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Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency. (English) Zbl 1394.78015

Summary: 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 [B. Zhang and H.-W. Zhang, J. Comput. Phys. 345, 58–73 (2017; Zbl 1378.35210)], 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 nonabsorbing impedance obstacle and the index of refraction \(n\) of the inhomogeneous medium is real-valued and satisfies that either \(n-1\geq c_1\) or \(n-1\leq-c_1\) 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.

MSC:

78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Citations:

Zbl 1378.35210
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References:

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