UC-NRLF llllllll B M 533 01b FRESNEL'S THEORY OF DOUBLE REFRACTION by W« Steadman Aldis — -~fncix.'t FRESNEL'S THEORY OF DOUBLE REFRACTION. W. STEADMAN ALDIS, M.A. TRINITY COLLEGE, CAMBRIDGE. SECOND EDITION. Digitized by the Internet Archive in 2007 with funding from Microsoft Corporation http://www.archive.org/details/chapteronfresnelOOaldirich v&- With the Authors Compliments. A CHAPTER !8T» \ Jf CALIFOB.... mmmn JUL IS 189 \\ OF CALIFORNIA. FRESNEL'S THEORY OF DOUBLE REFRACTION. 8102 BY W. STEADMAN ALDIS, MA. TRINITY COLLEGE, CAMBRIDGE. PRINCIPAL OF THE UNIVERSITY OF DURHAM COLLEGE OF PHYSICAL SCIENCE AT NEWCASTLE- UPON-TYNE AND PROFESSOR OF MATHEMATICS IN THE SAME, SECOND EDITION. CAMBRIDGE : DEIGHTON, BELL, AND CO. LONDON : G. BELL AND SONS. 1879 Gmfafogt : PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. Mr Griffin's tract on Double Refraction has been for some time quite out of print. The following pages are published with a view to supply the deficiency thus caused. It is hoped that they may serve as a useful companion to the latter part of the Astronomer Royal's treatise on the Undulatory Theory of Light. W. S. A. Cambkidge, April, 1870. Preface to the Second Edition, This Second Edition is an almost verbatim reprint of the first. It was the author s intention to incorporate this chapter in a larger work on the Wave Theory of Light. Unexpected hindrances have delayed the pro- gress of this work beyond his expectation, and this reprint is therefore issued to satisfy the demand of present students. Newcastle-upon-Tyne, April, 1879. 1—2 FEESNEL'S THEORY OF DOUBLE REFRACTION. 1. Fresnel's Theory of Double Refraction supposes that the phenomena of light are produced by the vibrations of par- ticles of ether under the influence of their mutual attractions. The hypothesis is first made that the particles of ether are arranged in such a manner that each of them is in stable equi- librium under the influence of the attractions of the others. Let — R be the potential of all the system of particles with respect to a point. Then the resolved parts of the force on the particle at this point parallel to axes arbitrarily assumed will be, if , ,, j. t. ,-, . , dR dR dR x, y, z be the co-ordinates 01 the point, -j- , -j- , -^- re- spectively, tending towards the origin. Hence we have dR dR dR m Let the single particle at #, y, z be displaced to a point os + u, y + v, z + w, while all the other particles remain at rest. Then if we suppose u, v y w so small that we may neglect their squares and higher powers, the force on this displaced particle parallel to the axes will be dR , d*R , df *> d^-^'dydz-^'difa-^'d^- we get, if X, Y } Z denote the forces parallel to the axes on the displaced particle, X=Au+C'v + B'w) Y=C'u+Bv + A'w\ .... (2). Z = B'u+A'v + Cw) Now if we construct the quadric whose equation is Ax 2 + By*+ Cz 2 + 2A'yz+2B'zoc + 2C , xy = l ... (3) the direction of the resultant force whose components are X; Y, Z is perpendicular to the plane which bisects all chords of the surface (3) parallel to the direction of displacement of the particle ; for the equation of this plane is £(Au+ C'v + ffw) + V (G'u + Bv + Aw) + f (B'u +A'v + Cw) =0. The resultant force on the particle will therefore not usually coincide with the direction of its displacement; and if we suppose the particle free to move under the action of this force it will not usually return to its old position. There will be however three directions of displacement with which the directions of the force of restitution will coincide, namely the directions of the three principal axes of the surface (3). If these directions be taken as axes of co-ordinates the equa- ' tion (3) reduces to . and the equations (2) reduce to X=Au, Y=Bv, Z=Cw. Now it is evident that if u, v, w are all positive, X, F, iTmust all tend towards the origin, since the equilibrium is stable, and A, B, G must be all positive. They are usually denoted by the letters a 2 , b 2 , c 2 . The equation (3) thus becomes aV + &y + cV = l (4). This surface is usually called the ellipsoid of elasticity, and its axes the axes of elasticity. It is assumed that the directions Theory of Double Refraction. 