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Geometric Optics Approximation and the Eikonal Equation Appendix A: Geometric optics approximation and the eikonal equation The basic concept in the geometric optics approximation in that in the optical range the propagation of waves can be represented (to a very high degree of accuracy) as the transfer of wave energy, which are described by geometric relations. Here we present a transition from the wave equation to an equivalent relation for rays. Consider the scalar wave equation for a light wave. This can be written as (1) where we have introduced the refractive index n=c\v of the medium relative to vacuum. lfn is a constant, for monochromatic waves we assume that ~ =Ij/(r)e -icol where co is the angular frequency. The above solution can be simplified by assuming that if ~ is a plane wave, A. A. i(k.r-OJt) 'f' ='f'.e - where k is the wave number ( = 27t = n ~ ) A. c Vasudevan Lakshminarayanan et al., Lagrangian Optics © Kluwer Academic Publishers 2002 200 Lagrangian Optics If the direction of k is assumed to be along the z direction, then ik (nz-ct) <I> = <I>.e • where leo is the wave number in vacuum {k=nko) The plane wave solutions are not physically possible in an inhomogeneous medium because the variation in refractive index in the direction ofpropagation will bend the wave. We assume that the solutions are very nearly plane waves and write: <I> =eA(r)+ik.(S(x)-CI) (3) where A and S are functions ofpositions. A is the amplitude ofthe wave and S would reduce to nz if n were a constant and can be recognized as being the optical pathlength. Both quantities are real. Taking the above solution, and substituting into the wave equation (I), we can write Since S and A are real measurable quantities the equation holds only ifthe two terms in the square brackets vanish: V2A+(VA)2 +k.(n2 _(VS)2)=0 (5) V2 +2VA·V ·S=O we make the assumption that n varies only slowly with distance, and is nearly a constant over distances ofthe order ofwavelength. This implies that A. < / where / is the dimension ofany change in the medium. This is the fundamental geometric optics approximation. 2 The third term k.2(n2 - (VS/) is the most important term since k. = 47t2 IA./ . This leads us to the fundamental equation of geometrical optics called the eikonal equation: Appendix A 201 (6) The gradient function S is directed along the normal to the surface S = constant. Therefore, the eikonal describes the constant-phase surfaces of a wave while grad A describes the propagation of light energy along a certain direction at a given point, and leads to the concept of a ray of light. The ray can formally be defined as a line the tangent to which at each point coincides with the vector grad S . The propagation oflight can therefore be considered as the motion oflight energy along the rays. The plane perpendicular to light rays (i.e., plane S = constant) is nothing but the wavefront. The geometric optics (ray) approximation is always valid when 'V2A/A is small in comparison to W..2. That is, the wave amplitude varies significantly only at distances greater that A.. Physically, this implies that this term descnbes the bending of light by material objects-diffraction effects. Without going into detail, it can be stated that "classical mechanics corresponds to geometric optics limit ofwave motion [1]" in which light rays orthogonal to wavefronts correspond to particle trajectories orthogonal to surfaces ofconstant S. Reference: 1. H. Goldstein, Classical Mechanics, Addison Wesley, Cambridge, MA, (1956) APPENDIX B: FERMAT'S PRINCIPLE FOR A GENERAL MEDIUM OF ARBITRARY ANISOTROPY Upto this point, we have been dealing with Fermat's principle by assuming an isotropic, non-dispersive medium. Arley (Ref. 1) first showed that Fermat's principle is also applicable to the general case of an anisotropic medium in his studies of the Maxwell equations in the geometrical optical result. A simpler proof was given by Newcomb (Ref. 2) and is derived here. Consider a simple monochromatic wave, with phase velocity cP =CO(IjI(X) - t] (1) where co is a constant and X = (x. y. z). We also define the vector (2) with components 204 Lagrangian Optics k 8Iv I 8Iv m 8Iv K =-=-,A.=-=-,f!