Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves

We consider a two-dimensional inverse scattering problem of determining a sound-soft obstacle from the far field pattern and, under a non-trapping condition, we establish uniqueness within a class of polygonal domains by means of a single incoming plane wave. Moreover, we will show similar uniqueness in the sound-hard case by means of two incoming plane waves. The key is the analyticity of the solution of the scattering problem and the reflection of solutions.