Uniqueness in inverse scattering with phaseless far-field data at a fixed frequency: the absorbing, unknown-type, and electromagnetic cases without reference objects

Time-harmonic inverse scattering asks whether an unknown obstacle or inhomogeneous medium can be recovered from far-field measurements of a scattered wave. When only the modulus (intensity) of the far-field pattern is measured — phaseless data — a single incident plane wave cannot even fix the scatterer's location, because the modulus is invariant under translations. Xu, Zhang and Zhang (SIAM J. Appl. Math. 78 (2018) 1737–1753) broke this invariance using superpositions of two plane waves at a single fixed frequency and proved the first uniqueness theorems with phaseless far-field data, but only for sound-soft or non-absorbing impedance obstacles and for non-absorbing media with real-valued, uniformly sign-definite refractive-index contrast; their proof rests on a limit law for the normalized eigenvalues of the far-field operator that is established only in these non-absorbing cases. The source paper's own list of open problems and the subsequent literature show that the goals were later achieved only by engineering the experiment (adding a known reference ball or extra point sources), and that the authors' 2018 Part II, 2020 Maxwell paper, and a 2021 review all explicitly leave the original configuration unresolved. The surviving open core is precise: decide, in the original configuration (moduli of far-field patterns generated by superpositions of two plane waves at one fixed frequency, no auxiliary scatterers), uniqueness for absorbing obstacles and media, for scatterers of unknown physical type, and for electromagnetic waves. The limit law for the normalized eigenvalues of the far-field operator in the absorbing case, where the known eigenvalue mechanism collapses, is the spectral question motivating the first of these goals rather than a separate deliverable.

Background

In time-harmonic acoustic scattering, an incident plane wave $u^i(x,d)=e^{ikx\cdot d}$ with wave number $k>0$ and direction $d$ on the unit sphere $S^2$ strikes an unknown scatterer, and the total field $u=u^i+u^s$ solves the Helmholtz equation $\Delta u+k^2u=0$ outside the scatterer, with the scattered field $u^s$ radiating to infinity. For an impenetrable obstacle — a bounded domain $D$ with $C^2$ boundary — the physical nature of the scatterer is encoded in a boundary condition: $u=0$ on $\partial D$ (a sound-soft obstacle), $\partial u/\partial\nu+\eta u=0$ with a continuous impedance function $\eta$ (an impedance obstacle; $\eta=0$ is the sound-hard or Neumann case), or a mixture on coated parts of the boundary. The impedance condition is called absorbing when $\mathrm{Im}(\eta)>0$ and non-absorbing when $\eta$ is real-valued (the standing physical hypothesis is $\mathrm{Im}(\eta)\ge0$). For a penetrable inhomogeneous medium, the refractive index $n\in L^\infty$ (with $n\equiv 1$ outside a compact set, $\mathrm{Re}(n)>0$) enters the equation $\Delta u+k^2 n u=0$; the medium is absorbing when $\mathrm{Im}(n)>0$. Far away, $u^s(x)=e^{ik|x|}|x|^{-1}\big(u^\infty(\hat x,d)+O(|x|^{-1})\big)$, and the complex-valued function $u^\infty(\hat x,d)$ on $S^2\times S^2$ is the far-field pattern. With full phase information, the classical theory gives uniqueness of the obstacle, its impedance, or the refractive index from $u^\infty$ for all incident and observation directions at one fixed frequency.

In many experiments only the intensity $|u^\infty|$ is measured reliably; such data are called phaseless. Phaseless far-field data have a fundamental obstruction: for a single incident plane wave, $|u^\infty|$ is invariant under translations of the scatterer, so the location cannot be recovered at all. Xu, Zhang and Zhang overcame this by using superpositions of two plane waves, $u^i(x;d_1,d_2)=e^{ikx\cdot d_1}+e^{ikx\cdot d_2}$ with distinct directions $d_1\ne d_2$, and proved the first uniqueness results for inverse scattering with phaseless far-field data at a fixed frequency: the moduli $|u^\infty(\hat x;d_1,d_2)|$ for all $\hat x,d_1,d_2\in S^2$ uniquely determine a sound-soft obstacle $D$; a non-absorbing impedance obstacle together with its impedance $\eta$; and a non-absorbing medium whose contrast $m=n-1$ is real-valued and uniformly sign-definite ($m\ge c_1>0$ or $m\le-c_1<0$ on its support). Their proof rests on a spectral result of independent interest. The far-field operator $F:g\mapsto\int_{S^2}u^\infty(\hat x,d)\,g(d)\,ds(d)$ has eigenvalues $\{\lambda_n\}$ whose normalized values converge: $\lambda_n/|\lambda_n|\to-1$ for a sound-soft obstacle, $+1$ for a non-absorbing impedance obstacle, and $\mathrm{sign}(m)$ for a non-absorbing sign-definite medium. This law reflects that the scattering operator $S=I+(ik/2\pi)F$ is unitary precisely in the non-absorbing case; for absorbing obstacles or media the eigenvalues of $F$ lie strictly inside the associated disk, are known only to accumulate at $0$, and no limit law for $\lambda_n/|\lambda_n|$ is known — the mechanism of the uniqueness proof collapses entirely. The paper closes by stating the problems left open: remove the non-absorbing assumption on $\eta$ and $n$; remove the a priori knowledge of the scatterer's physical type and of the sign-definiteness of $n-1$; and extend the framework from scalar acoustics to electromagnetic waves.

