very marked intensification of the phenomenon followed.
Although the spectrum as a whole was darker, the interferences stood out
from it, relatively much sharper, stronger, and broader than before. The
Fraunhofer lines were still quite clear. Thus the pattern, g, figure 17, was
now very common, both with sunlight and with sodium light. For a given
slit the phenomenon began with a strong burr c, figure 17, completely oblit-
erating and widening the superposed D2, D'z lines. When these lines were
moved apart, the striations followed them, as in figure 17, h and i, to a limit
depending on the width of the slit. A still more interesting pattern is shown
in figure 17 k, in which the interferences proper are strong and marked between
the two DI D'i doublets, but much fainter striations are also evident, reaching
obliquely across and obviously with the same period.
With this improvement I again tested the ether-gap phenomenon, using
the sodium arc, and to my surprise again succeeded. DI D\ lines of half
the breadth of the doublets apart induced strong fringes between them, and
the experiments were continued with the same results for a long time. Several
days after, however, with another adjustment, it in turn failed. Clearly
32 THE INTERFEROMETRY OF
there is some variable element involved that escaped me, and it will hardly
be worth while to pursue the question further with the given end in view,
without a radical change of method.
Screening middle parts of the grating (in relation to § 15) did not lead
to noteworthy results here, but such experiments will become of critical
importance below.
A word may be added in relation to Fresnellian interferences in the present
work. These would be liable to occur if the observations had been made outside
of the principal focus, with the sodium lines blurred. In all the experiments
on the excitation of a narrow ether gap, however, the D lines were clearly in
sight and sharp, so that the phenomena of non-reversed spectra and homogene-
ous light (in the next section) are not here in question. True, such interfer-
ences may often be found in the case of reversed spectra, when the sodium lines
are purposely blurred, by pushing the ocular toward the front or to the rear.
12. Experiments continued. Homogeneous light. — To turn to a second
class of experiments : very important results were obtained with homogeneous
light (sodium arc) on placing the DiD'i or D2D'Z lines in coincidence and then
broadening the slit indefinitely or even removing it altogether. A new type
of interferences was discovered, linear and parallel in character and inter-
secting the whole yellow field. These lines could (as above) be made to pass
from a grid of very fine, hair-like, nearly horizontal lines to relatively broad,
vertical lines, on changing the orientation of the grating G', figure 16. Small
changes of position of the grating produced a relatively large rotation and
enlargement of the lines of the interference pattern. The fringes, when verti-
cal and large, are specially interesting. The distances between successive
fringes obtained were about the same (accidentally) as the DiD2 distance of
the sodium lines. They are quiet in the absence of tremor. If DiD'i or D2D'Z
were only present, the field would be an alternation of yellow and black
striations; but as both doublets are present, the interferences overlap the
flat (non-interfering) yellow field of the lines not in coincidence. The fringes
are nevertheless quite distinct. A single homogeneous line (like the green
mercury line) would give better results. It is necessary that the line selected
(say DiD'i} should coincide horizontally and vertically before the slit is
broadened. Otherwise no fringes appear in the yellow ground, or at least
not in the principal focal plane. On using a thin mica compensator, it is
easy to make these fringes move while the mica film is rotated ; and they pass
from right to left and then back again from left to right, as the mica vane
passes through the normal position of minimum effective thickness. Thus
this is a new form of interferometer with homogeneous light. The fringes
remain identical in size, from their inception till they vanish, while the microm-
eter M, figure 1 6, passes (as above) over about 15,000 wave-lengths. In this
respect the new interferometer differs from all other types, the two air-paths,
GMG' and GNG', alone being in question. The condition of occurrence will
be investigated in paragraph 13.
REVERSED AND NON-REVERSED SPECTRA. 33
13. Experiments continued. Contrast of methods. — As these fringes were
produced with a concave reflecting grating, the question may be put whether
they would also appear in case of the plane reflecting grating, G', in the adjust-
ment of figure 1 6. The experiment was therefore repeated with a wide slit,
or with no slit at all, and there was no essential difference in the two classes
of results.
On the contrary, when the method of but one grating and sodium light
was used (fig. n), the interferometer fringes, in case of a very wide slit or
the absence of a slit, could not be produced over the yellow field, as a whole.
There appeared, however, an obviously pulsating flicker in parts of the field,
on reducing the width of the slit till the sodium lines were each about the
width of a DiDz space, with either D\D'\ or D2D'2 superposed. The sharply
outlined slit showed an irregular, rhythmic brightening and darkening over
certain parts of its length. These broad pulsations were very violent, very
much in character with the linear phenomenon above. This behavior is very
peculiar, recalling the appearance of a bright yellow ribbon undulating, or
flapping fore and aft, so as to darken parts of its length rhythmically. The
pulsations, moreover, were quite as active if seen at night, when the tremors
of the laboratory were certainly reduced to minimum. Nevertheless, I am
now convinced that such tremor only is in question.
Regarding the phenomenon as a whole, one may argue that in case of the
wide slit and single grating, in which the lines for both diffractions are there-
fore rigorously parallel, the interference fringes are on so large a scale as to
cover the whole field of view and thus to escape detection; i.e., that a single
vague, quivering shadow of a flickering field is all that may be looked for,
in the limited field of view of the eyepiece.
Returning to the case of two gratings and the wide vertical interference
fringes and, in turn, all but closing the slit (vertical interferences and sodium
arc light), the pulsating phenomenon simply narrowed in width. The two
or three sharp vibrating lines, alternating in black and yellow of the original
phenomenon (Chapter I), did not appear. The cause of this is now to be
investigated.
14. Experiments continued. Rotation, etc., of grating. — The method of
two gratings (fig. 16 or 20, plane transmitting and concave reflecting) was
first further improved by perfecting the fore-and-aft motion of the grating G'
(G' movable in the direction G'T on a slide), as well as the precision of the
independent rotation of G' normal to its face; i.e., around G'T. These adjust-
ments led to further elucidation of the phenomenon. To begin with the fore-
and-aft motion of the concave grating G' (i.e., displacements in the directions
G'T, fig. 20), it was found that the fringes, figure 21, a, b, c, d, e, in any good
adjustment, pass from extremely fine, sharp, vertical striations, which gradu-
ally thicken and incline to relatively coarse, horizontal lines, finally with
further inclination in the same direction into fine vertical lines again, while
G' continually moves (through about 5 cm.) on the slide normal to the face
34 THE INTERFEROMETRY OF
of the grating. It was not at all difficult to follow the continuous tilt of these
lines through the horizontal, occurring on careful and continuous front-and-
rear motions of the grating G' through the limiting positions. The fringes
usually vanish vertically merely because of their smallness.
Again, on rotating the grating G' around an axis normal to its face, the
fringes merely vary in size, without changing their inclination. Thus if the
horizontal fringes (which were here always closer than the inclined set) are
in view, these will pass from extremely small-sized, fine, hair-like striations,
through a maximum (which is a mere shadow, as a single fringe probably
fills the field) back into the fine lines again. Only a few degrees of rotation
of the grating suffice for the complete transformation. The maximum is
frequently discernible only in consequence of a flickering field. An oblique
set of fringes is equally available, remaining oblique as they grow continually
coarser and in turn finer with the continuous rotation of the grating.
When the very large horizontal fringes are produced by this method, the
change into vertical fringes by fore-and-aft motion of G' is very rapid, so
that relatively wide, nearly vertical forms may be obtained. All these effects
may be produced by solar or by arc light, around
the line of symmetry of the overlapping spectra;
or with sodium light when either DiD'i or D2D'z 2 1
coincide.
The fine vertical or inclined lines appear as & 6
such when the slit is widened, either in case of
white or of sodium light. These are the inter- O 2
ferometer fringes seen above (§ 6), coarse or
fine. With sodium light any width of slit, or
no slit at all, is equally admissible. The same is true for the narrow maxima.
Lines nearly horizontal were sometimes obtained, pointing, as a whole, toward
a center.
Finally (and this is the important result) the extremely large horizontal
maxima, when a single fringe fills the field, can not be seen apart from pulsa-
tions, in the case of a wide slit. With a very narrow slit, such as is suited for
the Fraunhofer lines, these horizontal fringes appear as intensely bright or
very dark images of the slit. In other words, the normal phenomenon of
overlapping symmetrical spectra as described in Chapter I is merely the
vertical strip of an enormous horizontal interference fringe, made sharp and
differentiated by its narrowness. This case occurs at once when the rulings
of the two gratings G and G' are all but parallel, and hence it is the regular
phenomenon when but a single grating is used for the two diffractions, as in
figures ii and 12.
In later experiments on the effect of the rotation of the grating, G', around
a normal axis, the above results were found to be incomplete. If the rotation
is sufficient in amount (a few degrees, always very small), it appears that,
after enlarging, the fringes also rotate. But the rotation in this case corre-
sponds to a vertical maximum, as indicated in figure 22, the vertical set being
REVERSED AND NON-REVERSED SPECTRA. 35
the coarsest possible for a given fore-and-aft position of the grating G'. In
the figure, the sequence a, b, c, d, e is obtained for a continuous rotation of
the grating (in one direction around a normal axis).
It now became interesting to ascertain how the vertical set c, figure 22,
would behave with the fore-and-aft motion. The experiments showed that
there was no further rotation, but that, while G' passes normally to itself
over about 1.5 cm. on the slide, the vertical fringes pass from extreme fine-
ness at the limit of visibility, through an infinite vertical maximum (a single
vague shadow pulsating in the field), back to extreme fineness again, without
any rotation. If the edges of the corresponding yellow strips (superposed
DI, D\ lines) did not quite coincide, the fringes were seen outside of the prin-
cipal focal plane, as usual. Probably the vertical and horizontal maxima are
identical in occurrence and appear in case of parallelism in the rulings of the
two gratings G and G', and the absence of path-difference. Hence if a single
grating is used, as in the original method, the interferometer fringes are not
obtainable. This is an important and apparently final result, remembering
that fore-and-aft motion is probably equivalent to a rotation around a vertical
axis, parallel to the grating.
With regard to the rotation in case of fore-and-aft motion of G', it is well
to remark that in approaching the position c, figure 24, it is apt to be very
rapid as compared with the displacement, precisely as in the case of the picket-
fence analogy.
Hence the original phenomenon, consisting of single lines, can not be mani-
folded by increasing the width of slit. It vanishes for a wide slit into an
indiscernible shadow. The phenomenon is a strip cut across an enormous
black or bright horizontal fringe, by the occurrence of a narrow slit. More-
over, the scintillations variously interpreted above are now seen to be due to
tremors, however different from such an effect they at first appear; i.e., the
enormously broad, horizontal fringe changes from dark to bright, as a whole,
by any half wave-length displacement of any part of the apparatus. It is
thus peculiarly sensitive to tremors. On the other hand, oblique or fine
vertical fringes are always recognizable for any size of slit. The inquiry is
finally pertinent as to why the phenomenon is so remarkably sharpened by
a narrow slit; but this must be left to the following experiments.
To be quite sure that the concave grating G' had no fundamental bearing
on the phenomenon, I again replaced it by the Michelson plane reflecting
grating (fig. 16, G transmitting, G' reflecting). In the same way I was able
to rotate the fringes, continuously, through a horizontal maximum of size
by fore-and-aft motion of G' . Rotation of G' in its own plane increased or
decreased the breadth and distance apart of the fringes through a maximum,
coinciding with the parallelism of the rulings of the two gratings. Here I
also showed decisively that as the rungs of the interference ladder (fig. 2 1 c)
thickened and receded from each other, the design passed, in the transitional
case, through the original phenomenon of the single vertical line dark or
brilliant yellow, for a slit showing the Fraunhofer lines clearly. The phenome-
36 THE INTERFEROMETRY OF
non vanishes with the spectrum lines as the slit is widened, but, on the other
hand, persists as far as the interference of light for a narrow slit. Finally,
the apparent occurrence of more than one line is referable to the presence of
more than one nearly horizontal wide band in the field of the telescope. Thus,
for instance, cases between b and c near c and between c and d near c, figure 24,
are the ones most liable to occur when both diffractions take place at a single
grating. This result will be used in paragraph 15.
15. Tentative equations. — In the first place, the actual paths (apart from
the theory of diffraction) of the two component rays, on the right and left
sides of the line of symmetry, II'Z, figure 23, will be of interest. The compu-
tation may be made for the method of two gratings at once, as the result
(if the distance apart of the gratings is C = o) includes the method with one
grating; i.e., the more complicated figure 23, where G is the transmitting and
G' the reflecting grating, resolves itself into a case of figure 24, with but one
grating, G. M and M' are the two opaque mirrors, I the normally incident
homogeneous ray. Supposing, for simplicity, that the grating planes G and
G' are parallel and symmetrically placed relatively to the mirrors M and M',
as in the figure, the ray Y diffracted at the angle 0i is reflected into X at an
angle 02-0i and diffracted into Z normally, at an angle 02, on both sides.
Under the condition of symmetry assumed X+Y — (X'+Y'} =o, or without
path-difference. Let N be the normal from / to M, and n the normal from
I' to M, with a similar notation on the other side. Hence if / be given an
inclination, di, 0i is incremented by ddi, Y-\-X passes into y \-\-y-\-x, Yf-\-X'
into y' \+y' +x' , decremented at an angle dQ\, while both are diffracted into
Z'. Since generally
sn 6i — sm = \ cos
for homogeneous light and the same di. Hence ddi = dd'i = dQ, say.
If 5 = 02— 0i, and «• = 0a+0i, the auxiliary equations
02 N — n
sin 8 cos (V2)
are useful. From a consideration of the yC and yx triangles, moreover,
the relations follow :
N N—n cos
y~T~y^ T „ fs/~ JQ\ y^ ~, „ /„. /,,\ „ _ /•/» i j/i\ ^ y
cos (5/2 -dO) ~cos(o-/2)cos(01+c?0) cos(92-dO)
and from the y\C and y'x triangles, similarly,
, , N , = N-n ,_ .cos (di-
" " 1- - "
COS (5/2+d0) COS ((T/2) COS (0i-d0) COS (02+d0)
Hence, after some reduction, the path on one side is
2Ncos(ff/2)_ N—n
cos (02 — d6) cos (cr/2) cos (02 —
REVERSED AND NON-REVERSED SPECTRA.
37
which may be further simplified to
N COS<T-{- n
-de) cos (a/2)
From this the path on the other side will be
>_i_ 'i i —
"
cos
COS (02+d0) COS (<7/2)
The path-difference, AP, thus becomes, nearly,
i \ _N_cos_ff-j-n 2 sin 02
cos (62— d 6)) cos (07 2) cos2 02
r cos 0-+w
COS (0-/2)\COS
This is perhaps the simplest form attainable. If, apart from diffraction,
this should result in interference, the angular breadth of an interference
fringe would be (AP = X)
X cos2 02 cos (0^/2)
2 sin 02 N cos ff-{-n
and if D is the grating space and sin 0 = X/Z)',
(D'2-X2) cos (cr/2)
zD'(N cos
25
22
dA JM
/u, 4-' Y'
6- a./ z> %
&~ ~~nE
<M
JL
or
23
In case of a single grating
cr/2 = 02=0 N =
cos2 0 2 sin
24
COS (7= 2 COS2 6—1
AP =
cos 0 cos2 0
a result which may be reduced more easily from figure 24. Hence, the
angular distance apart of the fringes would be (AP = X)
X D cos 0
de=
tan 0
4JV
if D is the grating space. To find the part of the spectrum (d\) occupied
by a fringe in the case postulated, since sin 0 = X/D,
de d\ _D_
cos 6~ D cos2 0~ 4A/T
38 THE INTERFEROMETRY OF
and from the preceding equations, finally,
where d\ would be the wave-length breadth of the fringe, remembering that
the fringes themselves are homogeneous light.
In the grating used
cr6 cm. X = 6oXicr6 cm. w=ioocm.
or
400
This is but 1/200 of the distance, cf\ = 6Xio"8 cm., between the D lines.
Hence such fringes would be invisible. Moreover, ddcci/N; the fringes, there-
fore, should grow markedly in size as N is made smaller. Experiments were
carried out with this consideration in view, by the single-grating and concave-
mirror method, N being reduced from nearly 2 meters to 20 cm., without any
observable change in the breadth or character of the phenomenon. It showed
the same alternation of one black and one or two bright linear fringes, or
the reverse, throughout. Hence, it seems improbable that the phenomenon,
i.e., the interference fringes, are referable to such a plan of interference as
is given in figure 24.
Similarly, for the case of two gratings, figure 23,
cos frr/a) (Z/2
2D'(N cos
where, if we insert the data 0i = g°4o' and 02 = i9°55'
Z/=i73Xio-6 cm. JV=i62cm. n = &2 cm. X = s8.9Xio-6 cm.
then
Hfi_ .967XlO-10X263 I*™-" radian*
de~ io-*X346X(i62X.87 + 82) -33Xio radians.
Thus dd is about of the same small order of values as above, i.e., less than
one-tenth second of arc or i/iooo of the DiD2 space, and thus quite inappre-
ciable. Some other source, or at least some compensation, must therefore
be found for the interferometer interferences seen with homogeneous light.
The full discussion of the effective path-difference in terms of the diffrac-
tions occurring will be given in § 27, in order not to interrupt the progress
of the experimental work here. It will then be obvious that the mere effect
of changing the obliquity of the incident homogeneous rays, I, introduces
no path-difference, or that the fringes observed are varied by the displace-
ments and rotations of the grating, G', and the mirrors M and N.
16. Experiments continued. Analogies. — With this possible case disposed
of, it now becomes necessary to inquire into the other causes of the phenome-
non, as described in paragraph 13. This is conveniently done with reference
to figure 26, where n and n' are the axes of the pencil of yellow light, reflected
from the opaque mirrors M and N, after arrival from the transmitting grating
REVERSED AND NON-REVERSED SPECTRA.
39
G. It is necessary to consider the three positions of the reflecting grating G';
viz, G', G\, and G'2. In the symmetrical position G', the pencils whose axes
are n and n' meet at a and are both diffracted along r. In the position G',
they are separately diffracted at b and bf in the direction r\ and r\, and they
would not interfere but for the objective of the telescope, or, in the other
case, of the concave mirror of the grating. In the position G'z, finally, the
pencils n and n' are separately diffracted at c and c' into r2 and r'2 and again
brought to interference by the lens or concave mirror, as specified.
Now it is true that the rays na and n'a (position G'}, though parallel in a
horizontal plane, are not quite collimated in a vertical plane. The pencils
are symmetrically oblique to a central horizontal ray in the vertical plane,
and their optical paths should therefore differ. But fringes, if producible in
this way here, have nothing to do with the rotation of the grating in its own
plane and may here be disregarded, to be considered later.
fJL
*¥
t--$
gr<
To take the rotation of the fringes first, it is interesting to note in passing
that the interferences obtained by rotation around a normal axis recall the
common phenomenon observed when two picket fences cross each other at a
small angle tp. It may therefore be worth while to briefly examine the rela-
tions here involved (fig. 27) where S' and 5 are two corresponding pickets of
the grating at an angle <p and the normals D' and D are the respective grating
spaces. The intersections of the groups of lines S' and S make the representa-
tive parallelogram of the figure (5 taken vertical), of which B is the large
and B' the small diagonal. The angles indicated in the figure are x-{-y = ip
and x'+y'+<p= 180°. As the bright band in these interferences is the locus
of the corners in the successive parallelograms, B is the distance between
two bright bands, while B', making an angle yr with 5, is the direction of
these parallel interference bands relative to the vertical. Let the free ends
of D and D' be joined by the line E' ; and if D is prolonged to the left and the
intercept is D in length, let this be joined with the end of D' by E. Then the
triangle DED' and S'BS, DE'D' and S'B'S, may be shown to be similar by
aid of the following equations :
r"i-» r~ir\r T->/ • T-> • r-/ • t~> • U *-* Sin X
SD=o D u sin <p = h sin x o sin <p = B sin y ~F7 = 75 ~^~
S B sin y
40 THE INTERFEROMETRY OF
If E is expressed in terms of D and D', and B in terms of S and 5', and
the first equation is used, then
5 sin x D'
from which, in the fourth equation,
E \/D*+D'2+2DD'cos<p
sin (p sin
Similarly,
E'
sn ip sin <p
Again, the angle y' is given from
sin y' D
or on reduction
. D sin <p
tan y' = -=-,
,
D —D COB <f>
If D = D', or 5 = 5', then
sin (<p/2) cos (<£>/2)
. sin <p
tan y = -r— -. — -r^ = cos (<p/2)/sm (<p/2). or tan y tan p/2 = i
sin (<p/2)
Thus if <p = o, tan y'= cot y' = 90°, or the fringes are horizontal and 5 = aS.
If 7' is nearly zero
changing very rapidly with <p.
If one grating of a pair, with identical grating spaces D, is moved parallel
to itself, in front of the other, the effect to an eye at a finite distance is to
make the grating spaces D virtually unequal ; or
cos
— — — j
2 COS (<f>/2)
so that for an acute angle <p, the fringe breadth is increased. Thus BQ is a
minimum in case of coincident gratings.
The analogy is thus curiously as follows: The fringes just treated rotate
with the rotation of either grating in its own plane and pass through a mini-
mum size with fore-and-aft motion; whereas in the above results the optical
grating showed a passage through a maximum of size with the rotation of
either grating in its own plane and a rotation of fringes with fore-and-aft
motion of the grating.
Returning from this digression to figure 26, if the grating G' is not quite
symmetrical, but makes a small angle <p with the symmetrical position as at
g', the fore-and-aft motion will change the condition of path-excess on the
right (position g'2) M path larger) to the condition of path-excess on the left
REVERSED AND NON-REVERSED SPECTRA. 41
(position g'i, N path larger) ; and if the motion is continuous in one direction,
#'2, g', g'i, the path-difference will pass through zero. No doubt the angle
<p is rarely quite zero, so that this variable should be entered as an essential
part of the problem. The resulting conditions are complicated, as there are
now two angles of incidence and diffraction and it will therefore be considered
later (§ 28). It is obvious, however, that if for a stationary grating G',
figure 26, the angle <p is changed from negative to positive values, through
zero, the effect must be about the same as results from fore-and-aft motion.
In both cases excess of optical path is converted into deficiency, and vice versa.
Hence, as has been already stated, the effects both of the fore-and-aft motion
and of the rotation of the grating G' around a vertical axis parallel to its
face conform to the interference fringes of figure 21, a to e.
It is common, moreover, if a concave grating is used (with parallel rays)
at G', to find the two sodium doublets due to reflection from M and N ap-
proaching and receding from each other in the field of view of the ocular
when the grating G' is subjected to fore-and-aft motion. This means that
although the axes of incident rays are parallel in two positions, whenever i
varies (as it must for a concave grating and fore-and-aft motion), the diffracted
rays from M and N do not converge in the same focus in which they originally
converged, but converge in distinct foci. For if sin i— sin 6 = \/D or cos idi =
cos 8d6, suppose that for a given i, 8 = 0; then cosidi = dd. But the devia-
tion, 5, of the diffracted ray from its original direction is now di+dQ, or
5 = di(i +cos i} = zdi cos2 1/2
Similarly, the principal focal distance p', for varying i, is not quite constant.
From Rowland's equation, if parallel rays impinge at an angle i and are
diffracted at an angle 8 = 0,
f = R = R
I+COS i 2 COS2 2/2
If 2 = 20°, then cos2 2/2 = .976, and p' = R/2, nearly but not quite.
I have not examined into the case further, as both the sodium doublets
are distinctly seen if the ocular follows them (fore and aft), and the lateral
displacement of doublets is of minor interest.
With the plane reflecting grating this discrepancy can not enter, since for
parallel rays the angles of incidence remain the same throughout the fore-and-
aft motion, and therefore the angles of diffraction would also be identical.
Two outstanding difficulties of adjustment have still to be mentioned,
though their effect will be discussed more fully in the next chapter. These
refer to the rotation of the grating G', around a vertical axis and around a
horizontal axis, in its own plane or parallel to it. The rotation around the
vertical axis was taken up in a restricted way above, in figure 13, Chapter II.
The effect (rotation of G') is to change the inclination of the fringes passing
from inclination to the left through zero to inclination towards the right.
The effect is thus similar to the fore-and-aft motion, as shown in figure 2 1 .
42 THE INTERFEROMETRY OF
It was here, with the ocular thrown much to the right (near M), that I
again encountered the arrow-shaped fringes of figure 2, D, Chapter I. Though
they are rarely quiet, the observation can not be an illusion. As seen with
white light and a fine slit they are merely an indication of fringes which, when
viewed with a broad slit and homogeneous light, will be horizontal.
Rotation around a horizontal axis parallel to the face of the grating must
also destroy the parallelism of the rulings. The usual effect was to change the
size of fringes (distance apart, etc.) ; but I was not able to get any consistent
results on rotating G', owing to subsidiary difficulties. On rotating the grating
G, however, a case in which fine rotation around a horizontal axis was more
fully guaranteed, the fringes passed with continuous rotation through a
vertical maximum, as in figure 22.
In figure 26, the central region a of the grating G' is found, on inspection,
to be yellow in the position G', red in the position G'\, and green in the posi-
tion G'z. The slit in this case must be very fine. For a wide slit and homo-
geneous light, the continuous change in the obliquity of pencils is equivalent
to the continuous change of wave-length in the former case. It is therefore
interesting to make an estimate of the results to be expected, if the vertical
fringes for the cases bbr or cc' were Fresnellian interferences, superposed on
whatever phase-difference arrives at these points. In the usual notation, if
c is the effective width at the concave grating, F its principal focal distance,
x the deviation per fringe, dd the corresponding angle of deviation, X the
wave-length of light,
x = \F/c ord9 = \/c
If c=i.6 cm., X = 6Xicr5 cm., then d0 = 3.7Xio-5, or about 7" of arc.
The corresponding deviation ddD equivalent to d\D of the DiD2 lines would
be (if the grating space is .D=i 73X10-" cm., 6 = 20°, nearly, the normal
deviations for yellow light),
. d\D 6XICT8 _ _ _4
^==--°-3-7>
Thus -77:- = 0.1, or, in this special case, there would be ten hair lines to the
duo
DiD2 space. As c is smaller or larger, there would be more or less lines.
This is about the actual state of the case as observed. Finally, if c is very
small, the fringes are large, since
dd _\D cos 6
dBD cd\o
Thus conditions for practical interferometry would actually appear, the
fringes being of DiP2 width for c = 0.17 cm., provided a wide slit and homo-
geneous light of at least D\ or D2 grade is used. Such an interferometer
seems to differ from other forms, inasmuch as the fringes remain of the same
size and distribution, from their entrance into the field to their exit ; or for a
motion of the opaque mirror M of about 6 mm.
REVERSED AND NON-REVERSED SPECTRA. 43
To resume the evidence thus far obtained, we may therefore assert that
in the case of homogeneous light and a wide slit, or the absence of a slit, the
field would either be bright or dark, as a whole. There is a single enormous
horizontal fringe in the field. Hence the pronounced flickering with half
wave-length displacements of any part of the apparatus. With the slit
narrowed until the Fraunhofer lines are seen sharply, the linear phenomenon
in question (Chapter I) appears. This may become ladder-like, but it always
remains very narrow (-3- DiD^) when the rulings of the two gratings are not
quite parallel.
17. Subsidiary diffractions. — The behavior of the linear phenomena some-
times suggests probable relations to the Fresnellian interferences, produced,
however, not within the telescope, as in §§ 27, 28, Chapter III (for the inter-
ferences are seen together with the Fraunhofer lines in the principal focal
plane), but outside of it, at the grating, as suggested by figure 26. If the
concave grating G' is screened off, until a width of strip parallel to the rulings
and not more than 5 mm. wide is used, the linear phenomenon is much en-
hanced, being both broader and stronger, without losing its general character.
Here the D lines are still visible. The ladder-like patterns show an equally
pronounced coarsening. So far as these phenomena go, it is obvious that the
resolving power of the grating must be in question, seeing that the total
number of rulings has been greatly reduced. The use of screens with narrower
slits carries the process farther; but after the opening is less than 2 mm. in
width the available light is insufficient for further observation. If a small
lens is used, the phenomena can still be seen over 2 meters beyond the principal
focus of the grating.
A screen was now made as in figure 280, with two slits about 2 mm. wide
and 2 mm. apart (6), and placed over the effective part of the grating. The
result, after careful trial as to position, was noteworthy. Oblique fringes
were widened to many times the DiDz space and coarsened, showing a definite
grid-like design, as in figure 286, whereas, on removing the screen, the original
pattern of a regular succession of brilliant dots (fig. 28 c) again appeared.
It was with the linear fringes, however, that the evidence obtained was
most striking; for these now showed all the Fresnellian interferences (fig. 28 d).
On removing the screen, the brilliant linear phenomenon (fig. 28 e), which
in all the experiments made had thus far resisted manifolding, appeared at
once. The pattern, d, moreover, when viewed with a small lens, within a
meter in the direction of the rays, showed very definite enlargement with
distance. Though a fine slit was needed, the resolving power of the grating
was now too small to show any Fraunhofer lines. Similar results were obtained
for a wire i or 2 mm. in diameter. With the screen, figure 28 a, and a bar, b,
i mm. wide, the fine interference grid due to the bar, and the coarse grid due
to the spaces (the fine lines being about twice as narrow as the coarse, but
all of the same inclination) were often obtained together (fig. 28/). A space
i cm. wide intersected by a bar 2 mm. wide gave similar results, fine grids or
44
THE INTERFEROMETRY OF
thick lines, according as one or both spaces were used. If either mirror, M
or N, is screened, the whole phenomenon vanishes.
It follows, then, if rv and v'r', figure 29, represent the two reversed, over-
lapping spectra at the grating, b the focus and aa'b the direction of the homo-
geneous diffracted rays condensed at b, that about 0.5 cm. of the spectrum, d'a'
and ad on either side of a, is chiefly active in modifying the resulting diffrac-
tion pattern. Within this the homogeneous rays, cc' and dd', are capable of
interference. Although the wave-fronts entering b are slightly spherical, their
radius is about r=i meter, and they may therefore be regarded plane. In
such a case the angular width d% of the illuminated strip at b, for a width of
screen dd' = i cm., between two extinctions, may be written
_,ff_dx_ \ _6oXio~6
~"~r~dd'~ ~
= 6Xio-5
whereas the angular breadth of the DiD2 doublets is about 37Xio~5; i e.,
the rays from d and d', if in phase, should cease to illuminate b at a breadth
of about one-sixth the distance between the sodium lines. The rays within
d & a
r.
a
28
I
6 c d & f
dd' would correspond to greater widths; those from cc', for instance, 0.5 milli-
meter apart, would illuminate twice the estimated width, so that a strip at
b, with a breadth of one-third the interval D\DZ, is a reasonable average. All
rays, however, would produce illumination at b. As the screens are nar-
rower, not only would the fringe be broader, but more lines would appear,
because there is less overlapping. All this is in accord with observation.
Excepting the occurrence of independent half wave-fronts, the phenomena
do not differ from the ordinary diffraction.
With regard to waves of slightly different lengths, focussed at b', each is
there superposed on a wave of different length from its own, and appreciable
interference ceases for this reason. If the slit is widened, the phenomenon
(with white light) also vanishes by overlapping. The case of the screen with
two spaces has already been treated in relation to figure 26. In general, these
are cases of the diffraction of a rod, or of a slit, which are possible only if the
colors, X, are symmetrically distributed to the right and to the left of it. Thus
they require both spectra and can not appear if single spectrum only is present.
To reveal the nature of the phenomenon, a wide slit and homogeneous light
must be resorted to, as has been done in the present paper, even if white light
and the fine slit totally change the aspect of the fringes.
REVERSED AND NON-REVERSED SPECTRA. 45
18. Conclusion. — To return, finally, to the original inference, it appears
that beating wave-trains have not been observed, but that the striking scin-
tillations are due to an exceptional susceptibility of the apparatus to labora-
tory tremors, when exhibiting the phenomenon in question. Of this I further
assured myself by observations made at night and on Sunday, though there is
some doubt in my mind. What has certainly been observed is the inter-
ference of a DI or DZ line with a reversed D'i or D'2 line, both having the same
source and longitudinal axis. One can only assert, therefore, that light of
the wave-length interval of the breadth of these lines is capable of interference,
when the line is reversed.
The phenomena, as a whole, are to be treated as diffractions of symmetrical
half wave-fronts, each of which may be separately controlled by the corre-
sponding micrometer.
CHAPTER III.
THE INTERFERENCES OF THE NON-REVERSED SPECTRA OF TWO GRATINGS,
TOGETHER WITH AN INTERPRETATION OF THE PHENOMENA IN
CHAPTERS I AND H.
19. Introduction. Method. — The chief purpose of the present paper is
the search for phenomena similar to those of Chapter II, but in which the
two spectra brought to interference are not inverted relatively to each other.
Incidentally the strong interferences may have a value on their own account.
It has been shown that the totality of the phenomena with spectra reversed
on a transverse or a longitudinal axis are quite complicated, and a series of
companion researches in which similar results are aimed at, in the absence
of inversion, is thus very desirable.
The apparatus (fig. 30) is a modification of that shown in figure 50, in the
next section, MM being the base of the Fraunhofer micrometer, 55 the slide,
E the micrometer screw. The brass capsules A and B are securely mounted
on the slide 5, free from the base M, and on the base M free from the slide
_
6
r
[=
V
~<]
r
/
r
,
<W CA
$0 g
-•
^
i
i
i--.
i
i
•i
*.::.•
<---&••>
=1^
•-&•>
c/M,
J' ^ 31
30
5, respectively. Each capsule is provided with three adjustment screws
relative to horizontal and vertical axes a, b, br, and c,d,dr> together with strong
rearward-acting springs, by which the gratings G and H at a distance e apart
may each be rotated slightly around a vertical or a horizontal axis (plane dot
slot mechanism) . The two gratings G and H must be identical, or very nearly
so, as to the number of lines per inch, and with their ruled faces toward each
other. These faces, as well as the ruled lines, are to be nearly in parallel.
To secure the latter adjustment a bolt, g, normal to the face of the grating
H, serves as an axis, and an available tangent screw and spring (not shown)
is at hand for fine adjustment. This device is of great importance in bringing
the longitudinal axes of the two spectra due to G and H into coincidence,
and a fine wire must be drawn across the slit of the collimator to serve as a
guiding-line through the spectrum. Any lack of parallelism in slit and rulings
rotates the fringes.
46
REVERSED AND NON-REVERSED SPECTRA. 47
The beam of light, L, either white or homogeneous, as the experiment may
require, is furnished by a collimator (not shown), which, with the telescope at
T (placed in plan, in figure 31, at T or D}, are the usual parts of a spectro-
scope. The collimator with slit is always necessary for adjustment. It may
then be removed if the phenomenon is to be studied in the absence of the
slit. The telescope is frequently replaced to advantage by a lens. White
light is to be furnished by the arc lamp (without a condenser), by sunlight,
or by an ordinary Welsbach burner. Both spectra are naturally very intense.
