Scattering of Partially Coherent Vector Beams by a Deterministic Medium Having Parity-Time Symmetry

Abstract

We study the scattering properties of the partially coherent vector beams with the deterministic media having the classic symmetric and parity-time (PT) symmetric scattering potential functions. The closed-form expressions for the intensity and polarization matrix of the far-zone scattered field are obtained, under first-order Born approximation, when the partially coherent vector beams are taken to be radially polarized and the deterministic media are assumed as the four-point scatterers. We demonstrate both analytically and numerically that the far-zone scattered field becomes noncentrosymmetric and the directionality appears in the scattering pattern when the scattering potential function is switched from classic symmetry to PT symmetry. We show the effect of spatial coherence of the incident partially coherent vector beam on the directionality in scattering. We find that by turning the symmetry property of the spatial coherence function of the incident beam, i.e., into PT symmetry, the directionality in the far-zone scattering can be suppressed or enhanced, depending on the joint effect from the symmetry of the scattering potential and the symmetry of the spatial coherence. Our findings may be useful in the application of dynamic control of the directionality in light scattering.

1. Introduction

Optical coherence and polarization are two fundamental properties of light fields, which however had been treated separately for historical reasons [1]. Today, it is well understood that optical coherence and polarization are closely connected. The electromagnetic theory of optical coherence [2,3,4,5,6], developed by Wolf, Gori, Friberg, and others, has shown that polarization can be viewed as a manifestation of the correlation involving the components of fluctuating electric field at a single point. Meanwhile, the optical coherence for a partially coherent vector light field can be regarded as the correlation involving the components of the fluctuating electric field between pairs of space-time or space-frequency points. The definitions for the degree of polarization [7] and electromagnetic degree of coherence [4], in some sense, are consistent. The changes of the degree of polarization and the polarization state of the partially coherent vector beams during propagation in free space can be viewed as direct evidence of the fact that coherence and polarization are interrelated [8,9,10]. Since the development of the unified theories for the optical coherence and polarization in time and frequency domains by Gori and Wolf, numerous efforts have been made on elucidating the fundamental properties and outline the potential applications of partially coherent vector beams, e.g., in free-space optical communications, remote sensing, optical imaging, and particle trapping [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

Meanwhile, optical scattering, as one of the most common optical phenomena in nature, has been used in many applications, including optical imaging, optical trapping, crystallography, and atmospheric optics [32]. However, due to the omnipresent randomness in electromagnetic fields and scattering media [33], the effect of optical coherence on light scattering may not be ignored, and sometimes it plays a key role in determining the scattered field [34]. Extensive attention has been paid to the effect of optical coherence of the incident beam on the light scattering by the deterministic and random media [35,36,37,38,39,40,41,42,43,44]. It has been shown that not only the spatial coherence width, but also the spatial coherence structure of the incident beam has significate impact on the scattered field. For example, by controlling the spatial coherence structure of the incident beam into Bessel-correlated shape, a cone-like scatterred field can be generated [45,46]. The joint effect from optical coherence and polarization of the partially coherent vector incident beam on the light scattering has also been studied extensively [47,48,49,50,51,52] since the scattering theory of scalar stochastic fields was generalized to electromagnetic domain by Tong and Korotkova in 2010 [53].

More recently, the scattering of partially coherent light by a deterministic medium with parity-time (PT) symmetric scattering potential has been studied [54]. The notion of PT symmetry is initially proposed in quantum mechanics that generalizes the real Hermitian systems into the complex domain [55,56]. The first experimental observation of the PT symmetry induced phenomena extends the study into the field of optics [57]. The PT symmetry in optics has found numerous applications, such as in unidirectional invisibility, anisotropic transmission resonances, single-particle detection, microlaser, robust wireless power transfer, and topological insulators [58,59]. The partially coherent Gaussian Schell-model beam scattered by the PT deterministic media, such as the two-point scatterers [54,60,61,62], localized continuous materials [63], and periodic materials with loss and gain [64] have been studied. It has been shown that the non-Hermitian property of the scattering potential induces anomalous spectral and spatial changes in the partially coherent scattered fields. More recently, Brandão and Korotkova introduced the concept of PT random media and gave a theoretical framework for the scattering of scalar radiation from the PT random media [65]. The detailed analysis on the light scattering from the random PT-symmetric particulate collections has been given by the same authors [66]. However, all these studies were confined to cases in which the incident field is scalar [65,66,67]. The effect of partially coherent vector beam on the light scattering with PT-symmetric scatterer has not yet been addressed.

In this work, we study the optical scattering of the partially coherent vector beam by a deterministic medium with its scattering potential having PT symmetry. We derive the closed-form expressions for the polarization matrix of the far-zone scattered field when the partially coherent vector beam is taken to be radially polarized and the scatterer is assumed to be weak. We show both analytically and numerically that the directionality appears in intensity and polarization of the far-zone scattered field by turning the scattering potential from classical symmetry to PT symmetry. The effect of spatial coherene wdith and spatial coherence symmetry is also discussed in detail. We find the directionality in far-zone scattering can be suppressed or enhanced by controlling the symmetry property of the incident partially coherent vector beam.

