Permeability tensor versus permittivity one in theory of nonreciprocal optical components

We discuss symmetry properties of the combined systems: electromagnetic field (TE and TM modes) + medium (nondiagonal permeability tensor [μ] and nondiagonal permittivity one [ϵ]) in 2D photonic crystals with transverse magnetization. We show that the origin of orthogonality of the pairs (TE mode + [ϵ]) and (TM mode + [μ]) is different parities of the corresponding fields with respect to the horizontal plane of symmetry of the crystal. As a result of this symmetry, a common use only of the permittivity tensor [ϵ] in the electromagnetic theory of nonreciprocal optical components, ignoring the permeability tensor [μ], leads to loss of a significant part of possible solutions related to TE modes.