Single-incidence uniqueness in inverse scattering from rough and periodic surfaces: the surviving open problems after Zhao-Hu-Yan (perfect conductors under one incident wave; finite-direction recovery of periodic sound-soft profiles)

Zhao, Hu and Yan (2019) proved uniqueness results for inverse acoustic and electromagnetic scattering from locally perturbed rough surfaces, including single-incidence uniqueness for polyhedral surfaces, and explicitly listed four unresolved issues: single-wave recovery of impedance-type polyhedral surfaces; transmission interface conditions; recovery of periodic sound-soft surfaces from multi-direction plane-wave data; and electromagnetic single-wave uniqueness for a bounded perfect conductor, including a perfectly conducting ball under electric-dipole illumination. The surviving open core consists of two single-incidence identifiability questions: (i) whether a bounded perfectly conducting obstacle, in particular a perfectly conducting ball illuminated by a single electric dipole, is uniquely determined by single-incident-wave data at a fixed frequency; and (ii) whether finitely many incident directions determine a general periodic sound-soft grating profile.

Background

Time-harmonic inverse obstacle scattering asks whether an unknown scatterer can be recovered from remote measurements of the waves it scatters. For the Helmholtz equation $\Delta u+k^2u=0$ in the exterior of an obstacle ($k>0$ is the wavenumber), a sound-soft obstacle imposes the Dirichlet condition $u=0$ on the boundary (a pressure-release surface in acoustics), a sound-hard obstacle imposes the Neumann condition $\partial_\nu u=0$ (a rigid surface), and an impedance or Robin condition $\partial_\nu u=ik\lambda u$ models a partially absorbing surface; a penetrable (transmission) interface instead couples the exterior field to a second field propagating inside the scatterer. The scattered field is required to satisfy the Sommerfeld radiation condition, which selects outgoing waves and yields the far-field pattern, the leading amplitude of the scattered field at large distance, which is the standard data of the theory. A classical argument of Schiffer (1967) shows that a sound-soft obstacle is uniquely determined by far-field patterns for infinitely many incident plane waves at one wavenumber, but uniqueness from a single incident wave is a famous open problem, settled only in special classes: polyhedral (flat-faced) scatterers, via reflection principles combined with path arguments (Liu-Zou 2006; Elschner-Yamamoto 2008), and balls by a single point-source wave (Hu-Liu 2014).

For unbounded surfaces the same questions are harder. A locally perturbed rough surface is a Lipschitz graph that coincides with the flat ground plane outside a bounded set, so the propagation domain is an unbounded half-space-like region. Zhao, Hu and Yan (2019) proved for such surfaces that sound-soft and sound-hard profiles are uniquely determined by far-field data for all (resp. infinitely many) incident directions, that polyhedral surfaces of Dirichlet or Neumann type are uniquely determined by the far field of a single point source or a single plane wave, and, electromagnetically, that a bounded or unbounded perfect polyhedral conductor is uniquely determined by the electric far-field pattern of a single electric dipole. Here a perfect conductor is the electromagnetic analogue of the sound-soft condition: the tangential component of the electric field vanishes, $\nu\times E=0$, on the boundary; an electric dipole is the point-source analogue for Maxwell's equations, a field with a prescribed polarization vector singular at a single source point. Their proofs combine a mixed reciprocity relation between plane-wave and point-source scattering in the half space with reflection and path arguments in polyhedral geometry.

The same paper states four open problems beyond the reach of its methods. First, single-wave uniqueness for impedance-type polyhedral surfaces was open, even for bounded obstacles. Second, its arguments did not handle transmission interface conditions. Third, for periodic sound-soft surfaces it seemed unknown how to obtain uniqueness even from near-field data excited by plane waves at different incident directions, because subtracting the incident plane wave from the total field no longer leaves a Sommerfeld-radiating field: for a periodic grating the natural radiation condition is the Rayleigh expansion, a discrete sum of propagating and evanescent plane-wave modes above the profile, and the field is quasi-periodic rather than decaying. Fourth, for a bounded perfect conductor of general (non-polyhedral) shape, single-wave uniqueness was open, including the concrete case of a perfectly conducting ball under illumination by a single electric dipole, in contrast to the acoustic case where a sound-soft ball is determined by one point source (Hu-Liu 2014).

