Dynamics and instability of nonlinear Fano resonances in photonic crystals

We employ an effective discrete model for the study of wave propagation in a photonic crystal waveguide side-coupled to a cavity with Kerr-type nonlinearity. Taking into account the linear coupling between guided and localized states and applying the time-dependent version of a Green's function formalism, we study and characterize analytically the scattering of continuous waves. The resonant reflectivity, which is tunable via the nonlinearity, takes the form of a nonlinear Fano resonance because the output field is composed of a linearly transmitted wave and a resonantly reflected contribution from the localized cavity. By studying the stability of the nonlinear Fano resonance, we reveal that the continuous-wave scattering may exhibit modulational instability near the resonance when the light intensity in the cavity starts growing in time. However, we demonstrate that this instability may be suppressed for Gaussian pulses, such that the bistable transmission curve can still be recovered in accordance with the analysis of the steady-state transmission. We demonstrate that our analytical results based on an effective discrete model are in excellent agreement with numerical results obtained by direct finite-difference time-domain simulations.