In recent years there have been a number of breakthroughs in the fabrication of
artificially structured materials with optical properties are of interest.
These include not only photonic crystals, but coupled cavity and coupled
microresonator structures as well. Because these structures are periodic or
nearly so in at least one dimension, their linear optical properties are
characterized by dispersion relations that can exhibit regions of high and low
group velocity, and high and low dispersion. Their nonlinear properties will
therefore be characterized by soliton and solitary-wave like effects, should be
of interest from the fundamental perspective of nonlinear dynamics, and may be
of use in switching operations for telecommunication applications.
Despite the fact that the index contrast of the constituent materials in these
structures can be quite large, and varies on the order of the wavelength of
light, realistic effective field theories for these materials can be developed
for the propagation of pulses of light whose length is many times the lattice
period. We describe one approach to constructing such theories, based on a
canonical description of the electromagnetic field, that allows for an easy
construction of conserved quantities and their relation to the symmetries of the
system.
Illustrations are given using structures of both fundamental and practical
interest.