Correlation between intensity fluctuations of electromagnetic waves scattered from a spatially quasi-homogeneous, anisotropic medium

Jia Li; Feinan Chen; Liping Chang

doi:10.1364/OE.24.024274

Optics Express
Vol. 24,
Issue 21,
pp. 24274-24286
(2016)
•https://doi.org/10.1364/OE.24.024274

Correlation between intensity fluctuations of electromagnetic waves scattered from a spatially quasi-homogeneous, anisotropic medium

Jia Li, Feinan Chen, and Liping Chang

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Jia Li, Feinan Chen, and Liping Chang, "Correlation between intensity fluctuations of electromagnetic waves scattered from a spatially quasi-homogeneous, anisotropic medium," Opt. Express 24, 24274-24286 (2016)

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- Coherence and Statistical Optics
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About this Article
History
- Original Manuscript: August 9, 2016
- Revised Manuscript: September 16, 2016
- Manuscript Accepted: September 21, 2016
- Published: October 11, 2016

Abstract

Within the validity of the first-order Born approximation, expressions are derived for the correlation between intensity fluctuations (CIF) of an electromagnetic plane wave scattered from a spatially quasi-homogeneous (QH), anisotropic medium. Upon establishing the correlation matrix of the scattering potential of the medium, we show that the CIF is the summation of Fourier transforms of the strengths and normalized correlation coefficients (NCCs) of the scattering potential matrix. Numerical results reveal that the CIF is susceptible to the effective width and correlation length of the medium, and degree of polarization of the incident electromagnetic wave. Our study not only extends the current knowledge of the CIF of a scattered field but also provides an important reference to the study of high-order intensity correlations of light scattered from a spatially anisotropic medium.

1. Introduction

During the past few decades, the investigations of the intensity fluctuation (IF) of the variety of random beams have attracted substantial interests in remote sensing, free space optical communication and ghost imaging [1–5]. Starting from the unified theory of coherence and polarization of beams, Wolf et al. indicated that the IF of a radiated beam is closely related to the degree of coherence of the source [6, 7]. With regard to the propagation of a stochastic electromagnetic beam in free space, it was shown that the CIF of the beam in the far field depends on the source’s degree of coherence and degree of polarization [8–11]. Other relevant reports demonstrated that the IF of an electromagnetic wave can be induced by the fluctuations of the two-point Stokes parameters or mutual degree of coherence [12, 13]. For the scattering cases, the IF properties of light scattered from a spatially random or deterministic medium were also studied in numerous literature. Unlike the free-space propagation case, the IF scattered from a medium of the Schell-model type is subjected to the degree of correlation of the medium [14]. The correlation between intensity fluctuation (CIF) of light scattered from a quasi-homogeneous (QH) medium was introduced and derived in analytic expressions [15], and later it was generalized to the case where the scattering of a stochastic electromagnetic beam from medium was studied [16]. In our previous research, we discussed how the CIF of a scattered field depends on the degree of correlation of a QH medium [17]. In addition, full considerations were also taken on the CIF scattered from a complex medium that incorporates the rough surface and turbulence [18–20].

Being similar to the spatially isotropic medium, the anisotropic medium have also attracted substantial interests during the past few decades. Typical medium of this kind include the solid and liquid crystals that have been frequently applied in sharping the spectral profiles of a weak-scattered light [21, 22]. Furthermore, the red blood cells of human, i.e. the elliptocyte behaves as the oval or enlongated shape that can be described by introducing the anisotropic models [23, 24]. Above studies have been paid close attentions due to their broad applications found in the biomedical diagnosis and biological imaging. Unlike the isotropic medium, the anisotropy-induced changes to the scattered spectrum in the far field may carry essential information of the anisotropic scatterer [25]. However, a question may arise: what will be the effect of the anisotropy of the medium on the CIF properties of the scattered field? We have noticed that such question has not been given any satisfactory answer so far. In this paper, we aim to solve the problem by introducing a 3 × 3 correlation matrix of the scattering potential (CMSP) that only occupies the diagonal elements. By using such correlation matrix, the anisotropy of a QH medium can be represented by setting up different coefficients for the matrix elements. Within the validity of the first-order Born approximation, we derive analytic expressions for the CIF scattered from the medium. Numerical results reveal that the CIF of the scattered field is susceptible to the degree of polarization of the incident electromagnetic wave, correlation lengths and effective widths of the medium. Finally, our results are compared with those obtained in the previous literature.

2. An electromagnetic wave scatters from a spatially QH, anisotropic medium

We start the analysis by reviewing the scattering theory of a 2D electromagnetic wave from a spatially QH medium. Without the loss of generality, we assume that the incident field is the plane wave that takes the following form

U_{j}^{(i)} (r', ω) = a_{j} (ω) \exp (i k s_{0} \cdot r'), (j = x, y),

where

a_{j} (ω)

is the spectral amplitude of the plane wave along the j axis,

k = ω / c

is the wave number,

r'

is the position vector and

s_{0}

is the unit vector that represents the propagation direction of the plane wave.

As shown in Fig. 1, an electromagnetic plane wave is incident on a spatially QH, anisotropic medium that occupies a volume D. Suppose that the interaction between the incident wave and the scatterer is extremely weak, so that the scattering process can be treated within the validity of the first-order Born approximation. In this regard, the three-dimensional (3D) scattered field components are given by the integral equations [26]

U_{x}^{(s)} (r s, ω) = \int_{D} F_{x} (r', ω) U_{x}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

U_{y}^{(s)} (r s, ω) = \cos^{2} θ \int_{D} F_{y} (r', ω) U_{y}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

U_{z}^{(s)} (r s, ω) = - \sin θ \cos θ \int_{D} F_{z} (r', ω) U_{y}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

where the superscripts (i) and (s) account for the incident and scattered fields, respectively, θ is the scattering angle, s denotes the unit vector that represents the propagation direction of the scattered wave. Upon deriving Eqs. (2)-(4), we have assumed that the spatial distribution of the far-zone scattered field is symmetric with respect to z axis. For such case, it means that the scattered intensity is independent of the polar angle ϕ. In Eqs. (2)-(4),

F_{j} (r', ω)

is the diagonal element of the scattering potential matrix of the medium [26]

F (r', ω) = (\begin{matrix} F_{x} (r', ω) & 0 & 0 \\ 0 & F_{y} (r', ω) & 0 \\ 0 & 0 & F_{z} (r', ω) \end{matrix}) .

