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Physical Review A

Analysis of multifrequency dispersive optical bistability and switching in nonlinear ring cavities with large medium-response times

Paweł Pliszka, Partha Pratim Banerjee

Published Jul 1, 1992• 3 citations
Publisher

Abstract

Using a simple model of a ring cavity comprising a cubically nonlinear medium, we analyze dispersive optical bistability in the presence of more than one spectral component. We show the phenomenon of so-called competition for resonance. In addition to presenting cavity characteristics for the cases of two and three different frequencies, we also discuss the general method for finding steady-state solutions and checking their stability. A simple and efficient algorithm, based on a relaxation method, is devised to find steady-state solutions satisfying appropriate boundary conditions. The relaxation dynamics is physically related to a finite response time of the medium.

DOI
10.1103/physreva.46.507

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Using a simple model of a ring cavity comprising a cubically nonlinear medium, we analyze dispersive optical bistability in the presence of more than one spectral component. We show the phenomenon of so-called competition for resonance. In addition to presenting cavity characteristics for the cases of two and three different frequencies, we also discuss the general method for finding steady-state solutions and checking their stability. A simple and efficient algorithm, based on a relaxation method, is devised to find steady-state solutions satisfying appropriate boundary conditions. The relaxation dynamics is physically related to a finite response time of the medium.
We analyze a simple model of dispersive optical bistability in a ring cavity filled with a Kerr medium in the presence of two and three (carrier and one sideband pair) different frequencies. The steady-state characteristics are shown and the condition for selective switching is studied. We show the phenomenon of the so-called competition for resonance between different frequencies. The characteristic phenomenon of locking of the output intensities is presented. The model of two frequencies may also be used to describe bistability in the presence of two different polarizations. For three frequencies our discussion is based on the rigorous propagation equations, which admit a real Hamiltonian description and are completely integrable. We also discuss a general numerical approach, based on a relaxation method, for finding steady-state solutions and checking their stability. The relaxation dynamics is physically related to a finite response time of the medium. The effect of this finite response time on the stability of the steady states and on the dynamics is also discussed.
In Chapter 9 we discussed the case of a nonlinear cavity containing an active medium. In this framework, the outstanding phenomenon which arises is laser emission, and we calculated the stationary solutions for the ring laser. In this chapter, instead, we turn our attention to the passive nonlinear cavity and in this framework the outstanding phenomenon is optical bistability , i.e. the existence of two distinct stable stationary states that coexist for the same fixed values of the parameters in play. This phenomenon can arise also in the case of an active system contained in a cavity, as we have already noted in Section 9.1 and as we will discuss further in connection with the laser with injected signal, but it became popular from the studies of nonlinear passive optical cavities in the 1970s and 1980s. In contrast to the laser case, in the passive configuration we use an input coherent field, which is injected into the cavity. We discussed this case in Chapter 8 and derived a general equation for the transmission of the cavity (Eq. (8.23)), which we used for an empty cavity and for a linear cavity in Sections 8.4 and 8.5, respectively. In this chapter we will discuss the full nonlinear configuration. It is customary to identify two extreme cases of optical bistability called absorptive optical bistability and dispersive or refractive optical bistability . In both cases an essential ingredient to obtain the bistable behavior is the feedback action exerted by the mirrors of the cavity. In the purely absorptive configuration, in which there is exact resonance among the frequencies of the input field, of the atoms and of the cavity, the other essential ingredient is the saturable nonlinearity of the absorption. In the dispersive case, instead, the main feature is the nonlinearity of the refractive index. These two cases will be illustrated in Sections 11.1 and 11.2, respectively, both “exactly” and in the low-transmission limit.
When trying to operate optically bistable systems as logical switches, one is confronted with the question of stability against noise in amplitude and frequency of the holding laser beam. It is generally accepted that this problem is governed by scaling laws which are, to a large extent, independent of the detailed description of the nonlinear medium. Therefore, we consider the case of a purely dispersive medium whose nonlinear susceptibility satisfies a Debye relaxation equation. The cavity is taken to be of moderate finesse and large bandwidth (short round-trip time).
A dynamical analysis of nonlinear optical resonators is performed in the framework of a single-mode formalism in the plane-wave regime, taking the finite response time of the nonlinear medium into account. We demonstrate that analytically solvable generic equations can be derived for the time evolution of the output intensity, in the two limiting cases where the medium's response time is either very large or very small as compared to the cavity buildup time. The stability picture and the characteristic time scales, important for applications in the field of optical information processing, are investigated analytically. This study reveals that a surprising effect may occur where an increase in the material-response time leads to a decrease of the relaxation time of the nonlinear device. This effect is called the speeding-up phenomenon. We also present a numerical study of the full nonlinear dynamics.
We consider the usual ring cavity with a nonlinear medium of which the response time is much longer than the period of the light frequency. It is assumed that two polarization modes affect this medium incoherently, i.e. on an additive power basis. We will show by heuristic physical reasoning, based partly on graphical construction, that a sufficient condition for jump type instabilities is that the sum of the inverse relative differential gains for the two modes not exceed unity. We will then briefly sketch how the same result can be obtained from a formal mathematical stability analysis.
We study the propagation of electromagnetic radiation through a ring-cavity device containing a dispersive nonlinear medium. Under conditions appropriate for introducing the slowly varying envelope approximation and for adiabatically eliminating the atomic variables, we calculate analytical expressions between the steady-state-output intensity and the input-field amplitude for two models of the nonlinear susceptibility, the two-level atomic medium and the Kerr medium. The stability of these curves is determined, with special attention given to changes in the thresholds by varying the return-trip phase shift, the linear absorption, and the reflection coefficients of the mirrors. We investigate the positive-branch instabilities resulting in a pulsing output intensity and calculate the bifurcation rate for the period-doubling hierarchy and the fractal dimension of our two-dimensional maps for the Kerr medium. The convergence of the bifurcation rate to a constant is slow, and we find, as in the publications of Snapp et al. [ Opt. Commun.40, 68 ( 1981)] and Carmichael et al. [ Phys. Rev. A26, 3408 ( 1982)], that this constant is consistent with that found for the class of one-dimensional maps with a quadratic maximum. This universal constant is observed over a large range of fractal dimensions for the chaotic attractor. Discussions of pulse-output bistability, intermittency, and anomalous switching are given.