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Optica Acta International Journal of Optics

Spatial Effects in Optical Bistability

J.A. Hermann

Published Feb 1, 1980• 34 citations
Publisher

Abstract

Within the framework of the theories of absorptive bistability recently proposed by various authors, the different sets of coupled differential equations describing the steady-state fields inside the Fabry-Perot cavity are solved analytically. Expressions are obtained for the bistability equation, which links the incident and transmitted fields, for different ranges of the mirror transmission coefficient. Comparison is made with the mean-field theory of Bonifacio and Lugiato. Conditions for stationary points in the bistability curve are derived.

DOI
10.1080/713820201

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Within the framework of the theories of absorptive bistability recently proposed by various authors, the different sets of coupled differential equations describing the steady-state fields inside the Fabry-Perot cavity are solved analytically. Expressions are obtained for the bistability equation, which links the incident and transmitted fields, for different ranges of the mirror transmission coefficient. Comparison is made with the mean-field theory of Bonifacio and Lugiato. Conditions for stationary points in the bistability curve are derived.
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The problem of dispersive optical bistability has so far been treated only in the mean-field approximation. A rigorous justification of the mean-field theory can only be obtained from exact solutions of the steadystate Maxwell-Bloch equations which retain the spatial dependence of the field. In this paper exact analytic solutions are presented for these equations. The authors demonstrate that the mean-field equation connecting the input and the output fields follows naturally from these solutions in the limits $T\ensuremath{\rightarrow}0$, ${\ensuremath{\delta}}_{F}\ensuremath{\rightarrow}0$, and $\ensuremath{\alpha}L\ensuremath{\rightarrow}0$ for the mirror transmission coefficient, the detuning of the field from the cavity resonance, and the linear absorption, respectively, with $\frac{\ensuremath{\alpha}L}{2T}$ and $\frac{{\ensuremath{\delta}}_{F}L}{2cT}$ remaining finite. The results are illustrated with the help of graphs showing the output versus input intensity for different values of the relevant parameters. The effect of these parameters on the phase shift of the output field is also displayed.
The transmission characteristics of a Fabry-Perot cavity containing a resonant absorber are described analytically and graphically for internal fields of arbitrary intensity. The multimode instability considered by Bonifacio and Lugiato [Opt. Commun. 19 , 72 (1976)] for the unidirectional ring laser is treated for the standing-wave cavity. In particular, an instability may exist when the medium is located at one of the ends of the cavity, but not if the medium fills the cavity. The results are explained in terms of induced side-mode gain and spatial hole burning.
We consider a gas of homogeneously broadened two-level atoms in a Fabry-Perot cavity injected with a coherent resonant electric field. Adopting a macroscopic birth-death description, we construct a stochastic master equation for the output intensity distribution. From this starting point we find the steady-state distribution and show it to be bimodal in the region of the optical bistability. We compare our results for the mean and variance with the microscopic results of Bonifacio, Gronchi, and Lugiato as a test of the stochastic-master-equation approach.
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.
In this work a theoretical model treating the optical bistability in a two state dissipative atomic medium inside a Fabry-Perot cavity is presented.The model is based on a semiclassical point of view to explain the field-matter interaction.The atoms of the medium are described by the "quantum polarization equation of motion" while the field is described by the "classical Maxwell's field equations".Applying the boundary conditions to the fields inside the Fabry-Perot cavity, the occurrence of optical bistability is deduced between the incident field intensity and both the intracavity and transmitted field intensities.Also the effect of building standing and nonstanding waves inside the cavity on bistability is discussed.