LECTURE III
APPLICATION OF INTERFERENCE METHODS TO MEASUREMENTS OF DISTANCES AND ANGLES
In the last lecture we considered the limitations of the telescope and microscope when used as measuring instruments, and showed how they may be transformed so that the diffraction and interference fringes which place the limit upon their resolving power may be made use of to increase the accuracy of measurements of length and of angle. We have named these new forms of instrument interferometers and illustrated many of the forms in which they may be made.
It has been found that the particular form of interferometer described on p. 40 is the most generally useful, and the principal subject of this lecture will be to illustrate the applications which have already been made of this instrument.
But before passing to the first application of the interferometer, we may make a little digression, and consider briefly the two theories which have been proposed to account for the various phenomena of light. One of these is the undulatory theory, which has already been explained; the other is the corpuscular theory, which for a long time held its ground against the undulatory theory, principally in consequence of the support of Newton.
The corpuscular theory supposes that a luminous body shines in virtue of the emission of minute particles. These corpuscles are shot out in all directions, and are supposed to produce the sensation of vision when they strike the retina. The corpuscular theory was for a long time felt to be unsatisfactory because, whenever a new fact regarding light was discovered, it was always necessary to make some supplementary hypothesis to strengthen the theory; whereas the undulatory theory was competent to explain everything without the addition of extra hypotheses. Nevertheless, Newton objected to the undulatory theory on the ground that it was difficult to conceive that a medium which offers no resistance to the motion of the planets could propagate vibrations which are transverse (and we know that the light vibrations are transverse because of the phenomena of polarization), for such vibrations can be propagated only in a medium which has the properties of a solid. Thus, if the end of a metal rod be twisted, the twist travels along from one end to the other with considerable velocity. If the rod were made of sealing wax, the twist would rapidly subside. If such a rod could be made of liquid, it would offer virtually no elastic resistance to such a twist.
Notwithstanding this, the medium which propagates light waves, and which was supposed to resist after the fashion of an elastic solid, must offer no appreciable resistance to such enormous velocities as those of the planets revolving in their orbits around the sun. The earth, for example, moves with a velocity of something like twenty miles in a second, has been moving at that rate for millions of years, and yet, as far as we know, there is no considerable increase in the length of the year, such as would result if it moved in a resisting medium. There are other heavenly bodies far less dense than the earth, e. g., the comets, and it seems almost incredible that such enormously extended bodies with such an exceedingly small mass should not meet with some resistance in passing through their enormous orbits. The result of such resistance would be an increase in the period of revolution of the comets, and no such increase has been detected. We are thus required to postulate a medium far more solid than steel and far less viscous than the lightest known gas.
These two suppositions are possibly not as inconsistent as they may at first seem to be, for we have a very important analogy to guide us. Consider, for example, shoemaker's wax, or pitch, or asphaltum. These substances at ordinary temperatures are hard, brittle solids. If you drop them, they break into a thousand pieces; if you strike them (so lightly that they do not break), they emit a sound which corresponds to the transverse vibrations of a solid. If, however, we place one of these substances on an inclined surface, it will gradually flow down the incline like a liquid. Or if we support a cake of shoemaker's wax on corks and place bullets on its upper surface, after a time the bullets will have sunk to the bottom, and the corks will be found floating on top. So in these cases we have a gross and imperfect illustration of the coexistence of apparently inconsistent properties such as are required in our hypothetical medium.[1] Nevertheless, it seemed impossible to Newton to conceive a medium with such incompatible properties, and this was, as stated above, a serious obstacle in the way of his accepting the undulatory theory. There were others, which need not now be mentioned.
For a long time after the various modifications that the corpuscular theory had to receive had been made, both theories were actually capable of explaining all the phenomena then known, and it seemed impossible to decide between them until it was pointed out that the corpuscular theory made it necessary to suppose that light traveled faster in a denser medium, such as water or glass, than it does in a rarer medium, such as air; while according to the undulatory theory the case is reversed. We may illustrate briefly the two cases: No matter what theory we accept, it is an observed fact that refraction takes place when light passes from a denser to a rarer medium, and consists in a bending of the incident ray toward the normal to the surface of the denser medium. Suppose we have a plate of glass, for example, and a ray of light falling upon the surface in any direction. According to the corpuscular theory, the substance
FIG. 41 below the surface exerts an attraction upon the light corpuscles. Such attraction can act only in the direction of the normal. If we separate it into two components, one in the surface and one normal to it, the normal one will be increased. These two components might be represented by OA and OB in Fig. 41, and the resultant of the two would be OC. In consequence of the presence of the denser medium, the normal component of the velocity of the particle is increased, and the resultant is now OC', which is greater than OC.