7 of these axes and the values of a, b, c are constant throughout the medium. A medium in which a, 6, c are all or any of them different is called a crystal. If all are unequal it is called a biaxal crystal. If two of them are equal and the third different it is called a uniaxal crystal. If the particle be displaced parallel to the axis of x and the other particles be undisturbed it will oscillate in a time — , for .... i ,i •• d*u o its motion is given by the equation -^ = — au. 2. It is then assumed that under these circumstances a particle so displaced will draw an adjacent particle into a pre- cisely similar state of displacement, and that this again will draw the next, and so on ; that thus a series of vibrations will be propagated through the medium, the velocity of propagation being connected with the constant a and the wave length by the simple relation X 2tt X - = — or v = tt- . a. V a Z7T For it is supposed that the wave travels over a wave length while one particle performs a complete oscillation. If the particle be displaced through a space p in a direction inclined at angles (a, ft, 7) to the axes of elasticity, the forces on it parallel to the axes are a?p cos a, Wp cos ft, c*p cos 7, respectively, and the force on it in the direction of displacement will be p (a 2 cos 2 a 4- 6 2 cos 2 ft + c 2 cos 8 7), and for its motion in that direction we have therefore d 2 p -rfe = — jp (a 2 cos 2 a + ¥ cos 2 ft + c 2 cos 2 7). If therefore the motion in that direction alone be considered the time of the particle's oscillation will be 2tt Va 2 cos 2 a + b 2 cos 2 ft + c 2 cos 2 7 ' and if a wave of such vibrations can be propagated through the medium its velocity of propagation will as above be propor- tional to Va 2 cos 2 a + ¥ cos 2 ft + c 2 cos 2 7. 8 Theory of Double Refraction. But if r be the central radius vector of the ellipsoid of elas- ticity drawn in the direction of this displacement, we have 1 r = a 2 cos 2 a + b 2 cos 2 j3 + c 2 cos 2 7. Hence the velocity of propagation of the wave corresponding to any given direction of displacement, if such a wave exist, is inversely proportional to the central radius vector of the ellipsoid' of elasticity drawn in that direction. 3. At this point it will be well to notice the important assumption made. The force on any particle is made to depend on its absolute displacement, and is supposed to be the same as if the other particles were undisplaced. It is evident that the real force will depend on the displacement of the particle relative to the surrounding particles, and quite a different equation of motion from that given above will arise. A particular case of the investigation is given in Airy's Undulatory Theory of Optics, Art. 103, and the general problem has been discussed by Cauchy. The simplicity and beauty of the Mathematical results of Fresnel's hypothesis probably more than counterbalance, from the point of view of a mathematical student, the possible superior accuracy of the more complicated hypothesis. 4. In considering the propagation of light through media of any kind, it is necessary to examine not the motion of one par- ticle alone, but to imagine a series of particles simultaneously vibrating similarly. The most simple hypothesis that can be made is that all the similarly displaced particles at any instant lie in a plane, the case ordinarily called a plane wave. It is evident that by the combination of a number of such plane Waves we can represent any other form of wave. A plane wave of light consists of vibrations of the particles of ether in the plane of the wave front, the displacements and velocities of all the particles in that plane being parallel and equal. This wave is propagated with a velocity which in a crystalline medium depends, as above explained, on the direction of the displacement of the particles. Theory of Double Refraction. 9 5. The fact that the vibrations which produce light are transversal to the direction of propagation, is deduced from the experimental result that two rays of light polarised in planes at right angles do not interfere. The methods of practically pro- ducing polarised light are explained in Airy's Undulatory Theory. We assume that polarised light consists of vibrations of the particles of ether in a fixed direction in the plane of the wave front, and that this direction is perpendicular to the plane of polarisation. The former assumption is sufficient to explain the experimental fact, the latter is usually accepted as true. 6. If a series of particles all lying in a plane within a crystalline medium be equally displaced in parallel directions, the force on each of these particles according to Fresnel's hypo- thesis will not usually be in the direction of displacement, or even in the plane. It may happen however that the resolved part of this force in the plane may coincide with the direction of displacement ; and we will prove presently that there are two directions of displacement for which this is the case. If the particles be displaced in either of these directions the force per- pendicular to the plane will produce vibrations perpendicular to that plane, which therefore do not produce light ; the other parts of the force will cause all the particles to oscillate equally in the plane front, and will thus produce a wave of light, if we assume that the particles oscillating in this plane immediately put in motion those in a contiguous parallel plane. The velocity of propagation of the wave will also, by what has preceded, be inversely proportional to the radius vector of the ellipsoid of elasticity drawn in the direction of the displacement. 