series of coloured patches and streaks will be formed, for light of the different colours which constitute white light will be destroyed by interference at different points, the thickness being given by the above expression, and the reflected light will, by the loss of the destroyed rays, appear coloured.

376. Newton's Rings.-When a convex lens of large radius is pressed on a flat piece of glass or on a concave glass surface of greater radius of curvature than that of the lens, the point where the lens touches the glass will be seen surrounded by a series of dark rings if the light is monochromatic, or of coloured rings if white light is used. These rings, which are known as Newton's rings, are produced by interference in the thin film of air enclosed between the two glass surfaces, and may be seen both in the reflected and in the transmitted light.

If SOPN (Fig. 357) represents a section of the sphere from which the lens may be supposed to be cut, and AB the glass plate, which the lens

N

C

R

A

FIG. 357

touches at O, then the thickness of the air film included between the lens and the plate is zero at O and increases as we pass out from 0.

Let Q be a point at a distancer from the point of contact, then the thickness of the air film at Q can be found as follows. Draw QP perpendicular to AB to meet the circle, and through P draw Ps parallel to AB, cutting the diameter, NO, of the circle in R. Then by a well-known property of the circle PR. RS=OR.RN= B. (ON-OR)OR.

Hence if R is the radius of curvature of the surface of the lens, and OR or PQ,

the thickness of the air film, is called T, we have

=(2R-T)T=2RT-T2.

Now since, when interference takes place, the thickness T of the air film is always very small compared to the radius of the lens R, the quantity T2 will be very small compared to 2RT, so that we may neglect T", and

We have already seen that in the case of reflected light interference will take place if T

ηλ
2μ cos β'

where was the wave-length in the

medium outside the film. If A, is the wave-length in the medium outside the film, and the wave-length in the film, then

where A is the wave-length in the film.

In the case of Newton's rings we are dealing with air as the film, and so is here the wave-length in air, so that we shall have a dark ring passing through Q if

or if

2R

πλη
2 cos B

If the lens and plate are in contact at the centre, we shall get interference, as we have already shown, and there will be a black spot at the centre; the radii of successive dark rings will be obtained by taking n equal to 1, 2, 3, &c., so that the squares of the radii of successive dark rings are proportional to the natural numbers, 1, 2, 3, &c. The angle ẞ is that which the light rays in the air film make with the plate, but in all practical cases the lens is of such small curvature, and the rings are only formed so near the centre, that we may regard the lens as a parallel plate, so that the rays in the air film will be parallel to the rays incident on the upper surface of the lens, and we may take as the angle of incidence of the rays on the lens.

If the light is incident normally ẞ=0, and

r2=nλR.

At the centre there is interference for all the colours, so that with white light the centre is black, as we pass out; if λg is the wave-length of violet light, then when is equal to R this violet light will be destroyed, and hence the remaining light will, along this circle, appear coloured red. A little further out, r is equal to R, so that the red light is destroyed, and the remaining light appears violet. When is equal to 2R, the violet will again be destroyed and the red left, while when is equal to 2R, the violet will be left. Thus with white light the central black spot will be surrounded by a series of coloured rings, each of which is red on the inside and violet on the outside.

Newton's rings are also formed in the light which is transmitted

through the lens and plate. If AB (Fig. 358) is the surface of the plate,

The ray IS, as it is nowhere reflected, undergoes no sudden change in phase; the ray IPQ'S', however, is reflected at P and at Q', and in each case at the surface of a denser medium, and loses at each half a wavelength, or a whole wave-length in all. Hence, as the loss or gain of a whole wave-length by one ray does not affect the interference phenomena between two rays, we have that interference will take place when the difference in the paths is equal to an odd multiple of the half wavelength. Hence there will be a dark ring passing through Q if

When n=o, r'=e, so that there will be a bright spot at the centre, as is also obvious since here the lens and plate are in contact, so that there is no air film, and the light is simply transmitted. By comparing the expressions for r in the case of reflected and transmitted light, it will be seen that where there is a dark ring for one, there will be a bright ring for the other.

