Transient Electromagnetic Waves in Nonlinear Media

This thesis is concerned with the propagation of transient electromagnetic waves in nonlinear media. It consists of a General Introduction and five scientific papers. The General Introduction gives a broad overview of nonlinear electromagnetic phenomena. The emphasis is on the representation of the constitutive functional modeling the material's response to electromagnetic excitation, and the methods employed to analyze the combination of the constitutive functional and the Maxwell equations. Some applications of nonlinear electromagnetics are also discussed. Paper I treats the inverse scattering problem for an isotropic, homogeneous, nonlinear slab, subjected to a normally incident field. It is shown that when both reflected and transmitted fields are measured, we can reconstruct both the nonlinear permittivity and permeability. When one of these functions is known, reflection data is sufficient to obtain the other. Paper II gives a formulation of transient electromagnetic fields, that can be used to analyze wave propagation in homogeneous media. The source free Maxwell equations are treated as an eigenvalue problem, from which we deduce the propagating waves and their wave speeds. The analysis is applied to the case of obliquely incident waves on a semi-infinite, bianisotropic, nonlinear medium. Paper III analyzes the propagation of electromagnetic waves in a waveguide filled with an isotropic, nonlinear material. The equations governing each waveguide mode are derived, and it is shown that the different modes couple to each other. This coupling is quantified, and a growth estimates is given for the induced modes. Paper IV deals with discontinuous electromagnetic waves, shock waves. It is shown that in order for these waves to be stable, they must satisfy a number of conditions, similar to Lax's classical shock conditions. These conditions permit us to classify electromagnetic shock waves as slow, fast or intermediate shock waves. Finally, Paper V investigates the uniqueness and continuous dependence on data for solutions of the quasi-linear Maxwell equations, when we also require them to satisfy an entropy condition. This condition is related to the second law of thermodynamics, that the energy that is not described by our model must be dissipated