Analytical transfer function for the nonlinear response of a resonant medium in the spatio-temporal Fourier-transform domain

The nonlinear response of a resonant medium has many applications. To model and find the response of such a medium requires solving the Schrödinger equation (SE), which is a computationally extensive task. In this paper, we develop an analytical model to find the response of a resonant medium due to an applied field by employing the spatio-temporal Fourier-transform (STFT)–domain-based transfer function. A key feature of this approach is the use of the resonant excitation approximation (REA), which amounts to assuming that a group of atoms (or other quantum systems) within a volume element in the STFT domain are excited by only the corresponding volume element in the STFT domain of the field. We first derive the one-dimensional transfer function using an inhomogeneously broadened atomic medium under the REA. Then, we develop the three-dimensional transfer function and show that the analytical model agrees closely with the results obtained via an explicit simulation of the atomic response. As a practical example of the analytical model, we show that it can be used to model a spatio-temporal-correlator-based automatic event recognition system at a speed that is many orders of magnitude faster than solving the SE.