Intensity-only (phaseless) far-field measurements at a single fixed frequency are known to determine a sound-soft obstacle, a nonabsorbing impedance obstacle, or a real-valued refractive index with uniformly sign-definite contrast, provided the incident fields are superpositions of two plane waves with distinct directions (Xu–Zhang–Zhang, 2018). Later work achieved uniqueness for absorbing obstacles/media, unknown boundary conditions, and general contrast only by modifying the experiment — adding a known reference ball to the scattering system — and proved the electromagnetic analogue in the same reference-ball framework, for perfectly conducting, real-nonnegative-impedance, and partially coated obstacles and for an isotropic medium with constant permeability. Still open is the original formulation: whether the modulus of the far-field pattern generated by all two-plane-wave superpositions at one wavenumber, in the unmodified scattering system, uniquely determines (i) an absorbing impedance obstacle (including its unknown physical type and impedance function), (ii) an absorbing inhomogeneous medium without any sign-and-magnitude condition on the contrast, and (iii) the same electromagnetic scatterer classes without the reference ball.

Background

Time-harmonic inverse scattering asks whether an unknown scatterer can be identified from wave measurements far away. For an incident plane wave ui(x)=eikx⋅du^i(x)=e^{ikx\cdot d} with wavenumber k>0k>0 and direction d∈S2d\in\mathbb{S}^2, the scattered field of an impenetrable obstacle DD solving the Helmholtz equation Δu+k2u=0\Delta u+k^2u=0 outside DD has the asymptotic form us(x)=eik∣x∣u∞(x^,d)/∣x∣+O(∣x∣−2)u^s(x)=e^{ik|x|}u^\infty(\hat x,d)/|x|+O(|x|^{-2}); the coefficient u∞u^\infty is the far-field pattern, observable on the unit sphere x^∈S2\hat x\in\mathbb{S}^2. The obstacle's physical nature enters through a boundary condition: sound-soft (Dirichlet, u=0u=0 on ∂D\partial D), sound-hard (Neumann), or impedance (∂u/∂ν+ηu=0\partial u/\partial\nu+\eta u=0 with impedance function η\eta); the impedance is called absorbing when Im⁡η>0\operatorname{Im}\eta>0, meaning the boundary dissipates energy. A partially coated obstacle carries a mixed boundary condition: ∂D=ΓD∪Π∪ΓI\partial D=\Gamma_D\cup\Pi\cup\Gamma_I is a Lipschitz dissection with ΓD,ΓI\Gamma_D,\Gamma_I disjoint relatively open subsets of ∂D\partial D sharing the common interface Π\Pi in ∂D\partial D, the sound-soft (Dirichlet) condition is imposed on ΓD\Gamma_D and the impedance condition on ΓI\Gamma_I (in the electromagnetic setting the perfectly conducting condition replaces the sound-soft one). For a penetrable inhomogeneous medium the total field solves Δu+k2nu=0\Delta u+k^2 n u=0 in R3\mathbb{R}^3 with refractive index nn; the medium is absorbing when Im⁡n>0\operatorname{Im} n>0, and the contrast is m=n−1m=n-1.

In many experiments (optics, X-ray, some radar/sonar settings) only intensities can be measured accurately, so one has access to the phaseless data ∣u∞∣|u^\infty| rather than u∞u^\infty. This creates a fundamental obstruction: the modulus of the far-field pattern produced by a single incident plane wave is invariant under translations of the scatterer, so location — and hence the scatterer — cannot be recovered, regardless of how many incident directions are used. Xu, Zhang and Zhang (SIAM J. Appl. Math. 78(3):1737–1753, 2018) broke this translation invariance by using incident fields that are superpositions eikx⋅d1+eikx⋅d2e^{ikx\cdot d_1}+e^{ikx\cdot d_2} of two plane waves with distinct directions d1,d2∈S2d_1,d_2\in\mathbb{S}^2, d1≠d2d_1\ne d_2, all at one fixed frequency, and proved the first uniqueness results with phaseless far-field data: the data ∣u∞(x^;d1,d2)∣|u^\infty(\hat x;d_1,d_2)| for all x^∈S2\hat x\in\mathbb{S}^2 and all d1≠d2d_1\ne d_2 determine a sound-soft obstacle, a nonabsorbing impedance obstacle, or a real-valued refractive index, under two restrictions: the physical type of the obstacle is known a priori, and the contrast satisfies m≥c1>0m\ge c_1>0 or m≤−c1<0m\le -c_1<0 throughout its support. Their proof rests on a spectral fact about the far-field operator FF (the operator mapping incident fields to far-field patterns): in the nonabsorbing cases considered there, FF is normal and has countably infinitely many eigenvalues λj\lambda_j accumulating only at 00, and the normalized non-zero eigenvalues λj/∣λj∣\lambda_j/|\lambda_j| tend to −1-1 for sound-soft obstacles and for real-valued indices with n−1≤−c1<0n-1\le -c_1<0, and to +1+1 for impedance obstacles with real-valued η\eta and for real-valued indices with n−1≥c1>0n-1\ge c_1>0; this class-dependent limit law substitutes for the missing phase information. The same authors note that for absorbing cases the limit of the normalized eigenvalues of FF is not known yet, a spectral gap that is one reason their method does not extend directly. (Rellich's lemma — a radiating field with vanishing far-field pattern vanishes identically — is the standard tool converting far-field agreement into pointwise field agreement.)