2. Method

2.1. Scattering Theory for Partially Coherent Vector Beam

We first consider a vector beam incident on a linear isotropic and nonmagnetic medium in a finite domain V. The scattered field at a point specified by position vector $r\mathbf{s}$ with $\mathbf{s}=(s_x,s_y,s_z)$ being a unit vector can be expressed in the form [32]

$${\mathbf{E}}^{\left(s\right)}(r\mathbf{s},\omega )=\mathbf{\nabla }\times \mathbf{\nabla }\times {\mathsf{\Pi }}_e(r\mathbf{s},\omega ),$$

(1)

where the superscript s denotes the scattered field, $\omega$ is the (angular) frequency, and ${\mathsf{\Pi }}_e(r\mathbf{s},\omega )$ is the electric Hertz potential, which is defined by the formula,

$${\mathsf{\Pi }}_e(r\mathbf{s},\omega )={\int }_V\mathbf{P}(\mathbf{r},\omega )G\left(\right|r\mathbf{s}-\mathbf{r}|,\omega ){\mathrm{d}}^3\mathbf{r}.$$

(2)

Above $\mathbf{P}(\mathbf{r},\omega )$ denotes the polarization of the medium and $G\left(\right|r\mathbf{s}-\mathbf{r}|,\omega )$ is a Green’s function that connects the polarization of the medium with the scattered field. We assume that the medium is a weak scatterer. Thus, the scattering problem can be treated within the framework of the first-order Born approximation. In such condition, the polarization of the medium can be expressed as

$$\mathbf{P}(\mathbf{r},\omega )=\frac{1}{k^2}F(\mathbf{r},\omega ){\mathbf{E}}^{\left(i\right)}(\mathbf{r},\omega ),$$

(3)

where $k=\omega /c$ with c being the speed of light in vacuum, ${\mathbf{E}}^{\left(i\right)}(\mathbf{r},\omega )$ is the electric field of the incident vector beam, and $F(\mathbf{r},\omega )$ denotes the scattering potential of the medium, which has the following relation with the refractive index, i.e.,

$$F(\mathbf{r},\omega )=\frac{k^2}{4\pi }[n^2(\mathbf{r},\omega )-1],$$

(4)

where $n(\mathbf{r},\omega )$ denotes the (complex) index of refractive. Here, we note the position $\mathbf{r}$ in the scattering potential function is within the scattering volume V. Otherwise, $F(\mathbf{r},\omega )=0$. Moreover, under the far-zone approximation, the Green’s function in free space can be written as [5]

$$G\left(\right|r\mathbf{s}-\mathbf{r}|,\omega )=\frac{{\mathrm{e}}^{\mathrm{i}kr}}{r}exp(-\mathrm{i}k\mathbf{s}·\mathbf{r}).$$

(5)

It follows from Equations (1)–(5) that the far-zone scattered field under first-order Born approximation can be expressed as

$${\mathbf{E}}^{\left(s\right)}(r\mathbf{s},\omega )=-\frac{exp\left(\mathrm{i}kr\right)}{r}\left\{\mathbf{s}\times \left[\mathbf{s}\times {\int }_{V}F(\mathbf{r},\omega ){\mathbf{E}}^{\left(i\right)}(\mathbf{r},\omega )exp(-\mathrm{i}k\mathbf{s}·\mathbf{r}){\mathrm{d}}^3\mathbf{r}\right]\right\}.$$

(6)

We now take the electric field ${\mathbf{E}}^{\left(s\right)}(r\mathbf{s},\omega )$ in Equation (6) to represent a field realization of the scattered field induced by a partially coherent incident vector beam. The $3\times 3$ spectral electric coherence matrix, characterizing all the second-order statistical properties of a partially coherent scattered field, can be written as [6]

$${\mathbf{W}}^{\left(s\right)}(r{\mathbf{s}}_1,r{\mathbf{s}}_2,\omega )=\langle {\mathbf{E}}^{\left(s\right)*}(r{\mathbf{s}}_1,\omega ){\mathbf{E}}^{\left(s\right)\mathrm{T}}(r{\mathbf{s}}_2,\omega )\rangle ,$$

(7)

where the asterisk, superscript T, and angular brackets denote complex conjugate, matrix transpose, and ensemble average, respectively. Taking Equation (6) into Equation (7) and doing the proper modifications, we obtain that

$$\begin{array}{cc}\hfill {\mathbf{W}}^{\left(s\right)}(r{\mathbf{s}}_1,r{\mathbf{s}}_2,\omega )=& \frac{1}{r^2}{\int }_V{\int }_V\mathbf{D}({\theta }_1,{\varphi }_1)\circ {\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )\circ {\mathbf{D}}^{\mathrm{T}}({\theta }_2,{\varphi }_2)F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )\hfill \\ \hfill & \times exp\left[\mathrm{i}k({\mathbf{s}}_1·{\mathbf{r}}_1-{\mathbf{s}}_2·{\mathbf{r}}_2)\right]{\mathrm{d}}^3{\mathbf{r}}_1{\mathrm{d}}^3{\mathbf{r}}_2,\hfill \end{array}$$

(8)

where ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )=\langle {\mathbf{E}}^{\left(i\right)*}({\mathbf{r}}_1,\omega ){\mathbf{E}}^{\left(i\right)\mathrm{T}}({\mathbf{r}}_2,\omega )\rangle$ is the $3\times 3$ coherence matrix for the partially coherent incident vector beam, the symbol `∘’ denotes the product of matrices, and the matrix $\mathbf{D}(\theta ,\varphi )$ can be written as

$$\mathbf{D}(\theta ,\varphi )=\left[\begin{array}{ccc}1-s_x^2& -s_x{s}_y& -s_x{s}_z\\ -s_y{s}_x& 1-s_y^2& -s_y{s}_z\\ -s_z{s}_x& -s_z{s}_y& 1-s_z^2\end{array}\right].$$

(9)

In the spherical coordinate frame, $s_x=sin\theta cos\varphi$, $s_y=sin\theta sin\varphi$, and $s_z=cos\theta$ with $\theta$ being the polar angle and $\varphi$ being the azimuthal angle. To simplify the representation in Equation (8), we define the following function

$$\mathbf{M}({\mathbf{s}}_1,{\mathbf{s}}_2,\omega )={\int }_V{\int }_V{\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )exp\left[\mathrm{i}k({\mathbf{s}}_1·{\mathbf{r}}_1-{\mathbf{s}}_2·{\mathbf{r}}_2)\right]{\mathrm{d}}^3{\mathbf{r}}_1{\mathrm{d}}^3{\mathbf{r}}_2.$$

(10)

