Unique determination of a perfectly conducting ball by finitely many electric far-field data: the size restriction $kR<\sqrt{2}$ and the minimality of four measurements

The perfectly conducting ball is the canonical test case for how little far-field information suffices to identify an obstacle in time-harmonic electromagnetic scattering. Hu, Liu and Zhang (J. Math. Anal. Appl. 352, 2009) proved that the radius of a perfectly conducting ball with known center is uniquely determined by a single electric far-field vector measured in the incident direction, and that radius and center together are determined by four electric far-field vectors generated by one incident plane wave, provided the size parameter satisfies $kR<\sqrt{2}$. Two questions arising from that work form this problem: (1) whether these uniqueness conclusions extend beyond $kR<\sqrt{2}$, and if so up to which critical size threshold, or whether non-uniqueness occurs for $kR\ge\sqrt{2}$ — the paper itself proves both theorems only under the explicit restriction $kR<\sqrt{2}$ and lists in its Introduction unique determination by a finite number of far-field patterns, given a priori information on the size of the scatterer, as an open question for electromagnetic scattering; and (2) the minimal number of electric far-field vector measurements needed to determine radius and center, for which the paper's heuristic remark that four data "seem to be the least" in three dimensions has never been proved. No work resolving either question for the electromagnetic ball with phased data is known; the problem remains open.

Background

In time-harmonic electromagnetic scattering one illuminates a bounded obstacle $D\subset\mathbb{R}^3$ with a plane wave $E^i(x)=p\,e^{ikx\cdot d}$ (wavenumber $k$, incident direction $d$, polarization vector $p\perp d$) and observes the scattered electric field far away, which is described by its asymptotic leading term $E_\infty(\hat x)$, the electric far-field pattern on the unit sphere $\mathbb{S}^2$. For a perfectly conducting obstacle the tangential component of the total electric field vanishes on the boundary ($\nu\times E=0$ on $\partial D$). Classically, an obstacle is known to be uniquely determined by the far-field patterns for all incident plane waves at a fixed frequency (Schiffer-type arguments, which compare two scatterers producing identical data and derive a contradiction from unique continuation of the governing elliptic equations; see Colton and Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 1998), but whether a single incident plane wave suffices is a famous longstanding open problem for general obstacles, solved only in special classes: for polyhedral scatterers (obstacles bounded by finitely many flat faces) in acoustics (Alessandrini and Rondi 2005; Liu and Zou 2006) and in electromagnetics (H. Liu 2008; Liu, Zhang and Zou 2009; Hu, Vashisth and Yang 2021), and for balls. For a ball the scattered field is explicit through the Mie series, an expansion in vector spherical harmonics whose coefficients are rational combinations of the spherical Bessel functions $j_n(t)$ and spherical Neumann functions $y_n(t)$, with $t=kR$ the size parameter. Two structural facts drive the source paper: first, the far-field of a translated ball obeys the translation relation $E_\infty(\hat x;B(x,R))=e^{ikx\cdot(d-\hat x)}E_\infty(\hat x;B(0,R))$, so the data moduli are independent of the center and the center enters only through phases; second, the radius can be read from the amplitude of a single far-field datum taken in the incident direction only as long as the relevant Mie coefficients depend monotonically on $t$, which in turn rests on uniform-in-order sign inequalities for spherical Bessel and Neumann function combinations. Hu, Liu and Zhang (2009) established, among others, that $y_n(t)+t\,y_n'(t)>0$ for all orders $n\ge1$ on $(0,\sqrt{2})$ (their own analysis leaves a gap here: an interlacing property of zeros that would sharpen the constants is not proved), and deduced that (i) the radius of a perfectly conducting ball centered at the origin is uniquely determined by the single electric far-field vector $E_\infty(d;d,p,k,R)$, evaluated at the observation direction $\hat x=d$ equal to the incident direction, whenever $kR<\sqrt{2}$, and (ii) the radius and the center of a ball with unknown center are uniquely determined by the four electric far-field vectors at observation directions $d_1=d$, $d_2$, $d_3$, $d_4$ with $d_2-d$, $d_3-d$, $d_4-d$ linearly independent, again under $kR<\sqrt{2}$; the phase factors at the four directions then solve for the center. Comparable one-datum results in the acoustic setting all carry restrictions of the same character: the radius of a sound-soft (Dirichlet boundary condition) ball with known center follows from the modulus of a single far-field datum — phaseless data, in which only the magnitude of the complex far-field value is measured — only for $kR<0.2303$ in 2D and $kR<\pi/4$ in 3D (X. Liu and B. Zhang 2010); for a sound-hard (Neumann boundary condition) ball with known center, for $kR<0.8838$ in 2D and $kR<1$ in 3D (Ye and Liu 2019); H. Liu and J. Zou (2007) could treat the resonance region (the regime of larger size parameter, beyond the first zeros of the relevant Bessel-function combinations, where the coefficients begin to oscillate) but only with known center, and a ball under an impedance boundary condition (a boundary condition of the third type, interpolating between sound-soft and sound-hard, that relates the field to its normal derivative through an impedance parameter) with unknown impedance parameter was treated by J. Liu and J. Zhang (2017) in the acoustic case; Hu and X. Liu (2014) proved uniqueness for balls from a single point-source wave rather than a plane wave. No electromagnetic analogue removing the smallness condition is known, and the source paper itself states that four measurements only seem to be the least number needed in three dimensions, without proof.

