Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/474

Rh 448 WAVE THEORY polarized components are not only each irregular, but there is no permanent phase-relation between them. No light derived from one can therefore ever interfere regularly with light derived from the other. If, however, we commence with plane-polarized light, we have only one series of irregularities to deal with. When resolved in two rectangular directions, the components cannot then interfere, but only on account of the perpendicularity. If brought back by resolution to the same plane of polarization, interference becomes possible, because the same series of irregularities are to be found in both components. 21. Double Refraction. The construction by which Huygens explained the ordinary and extraordinary refraction of Iceland spar has already been given (LIGHT, vol. xiv. p. 610). The wave-surface is in two sheets, com posed of a sphere and of an ellipsoid of revolution, inTcontact with one another at the extremities of the polar axis. In biaxal crystals the wave-surface is of a more complicated character, including that of Huygens as a particular case. Wave- It is not unimportant to remark that the essential problem of surface, double refraction is to determine the two velocities with which plane waves are propagated, when the direction of the normal to the wave-frout is assigned. When this problem has been solved, the determination of the wave-surface is a mere matter of geometry, not absolutely necessary for the explanation of the leading pheno mena, but convenient as affording a concise summary of the principal laws. In all cases the wave-surface is to be regarded as the envelope at any subsequent time of all the plane wave- fronts which at a given instant may be supposed to be passing through a particular point. Direc- In singly refracting media, where the velocity of a wave is the tion of same in all directions, the wave-normal coincides with the ray. In ray. doubly refracting crystals this law no longer holds good. The principles by which the conception of a ray is justified ( 10), when applied to this case, show that the centre of the zone system is not in general to be found at the foot of the perpendicular upon the primary wave-front. The surface whose contact with the primary wave-front determines the element from which the secondary dis turbance arrives with least retardation is now not a sphere, but whatever wave-surface is appropriate to the medium. The direc tion of the ray, corresponding to any tangent plane of the wave- surface, is thus not the normal, but the radius vector drawn from the centre to the point of contact. The velocity of propagation (reckoned always perpendicularly to the wave-front) may be conceived to depend upon the direction of the wave-front, or wave-normal, and upon what we may call (at any rate figuratively) the direction of vibration. If the velocity depended exclusively upon the wave-normal, there could be no double, though there might be extraordinary, refraction, i.e., refraction deviating from the law of Snell ; but of this nothing is known in nature. The fact that there are in general two velocities for one wave-front proves that the velocity depends upon the direction of vibration. Fresnel s According to the Huygenian law, confirmed to a high degree of views. accuracy by the observations of Brewster and Swan, 1 a ray polarized in a principal plane (i.e., a plane passing through the axis) of a uniaxal crystal suffers ordinary refraction only, that is, propagates itself with the same velocity in all directions. The interpretation which Fresnel put upon this is that the vibrations (understood now in a literal sense) are perpendicular to the plane of polarization, and that the velocity is constant because the direction of vibration is in all cases similarly related (perpendicular) to the axis. The development of this idea in the fertile brain of Fresnel led him to the remarkable discovery of the law of refraction in biaxal crystals. The hypotheses upon which Fresnel based his attempt at a mechanical theory are thus summarized by Verdet: (1) The vibrations of polarized light are perpendicular to the plane of polarization ; (2) The elastic forces called into play during the propagation of a system of plane waves (of rectilinear transverse vibrations) differ from the elastic forces developed by the parallel displacement of a single molecule only by a constant factor, independent of the par ticular direction of the plane of the wave ; (3) When a plane wave propagates itself in any homogeneous medium, the components parallel to the wave-front of the elastic forces called into play by the vibrations of the wave are alone operative ; (4) The velocity of a plane wave which propagates itself with type unchanged in any homogeneous medium is proportional to the square root of the effective component of the elastic force developed by the vibrations. Fresnel himself was perfectly aware that his theory was deficient in rigour, and indeed there is little to be said in defence of his second hypothesis. Nevertheless, the great historical interest of this theory, and the support that experiment gives to Fresnel s conclusion as to the actual form of the wave-surface in biaxal crystals, render some accoiint of his work in this field imperative. 