Volume Integral Equations for Scattering from Anisotropic Diffraction Gratings

We analyze electromagnetic scattering of TM polarized waves from a diffraction grating consisting of a periodic, anisotropic, and possibly negative-index dielectric material. Such scattering problems are important for the modelization of, e.g., light propagation in nano-optical components and metamaterials. The periodic scattering problem can be reformulated as a strongly singular volume integral equation, a technique that attracts continuous interest in the engineering community, but rarely received rigorous theoretic treatment. In this paper we prove new (generalized) G\r{a}rding inequalities in weighted and unweighted Sobolev spaces for the strongly singular integral equation. These inequalities also hold for materials for which the real part takes negative values inside the diffraction grating, independently of the value of the imaginary part.