On the theory of light propagation in crystalline dielectrics

A synoptic view on the long-established theory of light propagation in crystalline dielectrics is presented, where charges, tightly bound to atoms (molecules, ions) interact with the microscopic local electromagnetic field. Applying the Helmholtz-Hodge decomposition to the current density in Maxwell’s equations, two decoupled sets of equations result determining separately the divergence-free (transversal) and curl-free (longitudinal) parts of the electromagnetic field, thus facilitating the restatement of Maxwell’s equations as equivalent field-integral equations. Employing a suitably chosen basis system of Bloch functions we present for dielectric crystals an exact solution to the inhomogenous field-integral equations determining the local electromagnetic field that polarizes individual atoms or ionic subunits in reaction to an external electromagnetic wave. From the solvability condition of the associated homogenous integral equation then the propagating modes and the photonic bandstructure ω n ( q ) for various crystalline symmetries Λ are found solving a small sized matrix eigenvalue problem. Identifying the macroscopic electric field inside the sample with the spatially low-pass filtered microscopic local electric field, the dielectric tensor [ ε Λ ( q , ω ) ] ab of crystal optics emerges, relating the accordingly low-pass filtered microscopic polarization field to the macroscopic electric field, solely with the individual microscopic polarizabilities α ( R , ω ) of atoms (molecules, ions) at a site R and the crystalline symmetry as input into the theory. Decomposing the microscopic local electric field into longitudinal and transversal parts, an effective wave equation determining the radiative part of the macroscopic field in terms of the transverse dielectric tensor ε ab ( T ) ( q , ω ) is deduced from the exact solution to the field-integral equations. The Taylor expansion ε ab ( T ) ( q , ω ) = ε ab ( T ) ( ω ) + i ∑ c γ abc ( ω ) q c + ∑ c , d α abcd ( ω ) q c q d + . around q = 0 provides then insight into various optical phenomena connected to retardation and non-locality of the dielectric tensor, in full agreement with the phenomenological reasoning of Agranovich and Ginzburg in ‘Crystal Optics with Spatial Dispersion, and Excitons’ (Springer Berlin Heidelberg, 1984): the eigenvalues of the tensor ε ab ( T ) ( ω ) describing chromatic dispersion of the index of refraction and birefringence, the first order term γ abc ( ω ) specifying rotary power (natural optical activity), the second order term α abcd ( ω ) shaping the effects of a spatial-dispersion-induced birefringence. In the static limit an exact expression for the dielectric tensor is deduced, that conforms with general thermodynamic stability criteria and reduces for cubic symmetry to the Clausius-Mossotti relation. Considering various crystals comprising atoms with known polarizabilities from the literature, in all cases the calculated indices of refraction, the rotary power and the spatial-dispersion-induced birefringence coincide well with the experimental data, thus illustrating the utility of the theory.