456 WAVE THEORY angle of incidence for which there is no reflexion of the polarized light under consideration. As the angle of incidence passes through the polarizing angle, the reflected vibration changes sign, and increases in numerical value until it attains unity at a grazing incidence (0 = ^). Total We have hitherto supposed that the second medium (into which reflexion, the light enters at the refracting surface) is the denser. In the contrary case, total reflexion sets in as soon as sin = ^" 1 , at which point 0j becomes imaginary. We shall be able to follow this better in connexion with a mechanical theory. If light falls upon the first surface of a parallel plate at the polarizing angle, the refracted ray also meets the second surface of the plate at the appropriate polarizing angle. For if /u. be the index of the second medium relatively to the first, the tangent of the angle of incidence, which is also the cotangent of the angle of refraction, is equal to ju. At the second surface (the third medium being the same as the first) the angles of incidence and refraction are interchanged, and therefore the condition for the polarizing angle is satisfied, since the index for the second refrac tion is n~ l . Oblique The principal formulae apply to light polarized in, and perpen- polariza- dicular to, the plane of incidence. If the plane of polarization tion. make an angle a with that of incidence, the original vibration may be resolved into two, cos o polarized in the plane of incidence, and sin a polarized in the perpendicular plane. These components are reflected according to the laws already considered, and reconstitute plane-polarized light, of intensity - - ^ . 2 _tan 2 (0-0 1 ) . If the incident light be polarized in a plane making 45 with the plane of incidence, or be circularly-polarized ( 20), or be un polar ized, (5) applies to the reflected light, with substitution of for cos 2 o and sin 2 o. If /3 denote in the general case the angle between the plane of incidence and that in which the reflected light is polarized, a result the approximate truth of which has been verified by Fresnel and Brewster. Reflected The formula; for the intensities of the refracted light follow rays. immediately from the corresponding formulae relative to the re flected light in virtue of the principle of energy. The simplest way to regard the matter is to suppose the refracted light to emerge from the second medium into a third medium similar to the first without undergoing loss from a second reflexion, a supposition which would be realized if the transition between the two media were very gradual instead of abrupt. The intensities of the different lights may then be measured in the same way ; and the sup position that no loss of energy is incurred when the incident light gives rise to the reflected and refracted lights requires that the sum of the squares of the vibrations representing the latter shall be equal to the square of the vibration representing the former, viz., unity. We thus obtain, in the two cases corresponding to (1) and (3), sin 2 (0-0j) _ sin 20 sin 20j !- sin 2 (0 + 6j) 1- tan 2 (e- sin 20 sin 20 X tan 2 (0 + 0j) sin 2 (0 + 1 )cos 2 (0-0 1 ) A plate of glass, or a pile of parallel plates, is often convenient as a polarizer, when it is not necessary that the polarization be quite complete. At the precise angle of incidence (tan- J /x) there would be, according to Fresnel s formulae, only one kind of polarized light reflected, even when the incident light is unpolarized. The polar ization of the transmitted light, on the other hand, is imperfect ; but it improves as the number of plates is increased. Reflexion If we suppose that there is no regular interference, the intensity by a (r) of the light reflected from a plate is readily calculated by a plate. geometric series when the intensity (p) of the light reflected from a single surface is known. The light reflected from the first surface is p. That transmitted by the first surface, reflected at the second, and then transmitted at the first, is p (1 - p) 2 . The next component, reflected three times and transmitted twice, is p 3 (1 - p) 2 , and so on. Hence Pile of The intensity of the light reflected from a pile of plates has been plates. investigated by Provostaye and Desains. 1 If <(m) be the reflexion from m plates, we may find as above for the reflexion from (m + l) plates, + (1 - 2r}<f>(m) Ann. d. Chim. xxx. p. 159, 1850. By means of this expression we may obtain in succession the values of <(2), </>(3), &c., in terms of $>(!), viz., r. The general value is ,, N mr , m (f>()ll) = - -. r-r- Hwj as may easily be verified by substitution. The corresponding expression for the light transmitted by a pile of ?