LECTURE II
COMPARISON OF THE MICROSCOPE AND TELESCOPE WITH THE INTERFEROMETER
One of the principal objections which have been urged against the wave theory of light is the fact that light appears to travel in straight lines, whereas sound, which is known to be a wave motion, does not cast a shadow; in other words, the sound waves are capable of bending around an obstacle in the path of the waves.
We shall now not only try to show that both of these two statements are untrue, or, at least, only approximately true, but we shall actually show that sound waves do cast a shadow and that light waves do not move in straight lines. The effect, in fact, depends on the length of the wave, and we may say roughly that the reason why a sound shadow is not ordinarily observed is that the obstacles themselves are of the same order of magnitude as the length of the sound waves. If, therefore, we wish to cast a sound shadow, it will be necessary to use either very large screens or very short waves—that is, high sounds. Indeed, the effect will be most evident if we use sounds that are barely within the limits of audition, or in some cases higher than can be perceived by the ear; and it will be interesting to trace the relation between the definiteness of the sound shadow and the shortness of the sound wave.
I have here a whistle whose length is about one inch. It produces, therefore, a sound wave of the length of four inches. In order to show to an audience the effect of the whistle at different points on the other side of an obstacle, it is convenient to use what is termed a "sensitive flame." This flame is produced by allowing a jet of gas to issue under considerable pressure from a small nozzle, and by gradually increasing the pressure until the flame is on the point of flaring. On blowing the whistle, we observe that the flame ducks; it is lowered to perhaps one-third or one-fourth of its height, and broadens out at the same time. On placing the whistle behind an obstacle, we observe by the ducking of the flame that it responds to the whistle almost as readily as when no obstacle was present.
I now take a shorter whistle, half an inch long; which, therefore, produces a sound wave two inches long. The flame responds even more readily to this than to the longer whistle, and when the shorter whistle is sounded behind the obstacle the flame ducks, but to a much less marked degree than before.
I have here the means of producing still higher sounds. Strung on a piece of wire are a number of iron washers—rings of iron about an inch in diameter. When these are shaken they produce vibrations whose wave length is even shorter than that produced by the whistle just sounded. On shaking the rings you perceive the immediate response of the flame, and on shaking the rings behind the obstacle the flame responds still, but much more feebly. I take a new set of rings one-half inch in diameter. On shaking these the flame responds as before, but when I place the rings behind the obstacle the flame scarcely responds at all. I take a still smaller series of discs. These are approximately only one-fourth of an inch in diameter and produce a wave whose length is approximately one-half inch. On shaking the last set of discs outside the obstacle the flame responds not quite so strongly as before, because the total amount of energy in this case is very small; but, on shaking the discs behind the obstacle, the flame is absolutely quiescent, showing that the sound shadow is perfect. In moving the discs to and fro while shaking them, the geometrical limit of the shadow can be definitely marked to within something like half an inch; that is, a quantity of the same order as the length of the sound wave which is being used.
It is evident from the foregoing that, if we wish to investigate the bending of light waves around a shadow, we must take into account the fact which has already been established, namely, that the light waves themselves are exceedingly small—something of the order of a fifty-thousandth of an inch. The corresponding bending around an obstacle might, therefore, be expected to be a quantity of this same order; hence, in order to observe this effect, special means would have to be adopted for magnifying it.
The diffraction of sound waves is beautifully shown by the following experiment:[1] A bird call is sounded about ten feet from a sensitive flame, and a circular disc of glass about a foot in diameter is interposed. If the adjustment is imperfect, the sound waves are completely cut off; but when the centering of the plate is exact, the sound waves are just as efficient as though the obstacle were removed.
This surprising result was first indicated by Poisson, and was considered a very serious objection to the undulatory theory of light. It was naturally considered absurd to say that in the very center of a geometrical shadow there should not only be light, but that the brightness should be fully as great as though no obstacle were present. The experiment was actually tried, however, and abundantly confirmed the remarkable prediction.
