Abstract

We discuss the role of the symmetries in photonic crystals and classify them according to the Cartan–Altland–Zirnbauer scheme. Of particular importance are complex conjugation and time-reversal , but we identify also other significant symmetries. Borrowing the jargon of the classification theory of topological insulators, we show that is a “particle–hole-type symmetry” rather than a “time-reversal symmetry” if one considers the Maxwell operator in the first-order formalism where the dynamical Maxwell equations can be rewritten as a Schrödinger equation; The symmetry which implements physical time-reversal is a “chiral-type symmetry”. We justify by an analysis of the band structure why the first-order formalism seems to be more advantageous than the second-order formalism. Moreover, based on the Schrödinger formalism, we introduce a class of effective (tight-binding) models called Maxwell–Harper operators. Some considerations about the breaking of the “particle–hole-type symmetry” in the case of gyrotropic crystals are added at the end of this paper.