Optical bistability involving photonic crystal microcavities and Fano line shapes

The reflectivity of a single-channel waveguide mode upon resonantly coupling to a Kerr-active nonlinear resonant cavity is calculated analytically, including the effects of two-photon absorption. The resonant reflectivity takes the form of a Fano resonance because the solution includes linear reflections from perturbations downstream of the localized cavity. Instead of using a Hamiltonian formulation of the scattering problem, an intuitive set of basis states is used to expand the Green’s function of the electric field wave equation. All resulting overlap functions describing the linear coupling between guided and localized states, and the nonlinear renormalization of the material’s refractive index, are in terms of well-defined physical quantities. Although derived in the context of photonic crystal-based waveguides and cavities, the treatment is valid for any low-loss waveguide-resonator geometry that satisfies specific weak coupling criteria. For a cavity consisting of ${\mathrm{Al}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{As},$ hosting a localized mode at $1.55\mu \mathrm{m}$ with a Q of 4000 and a mode volume of $0.055\mu {\mathrm{m}}^3,$ we predict the onset of bistable reflection at incident powers of $\sim 40\mathrm{mW}.$ The downstream reflections lead to hysteresis loops in the reflectivity that are topologically distinct from conventional Lorentzian-derived loops characteristic of isolated Fabry-Perot cavities. We provide a stability argument that reveals the unstable branches of these unique hysteresis loops, and we illustrate some of the rich bistable behaviors that can be engineered with such downstream sources.