The experiment cannot be shown to an audience by projecting on a screen, but an individual need have no difficulty in observing the effect. The image of an arc light (or, better, of the sun) is concentrated on a pinhole in a card, and the light passing through is observed by a lens of two or three inches' focal length some twenty feet distant. About half-way a disc of about a quarter-inch diameter, and very smoothly and accurately turned, is suspended by three threads,[2] so that its center is accurately in line with the pinhole and the center of the lens. The field of the lens will now be quite dark, except at the center of the shadow, where a bright point of light is seen.
We shall now attempt to show the analogue of the sound-shadow experiment by means of light waves. The light is 
FIG. 18concentrated on a very narrow slit A (Fig. 18), which may be supposed to act as the source of light waves. Another slit B, about an inch wide, is placed at a distance of about eight feet, and beyond this a screen C receives the light which has passed through B. The borders bb of the shadow of the slit B are quite sharply defined (though a very slight bending of the light around the edges may be observed by means of a lens focused on b). But if the slit be made narrow, as at B', the sharp boundary which should appear at cc is diffuse and colored, the light being bent into the geometrical shadow as indicated by the dotted lines. The narrower the second slit is made, the wider and more diffuse will be the image on the screen; that is to say, the greater will be the amount of bending into the shadow. An interesting variation of the experiment is made by using two slits instead of the second slit B. In this case, in addition to the bending of the rays from their geometrical path, we have the interference of the light from the two slits, producing interference bands whose distance apart is greater the closer the two slits are together. If instead of two slits we have a very large number, such as would be produced by a number of very fine parallel wires, we have, in addition to the central, sharp image, two lateral, colored images, which, when carefully examined, show in their proper order all the colors of the spectrum. This arrangement is known as a diffraction grating, and, though mentioned here simply as an instance of diffraction or bending of the rays from their geometrical path, will be shown in a subsequent lecture to have a very important application in spectrum analysis.
We have thus shown that light consists of waves of exceeding minuteness, and have found approximate values of the lengths of the waves by measuring the very small interval between the surfaces at the thicker end of our air wedge. It can be shown also that this same measurement may be accomplished with a grating if we know the small interval between its lines. The question naturally arises: Might it not be advantageous to reverse the process, and, utilizing this extreme minuteness of light waves, make our measurements of length or angle with a correspondingly high order of accuracy? The principal object of these lectures is to illustrate the various means which have been devised for accomplishing this result.
Before entering into these details, however, it may be well to reply to the very natural question: What would be the use of such extreme refinement in the science of measurement? Very briefly and in general terms the answer would be that in this direction the greater part of all future discovery must lie. The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote. Nevertheless, it has been found that there are apparent exceptions to most of these laws, and this is particularly true when the observations are pushed to a limit, i. e., whenever the circumstances of experiment are such that extreme cases can be examined. Such examination almost surely leads, not to the overthrow of the law, but to the discovery of other facts and laws whose action produces the apparent exceptions.
As instances of such discoveries, which are in most cases due to the increasing order of accuracy made possible by improvements in measuring instruments, may be mentioned: first, the departure of actual gases from the simple laws of the so-called perfect gas, one of the practical results being the liquefaction of air and all known gases; second, the discovery of the velocity of light by astronomical means, depending on the accuracy of telescopes and of astronomical clocks; third, the determination of distances of stars and the orbits of double stars, which depend on measurements of the order of accuracy of one-tenth of a second—an angle which may be represented as that which a pin's head subtends at a distance of a mile. But perhaps the most striking of such instances are the discovery of a new planet by observations of the small irregularities noticed by Leverier in the motions of the planet Uranus, and the more recent brilliant discovery by Lord Rayleigh of a new element in the atmosphere through the minute but unexplained anomalies found in weighing a given volume of nitrogen. Many other instances might be cited, but these will suffice to justify the statement that "our future discoveries must be looked for in the sixth place of decimals." It follows that every means which facilitates accuracy in measurement is a possible factor in a future discovery, and this will, I trust, be a sufficient excuse for bringing to your notice the various methods and results which form the subject-matter of these lectures.
