Let CP, CP', CP, be the lines of intersection of the three planes of polarization with their respective wave-fronts; λ, μ, v; X', μ', '; Mr, M, v, the direction cosines of CP, CP, CP1. which is the condition that CP, CP', CP, should lie in the same. plane. The preceding theorem is a modification of one due to MacCullagh. 424. Let us now suppose, that the reflecting surface is a uniaxal crystal, and let the suffixes 1 and 2 refer to the ordinary and extraordinary rays respectively; then X1 = 0. If we suppose that the ordinary ray alone exists, I2 = 0, and we easily obtain from (49) the equations tan 0 = cos (ir) tan 01) tan o' = - cos (i+r) tan 01) (50). Since the angle of incidence is supposed to be given, r, and e, are known; and therefore (50) determine 0, ' which give the directions of vibration in the incident and reflected waves. Again, suppose that the extraordinary ray alone exists; putting I1 = 0, and writing ✪, O' for 0, 0', we obtain POLARIZING ANGLE. 371 Now we have shown that in order that the reflected light should be plane polarized, it is necessary that the two directions CO', CE' should coincide, in which case ''; we thus obtain from (50) and (51), cos (i+r) tan 0, − cos (i + r) tan 02 + which determines the polarizing angle i. sin2r, tan x cos sin (i-r2) = 0 (52), 425. A very elegant formula is given by MacCullagh for the polarizing angle, when the plane of incidence contains the axis of a uniaxal crystal, which is most simply obtained directly from (49), by determining the angle of incidence at which the intensity of the reflected light vanishes, when the incident light is polarized perpendicularly to the plane of incidence. We have I' = I1 = 0, 0 = 0′ = 01 = 1⁄2 π ; also if w is the angle which the extraordinary wave normal makes with the axis of the crystal where c and V are the velocities of propagation of the ordinary wave within the crystal, and in the medium surrounding the crystal, and r=r2. If λ be the angle which the optic axis makes with the reflecting surface, w +λ = 1⁄2π-r; whence multiplying (54) by tan r, we obtain = sin2r sin i cos i tan r- V-2(a2 - c2) sin (r+λ) cos (r+λ) sini tan r. But sin2 r = V−2 sin2 i {c2 + (a2 — c2) cos2 (r+X)}. Equating these two values of sin2 r, and reducing we shall obtain a2 cosλ+c2 sin2 tan r = V2 cot i + (a2 - c2) sin λ cos λ Substituting in (54), and reducing, we finally obtain which is the formula in question, which determines the polarizing angle i. Reflection at a Twin Plane. 426. We shall conclude this Chapter by giving an account of a peculiar kind of reflection, which is produced by iridescent crystals of chlorate of potash. The phenomena exhibited by the crystals in question were first examined by Sir G. Stokes', and the experimental results at which he arrived may be summed up as follows: (i) If one of the crystalline plates be turned round in its own plane, without altering the angle of incidence, the peculiar reflection vanishes twice in a revolution, viz. when the plane of incidence coincides with the plane of symmetry of the crystal. (ii) As the angle of incidence increases, the reflected light becomes brighter, and rises in refrangibility. (iii) The colours are not due to absorption, the refracted light being strictly complementary to the reflected. (iv) The coloured light is not polarized. It is produced indifferently, whether the incident light be common light, or light polarized in any plane; and is seen, whether the reflected light be viewed directly, or through a Nicol's prism turned in any way. (v) The spectrum of the reflected light is frequently found to consist almost entirely of a comparatively narrow band. When the angle of incidence is increased, the band moves in the direction of increasing refrangibility, and at the same time increases rapidly in width. In many cases the reflection appears to be almost total. 427. Sir G. Stokes has shown that the seat of the colour is a narrow layer about a thousandth of an inch in thickness, and he suggested that this layer consists of a twin stratum. The subject was subsequently taken up by Lord Rayleigh, who attributed the phenomena to the existence of a number of twin planes in contact with one another; and he has accounted for most of the phenomena by means of the electromagnetic theory of light. He has also shown, both from theory and experiment, that when the angle of incidence is sufficiently small, and the planes of incidence 1 On a remarkable Phenomenon of Crystalline Reflection, Proc. Roy. Soc., Feb. 26, 1885; see also Lord Rayleigh, Phil. Mag. Sep. 1888, p. 256; Proc. Roy. Institution, 1889. REFLECTION AT A TWIN PLANE. 373 and symmetry are perpendicular, reflection at a twin plane reverses the polarization; that is to say, if the incident light is polarized in the plane of incidence, the reflected light is polarized in the perpendicular plane and vice versa. This very peculiar law was not even suspected, until it had been obtained by theoretical considerations. 428. The easiest way of understanding what is meant by a twincrystal, is to suppose that a crystal of Iceland spar is divided into two portions by a plane, which is inclined at any angle a to the optic axis, and that one portion is turned through two right angles. The optic axes of the two portions will still lie in the same plane, but instead of being coincident, they will be inclined to one another at an angle 2a. Crystals whose structure is of this character, are called twin-crystals; and it is evident, that a crystal may possess more than one twin layer. 429. We shall now consider Lord Rayleigh's theory'. When the plane of incidence contains the optic axes of the two portions, and the light is polarized in the plane of incidence, the wave surfaces in both crystals are spheres of equal radii; and therefore the crystal will act like two isotropic media, whose optical properties are identical. Hence no reflection can take place, and the wave will pass on undisturbed. 430. We shall in the next place suppose, that the light is polarized perpendicularly to the plane of incidence. Let the axis of a be normal to the twin plane, and let the plane xy contain the optic axes of the two portions. Let Oy' be the axis of the upper portion, and let Ox' be perpendicular to Oy' in the plane ay. Let x'Ox = a; let f', g' be the electric displacements along Ox', Oy'; and let f, g, h be the displacements along Ox, Oy, Oz. In the upper portion, the wave surface for the extraordinary ray consists of the planetary ellipsoid x22/c2 + (y22 + z2)/a2 = 1. Accordingly by (5) we have P = 4πc2ƒ" cos a + 47a-g' sin a) (56). P=4π (Aƒ + Bg), Q=4π (Bf+ Сg), The equations of electric force for the obtained by changing the sign of a, whence R= 4πDh ... (57). lower medium are P1 = 4π (Aƒ1— Bg1), Q1=4π (- Bf+Cg1), R1 = 4πDh..........(58). Since none of the quantities are functions of z, it follows that if we substitute these values in (11), and put μ= 1, and recollect that h=h1 =0, we obtain Let the displacements in the incident, reflected, and refracted and S and S are obtained by changing p into p' and p1 respectively. Since p and q are proportional to the direction cosines of the incident wave, these equations satisfy the conditions that the displacement lies in the front of the wave. Substituting from the first of (60) in (59), we find that both equations lead to which is a quadratic equation for determining the two values of p corresponding to a given value of s. Changing the sign of B, we find for the second medium |