The uniqueness of the inverse transmission problem with phaseless far field data at a fixed frequency

In this paper, we establish the unique determination results for inverse acoustic scattering of a penetrable anisotropic obstacle by using phaseless far field data at a fixed frequency. It is well-known that the modulus of the far field pattern is invariant under translations of the scattering obstacle if only one plane wave is used as the incident field, so it is impossible to reconstruct the location of the underlying scatterers. Based on some new research results on the impenetrable obstacle and inhomogeneous isotropic medium, we develop four methods to break the translation invariance property. In the first part, we obtain two uniqueness results by the superposition of two plane waves as the incident field. Then we establish another two uniqueness results by taking the superposition of a plane wave and point sources with different scattering strengths as the incident field.