  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 charge 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 an electron moving with different velocities in a 2D
square-lattice photonic crystal. Our method demonstrates an excellent agreement
with numerically rigorous finite-difference time-domain calculations while
being less demanding on computational resources.
