  We use semiclassical Hamiltonian optics to investigate the propagation of
light rays through two-dimensional photonic crystals when slow spatial
modulation of the lattice parameters induces mixed stable-chaotic ray dynamics.
This modulation changes both the shape and frequency range of the allowed
frequency bands, thereby bending the resulting semiclassical ray trajectories
and confining them within particular regions of the crystal. The curved
boundaries of these regions, combined with the bending of the orbits
themselves, creates a hierarchy of stable and unstable chaotic trajectories in
phase space. For certain lattice parameters and electromagnetic wave
frequencies, islands of stable orbits act as a dynamical barrier, which
separates the chaotic trajectories into two distinct regions of the crystal,
thereby preventing the rays propagating through the structure. We show that
changing the frequency of the light strongly affects the distribution of stable
and unstable orbits in both real and phase space. This switches the dynamical
barriers on and off and thus modulates the transmission of rays through the
crystal. We propose microwave analogues of the photonic crystals as a route to
the experimental study of the transport effects that we predict.