Through writing Eq. (5), one might recall two analytical models for representing the anisotropic characteristics of a spatially anisotropic medium. For the first model, it stipulates that in the 3D Cartesian coordinate system, the correlation functions of the scattering potential have different coefficients within the scatterer volume [27–29]. In contrast with the first model, the second one requires that the dielectric susceptibility of the medium is characterized by a 3 × 3 matrix having only diagonal elements [26]. Both models are appropriate to describe the anisotropic property of a QH medium. In this paper, we utilize the second model to handle with the problem, i.e. scattering of an electromagnetic plane wave from an anisotropic medium. In Eqs. (2)-(4),

G (r s, r', ω)

is the outgoing free-space Green function. Typically, it has the asymptotic approximation in the far field

G (r s, r', ω) \approx \frac{\exp (i k r)}{r} \exp (- i k s \cdot r') .

In what follows, we will concentrate on deriving the CIF of an electromagnetic plane wave scattered from such medium. Prior to solving this problem, we first consider to introduce a pair of fields that are perpendicular to each other, as shown in Fig. 1

U_{α}^{(s)} (r s, ω) = U_{x}^{(s)} (r s, ω) e_{x},

U_{β}^{(s)} (r s, ω) = U_{y}^{(s)} (r s, ω) e_{y} + U_{z}^{(s)} (r s, ω) e_{z},

where

U_{α}^{(s)}, U_{β}^{(s)}

and s are mutually perpendicular to each other in the 3D Cartesian coordinate system

U_{α}^{(s)} (r s, ω) \times s = - U_{β}^{(s)} (r s, ω), U_{β}^{(s)} (r s, ω) \times s = U_{α}^{(s)} (r s, ω), U_{α}^{(s)} (r s, ω) \times U_{β}^{(s)} (r s, ω) = s .

The IF of the scattered field specified at a position vector

r s

is given by

Δ I^{(s)} (r s, ω) = I^{(s)} (r s, ω) - 〈 I^{(s)} (r s, ω) 〉 .

Hence, the two-point CIF of the scattered field results in the analytic form [15–17]

C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = 〈 Δ I^{(s)} (r_{1} s_{1}) Δ I^{(s)} (r_{2} s_{2}) 〉 = 〈 I^{(s)} (r_{1} s_{1}) I^{(s)} (r_{2} s_{2}) 〉 - 〈 I^{(s)} (r_{1} s_{1}) 〉 〈 I^{(s)} (r_{2} s_{2}) 〉 .

For simplicity, in Eq. (10) the frequency dependence of the CIF is omitted, and the angular bracket denotes the ensemble average over the scattered field. Based on Eqs. (7)-(9), Eq. (10) can be rewritten by the using following summation expansion

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = 〈 [U_{α}^{(s) *} (r_{1} s_{1}) U_{β}^{(s) *} (r_{1} s_{1})] \cdot [\begin{matrix} U_{α}^{(s)} (r_{1} s_{1}) \\ U_{β}^{(s)} (r_{1} s_{1}) \end{matrix}] \cdot [U_{α}^{(s) *} (r_{2} s_{2}) U_{β}^{(s) *} (r_{2} s_{2})] \cdot [\begin{matrix} U_{α}^{(s)} (r_{2} s_{2}) \\ U_{β}^{(s)} (r_{2} s_{2}) \end{matrix}] 〉 \\ - 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) + U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) 〉 \cdot 〈 U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) + U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉 \\ = 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) 〉 + 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉 \\ + 〈 U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) 〉 + 〈 U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉 \\ - 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) 〉 \cdot 〈 U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) 〉 - 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) 〉 \cdot 〈 U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉 \\ - 〈 U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) 〉 \cdot 〈 U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) 〉 - 〈 U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) 〉 \cdot 〈 U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉, \end{array}

where the asterisk denotes complex conjugate, the dot signs in the first line represent inner products of vectors. By substituting from Eqs. (2)-(4) and Eqs. (7)-(8) into Eq. (11), the CIF of the scattered field is represented by the integral form

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = {(\frac{S_{x}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{x}^{*} (r_{3}') F_{x} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{x}^{*} (r_{3}') F_{x} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉] \\ \times \exp [i k (s_{2} - s_{0}) \cdot r_{1}' - i k (s_{2} - s_{0}) \cdot r_{2}' + i k (s_{1} - s_{0}) \cdot r_{3}' - i k (s_{1} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{1} \cos^{2} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉] \\ \times \exp [i k (s_{2} - s_{0}) \cdot r_{1}' - i k (s_{2} - s_{0}) \cdot r_{2}' + i k (s_{1} - s_{0}) \cdot r_{3}' - i k (s_{1} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \cos^{4} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{y}^{*} (r_{1}') F_{y} (r_{2}') F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉 - 〈 F_{y}^{*} (r_{1}') F_{y} (r_{2}') 〉 \cdot 〈 F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{y}^{*} (r_{1}') F_{y} (r_{2}') F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉 - 〈 F_{y}^{*} (r_{1}') F_{y} (r_{2}') 〉 \cdot 〈 F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{z}^{*} (r_{1}') F_{z} (r_{2}') F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉 - 〈 F_{z}^{*} (r_{1}') F_{z} (r_{2}') 〉 \cdot 〈 F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{z}^{*} (r_{1}') F_{z} (r_{2}') F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉 - 〈 F_{z}^{*} (r_{1}') F_{z} (r_{2}') 〉 \cdot 〈 F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}', \end{array}

where

S_{j}^{(i)} (ω) = 〈 {| a_{j} (ω) |}^{2} 〉

(j = x, y) is the spectral density of the incident field along the j axis. It was shown that the 3 × 3 CMSP of a spatially QH, anisotropic medium has the form [26]

C_{i j}^{(F)} (r_{1}', r_{2}', ω) = 〈 F_{i}^{*} (r_{1}', ω) F_{j} (r_{2}', ω) 〉, (i, j = x, y, z) .

where each element satisfies the decomposition principle

C_{i j}^{(F)} (r_{1}', r_{2}', ω) \approx S_{i j}^{(F)} (\frac{r_{1}' + r_{2}'}{2}, ω) η_{i j}^{(F)} (r_{1}' - r_{2}', ω), (i, j = x, y, z) .

with

S_{i j}^{(F)}

and

η_{i j}^{(F)}

being the strength and NCC of the scattering potential of the medium, respectively. For simplicity, we assume that the scattering potential of the medium obeys the Gaussian statistics. Based on the Gaussian momentum theorem [15–18,27], the fourth-order moments of the scattering potentials can decompose into

\begin{array}{l} 〈 F_{i}^{*} (r_{1}') F_{j} (r_{2}') F_{u}^{*} (r_{3}') F_{v} (r_{4}') 〉 = 〈 F_{i}^{*} (r_{1}') F_{j} (r_{2}') 〉 \cdot 〈 F_{u}^{*} (r_{3}') F_{v} (r_{4}') 〉 + 〈 F_{i}^{*} (r_{1}') F_{v} (r_{4}^{'}) 〉 \cdot 〈 F_{u}^{*} (r_{3}') F_{j} (r_{2}') 〉, \\ (i, j, u, v = x, y, z) . \end{array}

Next, we substitute Eqs. (13)-(15) into Eq. (12) and change the integral variables

R_{s}^{+} = r_{1}' + r_{4}', R_{s}^{-} = r_{1}' - r_{4}', U_{s}^{+} = r_{3}' + r_{2}', U_{s}^{-} = r_{3}' - r_{2}' .

