Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory

1.   Introduction

Inverse scattering problems are fundamental in many scientific and industrial applications. Several exemplary scenarios involve sonar detection, radar sensing, medical imaging and geophysical exploration. Inverse scattering problems are concerned with the detection and identification of unknown targets from the knowledge of associated wave scattering data. In particular, inverse scattering of time-harmonic waves is of great significance. The typical time-harmonic inverse scattering problems are based on complex-valued data comprising phase and intensity/modulus. On the other hand, the phase information may be unavailable or extremely difficult to be detected accurately in practice. Hence, according to the accessibility to phase, the measured data in time-harmonic inverse scattering problems can be classified into two categories: phased/full data and phaseless or intensity-only/modulus-only data. Over the past several decades, the inverse scattering problems with full measured data received a great deal of attentions in the literature (see, e.g. [11, 12, 22] and the references therein). In fact, the phaseless measurements are usually more feasible in practice. Therefore, the phaseless inverse scattering problems have recently been intensively studied mathematically and numerically [1, 2, 4, 31, 32, 49, 57].

Due to the lack of phase information, the phaseless inverse scattering problems are in general very challenging. From the mathematical perspective, the first challenge is the uniqueness: can one uniquely identify the underlying target from the measured phaseless data? To compensate the missing phase and justify the uniqueness, some additional information should be incorporated into the scattering system. There are several strategies for supplementing more quantities, for instance, superposing incident waves and adding artificial reference/interfering objects. Using these techniques, the phaseless inverse scattering problems can be recast as "well-posed" problems in the sense of uniqueness justifications. The uniqueness results for phaseless inverse scattering problems usually rely on the existing uniqueness theorems concerning the phased data. We refer to the monograph [12] as an entry for the vast investigations on the uniqueness issues associated with full data.

Our studies on the uniqueness in phaseless inverse scattering problems are scattered in different papers [44, 50, 52, 55] and thus they are loosely coupled in certain sense. Accordingly, this paper aims to present a unified and consistent framework for the problem formulations, technical treatment and the uniqueness results. We would like to emphasize that the purpose of this short review is mainly to crystallize the major developments that we had participated in, that is, we made no effort to cover all the relevant topics or present a comprehensive investigation on the diverse literature. In fact we only consider the phaseless inverse acoustic scattering problems as models. Nevertheless, it paves the way for many extensions to the cases of electromagnetic and elastic waves, where analogous strategies can be employed but technical details might be more complicated. We refer to the recent survey paper [42] for the phaseless inverse problems for some typical wave equations.

The rest of this paper is arranged as follows. Section 2 is devoted to the reference ball technique for uniquely determining a bounded scatterer from phaseless far-field data. Then in section 3, we present the uniqueness results related to the superposition of exterior point sources and near-field measurements. Finally, the interior problem of imaging the cavity is discussed in section 4.

2.   The reference ball technique

We present some prototypical inverse acoustic scattering models and the reference ball technique for establishing the uniqueness.

2.1.   Model problems

In this subsection, we introduce the acoustic direct and inverse scattering models for an incident plane wave. Let the scatterer $D \subset\mathbb{R}^3$ occupy an open and simply-connected domain with $C^2$ boundary $\partial D$. Let $u^i(x,d) = \mathrm{e}^{\mathrm{i} k x\cdot d}$ be a given incident plane wave, where $d\in\mathbb{S}^2$ and $k>0$ are the incident direction and wavenumber, respectively.

Denote by $\nu$ the unit outward normal to $\partial D$. Then the forward obstacle scattering problem can be stated as: given $D$ and $u^i$, find the scattered field $u^s\in H_{\rm loc}^1(\mathbb{R}^3\backslash\overline{D})$ such that the total field $u = u^i+u^s$ satisfies the following boundary value problem (see [12]):

where $u^s$ denotes the scattered field and (3) is the Sommerfeld radiation condition. The boundary operator $\mathscr{B}$ defined in (2) is given by

where $\lambda$ is a real parameter signifying the physical impedance. The boundary condition (4) is rather general since it covers the usual Dirichlet/sound-soft boundary condition, the Neumann/sound-hard boundary condition ($\lambda = 0$) and the impedance boundary condition ($\lambda\neq 0$).

In contrast to the obstacle scattering problem dealing with an impenetrable scatterer $D$, if $D$ is penetrable with respect to wave propagation, then the forward scattering can be formulated as the medium scattering problem: given $D$ and $u^i$, determine the scattered field $u^s\in H_{\rm loc}^1(\mathbb{R}^3)$ such that the total field $u = u^i+u^s$ fulfills

where the refractive index $n(x)$ of the inhomogeneous medium is piecewise continuous such that $\mathrm{Re}(n)>0, \mathrm{Im}(n)\geq0$ and $1-n(x)$ is supported in $D$.

