1. Introduction

It is well-known that the correlation between intensity fluctuations (CIF) at two observation points in an optical field is also referred to the Hanbury Brown-Twiss effect [1,2], which is one of the celebrated experiments of modern physics that can accommodate equally classical and quantum interpretations. The original motivition of this effect is to determine angular diameters of radio stars, but since then it has become ubiquitous in different branches of physics, such as condensed matter physics [3] and nuclear physics [4], to name a few. Recently, the Hanbury Brown-Twiss effect has also been used in the domain of ghost imaging depending strongly on intensity fluctuations of light fields as a tool for retrieving an unknown object’s transmittance pattern [5].

A novel application of the Hanbury Brown-Twiss effect to weak scattering theory can be traced back to the study of the CIF of a scalar plane wave on scattering from a quasi-homogeneous random medium [6]. Such a pioneering work shed light on the relation between the CIF of the scattered field and the structural characteristics of the medium, and laid a foundation of the application of CIF to inverse scattering problem [7,8]. Shortly afterwards, this investigation was extended to the scattering of stochastic scalar fields [9] and electromagnetic light waves [10]. It is found that in addition to the structural characteristics of the medium, the coherence and polarization properties of incoming electromagnetic sources are also important elements for determining the CIF in the far-zone scattered field. In recent years, the CIF in scattered fields continues to be of great interest, among them [11–16], in which the CIF of light waves on scattering has been extended to a system of identical particles. However, as we know, a particulate system in which all particles are identical is actually a pretty idealized model, and it has no ability to characterize a more practical situation where particles in the collection have different sizes and shapes, as well as various distribution features of refractive index. Such a particulate collection is often encountered in practice, unfortunately, the CIF of light waves on scattering from a collection of particles of different types hasn’t been rigorously addressed.

In this work, a new pathway is paved within the first-order Born approximation to access the CIF of light waves on scattering from a collection of particles of $\mathcal {L}$ types. The PPM and the PSM will be introduced to jointly formulate the CIF of the scattered field. We will build a closed-form relation that associates the CIF of the scattered field with the PPM and the PSM. From this, the normalized CIF of the scattered field will be analyzed in detail in three special cases. The influence of the off-diagonal elements of the PPM and the PSM on the normalized CIF of the scattered field will be illustrated by analyzing the scattering of light waves from two special hybrid particulate systems, i.e., a collection of random particles with determinate density distributions and a collection of determinate particles with random density distributions.

2. Convolution representation of the scattering potential and its correlation function of a collection of particles of $\mathcal {L}$ types

For a collection of particles, there are usually $\mathcal {L}$ types of particles forming this system, $m(p)$ of each type, ($p=1,2,3,\cdot \cdot \cdot,\mathcal {L}$), located at points specified by position vectors $\mathbf {r}_{pm}$. We characterize the response of each particle to an incoming field by a scattering potential $f_{p}(\mathbf {r}^{\prime }, \omega )$, which is closely related to the refractive index of the particle ([17], Sec. 6.1). The scattering potential $F(\mathbf {r}^{\prime },\omega )$ of the whole collection is defined as [18]

For the sake of following discussions, we now rewrite the definition of the scattering potential of the collection in a slightly unfamiliar form, which can be expressed as

In general, the collection may be of deterministic or random nature. In the case when the collection is deterministic, its scattering potential $F(\mathbf {r}^{\prime },\omega )$ is a well-defined function of position. However, for a more involved case when the scattering potential of the collection is not deterministic, but varies randomly as a function of position. In this case, the spatial correlation function of scattering potential at two points $\mathbf {r_{1}}^{\prime }$ and $\mathbf {r_{2}}^{\prime }$ is given as ([17], Sec. 6.3.1)

It is seen in the transition from Eq. (4) to Eq. (5) that use has been made of the assumption that the average over the ensemble of the scattering potentials of particles and that over the ensemble of their density distributions are mutually independent ([17], Sec. 6.3.2).

3. Relation between the CIF of the scattered field and the PPM, the PSM

Assume now that a coherent polychromatic plane light wave, propagating in a direction specified by a real unit vector $\mathbf {s}_{0}$, is incident upon a statistically stationary collection of particles with $\mathcal {L}$ types (see Fig. 1), occupying a finite domain $\mathcal {V}$. The statistical property of the incident field at a pair of points $\mathbf {r_{1}}^{\prime }$ and $\mathbf {r_{2}}^{\prime }$ within the domain of the collection can be characterized by its cross-spectral density function with a form of ([17], Sec. 6.2)

 

Fig. 1. Illustration of notations.

