Detection of standing pulses in periodic media by pulse interaction

Creation and manipulation of standing light pulses would pave the way for manufacturing all-optical computational devices. The most promising material type that is believed to support such standing pulses are periodically structured optical materials, so-called, photonic crystals. A way of detecting or even manipulating the position of a standing pulse is by the interaction with a second moving pulse. In this work we give a mathematical justification of the formulas which have been derived for interaction effects such as a shift of the pulse carrier and envelope. A significant part of the analysis is devoted to the proper definition of these quantities. The analysis is carried out for a nonlinear wave equation with spatially periodic coefficients which can be derived as a model for the description of the underlying physical system. Our method yields a separate description of internal and interaction dynamics and can even be employed for the analysis of the interaction of NLS-scaled wave packets in general dispersive systems with spatially homogeneous and spatially periodic coefficients.