Problem Statement

Consider time-harmonic electromagnetic scattering in $\mathbb{R}^3$ at a fixed wavenumber $k>0$ by a perfectly conducting ball $B(z,R)$ (center $z\in\mathbb{R}^3$, radius $R>0$, both unknown), illuminated by a single plane wave $E^i(\xi)=p\,e^{ik\xi\cdot d}$ with field variable $\xi$, incident direction $d\in\mathbb{S}^2$ (the unit sphere of $\mathbb{R}^3$), and polarization vector $p\perp d$. The leading asymptotics of the scattered field at large distances define the electric far-field pattern $E_\infty(\hat x;B(z,R))$, a vector-valued function of the observation direction $\hat x\in\mathbb{S}^2$; for a ball it is given explicitly by the Mie series (the multipole expansion in vector spherical harmonics), and it obeys the translation relation $E_\infty(\hat x;B(z,R))=e^{ikz\cdot(d-\hat x)}E_\infty(\hat x;B(0,R))$. Throughout, the data are phased, i.e. complex-valued far-field vectors, not modulus-only. Because the center contributes to the far-field vectors only through the phases $e^{ikz\cdot(d-\hat x)}$, it cannot be recovered exactly from finitely many observation directions on its own; throughout, the center is taken to be known a priori to lie in a fixed bounded region, the same region for every measurement configuration, and unique determination of the center is meant within that region. Call two balls equivalent for a given configuration when they have equal radii and their centers differ by an element of the phase lattice $\Lambda=\{\delta\in\mathbb{R}^3:k\,\delta\cdot(d-\hat x_j)\in2\pi\mathbb{Z}\ \text{for every used direction}\ \hat x_j\}$; equivalent balls produce identical far-field vectors in that configuration. It is known (Hu, Liu and Zhang 2009, Theorems 3.1 and 4.1, both under the explicit restriction $kR<\sqrt{2}$, which arises from the inequality $y_n(t)+t\,y_n'(t)>0$ on $(0,\sqrt{2})$ in their Theorem 2.8) that the radius of a ball with known center is uniquely determined by the single electric far-field vector measured at the observation direction $\hat x=d$ (the incident, i.e. forward, direction), and that the radius and center of a ball with unknown center are uniquely determined by the electric far-field vectors measured at four observation directions $d_1=d,d_2,d_3,d_4\in\mathbb{S}^2$ with $d_2-d$, $d_3-d$, $d_4-d$ linearly independent. This problem asks to complete the uniqueness theory of the perfectly conducting ball with finitely many phased electric far-field data at one fixed frequency and one incident plane wave, by settling the following two linked open questions arising from that work. (A) Size threshold: determine, separately for the single-datum uniqueness (known center) and for the four-measurement uniqueness (unknown center), whether uniqueness persists for $kR\ge\sqrt{2}$; for each of the two results, either prove uniqueness on a strictly larger range of the size parameter and identify the sharp range of uniqueness (up to the largest size parameter for which uniqueness holds, if that range is bounded), or exhibit a rigorous counterexample: two balls with $kR\ge\sqrt{2}$ producing identical electric far-field data in the measurement configuration belonging to that result (the forward single-datum configuration for the known-center result, the four-direction configuration for the unknown-center result), where in the known-center case the two balls share the a priori known center and differ only in the radius, and in the unknown-center case the two balls are not equivalent for the four-direction configuration. (B) Data minimality: the source paper states twice, without proof, that four data "seem to be the least" in the three-dimensional case, since there are then four unknowns (the radius and the three components of the center; Introduction and Remark 4.2(i)). Within the size range $kR<\sqrt{2}$ of the two theorems recalled above, and judged by that same fixed a priori region, determine the least number $M$ of observation directions such that the $M$ electric far-field vectors from the single incident plane wave uniquely determine both the radius and the center of a perfectly conducting ball; in particular, either prove a lower bound showing that four measurements are necessary (no configuration of three or fewer observation directions can be injective), or prove that fewer than four well-chosen observation directions suffice. Exhibiting a single non-injective three-measurement configuration would not settle this question: by the translation relation, any center difference orthogonal to all difference vectors $d-d_j$ of the used observation directions leaves the measured far-field vectors unchanged, so three measurements are already non-injective whenever those difference vectors span at most a plane — in particular for every configuration containing the forward direction $\hat x=d$ together with only two other directions. The common objective is the sharp trade-off between the number of far-field measurements and the admissible size range for unique determination of the ball, the canonical model scatterer of the single-measurement uniqueness program; the broader open problem of unique determination of a general non-spherical obstacle by the far-field pattern of a single incident plane wave is context for this problem, not part of its completion boundary.

References

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