Subsequent work achieved these three goals only by modifying the measurement configuration. In Part II (SIAM J. Appl. Math. 78 (2018) 3024–3039) the same authors removed the a priori assumptions — and allowed absorbing impedance functions with $\mathrm{Im}(\eta)\ge0$ and absorbing media with $\mathrm{Re}(n)>0$, $\mathrm{Im}(n)\ge0$ — by adding a known reference ball to the scattering system and replacing the eigenvalue argument with Rellich's lemma and Green's representation formula; they explicitly state that proving the same results without the reference ball remains open. Zhang and Guo (Inverse Problems 34 (2018) 085002) obtained, with a reference ball and superpositions of a plane wave with point sources, uniqueness of the obstacle together with its general boundary condition and of the refractive index of an absorbing medium, reducing to classical phased uniqueness theorems; related reference-object and point-source-superposition schemes were developed by Ji, Liu and Zhang (SIAM J. Imaging Sci. 12 (2019) 1163–1189) and Sun, Zhang and Guo (Inverse Problems 35 (2019) 105007). The electromagnetic extension was carried out by Xu, Zhang and Zhang (IMA J. Appl. Math. 85 (2020)) — again with a reference ball — for perfectly conducting, impedance and partly coated obstacles and inhomogeneous media, using the moduli of the tangential components of the electric far-field pattern generated by superpositions of two electromagnetic plane waves with different directions and polarizations; they too state that the case without the reference ball is open and more challenging. Extensions to elastic waves, acoustic–elastic interactions, cavities, transmission problems, gratings and chiral media, and a 2021 survey by Zhang and Guo, all confirm the same pattern: every known phaseless uniqueness result introduces an additional degree of freedom into the measurement (a reference object, an additional point source, or a near-field measurement surface). The physically natural question — whether intensity-only far-field data at a single frequency can determine the scatterer without any engineered auxiliary object — remains unanswered beyond the original theorem's hypotheses.

Problem Statement

Work in the original measurement configuration of Xu–Zhang–Zhang (2018): a single fixed wave number $k>0$, incident fields that are superpositions of two plane waves $u^i(x;d_1,d_2)=e^{ikx\cdot d_1}+e^{ikx\cdot d_2}$ with distinct directions $d_1\ne d_2\in S^2$, and data consisting only of the moduli $|u^\infty(\hat x;d_1,d_2)|$ of the resulting far-field patterns for all observation directions $\hat x\in S^2$, with no reference ball, no additional point sources, and no other auxiliary scatterers in the system. Decide, by rigorous proof or explicit counterexample, uniqueness in this configuration for the regimes the 2018 theorem excludes, namely:

(1) Absorbing scatterers in acoustics. For impedance obstacles with continuous impedance $\eta$ satisfying $\mathrm{Im}(\eta)\ge0$ on $\partial D$ and not identically zero there (that is, $\mathrm{Im}(\eta)>0$ at least somewhere, so the obstacle is absorbing in the sense that the non-absorbing hypothesis $\eta$ real-valued fails), prove or disprove that the data determine the obstacle $D$ and its impedance $\eta$; and for inhomogeneous media with $n\in L^\infty$, $\mathrm{Re}(n)>0$, $\mathrm{Im}(n)\ge0$ and compactly supported contrast $n-1$, prove or disprove that the data determine $n$ — in particular, whether the uniform sign-definiteness of $n-1$ can be removed. This regime is tied to an open spectral question stated in the source paper: the original uniqueness proof rests on the non-absorbing limit law $\lambda_n/|\lambda_n|\to-1$, $+1$, $\mathrm{sign}(m)$ for the eigenvalues of the far-field operator $F$, and for an absorbing obstacle or medium no limit of the normalized eigenvalues is known.

(2) Unknown physical type. Prove or disprove the same uniqueness statements when it is not known in advance whether the scatterer is sound-soft, impedance (absorbing or not), or a penetrable medium, and when no sign condition on the contrast is assumed.

(3) Electromagnetic waves. Prove or disprove the analogue of (1)–(2) for the Maxwell equations: uniqueness of the obstacle (perfectly conducting, impedance, or partly coated) or of the inhomogeneous medium from the moduli of the tangential components of the electric far-field pattern generated by superpositions of two electromagnetic plane waves with different directions and polarizations, at a fixed frequency, without a reference ball.

The common objective of the three parts is to determine whether intensity-only far-field data at a single frequency can uniquely determine scatterers in the physically central regimes — absorbing media, unknown boundary type, vector waves — without engineering the experiment with auxiliary known objects. The problem is completely resolved when each of the three regimes admits a definitive answer: a uniqueness theorem covering the stated data class, or an explicit pair of distinct scatterers generating identical phaseless data (which would show the reference-ball workaround is essential).

References

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