Photonic crystal X-shaped waves

We study out-of-plane three-dimensional wave localization in two-dimensional photonic crystals (PCs) and predict the existence of two types of stationary X-shaped waves at frequencies corresponding to either a local top point of a band, where the effective in-plane diffraction turns out to be negative, or at band saddle points. In the former case the X wave is directed along the invariance direction of the PC, whereas in the latter case it lies in the PC plane and directed along one of the principal directions of the diffraction tensor. Numerical results of localized waves for a PC with a square lattice, obtained in the spectral domain by superposition of isofrequency Bloch modes, are presented and confirm the analytical predictions based on an effective mass approach.