Optical scattering by a nonlinear medium, I: from Maxwell's equations to numerically tractable equations

A new method to find the propagation equation system governing the scattering of an electromagnetic wave by a nonlinear medium is proposed. The aim is to let the effects appear spontaneously, deleting as far as possible the phenomenological ideas. In this way, we obtain propagation equation systems that encodes several nonlinear effects. Once these systems obtained, the numerical values of the tensors that characterize the answer of the medium to an electromagnetic perturbation give weights to the different effects. This aim is partly reached in this study, especially when treating harmonic generations. For this, we start from the Maxwell's equation and give rigorously all the hypothesis needed to attain equation systems that can be solved, at least from a numerical point of view. Finally, a symmetry on the susceptibility tensors that ensures that a medium is lossless from the electromagnetic point of view is worked out.