Energy balance for weak formulation of diffraction by lossy anisotropic inhomogeneous gratings

A general expression derived from Poynting´s theorem reports the well-posedness of energy conservation for a weak formulation of diffraction by lossy anisotropic inhomogeneous one- and biperiodic gratings. Formulas allow direct absorption calculus with the same rigor as solutions of Maxwell´s equations, i.e. via distributions employed to describe the field. Absorption integrals, valid for any rigorous method, are expressed using the near field and conductivity tensors in the volume or on the surface restricting a grating domain between uniform medias.