A sodium flame suffices for the work with homogeneous rays.
The adjustments in case of white light are simple and the interferences
usually very pronounced, large, and striking. Brilliant spectra, channeled
with vertical narrow black lines, are easily obtained when the longitudinal
axes are placed accurately in coincidence by rotating the plate h carrying
the grating H, on the plate /, around the axis g. If the gratings are quite
identical the sodium lines will also be in coincidence. Otherwise the two
doublets, DiD2 and D'lD'?, of the two spectra (nearly identical in all their
parts and in the same direction) are placed in coincidence by rotating either
grating around a vertical axis. Thereupon the strong fringes will usually ap-
pear for all distances, e, less than 2 cm. These fringes are nearly equidistant
and vertical and intersect the whole spectrum transversely. They are not
complicated with other fringes, as in the experiments of the next section.
They increase in size till a single shadow fills the field of view, in proportion
as the distance e is made smaller and smaller to the limit of complete contact.
With the two adjustments carefully made, finally, by aid of the fringes
themselves, further trials for parallelism are not necessary. Two film grat-
ings, or even films, give very good fringes. During manipulations great care
must be taken to keep the angle of incidence, i, rigorously constant; i.e.,
to avoid rotating both gratings together or the apparatus as a whole, as this
displaces the sodium doublets relative to each other and seriously modifies
the equations.
20. White light. Colored fringes. — The two sodium doublets seen in the
arc spectrum are usually equally brilliant, and but one set of strong fringes
is present in the field of the telescope. Relatively faint fringes may some-
times occur, due, no doubt, to reflection, as investigated in the next section.
If both gratings are rotated, changing the angle of incidence from o° to i°,
the fringes disappear from the principal focal plane, but reappear strongly
in another focal plane (ocular forward or rearward). In such a case the D
lines are no longer superposed. To be specific, let i and i', 6 and 6', be the
angles of incidence and diffraction at the two gratings in question, the angle
between their ruled faces being i-i'. Let D and D' be the two grating con-
stants, and nearly equal. Then for a given color, X, in relation to the individual
normals of the two gratings,
sin 6 — sin i = \/D sin 6' — sin i' = \/D'
48 THE INTERFEROMETRY OF
Now if B' is referred to the original normal it becomes B" '= Q'+i
or
If the sodium lines are to coincide, 6=9", or approximately
sin (Q—(i — i')} — sin i' = sin 9 — cos 9 . (i — i'} — sin i' =
or on eliminating sin 9
•i f • -i\ n X X
sim — sin 2 — (i — i ) cos 9 = ^ — jy
which is nearly
i-i' X
In case of the Wallace grating below
£=1.75X10-* X=58.93Xio-6 i-if= 10^X3-29 (D-D'}
Thus if the inclosed angle i — i' between the plates is i degree, or 0.0175
radian, D— D' = S-^Xio'"7, about 0.3 per cent of D and equivalent to about
43 lines to the inch. With adequate facilities for measuring i, this method
may be useful for comparing gratings, not too different, in terms of a normal
or standard, practically, since the finite equations may also be expanded.
In a similar way the slight adjustments of the longitudinal axes of the two
spectra may be made by rotating one grating around a horizontal axis ; but
this correction is less easily specified. Finally, one should bear in mind that
with film gratings there is liable to be an angle i-i' between the adjusted
plates. Fortunately this has very little bearing on the method below.
The range of displacement of grating within which the fringes may be
used with an ordinary small telescope extends from contact of the two gratings
to a distance of e = 2 to 3 cm. beyond.
In figure 31, which is a plan of the essential planes of the apparatus, G, G'
being the ruled faces of the gratings in parallel, 7, I', I", three impinging
rays of white light diffracted into D, Df, the points a, b, c, a , a" , b are in the
same phase, so that the path-difference of the rays from b at g and / is easily
computed. If the single ray I is diffracted into D and D' or I and I' into D,
I and I" into D', I' and /" into D and D', the equations for- these fringes
should be (if AP is the path-difference),
= <?(i-cos 0)=*(i-Vi-
where D is the grating space, e the distance apart, and X the wave-length.
Thus the micrometer value of a fringe for a color X should be, under normal
incidence,
For two colors X and X'
= e(i-cos 6}=eM («+w')X' = e(i— cos 6'}=eM'
REVERSED AND NON-REVERSED SPECTRA.
49
if n' is the number of fringes between X and X'. Thus
M'X-MX'
(3)
n = e-
XX'
or the number of fringes increases as e is greater.
Equation (2) does not, as a rule, reproduce the phenomenon very well.
Since the grating space D of the two gratings is rarely quite the same, the
air-plate inclosed, in case of apparent coincidence of the sodium lines, is
slightly wedge-shaped, as in figure 32. Hence the two diffractions take place
at incidences o° and a°, respectively, and the corresponding angles of diffrac-
tion will be 6 and tf . If we consider the two corresponding rays / and I",
diffracted at the first and second face, respectively, and coinciding at c in the
latter, the points a, b, and a' (ba' normal to ac), are in the same phase, and
we may compute the phase-difference at the coincident points at c. Since
the distance be is
cos a cos 9
e = e-
— a)
33
1 34
the path-difference is
whence
(4)
e cos a cos 0
cos (0— a)
— COS0)
X cos (6— a)
COS 0COS a • (i— COS 0)
which changes into equation (2) when a = o and iz = i. Fortunately this
correction is, as a rule, small. In case of the Wallace gratings (D= 1.7 5 X lo"4
cm.), for instance, if X= 58.93 Xicr6, then 0=19° 40' or 8e= i.oiXicr3; whereas
if a =5°, then 8e =1.04 Xio~3; if a =10°, then 8e =1.07 Xicr3, etc.
If the incidence is at an angle i and the plates are parallel, figure 33, the
inquiry leads in the same way to an equation of more serious import. If the
gratings G and G' are at a distance e apart and the incident rays are I and /',
the points a, b, c are in the same phase. Hence the two rays leaving d and
diffracted along D correspond to a path-difference
(5)
whence
COS I
i -cos (0-z))
X cos t
I— COS (0 — l)
50
THE INTERFEROMETRY OF
Table i and fig. 35 show the variation of fringes with the angle of incidence *,
equation (5). Hence if the angle of incidence is changed from —5° to + 5°,
be increases to nearly 3 times its first value. This, therefore, accounts for the
large discrepancies of be found in the successive data below. To secure in-
creased sensitiveness and to make the apparatus less sensitive to slight changes
of i, this angle should be about 25°, in which
case 8e is about three wave-lengths per fringe. But
normal incidence is frequently more convenient.
Finally, in figure 34, if the angle of incidence is i
and the two faces G and G' make an angle a with
each other and are initially at a distance e apart,
changing successively to e' and e", the points a,
b, c, being in the same phase, the two rays D and
D' leaving at d, at an angle 6—i, will have a path-
difference at d equal to
cos 0cos a
whence
(6)
,-(i — cos (6— i})
cosi cos (6— a)
X cost cos (6— a)
cos a cos 6 (i — cos (6 — i))
TABLE i. — Wallace gratings. £>=io-4Xi.75.
i
io3XSe
i
io3X5c
+ 19040'
OC
-5°
0.643
+ I5I
16.740
-10°
0-443
+ 10°
4.090
-15°
0.321
+5°
1.840
-20°
0.240
±0°
I.OI2
-25°
0.185
This equation reproduces the preceding equation (5) if a = o and the origi-
nal equation (2) if a = i = o. It shows that .a discrepancy or angle between
the plates is of minor importance. Hence the change of this angle may be
used to bring the sodium lines in coincidence when the gratings differ slightly
in their grating constants D. On the other hand, changes of incidence i are
of extreme importance.
Experiments made with the film grating showed that equation (2) not only
fits very badly, but that de per fringe is a fluctuating quantity. Table 2
gives some results obtained by measuring the successive values obtained for
5eXios, corresponding to 10 fringes. Fringes were distinctly seen within
3 cm. of displacement by an ordinary telescope.
TABLE 2.
Gratings 15,050 lines to inch; computed values io35e=o.94 cm. per fringe.
First sample: Second sample:
io»*= i .22 cm. 10^=0.98 cm.
1.16
I.2I
1. 06
I. II
(norma] inddence)
> (oblique incidence)
REVERSED AND NON- REVERSED SPECTRA. 51
TABLE 2. — Continued
Wallace gratings 14,050 lines to inch; computed value io35e=i.oi cm. per fringe.
io3Se= .22 cm. (large fringes); io35e=i.35 (different *')
24 1.32
.25 1-30
23 i-35
.23 (small fringes) 1.33
1.23 1-33
1.32
1-34
The reason for lack of accord is given in equations (5) and (6) and table
i. Any wedge effect of the glass plate is probably negligible. To show that
the irregularity of the above results is to be sought in the accidental varia-
tions of the angle of incidence i at both gratings, the rough experiments in
table 3 suffice.
TABLE 3.
i, negative (less than io°),l 5eXio3=o.77 cm.
Ocular drawn in, .78
focus changing. .74
z==tq; ocular set for ] 5eXio3= 1.18 cm.
principal focal plane. Na lines !•
in field and coincident, 1.12
i, positive (less than lo0),"! 5eXio3=i.69 cm.
ocular drawn out, / 1.66
Thus, as equation (6) implies, small variations of i produce relatively large
variations of 5e, and if i passes continuously through zero, from negative to
positive incidence, 5e increases continually and may easily be more than
doubled. If the phenomenon is in focal planes in front of the principal plane
(ocular in), de is small, and vice versa. Moreover, this enormous discrepancy
is quite as marked for thin glass (2 mm.) as for thick glass plates (8 mm).
Again, the rather stiff screw of the micrometer, which twisted the whole
apparatus slightly, was sufficient to introduce irregularity. Placing the tele-
scope close to the grating or far off made no difference. Hence the position of
the optical center of the objective does not affect the result.
An additional result was obtained by placing a plate of glass between the
two gratings G and G' . The effect was an unexpected enlargement of fringes,
increasing with the thickness of the glass plate (0.6 cm. or more). The reason
for this is given by equation (3), in paragraph 2, for the number of fringes
n' between two colors X and X',
- xv -
where M= i-cos0, M' = i-cos0'. Since n' is a number, the glass plate can be
effective only in changing 6 and 6' . As both are diminished by refraction, the
cosines are increased and i-cos 6, i-cos 6' are both decreased. Hence n' is
decreased or the number of fringes is decreased, and their distance apart is
thus larger.
It is obvious that when the sodium lines are not superposed the fringes
can not lie at infinity, but are found in a special focal plane, depending on
52 THE INTERFEROMETRY OF
the character of coincidence; i.e., whether the rays are convergent or diver-
gent. Finally, a slight rotation of the slit around the axis of the collimator
rotates the fringes in the opposite direction to the sodium lines, and it is
rather surprising that so much rotation of slit (10° or 20°) is permissible
without fatally blurring the image. The slightest rotation of one grating
relatively to the other destroys the fringes.
Naturally, the colored fringes vanish when the slit is widened or when it is
removed. To give them sharpness, moreover, the beam passing through the
grating must be narrow laterally. It is possible to see these colored fringes
with the naked eye; but the transverse and longitudinal axes must in this
case be slightly thrown out of adjustment, so that the fringes are no longer
visible in the telescope. To the eye they form a somewhat fan-shaped set of
colored fringes; i.e., narrower below than above. Neither are the lines quite
straight. If the collimating lens is removed, a slit about o.i cm. wide across
a white flame will also show (to the telescope or to the eye) fine, strong lines
rotating in opposite direction to the slit, according as the transverse and longi-
tudinal axes are differently placed. As has been already stated, it is with
the latter condition that the focal plane in which the fringes He varies enor-
mously.
Finally, when the sodium lines are superposed but the longitudinal axis of
the spectra not quite so, a second class of fringes appear, which, however,
are always more or less blurred. They rotate with great rapidity over 180°
when one grating rotates over a small angle relatively to the other and the
angle between the longitudinal axes of spectra passes through zero. In the
latter position the regular fringes appear in full strength in the principal
focus. To see the secondary interferences, the ocular must be drawn inward
(toward the grating) , and these fringes increase in size with the displacement
of the ocular away from its position when regarding the principal focal plane.
This secondary set of fringes is always accompanied by another very faint
set, nearly normal to them and apparently quivering. The quiver may be
due to parallax and the motion of the eye. These are probably the vestiges
of the regular set of fringes, out of adjustment.
21. Homogeneous light. Wide slit. Transverse axes coincident. — If there
is no color-difference, fringes of the same kind will nevertheless be seen in the
telescope, on widening the slit indefinitely. Path-difference is here due to
differences of obliquity in the interfering rays. As in the preceding case,
accurate adjustments of the longitudinal and transverse axes (in case of
sodium, Di and D\ or D2 and D'2 coincide horizontally and vertically) of
the homogeneous color-field are essential if strong fringes are to appear in
the principal focus. These fringes are, as a rule, well marked, and widening
the slit merely increases the width of the channeled, homogeneous field of
view. If, owing to slight differences of grating space, the sodium lines are
not quite superposed automatically, this may be corrected by rotating either
grating, or else the apparatus as a whole, until the fringes are strongest. The
REVERSED AND NON-REVERSED SPECTRA.
53
fringes may be made to vanish under inverse conditions. Table 4 shows their
close relation to the preceding colored set, so far as motion of the micrometer
is concerned.
TABLE 4.
Ives grating. 15,050 lines to inch, computed 5e=io-3Xo.94 cm. per fringe.
io35e=o.72 cm. (large fringes) io36e=o.83 cm. (small fringes)
.80
•77
.80
Wallace grating. Wide slit. Coincident Na lines. Fringes in principal focus,
very clear and strong.
io3de=i.i6 cm. io3<5e=i.o6 cm.
,, Ti.i6 1.08
Mnges |;:;7'
Wallace grating. Wide slit. Non-coincident Na lines.
io35e=i.34 cm. io3de=i.^8 cm.
(small fringes) 1.28 (large fringes) 1. 35
The fringes decrease in size as e increases and exhibit the same irregularity
of Se values, due, no doubt, to the same causes (equation 6). Moreover, be
is here below the normally computed value, supposing the angle i to be negli-
gible. In fact, figure 36 shows the optical center of the collimator C; so that
37
36
Ca and Ca' are the axes of parallel pencils, diffracted by the gratings G and G'
at the angles 6 for Ca and tf for Ca' . The rays are subsequently condensed
at F, the focus of the telescope, L being the principal plane of the objective.
The general path-difference is thus, by equation (5), e(i— cos (0'+*))/cos i,
which distributes the fringes from right to left with variation of i.
54 THE INTERFEROMETRY OF
If the grating G' is displaced be parallel to itself, however, the path-difference
will again be increased by X whenever
X cos i
fu? —
i-cos(0'-H)
Since i is small, this equation will not differ appreciably from equation (2),
with which it coincides for the central fringes.
If the sodium lines are not superposed, these fringes may still be seen, but
they are not in the principal focal plane and the new focal plane changes con-
tinually, as the fringes grow in size. Examples are given in table 4. The
large values of de show that i was not actually negligible. Experiments
similar to the above, bearing on the reason for the discrepancy (equation 6) ,
were tried with the thin Wallace gratings, and the results are given in table 5.
TABLE 5. — Thin Wallace grating.
i negative (within 10°) 5«'Xio3=2.6o cm. 5eXio3=2.4O cm.
Ocular in, 2.34 2.45
2-57
z'=±o, normal incidence, 5eXio3=i-48 cm. 5eXio3=i.37 cm j (small
Ocular set for principal 1.50 1.37 '\ fringes)
focal plane, 1.37 1.32 ((large
1.19 ^fringes)
i positive (within I o°) SeXio3=o.86 cm 8eXio3=o.<)6
Ocular out, .88 .86 cm,
•85
.87
.96
(small
fringes)
(large
fringes)
As before, the effect of i passing from negative (through zero) to positive
values is enormous, de increasing nearly threefold for a change of i estimated
as within 20.° Here, however, the drawn-out ocular (towards the observer)
corresponds to the small values of de, whereas above the reverse was the case.
This depends upon which of the spaces D or D' is the greater.
22. Homogeneous light. Fine slit. Transverse axes not coincident. — To
obtain this group of interferences, the two sodium lines from a very fine slit
are thrown slightly out of coincidence; i.e., by not more than the DiD2 dis-
tance. In the principal focal plane, therefore, these doublets are seen sharply,
while if the ocular is drawn sufficiently forward or reanvard, an interesting class
of fringes soon appears which resemble Fresnel's fringes for two virtual slits.
These fringes may be seen, however, on both sides of the focal plane and in-
crease in size with the distance of the plane of observation (focus of ocular)
in front or behind the principal focal plane. In figure 37 the two gratings,
G and G', are struck by parallel pencils from the collimator at different angles
of incidence (o° and i°). The two diffracted pencils of parallel rays are
caught by the objective L of the telescope and condensed at the principal
foci, F and Ff, appearing as two bright yellow lines. In front and behind the
plane FF', therefore, are two regions of interference, I and I', throughout
which the Fresnellian phenomenon may be seen in any plane parallel to
FF', observed by the ocular. When the electric arc is used with a very fine
REVERSED AND NON-REVERSED SPECTRA. 55
slit, these sodium fringes often appear at the same time as the colored fringes,
and, though they are usually of different sizes, their lateral displacement
with a change of distance apart of the gratings, 8e, is the same. The fringes
in question appear alone when the sodium burner is used. They may then
(at times) be observed with the naked eye, with or without a lens, and they
fail to appear in the telescope unless the objective is strengthened by an
additional lens. They are always vertical, but finer in proportion as the
DiD2 and D\D'z doublets are moved farther apart. They become infinite
in size, but still strong, when the doublets all but coincide, showing a ten-
dency to become sinuous or possibly horizontal. Rotation of either grating
G around an axis normal to itself and relative to the other produces greatly
enhanced rotation of the fringes, as in all the above cases, but they soon
become blurred.
Only in the case when the horizontal axes of the field coincide (parallel
rulings, etc.) do they appear strong. When the angle of incidence (or non-
coincidence) is increased for both gratings, the size of the fringes increases;
but when the e distance is increased by the micrometer, the fringes are appar-
ently constant as to size. However, after displacement of 4 mm. they are
liable to become irregular and stringy, though still moving. A fine slit is not
essential, particularly when e is small. They vanish gradually when the slit
is too wide. If a telescope with a strong objective is used, these fringes may
be seen, retaining their constant size long after those of the next paragraph
vanish. Examples of data are given in table 6, and 5e is too low in value as
compared with the computed datum for 4 = 0°. With the Wallace gratings,
these fringes were best produced by the aid of the sodium lines, in the ordi-
nary electric arc, simultaneously with the colored fringes and for the case of
a very fine slit. They were apparent both with an ocular drawn out or drawn
in. In the former case several successive groups were observed. Beginning
with the sharp sodium lines in principal focus (D2 and D'2 coincident) , a slight
displacement of the ocular outward showed the first group, this resembling a
grid of very fine striations. Further displacement outward produced a second
set, equally clear but larger. A third displacement of the ocular outward
showed the third set, and these now coincided with, and moved at, the same
rate as the colored fringes in the same field. Other groups could not be found.
No doubt, for these four successive steps the interference grids of DI and D\,
D2 and D'2 are coincident and superposed, until they finally find their place in
the colored phenomenon.
TABLE 6. — Ives grating. Homogeneous light. Fine slit. Sodium lines not coincident.
5eXio3=o.87 cm.
•77
.83
23. Homogeneous light. Slit and collimator removed. — Fringes similar
to those seen with the wide slit above may be observed to better advantage
by removing the slit altogether. The sodium flame is then visible as a whole ;
and if the adjustments are perfected it is intersected with strong, vertical black
56
THE INTERFEROMETRY OF
lines, visible to the naked eye or through a lens or a suitably strengthened
telescope. They decrease rapidly with increase in e, but vanish to the eye
before the preceding set in paragraph 22. The sodium lines need not be in
adjustment, but the longitudinal axes of the field must be, as usual. If diffuse
white light is present, faint colored fringes may be seen at the same time.
If the collimator only partly fills the field of view, these diffuse light fringes
and the preceding set may occur together. Both rotate markedly for slight
rotation of either grating in its own plane. There seems to be a double peri-
odicity in the yellow field, but it is too vague to be discerned. When magni-
fied with a lens, they admit of a play of e within about 0.6 cm. from contact.
When the sodium lines are not coincident, the focal plane continually changes
with e. Otherwise it remains fixed.
Some data are given in table 7.
TABLE 7. — Ives grating. Homogeneous light. Collimator and slit removed. Focus
Homogeneous light. Collimator and slit removed,
continually changing.
3eXio3=o.95 cm.
I.OI
1.04
1.02
0.96
38
Large fringes; ocular out; lens on <
Ocular in, lens on
Very small fringes, lens off
Wallace grating. Sodium lines coincident.
Principal focal plane SeX 10"= i .08 cm.
1.18
i. 20
These data are similar to the above and subject to the same discrepancy
whenever slight variation of the angle of incidence accidentally occurs. In
figure 38 the case of three rays from a given
flame-point F is shown corresponding to the
equation
_ X cos i
i— cos (0+0
when i passes from positive to negative values.
If either of the gratings is displaced and if they
are parallel, the focal plane will not change;
but if G and G' are not parallel, the focal plane
differs from the principal plane and now moves
with the grating.
24. Inferences. — The above data show that the equation underlying all the
interferences observed is the same. The interferences themselves may result
from different causes, but their variation in consequence of the motion of the
grating, Se, is due to one and the same cause. This is best seen by producing
them simultaneously in pairs. As a means of finding an accurate compari-
son of the number of lines per centimeter on any grating, in comparison
with those on the given grating, the method used in paragraph 20 deserves
consideration.
REVERSED AND NON-REVERSED SPECTRA.
57
If the fringes are to be used for practical purposes, great care must be taken
to keep the angle of incidence of the impinging light constant. This was not
done in the present paper, where the purpose is merely an identification of the
phenomena. Moreover, a micrometer with the screw running easily is essen-
tial, as otherwise the frame is liable to show appreciable twist (change of inci-
dence) during displacement of the fringes.
The fringes are not of the sensitive type, but they admit of a large range of
displacement and are therefore adapted to special purposes.
With regard to their bearing on the behavior of reversed spectra, for the
interpretation of which the present experiments were undertaken, it is obvious
that the interferences with homogeneous light and a wide slit (paragraph 21),
or in the absence of a slit (paragraph 23), are of analogous origin in both
cases. It makes no difference, therefore, whether one of the spectra is reversed
or not, except, perhaps, that in the former case (inversion), the coincidence
of longitudinal and transverse axes is a more insistent condition. The colored
fringes of paragraph 20 obviously can not be produced with reversed spectra.
There remain the fringes with the fine slit and homogeneous light (paragraph
22); in other words, the occurrence of a sort of generalized Fresnellian inter-
ferences, within the telescope, modified by causes which lie outside of it. Thus
DI and D'i or D2 and D\ may be placed sufficiently close together to pro-
duce a region of interference before and behind the principal plane in which
the sodium lines are in focus. If the DiD'i lines are o.oi cm. apart and the
fringes seen likewise at o.oi cm. apart, their position, measured from the
principal plane, will be at
YC
icr4 5
:— — 5=-cm.
6Xicr 3
or less than 2 cm. The ocular would then have to be displaced forward or
rearward by this amount. But there are two sodium doublets, each pair of
which is to interfere. Suppose that D2 and D'\ are in coincidence so that the
40
scheme is Di:D*D\: D'2( as in figure 39, where o is the principal plane of the
objective and DI to D'2 the principal focal plane. We should then have the
separate regions of interference I and I' and the combined regions I" and ]'" .
When the breadth of the latter is the whole number of fringes, the two pat-
terns clearly merge into a single pattern. The experiments show several of
these stages, terminating outermost in the focal plane of the colored fringes
under the given conditions. Since the fringes lie on hyperbolic loci the problem
58 THE INTERFEROMETRY OF
itself is beyond the present purposes ; but it appears that the colored fringes
will not appear until the corresponding DI and D2 lines are shared by the
whole of the two continuous spectra.
The final question at issue is the bearing of the present Fresnel phenome-
non on the reversed spectra. If in figure 40, 5 and s' represent the traces of
two reversed spectra in the principal focal plane, superposed throughout their
extent (i.e., in longitudinal coincidence), the rays aia'-JB, through the line of
symmetry a, a', are at once in a condition to interfere with a given difference
of phase; but so are all the symmetrically placed pairs of colors, c, c' , b, b', of
the two spectra (the distances cc' , bb', being arbitrary), provided the corre-
sponding rays meet. As they do not meet in the principal focus, they can
interfere only outside of this — b and b' at B, c and c' at C, etc. Similar con-
ditions must hold at B' and C' within the principal focal plane. The linear
interference is thus successively transferred to different pairs of wave-lengths.
The phenomena of this paper can not, therefore, be detected in case of reversed
spectra, because in the principal focal plane different wave-lengths are every-
where superposed, except at the narrow strip aa', which experiment shows to
be about one- third of the width of the sodium doublet, in apparent size.
Beyond the principal focus the corresponding conditions in turn hold for the
rays at B, C, etc., B', C', etc. Hence there can not be any Fresnellian inter-
ferences (paragraph 22), for there are not two virtual slits, but only a single
one, as it were, and the interferences are laid off in depth along the normal
C'C. The phenomenon may, in fact, be detected along this normal for 2 or
3 meters.
25. Rotation of colored fringes. Non=reversed spectra. — When the slit is
oblique, it effectively reproduces the wide slit, locally, and therefore does not
destroy the colored fringes. At every elevation in the field the slit is neces-
sarily linear, though not vertical. In figure 41, let the heavy lines, H, denote
the colored fringes for a fine vertical slit and white light, showing nearly the
same distance apart, throughout. Let the light lines, L, denote the fringes
for a wide vertical slit and homogeneous light, X. These fringes are due to
the successively increased or decreased obliquity of the rays in the horizontal
plane. Now let acb be the image of the oblique slit in homogeneous light. It
is thus merely an oblique strip, cut from the area of light lines or striations,
as it were, and consists of an alternation of black and bright dot-like vertical
elements in correspondence with the original striated field. We may suppose
ab to have rotated around c, so that the vertical through c is its position on
the colored field (white light and fine vertical slit).
A color, X' (near the one X) , corresponding to the field of the lines L in case
of a wide slit and homogeneous light X', will supply nearly the same grid, so
far as the distance apart of fringes is concerned. But the grid is displaced
laterally, in consequence of the different angle of diffraction, 6. This is shown
by the dotted lines D in figure 41, the effect being as if the slit had been dis-
placed laterally. If the wide slit for homogeneous light X' is now narrowed and
REVERSED AND NON-REVERSED SPECTRA.
59
inclined as before, an alternation of bright and dark elements will appear in
the image of the slit ed, corresponding to X'. If we suppose that for white
light and the fine vertical slit the position of the fringe (X') was at c', we may
again regard c' as an axis of rotation. To find the fringes such as//", it is then
only necessary to connect corresponding black elements on ab and ed. Their
inclination is thus opposite to ab and ed, or they have rotated in a direction
opposite to that of the slit. If, for instance, the slit image ab or ed is gradu-
ally moved back to the vertical, the points g and h will move with great
rapidity and in both directions toward infinity and the fringes // and /'/'
become vertical lines through c and c' , respectively.
It is interesting to inquire into the frequency of fringes, n, when the angle
of diffraction, 6, is changed. From the original equation e = n\/(i-cos 0),
since d\/dd = D cos 6, the rate of change
dn e i e
av D i+cos0 D+1/D2-X2
where e is the distance apart of films and D the grating space. Since cos 6
varies but slowly with 6 and is additionally augmented by i, dn/dd is nearly
constant and about equal to e/2D.
41
42
Ih,
The fringes and slit images are thus given by the two sides of the parallelo-
gram cgc'h for the two colors X and X'. The diagonal cc' represents dd; the
diagonal gh has no signification. On the other hand, the normal distances
apart, D' and D", of//" and/'/' and ab and ed are both important.
If D' and D" are the normal distances apart of the fringes and the slit
images, respectively, B and B', the two diagonals of the rectangle cgc'h, modi-
fied for convenience in figure 42 ,
D' =D"(cos <p-}-\/Bf2/D"2 — i sin <p)
which may be obtained from the two small triangles below c'. If B = D",
D' = D" cos (f>; and if <p = o, D' = D" = dd, remembering that c and c' lie on two
consecutive colored fringes obtained with white light and a fine slit. If the
slit images and fringes are symmetrical, each is at an internal angle, po-<p/2,
to the longitudinal axis of the spectrum.
60
THE INTERFEROMETRY OF
But these equations, though useful elsewhere, have very little immediate
value here, because the experimental variables, figure 41, are B', the distance
between two consecutive colored fringes and b" and bf the corresponding dis-
tance between the fringes in case of homogeneous light in each case X, X' ; and
the angle y', which indicates the inclination of the slit. Thus B'b'b" are
given by computation and y' is specified at pleasure. Obviously, if parallelo-
grams are to be obtained, figure 41, b' = b'f, appreciably. This is the case in
experiment. Hence if we evaluate the height in the triangle cgc' for each
angle it follows easily that
, tan y'
sin x =-
If
If B' = b', x' = go° for all values of y'; i.e., the fringes remain vertical.
B' is equal to 26, x' = y', the fringes and slit are symmetrically equiangular
with the longitudinal axis of the spectrum. This is nearly the case in figure
41 and frequently occurs in experiment. If b' differs from b", the fringes would
not be straight. This also occurs, particularly when the thickness e of the
air-film is very small.
26. Final treatment of reversed spectra. Hypothetical case. — To obtain an
insight into the cause of the interferometer fringes as obtained with reversed
spectra and two gratings, it is convenient to represent both gratings, figure
43, GG and G"G', as transmitting,
and suppose both diffracted beams,
ID' and ID", subsequently com-
bined in view of the principal plane
PP of an objective or a lens. It is
clear that this simplified device can
apply only for homogeneous light.
In the case of white light, the opaque
mirrors M and N (of the interfer-
ometer, above) return a divergent
colored beam or spectrum, so that
only for a single color can the second
incidence be the same as the first.
Again, if the constants of the two
43
gratings are different, it is the func-
tion of these mirrors to change the
&-
71'
incidence at the second grating correspondingly, so that for homogeneous
light the rays issue in parallel. Finally, no reference to the lateral displace-
ment OG" and OG' of rays need be made because, as more fully shown in
the next paragraph, this is eliminated by the theory of diffraction.
The motion of the opaque mirrors M and N (above), on a micrometer,
merely shortens the air-paths GG' or GG" in its own direction, and conse-
quently the same fringe reappears for an effective displacement of half a wave-
length, as in all interferometers.
REVERSED AND NON-REVERSED SPECTRA.
61
The case of a single grating, moreover, is given if the planes of the grating
GG and G'G" and their lines are rigorously parallel, the planes OG' and G"O
being coplanar. To represent the interferences of the two independent gratings
and with homogeneous light for the case of oblique incidence, it is necessary
to suppose the grating G'G" cut in two halves at 0, parallel to the rulings,
and to displace the parts OG' or OG" separately, normally to themselves.
Figure 43 shows that for normal incidence i = o, the displacement per fringe,
8e, would be
X
I— COS 0
or the fringes are similar to the coarse set of the present chapter.
If the rays impinge at an angle i, figures 43 and 46, they will be parallel after
the two diffractions are completed; for it is obvious that the corresponding
angles of incidence and diffraction are merely exchanged at the two gratings.
Hence the homogeneous rays I', impinging at an angle i, leave the grating at
D'i and D'\ in parallel, at an angle of diffraction i, and the rays unite into a
bright image of the slit. If, however, OG' be displaced to OiG\, parallel to
itself, as in figure 44, the paths intercepted are
6 €
.and - -.cos (Q — i)
cosz
cos*
and the path-difference per fringe, therefore,
X cos i
i— cos (0-i)
which reduces to the preceding equation if i — o. Hence a series of inter-
ference fringes of the color X must appear in the principal focus of the tele-
scope or lens, on either side of i = o. The theory of diffraction again annuls
the apparent path-difference between GG and G'G",
44
45
As to the number of fringes, n, between any two angles of incidence i and
i' , it appears that for homogeneous light of wave-length X,
_£/i— cos (0-t) _i — cos (0'-&')\
X V cos i cos i i
62 THE INTERFEROMETRY OF
where e is the distance apart of the two parallel halves of the grating G"0,
OG'. Hence n vanishes with e, or the fringes become infinitely large. Lateral
displacements are here without signification, as stated above.
If the grating G' is rotated over an angle <p, figure 43, and e = bp where 26
is half the virtual distance apart at the grating G' of the two corresponding
rays impinging upon it (Chapter II, fig. 26), the rotation of the grating per
fringe is thus
_ X cos i
^ = 6 i -cos (0-f)
or n (above) passes through zero as <p or b decreases from positive to negative
values. If b is considered variable there is a wedge-effect superposed on the
interferences.
It is this passage of n through zero that is accompanied by the rotation of
the fringes, as above observed.
In case of two independent gratings, GG and G'G" (G'G" to be treated as
consisting of identical halves, OG' and G"0), nearly in parallel, fringes may
be modified by rotating G'G" around the three cardinal axes passing through
the point of symmetry 0. The rotation of G'G" around an axis 0 normal to
the diagram is equivalent to the fore-and-aft motion of G'G' when mirrors
are used (fig. 26, Chap. II). The rotation around OT in the diagram and nor-
mal to the face of the grating requires adjustment at the mirrors around a
horizontal axis to bring the spectra again into coincidence. This is equiva-
lent to rotation around G"OG'. Both produce enlargement, and rotation of
fringes is already explained.
Let the grating G'G" be rotated over an angle <p into the position g'g", figure
45. Then the angle of incidence at the second grating, d, on one side is
increased to 6"=6-}-<f> and on the other decreased to 61 =d — <p. In such a
case the diffracted rays are no longer parallel. If 6' and B" are two angles of
diffraction on the right and on the left,
whence
sin #"+sin B' = 2 sin <p cos 6
or if 6 is the mean value of Q' and 6"
B = (p cos 0, nearly.
Similarly, since sin 0 = X/D, for i = o,
sin 6' -sin 6" = 2\ (i -cos <p)/D
Hence only so long as <p is very small, are the rays appreciably parallel on
rotating G'G" around O normal to the diagram; but this is usually the case,
as <p = o is aimed at, and fringes are thus seen in the principal focus.
To the same degree of approximation is it clear that on rotating the grating
into a position such as og" the rays emerge parallel to IT, figure 43.
