## 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].