Before the properties of lenses were known, linear measurements were made by the unaided eye, as they are at present in the greater part of the everyday work of the carpenter or the machinist; though in many cases this is supplemented by the "touch" or "contact" method, which is, in fact, susceptible of a very high degree of accuracy. For angular measurements, or the determination of direction, the sight-tube was employed, which is used today in the alidade and, in modified form, in the gun-sight—a fact which shows that even this comparatively rough means, when properly employed, will give fairly accurate results.
The question then arises whether this accuracy can be increased by sufficiently reducing the size of the apertures.
The answer is: Yes, it can, but only up to a certain limit, beyond which, apart from the diminution in brightness, the diffraction phenomena just described intervene. This limit occurs practically when the diameter of two openings a meter apart has been reduced to about two millimeters, so that the order of accuracy is about ×, or , for measurements of angle. Calling ten inches the limit of distinct vision, this means that about of an inch is the limit for linear measurement. An enormous improvement in accuracy is effected by the introduction of the microscope and telescope, the former for linear, the latter for angular measurements. Both depend upon the property of the objective lens of gathering together waves from a point, so that they meet again in a point, thus producing an image.
This is illustrated in Fig. 19. A train of plane waves traveling in the direction of the arrows encounters a convex lens. The velocity is less in glass, and since the lens is thickest at the center, the retardation is greatest there, gradually diminishing toward the edge. The effect is to change the form of the wave front from a plane to a spherical shell, 
FIG. 19which advances toward the focus at O, and produces at this point a maximum of light, which is the image of the point whence the waves started.
Fig. 20 illustrates the case where the convex waves diverging from a luminous point O are changed to concave waves converging to form the image at O'.
It can readily be shown that the luminous point and its image are in the same line with the center of the lens—sufficiently 
FIG. 20near for a first approximation. Accordingly, if we take separate points of an object, we can construct its image by drawing straight lines from these through the center of the lens, as shown in Fig. 21. The size of the image will be greater the greater the distance from the lens, so that the magnification is proportional to the ratio of the distances from object and image respectively to the center of the lens; hence in the microscope an error in determining the position of the image means a much smaller error in the determination 
FIG. 21of the position of the point source. This error could be diminished indefinitely by increasing the magnifying power, were it not for the attendant loss of light and the fact that the light waves, though very minute, are not infinitesimally small. In fact, the same diffraction effects again limit the possibility of indefinite accuracy of measurement. What, then, is the new limit?
Let p, Fig. 22, represent the center of the geometrical 
FIG. 22image of a luminous point. This will be a point of maximum brightness, because all parts of the concave wave which converges toward p reach this point at the same time, and therefore in the same phase. Let us consider an adjacent point q. The parts of the converging wave are no longer at equal distances from this point, and hence will not arrive in the same phase, and the brightness will be less than at p. At a certain distance pq there will be no light at all. This occurs when the difference of phase between the extreme ray and the central ray is half a wave, that is, calling the wave length l, when cq − bq = l; for these two pairs of rays destroy each other, and the same is true of every two such pairs of rays.
The same is equally true of every point about p at this same distance; hence there will be a dark ring about the bright image. This is succeeded by a bright ring, a second dark ring, and so on.
The radius of the first dark ring may be calculated as follows:
Draw qt at right angles to bp. Then cq − bq = l. But cq = cp, very nearly, and cp = bp, and bq = bt, so that bp − bq = pt = l.
But the triangles pqt and pbc are similar, whence pt:pq = bc:bp; or, calling r the radius of the first dark ring, F the focal length of the lens, and D the diameter of the lens, r = FD l; that is, the radius of the dark ring is greater than the length of the light wave, in the same proportion as the focal length of the lens is greater than its diameter.[3] For example, if the length of the light wave be taken as one fifty-thousandth of an inch, and the focal length of the lens as one hundred times the diameter, then this radius will be one five-hundredth of an inch—a quantity readily perceptible with a moderate eyepiece. The lack of distinctness of the image would be of the same order, and would be further aggravated by greater magnification, resembling a drawing made with a blunt point.
