CHAPTER VIII.

COLOURS OF CRYSTALLINE PLATES.

129. IN the present chapter, we shall discuss one of the most striking and beautiful phenomena in the whole science of Optics, viz. the production of coloured rings by thin crystalline plates. These rings were discovered by Arago1 in 1811, and we shall first give a general explanation of their formation.

When plane polarized light is incident upon a crystalline plate, the incident vibrations, upon entering the plate, are resolved into two components, which are polarized in perpendicular planes, and travel through the plate with different velocities; hence the phases of the two components upon emergence are different. If the angle of incidence is small and the crystal is thin, the two emergent rays are sensibly superposed; but since they are polarized in different planes, they are not in a condition to interfere. If however the emergent rays are passed through a Nicol's prism, each ray on entering the prism is again resolved into two components, which are respectively parallel and perpendicular to the principal section of the Nicol; the two latter components cannot get through the Nicol, whilst the two former components being brought into the same plane of polarization by the Nicol, and being already through the action of the crystalline plate in different phases, are in a condition to interfere. We thus perceive, how it is that coloured rings are produced by the action of a thin crystalline plate.

The apparatus (frequently a Nicol's prism), which is used to polarize the light which falls upon the crystal, is called the

1 Euvres Complètes, vol. x. p. 36.

polarizer, and the second Nicol is called the analyser. The planes of polarization of the light, which emerges from the polarizer and the analyser, are respectively called the planes of polarization and analysation.

130. We are now prepared to consider the mathematical theory of these rings.

Let OA, OB be the principal planes of the crystal at any point O, the former of which corresponds to the ordinary ray, and the latter to the extraordinary ray. Let OP be the direction of vibration of the incident light, so that if the light is polarized by a Nicol, OP is its principal section; and let OS be the principal section of the analyser. Let POA = a, POS = B; also let the vibration which is incident upon the crystal be represented by sin 2πt/T.

On entering the crystal, the incident vibration is resolved into

cos a sin 2πt/T

along OA, which constitutes the extraordinary ray, and

sin a sin 2πt/T

along OB, which constitutes the ordinary ray. The waves corresponding to these rays travel through the crystal with different velocities, and therefore on emergence, the two vibrations may be written

cos a sin 2π (t/T-EX), and sin a sin 2π (t/T - O/X) where A is the wave length in air, and O and E are the thicknesses of two lamina of air, such that light would occupy the same times in traversing them, as are occupied by the ordinary and extraordinary waves in traversing the crystalline plate.

On entering the analyser, only those vibrations can pass through which are parallel to OS; whence resolving the two vibrations in

INTENSITY OF EMERGENT LIGHT.

139

this direction and putting = t/r – O/λ, the resultant vibration on emerging from the analyser is represented by

cos a cos (a — ẞ) sin 2π {☀ + (0 − E)/λ} + sin a sin (a — ß) sin 2πÞ.

The intensity of the emergent light is therefore represented by I2 = {cos a cos (a – ẞ) cos 2π (0 – E)/λ + sin a sin (a – B}2

+ cos a cos2 (a-B) sin2 2π (0-E)λ

= cos ẞsin 2a sin 2 (a-B) sin2 π (0-E).

(1).

If in this expression we write + B for B, which amounts to turning the analyser through an angle of 90°, we obtain

whence

I'2 = sin2 ß + sin 2a sin 2 (a — ẞ) sin2 π (0 – E)/λ....... (2),

We therefore see, that the effect of turning the analyser through an angle of 90°, is to transform each colour into its complementary one.

131. Let us now suppose, that the incident light consists of a small pencil of rays converging to a focus, every ray of which makes a small angle with the normal to the crystalline plate.

Let P be the point of incidence, i the angle of incidence, PQ the front of the incident wave. At the end of time t, let TI, TO,

T

TE be the fronts of the incident, ordinary and extraordinary waves; and let v, u, u' be their velocities of propagation. Draw PO, PE perpendicular to TO, TE; and let r, r' be the angles which these perpendiculars make with the normal to the plate. Then

If the thickness of the crystal and the angle of incidence are small, the ordinary and extraordinary rays will be superposed on emergence. Hence if OM be the wave-front on emergence corre

sponding to the incident wave PQ, the difference between the times which the ordinary and extraordinary waves occupy in travelling from P to OM, is equal to

PO/u- PE/u' - EM/v,

which is equal to (0 – E)/v; whence

0-E=(v/u) PO-(v/u') PE-EM.

Accordingly if T be the thickness of the plate,

We thus see that the mathematical solution of the problem is reduced to the determination of the angles r, r', in terms of the angle of incidence and the optical constants of the crystal. Their values depend upon the particular kind of crystal under consideration, and the inclination of the face of the plate to the directions of the principal wave velocities in the crystal.

Uniaxal Crystals.

132. We shall now consider the coloured rings produced, when the crystalline plate consists of any uniaxal crystal, except crystals of the class to which quartz belongs.

PLATE PERPENDICULAR TO THE AXIS.

141

Plate cut perpendicularly to the Axis.

133. Let the axis of x be perpendicular to the faces of the plate; then in the present case, the wave surface in the crystal consists of the sphere

and the ellipsoid

x2 + y2+ z2 = b2

a2x2 + b2 (y2 + 22) = a2b2.

In negative crystals such as Iceland spar, the velocity of the ordinary wave is less than that of the extraordinary, and therefore a>b; hence the ellipsoidal sheet of the wave-surface lies outside the spherical sheet, and is planetary. The reverse is the case with positive crystals, for which the ellipsoidal sheet is ovary.

In the figure, P is the point of incidence, PB the axis of the crystal; let PQ be the front of the incident wave, and let TO, TE

Q

P

T

B

be the fronts of the ordinary and extraordinary waves at the end of unit of time. Then if p be the length of the perpendicular drawn from P on to TE,

a2 sin2 r' + b2 cos2 r' = p2= PT sin3 r' = v2 sin2 r'/sin2 i by (3) and (4); therefore

Similarly

whence

cot r' = (v2 - a2 sin2 ¿)3/b sin i.

b= PO = PT sin r = v sin r/sin i,

cot r= (v2-b2 sin i)/b sin i;

accordingly we obtain from (5)

0 - E = Tb-1 {(v2 — b2 sin2 ¿)3 — (v2 — a2 sin2 )1} ...... (6).

Now i the angle of incidence is a small quantity; if therefore we expand the right-hand side of (6), and neglect sini, we obtain 0 - E = (T/bv) (a2 — b2) sin2 i