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Appl. Math."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>This paper is concerned with uniqueness in inverse acoustic scattering with phaseless far-field data at a fixed frequency. In our previous work [ SIAM J. Appl. Math., 78 (2018), pp. 1737--1753], by utilizing spectral properties of the far-field operator we proved for the first time 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 a priori assumption that the obstacle is 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 $n-1\\ge c_1$ or $n-1\\le-c_1$ in the support of $n-1$ for some positive constant $c_1$. In this paper, we remove the a priori assumption on the obstacle and the index of refraction of the inhomogeneous medium by adding a reference ball to the scattering system together with a simpler method of using Rellich's lemma and Green's representation formula for the scattering solutions. Further, our new method is also used to prove uniqueness in determining a locally rough surface from the phaseless far-field patterns corresponding to infinitely many sets of superpositions of two plane waves with different directions as the incident fields at a fixed frequency.<\/jats:p>","DOI":"10.1137\/18m1196820","type":"journal-article","created":{"date-parts":[[2018,11,13]],"date-time":"2018-11-13T14:26:52Z","timestamp":1542119212000},"page":"3024-3039","source":"Crossref","is-referenced-by-count":25,"title":["Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency. 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