Analytical Solution of Maxwell's Equations in Lossy and Optically Active Crystals

Light propagation in isotropic materials and simple crystals is well understood classically in terms of solutions of Maxwell’s equations @1#. There exists today considerable interest in both experimental and theoretical work on optically active and other optically complex materials @2–9#. Analytic solutions of Maxwell’s equations in such materials are not readily available. Light propagation in lossy uniaxial crystals is discussed in detail in Ref. @10#. The problem is more formidable in lossy biaxial crystals, especially in the case where the real and imaginary parts of the dielectric tensor are not co-diagonal, and in materials where both natural and magnetically induced optical activities are present. Analytic solutions of Maxwell’s equation in lossy biaxial crystals are first given in the work of G. Szivessy @11# ~the original work is in German; we are not aware of translations in existence!; similar results have been obtained subsequently @12#. Light propagation in a special case of lossless biuniaxial materials with natural and magnetically induced optical activity is considered in Ref. @5#; the general solution even in the lossless case is not known. Solutions of Maxwell’s equations describe the optical eigenmodes of the system, obtained by solving the eigenvalue problem for the fields. The eigenvalues are inversely proportional to the square of the wave vector; the solution of the secular equation gives the dispersion relation. Since for the propagating modes the electric displacement is normal to the wave vector, the solution manifold is the plane normal to it. The secular equation for lossless materials is therefore biquadratic. For lossy materials, as has been shown @11#, the eigenvectors span the space normal to a complex wave vector. For optically active materials, the dielectric tensor is a function of the wave vector, leading to a nonstandard eigenvalue problem which has not been solved. Our contribution is to solve this problem for the case when the dielectric tensor depends linearly on the wave vector. In this paper, we give explicit analytical expression for the solutions to Maxwell’s equation in the most general homogeneous linear optical media: the lossy optically active biaxial crystal. Since our formalism allows a simple and