Thus, Equation (8) can be simplified as

$${\mathbf{W}}^{\left(s\right)}(r{\mathbf{s}}_1,r{\mathbf{s}}_2,\omega )=\frac{1}{r^2}\mathbf{D}({\theta }_1,{\varphi }_1)\circ \mathbf{M}({\mathbf{s}}_1,{\mathbf{s}}_2,\omega )\circ {\mathbf{D}}^{\mathrm{T}}({\theta }_2,{\varphi }_2).$$

(11)

The $3\times 3$ polarization matrix of the far-zone scattered field can be obtained by letting $r{\mathbf{s}}_1=r{\mathbf{s}}_2=r\mathbf{s}$ in Equation (11), i.e.,

$${\mathsf{\Phi }}^{\left(s\right)}(r\mathbf{s},\omega )=\frac{1}{r^2}\mathbf{D}(\theta ,\varphi )\circ \mathbf{M}(\mathbf{s},\mathbf{s},\omega )\circ {\mathbf{D}}^{\mathrm{T}}(\theta ,\varphi ),$$

(12)

where

$$\mathbf{M}(\mathbf{s},\mathbf{s},\omega )={\int }_V{\int }_V{\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )exp[\mathrm{i}k\mathbf{s}·({\mathbf{r}}_1-{\mathbf{r}}_2)]{\mathrm{d}}^3{\mathbf{r}}_1{\mathrm{d}}^3{\mathbf{r}}_2.$$

(13)

2.2. PT Symmetry in Scattering Medium

The notion of PT symmetry is initially proposed in quantum mechanics that generalizes the real Hermitian systems into the complex domain. Due to the similar mathematical structure of the Schrödinger equation for electrons and Maxwell’s theory for light under certain conditions, the complex potential in quantum mechanics can be viewed to be equivalent to a complex refractive index in optics [58]. The positive and negative signs of the imaginary part of the refractive index represent the optical gain and loss in the material, respectively. When the gain and loss are balanced equally, i.e., the imaginary part of the refractive index has odd symmetry,

$$n^{″}(-\mathbf{r})=-n^{″}\left(\mathbf{r}\right),$$

(14)

the material satisfies the PT symmetry. Here, double prime denotes the imaginary part. The real part of the refractive index of the PT-symmetric material obeys the even symmetry, i.e., $n^{\prime }(-\mathbf{r})=n^{\prime }\left(\mathbf{r}\right)$, where the prime denotes the real part. The complex index of refractive for the PT-symmetry medium thus can be expressed as

$$n(-\mathbf{r})=n^{*}\left(\mathbf{r}\right).$$

(15)

For the medium with classical symmetry, the refractive index obeys the geometrical symmetry, i.e., $n(-\mathbf{r})=n\left(\mathbf{r}\right)$.

Taking the symmetry property of the refractive index into scattering potential function (Equation (4)), we obtain that for the PT-symmetric material, the scattering potential obeys

$$F(-\mathbf{r},\omega )=F^{*}(\mathbf{r},\omega ),$$

(16)

and for the classic-symmetric material, the scattering potential obeys

$$F(-\mathbf{r},\omega )=F(\mathbf{r},\omega ).$$

(17)

The visualizations of the real and imaginary parts of the typical scattering potential functions having classic and PT symmetries are shown in Figure 1. It is found that both the real parts have even symmetry, while the imaginary part for the classic-symmetric potential is even and for the PT-symmetric potential is odd. We note that the conditions for the PT-symmetric potential function are exactly same for a Hermitian function [68].

We now take the symmetry property of the scattering potential function into the far-zone polarization matrix shown in Equation (12). It is found that the symmetry property is determined not only by the symmetry of $F(\mathbf{r},\omega )$, but also by the symmetry of the coherence matrix for the partially coherent incident vector beam.

2.3. PT Symmetry in Partially Coherent Vector Beam

We now discuss the symmetry property for the partially coherent vector beam. The random field realization ${\mathbf{E}}^{\left(i\right)}(\mathbf{r},\omega )$ for the partially coherent incident vector beam can be written as

$${\mathbf{E}}^{\left(i\right)}(\mathbf{r},\omega )={\mathbf{E}}^{\left(i\right)\prime }(\mathbf{r},\omega )+\mathrm{i}{\mathbf{E}}^{\left(i\right)″}(\mathbf{r},\omega ),$$

(18)

where prime and double prime denote the real and imaginary parts, respectively, as before. When the field realization obeys the PT symmetry, we have [69]

$${\mathbf{E}}^{\left(i\right)″}(-\mathbf{r},\omega )=-{\mathbf{E}}^{\left(i\right)″}(\mathbf{r},\omega ),$$

(19)

and

$${\mathbf{E}}^{\left(i\right)\prime }(-\mathbf{r},\omega )={\mathbf{E}}^{\left(i\right)\prime }(\mathbf{r},\omega ),$$

(20)

or in the complex form,

$${\mathbf{E}}^{\left(i\right)}(-\mathbf{r},\omega )={\mathbf{E}}^{{\left(i\right)}^{*}}(\mathbf{r},\omega ).$$

(21)

For the classical symmetric field realization, we have ${\mathbf{E}}^{\left(i\right)}(-\mathbf{r},\omega )={\mathbf{E}}^{\left(i\right)}(\mathbf{r},\omega )$. Taking the symmetry property of the field realization into the coherence matrix for the partially coherent vector light, we obtain that for the PT symmetry, the coherence matrix must meet the condition,

$${\mathbf{W}}^{\left(i\right)}(-{\mathbf{r}}_1,-{\mathbf{r}}_2,\omega )={\mathbf{W}}^{\left(i\right)*}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega ),$$

(22)

and for the classic-symmetry partially coherent vector light

$${\mathbf{W}}^{\left(i\right)}(-{\mathbf{r}}_1,-{\mathbf{r}}_2,\omega )={\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega ).$$

(23)

