1. Introduction

The correlation between intensity fluctuations (CIF), also known as Hanbury Brown-Twiss effect [1], sheds light on the higher-order coherence characteristics of the field and has attracted much interest in both the scientific and engineering communities. The traditional analysis of the CIF omits the polarization properties of light fields and thus a scalar treatment is usually enough [2–4]. With the extension of CIF to the domain of electromagnetic radiation, it has been shown that the CIF in stochastic electromagnetic fields does not depend only on the degree of coherence as in the scalar case, and the normalized CIF of stochastic electromagnetic fields is closely related to the degree of polarization of light fields [5,6]. Recent studies have shown that the knowledge of the degree of coherence and of the degree of polarization of light fields is not adequate to predict the CIF in stochastic electromagnetic fields. A new statistical quantity called the degree of cross-polarization is also needed [7–9]. This new quantity is a function of two observation points and it reduces to the usual degree of polarization when these two points coincide. The usefulness of the degree of cross-polarization was also later questioned because it may diverge [10]. Additionally, the propagation dynamics of the CIF of electromagnetic Gaussian Schell-model beams in free space [11] and in atmospheric turbulence [12] has been addressed, and the generalized Hanbury Brown-Twiss effect in partially coherent electromagnetic beams have also been executed [13–15].

The introduction of the concept of the CIF into the domain of weak potential scattering can be attributed to Xin et al, who were concerned with the fourth-order correlation statistics of a plane wave on scattering from a quasi-homogeneous medium [16]. This work has shown that the CIF of the scattered field depends on spatial Fourier transforms of both the intensity and degree of spatial correlation of scattering potentials of the medium, whereas its normalized version equals the squared modulus of the degree of spatial coherence of the scattered field. Inspired by this work, the problems of the CIF of the scattering of stochastic scalar stationary fields [17] and partially coherent plane-wave pulses [18] have been addressed, and in particular the CIF of electromagnetic light waves scattering from different media have been discussed extensively, for example, continuous media [19–23] and collections of identical particles [24]. Although these works have indicated that the polarization property of the incoming source is a crucial factor that affects the CIF of the scattered field, the qualitative and quantitative relation between the spectral degree of polarization $\mathcal {P}$ of the incident field and the normalized CIF of the scattered field hasn’t been fully exposed. On the other hand, the CIF of the scalar scattered field generated by a collection of particles of $\mathcal {L}$ types has been resolved by us [25] very recently. We have demonstrated that the CIF of the scattered field actually equals the squared modulus of the trace of the product of the PSM and the transpose of the PPM, and thus these two matrices provide the angular correlation information of the whole collection required to determine the CIF of the scattered field. Also, we have shown that the PPM and the PSM can reduce to two new matrices in three special cases, and the expression of the normalized CIF of the scattered field can have pretty interesting forms in these special cases. Our work can be viewed as a new approach to the CIF of scalar light scattering from a collection with different types of scatterers.

In the present paper, we are devoted to the vectorial extension of the work of [25]. We will develop a theoretical framework in the spherical polar coordinate system to systematically treat the CIF of electromagnetic light waves on scattering from a collection of particles of $L$ types, and establish a quantitative relation between the normalized CIF of the electromagnetic scattered field and the PPM, the PSM, and the spectral degree of polarization $\mathcal {P}$ of the incident field. Based on this, by a numerical example, we will put emphasis on how the normalized CIF of the electromagnetic scattered field evolves with $\mathcal {P}$, and how the evolution is influenced by the polar angle and the azimuthal angle. Finally, the difference between our findings and the existing results will be discussed in detail.

2. CIF of electromagnetic waves scattering from a collection with particles of $\mathcal {L}$ types in the spherical polar coordinate system

Consider now that an electromagnetic plane wave, propagating in the direction of a unit vector $\mathbf {s}_{0}$ along the $z$ axis, is incident on a collection of particles which occupies a finite domain $\mathcal {V}$ (see Fig. 1). The incident field at a point $\mathbf {r}^{\prime }$ can be characterized by a statistical ensemble $\left \{E_{i}(\mathbf {r}^{\prime },\omega )\exp {[-i\omega t}]\right \}$ of monochromatic realizations, all of frequency $\omega$, in the sense of coherence theory in the space-frequency domain. Here

 

Fig. 1. Illustration of notations. Here $\mathbf {r}_{pm}$ denotes the location of the scattering center of the $m$th-particle of type $p$.