Download Full Size | PDF

Within the validity of the first-order Born approximation, the cross-spectral density function of the scattered field at two points $r\mathbf {s_{1}}$ and $r\mathbf {s_{2}}$ in the far-zone out of the collection can be formulated as

The correlation of intensity fluctuations in the scattered field at a pair of points ${r\mathbf {s}}_{1}$ and ${r\mathbf {s}}_{2}$ is defined as

If we assume that the fluctuations of the scattered field obey the Gaussian statistics. It then follows, by use of the Gaussian moment theorem for complex random processes, that the fourth-order correlation can be calculated from the second-order one ([17], Sec. 7.2). In this way, the CIF of the scattered field can be simplified as

We now introduce two $\mathcal {L}\times \mathcal {L}$ matrices to jointly formulate the CIF of the scattered field generated by a collection of particles of $\mathcal {L}$ types. The first matrix is defined as

From Eq. (6) and Eq. (10), it readily follows that the diagonal elements of this matrix represent the angular self-correlations of the scattering potentials of particles of the same type, while the off-diagonal elements represent the angular cross correlations of the scattering potentials of each pair of particle types. The entire matrix contains all the information about the angular correlation properties of the scattering potentials of between particles within one type and across different types, it may be natural to call it a PPM, which has not been noticed before.

The second matrix is defined as

From Eq. (7) and Eq. (11), it is easy to see that the diagonal elements of this matrix stand for the angular self-correlations of the density distributions of particles of the same type, while the off-diagonal elements stand for the angular cross correlations of the density distributions of each pair of particle types. Notice that $\mathcal {G}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ is essentially the same as the pair-structure matrix introduced by Tong and Korotkova before [20], and one can see from here that the pair-structure factors and joint pair-structure factors are related to the angular correlations of the density distributions of particles within one type and across different types in the collection, respectively.

One may also need to notice that $\mathcal {F}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ and $\mathcal {G}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ are, in general, not Hermitian matrices since $\widetilde {C}_{{f}_{qp}}(-\mathbf {K_{2}},\mathbf {K_{1}},\omega )\neq \widetilde {C}_{f_{pq}}^{*}(-\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ and $\widetilde {C}_{g_{qp}}(-\mathbf {K_{2}},\mathbf {K_{1}},\omega )\neq \widetilde {C}_{g_{pq}}^{*}(-\mathbf {K_{1}},\mathbf {K_{2}},\omega )$. However, in many situations of practical interest, i.e., their elements have common Gaussian Schell-model distributions [21] or multi-Gaussian Schell-model distributions [22] or quasi-homogeneous distributions [23], $\mathcal {F}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ and $\mathcal {G}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ are symmetric with respect to $\mathbf {K}_{1}$ and $\mathbf {K}_{2}$, and thus they can be Hermitian.

On substituting from Eq. (9) into Eq. (14) and making use of the PPM and the PSM as well as the well-known trace operation of a matrix, the CIF of the scattered field can be formulated as

Equation (17) is one of the main result in the present work, which builds a closed-form relation that associates the CIF of the scattered field with the PPM and the PSM. It shows that in addition to a trivial factor $\bigr [S^{\text {(in)}}(\omega )\bigl ]^{2}/r^{4}$, the CIF of the scattered field exactly equals the squared modulus of the trace of the product of the PSM and the transpose of the PPM, and thus the CIF of the scattered field can be completely determined from the knowledge of these two matrices. At the same time, Eq. (17) is also the final expression to show how the CIF of the scattered field changes, depending on the angular correlation properties of the whole collection, which include two parts: one having to do with the angular correlations of the scattering potentials of particles within one type and cross different types, and the other with angular correlations of the density distributions of particles within one type and cross different types.