Considering the hermiticity of the coherence matrix for the partially coherent vector beam, i.e., ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_2,{\mathbf{r}}_1,\omega )={\mathbf{W}}^{\left(i\right)*}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )$, we further obtain that

$${\mathbf{W}}^{\left(i\right)}(-{\mathbf{r}}_1,-{\mathbf{r}}_2,\omega )={\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_2,{\mathbf{r}}_1,\omega ),$$

(24)

for the PT-symmetric coherence matrix, and

$${\mathbf{W}}^{\left(i\right)}(-{\mathbf{r}}_1,-{\mathbf{r}}_2,\omega )={\mathbf{W}}^{\left(i\right)*}({\mathbf{r}}_2,{\mathbf{r}}_1,\omega ).$$

(25)

When the symmetry properties for both the scattering potential function and the coherence matrix are obtained, we now consider the symmetry property for the polarization properties of the far-zone scattered field. Taking Equations (16), (17), (22) and (23) into Equation (12), we obtain that when both the scattering potential function and the coherence matrix obey the classic symmetry,

$${\mathsf{\Phi }}^{\left(s\right)}(-r\mathbf{s},\omega )={\mathsf{\Phi }}^{\left(s\right)}(r\mathbf{s},\omega ),$$

(26)

which indicates that the far-zone polarization properties are centrosymmetric.

However, for the PT symmetry cases, the far-zone polarization state is obtained as

$${\mathsf{\Phi }}^{\left(s\right)}(-r\mathbf{s},\omega )={\mathsf{\Phi }}^{\left(s\right)\prime }(r\mathbf{s},\omega )-\mathrm{i}{\mathsf{\Phi }}^{\left(s\right)″}(r\mathbf{s},\omega ),$$

(27)

where the terms ${\mathsf{\Phi }}^{\left(s\right)\prime }(r\mathbf{s},\omega )$ and ${\mathsf{\Phi }}^{\left(s\right)″}(r\mathbf{s},\omega )$ are induced by the real and imaginary parts of the coherence matrix ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )$ when the incident partially coherent beam obeys the PT symmetry, by the real and imaginary parts of the potential correlation $F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )$ when the scatterer obeys the PT symmetry, and by the real and imaginary parts of the function $\left[{\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )\right]$ when both the incident beam and the scatterer obey the PT symmetry. We note that in the first and the second cases, ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )$ and $F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )$ are complex functions, indicating that the imaginary part ${\mathsf{\Phi }}^{\left(s\right)″}(r\mathbf{s},\omega )$ is not vanished. Thus, the polarization properties of far-zone scattered field is non-centrosymmetic in the first two cases. In the last case, when both the incident beam and the scatterer obey the PT symmetry, the function ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )$ may be a real function (in the case when ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )$ and $F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )$ are complex conjugate, i.e., with opposite phases), which will make the imaginary part ${\mathsf{\Phi }}^{\left(s\right)″}(r\mathbf{s},\omega )$ vanishes. In such case, the far-zone scattered field becomes centrosymmetic again. In the last case, the non-centrosymmetic property can also be enhanced by modulating ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )$ and $F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )$ with the phases having the same direction.

3. Results

In this section, we give the simulation examples to show the symmetry properties for the intensity and polarization properties of the far-zone scattered field. Since the far-zone scattered field is a three-component field, the nine generalized Stokes parameters could be used to characterize the three-dimensional polarization properties. The nine generalized Stokes parameters can be expressed as [7]

$$\begin{array}{c}{\mathsf{\Lambda }}_0(r\mathbf{s},\omega )=\mathrm{tr}\left[{\mathsf{\Phi }}^{\left(s\right)}(r\mathbf{s},\omega )\right],\hfill \end{array}$$

(28)

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} {\mathsf{\Lambda }}_j(r\mathbf{s},\omega )=\frac{3}{2}\mathrm{tr}\left[{\mathbf{\lambda }}_j{\mathsf{\Phi }}^{\left(s\right)}(r\mathbf{s},\omega )\right],j=1,2,...,8,\hfill \end{array}$$

(29)

where the matrices ${\mathbf{\lambda }}_j$ are the Gell–Mann matrices or the eight generators of the SU(3) symmetry group. The matrices are Hermitian, trace orthogonal, and linearly independent. They can be expressed as

$$\begin{array}{cc}\hfill {\mathbf{\lambda }}_1& =\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right],{\mathbf{\lambda }}_2=\left[\begin{array}{ccc}0& -\mathrm{i}& 0\\ \mathrm{i}& 0& 0\\ 0& 0& 0\end{array}\right],{\mathbf{\lambda }}_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right],{\mathbf{\lambda }}_4=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],\hfill \\ \hfill {\mathbf{\lambda }}_5& =\left[\begin{array}{ccc}0& 0& -\mathrm{i}\\ 0& 0& 0\\ \mathrm{i}& 0& 0\end{array}\right],{\mathbf{\lambda }}_6=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],{\mathbf{\lambda }}_7=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& -\mathrm{i}\\ 0& \mathrm{i}& 0\end{array}\right],{\mathbf{\lambda }}_8=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -2\end{array}\right].\hfill \end{array}$$

(30)

The first Stokes parameter ${\mathsf{\Lambda }}_0(r\mathbf{s},\omega )$ is proportional to the total intensity of the field, ${\mathsf{\Lambda }}_1(r\mathbf{s},\omega )$ and ${\mathsf{\Lambda }}_2(r\mathbf{s},\omega )$ play a role analogous to parameters $S_2$ and $S_3$ in the 2D Stokes parameters, $[{\mathsf{\Lambda }}_4(r\mathbf{s},\omega ),{\mathsf{\Lambda }}_5(r\mathbf{s},\omega )]$ and $[{\mathsf{\Lambda }}_6(r\mathbf{s},\omega ),{\mathsf{\Lambda }}_7(r\mathbf{s},\omega )]$ play the same role, but in the $xz$ and $yz$ planes, respectively, ${\mathsf{\Lambda }}_3(r\mathbf{s},\omega )$ is analogous to $S_1$ in the 2D Stokes parameters, and ${\mathsf{\Lambda }}_8(r\mathbf{s},\omega )$ represents the sum of the excesses in the intensity in the x and y directions over that in the z direction. The 3D degree of polarization P can be obtained by

$$P^2(r\mathbf{s},\omega )=\frac{1}{3}\frac{{\sum }_{j=1}^8{\mathsf{\Lambda }}_j^2(r\mathbf{s},\omega )}{{\mathsf{\Lambda }}_0^2(r\mathbf{s},\omega )}.$$

