Propagation of light in a one-dimensional photonic crystal: analysis by the Floquet-Bloch function method

The problem of light propagation in a layered periodic medium with a step refractive index profile is considered. The exact solution of this problem is presented in the form of a nonuniform wave, for which the field amplitude distribution is written in an analytic form and the shape of its wave surfaces is determined. The reflection coefficient is obtained for a plane wave incident from the homogeneous medium at the boundary of a semi-infinite layered periodic medium and exciting a Floquet-Bloch wave. Critical conditions are found in which the Floquet-Bloch wave is infinite in the semi-infinite layered medium and exponentially decays in the adjacent homogeneous medium. Dispersion equations and field distributions of surface waves (modes) localised near the boundary of the semi-infinite layered medium are derived and conditions of their appearance are determined. The boundaries of admissible values of the refractive index of the adjacent medium depending on the parameters of the layered periodic medium are established. Dispersion relations for the surface modes in the semi-infinite layered periodic medium (bounded by a system of coupled waveguides) are obtained upon changing the thickness of the boundary layer.