# The Theory of the Interferometer

THE THEORY OF THE INTERFEROMETER The arrangement sketched in fig. 5A represents schematically the interferometer in its simplest form. B' represents the image of the mirror B in the mirror M, A and B' being supposed to be not quite coincident : it is evident that to the observer at E the effect of the whole combination is the same as that which would be produced by the surfaces A and B'. The light coming to E may then be considered as having come from two nearly coinci dent sources, which are the image of the source S in the two mirrors. These sources are therefore coherent, but not quite coin cident, and the form and visibility of the fringes clearly depend upon the degree of coincidence of the two sources. Exact coin cidence would be equivalent to a single source, and consequently there would be no fringes at all in such a case, but exact coincidence is not attained in practice, so that fringes are always visible. When the coincidence is nearly attained the fringes will clearly be broad; when the images are further apart the fringes will be narrow, and usually curved, as mentioned when the adjustment of the instru ment was being discussed. The visibility will depend largely upon the aperture, since for good visibility the fringes formed by the two images of one part of the source should be superposed as nearly as possible on those formed by the two images of another part of the source.

As the moving mirror is displaced backwards the two images separate, and the fringes move across the field. Every time the distance between the images increases by a wave length a whole fringe moves past a given fixed point. If the light was perfectly homogeneous the fringes would be visible with any degree of separation of the images, that is, with any difference of path length between the two beams. If, however, the light is not per fectly monochromatic, then clearly with large enough path differ ence the fringes formed by one component of the light will cancel those formed by a neighbouring component. The fact that even with the most homogeneous source fringes never show with a path length greater than 7o cm. proves that the light consists of trains of about this length.

Michelson has discussed the formation of the fringes in the following way. In fig.

5B let AB be the two equivalent mirrors shown in fig. 5A, the intersection of their planes being vertical, and consider the effect of the reflection of light from a source R to a point 0. Let t be the distance between the mirror surfaces at the point of incidence, co be the angle of incidence, which may be considered the same for both mirrors, the angle 4 between them being supposed to be very small (of the order of a second of arc, or less). Further let p be the perpendicular dis tance from 0 to the mirrors, the distance between the mirrors at the foot of this perpendicular, and i the angle between the horizontal projection of the incident ray and the normal. Then the difference in path of the two interfering pencils is seen to be expressed since is very small and so is i, the incidence being very nearly normal.

Hence The corresponding phase difference is 27r which with an un X limited aperture may clearly have so large a range of values that all traces of interference vanish. If, however, the cone of rays is limited, either by the pupil of the eye or, in the case of an observing telescope, by a diaphragm, then the range of angles may be small enough to show the phenomenon of inter ference.

Let us now enquire as to the distance p at which the fringes will be most clearly visible. This must occur when there is no change of 0 with co, the angle of incidence, that is, when o.

Now d _ — 2 sines pio) 2 coscwpO c = o, or since co, 4 dco dco It follows that for an unlimited beam, when i has various values, the different parts of the interference paths are not simulta neously in focus except, firstly, if = o, when p = o and the fringes are localised at the surfaces of the mirrors AB, or, secondly, if cp = o or the mirrors are parallel, when p = 00 and the fringes are localised at infinity.

If we call Ao the phase difference for normal incidence, co = o and if Lo — A = nnX, then Let 0 be the angle between the normal and the projection of the ray on the vertical plane containing the normal, and put and are the limiting values of i for the incident beam. This shows that if p = o, which, as we have already seen, in volves t= o if all parts of the interference patterns are to be simultaneously in focus, then l = i, which is independent of the aperture. For moderate aperture the visibility is independent of the aperture. This is also true if 4 =o.

An idea of the order of magnitude of the angular aperture permissible may be obtained in an elementary way as follows. If the differences in phase between the central and marginal rays be X the discordances will be considerable. In the case where 4)=o, and the mirrors are truly parallel, if f3 is the angular aperture of the objective, then difference in phase = 241 —cos /3 /2) =X whence /t as a first approximation. Thus if 1= 25X, then i = -1, and the angular aperture of the lens should certainly be less than 5.