The prior state of the art for gratings (periodic surfaces) was: global uniqueness for polygonal periodic structures with a minimal number of incident plane waves (Elschner-Schmidt-Yamamoto 2003; Elschner-Hu 2010) and unique determination of periodic polyhedral structures by scattered electromagnetic fields (Bao-Zhang-Zou 2011); for general smooth profiles no finite-direction uniqueness result was known. Subsequent literature resolved the impedance problem for polyhedral geometries (Hu-Vashisth 2020; Hu-Vashisth-Yang 2021; Cao-Diao-Liu-Zou 2022; Diao-Liu-Tao 2024), substantially addressed the transmission problem with point-source data (Wu-Yang 2024), and settled the periodic question for infinitely many incident directions (Xu-Hu-Zhang-Zhang 2023) while proving that a single direction cannot even determine the grating period. What survives is the single- and minimal-incidence core stated below.

Problem Statement

The problem is the surviving single- and minimal-incidence identifiability core of the open questions posed by Zhao, Hu and Yan (2019) for inverse scattering by rough and periodic surfaces, after the impedance and point-source transmission parts were resolved by later literature (see Background). Its common objective is to determine whether one, or finitely many, incident waves suffice to uniquely determine the scatterer in the two model settings below; each part is answered either by a uniqueness theorem covering the stated class or, negatively, by explicit constructions of two distinct admissible configurations producing identical data, which for Part 2 must be supplied for every finite set of distinct incident directions, since Part 2 asks whether some finite set of directions suffices. The overall objective is met when both parts are settled.

Part 1 (electromagnetic perfect conductor, single incident wave). Let $D\subset\mathbb{R}^3$ be a bounded perfectly conducting obstacle, i.e. the total electric field satisfies $\nu\times E=0$ on $\partial D$, and fix a wavenumber $k>0$. Part 1 comprises two single-incidence identifiability sub-questions, both of which must be settled: (1a) is $D$ uniquely determined by the electric far-field pattern (for all observation directions) generated by a single plane wave with a fixed incident direction and polarization at this one wavenumber, and (1b) is $D$ uniquely determined by the electric far-field pattern generated by a single electric dipole located at a fixed exterior point with fixed polarization at this one wavenumber? A negative answer to (1a) or (1b) must exhibit, for the corresponding incident field, two distinct admissible configurations producing identical far-field data. In particular, is a perfectly conducting ball with unknown radius and center uniquely determined by the electric far-field pattern of a single electric dipole? Positive results are known only for polyhedral conductors (Liu-Zhang-Zou 2009; Zhao-Hu-Yan, Theorems 3.1-3.2) and, in the acoustic analogue, for sound-soft balls with a single point source (Hu-Liu 2014); the smooth electromagnetic case, including the ball, is open.

Part 2 (periodic sound-soft surface, finitely many incident directions). In two dimensions, let $\Gamma$ be an $L$-periodic sound-soft curve (Dirichlet boundary condition), with the period $L$ possibly unknown, and let the scattered field above $\Gamma$ satisfy the Rayleigh expansion radiation condition. For a fixed lossless wavenumber $k>0$, do there exist finitely many incident plane waves with distinct directions whose phased near-field data, measured on a line segment above the grating (which, by analyticity of the Rayleigh expansion, determine all Rayleigh coefficients, propagating and evanescent, and in particular the finitely many propagating ones), determine $\Gamma$ uniquely within an explicitly stated general (non-polygonal, non-rectangular) profile class, such as $C^2$ or Lipschitz-graph periodic profiles? It is known that a single direction does not suffice (the period cannot even be recovered) and that infinitely many directions do suffice (Xu-Hu-Zhang-Zhang 2023); the finite-direction question is open outside the polygonal and rectangular profile classes.

The uniqueness questions are posed for the full classes stated in Parts 1 and 2 and for every fixed wavenumber $k>0$, including interior-resonance and Rayleigh-frequency values; they are not restricted to polyhedral shapes, small scatterers or gratings, lossy media, or a known period.

