of Fresnel's biprism, the interference of polarised light can be studied in the manner described in § 373. In this way it is found that two rays of light polarised in planes at right angles do not produce interference under circumstances in which two rays of ordinary light would interfere. Two rays polarised in the same plane do, however, interfere like two rays of ordinary light. The fact that rays polarised in planes at right angles do not interfere is a further proof that the direction of vibration in the case of light is transverse to the direction of propagation.

403*. Uniaxal and Biaxal Crystals.-We have, when speaking of double refraction in Iceland spar, mentioned that when a ray of light traverses the spar parallel to the optic axis there is only a single refracted ray. In Iceland spar there is only one direction in which this takes place, and therefore there is only one optic axis. Doubly refracting crystals which have only one optic axis are called uniaxal crystals. In other crystals there are two axes along which there is only a single refracted ray, and these are called biaxal crystals.

404*. Wave-Surface in Uniaxal Crystals.-If we have a disturbance produced at a point within an isotropic medium, the wave-surface at any moment will be a sphere with the point of disturbance as the centre, for the velocity of light being the same in all directions, the disturbance which originates at any point will in a given time spread to an equal distance in all directions. If, however, the body is not isotropic, and the velocity of light is different in different directions, the disturbance will, in a given time, travel further in some directions than in others, and so the wave-surface will no longer be a sphere.

Now, in the case of a crystal, the velocity with which light travels is not the same in all directions; and since there are in general two refracted rays there must be two wave-fronts. For the ordinary ray the refractive index is constant, and therefore the velocity of the ordinary ray in the crystal is constant, for we have seen in § 366 that the refractive index is equal to the ratio of the velocities in the two media (air-crystal). As the refractive index for the extraordinary ray varies with the direction

of the ray within the crystal, the velocity with which the extraordinary ray travels must depend on the direction of the ray in the crystal. The velocity in the case of the ordinary ray being constant, just as in isotropic bodies, the ordinary wave-surface must be a sphere. Huyghens, who first considered the subject, assumed that in uniaxal crystals the extraordinary wave-surface was a spheroid or ellipsoid of revolution, that is, the figure obtained by rotating an ellipse about one of its diameters, AB or CD (Fig. 385), and then verified the accuracy of this assumption experimentally. The axis about which the ellipse is rotated to form the extraordinary wave-surface coincides with the optic axis of the crystal. Hence the complete wave-surface for a disturbance

B

FIG. 385.

originating at a point within a uniaxal crystal consists of a sphere and a spheroid, both having their centres at the point; the axis of the spheroid being parallel to the optic axis of the crystal. Further, since when the ray travels in the crystal in a direction parallel to the optic axis, there is only one refracted ray, i.e. the ordinary and extraordinary rays travel with the same velocity, the sphere and spheroid must touch one another at points on the optic axis.

In the first place, the extraordinary

Two cases, however, may arise. ray may be more refracted than the ordinary ray, so that the velocity of the extraordinary ray

lies outside the sphere, Fig. 386 (b), again touching the circle on the optic axis xx'.

Uniaxal crystals in which the wave-surface is like Fig. 386 (a), and in which, except along the optic axis, the ordinary ray travels faster than the extraordinary ray, are called positive crystals. Quartz and ice are positive crystals.

Uniaxal crystals, in which the wave-surface is like Fig. 386 (b), are called negative crystals, and to this class belong Iceland spar and tourmaline.

405*. Huyghens's Construction for the Directions of the Refracted Rays in a Uniaxal Crystal.-Suppose we require to find the directions of the refracted rays in the case of Iceland spar. The spar being a negative crystal, the wave-surface is like Fig. 386 (6).

Let IQ or I'P (Fig. 387) be the direction of the light incident on the face of the crystal, and let the optic axis XQ lie in the plane of the paper, so that the paper is the principal plane for the face. Then QM is the wavefront of the incident wave.

When the wave reaches Q, we may consider that this point becomes a centre of disturbance within the crystal. If it takes the wave a time t to travel from M to P, and we describe the wave-surfaces in the crystal about the point Q for a time after the disturbance reaches Q, these wave-surfaces will represent the positions of the wave in the crystal when the wave in the air reaches P. Hence, if from P we draw PO and PE tangents to the ordinary and extraordinary wave-surfaces respectively,

PO will represent the ordinary wave-front in the crystal and PE the extraordinary wave-front, and the line Qo will represent the direction of the ordinary ray and QE the direction of the extraordinary ray.

If the plane of the paper had not been a principal plane, we should have had to draw through P a plane perpendicular to the plane of incidence to touch the spheroid, and it would not have touched it at a point in the plane of the paper; so that the extraordinary ray would not be in the plane of incidence, and thus would not have obeyed the first law of refraction as given in $ 341.

Two particular cases are worth examining: first, when the optic axis is parallel to the

face of the crystal and perpendicular to the plane of incidence; and second, when the optic axis is parallel to the face and also parallel to the plane of incidence.