7. Suppose that DPD' represents the central section of the ellipsoid of elasticity by a plane parallel to the wave front, and let C be its centre, CP the direction of displacement, CD the diameter of the section conjugate to (7P,and CQ the diameter of the ellipsoid conjugate to the plane PCD. Then the force of restitution is perpendicular to the plane QCD, since this is the plane to which CP is conjugate, and if the resolved part of this force in the plane of the wave front coincide with CP } we must have CP and CD at right angles, or CP must be an axis of the section 1—5 10 Theory of Double Refraction. DPD'. Hence the two directions of vibration with which the resolved part of the corresponding force in the plane coincides are the axes of the section of the ellipsoid of elasticity by the plane front, and the velocities of propagation of the corresponding waves are inversely proportional to the lengths of those axes. 8. If the equation of the plane front at first be lx + my + 7i£ = (1), and X, fi, v, the direction cosines of either axis of the section, we have (Aldis, Solid Oeom. Art. 56) the equations l\ + m/j, + nv — 0] \ {V-e) + j {J * + ^ + ^ = (4), y + Am+ jrzi?^ .....,..(5), 12 Theory of Double Refraction. z + An + Bn (6), Multiplying the first three of these equations by I, m, n respectively and adding, we get, v + A = 0. Transposing the third terms of these same equations, squaring and adding, we get, if r 2 = a? + y* + s\ r a + 2J.v + A ! = * 2 {(^^ m f + (« 1 -6 2 ) 2+ (^ 5 -err :r>-v* = ~,by(7); ij> 2 2 = nr } whence we get qV by cV _ where r 2 = x 2 + y 2 + £*. 10. If with the different points of the original wave front as centres we describe a series of equal wave surfaces it is evi- dent that the plane Ix + my + nz = v will touch them all. That is, the new wave front may be re- garded as the envelope of these wave surfaces. This is analogous to the case of propagation of light through a homogeneous medium, in which case the wave surfaces are spheres. We may also fairly suppose that the point in which the wave surface having any given point of the original wave front as centre touches the second wave front, is the point at which the disturbance in the second wave front is produced by the disturbance at the 1-7 14 Theory of Double Refraction. given point of the first front, and the line joining these points is the direction of the ray proceeding from the first point. A ray must be considered as a small portion of a wave separated from the rest. The existence of such rays must be accepted as a fact ; the theoretical explanation of the separation of a portion of a wave from the rest need not be considered here, belonging rather to the question of diffraction. It is not difficult to see that the reciprocal ellipsoid a *+ b * + c *- 1 has important properties relating to the ray velocities analogous to those which the ellipsoid of elasticity possesses with relation to wave velocities. These the student can develope for himself. 11. If a wave of light be incident from vacuum into a double refracting medium, we may suppose the vibration of each point of the incident wave to produce after a time, a vibration at some point of the wave surface described with the point of incidence as centre. Let the plane of the paper be the plane of incidence, and let AH be the trace of the front of the wave on the plane of the paper, AB the trace of the face of the crystal. Also let PQ be the wave surface to some point of which the disturbance produced by A has arrived when the disturbance at H has reached B. The vibrations at intermediate points will have reached points of wave surfaces similar and similarly situated to PQ, but sue- Theory of Double Refraction. 