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IEEE Journal of Quantum Electronics

Optical bistability in a radially varying mode

P. D. Drummond

Published Mar 1, 1981• 94 citations
Publisher

Abstract

The theory of optical bistability in an interferometer with a nonlinear absorber is treated, including radial variation in the mode function. Dispersive effects and inhomogeneous broadening are included in the resulting analytic state equations. These realistic equations requite much greater cooperativity (or atomic absorption), relative to the plane-wave theory, before bistability exists.

DOI
10.1109/jqe.1981.1071100

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The theory of optical bistability in an interferometer with a nonlinear absorber is treated, including radial variation in the mode function. Dispersive effects and inhomogeneous broadening are included in the resulting analytic state equations. These realistic equations requite much greater cooperativity (or atomic absorption), relative to the plane-wave theory, before bistability exists.
A general theory of optical bistability for a homogeneously broadened two-level system in a Fabry-Perot interferometer is presented. Both absorptive and dispersive mechanisms are included, and in the mean-field approximation a cubic state equation is derived relating transmitted intensity to all physical parameters. From the perspective of catastrophe theory, this equation describes an elementary cusp catastrophe. We show that bistable switching may follow not only from sweeping the incident laser intensity, but also with continuous variation of atomic or cavity detuning, or atomic density, with the laser intensity fixed. For the case having incident intensity as the variable control, the domain of bistability is studied in some detail as a function of the remaining fixed parameters. Stability criteria are considered for the general case including both absorption and dispersion.
A straightforward but general extension of the quantum theory of optical bistability is made to include spatial variations of the field mode in the good cavity limit. The analysis proceeds by dividing the field mode into small sections which are each microscopically large in terms of the atomic number to allow truncation of the generalized Fokker-Planck equation but which are macroscopically small to justify the assumption of constant field amplitude. In a linearized approximation, analytic expressions are obtained for the ratio of incoherent to coherent intensity and for the intensity correlation function of the transmitted field for the two particular examples of a Gaussian-mode field in a ring cavity and a plane-wave field in a standing-wave cavity. In the weak field limit the results of the plane-wave ring cavity are recovered independent of the form of the spatial dependence of the cavity mode. However, more generally a nonuniform distribution tends to suppress certain quantum features such as photon antibunching.
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.
The characteristics of the mode intensity and conditions for occurrence of optical bistability in a ring laser with a saturable absorber are considered. The rate equation approximation method is used to find analytic expressions for these characteristics. It is shown that changes in such physical parameters as the homogeneous and inhomogeneous broadening, coefficients of relaxation of amplifying and absorbing cells, etc. cause changes in the range of existence of optical bistability or a hysteresis cycle. The influence of inhomogeneous broadening and of spontaneous emission near the lasing threshold on the appearance of optical bistability is demonstrated for the single-mode regime.