Linear and Nonlinear Electrodynamics in Elastic Anisotropic Dielectrics

Publisher Website
Google Scholar
Abstract
We have constructed an ab initio macroscopic Lagrangian theory of linear and nonlinear electrodynamics for an anisotropic dielectric possessing acoustic, ionic, electronic, and other internal excitations. This theory can with modest effort predict the symmetry, frequency dispersion, and internal mechanisms for susceptibilities governing all orders of direct and indirect interactions between these excitations and the electromagnetic field. Application is made to optical harmonic generation, and the direct and indirect electro-optic and photoelastic effects. The theory starts with a Lorentzian microscopic formulation in terms of moving, massive point charges in a vacuum. It is converted to a long-wavelength macroscopic theory via a continuum limit. A Lagrangian is constructed from the vacuum electromagnetic Lagrangian, the usual interaction between the matter charge current and the electromagnetic field, a kinetic energy of the matter's internal motions and a matter stored-energy term. The stored energy must be invariant under arbitrary body rotations, displacements, and spatial reflections and is therefore a function of the basic invariants: the finite strain tensor and body components of the excitation fields. Parity-violating terms are also considered. The stored energy is expanded as a polynomial in these basic invariants with coefficients, called material descriptors (since they describe intrinsic properties of the crystal), which are restricted in form by the crystal space-group symmetry. Effective-local-field effects are shown to be included in the stored energy. Equations for the electromagnetic field, the acoustic field, and the internal excitations follow deductively from the Lagrangian. Elimination of the internal excitations leads to electromagnetic and acoustic equations with piezoelectric, photoelastic, electro-optic, frequency mixing, and other couplings. Among the linear results are a new eigenvector formulation of crystal optics, a generalized Lyddane-Sachs-Teller relation, and a dyadic Green's function for radiations in crystals of arbitrary anisotropy. Constitutive relations for linear and nonlinear susceptibilities in terms of the material descriptors indicate the mechanisms involved. All indirect contributions to any nonlinear interaction arise automatically, and their symmetry, which often differs from that of the direct interaction, is predicted. The nonlinear results include previously ignored rotational contributions to photo-elastic interactions of all orders.