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Journal of Henan Normal University

The Cusp Catastrophe of the Absorptive Optical Bistability Equation

Qun Li

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

By the methods of Catastrophe theory,the bifurcation analysis of the absorptive optical bistability equation has been done.We show that this equation is in the form of cusp catastrophe through an appropriate transformation.Finally,the bifurcation set of cusp catastrophe is obtained.

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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.
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The critical phenomenon analogy in an absorptive optical bistability system is studied. The asymptotic form of the family of potential functions for the system near threshold is obtained by using catastrophe theory. The well-known scaling hypothesis in the general homogeneous function form in critical phenomena is shown to be a characteristic of the asymptotic family. Four threshold exponents beta , delta , gamma , alpha and their four accompanied threshold amplitudes B, D, Gamma , A on both bistability and monotonic regions are estimated. The threshold exponents obey the same scaling laws as those in critical phenomena, while the threshold amplitudes obey the definite relations between exponents and amplitudes.