15 cessively diminishing in size. Any plane drawn through B perpendicular to the plane of the paper touching the surface PQ will touch all these other surfaces and will be a front of the refracted wave. There can be two such planes drawn, and thus one incident wave will produce two refracted waves. The corresponding refracted rays will be obtained by joining A with the points of contact of these planes with PQ. 12. The preceding Article gives the refracted rays when a ray passes from any homogeneous medium into a double re- fracting crystal. The following construction applies when a ray passes from any medium into any other. With the point of incidence of the ray on the common sur- face of the media as centre, describe in the second medium the half of the wave surface belonging to each medium. Produce the incident ray to cut the surface belonging to the first medium, and at the point of intersection draw a tangent plane. This tangent plane will cut the bounding plane of the media in a straight line. Through this line draw tangent planes to the wave surface of the second medium. The lines joining the point of contact of these tangent planes to the point of incidence of the ray will be the refracted rays. It would appear at first from this construction that a single ray passing from one double refracting medium into another would give rise to four rays, since the incident ray would meet the wave surface of the first medium in two points. We shall see however presently (Art. 15) that if a ray proceeding in any direction within a crystal have originally been refracted from air, it must be polarised in one or other of two definite planes according as it is considered to be proceeding to one or other of the points in which its direction cuts the wave surface; and thus if the given ray be polarised in either of these planes we must only take one of the points as the point to which the incident ray corresponds. If the given ray be either unpolarised or polarised in any other plane it must have arisen from two rays of common light, and must be considered to consist of two rays polarised in the required planes travelling with different velocities. We 16 Theory of Double Refraction. should in this case expect four rays, which the construction would give. The construction includes the last article as a particular case. 13. Returning to Art. 8, we see that for all ordinary positions of the wave front, there are two velocities of propaga- tion of the wave. These two will be equal if the wave front coincide with one of the circular sections of the ellipsoid of elasticity, and in that case, whatever be the direction of the vibrations in the plane of the front, only one wave will be propagated. The two lines perpendicular to these positions of the wave front are called the optic axes of the crystal, or the lines of equal wave velocity. The planes of polarisation of the two rays corresponding to any given wave front are connected with the optic axes by a very simple relation, which we will now investigate. Let CN be the normal to the wave front, CO, CO' the optic axes of the crystal. Then the planes of polarisation of the two rays are planes which contain CN and the axes of the sec- tion of the ellipsoid of elasticity by a plane perpendicular to CN. This section will evidently cut the circular section per- pendicular to CO in a line perpendicular to the plane OCN. Similarly it will cut the other circular section in a line perpen- dicular to the plane 0' CN. Hence the radii of the section by Theory of Double Refraction. 17 the wave front perpendicular to the planes OCN ) O'CN are equal and therefore they are equally inclined to the axes of the section. The planes of polarisation of the two rays are therefore planes through GN bisecting the angles between the planes OGN and O'CK 14. Again let v v v 2 be the velocities of the two waves corresponding to the same wave front. We can express these velocities in terms of the angles ON, and 0' CN, as follows. The equation of the ellipsoid of elasticity being aV + &Y + cV=l (1), the equations of the planes of circular section are xJa T =¥±zJF=c i = (2), and that of the wave front is Ix + my + nz = (3). Hence if we denote the angles OCN, O'CN by 6, & respectively, we have l*Ja 2 -b 2 -n>jV-c 2 COS as , l*J a 2 -b 2 + n*Jb 2 -c 2 cos & = — Va 2 - c W, 0' + cos0)\/a 2 -c 2 = 2l\/a 2 -b 2 ) , g) 6' - cos 0) *JaF=? * 2n \/#Wj ,\ (cos (cos Again v v v 2 are the roots of the equation „*_ a » ' v *-b 2 Hence v 2 + v 2 = I 2 (b 2 + c 2 ) + m 2 (c 2 + a 2 ) + n 2 (a 2 + b 2 ) = a 2 + c 2 -l 2 (a 2 -b 2 ) + n 2 {b 2 -c 2 ) since l 2 + m 2 + n 2 = 1 = a 2 + c 2 -(a 2 -c 2 )cos0cos0'by (4) (6). v 2 v 2 = nV + roW + n W, = aV - cT (a 2 - 6 2 ) + nV (b 2 - c 2 ) ; 18 Theory of Doable Refraction. .'. 