On Newton's Rings.

199. In the investigation of Newton's rings, which is given in Chapter III., the assumption is tacitly made, that the angle of incidence is less than the critical angle; but when these rings are formed between the under surface of a prism and the upper surface of a lens, there is no difficulty in increasing the angle of incidence on the under surface of the prism, until it exceeds the critical angle. Under these circumstances it is found, that as soon as the critical angle is passed, the rings disappear, but the central black spot remains. Now the theories of Green and Fresnel, and also the electromagnetic theory, as we shall see later on, show that when the angle of incidence exceeds the critical angle, a superficial wave exists in the second medium, which penetrates a few wave-lengths, and then disappears. In the experiment of Newton's rings, the stratum of air between the prism and the lens is so exceedingly thin, as to be comparable with the wave-length of light; accordingly, the existence of the central spot beyond the critical angle was attributed by Dr Lloyd1 to the disturbance in the second medium, which takes the place of that which, when the angle of incidence is less than the critical angle, constitutes the refracted light. We shall therefore proceed to give an investigation due to Stokes, in which the superficial wave is taken into account.

200. Adopting the notation for the colours of thin plates, let the amplitude of the incident vibration in the glass be taken as unity, let b, c be the amplitudes of the reflected and refracted vibrations, when light passes from glass to air, and let e and ƒ be the corresponding quantities, when light passes from air to glass. Also let us first suppose, that the stratum of air is bounded by two parallel planes at a distance D.

Taking the origin in the surface of the upper plate of glass, let the incident vibration in the glass be represented by

€TM, ( − x cosi+y sin i − V1t),

where 1 = 2, A being the wave-length in glass. Then the reflected vibration will be represented by

beki (x cosi+y sin i – V1t) ̧

1 Report on Physical Optics, Brit. Assoc. Rep., vol. 1. p. 310.

2 Trans. Camb. Phil. Soc. vol, vIII. p. 642; Math. and Phy. Papers, vol. 11. p. 56.

Since the refracted wave is a superficial wave, it may be represented by

Cεx+ (y sin i− Vt)

where x = 2π/, λ being the wave-length in air.

The quantities b and c are complex quantities, and their values as well as the value of a,, will depend partly upon the particular dynamical theory which we adopt, and partly also upon whether the light is polarized in or perpendicularly to the plane of incidence.

When the superficial wave is reflected at the second surface, the amplitude of the reflected wave will be ceq, where This wave will be again reflected and refracted at the first surface, and the amplitudes of the two vibrations will be ce q2 and cefq respectively; whence taking into account the infinite series of reflections, we obtain for the amplitude of the vibration which is finally refracted into the glass, the expression

b+cefq2 (1 + eq2 + e1q1 + ......) = b+

cefq

1 - e2q2*

..(48).

201. It will now be necessary to distinguish between light polarized in, and light polarized perpendicularly to the plane of incidence. Taking the first, it follows from Fresnel's theory that

a1 = (2π/λ) (u2 sini-1),

where X is the wave-length in air, and μ is the index of refraction from air to glass. Also from the same theory, see § 173,

Writing Te, the coefficient in (48) may be written

=

202. When the light is polarized perpendicularly to the plane of incidence, we must write e' for 0, where

tan 0′ = μ (μ2 tan2 i — sec2 i)1

.(52).

In either case the reflected vibration is accordingly equal to

and is equal to 0 or ', according as the light is polarized in or perpendicularly to the plane of incidence.

203. The corresponding formulæ for the transmitted light can be obtained in a similar manner. When the superficial wave arrives at the second surface, its amplitude is equal to cq; the amplitude of the refracted portion is cqf, and that of the reflected portion is cqe; whence after one reflection at the first plate, and one refraction at the second plate, the portion refracted at the latter becomes cq'ef. Hence the amplitude of the refracted portion is

q(01-↓)
b-1-q-b

In equations (54) and (55), tan 0 is determined by the righthand sides of (49) or (52), according as the light is polarized in or perpendicularly to the plane of incidence.

204. In examining these expressions, we shall suppose that the rings are formed by a lens in contact with a prism or a plate of glass. Hence at the point of contact, D=0, and therefore by (51) q= 1; accordingly p2 = 0, or there is absolute darkness. On receding from the point of contact, q decreases, but slowly at first, inasmuch as Dor, where r is the distance from the point of contact; whence the intensity ultimately varies as r, and therefore increases with extreme slowness. There is consequently a black spot at the centre. Further on q decreases rapidly, and soon becomes insensible; hence as we proceed from the black spot, the intensity at first increases rapidly, and then slowly again as it approaches its limiting value unity, to which it soon becomes sensibly equal.

205. We shall now consider how the intensity varies with the colour. Since μ is a function of X, it follows that 0 and ' depend upon the colour, but the variations in passing from one colour to another are so small, that they may be left out of account. From (51) we see, that as the distance from the point of contact increases, 7 decreases more rapidly when A is small; accordingly the spot must be smaller for blue light than for red light, and therefore the black spot must be bordered by a ring of blue light. On the other hand towards the edge of the bright spot, which is seen by transmission, the colours at the red end predominate.

206. We shall next consider, how the spot depends upon the nature of the polarization. Let s be the ratio of the transmitted light to the reflected; s1, s, the particular values of s, belonging to light polarized in and perpendicularly to the plane of incidence; then

4q sin2 20
(1 − q2)2

sin: 20
sin' 20'

=

4q' sin2 20'
(1 − q2)2

{(+1) sini-1}............(56).

Now the distinctness of the black spot, which is produced by reflection, depends upon s being large; and since in the neighbourhood of the critical angle, we have from (56), 88,μ, it follows that the spot is much more conspicuous for light polarized

perpendicularly to the plane of incidence, than for light polarized in that plane. As i increases, the spots seen in the two cases become more and more nearly equal, and become exactly alike when i', where cosec2 = (1 +μ2). When i becomes greater than, the order of magnitude is reversed, and when i=1, S=8, so that the inequality becomes large. It must however be recollected, that this statement refers to the relative magnitudes of the spots, for when the angle of incidence is nearly equal to π, the absolute magnitudes of the spots become very small.

All the conclusions deduced by the above theory have been verified experimentally by Stokes, and he has also discussed the case in which the incident light is polarized in a plane, making a given angle with the plane of incidence.

The Intensity of Light reflected from a Pile of Plates.

207. A common method of producing polarized light is by means of a pile of plates, and we shall now give an investigation, due to Stokes, of the intensity of the light reflected from, or transmitted through a pile of plates'.

The plates will be supposed to be formed of the same material, to be of equal thickness, and to be placed parallel to one another, and the plates and the intermediate layers of air, to be sufficiently thick to prevent the occurrence of the colours of thin plates.

In consequence of absorption, the intensity of light traversing an elementary distance da of a plate, will be reduced in the proportion of 1 to 1-qda, where q is a constant. Hence if I, I + dl be the intensities of the light on entering and emerging from a layer, whose thickness is da,

whence if is the thickness of the plate, r the angle which the direction of the light makes with the normal to the plate, and I, I, the intensities on entrance and emergence, we obtain from (57)

I1 = I ̧ € - 7q sec r = Ig (say).

0

The constant q, which may be called the coefficient of absorption, depends upon the material of which the plate is made; also since the amount of light absorbed varies with its refrangibility, also depends upon the colour.

1 Stokes, Proc. Roy. Soc. vol. xi. p. 545.