Beyond the mean-field theory of dispersive optical bistability

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 $T\to 0$, ${\delta }_F\to 0$, and $\alpha L\to 0$ for the mirror transmission coefficient, the detuning of the field from the cavity resonance, and the linear absorption, respectively, with $\frac{\alpha L}{2T}$ and $\frac{{\delta }_{F}L}{2cT}$ 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.