Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency. II

Abstract

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), 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 non-absorbing impedance obstacle and the index of refraction n of the inhomogeneous medium is real-valued and satisfies that either n − 1 ≥ c 1 or n − 1 ≤ −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.

Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency

“…In this paper, by adding a given reference ball into the electromagnetic scattering system, we established uniqueness results for inverse electromagnetic obstacle and medium scattering with phaseless electric farfield data generated by infinitely many sets of superpositions of two electromagnetic plane waves with different directions and polarizations at a fixed frequency for the first time. These uniqueness results extend our previous results in [35] for the acoustic case to the electromagnetic case. Our method is based on a simple technique of using Rellich's lemma and the Stratton-Chu formula for the radiating solutions to the Maxwell equations.…”

“…Throughout this paper, we assume that the wave number k is arbitrarily fixed, i.e., the frequency ω is arbitrarily fixed. Following [34,35,38,39], we make use of the following superposition of two plane waves as the incident (electric) field:…”

“…Note further that there are certain studies on uniqueness for phaseless inverse scattering problems with using superpositions of two point sources as the incident fields (see [30,33,37,42,43]). The purpose of this paper is to establish uniqueness results in inverse electromagnetic scattering problems with phaseless far-field data at a fixed frequency, extending the uniqueness results in [35] for the acoustic case to the electromagnetic case. Different from the acoustic case considered in [35], the electric far-field pattern is a complex-valued vector function, so the measurement of the phaseless electric far-field pattern is more complicated.…”

“…In practice, one usually makes measurement of the modulus of each tangential component of the electric total-field or electric far-field pattern on the measurement surface (see, e.g., [10,28,31,44]). Motivated by this and the idea in [34,35], we make use of superpositions of two electromagnetic plane waves with different directions and polarizations as the incident fields and consider the modulus of the tangential component in the orientations e φ and e θ , respectively, of the corresponding electric far-field pattern measured on the unit sphere as the measurement data (called the phaseless electric far-field data). We then prove that, by adding a known reference ball into the electromagnetic scattering system, the impenetrable obstacle or the refractive index of the inhomogeneous medium (under the condition that the magnetic permeability is a positive constant) can be uniquely determined by the phaseless electric far-field data at a fixed frequency.…”

Uniqueness in inverse cavity scattering problems with phaseless near-field data

“…We would like to point out that the idea of adding a reference ball to the scattering system was first proposed in [31] to numerically enhance the resolution of the linear sampling method. The reference ball technique was used in [47] to alleviate the requirement of the a priori assumptions in [46]. Similar strategies of adding reference objects or sources to the scattering system have also been extensively applied to the theoretical analysis and numerical approaches for different models of phaseless inverse scattering problems [13,14,19,20,21,53].…”