Uniqueness of the scatterer for electromagnetic field with one incident plane wave

In this paper, we solve a longstanding open problem for determining the shape of an obstacle from the knowledge of the electric (or magnetic) far field pattern for the scattering of time-harmonic electromagnetic field. We show that the electric (or magnetic) far field patten ${\mathbf{E}}^\infty(\boldsymbol{\beta}, {\boldsymbol{\alpha}}_0, k_0)$ (or ${\mathbf{H}}^\infty (\boldsymbol{\beta}, {\boldsymbol{\alpha}}_0, k_0)$), known for all $\boldsymbol{\beta}\in {\mathbb S}^2$, where ${\mathbb {S}}^2$ is the unit sphere in ${\mathbb{R}}^3$, ${\boldsymbol{\alpha}}_0\in {\mathbb{S}}^2$ is fixed, $k_0>0$ is fixed, determines the obstacle $D$ and the boundary condition on $\partial D$ uniquely. The boundary condition on $\partial D$ is either the perfect conductor or the impedance one.