In most cases these diffraction rings are so small that they escape notice, unless the air is unusually quiet and the lens exceptionally good. If these conditions are satisfied, and the instrument is focused on a very small or distant bright object (a star, or a pinhole in front of an electric arc), the rings are readily visible
FIG. 23 with a sufficiently high-power eye-piece. They may be much more readily observed, however, if the ratio of diameter to focal length be diminished by placing a circular aperture before the lens. The smaller the aperture, the larger will be the diffraction rings. Fig. 23 is a photograph of the phenomenon, showing the appearance of the rings when the diameter of a lens of five meters' focal length has been reduced to one centimeter.
In the case of a telescope the corresponding limiting angle is the angle subtended by r at the distance F, i. e., rF, and this, by the formula, is the same as the angle subtended by the light wave at the distance D—the diameter of the objective. This limiting angle for a five-inch lens would, therefore, be of an inch, i. e., about the size of a quarter of a dollar viewed at the distance of a mile. This could be measured to within one-fifth of its value, so that the accuracy of measurement in this case corresponds to as against without the lens; i. e., the order of accuracy is increased about five hundred times.
For a microscope it will be simpler to proceed a little differently. The magnification increases as the object approaches the front of the objective lens. Suppose it is almost in contact. The waves from p (Fig. 24) reach o in the same phase, but those from q reach o more quickly through the upper half of the lens than through the lower half. Let the difference in the paths qao and qbo be l, that is, one of the light waves. Then there will be darkness at o so far as the
FIG. 24 point q is concerned; i. e., the dark ring in the image of q will lie at o and will thus coincide with the bright center of the image of p. This condition of affairs corresponds to a displacement pq = l. Hence, if there were two luminous points at a distance pq = l apart, their diffraction images would overlap so as to be indistinguishable from each other. Hence l , or of an inch, is the "limit of resolution" in any microscope, as against of an inch with the naked eye. So that here again the increase in accuracy is about four hundred times.
These theoretical deductions are amply confirmed by actual observation, and since in this investigation we have supposed a theoretically perfect lens, these results show that our present microscopes and telescopes, when operated under proper conditions, are almost perfect instruments.
Thus, Fig. 25 shows a micro-photograph of the specimen called Amphipleura pellucida, whose markings are about 100,000 to the inch. This is about the theoretical limit for blue light. By using the portion of the spectrum beyond the violet it might be possible to go still farther.
Doubtless by a much higher magnification a much more accurate setting on a given phase of the fringes could be made, and hence a corresponding increase of accuracy of measurement could be attained. But this involves a great loss of light, since the intensity varies inversely as the square of the magnification. Consequently, even with a threefold magnification the intensity is diminished ninefold, so that it would be difficult to see the image unless the illumination were so powerful as to endanger the specimen, or to introduce temperature variations which would vitiate the results of the measurement.
It is apparent from all that precedes that in all measurements by the microscope or the telescope we are, in fact,
FIG. 26 making use of the interference of light waves. Let us see, then, if we are making the best use of this interference, or whether it may not be possible to increase the high degree of accuracy already attained.
It has just been shown that, in the case of a telescope, the angular magnitude of the diffraction rings, and with this the accuracy of measurement of the position of the luminous point, depends only on the diameter of the objective. Now, the form of the fringes will of course vary with the form of the aperture, and if this be square instead of circular, the diffraction image will be represented by Fig. 26, which may be compared with Fig. 23. The width of the fringes is but little altered, while there is a perceptible increase in distinctness. Let the middle part of the aperture now be covered up, as in Fig. 27, so that the light can pass through the uncovered portions, a and b, only.
FIG. 27 Fig. 28 shows the appearance of the fringes in this case. The distribution is somewhat different, but the distinctness is considerably increased, so that the position of the center of any fringe (the central bright fringe, for instance) may be measured with a decided increase in accuracy. The utilization of the two portions of a lens, at opposite ends of a diameter, converts the telescope or microscope into an interferometer.
