Fabry–Perot cavities made easy

The theoretical investigations on the dynamics and the instabilities in nonlinear optical systems have considered almost exclusively ring cavities, because the field propagation is unidirectional, whereas in Fabry–Perot cavities there are two field envelopes propagating in opposite directions, which constitutes a much more complex context. In the recent years, we have introduced a traveling wave formalism for Fabry–Perot cavities, obtained by considering a cavity with length doubled with respect to the original Fabry–Perot cavity and appropriately defining the variables in play in the additional interval. In this formalism, the dynamical equations are formulated in terms of the forward propagating envelope only. Moreover, in the doubled cavity the forward propagating envelope obeys periodic boundary conditions, which allows to consider traveling waves instead of standing waves, contrary to what is usually done. These circumstances provide a substantially simplified context, which allows to shorten the calculations by orders of magnitude with respect to the standard context. This turns out to be especially useful in the case of adiabatic elimination of the atomic variables or of the atomic polarization only. We first apply this procedure to the standard Maxwell–Bloch equations and extend it to the effective semiconductor Maxwell–Bloch equations. Remarkable is that it is possible to demonstrate that in Fabry–Perot cavities one can identify parametric conditions in which multimode instabilities arise near threshold. This is in sharp contrast with ring cavities, in which instabilities arise only nine times above threshold.