(31)

For a genuine 3D field, the degree of polarization is bounded as $0\le P(r\mathbf{s},\omega )\le 1$ with the upper and lower limits corresponding to fully polarized and completely 3D unpolarized, respectively. Meanwhile for a 2D field, the degree of polarization is bounded as $0.5\le P(r\mathbf{s},\omega )\le 1$ with the lower limit corresponding to the 2D unpolarized field.

3.1. Four-Point Scatterer

In this work, our aim is to show the effect of scatterer’s symmetry on the polarization properties of the far-zone scattered field. Therefore, the polarization of the incident beam located on the scatterer must be nonuniform. For a radially polarized incident beam we consider in the simulation (see Section 3.2), the fields on the two-point positions of a dipole scatterer considered in Refs. [54,60,61,62] have the same polarization. Thus, in this work we consider that the scatterer is composed by two pairs of point dipole. The four point scatterers are placed at ${\mathbf{P}}_1=(a,a)$, ${\mathbf{P}}_2=(a,-a)$, ${\mathbf{P}}_3=(-a,a)$, and ${\mathbf{P}}_4=(-a,-a)$, respectively. The scattering potential function thus can be expressed as

$$\begin{array}{c}F(\mathbf{r},\omega )=\delta \left(z\right)[(\sigma +\mathrm{i}\gamma )\delta (x-a)\delta (y-a)+(\sigma -\mathrm{i}\gamma )\delta (x-a)\delta (y+a)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+(\sigma +\mathrm{i}\gamma )\delta (x+a)\delta (y-a)+(\sigma -\mathrm{i}\gamma )\delta (x+a)\delta (y+a)],\end{array}$$

(32)

where $\delta (·)$ denotes a Dirac delta function, $\sigma$ and $\gamma$ are real parameters, which represent the gain and loss properties of the scatterer, and the real factor a controls the distance between the point scatterers. We note that the condition $F(-\mathbf{r},\omega )=F^{*}(\mathbf{r},\omega )$ holds in Equation (32), which indicates that the four-point scatterer described by Equation (32) obeys the PT symmetry (in the case when $\gamma \ne 0$). In the case when $\gamma =0$, $F(-\mathbf{r},\omega )=F(\mathbf{r},\omega )$ holds, indicating the scatterer becomes a classic-symmetric material.

3.2. Partially Coherent Vector Beam with Spatially Nonuniform Radial Polarization

The incident partially coherent vector beam is considered as a partially coherent vector beam with radial polarization. Thus, its coherence matrix can be expressed as [18]

$${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )=\left[\begin{array}{ccc}{\tau }_x^{*}\left({\mathbf{r}}_1\right){\tau }_x\left({\mathbf{r}}_2\right)& {\tau }_x^{*}\left({\mathbf{r}}_1\right){\tau }_y\left({\mathbf{r}}_2\right)& 0\\ {\tau }_y^{*}\left({\mathbf{r}}_1\right){\tau }_x\left({\mathbf{r}}_2\right)& {\tau }_y^{*}\left({\mathbf{r}}_1\right){\tau }_y\left({\mathbf{r}}_2\right)& 0\\ 0& 0& 0\end{array}\right]\mu ({\mathbf{r}}_1,{\mathbf{r}}_2)exp[-ik{\mathbf{s}}_0·({\mathbf{r}}_2-{\mathbf{r}}_1)],$$

(33)

where

$$\begin{array}{cc}\hfill {\tau }_x\left(\mathbf{r}\right)& =\frac{x}{w_0}exp\left(-\frac{x^2+y^2}{w_0^2}\right),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\tau }_y\left(\mathbf{r}\right)& =\frac{y}{w_0}exp\left(-\frac{x^2+y^2}{w_0^2}\right),\hfill \end{array}$$

(35)

with $w_0$ being the beam waist, ${\mathbf{s}}_0=(0,0,1)$ being a unit vector representing the normally incident direction of the beam, and $\mu ({\mathbf{r}}_1,{\mathbf{r}}_2)$ denotes the degree of coherence of the partially coherent incident beam at points ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$. In the simulation, we consider two types of degrees of coherence. One obeys the classical symmetry and the other obeys the PT symmetry.

3.2.1. Degree of Coherence with Classical Symmetry

The Gaussian Schell-model (GSM) correlation function is a typical coherence function with classical symmetry. Its degree of coherence can be expressed as

$$\mu ({\mathbf{r}}_1,{\mathbf{r}}_2)=exp\left[-\frac{{(x_1-x_2)}^2+{(y_1-y_2)}^2}{2{\xi }_0^2}\right],$$

(36)

where ${\xi }_0$ denotes the spatial coherence width of the beam.