The next question at issue is the rotation of fringes with fore-and-aft mo-
tion, or rotation around an axis 0 normal to the diagram, as shown in figure 26,
REVERSED AND NON-REVERSED SPECTRA. 63
Chapter II. In other words, when e, the virtual distance apart, is zero, since
ncce/\, the fringes are infinitely large horizontally. The collimator, how-
ever, furnishes a pencil of rays which are parallel in a horizontal sectional
plane only. They are not collimated or parallel in the vertical plane (parallel
to the length of the slit) . Hence when the fringes are reduced to a single one
of infinite size horizontally, this is not the case vertically; i.e., from top to
bottom of the spectrum the path-difference still regularly varies. The adjust-
ment around an axis through 0, G'OG", normal to the rulings, is still out-
standing. It does not seem worth while to enter the subject further because
much of the rotational phenomenon will depend upon whether the axes used
are, in fact, truly vertical or parallel to the slit. In my apparatus this was
not quite guaranteed, and the quantitative results obtained may therefore be
due to mixed causes. Also, a rotation around an axis normal to 0 always
requires an adjustment for superposition of the longitudinal axes of the spectra,
and this introduces path-difference.
Finally, the case of figure 21, Chapter II, or the rotation around an axis
parallel to IT in the present figure 43, is to be considered. This has already
been given in terms of colored fringes (white light), but it occurs here for
homogeneous light, in which case the above explanation is not applicable.
Seen in the principal focal plane with telescope and wide slit, the non-reversed
spectra would require careful adjustment of longitudinal and transverse axes ;
otherwise they vanish. Nothing will rotate them.
Figure 43 shows that if G'G" is rotated about IT, the effect is merely to
destroy the fringes, since the coincidence of the longitudinal axes of the spectra
is here destroyed. No effect is produced so far as path-difference is concerned.
To restore the fringes, therefore, either of the opaque mirrors M or N of the
apparatus must be rotated on a horizontal axis until the two spectra are again
longitudinally superposed. It is this motion that modifies the path-difference
of rays in a vertical plane. In other words, when the fringes corresponding
to any virtual distance apart, e = b <p, of the halves of the grating G'G",
have been installed, the rays as a whole may still be rotated at pleasure
around a horizontal axis. In this way a change in the number of fringes inter-
sected by a vertical line through the spectrum is produced. The number of
intersections will clearly depend on the obliquity of the rays (axes of vertical
pencils) , and will be a minimum when the center of the field of view corre-
sponds to an axis of rays normal to the grating G'G". In other words, the
vertical maximum in figure 22 occurs under conditions of complete symmetry
of rays in the vertical plane. If, therefore, e or the virtual distance apart
of the half gratings, G"0 and OG', is also zero, the field will show the same
illumination throughout.
In conclusion, therefore, to completely represent the behavior of fringes, it
will be sufficient and necessary to consider that either grating, G'G" for
instance, is capable of rotation, not only around a vertical axis through 0,
but also through a horizontal axis through 0 parallel to the grating. The
last case has been directly tested above, Chapter II, § 16. But a rotation
64 THE INTERFEROMETRY OF
around these two axes is equivalent to a rotation around a single oblique
axis, and the fringes will therefore in general be arranged obliquely and
parallel to the oblique axis.
Thus if <pv and <ph are the angles of rotation of the grating (always small)
around a vertical and a horizontal axis, respectively, and if %' is the angle of
the interference fringes with the horizontal edge or axis of the spectrum
j. / <?V
tan x = -
<fh
so that if <pv = o, x' = o; if <ph = o, x' = go°. This recalls the result obtained
above for the interferences of two coarse grids. In other words, for a rotation
of grating around a vertical axis (parallel to slit) the fringes of maximum size
will be horizontal (Chapter II, fig. 21), because the adjustment around the
horizontal axis remains outstanding and the residual fringes (large or small)
are therefore parallel to it. For a rotation of grating around a horizontal
axis, the fringes of maximum size will be vertical (Chapter II, fig. 22), for the
vertical adjustment is left incomplete. When both adjustments are made, a
single fringe fills the whole infinite field, and this result follows automatically
if but a single grating is used to produce the fringes, as in the original method
(Chapter I).
To deduce equations it is convenient to regard both gratings as trans-
mitting and to suppose one of them to be cut into independent but par-
allel halves, either by a plane through its middle point and parallel to the
rulings (vertical axis of rotation), or by a plane through the same point and
normal to the rulings (horizontal axis of rotation). The parallel halves of
the grating are then displaced along the normal, e, to both.
27. Case of reflecting grating. Homogeneous light. — The results exhibited
in figure 43 for transmitting gratings are shown in figures 47 and 48 for the
combination of one transmitting grating G and one reflecting grating G' (the
adjustment used in Chapter II), for which the direct path-lengths of rays
were computed (cf. figs. 23 and 24, Chapter II). The path-differences
obtained were inadmissible. It is now necessary to completely modify the
demonstration.
In figure 47 the rays are shown for the case of complete symmetry of all
parts, gratings at G and G' vertical and parallel, opaque mirrors at MI and
Ni, telescope or lens at T. The incident ray I at normal incidence is diffracted
and reflected into Y, X, T, and Y', Xf, T, respectively; the incident ray I' at
an angle of incidence di into YI, Xi, etc., and Y'i, X'i, etc., respectively; both
at a mean angle of diffraction dd (nearly) to the right, corresponding to di.
The angles of diffraction (di=*o) are 0i, and 02; the double angles of reflec-
tion, therefore, 8 = 6*— 6it on both sides; the double angles of the grating G'
with the mirrors MI and A/\, symmetrically, <r= 0i+02-
The normal from the point of incidence at G and at G' , N, and n makes
angles 5/2 with Y and A', respectively, on both sides. The method of treat-
REVERSED AND NON-REVERSED SPECTRA.
65
ment will consist in reflecting G' in M\ and Ni, producing the planes G' \ and
G'i (virtual images), and then rotating M\ and G'\ 180° around IT (axis of
symmetry) into coincidence with N\ and G'z (interference). Then the rays
prolonged into a and /3 coincide with the rays prolonged into a and 8' and
the (virtual) diffracted rays T\ and Tz become T'i and T'z. The ray on the
left, prolonged into s, is diffracted into Ts. Then the interferences will all be
given by discussing the left half of this diagram, which is amplified in figure 48.
47
Since the distance GGr, figure 47, is very large, the rays are nearly parallel.
Thus the arc d'y, with its center at G, is practically a plane wave-front,
perpendicular to the rays in 5', /3', 7, and the diffracted rays T', T'z, and T'3
are also practically parallel. Hence in the case of symmetry and coincidence
of Mi and Ni the points 5', /3', 7, 5', a', and s are in the same phase (diffrac-
tion). In other words, there is no path-difference between Y+X and Y'-}-X',
whether the angle of incidence is zero or not (Yi+Xi and Y\-\-X'i). The
whole field in the telescope must therefore show the same illumination (homo-
geneous light, wide slit) between the maximum brightness and complete
darkness. Interference fringes can occur only when the opaque mirror, MI,
is displaced parallel to itself out of the symmetrical position. If MI and Ni are
symmetrical, as in figure 47, the displacement of G', fore and aft, parallel to
itself, is without influence.
This reduces the whole discussion to the normal displacements of the sys-
tem G', Mi, Ari, given in figure 48. Let the mirror MI be displaced over a
normal distance em to the position M3, Ni remaining in place. Then the
image of G' will be at G'3t at a perpendicular distance, e, from its original posi-
66 THE INTERFEROMETRY OF
tion G'I. The path-difference so introduced, since a and b (ab normal to the
ray Yz impinging on M3 at c and reflected to b) are in the same phase, is
zem cos 6/2
and the displacement per fringe1 will be
X
wt/m — <• /
2 COS 6/2
which is nearly equal to X/2, as in most interferometers, remembering that
em and dem refer to the displacement of the mirror Mi. Two interfering rays
will be coincident at b.
In the next place e and 8e may be reduced from the corresponding displace-
ments em and dem of the mirror Mi. In figure 48 the figure fdbe is approxi-
mately a parallelogram with the acute angles 6/2. Hence, since 02 = ( 5+00/2
2Cm COS (7/2 =6
as is also otherwise evident. Thus per fringe, if the length £g = c
\ = de cos 6 2+ 8c sin 02
since dc = 2 5em sin (7/2 .
If G' is displaced parallel to itself, 8e will not be modified, since each virtual
image G\ and G's moves in parallel, in the same direction, by the same amount.
If then the grating G' is rotated around an axis at G', perpendicular to the
diagram, figure 47, over a small angle, <p, the result (apart from the super-
posed rotational effect) is equivalent to a displacement of the mirrors MI and
Ni in opposite directions, producing a virtual distance apart e and the cor-
responding interference fringes. In other words, the rotational effects may be
explained here in the same way as in the preceding paragraph.
The angle zdd, within which the interference rays lie, per fringe, is sub-
tended by 8em, and this may be put roughly (N= 162 cm., normal distance)
2dd=(28em sin 5/2)/JV= (X tan 8/2)/N
This angle is very small, scarcely icr8X3.2 radians, or less than o.oi second
of arc. Hence all pencils consist of practically parallel rays.
An important result is the angular size of the fringes; i.e., if em and X are
given
d62_ X _ Dz sin 02
dn em sin 6/2 em sin 6/2
D2 being the grating space.
Thus they become infinitely large when em passes through zero. The angu-
lar size is independent of the distance between the gratings. It ought, there-
fore, to be easy to obtain large interference fringes, which is not the case.
The reason probably lies in this : that the two opaque mirrors are not quite
1 The differential symbol 5 is unfortunately also used to designate the double angle of
reflection 5. But it is improbable that this will lead to confusion.
REVERSED AND NON-REVERSED SPECTRA.
67
symmetrical, so that in figure 47, on rotation of MI 18 ° on GG' , the trace of
Mi crosses Ni at an angle. If dd/dn = $.'jX'io-4, the distance apart of the
sodium lines, and DZ = 1 73 X lo"6 cm.,
e=i.S cm., about
i.e., path-lengths on the two sides should differ by about 2 centimeters, if the
mirrors were quite symmetrical.
28. Non=symmetrical positions. Fore=and=aft motion. — It remains to
account for the marked effect produced on displacing the grating G' in a direc-
tion nearly normal to it-
self. If the displacement
is symmetrical, or even if
the grating and mirrors -.^
are reciprocally non-sym-
metrical (i.e., the former fl-
at an angle <p to the trans-
verse line of symmetry
gg' , the latter inclosing
an angle a, fig. 49), no
effect results from the
displacement of G', if the
mirrors MI and NI are
so placed that the vir-
tual images Gm and Gn
are parallel and the dif-
fracted rays, therefore,
also parallel. In such a
case Gm and Gn are dis-
placed by the same amount, normally, their distance apart is constant, and
the intercepts of rays equal.
If, however, this compensation does not occur; if the grating G', the mirrors
NI and Mi make angles <p,ff/2, r/2, respectively (za= r — <r), with the trans-
verse line of symmetry gg' , the fore-and-aft motion of G' is more effective as
the angle a — <p is greater. The diffracted rays are then no longer parallel,
but make angles of incidence at the second grating, 0'2 for the N\ side and
02 for the Mi side, and of diffraction i' and i, respectively, as shown in figure
49, at rn and Tm. The following relations between the angles are apparent:
49
If at the first grating 0i= 0'i and a is the angle between the mirrors,
2<X= T — ff= 62— 0'2
The images are at an angle /3, where
68 REVERSED AND NON-REVERSED SPECTRA.
If G'G' is displaced to G\G\ over a normal distance e, or e/cos <p along the
line of symmetry GT, the virtual images Gm and Gn will be displaced to G'm
and G'n over the same normal distance e. This is obvious, since the quadri-
laterals ab and a'b' are rhombuses by the law of reflection, and hence the
perpendicular distances e between the (equal) sides all identical.
If D 2 is the grating space of G',
(1) sin 02+sint = X/Z}2 sin 0'2-{-sini' = X/.D2
or if i and i' are very nearly equal and both small, as in the experiment,
(2) cos 6zdO= —cos idi
Again, in case of the displacement e of Gf, the paths are shortened at Gm
by e/ cos 02, at Gn by e/ cos 0'2, resulting in the path-difference AP, or
(3) AP = e(sec 02-sec 0'2)
Since 02 and 0'2 are nearly the same, this may be adequately simplified by
differentiation. Putting
(4) dd= 02— 0'2 = 2(a — <p) AP = 2(a — <p)e tan 02 sec 02
Hence per fringe, apart from sign,
(5) = Xcos°- 02
2(0: — ^>) sin 02
Thus, if
X = 6Xio~5 a — (f>= i° = o.oi75 02 = 2O°
then
6Xio-5Xo.88
d£ = — — - =0.004, 4 cm.
2X0.0175X0.342
per fringe for each degree of arc of non-symmetry, a—<p.
The effectiveness of the fore-and-aft motion, according to this equation,
is evidence of a residual angle, a — <p, of non-symmetry. This is not improb-
able, as my apparatus was an improvised construction, lacking mechanical
refinement. Further, the wedge effect due to a, which makes em variable,
would be superposed on the interferences, and hence these could not be in-
creased in size above a certain maximum. This is also quite in accord with
observation.
If a = <p, 18 = 2(0: — <p) =o and 02= 0'2; i.e., the virtual images Gm and Gn and
the diffracted rays are parallel and 8e = co . In other words, the fore-and-aft
motion has no effect. If a = o, /3 = 2<p; or if <p = o, fi = 2a. In either case de
is finite, and fore-and-aft motion is effective. If the mirrors and grating were
rotated in counter-direction so that tp is negative, de will depend on a-\-<p, and
the fore-and-aft effect will be correspondingly marked. Moreover, the inter-
ference will not in general appear in the principal focus, but usually suffi-
ciently near it for adjustment.
If 5eg is the actual displacement of the grating G' in the line of symmetry,
8ea = 5e/ cos if>, so that the angle <p enters equation (5) again, but only to a
small extent.
CHAPTER IV.
THE DISTANCE BETWEEN TWO PARALLEL TRANSPARENT PLATES.
29. Introductory. — The problem of finding the distance separating two
parallel glass disks, as well as their degree of parallelism, is frequently one of
practical importance. Thus, in my work on the repulsion of two such disks,
it would enter fundamentally, and it has long been my intention to repeat
that work with two half-silvered glass disks, for comparison with the case of
metallic disks. It has since occurred to me that the method devised by my
son, Mr. Maxwell Barus, and myself* would probably be ideal for the purpose,
both for very small distances (within o.i mm.) as well as for distances ten or
more times larger. This method admits of use of the film grating, and there
are three types of interferences of successive orders of fineness, the first virtu-
ally involving the colors of thin plates (resolved spectroscopically), the other
two being dependent on diffraction. To measure the thickness of the air-space
it would be necessary to count the number of fringes between two definite
Fraunhofer lines only, supposing the constants of the grating to be given.
30. Apparatus. — The apparatus has been designed for transmitted light,
in preference, though the case of reflection is also available.
A
^r
cJ
Jl
^
J^
c/M.
5Q
5!
MM, figure 50, is the base of a Fraunhofer micrometer, firmly attached
below to a massive tripod (not shown) . 55 is its raised slide, and E the head
of the micrometer screw, reading to icr4 cm. The open case A is screwed to
the slide 55 and contains the glass plate H half-silvered on the right. H is
attached to a plate of brass, on the plane-dot-slot principle, and may there-
fore be rotated around the vertical and horizontal axis by aid of a rearward
spring mechanism (not shown) and the adjustment screws a, b, b' (the last
not visible). The grating G, with a ruled face on the left, is similarly carried
by the open rectangular case B, screwed down to the base M of the micrometer.
Thus B is stationary, while A moves. Three adjustment screws, c, d, d' (df
not shown), and a spring pulling to the right suffice to rotate G around the
* C. Barus and M. Barus, Carnegie Inst. Wash. Pub., No. 149, Part I, Chapters II and
III. 1911.
69
70 THE INTERFEROMETRY OF
vertical and horizontal axis. The thickness of the efficient air-film is thus e-
and H and G may be brought to touch or to recede from each other several
centimeters. L is the collimator (slit and lens), furnishing intense white sun,
light or arc light, and the beam, after traversing the system, is viewed by
the telescope T (direct beams, fig. 51), or D (diffracted beams).
The plate H is half-silvered, but the grating G is left clear. In this case,
however, only the fine fringes are seen strongly on transmission. The others
appear on reflection at G, preferably in the second order of spectra. Fine
fringes are not well reflected, but the medium and coarse fringes are very
strong and clear, and the first observations were made by means of them.
Thereafter the ruled face of the grating was half -silvered. This largely
destroys the reflected field, Df, except the fine fringes, but the transmitted
field D is now strong, particularly in the second order of spectra, for all the
three sets of fringes in question. Mr. Ives's direct-vision prism grating
shows the fine fringes well in the direct beam T. The lines are always rigor-
ously straight, so far as they can be observed; i.e., it is impossible to bring
H and G rigorously in contact, not only because of dust, but since the grating
(at least) is not optical plate. The fine fringes may always be found in the
principal plane of a telescope, but the medium and coarse fringes usually lie
in other focal planes differing from each other. By placing the ocular it is
thus possible to eliminate any of the interferences or to show a single set in
the field only.
To find the fringes, the direct white-slit images are made to coincide through-
out their extent, and the same may be done with a pair of spectrum lines in
the superposed spectra. The proper e is then to be sought. Owing to imper-
fect plane parallel plates, it may be necessary to correct this by the adjustment
screws on the mirror until sharp, strong fringes are seen in the corresponding
focal plane.
31. Equations.— The equations for the three useful interferences in ques-
tion are for r<6m and a similar group for r > 6m
(1) n\ = 2e/j.cosr
(2) n\ = 2en cos d'm
(3) n\ = 2£Ai(cos r — cos 0''m)
where X is the wave-length of the color used, n the order of the interference,
e the thickness of the sheet to be measured, and (j, index of refraction, if i is
the angle of incidence of the white light on the grating, r the angle of refrac-
tion in the plate (ju) , and Q'n the angle of diffraction of the mth order of spec-
tra therein. If the sheet is an air-space, these equations become simplified,
since n = i and r is replaced by i, tfm by 0m, the angle of diffraction in air.
Thus, since positive values are in question,
(4) n\ = 2ecosi
(5) n\ = 2e cos dm
(6) n\ = 2e(cosi — cos 0m)
REVERSED AND NON-REVERSED SPECTRA.
71
In the present apparatus I have made i = r = o, a more convenient plan
of testing the method, though not necessary and, in fact, often inconvenient
in practice. The equations are, finally,
(7) n\ = ze
(8)
(9)
-cos 0m = 2<?i - -
n\ =
if D is the grating space, and the interference in question is due to the grating
spectrum of the mih order.
The meaning of the equations (7), (8), and (9) is given in figure 52. The
case of equation (7) may be seen in the direct white ray, figure 52 a, provided
the light of the focussed slit-image is resolved by direct-vision spectroscope.
For this purpose Mr. Ives's grating with attached direct grating prism may
conveniently be placed in front of the telescope T, figure 50, focussed on the
slit. After adjustment these fringes appear strong. Of course, H and G
a
I
d
V\
ge
52
53
must be parallel and all but touch. Under the same conditions the fringes
may be seen laterally in any order of spectrum, as in figure 52 b. Figure
52 c illustrates equation (8) and figure 52 b equation (9). Figure 53, finally,
illustrates the general case of incidence, i.
The first and second orders of spectra are alone intense enough to pro-
duce marked effects. In case oii — o, a double diffraction of the first order,
6' reinforces a single diffraction of the second order, 02, since
\/D = (sin 9' — sin 6} =sin 02/2,
(sin 0'-X/£>) = (2\/£>)/2 or sin 6' = 2\/D
Probably for this reason they are visible. The general case, equations (4),
(5), and (6), is illustrated in figure 53, the rays /, I', and /" being incident,
R reflected, and D diffracted. The retardations are ef and df, respectively.
If the diffractions differ by a whole number of wave-lengths the total diffrac-
tion is obtained. One would be tempted to resolve the case by aid of a wave-
front ab, in which case the equations would be different; but they do not
reproduce the phenomenon.
72 THE INTERFEROMETRY OF
32. Method. — Suppose, now, two Fraunhofer lines, X and X' of the spectrum,
are selected as the rays between which interference fringes are to be counted.
Then, in case of equation (7), if n' is the number of interference rings between
X and X',
(10) n\=(n-\-n'}\' = 2e
(n) « = tt'X'/(X-X')
(12) 2£ = n'X
In order to measure e, therefore, it is necessary to count the number of fringes
n' between X and X', and e varies directly as n'.
If the mean D and magnesium b lines be taken as limiting the range, io6X =
58.93 cm., io6X'=si.75 cm., Ci= 10^X4-25; then
11'= i io3£= 0.21 cm.
= 10 = 2.1
= 100 = 21 etc.
As it will not be convenient to count more than 100 lines ordinarily, the
method is thus limited to air-spaces below 0.2 mm. and becomes more avail-
able as the film is thinner. Of course, in case of plates which contain specks
of dust or lint, or are not optically flat on their surfaces, it is extremely diffi-
cult to get e down below 0.002 cm., so that ten fringes between D and b would
require very careful preparation.
If equation (8) is taken, X is to be increased to
L = X/ cos Om = X/\/i - (m\/DY
where m is the order of the grating spectrum, whose rays interfere. Thus
equations (n) and (12) now become, since nL = (n-\-n'}L' = 2e
(13) « = n'L/(L-L')
(14) 2e = n'LL'/(L-L'}=C,n'
If first order of diffractions are in question, m= i, io6L = 59.n,
€'2=10^X4.20. Thus for
n= i io3e= 0.21 cm.
10 02. i
100 021.
scarcely differing from the preceding case, so that one would not know in
which series one is working.
If the diffractions occur in the second order, m = 2 ,
io6L2 = 62.56 io6L'2 = 54.i5 c\= 10^X4-03
thus again differing but slightly from the above.
If we inquire into the condition of coincidence and opposition of these
fringes, the following results appear: Let the spectrum distance between
the G and b line be taken as unity, and let there be n\ and nz first-order fringes
in this distance. Then in is the difference of distance per fringe. Let
Hi «2
REVERSED AND NON-REVERSED SPECTRA. 73
% be the number of long fringes, to restore the original coincident phase; i.e.,
let x longer fringes gain one long fringe on the x shorter fringes. Then
that is, x fringes constitute a new period. From the above data
It follows that the length of coincident strips is subject to
e = CI(HI — i ) = Cznz = Cnz/x or C = 7
C\ — Cz
where C is the new constant. This would place the fringes beyond the coarse
group below, but naturally C is enormously dependent on small errors in C\
and Cz.
Finally, if equation (9) be taken, the X is to be increased to
M = X/(i -cos 6>m) =X/(i - Vi - (wX/D)8
in order that equations similar to the above may apply. Thus
n'M MM'
In the diffractions of the first order of spectra m = i and
io3M' = 4.747 C'3 = 0.0330
These are the coarse order of fringes, so that
n= i 2^ = 0.033
10 0.33
100 3.3 , etc.
Fringes are thus still strongly available, even if the distance apart of the
plates is over 2 cm.
If the diffractions are of a second order of spectra, m = 2 ,
ioW=i.oi6 io3M'=i.i6s C"'3=io-3X7.85
These fringes are therefore of intermediate order, since
n= i 2£ = 0.0078 cm.
10 0.0785
100 0.785
They would be enhanced, since they cooperate with the double diffractions
of the first order.
33. Observations and corrections. Preliminary work. — The following
work was done merely with a view to testing the equations and with no
attempt at accuracy. The grating was left unsilvered, so that the ruled sur-
74
THE INTERFEROMETRY OF
faces confronted the half -silvered surface on ordinary plate glass. Conse-
quently, the fine fringes were observed by transmitted light behind, and the
medium and coarse fringes by reflected light in front. The micrometer was
a good instrument for general purposes, but hardly equal to the present work,
where the slightest rocking of the slide introduces annoyances.
To count the number of fringes between D and b, since the fringes were not
generally seen in the principal focal plane of the telescope, it was considered
sufficient to rotate the cross-hair into an oblique position, until its ends ter-
minated in the D and b lines, respectively, and then to count the number of
fringes on running the eye down the wire from end to end. When there are
many fringes, 25 to 50, the eye is apt to tire before reaching its destination,
so that several counts must be made and the mean taken.
50
0
0 -01
The results are given as a whole in figure 54, where the distance between
plates, measured in centimeters on the micrometer, beginning at an approxi-
mate zero, is laid off horizontally and the number of fringes vertically, in case
of each of the three series. The computed line e = Cn'/2 is drawn in full and
the observations laid off with regard to it. The zeros do not quite correspond,
as very small distances here are significant. With the fine fringes I did not
spend much time, as they are virtually colors of thin plates seen by diffrac-
tion. The chief difficulty with these small distances is that the plates touch
and a complete readjustment is necessary. After touching, the micrometer
acts like a forcing screw and its reading is too low. This is the meaning of
the data in the curves a and a', the latter with its horizontal scale magnified
ten times. The object of this series is chiefly to locate the position of the
line in relation to the other lines.
REVERSED AND NON-REVERSED SPECTRA. 75
The observations for medium and fine fringes were made together, so that
a single micrometer reading suffices. Beginning with very small distances
apart, called zero, this was rapidly increased to nearly i cm. The fine fringes
soon vanished, later the medium fringes vanished, finally (when e is several
centimeters) the coarse fringes also vanish. The three together, therefore,
cover with accuracy a relatively enormous range of displacement for measure-
ments of this kind.
The curves b and c show that the observations are not completely repro-
duced by the line. Mean lines drawn through the observations indicate that
the zeros do not correspond sufficiently for the two lines b and c to locate the
common zero. This is inevitable, since the micrometer begins to count at a
small distance as specified, which is otherwise arbitrary. In fact, it should
be noticed, as an accessory property of this interferometry, that the two lines
for finding the zero determine the absolute reading of the micrometer, mutu-
ally, and these readings are here 0.22 mm. too large. But even if the zeros
were horizontally to coincide, the observations would not adequately conform
to the computed lines. All that can be affirmed is that the angle between
the observed and computed loci is about the same.
The main reason for the divergence is referable to the fact that the air-space
is not quite plane parallel, but slightly wedge-shaped, so that the effect of the
angle of the wedge is superposed on the interferences. Any slight unsteadi-
ness of the micrometer slide, for instance, would already introduce the wedge
discrepancy, without necessarily interfering with the sharpness of visibility ,
while any attempt to readjust would destroy the continuity of measurement.
There will also be many secondary reasons for divergence, as, for instance, the
three separate focal planes in which the fringes lie and the fact that the glass
plates which limit the air-space are themselves wedge-shaped; other, but
fainter, fringes are marching through
the spectrum, such cases as coincidence
and opposition, for instance, as were
pointed out above, etc. But the ade-
quate reason for the discrepancies in
this paper is the incidental change of
the angle of incidence, i.
If the film is wedge-shaped, very
little disturbance results; but the cor-
rection to the second order of small
quantities is unfortunately somewhat
cumbersome. Let the edge of the wedge of air be vertical and subtend a small
angle, <p, figure 55, between the two faces A and B. Let I and /' be the two
corresponding rays incident at the angle i at the first face and at the angle
i-\-<p at the second face, n and n' being the normals. Let e and e' be the con-
secutive thicknesses of the air-plate, taken normal to B for convenience. Then
the I rays R' will issue at the A face, after reflection, at an angle i-\-z<p, and
will interfere with the I' rays R, if the objective of the telescope is sufficiently
76 THE INTERFEROMETRY OF
large to converge both to the same point of the image, spectroscopically
resolved. If the wave-front ab is drawn and e' prolonged, it follows at once
that
sin i+< sin
cos
Hence
/ , sin (i-\-(p) sin <p\
n\=2e[cos(i+<p)-}— A—
V cos i I
If i = o and $ very small, this becomes
<A
/
If in the first equation i is replaced by the angle of diffraction 6, the equation
for the diffracted fringes, as far as ^2, may be reduced to
= 2e\cos 6+<p sin 0— ( - - — 7 cos 0}
L 2 Vcos 6 / J
so that <p sin 6 is the chief correction.
Finally, the equation for the coarse fringes becomes
f <£>V 4 M
n\ = 2e\ i —cos 6 — <p sin 0+— I 3+— —n— 7 cos 6}
I 2 \ cos v / J
with a similar equation for the medium fringes.
If we neglect the second order of small quantities (<p2), the last equation
for the medium fringes may be put in another form, since
2L
2\/D = sin 6 and \/L = i — cos 6 sin 6 = — =L(I — cos 6)
u
whence
nL (n+n')Lr
i-2<pL/D i-2<pL'/D
D being the grating space and X the wave-length. Hence if n be eliminated,
LL' , LL'
2 C fl f ft / / T-\ \/TT/ TT/\ '*"
L-L'-(2<p/D}(LL'-LL1) L-L'
In other words, if <p is small so that <p- may be neglected, the relation of e and
n' is independent of <p ; or a slightly wedge-shaped air-film will show the same
result as a plane-parallel film. Experiments made by turning the adjust-
ment screws seem to bear this out, provided the mean thickness remains
unchanged.
To give the whole subject further study, I have since half-silvered the
grating as specified, so that all the fringes may be seen by transmitted light,
preferably in the second order, since there is an abundance of light available.
The apparatus in such a case takes a good shape and is convenient for manip-
ulation. But these details will have to be given at some other time, and it
is the chief purpose of this paper to exhibit the phenomenon as a whole.
REVERSED AND NON-REVERSED SPECTRA.
77
In conclusion, I may recall that if we regard 100 fringes between the D
and b lines as still available for counting under proper facilities, the succes-
sive ranges of measurements will be roughly as follows :
e=o.O2i cm.
£•=0.392 cm.
e= 1.65 cm.
Fine fringes
n'= 100
Medium fringes
n'= 54.
n'= 100
Coarse fringes
. °^
»— !.•*
n'= 2^.8
n'=ioo
The transition from fine fringes to medium is a little abrupt. Otherwise, in
cases where manual interference is not permissible, all thickness of air-films,
from a fraction of a wave-length of light to nearly 2 cm., may be adequately
measured in this way to advantage. It is probable, moreover, that it would
be advisable to observe the fine fringes by transmitted light, but to leave the
grating (which may be a film grating) clear, and to observe the medium and
coarse fringes by reflected light. A concave mirror and lens (reflecting tele-
scope) should be used for this purpose, as this will put the observer behind
the plates in all cases.
CHAPTER V.
ynf
INTERFEROMETERS FOR PARALLEL AND FOR CROSSED RAYS.
34. Introduction. Methods. — To exchange the component beams of the
interferometer, to mutually replace the two pencils which interfere, is not an
unusual desideratum, for instance, in the famous experiment of Michelson
and Morley. To replace two pencils of component rays, traveling more or
less parallel to each other, by pencils moving more or less normal to each other,
or to be able to operate upon pencils of corresponding rays (from the same
source, crossing each other at any angle) at their point of intersection, may
be of interest in a variety of operations to which the interferometer lends
itself, or may even suggest novel experiments. The facility with which this
may be done, or at least partially done, with the above types of spectrum
interferometers, particularly when homogeneous light is used, has tempted
me to investigate a number of cases.
Let us begin with the above diagram-
matic method, using two transmitting
gratings, G and G', figure 56, with the same
(or in general with different) grating con-
stants. Let L be the incident beam of
collimated homogeneous light, m, n, m', n' ,
four opaque mirrors on vertical and hori-
zontal axes parallel to their faces. The
ruled faces of the gratings are to be toward
each other. Then the beams Gm and Gn
may be reflected either across each other,
as shown at win' and nm', thence along n'G'
and m'G', and, after a second diffraction at
G', unite to enter the telescope at T; or
they may be reflected along m, m', and
n, nf, parallel to each other, and thereafter take the same course. In the
first case homogeneous light is apparently not necessary. It will be seen
that the path of the rays is the same, except for the branches mn' and nm' ',
and mm' and nn', respectively normal and parallel to each other; moreover,
that the rays are exchanged, a and 6 left and right combining at G' in one case,
b and a left and right in the other. The rays cross at c in free space and are
available there for experiments. Direct light is to be screened off. The ques-
tion is whether the mirrors m and n, m' and n', can be adjusted mechanically
to move symmetrically toward each other on a vertical axis with sufficient
precision to guarantee replacement. This is a matter of trial, though a
successful issue is, of course, problematical. It would be advantageous to
arrange the experiment so that only one pair of mirrors — e.g., m and n — need
78
REVERSED AND NON-REVERSED SPECTRA. 79
be moved, whereas the others, m' and n', are ends of the same rigid plate.
Gratings of different constants may advantageously contribute to this end.
Beyond this, the paths mn' and nm' and mm' and nn' may be increased to
any length, either directly or by multiple reflections from a special system.
Many other modifications are suggested. If white light is used, the phe-
nomenon is confined to a narrow strip of spectrum and the fringes must be
horizontal.
As I did not have two ruled transmitting gratings and as film gratings
seemed unpromising for work of this kind, the method of figure 57 represents
a simple disposition of reflecting gratings, of which several were available.
The ruled faces of the gratings G' and G face away from each other.
58
The former receives the collimated pencil of homogeneous light, L, and
after diffraction the partial beams pass to the pair of opaque mirrors m and
n (symmetrically placed), and thence by reflection to a similar pair of mirrors,
M and N. From here the pencils reach the second grating, G', where each
is again diffracted into the common ray G'T, entering the telescope T. The
grating G' may be concave with the lens at T beyond the principal focus.
If the mirrors M and N are symmetrically rotated, the parallel component
pencils Nn and Mm may be replaced by the pencils Mn and Nm, crossed at
any angle. Homogeneous light is preferable. Simultaneously the rays are
exchanged. The pencils, Mm, etc., may be of any length, and in general the
remarks in the preceding paragraph apply.
A more flexible design also suggests itself, with four fixed mirrors, m, n,
m', n', four movable mirrors, M, N, M', N', rotating symmetrically around
vertical axes parallel to the faces of the gratings G and G', these being parallel
to each other, as in figure 57. On rotating M, N, M', N', the rays may be
exchanged. Here M . . . . N' should be a near system, m . . . . n' a fixed
and far system of mirrors. Other methods will presently be described.
35. Experiments. Reflecting gratings. Parallel rays. — The experiments
were begun with the apparatus as in figure 57, G being a Michelson grating
and G' a Rowland grating, each with somewhat less than 15,000 lines to the
inch. The distance of G from the mirrors m and n was about 22 cm., of G
from G' about 60 cm., and of G' to the focal point just ahead of the lens (or
the line of mirrors M and N) about 90 cm. The latter were about 50 cm.
apart. In the absence of sunlight, the arc lamp was used, and the fringes for
reversed spectra were found without great difficulty. It was also easy to
80 THE INTERFEROMETRY OF
erect them by rotating G' on an axis normal to its face. A difficulty, however,
existed in retaining the fringes with a flickering arc. It will be seen that in
this case the line LG moves over a small angle in all directions with the bright
spot on the positive carbon, so that the angle of incidence is varied, and with
it the angle of diffraction 6 at G. All this is magnified by reflection from the
miiTors. Moreover, unless the collimator lens is very near G, the illuminated
part or bright line on G is displaced right and left. Path-difference between
GnNG' and GmMG' is thus modified. If the faces of the mirrors are not all
quite in a vertical plane or parallel to the same plane, the up-and-down play
of the arc will mar the longitudinal coincidence of the two superposed spectra,
and hence the interferences will vanish. Thus they appear and disappear
periodically, depending on the accidental position of the bright spot of the
arc; and if this annoyance is to be avoided, sunlight or a steady light must
be used. The phenomenon and the spectra were not nearly so bright as
when observed with the transmitting grating, a result probably due both
to the additional reflections (particularly those at the grating) and to the
high dispersion.
