Into the wild of nonlinear electromagnetism-a course on nonlinear electromagnetism, not quite from scratch, part III:...

This tutorial is aimed at introducing in a natural fashion the propagation equations system governing the scattering of an electromagnetic wave by a nonlinear medium. The purpose is first to obtain the equations showing the most common nonlinear effects such as the Kerr effect or second and third harmonic generation by avoiding conventional recipes and trying to arrive at these equations with a minimum of assumptions. For this, we start from the general Maxwell's equations involving the fields E, B, D, and H, and we rigorously provide all the hypotheses needed to attain the nonlinear systems of partial differential equations involving the different complex amplitudes of the different fields associated with the different frequencies at stake. In part II, the difficult question of energy transfer between fields emitted at the various frequencies involved is discussed in detail. We then examine the tensorial nature of susceptibilities and, using Neumann's principle, show how the number of their independent components can be significantly reduced. In part III, numerical examples of scattering by nonlinear materials are given and discussed.