✓ 377. Stationary Waves.-Lippmann's Colour-Photography.— Suppose a beam of parallel rays, or, in other words, a series of plane waves, is incident normally on a plane mirror, then the waves will be reflected at the mirror, and we may, as in

P: P

the case of water waves (§ 275), have sta- A C C' tionary waves set up owing to the interference between the direct and reflected waves. Consider a point P (Fig. 359), at 0 a distance from the mirror, then we may consider that at P we have two series of waves, one the direct waves and the other a series, which, starting in the same phase as the direct waves, has travelled along a path which exceeded that of the direct waves by 2x. We must also add half a wave-length, for the reflection at O takes place at the surface

B D

D'

FIG. 359.

of a denser medium. Hence the difference of path is really 2r+ 2 There will be interference at P, if this difference in path is equal to an odd number of half wave-lengths, or if

If then P is a point such that PO=X/2, there will be interference throughout a plane CD, drawn through P parallel to the reflecting surface; there will also be interference throughout the plane C'D', which is at a distance of A from AB, and so on. The distance between the planes in which interference occurs will vary with the wave-length of the light, being smaller for violet light than for red light. The distance between consecutive planes, even for red light, is, in the case of normal incidence, excessively small, being only 3.8 × 105 cm. for the red (A line).

The formation of these planes, over which interference takes place, in the neighbourhood of a plane mirror has been utilised by Lippmann in his excessively beautiful method of obtaining photographs in natural colours. If the front surface of the mirror is coated with a sensitive photographic emulsion, then when light of wave-length X' is incident on the mirror the light will be destroyed in planes which are at a distance of X'/2, 2X′/2, 3\/2, &c., from the mirror, so that the emulsion will not be affected on these planes. At the planes at distances X'/4, 2X/4, 3/4, &c., the incident and reflected light strengthen each other, and the emulsion will be acted upon, so that on development the silver of the emulsion will be reduced on these planes, and thus a number of parallel planes at a uniform distance apart will be produced within the film.

If now a beam of white light is incident on the developed plate, the interval between each of the planes in which the silver has been deposited

will act as a thin film producing interference in the manner considered in § 375 between the light reflected from two consecutive planes in which the silver is deposited. The difference in phase between the light

reflected at plane 1 (Fig. 360), and that reflected at plane 2, will be equal to twice the distance between the planes. Hence, since the distance between the planes is X'/2, the difference in the paths is X', so that, in the case of the component of the incident white light which has the wave-length X, the two reflected rays combine to strengthen each other. For light of all other wavelengths the two reflected rays will differ in phase, and will therefore more or less interfere. If there were only two planes the selective strengthening of the reflected light of one wave-length would not be very marked; when, however, there are hundreds of planes placed one after the other, the final result is that practically only light of wave-length is reflected, that is, light of the same colour as that originally incident on the sensitive film.

FIG. 360.

If then, instead of using homogeneous light to act on the sensitive film, light of different colours in different parts is used, such as would be obtained if the image of a party-coloured object formed by a lens is thrown on the film, then on development the silver will be so deposited that, when the film is afterwards illuminated by white light, the light reflected from different parts of the film will be of the same colour as the corresponding part of the original image, and hence of the object, and we shall thus get a photograph in natural colours.

378. Michelson's Interference Apparatus.-In the cases of interference which we have hitherto considered, the difference in the length of the paths of the interfering waves has only amounted at most to a few hundred wave-lengths. Michelson has, however, obtained interference when the paths differed by as much as 20 cm., i.e. about 400,000 times the wave-length. His apparatus consists of two parallel-sided plates of glass, G1 and G2 (Fig. 361), of equal thickness, and two plane mirrors, M1 and M, arranged as shown in the figure. The surface of the glass plate G1, which is turned towards the mirror M1, is lightly silvered, so that when light is incident at an angle of 45° on this surface, half the light is reflected and half is transmitted through the thin coating of silver.

If a parallel beam of light is incident on the plate G, along the direction 10, the greater part will be refracted and traverse the plate. When this light meets the silvered surface half will be reflected, and after again traversing the glass plate will be incident normally on the mirror M.; the other half will be transmitted, and after traversing the plate G will be incident normally on the mirror M. The light which falls on the mirror M, will be reflected back along its path, will again traverse the plate G, and will then be partly reflected at the silvered surface of G1 along O'R1. The light which falls on the mirror M, will be reflected back