| We begin by deriving a set of equations that describe pulse propagation in one dimensional, periodic media, in the presence both of birefringence and a Kerr nonlinearity. We use these equations to interpret the results of a series of experiments performed in fiber gratings, which, with the appropriate approximations, can be considered to have one effective dimension.; We next turn to the consideration of two channel waveguides coupled by a sequence of periodically spaced microresonators, in the presence of a Kerr nonlinearity. We show that two distinct types of gaps open in the dispersion relation of the device. The frequency of one type of gap is related to the spacing of the resonators. The frequency of the other type of gap is related to the radius of the resonators. We derive a set of coupled nonlinear Schrödinger equations (NLSE) to describe the propagation of light in the system. We show that the properties of the dispersion relation in the vicinity of the two types of gaps are markedly different, and that near the gap associated with the radius of the resonators, the group velocity dispersion experienced by a pulse is very small. We then demonstrate that a gap soliton should be observable at much lower intensities in this latter gap than in a Bragg gap of the same frequency width.; The study the operation of a grating-waveguide structure (GWS), where a grating is used to coupled an incident plane wave into a guided mode of a layered medium. We derive equations that determine the field everywhere in the presence of a grating of arbitrary thickness, and a Kerr nonlinearity. We demonstrate that the GWS can be used as a low-loss, narrow-band reflector, or as an all optical switch.; Finally, we construct a Hamiltonian formulation for pulse propagation equations in a one dimensional, Kerr nonlinear, periodic medium. In doing so, we clear up some confusion in the literature surrounding the nature of the conserved quantities associated with Kerr nonlinear pulse propagation equations. |