Comparative Place of Interferometer as an Optical In strument.—It had been pointed out by Michelson that the in terferometer can be regarded as a modification of the ordinary instrument, telescope or microscope, consisting of lenses or mir rors, and that prisms and gratings as well have their analogies in interferometers. We will consider first the case of a converging lens, of supposedly perfect optical construction, that is, for which the optical distance from point to point is equal for all paths. Such a lens, considered as an optical instrument, may be employed to form an accurate image of the object, as in an astronomical observing telescope, but it may also be used for accurate measure ment of a displacement, or angle, where a representation of the object is not necessary, so long as there is some recognizable feature of the image on which a setting can be made. It is well known that the image of a luminous point formed by a lens is a diffraction pattern consisting of a bright spot, surrounded by rings, and that the resolving power of the instrument depends upon the extent of the overlapping of the diffraction rings due to two close object-points. (See LIGHT, section on Resolving Power.) The size of the diffraction pattern is inversely proportional to the angular aperture of the lens, and the definition and the re solving power both increase as this aperture is increased. With a given aperture, however, the indistinctness of the image becomes more and more pronounced with increasing magnification, since all the imperfections of the image are subject to the same enlarge ment as the size itself, while the brightness diminishes. The in terference fringes of the diffrac tion pattern, however, are more accurately measurable if the pat tern is magnified, since, gen erally speaking, setting a cross wire is possible to within a fixed fraction of the fringe width.

For the purpose of accurate measurement of position or angle we may therefore block out the central portion of the lens alto gether, leaving merely a circular annulus which will produce sharp rings, or, better still, we can merely isolate two small portions of the lens at opposite ends of a diameter. We shall in this way obtain an exceedingly bad image of the luminous object, but we shall have interference fringes which can be increased in size up to practically any limit without affecting the amount of light. The result is exactly the same, as far as measuring position is con cerned, as could be obtained with a perfect microscope of infinitely great magnifying power with an infinitely bright source. This device of isolating two small strips of lens at the opposite end of a diameter is, in fact, that adopted in Rayleigh's interferom eter, already described.. Regard ing the lens as an optical instru ment we sacrifice resolution and definition, with a gain of accuracy of measurement. We may then say that, since all optical phe nomena ultimately depend upon interference in its most general conception, there is no fundamental difference between the inter ferometer and ordinary lens system. The analogies will now be considered in a little more detail.

The Rayleigh interferometer has already been considered. The close analogy between the opti cally perfect convex lens and an interferometer consisting of four mirrors is illustrated in fig. 6. Thus in fig. 6A the image of a source (a slit A, or a fine line ruled by a diamond on a smooth glass or metal surface) is formed at D (the result of the "combina tion" of all the rays which fall on the lens BC), where it may be observed, as in the telescope or the microscope, by an eyepiece. In fig. 6B the source is re placed by the surface A, whence two of the pencils (one trans mitted and the other reflected) are bent by the prisms (or mir rors) at B and C so that they meet at the surface D, proceeding thence to the eye or the observing telescope. In fig. 7 the same analogy is illustrated when the lens is replaced by a mirror.

Thus it appears that the essential difference between lenses or mirrors, on the one hand, and the interferometer, on the other, is that in the former all the rays from the source which fall on the lens unite in the focal plane to form an image; whereas in the interferometer of the type in question there are only two interfering pencils. The advantage of this has already been pointed out. In the illustrations just given, the microscope or telescope, and the analogous forms of interferometers, may be applied to the measurement of distances or of angles But prisms and gratings are employed in what seems at first sight to involve different principles, and for a different purpose, namely, the analysis of light into its component constituents. The analogy still holds, however, as is shown in fig. 8A and B Thus in figure 8A, A represents the slit source of light and BC a grating which diffracts the light back to A (part being thrown on one side by the plane-parallel plate P for observation or for photography) while in figures 8B and 8C, the interferometer shows a similar light-path, but only for the two limiting pencils of light, 8C being produced from 8B by a slight rearrangement of the mirrors.

If in this arrangement one of the mirrors, say C, is movable, and the incident light monochromatic of wave-length and if n is the number of maxima (or minima) corresponding to d, the measured difference in path, then the wave-length is given by X =d/n; and, as is described un der Spectroscopy, this can be measured with far greater accu racy than is possible by the use of prisms or gratings.