The forward scattering problems (1)–(3) and (5)–(6) admit a unique solution (see, e.g., [6, 12, 37]), respectively. Moreover, the scattered wave $u^s$ has the following asymptotic behavior

uniformly in all observation directions $\hat{x} = x/|x|$. Denote by $\mathbb{S}^2: = \{x\in\mathbb{R}^3: |x| = 1\}$ the unit sphere in $\mathbb{R}^3$. Then the complex-valued analytic function $u^{\infty}(\hat{x}; d)$ is defined on $\mathbb{S}^2$ and known as the scattering amplitude or the far field pattern (see [12]). In what follows, we shall also employ $u^{\infty}_D(\hat{x}; d)$ to indicate the dependence of the far field pattern $u^\infty$ on the observation direction $\hat{x}$, the incident direction $d$, the obstacle or medium $D$ and a fixed wavenumber $k$.

A typical phaseless inverse scattering problem is stated as follows.

Problem 2.1. Given phaseless far field data $|u^{\infty}_D(\hat{x}; d)|$ for $\hat{x}, d\in \mathbb{S}^2$ and a fixed $k>0$.

(i) Suppose that $D$ is an impenetrable obstacle, determine the location and shape $\partial D$ as well as the boundary condition $\mathscr{B}$.

(ii) Suppose that $D$ is a penetrable inhomogeneous medium, determine the refractive index $n_D$ for the medium inclusion.

There is a well-known obstruction of Problem 2.1 due to the translation invariance. Specifically, for the shifted domain $D^h: = \{x+h: x \in D\}$ with a fixed vector $h\in \mathbb{R}^3$, the far field pattern $u^{\infty}_{D^h}(\hat{x}; d)$ satisfies the relation [29, 34]

Hence the location of the scatterer cannot be uniquely determined from the intensity-only far-field data. More notoriously, this intrinsic ambiguity cannot be remedied by simply using finitely many incident waves with different wavenumbers or distinct directions of incidence. Nevertheless, it is still possible to reconstruct the shape without such phase information. In fact, a great number of inversion schemes have been proposed to recover the shape of scatterer from the intensity-only far-field data with a single incident plane wave, see [17, 18, 19, 30, 31, 32]. We also refer to [10, 16] for the relevant numerical investigations.

2.2.   Uniqueness with a reference ball

To compensate the lack of phase and thereby break the translation-invariance obstruction, some supplementing information is required to be incorporated into the scattering system. We made an attempt in [50] to tackle the translation invariance by adding a reference ball in conjunction with the superposition of incident waves. In this subsection, we shall give a brief outline of the reference ball technique for phaseless inverse scattering problems.

Definition 2.1 (Reference ball) For the inverse scattering problem of recovering $D$, let $B\subset\mathbb{R}^3$ such that $\overline{D}\cap\overline{B} = \emptyset$. The scatterer $B$ is called a reference object if

(i) The geometrical (location, shape, etc.) and physical properties (boundary conditions, refractive index, etc.) of $B$ are assumed, a priori, to be known.

(ii) Determination of the target $D$ relies on the scattering data produced by $D\cup B$.

Clearly, a simplest candidate for the reference object is the reference ball $B = B(x_0,R): = \{x\in \mathbb{R}^3: |x-x_0|<R\}$.

Due to the nonlinearity of typical inverse scattering problems, one should bear in mind that extracting the contribution of $D$ from the data due to $D\cup B$ is not straightforward in most cases.

We also need the following admissible source location.

Definition 2.2 (Admissible source location) Assume that $P$ is a simply-connected convex polyhedron with boundary $\Pi: = \partial P$ such that $P \subset\mathbb{R}^3\backslash (\overline{D}\cup \overline{B})$ and $k^2$ is not a Dirichlet eigenvalue of $-\Delta$ in $P$.

Then let us consider the superposition of a plane wave $u^i$ and a point source $v^i$ as the incident wave:

where $z \in \Pi$ denotes the location of point source, and

is the fundamental solution to the Helmholtz equation.

Let the $u^{\infty}_{D\cup B}(\hat{x}; d)$ and $v^{\infty}_{D\cup B}(\hat{x}; z)$ be the far-field pattern generated by $D\cup B$ corresponding to the incident field $u^i$ and $v^i$, respectively. Then, by the linearity of direct scattering problem, the far-field pattern generated by $D\cup B$ and the incident wave $u^i+v^i$ defined in (8) is given by $u^{\infty}_{D\cup B}(\hat{x}; d)+v^{\infty}_{D\cup B}(\hat{x}; z), \hat{x}\in \mathbb{S}^2$, respectively.

