Mathematical models for dispersive electromagnetic waves: an overview <s> An augmented self-adjoint PDE model for Maxwell's equations in passive media <s> We investigate the limiting case of the Drude-Lorentz model for the complex electric permeability epsilon(omega)=1+chi;(omega) as the damping tends to zero. We find that epsilon(omega) becomes real except for a number of discrete frequencies. The Kramers-Kronig relations connecting the real and imaginary parts of chi;(omega) remain valid. <s> BIB001 </s> Mathematical models for dispersive electromagnetic waves: an overview <s> An augmented self-adjoint PDE model for Maxwell's equations in passive media <s> We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively determine if a given time dispersive system can be extended to a conservative one; (ii) to construct that very conservative system—which we show is essentially unique. We illustrate the method by applying it to the spectral analysis of time dispersive dielectrics and the damped oscillator with retarded friction. In particular, we obtain a conservative extension of the Maxwell equations which is equivalent to the original Maxwell equations for a dispersive and lossy dielectric medium. <s> BIB002 </s> Mathematical models for dispersive electromagnetic waves: an overview <s> An augmented self-adjoint PDE model for Maxwell's equations in passive media <s> We develop a Hamiltonian theory for a time dispersive and dissipative (TDD) inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The canonical Hamiltonian constructed here exactly reproduces the original dissipative evolution after integrating out auxiliary fields. In particular, for a dielectric medium we obtain a simple formula for the Hamiltonian and closed form expressions for the energy density and energy flux involving the auxiliary fields. The developed approach also allows to treat a long standing problem of scattering from a lossy non-spherical obstacle and, more generally, wave propagation in TDD media. <s> BIB003 </s> Mathematical models for dispersive electromagnetic waves: an overview <s> An augmented self-adjoint PDE model for Maxwell's equations in passive media <s> We develop a Hamiltonian theory of a time dispersive and dissipative inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The proposed Hamiltonian couples the given system to auxiliary fields, in the universal form of a so-called canonical heat bath. After integrating out the heat bath the original dissipative evolution is exactly reproduced. Furthermore, we show that the dynamics associated to a minimal Hamiltonian are essentially unique, up to a natural class of isomorphisms. Using this formalism, we obtain closed form expressions for the energy density, energy flux, momentum density, and stress tensor involving the auxiliary fields, from which we derive an approximate, ``Brillouin-type,'' formula for the time averaged energy density and stress tensor associated to an almost mono-chromatic wave. <s> BIB004 </s> Mathematical models for dispersive electromagnetic waves: an overview <s> An augmented self-adjoint PDE model for Maxwell's equations in passive media <s> We study the linear phenomenological Maxwell’s equations in the presence of a polarizable and magnetizable medium (magnetodielectric). For a dispersive, nonabsorptive, medium with equal electric and magnetic permeabilities e(ω) and μ(ω), the latter can assume the value of −1 (+1 is their vacuum value) for a discrete set of frequencies ±ωn, i.e., for these frequencies the medium behaves as a negative index material (NIM). We show that such systems have a well-defined time evolution. In particular, the fields remain square integrable (and the electromagnetic energy finite) if this is the case at some initial time. Next we turn to Green’s function G(x,y,z) (a tensor), associated with the electric Helmholtz operator for a set of parallel layers filled with a material. We express it in terms of the well-known scalar s and p ones. For a half space filled with the material and with a single dispersive Lorentz form for e(ω)=μ(ω), we obtain an explicit form for G. We find the usual behavior for NIMs for ω=±ω, th... <s> BIB005
We state below the generalization of theorem 3.24 for passive non-Lorentz materials. The idea of augmented models, with an auxiliary unknown depending on a (a priori) continuum of auxiliary variables was developed in BIB001 , BIB005 , BIB002 , BIB003 , BIB004 . The pioneering idea goes back to Lamb ([31] ) in 1900.