separation LL. The velocity of light in Canada balsam being intermediate between that of the two sets of waves in Iceland Spar, the inclination of LL' may be adjusted so that the ordinary ray is totally reflected, while the extraordinary ray passes through the combination. Fig. 124 shows in perspective how the plane of division is cut through

D

B

B'

Fig. 128.

the rhomb. When the end face ABCD of the rhomb is a parallelogram and parallel to one of the cleavage planes, the inclination of the section must be such that the side BB' of the rhomb is about 37 times as long as one of the sides of the end faces. It is difficult to secure crystals of Iceland Spar which are sufficiently long to give, under these conditions, a beam of such cross section as is generally required in optical work. The angular space through which the Nicol prism is effective in polarizing light is determined by the fact that if the incidence on the face LL' is too oblique, the extraordinary ray is totally reflected as well as the ordinary ray, and if not oblique enough, the ordinary ray can pass through. The field of view containing the angular space thus limited when the prism is cut according to the above directions, is about 30°. If it is not necessary to have so wide a field of view, shorter lengths of crystals can be used by cutting the end face ABCD, so as to be more nearly perpendicular to the length. Sometimes that face is even inclined the other way. A field of view of 25° may thus be secured with a ratio of length to breadth of 2 to 5. Artificial faces at the end have, however, the disadvantage of deteriorating more quickly than cleavage planes.

Foucault constructed a rhomb in which a small thickness of air is introduced in place of the Canada balsam. The prism need then be barely longer than broad, but the field of view is reduced to 7°.

100. Double Image Prisms. It is sometimes convenient to have two images of a source near together, achromatic as far as possible, and polarized perpendicularly to each other. An ordinary prism made of Iceland spar or quartz cannot be used on account of the colour

Р

M

dispersion, but if a prism of quartz be achromatised by means of a prism of another material, the desired result may be obtained. If glass is chosen for the material of the second prism, the achromatism is only complete for one of the images, but for many purposes it is sufficiently perfect for the second image also. The purpose is better obtained by prisms, like that of Rochon, in which the same material is used for both prisms, but turned differently with respect to the optic axis. In Rochon's

B

Fig. 129.

arrangement the optic axis of the first prism ABC (Fig. 129) is parallel to the normal BC, this being indicated in the figure by the direction of the shading. A ray LP incident normally is propagated without change of direction. The axis of the second prism ACD is at right angles to the plane of the figure, and

The

double refraction takes place at K, one ray being propagated in the normal direction as before, but the extraordinary ray being refracted along KQ and, on passing out of the prism, along QM. achromatism is complete for the image formed by the ordinary ray, and nearly complete for the other. In the prism of Wollaston (Fig. 130), the Fig. 130. axis of the first prism is parallel to AB and that of the second at right angles to the plane of the figure; the path of the rays is indicated in the figure.

101. Principal Refractive Indices in biaxal crystals. If refraction takes place at the surface of a biaxal crystal, and the plane of incidence is one of the principal planes (e.g. the plane of YZ), both rays lie in the plane of incidence. A plane wave-front incident at O must, after refraction, touch a circle of radius a, and an ellipse of semiaxes b and c which form the intersection of the wavesurface with the plane of YZ. One of the rays follows the ordinary law of refraction, while the angle of refraction of the other ray may be obtained as in case (c), Art. 96. The refractive index of the rays belonging to the circular section is 1/a; similarly for planes of incidence coincident with the planes of XZ and of YZ, we should have always one ray following the ordinary law, the corresponding refractive indices being 1/6 and 1/c. These three quantities are therefore called the principal refractive indices. Denoting these by M1, M2, M3, we may express all quantities relating to the wave-surface in terms of them. Thus for the direction cosines of the optic axes, we may put, using (5),

Fig. 131.

and for the direction cosines of the rays of single ray velocity, using (27),

102.

Conical Refraction.

Two cases of refraction in 'biaxal crystals have a special interest. If a wave-front WF is incident on

a plate cut out of the crystal at an angle such that the refracted wave-front HKLM is normal to an optic axis, the ray PD may, according to the direction of vibration, be refracted along any direction lying on the surface of the cone investigated in Art. 93, the cone intersecting the wave-front inside the crystal in a circle (Fig. 132).