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The cross-spectral density matrix of the incident field at a pair of points $\mathbf {r}^{\prime }_{1}$ and $\mathbf {r}^{\prime }_{2}$ is defined as ([26], Sec. 9.1)

The cross-spectral density matrix of the incident field in this case is diagonal, and thus the spectral degree of polarization $\mathcal {P}$ of the incident field has a simple form ([26], Sec. 9.2)

Let us now focus on the particulate collection composed of $\mathcal {L}$ types of particles, $m(p)$ of each type, ($p=1,2,3,\cdot \cdot \cdot,\mathcal {L}$). We suppose that the $\mathcal {L}$ types of particles are located at points specified by position vectors $\mathbf {r}_{pm}$, and have scattering potentials $f_{p}(\mathbf {r}^{\prime },\omega )=k^{2}[n^{2}_{p}(\mathbf {r}^{\prime },\omega )-1]/4\pi$, with $n_{p}(\mathbf {r}^{\prime },\omega )$ being the refractive index of the $p$th-type particle ([26], Sec. 6.1). The scattering potential of the whole collection is defined as [28]

If the medium is a weak scatterer so that the scattering can be analyzed within the validity of the first-order Born approximation [29]. The electric vector of the scattered field in the far zone out of the scatterer is formulated as

From Eq. (7), one can notice that $\mathbf {s}\cdot \mathbf {E}^{(\text {s})}(r\mathbf {s},\omega )=0$, i.e., only two transverse components of the scattered field are important, and the global behavior of the scattered field in the far zone is essentially of an outgoing spherical wave. In this sense, the scattered field can be greatly simplified if one expresses it in the spherical polar coordinate system [30], viz.,

The second-order coherence and polarization properties of the scattered field at two points $r\mathbf {s}_{1}$ and $r\mathbf {s}_{2}$ can also be characterized by a $2\times 2$ cross-spectral density matrix, with a form of ([26], Sec. 9.1)

We now consider the intensity of a single realization of the scattered field at a point $r\mathbf {s}$ at frequency $\omega$, which can be expressed as

The intensity dispersion from its mean value at the same point $r\mathbf {s}$ is given as

Thus the correlation of intensity fluctuations at two points ${r\mathbf {s}}_{1}$ and ${r\mathbf {s}}_{2}$ is defined as

If the fluctuations of the scattered field are Gaussian, and then the fourth-order correlation can be calculated from the second-order one ([26], Sec. 7.2), by use of the Gaussian moment theorem for complex random processes. Thus the CIF of the scattered field can be simplified as

In many situations of practical interest, one often takes into account the normalized version of the CIF of the scattered field, which can be defined in terms of Eq. (13) as the following form ([26], Sec. 7.2)

On substituting from Eqs. (1) and (8) to Eq. (9) first, and then into Eq. (14), after some pretty long but straightforward calculations, the normalized CIF of the scattered field finally can be computed as

