Abstract
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 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 of crystal optics emerges, relating the accordingly low-pass filtered microscopic polarization field to the macroscopic electric field, solely with the individual microscopic polarizabilities 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 is deduced from the exact solution to the field-integral equations. The Taylor expansion 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 describing chromatic dispersion of the index of refraction and birefringence, the first order term specifying rotary power (natural optical activity), the second order term 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.
Highlights
When optical signals traverse a transparent dielectric, for example a fused quartz prism, the light travels at different speed depending on frequency f = ω 2π, so that the shape of a wave packet, say composed of mixed frequencies |f − fc| ≤∆fc 2 around a carrier frequency fc in a frequency interval of width∆fc, tends to spread out.This is the well known chromatic dispersion effect resulting from the frequency dependence of the refractive index n = n (ω).Microscopic considerations based on first principles reveal, that the frequency dependence of the refractive index n (ω) is directly connected to the retarded response of the polarizable constituents of matter, the latter distinguishing themselves as atoms, molecules or ions
For light propagating along the diagonal of the x-y plane, i.e. q = √|q2| e(x) + e(y), the dielectric tensor εa(Tb ) (q, ω) reveals two transversal modes capable to propagate with slightly different speeds inside the crystals mentioned above, causing an intrinsic birefringence ∆n (ω) induced by spatial dispersion
The field-integral equation approach presented in this article differs from traditional presentations of crystal optics, for example [12, 13, 15]
Summary
When optical signals traverse a transparent dielectric, for example a fused quartz (silica) prism, the light travels at different speed depending on frequency f. In what follows we conceive the optical transition frequencies ων R(j) , the life-time parameter τν R(j) and the oscillator strengths fa,a′ ν, R(j) of each atom species positioned at a site R(j) = R + η(j), with R ∈ Λ a lattice vector and η(j) indicating a position of an atom (ion, molecule) inside a unit cell CΛ of the crystal Λ, as fitting parameters, so that the optical properties as calculated from the dielectric tensor εΛ (q, ω) of the crystal coincide with experiment How this objective can be accomplished, and in particular how εΛ (q, ω) depends on the individual atom polarizabilities αa,a′ η(j), ω , and via (2) on the atom individual multi-electron spectrum, we shall elaborate in what follows. For the special case of cubic symmetry the long-known Clausius-Mossotti relation is recovered
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