distortional wave of homogeneous type, are the same as those due to central attracting forces. It is also clear that a plane polarized wave cannot be transmitted as a single wave unless the force of restitution is in the direction of the displacement. If we disregard longitudinal waves as having no reference to the phenomena of light, we need only consider that component of the force which acts in the plane of the wave. This consideration leads to Fresnel's construction. For if we take the ellipsoid

a2x2 + b2y2 + c2x2 = 1,

which, as we now see, is quite appropriately called the ellipsoid of elasticity, a central section parallel to the wave-front gives an ellipse which, by its principal axes, indicates the two directions of displacement which are compatible with a transmission of a single plane wave. The periods of oscillation are proportional to the axes of this section, and as for a given wave-length the periods of oscillation are inversely proportional to the velocity of transmission, it follows that the velocities of the plane waves parallel to the section are inversely proportional to the axes of the ellipse of intersection. We have thus arrived at the construction which has formed the starting point of our discussion of the phenomena of double refraction (Art. 81).

The direction of the elastic force for any displacement being parallel to the normal to the ellipsoid of elasticity, drawn at the point at which the direction of the displacement intersects the ellipsoid, the proposition proved in Art. 86 shows that the four vectors representing the direction of vibration, the elastic force, the ray and the wavenormal are coplanar.

CHAPTER IX.

INTERFERENCE OF POLARIZED LIGHT.

104. Preliminary Discussion. If a plane unpolarized wave enters a plate of a doubly refracting substance, the two waves inside the crystal travel with different velocities and in slightly different directions, but on emergence both waves are refracted so as again to become parallel to their original directions. If the wave was originally polarized in one of two definite directions, it is observed that there is only one refracted wave. The two planes of polarization for which. this is the case, are observed to be at right angles to each other. Inside the crystal the two waves are polarized in directions nearly though not quite at right angles to each other. After emergence the planes of polarization are at right angles to each other, not only approximately, but strictly. This follows from the principle of reversion, assuming the above-mentioned result of observation.

In general, a wave of polarized light incident on a doubly refracting plate becomes polarized elliptically. The axes of the ellipses vary with the wave-length and the thickness of material travelled through, hence also with the direction of the incident light, and the ellipse may, in particular cases, become a straight line or a circle. If the emergent light is examined through a Nicol prism or any arrangement which transmits oscillations in one direction only, colour effects are observed

which we shall have to explain in greater detail. It is clear, however, that any interference effect must depend on the difference of phase in the two overlapping emergent waves. Let LO (Fig. 139) be an incident ray, forming part of a parallel beam, OA and OB the refracted wave normals, AS and BT the emergent wave normals. Draw AK at right angles to BT and BH at right angles to OA. Imagine a second incident ray, parallel to the first, and at such a distance that the wave normal which

K

Fig. 139.

is parallel to OA passes through B, and is refracted outwards along BT; then from the principle of wave transmission it follows that the optical length of BK is the same as that of AH. In the emergent OB он 21 V2

wave-front, the difference in optical length is therefore. where v1 and v2 are the velocities of the waves along OB and OH respectively. (The unit time is still taken to be such that the velocity of light in vacuo is one.) The angle between OB and OA is small, and if we neglect its square, we may write OB = OH. The difference in ρ is the length of that

optical length is therefore p (-1), where

wave normal inside the plate, which lies nearest to the plate normal. Unless the incidence is very oblique, it makes no difference, to the degree of approximation aimed at, along which wave normal p is measured, but for the sake of definiteness, we adhere to the specified meaning of p. If OB and OA represent the refracted rays, we argue similarly that by Fermat's principle, optical lengths may be measured along a path near the real one, committing only an error of the second order. The optical length for the ray of velocity s might therefore be measured either along its real path OA or along its neighbour OB + BK, ending, of course, in the same wave-front. We may there

fore also express the difference in optical length as t

where t

is the length of ray inside the crystal and 81, 82, are the ray velocities. We may, according to convenience, use either one or the other two forms, which are both approximate only. Which of these is the more accurate in a particular case depends on the question as to whether the angle between the two ray velocities or between the two wave normals is the smaller. In the neighbourhood of the optic axes, it is preferable to refer the relative retardation to the wave normals.