Let us next consider refraction according to the wave theory. A wave front ab (Fig. 42) is approaching the surface ac of a denser medium in the direction bc. This direction is changed by refraction to ce, and the corresponding direction of the new wave front is cd. During the time that the wave ab moves through the distance bc in the rarer medium, it moves through the smaller distance ad in the denser. Thus the results, according to the two theories, are exactly reversed.
Hence, if we could measure the enormous speed of light—about 400,000 times as great as that of a rifle bullet—it would be possible to put the two theories to the test. In order to accomplish this we must compare the velocities of light in air and in some denser, transparent medium—say water. Now, the greatest length of a column of water which still permits enough light to pass to enable us to measure the very small quantities involved is something like thirty feet.
FIG. 42 We should therefore have to determine the time it takes the light to pass through thirty feet of water, at the rate of 150,000 miles a second. This interval of time is of the order of one twenty-millionth of a second. But we must measure a time interval even smaller than this, for we have to distinguish between the velocity in water and the corresponding velocity in the air, i. e., to determine the difference between two time intervals, each of which is of the order of one twenty-millionth of a second. This, at first sight, seems beyond the possibility of any physical experiment; but, notwithstanding this exceedingly small interval of time, by the combined genius of Wheatstone, Arago, Foucault, and Fizeau the problem has been successfully solved. The method proposed by Wheatstone for measuring the velocity of electricity was this: A mirror was mounted so that it could be revolved about an axis parallel to its surface at a very high rate, and the light from the spark produced by the discharge of a condenser was allowed to fall on the mirror. The images of two sparks were observed in the revolving mirror; the second spark passed after the electric current which produced it had passed through a considerable length of wire—perhaps several miles; the first, after it had passed through only a few feet of wire. If the mirror in this interval had turned through a perceptible angle, the reflected light would have moved through double that angle; and, knowing the velocity of rotation of the mirror, and measuring this small angle, the velocity of electricity could be determined. Arago thought this same method might be adapted to the measurement of the velocity of light.
The principle of Arago's method may be illustrated as follows: Suppose we have a mirror R (Fig. 43), revolving in the direction of the arrows, s is a spark from a condenser, which sends light directly to the mirror R, and also to the distant mirror M, whence it returns to R, and both rays are reflected in the direction s1. If, however, the light takes an appreciable time to pass from s to M and back, this light will reach the mirror R later, and the mirror will have turned in the interval so as to reflect the light to s2.
If the angle s1Rs2 can be measured, the angle through which the mirror moves is one-half as great; and, knowing the speed of the mirror, we know also the time it takes to turn through this angle; and this is the time required for light to traverse twice the distance sM, whence the velocity of light.
The principle of Arago's method is sound, but it would be extremely difficult to carry it into practice without an important modification, due to Foucault, which is illustrated in Fig. 44. Light from a source s falls on the revolving mirror R, and by means of a lens L forms an image of s at the surface of a large concave mirror M. The light retraces its path and forms an image which coincides with s if the mirror R is at rest or is turning slowly. When the rotation is sufficiently rapid the image is formed at s1, and the displacement ss1 is readily measured.
If the distance LM is occupied by a column of water, the displacement would be less if the velocity of light is greater in water than in air, as it should be according to the corpuscular theory; and if the undulatory theory is correct, the displacement would be greater. Foucault found the displacement greater, and thus the corpuscular theory received its death-blow.
It remained for subsequent experiment to determine whether the undulatory theory was true, because it was not sufficient to show that the velocity was smaller in water; it was necessary to show that the ratio of the two velocities was equal to the index of refraction of the water, which is 1.33. Experiments showed that the ratio of the two velocities is almost identical with this number, thus furnishing an important confirmation of the undulatory theory.
Ordinarily the index of refraction is found by measuring the amount of bending which a beam of light experiences in passing from air into the medium in question. But if this number is identical with the ratio of the velocities, the index would evidently be determined if we knew the ratio of the wave lengths, since the wave lengths are also proportional to the velocities. This can be obtained by the interferometer. 
FIG. 45 In fact, the original name of the instrument is "interferential refractometer," because it was first used for this purpose by Fresnel and Arago in 1816. This name, however, is as cumbersome as it is inappropriate, for, as we shall see, the range of usefulness of the instrument is by no means limited to this sort of measurement.
The interferometer being adjusted for white light, the colored interference fringes are thrown on the screen. If, now, the number of waves in one of the paths be altered by interposing a piece of glass, the adjustment will be disturbed and the fringes will disappear; for the difference of path thus introduced is several hundreds or thousands of waves; and, as shown in the preceding lecture, the fringes appear in white light only when the difference of path is very small.