Additionally, one also often takes into account the normalized version of the CIF of the scattered field, which is defined as ${\mathcal {C}_{n}}\left ( {r{\mathbf {s}}}_{1}{,r{\mathbf {s}_{2}},\omega } \right )\equiv {\mathcal {C}}\left ( {r{\mathbf {s}}}_{1}{,r{\mathbf {s}_{2}},\omega }\right )/W^{(\text {s})}(r{\mathbf {s}_{1}},r{\mathbf {s}_{1}},\omega )W^{(\text {s})}(r{\mathbf {s}_{2}},r{\mathbf {s}_{2}},\omega )$ ([6]; [17], Sec. 7.2). By the aid of Eq. (9) and Eq. (17) and making use of the PPM and the PSM, the normalized CIF of the scattered field is finally computed as the following form

In what follows, we will show that Eq. (18) can have pretty compact and profound forms in three special cases:

(I) The spatial distributions of the scattering potentials of particles of different types are similar, i.e., $\widetilde {C}_{f_{pq}}(-\mathbf {K},\mathbf {K},\omega )\approx \widetilde {C}_{f}(-\mathbf {K},\mathbf {K},\omega )$. In this case, Eq. (18) simplifies to the form

We note that a new matrix, i.e., $\mathcal {U}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$, appears in Eq. (19). Its elements $\mathcal {U}_{pq}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ quantify the degree of angular cross correlation of the scattering potentials of particles of different types (if $p\neq q$) or the degree of angular self-correlation of the scattering potentials of particles of the same type (if $p=q$). The normalization factors in the denominator of Eq. (19) are now equal to the trace of $\mathcal {I}\cdot \mathcal {G}(\mathbf {K_{1}},\mathbf {K_{1}},\omega )$ and the trace of $\mathcal {I}\cdot \mathcal {G}(\mathbf {K_{2}},\mathbf {K_{2}},\omega )$, respectively. The normalized CIF of the scattered field in this case is proportional to the squared modulus of the trace of $\mathcal {U}^{^\top }(\mathbf {K_{1}},\mathbf {K_{2}},\omega )\cdot \mathcal {G}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$.

(II) The spatial distributions of the densities of particles of different types are similar, i.e., $\widetilde {C}_{g_{pq}}(-\mathbf {K},\mathbf {K},\omega )\approx \widetilde {C}_{g}(-\mathbf {K},\mathbf {K},\omega )$, Eq. (18) reduces to

Likewise, we can see that Eq. (22) also contains a new matrix, i.e., $\mathcal {K}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$. Its elements $\mathcal {K}_{pq}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ quantify the degree of angular cross correlation of the density distributions of particles of different types (if $p\neq q$) or the degree of angular self-correlation of the density distributions of particles of the same type (if $p=q$). The normalization factors in the denominator of Eq. (22) now become the trace of $\mathcal {F}(\mathbf {K_{1}},\mathbf {K_{1}},\omega )\cdot \mathcal {I}$ and the trace of $\mathcal {F}(\mathbf {K_{2}},\mathbf {K_{2}},\omega )\cdot \mathcal {I}$, respectively. The normalized CIF of the scattered field in this case is proportional to the squared modulus of the trace of $\mathcal {F}^{^\top }(\mathbf {K_{1}},\mathbf {K_{2}},\omega )\cdot \mathcal {K}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$.

(III) Both the scattering potentials and the densities of particles of different types are similarly distributed in space, i.e., $\widetilde {C}_{f_{pq}}(-\mathbf {K},\mathbf {K},\omega )\approx \widetilde {C}_{f}(-\mathbf {K},\mathbf {K},\omega )$ and $\widetilde {C}_{g_{pq}}(-\mathbf {K},\mathbf {K},\omega )\approx \widetilde {C}_{g}(-\mathbf {K},\mathbf {K},\omega )$, we find from Eq. (18) that

In comparison Eq. (25) with Eq. (19) and Eq. (22), we can see that the normalized CIF of the scattered field in the present case has a more compact and profound form. Not only $\mathcal {U}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ and $\mathcal {K}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ appear simultaneously in Eq. (25), but the number of species of particles also appears as a scaled factor to ensure the normalization of the CIF of the scattered field. Clearly, the dependence of the normalized CIF of the scattered field on angular positions $\mathbf {K_{1}}$ and $\mathbf {K_{2}}$ in this special case is entirely up to the squared modulus of the trace of $\mathcal {U}^{^\top }(\mathbf {K_{1}},\mathbf {K_{2}},\omega )\cdot \mathcal {K}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$.

In the following, we will take two special hybrid particulate systems as examples to show how the non-zero cross correlation between particles of different types affects the normalized CIF of light scattering from collections of particles of different types, i.e., we will mainly pay attention to the effects of the off-diagonal elements of the PPM and the PSM on the normalized CIF of the scattered field.

