Nonlinear gain models in a quantum cascade laser

Density matrix analysis of a three-state model of quantum cascade laser (QCL) reveals that in this device, the optical gain is composed of the linear part (proportional to population inversion Δn) and the remaining nonlinear part. The nonlinear component non-negligibly contributes even to the small-signal response of the medium. In many attempts to modeling QCLs, the common practice to account for nonlinear gain components is to complement the equation for the gain, g = g_cΔn, g_c is the gain cross-section, by a compression factor f. In this paper, improved (but still simple) models of the optical gain in QCL are proposed, which preserve the two-component gain structure. With these models, there is no need to solve the Hamiltonian with time-dependent potentials, so that extraordinary numerical loads can be avoided, but simultaneously the essential physics of the phenomena is kept. The improved gain models defined by Eqs. (12), (15) and (16) enable accounting for its nonlinear components while preserving the load-saving, scattering-like approach to light-matter interaction. It is also shown that as long as the populations and dc coherences are determined such that they account for the interaction with the optical field, the small-signal formulation of the gain gives its realistic estimate also for a large optical signal. This conjecture validates the use of non-equilibrium Green's function-based approaches, in which the interaction with the optical field is included through electron-photon selfenergies. The small-signal formulation of the gain can be used in this approach to monitor the saturation process, estimate the clamping flux and the light-current characteristic.