Some mathematical properties of Maxwell's equations for macroscopic dielectrics
We consider a number of mathematical properties of Maxwell's equations for linear dispersive and absorptive dielectric media using the auxiliary field method developed earlier by the author [A. Tip, Phys. Rev. A 57, 4818 (1998)]. Here the fields are interpreted as square integrable functions of x ∈ R3. In case the susceptibility ?(x,t) is piecewise constant in x, we show rigorously that a decomposition into independent equations for longitudinal and transverse fields can be made. We point out its relevance for the study of spectral properties of photonic crystals. Again, for the piecewise constant case we discuss the usual boundary conditions at interfaces and discuss the different nature of those for the longitudinal and transverse fields. Then we consider energy conservation for dispersive, nonabsorptive, media. We show that additional contributions to the free field energy density, as given in the literature, are associated with the energy stored in the auxiliary field modes. Finally, we show that also for nonlinear dielectrics it is possible to obtain a conserved energy by introducing auxiliary fields.