Dispersive optical bistability in a nonideal Fabry-Pérot cavity

A stability analysis is performed for optical bistability in a Fabry-Pérot cavity with mirrors of arbitrary transmission coefficient. The mixed absorptive and dispersive régime is covered. In order to describe the system we use the Maxwell0Bloch equations formulated in terms of slowly varying envelopes. Standing-wave effects are completely taken into account by refraining from a truncation of the harmonic expansions for the polarization and the inversion density. We represent the solutions of the linearized Bloch hierarchy in terms of Chebyshev polynomials depending on the stationary electric field envelopes. In this way, we reduce the stability problem to a four-dimensional set of linear differential equations. Together with a couple of boundary conditions these equations govern the spatial behaviour of the deviations of the forward and the backward electric field envelopes. Our final stability problem becomes much simpler in the uniform-field limit and in the adiabatic limit. If we choose the stationary backward electric field equal to zero we recover results that were derived earlier for the case of a ring cavity.