The exact number of waves introduced can readily be shown to be 2(n−1)tl; that is, twice the product of the index less unity by the thickness of the glass divided by the length of the light wave. Thus, if the index of the glass plate is one and one-half and its thickness one millimeter, and the wave length one-half micron, the difference in path would be two thousand waves.
Let us take, therefore, an extremely thin piece of mica, or a glass film such as may be obtained by blowing a bubble of glass till it bursts. Covering only half the field with the film, the fringes on the corresponding side are shifted in position, as shown in Fig. 45, and the number of fringes in the shift is the number of waves in the difference of path, from which the index can be calculated by the formula.[2]
FIG. 46The interferometer is particularly well adapted for showing very slight differences in the paths of the two interfering pencils, such, for instance, as are produced by inequalities in the temperature of the air. The heat of the hand held near one of the paths is quite sufficient to cause a wavering of the fringes; and a lighted match produces contortions such as are shown in Fig. 46. The effect is due to the fact that the density of the air varies with the temperature; when the air is hot its density diminishes, and with it the refractive index.
It follows that, if such an experiment were tried under proper conditions, so that the displacement of the interference fringes were regular and could be measured—which means that the temperature is uniform throughout—then the movement of the fringes would be an indication of temperature. Comparatively recently this method has been used to measure very high temperatures, such as exist in the interior of blast furnaces, etc.
In one of the preceding lectures an image of a soap film was thrown on the screen, and it was shown that the thickness of the film increased regularly from top to bottom, and that where the thickness was sufficiently small the interference fringes enable us to deduce the thickness of the film. It was also shown that at the top of the film, where the thickness was very small, a black band appears, its lower edge being sharply defined as though there were here a sudden change in thickness, as illustrated in Fig. 47.
Now, this "black spot" may be observed sufficiently long 
FIG. 47to measure the displacement produced in interference fringes when the film is placed in the interferometer. It is probable that over the area of the "black spot" the two surfaces of the film are as near together as possible; and if the water is made up of molecules, there are very few molecules in this thickness—possibly only two—so that a measurement of this thickness would give at least an upper limit to the distance between the molecules.
A soap solution of slightly different character from that used in the last lecture is more serviceable for this purpose.[3] With such a solution the film lasts a remarkably long time. It is interesting to note that some time after the "black spot" has formed, portions of its surface reflect even less light than the rest, and these portions gradually increase in size and number till the whole surface almost entirely vanishes.
It is found on placing such a film as this in the interferometer that there is no appreciable change in the fringes. The film is so thin that we cannot observe any displacement at all; if we place two films in the interferometer, the displacement should be twice as great; but even then it is inappreciable. To obtain a measurable displacement it was found necessary to use fifty such films. The arrangement of the interferometer for this experiment[4] is shown in Fig. 48. The films are introduced in the path AC, as indicated at 
FIG. 48F. Yet even fifty films produced a displacement of only about half a fringe, as shown in Fig. 49. Since the light passed through each film twice, this displacement of half a fringe is what would be produced by a single passage through one hundred films. One film would therefore produce a displacement of one two-hundredths of a fringe. A simple calculation tells us that the corresponding distance between the water molecules is not greater than six millionths of a millimeter. It may be much less than this.
The interferometer is especially useful whenever it is necessary to measure small changes in distance or angle. 
FIG. 49One rather important instance of such a measure is that of coefficient expansion. Most bodies expand with heat—certainly a very small quantity: one or two parts in ten thousand for a change of temperature of a single degree.
In some cases it may be necessary to experiment upon a very small specimen of the material in question, and in such cases the whole change to be measured may be of the order of a ten-thousandth part of an inch—a quantity requiring a good microscope to perceive; but such a quantity is very readily measured by the interferometer. It means a displacement amounting to several fringes, and this displacement may be measured to within a fiftieth of a fringe or less; so that the whole displacement may be measured to 
FIG. 50within a fraction of 1 per cent. Of course, with long bars the attainable degree of accuracy is far greater.
Figs. 50 and 51 represent a piece of apparatus designed by Professors Morley and Rogers,[5] based on this principle. b and c (Fig. 50) are the two plane-parallel plates of the interferometer, and the two mirrors are at a and a'. Each mirror is divided into two halves as at aa, so that a motion of each end of the bar to be tested can be observed. The jackets gg serve to keep the bars at any desired temperatures. One side of the instrument, as aa, being kept at a constant temperature, a change in the temperature of a'a' will cause the fringes to move, and from this motion of the fringes the change in length, which is caused by the change in temperature, can be very accurately determined. Fig. 51 shows a perspective view of the apparatus.