4. Numerical examples

Firstly, we consider a system of random particles with determinate density distributions. A representation of this model can be a collection of particles suspended in a atmospheric mass, where irregular fluctuations in temperature and pressure of atmosphere usually lead the refractive indices of particles in different locations to be different and to be random functions of position space. If these particles move slowly, at least there will be no appreciable changes in their locations during the whole scattering process, and thus their density distributions may be determinate in space ([17], Sec. 6.3.1). For simplicity, we consider the situation where only two types of particles are contained in the system, and assume that both 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 have the following Gaussian-Schell models, i.e.,

The self-correlation functions of the density distributions of particles of the same type and the cross correlation functions of the density distributions of particles of different types have the following forms

In this case, from Eq. (10) and Eq. (11), the elements of the matrices $\mathcal {F}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ and $\mathcal {G}(\mathbf {K_{1}},\mathbf {K_{2}},\omega )$ can be readily calculated as

Once these matrix elements are known, the normalized CIF of the scattered field is straightforward from Eq. (18).

We now concern ourselves with the effects of the off-diagonal elements of the PPM on the normalized CIF of the scattered field. Figure 2(a) depicts the behaviors of the normalized CIF of the scattered field for different effective widths $\sigma _{12}$. It is found that when the effective width $\sigma _{12}$ decreases, the normalized CIF can be improved greatly, which means that even if the cross correlation between the scattering potentials of particles of different types are weak, it can still affect the CIF of the scattered field strongly. In comparison to the effective width $\sigma _{12}$, the effective correlation width $\eta _{12}$ can have a negligible influence on the normalized CIF of the scattered field, as can be seen from Fig. 2(b). This is because the effective width $\sigma _{12}$ and the effective correlation width $\eta _{12}$ meet the relation $\sigma _{12}/\eta _{12}\gg 1$ in our numerical calculations. In this case, the Gaussian Schell-model distributions in Eq. (26) can reduce to quasihomogeneous distributions, in which the well-known reciprocity relation holds [7,24], leading the effective correlation width $\eta _{12}$ to have no consequence on the normalized CIF of the scattered field.

 

Fig. 2. (a) Plots of the influence of the effective width $\sigma _{12}$ on the normalized CIF of the scattered field. 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. $\mathbf {s}_{0}=(0,0,1)$, $\mathbf {s}_{1}=(\sin {\theta _{1}}\cos {\phi _{1}},\sin {\theta _{1}}\sin {\phi _{1}},\cos {\theta _{1}})$, $\mathbf {s}_{2}=(\sin {\theta }\cos {\phi },\sin {\theta }\sin {\phi },\cos {\theta })$. $\phi =\phi _{1}=\pi /2$, $\theta _{1}=0$, $\lambda =632.8nm$. The parameters for calculations are $\sigma _{11}=\sigma _{22}=0.1\lambda$, $\eta _{11}=\eta _{22}=0.01\lambda$, $\eta _{12}=\eta _{21}=0.03\lambda$; (b) Plots of the influence of effective correlation width $\eta _{12}$ on the normalized CIF of the scattered field. $\sigma _{12}=\sigma _{21}=0.4\lambda$, and the other parameters for calculations are the same as Fig. 2(a).

Download Full Size | PDF

Next let us turn to the second model, i.e., a system of determinate particles with random density distributions. This model has been taken into account preliminarily in [20], where use has been made of the assumption that the density distributions of particles of different types are similarly distributed in space. Here this constraint will be relaxed and a more general case is focused, viz.,

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 have the following forms

We now focus on the effects of the off-diagonal elements of the PSM on the normalized CIF of the scattered field. Figure 3(a) presents the behaviors of the normalized CIF of the scattered field for three effective widths $\gamma _{12}$. It is revealed that the smaller the effective width $\gamma _{12}$, the more obvious of changes of the normalized CIF, and the normalized CIF in the current model can still has an appreciable value even if the scattered field is observed at the large scattering polar angle, compared to the first model. Figure 3(b) shows the distributions of the normalized CIF of the scattered field for three different effective correlation widths $\delta _{12}$, where $\delta _{12}$ now is comparable to $\gamma _{12}$. From Fig. 3(b) it follows that the effective angular width of the normalized CIF increases with the increase of the cross correlation width of the correlation function in Eq. (30), which demonstrates intuitively that the influence of the effective correlation width $\delta _{12}$ on the normalized CIF of the scattered field can be observed provided that the constraint $\gamma _{12}/\delta _{12}\gg 1$ is relieved.

