Nonlinear optical interactions of wave packets in photonic crystals: Hamiltonian dynamics of effective fields

We develop an effective-field formalism that is suitable for describing nonlinear interactions of multiple wave packets in photonic crystals of arbitrary dimensionality. The theory is valid for "high-contrast" variations of the refractive index in the photonic crystal, provided dispersion and absorption effects can be neglected; it is based on a Hamiltonian formulation of the underlying Maxwell equations. We show that the dynamical equations for the effective fields are similar to those commonly used in nonlinear optics of homogeneous media, but with coefficients determined from the photonic band structure. We can introduce an effective energy density and an effective Poynting vector, expressed in terms of the effective fields, that satisfy a continuity equation. We illustrate our approach with a solution of the problem of degenerate optical parametric amplification in the undepleted pump approximation, and by considering the linear electro-optic effect as a quadratic nonlinear optical process.