As a result, Eq. (12) can be rewritten as the integral form

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = \frac{1}{4} {(\frac{S_{x}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{x x}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x x}^{(F)} (R_{s}^{-}) S_{x x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{x x}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{x y}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x y}^{(F)} (R_{s}^{-}) S_{y x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{y x}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{x z}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x z}^{(F)} (R_{s}^{-}) S_{z x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{z x}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} S_{x y}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x y}^{(F)} (R_{s}^{-}) S_{y x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{y x}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{2} - s_{1}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{2} - s_{1}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{1} \cos^{2} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} S_{x z}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x z}^{(F)} (R_{s}^{-}) S_{z x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{z x}^{(F)} (U_{s}^{-}) \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \cos^{4} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{y y}^{(F)} (\frac{R_{s}^{+}}{2}) η_{y y}^{(F)} (R_{s}^{-}) S_{y y}^{(F)} (\frac{U_{s}^{+}}{2}) η_{y y}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{2} - s_{1}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{2} - s_{1}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{y z}^{(F)} (\frac{R_{s}^{+}}{2}) η_{y z}^{(F)} (R_{s}^{-}) S_{z y}^{(F)} (\frac{U_{s}^{+}}{2}) η_{z y}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} S_{z y}^{(F)} (\frac{R_{s}^{+}}{2}) η_{z y}^{(F)} (R_{s}^{-}) S_{y z}^{(F)} (\frac{U_{s}^{+}}{2}) η_{y z}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{z z}^{(F)} (\frac{R_{s}^{+}}{2}) η_{z z}^{(F)} (R_{s}^{-}) S_{z z}^{(F)} (\frac{U_{s}^{+}}{2}) η_{z z}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} . \end{array}

It can be noticed that the integrations in Eq. (17) can be performed by the Fourier transforms of the strength and NCC of the scattering potential. After performing tedious but straightforward integrations, we can obtain the CIF of the scattered field as the summation of Fourier transforms of the scattering potentials of the medium

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = \frac{1}{4} {(\frac{S_{x}^{(i)} (ω)}{r_{1} r_{2}})}^{2} {\tilde{S}}_{x x}^{(F)} [k (s_{2} - s_{1})] {\tilde{S}}_{x x}^{(F)} [k (s_{1} - s_{2})] {{\tilde{η}}_{x x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{2} {\tilde{S}}_{x y}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{x y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{y x}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{y x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{2} \cos^{2} θ_{2} {\tilde{S}}_{x z}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{x z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{z x}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{z x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{1} {\tilde{S}}_{x y}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{x y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{y x}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{y x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{1} \cos^{2} θ_{1} {\tilde{S}}_{x z}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{x z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{z x}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{z x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \cos^{4} θ_{2} {\tilde{S}}_{y y}^{(F)} [k (s_{2} - s_{1})] {\tilde{S}}_{y y}^{(F)} [k (s_{1} - s_{2})] {{\tilde{η}}_{y y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} {\tilde{S}}_{y z}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{y z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{z y}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{z y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{2} \sin^{2} θ_{1} \cos^{2} θ_{1} {\tilde{S}}_{y z}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{y z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{z y}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{z y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \sin^{2} θ_{1} \cos^{2} θ_{2} {\tilde{S}}_{z z}^{(F)} [k (s_{2} - s_{1})] {\tilde{S}}_{z z}^{(F)} [k (s_{1} - s_{2})] {{\tilde{η}}_{z z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2}, \end{array}

where

{\tilde{S}}_{i j}^{(F)} (k s) = \int_{D} S_{i j}^{(F)} (R_{s}^{+}) \exp (i k R_{s}^{+} \cdot s) d^{3} R_{s}^{+},

{\tilde{η}}_{i j}^{(F)} (k s) = \int_{D} η_{i j}^{(F)} (R_{s}^{-}) \exp (i k R_{s}^{-} \cdot s) d^{3} R_{s}^{-},

are the 3D Fourier transforms of

S_{i j}^{(F)}

and

η_{i j}^{(F)}

, respectively. Notice that

S_{i j}^{(F)} (R_{s}^{+})

must be real and satisfies the following property

{\tilde{S}}_{i j}^{(F)} [k (s_{2} - s_{1})] = {\tilde{S}}_{j i}^{(F)} [k (s_{2} - s_{1})] = {{\tilde{S}}_{i j}^{(F)} [k (s_{1} - s_{2})]}^{*} .

Whereas

η_{i j}^{(F)} (R_{s}^{-})

varies rapidly with

R_{s}^{-}

and may take a complex value. Hence, the following approximation must be satisfied

{{\tilde{η}}_{i j}^{(F)} [k (\frac{s_{1} + s_{2}}{2} - s_{0})]}^{*} \approx {{\tilde{η}}_{i j}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{*} = {\tilde{η}}_{j i}^{(F)} [k (\frac{s_{1} + s_{2}}{2} - s_{0})] .

By substituting from Eq. (21) and (22) into Eq. (18), the CIF of the scattered field can be simplified to the following expression

Fig. 1 Schematic diagram for the scattering of an electromagnetic wave from a spatially QH, anisotropic medium that occupies the spatial volume D. $s_{0}$ is the unit vector that represents the propagation direction of the incident wave, $s$ is the unit vector that accounts for the propagation direction of the scattered wave, θ is the scattering angle. $U_{y}^{(i)}$ denotes the incident field component along the y axis. $U_{x}^{(s)}, U_{y}^{(s)}, U_{z}^{(s)}$ are the 3D components of the scattered field. $U_{α}^{(s)}$ and $U_{β}^{(s)}$ are the mutually perpendicular field components that remain orthogonal to $s .$ The sign “•” means that the field component is perpendicular to the y-z plane.

