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Reports (Research Report) Year : 2015

Discontinuous Galerkin methods for the numerical solution of the nonlinear Maxwell equations in 1d

Abstract

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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Dates and versions

hal-01114155 , version 1 (07-02-2015)
hal-01114155 , version 2 (11-02-2015)

Identifiers

  • HAL Id : hal-01114155 , version 2

Cite

Loula Fezoui, Stéphane Lanteri. Discontinuous Galerkin methods for the numerical solution of the nonlinear Maxwell equations in 1d. [Research Report] RR-8678, Inria. 2015. ⟨hal-01114155v2⟩
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