refracted light be denoted by 1, r, s, respectively, we may as in Art. 140 write The surface conditions are the same as for transparent media, hence the previous investigations apply as far as the analytical expressions of r and s are concerned. When the incident light is polarized in the plane of incidence, the amplitude r. was found to be sin (0-0) and for light polarized perpendicularly to the plane of incidence .(61), But 0, being now complex, r, and r are complex quantities from which we may separately deduce the real amplitude and the change of phase. Writing rhes and substituting in (59) we see that h, is the real amplitude, and 8 denotes the change of phase. It therefore becomes a problem of algebraic transformation to change (61) and (62) into the standard form hes. Little experimental work has been done to measure the intensity of the reflected light and we may therefore confine ourselves to calculating the value of h for perpendicular incidence for which, as for transparent media, If this quantity be called P, and Q be that obtained from P by reversing the sign of i, the proposition proved in Art. 8 shows that It will be noticed that great absorbing power means a large intensity of reflected light, for if κ, is large compared with v。 + 1, h is nearly 1, and the light is almost totally reflected. The absorbing power therefore is ко active not only in transforming the energy that enters, but also in preventing the light from entering. There is also nearly total reflexion when v is either large or small compared with 1, but this effect is not confined to opaque substances. When light polarized at an angle & to the plane of incidence falls on a metal surface, it may be considered to be made up of light having amplitude cos polarized in that plane, and light of amplitude sin & polarized at right angles. We write R, and R, for the reflected amplitudes, which are respectively equal to r, cos and r, sin o, where r, and r, are defined by (61) and (62). Putting and writing 82-81, for which we shall write 8, represents the difference in phase of the two components of reflected light. That light is therefore polarized elliptically. If by some compensating arrangement the difference of phase be destroyed, and the reflected light restored to plane polarization, the angle x which the plane of polarization of the reflected light forms with the plane of incidence, is determined by tan x = tan tan y. determined. and Ko in The quantities x and may be measured, and hence We must endeavour now to express the optical constants terms of 8 and 4. For this purpose, we have from (61) and (60), cot 0, sin - tan sin = cot 0, sin + tan sin 01 This expression is of the same form as (63), and by subjecting it to the same transformation, we find When the difference in phase, 8, is equal to a right angle, tan & is infinitely large, and hence in that case sin2 0 tan2 0 = v2 + k2. This particular value of 0 is called the principal angle of incidence, and, for κ=0, corresponds to the polarizing angle. The problem as it generally presents itself, consists in determining the optical constants of the metal from observation of and 8; for this purpose (66) and (67) are not convenient, and we must transform (65) in a different manner. and the right side of this equation may be transformed in the same manner as (63). We thus find If instead of using a compensator similar to Babinet's, the elliptic path of the disturbance of the reflected ray is analysed by a quarter waveplate or Fresnel's rhomb, the quantities measured are the ratio of the axes of the ellipse and the inclination of these axes to the plane of incidence. Calling tan the ratio of the minor to the major axis, and y the angle between the major axis and the plane of incidence, we obtain with the assistance of (18) (19) and (20) of Chapter I., Having obtained m and we determine the optical constants v Solving these equations for v, and, we obtain 2v.2 = √√(m2 — k2 + sin2 0)2 + 4m2x2 + (m2 — «2 + sin2 0) 2k2 = √(m2 — k2 + sin2 6)2 + 4m2Ê2 − (m2 — ×2 + sin2 0) and ...(71). Equations (68), (69) and (71) constitute the solution of our problem in the form in which Ketteler* first gave it. This form is to be preferred to the earlier one given by Cauchy, whose solution did not directly lead to the separation of the constants v。 and K, but only to a set of equations which involved intermediate constants and variables, having no physical meaning. In the case of metals, m2+2 exceeds sin20 sufficiently to allow us generally to neglect the square of sin2 0/(m2 +2). Under these circumstances, expressing the square root which occurs in equations (71) as a series proceeding by powers of sin2 0, and neglecting all powers higher than the first, we find showing that as a first approximation, and especially when the angle of incidence is small, m and κ may be taken to be equal to v, and 。. 156. The Optical Constants of Metals. We owe to Drude † the best determination of the optical constants of metals. After a careful investigation of the effects of the condition of the surface and the reflexion of surface films, results were obtained which are reproduced in Table XI. The measurements refer to sodium light. K2 I have added the third and fourth columns giving the values of 2 and vк cos p, the two invariants of metallic refraction. The column headed 0 gives the angle of principal incidence; the last column, the calculated reflected intensity for normal incidence. - The table shows the remarkable fact that v2-k2 is negative for all metals. In the older theories of refraction in which the sympathetic vibrations within the molecule were neglected, this appeared to be an anomaly, for turning to equation (38), 2-2 could not be negative independently of A and B unless were negative, which has no meaning. |