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\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = \frac{1}{4} {(\frac{S_{x}^{(i)} (ω)}{r_{1} r_{2}})}^{2} {| {\tilde{S}}_{x x}^{(F)} [k (s_{2} - s_{1})] |}^{2} {{\tilde{η}}_{x x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} (\cos^{4} θ_{1} + \cos^{4} θ_{2}) {| {\tilde{S}}_{x y}^{(F)} [k (s_{2} - s_{1})] |}^{2} {| {\tilde{η}}_{x y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] |}^{2} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} (\sin^{2} θ_{1} \cos^{2} θ_{1} + \sin^{2} θ_{2} \cos^{2} θ_{2}) {| {\tilde{S}}_{x z}^{(F)} [k (s_{2} - s_{1})] |}^{2} {| {\tilde{η}}_{x z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] |}^{2} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} {\cos^{4} θ_{1} \cos^{4} θ_{2} {| {\tilde{S}}_{y y}^{(F)} [k (s_{2} - s_{1})] |}^{2} {{\tilde{η}}_{y y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} \\ + \frac{1}{4} \cos^{2} θ_{1} \cos^{2} θ_{2} (\cos^{2} θ_{1} \sin^{2} θ_{2} + \cos^{2} θ_{2} \sin^{2} θ_{1}) {| {\tilde{S}}_{y z}^{(F)} [k (s_{2} - s_{1})] |}^{2} {| {\tilde{η}}_{y z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] |}^{2} \\ + \frac{1}{4} \sin^{2} θ_{1} \cos^{2} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} {| {\tilde{S}}_{z z}^{(F)} [k (s_{2} - s_{1})] |}^{2} {{\tilde{η}}_{z z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} . \end{array}

3. Numerical results and discussions

Equation (23) is the primary result that helps to study the CIF properties of light scattered from a spatially QH, anisotropic medium. It is shown that the two-point CIF is the summation of Fourier transforms of the strength and NCC of the scattering potential. Hence, we may conclude that the anisotropy of a QH medium is capable to affect the CIF distributions of the scattered field. Without the loss of generality, Eq. (23) is the generalization of the analytic result for the CIF scattered from a spatially isotropic medium [17]. When we compare Eq. (23) with Eq. (25) of [17], it is found that the CIF expression for the anisotropic medium case is more tedious than that scattered from the isotropic medium. This result is induced by the fact that the anisotropic medium is represented by the 3 × 3 CMSP rather than the scattering potential function. Accordingly, the strength and NCC of the scattering potential of the anisotropic medium are also given by the 3 × 3 matrices that compose only the diagonal elements. Therefore, the CIF of light scattered from the anisotropic medium contains both the self-correlation and cross-correlation terms. Furthermore, if we assume that the QH medium is spatially isotropic in distribution, the strength and NCC of the CMSP elements have the following properties

\begin{array}{l} {\tilde{S}}_{i j}^{(F)} [k (s_{2} - s_{1})] = {\tilde{S}}^{(F)} [k (s_{2} - s_{1})], {\tilde{η}}_{i j}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] = {\tilde{η}}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})], \\ (i, j = x, y, z) . \end{array}

By substituting from Eq. (24) into Eq. (23), we can obtain the CIF for the isotropic medium case

C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = \frac{1}{r_{1}^{2} r_{2}^{2}} [S_{x}^{(i)} (ω) + \cos^{2} θ_{1} S_{y}^{(i)} (ω)] \cdot [S_{x}^{(i)} (ω) + \cos^{2} θ_{2} S_{y}^{(i)} (ω)] {| {\tilde{S}}^{(F)} [k (s_{2} - s_{1})] |}^{2} {{\tilde{η}}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} .

Notice that Eq. (25) is consistent with Eq. (25) of [17]. In what follows, numerical results are shown to reveal the CIF properties of the scattered wave from an anisotropic QH medium. For simplicity, we suppose that the strength and NCC of the CMSP elements obey the Gaussian distributions

S_{i j}^{(F)} (R_{s}^{+}) = \frac{A}{{(2 π σ_{i j}^{2})}^{3 / 2}} \exp [- \frac{{(R_{s}^{+})}^{2}}{2 σ_{i j}^{2}}], η_{i j}^{(F)} (R_{s}^{-}) = \frac{B}{{(2 π δ_{i j}^{2})}^{3 / 2}} \exp [- \frac{{(R_{s}^{-})}^{2}}{2 δ_{i j}^{2}}], (i, j = x, y, z),

where A and B are positive constants,

σ_{i j}

and

δ_{i j}

denote the effective width and the correlation length of the scattering potential of the medium, respectively. These parameters are positive and may depend on frequency. Also,

σ_{i j} > > δ_{i j}

is satisfied to validate the spatial homogeneity of the 3D scatterer. By substituting from Eq. (26) into Eq. (23) and letting

θ_{1} = - θ_{2} = θ,

r_{1} = r_{2} = r,

the CIF specified at two scattering angles is obtained by performing the Fourier transforms of the strength and NCC of the CMSP elements

\begin{array}{l} C I F^{(s)} (θ, - θ, r) = \frac{1}{4} {(\frac{S_{x}^{(i)} (ω)}{r^{2}})}^{2} A^{2} B^{2} \exp [- 4 k^{2} σ_{x x}^{2} \sin^{2} θ - k^{2} δ_{x x}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{2 r^{4}} A^{2} B^{2} \cos^{4} θ \exp [- 4 k^{2} σ_{x y}^{2} \sin^{2} θ - k^{2} δ_{x y}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{2 r^{4}} A^{2} B^{2} \sin^{2} θ \cos^{2} θ \exp [- 4 k^{2} σ_{x z}^{2} \sin^{2} θ - k^{2} δ_{x z}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r^{2}})}^{2} A^{2} B^{2} {\cos^{8} θ \exp [- 4 k^{2} σ_{y y}^{2} \sin^{2} θ - k^{2} δ_{y y}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{1}{2} \cos^{6} θ \sin^{2} θ \exp [- 4 k^{2} σ_{y z}^{2} \sin^{2} θ - k^{2} δ_{y z}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{1}{4} \cos^{4} θ \sin^{4} θ \exp [- 4 k^{2} σ_{z z}^{2} \sin^{2} θ - k^{2} δ_{z z}^{2} {(\cos θ - 1)}^{2}]} . \end{array}

Figures 2-4 show the CIFs of the electromagnetic plane wave scattered from the anisotropic medium. The numerical parameters are chosen as σ_xx = σ_yy = σ_zz = σ, δ_xx = δ_yy = δ_zz = δ, unless otherwise specified. When plotting these figures, we normalize the CIF expression, i.e. Equation (27) by dividing the constant factor $\frac{{[S^{(i)} (ω)]}^{2}}{r^{4}} A^{2} B^{2} .$ Fig. 2 exhibits the dependence of the CIF on the effective widths of the medium, and the incident plane wave is unpolarized, i.e., $S_{x}^{(i)} (ω) = S_{y}^{(i)} (ω) .$ It is found that by changing the effective widths of the medium, it induces the different CIF distributions of the scattered field. In addition, the reductions of σ_xx, σ_yy, σ_zz, and σ_xy narrow down the spatial distributions of the CIF. By contrast, changes of σ_yz and σ_xz, hardly affect the CIF distributions when these parameters are sufficiently small (see Figs. 2(c) and 2(d)).

