Effective permittivity and permeability of two-dimensional photonic crystals

We present the results of rigorous numerical calculations of the dependence of the reflection coefficient of a semi-infinite two-dimensional photonic crystal on the angle of incidence of the incoming plane wave. We show that, contrary to some results published earlier. this coefficient is not strictly real even outside the crystal banduaps. We also propose a definition of the effective permittivity c and permeability mu of a truncated photonic crystal and specify the symmetry conditions to be satisfied by the truncation plane and the dominant crystal eigenmode to assure continuity of epsilon and mu when the mode character changes from propagating to evanescent. The value of the reflection coefficient obtained by treating the crystal as a homogeneous medium with epsilon and mu defined in the proposed way is shown to be a good approximation to the rigorous value in a wide range of angles of incidence, extending beyond that corresponding to propagating crystal modes.