Electromagnetic scattering

In this example we will showcase the ability of the medusa library to solve coupled domain problems. We will solve the problem of electromagnetic scattering on an anisotropic cylinder.

Anisotropic cylinder

Let us first quickly derive the problem we are about to solve. Beginning with the electromagnetic wave equation in anisotropic media

\( \label{eq:frekaniwave} \nabla \times \left( \underline{\varepsilon}^{-1} \nabla \times \boldsymbol{H} \right) = \omega^2 \mu_0 \varepsilon_0 \underline{\mu} \boldsymbol{H}, \)

where \(\underline{\varepsilon}\) is the relative dielectric tensor, and \(\underline{\mu}\) is the magnetic permeability tensor of the anisotropic material

\( \label{eq:relepsmi} \underline{\varepsilon} = \begin{pmatrix} \varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\ \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\ \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \end{pmatrix} , \qquad \underline{\mu} = \begin{pmatrix} \mu_{xx} & \mu_{xy} & \mu_{xz} \\ \mu_{yx} & \mu_{yy} & \mu_{yz} \\ \mu_{zx} & \mu_{zy} & \mu_{zz} \end{pmatrix}. \)

Let $D \subset \R^2$ be the cross section of an infinitely long anisotropic dielectric cylindrical scatterer with its axis alligned with the $z$-axis, surrounded by a free space, with an outward normal $n$ on boundary $\partial D$. The scatterer is excited by an $e^{i \omega t}$ time-harmonic plane wave with $\b{TM}^z$ polarization, with $\omega$ standing for its angular frequency. Let $v \in C^2(\C)$ denote the complex valued field inside the scatterer and $u \in C^2(\C)$ the field outside of the scatterer. Field $u$ can be further decomposed into the incident $u^i$ and the scattered field $u^s$.