Fig. 2 Dependence of the CIF of the scattered field on the effective widths σ, σ_xy, σ_yz, and σ_xz of the QH, anisotropic medium. The unpolarized plane wave on incidence is considered, i.e. $S_{x}^{(i)} (ω) = S_{y}^{(i)} (ω),$ the uniform parameters are chosen as δ = 0.3λ, δ_xy = δ_yz = 0.2λ, δ_xz = 0.1λ. (a) σ_yz = 3λ, σ_xy = σ_xz = 4λ, (b) σ = 5λ, σ_yz = 3λ, σ_xz = 4λ, (c) σ = 5λ, σ_xy = σ_xz = 4λ, (d) σ = 5λ, σ_xy = 4λ, σ_yz = 3λ.

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Fig. 3 Dependence of the CIF of the scattered field on the correlation lengths δ, δ_xy, δ_yz, and δ_xz of the QH, anisotropic medium. The unpolarized incident plane wave on incidence is considered, i.e., $S_{x}^{(i)} (ω) = S_{y}^{(i)} (ω),$ the uniform parameters are chosen as σ = 5λ, σ_xy = σ_xz = 4λ, σ_yz = 3λ. (a) δ_xy = δ_yz = 0.2λ, δ_xz = 0.1λ, (b) δ = 0.3λ, δ_xz = 0.1λ, δ_yz = 0.2λ, (c) δ = 0.3λ, δ_xy = 0.2λ, δ_xz = 0.1λ, (d) δ = 0.3λ, δ_xy = δ_yz = 0.2λ.

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Fig. 4 Effects of the polarization of the incident plane wave on the CIF distributions of the scattered field for different correlation lengths of the medium. Linear polarization along x axis: $S_{y}^{(i)} (ω) = 0;$ unpolarized wave: $S_{x}^{(i)} (ω) = S_{y}^{(i)} (ω);$ linear polarization along y axis: $S_{x}^{(i)} (ω) = 0.$ (a) δ = 0.1λ, (b) δ = 0.6λ, (c) δ_xy = 0.2λ, (d) δ_xy = 0.6λ. Other numerical parameters are the same as those in Fig. 3.

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Figure 3 reveals the effects of the correlation lengths δ, δ_xy, δ_yz and δ_xz on the CIF of the scattered field. Compared with the effective widths of the medium, it is shown that the CIF distribution is more susceptible to the changes of the correlation lengths. Moreover, it is distinguishable that the CIF distribution is more sensitive to the cross-correlation lengths (δ_yz and δ_xz) than the self-correlation length (δ).

Figure 4 presents the effects of the polarization of the incident plane wave on the CIF distributions of the scattered field. For Figs. 4(a)-4(d), we choose different correlation lengths in the numerical simulation. Results indicate that the CIF value is susceptible to the changes of the polarization of the plane wave. Compared with the linear polarization, an unpolarized electromagnetic plane wave induces larger CIF of the scattered field, except when δ = 0.6λ is considered (see Fig. 4(b)). Compared with the incident wave whose polarization is along x axis, the incident plane wave with linear polarization along y axis results in larger CIF of the scattered field.

Furthermore, we make comparisons between the obtained results and those from the previous literature. From observing Eq. (23) and Eq. (27), it is shown that the CIF composes the self-correlation and cross-correlation components that are induced by the anisotropy of the scattering potential in 3D spatial domain. These correlation effects narrow down or broaden the CIF distributions, depending on the global choice of the correlation lengths of the medium. The CIF of the scattered field not only depends on the scattering angle and polarization of the incident wave but also associate with the effective widths and correlation lengths of the medium. This is why Eq. (23) entirely differs from either Eq. (25) of [17] or Eq. (15) of [15].

To further highlight the importance of our work, we generalize the potential applications of the results into three aspects:

1. The results extends the current knowledge of the Hanbury Brown–Twiss theorem [13] to the scattering case where the anisotropic medium is addressed. The CIF of the electromagnetic wave scattered from a QH, anisotropic medium helps to reconstruct the CMSP. As a result, structural parameters of the medium, for instance the effective widths and correlation lengths can be obtained from measuring the scattered CIF. By utilizing Eq. (23), further assumptions can be made to the CMSP of the anisotropic medium, for example, it can compose non-zero non-diagonal elements. It might be an interesting work that deserves to be done in future.
2. The results are useful for studying high-order intensity correlations of scattered random beams. For example, high-order moments (typically larger than two) of a statistically stationary optical field are significant for the determination of an unknown anisotropic scatterer [2]. In such case, the CIF of the scattered field can be obtained from Eq. (23) or Eq. (27).
3. The results provide essential references for the biomedical diagnosis and biological imaging. By using the theoretical model introduced in this paper, the intensity correlation of light scattered from an anisotropic tissue layer can be calculated and compared with the experimental results.

4. Conclusion

Within the validity of the first-order Born approximation, analytic expressions are derived for the CIF of light scattered from a spatially QH, anisotropic medium. We show that the CIF composes the self-correlation and cross-correlation components that are produced by the anisotropic medium. It is found that the analytic form of the CIF scattered from the medium is more complicated than that from an isotropic medium. Results also show that the CIF not only depends on the polarization of the incident wave but also closely relates to the effective widths and correlation lengths of the medium. Our results have potential applications to the determination of an unknown anisotropic object.

Funding

J. Li and L. Chang acknowledge the support from the National Natural Science Foundation of China (NSFC) (61205121, 61304124) and the Natural Science Foundation of Zhejiang Province (LY13F010009, LY15F050012). F. Chen’s research is supported by the NSFC (11504383).

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Fig. 1 Schematic diagram for the scattering of an electromagnetic wave from a spatially QH, anisotropic medium that occupies the spatial volume D.

s_{0}

is the unit vector that represents the propagation direction of the incident wave,

s

is the unit vector that accounts for the propagation direction of the scattered wave, θ is the scattering angle.