=-=- ro Bx ro By ro Bz acp acp acp k=-,l=-,m=- (3) Bx By Bz 5~ ro=-- 5t and 'l'satisfies the equation F=(K,z)=O (4) where (5) This is known as the "reduced eikonal" equation. In the above equation a and b are the principal phase velocities. In a spatially inhomogeneous medium in the z direction they are functions of z. We discard polarization in this analysis. In general ¢ will satisfy a first-order partial differential equation of the form G(ro,K, X) = 0 (6) The X dependence takes into account spatial inhomogeneities in the medium, and K(k, I, m) = V¢.The reduced eikonal equation becomes F(ro,K, X) =0 (7) or F(K, X) =0 since we are dealing with a purely monochromatic wave. Anisotropic Medium 205 It is well-known that the solution of a first order differential equation such as that given above can be reduced to an equivalent set of ordinary differential equations, the characteristic equations: (8) where Fk and Fx or the partial derivatives ofFor X'(s) = vFk (K, X) (9) K'(s) = -vFx (K,X) (10) where v is an arbitrary multiplier which depends on the choice ofthe parameter s. The solution curves X =X(s) are the rays. We can also define the ray direction as the direction of signal propagation. Consider the interference between two neighboring solutions of the wave equation corresponding to the same value of co. The two solutions are given by exp(ico 'If) and exp[{ico( ",+ o'lf)] where ",satisfies F(K, X) =0 and 0'"at the same portion satisfies (11) The condition for constructive interference occurs at any part 0"'= 0 or alternately in terms ofray direction dX (12) This above equation should be satisfied by any admissible oK which satisfies the expression for of. This implies that the vectors dX and FK are parallel. Next, consider the following (use is made ofthe symmetry ofthe tensor VK = VV'If). VF= Fx +VKFK =0 (13) 206 Lagrangian Optics K' = X'.VK = vFK.VK = vVKFK = -vFx (14) This is nothing but a generalized form of the law governing refraction of waves by spatial inhomogeneities in the medium. Consider the case ofa uniaxial crystal, with optic axis in the z-direction. If, as before, a and b are the principal phase velocities and ¢J the wave phase, the eikonal equation for the E wave is (15) where CJ) = aep Ja t. The characteristic equations are: Eliminating K, A. and JJ and putting F = 0, we get (x' / ~2)' =(y' /~2)' =0 (16) (17) 2 2]1 where j.l = [(,X'2 + y'2)/a + z,2/b 12 . These are the differential equations which describe the rays. If we make a suitable choice ofthe parameter s such that b2(x'2+ y'2) + a2z'2 = b2/a2 ~j.la2=1 Anisotropic Medium 207 The ray equations reduce to x' = y' = 0 (18) (19) The ray direction is in the direction ofthe Poynting vector. The wavefront is defined as a moving contour surface of constant phase ¢ . The moving wavefront successively coincides with each of the stationary surfaces of constant lfJ{x) . This means dljl=dt dljl dt (20) For the change in '¥ in the time dt following the wavefront. This can also be written as: dt =dx.VIjI =K.X 'ds (21) Let Ao and A 1 be any two fixed points on a particular ray R, to which we arbitrarily give the parameter values S = 0 and S = 1, and P denote the intersection of the moving wavefront with R the ray. The time required for passage of P from Aoand A1 is given by I T = jK.X'(S) ds (22) o It should be noted that R is the path of stationary T. R is determined as the path of minimum T between the given end points. This is nothing but Fermat's principle. Note that the "time ofpassage" in the Fermat prinbciple is specifically that ofa point 208 LagrangianOptics on the ray carryinga constantvalueofthe phase ,p. As a result of dispersion, it will in generaldiffer from the measuredtraveltime ofa wavesignal fromAoandA. (Ref. 3). By solving the system of equations (23) We can representk and v as functions of X and X', as K.X' = L(X,X') (24) Therefore,the integrandof T is expressibleas a functionof X and X'. Fermat's principlerequiresus to computethe integralnot only along the ray, but also along various neighboring paths which are not themselves rays. How can this be done? The value of K to be used at a particular point is that which is appropriate to the ray having the same tangent vector, at the point in question as the given path. If we do a total differentiation of F = O. dF = 0 = Fk .dk + Fx.dx (25) dL = Lx.dx = L, .dx' = d(k.x')=K.dx' + X' .dK = k.dx' + vF...dK = K .dx' - vFx'dx (26) The secondorder equationfor x(s) to whichK'(s) = -vFx(K, X) reduces in general: (27) This is nothingbut the Euler-Lagrangeconditionfor stationaryT' . Anisotropic Medium 209 In the example ofthe uniaxial crystal given above, (28) A differential geometric analysis of light transmission in an anisotropic medium. For the anisotropic medium, Shen et al [4] have generalized Fermat's principle and the optical metric and have developed a Riemannian geometrical description ofthe space geometry.
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