Hamiltonian formulation of the nonlinear coupled mode equations

We derive a canonical Hamiltonian formulation of the nonlinear coupled mode equations (CME) that govern the dynamics of pulse propagation in a one-dimensional, periodic Kerr medium when the frequency content of the pulse is in the vicinity of a photonic band gap, and sufficiently narrow relative to a carrier frequency. The Hamiltonian is equal to the energy in the electromagnetic field. We show that even for large photonic band gaps (25% of the Bragg frequency), the CME give an excellent approximation to the dispersion relation of the linear, periodic medium. This suggests that two- and three-dimensional photonic band-gap materials, which necessarily have large index contrasts, might be effectively described by a set of generalized CME.