105. Intensity of illumination in transmitted light. Consider polarized light with its direction of vibration along OP (Fig. 140), falling normally on the surface of a crystal which divides the wave into two portions, one vibrating along OX and one along. OY. After traversing the thickness of the plate, the two waves emerge normally with a difference of phase 8 depending on the difference in optical length of the two wave normals inside the crystal. If the amplitude of the incident light is one, the emergent waves have amplitudes cos a, sin a, if a is the angle between OP and OX, there being a difference in phase & between them. If now the emergent beam be examined through a Nicol prism called the analyser," transmitting light only which vibrates along OA, the

66

Fig. 140.

component k1 of the transmitted light due to that portion which in the crystal had OX for its direction of vibration, is k1 = cos ẞ cos a ; similarly k2 = sin ẞ sin a is that component of the light which, having OY for the direction of vibration inside the crystal, is capable of traversing the analyser.

Two rays of amplitude k, and k1⁄2 and phase difference 8, polarized in the same direction, have a resultant, the intensity of which is

Substituting the values of k, and k2 the intensity of the emergent beam becomes

I = cos2 (Ba) – sin 2a sin 2ẞ sin2

2

(1).

All colour or interference effects shown by crystalline plates when examined by polarized light, depend on the application of this formula. So long as there is only one parallel beam, the plate having the same thickness everywhere, all the quantities are constant, and the plate appears uniformly illuminated. Important particular cases are those in which the Nicols are either parallel (a = 6), or crossed at right angles

This relation is a particular case of the general law that if for any value of a and B, II, and if I becomes I on turning either the analyser or polarizer through a right angle, then I1+ I = 1.

A

We may convince ourselves that this is true without having recourse to the equations. The light falling on the analysing Nicol is partly transmitted and partly deviated to one side, the two portions making up together the incident light which is supposed to be white. On rotating the Nicol through a right angle the transmitted and deviated portions are interchanged so that the complementary effect must be observed.

When white light passes through the plate, the relative proportion of different colours is not in general preserved because & depends on the wave-length. If a is the amplitude of light of a particular wavelength, so that white light may be represented by Za2, the light transmitted through the system is

inc).

cos2 (a – ß) Za2 – sin 2a sin 2ẞ Σ (a2 sin2

The first term represents white light of intensity proportional to cos2 (a B), and the second term represents coloured light. The relative proportion of the different wave-lengths is not affected by a change in a or ẞ, but the total colour effect may change because the product sin 2a sin 2ẞ may be either positive or negative. In the first case, we get a certain colour, in the second, white light minus that colour, i.e. the complementary colour. We distinguish two special cases.

have

Case 1. The Nicols are crossed so that a В

I = sin3 2a Σ (a2 sin2o).

The colours are most saturated in this case, because there is no admixture of white light. As the axes of x and y are fixed in the crystal, we may vary a without change of a-ẞ by turning the crystalline plate in its own plane. There will then be four places of maximum intensity at which a=45° or an odd multiple thereof, and four places of zero intensity at which a is a multiple of 90°.

Case 2. The Nicols are parallel so that a =

B. Here we have

Σ (a
(a2 sin2o).

I = Σα? - sin2 2α Σ

The colour here is always complementary to that in the previous case for the same value of a, the light being white when a is a multiple of a right angle, and most saturated when a is an odd multiple of 45°.

If for any value of a and ẞ, the crystal is turned in its own plane, there are eight positions at which sin 2a sin 2ẞ vanishes; these occur whenever one of the axes OX and OY coincides with the principal planes of either the polarizing or analysing Nicol. In these positions of the crystal, the light is white, and on passing through these positions, the colour changes into its complementary.

106. Observations of colour effects with parallel light. The general experimental arrangement by means of which the colour effects of polarized parallel light may be shown, is sketched dia