Substituting Equations (32)–(36) into Equation (13), we obtain that the $\mathbf{M}(\mathbf{s},\mathbf{s},\omega )$ matrix can be expressed analytically as

$$\begin{array}{cc}\hfill \mathbf{M}(\mathbf{s},\mathbf{s},\omega )=& \sum _{m=1}^4\sum _{n=1}^4\left[\begin{array}{ccc}{b}_m{b}_n& b_m{(-1)}^{n+1}& 0\\ {(-1)}^{m+1}{b}_n& {(-1)}^{m+1}{(-1)}^{n+1}& 0\\ 0& 0& 0\end{array}\right]\hfill \\ \hfill & \times exp\left\{-\frac{\left[(b_m-{(-1)}^{m+1})\right]+\left[b_n-{(-1)}^{n+1}\right]}{2{\xi }_0^2}\right\}\hfill \\ \hfill & \times exp\left[-\frac{4a^2}{w_0^2}\right]\left[\sigma +{(-1)}^{n+1}\mathrm{i}\gamma \right]\left[\sigma +{(-1)}^m\mathrm{i}\gamma \right]\hfill \\ \hfill & \times exp\left\{\mathrm{i}ka\left[b_m{s}_x+{(-1)}^{m+1}{s}_y\right]+\mathrm{i}ka\left[b_n{s}_x+{(-1)}^{n+1}{s}_y\right]\right\},\hfill \end{array}$$

(37)

where $b_1=b_2=1$ and $b_3=b_4=-1$. The polarization properties of the far-zone scattered field thus can be obtained by following Equations (12) and (37).

In Figure 2, we show the simulation results for the total intensity ${\mathsf{\Lambda }}_0(r\mathbf{s},\omega )$ of the far-zone scattered field by the four-point scatterers having classic-symmetric and PT-symmetric potentials. The effect of spatial coherence of the incident beam on the far-zone intensity is also considered in Figure 2. In the simulation, the parameters used for the incident beam are the beam waist $w_0=2/k$ and the spatial coherence width $\xi =100/k,2/k$, and $0.2/k$ to represent the highly coherent, moderately coherent, and lowly coherent incident beam, respectively. $\sigma =1$, $\gamma =0$, and $a=1/k$ are used for the classic-symmetric potential function, while $\sigma =1$, $\gamma =1$, and $a=1/k$ are used for the PT-symmetric potential function. The simulation results in Figure 2 show that when the potential function obeys classical symmetry, the far-zone intensity is centrosymmetric. The dark-hollow beam profile in the far-zone intensity (Figure 2a) for the highly coherent case is induced by the radial polarization of the incident beam. We further find with the decrease of the spatial coherence, the dark-hollow beam profile vanishes and becomes a solid Gaussian beam spot for the lowly coherent case (Figure 2c). The reason behind is due to the decreased degree of polarization for the lowly coherent case, which will be shown in Figure 5.

At the same time, we find when the scattering potential becomes PT symmetry, the far-zone scattered intensity becomes asymmetric and shows the directionality in the scattering pattern for the highly coherent case (Figure 2g,j). With the decrease of the spatial coherence of the incident beam, the effect of directionality in scattering becomes weaker. When the spatial coherence of the incident beam is effectively low (Figure 2i,l), the directionality effect disappears and the far-zone beam spot becomes the same as that for the classic-symmetric scatterer. This phenomenon can be explained as follows. For the effectively low spatial coherence, the degree of coherence can be reduced to $\mu ({\mathbf{r}}_1,{\mathbf{r}}_2)\to \delta ({\mathbf{r}}_1-{\mathbf{r}}_2)$. Taking the delta function into Equation (13), we obtain that

$$\mathbf{M}(\mathbf{s},\mathbf{s},\omega )={\int }_V{\mathbf{W}}^{\left(i\right)}(\mathbf{r},\mathbf{r},\omega ){\left|F(\mathbf{r},\omega )\right|}^2{\mathrm{d}}^3\mathbf{r}.$$

(38)

It is found from Equation (38) that the phase information in $F(\mathbf{r},\omega )$ will not affect the far-zone polarization matrix. Thus, the PT symmetry involved in the imaginary part of $F(\mathbf{r},\omega )$ has no effect on the far-zone intensity for the lowly coherent case.

We now turn to examine the polarization properties of the far-zone scattered field. In Figure 3 we show the simulation results for the nine generalized Stokes parameters for the far-zone scattered fields generated by the classic-symmetric and PT-symmetric scatterers. In the simulation, we take the incident beam to be highly coherent, i.e., the spatial coherence width ${\xi }_0=100/k$. It is found that the symmetry property of the scattering potential has a significant effect on the far-zone polarization distribution in such case. For the scattering potential with classical symmetry, we find from the spatial distributions of ${\mathsf{\Lambda }}_1(r\mathbf{s},\omega )$, ${\mathsf{\Lambda }}_2(r\mathbf{s},\omega )$, and ${\mathsf{\Lambda }}_3(r\mathbf{s},\omega )$ that the far-zone scattered field in the $xy$ plane is still radially polarized. However, when the scattering potential becomes PT symmetry, the spatial distributions of ${\mathsf{\Lambda }}_1(r\mathbf{s},\omega )$, ${\mathsf{\Lambda }}_2(r\mathbf{s},\omega )$, and ${\mathsf{\Lambda }}_3(r\mathbf{s},\omega )$ becomes asymmetry and the radial polarization distribution in the $xy$ plane disappear. In addition, it is found from the spatial distributions of ${\mathsf{\Lambda }}_4(r\mathbf{s},\omega )$ to ${\mathsf{\Lambda }}_8(r\mathbf{s},\omega )$ that the z-component field appears in the far-zone scattered field due to the nonparaxial scattering effect. The symmetries in ${\mathsf{\Lambda }}_4(r\mathbf{s},\omega )$ to ${\mathsf{\Lambda }}_8(r\mathbf{s},\omega )$ are broken as well for the PT-symmetric scattering potential. Similar to the total intensity, with the decrease of the spatial coherence of the incident beam, less effect of the scattering potential symmetry impacts on the polarization properties of the far-zone scattered field. As shown in Figure 4, when the spatial coherence width is ${\xi }_0=0.2/k$ (lowly coherent case), the nine generalized Stokes parameters for the far-zone scattered fields generated by the classic-symmetric and PT-symmetric scatterers have the same spatial distribution. Such effect can also be explained by the result shown in Equation (38).