U_{y}^{(i)}

denotes the incident field component along the y axis.

U_{x}^{(s)}, U_{y}^{(s)}, U_{z}^{(s)}

are the 3D components of the scattered field.

U_{α}^{(s)}

and

U_{β}^{(s)}

are the mutually perpendicular field components that remain orthogonal to

s .

The sign “•” means that the field component is perpendicular to the y-z plane.

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Fig. 2 Dependence of the CIF of the scattered field on the effective widths σ, σ_xy, σ_yz, and σ_xz of the QH, anisotropic medium. The unpolarized plane wave on incidence is considered, i.e.

S_{x}^{(i)} (ω) = S_{y}^{(i)} (ω),

the uniform parameters are chosen as δ = 0.3λ, δ_xy = δ_yz = 0.2λ, δ_xz = 0.1λ. (a) σ_yz = 3λ, σ_xy = σ_xz = 4λ, (b) σ = 5λ, σ_yz = 3λ, σ_xz = 4λ, (c) σ = 5λ, σ_xy = σ_xz = 4λ, (d) σ = 5λ, σ_xy = 4λ, σ_yz = 3λ.

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Fig. 3 Dependence of the CIF of the scattered field on the correlation lengths δ, δ_xy, δ_yz, and δ_xz of the QH, anisotropic medium. The unpolarized incident plane wave on incidence is considered, i.e.,

S_{x}^{(i)} (ω) = S_{y}^{(i)} (ω),

the uniform parameters are chosen as σ = 5λ, σ_xy = σ_xz = 4λ, σ_yz = 3λ. (a) δ_xy = δ_yz = 0.2λ, δ_xz = 0.1λ, (b) δ = 0.3λ, δ_xz = 0.1λ, δ_yz = 0.2λ, (c) δ = 0.3λ, δ_xy = 0.2λ, δ_xz = 0.1λ, (d) δ = 0.3λ, δ_xy = δ_yz = 0.2λ.

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Fig. 4 Effects of the polarization of the incident plane wave on the CIF distributions of the scattered field for different correlation lengths of the medium. Linear polarization along x axis:

S_{y}^{(i)} (ω) = 0;

unpolarized wave:

S_{x}^{(i)} (ω) = S_{y}^{(i)} (ω);

linear polarization along y axis:

S_{x}^{(i)} (ω) = 0.

(a) δ = 0.1λ, (b) δ = 0.6λ, (c) δ_xy = 0.2λ, (d) δ_xy = 0.6λ. Other numerical parameters are the same as those in Fig. 3.

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Equations (27)

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U_{j}^{(i)} (r', ω) = a_{j} (ω) \exp (i k s_{0} \cdot r'), (j = x, y),

U_{x}^{(s)} (r s, ω) = \int_{D} F_{x} (r', ω) U_{x}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

U_{y}^{(s)} (r s, ω) = \cos^{2} θ \int_{D} F_{y} (r', ω) U_{y}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

U_{z}^{(s)} (r s, ω) = - \sin θ \cos θ \int_{D} F_{z} (r', ω) U_{y}^{(i)} (r', ω) G (r s, r', ω) d^{3} r',

F (r', ω) = (\begin{matrix} F_{x} (r', ω) & 0 & 0 \\ 0 & F_{y} (r', ω) & 0 \\ 0 & 0 & F_{z} (r', ω) \end{matrix}) .

G (r s, r', ω) \approx \frac{\exp (i k r)}{r} \exp (- i k s \cdot r') .

U_{α}^{(s)} (r s, ω) = U_{x}^{(s)} (r s, ω) e_{x},

U_{β}^{(s)} (r s, ω) = U_{y}^{(s)} (r s, ω) e_{y} + U_{z}^{(s)} (r s, ω) e_{z},

U_{α}^{(s)} (r s, ω) \times s = - U_{β}^{(s)} (r s, ω), U_{β}^{(s)} (r s, ω) \times s = U_{α}^{(s)} (r s, ω), U_{α}^{(s)} (r s, ω) \times U_{β}^{(s)} (r s, ω) = s .

C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = 〈 Δ I^{(s)} (r_{1} s_{1}) Δ I^{(s)} (r_{2} s_{2}) 〉 = 〈 I^{(s)} (r_{1} s_{1}) I^{(s)} (r_{2} s_{2}) 〉 - 〈 I^{(s)} (r_{1} s_{1}) 〉 〈 I^{(s)} (r_{2} s_{2}) 〉 .

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = 〈 [U_{α}^{(s) *} (r_{1} s_{1}) U_{β}^{(s) *} (r_{1} s_{1})] \cdot [\begin{matrix} U_{α}^{(s)} (r_{1} s_{1}) \\ U_{β}^{(s)} (r_{1} s_{1}) \end{matrix}] \cdot [U_{α}^{(s) *} (r_{2} s_{2}) U_{β}^{(s) *} (r_{2} s_{2})] \cdot [\begin{matrix} U_{α}^{(s)} (r_{2} s_{2}) \\ U_{β}^{(s)} (r_{2} s_{2}) \end{matrix}] 〉 \\ - 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) + U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) 〉 \cdot 〈 U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) + U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉 \\ = 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) 〉 + 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉 \\ + 〈 U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) 〉 + 〈 U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉 \\ - 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) 〉 \cdot 〈 U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) 〉 - 〈 U_{α}^{(s) *} (r_{1} s_{1}) U_{α}^{(s)} (r_{1} s_{1}) 〉 \cdot 〈 U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉 \\ - 〈 U_{α}^{(s) *} (r_{2} s_{2}) U_{α}^{(s)} (r_{2} s_{2}) 〉 \cdot 〈 U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) 〉 - 〈 U_{β}^{(s) *} (r_{1} s_{1}) U_{β}^{(s)} (r_{1} s_{1}) 〉 \cdot 〈 U_{β}^{(s) *} (r_{2} s_{2}) U_{β}^{(s)} (r_{2} s_{2}) 〉, \end{array}

