Periodic waves in fiber Bragg gratings
K. W. Chow, Ilya M. Merhasin, Boris A. Malomed, K. Nakkeeran, K. Senthilnathan, and P. K. A. Wai

We construct two families of exact periodic solutions to the standard model of the fiber Bragg grating (FBG) with the Kerr nonlinearity. The solutions are named ``\textrm{sn}" and ``\textrm{cn}" waves, according to the elliptic functions used in their analytical representation. The \textrm{sn} wave exists only inside the FBG's spectral bandgap, while waves of the \textrm{cn} type may only exist at negative frequencies ($\omega &lt0$), both inside and outside the bandgap. In the long-wave limit, the \textrm{sn} and \textrm{cn} families recover, respectively, the ordinary gap solitons, and (unstable) anti-dark and dark solitons. Stability of the periodic solutions is checked by direct numerical simulations, and, in the case of the \textrm{sn} family, also through the calculation of instability growth rates for small perturbations. Although, rigorously speaking, all periodic solutions are unstable, a subfamily of practically stable \textrm{sn} waves, with a sufficiently large spatial period and $\omega &gt0$, is identified. However, the \textrm{sn} waves with $\omega &lt0$, as well as all \textrm{cn} solutions, are strongly unstable.

© 2008 The American Physical Society.