Time-harmonic electromagnetic scattering by a bounded obstacle or an inhomogeneous medium is described by the electric far-field pattern, which is complex-valued; intensity-only (phaseless) experiments can measure only its modulus. Xu, Zhang and Zhang proved that if a known reference ball is added to the scattering system, the moduli of the two tangential components of the electric far-field pattern on the unit sphere, generated by infinitely many superpositions of two plane waves at a single fixed frequency, uniquely determine the obstacle together with its boundary condition or the refractive index of the medium. The authors explicitly left open whether the same uniqueness holds without inserting any known reference object. This problem asks for a proof, or a counterexample, of uniqueness for this reference-ball-free phaseless far-field inverse problem at the full generality of the reference-ball theorem. Later work (2023) settled the adjacent question affirmatively for a different measurement enrichment (superpositions of electric dipoles), but the original plane-wave setup without a reference object remains open.

Background

Consider scattering of a time-harmonic electromagnetic plane wave, with wavenumber kk fixed once and for all, by either an impenetrable obstacle DD or an inhomogeneous medium in R3\mathbb{R}^3. The total electric field EE solves the Maxwell equations outside the scatterer, subject to a boundary condition encoding the physical nature of the obstacle: ν×E=0\nu\times E=0 on ∂D\partial D for a perfectly conducting obstacle (ν\nu the unit outward normal), ν×curl E−iλ(ν×E)×ν=0\nu\times\mathrm{curl}\,E-i\lambda(\nu\times E)\times\nu=0 for an impedance obstacle with impedance function λ∈C(∂D)\lambda\in C(\partial D), λ≥0\lambda\ge 0, or a mixed condition of these two types on the two parts of a partly coated boundary; for a medium, the refractive index nn (with constant magnetic permeability, Re n>0\mathrm{Re}\,n>0, Im n≥0\mathrm{Im}\,n\ge 0, n−1n-1 compactly supported) enters the constitutive relation. At large distance the scattered field has the asymptotic form Es(x)=eik∣x∣E∞(x^)/∣x∣+O(∣x∣−2)E^s(x)=e^{ik|x|}E^{\infty}(\hat x)/|x|+O(|x|^{-2}), where x^=x/∣x∣\hat x=x/|x| and E∞E^{\infty} is the electric far-field pattern, a tangential vector field on the unit sphere S2S^2. Classical uniqueness theory (Colton–Kress and successors) recovers the scatterer from the phased far-field pattern for all incident plane waves. Measuring the phase of high-frequency fields is, however, experimentally difficult, and several technologies deliver only intensity-only data. The natural phaseless data studied here are the moduli of the two tangential spherical components of the electric far-field pattern, ∣em(x^)⋅E∞∣|e_m(\hat x)\cdot E^{\infty}| with m∈{ϕ,θ}m\in\{\phi,\theta\} and eϕ,eθe_\phi,e_\theta the spherical unit vectors, measured for all observation directions x^∈S2∖{N,S}\hat x\in S^2\setminus\{N,S\}, where NN and SS are the north and south poles, excluded because the spherical parametrization of the tangential basis is singular there, and for all incident fields that are superpositions of two plane waves with directions d1,d2∈S2d_1,d_2\in S^2 and polarizations p1⊥d1p_1\perp d_1, p2⊥d2p_2\perp d_2. For a single incident plane wave the modulus of the far-field pattern is invariant under translations of the scatterer, so the scatterer's location cannot be recovered; Zhang and Zhang (2017) showed that superpositions of two plane waves with an interval of frequencies break this translation invariance. Building on this, Xu, Zhang and Zhang proved in two papers (SIAM J. Appl. Math. 78, 2018) that, in the acoustic case, the phaseless far-field data of infinitely many two-plane-wave superpositions at a fixed frequency uniquely determine the scatterer either (Part I) under a priori assumptions (sound-soft or nonabsorbing impedance obstacle; real-valued refractive index with n−1n-1 of one sign), or (Part II) in full generality after inserting a known reference ball — a small perfectly conducting ball (or a ball filled with a known refractive index) placed at a known position disjoint from the search region, whose known scattered field serves as a phase anchor; kk must not be a Maxwell eigenvalue of the reference ball, that is, not an interior cavity resonance frequency of the ball (respectively, for the known-index ball, not an interior transmission eigenvalue, a frequency at which the interior transmission problem coupling the fields inside the ball at indices 11 and n0n_0 admits a nontrivial solution). Xu, Zhang and Zhang (IMA J. Appl. Math. 2020) extended the reference-ball result to electromagnetism: with a known perfectly conducting reference ball BB in the system and both scatterers assumed to lie in a common a priori known ball BRB_R disjoint from BB, equality of the data ∣em(x^)⋅E∞(x^;d1,d2,p1,p2)∣|e_m(\hat x)\cdot E^{\infty}(\hat x;d_1,d_2,p_1,p_2)| for all admissible parameters implies that the two obstacles coincide (D1=D2D_1=D_2) together with their boundary conditions, including the impedance functions, and analogously that two admissible refractive indices coincide (n1=n2n_1=n_2) for media. Their proof combines Rellich's lemma (a uniqueness-via-energy-integral device for the Helmholtz/Maxwell equations) with the Stratton–Chu representation formula for radiating Maxwell fields. The paper's conclusion states verbatim that the authors hope to show the same uniqueness results without using the reference ball, which they describe as more challenging. Subsequent literature has circumvented, but not answered, this question: Niu, Lv and Wu (2023) proved the same type of uniqueness without any reference object by changing the incident fields to superpositions of two electric dipoles (point sources at known positions with known polarizations), and Gao, Lv and Li (2025) did the same for locally rough surfaces with a dipole-plus-plane-wave incident field; other works retain the reference ball for further electromagnetic models (Zhu, Guo and Wang 2023) or switch to phaseless near-field data, for which translation invariance fails (Xu 2024); in the Schrödinger/medium setting the Novikov school likewise compensates for the missing phase by known background scatterers (Novikov–Sivkin 2021, Hohage–Novikov–Sivkin 2024). No work found establishes uniqueness for the original plane-wave-superposition far-field setup without any reference object and without the a priori assumptions that the reference ball was introduced to remove.