4^V=4aV-(a 2 -c 2 ) {c 2 (cos^+cos6>) 2 -a 2 (cos^-cos^) 2 }by (5), = 4aV+ (a 2 -c 2 ) 2 (cos 2 0+ cos 2 0') -2 (a 4 -c 4 ) cos0cos<9'. . . (7). Hence squaring (6) and subtracting (7) we get w - \v*~vf~(a* r <$m$mff (8). From (6) and (8) we easily deduce by adding and subtracting o o • o 6 + U o J + P *V = a 2 sin 2 -=— + c 2 cos 2 -=k— v* = a 2 sin 2 — - — + c 2 cos 2 — ^— (9). The results of equations (8) and (9) are easily seen to coin- cide with those deduced in a different manner in Salmon's Solid Geometry, Art. 245. Analogous results can be obtained for ray velocities from the reciprocal ellipsoid. (Lloyd, Wave Theory of Light, Art. 186.) 15. The formulae of the last article enable us to determine completely the circumstances of the vibrations of the two rays corresponding to the same wave front in the crystal. They do not however determine the plane of polarisation if we are only given the direction in which the ray proceeds within the crystal. For this purpose we must revert to the wave surface of Art. 9. Let a ray meet the wave surface at the point x, y, z, let I, m, n be the direction cosines of the normal to the wave front to which the ray belongs, and \, //,, v the direction cosines of the direction of vibration of the particles in the ray. Then we have, if v be the corresponding wave velocity, . v * = a\ 2 + by + cV, where I (b* - c 2 ) + - (c 2 - a 2 ) + - (a 2 - V) = 0, \ fJb v and l\ + nifi + nv= 0. Theory of Doable Refraction. 19 Whence eliminating n we get or l - [b 2 - W - fSb 2 - »VJ - - {a 2 - * 2 c 2 - /* 2 & 2 - \ 2 a 2 } = 0, I m ^ i by symmetry. "\(a 2 -v 2 ) fi(b 2 -v 2 ) v(c 2 -v 2 ) These equations determine \, fi, v in terms of v. Combining these results with the equations (8) of Art. (9), we easily obtain X (V - a 2 ) = fi (r 2 - b 2 ) ^_ v (f - c 2 ) x y z * which give the direction of vibration in the ray proceeding to any given point (x, y, z). A geometrical interpretation can be given to these equations. The co-ordinates of the foot of the perpendicular on the tangent plane to the wave surface at x, y, z are with our previous nota- tion, Iv, mv, nv, and the direction cosines of the line joining this point with the point of contact are proportional to x — lv, y — mv, z — nv. But we have by equations (8) of Art. (9), IV- v 2 - a 2 r 2 - ■a 2y . X- -lv- x (r 2 ■ r 2 - -v 2 ) ■a 2 y- •mv _y{r 2 - r 2 - -v 2 ) ■b 2 z ■ - nv z(r 2 - r 2 - -v 2 ) -c 2 Similarly Hence \, /jl, v are proportional to x — lv, y — mv, z — nv, or the direction of the vibration constituting any ray is the projection of the ray on the tangent plane to the wave surface at the point 20 Theory of Double Refraction. where it meets it. The plane of polarisation is of course per- pendicular to this. The plane of polarisation of a ray proceed- ing in a double refracting medium is usually taken to be the plane containing the normal to the wave front and perpendicular to the direction of vibration. This result may be otherwise obtained. If \, jul, v are the direction cosines of the direction of displacement of a particle, those of the resultant force are proportional to a 2 X, b 2 fi, cV Hence the direction of displacement is perpendicular to the tangent plane drawn to the reciprocal ellipsoid at the point where the line of the resultant force meets it. From this, since the wave surface is the apsidal surface of the reciprocal ellipsoid, it follows by Arts. 461, 462 of Salmon's Solid Geo- metry that the direction of displacement, the direction of the resultant force, the normal to the wave front and the ray all lie in one plane. Hence the direction of displacement is the projection of the radius vector on the tangent plane to the wave surface. It also follows that the ray and the direction of the re- sultant force are at right angles. 16. The wave surface has thus been shown to possess the following properties. Its tangent planes give the positions of the wave fronts after a given time, and the perpendiculars on those tangent planes from the centre represent the velocities of wave fronts in different directions. Its radii vectores from the centre to the points of contact are the directions of the rays corresponding to the different wave fronts, and the lengths of these radii vectores represent the cor- responding ray velocities. The projections of these radii on the tangent planes give the directions of the vibrations in the corresponding rays. 17. The equation of the wave surface becomes by multiply- ing up ^ ( r * _ y) (f _ c ') + f (f - c 2 ) (r 2 - a 2 ) + s 2 (r 2 - a 2 ) (r 2 - L') Theory of Double Refraction. 21 Its trace on the plane of xy is obtained by putting z — ; we then get (* 2 + f - c 2 ) {* 2 (* 2 +jf- 6 2 ) +y (* 2 + 2/ 2 - a 2 )} = (^ 4 Y-0(* 2 +# 2 -& a )(* 2 +y'-c 2 ) whence a? 2 + y 2 = c 2 , or oV + &Y = a 2 6 2 . That is, the trace consists of a circle whose radius is c and an ellipse whose axes are 26 and 2