In other respects the behavior was the same as that described in Chapters
I and II, though the strip of fringes for reversed spectra seemed to be some-
what broader, probably owing to the increased dispersion and hence the greater
breadth of adequately homogeneous spectrum light. The linear phenome-
non, moreover, consisted of two or more black lines alternating with bright,
whereas a single black line was the characteristic feature above. When dif-
ferent strips of the grating G are used (the illumination should not be more
then 0.5 cm. wide), considerable fore-and-aft displacement at the mirror M
is necessary.
The adjustment for crossed rays Mn and Nm, figure 57, is subject to new
conditions. In case of white light and a narrow slit, the dispersion produced
by G is at least partially annulled by G' instead of being incremented ; for the
change of the angle of incidence here compensates the changes of the angle
of diffraction. Thus if sini'v— sin dv = \v/D for violet and sin^Y — sin 6r =
\T/D for red, and if sin iv=\v/D and sin ir = \r/D, then sin 0 =sin 9r = o.
A sharp, white image of the slit may thus be seen for the reflection from each
mirror M and N, or the images may be colored if but a part of the spectrum
is reflected from M and N. The system of two gratings, G and G', tends to
become achromatic. It would seem to follow, therefore, that in general
homogeneous light and a wide slit would have to be used, but this introduces
additional annoyances, inasmuch as the transverse axes of the spectra (sodium
lines), which are to coincide, are not visible, but must be replaced inade-
quately by the edges of the slit. The experiment is thus (particularly in view
of the faint illumination seen in the telescope) difficult, and in a laboratory
not free from agitation, or in the absence of a good mercury lamp of intense
homogeneous light, it did not seem worth while to spend much time on it.
Moreover, a similar investigation will presently be made with a transmitting
grating.
REVERSED AND NON-REVERSED SPECTRA. 81
In other words, in case of the rays nM, the violet is incident at a larger
angle at G' than the red, and but one color (yellow) can be diffracted along
G'T, whereas in case of the rays mN violet is incident at G' at a smaller angle
than red, and G' may thus be so placed that all rays are diffracted along G'T,
supposing the two gratings to be nearly identical as to dispersion. Figure
58, presently to be described, suggests the inclination of the successive verti-
cal planes in figure 57.
One curious result deserves special mention. Each separate spectrum
(a or b, fig. 57, without superposition) shows very definite coarse stationary
interferences; i.e., the usual appearance of channeled spectra. The cause of
this long remained obscure to me, but will be explained in Chapter VI. The
gratings being of the reflecting type and the mirrors silvered on the front face,
there is no discernible cause for interferences. No film or set of parallel plates
enters into the experiments. If in figure 5 7 the grating G' is reflected at M
into G'i, and this image reflected in m into G'z, the phenomenon may be treated
as if the gratings were transmitting in a manner shown in figure 58. Here the
direction of the traces of the grating G and G', the mirrors m and M only
are given, together with the direction of the reflected images of G' in M
(G'i), and in m (£'2). Then the violet (v) and red (r) rays from G impinge
on G'z virtually with a greater angle for v and a smaller one for r, as already
suggested. An enhanced spectrum must be produced beyond G'z- This
second spectrum is channeled.
36. Experiments. Transmitting grating. Parallel rays. — The chief diffi-
culty in the preceding experiments was the absence of sufficiently intense
homogeneous light. This may be obviated by using the transmitting grating.
But as two samples were not available (as in fig. 56), the simplified method of
figure 59 was tested, where but a single grating G is used. Here the light L
from collimator and slit impinges on the grating G and is diffracted to the
opaque mirrors M and AT. From here it is reflected to the corresponding
opaque mirrors m and n, to be again reflected to the grating G, and finally
diffracted along the line GT. The interferences are observed by the telescope
at T. In order that the undeviated white beam may not enter the telescope
annoyingly, the diffraction LG takes place in the lower half of the grating and
the mirrors are slightly inclined upward, so that the second diffraction GT
may occur in the upper half of the grating. To obviate glare in the field, the
beam LG is carried to the grating in an opaque tube and all undeviated light
is suitably screened off. The distances mn to G and G to MN were about a
meter each.
The interferences were easily found. They are usually at an angle to the
vertical, but may be erected by rotating the grating on an axis normal to its
face. They were linear and exactly like the cases of Chapter I, probably in
consequence of the low dispersion of the grating used. Considerable mag-
nification at the telescope is thus admissible.
The horizontal fringes traveling up or down are available for interferometry,
82
THE INTERFEROMETRY OF
and the independent and separated component beams Mm and Nn are con-
veniently accessible.
The experiments with homogeneous light (sodium arc) gave perfectly
regular striations covering the whole of the wide slit image, uniformly. With
glass compensators 0.6 to o.i cm. or more thick on both sides, the striations
became somewhat smaller, as was to be anticipated. Fringes could be erected
and enlarged by rotating the grating on an axis normal to its face and by
other corresponding rotations. The fringes, as a whole, were large and
splendid and suitable for general purposes in interferometry.
37. Experiments. Transmitting grating. Crossed rays. — The second posi-
tion of this apparatus was now tested, the rays passing along the diagonal
of the rectangle (fig. 59) and crossing at G in the grating. The interfering
pencils were thus GNGmG and GMGnG. The slit
should be quite wide. Seen in the telescope at T,
therefore, the dispersion is reduced in virtue of
double diffraction, the tendency being toward white
slit images, as already explained. A variety of very
interesting results were obtained after the interfer-
ences had been found. The outgoing and returning
paths are coincident, and both component rays pass
through the grating two times, the ruled face being
towards the telescope.
The adjustment is at first somewhat difficult.
Having made a rough setting of the mirrors as to
distance, etc., by the aid of sunlight or arc light, so
that the spectra may be seen, two wide slit images
will appear in the telescope T, but they will usually
be differently colored. The mirrors m and n are then to be rotated around
vertical axes (fine-screw motion) until both slit images are identically colored
and coincide. After this, homogeneous light (sodium arc) must be used and
the rotation of mirrors on the vertical and horizontal axes repeated until both
fields are identically yellow on coincidence. The sharply focussed edges of
the wide slit are now the vertical and horizontal guide-lines for adjustment.
All corresponding lines must coincide if the phenomenon is to be obtainable.
Thereafter the micrometer at M, actuating the mirror fore-and-aft parallel
to itself, is manipulated till the fringes appear.
Two types of interference may be observed. The first are variations of
nearly equidistant fringe patterns, obtained with homogeneous light only
and covering the whole wide slit image on good adjustment. They would
appear equally well in the absence of the slit. The second type is obtained
in the presence of white light, or of the mixture of white light with the homo-
geneous light. It is a linear phenomenon, identical in appearance with the one
described in Chapter I, though occurring here in the case of a wide slit. Both
are very vivid, and the latter particularly, when at its best, in violent tremor.
REVERSED AND NON-REVERSED SPECTRA. 83
It is convenient to describe the homogeneous fringes first. White light
must be absent, the wide field full yellow, and the longitudinal and side edges
of the two slit images sharply superposed. When the fringes appear they will
usually be oblique; but they may be made vertical by rotating the grating
on an axis normal to its face. If the grating is in the symmetrical position of
figure 59, the size of fringes is an intermediate minimum. To enlarge them,
curiously enough, the grating must be slightly rotated, either way, on a ver-
tical axis. The fringes then pass through a maximum of size at a definite
angle on either side of the minimum. In such a case they also appear rapidly
to become irregular and their perturbation is naturally enhanced. They con-
tain a double periodicity, which will presently be carefully examined.
Fore-and-aft motion of the grating has no effect. In displacing the mirror
at M on the micrometer, the fringes remain visible for an excursion of at least
0.7 cm. In fact, in case of a strong telescope and wide slit they were not lost
for a micrometer displacement of over i cm., i.e., much over 30,000 wave-
lengths of path-difference. As a rule, the fringes are strong only in part of
the yellow field, and in such a case the center of intensity moves with the
displacement of M across the slit image, to disappear at the edges, as in the
usual cases of displacement interferometry. Slight non-coincidence of the
horizontal edges of the slit images slightly rotates the fringes, but they soon
vanish completely. Slight rotation of the grating around the vertical axis
distributes the fringes more evenly over the field, the proper setting being
determined by trial. Displacement by aid of a compensator of glass gave the
usual results.
Later I returned to the experiments with sodium light and with the grating
rotated around a vertical axis to the right or left and out of the symmetrical
position of figure 59. In each case the fringes passed through maximum size
at an angle of asymmetry of about 5° or 10° from a normal position. Beyond
or below this they diminish in size. Naturally, to bring the fringes to the
center of the field, the micrometer screw at M or N had to be adjusted for
path-difference, as in displacement interferometry generally.
The details of the interference patterns obtained were in astonishing variety.
Suppose that by rotating the grating around an axis normal to its face the
fringes are made nearly but not quite vertical at the beginning. Then on rota-
ting the grating around a vertical axis into the position for maximum size
just specified, the standard type of large fringes seen are of the appearance
shown in figure 6oa. In other words, they look and behave like independent,
thick, twisted cords, hung side by side. The evolution of these independent
parallel striations of fringes may be detected on rotating the mirror M or N
around a vertical axis, thus moving one slit image in definite amounts, micro-
metrically, over the other, horizontal edges remaining superposed. As the
one slit image passes in this way across the other, the original type, figure 606,
apparently continuous, breaks up and enlarges into the type c by the rotation
of its parts. Thus the successive lengths of the continuous fringe b behave
like a series of magnetic needles, each rotating on its own pivot. These may
84
THE INTERFEROMETRY OF
again correspond and appear as a single striated field; but more frequently
the form figure 6oa is in evidence, though sometimes quite irregular. In fact,
there are many variations of this design. Families of curves, intersecting each
other nearly orthogonally, may even appear.
If the fringes are originally quite vertical, there seems to be no rotation,
but two sets of vertical fringes apparently pass through each other as the
mirror M is rotated micrometrically on a vertical axis. These fringes at inter-
vals again unite into an apparently simple striation. One slit image may be
broader than the other. Fringes of different sizes then appear, so long as the
smaller is within the larger, and are most intense when the vertical edges meet.
In general, therefore, the interference patterns of originally nearly vertical
fringes consist of a succession of strands, nearly in parallel, which behave alike
but independently.
60
61
a.
If the grating is rotated on an axis normal to its face until the fringes are
nearly horizontal, a correlative series of interesting phenomena may be
observed. When the grating is normal to the incident pencil, the fringes are
usually arranged in parallel strands. They are equidistant in each strand; but
these strands are separated by a narrow band of even color, so that the phe-
nomenon looks as if thick, twisted, yellow cords were hanging apart, side by
side. Usually the central or the two central cords are more intense, and there
may be four to six in all, filling the whole of the wide-slit image. On rotating
the mirror, M or N, micrometrically, on a vertical axis, the fringes of the
strand may be made to correspond, so as to fill the field with uniform stria-
tions and without apparent vertical separation. This is particularly the case
when the fringes are very fine.
On rotating the grating to the right or to the left about 20°, on the vertical
axis from the symmetrical position of figure 58, the fringes reach a maximum
of size, after which (on further rotation to about 30°) they diminish indefi-
nitely. These maximum cases are shown in figure 61, a and b, and their ap-
pearance is now that of a string of elongated beads, hung vertically and equi-
distant. On rotating N about a vertical axis, slightly, the nodules become
quite horizontal. They are continually in motion, up and down, and quiver
about the horizontal position like small disturbed magnetic needles. At times
the field appears reticulated (indicated in the figure) , as if two sets of nearly
horizontal fringes intersected at a small angle. It is now difficult to obtain
REVERSED AND NON-REVERSED SPECTRA. 85
continuous striations on rotating N, but the whole field may easily be filled
with nodules. The occurrence of two maxima is probably an incidental
result, as in other adjustments but a single one appeared. Naturally the rota-
tion of the grating or of the mirrors M and N changes the path-difference of
the pencils crossing within it, so that the micrometer screw //, / » ^ \\\
at the mirror M must be moved in compensation. Thus ///
this is another method of displacement interferometry and y/f
the usual equation suffices. $ ^ v W
The following rough experiments were made: Placing
the strong fringes in the center of the field (slit image),
the reading of the micrometer was taken. Then a thick glass plate, 0 = 0.71
cm., was inserted in one beam, nearly normally, and the micrometer displace-
ment, AN, was found when the fringes were brought back to the center of the
field again. The results were (for instance)
o.375 0.393 cm.
The displacement equation is (n being the index of refraction of the plate)
where the correction for dispersion may be put 2B/\2 — 0.026. Hence IJL =
1.50, 1.52, as was anticipated. On using white light, where there is but a
single strand, a cross-hair, and greater care as to the normality of the plate
compensator, etc., there is no reason why results of precision should not be
obtained.
38. The same. The linear phenomenon. — The occurrence of the linear
phenomenon reciprocally with the fringes for homogeneous light is interesting.
It usually appears when there is a flash of the arc lamp, i.e., a displacement
of the crater, introducing white light into the sodium arc. It is thus undoubt-
edly due to the reversed spectra for white light and may, in fact, be produced
by using the white arc or sunlight in place of the sodium arc. When the
mirror M is displaced on the micrometer parallel to itself, the linear pattern
moves through the wide-slit image from right to left; or the reverse. It does
so also when either mirror, M or N, is slightly rotated on a vertical axis. The
change in appearance during this transfer is very striking. In the middle,
between the extreme right and left positions, the linear phenomenon is excep-
tionally strong and fairly tumbling in its mobility. Toward the right or left
from the center it becomes gradually less intense, and on one side merges into
the homogeneous striations which then appear. On the other side it seems
merely to vanish. Doubtless the linear phenomenon is found, as usual, at
the line of symmetry of two reversed spectra; but, as both spectra are
shrunk to very small lateral dimensions, many colors probably adequately
coincide. In an achromatic reproduction of the slit all colors will coincide.
It is thus not necessary that the edges of the slit images should be super-
posed to produce the linear phenomenon. What is still more curious is the
86 THE INTERFEROMETRY OF
result that not even the longitudinal axes of the spectra need be quite in coin-
cidence, though, of course, the phenomenon appears most intensely for the
case of precise superposition. The angle of admissible separation of longitu-
dinal axes is, however, much larger here than in the usual cases above, so that
one of the longitudinal guide-lines of the two spectra may be appreciably
above the other.
The last result and the fact that the linear phenomenon appears here with
an indefinitely wide slit are new features. The cause of the latter has just
been referred to the exceptionally reduced width of spectrum resulting from
the double diffraction. If the dispersion were quite reduced to zero, all colors
in a definite narrow, transverse strip of the white slit image would be in a condi-
tion to interfere. This strip contains the superposed images of an indefinitely
fine slit. The slit in any other position, right or left, would have two non-
coincident images. Hence, when one wide-slit image moves over the other,
there is also a shift of the linear phenomena.
To produce the linear phenomenon with sunlight is difficult. The inter-
ferences should first be produced with the sodium arc, strongly, and the arc
thereafter replaced with sunlight entering the slit at the same angle. Further-
more, the pencil leaving the collimator should be a narrow, vertical blade of
light, and at the mirrors, M and N, red and green light should be screened
off, retaining only a narrow strip of yellow light for each. Finally, to avoid
glare, the slit is not to be too broad nor too narrow to cut off the yellow field
of the telescope.
Under these circumstances of completed adjustment, the linear phenomenon
usually appears strongly. Its form may be greatly modified by rotation of
either mirror, M or N, micrometrically, around the vertical axis, as already
suggested. The types are given in figure 62, quite fine, nearly vertical lines,
q, changing to moving, coarser forms, m, and these into the tumbling variety,
t, very coarse and nearly horizontal. The latter change by rotation and dimi-
nution into m' and q', while N is being continually rotated over a very small
angle, sliding one slit image continuously over the other. In the condition t,
the fringes rotate with astonishing rapidity, and this rotation is nearly 180°;
i.e., if the angle between m and m' is a, the angle of rotation has been 180° — a,
so that between q and q' there is about 180° of rotation. At the stage t, with
fine micrometric adjustment, the fringes may be made quite horizontal, and
they are then relatively large and square, or at times shaped like blunt arrow-
heads. This rapid rotation of fringes near / accounts for their turbulence,
since tremors have the effect not merely of raising and lowering them, but
also of producing the rotary motion in question. They may also be rotated,
of course, on slightly tilting the grating about an axis normal to its face.
Rotating the latter on an axis parallel to its face places the phenomenon in
different parts of the superposed yellow field.
Since a preponderance of yellow homogeneous light is present in the whole
of the superposed wide-slit images in the telescope, it is not difficult to suggest
the cause for the variations of the interference pattern when one image passes
REVERSED AND NON-REVERSED SPECTRA.
87
horizontally over the other. The forms, t, correspond to minimum path-
difference, remembering that in accordance with figure 59 all rays pass the
plate of the grating twice.
Further experiments were made with sunlight to detect the changes which
befall the phenomena in different focal planes. The ocular of the telescope
was gradually drawn out from an inner extreme position to an outer extreme
position, through the normal position for principal focal plane. In this case
a variation of form corresponding closely to figure 62 was also observed. The
characteristic feature, however, was the prevalence of arrow-head or caret-
shaped lines, both in the case of the extremely fine striations and of the coarser
nodules. In the former case these roof -like designs were closely packed from
end to end of the phenomenon and usually pointed upward. They recall the
top edges of extremely eccentric ellipses in displacement interferometry, and
in view of their lateral motion with the micrometer M and the decreased dis-
persion due to double diffraction, their origin may be similar.
39. The same. Inferences. — When the pencils, Mm and Nn, figure 59, are
parallel and sodium light is used, the whole field is uniformly striated, whether
the striations are made fine or coarse. I have found it impossible, on placing
plate compensators (0.5 to 1.5 cm.) in both beams and rotating these to any
degree whatever, to produce any suggestion of a secondary periodicity in the
field. The fringes for a thick compensator, slightly wedge-shaped, merely
become a little finer. Films of mica are liable to blur the field. In general,
moreover, reflections would be relatively weak and thus inappreciable. They
would require a separate adjustment for coincidence and not appear with
the principal phenomenon. Hence the strands of interferences obtained in
case of crossed rays are in a measure unique. The second periodicity is not
stationary, but a part of the phenomenon. The glass plate of the grating
produces an effect in virtue of its thickness, precisely as in the case of the dis-
placement interferometry of my earlier papers.
Experiments made with polarized light proved to be entirely negative.
The phenomenon appears between a polarizer and an analyzer so long as
sufficient light is present to ex-
i'Lf 'ru fy) /f)'7 fy}
hibit it. Observation with a
nicol, in the absence of the polar-
izer, showed nothing but the ob-
vious effect of reflection.
The occurrence of these par-
allel strands for crossed rays and
homogeneous light is thus diffi-
cult to explain. I have tried a great variety of methods of superposing
special interferences, etc., to produce the nodules with parallel rays, niM
and nN, or to break them with crossed rays, mN and nM, without avail.
There is no focal plane effect, nor any polarization effect. It is therefore
necessary to confront the case at its face value, as in figure 63. Here 5
_
^_
V
a
V-
a
- &
<£>
0(f<~
— ).
< —
— >.
i
' \
63
88 THE INTERFEROMETRY OF
and 5' are the traces of two longitudinally coincident reversed spectra, drawn
apart for distinction, the region of the D lines only being used. The light is
homogeneous to this extent and the slit wide, so that there is oblique inci-
dence. Then every point of S should (on adjustment) interfere with every
point of 5', the result showing a uniformly striated field in the telescope.
This is emphatically the case for the parallel rays, mM, nN; but with the
crossed rays, mN, nM, the interference is confined to the rays in the equi-
distant positions, n, in figure 63, and midway between them the field is a
neutral yellow. In other words, between the rays n the rays are displaced,
as shown by the arrows, recalling the arrangement of nodes in acoustics.
Corresponding rays a and a' (for instance) do not coincide and hence can
not interfere, the region aa' remaining neutral. In figure 64 the rays crossing
at c (fig. 57) have been shown for three nodes and the transverse arrows indi-
cate the directions in which the rays have been urged laterally. Naturally, I
am merely stating the case as immediately suggested by the results. One may
argue that there may be a secondary periodicity in the grating. But why
does it not appear at all in the case of parallel pencils, when it is so obtrusive in
the case of crossed pencils of rays ? Again, the interferences are unquestion-
ably due to Di and D2 light, simultaneously. If the grids for these two wave-
lengths should be at a slightly different angle to each other, their superposition
would give something like the observed phenomenon, apart from details. Thus
in figure 65 the two grids due to DI and D2, intersecting at a small angle, may
be interpreted as appearing strand or cord like at N, and neutral at I and I'.
With white light the linear phenomenon would eventually become achromatic.
But, again, why should lines so close together as DI and D2 show any appre-
ciable difference of angle or rotational phase-difference in their interference
pattern? Intersecting grids, moreover, can be produced by other methods
and nearly always betray their origin. The final inference is that suggested
by figures 63 and 64, that homogeneous rays on crossing (here in a medium
of plate glass) may exert a lateral influence on each other, to the effect that
identical rays emerging from the crossing are arranged in equidistant nodal
planes according to figure 63.
40. Experiments. Reflecting grating. Crossed rays. — In the preceding
experiments the remarkable phenomenon of double interferences was ob-
tained with glass-plate apparatus. It is improbable that any secondary inter-
ference can have been produced by the presence of reflected light, since the
reflected pencils will be weak as compared with the primary pencils and
REVERSED AND NON-REVERSED SPECTRA. 89
differently situated. It is nevertheless necessary to forestall all misgivings
by avoiding glass plates altogether and adapting the methods of figure 57,
where reflecting surfaces (front faces) only are present, to the experiment
for crossed rays mN and nM.
In the apparatus as finally perfected, G, figure 57, was a Michelson plate
grating and G' a Rowland concave grating, each with about the same grat-
ing constant. A strong lens was placed at T for observation at the focus of
the concave mirror of G'. The latter was capable of fore-and-aft motion, of
rotation about a vertical axis in its own plane and about an axis normal to that
plane; G was capable of rotation about a horizontal axis parallel to its plane.
Thus the possibility of fore-and-aft motion and the three cardinal rotations
for the gratings, together with a micrometric fore-and-aft motion of M, was at
hand, as well as the rotation of M and N about horizontal and vertical axes.
The interferences were found after establishing the coincidence of the yellow
homogeneous fields, in the manner described in the preceding paragraph.
The fringes were at first small and apparently single, but they could be
enlarged at pleasure and the two definite systems separated by fore-and-aft
motion of G'. They occupied only a part of the wide yellow slit image, the
sodium arc being used. On actuating the micrometer at M there was dis-
placement of the interference pattern as a whole, so that the conditions of
displacement interferometry are here also implied, though the equations are
liable to be different. On rotating M, micrometrically, about a vertical axis,
the structure of the interference reticulations changed and was at times
reduced to a single set.
Whenever the arc flashed, or when white light was used, the linear phenom-
enon appeared alone, either cross-hatched or longitudinal, depending upon
the character of the reticulated pattern for homogeneous light. With sun-
light, even after narrowing the blade from the eollimator and screening off
red and green light, the phenomenon was faint and hard to find, unless it
was produced alternately with sodium arc.
With the arc freshly charged with sodium, but a single set of interferences
or else the linear phenomenon appears, since the broadened sodium lines are
equivalent to a continuous spectrum in this region. Not until the excess of
sodium has all been evaporated and the sodium lines are normal does the true
reticulation show itself. It is interesting to describe two cases of this double-
interference pattern, obtained by gradual and successive fore-and-aft motion
of the grating G', between limits, while the edges of the two wide-slit images,
respectively horizontal and vertical, are kept in contact throughout.
Suppose the original fine fringes to be nearly vertical ; then the apparently
simple fringes, a, figure 66 (their appearance, however, would lead one to sus-
pect their simplicity), change to the cord-like strands b, appearing like helices
of a very large pitch. Both interference fringes are still nearly parallel, and
they cover the whole wide-slit image uniformly. These eventually pass into
the square or rectangular reticulation, c, with both systems equally strong.
Probably intermediate forms have here been skipped. The system, e, occurs
90
THE INTERFEROMETRY OF
very soon afterward, in which the difference in size of fringes has become
enormous. Following e, the procession is reversed in g, h.
Both systems (a and /3 systems, say) have passed through maxima, but
not at the same time, or not for the same fore-and-aft adjustment. Both sys-
tems have rotated, the rotation being very rapid near the maximum. The
reticulations quiver and look
precisely like capillary waves
in a rectangular trough of mer-
cury, except that they are usu-
ally at an angle to the bound-
ing edges of the superposed
slit images.
In this quivering system of
two identically strong fringes
it is difficult to make out the
rotations, but after consider-
able revision the sequence in
figure 67 was definitely ascer-
tained. Beginning with the
extreme fore-and-aft position of G', and moving it successively forward in steps
of i or 2 millimeters, the apparently single grid, i changes to 2, where the
two systems a and /3 can be disentangled, a expanding and rotating more
rapidly, so that 3 and 4 follow. Here a is horizontal and probably of maxi-
mum size, /3 is still nearly vertical and but slightly expanded. Therefore, while
the a-effect wanes the /3-effect waxes, and the squared or orthogonal type, 5, is
produced. The lines are here equally strong and it is the symmetrical figure
of the series. Thereafter in 5, 6, 7, 8, and 9 the chief expansion and rotation
is transferred to the /3 system, with which the a. system has changed functions.
Hence both systems rotate nearly 180° in the same direction and pass through
maximum size; but the maximum is retarded in rotational phase for one as
compared with the other. Rotation and growth are accelerated near the
maximum. The total displacement of the grating G' between the cases i and
9 (fig. 67) was about 2 cm. ; but this depends upon the obliquity of the grating
and incidental conditions, as explained above.
Suppose, in the second place, that the original fringes, i, figure 67, were
nearly horizontal ; in such a case the evolution is much the same, but the sym-
metrical form number 5 becomes smaller and more and more flatly rhomboidal
horizontally. Probably the scheme of rotation is the same, but is much harder
to ascertain in view of the flat forms. On the other hand, the field now abounds
in vertical strands of interferences, like those of the preceding paragraph, and
nodules are often in evidence, as before.
If the original lines are quite vertical, they do not seem to rotate with fore-
and-aft motion of G', but form intersecting, vertical, apparently simple sys-
tems throughout the motion. Slight departure from the vertical produces
rhomboids very long vertically and often very coarse.
REVERSED AND NON-REVERSED SPECTRA. 91
41. The same. Compensators. — A compensator of ordinary plate glass,
at the intersection c, figure 57, produces no effect, if symmetrical to both beams.
If not symmetrical, the interferences are displaced to right or left in the field
of the telescope, as in any case of displacement interferometry, depending on
which component beam receives the longer glass-path. Thus this adjustment
corresponds to the grating in the preceding paragraph, the difference being
that in the latter case the same ruling is used for both diffractions. Hence
the interference figures obtained are simpler, showing vertical strands only.
In the present case strands occur in all directions. The maxima for oblique
positions of the glass plate were not found with reflecting gratings.
If the compensator is within i inch in thickness, its introduction occasions
no difficulty. The interference pattern may be changed, but it remains the
same during the rotation of the compensator ; but if the latter is thicker than
2 inches, the figure is usually so small as to be found with difficulty, unless
the grating G' is brought forward, to allow for the mutually inward refraction
of the rays. If this is done, the same figure may be reproduced. On advancing
the grating, plate compensators much over 3 inches thick were tested without
the slightest annoyance. Lenticular compensators require special adjust-
ment and are very difficult of use.
The effects of rotating the grating about the three cardinal axes have all
been considered above. In the present instance two sets of fringes are sym-
metrically rotated, subject to the same conditions. Rotation of G' around a
horizontal axis requires an elevation or depression of the arc lamp, if the
fringes are to remain in the field. Rotation around a vertical axis separates
the slit images, and a readjustment for superposition is necessary. Results
so obtained are therefore complicated and were not studied.
42. Miscellaneous experiments. Fringes with mercury light. — A few
random experiments made with the sodium arc, in the presence and absence
of the magnetic field, showed no results; nor was this to be expected, as a
reasonably strong field would blow out the arc. Again, the insertion of a
glass compensator, 0.7 cm. thick, in one of the component beams, developed no
maximum on rotating the compensator about a vertical axis. Thus with reflect-
ing gratings the peculiar behavior of the transmitting grating, showing a maxi-
mum on either side of a symmetrical minimum (§36, 37), is not reproduced.
The effect of rotating the first reflecting grating G on a vertical axis is only
to throw the sodium light out of one side or the other of the (superposed)
slit images. No available means of enlarging the fringes indefinitely was
found. It is probable that this would require fine adjustment for symmetry.
The field of interference, as a whole, is within a spot-like area which may be
moved up and down, or right and left, by the vertical and horizontal adjust-
ment screws on the mirror M. Coincidence at the two sides of the slit favor
different interferences. The case is always as if, at a single point of the field
only, there were actual coincidence, and that the interference pattern is
grouped closely around it.
92 THE INTERFEROMETRY OF
With the use of an ordinary glass mercury lamp (27 storage cells, 5 amperes)
the fringes are found with difficulty when the beam at the first grating is
wide. On using a vertical blade of light the definition was improved. The
fringes are faint, very susceptible to motion, and at times even absent. They
occur, however, as a single set, as was anticipated, showing that the above
duplicated fringes are actually due to the two sodium lines. The mercury
fringes are easily rotated and pass through a horizontal maximum with fore-
and-aft motion. Rotating G about a normal axis may further increase this
maximum size to a limit at which the fringes appear irregular or sinuous. A
displacement of the mirror M over 0.7 cm. was easily permissible, without
destroying the fringes. They occur, as above stated, within a certain adjusted
spot area of the field of view. An attempt was again made to detect a Zeeman
effect by placing the poles of an electromagnet on the two sides of the lamp ;
but here again no difference was discernible on opening and closing the electric
circuit. The field, however, for incidental reasons, could not be made strong
enough for a critical experiment.
43. Inferences. — After these experiments (made with the apparatus figure
57, free from glass plates and depending on reflections only) the cause of the
phenomenon is no longer obscure. Obviously one of the paired grids in figure
66 or 67 belongs to each sodium line. The retardation of one phenomenon,
rotationally, as compared with the other, is due to the difference in wave-
length between D\ and D%- The phase-difference between numbers 4 and 6
(fig. 67) is thus equivalent to 6 Angstrom units. If the displacement of G'
is about 0.3 cm., there should be about 0.5 mm. displacement, fore and aft,
for i Angstrom unit. If the grating, G', is on a micrometer, this should be a
fairly sensitive method of detecting small differences of wave-length, or give
evidence of doublets lying close together. The sensitiveness clearly increases
with the length of path of the component rays and may thus be increased.
With this definite understanding of the phenomenon, it is desirable to deduce
the equations, which in the occurrence of parallel rays would not differ essen-
tially from those of Chapter II or III. It is useful, however, to treat the new
case of crossed rays. In figure 69 the angles of diffraction are 61 and 62, if the
incidence of light, L, is normal at G and at an angle iz at G', G and G' being
parallel. The mirrors are set symmetrically at angles a\ and <r2 to the normal
in question, and the diffracted rays are reflected at angles 0:1/2 and 0:2/2,
respectively. The reflected rays cross the normal at an angle /3. Then
sin 6 'i = \/D\ sin 02 = X/Dz — sin iz
where Di and Dz are the grating constants. From the figure
"1/2 = 01+0-1 — 90° o:2/2 = 02+0"2 — 90° 0i = o:i-f-/3 02
From these equations,
Dz sin i = Dz sin (202+2(r2+j8)— DI sin (••••)
If Di = D2, then 6i=62, ffi = ffz, and therefore iz = o.
REVERSED AND NON-REVERSED SPECTRA.
93
Thus the relations are quite complicated, but if Di = Dz, or the gratings
have the same constant, rays of all wave-lengths should, after double diffrac-
tion, issue normally to the grating G', and the arrangement is therefore
achromatic. If DI is not quite the same as D%, but nearly so, an adjustment
of a would probably meet the case approximately. If the original incidence is
at an angle i\, DI sin i\ would have to be subtracted from the first member,
but the diffractions would now differ on the two sides of the apparatus.
The relations of the rotations of the striations of D\ and D2 light to
the fore-and-aft motion is next to be considered. It will be convenient to
make use of figure 68 for this pur-
pose, the notation being the same
as in figure 69. The two rays, i
and 2 (Di and D2), have both
been introduced, and the position
of G' is such that the DI rays
intersect in its face and are dif-
fracted into TV In such a case
the combined pencil is divergent,
DI rays will undergo an earlier ^
intersection, and consequently be
separately diffracted into TI and
T'i. Hence DI and D2 are differently circumstanced in relation to the fore-
and-aft motion, and the rotation produced will thus be advanced in one
case, as compared with the other, for the reason discussed in Chapter III,
paragraph 26. It is also clear that the difference of phase in the two rotations
mil be greater, as the total path of rays between G and G' is greater, so that
the large distances used in the present experiments (nearly 3 meters) account
for the astonishing sensitiveness of the phase of rotation to the wave-length
difference. In fact, D-2 will be in the same phase as DI, if the grating is moved
forward from G' to g', figure 68, since in both cases the rays intersect in the
normal. Hence if R is the total path GmNG', and if the angle of dispersion
between DI and D* is d6, 62 the angle of diffraction at G', and h the displace-
ment from G' to g', D the grating space,
69
or
and the resolving power
h\
''DR
d\ h
Dh
X R cos 62 R]/ D2 — ]
In the given adjustment, roughly,
whence
h = '
0.34
==
94 REVERSED AND NON-REVERSED SPECTRA.
As the resolving power is, roughly, h/R, and if h = 0.003 cm. is still appreciable,
3Xio2
i.e., lines i/ioo of the distance apart of the sodium lines should be rotationally
separated.
Again, the displacement, fore and aft, between like rotational phases of
DI and Di should be about 3 mm., and this agrees fairly well with the order
of values found.
