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

Abstract

This paper is concerned with uniqueness in inverse electromagnetic scattering with phaseless farfield pattern at a fixed frequency. In our previous work [SIAM J. Appl. Math. 78 (2018), [3024][3025][3026][3027][3028][3029][3030][3031][3032][3033][3034][3035][3036][3037][3038][3039], by adding a known reference ball into the acoustic scattering system, it was proved that the impenetrable obstacle and the index of refraction of an inhomogeneous medium can be uniquely determined by the acoustic phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency. In this paper, we extend these uniqueness results to the inverse electromagnetic scattering case. The phaseless far-field data are the modulus of the tangential component in the orientations e φ and e θ , respectively, of the electric far-field pattern measured on the unit sphere and generated by infinitely many sets of superpositions of two electromagnetic plane waves with different directions and polarizations. Our proof is mainly based on Rellich's lemma and the Stratton-Chu formula for radiating solutions to the Maxwell equations. 1 generated by one plane wave is invariant under the translation of the scatterers. This implies that it is impossible to recover the location of the scatterer from the phaseless far-field data with one plane wave as the incident field. Several iterative methods have been proposed in [12,13,14,21] to reconstruct the shape of the scatterer. Under a priori condition that the sound-soft scatterer is a ball or disk, it was proved in [22] that the radius of the scatterer can be uniquely determined by a single phaseless far-field datum. It was proved in [23] that the shape of a general, sound-soft, strictly convex obstacle can be uniquely determined by the phaseless far-field data generated by one plane wave at a high frequency. However, there is no translation invariance property for phaseless near-field data. Therefore, many numerical algorithms for inverse scattering problems with phaseless near-field data have been developed (see, e.g., [3,4,6,18,32,36] for the acoustic case and [5] for the electromagnetic case). Uniqueness results and stability have also been established for inverse scattering problems with phaseless near-field data (see [16,17,19,24,26,27,30,37,42,43] for the acoustic and potential scattering case and [29,37] for the electromagnetic scattering case).Recently in [38], it was proved that the translation invariance property of the phaseless far-field pattern can be broken by using superpositions of two plane waves as the incident fields with an interval of frequencies. Following this idea, several algorithms have been developed for inverse acoustic scattering problems with phaseless far-field data, based on using the superposition of two plane waves as the incident field (see [38,39,40]). Further, by using the spectral properties of the far-field operator, rigorous uniqueness results have also been established in [34] for inverse acousti...

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

“…On the other hand, a fast imaging algorithm was developed in [40] to numerically recover the scattering obstacles from phaseless far-field data at a fixed frequency. Furthermore, it was rigorously proved in [37] 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 at a fixed frequency under the a priori assumption that the obstacle is a sound-soft obstacle or an impedance obstacle with a real-valued impedance function and the refractive index n of the inhomogeneous medium is real-valued and satisfies the condition that either n − 1 ≥ c 1 or n − 1 ≤ −c 1 in the support of n − 1 for some positive constant c 1 . The purpose of the present paper is to remove the a priori assumption on the obstacle and the refractive index of the inhomogeneous medium by adding a known reference ball to the scattering system in conjunction with a simple technique based on Rellich's lemma and Green's representation formula for the scattering solutions.…”

“…In contrast to the case with phaseless near-field data, inverse scattering with phaseless far-field data is more challenging due to the translation invariance property of the phaseless far-field pattern (see [22,27,37,38]). Thus, it is impossible to reconstruct the location of the scatterers from the phaseless far-field pattern with one plane wave as the incident field.…”

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

“…This idea leads to the multi-frequency Newton iteration algorithm [49,50] and the fast imaging algorithm at a fixed frequency [51]. Further, by the superposition of two incident plane waves, uniqueness results were established in [46] under some a priori assumptions.…”

“…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].…”

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

“…The purpose of this paper is to propose a new approach to establish uniqueness results for inverse acoustic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work [39], where uniqueness results have been proved for inverse acoustic scattering with phaseless far-field data corresponding to superpositions of two plane waves as the incident fields at a fixed frequency, we consider to utilize the superposition of two point sources at a fixed frequency as the incident field. However, the idea of proofs used in [39] can not be applied directly to the inverse scattering problem with phaseless near-field data.…”