Based on Definitions 2.1 and 2.2, we are now in the position to introduce the phaseless datasets.

Definition 2.3 (Far field data with a reference ball) For a fixed wavenumber $k>0$ and a fixed $d_0\in \mathbb{S}^2$, define

The incursion of the reference ball and superposition of incoming waves lead to the following reformulation of phaseless inverse scattering problems:

Problem 2.2 (Inverse obstacle scattering with a reference ball and far-field data). Let $D$ be the impenetrable obstacle with boundary condition $\mathscr{B}$. Given the phaseless triple $\{\mathbb{D}^{\rm FB}_1, \mathbb{D}^{\rm FB}_2, \mathbb{D}^{\rm FB}_3\}$ due to $D$ and a sound-soft reference ball $B$, determine the location and shape $\partial D$ as well as the boundary condition $\mathscr{B}$ for the obstacle.

Problem 2.3 (Inverse medium scattering with a reference ball and far-field data). Let $D$ be the inhomogeneous medium with refractive index $n$. Given the phaseless triple $\{\mathbb{D}^{\rm FB}_1, \mathbb{D}^{\rm FB}_2, \mathbb{D}^{\rm FB}_3\}$ due to the inhomogeneity $D$ and a penetrable reference ball $B$, determine the refractive index $n$ for the medium inclusion.

We refer to Figure 1 for an illustration of the geometry setup of Problems 2.2 and 2.3. The following theorem shows the uniqueness results for Problem 2.2 and Problem 2.2, i.e., the scatterer can be uniquely determined from the consolidated phaseless data $\{\mathbb{D}^{\rm FB}_1, \mathbb{D}^{\rm FB}_2, \mathbb{D}^{\rm FB}_3\}$.

Theorem 2.1. [50] Given a fixed $d_0 \in \mathbb{S}^2$ and a reference ball $B$, the far-field pattern with respect to the incident field $u^i(x; d_0)$ and $v^i(x; z)$ are denoted by $u^\infty_{D_j\cup B}(\hat{x}; d_0)$, and $v^\infty_{D_j\cup B}(\hat{x}; z), j = 1,2$, respectively. Suppose that

Then we have

(i) If $D_1$ and $D_2$ are two obstacles with boundary conditions $\mathscr{B}_1$ and $\mathscr{B}_2$ respectively, and $B$ is sound-soft, then $D_1 = D_2$ and $\mathscr{B}_1 = \mathscr{B}_2$.

(ii) If $D_1$ and $D_2$ are two inclusions with refractive indices $n_1$ and $n_2$ respectively, and $B = B(x_0, R)$ is a homogeneous medium with refractive index $n_0>0$ such that $2k R(n_0+1)<\pi$, then $n_1 = n_2$.

To resolve non-uniqueness issues, the idea of utilizing the superposition of distinct incident plane waves was proposed in [56], which led to the multi-frequency Newton iteration algorithm [56]. Moreover, by the superposition of two incident plane waves, uniqueness results were established in [46] under some a priori assumptions.

We would like to point out that the idea of adding a reference ball to the scattering system in [50] was motivated by [33] and [38, 39]. Later, the reference ball technique was used in [47] to alleviate the requirement of the a priori assumptions in [46]. Recently, similar strategies of adding reference objects or sources to the scattering system have also been intensively applied to the theoretical analysis and numerical approaches for different models of phaseless inverse scattering problems [15, 13, 20, 21, 51, 53, 14].

3.   Superposition of point sources

Next we will deal with the uniqueness issue concerning the inverse acoustic scattering problems with point excitation sources and associated phaseless near-field measurements. In optics and engineering areas, the phaseless inverse scattering with near-field data is also known as phase retrieval problem [35]. The numerical studies on the inverse scattering problems with phaseless near-field data are intensive (see, e.g., [8, 7, 10, 9, 40, 45]), whereas few theoretical investigations have been made on the uniqueness aspects.

A uniqueness result was established in [25] to the reconstruction of a potential with the phaseless near-field data for point sources on a spherical surface and an interval of frequencies, which was extended to the determination of wave speed in generalized 3-D Helmholtz equation [26]. The uniqueness of a coefficient inverse scattering problem with phaseless near-field data was established in [28]. We also refer to [27, 38, 39] for the inversion algorithms for the inverse medium scattering problems with modulus-only near-field data. The stability analysis for linearized near-field phase retrieval in X-ray phase contrast imaging was given in [36].