Any arrangement of mirrors and lenses has its analogue in a possible interferometer. Fig. 9 shows three different disposi tions of optical apparatus and three corresponding interferom eters. The diagrams are self-explanatory, A being a concave mir ror with the source at the centre of curvature, and hence a single plane mirror in place of the two mirrors of fig. 7: C a double convex lens with equidistant ob ject and image, a variation on fig. 6 ; and B a concave mirror and two plane mirrors with the source at the focus of the mirror. The interferometer diagram matically represented in Ciii. is a slight variation on that of Cii. obtained if the mirrors B and C are separated until AB and AC are at right angles.

Measurement of the Stand ard Metre in Terms of Light Waves.—The standard metre is defined as the distance between two fine lines on a particular iridio-platinum bar kept at Paris. It is, however, clearly desirable to have some absolute standard of length, not depending upon a particular piece of material, since not only are secular changes possible in the material which will produce minute changes in length, but also, in the remote chance of an accident to the bar, it should be possible to replace it. Long ago two proposals were made to fill this need, the first being to adopt as a standard the length of a pendulum which swings once a second at Paris. It was found on trial that the error of measurement was considerably greater than expected. The second proposal was to use the earth's circumference as a standard, and, in fact, the original metre was intended to be one forty-millionth of this length. As a result of several of the very costly investigations of the measurement of a given arc of the meridian it proved, however, that this measure ment was too inaccurate to serve. The interferometer with large differences of light paths enables us to measure a length of many centimetres in terms of the wave length of a specified light, and so gives us a standard which can be easily reproduced at any time or place. The International Committee on Weights and Measures decided, in fact, in 1923 on the adoption of such a standard, so that the measurement of the standard metre in terms of a wave length is of the highest importance.

If a wave length is to be a standard length it is, of course, essential that the light should be very homogeneous; among the hundreds of radiations examined none answered the requirements so well as the red line of cadmium vapour. With this light inter ference fringes are still measurable with a path difference of 22 cm., which distance contains about 35o,000 waves, or, say, 700, 00o fringes. The optical error of measurement will depend some what on the visibility of the fringes, but an estimate of one-tenth of a fringe width is quite conservative, and thus indicates the possibility of making such a measurement to an order of accuracy of one in ten million. The general principle of the method is sim ply to move the mirror of the interferometer through a distance given by the standard, and to count the number of fringes, but to make sure that the mirror has actually been moved through the standard distance, and to extend the measurements to a length of a metre a number of special devices are necessary.

Since fringes cannot be obtained with a path difference of one metre, and even distances of several centimetres are troublesome, as they involve the counting of some hundred thousand fringes, a standard decimetre and a series of substandards were prepared. The standard decimetre was afterwards compared with the metre in a way which will be described. The intermediate standards, by the help of which the number of fringes was reduced to a countable one, consisted of a 5 cm. one, a 2.5 cm. one, and so on, each successive standard be ing half the length of the previous one, the ninth and last being .39 mm., which contains only 600 red light waves, or 1,200 fringes in the doubled distance between the parallel surfaces which consti tute the standard. This number of fringes can readily be counted without fear of error. The construction of those intermediate standards is made clear by the example shown in fig. i o. It consists of two plane-parallel glasses A, A shown on the front surfaces, and held in contact with three brass pins, which are filed and polished until the two surfaces are as nearly parallel as required. The distance between the planes of the front surfaces of these mirrors, one of which stands, as will be seen, higher than the other, constitutes the standard length.

The first task is to determine the exact number of wave lengths in the smallest standard, which we will call standard I. Fig. i i represents the interferometer set up for this purpose: d is the moveable mirror, m m' are the two mirrors of the standard (the distance between them being ex aggregated in the diagram) and n is a stationary auxiliary mirror This mirror is used for the purpose of counting the fringes that pass when d is displaced, that is, for measuring the displacement of d in wave lengths. To start, the front surface of m is made to fall in line with the image of the reference plane d, making, however, a very small horizontal angle with it, so that with white light a series of vertical interference bands is formed, the central band being achromatic and therefore readily distinguishable. The mirror d is now steadily moved back, and the succession of cir cular fringes formed by the cadmium light on n is counted. The motion is continued until d coincides with the rear surface m', a coincidence detected once more by the white light fringes, the achromatic band being brought to the same position on m' as it had before on m. The motion of d is thus measured in terms of wave lengths of cadmium light, and the limits of its motion, given by the distance between m and m', determined by the use of white light fringes to establish coincidence.