Equation (15) establishes a closed-form relation that associates the normalized CIF of the electromagnetic scattered field with the PPM, the PSM, and the spectral degree of polarization $\mathcal {P}$ of the incident field. The first line in Eq. (15) is suitable for the case where the incoming source is polarized along the $x$ axis ($A_{x}> A_{y}$), and the second line is applied to the situation where the incoming source is polarized along the $y$ axis ($A_{x}<A_{y}$). The transition from the first line to the second line will occur when $\mathcal {P}$ is replaced by $-\mathcal {P}$, therefore, the normalized CIF of the scattered field produced by an electromagnetic wave polarized along $y$ axis can be easily predicted from the normalized CIF of the scattered field produced by an electromagnetic wave polarized along $x$ axis, and vice versa. The fact that the normalized CIF of the scattered field splits into two parts, one having to do with the spectral degree of polarization $\mathcal {P}$ of the incident field and the other with the PPM and the PSM, is not accidental but we have assumed that the scattering is weak ([26], Sec. 6.3). The first part shows that the dependence of the normalized CIF of the scattered field on $\mathcal {P}$ is only determined by the polar angle $\theta$ and the azimuthal angle $\phi$. As we will see, depending on $\theta$ and $\phi$, the normalized CIF itself can be monotonically increasing or be nonmonotonic with $\mathcal {P}$ in the region $[0,1]$. Furthermore, the vertoral extension of the result of the scalar case in [25] has been clearly shown in Eq. (15), where the spectral degree of polarization $\mathcal {P}$ of the incident field is incorporated. In what follows, we will illustrate the dependence of the normalized CIF itself on the spectral degree of polarization $\mathcal {P}$ of the incident field by a specific numerical example.

3. Numerical example

Let us consider a system of random particles with determinate density distributions. For simplicity, we assume that there are only two types of particles in the system, i.e., $p,q=1,2$, and further assume that the self-correlation functions of the scattering potentials of particles of the same type and the cross-correlation functions of the scattering potentials of particles of different types both obey the so-called Gaussian-Schell model. In this case, the elements of the matrices $\mathcal {F}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ and $\mathcal {G}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ are calculated as [25]

Figure 2(a) displays the surface plot of the normalized CIF of electromagnetic light waves polarized along the $x$ axis on scattering from a collection of random particles with determinate density distributions, as a function of the spectral degree of polarization $\mathcal {P}$ of the incident field and the dimensionless polar angle $\theta /\pi$. As shown in Fig. 2, the normalized CIF of the scattered field is a monotonically increasing function of $\mathcal {P}$ in the region $[0, 1]$ within the whole polar angle, which means that the more polarized the incident field, the more intense the CIF of the scattered field. What’s more, the rate (i.e., the slope $\partial \mathcal {C}_{n}/\partial \mathcal {P}$) at which the normalized CIF of the scattered field grows with $\mathcal {P}$ depends largely on $\theta /\pi$. To see this well, the normalized CIF of the scattered field for three selective values of $\theta /\pi$ is separately plotted in Fig. 2(b). It can be seen that the dependence of the normalized CIF of the scattered field on $\mathcal {P}$ transits gradually from nonlinear to linear with the increase of $\theta /\pi$. The appearance of the linear relation between the normalized CIF of the scattered field and $\mathcal {P}$ in the large polar angle results from the fact that the slope $\partial \mathcal {C}_{n}/\partial \mathcal {P}$ in the neighborhood of $\theta /\pi =0.5$ is independent of $\mathcal {P}$, i.e., $\frac {\partial \mathcal {C}_{n}}{\partial \mathcal {P}}\bigl |_{\theta /\pi \approx 0.5}\propto \frac {1}{2}$, which can be readily verified from the first line of Eq. (15) [31]. The obvious change of the slope $\partial \mathcal {C}_{n}/\partial \mathcal {P}$ is a manifestation that the sensitivity of the CIF of the scattered field at the small polar angle and the large polar angle to the spectral polarization of the incident field is different. In this way, the linear dependence of $\mathcal {C}_{n}$ on $\mathcal {P}$ at the large polar angle originates from the physical fact that the CIF of the scattered field is equally sensitive to highly polarized and weakly polarized incident fields.

 