In this section, we will be concerned the uniqueness via superposition of incident point sources, which does not rely on any additional reference/interfering scatterer. Toward establishing the uniqueness for exterior inverse scattering problems, superposing certain point sources has the capability of providing more information and overcoming the indispensability of incorporating a reference ball in the previous section. By introducing a general admissible surface/curve, along with the superposition of point sources, we proved that the bounded scatterer (impenetrable obstacle or medium inclusion) and the locally perturbed half-plane (locally rough surface) could be uniquely determined from the phaseless near-field data [52]. The uniqueness can be also established by the superposition of incident point sources and phaseless far-field data, see [44]. For the uniqueness of inverse scattering by locally rough surfaces with phaseless far-field data, we refer to [47]. The crux of our study is the utilization of limited-aperture phaseless near-field data co-produced by the scatterer and point sources. A related study on the electromagnetic case can be found in [48].

The following definition of admissible surfaces is needed.

Definition 3.1 (Admissible surface) An open surface $\Gamma$ is called an admissible surface with respect to domain $\Omega$ if

(i) $\Omega\subset\mathbb{R}^3\backslash\overline{D}$ is bounded and simply-connected;

(ii) $\partial \Omega$ is analytic homeomorphic to $\mathbb{S}^2$;

(iii) $\Omega$ is non-resonant, i.e., $k^2$ is not a Dirichlet eigenvalue of $-\Delta$ in $\Omega$;

(iv) $\Gamma\subset\partial\Omega$ is a two-dimensional analytic manifold with non-vanishing measure.

For a generic point $z\in\mathbb{R}^3\backslash\overline{D}$, recall that the incident field due to the point source located at $z$ is given by

Denote by $v^s_D(x; z)$ and $v^\infty_D(\hat{x}; z)$ the near-field and far-field pattern generated by $D$ corresponding to the incident field $\Phi(x; z)$. Define

and

where $\Phi^\infty(\hat{x},z): = \mathrm{e}^{-\mathrm{i} k \hat{x}\cdot z}/(4\pi)$ is the far-field pattern of $\Phi(x; z)$.

For two generic and distinct source points $z_1, z_2\in\mathbb{R}^3\backslash\overline{D}$, we denote by

the superposition of these point sources. Then, by the linearity of direct scattering problem, the near- and far-field co-produced by $D$ and the incident wave $v^i(x; z_1,z_2)$ are respectively given by

3.1.   Bounded scatterer with far-field data

We first outline the results for recovering a bounded scatterer from phaseless far-field data. Following [44], we would like to remark that the phaseless data $|v^{\infty}_D(\hat{x},z)+\Phi^\infty(\hat{x}, z)|$ is usually obtainable in practice, while the phaseless data $|v^{\infty}_D(\hat{x},z)|$ cannot be directly measured. Hence, in our opinion, it would be more meaningful to use modulus-only data $|v^{\infty}(\hat{x},z)| = |v^{\infty}_D(\hat{x},z)+\Phi^\infty(\hat{x}, z)|$.

Definition 3.2 (Far-field datasets) For a fixed wavenumber $k>0$ and a fixed $z_0\in\mathbb{R}^3\backslash(\overline{D}\cup\Gamma)$, define

where $\Gamma$ is an admissible surface.

The phaseless inverse scattering problems are listed in Problems 3.1 and 3.2. These problems are depicted in Figure 2 as an illustration.

Problem 3.1 (Inverse obstacle scattering with far-field data). Let $D$ be the impenetrable obstacle with boundary condition $\mathscr{B}$. Given the phaseless far-field data triple $\{\mathbb{D}^{\rm F}_1, \mathbb{D}^{\rm F}_2, \mathbb{D}^{\rm F}_3\}$ due to the obstacle $D$, determine the location and shape $\partial D$ as well as the boundary condition $\mathscr{B}$ for the obstacle.

Problem 3.2 (Phaseless inverse medium scattering with far-field data). Let $D$ be the inhomogeneous medium with refractive index $n$. Given the phaseless far-field data triple $\{\mathbb{D}^{\rm F}_1, \mathbb{D}^{\rm F}_2, \mathbb{D}^{\rm F}_3\}$ due to the inhomogeneity $D$, determine the refractive index $n$ for the medium inclusion.

The following theorem tells us that Problem 3.1 (resp. Problem 3.2) admits a unique solution, namely, the geometrical and physical information of the obstacle (resp. the refractive index for the medium) can be uniquely determined from the far-field data triple $\{\mathbb{D}^{\rm F}_1, \mathbb{D}^{\rm F}_2, \mathbb{D}^{\rm F}_3\}$.

