Reflextion On The Elastic Solid Theory
reflexion plane waves wave media formula incidence fresnel
REFLEXTION ON THE ELASTIC SOLID THEORY. On the theory which assimilates the father 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 reflexiou and refraction can be explained, at least in their principal features, on a perfectly definite and intelligible theory, which, if not strictly applicable to the tether, has at any rate a distinct mechanical significance. The refracting surface and the The intensity of the light reflected from a pile of plates has been wave-fronts may for this purpose be supposed to be plane. investigated by Provostaye and Besains.' If 0(in) be the reflexion When the vibrations are perpendicular to the plane of incidence from 9n plates, we may find. as above for the reflexiou from (st +1) (z= 0), the solution of the problem is very simple. We suppose plates, that the refracting surface is x=0, the rigidity and density in the - - - - ' - first medium being N, D, and in the second N„ D„ The displacements in the two media are in general denoted by E, n, C; 6, nr, Ci; but in the present ease Z, n, r r, 77, all vanish. Moreover C, Cr are independent of The equations to be satisfied in the interior of the media are accordinalv (5 24) The formula for intensity of the reflected wave is here obtained on the supposition that the waves are of harmonic type ; but, since it does not involve x and there is no change of phase, it may be extended by Fourier's theorem to waves of any type whatever. It may be remarked that when the first and second media are interchanged the coefficient in (12) simply changes sign, retaining its numerical value.
The amplitude of the reflected wave, given in general by (12), assumes special forms when we introduce more particular suppositions as to the nature of the difference between media of diverse refracting power. According to Fresnel and Green the rigidity does not vary, or N= N1. In this case supposed to vibrate normally to the plane of polarization ; while, if 1)- D„ the vibrations are parallel to that plane.
An intermediate supposition, according to which the refraction. is regarded as due partly to a difference of density and partly to a difference of rigidity, could scarcely be reconciled with observation, unless one variation were very subordinate to the other. But the most satisfactory argument against the joint variation is that derived from the theory of the diffraction of light by small particles (§ 25).
We will now, limiting ourselves for simplicity to Fresnel's sup-' position (N1-N), inquire into the character of the solution when total reflexion sets in. The symbolical expressions for the reflected and refracted waves are In this case there is a refracted wave of the ordinary kind, conveying away a part of the original energy. When, however, the second medium is the rarer (V1>V), and the angle of incidence exceeds the so-called critical angle [sin-1(V/V1)], there can be no refracted wave of the ordinary kind. In whatever direction it may be supposed to lie, its trace must necessarily outrun the trace of the incident wave upon the separating surface. The quantity al, as defined by our equations, is then imaginary, so that (13) and (14) no longer express the real parts of the symbolical expressions tril and fin_ tan E tan'20 - sec20(r/V12)) . . . . (21).
The principal application of the fornrulaebeing to reflexions when the second medium is air, it will be convenient to denote by g the index of the first medium relatively to the second, so that pc-IT,/V.. Thus tan E= tan2 - see?. 0/y21 . . . . (22).
The above interpretation of his formula sin (0,- 0)/sin (01+0), in the case where 01 becomes imaginary, is due to the sagacity of Fresnel. His argument was perhaps not set forth with full rigour, but of its substantial validity there can be no question. By a similar process Fresnel deduced from his tangent formula for the change of phase (2e') accompanying total reflexion when the vibrations are executed in the plane of incidence, The phase-differences represented by 2E and 2E' cannot be investi gated experimentally, but the difference (2e' - 2E) is rendered evident when the incident light is polarized obliquely so as to contribute components in both the principal planes. If in the act of reflexion one component is retarded more or less than the other, the resultant light is no longer plane but elliptically polarized.
The most interesting case occurs when the difference of phase amounts to a quarter of a period, corresponding to light circularly polarized. If, however, we put cos (2E' - 20-0, we find The coefficient of t is necessarily the same in all three waves on account of the periodicity, and the coefficient of y must be the same since the traces of all the waves upon the plane of separation must move together. With regard to the coefficient of x, it appears by substitution in the differential equations that its sign is changed in passing from the incident to the reflected wave ; in fact c2-V2((±a)2+b21=V12012+1,21 . . . . (7), where V, V, are the velocities of propagation in the two media given by V2= N/D , Vi2= (8) Now bl1/(a2+7)2) is the sine of the angle included between the axis of x and the normal to the plane of waves - in optical language, the sine of the angle of incidence - and bh/(a,2-1-b2) is in like manner the sine of the angle of refraction. If these angles be denoted (as before) by 0, 0„ (7) asserts that sin 0 : sin 01 is equal to the constant ratio V : V1, the well-known law of sines. The laws of reflexion and refraction follow simply from the fact that the velocity of propagation normal to the wave-fronts is constant in each medium, that is to say, independent of the direction of the wave-front, taken in connexion with the equal velocities of the traces of all the waves on the plane of separation nr/sin 8= exceed 3 +N/8. So large a value of /22 not being available, the conversion of plane-polarized into circularly-polarized light by one reflexion is impracticable.