/i plates is The investigation has been extended by Stokes so as to cover the case in which the plates exercise an absorbing influence. 2 The verification of Fresnel s formulae by direct photometric measurement is a matter of some difficulty. The proportion of perpendicularly incident light transmitted by a glass plate has been investigated by Rood ; 3 but the deficiency may have been partly due to absorption. If we attempt to deal directly with the reflected light, the experimental difficulties arc much increased; but the evidence is in favour of the approximate correctness of Fresnel s formulae when light is reflected nearly perpendicularly from a recently polished glass surface. When the surface is old, even though carefully cleaned, there may be a considerable falling off of reflecting power. 4 We have seen that according to Fresnel s tangent-formula there would be absolutely no reflexion of light polarized perpendicularly to the plane of incidence, when the angle of incidence is tan~ V> or, which comes to the same thing, common light reflected at this angle could be perfectly extinguished with a Nicol s prism. It was first observed by Airy that in the case of the diamond and other highly refracting media this law is only approximately in accordance with the facts. It is readily proved by experiment that, whatever be the angle of incidence, sunlight reflected from a plate of black glass is incapable of being quenched by a Nicol, and is therefore imperfectly plane-polarized. This subject has been studied by Jamin. The character of the Jamin s reflected vibration can be represented, as regards both amplitude observa- and phase, by the situation in a plane of a point P relatively to tions. the origin of coordinates 0. The length of the line OP represents the amplitude, while the inclination of OP to the axis of x repre sents the phase. According to Fresnel s formula appropriate to light polarized perpendicularly to the plane of incidence, P is situated throughout on the axis of x, passing through when the angle of incidence is tan " 1 /j.. Jamin found, however, that in general P does not pass through 0, but above or below it. When P is on the axis of y, the amplitude is a minimum, and the phase is mid way between the extreme phases. For one class of bodies the phase is in arrear of that corresponding to perpendicular incidence, and for another class of bodies in advance. In a few intermediate cases P passes sensibly through ; and then the change of phase is sudden, and the minimum amplitude is zero. In the case of metals the polarization produced by reflexion is Metallic still more incomplete. Light polarized perpendicularly to the reflexion, plane of incidence is reflected at all angles, the amount, however, decreasing as the angle of incidence increases from to about 75, and then again increasing up to a grazing incidence. The most marked effect is the relative retardation of one polarized component with respect to the other. At an angle of about 75 this retarda tion amounts to a quarter period. The intensity of reflexion from metals is often very high. From silver, even at perpendicular incidence, as much as 95 per cent, of the incident light is reflected. There is reason for regarding the high reflecting power of metals as connected with the intense absorption which they exercise. Many aniline dyes reflect in abnormal proportion from their surface those rays of the spectrum to which they are most opaque. The peculiar absorption spectrum of permanganate of potash is reproduced in the light reflected from a surface of a crystal. 5 27. Reflexion on the Elastic Solid Theory. On the theory which assimilates the aether to an elastic solid, the investigation of reflexion and refraction presents no very serious difficulties, but the results do not harmonize very well with optical observation. It is, however, of some importance to understand that reflexion and refraction can bo explained, at least in their principal features, on a perfectly definite and intelligible theory, which, if not strictly applicable to the aether, has at any rate a dis tinct mechanical significance. The refracting surface and the wave-fronts may for this purpose be supposed to be plane. When the vibrations are perpendicular to the plane of incidence (2 = 0), the solution of the problem is very simple. We suppose that the refracting surface is # = 0, the rigidity and density in the first medium being N, D, and in the second N 1; D r The displace- 2 I roc. Roy. Sac., xi. p. 545, 1862. 3 Am. Jour., vol. 1., July 1870. 4 Ou the Intensity of Light reflected from Certain Surfaces at nearly Per pendicular Incidence," Proc. Hoy. Soc., 1886. 5 Stokes, " On the Metallic Reflection exhibited by Certain Non-Metallic Substances," I hil. Mag., Dec. 1853. Vibra tions per pendicu- lar to plane of
incidence