Inverse Acoustic Scattering with Phaseless Far Field Data: Uniqueness, Phase Retrieval, and Direct Sampling Methods

The well-known translation invariance property of the phaseless far field patterns with incident plane waves make it is impossible to reconstruct the location of the underlying scatterers even for multiple incident directions and frequencies. To overcome this difficulty, we take the superpositions of plane waves and point sources as the incident fields. The point sources have fixed source location with at most three different scattering strengths. By adding a reference sound-soft obstacle into the scattering system, we show some uniqueness results with phaseless far field data. Of possibly even more interest are the phase retrieval technique and direct sampling methods, without using the reference obstacle. With the help of the point sources, we propose a simple and stable phase retrieval algorithm and design some novel direct sampling methods using the phaseless data directly. The novel phase retrieval algorithm can then be combined with the classical sampling methods for scatterer reconstruction. Finally, we present a variant of numerical examples in two dimensions to verify the effectiveness and robustness of the proposed algorithms.