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.
Categories: math.AP math-ph math.MP