Recovering scattering obstacles by multi-frequency phaseless far-field data

It is well known that the modulus of the far-field pattern (or phaseless far-field pattern) is invariant under translations of the scattering obstacle if only one plane wave is used as the incident field, so the shape but not the location of the obstacle can be recovered from the phaseless far-field data. This paper aims to devise an approach to break the translation invariance property of the phaseless far-field pattern. To this end, we make use of the superposition of two plane waves rather than one plane wave as the incident field. In this paper, it is mathematically proved that the translation invariance property of the phaseless far-field pattern can indeed be broken if superpositions of two plane waves are used as the incident fields for all wave numbers in a finite interval. Furthermore, a recursive Newton-type iteration algorithm in frequencies is also developed to numerically recover both the location and the shape of the obstacle simultaneously from multi-frequency phaseless far-field data. Numerical examples are also carried out to illustrate the validity of the approach and the effectiveness of the inversion algorithm.