Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency

Xiaoxu Xu; Bo Zhang; Haiwen Zhang

doi:10.3934/ipi.2020023

Article Contents

2020, Volume 14, Issue 3: 489-510. Doi: 10.3934/ipi.2020023

This issue Previous Article Uniqueness and stability for the recovery of a time-dependent source in elastodynamics Next Article Statistical characterization of scattering delay in synthetic aperture radar imaging

Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency

1.
Beijing Computational Science Research Center, Beijing 100193, China
2.
LSEC, NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
4.
NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
5.
Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany

^* Corresponding author: Haiwen Zhang
^* Corresponding author: Haiwen Zhang

Received: July 2019

Revised: November 2019

Published: March 2020

This work is partly supported by the NNSF of China grants 91630309, 11871466 and 11571355, the NSAF grant U1930402, and the China Postdoctoral Science Foundation grant 2019TQ0023.

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Abstract

This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. We use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called phaseless acoustic near-field data) on two spheres containing the scatterers generated by such incident fields on the two spheres. Based on this idea, we prove that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined from the phaseless acoustic near-field data at a fixed frequency. Moreover, the idea is also applied to the electromagnetic case, and it is proved that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined by the phaseless electric near-field data at a fixed frequency, that is, the modulus of the tangential component with the orientations $ \boldsymbol{e}_\phi $ and $ \boldsymbol{e}_\theta $, respectively, of the electric total-field measured on a sphere enclosing the scatters and generated by superpositions of two electric dipoles at a fixed frequency located on the measurement sphere and another bigger sphere with the polarization vectors $ \boldsymbol{e}_\phi $ and $ \boldsymbol{e}_\theta $, respectively. As far as we know, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data.

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Mathematics Subject Classification: Primary: 78A46; Secondary: 35P25.

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Figure 1. Acoustic scattering by an obstacle (left) or a medium (right)

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Figure 2. Electromagnetic scattering by an obstacle (left) or a medium (right)

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