This term is used to denote any arrangement which separates a beam of light into two parts and allows them to reunite under conditions to produce interference. The path of the separated pencils may be varied in every possible way;
FIG. 28 for instance, by interposing prisms or mirrors, provided the optical paths are nearly equal and the angle between the two final directions very small. The first condition is essential only when the light is not homogeneous. The reason will be apparent when it is remembered that the width of the interference bands depends on the wave length of the light employed. If the light is composite, as in the case of white light, each component will form interference bands whose width is proportional to the wave length.
This is illustrated in Fig. 29, where the fringes due to red, yellow, and blue light respectively are separated. In the actual experiment, however, they are all superposed. At the middle point, where the two paths are equal, all the
FIG. 29 colors will be superposed, the result being a white central band. At no other point will this be true, and the result will be a series of 1 colored fringes symmetrically disposed about the central white fringe, the succession of colors being exactly the same as in the case of thin films (cf. Plate II).
The breadth of the fringes is determined by the smallness of the angle under which the two pencils meet. This is shown in Fig. 30. In the right-hand figure the angle between the pencils is smaller than in the other, while the breadth of the fringes is correspondingly greater in the former than in the latter. The exact
FIG. 30 relation is readily obtained. We have only to note that ac is the wave length l (very nearly) and bc is (very nearly) the width b of a fringe; whence, if c is the very minute angle at b (which is the same as the angle between the directions of the interfering pencils), b = le; or, in other words, the width of the fringes is proportional to the wave length of the light, and inversely proportional to the angle between the pencils.
Thus, if the pencils converge from two apertures a quarter of an inch apart, and meet at a screen ten feet away, the breadth of the fringes will be one-hundredth of an inch.
The importance of using a very small angle will be noted.
In this simple form of interferometer the angle can be made small only by bringing the two apertures very near together, which seriously diminishes the efficiency of the instrument; or by increasing the distance from the openings to the fringes, or by using a high magnification, which enfeebles the light, already very faint in consequence of having to start from a pinhole or a narrow slit s (Fig. 31) and to pass through the narrow apertures a and b. There is, therefore, but little advantage in this form of interferometer over the corresponding older analogues (microscope and telescope).
An important improvement may be effected by bending one or both the rays ap, bp by reflections in such a way as to diminish the angle at p, as shown in Fig. 32.
A further improvement is effected by replacing the apertures a and b by mirrors; and, finally, by replacing the slit s by a plane surface. The interferometer is now changed into the form illustrated in Fig. 33. It will now be noted that the source need no longer be a point or a slit, but may be a broad flame; and the object whose position is to
FIG. 32 be measured is no longer a fine line or a slit, but a flat surface. The width of the fringes may be made as great as we please without any sacrifice in the brightness of the light. The corresponding increase in accuracy is from twenty to one hundred fold. We may conveniently restrict the term interferometer to this arrangement, in which the division and the union of the pencils of light are effected by a transparent plane parallel plate. It is important to note that the path of the two pencils
FIG. 33 after their separation by the first plate is entirely immaterial; for example, either or both pencils may suffer any number of reflections or refractions before they are reunited by the second plate, without affecting in any essential point the efficiency of the interferometer, provided that the difference in the path of the two pencils is not too great, and
provided that the two pencils are reunited at a sufficiently
small angle. By altering these conditions of reflection or
FIG. 34 refraction we may obtain a very considerable number of variations of form, as illustrated in Figs. 34, 35.
One of these types, enlarged in Fig. 36, has been arranged
FIG. 35 in such a way as to show the extreme delicacy of the interferometer in measuring exceedingly small angles. For this purpose two of the
FIG. 36 mirrors, C and D, have been mounted on a piece of steel shafting P two inches in diameter and six inches in long. When the length of the paths of the two pencils is the same to within a few hundred thousandths of an inch, the interference fringes in white light are readily observed, or may be projected on the screen. If, now, the steel shafting be twisted, one of the paths is lengthened and the other diminished, and for every
FIG. 37 movement of one two-hundred-thousandth of an inch there would be a motion of the fringes equal to the width of a fringe. Now, taking the end of the steel shafting between thumb and forefinger, the exceedingly small force which may thus be applied in this way is sufficient to twist the solid steel shafting through an angle which is very readily observed by the movement of the fringes across the field.
