Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency

Abstract: This paper is concerned with uniqueness in inverse acoustic scattering with phaseless far-field data at a fixed frequency. The main difficulty of this problem is the so-called translation invariance property of the modulus of the far-field pattern generated by one plane wave as the incident field. Based on our previous work (J. Comput. Phys. 345 (2017), 58-73), the translation invariance property of the phaseless far-field pattern can be broken by using infinitely many sets of superpositions of two plane waves as the incident fields at a fixed frequency. In this paper, we prove that the obstacle and the index of refraction of an inhomogeneous medium can be uniquely determined by the phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency under the condition that the obstacle is a priori known to be a sound-soft or non-absorbing impedance obstacle and the index of refraction $n$ of the inhomogeneous medium is real-valued and satisfies that either $n-1\ge c_1$ or $n-1\le-c_1$ in the support of $n-1$ for some positive constant $c_1$. To the best of our knowledge, this is the first uniqueness result in inverse scattering with phaseless far-field data. Our proofs are based essentially on the limit of the normalized eigenvalues of the far-field operators which is also established in this paper by using a factorization of the far-field operators.