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = {(\frac{S_{x}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{x}^{*} (r_{3}') F_{x} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{x}^{*} (r_{3}') F_{x} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉] \\ \times \exp [i k (s_{2} - s_{0}) \cdot r_{1}' - i k (s_{2} - s_{0}) \cdot r_{2}' + i k (s_{1} - s_{0}) \cdot r_{3}' - i k (s_{1} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{1} \cos^{2} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉 - 〈 F_{x}^{*} (r_{1}') F_{x} (r_{2}') 〉 \cdot 〈 F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉] \\ \times \exp [i k (s_{2} - s_{0}) \cdot r_{1}' - i k (s_{2} - s_{0}) \cdot r_{2}' + i k (s_{1} - s_{0}) \cdot r_{3}' - i k (s_{1} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \cos^{4} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{y}^{*} (r_{1}') F_{y} (r_{2}') F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉 - 〈 F_{y}^{*} (r_{1}') F_{y} (r_{2}') 〉 \cdot 〈 F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{y}^{*} (r_{1}') F_{y} (r_{2}') F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉 - 〈 F_{y}^{*} (r_{1}') F_{y} (r_{2}') 〉 \cdot 〈 F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{z}^{*} (r_{1}') F_{z} (r_{2}') F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉 - 〈 F_{z}^{*} (r_{1}') F_{z} (r_{2}') 〉 \cdot 〈 F_{y}^{*} (r_{3}') F_{y} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}' \\ + {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} [〈 F_{z}^{*} (r_{1}') F_{z} (r_{2}') F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉 - 〈 F_{z}^{*} (r_{1}') F_{z} (r_{2}') 〉 \cdot 〈 F_{z}^{*} (r_{3}') F_{z} (r_{4}') 〉] \\ \times \exp [i k (s_{1} - s_{0}) \cdot r_{1}' - i k (s_{1} - s_{0}) \cdot r_{2}' + i k (s_{2} - s_{0}) \cdot r_{3}' - i k (s_{2} - s_{0}) \cdot r_{4}'] d^{3} r_{1}' d^{3} r_{2}' d^{3} r_{3}' d^{3} r_{4}', \end{array}

C_{i j}^{(F)} (r_{1}', r_{2}', ω) = 〈 F_{i}^{*} (r_{1}', ω) F_{j} (r_{2}', ω) 〉, (i, j = x, y, z) .

C_{i j}^{(F)} (r_{1}', r_{2}', ω) \approx S_{i j}^{(F)} (\frac{r_{1}' + r_{2}'}{2}, ω) η_{i j}^{(F)} (r_{1}' - r_{2}', ω), (i, j = x, y, z) .

\begin{array}{l} 〈 F_{i}^{*} (r_{1}') F_{j} (r_{2}') F_{u}^{*} (r_{3}') F_{v} (r_{4}') 〉 = 〈 F_{i}^{*} (r_{1}') F_{j} (r_{2}') 〉 \cdot 〈 F_{u}^{*} (r_{3}') F_{v} (r_{4}') 〉 + 〈 F_{i}^{*} (r_{1}') F_{v} (r_{4}^{'}) 〉 \cdot 〈 F_{u}^{*} (r_{3}') F_{j} (r_{2}') 〉, \\ (i, j, u, v = x, y, z) . \end{array}

R_{s}^{+} = r_{1}' + r_{4}', R_{s}^{-} = r_{1}' - r_{4}', U_{s}^{+} = r_{3}' + r_{2}', U_{s}^{-} = r_{3}' - r_{2}' .

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = \frac{1}{4} {(\frac{S_{x}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{x x}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x x}^{(F)} (R_{s}^{-}) S_{x x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{x x}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{x y}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x y}^{(F)} (R_{s}^{-}) S_{y x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{y x}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{x z}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x z}^{(F)} (R_{s}^{-}) S_{z x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{z x}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} S_{x y}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x y}^{(F)} (R_{s}^{-}) S_{y x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{y x}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{2} - s_{1}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{2} - s_{1}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{1} \cos^{2} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} S_{x z}^{(F)} (\frac{R_{s}^{+}}{2}) η_{x z}^{(F)} (R_{s}^{-}) S_{z x}^{(F)} (\frac{U_{s}^{+}}{2}) η_{z x}^{(F)} (U_{s}^{-}) \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \cos^{4} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{y y}^{(F)} (\frac{R_{s}^{+}}{2}) η_{y y}^{(F)} (R_{s}^{-}) S_{y y}^{(F)} (\frac{U_{s}^{+}}{2}) η_{y y}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{2} - s_{1}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{2} - s_{1}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{y z}^{(F)} (\frac{R_{s}^{+}}{2}) η_{y z}^{(F)} (R_{s}^{-}) S_{z y}^{(F)} (\frac{U_{s}^{+}}{2}) η_{z y}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \int_{D} \int_{D} \int_{D} \int_{D} S_{z y}^{(F)} (\frac{R_{s}^{+}}{2}) η_{z y}^{(F)} (R_{s}^{-}) S_{y z}^{(F)} (\frac{U_{s}^{+}}{2}) η_{y z}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} \int_{D} \int_{D} \int_{D} \int_{D} S_{z z}^{(F)} (\frac{R_{s}^{+}}{2}) η_{z z}^{(F)} (R_{s}^{-}) S_{z z}^{(F)} (\frac{U_{s}^{+}}{2}) η_{z z}^{(F)} (U_{s}^{-}) \\ \times \exp [i k \frac{R_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k R_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0}) - i k \frac{U_{s}^{+}}{2} \cdot (s_{1} - s_{2}) + i k U_{s}^{-} \cdot (\frac{s_{1} + s_{2}}{2} - s_{0})] d^{3} R_{s}^{+} d^{3} R_{s}^{-} d^{3} U_{s}^{+} d^{3} U_{s}^{-} . \end{array}

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = \frac{1}{4} {(\frac{S_{x}^{(i)} (ω)}{r_{1} r_{2}})}^{2} {\tilde{S}}_{x x}^{(F)} [k (s_{2} - s_{1})] {\tilde{S}}_{x x}^{(F)} [k (s_{1} - s_{2})] {{\tilde{η}}_{x x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{2} {\tilde{S}}_{x y}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{x y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{y x}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{y x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{2} \cos^{2} θ_{2} {\tilde{S}}_{x z}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{x z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{z x}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{z x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \cos^{4} θ_{1} {\tilde{S}}_{x y}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{x y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{y x}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{y x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} \sin^{2} θ_{1} \cos^{2} θ_{1} {\tilde{S}}_{x z}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{x z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{z x}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{z x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \cos^{4} θ_{2} {\tilde{S}}_{y y}^{(F)} [k (s_{2} - s_{1})] {\tilde{S}}_{y y}^{(F)} [k (s_{1} - s_{2})] {{\tilde{η}}_{y y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} {\tilde{S}}_{y z}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{y z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{z y}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{z y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \cos^{4} θ_{2} \sin^{2} θ_{1} \cos^{2} θ_{1} {\tilde{S}}_{y z}^{(F)} [k (s_{1} - s_{2})] {\tilde{η}}_{y z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] {\tilde{S}}_{z y}^{(F)} [k (s_{2} - s_{1})] {\tilde{η}}_{z y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} \sin^{2} θ_{1} \cos^{2} θ_{1} \sin^{2} θ_{1} \cos^{2} θ_{2} {\tilde{S}}_{z z}^{(F)} [k (s_{2} - s_{1})] {\tilde{S}}_{z z}^{(F)} [k (s_{1} - s_{2})] {{\tilde{η}}_{z z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2}, \end{array}

