CHAPTER VII.

FRESNEL'S THEORY OF DOUBLE REFRACTION.

100. WHEN the disturbance which produces light is excited at any point of an isotropic medium, a spherical wave is propagated from the centre of disturbance with constant velocity; but we have pointed out in the preceding chapter, that when the disturbance is excited in a doubly refracting medium, two waves are propagated with different velocities, and that when the medium is a biaxal crystal, the velocity in any given direction is a function of the inclination of this direction to the optic axes of the crystal.

The laws regulating the propagation of light in crystals, were first investigated mathematically by Fresnel, who showed that the wave-surface in biaxal crystals, is a certain quartic surface, which reduces to a sphere and an ellipsoid of revolution in the case of uniaxal crystals. The theory by means of which Fresnel arrived at this result, cannot be considered to be a strict dynamical theory; but on account of its historical interest, and also owing to the fact that experiment has proved that Fresnel's wave-surface is a very close approximation to the true form of the wave-surface in biaxal crystals, we shall proceed to explain its leading features, and afterwards discuss the geometry of this surface.

101. The theory of Fresnel depends upon the following four hypotheses, which are thus summarized by Verdet1.

(i) The vibrations of polarized light are perpendicular to the plane of polarization.

(ii) The elastic forces which are produced by the propagation of a train of plane waves, whose vibrations are transversal and

1 Leçons d'Optique Physique. Vol. 1. p. 465.

FUNDAMENTAL ASSUMPTIONS.

113

rectilinear, are equal to the product of the elastic forces produced by the displacement of a single molecule of that wave, into a constant factor, which is independent of the direction of the wave.

(iii) When a wave is propagated in a homogeneous medium, the component of the elastic forces parallel to the wave-front, is alone operative.

(iv) The velocity of propagation of a plane-wave, which is propagated in a homogeneous medium without change of type, is proportional to the square root of the effective component of the elastic forces developed by the vibrations of that wave.

102. We have stated in the preceding chapter, that according to the generally received opinion, the vibrations of the ordinary wave in a uniaxal crystal are perpendicular to the plane containing the direction of propagation and the optic axis, whilst the vibrations of the extraordinary wave are executed in the corresponding plane. Up to the présent time no experiments have been described which prove that this is the case, and consequently for all we know to the contrary, the vibrations of the ordinary wave night take place in the principal plane, whilst those of the extraordinary wave might be perpendicular to that plane. We shall hereafter show, that there are strong grounds for supposing, that the vibrations of polarized light are perpendicular to the plane of polarization; but for the present Fresnel's first hypothesis must be regarded as an assumption.

103. The second and third hypotheses require careful consideration, and it will be convenient to discuss them together.

Since the motions of the ether which constitute light are of a vibratory character, it follows that the ether when undisturbed, must be in stable equilibrium. Hence if F (x, y, z) = V be its potential energy at any point x, y, z; and if a particle situated at this point be displaced to the point x + u, y + v, z+w, it follows that dV/dxdV/dy = d V/dz = 0; and therefore expanding by Taylor's theorem,

V = V2+ {(Au2 + Bv2 + Cw2 + 2A'vw + 2B'wu + 20'uv), where V, is the constant potential energy when in equilibrium, and A, B... are positive constants. By properly choosing the axes, the products may be made to disappear; whence omitting the

B. O.

8

constant term Vo, which contributes nothing to the forces, the value of V may be written

whose centre is 0, and if we draw a radius OP parallel to the direction of displacement and meeting the ellipsoid in P, and OY be the perpendicular from O on to the tangent plane at P, then OY will be the direction of the resultant force.

The ellipsoid (3) is called the ellipsoid of elasticity; and it follows from the preceding construction, that the resultant force will not be in the direction of displacement, unless the displacement is parallel to one of the principal axes of this ellipsoid.