The companion paper Part II (SIAM J. Appl. Math. 78(6):3024–3039, 2018) showed that all these restrictions can be dropped — absorbing impedances (Im⁡η≥0\operatorname{Im}\eta\ge 0), absorbing media (Re⁡n>0\operatorname{Re} n>0, Im⁡n≥0\operatorname{Im} n\ge 0), unknown boundary condition, no sign condition — but only after inserting a known reference ball into the scattering system, which changes the measured data; Zhang and Guo (Inverse Problems 34:085002, 2018) reached similar generality with a reference ball plus point-source superpositions, and the electromagnetic (Maxwell) case was settled in the same reference-ball framework by Xu–Zhang–Zhang (IMA J. Appl. Math., 2020), though only for real nonnegative impedance functions. Those authors explicitly state that proving the same results without the reference ball remains open and is more challenging. Thus everything beyond the original 2018 hypotheses currently requires an artificial auxiliary scatterer in the measurement process, and the spectral limit law underlying the original method has no known absorbing-case counterpart.

Problem Statement

In the unmodified acoustic scattering system — no artificial reference ball or auxiliary point scatterer — fix the wavenumber k>0k>0 and take as data the moduli ∣u∞(x^;d1,d2)∣|u^\infty(\hat x;d_1,d_2)| of the far-field patterns produced by all superpositions eikx⋅d1+eikx⋅d2e^{ikx\cdot d_1}+e^{ikx\cdot d_2} of two plane waves, for all observation directions x^∈S2\hat x\in\mathbb{S}^2 and all pairs of incident directions d1,d2∈S2d_1,d_2\in\mathbb{S}^2 with d1≠d2d_1\ne d_2. Establish, or disprove by explicit counterexample, the following:

  1. (Absorbing obstacles, unknown type) An obstacle whose physical type (sound-soft, sound-hard, impedance, partially coated) is not known a priori — and which, in the case of an impedance boundary condition, has a continuous impedance function satisfying Im⁡η≥0\operatorname{Im}\eta\ge 0 (including strictly absorbing Im⁡η>0\operatorname{Im}\eta>0) — is uniquely determined, in its shape, its physical type, and its impedance function, by these data, without any a priori knowledge of the scatterer type.

  2. (Absorbing media without sign conditions) A refractive index n∈L∞n\in L^\infty with Re⁡n>0\operatorname{Re} n>0 and Im⁡n≥0\operatorname{Im} n\ge 0, whose contrast n−1n-1 has compact support but satisfies no uniform sign or magnitude condition, is uniquely determined by these data.

