Effects associated with a saddle point of the dispersion surface of a photonic crystal

We consider a one-dimensional photonic crystal consisting of alternating dielectric layers of two types. The dispersion relation for such a crystal gives the dependence of the frequency on the transverse wave vector and the quasi-momentum. If the frequency of the incident wave coincides with the frequency of the saddle point, the behavior of the envelope of the wave field is determined by the hyperbolic equation, where the longitudinal coordinate plays the role of time. Depending on the parameters of the isofrequency contour, the canalization or localization of the wave field may occur. If the parameters correspond to the localization, it can be achieved by a proper choice of the field distribution on the surface of the crystal.