Stability of higher-order Bragg interactions in active periodic media

Abstract

The stability of waves in unbounded, longitudinally periodic media is studied for index and gain coupling. Time-independent periodic media are found to support both stable and absolutely unstable waves. The wave characteristics depend upon average gain or loss, coupling type, and Bragg order. The extended coupled waves (ECW) equations provide explicit values of threshold, frequency, and temporal growth rate for instabilities at all Bragg resonances through the dispersion relation. The first and second Bragg resonances are studied in detail since they are the archetypes for all odd and even resonances. Applications to multiharmonic periodicities and complex couplings are briefly discussed with particular not taken of possible reductions of the stability thresholds and removal of threshold degeneracies. Comparisons are made to the longitudinally bounded case of distributed feedback (DFB) lasers.