 

Fig. 3. (a) Plots of the influence of the effective width $\gamma _{12}$ on the normalized CIF of the scattered field. The parameters for calculations are $\phi =\pi /2$, $\theta _{1}=0$, $\phi _{1}=\pi /2$, $\zeta _{1}=0.2\lambda$, $\zeta _{2}=0.1\lambda$, $\gamma _{11}=\gamma _{22}=0.1\lambda$, $\delta _{11}=\delta _{22}=0.02\lambda$, $\delta _{12}=\delta _{21}=0.01\lambda$; (b) Plots of the influence of the effective correlation width $\delta _{12}$ on the normalized CIF of the scattered field. $\gamma _{12}=\gamma _{21}=1\lambda$, $\delta _{11}=\delta _{22}=0.2\lambda$. The other parameters for calculations are the same as Fig. 3(a).

Download Full Size | PDF

5. Conclusions

In summary, we have paved a new pathway to access the CIF of light waves on scattering from a collection of particles of $\mathcal {L}$ types. The PPM and the PSM have been introduced to jointly formulate the CIF of the scattered field, and we have demonstrated that the CIF of the scattered field equals the squared modulus of the trace of the product of the PSM and the transpose of the PPM. This means that these two matrices not only contain all the angular correlation information of the whole collection required to determine the CIF of the scattered field, but also largely simplify the theoretical representation for the scattering of light from complex collection of scatters. We have also analyzed in detail the normalized CIF of the scattered field in three special cases, and found that the PPM and the PSM in these cases can reduce to two new matrices. Finally, two special hybrid particulate systems were taken as examples to illustrate the effects of the off-diagonal elements of the PPM and the PSM on the normalized CIF of the scattered field. It was found that the non-zero cross correlation between particles of different types has a strong impact on the normalized CIF of the scattered field. These findings contribute to reconstruct the structural characteristics of systems of particles of different types, such as the cross correlation information of the scattering potentials of each pair of particle types and that of their density distributions, from the measurement of the CIF of the scattered field.

It is worthwhile to mention that our new approach that the PPM and the PSM are jointly used to characterize the scattering of light waves from a collection of particles of $\mathcal {L}$ types can be viewed as an important addition to that developed by Tong and Korotkova [20] where only the PSM was used. Our approach is not limited to whether the randomness of the scattering potentials of particles or of their density distributions is invoked or not. This is because the PPM and the PSM themselves are independent of that, unlike the method in [20], where the validity of the column vector formed by the scattering potentials of individual particles is up to such a randomness. Very recently, the importance of our approach and how it differs from the previous attempt has been discussed exclusively in our another paper [25], where we only showed that the second-order coherence characteristic of the scattered field, i.e., spectral density and spectral degree of coherence, can be determined from the PPM and the PSM. Here the formalism in [25] has been extended to treat the CIF of the scattered field, and we have shown how the higher-order coherence characteristics of the scattered field can also be determined from the PPM and the PSM. Such a generalization gives a new protocol to investigate the higher-order coherence effects of the scattered field and have potential applications in ghost imaging and ghost scattering.

Finally, we also would like to mention that the intensity fluctuations in transmission of light through other types of scattering media have also been studied extensively, for example, turbulent atmosphere [26–28] and diffusive media [29–31]. In diffusive media, fluctuations in the total transmitted flux originate from the wave trajectories reflected from the medium (no reflection means no fluctuations). In [31], Kondrat’ev et al found, for a plane slab of a highly forward scattering medium, that a great enhancement of the fluctuations in the total transmitted flux is attained in the crossover from the ballistic to the diffusive regime at grazing angles, where small-angle multiple scattering leads the light to be effectively reflected. Here, we discussed the CIF of the single scattering of light waves from random collections of particles, and assumed that the statistical correlations of between particles within one type and across different types obey the so-called Gaussian-Schell models (Eq. (26) and Eq. (30)). Although these simplifications cannot fully explain detailed characteristics of light propagating through actual scattering media [32], this is enough to illustrate our formalism developed here and to show how the non-zero cross correlation between particles of different types affects the CIF of the scattered field. Also, the Gaussian-Schell models are appropriate for the descriptions of the troposphere and confined plasmas [17]. The problem of extending our results to a more realistic medium with a power-law fluctuation spectrum, for instance Kolmogorov-Obukhov power spectrum [33], is of great interest and is being addressed in the following work.