Problem Statement

Fix a wavenumber k>0k>0 and consider the time-harmonic Maxwell scattering system with no reference object of any kind present. The unknown scatterer is either an impenetrable obstacle DD that is perfectly conducting, impedance (with impedance function λ∈C(∂D)\lambda\in C(\partial D), λ≥0\lambda\ge 0), or partly coated (mixed Dirichlet/impedance boundary condition), or an inhomogeneous medium with refractive index n∈C2,γ(R3)n\in C^{2,\gamma}(\mathbb{R}^3), Re n>0\mathrm{Re}\,n>0, Im n≥0\mathrm{Im}\,n\ge 0, n−1n-1 compactly supported, and constant magnetic permeability. Illuminate the scatterer by superpositions of two plane waves with directions d1,d2∈S2d_1,d_2\in S^2 and polarizations p1,p2∈R3p_1,p_2\in\mathbb{R}^3 satisfying p1⊥d1p_1\perp d_1 and p2⊥d2p_2\perp d_2, and measure the phaseless electric far-field data

∣em(x^)⋅E∞(x^; d1,d2,p1,p2)∣,m∈{ϕ,θ},\bigl|e_m(\hat x)\cdot E^{\infty}(\hat x;\,d_1,d_2,p_1,p_2)\bigr|,\qquad m\in\{\phi,\theta\},

defined for all observation directions x^∈S2∖{N,S}\hat x\in S^2\setminus\{N,S\} and all admissible incident parameters (d1,d2,p1,p2)(d_1,d_2,p_1,p_2) at this single fixed frequency, where E∞E^{\infty} is the electric far-field pattern and eϕ,eθe_\phi,e_\theta the tangential spherical unit vectors. Assume only that both scatterers of any comparison pair are contained in a common a priori known bounded ball. Prove or disprove: (a) if two obstacles with their boundary conditions (perfectly conducting, impedance, or partly coated, including the impedance function) give rise to identical phaseless data for all admissible parameters, then the obstacles and their boundary conditions coincide; and (b) if two admissible refractive indices give rise to identical phaseless data for all admissible parameters, then the indices coincide. The problem is posed throughout with no known reference ball or other reference scatterer, no phased data, and no restriction to a priori boundary classes or sign conditions on n−1n-1 beyond the admissibility stated above.

References

  1. Xiaoxu Xu, Bo Zhang, Haiwen Zhang, Uniqueness in inverse electromagnetic scattering problem with phaseless far-field data at a fixed frequency, IMA Journal of Applied Mathematics (2020), DOI 10.1093/imamat/hxaa024, https://doi.org/10.1093/imamat/hxaa024 (preprint arXiv:1906.08622, https://arxiv.org/abs/1906.08622)
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