## Invisibility in waveguides

**Perfect invisibility.** We represent the total field (acoustic pressure) in the reference geometry (bottom) and in the pertubed waveguide (top).
At \(x=\pm\infty\) in the perturbed waveguide, the field is the same as in the reference geometry.
The reflection coefficient \(R\) and the transmission coefficient \(T\) satisfy \(R=0\), \(T=1\). More details [PDF].

Another geometry where \(R=0\) and \(T=1\). More details

[PDF].

In the example below, we played with the height of the thin chimneys. More details

[PDF].

**Complete reflectivity.** We represent the total field (acoustic pressure). The goemetries have been designed so that \(T=0\). All the energy is backscattered.
More details [PDF].

## Plasmonic and metamaterials

We consider the problem

\begin{array}{|rl}
-\mbox{div}(\sigma\nabla u)=f &\mbox{ in }\Omega\\
u=0&\mbox{ on }\partial\Omega
\end{array}

with a sign changing \(\sigma\).

**Singular behaviour.** For certain values of \(\sigma\), very singular behaviours can appear which are not met with positive materials. Here we represent \(t\mapsto \Re e\,(u(x,y)e^{-i\omega t})\) for a given \(\omega>0\).
Everything happens like if a wave was absorbed by the corner. The wave propagates to the corner but never reaches it. We talk about "black-hole phenomenon". More details [PDF], [PDF].

**Rounded corner.** The solution can be very sensitive to the geometry. Here we represent \(\delta\mapsto u^{\delta}\) as \(\delta\to0\), where \(\delta\) is the radius of the inner circle. More details [PDF], [PDF].

**Mesh refinement.** We solve numerically the above problem with a usual P1 finite element method for different meshes (we refine the mesh).

For \(\sigma_2/\sigma_1\in[-3;-1/3]\), in general the solution does not converge (below left \(\sigma_2=-1.0001\)).

For \(\sigma_2/\sigma_1\in(-\infty;0)\setminus[-3;-1/3]\), the solution does converge (below right \(\sigma_2=-3.001\)). More details [PDF].