In Figure 5, we show the simulation results for the 3D degree of polarization of the far-zone scattered field generated by the four-point scatterers having the classic-symmetric and PT-symmetric potentials. The effect of spatial coherence of the incident beam is also considered in Figure 5. We find from the results that with the decrease of the spatial coherence of the incident beam, the far-zone scattered field becomes less polarized. Thus, the dark-hollow intensity induced by the radial polarization gradually disappears as shown in Figure 2. Moreover, by switching the potential symmetry from classic to PT, the spatial distribution of 3D degree of polarization becomes asymmetry as well for the highly coherent case. For the lowly coherent case shown in Figure 5c,f, the spatial distributions of 3D degree of polarization for the classic-symmetric and PT-symmetric potentials become identical. We also find that the values of the 3D degree of polarization for the far-zone scattered field in all cases are bounded as $0.5\le P(r\mathbf{s},\omega )\le 1$, which indicates that the far-zone scattered field is a 2D field, i.e., the field along the radial direction ${\widehat{\mathbf{e}}}_r$ in the spherical coordinate frame vanishes, no matter how incoherent of the incident beam is.

3.2.2. Degree of Coherence with PT Symmetry

We now turn to consider the incident partially coherent vector beam with the PT-symmetric degree of coherence. Since the scatterer is a four-point scatterer, we only need to know the degree of coherence between two different points located at the positions of the four point scatterer. In this work, we consider a general degree of coherence, i.e.,

$$\begin{array}{c}\hfill \mu \left({\mathbf{r}}_1,{\mathbf{r}}_2\right)=\left\{\begin{array}{c}1,{\mathbf{r}}_1=(a,a),{\mathbf{r}}_2=(a,a)\hfill \\ {\alpha }_1+i{\beta }_1,{\mathbf{r}}_1=(a,a),{\mathbf{r}}_2=(a,-a)\hfill \\ {\alpha }_2+i{\beta }_2,{\mathbf{r}}_1=(a,a),{\mathbf{r}}_2=(-a,a)\hfill \\ {\alpha }_3+i{\beta }_3,{\mathbf{r}}_1=(a,a),{\mathbf{r}}_2=(-a,-a)\hfill \\ {\alpha }_1-i{\beta }_1,{\mathbf{r}}_1=(a,-a),{\mathbf{r}}_2=(a,a)\hfill \\ 1,{\mathbf{r}}_1=(a,-a),{\mathbf{r}}_2=(a,-a)\hfill \\ {\alpha }_3-i{\beta }_3,{\mathbf{r}}_1=(a,-a),{\mathbf{r}}_2=(-a,a)\hfill \\ {\alpha }_2+i{\beta }_2,{\mathbf{r}}_1=(a,-a),{\mathbf{r}}_2=(-a,-a)\hfill \\ {\alpha }_2-i{\beta }_2,{\mathbf{r}}_1=(-a,a),{\mathbf{r}}_2=(a,a)\hfill \\ {\alpha }_3+i{\beta }_3,{\mathbf{r}}_1=(-a,a),{\mathbf{r}}_2=(a,-a)\hfill \\ 1,{\mathbf{r}}_1=(-a,a),{\mathbf{r}}_2=(-a,a)\hfill \\ {\alpha }_1+i{\beta }_1,{\mathbf{r}}_1=(-a,a),{\mathbf{r}}_2=(-a,-a)\hfill \\ {\alpha }_3-i{\beta }_3,{\mathbf{r}}_1=(-a,-a),{\mathbf{r}}_2=(a,a)\hfill \\ {\alpha }_2-i{\beta }_2,{\mathbf{r}}_1=(-a,-a),{\mathbf{r}}_2=(a,-a)\hfill \\ {\alpha }_1-i{\beta }_1,{\mathbf{r}}_1=(-a,-a),{\mathbf{r}}_2=(-a,a)\hfill \\ 1,{\mathbf{r}}_1=(-a,-a),{\mathbf{r}}_2=(-a,-a)\hfill \end{array}\right.\end{array}$$

(39)

where ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are real factors that control how coherent the field is. In general, with the decrease of these factors, the spatial coherence of the incident beam decreases. In addition, the factors ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ control the symmetry of the beam. For ${\beta }_1={\beta }_2={\beta }_3=0$, the beam reduces to the classical symmetry. While one of the three factors does not vanish, the incident beam becomes PT symmetry.

Taking Equations (32)–(34) and the degree of coherence in (39) into Equation (13), we obtain that the $\mathbf{M}(\mathbf{s},\mathbf{s},\omega )$ matrix can be expressed analytically as

$$\begin{array}{cc}\hfill \mathbf{M}(\mathbf{s},\mathbf{s},\omega )=& \sum _{m=1}^4\sum _{n=1}^4\left[\begin{array}{ccc}{b}_m{b}_n& b_m{(-1)}^{n+1}& 0\\ {(-1)}^{m+1}{b}_n& {(-1)}^{m+1}{(-1)}^{n+1}& 0\\ 0& 0& 0\end{array}\right]\hfill \\ \hfill & \times exp\left[-\frac{4a^2}{{\omega }^2}\right]\left[\sigma +{(-1)}^{n+1}i\gamma \right]\left[\sigma +{(-1)}^{m}i\gamma \right]\mu ({\mathbf{P}}_n,{\mathbf{P}}_m)\hfill \\ \hfill & \times exp\left\{ika\left[b_m{s}_x+{(-1)}^{m+1}{s}_y\right]+ika\left[b_n{s}_x+{(-1)}^{n+1}{s}_y\right]\right\},\hfill \end{array}$$

(40)

where ${\mathbf{P}}_1=(a,a)$, ${\mathbf{P}}_2=(a,-a)$, ${\mathbf{P}}_3=(-a,a)$, and ${\mathbf{P}}_4=(-a,-a)$ are the position vectors for the four-point scatterer, as before. The far-zone intensity and polarization properties thus can be obtained by following Equations (12) and (40). In Figure 6, we show the simulation results for the total intensity of the far-zone scattered field generated by scattering the partially coherent radially polarized beam having degree of coherence shown in Equation (39) with the four-point scatterer having classic-symmetric and PT-symmetric scattering potential functions. In Figure 6a, ${\beta }_1={\beta }_2={\beta }_3=0$, indicating that the incident beam is classical symmetry, the far-zone intensity is centrosymmetric. When the beam becomes PT symmetry, i.e., ${\beta }_1$ and ${\beta }_3$ are taken to be $-1$, we find in Figure 6b that the far-zone intensity shows the directionality effect in the scattering pattern similar to that induced by the PT-symmetric scatterer (c.f., Figure 2j). However, in such case the asymmetric pattern is purely induced by the PT-symmetric coherence function of the incident beam. When we switch the sign of ${\beta }_1$ and ${\beta }_3$ into $+1$, it is found in Figure 6c that the directionality switches as well, which indicating a flexible way to control the directionality scattering.

