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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. 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), pp. 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 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 $c_1$. 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.<\/jats:p>","DOI":"10.1137\/17m1149699","type":"journal-article","created":{"date-parts":[[2018,6,19]],"date-time":"2018-06-19T12:53:00Z","timestamp":1529412780000},"page":"1737-1753","source":"Crossref","is-referenced-by-count":38,"title":["Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency"],"prefix":"10.1137","volume":"78","author":[{"given":"Xiaoxu","family":"Xu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bo","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4035-4754","authenticated-orcid":true,"given":"Haiwen","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,6,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/15M1043959"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAA.30.000293"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/32\/8\/085002"},{"key":"atypb4","unstructured":"E. Blasten,\n                      The Inverse Problem of the Schro\u0308dinger Equation in the Plane\n                      , Licentiate Thesis, University of Helsinki, 2011; preprint,https:\/\/arxiv.org\/abs\/1103.6200, 2011."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1515\/jiip.2008.002"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21432"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2015.2399924"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1109\/TGRS.2003.815676"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/15M1053475"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.4208\/nmtma.2017.m1617"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"D. Colton and R. Kress,\n                      Inverse Acoustic and Electromagnetic Scattering Theory\n                      , 3rd ed., Springer, New York, 2013.","DOI":"10.1007\/978-1-4614-4942-3"},{"key":"atypb12","unstructured":"L. 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Zhang,\n                      Fast Imaging of Scattering Obstacles from Phaseless Far-Field Measurements at a Fixed Frequency\n                      , preprint,https:\/\/arxiv.org\/abs\/1805.09046, 2018.","DOI":"10.1088\/1361-6420\/aad81f"}],"container-title":["SIAM Journal on Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1149699","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:40:27Z","timestamp":1787337627000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1149699"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1149699"],"URL":"https:\/\/doi.org\/10.1137\/17m1149699","relation":{},"ISSN":["0036-1399","1095-712X"],"issn-type":[{"value":"0036-1399","type":"print"},{"value":"1095-712X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}