Dispersive optical bistability in a nonideal Fabry-Perot cavity I. Stability analysis of the Maxwell-Bloch equations

A stability analysis is performed for optical bistability in a Fabry-Perot cavity with mirrors of arbitrary transmission coefficient. The mixed absorptive and dispersive regime is covered. In order to describe the system we use the Maxwell-Bloch 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. The authors represent the solutions of the linearized Bloch hierarchy in terms of Chebyshev polynomials depending on the stationary electric field envelopes. In this way, they 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. The final stability problem becomes much simpler in the uniform-field limit and in the adiabatic limit. If one chooses the stationary backward electric field equal to zero one recovers results that were derived earlier for the case of a ring cavity.