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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Optical bistability involving photonic crystal microcavities and Fano line shapes

A.R. Cowan, Jeff F. Young

Published Oct 20, 2003• 124 citations
Publisher

Abstract

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 Al0.18Ga0.82As, hosting a localized mode at 1.55 microm with a Q of 4000 and a mode volume of 0.055 microm(3), we predict the onset of bistable reflection at incident powers of approximately 40 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.

DOI
10.1103/physreve.68.046606

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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 Al0.18Ga0.82As, hosting a localized mode at 1.55 microm with a Q of 4000 and a mode volume of 0.055 microm(3), we predict the onset of bistable reflection at incident powers of approximately 40 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.
We present an analytical model and numerical experiments to describe optimal bistable switching involving photonic crystals nonlinear cavity. The involving photonic crystals nonlinear cavity is particularly suitable for large-scale all-optical integration. It is interesting to show that the refractive index of the system can be altered by the incident intensity when the incident frequency is tuned to the evanescent mode and the cavity contains a nonlinear Kerr medium. The response of this system to high-intensity incident waves is studied in detail. A cavity host a localized mode at 1.55 um with a Q of 4000 and a mode volume of 0.055 um3, we predict the onset of bistable reflection at incident powers of ~40 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, this fact may be used to obtain more flexibility in designing nonlinear devices.
We analyze the resonant linear and nonlinear transmission through a photonic crystal waveguide side-coupled to a Kerr-nonlinear photonic crystal resonator. First, we extend the standard coupled-mode theory analysis to photonic crystal structures and obtain explicit analytical expressions for the bistability thresholds and transmission coefficients which provide the basis for a detailed understanding of the possibilities associated with these structures. Next, we discuss limitations of standard coupled-mode theory and present an alternative analytical approach based on the effective discrete equations derived using a Green's function method. We find that the discrete nature of the photonic crystal waveguides allows a geometry-driven enhancement of nonlinear effects by shifting the resonator location relative to the waveguide, thus providing an additional control of resonant waveguide transmission and Fano resonances. We further demonstrate that this enhancement may result in the lowering of the bistability threshold and switching power of nonlinear devices by several orders of magnitude. Finally, we show that employing such enhancements is of paramount importance for the design of all-optical devices based on slow-light photonic crystal waveguides.
The authors observe experimentally optical bistability enhanced through Fano interferences in high-Q localized silicon photonic crystal resonances (Q∼30000 and modal volume ∼0.98 cubic wavelengths). This phenomenon is analyzed through nonlinear coupled-mode formalism, including the interplay of χ(3) effects such as two-photon absorption and related free-carrier dynamics, and optical Kerr as well as thermal effects and linear losses. Experimental and theoretical results demonstrate Fano resonance based bistable states with switching thresholds of 185μW and 4.5fJ internally stored cavity energy (∼540fJ consumed energy) in silicon for scalable optical buffering and logic.
We observe experimentally optical bistability enhanced through Fano interferences in high-Q localized silicon photonic crystal resonances (Q ~ 30,000 and modal volume ~ 0.98 cubic wavelengths). This phenomenon is analyzed through nonlinear coupled-mode formalism, including the interplay of chi(3) effects such as two-photon absorption and related free-carrier dynamics, and optical Kerr as well as thermal effects and linear losses. Our experimental and theoretical results demonstrate for the first time Fano-resonance based bistable states with switching thresholds of 185 micro-Watt and 4.5 fJ internally stored cavity energy (~ 540 fJ consumed energy) in silicon for scalable optical buffering and logic.
This paper presents theoretical investigations on a new ring cavity-coupled scheme of a Michelson interferometer with a linear loop mirror, which has the attraction of low loss and high stability due to the closed geometry. The optical transmission response is modeled using a scattering matrix approach and analytical expressions are derived. This single ring cavity-coupled system exhibits optical Fano resonance features and significantly enhances the switching sensitivity to the cavity detuning required for a large ON/OFF contrast ratio. The evolution of Fano-like interference and the effect of ring cavity confinement on the Fano-like resonance are studied numerically. Furthermore, optical bistability and multi-bistability are demonstrated by introducing Kerr nonlinearity inside the ring cavity. The shape and polarity of hysteresis loops are easily controllable by adjusting the phase compensation of the direct channel, while the switching thresholds are affected remarkably by the linear phase shift of the ring cavity. The simple structure of the single cavity-coupled interferometer will be important for practical applications in optical sensors, all-optical switches and logic function elements.
We derive an equation that describes the nonlinear operation of a Fabry-Perot resonator with a large group index waveguide. Specifically, a silicon photonic crystal microcavity with two-photon-excited free carrier nonlinearity and Kerr nonlinearity is assumed. The equation clearly explains the bistability of the device and the reduction of the required pump energy for a specific nonlinear phase shift at an appropriate phase detuning from the resonance. We present a simple procedure to predict the required optical pump energy for the modulation and the resulting modulation depth by use of the equation and the device parameters.