The case of the transmitting grating (fig. 59) is thus also elucidated, though
it is not clear to me why the duplication of fringes is so efficiently concealed
in the nodular forms observed. The reason for the minimum of size, for the
symmetrical position i = o, and the two maxima for oblique positions of the
grating (2 =±20° about), suggests an explanation similar to that given in
Chapter II. In other words, in the oblique position the short path-length
is compensated by the increased thickness resulting from the greater obliquity
of grating, whereas the long path-rays traverse the plate of the grating more
nearly normally. In this way the path-difference is reduced as compared
with the symmetrical position, and the fringes are therefore larger. The
oblique grating acts as a compensator in both of the component beams, and the
fringes may be visible, even if in the original position (fig. 59) they are all but
invisible. If, however, the apparatus (fig. 57) is used with a plate-glass com-
pensator symmetrical at c, there are no maxima or minima for any obliquity.
Hence the tentative explanation for the case of figure 59 is not warranted.
The fore-and-aft motion of the plate grating (fig. 59) produces no effect,
since the rays are reflected back so as to retrace their paths. They are also
reflected between parallel mirrors Ar, m and n, M. Thus the path-difference
is not modified. The result is merely a decrease of the distance M, N, and a
corresponding increase of m, n, and vice versa.
The marked effects produced by rotating the transmitting grating around a
normal axis, finally, follow the explanations given for the rotation of fringes
of non-reversed spectra in Chapter III, paragraphs 25 and 26.
In conclusion, an interesting application of the apparatus (fig. 56) or the
other similar types may be suggested. By half -silvering the mirrors and pro-
viding a similar opaque set beyond them, there should be no difficulty (in
the case of homogeneous light) of bringing the interferences due to crossed rays,
c, and to parallel rays, a'b', into the field of the telescope together. Strictly
homogeneous light (mercury arc) would be needed to obviate the duplication
of the sodium arc. In such a case, therefore, the parallel fringes could be used
after the manner of a vernier on the crossed fringes, with a view to a repetition
of the experiment of Michelson and Morley, if this experiment had not been
so thoroughly carried out by the original investigators. However, the plan
would be to rotate the apparatus, as a whole, so that the two crossed rays
would be alternately in and at right angles to the earth's motion, whereas the
two parallel rays would preserve the same relation to that motion. Naturally,
the parallel and crossed paths would in such a case have to be lengthened by
multiple reflections.
CHAPTER VI.
CHANNELED SPECTRA OCCURRING IN CONNECTION WITH THE DIFFRAC-
TIONS OF REFLECTING GRATINGS.
44. Introductory. — -Throughout the preceding work I had noticed that
the spectrum due to either of the component beams, after successive reflection
from two reflecting gratings, was often regularly furrowed by transverse
black bands, before the two spectra were brought to interfere. As these
fringes are stationary, they do not modify the phenomenon investigated ; but
questions now arise as to whence these reflected fringes of a single beam come.
They are not strong, as a rule, and I was therefore inclined to attribute them
to some imperfection of the silvering of the opaque mirrors, but this proved
not to be the case, so that it seemed worth while to examine them by special
experiments.
45. Apparatus. — The apparatus for this purpose, as one beam only is
wanted, is quite simple. In figure 70, L is a vertical blade of parallel rays of
white light from a collimator and slit. These rays impinge on the plane grat-
ing G, whence the orders 1,2,
3, etc., of spectra are reflected. && -j % ^'
Either of these pencils may be
received by the second grat-
ing Gf, plane or concave, from
which spectra of any order
are available . If o denotes the
reflected pencils, the groups
from two gratings may be dis-
tinguished as (3, i), (3, o),
(3. — i), (3. -2), etc., as in
the figure. Any of these two
different pencils is to be examined at T by a lens or telescope, for instance,
and the latter (with strengthened objective where needed) is more convenient,
even when the concave grating is used. A wide slit S, revolvable about G,
is often useful for screening off spectra or parts of spectra. In some experi-
ments the grating G' may be replaced by an opaque mirror.
The gratings are provided with the usual adjustments for parallelism of
rulings and slit. G' and T must be capable of considerable right-and-left
motion, and G, in particular, of controllable fore-and-aft motion.
46. Scattering. — An interesting result of this work is the evidence and
spectroscopic quality of scattered rays, incidentally encountered. For instance
in figure 70, if the slit 5 is narrow, it cuts off all the rays but the orange yellow
of the third order, and the reflected spectra (3, i), (3, — i), etc., will largely
95
70
96 THE INTERFEROMETRY OF
consist of orange-yellow light. Associated with each of these reddish-yellow
patches, however, are vividly violet-blue patches, each separated from the
reddish yellow by an almost total absence of green, relatively speaking. If
the light is very intense, the connecting part of the spectrum also appears,
but it is always far less vivid than the ends of the spectrum in question.
Inasmuch as all violet radiation proper has been screened off at S, it is
obvious that violet light must have been scattered in all directions from G,
a part of which, therefore, passes the slit and is resolved by the second grating
G'. Moreover, as the scattering lines of the grating are equidistant, the
scattered light has a regular wave-front. (Cf. Cam. Inst. Wash. Pub. No.
229, 1915, pp. 100— 102.)
The correlative experiment of detecting the reddish light transmitted after
scattering was also tested. For this purpose the reflecting grating G may be
replaced by a transmitting grating, slit 5 placed beyond, and the light then
analyzed by a second grating G' behind the slit and diffracting toward it on
one side. But no results of value were obtained.
47. Fringes with white light. — The experiments with the apparatus (fig. 70)
were commenced with sunlight and (what is essential) a fine slit. Fringes
are found in all combinations of doubly diffracted pencils (3, +i), (3, — i),
(3, —2), etc.; (2, i), (2, — i), etc.; (i, — i), (i, i), etc., but none in the
reflected pencils (3, o), (2, o), (i, o), etc., as a rule. Whether the grating G'
be concave or plane, it is best to use a telescope at T, because (when provided
at the objective with an auxiliary concave or a convex lens) it more easily
offers a wide range of observation along its axis than an ocular. The latter
must be wide and has to be shifted bodily; but both methods were used. A
concave grating at G and plane grating at G' gave no results. The concave
grating is usually more free from channeled spectra.
Of the great variety of fringes obtained, I shall give only two typical
examples. The second order of spectra for G (plane) was separated from the
others by the slit S and diffracted into G' (fig. 70). The successive fringes
appear as the ocular is drawn outward from the principal focus.
Combination £2, G'o : Only a good sodium doublet, which became washed
on drawing out the ocular of T, was obtained ; no fringes appeared.
Combination Gz, G'--i: Just outward from the principal focus a large,
coarse, irregular set of fringes appeared; next (ocular farther out) a large
regular set, somewhat diffuse, possibly double and superposed; then a finer,
half-size, very regular set, possibly decreasing. After this the mottled sur-
faces of the gratings were successively in focus. A weak spectacle lens was
now added to the objective of T, whereupon very large regular fringes were
seen when the ocular was far out.
Combination £2, G— 2: The ocular moving outward from the principal
focus, the fringes seen in succession were as follows: large, regular, vague;
half -size sharp; surfaces vertically striated; (lens on) fine regular set in red;
doubled regular set in green.
REVERSED AND NON-REVERSED SPECTRA. 97
Combination G2, G— 3 : Fine set just before the surfaces appeared, which
were delicately striated; fine regular set; coarse set, both close to surface;
(with lens on) fine regular set; doubled, strong regular set.
Different distances between G and G' had very little influence on the size
of the phenomena. A few examples may be given, which are observed when
the ocular is moved outward.
£3, Gi : Distance 10 cm. — Fringes, faint regular; strong irregular; faint
regular; flat field; surfaces visible; faint regular.
Distance 25 cm. — Strong irregular; faint regular; small regular; large
(double) irregular; lines slit into fine fringes; large faint regular.
Distance 46 cm. — Large strong, with two absorption bands; fine regular;
double-sized faint ; surfaces with fine striations ; alternations of fine and coarse
lines; faint, regular, large, etc.
Fringes of different color are often in different focal planes. When a lens
is used with the concave grating, observations must sometimes be made 2
meters off to get the large regular fringes. Red fringes may be narrower than
the corresponding violet set.
If the grating G is moved fore and aft, parallel to itself, the fringes are
shifted across the stationary sodium line, as in displacement interferometry.
Whereas in the positive combination (3, i), (3, 2), etc., the spectra widen,
they tend to close up for the negative combinations (3, — i), (3, --2), etc.
With two identical plate gratings they may image the white slit. But this
seems to have little effect on the fringes seen as a whole when the ocular is
out of focus.
When white light is used and the grating G' replaced by an opaque mirror,
or in case of combinations which involve direct reflection (2, o; 3, o; etc.) at
G', there seem to be no fringes.
48. Fringes with sodium light. — While there is some difficulty in obtaining
the fringes with white light, fringes with homogeneous light are obtained at
once, provided the light is sufficiently intense. A sodium arc lamp, or a
mercury lamp, with a fine slit, must therefore be used. In this case, moreover,
the grating G' may often be replaced by an opaque mirror, or the fringes of
the order £26*0, GzGo, etc., may be produced with entire success. On moving
G fore and aft, they again travel across the sodium line. Often, in fact, two
sets of fringes seem to be shifted. A few examples again may be given of
the great variety in this display while the ocular is being drawn out :
Gi,G' — 2: Sodium lines DiDz single size ; large strong fringes, lines split.
Gi, G'—i: Closed spectrum; striations continuous.
Gi, G'o: Reflection; DiD2 single size; surfaces of gratings finely
striated.
Gi, G'I
Gi, G'2
G2, G'o
DiD2 double size; strong grid seen very near the surface of G'.
D\Dz treble size, out of reach.
Reflection; no fringes.
98 THE INTERFEROMETRY OF
Gz, G'—i: Distance 12 cm. — Coarse irregular; with lens fine regular
set, near and beyond the surfaces.
G2, G'—i: Distance 45 cm. — Surfaces with doubled fine striations;
with lens finally strong and regular.
G2, G'—i: Distance 60 cm. — Regular faint; irregular double, very
strong; surfaces striated; with lens strong double irregular;
finally regular small.
£3, G'I: Regular; regular line split; irregular coarse; surfaces finely
striated, G coarser; fringes grow continually larger without
vanishing.
On moving G fore and aft, two grids seem to travel through each other in
opposite directions. This probably accounts for the occurrence of irregular
fringes. The size of fringes seems to be a minimum for a conjugate focus
near the surfaces. The whole phenomenon is continuous. Irregular fringes,
probably superpositions, become regular in other focal planes.
£3 , Gf2 : About the same ; minimum size at the surfaces, increasing
about three times as the ocular is drawn either way.
6*3, G'Q\ also 6*3, mirror: About the same results, only brighter and
better. Hence in case of large dispersion two gratings are not
needed. The two sodium lines, when the ocular is drawn out
of focus, multiply themselves at regular intervals, so that the
grids are sometimes distinct, sometimes partially superposed.
Thus the classic diffraction phenomena of a slit suggest them-
selves as the starting-point for an explanation of the present
phenomena as a whole.
£3, G'o, produced alternately with sodium light and sunlight, showed
the same sequence of fringes (the large ones with a tendency
to split) in the former case, while nothing appeared in the case
of white light.
49. Grating on a spectrometer. — It seemed necessary, therefore, to con-
sider the diffraction of a fine slit, when seen in the telescope, somewhat in
detail. In Chapter III the production of beautiful Fresnellian interferences
from two identical slit images and homogeneous light was demonstrated; but
an equally clear manifestation of the diffraction of a slit image, when the
ocular is out of focus, does not seem to occur. The broad image of the slit
out of focus shows a stringy structure only, but no separation is easily obtain-
able. Fringes, as such, are quite absent when the ocular is drawn out.
The light of the sodium arc was now passed through a very fine slit and
collimator and reflected from a plate grating. The above intermittently
regular and irregular fringes were strikingly obtained with the ocular out of
focus. As this is successively more and more drawn out, fine lines become
coarser, and then seem to subdivide, giving the structure a fluted appear-
ance, frequently regular. There is, in other words, a double periodicity. In
REVERSED AND NON-REVERSED SPECTRA. 99
the case of highly diffracting grating (D=io"6Xi75), the results appear best
in the second order.
The same beautifully duplicated fringes were obtained with a transmitting
film grating of about the same dispersion, particularly well in the first order.
The sodium flame gives too little light for the present purposes, but the
phenomenon is seen.
Believing that some irregularity might be introduced by the double-sodium
line, I installed a mercury lamp for comparison. In the first experiment a
film grating (D= 173X1 o"6) was used, the ocular traveling outward from the
principal focus. Both the green and the double yellow mercury lines enlarged
and showed fringes of increasing size and number together. The green field
had a darker band, the yellow a bright band in the middle. As the fringes
enlarged, each split up into secondary fringes, 4 or 5 eventually, and this
again occurred for both green and yellow fields.
Rotating the grating around a vertical axis seemed to shift the primary
fringes laterally over the stationary secondary fringes. A concave lens for
positions anterior to the principal focus and a convex
lens for posterior positions (toward the eye) were
successively added to increase the range of observa-
tion. On both sides of the principal focal plane
(fig. 71) fringes occur, which enlarge with the dis-
tance x from that plane. As they enlarge, each fringe
splits up into secondary fringes, which in turn enlarge.
Sometimes the arrangement is irregular. Green and yellow fields may
overlap, but they do not do so conformably.
The undeviated ray, however fine the slit may be, merely shows a stringy
field, sometimes suggesting structure, but never showing clear-cut fringes.
The same kind of results were obtained with a reflecting grating of about
the same dispersive power. In the second order the fringes were particularly
clear and regular. Primary fringes, finally, carried three to four secondary
fringes each.
Next, a ruled transmitting grating of less dispersive power (grating constant
352Xio~6 cm.) was adjusted for mercury light. Here in the undeviated ray
and in the first order no clearly separated fringes were obtained. In the second
and third orders, however, they were very perfect, and followed the above
rules, showing sharp secondary fringes.
It follows, therefore, that a certain degree of dispersion is needed to resolve
the fringes, which is inadequate in amount in the order zero, in this case,
and scarcely so in the first order. In the higher orders the conditions are met.
Using a very fine slit, however, I later just succeeded in separating the fringes
in the first order.
Finally, I returned to the endeavor of detecting diffraction fringes in the
undeviated image, using a micrometer slit, a good achromatic lens (or no lens),
and a distant (2 meters), moderately strong telescope. In this case separated
and distinct diffraction fringes, white throughout, were undoubtedly obtained.
They moved with the eye so as rarely to be stationary and in the same direc-
100 THE INTERFEROMETRY OF
tion if the ocular is drawn out, or the reverse if it is thrust in. On close
examination two sets, in different focal planes, seemed to be present, one
stationary and the other moving as described, and accounting for the observed
pronounced parallax. Suggestions of movable fringes accompanying the
stationary are also present when the latter are produced by the grating. In
this case the stationary fringes are strong ; in the case of simple diffraction the
movable fringes are more prominent.
50. Inferences.— -There can be no doubt that the great variety of chan-
neled spectra obtained, when white light is successively diffracted by two
gratings, is referable to the fringes obtained in the diffraction of homogeneous
light, observed outside the principal focal plane, on a spectrometer. In other
words, if light of a given pure color (sodium, mercury) is used, a single grating
suffices. Each line of the spectrum is resolved into well-defined groups of
fringes, if it is observed either in front of or behind the principal focal plane.
The arrangement of fringes varies in marked degree with the distance of the
plane observed from the latter (x, fig. 71). If reflecting gratings are used,
there is no other possible source of interferences ; but reflecting and transmit-
ting gratings show the phenomenon equally well.
After finding how easily the Fresnellian interferences of two virtual slits
could be reproduced in the telescope (Chapter III) and observed on either
side of (before or behind) the sharp images, it seemed reasonable to suppose
that the diffraction of a slit could also be produced and exhibited in this way ;
but the availability of this anticipation is attended with much greater diffi-
culty. The image of a very distant slit does indeed show separated diffraction
fringes on either side of the principal focal plane in the observing telescope.
But they move right and left with the eye, in the same direction if the ocular
is drawn outward from the principal focal plane, and in the direction opposite
to the eye if the ocular is thrust in. Hence, in this respect, the fringes do not
at once recall the phenomena under consideration. Usually the blurred image,
out of focus, is stringy, without definite structure. It is resolved in a single
focal plane only.
To obtain sharp stationary fringes from an image of the slit, this image must
be produced by the diffraction of a grating having a dispersing power above a
certain minimum. Thus in a grating of about 7,000 lines to the inch the un-
deviated slit image and the image of the first order are not clearly resolved,
unless the slit is very fine. In the second and higher orders, however, the res-
olution is very pronounced and the fringes stationary.
The resolution of fringes is equally manifest in front of or behind the prin-
cipal focal plane, so that if a weak convex lens is added to the objective of the
telescope, the succession of fringes is found with an outgoing ocular; if a weak
concave lens is added to the objective, the succession is found with an ingoing
ocular, starting in each case near the principal focus. As the fringes increase
in size they in turn subdivide, sometimes irregularly, as if each fringe were a
new slit image, capable of undergoing secondary diffraction. Beyond these
secondary fringes no further resolution was detected.
REVERSED AND NON-REVERSED SPECTRA. 101
Returning to the work with two successive gratings and white light, the
channeled spectra obtained are too complicated for concise description. A
very interesting result, however, is the passage of the fringes across the sta-
tionary sodium line, when the first grating G is moved fore and aft in a direc-
tion normal to its plane. The region of the D line is thus alternately dark and
bright. The direction of these rays remains unaltered while the illumined strip
is shifted horizontally across the ruled space (fig. 70) of the second grating.
Usually it is difficult to see the D line in the focal plane of the fringes. When
homogeneous light is used this fiducial mark is necessarily absent and the
cross-hairs of the ocular must be supposed to replace it. The shift of the
fringes is then equally obvious, and sometimes (sodium light) different groups
seem to travel in opposite directions while the grating G moves in one direction.
In case of homogeneous light and two gratings, moreover, the fringes seem to
be of minimum size in the conjugate focal plane of the gratings. They
increase in size and in turn split up in focal planes before and behind this.
An insight into these occurrences was finally obtained in observation with
homogeneous light in the spectrometer by shifting the grating (transmitting)
in its own plane, right and left. The fringes in such a case move bodily across
the field of the telescope, new groups entering on one side for those which
leave on the other. These fringes, even if quite distinct, are differently
arranged in coarse and fine series and are frequently accompanied by dark or
bright bands. This probably also accounts for the effect of the fore-and-aft
motion of the grating, mentioned above. Moreover, it would be interesting
to search for repetitions of given groups of fringes while the grating is being
shifted parallel to itself, from end to end, as this might indicate the residual
imperfections of the screw with which the grating was ruled. If the ocular is
drawn and set outward from the principal focal plane (at which the slit image
is quite sharp) into a different position, the fringes move in a direction opposite
to the grating. If the ocular is set inward from the principal focal plane, they
move in the same direction as the grating. This would not be unexpected ; but
secondary fringes or something else in the field seem to remain stationary.
Successive fields may be quite different as to arrangement of fine and coarse
lines, but all plane gratings exhibit the same phenomena. Thus it is obvious
that the fringes of the present paper result from a residual irregularity in the
rulings of the grating. Micrometrically, the successive strips of a slit image,
however fine, are of unequal intensity. Between these there is diffraction, as
may be tested by examining the clear glass at the edge of the ruled space.
To attempt a theory of these phenomena seems premature ; but it is obvious
that in the otherwise indistinguishable images of a slit in homogeneous light,
however sharp or however narrow, the nature of its origin still persists and
may be detected by observations outside of the principal focal plane. A fine
slit is in all cases presupposed, and all the phenomena vanish for a wide slit.
On the other hand, the width of the pencils of parallel rays may be far greater
than is necessary to show the strong Fraunhofer lines, if indeed there is any
limitation to this width.
CHAPTER VII.
PRISMATIC LONG-DISTANCE METHODS EN REVERSED AND NON-REVERSED
SPECTRUM tNTERFEROMETRY.
51. Purpose. — It is preliminarily the object of the present paper to examine
a variety of new methods for the production of interferences with spectra,
with a view to the selection of as simple a design as possible for practical pur-
poses. Some interesting differences appear in the results, so that the sim-
plicity of construction does not necessarily recommend the apparatus for use.
In the second place, the endeavor will be made to assemble appurtenances
in such a way that the extremely mobile phenomena may be under control,
even in a moderately agitated laboratory. In case of the early interferometer
experiments, the interferences disappeared on merely touching the apparatus,
and are rarely or never at rest; whereas it is, of course, necessary that they
should remain visible while the micrometer is being moved. These experi-
ments are now nearly completed, but will preferably be described in a succeed-
ing report.
52. Methods and apparatus. — Some prismatic methods were tested in the
earlier volume, but not developed; for the plan of using a transmitting grating
twice, or two gratings in succession, seemed to contain greater promise. The
prism method is, however, more sim-
ple than any of the others and there-
fore deserving of special study.
In figure 72 the large right-angled
prism P, with its faces silvered, re-
ceives the pencil of parallel white
rays, L, on its orthogonal faces and
reflects them to the plane opaque
mirrors n and m. From here the rays
are further reflected, either nearly in
parallel, as in the figure, or crossed,
as at c, c', to the remote opaque
mirrors N and M, which in turn re-
flect them to the plane or concave
grating G. If the rays converge at the
appropriate angle of diffraction, 6, a
selected color will be diffracted in the direction of the normal to G in each
case. If the two paths are nearly equal, these rays will therefore interfere
in the axis GT and the results may be observed by a telescope or a lens at T.
In my apparatus the distances mM and nN were of the order of 2 meters.
In consequence of the three successive reflections, it is somewhat difficult to
102
73
REVERSED AND NON-REVERSED SPECTRA. 103
obtain spectrum lines normal to the axis of the spectrum, so that if the latter
are superposed the lines will be at an angle. But if this is small, it does not
seriously interfere with the occurrence of fringes, as they extend from top
to bottom of the spectrum.
The appearance in general is of the linear character heretofore described.
They pass symmetrically from extreme fineness, through a maximum size, to
fineness again, with the fore-and-aft motion of the grating G, and they usually
rotate near the maximum.
If the mirror M is displaced nearly in a direction normal to itself, on a
micrometer, the fringes undergo the same evolution, and in this respect differ
from the case where the primary differentiator, P, was also a grating. In
this case the displacement of M showed no discernible modifications of the
size or character of the fringe pattern. The fringes merely moved. In figure
72 the effect of moving G or M fore and aft is similar, since it throws the
point of convergence of the rays NG and MG in front of or behind the grating.
The result is therefore different when white light impinges on G from what it
is when the light is already nearly homogeneous.
The limit of visibility is also inferior to the double-grating method heretofore
used, for the fringes passed between the limits of visibility through the maxi-
mum size, for a displacement of M of only about 3 mm. Smaller ranges
may occur. On limiting the incident beam at L to a breadth of about 0.5 cm.,
the fringes became much broader and relatively intense.
There is, of course, an abundance of light, so that the screening of the
incident beam is not disadvantageous. In this case, when the fore-and-aft
position (illuminated strips on the grating coincide, as in figure 72) and the
position of the grating relative to its normal axis were carefully adjusted,
large arrow-headed fringes, as in figure 73, were obtained, usually less closely
packed vertically. Apart from tremors, these move slowly up and down
(breathing), as a result, no doubt, of changes of temperature in the air-
paths. A mica film inserted into one beam and slowly rotated produced
similar motion, besides introducing its own grid of vertical and parallel fringes.
The reason for the occurrence of these arrows is not quite clear to me, though
they are associated with horizontal fringes and homogeneous light, the doubly
inflected forms belonging to inclined fringes and homogeneous light.
In the endeavor to reproduce these fringes with the sodium arc, I failed
after long trials. The reason may be sought in the flicker of the arc, whereby
the beam passes from one side to the other of the edge of the prism P, but it
is probably due to the inadmissibility of a wide slit.
53. The same. Crossed rays. — The present method, using four mirrors,
has, nevertheless, the advantage of admitting the use of either parallel or
crossed rays. Inasmuch as these rays are white until they leave the grating,
the method is interesting. On being tested it showed the same peculiarities
as the preceding. The crossed rays (cc', figure 72) are more nearly normal
to the mirrors M and N; nevertheless the range within which the interfer-
104
THE INTERFEROMETRY OF
ences are visible is not above 2 mm. of displacement of M. The fringes may,
as usual, be made as large as possible, by first superposing the two illuminated
strips on the grating G (by fore-and-aft motion) and then rotating the grating
on an axis normal to its face until the best conditions appear. Both spectra
are very bright, but liable to be in different focal planes from inadequate
planeness of the reflecting system. If work of precision is aimed at, this con-
dition is of foremost importance.
54. Another method. — If the opportunity of using crossed pencils of white
light is to be dispensed with, the prism method may be simplified, as shown
in figure 74. Here P is a prism with silvered sides and a prism angle of less
than 30°. It receives horizontal white rays L from a collimator, which, after
reflection from the opaque mir-
rors M and N, impinge on the
grating G, plane or concave, and
are observed at T by a telescope
or lens.
If <p is the prism angle and 6
the angle of diffraction, it is
easily seen that the angle be-
tween the rays reflected at M
or N is
d=6-<p
Hence, if P is a 30° prism, the
observations can be made only
in the second-order spectra. If cJlf
observations in the first order
are desired because of the greater illumination, <^ must be less than 20°, as a
rule, for a grating of about 15,000 lines to the inch. The mirrors M and N
make an angle of a/ 2 = (<p-\-d)/2 with the line MN.
The first experiments were made with a 30° prism and second-order spectra
from a concave grating (.0=177X10-* cm.). Sunlight was used. The two
superposed spectra were magnificent, with abundance of light and high disper-
sion ; but the spectra were of unequal intensity and in different focal planes,
so much so that the images of the guiding horizontal thread of the spectra
could scarcely be seen together. This made the adjustment for coincident
longitudinal axes very difficult, and the interferences were not found until after
long trial. The reason for this is the probable concavity or convexity of one
or more of the reflecting surfaces. Another difficulty was the distance apart
of the mirrors M and N (roughly, 150 cm. for a distance of about 2 meters
from P to T), so that it was inconvenient to observe and actuate the mirror
micrometer at M. Further attempt at improvement was therefore abandoned.
This prism was now replaced by one of less angle than <p = 2o°, also well
silvered. In the first experiments the adjustment did not admit of a coinci-
dence of light, except near the C line of the red; but M andN" were now less
than oo cm. apart, while the distance between G and T was about no cm., and
REVERSED AND NON-REVERSED SPECTRA.
105
between G and P about 10 cm. In this case the focal planes were nearly iden-
tical and the interferences easily found in the red region between the two C
lines. They appeared as small red pearls, very vivid on limiting the lateral
extent of the pencil L to about 5 mm., but, to my astonishment, they very
soon vanished on displacing M in a direction normal to itself i or 2 mm.
55. Methods using prismatic dispersion. — The small range of displace-
ment available in the prismatic reflection methods induced me to devise
corresponding refraction methods, to see whether these would show any
advantage in this respect. Accordingly the interferometer (fig. 75) was in-
stalled and the fringes found without much difficulty. Here P is the symmet-
rical prism, receiving the collimated beam of incident white light on the faces
meeting at the obtuse edge and refracting them in relation to the smaller
prism angle <p. This must be less than 45°, for convenience in observation,
as otherwise the dispersed beams meeting the opaque mirrors M and N will
c/J-
76
be too far apart for manipulation, supposing, of course, that the distance
PM and PN are over a meter. I used an equilateral 90° prism for want of a
better. The spectra reflected from M and N respectively impinge on the
grating G, concave or plane, and are viewed at T with a lens or telescope.
In consequence of the large angle 9, second-order spectra were used, without
apparent disadvantage. The dispersion of P and G being summational, the
total is very large.
To return to the angles again, if <£ denotes the obtuse prism angle, and r the
angle of refraction, the angle of incidence is 90° — <£/2, or
(1) cos0/2=/i sin r
Again,
(2) sin i' — fj. cos (0/2 +r)
when i' is the angle of emergence. Hence
M2 — cos2 0/2— sin </>/2)
106 REVERSED AND NON-REVERSED SPECTRA.
If 0 = go°, then sin «'= (1/2) (\/2/x2— i — i). Thus if n= i. 55, then sin ^ = 0.47 5,
and £' = 28.4°. Now, since i+5= 6, the angle 6 will obviously have to be in
the second order of the spectra of the grating G.
Although the two spectra obtained in this way were highly dispersed and
very brilliant, the interference phenomenon itself was not much superior to
the case where reflection from the (silvered) faces of the prism was employed.
The fringes disappeared, in fact, for a displacement of i or 2 mm. of the mirror
M, showing the usual inflation of form just before vanishing. The details
also were of the same nature, the large arrow-shaped forms being obtained
when illuminated strips on the grating were superposed and the latter slightly
rotated until the maximal conditions appeared.
To increase the range, the angle 5 must be reduced, as far as practicable.
This is possible in the present method, since the points of intersection at a
and G may be made to all but coincide. Reflection from the mirrors M and
N would then be normal. To attain this end it will be necessary either to
have the grating constant or the prism angle $ predetermined, or to use rays
of suitable divergence at L.
56. Methods with paired prisms. — White light (fig. 76, L) from a collimator
is reflected in turn from the silvered sides of the sharp prism P, from the
opaque mirrors M and N, and from the silvered blunt prism P', as shown by
the component beams abc and a'b'c'. Thereafter the white beams are diffracted
by an Ives film grating G, with attached prism p, and observed in a telescope
at 7\ Interference, therefore, takes place in the focal plane of the telescope
and would not (for the case in fig. 76) occur in its absence. Very interesting
results were obtained with this apparatus. The spectra are non-reversed or
else (if slit and grating are rotated 90°) inverted. The work, however, is
still in progress and will be described elsewhere. I will merely add, in this
place, that the work with prisms is important, inasmuch as it shows the essen-
tial part played by the diffraction of the slit of the collimator, in its bearing
on the phenomena of the present report. It is the function of the prism P to
cleave the diffracted field which leaves the collimator. For this reason pencils
identical in source are found on both sides of P. The experiments thus fur-
nish the final link in the theory of the phenomena.
Furthermore, as the above results already show, the range of displace-
ment of either opaque mirror (M, N) within which interference fringes are
visible, increases in marked degree with the dispersion to which the white ray
is subjected on separation and before the resulting partial rays reach their
final recombination. These ranges increase from a fraction of a millimeter
to almost a centimeter, while the width of the strip of spectrum carrying
the interference fringes, caet. par., remains the same. This also has a fun-
damental bearing on the phenomenon and is under investigation. The ques-
tion at issue is whether increase of range of displacement results simply
from the geometry of the optic system, or whether wave-trains are actually
uniform throughout greater lengths, in proportion as they have been more
highly dispersed.
CHAPTER VIII.
THE LINEAR TYPE OF DISPLACEMENT INTERFEROMETERS.
57. Introductory. — This apparatus will be referred to in various places in
this book and presents certain interesting features. The incidence of the
grating is normal (I = R = o), and both component rays in their vertical pro-
jection lie strictly in the same plane. To make the horizontal projection also
collinear is not quite possible in practice, because the direct or unreflected
rays and the corresponding spectra would overlap with the spectra of the
interferometer. As the former are much more intense, the interference pat-
terns would scarcely be visible in the combination. To avoid this, the rays
diverge slightly (a few degrees, depending on the distance between grating
and opaque mirrors) in a vertical plane. But this is of no consequence, as the
horizontal projections only are used in the measurements. One may note,
in passing, that this avoidance of coincidence with undesirable spectra secured
by tilting the grating and the corresponding opaque mirror in the same
direction is, in general, one of the essentials of the adjustments.
The advantage of the linear displacement interferometer is this: that it
can be built on a rail and mounted along a wall or a pier. If the rail is tubular,
a current of water may be passed through it from the middle toward both
ends, to insure constancy of temperature.
58. Apparatus. — The apparatus was constructed as follows and gave good
results at once, showing strong interferences. The ellipses were, in fact,
oblate in the red, circular in the yellow, and prolate in the blues, but clear
throughout.
<l
nm
©^=W
77
Light enters from an arc lamp, A, or Nernst burner, or the sun, at the slit
5, and is collimated by the lens L. Then the parallel rays pass the grating G
with its ruled side toward L. From the grating the reflected beam returns
to the opaque mirror N, and is then reflected into the auxiliary or adjustment
telescope, T. The component beam transmitted at G is reflected from the
opaque mirror M, returned to the ruled side of G, and thus also reflected
into T coincidently with the other beam.
107
108 THE INTERFEROMETRY OF
Figure 77 shows that the entering undivided beam LG passes just
above the mirror M, and is reflected just below this from the top of N.
Similarly, the reunited beam GT passes just above M, but is reflected from
the top of M , the object being to make the vertical angle at G as small
as possible.
The mirrors M and N and the grating G are on adjustable bases, a, a', a",
each controlled by three leveling screws on a plane-dot-slot arrangement in
the tablets m, mf, m", the axis of rotation being horizontal and normal to the
diagram. The tablets, furthermore, may be revolved and raised or lowered
by the rods n, n', n", which are attached by ordinary clamps to the large,
tubular, horizontal rail, RR, in question, admitting of a circuit of water. The
latter is secured to the pier.
The angles of inclination of the figure are much exaggerated, since the dis-
tance MG = GN (nearly) is from one-half to several meters in extent.
The mirror M is on a Fraunhofer micrometer suggested at m. The bases,
a, a', a", are drawn to the tablets, m, m', m", by firm springs, preferably run-
ning into the tubes below them.
The axis of the adjustment telescope, T, lies in the plane of the figure and
serves the purpose of bringing the direct slit images into horizontal and vertical
coincidence. When this is done it may be removed, if desirable, as the ray
GT is not thereafter used. T should not be attached to the rail, but placed
on an independent table, or standard, so as not to be an integrant part of the
interferometer. The telescope, T (not shown), for the observation of the
interferences, should be independently mounted on the same table. This tele-
scope lies outside of the diagram, to the right or the left of it, to catch either
of the two diffraction spectra selected. It will be seen that these lie quite
above the direct diffraction spectra of the ray LGM. Otherwise, as this is
much more intense, it would completely wipe out the interference spectra
and their combination. The latter, when seen alone, are very brilliant, black
and colored patterns, running through the spectrum when the micrometer,
m, is manipulated. If the distance GN is large and the grating G, as usual,
slightly wedge-shaped, the superfluous rear reflection from G may be blotted
out at N by a small screen. It is easily recognized, as it is brown from
scattered light.
The installation is simple. The parts being adjusted nearly symmetrically,
the undivided ray from a wide slit is brought to the top of M by raising or
lowering the lamp. This should first be done roughly with the lens and slit
removed. N has at the same time been placed just below the beam, and this
passes through the middle part of G. The latter is then inclined by the adjust-
ment screws until the component beam GN strikes the top of N, symmetri-
cally. Next N is inclined and rotated (vertical axis) until the reflected beam
enters the telescope, T. Finally, M is inclined and rotated (vertical axis)
until the reflected rays MG and GT also enter the telescope, the final sharp
adjustment being made with a narrow slit and the eye at the telescope.
