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Physical Review A

Multimode instability criterion in optical bistable systems

Gang Xiang Hu, Hermann Haken

Published Aug 1, 1989• 2 citations
Publisher

Abstract

An analytical function H is derived in optical bistable systems, which serves as a criterion to characterize the multimode instability. Based on the criterion, the multimode instability boundary can be readily drawn. It is proved and verified by numerical results that in long-cavity systems, multimode instability always arises before the single-mode instability.

DOI
10.1103/physreva.40.1899

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An analytical function H is derived in optical bistable systems, which serves as a criterion to characterize the multimode instability. Based on the criterion, the multimode instability boundary can be readily drawn. It is proved and verified by numerical results that in long-cavity systems, multimode instability always arises before the single-mode instability.
A single-mode dynamic equation is obtained from the Maxwell equation with the slowly-varying amplitude approximation, and the boundary conditions of the cavity. Here the mean-field limit and the good-quality cavity case have been used. This equation is useful in describing the dynamic behavior of intrinsic optical bistable systems and/or lasers with injected signals for any kinds of materials, including two-level atoms. Furthermore, according to the linear stability analysis, a generalized instability criterion is derived.
A single-mode dynamical equation is obtained from the Maxwell equation with the slowly varying amplitude approximation and the boundary conditions of the cavity, here the mean field limir and the good quality cavity case have been used. This equation is useful in describing the dynamical behavior of intrinsic optical bistable systems and/or lasers with injected sig-nals for any kinds of materials including two-level atoms. Furthermore, according to the li-near stability analysis, a generalized instability criterion is derived.
A single-mode dynamical equation is obtained from the Maxwell equation with the slowly varying amplitude approximation and the boundary conditions of the cavity, here the mean field limir and the good quality cavity case have been used. This equation is useful in describing the dynamical behavior of intrinsic optical bistable systems and/or lasers with injected sig-nals for any kinds of materials including two-level atoms. Furthermore, according to the li--near stability analysis, a generalized instability criterion is derived.
Although the discovery of oscillatory instabilities (both single-mode and multimode) in free-running lasers was achieved in the 1960s, investigations on oscillatory instabilities in optical bistability did not take place until the 1970s and 1980s. The first section of this chapter does not deal with oscillatory instabilities, but instead concerns an interesting dynamical phenomenon in optical bistability that arises when the input field approaches a boundary of the bistable domain, and is called critical slowing down . This has been investigated both theoretically and experimentally. The remainder of this chapter is entirely devoted to oscillatory instabilities, following a historical order. With an inverse approach with respect to the laser case, we discuss multimode instabilities first. In Section 24.2.1 we start with the description of the instability which arises under exactly resonant conditions. While this is the counterpart in optical bistability of the resonant multimode laser instability discussed in Section 22.5, a striking difference from the laser case is that in the passive configuration there is no corresponding single-mode instability. This difference disappears, however, when we consider the generalization of the multimode instability of optical bistability to the detuned configuration. Finally we show that in the detuned case it has been possible to achieve a convincing experimental observation of the multimode instability in a long microwave cavity. After some relevant miscellaneous considerations in Section 24.2.2, in Section 24.2.3 we address the subject of the multimode Ikeda instability which leads to period doubling and chaos and aroused a lot of interest in optical chaos. We emphasize also the intrinsic connections with the standard multimode instability discussed in Section 24.2.1, showing, in particular, that the Ikeda instability requires conditions far from the low-transmission limit, i.e. that ∝L is not small. In Section 24.3 we discuss single-mode instabilities in optical bistability. In Section 24.3.1 we illustrate a configuration for the single-mode instability that includes also the Gaussian shape of the electric field in the beam section, and leads to periodic, close-to-sinusoidal self-pulsations.
We compare theoretical predictions and experimental data for the multimode instability of optical bistability, which arises from the coherent dynamics of a system of two-level molecules contained in an optical cavity and driven by an external stationary laser field. Considerable insight into this phenomenon is provided by two ``rules of thumb'' that govern the behavior of the self-pulsing frequency, and establish its relation with the Rabi frequency of the intracavity field. We describe in detail the experimental apparatus and the main features of the experimental findings. The theoretical results are based on the Maxwell-Bloch equations for a ring cavity in the plane-wave approximation. Despite the crudeness of this model, the numerical data display a satisfactory qualitative agreement with the experimental results. Additional period-doubling and chaotic phenomena, predicted by the theory and not observed in the experiment, are presumably washed out by the longitudinal and radial variation of the electric field in the cavity.