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Starting from the microscopic Maxwell equations, a self-contained theory of the local electromagnetic field in crystalline dielectrics and its relation to macroscopic electrodynamics is established. Applying the Helmholtz decomposition theorem to the microscopic Maxwell equations, two independent sets of equations determining the solenoidal (transverse) and irrotational (longitudinal) contributions of the local electromagnetic field are initially identified. This enables to restate the microscopic Maxwell equations in terms of equivalent inhomogeneous integral equations, where the entire matter and its response to the local electromagnetic field is fully taken into account by the current density. Implementing a phenomenological material model into this current density, where the local electric field is assumed to polarize individual atoms/ions or subunits of the material in reaction to an externally applied electric field, the inhomogeneous integral equation determining the local electric field is specified first with respect to crystalline dielectrics, solely with the crystal structure and the individual polarizabilities as an input into the theory. Afterwards, it is solved exactly by making use of a newly discovered orthonormal and complete system of non-standard Bloch eigenfunctions of the crystalline translation operator. The propagable modes within the dielectric crystal as well as their dispersion relations \omega_{n}\left(\mathbf{q}\right) , also called photonic band structure, are then obtained from the corresponding solvability condition of the associated homogeneous integral equation. Additionally, the impact of radiation damping and of a static external magnetic induction field on \omega_{n}\left(\mathbf{q}\right) is elucidated, followed by a discussion on the characteristics of the local electromagnetic field in crystalline dielectrics. Subsequently, the macroscopic electromagnetic field is consistently derived from the local electromagnetic field by applying a spatial low-pass filter to the latter one to eliminate all spatial variations that occur on length scales comparable to the lattice constant. The same filtering procedure is then deployed to the microscopic polarization, so that the dielectric tensor \varepsilon_{\Lambda}\left(\mathbf{q},\omega\right) can be introduced in a tried and tested way as that quantity that relates the low-pass filtered microscopic polarization with the macroscopic electric field. Next, the differential equations satisfied by the longitudinal and transverse parts of the macroscopic electric field describing electric field screening or wave propagation are deduced in terms of the longitudinal and transverse dielectric tensor \varepsilon^{\left(\text{L}\right)}\left(\mathbf{q},\omega\right) and \varepsilon^{\left(\text{T}\right)}\left(\mathbf{q},\omega\right) respectively, followed by a comparison of the local and macroscopic radiation field in dielectric crystals. Finally, the dielectric tensor and its transverse counterpart are discussed. It is shown, that the derived expression for the dielectric tensor \varepsilon_{\Lambda}\left(\mathbf{q},\omega\right) conforms in the static limit with general principles such as causality and thermodynamic stability and eventually reduces to the well-known Clausius-Mossotti equation when monatomic simple cubic crystals are considered. Furthermore, its frequency dependence is proven to comply well with the Lyddane-Sachs-Teller relation. In the end, the Taylor expansion of the transverse dielectric tensor \varepsilon^{\left(\text{T}\right)}\left(\mathbf{q},\omega\right) up to second order around \mathbf{q}=\mathbf{0} gives insight into various optical effects or quantities, that are related to non-locality (spatial dispersion) and retardation (chromatic dispersion), in full agreement with the phenomenological theory of Agranovich and Ginzburg. This includes the index of refraction, natural optical activity and spatial dispersion induced birefringence as well as their respective frequency dependencies. The utility of the presented theory is then proven by demonstrating, that the calculations regarding the previously mentioned optical effects or quantities coincide well with experimental data for a variety of highly diverse and complex crystal structures. In particular, the disruptive influence of spatial dispersion induced birefringence in cubic crystals is highlighted in view of the design of optical imaging systems for lithographic applications in the deep ultraviolet spectral region. |
