Discontinuous Galerkin methods for the numerical solution of the nonlinear Maxwell equations in 1d
Résumé
The system of Maxwell equations describes the evolution of
the interaction of an electromagnetic field with a propagation
medium. The different properties of the medium, such as isotropy,
homogeneity, linearity, among others, are introduced through {\it
constitutive laws} linking fields and inductions. In the present
study, we focus on nonlinear effects and address nonlinear Kerr
materials specifically. In this model, any dielectric may become
nonlinear provided the electric field in the material is strong
enough. This is even the case in vacuum but then the minimal amount
of energy necessary to observe nonlinear effects is similar to the
total energy produced by the sun in one second. We nonetheless use
the vacuum as one of the two dielectrics in our numerical
simulations. The other one is the air wherein the minimal electric
field magnitude for observing nonlinear effects is $10^6$~V/m, which
can be achieved by lasers in production today. We will see that in
some situations, such as when an oscillating electric dipole
radiates in open space, frequency may also impact and even increase
nonlinear effects. The work presented here is restricted to one
dimensional space in order to compare numerical results to analytic
solutions when possible, but all physical details and numerical
techniques extend to higher dimensions without much difficulty.
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