The first equation, with the help of the second, becomes

p2 = b (λx + vz).

Hence the locus of the foot of the perpendicular from 0 to the tangent planes at R is the circle formed by the intersection of a sphere with a plane. The equation of the plane shows that it is parallel to OY and touches at R the ellipse ARC, Fig. 115.

95. Wave-surface in uniaxal crystals. The wave-surface in uniaxal crystals takes the shape already indicated by Huygens. If bc equation (17) reduces to

(y2 — c2) (a2x2 + c3y2 + c2x2 - a2c2) = 0.

The surface splits up therefore into the sphere of radius c and the spheroid

X

C

Fig. 118.

X

Similarly, if ab, the equation of the wavesurface splits up into a sphere of radius a and Z into the ellipsoid

The positions of take a, b, c to be

the axis of X and in the second case the axis of Z. the axes are determined if we always in descending order, but if we drop that supposition, we may take the optic axis of uniaxal crystals to be at our choice either in the direction of X or in the direction of Z. The spheroid is formed by the revolution of the ellipse and circle round the optic axis. The two types of wave surfaces, one having an oblate and the other a prolate spheroid, according as the generating ellipse is made to rotate about its shorter or longer diameter, are illustrated by the case of Iceland Spar and Quartz. The term positive and negative crystals, as applied to crystals similar to Quartz and Iceland spar respectively, is confusing and should be avoided. We may speak instead of crystals which are optically prolate, or optically oblate, and in a discussion relating to optical properties only, where no confusion is possible, we may call them shortly prolate and oblate crystals.

Fig. 119.

96. Refraction at the Surface of Uniaxal Crystals. The refracted waves may, in crystalline media, be constructed exactly as in

isotropic bodies, but as the wave-surface consists of two sheets, there are in general two refracted rays. In uniaxal crystals, one sheet of the wave-surface is always a sphere, and hence one of the rays follows the ordinary laws of refraction. This ray is called the ordinary ray, and the ratio of the sines of the two angles of direction is called the ordinary refractive index.

As regards the second ray, it will not in general follow any simple

H

B

law, and may or may not lie in the plane of incidence. If the optic axis is inclined to the plane of incidence, the point of contact of the spheroid with the sphere does not lie in that plane, Fig. 120, and if the plane BT is drawn in the usual way at right angles to the plane of incidence, to touch the spheroidal sheet of the wavesurface, the ray being the line joining O to the point of contact does not lie in that plane. The ordinary ray OK is found in the usual way and always remains in the plane of incidence.

T

K

Fig. 120.

In Fig. 120 HB is drawn parallel to the incident ray and at such a distance from it that HB is unit length. The scale of the wave-surface is such that it represents the locus of the disturbance spread out from O in unit time, which for the present purpose has been chosen to be such that the velocity of light in vacuo is unity. BT represents the trace of the refracted wave which touches the ellipsoid at a point outside the plane of incidence. The refracted ray does not in this case lie in that plane.

It is not necessary to obtain the general equation, giving the direction of the refracted ray, and we may treat a few special cases separately.

M

Μι

B

(a) The optic axis of the crystal at right angles to the plane of incidence. The trace of the wave-surface on the plane of incidence is in this case a circle, and the refracted ray may by symmetry be seen to lie in the plane of incidence. Hence the rays follow the ordinary law of refraction. In oblate crystals the outer circle of radius a belongs to the extraordinary ray, and its angle of refraction is greater. The reverse holds for prolate crystals. For oblate crystals, the ratio of the sines for the extraordinary ray is with the unit time chosen 1/a. Calling this μe we may write for the equation to the wave-surface

Fig. 121.

where is the refractive index of the ordinary ray. The extraordinary

Мо

refractive index μe has obtained its name and significance from the optic behaviour of the extraordinary ray in the general case we are now considering. In the case of prolate crystals, the equation of the wave-surface in terms of the principal refractive indices becomes

2

μ2 (x2 + y2) + μ2 ≈2 = 1,

the axis of being now the optic axis.

