In time-harmonic inverse electromagnetic scattering, it is a long-standing open problem whether a single scattering experiment — the electric (or magnetic) far-field pattern produced by one incident plane wave at fixed wavenumber, fixed direction, and fixed polarization, observed in all directions — uniquely determines the scatterer. Originating from the reflection-principle paper of Liu, Yamamoto and Zou (2007), the problem asks for uniqueness within general obstacle classes (non-polyhedral perfect conductors; impedance or partially coated scatterers; penetrable inhomogeneous media) and for the associated stability theory under very limited data. All known uniqueness results to date are restricted to polyhedral-type or other special geometries (balls, convex polyhedra, planar screens, rectangular penetrable scatterers in two dimensions), and the only claimed general proof (arXiv:1606.06529) is withdrawn by its author's note that the proof contains a gap. The problem remains open, in both directions: no general uniqueness theorem and no non-uniqueness example is known.

Background

Time-harmonic electromagnetic scattering in R3\mathbb{R}^{3} is governed by the Maxwell equations ∇×E−ikH=0\nabla\times E-\mathrm{i}kH=0, ∇×H+ikE=0\nabla\times H+\mathrm{i}kE=0 at wavenumber k>0k>0. An incident plane wave Ei(x)=p eikx⋅dE^{i}(x)=p\,\mathrm{e}^{\mathrm{i}kx\cdot d}, Hi(x)=(d×p) eikx⋅dH^{i}(x)=(d\times p)\,\mathrm{e}^{\mathrm{i}kx\cdot d}, with propagation direction d∈S2d\in\mathbb{S}^{2} and polarization p∈S2p\in\mathbb{S}^{2}, p⊥dp\perp d, impinges on a bounded obstacle DD; the scattered field (Es,Hs)(E^{s},H^{s}) solves the Maxwell equations outside Dˉ\bar D and satisfies the Silver–Müller radiation condition, which yields the asymptotic expansion Es(x)=eik∣x∣∣x∣E∞(x^)+O(∣x∣−2)E^{s}(x)=\frac{\mathrm{e}^{\mathrm{i}k|x|}}{|x|}E^{\infty}(\hat x)+O(|x|^{-2}) as ∣x∣→∞|x|\to\infty. The coefficient E∞(x^)E^{\infty}(\hat x), defined for all observation directions x^∈S2\hat x\in\mathbb{S}^{2}, is the electric far-field pattern; it encodes all remotely measurable information about the scatterer. For a perfectly conducting (PEC) obstacle the boundary condition is ν×E=0\nu\times E=0 on ∂D\partial D (ν\nu the outward normal, E=Ei+EsE=E^{i}+E^{s} the total field); impedance (Leontovich) boundary conditions ν×(∇×E)+iλ ν×(ν×E)=0\nu\times(\nabla\times E)+\mathrm{i}\lambda\,\nu\times(\nu\times E)=0 on ∂D\partial D model coated or partially conducting surfaces (a partially coated scatterer carrying the perfect-conductor condition on one part of ∂D\partial D and the impedance condition on the remaining part), and penetrable scatterers are inhomogeneous media in which the material coefficients change across an interface. Classical theory (Colton and Kress, Inverse Acoustic and Electromagnetic Scattering Theory) gives uniqueness of the obstacle when the far-field patterns are known for all incident directions and polarizations; by mixed reciprocity (the identity relating far-field patterns for plane-wave incidence to derivatives of the point-source Green function) and Rellich's lemma (the identity principle stating that a radiating solution of the Maxwell equations whose far-field vanishes identically must vanish), this extends to all but possibly a discrete set of wavenumbers. The situation with very little data is radically different. Liu, Yamamoto and Zou (2007) established a reflection principle for the time-harmonic Maxwell equations — an extension formula for solutions vanishing appropriately across a plane — and used it to prove that a polyhedral-type perfect conductor (finitely many solid polyhedra together with subsets of planes) is uniquely determined by the far-field pattern for plane-wave incidence with one direction and two polarizations. Successive works pushed single-measurement uniqueness further within special classes: general polyhedral PEC conductors with a single incident wave (Liu 2008), mixed PEC/PMC (perfect magnetic conductor, the dual boundary condition ν×H=0\nu\times H=0) polyhedral scatterers with a single electric or magnetic far-field measurement (Liu, Zhang and Zou 2009), coated convex polyhedra with a single electric far-field pattern via a reflection principle for the Leontovich condition (Hu, Vashisth and Yang 2021), penetrable rectangular scatterers in the two-dimensional TE (transverse electric) polarization with a single incoming wave (Hu, Li and Zou 2019), and bounded planar screens (Ola, Päivärinta and Sadique 2023). Parallel results for impedance obstacles and diffraction gratings in polyhedral geometries were obtained by Cao, Diao, Liu and Zou (2022). On the quantitative side, stability of the recovery from a single or a minimal number of measurements is understood only within polyhedral-type classes (Rondi 2008; Liu, Petrini, Rondi and Xiao 2017 in the acoustic case; Liu, Rondi and Xiao 2019 for electromagnetic stability via Mosco convergence (a variational notion of convergence of function spaces) of H(curl)H(\mathrm{curl}) spaces, the Sobolev space of square-integrable vector fields with square-integrable curl). All these techniques fundamentally exploit flat boundary pieces — the reflection principle needs a reflecting plane — and none extends to general curved obstacles. Hu, Vashisth and Yang (2021) state explicitly that beyond such restrictive shapes (balls, convex polyhedra, polyhedral-type sets) it is 'widely open' whether a general impenetrable or penetrable scatterer can be uniquely determined from a single far-field pattern. A preprint of Genqian Liu (2016, last revised 2021) claimed the general single-incident-plane-wave uniqueness for PEC or impedance boundary conditions, but its author attached the note 'there is a gap in the proof of main theorem', and the paper was never published; no accepted resolution exists. The acoustic analogue — uniqueness of a general sound-soft obstacle from a single far-field pattern — has been open since the 1980s, and the electromagnetic problem shares its structure while adding the coupling of polarization and the vectorial boundary conditions.

