## Section: New Results

### Scattering in Complex Media

#### The Electromagnetic Lippmann-Schwinger Equations

Participant : Armin Lechleiter.

The Lippmann-Schwinger integral equation describes scattering
electromagnetic waves from penetrable objects.
If the modeling of the inhomogeneous medium involves space
dependent coefficients in the highest order terms of the underlying
partial differential equation, then the corresponding integral
operators typically fail to be compact.
In a joint work with Andreas Kirsch we investigate such cases and study
the arising integral equations in weighted spaces of
square integrable functions. The
two examples we treat are acoustic scattering from a medium with a space
dependent material density and electromagnetic medium scattering where
both the electric permittivity and the magnetic permeability vary. In these
cases, Riesz theory is not applicable for the solution of the arising integral
equations of Lippmann-Schwinger type. Therefore we show that positivity
assumptions on the relative material parameters allow to prove positivity of
the arising volume potentials in tailor-made weighted spaces of square
integrable functions. This result merely holds for imaginary wavenumber and
we exploit a compactness argument to conclude that the arising integral
equations are of Fredholm type, even if the integral operators themselves
are not compact. Finally, we explain how the solution of the integral equations
in L^{2} affects the notion of a solution of the scattering problem and illustrate
why the order of convergence of a Galerkin scheme set up in L^{2} does not
suffer from our L^{2} setting, compared to schemes in higher order Sobolev
spaces [17] .

#### Spectral Methods for the Lippmann-Schwinger Equation

Participants : Armin Lechleiter, Dinh Liem Nguyen.

Waves in inhomogeneous media with variable refractive index can be described by the Lippmann-Schwinger integral equation. We investigate spectral methods for these integral equations that go back to an idea of Vainikko. In our analysis, we are especially interested in media with non-smooth physical characteristics and analyze the convergence order of adapted numerical schemes for this problem. We are also interested in special ways of discretizing the corresponding integral equations for multiple distant scattering objects. In the future, we aim to apply such spectral techniques to electromagnetic scattering problems in optics.

#### Scattering from Rough Unbounded Penetrable Layers

Participants : Houssem Haddar, Armin Lechleiter.

Scattering of electromagnetic waves from the surface of ground are often modelled by a time-harmonic scattering problem involving unbounded scattering objects. We are interested in theoretical and numerical studies of this type of problems via variational formulations.

In a first work, A. Lechleiter and S. Ritterbusch considered the scalar problem in dimension two and three. The refractive index describing physical properties of the medium can be real or (partially) complex valued and is allowed to jump across interfaces. However, the index needs to satisfy a non-trapping condition, which requires, roughly speaking, monotonicity in the direction normal to the layer. In the half space above and below the rough layer a radiation condition is set up using the angular spectrum representation. Due to the unbounded setting, integral formulas similar to Rellich's identity are derived to obtain a priori bounds for a variational solution of the rough layer scattering problem. This a-priori bound is the basis for formulated existence result. Regularity theory and bounds on its frequency dependence are also provided [20] .

We are also investigating extensions of such approach to the more complicated 3-D electromagnetic problem. In that perspective we considered the scattering of time-harmonic electromagnetic waves from a metallic plate coated with a dielectric layer. This problem occurs for instance when monochromatic light propagates through photonic assemblies mounted on a plate. We first established a variational framework using the DtN map for Maxwell equation in half space. As opposed to the scalar case, the real part of this operator does not have a fix sign, which induces difficulties is establishing existence of solutions. The latter is done using an appropriate limiting absorption principle combined with a priori estimates derived from Rellich type identities. Our analysis only apply to small perturbation of stratified parallel layers. We are now interested in cases where the perfect conductor has a rough surface and also in widening the range of admissible material configurations.