Abstract

It is well-known that periodic Bragg structures induce gaps in the linear dispersion relation where the propagation is forbidden. Thanks to the nonlinearity, the e.m. energy can be localized at these frequencies in the form of gap solitons [1-5]. Their most intriguiging feature is the possibility to travel with velocities much slower than the light velocity, or remarkably even at zero velocity (stationary trapping) in the limit case. Propagation of slow solitonic envelopes was recently demonstrated experimentally operating near Bragg wavelength of a fiber grating [5]. However no experimental data in the region of zero (or extremely small) velocities has been reported to date. To address the observability of stationary localization, a fundamental prerequesite to be fullfilled is the stability of the gap solitons. In spite of its importance the stability problem for gap solitons was left practically unaddressed, except for few numerical simulations. Here our purpose is to derive an analytical stability criterion, starting from the usual coupled-mode formulation of the propagation [1]. This involves derivatives of the invariants similarly to a well-known criterion for equations with second-order dispersion (i.e., nonlinear Schrödinger type [6]), recently extended to quadratic solitons [7,8]. Specifically, we consider the Lorentz-invariant Hamiltonian equations which rule the propagation of forward (+) and backward (−) envelopes u_± at Bragg or gap-center carrier frequency with cubic nonlinearity [1,2] where H is the conserved Hamiltonian of Eqs. (1).

© 1998 Optical Society of America