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Similar to the obstacle or medium scattering problems, an important propertyof the phaseless far field patterns for source scattering problems is thetranslation invariance. Thus it is impossible to reconstruct the location ofthe underlying sources. Furthermore, the phaseless far field pattern is alsoinvariant if the source is multiplied by any complex number with modulus one.Therefore, the source can not be uniquely determined, even the multifrequencyphaseless far field patterns are considered. By adding a reference point sourceinto the model, we propose a simple and stable phase retrieval method andestablish several uniqueness results with phaseless far field data. We proceedto introduce a novel direct sampling method for shape and locationreconstruction of the source by using broadband sparse phaseless data directly.We also propose a combination method with the novel phase retrieval algorithmand the classical direct sampling methods with phased data. Numerical examplesin two dimensions are also presented to demonstrate their feasibility andeffectiveness.

Phaseless inverse source scattering problem: phase retrieval, uniqueness and direct sampling methods