The desired object may, however, be attained by two successive reflexions. The angle of incidence may be so accommodated to the index that the alteration of phase amounts ton period, in which case a second reflexion under the same conditions will give rise to light circularly polarized. Putting (2€ - 2€1=1/r, we get au equation by which 0 is determined when ,u is given.
It appears that, when 0=48° 37' or 54° 37'.
These results were verified by Fresnel by means of the rhomb shown in fig. 27.
The problem of reflexion upon the elastic solid theory, when the vibrations are executed in the plane of incidence, is more complicated, on account of the tendency to form waves of dilatation. In order to get rid of these, to which no optical phenomena correspond, it is necessary to follow Green in supposing that the velocity of such waves is infinite, or that the media are ineompressible.1 Even then we have to introduce in the neighbourhood of the interface waves variously called longitudinal, pressural, or surface waves ; otherwise it is impossible to satisfy the conditions of continuity of strain and stress. These waves, analogous in this respect to those occurring in the second medium when total reflexion is in progress (19), extend to a depth of a few wave-lengths only, and they are so constituted that there is neither dilatation nor rotation. On account of them the final formulre are less simple than those of Fresnel. If we suppose the densities to be the same in the two media, there is no correspondence whatever between theory and observation. In this case, as we have seen, vibrations perpendicular to the plane of incidence are reflected according to Fresnel's tangent-formula ; and thus vibrations in the plane of incidence should follow the sine-formula. The actual result of theory is, however, quite different. In the ease where the relative index does not differ greatly from unity, polarizing angles of 22i° and 671° are indicated, a result totally at variance with observation. As in the case of diffraction by small particles, an elastic solid theory, in which the densities in various media are supposed to be equal, is inadmissible. If, on the other hand, following Green, we regard the rigidities as equal, we get results in better agreement with observation. To a first approximation indeed (when the refraction is small) Green's formula coincides with Fresnel's tangent-formula ; so that light vibrating in the plane of incidence is reflected according to this law, and light vibrating in the perpendicular plane according to the sine-formula. The vibrations are accordingly perpendicular to the plane of polarization.
The deviations from the tangent-formula, indicated by theory when the refraction is not very small, are of the same general character as those observed by Jamin, but of much larger amount. The minimum reflexion at the surface of glass (i.,-41) would be iiy,2 nearly the half of that which takes place at perpendicular incidence, and very much in excess of the truth. This theory cannot therefore be considered satisfactory as it stands, and various suggestions have been made for its improvement. The only variations from Green's suppositions admissible in strict harmony with an elastic solid theory is to suppose that the transition from one medium to the other is gradual instead of abrupt, that is, that the transitional layer is of thickness comparable with the wave-length. This modification would be of more service to a theory which gave Fresnel's tangent-formula as the result of a sudden transition than to one in which the deviations from that formula are already too great.
It seems doubtful whether there is much to be gained by further discussion upon this subject, in view of the failure of the elastic solid theory to deal with double refraction. The deviations from Fresnel's formtffie for rcflexion are comparatively small ; and the whole problem of reflexion is so much concerned with the condition of things at the interface of two media, about which we know little, that valuable guidance can hardly be expected from this quarter. It is desirable to bear constantly in mind that reflexion depends entirely upon an approach to discontinuity in the properties of the medium. It' the thickness of the transitional layer amounted to a few wave-lengths, there would be no sensible reflexien at all.
Another point may here be mentioned. Our theories of reflexion take no account of the fact that one at least of the media is dispersive. The example of a stretched string, executing transverse vibrations, and composed of two parts, one of which in virtue of stiffness possesses in some degree the dispersive property, shows that the boundary conditions upon which refiexion depends are thereby modified. We may thus expect a finite reflexion at the interface of two media, if the dispersive powers are different, even though the indices be absolutely the same for the waves under consideration, in 'which case there is no refraction. lint a knowledge of the dispersive properties of the media is not sufficient to determine the reflexion without recourse to hypothesis.3