Theorem 3.1. [44] For two scatterers $D_1$ and $D_2$, suppose that the corresponding far-field patterns satisfy that

for an admissible surface $\Gamma$ and an arbitrarily fixed $z_0\in\mathbb{R}^3\backslash(\overline{D}\cup\Gamma)$. Then we have

(i) If $D_1$ and $D_2$ are two impenetrable obstacles with boundary conditions $\mathscr{B}_1$ and $\mathscr{B}_2$ respectively, then $D_1 = D_2$ and $\mathscr{B}_1 = \mathscr{B}_2$.

(ii) If $D_1$ and $D_2$ are two medium inclusions with refractive indices $n_1$ and $n_2$ respectively, then $n_1 = n_2$.

3.2.   Bounded scatterers with near-field data

We then talk about the inverse scattering problems for a bounded scatterer with near-field data. Let $D$ be the impenetrable obstacle with boundary condition $\mathscr{B}$ or the inhomogeneous medium with refractive index $n$. The mathematical formulation of forward scattering problems can be found in Section 2.

Definition 3.3 (Near-field datasets for a bounded scatterer). Assume that $\Gamma$ and $\Sigma$ are admissible surfaces with respect to $\Omega$ and $G$, respectively, such that $\overline{\Omega}\cap \overline{G} = \emptyset$. Define

for a fixed wavenumber $k>0$ and a fixed $z_0\in\mathbb{R}^3\backslash(\overline{D}\cup\Gamma\cup\Sigma)$.

With these configurations, we formulate the phaseless inverse scattering problems as follows.

Problem 3.3 (Inverse obstacle scattering with near-field data). Let $D$ be the impenetrable obstacle with boundary condition $\mathscr{B}$. Given the phaseless near-field triple $\{\mathbb{D}^{\rm N}_1, \mathbb{D}^{\rm N}_2, \mathbb{D}^{\rm N}_3\}$ due to the obstacle $D$, determine the location and shape $\partial D$ as well as the boundary condition $\mathscr{B}$ for the obstacle.

Problem 3.4 (Inverse medium scattering with near-field data). Let $D$ be the inhomogeneous medium with refractive index $n$. Given the phaseless near-field triple $\{\mathbb{D}^{\rm N}_1, \mathbb{D}^{\rm N}_2, \mathbb{D}^{\rm N}_3\}$ due to the inhomogeneity $D$, determine the refractive index $n$ for the medium inclusion.

We refer to Figure 3 for an illustration of the geometry setting of Problems 3.3 and 3.4.

Now we present the uniqueness results on phaseless inverse scattering. The following theorem asserts that Problem 3.3 and Problem 3.3 admits a unique solution respectively, namely, the geometrical and physical information of the scatterer boundary or the refractive index for the medium can be simultaneously and uniquely determined from near-field data $\{\mathbb{D}^{\rm N}_1, \mathbb{D}^{\rm N}_2, \mathbb{D}^{\rm N}_3\}$. For similar studies on phaseless inverse problem with superimposed/interfering waves, we refer to [43, 48].

Theorem 3.2. [52] Assume that $\Gamma$ and $\Sigma$ are admissible surfaces with respect to $\Omega$ and $G$ respectively, such that $\overline{\Omega}\cap \overline{G} = \emptyset$. For two scatterers $D_1$ and $D_2$, suppose that

for an arbitrarily fixed $z_0\in\mathbb{R}^3\backslash(\overline{D}\cup\Gamma\cup\Sigma)$. Then we have

(i) If $D_1$ and $D_2$ are two impenetrable obstacles with boundary conditions $\mathscr{B}_1$ and $\mathscr{B}_2$ respectively, then $D_1 = D_2$ and $\mathscr{B}_1 = \mathscr{B}_2$.

(ii) If $D_1$ and $D_2$ are two medium inclusions with refractive indices $n_1$ and $n_2$ respectively, then $n_1 = n_2$.

Remark 1. In fact Theorem 3.2 (ii) is concerned with the uniqueness for a phaseless coefficient inverse problem of determining the coefficient $n$ in the Helmholtz equation. It is well known that among all inverse problems, with phase and without phase information, the problems about finding coefficients of PDEs are the most challenging ones. So, Theorem 3.2 does not come for "free". In the sense of cardinality counting, the main simplifying assumption in it is of course the over-determination of data: the data depend on more variables than the unknown coefficient. In this regard, the unique method at this point of time allowing to prove uniqueness for coefficient inverse problems with non-overdetermined data is the Bukhgeĭm-Klibanov method [5], also see a survey in [23]. As to the phaseless case, the single paper addressing this issue is the paper [24]. What is done in [24] is that first, the phase is determined. And, second, reference to [23] is given.

