CRITICISMS ON FRESNEL'S THEORY.

127

(i) It is assumed that the potential energy of an elastic medium, which is displaced from its position of equilibrium, is a quadratic function of the component displacements; whereas it will be shown in a subsequent chapter, that the potential energy of an elastic medium, which is symmetrical with respect to three rectangular planes, is a certain quadratic function which involves the space variations of the displacements, and not the displacements themselves.

(ii) The component of the force of restitution, perpendicular to the direction of propagation of the wave, is altogether neglected. And although attempts may be made to justify this by arguing, that the effect of this force, whatever it may be, cannot give rise to vibrations which affect the eye, yet the argument is fallacious; inasmuch as if such forces existed, they would produce waves of longitudinal vibrations, which would give rise to transversal vibrations, when light passes from a crystalline medium into another medium, and thus the sensation of light would be produced by something which is not light.

(iii) It is not a legitimate way of dealing with the motion of an elastic medium, to treat a wave as if it were composed of a number of distinct particles, each of which is acted upon by a force depending on its displacement. The rigorous theory of æolotropic elastic media is due to Green, and will be considered in a subsequent chapter; but although this theory is rigorous as far as its dynamics are concerned, it does not offer a satisfactory explanation of double refraction.

On the Methods of producing Polarized Light.

124. When light falls upon a plate of Iceland spar, it is divided into two rays within the crystal, which are polarized in perpendicular planes, and on emerging from the plate, two streams of plane polarized light are obtained, which are parallel to the incident rays; but unless the thickness of the plate is considerable, these two streams overlap. Since the velocities of the two streams within the crystal are unequal, their phases on emergence are different, and consequently the emergent beam is elliptically polarized, unless the difference of phase amounts to a quarter of a wave-length, in which case it is circularly polarized.

125. A very convenient method of producing plane polarized light consists in passing common light through a Nicol's prism, so called after the name of its inventor, the construction of which we shall proceed to explain.

There is a transparent substance called Canada balsam, whose index of refraction is intermediate between the ordinary and extraordinary indices of refraction of Iceland spar. If therefore two rhombs of Iceland spar are cemented together with this substance, it is possible for the ordinary ray to be totally reflected at the surface of the balsam, so that the extraordinary ray is alone transmitted.

Let AC be the optic axis, ACOF, ACEH the spherical and spheroidal sheets of the wave-surface; and let the plane of the paper be the plane of incidence, which is supposed to contain the optic axis. Let AO, AE be the ordinary and extraordinary rays, corresponding to a ray incident at A.

Let ABG be the wave-surface of the balsam; then since the index of refraction of the latter is intermediate between the ordinary and extraordinary indices of the spar, ABG will be a sphere, whose radius is intermediate between the polar and equatorial axes of the spheroid.

In order to obtain the directions within the balsam of the ray corresponding to the ordinary ray, draw a tangent at O meeting the face of the spar in T, and from T draw a tangent to ABG, and join the point of contact with A; if however T lies between F and G, it will be impossible to draw this tangent, and the ordinary ray will be totally reflected.

ray,

To obtain the directions within the balsam of the extraordinary

draw a tangent at E meeting the face of the crystal in S, from S draw a tangent to ABG meeting it in P, and join AP. If S lie beyond G, it will be possible to draw this tangent, and AP will be the ray corresponding to the extraordinary ray within the balsam. If this ray is not totally reflected at the second rhomb, it will be transmitted, and the emergent beam will be plane polarized.

126. In order to construct a Nicol's prism, a rhomb of Iceland spar is taken, whose length is about double its thickness, and is cut in two by a plane PE, and the two parts are then cemented together with Canada balsam. The plane ABCD contains the optic axis, and is therefore a principal plane; and the plane of section is inclined to BC at such an angle, that when a ray is incident at I parallel to BC, the ordinary ray is totally reflected by the balsam. The extraordinary ray IE is therefore alone transmitted, and emerges at M parallel to its original direction.

The vibrations of the emergent light accordingly lie in the plane ABCD, which is called the principal section of the Nicol.

127. A second method of producing plane polarized light is by means of a plate of tourmaline. Tourmaline is a negative uniaxal crystal, which possesses the property of absorbing the ordinary ray, even when the thickness of the crystal is small. If therefore we take a plate of tourmaline cut parallel to the axis,

B. O.

9

and pass common light through it, the emergent light will be completely polarized perpendicularly to the principal section of the plate.

128. A third method consists in using a pile of plates. When common light is incident upon a plate of glass at an angle equal to tan1μ, where μ is the index of refraction, it appears both from theory and experiment, that the reflected light is nearly, but not entirely, polarized in the plane of incidence; and by employing a pile of plates so as to cause the light to undergo successive reflections, the component vibrations in the plane of incidence may be entirely got rid of, and the resulting light becomes plane polarized.

EXAMPLES.

1. In a biaxal crystal, prove that the cosine of the angle between the ray axis and the optic axis is

ac + b2 b(a+c)

2. In a biaxal crystal, prove that if v be the velocity of wave propagation, a, b, c the principal wave velocities in descending order of magnitude, , the angles which the direction of vibration makes with the two optic axes,

v2 = b2 — (a2 — c2) cos y cosy.

3. Prove that the velocity of propagation of the wave in a biaxal crystal may be expressed in the form

where a, b, c are the principal wave velocities, and

0 = x2 + y2 + z2, $= a2x2 + b2y2 + c222,

x, y, z being the components of the ray velocity parallel to the axes of the crystal.

4. Light falls normally through a very small hole on a plate of biaxal crystal, of which the parallel faces are perpendicular to one of the circular sections of the surface of elasticity; show that if t be the thickness of the plate, and the semi-axes of the surface of

elasticity are proportional to λ, 1, λ' respectively, the area of the transverse section of the emergent cylinder of rays will be ‡π (λ2 – 1) (1 − λ'2) t2.

5. If v1, v2 be the velocities of propagation through a biaxal crystal, of the two waves corresponding to a plane wave-front, whose direction cosines are l, m, n; prove that

6. If one of the directions of vibration in a plane wave inside a biaxal crystal, make angles a, ß, with the two optic axes, and the other make angles y, 8; prove that

cos a cos d+cos B cos y = 0.

7. If a ray be incident on the face of a biaxal crystal in a plane passing through one of the optic axes, prove that the directions of vibration within the crystal will be either perpendicular to this axis, or will lie on the surface of a cone of the second degree.

8. A prism of angle i is cut from a biaxal crystal, whose principal wave velocities a, b, c are known. Prove that the position of either face of the prism relatively to the principal axes of elasticity of the crystal, may be ascertained thus; let a pencil of rays be incident normally on the face, and measure the deviations 01, 0, of the two rays emergent from the prism, then will

2

sin'i

sin (i+0) sin (i + 02)

λ2 (b2 + c2) + μ2 (c2 + a2) + v2 (a2 + b2) =

sin'i

sin'i

+ sin (i+0,) sin (i+0)' where λ, μ, v are the direction cosines of the face referred to the principal axes.

9. If a biaxal crystal be cut in the form of a right-angled prism, two of whose faces are principal planes, find how a ray must be incident at one face, so that the extraordinary ray may emerge at right angles to the other face. Show also that the minimum deviation of the extraordinary ray

is

sin−1 (u2 — a2)3 (u2 — b2)3/ab,

where a, b are the principal wave velocities in the plane of incidence, and u is the wave velocity in the medium surrounding the crystal.