The mirror M must also have a fine vertical adjustment (not shown). If the
REVERSED AND NON-REVERSED SPECTRA. 109
distances NG (face toward the light) and MG are equal, the interferences
are then easily found by moving the mirror M on the micrometer toward
the grating.
As compared with the other non-linear interferometers used under like
conditions, the present instrument, even when mounted on a y^-inch gas-
pipe, RR, showed itself remarkably steady, so that rings could be observed
in spite of the tremors of the hill on which the laboratory is built.
59. Film=grating adjustment. Michelson's interferometer. — If the grat-
ing G is a film grating, like those in the market, with 14,000 lines to the inch,
it should be mounted smoothly on the unruled side, on a thick glass plate,
with Canada balsam, and without a cover plate for the ruled side. It is to be
adjusted with the glass side toward the source of light, so that the reflection
taken may be from this side only (see r, fig. 78). In the telescope, T, directed
toward the reflected beams, two slits (one for each component beam) only
appear, as the glass plate does not reflect on the side covered by the grating
(g in fig. 78). The slits placed in coincidence will then show the elliptic inter-
80 78
ferences in the diffracted beam D at the proper distances. With so large a
dispersion as the above, the ellipses are usually too large. They should then
be reduced in size by a compensator placed in the beam on the ruled side of
the grating; or, preferably, the grating may be mounted on a plate of glass
fully i cm. (or more) thick, as in figure 78. This thick plate has the additional
advantage of eliminating the stationary interferences due to the front and
rear faces of the grating. In case of thin glass plates (2 or 3 mm.), these
stationary interferences are very strong, coarse, vertical lines and exceed-
ingly annoying.
If the film grating is carefully mounted in this way, it is nearly as good as
a ruled grating. There is, however, one insuperable objection, inasmuch as
the ruled face, though it does not reflect sharply, does diffract, and this more
strongly than the other. Thus there are always 3 superposed spectra in
the telescope, the third coming from the film side only, whereas the other two
are produced by the rays coming coincidently from r on the unruled side of
the grating. Hence the velvety blackness of the interferences in case of the
ruled gratings can not be reproduced by the film grating, since the interfer-
ences are spread out on a colored ground. They are, however, quite strong
enough for all practical purposes, and the lines are sharply and symmetrically
110 THE INTERFEROMETRY OF
traced. A vertical wire 2 or 3 mm. thick, placed symmetrically in front of
the objective of the telescope, makes the interference relatively strong and
sharp, by blotting out the third spectrum partially; but it at the same time
diminishes the light available. A wide slit in front of the objective subserves
the same purposes better.
If the distance apart of the mirrors M and N and the grating G is large, it
is best to dispense with the rail RR altogether, and to mount the mirrors and
grating directly on the pier or wall. This has the additional advantage of a
large free space between M and G or G and N, so that spacious apparatus like
a fog-chamber may be independently mounted there. This was the case in
the optic experiments on the thermal coefficients of the refraction of air, etc.,
below, where the distance between MG and GN was nearly 2 meters. In such
a case, moreover, in addition to the usual three adjustment screws of the
mirror M at the micrometer, it is desirable to have two others bearing on the
rigid parts of the support, so that the final adjustment may be made elastically.
By devising a tetrahedral plan of bracing M, G, N, independent of each other,
using short rods and clamping all parts on relatively short stems, I eventually
obtained a mounting which was almost free from tremors, even amid the dis-
turbances of the surrounding laboratory. In figure 79 one of these mount-
ings is suggested: a and b are ^-inch gas-pipes (about a foot long), sunk into
the wall of the pier at their rear ends ; cd is a cross-rod of same size and material,
clamped in place, and supporting the grating (or a micrometer) G. The
screw h abutting against the wall gives the horizontal elastic adjustment.
The braces e and /, which may be adjusted by rotation (screw) abutting in g
at the wall, give the grating vertical elastic adjustment. Thus h, e, and /,
rotate G around vertical and horizontal axes, respectively.
60. Michelson's interferences. — If the collimator, SL, is removed and
replaced by a strong sodium flame provided with a condenser, Michelson's
interferences will appear at T when the instrument is in adjustment. It is
rather surprising that, even in case of a film grating adjusted as above, they
are well-defined circles covering the whole field of the telescope. If the col-
limator SL is retained and the sodium light introduced from the side by aid
of a reflecting mirror, placed between the grating G and the collimating lens
L, both interferences may be observed at the same time in corresponding
telescopes. The mirror introducing the homogeneous light should in such a
case be provided with a clear space (silver removed), through which the white
beam, SL, may pass without obstruction. In a vertical plane the interferences
have the same size and character at the sodium line. Horizontally the spec-
trum interferences vary with the dispersion.
If an apparatus constructed of gas-pipe is employed, however, it is far too
frail for the practical use of the Michelson interferences. Vibrations within
the apparatus are excited on merely touching it. For the purpose of displace-
ment interferometry, however, such an apparatus is quite adequate; for the
measurements are taken when the tremors have vanished.
REVERSED AND NON-REVERSED SPECTRA. Ill
61. Film grating. Another adjustment.— The supernumerary spectra may
be gotten rid of altogether by using the method shown in figure 80. Here the
impinging vertical sheet of white light, L, from the collimator, falls upon the
clear or unruled part p of the plate of the grating, the film extending out as
far as shown at G. If M and N are the opaque mirrors, the reflected rays a and
b passing G are additionally reflected into a' and b', and thence, after leaving
the grating, into c and d. As both of the latter pass through the film, both
produce spectra; but b' and e may be blotted out by a screen at the mirror
N. This leaves only d beyond the grating. Again, the transmitted ray from
L, after reflection at M, is again reflected into c and d1 ', which is made coinci-
dent with d. But c, being reflected from the unruled side, has no spectrum.
Thus the spectra due to the two rays d alone interfere.
Had the grating been reversed, caet. par., then the ray c would have pro-
duced the strongest spectrum, and superposed on the other two it would
have greatly diminished the clearness.
In the telescope, whereas the ray a' prolonged is white, the ray d' from M
and reflected from the film is strongly azure blue, due to regularly scattered
light. This blue image is apt to be less sharp, unless very flat parts of the
film are found. The two spectra, however, are good and the interferences
satisfactory. The sodium line is sufficiently indicated, though, like the blue
image, not quite sharp.
This method of using the unruled edge of the plate of the grating for reflec-
tion is, of course, equally applicable and advantageous in the case of the
ruled grating. Only the two interfering spectra and no diffused light are
present in the field of the telescope, and if sunlight is used the Fraunhofer
lines are beautifully sharp.
62. Equations. — The equations for Ne/e, for normal incidence I=R = o,
takes its simplest form as
(i) Nc/e = M - Xd/z/dX =A+ 3-S/X2, nearly
where Nc is the coordinate of the center of a given ellipse on the micrometer
M, for the thickness of glass grating e, index of refraction ju, and color of wave-
length X.
Hence if two different wave-lengths, X and X', are in question (5 refers to
differences),
M
8Ne being the displacement of the micrometer to pass the center of ellipses
from line X to line X'.
If n=A+B/\* and \d^/d\=-2B/\\ then
(3)
112 REVERSED AND NON-REVERSED SPECTRA.
from which B may be obtained without further measurements. If greater
approximation is necessary, so that two constants, B and C, enter the disper-
sion equation,
(4)
so that observations at three spectrum lines, X, X', X", would be necessary.
The amount of displacement corresponding to the thickness e of glass is,
at a given spectrum line X,
where 2J5/X2 is constant for all values of e, or
^
It is therefore not possible to obviate the term in B, determined as shown,
if /z is to be measured.
If equal distances are cut off at M and N, the interference pattern, of course,
remains stationary in the spectrum. It is interesting to inquire to what degree
this may be guaranteed. Equation (3) is available for the purpose, and, since
X and X' are nearly the same, X' — X = 5X and
6eB .,
Let 5X be the width of the sodium lines:
5X = 6Xicr8 cm. X = sgXicr6 cm. e = 0.68 cm.
data for the above grating and sodium light. Hence
6Xo.68X4-6Xio-HX6Xio-s
(59)'Xio-» - .SXio-o cm.
i.e., about a half of lo"4 cm. This would be equivalent to the space on a grating
with about 20,000 lines to the centimeter, or 50,000 to the inch. The ellipses
can not be set as closely as this, but the order of sensitiveness is within that
of a good micrometer.
It is interesting to inquire whether the sensitiveness will change markedly
for larger angles of incidence /. If n is the index of refraction, the largest
angle R obtainable at grazing incidence, 1 = 90°, would be sin R—I/H. It
may then be shown that
d\ X3
Putting (JL= 1.5 and the other data as above, where d\ = 6X io~s cm.,
_ i.34X46X.o- 9.aX.y"/(59)'X.o-"+4.S _ „ x IO, cm.
(59)3Xio-18 1. 12
The datum is of the same order as above, so that the sensitiveness changes
but very little for different angles of incidence. Thus there is no disadvan-
tage in using I = o.
CHAPTER IX.
THE USE OF COMPENSATORS, BOUNDED BY CURVED SURFACES, IN
DISPLACEMENT INTERFEROMETRY.
63. Introduction. — -The method of increasing the sensitiveness of the dis-
placement interferometer by increasing the dispersion of the grating readily
suggests itself, but unfortunately the interference pattern loses sharpness in
the same ratio and ultimately becomes too diffuse for practical purposes.
Similar sensitiveness is secured when the air-paths and the glass-paths of the
component beams of light are respectively identical, with the same inadequacy
in the huge mobile figures, for the purpose of adjustment. In fact, if for sim-
plicity we consider the incidence normal (I = R = o, linear interferometer),
the sensitiveness becomes
de/dn = \z/[2eD cos 0.
where 6 is the angle of diffraction for the wave-length X, e the thickness of the
plate of the grating, /* its index of refraction, D the grating space, n the order
of the fringe, and b, N, constants. Hence, other things being equal, dQ/dn
increases as D and e grow smaller, where e = o is obtained by a compensator
counteracting the thickness of the plate of the grating.
It occurred to me that the difficulty of diffuse interference patterns might
be overcome, in part, by the use of compensators with curved faces, when the
case would become similar to the conversion of the usual interference colors
of thin plates into Newton's rings. Naturally a cylindric lens with its elements
normal to the slit is chiefly in question, though an ordinary lens also presents
cases of interest, chiefly because of the easy conversion of elliptic into hyper-
bolic patterns, and the lens is more easily obtained.
Other methods were tried. For instance, on using a Fresnel biprism with
its blunt edge normal to the slit, two sets of interference patterns, one above
the other in the spectrum, are obtained. When the blunt edge is parallel to
the slit, either side of the prism gives its own interferences, but they can
not be made clearly visible at the same time. A doubly reflecting plate or a
thin sheet of mica covering one half of the beam will produce two intersecting
patterns, but these also are of little use for measurement.
64. Lens systems. — -If but a single compensator is to be used, i.e., compen-
sation in one of the component beams only, the lens in question must be of
very small focal power; otherwise the adjustment will be impossible, as the
two direct images of the slit will be in very different focal planes. Moreover,
the focal power should be variable. All this makes it necessary to use a
doublet, preferably consisting of lenses of the same focal power, respectively
convex and concave. If these lenses are themselves weak, say i meter in
focal distance, both slit images may easily be seen in the telescope and be
113
114
THE INTERFEROMETRY OF
sufficiently sharp for adjustment. If the lens first struck by light is convex
and the second concave, their focal distances f\ and/2, respectively, and their
distances apart D, the focal power i/F of the combination used is
since j\ =/2 =/. The position of the equivalent lens is d = DF/fi =/2 =/. D, d
are both measured from the second or concave lens to the convex lens, and D
would always be smaller than /. If the lens system is reversed, F remains
the same as before for the same D, the system being again convex, but d is
reversed. The equivalent lens again lies toward the convex side of the system.
In other words, the equivalent lens generally lies on the same side of the
doublet as the convex lens.
In the actual experiment, however, the rays go through the lens system
twice. In this case it is perhaps best to compute the distances directly. Of
the two adjustments, the one with the concave lens toward the grating and
the convex lens toward the mirror has much the greater range of focus relative
to the displacement D. Supposing the mirror appreciably in contact with a
convex lens, therefore, if & is its principal focal distance measured from the
concave lens, b-\-D = M its principal focal distance from the convex lens or
mirror,
i= 2 //2- !/(/:+£>) i
b I-D(2/fz-i/(fl+D)) fi
(2)
where f\ is the (numerical) focal distance of the concave and /"2 that of the
convex lens. If we now write
(3) 6 = 5(i-D(2//1-
equation (2) is easily converted into
h fl />/2
so that the usual value of the principal focal distance has been halved rela-
tively to the new position of the equivalent lens. If, as in the present case,
fl =/2 =/
b =
f f2-
f+D
The following table shows roughly the corresponding values of D and M in
centimeters :
D
M=C+D
2B
d
2
2450
2500
49
5
950
IOOO
47
10
455
500
45
15
292
333
4i
20
212
250
38
25
165
200
35
REVERSED AND NON-REVERSED SPECTRA.
115
As b is smaller than B by equation (3), the equivalent lens is on the side of the
convex lens and at a distance
behind the mirror, or
B-b=f(f+2D)/2(f+D)
behind the concave lens.
If the system is reversed, j\ and f% are to be replaced by — j\ and — /j, whereas
D remains positive. Thus the equations become successively
jc_ i/(/i-£0-2//2 __i
it \ i i
I _ 2 I ,
~
If /i =/«=/, then
b =
f f-
2D f-D
fl-\-2D*
K» h f f~2D
B -b-~J=D
2 —
Hence the equivalent lens has the same focal distance as before, but it is
now placed in front of the system, at a greater distance than it was formerly
behind it. Measured from the mirror (mirror distances, M) the data (in
millimeters) are roughly as follows :
D
2B
B-M
2
2500
-5i
5
IOOO
-52
10
500
-54
15
333
-56
20
250
-57
25
200
-58
The total displacement of the equivalent lens on reversal is about i meter,
falling off to 96 cm. in the extreme case. The image is larger if the convex
lens is nearer the grating and the concave lens nearer the mirror.
65. Effective thickness of the lenticular compensator. — The compensator
with curved faces may change the interference pattern in two ways; viz, by
changing the angle of incidence and refraction of the rays at the grating, and
by changing the path-difference of successive rays passing through it. Both
conditions are virtually the same, or at least occur simultaneously. If there
is but one compensator, as above, the two effects must be small, since the rays
reflected from each of the opaque mirrors, M and N, of the interferometer,
must eventually enter the telescope, to unite in two nearly identical images
of the slit. It was rather unexpected to observe that the interferences are
still obtained, even when the two slit images are quite appreciably different
in size, but they are then confined to a single plane, as will be shown in § 69.
116
THE INTERFEROMETRY OF
Since the beam of light coining out of the collimator and traversing the
grating is a vertical ribbon of light, several centimeters high vertically, but
very thin in comparison (a few millimeters) horizontally, it is relative to the
vertical plane that the marked effect must be expected. In figure 81, G is
the grating, cc the principal plane of the concave, cv that of the convex lens,
M the opaque mirror. If the beam consists merely of the axial pencil c, the
distorting effect due to the introduction of the lens doublet is slight for any
value of their distance apart, D. The two lenses are practically equivalent
to a plate. If a broad beam dd is in question and the rays retrace their path,
the same is still true. But if, on changing D, the rays do not retrace their
path, so that the equivalent lens is convergent or divergent, then the rays
after leaving M re-impinge on the grating at different angles than before and
the interference pattern is correspondingly changed, principally in its vertical
relations.
Thus it is the lens system which changes the obliquity of rays lying in a
vertical plane and passing through the grating, to the effect that the axial rays
may represent a case of either maximum or minimum path-difference. The
latter will be the case when the divergent pencil which usually traverses the
grating becomes convergent in consequence of a sufficiently large value of the
D of the lens system.
81
83
66. Observations largely with weak lenses and short interferometer. — The
film grating used (Wallace, 14,500 lines to the inch) was cemented with Canada
balsam to a thick piece of plate glass, so that the total thickness of plate at the
grating was 1,734 cm. This introduces a large excess of path in one of the
component beams; but it is generally necessary, if the stationary interferences,
due to the reflection at the two faces of the plate of the grating, are to be obvi-
ated and if the ellipses produced are to be reasonably large for adjustment (cf .
§ 69). The lens doublet was to be added on the same side as the glass speci-
fied, so that the excess of glass thickness on one side was further increased by
about 0.19 cm., on the average. Under these circumstances the ellipses were
strong, but (in view of the large dispersion) with inconveniently long horizon-
tal axes.
On inserting the doublet (convex and concave lens, each i meter in focal
distance) with its concave lens at the mirror and gradually increasing the
distance D by moving the convex lens toward the grating, a series of forms
REVERSED AND NON-REVERSED SPECTRA. 117
was obtained which passed from the initial horizontally long ellipse, through
circles, vertically long ellipses, vertical lines, into hyperbolic forms of increas-
ing eccentricity, as recorded in figure 82.
On reversing the system, keeping the convex lens fixed near the mirror and
increasing the distance D by moving the other lens toward the grating, the
original ellipse usually flattened out further, as shown in figure 83. Moving
the lenses sideways parallel to themselves had no definite effect; moving
them fore and aft together (D constant) produced results similar to the above.
The vertical lines of figure 82 are liable to be sinuous or to resemble the grain
of wood around a knot. In case of figure 82, as the equivalent lens lies in
front of the mirror, the rays reaching the grating are thus necessarily converg-
ing. In figure 83 the equivalent lens lies behind the mirror, so that the rays
at the grating are more convergent. Both positions furnish essentially
convergent rays.
If corresponding to figure 82, the convex lens is kept fixed near the grating
and the concave lens gradually moved up to it, the order of forms is reversed,
but not quite completely. They usually terminate in long, vertical ellipses,
before reaching which the wood-grained forms are sometimes passed. The
same is similarly true for the case of figure 83.
With cylindrical lenses (respectively convex and concave, each i meter in
focal distance) very little effect was observed when the axes of the cylinders
were parallel to the slit. With the axes perpendicular to the slit, the effects of
spherical lenses were virtually reproduced, except that the central fields
partook of a more rectangular character.
To carry out the purposes of the present paper with strong lenses, respec-
tively convex and concave, the vertical sheet of light from the slit must be
diverged into a wedge by the concave lens and then collimated by the convex
lens. The mirror, normal to the rays, reflects them, so that they retrace their
path and become a sheet of light before the final reflection and diffraction at
the grating. The following experiments were made with strong lenses:
At first lenses of double the preceding focal power, /= ± 50 cm., were tried,
but with no essential difference in the results. Thereupon strong lenses
of focal distances /:= —73 cm. and /2= 13.1 cm. were used together, the
convex lens being, as usual, near the mirror. For .0 = 7.5 cm., about, these
gave fairly clear images of the slit and it was easy to find the ellipses, which
were now very eccentric, almost spindle-shaped in form. They could be
obtained strong and clear without difficulty, and the nearly horizontal lines
filled the whole spectrum. Reversal of lenses practically failed to give results,
the rays after reflection being too divergent.
On the large interferometer, where the distances between mirror and grating
are nearly 2 meters, adjustment was more difficult and the result (if parallel
rays are retained) less satisfactory, because the slit images are not in focus at
the same time. This is particularly the case when the convex lens is nearest
the mirror and the concave lens toward the grating. Thus when/= ± 100 cm.
and D = 1 5 cm., the modified slit image may be twice as large as the other and
118 THE INTERFEROMETRY OF
the interferences in the principal focal plane of the telescope are only just seen.
At D = 5 cm., however, the results are acceptable. When the concave lens is
nearest to the mirror and the convex lens toward the grating, the modified slit
image is smaller than the other. Adjustment is then easier and the usual
elliptic and hyperbolic forms may be observed without trouble. In both cases
the flickering of the arc lamp used passes the rays through different parts of
the lenses relatively to the center, and the adjustment is thus easily destroyed.
If the spectra from M and N, however, are observed, not in the principal
focal plane but in advance of it (toward the eye), interferences of great interest
will be observed, to be discussed in § 69.
67. Remarks. — A few explanatory observations may here be inserted. The
occurrence of the elliptic or oval and the hyperbolic type of fringes may be
most easily exhibited by laying off the order of the fringe in terms of the dis-
tance (in arbitrary units) above and below the center of the image of the slit.
If we call the latter y and consider the allied colors of thin plates, for instance,
n = 2en cos r/X or more generally n = (0ju A)/ (y, 0
(where e is the thickness of the plate, /* its index of refraction, X the wave-
length of light in case of a dark locus of the order n) is to be expressed in terms
of y, which itself determines e cos r, r being the angle of refraction in the plate
of the grating. The phenomenon will thus be coarser for red light than for
violet light, since n decreases when X increases, and any two curves, r and v,
figure 84, may be assumed as the loci of the equation in question. If, now,
horizontal lines be drawn for n=i, 2, 3, etc., they will determine the number
of dark bands in the spectrum for any value of y.
If the central ray is also a line of symmetry and intersects the grating nor-
mally, it must correspond to a maximum or a minimum of n. These conditions
are shown in the diagram at M, where the maximum number of bands occurs,
and at m, where the reverse is true. The question is thus referred to two sets
of loci, rr' and w', or r'r" and v'v", etc. In the former case e cos r varies with y
in the same sense as n/\ ; in the latter in the opposite sense and is preponder-
ating in amount. Both may vary at the same rates in the transitional case, in
which, therefore, the two curves r and v are at the same distance apart for all
values of y.
Suppose, furthermore, that the same phenomenon is exhibited in terms of
wave-length X, as in the lower part of the diagram, the spectrum being now
equally wide for all values of y, while at any given y the upper diagram still
shows the number of dark points (bands) between r and v. If now, we suppose
that under any conditions these dark points are grouped symmetrically with
reference to any given color (which is probable, for a maximum or a minimum
of any value of y will be so for all values), and that the successive dark points
have been connected by a curve, the interference pattern will be of the elliptic
type in case of aa', a" a'", and of the hyperbolic in the case of a' a".
The other features of the phenomenon are secondary and therefore left out
REVERSED AND NON-REVERSED SPECTRA.
119
of the diagram. Thus, for instance, the distance apart of the bands shrinks
from red to violet, and the ovals, etc., are only appreciably symmetric, because
they occupy so small a part of the spectrum. The horizontal distribution of
dark bands around the center is determined by variations e cos r and is not
linear. Whether the long axes of the ellipses are horizontal or vertical depends
upon the slope of the lines r and v. Maxima and minima will not, as a rule,
occur close together, though in certain wood-grain-shaped patterns this seems
to be the case.
In conclusion, therefore, the main feature in modifying the type of inter-
ference pattern is the varying thickness of the compensator. For oval types
the preponderating lens is convex; for the hyperbolic type it is concave.
Neither of these lenses is here appreciably affected in modifying the horizontal
distribution of path-difference, because the dispersion of the grating requires
a horizontally parallel system of rays.
68. Observation with lens systems on both sides. — The method shown in
plan in figure 85 (L and L' convex lenses, G grating, M and N mirrors, telescope
at T} was tested. The outcome can not at once be foreseen, since the focal
distances for different colors is different and since slight displacements of
either lens must greatly modify the interference pattern. The latter, however,
as obtained in every case, proved to be exceedingly fine lines, tipping in the
120
THE INTERFEROMETRY OF
usual way with the motion of the micrometer and indicating a center of ellipses
very distant in the field of the spectrum. In other words, the interference
pattern is no longer automatically centered and is therefore useless.
A modification of this plan is the method shown in figure 86 (horizontal
section), where B is the beam from the collimator, L, L',L",L"', four condens-
ing lenses of the same power (/= 50 cm.), G the grating, Mand N opaque plane
mirrors, T the telescope. In all the above cases the horizontal rays from the
collimator traverse the grating in parallel and eventually condense to a single
point in the field of the telescope. The same is true of all rays having the same
angle of altitude. These rays, therefore, act as a whole, since they pass through
the plate of the grating at the same angle of incidence. On the other hand,
85
86
relative to a vertical plane, the rays traverse the grating at different angles,
each angle corresponding to a horizontal strip of the spectrum. It is by the
easy modification of this obliquity that the curved compensator becomes effec-
tive. In figure 86 the rays are also oblique relative to a horizontal plane; but
the result, unfortunately, is not available, since each of these oblique rays must
have its own complete spectrum. Consequently the diffracted pencil will con-
sist of an infinite number of overlapping spectra, the extreme cases lying within
the same angle a shown in the figure. A large telescopic objective would then
reunite these spectra into a white image of the slit, while a small objective will
show colored slit images, passing from impure red to impure violet. Naturally,
the interferences will also overlap, and therefore vanish.
69. Telescopic interferences. — If interference patterns of small angular
extent are to be obtained, it is essential that the rate at which obliquity
increases from ray to ray be made as large as practicable. Probably, therefore,
REVERSED AND NON-REVERSED SPECTRA. 121
an opportunity for realizing these conditions will be found within the telescope ;
i.e., after the rays pass the objective. The endeavor would therefore be
directed to bringing two spectra, focussed in two planes, one of which is behind
the other and consequently of different sizes, both vertically and horizontally,
to eventual interference.
The experiment was made on the long interferometer (fig. 87), the distances
between mirror M and grating G and from the latter to the mirror N being
nearly 2 meters each. C is the lenticular compensator, consisting of two lenses,
respectively concave and convex, each having the same focal distance, /= ± 50
cm. The distances apart, D, of the lenses may be varied. The glass plate C',
which is revolvable about the vertical, is thick enough to exactly counterbal-
ance, if necessary, the thickness of the glass plate of the grating and of the lens
system C. A sharp wedge sliding transversely may also be used. It is best to
replace C' by two plates of glass, one thick and the other thin, so that the lat-
ter may be removed.
The telescope directed along the axis R will therefore, in general, see two
white slit images, A and A' (fig.88), not both in focus at once, A' coming from
cA- n a'
a
"6f
C'
C
87 88
M being larger, A from N (parallel rays) smaller. The focal plane of A' will be
towards the grating as compared with A , and A' is larger than A , in proportion
as the distance apart of the lenses C is larger. Similarly, the two spectra
are observed along the diffraction axis, D, not in focus at once and of
different areas.
To obtain the interferences the slit image A must be placed anywhere within
A', and they will occur at the top of the spectrum if a and a' are vertically in
coincidence; in the middle if b and b' coincide, etc.
The plane of the new interferences is no longer the principal focal plane, con-
taining the Fraunhofer lines, but lies in front of it; i.e., towards the eye of the
observer and away from the grating. This distance, measured along D for the
given small telescope used, was fully i cm. The focal planes of the two spectra
are usually not so far apart. A' corresponds to a virtual object behind the
observer.
If the vertical plane in which the interferences lie be taken as the image, the
object would be situated about 3 meters beyond the objective of the telescope
used. This would place it 30 cm. in front of the mirror M or N, where there is
but a single beam in each case. In fact, the telescope may be brought quite
122
THE INTERFEROMETRY OF
up to the grating. Hence interference is produced in the telescope itself, where
rays are relatively very divergent, a condition which accounts for the smallness
of the interference pattern. This understanding of the case is tentatively
shown in figure 89, where 0 is the objective of the telescope, M the larger image
from the mirror with the lens compensator, and N the image from the other
mirror (parallel rays) . If the corresponding rays be drawn through the extrem-
ity of M and N, their fields of interference, F and F', would begin in the plane
I and /'. For axial rays it would be at i. Thus the locus as a whole would not
be a plane, and this seems to be the case. If the telescope moves toward the
grating, II' moves toward the right in the figure, as though the virtual object
beyond the grating were fixed in position. At all events, the problem is to find
the interference diagram of two symmetrical plane parallel spectra, of different
areas and placed at definite distances apart.
0
The appearance of the fringes is indicated in figure 90, where S is the height
of the spectrum, usually quite out of focus. There are many more lines than
could be drawn in the sketch. The ends a and a' seem to surround small ellipses,
but these are not quite closed on the outer edge. The center of symmetry
is at C. The demarcations are stronger and broader vertically if the distance
apart of the lenses C (fig. 87) is small; fainter, but nevertheless clear and nar-
rower, if this distance is large. Horizontally the fine lines thread the spectrum.
The best results were obtained when the lenses C are less than i cm. apart, the
middle band being about half as high as the spectrum. Two contiguous lenses
gave a design which nearly filled the spectrum vertically. For practical pur-
poses the lens compensator C is to be attached to the mirror M, just in front
of and moving with it. It makes little difference here whether the concave lens
or the convex lens of the doublet C is foremost.
If the micrometer M is moved, or if the telescope is slid to the right or left,
or forward, so as to take in other parts of the spectrum, the nearly closed lines
at a and a' become finer and finer crescent-shaped lines,
always open outward, till they pass beyond the range of
vision. The whole phenomenon remains on the same level
of the spectrum. On moving the telescope forward as far
as G (fig. 87), the ocular has to be drawn outward (towards
the eye) till it is fully 2 cm. beyond the position of the
principal focal plane. The whole spectrum is now seen
with the interferences from red to violet (no ellipses), but
having the same relative position as before. The central
horizontal band measures about one-fifth the height of the spectrum, while
the fine parallel horizontal lines extend to the upper and lower edges. The
REVERSED AND NON-REVERSED SPECTRA. 123
appearance is now curiously like a blunt wedge (fig. 91), with a band at b
nearest the eye, and the lines dd extending quite to the rear. This impression
is probably an illusion, due to the shading; the lines grow finer and are more
crowded toward the bottom and top of the spectrum. The illusion of a
reentrant wedge is not possible.
To use this interference pattern for measurement, the cross-hair is supposed
to pass through the region c (fig. 90) symmetrically. Very slight motion of the
micrometer mirror M then throws c either to the right or the left of the cross-
hair. In this case the lens doublet, C, is attached to the mirror and moves with
it, as stated. To obtain the extreme of sensitiveness, the path-difference of NG
and GM must be all but zero; i.e., the grating plate G and the lens doublet C
(fig. 87) must be all but compensated for equal air-distances by the compen-
sator C'. In this case of full compensation, the interference pattern, in the
absence of a doublet C, would be enormous and diffuse, seen preferably in the
principal plane of the telescope, but useless for measurement. The introduc-
tion of a lenticular compensator, balanced by a compensator in GN, transforms
the huge pattern into the small interference fringes in question, with the advan-
tage that the high mobility of the coarse design has been retained. In other
words, an index suitable for adjustment has been found, compatible with
extreme sensitiveness. In fact, it is difficult to place the micrometer mirror
M so that the region c (fig. 90) is exactly bisected. As the plane in which these
interferences are seen most distinctly is i cm. or more anterior to the principal
focal plane, the Fraunhofer lines are unfortunately blurred and a cross-hair is
needed as a line of reference.
I may in conclusion refer to a similar series of experiments now in prog-
ress, in which the compensators placed in the M and N pencils (fig. 87,
C, C'}, instead of being of different shapes as above, are plates of different
kinds of glass (crown and flint, for instance). Here the successive differ-
ences of dispersive power, from wave-length to wave-length, produce effects
closely resembling those discussed, with the advantage that difficulties
inherent in the curved system are avoided.
CHAPTER X.
THE DISPERSION OF AIR.
70. Introduction. — In view of the long-armed interferometer available, it
seemed interesting to test the refraction of air at different wave-lengths, X. An
iron tube of inch gas-pipe, 138 cm. long, was therefore placed in one or the other
of the component beams. The tube was closed at both ends by glass plates,
about one-eighth of an inch thick, kept in place with resinous cement. A
lateral tube communicated with an air-pump and drying train, so that the
tube could be alternately exhausted and refilled with air. By using sun-
light, the different lines of the spectrum were obtained with sufficient clear-
ness, and the method consisted in finding the reading of the micrometer for
successive Fraunhofer lines, both for the case of a plenum of air and for a
vacuum. If AN is the (monochromatic) displacement of micrometer corre-
sponding to the latter difference of pressure, /j. being the index of refraction
of air, e the thickness,
To determine jux, we must know a/*x/ d\. It has been omitted above, because
it enters differentially and because of its small value. It appears as a con-
stant decrement of AATX, as X is constant and d/zx/ dX is negative. In the present
case, where /j. is actually to be measured, cfyix/ dX enters directly and is essen-
tial; but it follows from any two experiments when p is found for different
colors.
TABLE 8. — Values of B. Inch iron gas-pipe, 138.0 cm. long. D line.
t
Bar
P
iozAN
22.0°
75-o
38.0
76.20!
20° /
22.3°
74-5
37-9
75-831
37-7
20° /
37-6
37-9
37-7
37-6
37-6
37-7
37-5
37-7
Mean
•37 fin
O/ •":/
71. Observations with arc lamp. — In table 8 results are given as obtained
with the electric arc, in which the sodium line usually appears with sufficient
distinctness in the spectrum to be available as a line of reference for measure-
124
REVERSED AND NON-REVERSED SPECTRA. 125
ment. Disregarding earlier results, the following are mean values of the ten
independent data for AN (each comprising a reading for vacuum and for
plenum) :
= 58.gcm. t = zz. 3° £ = 74.5011. 10^^ = 37.69 cm. /=i3S.ocm.
Thus
where ^0 refers to normal pressure and absolute temperature (r). If ju0 is
given for the D line, dp/ d\ is determinable. It will be sufficient for the present
purposes to put ^0 = .4+5/X2, or \.dn/d\= — 2.5/X2
5 referring to r and £. Mascart's * value for /z0— i (agreeing with Fabry's)
is 10^X292.7, whence
2?=io-14Xi.34 at r and p
If the value B be computed from Mascart's observations between C and
£, D and F, respectively,
so that the mean value 10 "5=1.65 maY be taken. Since the last decimals
of M are in question, it will not be correct to more than 5 to 10 per cent.
The value found above (io142? = i.34) is therefore somewhat too small.
True, since from equation (3)
(4)
an error of lo"4 cm. in AN is an error of 0.13 X icr14 or 10 per cent in B. Very
close agreement can not therefore be expected in either result. One is tempted
to refer the present low value of B to flexure of the glass end plates of the
tube, which, when the tube is exhausted, become slightly saucer- shaped and
introduce a sharp concentric wedge of glass into the component beam, whereby
the interference pattern is changed, probably in the direction of smaller
values, as found. But the direct experiments below do not show this. In
any case, the measurement of B lies at the limits of the method. An advan-
tage may possibly be secured by using two identical tubes, one in each com-
ponent beam, the tubes to be exhausted alternately. The sensitiveness
would then be doubled.
72. Observations with sunlight. Single tube. — These observations are
given in table 9, the exhaustion throughout being 75 cm. and the temperature
about 1 6°. In the first set sunlight was used without a condensing lens; in
* See excellent summary in Landolt and Boernstein's Tables, 1905, p. 214.
126
THE INTERFEROMETRY OF
the second set the sun was focussed with a weak lens (0.5 meter in focus) at
the point formerly occupied by the electric arc. The spectrum (particularly
in the second case) was brilliant and the lines clear. The focus of sunlight is to
be placed just outside the focus of the collimator lens, in order that a nearly
linear pencil may be available to penetrate the long refraction tube twice.