3.3.   Locally perturbed half-planes

This subsection deals with the model scattering problem of a locally perturbed half-plane. Assume that the real-valued function $f\in C^2(\mathbb{R})$ has a compact support. Let $\Gamma = \{x = (x_1, x_2)\in \mathbb{R}^2 |\ x_2 = f(x_1), x_1\in \mathbb{R} \}$ be the locally perturbed curve and $D = \{x\in \mathbb{R}^2|\ x_2>f(x_1), x_1\in \mathbb{R} \}$ be the locally perturbed half-plane. Denote by $\Gamma_c = \{x\in \mathbb{R}^2 |\ x_2 = 0\}$ and then $\Gamma_p = \Gamma\backslash\Gamma_c$ is the local perturbation. For a generic point $z\in D$, the incident field $u^i$ due to the point source located at $z$ is given by

which is the 2D fundamental solution to the Helmholtz equation. Here $H_0^{(1)}$ signifies the Hankel function of the first kind and order zero. Then the scattering problem can be formulated as: find the scattered field $u^s$ such that

where $u = u^i+u^s$ denotes the total field. The boundary operators $\mathscr{B}_c$ and $\mathscr{B}_p$ appeared in (10)-(11) are defined by

where $\nu$ is the unit normal on $\Gamma$ directed into $D$, $\Gamma_{p,_D}\cup \Gamma_{p, I} = \Gamma_p$, $\Gamma_{p,_D}\cap \Gamma_{p, I} = \emptyset$, $\lambda\in C(\Gamma_{p, I})$ and $\mathrm{Im}\lambda\geq0$. It can be seen that the mixed boundary condition (13) is rather general since it covers the usual Dirichlet/sound-soft boundary condition ($\Gamma_{p, I} = \emptyset$), the Neumann/sound-hard boundary condition ($\Gamma_{p, D} = \emptyset$ and $\lambda = 0$), and the impedance boundary condition ($\Gamma_{p, D} = \emptyset$ and $\lambda\neq 0$).

The well-posedness of scattering problem (9)–(12) in various specific scenarios can be established by the variational method or the integral equation method [3, 41, 54]. We now consider the inverse scattering problem by the locally perturbed half-plane for incident point sources with limited-aperture phaseless near-field data. Analogous to Definition 3.1, the admissible curves need to be introduced [52].

Definition 3.4 (Admissible curve) An open curve $\Lambda$ is called an admissible curve with respect to domain $\Omega$ if

(i) $\overline{\Omega}\subset D$ is bounded and simply-connected;

(ii) $\partial \Omega$ is analytic homeomorphic to the unit circle $\mathbb{S}$;

(iii) $\Omega$ is non-resonant, i.e., $k^2$ is not a Dirichlet eigenvalue of $-\Delta$ in $\Omega$;

(iv) $\Lambda\subset\partial\Omega$ is a one-dimensional analytic manifold with non-vanishing measure.

Similar to the arguments in the previous section, for two generic and distinct source points $z_1, z_2\in D$, we denote by

the superposition of them. By the linearity of direct scattering problem, the total near-field is expressed by

The phaseless datasets are analogous as well.

Definition 3.5 (Near-field datasets for locally perturbed half-plane) Assume that $\Gamma$ and $\Sigma$ are admissible surfaces with respect to $\Omega$ and $G$, respectively, such that $\overline{\Omega}\cap \overline{G} = \emptyset$. Define

for a fixed wavenumber $k>0$ and a fixed $z_0\in D\backslash( \Lambda\cup\Sigma)$.

The formulation of the phaseless inverse scattering problems under consideration is given by the following problem, which is geometrically illustrated in Figure 4.

Problem 3.5 (Inverse scattering by locally perturbed half-planes). Let $\Gamma$ be the locally perturbed curve with boundary condition $\mathscr{B}_c$ and $\mathscr{B}_p$. Assume that $\Lambda$ and $\Sigma$ are admissible curves with respect to $\Omega$ and $G$, respectively. Given the near-field data triple $\{\mathbb{D}^{\rm NL}_1, \mathbb{D}^{\rm NL}_2, \mathbb{D}^{\rm NL}_3\}$, determine the locally perturbed curve $\Gamma$ as well as the boundary condition $\mathscr{B}_c$ and $\mathscr{B}_p$.

