Periodic waves in fiber Bragg gratings

Abstract

We construct two families of exact periodic solutions to the standard model of fiber Bragg grating FBGwith Kerr nonlinearity. The solutions are named “sn” and “cn” waves, according to the elliptic functions usedin their analytical representation. The sn wave exists only inside the FBG’s spectral bandgap, while waves ofthe cn type may only exist at negative frequencies 0, both inside and outside the bandgap. In thelong-wave limit, the sn and cn families recover, respectively, the ordinary gap solitons, and unstable antidarkand dark solitons. Stability of the periodic solutions is checked by direct numerical simulations and, in the caseof the 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 sn waves, with asufficiently large spatial period and 0, is identified. However, the sn waves with 0, as well as all cnsolutions, are strongly unstable.