1 research outputs found
Coherent Structures and Carrier Shocks in the Nonlinear Periodic Maxwell Equations
We consider the one-dimensional propagation of electromagnetic waves in a
weakly nonlinear and low-contrast spatially inhomogeneous medium with no energy
dissipation. We focus on the case of a periodic medium, in which dispersion
enters only through the (Floquet-Bloch) spectral band dispersion associated
with the periodic structure; chromatic dispersion (time-nonlocality of the
polarization) is neglected. Numerical simulations show that for initial
conditions of wave-packet type (a plane wave of fixed carrier frequency
multiplied by a slow varying, spatially localized function) very long-lived
spatially localized coherent soliton-like structures emerge, whose character is
that of a slowly varying envelope of a train of shocks. We call this structure
an envelope carrier-shock train.
The structure of the solution violates the oft-assumed nearly monochromatic
wave packet structure, whose envelope is governed by the nonlinear coupled mode
equations (NLCME). The inconsistency and inaccuracy of NLCME lies in the
neglect of all (infinitely many) resonances except for the principle resonance
induced by the initial carrier frequency. We derive, via a nonlinear
geometrical optics expansion, a system of nonlocal integro-differential
equations governing the coupled evolution of backward and forward propagating
waves. These equations incorporate effects of all resonances. In a periodic
medium, these equations may be expressed as a system of infinitely many coupled
mode equations, which we call the extended nonlinear coupled mode system
(xNLCME). Truncating xNLCME to include only the principle resonances leads to
the classical NLCME.
Numerical simulations of xNLCME demonstrate that it captures both large scale
features, related to third harmonic generation, and fine scale carrier shocks
features of the nonlinear periodic Maxwell equations.Comment: 18 figures, added references, fixed typo