Fig. 2. (a) Surface plot of the normalized CIF of electromagnetic light waves polarized along the $x$ axis on scattering from a collection of random particles with determinate density distributions, as a function of the spectral degree of polarization $\mathcal {P}$ of the incident field and the dimensionless polar angle $\theta /\pi$. The coordinates for the first kind of particles are set to be $(0,0.1\lambda,0)$ and $(0,-0.1\lambda,0)$, and $(0,0.2\lambda,0)$ and $(0,-0.2\lambda,0)$ for the second kind of particles. $\mathbf {s}_{0}=(0,0,1)$, $\mathbf {s}_{1}=(\sin {\theta _{1}\cos {\phi _{1}}},\sin {\theta _{1}\sin {\phi _{1}}},\cos {\theta _{1}})$, $\mathbf {s}_{2}=(\sin {\theta \cos {\phi }},\sin {\theta \sin {\phi }},\cos {\theta })$. The parameters for calculations are $\theta _{1}/\pi =0$, $\phi _{1}/\pi =\phi /\pi =1/2$, $\lambda =0.6328\mu m$, $\sigma _{11}=\sigma _{22}=0.9/k$, $\sigma _{12}=\sigma _{21}=0.8/k$, $\eta _{11}=\eta _{22}=0.01/k$, $\eta _{12}=\eta _{21}=0.03/k$. (b) Behaviors of the normalized CIF of the scattered field for three selective values of $\theta /\pi$.

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However, $\mathcal {C}_{n}$ won’t be always monotonically increasing with $\mathcal {P}$ in the region $[0, 1]$ within the whole polar angle, when the azimuthal angle $\phi$ changes. The surface plot of the normalized CIF of the electromagnetic scattered field in the case of $\phi /\pi =1/60$ is plotted in Fig. 3(a), as a function of $\mathcal {P}$ and $\theta /\pi$. One may see that only within very small polar angle can $\mathcal {C}_{n}$ be monotonically increasing functions of $\mathcal {P}$, otherwise will it be non-monotonic with $\mathcal {P}$ in the region $[0, 1]$. To see this more clearly, the normalized CIF of the scattered field for four selective values of $\theta /\pi$ is separately plotted in Fig. 3(b). It is shown that, within the large polar angle, the normalized CIF of the scattered field has a manifestation of decreasing first and then increasing gradually with the growth of $\mathcal {P}$, and the decay range of $\mathcal {C}_{n}$ at the $\mathcal {P}$ axis will be greatly extended as $\theta /\pi$ raises. This tells us a truth that even if the incident field is weakly polarized, it can still produce the scattered field with appreciably intense CIF. When $\theta /\pi$ becomes large enough, for example, $\theta /\pi =0.48$, $\mathcal {C}_{n}$ decays first in an obvious linear fashion and then grows rapidly in a nonlinear fashion within the neighborhood of $\mathcal {P}=1$. Such a dramatic change happening to $\mathcal {C}_{n}$ can have a well-defined explanation from its slope, which can be approximately computed as $\frac {\partial \mathcal {C}_{n}}{\partial \mathcal {P}}\bigl |_{\mathcal {P}\approx 1}\propto \frac {1}{1+\cos {2\theta }}$. The denominator $1+\cos {2\theta }$ will tend to zero with the approach of $\theta /\pi$ to $0.5$, naturally leading the growth of $\mathcal {C}_{n}$ with $\mathcal {P}$ at this region to be very dramatic. Physically, this dramatic change originates from the fact that the CIF at the large polar angle of the scattered field produced by the highly polarized incident field is much more sensitive to the spectral polarization of the incident field than that at the small polar angle.

 

Fig. 3. (a) Surface plot of the normalized CIF of the electromagnetic scattered field as a function of $\mathcal {P}$ and $\theta /\pi$. $\phi /\pi =1/60$, the other parameters for calculations are the same as Fig. 2(a). (b) Plots of the normalized CIF of the scattered field for four selective values of $\theta /\pi$.