The form of interferometer which has proved most generally useful is that shown in Fig. 38. The light starts
FIG. 38 from source S and separates at the rear of the plate A, part of it being reflected to the plane mirror C, returning exactly, on its path through A, to O, where it may be examined by a telescope or received upon a screen. The other part of the ray goes through the glass plate A, passes through B, and is reflected by the plane mirror D, returns on its path to the starting-point A, where it is reflected so as nearly to coincide with the first ray. The plane-parallel glass B is introduced to compensate for the extra thickness of glass which the first portion of the ray has traversed in passing twice through the plate A. Without it the two paths would not be optically identical, because the first would contain more glass than the second.
Some light is reflected from the front surface of the plate A, but its effect may be rendered insignificant by covering the rear surface of A with a coating of silver of such thickness that about equal portions of the incident light are reflected and transmitted.
The plane-parallel plates A and B are worked originally in a single piece, which is afterward cut in two. The two pieces are placed parallel to each other, thus insuring exact equality in the two optical paths AC and AD.
The foregoing principles are applied in concrete form in the instrument shown in Figs. 39, 40. A rigid casting serves as the bed of the instrument. One end of this bed has fastened to it a heavy metal plate H, which carries the three glass plates A, D, and B. The plate A is held in a metal frame which is rigidly fastened to the plate H. The frame which holds B can be turned slightly about a vertical axis to allow of adjusting
FIG. 39 B so that it is parallel to A. The mirror D is held by springs against three adjusting screws which are set in a vertical plate attached to the end of the plate H. Both C and D are silvered on their front faces. The frame which holds the mirror C is firmly mounted on a metal slide which can be moved by the screw S along the ways EF. One very essential feature of the apparatus is that these ways shall be so true that the mirror C shall remain parallel to itself as it is moved along. The accuracy of the ways must be so great that the greatest angle through which the mirror C turns in passing along them is less than one second of an arc. This accuracy cannot be attained by the instrument maker, but the final grinding must be done by the investigator himself.
To adjust the instrument so that fringes are formed, a small object like a pin is held between the source and the plate A. Two images of this pin will be seen by an observer at O—one formed by the light which is reflected from C, and the other by that reflected from D. The fringes in monochromatic light will appear when these two images have been made to coincide with the help of the adjusting screws ss. The fringes in white light appear only when the lengths of the two paths AD and AC are the same. The
FIG. 40 width and the position of the fringes in the field of view can be varied by slightly moving the adjusting screws. We shall have occasion to discuss this particular form of interferometer in a subsequent lecture.
SUMMARY
1. The objection to the wave theory of light, that light moves in straight lines while sound waves can bend around an obstacle, is shown to be groundless, since we have seen that if the sound waves are sufficiently short they cast a sound shadow, while by devices which take into account the extreme minuteness of light waves their bending around obstacles may be readily observed.
2. The extreme minuteness of light waves renders it possible to utilize the microscope and the telescope as instruments of great precision. These instruments depend on the property of the objective of gathering together waves from a point so that they are concentrated in the diffraction pattern which is called the image.
3. The accuracy of measurement is still further increased by modifying the telescope or microscope so as to utilize only two pencils, thus converting these instruments into interferometers.
4. By the device of separating the two pencils and reuniting them by reflections from plane-parallel surfaces, the fringes may be made as large as we please without diminishing the brightness of the light, and hence the accuracy of measurement may be correspondingly increased.
- ↑ Exhibited by Lord Rayleigh at the Royal Institute.
- ↑ The disc may be glued to a piece of optical glass, care being taken that no trace of glue appears beyond the edge of the disc.
- ↑ Strictly, this is about one-fourth greater on account of the fact that the aperture is circular instead of rectangular.