Let $\Gamma_j = \{x\in \mathbb{R}^2: x_2 = f_j(x_1), x_1\in \mathbb{R} \}$ be the locally perturbed curve with compactly supported function $f_j\in C^2(\mathbb{R}), j = 1,2$. Denote by $D_j = \{x\in \mathbb{R}^2: x_2>f_j(x_1), x_1\in \mathbb{R} \}$ the corresponding domain above $\Gamma_j, j = 1,2$, and by $D_0 = D_1\cap D_2$. Denote by $u_j^s$ and $u_j$ the scattered field and the total field produced by $\Gamma_j$, respectively, corresponding to the incident field $u^i(x; z), j = 1,2$. The following theorem shows that Problem 3.5 admits a unique solution, in other words, the geometrical and physical information of the locally perturbed plane can be simultaneously and uniquely determined from the phaseless near-field data $\{\mathbb{D}^{\rm NL}_1, \mathbb{D}^{\rm NL}_2, \mathbb{D}^{\rm NL}_3\}$.

Theorem 3.3. [52] Let $\Gamma_1$ and $\Gamma_2$ be two locally perturbed curves with boundary conditions $\mathscr{B}_{c,1}, \mathscr{B}_{p,1}$ and $\mathscr{B}_{c,2}, \mathscr{B}_{p,2}$, respectively. Assume that $\Lambda$ and $\Sigma$ are admissible curves with respect to $\Omega$ and $G$, respectively, such that $\overline{\Omega}\subset\subset D_0$, $\overline {G}\subset\subset D_0$ and $\overline{\Omega}\cap \overline {G} = \emptyset$. Suppose that the corresponding total near-fields satisfy that

for an arbitrarily fixed $z_0\in D_0 \backslash(\Lambda\cup\Sigma)$. Then we have $\Gamma_1 = \Gamma_2$, $\mathscr{B}_{c,1} = \mathscr{B}_{c,2}$ and $\mathscr{B}_{p,1} = \mathscr{B}_{p,2}$.

4.   An interior problem

In this last section, a recent result concerning phaseless inverse cavity scattering problems will be presented [55]. To our knowledge, this is the first uniqueness result in inverse cavity scattering problems with phaseless data. As mentioned in the previous sections, the reference ball approach and the superposition of emanating point sources are crucial ingredients for the success of establishing uniqueness for phaseless exterior inverse scattering problems. For the interior problem of an impenetrable cavity, it is reasonable to combine these two strategies in order to guarantee the uniqueness. To this end, we bring together four main ingredients in our analysis: utilization of the reference ball technique, superposition of point sources, the reciprocity relations and the singularity of the total fields. In particular, an impedance reference ball plays an exceptionally important role in our analysis and thus irreplaceable. We refer the interested reader to [55] for more details.

We first formulate the model cavity scattering problem. Let $D \subset\mathbb{R}^3$ be an open and simply connected domain with $C^2$ boundary $\partial D$. Then, the interior scattering problem for cavities can be stated as: given the incident field $u^i$, find the scattered field $u^s$ which satisfies the boundary value problem:

where $u = u^i+u^s$ denotes the total field and $k>0$ is the wavenumber. Here $\mathscr{B}$ in (15) is the boundary operator defined by

where $\nu$ is the unit outward normal to $\partial D$, and $\lambda\in C(\partial D)$ is the impedance function satisfying $\Im(\lambda)\geq 0$. This boundary condition 16 covers the Dirichlet/sound-soft type, the Neumann/sound-hard type ($\lambda = 0$), and the impedance type ($\lambda\neq 0$). The existence of a solution to the direct scattering problem 14–15 is well known (see, e.g., [12]).

To introduce the interior inverse scattering problem for incident point sources with limited-aperture phaseless near-field data, we again employ a reference ball $B$ as an extra artificial object to the scattering system such that $\overline{B}\subset\subset D$ with the impedance boundary condition

where $\lambda_0$ is a positive constant. Similar to the previous sections, we also need Definition 3.3 as an interior admissible surface.

The superposition of point sources are almost identical to the previous section. For a generic point $z\in D\backslash\overline{B}$, the incident field $u^i$ due to the point source located at $z$ is given by $u^i (x; z) = \Phi(x; z),\ x\in D\backslash(\overline{B}\cup\{z\})$. Denote by $u^s(x; z)$ the near-field generated by $D$ and $B$ corresponding to the incident field $u^i(x; z)$. Let $u(x; z) = u^s(x; z)+u^i(x; z), x\in D\backslash(\overline{B}\cup\{z\})$ be the total field.