References

1. Yu Zhao, Guanghui Hu, Baoqiang Yan, 'Uniqueness to inverse acoustic and electromagnetic scattering from locally perturbed rough surfaces', Applicable Analysis 100(2):361-382 (online 2019; print issue 2021), DOI: 10.1080/00036811.2019.1603371, https://doi.org/10.1080/00036811.2019.1603371; preprint arXiv:1812.09009, https://arxiv.org/abs/1812.09009.
2. Guang-Hui Hu, Manmohan Vashisth, 'Uniqueness to inverse acoustic scattering from coated polygonal obstacles with a single incoming wave', Inverse Problems 36(10):105004 (2020), DOI: 10.1088/1361-6420/abb61c, https://doi.org/10.1088/1361-6420/abb61c; preprint arXiv:2005.04676, https://arxiv.org/abs/2005.04676.
3. 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(1):25-46 (2021), DOI: 10.1137/20M1330798, https://doi.org/10.1137/20M1330798.
4. 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(6):1501-1528 (2022), DOI: 10.3934/ipi.2022023, https://doi.org/10.3934/ipi.2022023; preprint arXiv:2111.13886 (preprint title: 'Two unique identifiability results for inverse scattering problems within polyhedral geometries'), https://arxiv.org/abs/2111.13886.
5. Huaian Diao, Hongyu Liu, Longyue Tao, 'Stable determination of an impedance obstacle by a single far-field measurement', Inverse Problems 40(5):055005 (2024), DOI: 10.1088/1361-6420/ad3087, https://doi.org/10.1088/1361-6420/ad3087; preprint arXiv:2302.01083, https://arxiv.org/abs/2302.01083.
6. Huaian Diao, Hongyu Liu, Longyue Tao, 'Stably Determining a generalised Impedance Obstacle from a Single Far-Field Pattern', arXiv:2604.12835 (2026), https://arxiv.org/abs/2604.12835.
7. Xiaoxu Xu, Guanghui Hu, Bo Zhang, Haiwen Zhang, 'Uniqueness in Inverse Diffraction Grating Problems With Infinitely Many Plane Waves at a Fixed Frequency', SIAM Journal on Applied Mathematics 83(1):302-326 (2023), DOI: 10.1137/22m1480963, https://doi.org/10.1137/22m1480963; preprint arXiv:2202.13280, https://arxiv.org/abs/2202.13280.
8. Chengyu Wu, Jiaqing Yang, 'Direct and inverse acoustic scattering by global rough surfaces', arXiv:2406.06129 (2024, v2 2025), https://arxiv.org/abs/2406.06129.
9. Hongyu Liu, Hai Zhang, Jun Zou, 'Recovery of polyhedral scatterers by a single electromagnetic far-field measurement', Journal of Mathematical Physics 50(12):123506 (2009), DOI: 10.1063/1.3263140, https://doi.org/10.1063/1.3263140.
10. Guanghui Hu, Xiaodong Liu, 'Unique determination of balls and polyhedral scatterers with a single point source wave', Inverse Problems 30(6):065010 (2014), DOI: 10.1088/0266-5611/30/6/065010, https://doi.org/10.1088/0266-5611/30/6/065010.
11. Hongyu Liu, 'Solving Schiffer's problem in inverse scattering theory almost surely', arXiv:2408.10132 (2024), https://arxiv.org/abs/2408.10132.
12. Johannes Elschner, Guanghui Hu, 'Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves', Inverse Problems 26(11):115002 (2010), DOI: 10.1088/0266-5611/26/11/115002, https://doi.org/10.1088/0266-5611/26/11/115002.
13. J. Elschner, G. Schmidt, M. Yamamoto, 'An inverse problem in periodic diffractive optics: global uniqueness with a single wavenumber', Inverse Problems 19(3):779-787 (2003), DOI: 10.1088/0266-5611/19/3/318, https://doi.org/10.1088/0266-5611/19/3/318.
14. Gang Bao, Hai Zhang, Jun Zou, 'Unique determination of periodic polyhedral structures by scattered electromagnetic fields', Transactions of the American Mathematical Society 363(9):4527-4551 (2011), DOI: 10.1090/S0002-9947-2011-05334-1, https://doi.org/10.1090/S0002-9947-2011-05334-1.
15. Jiahui Gao, Junliang Lv, Shuxin Li, 'A Uniqueness Result for the Inverse Electromagnetic Scattering by Locally Rough Surfaces With Phaseless Far-Field Data', Mathematical Methods in the Applied Sciences 48(13):12790-12800 (2025), DOI: 10.1002/mma.11062, https://doi.org/10.1002/mma.11062.
16. Jiahui Gao, Junliang Lv, Tian Niu, 'Uniqueness in Inverse Electromagnetic Scattering by Locally Rough Surfaces with Phaseless Near-Field Data', Communications in Mathematical Research 41(2):189-208 (2025), DOI: 10.4208/cmr.2025-0007, https://doi.org/10.4208/cmr.2025-0007.
17. Jianliang Li, Jiaqing Yang, Bo Zhang, 'A linear sampling method for inverse acoustic scattering by a locally rough interface', Inverse Problems and Imaging 15(5):1247-1267 (2021), DOI: 10.3934/ipi.2021036, https://doi.org/10.3934/ipi.2021036.
18. Hongyu Liu, Jun Zou, 'Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers', Inverse Problems 22(2):515-524 (2006), DOI: 10.1088/0266-5611/22/2/008, https://doi.org/10.1088/0266-5611/22/2/008.
19. Johannes Elschner, Masahiro Yamamoto, 'Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave', Inverse Problems 24(3):035004 (2008), DOI: 10.1088/0266-5611/24/3/035004, https://doi.org/10.1088/0266-5611/24/3/035004.