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We next turn to how the CIF of the electromagnetic scattered field evolves with the azimuthal angle and the spectral degree of polarization of the incident field. Figure 4 plots the normalized CIF of the scattered field as a function of $\mathcal {P}$ and $\phi /\pi$. As we can see, the normalized CIF of the scattered field is a monotonically increasing function of $\mathcal {P}$ in the region $[0, 1]$ and the form of dependence of the normalized CIF on $\mathcal {P}$ is the same throughout the azimuthal angle, which is quite different from the evolution of the normalized CIF with $\mathcal {P}$ at the polar angle, as shown in Fig. 2(a). However, the situation will change dramatically when the polar angle varies. $\mathcal {C}_{n}$ will become a non-monotonic function of $\mathcal {P}$ in the range $[0, 1]$, except at some large azimuthal angles, as shown in Fig. 5(a). To see this well, the normalized CIF of the scattered field for three selective values of $\phi /\pi$ is exclusively plotted in Fig. 5(b). When the azimuthal angle is very large, $\mathcal {C}_{n}$ will increase linearly with $\mathcal {P}$. Such a linear dependence of $\mathcal {C}_{n}$ on $\mathcal {P}$ at the large azimuthal angle reflects well the physical fact that the sensitivity of the CIF of the scattered field to highly polarized and weakly polarized incident fields is the same. As the azimuthal angle becomes small, $\mathcal {C}_{n}$ will decline first and then raise gradually with the growth of $\mathcal {P}$, and the decay range of $\mathcal {C}_{n}$ at the $\mathcal {P}$ axis will be greatly extended as $\phi /\pi$ decreases. When $\phi /\pi$ becomes small enough, such as $\phi /\pi =0.01$, $\mathcal {C}_{n}$ decays linearly first and then grows very fastly at the vicinity of $\mathcal {P}=1$. This intense change in $\mathcal {C}_{n}$ at the vicinity of $\mathcal {P}=1$ can also be explained from its slope, which can be calculated as $\frac {\partial \mathcal {C}_{n}}{\partial \mathcal {P}}\bigl |_{\mathcal {P}\approx 1}\propto \frac {1}{1-\cos {2\phi }}$. The denominator $1-\cos {2\phi }$ will approach to zero with $\phi /\pi$ tending to $0$, which results in the intense raise of $\mathcal {C}_{n}$ with $\mathcal {P}$ at this region. Likewise, this intense change can be explained physically as follows: the CIF at the small azimuthal angle of the scattered field produced by the highly polarized incident field is much more sensitive to the spectral polarization of the incident field than that at the large azimuthal angle.

 

Fig. 4. Surface plot of the normalized CIF of the electromagnetic scattered field as a function of $\mathcal {P}$ and $\phi /\pi$. $\theta /\pi =1/60$, the other parameters for calculations are the same as Fig. 2(a).

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Fig. 5. (a) Surface plot of the normalized CIF of the electromagnetic scattered field as a function of $\mathcal {P}$ and $\phi /\pi$. $\theta /\pi =1/2$, the other parameters for calculations are the same as Fig. 2(a). (b) Plots of the normalized CIF of the scattered field for three selective values of $\phi /\pi$.

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4. Conclusions

In conclusion, we have developed a theoretical framework in the spherical polar coordinate system to access the CIF of electromagnetic light waves on scattering from a collection of particles of $\mathcal {L}$ types. We have built the quantitative relation between the normalzied CIF of the electromagnetic scattered field and the PSM, the PPM, and the spectral degree of polarization $\mathcal {P}$ of the incident field, and thus we have completed the vectorial extension of our recent work [25]. In particular, we have paid much attention to the dependence of the normalized CIF of the scattered field on $\mathcal {P}$, and have discussed how the polar angle $\theta$ and the azimuthal angle $\phi$ affect this dependence. It was found that the normalized CIF can be a monotonically increasing function of $\mathcal {P}$ or be nonmonotonic with $\mathcal {P}$ in the range $[0,1]$, determined by $\theta$ and $\phi$. The former means that the more polarized the incident field, the more intense the CIF of the scattered field, while the latter indicates that even though the incident field is weakly polarized, it can still produce the scattered field with appreciable CIF. Additionally, it was also found that the distributions of the normalized CIF of the scattered field with $\mathcal {P}$ at polar angles and azimuthal angles are quite different, such as the nonmonotonic variations of the normalized CIF with $\mathcal {P}$ appear at small azimuthal angles, which are inverse to the polar situations. Also, the form of dependence of the normalized CIF of the scattered field on $\mathcal {P}$ can be independent of $\phi$. To explain these findings, we have determined the slope function $\partial \mathcal {C}_{n}/\partial \mathcal {P}$, and the corresponding physical explanations have been given.

