Difference Differential Equations for a Resonator with a Very Thin Nonlinear Medium

We derive from the classic Maxwell-Bloch equations a set of difference-differential equations valid, in general, when the length of the nonlinear medium in the optical cavity is much smaller than a wavelength. Such equations provide an elegant and simple framework in which the case of Fabry-Perot and ring cavity can be discussed in a unified way. We outline a complete scenario for the multimode laser instability in the Fabry-Perot case, illustrating the results for parameter values appropriate to quantum cascade lasers. Our approach can have a relevant impact also on the study of dynamical instabilities in external cavity semiconductor lasers, including multiple quantum well or quantum-dot structures.