The distance of the collimator lens to the grating was about 2 meters. The
spectrum is then a bright band in the telescope, the width being limited by the
height of the ruled part of the grating. The strip of white light on the grating
should not be more than a few millimeters wide. It must therefore be nar-
rowed by an opaque screen (wide slit of the given width) in the path of the
beam (see fig. 92 below).
TABLE 9. — Dispersion of air. Tube /= 138.0 cm. Bar. 77.25 cm. at 19.5°. P=75-O cm
Line.
Temp.
io3AN
io+145
C
D
b
16.0°
II
II
38.51
39- \
394J
i-5
C
D
b
16.0°
ii
u
38.51
38.9
39-3J
1.4
Improved seeing, weak condensing lens.
C
D
16.4°
1C
38.8!
39-2J
1.51
C
D
b
F
16.4°
it
It
«
38.9]
39-1
?39-3|
40. 1 J
1.65
1.4
C
F
16.4°
ii
38.81
40. 1 /
1.61
The equation for B in this case, if the symbol 8 refers to differences for two
given values of X, is
_ 8&N/e 76 r
*i P 273
if the value of B is to hold for normal conditions.
The data are shown in table 9, series i and 2 being obtained without con-
densing lens. These are inferior, as regards definition of lines, to the subse-
quent set, in which condensed sunlight was used. In all cases there is some-
times an irregularity (marked by ? in table 9) in which the observation is
obviously discordant, but the reason could not be found. Possibly values of
T and p taken were not the actual values. The data for BX io14 given in the
table are mean values. Some of these are low. Later values, where the F
line is included, come out larger, the range being from 1.3 to 1.8 or 1.5, on
the average. It is desirable to use the whole of the available range of the
spectrum (sufficiently luminous from C to F) to obtain an acceptable value
of the coefficient B and additionally to improve the method by using two
REVERSED AND NON-REVERSED SPECTRA.
127
identical tubes, alternately exhausted as suggested above. The attempted B
measurement is at the limits of the method, as has already been instanced
in the discussion of errors in the preceding paragraph, and it is not to be con-
cluded that data which happen to agree with Mascart's result from a correct
application of the present method. In fact, there is no reason for excluding the
exceptional values, and the present results are to be regarded as preliminary.
73. Two (differential) refraction tubes* — In the following experiments two
identical iron tubes (138 cm. long, of inch gas-pipe) were installed, one being
placed in each of the component beams of light, which subsequently interfered,
and the tubes were exhausted alternately. There are apparently three advan-
tages in this arrangement. In the first place, the sensitiveness is doubled; in
the second, the flexure of glass plate should be the same at each tube, in each
experiment, and thus fail to disturb the interference pattern. Furthermore,
by using the tubes in parallel (i.e., exhausting both at the same time), any
irregularity of flexure effect, etc., should be determinable, as the air in both
tubes will be identically circumstanced. Finally, the air being inclosed in a
thick metallic envelope at both beams is not subject to incidental disturb-
ances. An unexpected difficulty, however, was encountered ; for there is reflec-
tion of direct spectra from the eight glass surfaces, and this must be specially
met. The direct spectrum is easily eliminated by inclining the grating until
the reflected interference spectra are at a different level; but reflections of
this spectrum are not so easily dealt with. Fortunately they are weak. Even
so, they are very annoying, as they overlap the interference pattern and dull
it. They could be eliminated by attaching the glass plates obliquely to the
axis of the pipes, but this remedy was not thought of at the outset.
Figure 92 is a diagram of the disposition of the parts of the apparatus.
L is the beam of white sunlight from the collimator limited laterally by the
wide slit (i inch) 5. G is the grating, T and T' the two refraction tubes,
M (micrometer) and N the opaque mirrors, R the refracted and D the dif-
fracted (spectrum) beam of light. C is virtually a four-way stopcock (or two
3 -way glass stopcocks) leading respectively to the exhaust pump E and dry
128
THE INTERFEROMETRY OF
air supply A, from the tubulures e and e' of both refraction tubes T and T'.
These are therefore alternately exhausted.
Preliminary results are given in table 10, the arc lamp with its sodium line
being used in the absence of sunlight. It will be seen that A2V, apart from
temperature (which is here higher than above), has been doubled. The
deflections were symmetrical within 0.15 X icr3 cm.
TABLE 10. — Dispersion of air. Differential tubes, each 138 cm. long. D line in electric arc.
£=74 cm.
Barometer.
Temp.
io3AW
(//o-i)Xio8
£oXio14
75.93 cm., 22°
2i-5°
754
291.6
1-47
75-3
75-4
76.02 cm., 19.5°
18.0°
76.3
291.8
1.49
76.4
76.4
(Single lube)
23.0°
37-8
293.6 1. 80
The values of B found in the first two series of this table, if the standard
value of /x0— i =0.0002927 is assumed, is somewhat small, but as near to the
true values as may be expected. Again, if 5Xio14=i.6s is assumed, the
At0— i values given in the table are similarly small, being 0.3 per cent short of
standard values. A single-tube experiment made for comparison (series 3),
similarly, came out too large in each case. It follows from this that p and t
observations are not sufficiently guaranteed. It is hardly probable, however,
that with a micrometer reading to io~3 cm. and estimated to lo"4 cm. (vernier)
the precision can be much enhanced ; for since
D_\2AAf 76 T
-—(jie—i)
dB0
273 P
76 r
dB0
dp 2 e 273 pz
or at the C line and D lines, respectively,
8P= i cm. 5£ = 0<°-8*IOT4
\o-7Xio-1
2 e 273 p
6r=ic
0.2 Xio-14
0.2 XlO-14
Now, unless the measurement can be made in terms of rings, it is difficult
to detect a few millimeters of pressure difference by displacement only.
The interesting question now occurs whether the two tubes, if identical
for a plenum of air, remain identical (no shift of the interference pattern)
throughout all the stages of identical exhaustion. On trial, nothing could be
detected, the fringes remaining stationary during the whole period of exhaus-
tion, or during the influx of air following a high vacuum. Hence there is no
perceptible difference effect of flexure of the glass ends, and the ultimate
question of accuracy depends on the measurement of r and p. To eliminate
the possible effect of flexure, an air column of negligible length, in which the
glass effect only is present, will have to be tested; otherwise there is no possi-
bility of separating the air and glass effect.
REVERSED AND NON-REVERSED SPECTRA.
129
74. Differential and single refraction tubes. Sunlight. — The direct experi-
ments for the coefficient B were now resumed and conducted with sunlight,
with the results given in table 1 1 . The first two series were made with the
two identical tubes specified, exhausted alternately, one tube containing a
plenum of air, while the other was nearly empty in each experiment. The C
and F lines alone were used for measurement. In spite of the large displace-
ment (AN = 0.07 6 to 0.078 cm.), the results were not as satisfactory as was
expected, owing to the fact that sharpness of vision is made difficult by the
stray reflected spectra to which reference has already been made. But the
data for B obtained with one exception (No. 2 in the first series) are consistent
and reasonably good. In every other respect the work was satisfactory and
could have been improved by using oblique cover-glasses. The values of B
obtained are therefore disconcerting.
TABLE n. — Dispersion of air. First and second series, Differential Tubes, each 138.0 cm.
long. C and F lines. Third and fourth series, Single Tube, good adjustment.
Barom. p
Line
io3&N
IO"B
Temp.
Barom. p
Line
io3A7V
io»B
Temp.
cm.
77.05! 74.0
22° )
74.0
C
F
C
F
cm.
76.4
78.0
76.1
78.2
I.O
i-3
°C
17.4
17.4
cm.
77-22! 74.0
21°)
74.0
C
F
C
F
cm.
384
39-2
38.2
•zq.i
0.99
i. ii
°C
18.0
18.0
74-0
74-0
C
F
C
F
76-3
78.0
76.2
78.0
i.i
i.i
17.4
17.4
74.0
74.0
C
F
C
F
38.3
39-3
38-2
39-o
i.iS
i. 06
18.0
18.0
76.27! 74.0
23° /
74.0
74.0
C
F
C
F
C
F
75-8
774
754
77.2
754
77.1
I.O
i.i
i.i
21.4
21.4
21.4
77.22! 76.0
21° 1
76.0
76.0
C
F
C
F
C
F
39-3
40.1
39-2
40.0
39-0
4.0.0
0.99
•99
•99
18.1
is.i
18.2
76.0
C
F
39-2
40.0
•99
18.2
I then went back to the single-tube experiments (in series 3 and 4), and
these are the smoothest results obtained. The C and F lines were used as
before. In the last series, for instance, the micrometer reading is the same
to io"4 cm. throughout. In spite of this satisfactory behavior, the value of
B obtained is again of the same low order, all the data, both for double and
single tubes, being consistent throughout in this respect. Series 3 and 4 agree,
although p is changed from 74 cm. to 76 cm. of mercury.
In table 12 I have summarized the data in comparison with the standard
results, computing /J.Q—I and B0, for each of the cases, reducing all values to
o°C. and 76 cm. of mercury. The difference of no— i for the F and C lines,
which is 3.2 X io"6 for the standard data, is but 2.3 X io~6 on the average in the
present results. Similarly, the mean values of the latter are 10^X4.1 and
10^X3.2 larger for the C and F lines, respectively, than the standard results.
130
THE INTERFEROMETRY OF
These conditions are particularly puzzling, since in §73, with the use of the
arc lamp, both results were nearly normal. I therefore endeavored to detect
the causes for this difference of behavior.
TABLE 12. — Summary of Table n. "St." refers to standard data. A=(HQ— i).
Line, etc.
XX i o6
St.
ios A
Series.
Ser.
(i) to (4)
50Xio14
(MO- i) St.
(O
i&A
(2)
ioeA
(3)
io6A
(4)
io*A
S0Xio14C.
B<>X io14F.
C
65-63
291.8
295.0
-3-2
297-3
-2.3
2.3
296.0
-4.2
298.0
-3-3
2-3
296.3
-4-5
298.7
-3-7
2.4
296.5
-4-7
298.6
-3-6
2.1
1.18
1.20
1.28
1. 08
1.85
2-37
2.22
2-55
1.33
1.56
1.70
1.50
io« Diff. from St.
F
48.61
295.0
io9Diff. from St.
I o8 Diff. FandC.
3-2
The standard of length was first compared with a normal meter, showing
that the i = 138.0 cm. for the M tube should be replaced by 137.75 cm. and
for the N tube by 137.59 cm. As this correction affects all the results, MO— i
and B, in the same ratio, it contributes nothing to modify the discrepancy
in question. The correctness of the micrometer screw was assumed, as it was
of good manufacture.
The test next was made to see if the lines taken as C and F actually had
the accepted wave-lengths. A revolvable arm, with its axis at the grating
and 125.5 cm- l°ng> was therefore installed for the direct measurement of the
diffraction of the grating. The results obtained for the wave-lengths of the
lines taken showed that no mistake had been made in their selection.
To endeavor to obtain further evidence, the values of B were computed for
the mean data of table n, by using the standard values for MO— i in case of
the C and F lines and the AN/e given by the observations. The results so
obtained are given in the last columns of table 12, for each of the four series
and the C and F lines at normal temperature and pressure. The mean results
are thus —
#oXio14=i.i9 from observations with sunlight directly
between C and F lines.
.Z?oXio14 = 2.25 from standard MO— i = .0002918 at C line.
50Xio14=i.52 from standard MO— i = .0002950 at F line.
The B values of table 10 show a march to be referred to temperature and
pressure. So the present unsatisfactory differences are probably pressure-
temperature effects beyond the discrimination of the method.
One reason for this discrepancy which suggests itself is the possible distor-
tion of the glass plates at the end of the exhaust tubes during the exhaustion.
There may be a residual temperature effect due to the heating of the air by
the beam which passes twice through it, above the indicated temperature of
the surrounding tube of iron. But as MO— i is already too large compared
with standard values, this would make the case worse. Similarly, a larger
thermal coefficient than the normal value (1/273) would further increase
REVERSED AND NON-REVERSED SPECTRA.
131
the data for /IQ— i. For the case of the F lines, the B values found by com-
parison with the standard juo — i (last column of table 12) might be taken as
correct within the error of method. Nothing, however, has been found to
account for the correspondingly large values of BO for the C line.
75. Distortion of glass absent. — To test the effect of possible distortion of
the end plates of the tube, a shallow cell was constructed but 0.8 cm. deep,
closed by plates of the same glass. The diameter of the tube was identical
with that of the long refracting tubes. Tests made with the electric arc and
sodium line gave the mean values
Plenum io3A7V=5.47; 6.6 cm.
Vacuum 5-25; 6.4 cm.
Thus the effect of exhaustion is 0.0002 1 cm. The long tube gave, on the aver-
age, 0.039 cm. for 138 cm. of length. Hence the air effect should be
0.039X0.8/138 = 0.00022
which is practically identical with the value found. Hence there is no per-
ceptible distortion referable to the glass plates.
TABLE 13. — Dispersion of air. D and F lines. Single Tube, length 138 cm. Barom., 75.40
at 28°. £=74 cm. Temp. 20°. St. refers to standard data. A = ^— i.
T • ,
BA.VO
St.
Mean
St. /
lg—1
Line.
lO^A/V
IOh
e
I014r>o
Line, etc.
XX i o6
io6.4
io6 A
10UB0,F
\OuBo,D
F
D
cm.
39-4
38.2
312.8
101.8
2.24
F
io6 Diff. from St.
cm.
48.61
295.0
304.0
— Q.O
2.IO
19.2
F
D
F
39-o
38.3
1Q.O
310.1
304.6
aio.S
1.38
1.4.0
D
io6 Diff. from St.
58.93
292.7
299.4
-6.7
I.78
2.06
D
•58.-?
104.6
i o6 Diff. FandD
2.1
4.6
1. 81
2.O6
*
76. Further observations with sunlight. — In the absence of other than
inferential reasons to account for the difficulties met with, a final series of
observations was made between the D and F lines and a single tube, with
the results given in table 13. The mean value of B0 found directly, viz, 1.70
Xicr14, would be admissible; but the corresponding values of Mo— i as com-
pared with the standard values are again too large and worse than above.
The same is true of the values of B0 found by comparison of AN/e with the
standard values of (JLO— i, and their coefficients come out differently for the
C and F lines. In fact, the discrepancy of MO— i is now about 3 per cent,
whereas observations for AN/e should not be in error more than (2 X 1/400 =
0.0025) 0.5 per cent. There is thus something variable at the limit of appli-
cation of the present method which has persistently escaped detection. I
have thought that a distortion associated with the form of the interference
pattern in passing from C to the F line may be in question, as the discrepancy
132 REVERSED AND NON-REVERSED SPECTRA.
varies in different, otherwise satisfactory experiments; or the failure to com-
pletely exhaust the tube may leave a small error which becomes appreciable
in B.
77. Conclusion. — If allowance is made for the fact that AN at the
micrometer is measured for air, at barometric pressure p' and absolute tem-
perature T, the equation for ju0 — i at normal conditions would be
_ _AAT 76 T i 2B0
~e~~$^n i-p'&N/ep "X*"
where the correction factor i—ANp'/pe would not appreciably modify the
results.
It is difficult to see, therefore, why the promising results of §4, which are
quite as near the standard data as the method warrants, did not bear consis-
tent fruit in the sequel. The direct values of B0X io14 are usually too small,
sometimes too large, and range from i to 2. On the other hand, MO— i usually
comes out too large, whereas it should be correct to a few tenths percentage.
None of the causes examined, temperature, pressure, thermal coefficient,
flexure of glass, etc., quite account for such a result. If BoX io14 is computed
from standard results for HQ—I and observed at different spectrum lines, the
data are nearly correct for some lines, but too large for other lines, so that a
single constant does not reduce the series. It does not seem probable, however,
that equation (i) is inadequate; for the results obtained with equal care at dif-
ferent times for the same HQ— i or B0 are not in accord. The discrepancies, in
other words, are not persistent in value and are therefore due to some inci-
dental cause which has not been detected. It has seemed to me that the change
in shape of the interference pattern on passing from red to violet, which in
case of ordinary glass mirrors is marked, may be responsible for some of the
difficulties encountered. This pattern, which for optically flat surfaces would
remain elliptical, becomes more and more irregular as the distances, e, of the
mirror and grating are increased. The distorted image shrinks laterally from
red to violet fully one-half, so that it is not certain that the center of figure
is actually a fiducial point. The question, however, would have to be tested.
CHAPTER XI.
THE CHANGE OF THE REFRACTION OF AIR WITH TEMPERATURE.
78. Apparatus. — In the earlier report (Carnegie Inst. Wash. Pub. 149, III,
Chap. 15, p. 223) I began some experiments on the change of the refrac-
tive index of air with rise of temperature. The question is interesting, inas-
much as the temperature coefficient has not in most investigations been
found identical with the coefficient of expansion of air, as Lorentz had obtained
it and as would otherwise be anticipated; but a value, over 3 per cent larger,
first put forward by Mascart, seems preferable. My earlier work was left
unfinished, however, because the design of the apparatus, in which the refrac-
tion tube was heated in an independent annular steam-bath, was unsatis-
factory. It seemed to be impossible to reach the temperature of the steam
in that way, even after half a day's waiting. In the present work, therefore,
the apparatus is modified, so that the steam may play directly on the long
refraction tube. In this way the temperature difficulty was quite eliminated.
93
J
s
,a
w
^3
rt:
•s^Jt'* ''
^a
The tube containing the air column was made of inch brass gas-pipe, 71.7
cm. long (between windows) and 2.5 cm. in internal diameter (A, fig. 93,
which shows one end of the apparatus) . The ends were closed with the usual
brass caps a, in which round windows, about 2 cm. in diameter, had been cut
on the lathe. The ends were closed by plates of glass g, secured between two
jackets of rubber and " vulcanized" fiber. L shows the axis of the beam of
light.
BB is the steam chamber, steam entering at S and leaving by a similar tube
at the other end of the apparatus. Steam is thus directly in contact with the
tube. The projecting end of A is inclosed by a recess packed with wadding,
133
134
THE INTERFEROMETRY OF
CC. As the walls of the brass pipe were thick and the ends relatively short,
there seemed to be no objection to this arrangement. Care was taken to
conduct the escape steam and hot gases away from the interferometer.
The displacement interferometer was of the linear type described above,
the mirrors M and N and the grating G being attached directly to the wall of
the pier and without an intervening rail. Unfortunately the pier in a large
city is also in incessant vibration, so that the interference patterns quiver.
It is this insuperable difficulty which has prevented me from reaching results
as accurate as were anticipated. A few of the data, however, will be added
as an example of the efficiency of the method.
TABLE 14. — Refraction of air at different temperatures. Tube, 71.7 cm. long,
2.5 cm. in diameter.
Barometer.
Temp.
P.
io3XAN
Barometer.
Temp.
P-
io3XA7V
77.04 cm.
19.7°
75-3
19-3
75.12 cm.
19.9°
74-3
19-3
22°
19.4
20°
19.1
I
19-3
VI
19.0
19.2
76.25 cm.
21.7°
75-5
19.1
20°
19-3
76.98 cm.
100.4°
75-5
15.0
II
19.2
18.7°
15.0
19.4
VII
15.0
19.4
19-3
77.22 cm.
19.1°
75-7
19.8
76.55 cm.
100.2°
75-3
15.0
20°
19.7
23°
15-0
IV
19.8
III
15-4
19.7
19.7
76.25 cm.
22.2°
75-5
19.1
75.12 cm.
99-7°
74-3
15.0
20°
19.2
20°
15.0
V
19.4
VIII
14.9
19.6
15.0
19-5
I5-I
79. Observations. — The data are given in table 14, where the temperature
and barometric pressure are shown in the first column, the differences in the
pressure p between the plenum of air and the exhausted air in the refraction
tube in the second column, while the third shows the values of A-/V, the dis-
placement of the micrometer corresponding to p, as found in successive inde-
pendent experiments at the temperature given. For such long distances
between grating and mirrors the ellipses are visually distorted, and much
depends on finding a satisfactory sharp interference pattern. This was the
case, except in series 3 and 5, when for incidental reasons (outside tremors)
the patterns were disagreeably flickering. The observations are usually for
room air, as the special drying of air in series 3 and 5 made no perceptible
difference. At 100° care must be taken to obviate convection currents of
air, so far as possible. The endeavor was made to keep p as nearly as possi-
ble at the same value, apart from the barometer pressure, which does not
enter into the equations. In series 4 the values of AN" are relatively large,
REVERSED AND NON-REVERSED SPECTRA. 135
but quite consistent with each other. The reason for this could not be made
out. But for the inevitable tremors the observations would all have been
acceptable.
80. Computation. — Since the ends of the air-tube are perpendicular to
the beam of light
(i)
where AN is the difference of the displacements of the micrometer in the
presence and absence of air in the tube, e the effective length of the air column,
and n the index of refraction of the air for the given wave-length X. The
equation presupposes a knowledge of the dispersion of air O/JL/ dX; but, as this
is small, the term may be temporarily omitted. If X is constant, it corre-
sponds to a constant correction of AN throughout the experiments.
Again, if we have an equation of the form of Mascart's, nn referring to
o° C. and normal barometer, and N0 to the absence of air in the tube,
M-i p i + /3(ft-76)=JVp-JV0 = AJVp
jU7((-l 76 i + crf Nn-No ANn
where a and /3 are two constants. If the tube is not quite exhausted (8B
remaining), the observations for a plenum (barometric pressure, B) and
exhausted air being made at the same temperature,
or nearly
B ~ NB-NO ~ ANB
Thus if one neglects the small correction i — fiB of SB
T)
(3) ANB=ANB-SBg^g
the micrometer displacement ANB in case of complete exhaustion at the
barometric height, B, and the displacement ANB-5B corresponding to partial
exhaustion B — dB, are proportional to those pressures. Since dB was quite
small, this equation was assumed, and p — 8B is thus nearly the height of the
mercury column of the partially exhausted tube. In the table this is briefly
called p, and differs from the barometric height.
Finally, for two partial pressures p and pf and temperatures i and t' of the
air
AAT and AN7 being the micrometer displacement corresponding to p, t, and
p', t', respectively. Hence if care be taken to make p = p', nearly,
i + at' _ i + at _ a(t'-t) =a8t
AN '' AN' '~~AN-AN'Nl>
136 THE INTERFEROMETRY OF
if, for brevity, t'-t = St and AN-AN' = dN; or
If £ is not quite equal to p', $(p' — p) may still be neglected, but bN/p and
&N'/p' must replace AN and AA/"', orAN(i — 8p/p) replace AJV where 8p =
p-p'.
On applying equation (5) to series i, 2, 3, for which p is nearly constant,
= 0.00423 01 = 0.00380
applying it to series 4, 5, 7, similarly,
52 = 79.8° SJV = 0.00404 01 = 0.00404
The mean value is thus 01 = 0.00392. The reason of this difference is found in
series 4, where AAT is excessive. In fact, if we compare percentage errors of
a and SN
so that an error of 5 per cent in 8N would be an error of over 5 per cent in a.
For the case where the fringes tremble this is inevitable. If the mounting were
without tremor, however, dN should be guaranteed to sXio-5 cm., corre-
sponding to the evanescence of a single interference ring, so that a should be
determinable to i per cent, even in case of a tube of the length 71.7 cm. given.
If *>t is small or t' small, equation (5) becomes, approximately,
= at or
This equation may be used to find the successive values of AAf in the table,
if the second, for instance, is supposed to be correct. It appears that the
first and fifth differ about equally ( = 0.0001 cm.) from the second, but the
error of the fourth ( — 0.00028) is excessive. Hence if this second datum be
taken as the mean of series i, 2, 5, and combined with the two data for 100°,
AAT = 19.28 5AT = o.42i 6^ = 78. 6° *'= 100.3° 0 = 0.00385
This is the more probable result of table 14 and would agree with Mascart's
value, 0.00382.
Somewhat later, the independent series of observations 6 and 8 were carried
out. The interference pattern at 99.7° was exceptionally quiet and clean, but
at lower temperatures this was not better than usual. The results are
4.i5 </ = 79. 0 = 0.00372
somewhat below the preceding value.
81. Final experiments at 100°. — Somewhat later, at a time when the labo-
ratory was relatively quiet and after the same effective improvements had
been made in the mounting of the interferometer mirrors, the experiments
REVERSED AND NON-REVERSED SPECTRA.
137
at 100° were repeated. The optical measurements were satisfactory, or at
least just short of the counting of interference rings for measurement. The
arc lamp, moreover, which is unsteady, would scarcely suffice for this purpose.
The results obtained were as follows (table 15) :
TABLE 15. — Refraction of air at different temperatures.
Bar.
Temp.
P
io3AN
Bar.
Temp.
P
I03AAT
75.6 cm.
21.8°
74.0
19-3
76.72 cm.
100.3°
74.0
I5-I
19.2
20.5°
154
19-3
15.0
19.1
I5-I
76.65 cm.
21.8°
74.0
19.6
15-5
19.4
15.0
19-5
15-2
19.2
15-4
19-3
15-2
15-3
15-2
If the mean values of AN and AN' be taken and a computed
the result is
01 = 0.00361
As these experiments were the smoothest and were made under the most
satisfactory conditions, they are probably the most trustworthy. I have not,
therefore, been able to obtain evidence for a value of a. (between o° and 100°)
greater than the coefficient of expansion of gases, though it must be confessed
that the method in its present surroundings is not sufficiently sensitive to
furnish a definite criterion.
Later results at low temperatures (series 3) like the above series 4, table
14, again gave a high result for AN, in each case consistently. It is probable
that the interference pattern changes between the case of a plenum and of
highly exhausted air, owing either to flexure of the glass ends or to some other
cause, or possibly depending only on the form of the pattern which happens
to appear. In such a case the lines of symmetry for N (plenum) and N (ex-
haustion) would differ, introducing a systematic error very difficult to obviate.
Thus different values of AN often follow a difference of adjustment of the
mirror at the micrometer, while all cases for the same adjustment are practi-
cally identical.
82. Experiments at red heat.— To investigate the feasibility of such experi-
ments, an inch steel tube (bicycle tube), 68 cm. long, with flanges brazed on
at the ends, and an exhaustion tube near the middle, was heated in an organic
combustion furnace to low red heat. The ends just projected outside the
furnace and were closed by plate-glass windows with a jacket of asbestos
between (applied wet and dried); or, finally, with a jacket of aluminum
cement, clay, plaster, etc. These short but relatively cold ends are, of course,
138 THE INTERFEROMETRY OF
an objection to the method, but no better device was found. Even so, the
windows frequently cracked and had to be replaced. Such an apparatus
naturally leaks, particularly at low temperatures, where the viscosity of air
is relatively small, so that the experiments as a whole are merely tentative.
To maintain the exhaustion as high as 70 cm., it was necessary to keep the
air-pump at work. To reduce this annoyance the exhaustions were at first
not carried above 60 cm. of mercury. With the interference fringes, however,
no serious difficulty was experienced after the tube had taken definite shape.
Distortion of fringes was inevitable, but centers of symmetry for measure-
ment were always available.
The first experiments were made without exhaustion, at low and high
temperature (low red heat). The difference of displacement 8N between
cold (25°) and hot was (for instance) in two different experiments
25° loW =35.0 cm. 35.2 cm.
red hot ioW =28.5 28.6
or ioW= 6.5 6.6
at atmospheric pressure. The 8 N so obtained makes no allowance for the
change of refractive index of the hot glass ends, nor for any displacement or
rotation or warping of the ends during the course of the experiment, which
required a lapse of an hour or two.
In the next experiment, therefore, the method of exhaustion was attempted,
the partial vacuum used being about 16.6 cm. when the full barometer read
76.64 cm. Thus p = 6o cm. An example of the results obtained is given in
the following data.
Cold Tube.
Pressure 76.6 cm. ioW =34.8 cm. 34.7 cm.
Pressure 16.6 22.7 22.5
p= 60.0 10^^=12.1 12.2
Red-hot Tube.
Pressure 76.6 cm. ioW =24.6 24.5 26.8 26.0 cm.
Pressure 16.6 20.1 19.4 20.0 20.0
p= 60. io3AA/" 4.5 5.1 6.8 6.0
In the two experiments at the end readjustment was necessary, as the red-hot
tube warped during the exhaustion. In the last case the glass cracked. The
first two data should therefore be taken, so that
io3AA/"=i2.i cm. io8A/\T'=4.8cm. io3 5^=7.3 cm. / = 25°£ = 6
If equation (5) above is solved for t' the result is
or if a =1/2 73
/'
This result is certainly small, as one would estimate the temperature (red
heat) at several hundred degrees higher. Unfortunately the relatively cold
REVERSED AND NON-REVERSED SPECTRA.
139
ends of the tube and the leakage at the windows both contribute to a low
value of t' ', But these do not seem to be adequate reasons. It is more probable
that the longitudinal radiation of the air on the one hand and the value of
i /a = 2 73 assumed (if this is too small) may be the chief causes for the low
value of t'. It is not, of course, possible to come to any further decision; but
the experiments are distinctly unfavorable to the large value of a (small
i /a) above considered.
The method is not adapted for very high temperatures, since equation (7)
may be written
and therefore, since r'AAf' =
where (T referring to absolute temperature) AN' rapidly reaches the limit of
accurate measurement.
83. Further experiments at high temperatures. — A variety of experiments
were now made to obtain a more nearly tight joint at the ends, by using
various clays, aluminum, etc., as cements, but
without success. Finally, an improvement was
obtained by using plaster of paris in the way
shown in figure 94. A is the end of the hot
tube in the combustion furnace F. The flange /
is set somewhat back, so that packing of plaster
p may secure the window g to the end of the
The plaster is put on wet and allowed to
94
tube.
dry thoroughly. Lying outside of the furnace,
it is never heated to redness. The joint is at
first fairly good, though it gradually deteriorates at high temperatures, and
must be replaced. In this way the following results were found:
Just below red heat
p.
74.5
12.4
12.5
12.0
TABLE 16.
Cold tube (22°)
p. I03XAAT
74.0 17.6 Low red heat
17-7
17.8
18.1
17.8
p.
73.5
8.9
8.5
8.6
8.8
Thus, from the first and second series, t1 '= 154°; from the first and third series,
£' = 330°. As in the first experiments tried, both of these data are much too
low. Here they can hardly be referred to the leak, since this was smaller.
The ends are exposed not more than i or 2 cm. each, or a total length of about
70 cm. of tube.
Some adjustment is needed at the mirrors, to place the slit images in coinci-
dence for the case of an exhaustion, as compared with a plenum of air. This
adjustment is slight, but unfortunately its effect on AN' can not be estimated.
140 THE INTERFEROMETRY OF
Cooling of gas as resulting from longitudinal radiation might be suggested,
but, as it was not encountered in the case of the steam tube, it would not
seem to be menacing here.
Finally, it will be seen from equation (8) that the effect of a leak is to make
AAP too small. It will be larger as the vacuum is more perfect. Hence t'
should be too large for this reason. A small /' can not be due to a leak. The
exhaustion effect, since the gas expands into a vacuum, can not be serious.
None of these incidental difficulties seem adequate to account for the large
temperature discrepancies consistently obtained. All things considered, it
seems to me most probable that the temperature coefficient, as the gas enters
the region of red heat more fully, continually decreases, and that this is the
real explanation of the low temperature values obtained.
The apparatus was now taken apart and provided with a fresh jacket.
After drying, the cold apparatus again appeared in good condition. The
results with the barometer at 75.55 cm. were
Cold (22°) p iosAN Red hot p io3A7V'
73.0 17.7 73.0 8.0
17.8
17.8
Unfortunately the glass cracked after the first experiment at red heat.
The data for AAT (cold) agree almost exactly with the preceding results. The
high temperature would be t' = 383°, again enormously too low. Nevertheless,
if the values of a. were in question, as the temperature must have been at
least 850°, this would come out as low as 01 = 0.0015. The misgivings already
enumerated apply here as before. As the experiments are very laborious
they were abandoned at this point, for it did not seem that further work
would materially enhance the result ; nor was it thought necessary to actually
measure the high temperatures.
84. Flames. — In the earlier report on the refraction of flames an abnormally
low result of ju was obtained for the ignited gases. I have since repeated this
work with additional improvements. It appears that it is quite possible to
look through the peak of the blue case (symmetrically) without destroying
the interference pattern as a whole, though this naturally quivers excessively.
The last of the new results showed for the presence (A/"') and (A/) of the flame
the micrometer readings :
N', flame 0.029 0.029 0.029 0.029 0.029
N, air .0297 .0295 .0294 .0296 .0296
Hence the mean difference is 0.00056 cm., or per centimeter of breadth
(2.3 cm.),
dN = 0.00024 cm.
If the space occupied by flames were vacuum, the difference would have been
0.000268 per linear centimeter. Thus AAf' = 0.00002 cm., which lies within
the error of observation, but is otherwise quite of the order to be expected
for the hot gases in question.
REVERSED AND NON-REVERSED SPECTRA. 141
85. Conclusion. — Though the experiments made are of a tentative charac-
ter, the inference seems warranted that, so far as my work goes, the tem-
perature coefficient a. of air at low temperature is identical with the coeffi-
cient of expansion of gases. At high temperatures the value of a seems to
decrease rapidly, in proportion as the gas is more highly ionized at red heat.
It has occurred to me that such ionization might load the gas in relation
to the light-wave passing through it, and that the observed excess of index
of refraction over the value anticipated at high temperatures might be
explained in this way. But air ionized by the X-rays shows no such effect.
Neither does the refraction of flames at high temperatures, so far as can be
made out, show a large value of the refractive index of the ignited gases.
It is difficult to see how the experiment at red heat can be improved, unless
a quartz tube is made for the purpose. But even here the difficulty of obtain-
ing adequately plane parallel ends and a tube of sufficient breadth is formid-
able. The attempt to grind in reentrant glass cylinder-like stoppers at the
end of the tube was thought of, but did not succeed.
CHAPTER XII.
ADIABATIC EXPANSION OBSERVED WITH THE INTERFEROMETER.
86. Introductory. — In the preceding report1 I tested a number of receivers
in which air was expanded adiabatically, by passing one of the component
beams of the displacement interferometer through the air contained. The
vessels then used were not very satisfactory, being, as a rule, not long or
capacious enough to insure trustworthy results. Moreover, the interferometer
did not at that time admit of the introduction of long or bulky apparatus,
whereas in the new form a length of almost 150 cm. is available. The main
purpose of the research will thus be to ascertain how long and thin a tube may
be made to be serviceable for expansion experiments. Furthermore, it seemed
worth while to repeat the work preliminarily with a large, staunch tank since
found in the laboratory. This was a heavy cylinder of cast brass, about 27.1
cm. (inside) and closed by plates of heavy glass, each 0.56 cm. thick and 20.3
cm. apart (inside), the whole containing a volume of air of about 11,713
cubic centimeters, to be increased to 12,800 cubic centimeters, because of the
efflux pipe. The expansion pipe was 2 inches in diameter and closed by a
-2.^/2- inch brass stopcock, with a plug practically floating in oil to prevent the
ingress of air from without. The glass plates were secured by iron bolts, a
layer of resinous cement (equal parts of beeswax and resin) between glass and
the flat end faces of the cylinder being introduced to prevent leakage.
