Volume 33, number 1 OPTICS COMMUNICATIONS April 1980 DISPERSIVE OPTICAL BISTABILITY IN A FABRY-PEROT CAVITY E. ABRAHAM, S.S. HASSAN * and R.K. BULLOUGH Department o f Mathematics, UMIST, P.O. Box 88, Manchester M60 1QD, England Received 3 January 1980 This is a numerical analysis of the steady-state Maxwell-Bloch equations in a Fabry-Perot cavity with standing waves, atomic detuning and cavity mistuning; the atoms are homogeneously broadened. In the Mean-Field Approximation (MFA) limit, the resulting output-input characteristics agree exactly with the analytical predictions for a ring cavity. This leads to the derivation of a state equation for the Fabry-Perot cavity in the MFA limit identical to that for a ring cavity. Agreement with the MFA within 4% are obtained when aL < 3, the mirror reflectivity is above 90%, and the cavity mistuning below 4% of its free spectral range. Simulations beyond the scope of the MFA are also made and the case of purely dispersive optical bistability is also considered. 1. Introduction Optical bistability (OB) arises from the non-linear interaction between a laser field and a collection of atoms placed in a cavity [1-3]. It was first proposed independently by Seidel [1] and Sz6ke et al. [1] who predicted OB based on absorption only. It was not until the first successful experiment by Gibbs, McCall and Venkatesan [4] that the role of the non- linear refractive index in the phenomenon, was proper- ly appreciated. This had many experimental implica- tions - for example the possibility of achieving OB at lower powers, and in shaping the characteristic curves (curves of output versus input intensity) through the interplay of the atomic detuning and cavity mistuning. Gibbs et al. [4] also gave a simple though illuminating theory of purely dispersive OB and presented numerical results based on the Maxwell --Bloch equations including the effect of both absorp- tion and dispersion [5]. In the present paper a model consisting of two- level atoms homogeneously broadened in a Fabry- Perot (FP) cavity is considered. The incident field is a CW laser off-resonance with both atoms and cavity. The system is described by Maxwell-Bloch (MB) equations and both components of polarization are * On leave from: Ain Shams University, Faculty of Science, Applied Mathematics Department, Cario, Egypt. allowed for - consequently the model is not a purely dispersive one like that in ref. [9] or that of [11]. The purpose of this work is to compare numerical results for a FP cavity with analytical expressions derived for a ring cavity by Bonifacio and Lugiato [12] and Bonifacio et al. [13]. Independently, Hassan et al. [14] have given quantum theories (see also [12]) of the dispersive effects in OB for both homogeneously and gaussian inhomogeneously broadened atomic sys- tems placed in a ring cavity. Their results in the case of homogeneous broadening and no inhomogeneous broadening agree with those of ref. [12] so that the numerical results for the FP cavity presented here compare also with this work on the ring cavity. In both [12] and [14] a state equation is obtained for the ring cavity. We show below that in the triple limit of the mean-field approximation (MFA) [12], it is also possible to find the same state equation for the FP cavity. The paper is organised as follows: in section 2 the time-dependent and steady state MB equations together with appropriate boundary conditions are exhibited. The numerical results based upon them are presented in section 3. A state equation is derived analytically in section 4 using the MFA. A comparison with the results of numerical integration is then made there both within and outside the limits of validity of the analyti- cal expressions. The section 5 is a summary of the re- suits of the paper. 93