{\tilde{S}}_{i j}^{(F)} (k s) = \int_{D} S_{i j}^{(F)} (R_{s}^{+}) \exp (i k R_{s}^{+} \cdot s) d^{3} R_{s}^{+},

{\tilde{η}}_{i j}^{(F)} (k s) = \int_{D} η_{i j}^{(F)} (R_{s}^{-}) \exp (i k R_{s}^{-} \cdot s) d^{3} R_{s}^{-},

{\tilde{S}}_{i j}^{(F)} [k (s_{2} - s_{1})] = {\tilde{S}}_{j i}^{(F)} [k (s_{2} - s_{1})] = {{\tilde{S}}_{i j}^{(F)} [k (s_{1} - s_{2})]}^{*} .

{{\tilde{η}}_{i j}^{(F)} [k (\frac{s_{1} + s_{2}}{2} - s_{0})]}^{*} \approx {{\tilde{η}}_{i j}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{*} = {\tilde{η}}_{j i}^{(F)} [k (\frac{s_{1} + s_{2}}{2} - s_{0})] .

\begin{array}{l} C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = \frac{1}{4} {(\frac{S_{x}^{(i)} (ω)}{r_{1} r_{2}})}^{2} {| {\tilde{S}}_{x x}^{(F)} [k (s_{2} - s_{1})] |}^{2} {{\tilde{η}}_{x x}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} (\cos^{4} θ_{1} + \cos^{4} θ_{2}) {| {\tilde{S}}_{x y}^{(F)} [k (s_{2} - s_{1})] |}^{2} {| {\tilde{η}}_{x y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] |}^{2} \\ + \frac{1}{4} \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{{(r_{1} r_{2})}^{2}} (\sin^{2} θ_{1} \cos^{2} θ_{1} + \sin^{2} θ_{2} \cos^{2} θ_{2}) {| {\tilde{S}}_{x z}^{(F)} [k (s_{2} - s_{1})] |}^{2} {| {\tilde{η}}_{x z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] |}^{2} \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r_{1} r_{2}})}^{2} {\cos^{4} θ_{1} \cos^{4} θ_{2} {| {\tilde{S}}_{y y}^{(F)} [k (s_{2} - s_{1})] |}^{2} {{\tilde{η}}_{y y}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} \\ + \frac{1}{4} \cos^{2} θ_{1} \cos^{2} θ_{2} (\cos^{2} θ_{1} \sin^{2} θ_{2} + \cos^{2} θ_{2} \sin^{2} θ_{1}) {| {\tilde{S}}_{y z}^{(F)} [k (s_{2} - s_{1})] |}^{2} {| {\tilde{η}}_{y z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] |}^{2} \\ + \frac{1}{4} \sin^{2} θ_{1} \cos^{2} θ_{1} \sin^{2} θ_{2} \cos^{2} θ_{2} {| {\tilde{S}}_{z z}^{(F)} [k (s_{2} - s_{1})] |}^{2} {{\tilde{η}}_{z z}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} . \end{array}

\begin{array}{l} {\tilde{S}}_{i j}^{(F)} [k (s_{2} - s_{1})] = {\tilde{S}}^{(F)} [k (s_{2} - s_{1})], {\tilde{η}}_{i j}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})] = {\tilde{η}}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})], \\ (i, j = x, y, z) . \end{array}

C I F^{(s)} (r_{1} s_{1}, r_{2} s_{2}) = \frac{1}{r_{1}^{2} r_{2}^{2}} [S_{x}^{(i)} (ω) + \cos^{2} θ_{1} S_{y}^{(i)} (ω)] \cdot [S_{x}^{(i)} (ω) + \cos^{2} θ_{2} S_{y}^{(i)} (ω)] {| {\tilde{S}}^{(F)} [k (s_{2} - s_{1})] |}^{2} {{\tilde{η}}^{(F)} [k (s_{0} - \frac{s_{1} + s_{2}}{2})]}^{2} .

S_{i j}^{(F)} (R_{s}^{+}) = \frac{A}{{(2 π σ_{i j}^{2})}^{3 / 2}} \exp [- \frac{{(R_{s}^{+})}^{2}}{2 σ_{i j}^{2}}], η_{i j}^{(F)} (R_{s}^{-}) = \frac{B}{{(2 π δ_{i j}^{2})}^{3 / 2}} \exp [- \frac{{(R_{s}^{-})}^{2}}{2 δ_{i j}^{2}}], (i, j = x, y, z),

\begin{array}{l} C I F^{(s)} (θ, - θ, r) = \frac{1}{4} {(\frac{S_{x}^{(i)} (ω)}{r^{2}})}^{2} A^{2} B^{2} \exp [- 4 k^{2} σ_{x x}^{2} \sin^{2} θ - k^{2} δ_{x x}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{2 r^{4}} A^{2} B^{2} \cos^{4} θ \exp [- 4 k^{2} σ_{x y}^{2} \sin^{2} θ - k^{2} δ_{x y}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{S_{x}^{(i)} (ω) S_{y}^{(i)} (ω)}{2 r^{4}} A^{2} B^{2} \sin^{2} θ \cos^{2} θ \exp [- 4 k^{2} σ_{x z}^{2} \sin^{2} θ - k^{2} δ_{x z}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{1}{4} {(\frac{S_{y}^{(i)} (ω)}{r^{2}})}^{2} A^{2} B^{2} {\cos^{8} θ \exp [- 4 k^{2} σ_{y y}^{2} \sin^{2} θ - k^{2} δ_{y y}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{1}{2} \cos^{6} θ \sin^{2} θ \exp [- 4 k^{2} σ_{y z}^{2} \sin^{2} θ - k^{2} δ_{y z}^{2} {(\cos θ - 1)}^{2}] \\ + \frac{1}{4} \cos^{4} θ \sin^{4} θ \exp [- 4 k^{2} σ_{z z}^{2} \sin^{2} θ - k^{2} δ_{z z}^{2} {(\cos θ - 1)}^{2}]} . \end{array}

Abstract

1. Introduction

2. An electromagnetic wave scatters from a spatially QH, anisotropic medium

3. Numerical results and discussions

4. Conclusion

Funding

References and links

Cited By

Figures (4)

Equations (27)

Optics Express