(b) The optic axis is in the surface and plane of incidence. The refracted rays are both in the plane of incidence.

an equation which holds for both prolate and oblate crystals.

If the angles of the refraction of the rays, OML and OM1L, be denoted by r and r1, we obtain similarly

But

(c) The optic axis is perpendicular to the refracting surface. If

and if, for OM1, we may put its value a, and for OB, 1/sin i,

And similarly if r, is the angle of refraction of the extraordinary ray,

N M

Fig. 124.

H

(d) The incident wave-front is parallel to the surface. Let the plane of the paper (Fig. 124) contain the optic axis. The refracted extraordinary ray lies along OM, where Mis the point of contact of the spheroidal portion of the wave-surface with a plane drawn parallel to the surface. The ordinary ray coincides, of course, with the normal ON. To determine the angle between the two rays, which is also the angle of refraction of the extraordinary ray, we must obtain an expression for the angle between the radius vector OM of an ellipse, and the normal to its tangent at M. If @ be the angle between the optic axis and the surface which is equal to the angle between ON and OH, the major axis of the ellipse, and y be the angle between OM and OH, we have for the properties of the ellipse,

Hence if r is the angle of refraction of the extraordinary ray,

97.

=

Direction of vibration in uniaxal crystals. The rule that the direction of vibration is in the direction of the projection of the ray on the wave-front shows at once that on the spheroidal portion of the wave-front, the direction of vibration must be in a plane containing the optic axis. As the condition (Art. 90) under which the two vibrations along the same ray are at right angles to each other, always holds in uniaxal crystals, we may say that the ordinary ray is always polarized in a principal plane, and the extraordinary ray at right angles to that plane.

B

98. Refraction through a crystal of Iceland Spar. A crystal of Iceland Spar is a rhomb (Fig. 125). The parallelograms forming its six faces have sides which include angles of 102 and 78° respectively. The faces are inclined to each other at augles of 105° and 75°. There are two opposite corners A and B at which the three edges all form obtuse angles of 102° The optic axis is parallel to the line drawn through one of these corners A, and equally inclined to the three faces.

Fig. 125.

A

with each other.

D

S

A

NM

T

B

Fig. 126.

C

Double refraction may easily be exhibited by placing such a rhomb on a white sheet of paper on which a sharp mark is drawn. When this mark is looked at from above through the crystal, it appears double, and if the crystal be turned round, one image seems to revolve round the other. Let O, Fig. 126, be the mark, the images of which are observed. To trace the image formed by the extraordinary ray, construct a wave-surface to such a scale that the spheroid touches the face DC. If T is the point of contact, a ray OT is refracted outwards along the normal TM, because at T the tangent plane to the wave-surface and the surface are coincident. The refraction is therefore the same at that point as for a wave incident normally.

A ray OS parallel to the optic axis intersects the face at a point E, and is refracted along some direction EK. Disregarding aberrations, the intersection Q of KE and TM gives the extraordinary image. As there can be no distinction between an ordinary and an extraordinary ray along the optic axis, the ordinary image P is obtained by the intersection of the same line EK with the normal ON, on which the ordinary image must lie. The figure shows that this ordinary image lies nearer to the surface than the extraordinary one, and if the crystal be turned round the point 0, the image Q travels in a circle round P. The vertical plane containing P and Q contains also the optic axis, and the ordinary image is therefore polarized in the plane which passes through the two images, the extraordinary image being polarized at right angles to it.

99. Nicol's Prism. A Nicol's prism, or, as it ought to be more appropriately called, a Nicol's rhomb, is one of the most useful appliances we have for the study of polarization. Let Fig. 127

represent the section of a long rhomb of Iceland Spar, passing through the optic axis, and LL' an oblique section through it. If the rhomb be cut along this section and then recemented together by means of a thin layer of Canada balsam, only rays polarized at right angles to the principal plane are transmitted through it, if the inclination of the section LL' has been properly chosen. An unpolarized ray is refracted at the surface, and separated into two, the extraordinary ray being bent less away from the original direction. The ordinary ray falls therefore more obliquely on the surface of

Fig. 127.

L