The next step is to compare standard II. with the one already measured. Now n represents the front mirror of standard the rear mirror. The two mirrors m and n are brought into the same plane with the help of white light fringes. The mirror d is now moved to coincide with m', and standard I. is moved until fringes reappear on m, so that m' and n' are now very nearly in the same plane. The small distance between m' and n' in this position is determined in fractions of a fringe of cadmium light by moving d, so that the correction to be applied to standard II. is determined. Standard III. is checked against II. in the same way, and so on, until the decimetre is reached.

The table gives the results of three independent measurements of the number of light waves in the (doubled) length of standard IX., the decimetre. The fact that these measurements were made, at different times, months apart, and by different individuals, and still give the same result to a few hundredths of a light wave, gives confidence in the accuracy of the result. Besides the red line of cadmium, the green and the blue line were used, as recorded.

The final operation is the comparison of the decimetre with the standard metre. For this purpose an auxiliary metre X. was provided with two diamond scratches at a distance apart very nearly equal to a metre. An arm extending at right angles from the decimetre has a similar mark which is placed as nearly as possible in coincidence with one of the metre marks. The stand ard decimetre is then "stepped off" ten times by the help of the interferometer fringes. The resulting error is, however, multiplied by ten, instead of by two as in the comparison of the smaller standards. It is estimated that the error of separate determina tions of the metre may be of the order of one-half of a light wave, but the mean of all measurements is doubtless much less. To this error must, however, be added the errors of the micrometric measurements of the "coincidences" at both ends of the metre bar, and finally the error of the comparison of the auxiliary metre with the standard.

The final results are as follows: close luminous point will likewise produce a pattern of fringes overlapping the first pattern. If we consider the two points as originally coincident, and then gradually separate them, the su perposed pattern will give a certain pattern of fluctuating in tensities depending upon the separation, but ultimately a separa tion will be reached such that the darkest parts of one pattern fall exactly upon the brightest parts of the other pattern, and we shall have uniform illumination. The argument can be ex tended to an uniformly illuminated disc, for which it can be proved that the fringes vanish when where S is the separation of the two slits, X the wave length of the light used, and a the angle subtended at the telescope by the diameter of the disc.

This principle has been applied by Michelson to the measure ment of stellar diameters. It is not feasible to make a lens of large enough diameter to hope for vanishing with the extremely small a provided by a star disc, but Michelson gets over the diffi culty by receiving the light from the star on two mirrors set at from which it passes to two other 45° mirrors which throw the light in two pencils at oppo site ends of a diameter of the lens or the mirror of the telescope. The general arrangement of the mirrors on the great reflecting telescope at Mount Wilson is shown in fig. 12. The separation of the first two mirrors, which, with the second two, are mounted on an arm set normally across the telescope at the aperture and, is limited only by mechanical considerations, and, in Michelson's first instrument, was variable and could be made as great as 20 ft. The method of mounting of the mirrors is indicated in fig. 13. With the red star Betel geuse a vanishing of the fringes was obtained with a separation of about i o feet (306 cm.), which, taking the effective X as 5.7 5 X cm., gives the angular diameter a to be -047 seconds of arc. Using a parallax of •oi8 we obtain an estimate of 240 million miles for the linear diameter of this star. The diameters of a few other stars have been measured by this method. In 1928 an The metre rod is in air at i 5 ° C and 76o mm. of mercury pres sure. It is estimated that these results are correct to about one part in two million.

## Measurement of Stellar Diameters.

Most of the applica tions of the interferometer have been mentioned at one part or another of this article. The chief of them are :—the measurement of standard wave length, or, differently put, the measurement of the meter in terms of the light wave; the measurement of the maximum distance over which interference can be produced, or the coherence of a wave train; the measurement of refractive in dices of gases and weak solutions; and the resolution of close spectral lines. Reference must also be made to the recent appli cation of interferometric methods for the measurement of stellar diameters. Suppose we consider the whole of the lens of a tele scope stopped out, except for two slits at opposite ends of a diameter, as in the Rayleigh interferometer. A single luminous point will produce a pattern of interference fringes. A second
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