If l, m, n be the direction cosines of the normal to the wavefront; λ, u, v those of the direction of displacement, it follows that the resultant force of restitution will not be in the plane of the wave-front. This force may however be resolved into two components, one of which is in the plane of the wave, and the other is perpendicular to it. The latter component according to the third hypothesis will not give rise to vibrations which produce light, and therefore need not be considered. The former component will give rise to vibrations which produce light; but it will not coincide with the direction of displacement, unless the latter coincides with that of one or other of the principal axes of the section of the ellipsoid of elasticity by the plane lx + my + nz = 0. For the direction cosines of the force are proportional to aλ, b3μ, cv; and the condition that this line, the direction of the displacement, and the normal to the wave-front should lie in the same plane, is

which is the condition, that the line λ, μ, v should be a principal axis of the section of the ellipsoid of elasticity by the wave-front.

VELOCITY OF PROPAGATION.

115

Let us now suppose, that plane waves of polarized light are incident normally upon a crystalline plate, the direction cosines of whose face, with respect to the principal axes, are l, m, n. Let OA, OB be the directions of the axes of the section of the ellipsoid of elasticity made by the surface of the plate, and OP the direction of vibration of the incident light. If the second medium were isotropic instead of crystalline, a single refracted wave would be propagated, consisting of light polarized in a plane perpendicular to OP and the surface of the plate; but if the second medium is a crystal, a single wave whose vibrations are parallel to OP is incapable of being propagated, and it is necessary to suppose that the incident vibrations are resolved into two sets of vibrations, which are respectively parallel to OA, OB. These two sets of vibrations are propagated through the crystal with different velocities (unless the normal to the surface of the plate is parallel to one of the optic axes), and thus give rise to two waves of polarized light, whose planes of polarization are at right angles to one another.

104. If q be the displacement of a particle of ether in either of the waves, the equation of motion of that particle will be

d2q dt2

=

- (a2x2 + b2μμ2 + c2v2) q;

and therefore if 7 be the time of oscillation,

T

2π/τ = (α2X2 + b2μ2 +c2v) = 2πv/X';

where v is the velocity of propagation, and X' is the wave length. Hence if we write 27a/' &c. for a, b, c, where a, b, c now denote the three principal wave velocities, we obtain

From (5) it appears, that the force of restitution aaλ, b3μ, c2v corresponding to a displacement unity, is equal to a force v2 along the direction of displacement, together with some force P along l, m, n, the normal and the wave-front; whence resolving parallel to the axes, we obtain

which determines the velocities of propagation of the two waves,

whose direction cosines are l, m, n.

105. Before proceeding to discuss Fresnel's wave-surface, it will be convenient to consider some preliminary propositions.

We have shown that when polarized light is incident normally upon a crystal, the incident vibration must be conceived to be resolved into two components, which are parallel to the principal axes of that section of the ellipsoid of elasticity, which is parallel to the surface of the crystal; and that these two sets of vibrations give rise to two waves within the crystal. Now if the surface of the crystal is parallel to either of the circular sections of the ellipsoid of elasticity, every direction will be a principal axis, and therefore the component force parallel to the wave-front will be in the direction of displacement; hence only one wave will be propagated through the crystal. These two directions are the optic axes of the crystal, and therefore the optic axes are perpendicular to the two planes of circular section of the ellipsoid of elasticity.

106. We can now prove the following propositions :

The planes of polarization of the two waves corresponding to the same wave-front, bisect the angles between the two planes passing through the normal to the wave-front and the optic axes.

N

B

S

B

Let BAB' be the section of the ellipsoid of elasticity by the wave-front, ON the wave normal, and OS the intersection of one of the planes of circular section with the wave-front. The optic axis corresponding to the circular section through OS is perpendicular to OS, and therefore the plane through it and ON cuts the plane BAB' in a line OQ, which is perpendicular to OS. Similarly, if OS' be the intersection of the other plane of circular section with BAB', and OQ be the projection of the other optic axis, OQ'