CRYSTALLINE REFLECTION AND REFRACTION.

365

When the second medium is isotropic, a = c = p, whence (35)

418. Returning to (35), we observe that the intensity of the reflected light vanishes, when

419. Let us now suppose, that light polarized perpendicularly to the plane of incidence, is internally reflected at the surface of the crystal in contact with air. Then

and therefore since V > a, cos r will become imaginary, when

The right-hand side of this inequality determines the tangent of the critical angle, for light polarized perpendicularly to the plane of incidence.

Under these circumstances, the right-hand sides of (35) become complex, and it can be shown by the same methods as have already been employed, that the reflection is total, and is accompanied by a change of phase e, whose value is determined by the equation

tan eλ = V {(V-a) tan2 i - c2)/c2..........(38).

When the second medium is isotropic, so that a = c, we fall back on Fresnel's formulæ.

The corresponding results for a uniaxal crystal cut parallel to the axis, may be obtained by interchanging a and c.

Crystalline Reflection and Refraction1.

420. We shall now pass on to the general case in which the first medium is isotropic, whilst the second medium is a biaxal crystal.

=

Let O be the point of incidence, and let the axis of a be the normal to the reflecting surface, and let the axis of z coincide with the trace of the waves. Let COP be the front of the incident wave, OP, OQ the directions of the electric and magnetic displacements; also let COP = 0, so that COQ+0. Let COP', COP1, COP, (not drawn in the figure) be the fronts of the reflected and the two refracted waves; i, r1, r, the angles which the normals to the incident and two refracted waves make with the normal to the reflecting surface; also let OP', OP1, OP, be the directions of the electric, and OQ, OQ1, OQ, be those of the magnetic displacements in these waves.

Since the terms involving the suffix 2 are of the same form as those involving the suffix 1, they may be omitted during the work, and can be supplied at the end of the investigation.

1 Glazebrook, Proc. Camb. Phil. Soc. vol. iv. p. 155,

CRYSTALLINE REFLECTION AND REFRACTION.

The continuity of electric displacement along OA, gives

367

A cos AP + A' cos AP' = A, cos AP1 ...........(39).

The continuity of electric force parallel to OB and OC, give VA cos BP + V3A' cos BP' = VA, cos BP, + VA, tan X1 sin r1

and

V2A cos CP+ VA' cos CP' VA, cos CP, cos

=

Χι

...(40),

....(41),

where Χι is the angle between the refracted ray and the wave normal.

The continuity of magnetic induction along OA, gives

VA cos AQ+ VA' cos AQ V,A, cos AQ, ......(42).

=

The continuity of magnetic force, parallel to OB and OC, give VA cos BQ+ VA' cos BQ V1A, cos BQ, ......(43),

=

VA cos CQ+ VA' cos CQ VA, cos CQ,.....(44).

V

sin i

=

and

Now

= ·

V1 sin r1

also

=

cos AP sin i sin 0,

cos CQ = - sin 0;

whence (39) and (44) reduce to

which proves the equivalence of (39) and (44).

(A cos 0 + A' cos 0') sin3 i = A, cos 01 sin2 r1 ....(46).

cos BP

(40) and (43) become,

= cos i sin 0, cos BQ= cos i cos 0,

........

(A sin 0-A'sin ') sin2 i cos i A, (cos r, sin 0, + sin2r, tan x1) sin2 r

and

=

(A cos 0 - A' cos 0') sin i cos i = A1 sin r1 cos r1.....

.(47),

.(48).

Recollecting that if I, I, I, are the square roots of the

and restoring the terms in A,, we finally obtain from (46), (48), (45) and (47)

(I cos + I' cos 0') sin i = I, cos 0, sin r1 + I, cos 0, sin r2

2

(I cos 0- I' cos 0') cos i = 1, cos 0, cos r1 + I, cos 2 cos r2 I sin + I' sin e' I, sin 0, + I, sin 0,

(I sin - I sin O') sin 2i

=

= I, (sin 6, sin 2r, + 2 sin2 r1 tan X1)
+ I, (sin 0, sin 2r, + 2 sin2 r, tan X2)

(49).

When the angle of incidence is given, 01, 02, T1, T2, X1, X2 are known from the properties of the wave surface, hence these equations are sufficient to determine the unknown quantities I', I1, I, and e'.

421. Equations (49) are the same as those obtained by means of Lord Kelvin's modification of Lord Rayleigh's theory-see (33) of § 270; and it is also remarkable, that they are the same as those obtained in 1835 by MacCullagh1 by means of an erroneous theory. MacCullagh discussed these equations, and compared the results obtained from them with the experiments of Brewster, and found that they agreed fairly well. Accordingly, although we cannot at the present time accept the assumptions, upon which MacCullagh based his theory, as sound, yet most of the results of his first paper, with certain modifications necessitated by his having supposed that the vibrations of polarized light are parallel to the plane of polarization, are applicable to the electromagnetic theory; and thus MacCullagh's investigations regain their interest.

422. The discussion of (49) may be facilitated by a device invented by MacCullagh3.

When polarized light is incident upon a crystalline reflecting surface at a given angle, it is known both from experiment and theory, that it is always possible by properly choosing the plane of polarization of the incident light, to make one or other of the two refracted rays disappear. The two directions of vibration for which this is possible, are called by MacCullagh uniradial di

1 Trans. Roy. Irish Acad. vols. xv. p. 31 and xxi. p. 17.

2 Phil. Trans. 1819, p. 145; Seebeck, Pogg. Ann. vol. xxi. p. 290; xxш. p. 126; XXXVIII. p. 230; Glazebrook, Phil. Trans. 1879, p. 287; 1880, p. 421.

3 Trans. Roy. Irish Acad. vol. xv. p. 31.

UNIRADIAL DIRECTIONS.

369

rections. In the figure, let CA be the line of intersection of the plane of incidence with the plane of the paper, and let CO be the

direction of vibration of the incident light, when the ordinary ray alone exists, and CE the corresponding direction when the extraordinary ray alone exists; also let CO', CE' be the directions of the vibrations in the reflected waves corresponding to CO and CE.

Now whatever may be the character of the incident light, the vibrations may always be conceived to be resolved along the two uniradial directions CO, CE; and these two vibrations will give rise to the vibrations CO', CE in the reflected wave. If the incident light is plane polarized, the vibrations CO, CE, and also the vibrations CO', CE' will be in the same phase, and therefore the reflected light will be plane polarized, although its plane of polarization will not usually coincide with that of the incident light. If however the incident light be not plane polarized, the phases of the vibrations CO, CE, and therefore of CO', CE' will be different; hence the reflected light will not usually be plane polarized. It is however usually possible by properly choosing the angle of incidence, to make the two reflected vibrations CO', CE' coincide; and whenever this is possible, the reflected light will be plane polarized, and the angle of incidence at which this takes place is therefore the polarizing angle. These considerations, as we shall presently show, greatly simplify the problem of finding the polarizing angle.

423. The four vibrations CO, CE, CO', CE' do not usually lie in the same plane; we can however show, that when one of the refracted rays is absent, the lines of intersection of the planes of polarization of the three waves with their respective wave fronts lie in a plane.

B. O.