Although there have been many studies on the influence of polarization property of the incident field on the CIF of the scattered fields generated by different scattering media [19–24], but we still would like to emphasize the difference between our results and theirs. Firstly, the previous efforts to explore the CIF of the scattered field were mainly confined within the so-called scattering plane [32], that is, the effect of $\phi$ on the CIF of the scattered field was usually not taken into account. However, as we have seen from Eq. (15) or Figs. 2–5, $\phi$ is a nontrivial element that determines the dependence of the normalized CIF of the scattered field on $\mathcal {P}$. Secondly, Eq. (15) is the final expression to show how the normalized CIF of the scattered field changes, depending on $\mathcal {P}$. In the previous works, the normalized CIF of the scattered field was always shown as a function of polarization amplitudes $A_{x}$ and $A_{y}$, or as a function of $a^{2}_{y}(\omega )/a^{2}(\omega )$ (please see Fig. 4 in [20]), which is still not the spectral degree of polarization $\mathcal {P}$. The normalized CIF of the scattered field as a function of $a^{2}_{y}(\omega )/a^{2}(\omega )$ is monotonically decreasing in the range $[0,0.5]$ when the incoming electromagnetic light source is polarized along the $x$ axis [33]. However, the relationship between the normalized CIF of the scattered field and the polarization property of the incident field actually has more complicated behaviors even if the incident field is polarized along the $x$ axis, for example, we have shown that the normalized CIF of the scattered field can decay first in an obvious linear fashion with the raise of $\mathcal {P}$ in the range $[0,1]$ and then grows rapidly in a nonlinear fashion within the neighborhood of $\mathcal {P}=1$. Also, we have found that when $\theta _{1}=\theta =0$, the normalized CIF of the scattered field usually won’t remain constant no matter what $\mathcal {P}$ the incident field carries, unlike the results in [20], where the normalized CIF of the scattered field will remain unchanged as $a^{2}_{y}(\omega )/a^{2}(\omega )$ varies from $0$ to $0.5$ (also Fig. 4). We have also found that when $\theta _{1}=0$, the dependence of the normalized CIF of the scattered field on $\mathcal {P}$ will gradually transit from nonlinear to linear with the increase of $\theta /\pi$ to the vicinity of $0.5$ instead of a constant linear dependence. These elaborate analyses on the relation between the normalized CIF of the scattered field and $\mathcal {P}$ in this work may have potential applications in ghost scattering [34] and ghost imaging [35,36], especially when these phenomena happen in natural environments where collections of particles of different types are often encountered. Our results may be more conducive to retrieve a high quality of ghost image via collecting intensity correlation information at proper observation angles, when an electromagnetic light wave with a certain $\mathcal {P}$ is used as the incoming source.

Finally, it deserves to point out that the examples presented in the paper are illustrative of the relationship between the normalized CIF of the scattered field and the spectral degree of polarization $\mathcal {P}$ of the incident field, but not exhaustive, that is one may expect that other forms of the dependence of the normalized CIF on $\mathcal {P}$ can be shown through further numerical simulations from Eq. (15) with other different choices of the polar angle and the azimuthal angle. Furthermore, here we have assumed that the $x$ and $y$ components of the incoming source are uncorrelated for simplicity ([13]; [26], Sec. 9.4.3; [28]). Although this assumption somewhat limits the application of our results to a more general situation, this is enough for our current purpose—to preliminarily show how the normalized CIF of the scattered field is related to the spectral degree of polarization $\mathcal {P}$ of the incident field and how the normalized CIF of the electromagnetic scattered field evolves with $\mathcal {P}$, as well as how the evolution is influenced by the polar angle and the azimuthal angle. When our particulate collection is illuminated by a more general source, such as an electromagnetic Gaussian Schell-model source [26,37], the corresponding scattering problem will become complicated but can still be addressed by invoking the method of angular spectrum to decompose the incoming optical source. This allows us to discuss the effects of both spatial coherence and polarization of the incident beam on the intensity correlation of the scattered field, which will be reported in detail elsewhere.