Figure 6d–f show the simulation results when the scattering potential function becomes PT symmetry ($\sigma =2$ and $\gamma =1$). It is found that the directionality scattering pattern induced by the PT-symmetric scatterer shown in Figure 6d can be suppressed (see in Figure 6e) or turned into a inverse direction (see in Figure 6f) by introducing the PT symmetry into the spatial coherence function of the incident beam. This phenomenon can be explained by Equation (27) that the asymmetry part ${\mathsf{\Phi }}^{\left(s\right)\prime \prime }(r\mathbf{s},\omega )$ is induced by the imaginary part of the joint function ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )$. By turning the symmetries of the coherence function and the scattering potential to let the phases of the coherence function ${\mathbf{W}}^{\left(i\right)}({\mathbf{r}}_1,{\mathbf{r}}_2,\omega )$ and the potential correlation $F^{*}({\mathbf{r}}_1,\omega )F({\mathbf{r}}_2,\omega )$ match well, the asymmetry intensity can be suppressed or enhanced. The effect of spatial coherence of the incident beam on the far-zone intensity are shown in Figure 7. In the simulation, the spatial coherence of the incident beam is controlled by the values of ${\beta }_1$, ${\alpha }_2$, and ${\beta }_3$. We let ${\beta }_1={\alpha }_2={\beta }_3=d$. With the decrease of $\left|d\right|$, the spatial coherence of the incident beam decreases as well. In Figure 7, we find that the directionality scattering effect becomes weaker with decrease of the spatial coherence, which is consistent with the results shown in Figure 2g–i.

In Figure 8, we display the simulation results for the nine Stokes parameters of the far-zone scattered field with the classic-symmetric and PT-symmetric scatterers. The parameters used in the simulation are the same as those in Figure 6c,f, respectively. We find in the simulation results that the polarization state becomes asymmetry with the introduction of the PT symmetry into the spatial coherence of the incident beam. We note that the asymmetry properties in Figure 8a–i are purely induced by the PT symmetry in the partially coherent incident beam, while the asymmetry properties in Figure 8j–r are the joint effect from the PT symmetries in the incident light and the scatterer. Finally, the simulation results for the 3D degree of polarization of the scattered field are shown in Figure 9, we find for ${\beta }_1={\alpha }_2={\beta }_3=1$ the scattered field is fully polarized for both the classic-symmetric and PT-symmetric scatterers since the incident fields among the positions of four point-scatterers are completely correlated. With the decrease of the spatial coherence of the incident beam, it is found that the 3D degree of polarization decreases as well, and at the same time, the asymmetry property appears in its spatial distribution due to the PT symmetry in the spatial coherence of the incident. In addition, we find with the PT-symmetric incident light, the 3D degree of polarization for the far-zone scattered field is still bounded as $0.5\le P(r\mathbf{s},\omega )\le 1$, indicating that the far-zone scattered field is a 2D field.

4. Discussion

In this work, we have studied the polarization properties of the far-zone scattered field generated by interacting a partially coherent nonuniformly polarized beam with a deterministic scatterer having PT-symmetric potential function. We have found the important roles of the symmetry properties in both the scatterer’s potential function and the incident beam’s coherence function on the far-zone polarization distribution. We have showed that the PT symmetry in both functions can induce the polarization directionality in the far-zone scattered field. Moreover, the directionality can be suppressed or enhanced by controlling the phase difference between scattering potential function and the incident beam’s coherence function. We note that the previous studies, e.g., in Refs. [54,60,61,62,63,64], were focused on the scalar incident beam, i.e., no polarization properties are considered. As far as we know, this paper has established the first framework for the weak scattering of partially coherent vector light with the PT-symmetric medium. Further, in the previous studies, the incident partially coherent beams have the classic-symmetric coherence function, leading to the limited degrees of freedom to govern the directionality in the far-zone scattered field. In our work, we can control both the degrees of freedom in the incident beam and in the scattering potential function. Finally, the partially coherent vector beam with PT-symmetric coherence function may be synthesized with the help of the random-mode superposition method [27], in which all the spatially coherent random modes are control to have the PT-symmetric electric field by the spatial light modulator or a digital mirror device.

5. Conclusions

In summary, we have studied the scattering properties of a partially coherent radially polarized vector beam by a deterministic medium having PT-symmetric scattering potential. We obtained the explicit analytic expression for the intensity and polarization of the far-zone scattered field under the first-order Born approximation. The four-point deterministic scatterers with classic-symmetric and PT-symmetric potential functions have been considered in our simulation. We showed that by turning the scattering potential function from classic to PT symmetry, the intensity and polarization of the far-zone scattered field becomes asymmetry and the directionality appears in the scattering pattern. However, with the decrease of the spatial coherence of the incident beam, it is found that the effect of directionality becomes weaker as the role of the phase information in the scattering potential decreases with the spatial coherence of the incident beam. Meanwhile, we found that by modulating the symmetry property of the spatial coherence function of the incident beam, the directionality in scattering can be suppressed or enhanced due to the joint effect from both the PT-symmetry of the spatial coherence and the PT symmetry of the scattering potential. Our findings may find applications, e.g., in controlling the directionality in light scattering.