Theory of modulational instability in Bragg gratings with quadratic nonlinearity

Modulational instability in optical Bragg gratings with a quadratic nonline arity is studied. The electric field in such structures consists of forward and backward propagating components at the fundamental frequency and its s econd harmonic. Analytic continuous wave (CW) solutions are obtained, and t he intricate complexity of their stability, due to the large number of equa tions and number of free parameters, is revealed. The stability boundaries are rich in structures and often cannot be described by a simple relationsh ip. In most cases, the CW solutions are unstable. However, stable regions a re found in the nonlinear Schrodinger equation limit, and also when the gra ting strength for the second harmonic is stronger than that of the first ha rmonic. Stable CW solutions usually require a low intensity. The analysis i s confirmed by directly simulating the governing equations. The stable regi ons found have possible applications in second-harmonic generation and dark solitons, while the unstable regions maybe useful in the generation of ult rafast pulse trains at relatively low intensities. [S1063-651X(99)03005-6].