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Applied Physics Letters

Bistability and switching in nonlinear prism coupling

G. I. Stegeman, G. Assanto, Raymond Zanoni, C. T. Seaton, E. Garmire, A. A. Maradudin, R. Reinisch, Guy Vitrant

Published Mar 14, 1988• 71 citations
Publisher

Abstract

We demonstrate that when a nonlinear coupler is operated in the switching regime, small amounts of either nonlocality in the optically induced refractive index change or of longitudinal feedback can lead to optical bistability. Outside the switching regime, there is a threshold for the amount of nonlocality or feedback required for bistability. The diverse predictions of previous theoretical formulations are therefore reconciled.

DOI
10.1063/1.99257

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We demonstrate that when a nonlinear coupler is operated in the switching regime, small amounts of either nonlocality in the optically induced refractive index change or of longitudinal feedback can lead to optical bistability. Outside the switching regime, there is a threshold for the amount of nonlocality or feedback required for bistability. The diverse predictions of previous theoretical formulations are therefore reconciled.
The salient question which arises when light is coupled via a grating or a prism into a waveguide exhibiting an intensity-dependent refractive index is whether switching occurs with or without bistability under appropriate detuning conditions. There are two prevailing theories. In the first1, a plane wave input is assumed leading to a uniform optically induced index change in the coupling region. In the second2, a finite width incident beam is assumed and the growth of the guided field under the coupler leads to an index change proportional to the local guided wave intensity. The first approach leads to optical bistability under certain detuning conditions, whereas the second results in switching only. Although it is known that plane wave analyses are inadequate when dealing with linear-nonlinear interfaces, bistability has been observed experimentally with nonlinear prism couplers3. Fundamentally the finite beam theoretical approaches are unable to predict bistability because no longitudinal feedback is included. Recently we showed that when non-locality is introduced into the optically induced refractive index change, longitudinal feedback occurs and bistability is predicted for the growing mode case4. However, the model used for the non-locality was artificial and resulted only in qualitative predictions. In this paper the nonlocality is modelled by the same equation used to describe heat transfer and carrier diffusion, leading to a quantitative estimate of the minimum diffusion length required for bistability.
In the last chapter, we examined the effect of cubic nonlinearity on selfrefraction of an optical beam. We have also seen that the presence of a quadratic nonlinearity can also contribute to change in the profile of a Gaussian beam in a crystal. In this chapter we will see how the effect of nonlinearity, alongwith the proper feedback, can give rise to optical bistability and hysteresis. This is not unexpected as hysteresis and bistability are also observed in nonlinear electronic circuits with feedback, such as in the Schmitt trigger, as well as in hybrid optical devices, such as an acoustooptic device with feedback (Banerjee and Poon, 1991). In what follows, we will examine two different types of optically bistable devices based on optical nonlinearities. The first is a nonlinear ring resonator comprising a two-level gain medium as the nonlinear element. We will treat both absorptive and dispersive bistability. Absorptive bistability is the case when the incident optical frequency is close to or equal to the transition frequency of the atoms from one level to another. In this case, the absorption coefficient becomes a nonlinear function of the incident intensity. On the other hand, if the frequencies are far apart, the medium behaves like a Kerr-type material and the system exhibits what is called dispersive bistability. In this case the material can be modeled by an effective n2 or v (3), meaning that the refractive index becomes a nonlinear function of the optical intensity. The second configurationwewill examine is a linear-nonlinear interface. In this case, the interface provides the feedback due to refractive index mismatch. In fact, the reflection coefficient (as well as the transmission coefficient) is modified due to the nonlinearity, which, as we shall see later, is responsible for demonstrating optical switching and bistability.
In the nonlinear directional coupler, the optical field and the nonlinear material usually interact in a spatially and temporally local fashion and the device exhibits a nonlinear transmission characteristics, but it is not bistable. In this paper, it is shown theoretically that optical bistability in the input optical power is possible for the nonlinear directional coupler if the local phase change of the optical mode propagating through tile coupler is influenced by the excitation over the whole coupler. This condition can be satisfied in a coupler based on biased-pin-diodes comprising coupled asymmetric quantum wells in the i-region, Calculations show that in this configuration, the nonlinear directional coupler can exhibit sharp switching characteristics, low switching optical powers, optical bistability or bifurcation.
We report the optical bistability in two core oppositely directed coupler with negative index material channel. Using Langrangian variational method and Jacobi elliptic functions, we construct the solutions of the coupled nonlinear Schrödinger equations. The bistability arises due to the effective feedback mechanism as a result of opposite directionality of the phase velocity and energy flow in the negative index material channel. We report the various ways to control and manipulate the bistability threshold and hysteresis loop, which could be useful in the design and development of fast and low-threshold optical switches.
We deal with second-harmonic generation in chi((2)) nonlinear-optical resonators such as prism or grating couplers. The theoretical study is performed within the framework of a recently developed coupled-mode analysis leading to a set of equations governing the amplitudes of pump and second-harmonic frequency fields. We predict analytically that second-harmonic generation in prism or grating couplers may lead to optical bistability. We believe this to be the first demonstration of such an effect in chi((2)) optical resonators.