For two generic and distinct source points $z_1, z_2\in D\backslash\overline{B}$, we denote by

the superposition of these point sources. Once again, by the linearity of direct scattering problem, the near-field co-generated by $D$, $B$ and the incident wave $u^i(x; z_1,z_2)$ is written as

Definition 4.1 (Near-field datasets for the cavity) Assume that $\Gamma$ and $\Sigma$ are admissible surfaces with respect to $\Omega$ and $G$, respectively, such that $\overline{\Omega}\subset\subset G$ and $\overline {G}\subset\subset D\backslash\overline{B}$. Define

for a fixed wavenumber $k>0$ and a fixed $z_0\in D\backslash(\overline{B}\cup\Gamma\cup\Sigma)$.

With these preparations, the inverse problem under consideration is formulated as the following phaseless interior problems. The illustration is shown in Figure 5.

Problem 4.1 (Inverse scattering for cavities). Let $D$ be the impenetrable cavity with boundary condition $\mathscr{B}$. Given the data triple $\{\mathbb{D}^{\rm C}_1, \mathbb{D}^{\rm C}_2, \mathbb{D}^{\rm C}_3\}$, determine the location and shape $\partial D$ as well as the boundary condition $\mathscr{B}$ for the cavity.

A rigorous proof shows that Problem 4.1 admits a unique solution, namely, the geometrical and physical information of the cavity can be simultaneously and uniquely determined from the phaseless data $\{\mathbb{D}^{\rm C}_1, \mathbb{D}^{\rm C}_2, \mathbb{D}^{\rm C}_3\}\}$. To sum up, we end this section by the following uniqueness theorem.

Theorem 4.1. [55] Let $D_1$ and $D_2$ be two impenetrable cavities with boundary conditions $\mathscr{B}_1$ and $\mathscr{B}_2$, respectively. Assume that $\Gamma$ and $\Sigma$ are admissible surfaces with respect to $\Omega$ and $G$, respectively, such that $\overline{\Omega}\subset\subset G$ and $\overline {G}\subset\subset (D_1\cap D_2)\backslash\overline{B}$. Suppose that the corresponding near-fields satisfy that

for an arbitrarily fixed $z_0\in (D_1\cap D_2) \backslash(\overline{B}\cup\Gamma\cup\Sigma)$. Then we have $D_1 = D_2$ and $\mathscr{B}_1 = \mathscr{B}_2$.

5.   Conclusion

The phaseless inverse scattering problems are theoretically and practically important. Nevertheless, being lack of phase information in measured data makes it difficult to resolve the uniqueness issue. This paper briefly reviews some recent developments in this field by summarizing the results obtained so far in a series of our publications. Intrinsically speaking, our proofs of the aforementioned uniqueness results hinge on the crux of decoupling the phaseless data via a polarization identity and with an introduction of an additional degree of freedom in the measurement, e.g., with a reference ball, with another point source/plane wave or measurement surface. Consequently, the desired uniqueness can be established by the existing uniqueness results with phase information.

In our view, there are still some open problems in this intriguing and challenging field. Therefore, we finally propose several more topics from our perspective that are interesting for further investigation.

Problem 5.1. Can one uniquely determine the obstacle using the phaseless far-field data due to incident plane waves with distinct directions with a fixed wavenumber and a reference ball?

Although there exist numerical results showing that the target obstacle can be accurately reconstructed without any superposition of waves [15], a rigorously mathematical answer to Problem 5.1, to our best knowledge, is still open.

Problem 5.2. Given the a priori information that the obstacle is of general sound-soft or sound-hard polygon or polyhedron type, can the obstacle be uniquely determined using less phaseless data due to point sources/plane waves in conjunction with a reference object?

Under the assumption of physical optics approximation, one can indeed determine the exterior unit normal vector of each side/face of the polygonal obstacle from phaseless backscattering data corresponding to a few incident plane waves with suitably chosen incoming directions [31, 32]. By incorporating a reference ball into the scattering system, the phaseless data triple such as the one defined in Definition 2.3 would probably involve redundant data for recovering a polygon/polyhedron. Thus it would be interesting to investigate the proper choice of the dataset so as to establish the corresponding uniqueness.

Problem 5.3. For the determinations of periodic structures in inverse diffraction grating problems, is it possible to develop similar reference object techniques to obtain the uniqueness from associated amplitude information of scattering data?

A number of inversion methods have been proposed to numerically reconstruct the grating profiles from phaseless data, e.g., [2, 4, 57] but the theoretical study on the uniqueness seems to be very limited. Whilst various inversion approaches for imaging bounded scatterers can be modified to find unbounded structures, it is hence plausible to consider devising novel methodologies with the techniques presented in this paper.

Acknowledgments

We would like to thank Prof. Hongyu Liu for his thoughtful and valuable suggestions on improving the details in this paper. We would also like to thank the anonymous referees for their insightful comments and constructive suggestions on our manuscript, which have led to significant improvements on the presentation of this paper.