1 Edinb. Trans., vol. xvi. p. 375. The potential energy of displacement of a single molecule from Energy of its position of equilibrium is ultimately a quadratic function of the displace- three components reckoned parallel to any set of rectangular axes. meut. These axes may be so chosen as to reduce the quadratic function to a sum of squares, so that the energy may be expressed, V = i 2 | 2 + ^V + ic 2 C 2 ...... (1), where |, T;, C are the three component displacements. The corre sponding forces of restitution, obtained at once by differentiation, are X = 2, Y-Zoj, Z = c 2 C. . . (2). The force of restitution is thus in- general inclined to the direction of displacement. The relation between the two directions X, Y, Z and |, ?], is the same as that between the normal to a tangent plane and the radius vector p to the point of contact in the ellipsoid 2 | 2 + &V + c 2 2 ^l ...... (3). If a 2, & 2 , c 2 are unequal, the directions of the coordinate axes are the only ones.in which a displacement calls into operation a parallel force of restitution. If two of the quantities a&quot;, b&quot;, c 2 are equal, the ellipsoid (3) is of revolution, and every direction in the plane of the equal axes possesses the property in question. This is the case of a uniaxal crystal. If the three quantities a?, b&quot;, c 2 arc all equal, the medium is isotropic. If we resolve the force of restitution in the direction of displace ment, we obtain a quantity dependent upon this direction in a manner readily expressible by means of the ellipsoid of elasticity (3). For, when the total displacement is given, this quantity is proportional to that is to say, to the inverse square of the radius vector p in (3). We have now to inquire in what directions, limited to a particular Uirec- plane, a displacement may be so made that the projection of the tions of force of restitution upon the plane may be parallel to the displace- yibra- meut. The answer follows at once from the property of the ellip- tion. soid of elasticity. For, if in any section of the ellipsoid we have a radius vector such that the plane containing it and the normal to the corresponding tangent plane is perpendicular to the plane of the section, the tangent line to the section must be perpendicular to the radius vector, that is, the radius vector must be a principal axis of the section. There are therefore two, and in general only two, directions in any plane satisfying the proposed condition, and these are perpendicular to one another. If, however, the plane be one of those of circular section, every line of displacement is such that the component of the force, resolved parallel to the plane, coincides with it. According to the principles laid down by Fresnel, we have now complete data for the solution of the problem of double refraction. If the direction of the wave-front be given, there are (in general) only two directions of vibration such that a single wave is propa gated. If the actual displacements do not conform to this condi tion, they will be resolved into two of the required character, and the components will in general be propagated with different velocities. The two directions are the principal axes of the section of (3) made by the wave-front, and the velocities of propagation are inversely proportional to the lengths of these axes. The law connecting the lengths of the axes with the direction (/, m, n] of the plane is a question of geometry ; 2 and indeed the whole investigation of the wave-surface may be elegantly carried through geometrically with the aid of certain theorems of Mac- Cullagh respecting apsidal surfaces (Salmon, ch. xiv.). For _this, however, we have not space, and must content ourselves with a sketch of the analytical method of treatment. If v be the velocity of propagation in direction I, in, n, the wave- surface is the envelope of planes lx+my + nz = v ....... (4), where v is a function of I, m, n, whose form is to be determined. If (A, n, i&amp;gt;) be the corresponding direction of vibration, then =0 ....... (5). According to the principles laid down by Fresnel, we see at once that the force of restitution ( 2 A, b 2 /*, c~v), corresponding to a displacement unity, is equivalent to a force v 2 along (A, /u, v), to gether with some force (P) along (I, m, n). Resolving parallel to the coordinate axes, we get I P = a- A - ir, mT = b-fj. - v&quot;fji , P = c&quot;t&amp;gt; - v 2 v , A 22, /x=,- o, &quot;=-., -o - (6). or tr 6- - ?r c- - tt~ Multiplying these by I, m, n respectively, and taking account of Law of (5), we see that velocity. Z 2 in 2 n 2 ,. -5+,., +r5-r.o=0 (/) V.2 _, ? *2 i 2 - v 2 is the relation sought for between v and (I, in, n). In this equa tion b, c are the velocities when the direction of propagation is 2 See Salmon s Analytical Geometry of Three Dimensions, Dublin, 1882, 102.