Problem Statement

Fix a wavenumber k>0k>0 and a single incident plane wave with direction d∈S2d\in\mathbb{S}^{2} and polarization p∈S2p\in\mathbb{S}^{2}, p⊥dp\perp d. For a bounded obstacle D⊂R3D\subset\mathbb{R}^{3} (with connected exterior, within a natural regularity class such as Lipschitz obstacles) satisfying the perfect-conductor boundary condition ν×E=0\nu\times E=0 on ∂D\partial D, let E∞(x^;D,d,p,k)E^{\infty}(\hat x;D,d,p,k), x^∈S2\hat x\in\mathbb{S}^{2}, denote the electric far-field pattern of the scattered field. Prove, or disprove by an explicit counterexample, the uniqueness conjecture: if E∞(x^;D1,d,p,k)=E∞(x^;D2,d,p,k)E^{\infty}(\hat x;D_{1},d,p,k)=E^{\infty}(\hat x;D_{2},d,p,k) for all observation directions x^∈S2\hat x\in\mathbb{S}^{2}, then D1=D2D_{1}=D_{2}. This conjecture is the core of the task and can be settled on its own; the companion questions listed below are also open and are not preconditions for completing the primary conjecture, and each of them is a separate partial-resolution target except item (i), which is the same question restated in magnetic form. The same question is open, and asked, for: (i) the magnetic far-field pattern alone, which is equivalent to the electric case rather than an independent question, since the Silver–Müller radiation condition forces the far-field patterns of every radiating solution to satisfy H∞(x^)=x^×E∞(x^)H^{\infty}(\hat x)=\hat x\times E^{\infty}(\hat x) and the magnetic and electric far-field patterns therefore carry identical information about DD; (ii) impedance (Leontovich) or partially coated boundary conditions, including simultaneous recovery of the obstacle and the boundary condition or impedance function; (iii) penetrable inhomogeneous-medium scatterers; and (iv) the companion quantitative problem of whether stable (e.g., logarithmic) recovery of the scatterer from such a single far-field pattern is possible for general obstacle classes, as established so far only for polyhedral-type scatterers. No restriction to polyhedral, convex, or otherwise special geometries is admissible: the open problem is precisely the general case.

References

  1. Hongyu Liu, Masahiro Yamamoto, Jun Zou, Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering, Inverse Problems 23 (2007) 2357–2366. DOI: 10.1088/0266-5611/23/6/005, https://doi.org/10.1088/0266-5611/23/6/005
  2. Hongyu Liu, A global uniqueness for formally determined inverse electromagnetic obstacle scattering, Inverse Problems 24 (2008) 035018. DOI: 10.1088/0266-5611/24/3/035018, arXiv:0801.2623, https://arxiv.org/abs/0801.2623
  3. Hongyu Liu, Hai Zhang, Jun Zou, Recovery of polyhedral scatterers by a single electromagnetic far-field measurement, Journal of Mathematical Physics 50 (2009) 123506. DOI: 10.1063/1.3263140, https://doi.org/10.1063/1.3263140
  4. Guang-Hui Hu, Manmohan Vashisth, Jiaqing Yang, Inverse time-harmonic electromagnetic scattering from coated polyhedral scatterers with a single far-field pattern, SIAM Journal on Applied Mathematics 81 (2021) 25–46. DOI: 10.1137/20M1330798, arXiv:2004.04392, https://arxiv.org/abs/2004.04392
  5. Guanghui Hu, Long Li, Jun Zou, Unique determination of a penetrable scatterer of rectangular type for inverse Maxwell equations by a single incoming wave, Inverse Problems 35 (2019) 035006. DOI: 10.1088/1361-6420/aafe96, https://doi.org/10.1088/1361-6420/aafe96
  6. Xinlin Cao, Huaian Diao, Hongyu Liu, Jun Zou, Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries, Inverse Problems and Imaging 16 (2022), no. 6, 1501–1528. DOI: 10.3934/ipi.2022023 (preprint arXiv:2111.13886, posted under the title 'Two unique identifiability results for inverse scattering problems within polyhedral geometries'), https://doi.org/10.3934/ipi.2022023
  7. Petri Ola, Lassi Päivärinta, Sadia Sadique, Unique determination of a planar screen in electromagnetic inverse scattering, Mathematics 11 (2023), no. 22, article 4655. DOI: 10.3390/math11224655, https://doi.org/10.3390/math11224655
  8. Genqian Liu, Uniqueness of the scatterer for electromagnetic field with one incident plane wave, arXiv preprint arXiv:1606.06529 (v1 2016; v5 2021 is a withdrawal note carrying the author's comment 'there is a gap in the proof of main theorem'), https://arxiv.org/abs/1606.06529
  9. Hongyu Liu, Luca Rondi, Jingni Xiao, Mosco convergence for H(curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, Journal of the European Mathematical Society 21 (2019) 2945–2993. DOI: 10.4171/JEMS/895, https://doi.org/10.4171/JEMS/895
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  11. Luca Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, Indiana University Mathematics Journal 57 (2008) 1377–1408. DOI: 10.1512/iumj.2008.57.3217, https://doi.org/10.1512/iumj.2008.57.3217
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