Exact analytical expressions for the dispersion relation of one-dimensional chiral photonic crystals

We consider one-dimensional photonic crystals made of alternating layers of two kinds of isotropic chiral media. The two layers are characterized by the dielectric permittivities mu(1) and mu(2), the magnetic permeabilities mu(1) and mu(2), the chirality parameters gamma(1) and gamma(2) and the thicknesses d(1) and d(2). By diagonalizing an 8 x 8 matrix derived from the two coupled wave equations satisfied by our system, we obtain exact analytical expressions for the dispersion relation for plane electromagnetic waves propagating at an arbitrary angle with respect to the layers. We present a detailed analysis of our dispersion relation and explore the evolution of the photonic band structure as the frequency, the chirality parameter and the incident angle vary, with a special emphasis on the generation of the cross- and co-polarization bandgaps.