In the first case (Fig. 388) the optic axis is perpendicular to the plane of the paper, and hence the sections of the wave-surfaces consist of two

α

I'M
N

FIG. 388.

E

circles, the inner one, since the crystal is negative, corresponding to the ordinary ray. The reason the section of the extraordinary wave-surface is a circle is that this surface is obtained by rotating an ellipse about the optic axis, so that all sections perpendicular to the axis must be circles.

If a is the velocity of the ordinary ray and b

the velocity of the extraordinary ray in a plane at right angles to the optic axis, the radius of the spherical wave-surface in the crystal being taken as a, the major axis of the spheroid will be b. Hence in Fig. 388, if Qo is a, QE will be equal to b. If the velocity of light in air is c, the refractive index for the ordinary ray is c/a, and that for the ordinary ray in a plane at right angles to the optic axis is c/b. Now b or QE is

constant for all angles of incidence in a plane at right angles to the optic axis, and hence the extraordinary refractive index is constant in this plane, and the extraordinary

ray obeys the ordinary laws of refraction. By cutting a prism of Iceland spar with its refracting edge parallel to the optic axis two refracted rays will be obtained, and the refractive index (c/a and cb) corresponding to each of these can be measured. In this way it can be proved that the extraordinary refractive index (c/b) in a plane at right angles to the optic axis is constant. Hence b or QE must be constant, and so it is proved that the section of the extraordinary wave-surface perpendicular to the axis is a circle.

AXIS

FIG. 389.

M

The construction for finding the directions of the refracted rays when the optic axis is parallel to the face of the crystal, and in the plane of incidence, is shown in Fig. 389.

406. Nicol's Prism.-As a means of obtaining plane polarised light, a tourmaline plate is, for many purposes, unsuited, for, as has been mentioned, the light transmitted by

tourmaline is coloured green.

Since,

when a beam of light is passed through
a crystal of Iceland spar, two refracted
beams are obtained, each of which is
plane polarised, but in planes at right
angles, if by any means we could inter-
cept one of these refracted beams, the
other would give us plane polarised
light. Since the angular separation A
between the ordinary and extraordinary
rays is not very great, it is not possible
to stop one of the beams with a screen,
unless only a very narrow beam is em-
ployed, or we use a very thick crystal.

D

The most convenient method of getting rid of one of the rays is to make use of total internal reflection for this purpose. A rhomb of Iceland spar is taken and cut in two by a plane, AC (Fig. 390), perpendicular to the principal plane for the face AB.

FIG. 390.

Ε'

The two

surfaces are then polished and cemented together in their original position by means of a thin film of Canada balsam.

Now the refractive index of Canada balsam (1.55) is greater than the minimum value for the extraordinary ray (1.486) in Iceland spar, and less than that for the ordinary ray (1.658). As total reflection can only occur when light is passing from a media of greater to one of less refractive index, we can never get total reflection in the case of the extraordinary ray when passing from spar to balsam, so long as the ray passes in such a direction that the refractive index is less than 1.55. In the case of the ordinary ray, however, if the incidence is sufficiently oblique we shall obtain total reflection. Hence if the plane AC is suitably inclined, the ordinary ray, PO, will be incident on the surface AC at an angle greater than the critical angle, and will therefore be totally reflected along Oo', while the extraordinary ray, PEE', will pass through the prism.

The light transmitted by such a rhomb of Iceland spar, which is called a Nicol's prism, will therefore be plane polarised, and since it is the extraordinary ray which is transmitted, the plane of polarisation is perpendicular to the principal plane, ie. is a plane perpendicular to the paper in Fig. 390.

A Nicol's prism may be used, not only for producing plane polarised light, when it is called a polariser, but also for detecting whether light is plane polarised, and, if so, determining the plane in which it is polarised, when it is said to be used as an analyser.

If the light incident on the Nicol is unpolarised, then the intensity of the transmitted light will remain the same when the Nicol is rotated round the light ray as an axis, the intensity of the transmitted light being practically half that of the incident light. There is, however, a very slight loss due to reflection at E (Fig. 390), and where the ray leaves the crystal.

If the incident light is plane polarised, the intensity of the transmitted light varies as the analyser is rotated. When the principal plane of the Nicol is parallel to the plane of polarisation of the incident ray, then (§ 401) there will be only an ordinary ray in the spar, and this ray is totally reflected, so that no light will be transmitted. When the principal plane of the Nicol is perpendicular to the plane of polarisation of the incident light, only an extraordinary ray will be produced in the spar, and this will be transmitted undiminished, so that in this case the intensity of the transmitted light is equal to that of the incident light. If the principal plane of the Nicol is inclined at an angle a to the plane of polarisation, it can be shown, exactly as in § 401, that the intensity of the extraordinary ray, and hence that of the transmitted light, is I sin2 a, where I is the intensity of the incident light. Thus when a=0 or 180° the intensity of the transmitted light is zero, and when a=90° or 270° the intensity of the transmitted light is I.

407. Polarisation by Reflection.-If the light reflected from a non-metallic surface, such as glass, is examined with an analysing Nicol,