Theory of Cherenkov radiation in periodic dielectric media: Emission spectrum

The Cherenkov radiation is substantially modified in the presence of a medium with a nontrivial dispersion relation. We consider Cherenkov emission spectra of a point or line charge, respectively, moving in general, three- (3D) and two-dimensional (2D) photonic crystals. Exact analytical expressions for the spectral distribution of the radiated power are obtained in terms of the Bloch mode expansion. The resulting expression reduces to a simple contour integral (3D case) or a one-dimensional sum (2D case) over a small fraction of the reciprocal space, which is defined by the generalized Cherenkov condition. We apply our method to a specific case of a line source moving with different velocities in a 2D square-lattice photonic crystal. Our method demonstrates a reasonable agreement with numerically rigorous finite-difference time-domain calculations while being less demanding on computational resources.