Evidently the same kind of instrument is suitable for experiments in elasticity, and one of these was shown in the last lecture, where a steel axle was twisted (cf. Figs. 36 and 37, p. 39). If we measure the couple producing the twist, and the number of fringes which pass by, we can find the corresponding angle of twist, and a simple calculation gives us the measure of our coefficient of rigidity.
The interferometer in this second form has also been 
FIG. 51applied to the balance. Fig. 52 shows such an arrangement. The mirrors of the interferometer are on the upright metal plate, the two movable mirrors being fastened to the ends of the arms of a balance which is just visible within the horizontal box. The object of this particular experiment was to determine the constant of gravitation; in other words, to find the amount of attraction which a sphere of lead exerted on a small sphere hung on an arm of the balance. The amount of this attraction, when the two spheres are as close together as possible, is proportional to the diameter of the large sphere, which was something like eight inches. The attraction on the small ball on the end of the balance was thus the same fraction of its weight as the diameter of the large ball was of the diameter of the earth, i. e., something like one twenty-millionth.[6] So the force to be measured was one twenty-millionth of the weight of this small ball. This force is so exceedingly small that it is difficult to measure it by an ordinary balance, even if the microscope is employed. But by the interference method the approach of the large ball to the small one produced a displacement of seven whole fringes. The number of fringes can be determined 
FIG. 52to something of the order of one-twentieth of the width of one fringe. We therefore have with this instrument the means of measuring the gravitation constant, and thence the mass of the whole earth, to within about of the whole. By still more sensitive adjustment it would be possible to exceed this degree of accuracy.
An instrument in which the interferometer is used for testing the accuracy of a screw is shown in Fig. 53. The screw which was to be tested by this device was intended to be used in a ruling engine for the manufacture of diffraction gratings. Now, it is necessary, in ruling gratings, to make the distance between the lines the same to within a small fraction of a micron. The error in the position of any of the lines must be less than a ten-millionth part of an inch. Ordinarily a screw from the best machinists has errors a thousand times as great. The screw must then be tested and corrected. The testing is often done with the microscope, but here the microscope is replaced by the interferometer, with a corresponding increase in the delicacy of the test.
I will conclude by showing how to measure the length of light waves by means of the interferometer. By turning 
FIG. 53the head attached to the screw, one of the interferometer mirrors (namely C, Fig. 39) can be moved very slowly. This motion will produce a corresponding displacement of the interference fringes. Count the number of interference fringes which pass a fixed point while the mirror moves a given distance. Then divide double the distance by the number of fringes which have passed, and we have the length of the wave. Using a scale marked from 0 to 10, made of such a size and placed at such a distance that, when a beam of light reflected from a mirror attached to the screw moves over one division, a difference in path of one-thousandth of a millimeter has been introduced, and projecting the interference fringes upon the screen, it will be noted that while ten or twelve of these fringes move past the fiducial line the spot of light will move over a corresponding distance on the scale. In moving through ten fringes the spot of light moves through six of the divisions, and therefore the length of one wave would be six-tenths of a micron, which is very nearly the wave length of yellow light. If the light passes through a piece of red glass, and the experiment is repeated, the wave length will be greater; it is nearly sixty-seven hundredths. It is easy to see how the process may be extended so as to obtain very accurate measurements of the length of the light wave.
SUMMARY
1. A comparison between the corpuscular and the undulatory theories of light shows that the speed of light in a medium like water must be greater than in air according to the former, and less according to the latter. In spite of the inconceivable swiftness with which light is propagated, it has been possible to prove experimentally that the speed is less in water than in air, and thus the corpuscular theory is proved erroneous.
2. A number of applications of the interferometer are considered, namely, (a) the measurement of the index of refraction; (b) the coefficient of expansion; (c) the coefficient of elasticity; (d) the thickness of the "black spot;" (e) the application to the balance; (f) the testing of precision screws; (g) the measurement of the length of light waves.
- ↑ The specialization of the undulatory theory known as the electro-magnetic theory does not remove this difficulty; for it is even more difficult to account for the properties of a medium which is the seat of electric and magnetic forces.
- ↑ For quantitative measurements it is necessary to employ monochromatic light. The shifting of the central band of the colored fringes in white light does not give even an approximately accurate result.
- ↑ This solution is made of caustic soda 1 gm., oleic acid 7 gm., dissolved in 600 c.c. of water.
- ↑ E. S. Johonnott, Phil. Mag. (5), Vol. XLVII (1899), p. 501.
- ↑ Morley and Rogers, Physical Review, Vol. IV (1896), pp. 1, 106.
- ↑ This ratio takes into account the increased attraction due to the greater density of the lead sphere.