If the wave-front WF contains a number of coincident rays, having their planes of polarization symmetrically distributed in all directions, the refracted rays form the surface of a cone of the second degree which becomes a cylinder on emergence at the upper surface. This interesting result was first deduced theoretically by Sir Wm Hamilton, from the shape of the wave-surface, and was afterwards experimentally verified by Lloyd. To illustrate it experimentally, we may take a plate (Fig. 133), cut so that its face is equally inclined to both axes. opaque plate PQ with a small aperture O, covers the side on which the light is incident. A second plate P'Q' transmits light through a small hole at O', which, if properly illuminated, may be considered

An

to act as a source of light. If now PQ be moved along the face of the crystal, a direction O'O may be found such that if the original light is unpolarized, the ray O'O splits into a conical pencil, which may be observed after emergence at AB. This phenomenon is called "internal conical refraction" to distinguish it from another similar effect which takes place when a ray travels along an axis of single ray velocities.

Fig. 133.

We may always follow the refraction of a ray belonging to a certain wave-surface and incident internally on the face of a crystal by considering it to be part of a parallel beam. The wave-front belonging to this parallel beam would be the plane which touches

the incident wave-surface at the point of incidence. If now a ray HR (Fig. 134) travels inside a crystal along the axis of single ray velocity, there is an infinite number of tangent planes to the wave-surface at the point R, the normals of the tangent planes forming a cone HKL with a circular section at right angles to HR.

H

L

Ꭱ

To each of these normals corresponds a separate ray on emergence and each ray has its own plane of polarization. The complete cone can only be obtained on emergence if all directions of vibration are represented in the incident ray.

Fig. 134.

Fig. 135 shows how the phenomenon of external conical refraction may be illustrated experimentally. A plate of arragonite has its surfaces covered by opaque plates, each having an aperture. If one of these plates be fixed and the other is movable, a position may be found of the apertures O and O' such that only such light can traverse

the plate as passes along the axis of single ray velocity. The rays on emergence are found to be spread out and to form the generating lines of a cone. But as any ray after passing through a plate must necessarily be parallel to its original direction, it follows that to obtain the emergent cone, the incident beam must also be conical. This may be secured by means of a lens LL arranged as in the figure. Those parts of the incident beam forming a solid cone which are not required, do not travel inside the crystal along 00' and hence are cut off by the plate covering the upper surface.

L

L'

Fig. 135.

103. Fresnel's investigation of double refraction. Fresnel's method of treating double refraction which led him to the discovery of the laws of wave propagation in crystalline media, though not free from objection, is very instructive, and deserves consideration as presenting in a simple manner some of the essential features of a more complete investigation.

Consider a particle P attracted to a centre O with a force a2 when the particle lies along OX, and a force by when it lies along OY. The time of oscillation, if the particle has unit mass, is, by Art. 2, 2π/a or 2π/b according as the oscillation takes place along the axis of X or along the axis of Y. When the displacement has components both along OX and along

Ꭱ

Fig. 136.

t

OY, the components of the force are ar and by, and the resultant force is R = √a^x2 + b1y2.

The cosines of the angles which the resultant makes with the coordinate axes are a2x/R and by/R. The direction of the resultant force is not the same as that of the displacement, the direction cosines of which are x/r and y/r. The cosine of the angle included between the radius vector and the force is found in the usual way to be

and the component of the force along the radius vector is

(a2x2 + b2y2)/r.

If we draw an ellipse a2x2 + b2y2 = k2, where k is a constant having the dimensions of a velocity, the normal to this ellipse at a point P, having coordinates a and y, forms angles with the axes, the cosines

Y

Ꭱ

of which are in the ratio a2x to b2y, hence the force in the above problem acts in the direction. ON of the line drawn from O at right angles to the tangent at P. The component of the force along the radius vector is k2/r, and the force per unit distance is k2/r2, so that if the particle were constrained to move on the radius vector OP, its period would be 2r/k. The ratio r/k depending only on the direction of OP our result is independent of the particular value we attach to k.

Fig. 137.

If we extend the investigation to three dimensions, the component of attraction along OZ being cz, we obtain the same result, and the component of force acting along any radius vector OP per unit length is k2/2, where r is the radius drawn in the direction of OP to the ellipsoid

a2x2 + b2y2 + c2z2 = k2.

If the displacement is in any diametral plane HPK of this ellipsoid (Fig. 143), the normal PN does not in general lie in this plane, and the projection of PN on the plane does not pass through 0, unless OP is a semiaxis of the ellipse HPK. In the latter case, PL the tangent to the ellipse in the diametral plane, is at right angles to PO and to PN, and hence the plane containing PO and PN is normal to the plane of the section.

K

H

Fig. 138.

Fresnel considers the condition under which a plane wave propagation is possible in a crystalline medium. The investigation in Art. 11 has shown that the accelerations of any point in a plane