  3. (Electromagnetic extension) The same fixed-frequency uniqueness holds without a reference ball in Maxwell scattering, for the scatterer classes treated by Xu–Zhang–Zhang (IMA J. Appl. Math., 2020) in the reference-ball framework. The data are the moduli ∣em(x^)⋅E∞(x^,d1,d2,p1,p2)∣|\bm e_m(\hat x)\cdot E^\infty(\hat x,d_1,d_2,p_1,p_2)|, m∈{ϕ,θ}m\in\{\phi,\theta\}, of the two tangential components (with respect to the orthonormal spherical tangential frame eϕ,eθ\bm e_\phi,\bm e_\theta) of the electric far-field patterns produced by all superpositions Ei(x,d1)p1+Ei(x,d2)p2E^i(x,d_1)p_1+E^i(x,d_2)p_2 of two electromagnetic plane waves, for all x^∈S2∖{N,S}\hat x\in\mathbb{S}^2\setminus\{N,S\} (the unit sphere minus the north and south poles), all directions d1,d2∈S2d_1,d_2\in\mathbb{S}^2 and all polarizations p1,p2∈R3p_1,p_2\in\mathbb{R}^3 with dj⊥pjd_j\perp p_j, at the same fixed wavenumber. The question is whether these data uniquely determine a perfectly conducting, an impedance (with impedance function λ∈C(∂D)\lambda\in C(\partial D), λ≥0\lambda\ge 0), or a partially coated obstacle, in shape and physical type, as well as an isotropic inhomogeneous medium with constant positive permeability μ=μ0\mu=\mu_0 and refractive index n∈C2,γn\in C^{2,\gamma} with Re⁡n>0\operatorname{Re} n>0, Im⁡n≥0\operatorname{Im} n\ge 0 and compactly supported contrast n−1n-1, in the unmodified electromagnetic scattering system.

The three components are independent sub-claims: each may be established or refuted on its own, none of them implies another, and a counterexample to one refutes only that component while leaving the others open, so resolving any single component settles that component alone and the problem is settled only when all three are resolved. Answers to any component must be evaluated against this exact data set and system; resolving a component only after inserting a known reference scatterer (as in existing work) answers a different, derived problem and does not settle the question as stated.

References

  1. Xiaoxu Xu, Bo Zhang, Haiwen Zhang, 'Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency', SIAM Journal on Applied Mathematics 78(3):1737–1753, 2018. DOI: 10.1137/17M1149699, https://doi.org/10.1137/17M1149699
  2. Xiaoxu Xu, Bo Zhang, Haiwen Zhang, 'Uniqueness in Inverse Scattering Problems with Phaseless Far-Field Data at a Fixed Frequency. II', SIAM Journal on Applied Mathematics 78(6):3024–3039, 2018. DOI: 10.1137/18M1196820, https://doi.org/10.1137/18M1196820 (preprint arXiv:1806.09127, https://arxiv.org/abs/1806.09127)
  3. Deyue Zhang, Yukun Guo, 'Uniqueness results on phaseless inverse acoustic scattering with a reference ball', Inverse Problems 34:085002, 2018. DOI: 10.1088/1361-6420/aac53c, https://doi.org/10.1088/1361-6420/aac53c
  4. 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)
  5. Deyue Zhang, Yukun Guo, 'Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory', Electronic Research Archive 29(2):2149–2165, 2021. DOI: 10.3934/era.2020110, https://doi.org/10.3934/era.2020110
  6. Xia Ji, Xiaodong Liu, Bo Zhang, 'Inverse Acoustic Scattering with Phaseless Far Field Data: Uniqueness, Phase Retrieval, and Direct Sampling Methods', SIAM Journal on Imaging Sciences 12(2):1163–1189, 2019. DOI: 10.1137/18M1236022, https://doi.org/10.1137/18M1236022
  7. Xiaoxu Xu, Bo Zhang, Haiwen Zhang, 'Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency', Inverse Problems and Imaging 14(3):489–510, 2020. DOI: 10.3934/ipi.2020023, https://doi.org/10.3934/ipi.2020023 (preprint arXiv:1906.05116, https://arxiv.org/abs/1906.05116)
  8. Xianghe Zhu, Jun Guo, Haibing Wang, 'Uniqueness Results for Some Inverse Electromagnetic Scattering Problems with Phaseless Far-Field Data', Axioms 12(12):1069, 2023. DOI: 10.3390/axioms12121069, https://doi.org/10.3390/axioms12121069
  9. Xia Ji, Xiaodong Liu, 'Inverse elastic scattering problems with phaseless far field data', Inverse Problems 35:114004, 2019. DOI: 10.1088/1361-6420/ab2a35, https://doi.org/10.1088/1361-6420/ab2a35
  10. Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen, 'A Coefficient Inverse Problem with a Single Measurement of Phaseless Scattering Data', SIAM Journal on Applied Mathematics 79(1):1–27, 2019. DOI: 10.1137/18M1168303, https://doi.org/10.1137/18M1168303