To expand the gas in the receiver, the 2 -inch pipe communicated with a
tall, galvanized iron boiler used as a vacuum chamber, 29.4 cm. in diameter
and 147 cm. high, thus containing a volume of 99,800 cubic centimeters, or
100,200 cubic centimeters with the influx pipe. It was in communication
with a large air-pump and provided with a mercury gage for the measurement
of the partial vacuum produced by the pump. The air flowing into the air-
chamber after exhaustion was dried in the usual way and the influx controlled
by a fine screw stopcock. There was a special opening for a thermometer.
Vacuum and air-chamber were rigidly connected by a brass union with a
rubber washer. There was no appreciable leakage so far as the atmosphere
without was concerned. The 2 -inch stopcock, however, was not quite tight
within, so that air passed very slowly from the air to the vacuum chamber,
in proportion as their pressures were different; but as the air-chamber is in
service, either at atmospheric pressure (the influx cock being open) or, after
exhaustion, at approximately the same pressure as the vacuum chamber, this
leakage was of no appreciable consequence. Otherwise the interference pat-
tern would not have been stationary.
While this apparatus was not long enough to fully realize the advantages
of the method of displacement interferometry for the purposes in question,
Carnegie Inst. Wash. Pub. 149, Part II, Chapter IX, 1912.
142
REVERSED AND NON-REVERSED SPECTRA.
143
it was useful for testing the ring method in comparison with the former. The
equivalent of a vanishing interference ring is here not immediately given in
terms of the wave-length of light, since the rings move through the spectrum.
With the exception of a few incidental experiments of my own, optic
methods of the present kind have not hitherto been used. They are here par-
ticularly applicable, since the number of the rings vanishing in a given region
of the spectrum has merely to be counted after the sudden exhaustion and
during the period of slow influx of air.
Succeeding parts of the chapter will refer to other available forms of ap-
paratus with similar ends in view, and the additional purpose of ascertaining
how long and narrow an apparatus may be shaped, without seriously inter-
fering with the adiabatic measurements; for if the apparatus is increased
indefinitely in length and diameter, it is obvious that the suddenness of the
exhaustion through any available pipe will be more and more impaired. The
same is true if the apparatus, for a given (sufficient) length, is too narrow,
though for a different reason.
TABLE 17. — Values of y. Bulky air chamber, 7=99,800 cub. cm., v= 11,620 cub. cm.
(F+tO/F=i.ii6. C=952.6; 1+*= 1.0341; 0=20.3 cm-
Series.
t
Po
P
io3AN
7
Number
of rings.
1
7'
I
°c.
22.4
cm.
75-88
cm.
56.38
56.88
cm.
I.I5
•95
.02
•15
•39
.44.
30
29
20
1.49
1.50
I. SO
I. O2
.28
20
I. SO
II
22.4
75-88
47-68
i-53
i.'?'*
.29
• SI
46
46
1.58
I.S8
1.4.5
• •^8
46
I.S8
1.4.7
r.^6
46
I C-2
III
22.8
2^.2
75-70
38-50
1-95
2.2S
1-38
1. 14
*6i
J6o
1-54
I.S4
23-6
24.O
2.05
I.QI
1.29
I.A2
6l
62
i-53
I. SO
IV
222.5
222.5
2^.6
76.75
30.35
2.65
2.2O
2.7'?
1.28
1.65
1.21
77
76
78
1.58
1-59
I.S4.
2^.6
2. .10
I.3Q
78
I.S4
1 Count broken owing to flicker of arc; obtained from rhythm. 2 Sunlight.
87. Experiments with short, bulky air=chambers. — An example of the data
obtained is given in table 17, where the ratio of specific heats, 7, computed
directly both from displacement of ellipses and 7' from interference rings, is
shown in detail. The original pressure of the air-chamber is that of the barom-
eter, pQ. The pressure of the vacuum chamber is given under p. The dis-
placement, AN, from four independent observations in each case and the
number of interference rings vanishing from exhaustion to plenum are the
data chiefly of interest. It has not been possible, according to the table, to
*Carnegie Inst. Wash. Pub. 149, Part II, §83, p.129; §85, p. 135. 1912.
144 THE INTERFEROMETRY OF
place the micrometer with an accuracy of more than 0.0002 cm. or 0.0003
cm. in successive cases, A./V being the difference of two readings, each uncer-
tain to lo"4 cm. But the effect of this is to throw out 7 by about the same
number of tenths, so that the roughness of values in the table is inevitable.
On the other hand, however, 7 obtained by displacement is usually too small,
whereas the value computed from the evanescence of rings is always much
too large. Thus in the first series there should have been an evanescence of 3 1
rings, in the second of about 50 rings, in the third of 64 rings, in the fourth of
85 rings. The reason for this discrepancy is very hard to determine, but will
be considered in the next paragraph. The mean values of 7 from displace-
ment and from rings are usually more nearly correct than either, as if the
errors were equal and opposite in the two cases. The error is, in some way
which has not been made out, associated with the placing of the micrometer.
Thus, without apparent cause, the micrometer reading with a plenum of air
may differ by several io~4 cm., so that if these discrepancies are in opposite
directions the value of 7 shows such large divergences as in series 4, for in-
stance. In other words, the error appears to be extraneous to the method of
experiment.
It has been suggested that the number of vanishing rings observed is approx-
imately about 10 per cent too small throughout, and that the corresponding
data for 7, though excessive, are nevertheless of the same order of value.
Experiments were made to determine whether the change of wave-length, X,
influenced this result. This was done by allowing the center of ellipses in
one case to move from the D line towards the red, in the other from the yellow
into the D line. The mean wave-length would in the last case be smaller, and
one may estimate the former as
c-Xp AAT
where AAfo is the displacement of mirror which passes the center of ellipses
from the C to the D line. This was found to be io3A7V0 = 28.1 cm. Hence
A/V
X = XD+3-35Xio-6X
O.O2OI
Even in the final case, therefore, where io3A]V=2.5, \D would not be in error
by more than 0.5 per cent. Using sunlight and at
£0 = 76. 75 cm. £ = 48. 65 cm. T = 2Q4.5° (abs. temp.)
the number of rings R were counted when the ellipses traveled into the D
line and from the D line, respectively, with results of which the following
are examples:
From D line, R = 47 46 46 Mean ^ = 46.3
Into D line, R = 4S 47 46.5 46 Mean ^ = 46.1
Indifferent, R = 46 46 Mean R = 46.0
These results agree with the second series of table 17, and there is thus no
appreciable difference.
REVERSED AND NON-REVERSED SPECTRA. 145
One may note that the results for 7, when rings are counted, are consistently
too large, but always of the same order. In fact, if R were increased by the re-
duction factor To76/2 73^0, the values of 7 would all be nearly correct ;but there
is no reason for such a correction. Moreover, since the data for 7 obtained
from AW (ellipses brought back to fiducial position) and from R (ellipses dis-
placed) are each separately consistent with each other, the discrepancy can
not be due to leakages of air, as these would affect both measurements in the
same way. The only source of error which is not common to both (apart from
the displacement of ellipses) is the possible distortion of the glass upon exhaus-
tion ; for, in case of A/V, measurement is made at a plenum and at maximum
exhaustion only, but at varying pressures for the case of rings. Thus if the
rings needed are supposed to increase in the ratio of
pa/p=i.$ 1.6 2.0 2.5
roughly, an approximate adjustment of the two sets of observations would
also be obtained. Moreover, the effect of flexure would be an increase of the
path of the beam in glass and so counteract the negative effect of decreased
density.
88. Effect of strained glass. — To detect the possible effect of the inward
flexure of the two plates of glass, a metallic ring about 25 cm. in internal
diameter was provided. To this, two glass plates of about the same thickness
(0.8 cm. each) as in the above vessel were cemented free from leakage and
kept in place by clamps. The distance apart of the two plates within was
but 1.8 cm., so that the micrometer displacement due to exhaustion of air
was reduced to a small value. Hence, if the flexure of the glass plates due to
exhaustion and the reverse were optically appreciable, it should here be
detected.
To compute the residual air effect for the lamella of air, e= 1.8 cm. thick,
we may write
(1) C#O=£/(M-I)=A/(JUO-I)
where (7 = 952.6, t?o is the temperature of the isothermal experiment, n and
/z0 the index of refraction of air at the pressures p and p0. Furthermore,
(2) M— I=MO— i— A-/V/2
if A1V is the micrometer displacement for the pressure difference p— p0 at #0.
Finally, if n is the number of rings vanishing or of fringes passing at the
sodium line, then
(3) AAT = n^
2Xl06
Thus if p-p» = dp, then
6
= o. 216
2Xio6 e
58.93
146
THE INTERFEROMETRY OF
so that n and 5p are proportional quantities. The following results were
found :
sp
No. of
rings
observed.
No. of
rings
computed.
Middle of glass plate . .
4 cm. above middle... .
8 cm. above middle.. . .
At edge
(30 cm.
45
[60
45
45
4S
7-2
IO.O
13.2
9.8
IO.O
io.s
6-5
9-7
13.0
9-7
9-7
Q.7
The observed data are the means of 5 or 6 trials. As it is difficult to observe
the rings without interruption in an agitated laboratory, there is no doubt
that observed and computed values are coincident. The first and last rings
are not easily counted, and individual data were found to agree with the
computed results perfectly. Finally, if the glass strain were effective (for
there is actual flexure), it would be shown in the observations made by pass-
ing the beam through different parts of the plate of glass, between the center
and the edge. No consistent difference was found.
Hence an appreciable strain effect is also absent, and the reason for the
discrepancy in the two sets of values from AA7" and from n, in table 1 7 , remains
outstanding.
89. Equations.— In the preceding report * the equation for the value of 7
is deduced as
log
7 =
or
Here p0 and p are the pressures in the air-chamber (barometric) and the
vacuum chamber respectively, before exhaustion, #0 the original tempera-
ture of the air, AAf the displacement of the micrometer corresponding to the
shift of the ellipses on exhaustion. If the air-chamber is quite tight, AN" may
be taken at any time. C and i -\-x are the optic constants
v*— 1)= 952.6
for dry air, being the optic gas constant, if /JLO— i replaces p0, the normal
density of the gas. To allow for the dispersion of air an empirical equation
(convenient in the present calculation),
= 0.0002 ioi/X°341
was constructed. The deduction assumes that the centers of ellipses are
brought back again to the fiducial line D, of the spectrum, the micrometer
displacement in question being AN.
* Carnegie Inst. Wash. Pub. 149, Part n, pp. 166-168.
REVERSED AND NON-REVERSED SPECTRA. 147
In the case where rings are counted, however, the center of ellipses leaves
the D line by a short distance, less than one-tenth of the interval between the
C and D lines. In such a case, if v& = Ati-\-ba,/\2 for air and M8=<4g+&gA2
for glass, the micrometer displacement to bring the ellipses back again from
X' to X should be
and eg being the lengths of air and glass* in the beam. Here
6a = icr14Xi.65 ejo.. = icr14X33-5
ee= 2 cm. &g = io~12X 48 eebe = io-
so that the effect of air, where 6a is variable with pressure, is but 0.3 per cent
of the glass effect and may in the first approximation be neglected. The
equation may therefore be written :
X2 X3
If the mean data from series I be inserted (dN = 960 Xio"6 when X refers to
the D line)
_a = 9AoXio^X^Xo.3473 = IO_2Xa^
For the case of the C and D lines 6XA = 3. 3 5/58. 9 = 0.05 7, roughly, about ten
times the preceding distance.
In fact, the observations made for the estimate given in the preceding para-
graph (semi-displacement) ,
_AX , .=2Xngi
X2 58.9
compared with the present
$X_io-8Xo.34733JV_
~T= 576X10-"
are quantities of the same order, though one would have expected closer
coincidence.
The discrepancy observed between the method of measurement in terms
of the displacement (AN to bring the ellipses back to the fiducial position)
and the method of counting rings can not, therefore, be explained as the
result of a change of wave-length X in the latter case; i.e., the equation
AAr=(w0— w)X/2
where n0— n is the number of vanishing rings of the mean wave-length X, is
at fault for some other reason. Curiously enough, the ring method is essen-
tially simple, as it reduces to 7 = , — / / \, if «o and n are the number of rings
* Thickness of glass plates of air-chamber, 1.3 cm.; of the plate of the grating, 0.7 cm.
148 THE INTERFEROMETRY OF
vanishing when a plenum of air and the adiabatically exhausted air, respec-
tively, are introduced into one of the beams. Since
/4) — I = Ho\/2e =
this is equivalent to
90. Experiments with long tubes. Diameter, one inch. — The difficulty
encountered in the case of the preceding experiments was the small value of
the displacement AN obtained. As a consequence, every little incidental
disturbance produced a large effect in 7. It is the purpose of the present
experiments to remedy this defect by using long tubes by which AN" may be
increased over ten times. It was particularly of interest, moreover, to begin
with relatively thin tubes, and inch gas-pipe suggested itself for the purpose.
The value of 7 to be expected will necessarily be too small, as the air must
undergo reheating before the exhaust cock can be closed. The question, how-
ever, is whether consistent values of 7 will be found, even for these extreme
conditions and for large variations of pressure. Obviously the window plates
will not produce discrepancies, as has been directly shown in paragraph 88.
The gas-pipe installed was 143.4 cm. long within. To make the junction
with the vacuum chamber, a straight pipe of the same diameter and about
75 cm. long was needed between the main pipe and the 2^-inch stopcock.
The connecting pipe, together with the tube itself, is probably the chief cause
of the resistance to flow and the low value of 7 found, but it was not possible
to shorten it.
The large stopcock inevitably leaked slightly when the pressures were
different in the two chambers; but immediately after exhaustion this made
no appreciable difference, as the two pressures are then nearly the same. In
fact, no rings vanish from the spectrum from this cause. Just before exhaus-
tion, however, after closing the gas-pipe by the fine influx stopcock, appreciable
leakage is shown by the spectrum. Hence the exhaustion must be made
immediately after the influx cock is closed. Some low results at the outset
are referable to this difficulty.
The tube was, as usual, filled with dry air after exhaustion. The results
are given in table 18, in the same way as in the preceding case. The experi-
ments themselves were throughout satisfactory, no difficulty being encountered
at the interferometer. The work, moreover, is equally trustworthy at low
and at high exhaustions, a result which is rather surprising. In the latter
case, as the total displacement, AN, is over 0.0276 cm., the 7 contained should
be correct within i per cent.
Only one attempt was made to find AN by the march of the interference
fringes. Fully 276 were observed, and it is here necessary to count the fringes
passing the D line, since the ellipses are displaced throughout the greater part
of the length of the spectrum ; but this introduces no inconvenience whatever.
The difficulty is due to the time needed in counting so many evanescences;
for during this interval the electric lamp is liable to flicker seriously, or some
REVERSED AND NON-REVERSED SPECTRA.
149
commotion will occur in the laboratory or without, tending to make the count
uncertain. The rings disappear temporarily during the tremor. In a quiet
laboratory, however, and with sunlight replacing the arc light, this would be a
method of precision. Thus, for instance, at the highest exhaustions used, over
goo fringes would have to pass the D line, a datum from which 7 could be
accurately obtained.
TABLE 18. — Values of 7. Iron gas-pipe, i inch internal diameter. £=952. 6.
e= 143.4 cm. i+*=l.O34l.
Series.
/
Po
P
io3A,/V
7
No. of
rings.
7'
I
°C.
19.2
cm.
76.84
cm.
57.84
cm.
8-75
8.60
1.18
1. 2 1
276
1.28
8. IS
1.24.
8.AO
1.24.
II
19.1
76.28
48.08
12-95
n.iS
1. 2O
1.18
....
. . • . .
I3-I5
13-25
13.10
I-I.2S
1.18
1.17
1.19
1. 17
III
19.2
76.28
38.98
18.05
18.10
17-95
I7.8O
I.I4
I.I4
I-I5
1. 17
IV
19-3
76.28
29.78
22.70
22.8O
I-I5
1. 14.
22.6S
I.I S
22.80
1. 14.
V
19-3
76.28
20.88
27.85
27.60
27.6O
1. 12
I.I4
1. 14
If we compare the mean results for 7 with the exhaustion used (pressure p
in the vacuum chamber, full barometric pressure po in the air-chamber), the
results decrease slightly as the vacuum is higher. Thus
If p= 58 cm.
Then 7= 1.23
48 cm. 39 cm. 30 cm.
1.18 1.15 1.14
21 cm.
which is what might have been expected, except that the rate of decrease is
much less than would be surmised. There seems thus to be no objection to
the use of high exhaustions, which in turn give a better value of 7 from the
large range of AJV obtained.
The low mean value of 7 obtained has been referred to the resistance of
the inch piping to the outflow of air. It is probably not due to the stop-
cock, as incidental differences in the speed of opening and closing would
otherwise have shown a marked effect. One may conclude that the air in
the long inch gas-pipe expands adiabatically with a coefficient 7 between i . i
and 1.2, in case of such exhaustions as the above.
150
THE INTERFEROMETRY OF
91. The same. Diameter of tube, two inches. — The experiments were now
continued by enlarging the diameter of the tube to 2 inches. Brass gas-pipe,
1.35 cm. long, to be closed with thick glass plates, was at hand. To connect
the same with the vacuum chamber, a similar 2-inch pipe, 115 cm. long, as far
as the 2^2 -inch stopcock, was necessary. Moreover, as this was in the way of
the light received from the grating, the beam was reflected by an offset con-
sisting of two silver mirrors in parallel. No difficulty was found with this
arrangement, and the sodium line was in view to give evidence if any acci-
dental displacement should occur.
Unfortunately, the ellipses obtained were somewhat irregular open forms
(i.e., half ellipses), and the endeavor to secure small closed patterns did not
succeed. This annoyance depending chiefly on the parts of the mirror and
grating used, and on shifting accessories, is not easily controlled. The indi-
vidual measurements of AAf are therefore not as good as those recorded in
table 1 8, where a displacement of icr4 cm. was assured. They suffice, how-
ever, for the present purposes.
The new data are given in table 19, t being the temperature of both cham-
bers, po the initial normal pressure of the air-chamber (2-inch pipe), and p
that of the vacuum chamber.
TABLE 19. — Values of 7. Brass gas-pipe, 2 inches internal diameter.
C= 952.6 ; i+x= 1.0341. 6=135.3 cm. (7+»)/»=i.049.
Series.
t
Po
P
I03A7V
7
I
°c.
17.0
cm.
75.66
cm.
56.46
cm.
7-15
7-^S
1-35
I.T,O
7.2S
1.1,1,
7-SS
1.27
II
17.0
75-66
47.36
10.85
IO.QS
1-33
i-3i
II. IO
I.2Q
1 1. OS
I.^O
III
16.1
75.86
38.36
14.93
IS. IS
I-3I
1.28
IS-IO
I.2Q
15.38
1.26
IV
16.2
75-86
29-36
19.53
19.57
1.25
1.25
IQ.27
1.28
IQ.7S
1.2^
V
16.4
75-86
20.36
23-71
2VQS
1.27
I.2S
2^.7O
1.27
23-75
1.26
The effective value of 7 in these experiments is, for the lower exhaustions,
above 7 = 1.3, showing a considerable improvement over the data for the inch
tube, which were not much above 7 = 1.1. This was to be inferred, of course;
but it was not expected that the increment of 7 due to increased diameter
REVERSED AND NON-REVERSED SPECTRA.
151
would be so rapid. It would seem to be probable, therefore, that if a 4-inch
tube were used the conditions for obtaining a trustworthy value of 7 would
be nearly met.
As the exhaustions in a successive series are gradually increased (initial
partial vacua from £ = 56.46 cm. to £ = 20.36 cm. in the vacuum chamber),
the observed values of 7 gradually but slowly decrease, the mean values being
Oo = 75-7 cm. to 75.9 cm.)
£ = 56.46
7= i-32
38.36
1.29
29.36
1.25
20.36 cm.
1.26
where the fourth value is too small, for incidental reasons. This general
result is also to be expected ; but it is rather remarkable that with such high
exhaustions as those finally used the decrease of 7 is not more marked.
The work, as a whole, progressed smoothly throughout, the only interfer-
ence with precision being the incidental occurrence of open ellipses. To obtain
other patterns would have required longer additional adjustment than the
work at the present stage seemed to warrant.
92. The same. Diameter of tube, four inches. — The first experiments made
with the 4-inch tube are given in table 20. The completed apparatus showed
a slight leak, which could not be detected after long searching. The tube
was therefore admitted for a tentative series of experiments. The exhaust
pipe here, as above, was rigid and straight, but only 2 inches in diameter,
with a 2^-inch stopcock. To exhaust the air-chamber, the handle of the
cock was suddenly jerked over an angle 180° between the two closed positions.
The plug virtually floated in oil, as shown elsewhere.
TABLE 20. — Values of -y. Brass pipe, 4 inches internal diameter.
C=952. 6; i+x= 1.0341; e= 126.9. (V+v)/V= 1.119. Small leak in apparatus.
Series.
t
P°
P
io3AN
7
I
°C.
19.9
cm.
76.15
cm.
57-00
cm.
5-90
6.10
1.42
1.1,7
6.2^
I.-jj.
....
6.15
1.36
II
23.0
75-79
38.39
12.40
12.7^
1-36
1.^2
12. 6O
I.^O
12. 7">
1. 71
III
23.2
75-79
29-39
I5-50
IS- 71
1.40
1.^7
IS. 77
1.^7
As a whole, the results are disappointing; and they are irregular, for mean
readings could not be made because of the leak. They are, nevertheless,
interesting, inasmuch as with some of the above data they point out a special
source of discrepancy. It will be seen that the 7 values tend to decrease in
successive measurements, beginning with a high value, which is here nearly
152
THE INTERFEROMETRY OF
correct. This can not be referred to the temperature of the 4-inch tube,
because the initial optic density is necessarily measured. It must therefore
be due to the temperature of the vacuum chamber. It follows, therefore,
that the time allowed in these experiments, between observations, though
sufficient for establishing the initial temperature of the air-chamber, is not
sufficient for the much larger vacuum chamber. The two chambers are thus
no longer at the same temperature, a condition which the equations implicitly
assume.
The apparatus was now taken apart and thoroughly overhauled. After
reassembling the parts, the chamber was found free from leakage. As the
exhaust pipe was in the way of the beam of light entering the telescope, the
offset, consisting of two parallel mirrors firmly adjusted, was used without
annoyance, here as above. The work throughout progressed smoothly, though
the ellipses were again not as satisfactory in form as would have been desirable.
TABLE 21. — Values of 7. Data as in Table 11,4" brass pipe.
Series.
/
Po
P
IOSAAT
T
I
°c.
16.5
cm.
75-90
cm.
56.80
cm.
6-34
6.15
•33
•37
6.27
•34
6.30
•35
II
16.6
75-90
47.60
9-25
0.4.0
-38
.36
9.^0
•34
....
9.20
9-SS
•39
1.33
16.7
76.19
47.89
9-31
9-45
1-37
1.35
III
16.9
76.19
38.69
12.87
12. 7O
1-34
1.36
12.80
1.35
12.76
1.35
IV
17.0
76.19
29.69
16.26
16.23
i-35
• 3S
16.25
•35
16.04
.38
V
17.1
76.19
20.69
20.00
19.60
19-73
19.60
•35
.40
-38
• 39
Table 21 contains the results. Changes in the values of AAT in a given
series are most likely referable to the form of the interference pattern, indi-
rectly to the flickering of the electric lamp. There seems to be no evidence
to associate them with the manner in which the 2^-inch stopcock is opened
and closed. This was merely jerked around 180°, between the two closed
positions of the plug, and, so far as can be seen, the rate of motion is adequate.
The successive observations show no consistent difference, as was the case
REVERSED AND NON-REVERSED SPECTRA.
153
in the preceding table. Hence this discrepancy has been eliminated. What
is most interesting is that the 4-inch tube shows no consistent difference in
the 7 values for high or low exhaustion. Thus the mean values under increas-
ing exhaustion, p, are
£ = 56.8
7= 1-35
47.6
1.36
38.7
i-3S
29.7
1.36
20.7
1.38
Accidentally the highest value of 7 belongs to the highest exhaustion.
The chief anticipation of the work (i.e., that with a 4-inch tube the true
value of 7 would appear) has not been fulfilled. The value obtained is still
much below normal, successive results ranging as follows:
Diameter of tube 2.5 5.0 10.0 cm.
Mean 7 1.17 1.29 1.36 cm.
Diameter of exhaust pipe 2.5 5.0 5.0011.
The relatively small increase between the tubes 5 cm. and 10 cm. in diam-
eter is disappointing. At the rate obtained from the first two experiments
(see fig. 95) a 3-inch tube should have been nearly sufficient. At the rate
established by the last two observations, however, a tube at least 5.5 inches
1-0
0
95
in diameter would be needed to obtain trustworthy values of 7. These differ-
ences are possibly due to the exhaust pipe, which in case of the last observation
does not increase in size. Hence a 3-inch pipe with a 4-inch stopcock may
be estimated as being adequate for 7 measurement, provided the exhaust pipe
is straight and clear throughout.
The observations were broken off at this point, with the object of searching
for some means of obtaining a more sensitive and regular interference pattern.
If the method is to be ultimately successful, then icr4 cm. on the micrometer
must be guaranteed. If the ellipses are not quite regular or not closed, this
is not the case. A more sensitive method of defining optic density is thus in
question.
CHAPTER XIII.
MISCELLANEOUS EXPERIMENTS.
93. Effect of ionization on the refraction of a gas. — It seemed interesting to
test this question carefully, although a negative result was to be expected.
Accordingly one component beam was surrounded by a thick iron tube, while
the other was allowed to travel freely in air, along a path energized by the
X-rays. For this purpose the X-ray bulb was placed near the grating and the
radiation directed toward the mirror N, the beam GM being inclosed. A thick
sheet of lead, i foot square, was placed behind the bulb to additionally screen
off radiation along GM. Under these circumstances the ionization along GN
must have been enormous by comparison with GM. Quiet ellipses were pro-
duced in the interferometer, and the effect of opening the X-ray current and
closing it again, alternately, was observed. Not the slightest deformation of
the ellipses or any motion of the fringes could be detected. An ionization effect
is therefore wholly absent. It might have been supposed, for instance, that
the ions present might load the wave of light and produce an appreciable result
in the interferometer (cf. fig. 92).
Since a shift of o.i of a ring would probably have been detected, AAT =
0.000005 cm. would have produced a perceptible effect. Hence, since n— i
is, roughly, equal to A N/e, the value of the ionization effect could not exceed
The ionization effect can not, therefore, exceed o.oi per cent of /*— i.
To further test this question, the iron tube, i inch in diameter and 138 cm.
long, was provided with a fine axial wire about 0.02 cm. in diameter, passing
through central holes in the glass plates at the end. The ends of the wire were
drawn tight by hard-rubber rods on the outside, so that the tube became
a cylindrical condenser. All holes were sealed hermetically with resinous
cement. The interference fringes were clearly producible.
The poles of an induction coil were now connected with the inner wire and
the tube, respectively, to alternately change the condenser and discharge it,
with the object of strongly ionizing the air within. On partial exhaustion the
whole tube became luminous, on account of the discharge, in the usual way.
The best results were obtained with a plenum of air when but two storage
cells actuated the coil. Under these circumstances no sparks passed from core
to shell of the iron condenser tube, while the air within was intensely ionized
by the silent discharge. On closing the current, from 0.5 to i per cent of the
rings was swept inward at once. On opening it, the rings again emerged. This
inward motion, however, was in the same sense as the effect of a decrease of
154
REVERSED AND NON-REVERSED SPECTRA. 155
density, such as would result, for instance, from rise of temperature or from
partial exhaustion. Hence the effect observed, though very definite, would
correspond to a temperature effect due to electrical currents traversing the air.
One should expect the effect of ionization, if appreciable, to be the reverse of
this. With voltages high enough to produce sparks in the tube, the inter-
ference figures naturally show violent agitation or quiver. If the displacement
in question is one ring and S denotes differences, 5(AA/") = 3oXio~6 cm.
If only temperature changes, one may write, roughly, A N.T = constant,
T referring to absolute temperature, whence
8r =
40X10°
if results found for a similar tube, above, be taken.
Thus 5r = 2.2Xicr7 degrees centigrade is the average temperature incre-
ment, for the whole length of the tube.
When but a single cell was used to energize the coil, no effect could be recog-
nized. In case of two cells, moreover, when the plenum of air was replaced
by a partial vacuum of i cm. or less, so that an arc was seen, no effect was
observable, although the reddish light colored the field of the telescope.
There are two points of view, however, from which the assumption of a tem-
perature effect is not admissible. If the pipe is closed, so that the density of
the air contained remains unchanged, there is no difference in the phenomenon.
But there should not, for the case of constant density, be any effect, unless the
nature of the gas is changed. Again, the effect is instantaneous and not
increased on keeping the circuit closed. The simple explanation in terms of
temperature made above must therefore be taken with reservation. At all
events, the effect of ionization would be small and equivalent to a dilution of
the gas of but
-X2.2Xio~7 or about io~9
273
of its density, when sparks are about to occur.
94. Mach's interferences. — It is frequently necessary to use the interferom-
eter in such a way that but one ray passes in a given direction ; i.e., the rays are
not to retrace their path. Interferometers of this kind are treated above, but
Mach's design offers advantages, which will be presently pointed out. As a
rule, in using these interferometers, the center of the elliptic interference pat-
tern is remote and the lines are hair-like and found with great difficulty. These
annoyances are overcome when the apparatus is put together as follows :
In figure 96, L is the vertical sheet of light from a collimator impinging on
the strip of plate glass gg, half -silvered on one side, toward or near the ends.
The pencil L is thus reflected to the opaque mirror N and transmitted to the
opaque mirror M (on a micrometer) , and then reflected to the other end g' of
the glass strip gg' . Thereafter, both the pencils, Mg' and Ngf, are available;
156 THE INTERFEROMETRY OF
but it is generally more convenient to use the former (Mg') , reflecting it from
the plane opaque mirror m to the telescope at T. When L came from sunlight,
or from an arc light, etc., the white images of the slit were very bright. After
putting them in coincidence, horizontally and vertically, by aid of the three
adjustment screws on the mirror M, Ives prism-grating G may be placed in
front of the objective of the telescope. A very brilliant spectrum thus appears,
and the fringes are easily found by moving the micrometer slide which carries
M to the proper position. In my apparatus gg' was about 50 cm. long and
gM = gN about 2 meters. The telescope is sufficiently near M to manipulate
the micrometer, the mirror m being so placed that the beam just misses the
strip gg'.
96
The interference pattern, found at once and satisfactorily centered, consisted
of large, broad circles. On moving the micrometer M from evanescence on one
side of the center to evanescence on the other, the slide was found to have
moved over about 2 mm. With a stronger telescope to magnify the fine, hair-
like fringes, this distance would have been larger. It is interesting to compare
this datum displacement with the datum found in the case of the phenomenon
above, where a range of over 0.5 cm. (double path-difference) was observed.
In the present experiment the range is smaller, because the interference pattern
falls below the limit of visibility before the possibility of interference is
exhausted. Mg' slides along g' when M moves.
95. A Rowland spectrometer for transmitting and reflecting gratings, plane
or concave. — In the above experiments I had occasion to examine a variety of
gratings, and it was therefore desirable to devise a universal instrument by
which this could be accomplished without delay. The method chosen is sim-
ilar to that previously described,* but its details have been greatly simplified,
on the one hand, and made more generally applicable, on the other. It seems
permissible, therefore, to give a brief description.
* Carnegie Inst. Wash. Pub. No. 149, Chapter I, 1911.
REVERSED AND NON-REVERSED SPECTRA.
157
In figure 97, GG' and HH' are double slides like the carriage bed of a lathe,
each about 1.5 to 2 meters long and 10 cm. wide, rigidly fastened together.
They are placed at right angles to each other on a fiat table, the vacant distance
between G' and HH' being less than a meter. For ordinary purposes they need
not be screwed down. A, B, D, K, are flat carriages, or tables, provided with
screw sockets for supporting the different standards, and capable of sliding to
and fro with a minimum of friction. A carries the micrometer slit S. B and C
are joined by the Rowland rail R, whose length is thus equal to the radius of
the concave grating to be examined, or nearly so, so that the ends of R are on
vertical axes at b and d. B also supports the table C (somewhat enlarged in
the side elevation, fig. 98), on which the table t of the grating g may be adjusted
on its leveling screws. To secure a common axis, 6, e, the rod at ace is twice
bent at right angles. Moreover, if c is turned to one side, the supporting rod e
may be screwed into the vacant socket b at the end of R. For the case of fig-
ure 98, the angle of diffraction 6 is varied and \ = D sin 6, where D is the grat-
ing space. For the other case (c being turned aside and C screwed into and
turning with b) the angle of incidence is varied and X = D sin i. This is much
simpler in form than the early method used.
97
Finally, the table C carries the essentially new addition to the apparatus
(shown in front elevation in fig. 99), viz, the long slot ff, adapted to support
the right-angled reflecting prism E and at the same time to allow free play to
the rail R within ./f. Figure 99 then shows the progress of the rays (turned 90°
to the front in a horizontal plane) from the slit or collimator, 5. They are
doubly reflected at E, return in a vertical plane and then impinge on the grat-
ing at G. The rays thereafter pass along the rail R (fig. 97) and are examined
by a strong eyepiece at d (not shown), rigidly but adjustably attached to the
near end of the rail.
The displacement of K along HH ' is accurately measurable on a parallel
scale with vernier (not shown) . If Xi and xz are the two symmetrical readings
on opposite sides of the virtual slit image at S (fig. 97), and R the radius of
the concave grating, and x = xz — x\
sn
= x/zR, or sin i = x/z
158 REVERSED AND NON-REVERSED SPECTRA.
If a plane grating is used, a weak lens L is attached to the rail R and moves
with it, so that its focus is in front of the ocular d (with cross-hairs) . In this
case 5 is a collimator. If a transmitting grating is examined, the collimator
5 (fig. 99), etc., are merely to be lowered, and the prism E is superfluous. It
need not even be removed. Naturally, it is in the interest of accuracy to have
all the standards like e and h as short as possible.
Dx
Finally, in the equation X = — „, if D = io6d, the values d and R are usually
of the same order (175 cm.) for gratings with about 15,000 lines to the inch.
In this case we may make the rail length R = d, whence
Even in case of the concave grating, when ultimate precision is not aimed
at, some variation of the distance SS' = 2SE, nearly, is admissible without
destroying the definition. The carriage D with the prism E may be moved
fore and aft on the slides GG' until the focus at d is sharp. The values of x are
usually of the order of 100 to 125 cm., so that an accuracy of Angstrom units
is easily obtainable without special refinement.
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