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

## TL;DR

This paper proves that the shape and material properties of obstacles or media can be uniquely identified using phaseless near-field data at a fixed frequency in both acoustic and electromagnetic inverse scattering problems, employing superpositions of two point sources or electric dipoles.

## Contribution

It introduces a novel approach using superpositions of two point sources or electric dipoles to achieve uniqueness in inverse scattering with phaseless near-field data at a fixed frequency, extending previous far-field results.

## Key findings

- Unique determination of obstacles or media from phaseless near-field data.
- Extension of uniqueness results to electromagnetic scattering.
- First known uniqueness proof for 3D inverse electromagnetic scattering with phaseless near-field data.

## Abstract

This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work ({\em SIAM J. Appl. Math. \bf78} (2018), 1737-1753), where uniqueness results were proved for inverse acoustic scattering with phaseless far-field data generated by superpositions of two plane waves as the incident waves at a fixed frequency, in this paper, 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 enclosing 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 extended 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 $\mathbf e_\phi$ and $\mathbf e_\theta$, 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 $\mathbf e_\phi$ and $\mathbf e_\theta$, respectively. As far as we know, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data.