Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces

Geng Zhang; Zhensen Wu

doi:10.1364/OE.20.001491

1. Introduction

The investigation on the properties of the scattering electromagnetic waves and optics pulse from randomly rough surfaces has always been a very important topic, such as in radar, SAR remote sensing, surface detection and target recognition [1–6]. In order to study the properties better, Ishimaru employed the two-frequency mutual coherence function to study pulse scattering from rough surfaces and discuss the pulse broadening and the enhanced backscattering effect [7]. Phu experimentally studied the dependence of the angle and frequency of the scattered wave on the surface roughness [8]. Chen used Kirchhoff and physical optics approximation to derive the two-frequency mutual coherence function of the pulse plane wave and pulse beam [9]. Schertler and George derived the formulas of the two-frequency mutual correlation function of the backscattering from roughened sphere and roughened disk, but didn’t give the final numerical results [10, 11]. We also studied the two-frequency mutual coherence function of the scattering from arbitrarily shaped rough objects, and gave the numerical results for rough spheres and cylinders, analyzed the dependence of the function on the shape and the size of the objects and on the roughness of the surface [4]. Besides the two-frequency mutual coherence function, the angular correlation function (ACF) has also been presented, because observing the targets from different angles of view is needed to identify targets better. Michel and Donnell studied theoretically the angular dependence of the correlation functions of the scattering field from a one-dimensional, perfectly conducting rough surface, and demonstrated that the correlation functions exhibit two distinct and equal maxima, One of these is an autocorrelation peak, and the second peak arises from the cross correlation between two distinct scattered fields related by a reciprocity condition, namely, the angular memory effect [12]. Zhang and Tsang also presented detailed numerical studies of the ACF of the scattered light from rough surfaces with and without a buried object, indicating that the ACF is superior to the radar cross section (RCS) in the detection of buried objects [13]. Zhang and Tsang studied the angular correlation function (ACF) of the wave scattering from a buried object under a two-dimensional rough surface at two or more different incident and scattered angles and found that the cross-polarization components of ACF is more useful for the detection of the buried object [14]. Using the second-order Kirchhoff approximation and propagation shadowing functions, Le et al. derived the angular correlation function of scattering field and made a comparison with experiments [15, 16]. Le and Ishimaru provided a technique to obtain the mean topographic height through the angular memory effect in ACF and the two-frequency mutual coherence function, and conducted the experiments for rough surfaces of different statistics and scattering media of different type to prove the effectiveness of the technique [17].

The properties of the scattering from rough surfaces discussed above are the second-order moment of the scattered field, however, the properties of the scattering intensity, i.e., the fourth-order moment of the scattering field, are of considerable interest, since the intensity is the directly detected parameter by experiments not the scattered field. The solution for the fourth moment equation of waves in random media was given by Xu et al [18]. Experimental results were presented for the angular correlation functions of far-field intensity scattered by a conducting, one-dimensionally rough surface that produces backscattering enhancement [19]. The angular intensity correlations of the light multiply scattered from random rough surfaces with high slopes were studied, also the so-called memory effect was discussed [20]. The frequency spectrum of the intensity fluctuations of the scattered field was used to measure the parameters of vibrations and the surface roughness, also the experimental results were presented [21].According to Kirchhoff approximation, the fourth order moment statistical characteristics of the wave scattering from random rough surfaces were derived and the two-frequency mutual coherence function of the scattered intensity was investigated numerically [22]. The correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media was analytically derived, and its dependence on spatial Fourier transforms of both the intensity and degree of spatial correlation of scattering potentials were analyzed [23]. The expressions for the fourth-order moment of a random field incident upon deterministic and random scatterers were derived which can serve as a rigorous analytic prediction of a scattered field by scatterers and demonstrated how the intensity–intensity correlations were affected at various points in the far field [24]. The fourth- order moment of the scattered field discussed above were simplified as the square of the module of the correlation function of the scattered field, since it was assumed that the surface was very rough whose root-mean-square (RMS) height was greater than the wavelength. Based on Gaussian moment theorem, a general expression of the intensity correlation scattered from a weakly one-dimensional rough surface whose RMS of the surface was smaller than the wavelength were presented, and the results for the intensities with two different wavelengths were given [25], and the results can be easily reduced to the very rough surface.

In this paper, the correlation function of the intensity fluctuation from two-dimensional weakly homogeneous and isotopic rough surface whose fluctuation obeys Gaussian distribution is investigated. Based on Beckmann theory and Gaussian moment theorem, the expression for the correlation function is obtained, and then the results are presented. The numerical results for the two special cases, two-frequency correlation and angular correlation, are discussed in detail to demonstrate the dependence on the surface characteristic parameters and other conditions.

2. Normalized fluctuation correlation of the scattered intensity from two-dimensional weakly rough surfaces

The geometry of the scattering from a homogeneous and isotropic rough surface is shown in Fig. 1 . A coherent plane laser beam with a wavelength $λ$ illuminates a two-dimensional Gaussian rough surface $ζ (x, y)$ whose roughness is less than or equal to the wavelength $λ$ , and the incident angle is $θ_{i}$ , while the scattering angle is $(θ_{s}, φ_{s})$ . ${\vec{k}}_{i}, {\vec{k}}_{s}$ are the incident wave vector and the scattering wave vector, respectively. The time dependence of the field and the shadowing effect are omitted.

Fig. 1 Geometry of rough surface scattering.

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According to Beckmann theory for the scattered wave from rough surfaces [2], the far scattered field $E_{s} ({\overset{⇀}{k}}_{i}, {\overset{⇀}{k}}_{s})$ is given by

E_{s} ({\vec{k}}_{i}, {\vec{k}}_{s}) = K (ω) F (θ_{i}; θ_{s}, φ_{s}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} p (x, y) \exp (i \vec{υ} \cdot \vec{r}) d x d y

where

K (ω) = - i k \exp (i k R) / 2 π R

,

k = 2 π / λ

is the wave number of the light in free space, and

F (θ_{i}; θ_{s}, φ_{s}) = (1 + \cos θ_{i} \cos θ_{s} - \sin θ_{i} \sin θ_{s} \cos φ_{s}) / (\cos \cos θ_{i} + \cos θ_{s})

. To simplify our derivation in the following, an aperture function

p (x, y) = \exp [- (x^{2} + y^{2}) / D^{2}]

is introduced to expand the integral domain to infinite, and

D

is its dimension.

In Eq. (1), the vector $\vec{υ}$ is the difference between the incident wave vector and the scattering wave vector, which can be expressed as

\vec{υ} = {\vec{k}}_{i} - {\vec{k}}_{s} = υ_{x} \hat{x} + υ_{y} \hat{y} + υ_{z} \hat{z}

and

\vec{υ} \cdot \vec{r} = υ_{x} x + υ_{y} y + υ_{z} ζ (x, y)

where

υ_{x} = - k (\sin θ_{s} \cos φ_{s} - \sin θ_{i}), υ_{y} = - k \sin θ_{s} \sin φ_{s}, υ_{z} = - k (\cos φ_{s} + \cos θ_{s})

.

The scattered field $E_{s} ({\overset{⇀}{k}}_{i}, {\overset{⇀}{k}}_{s})$ can be expressed as

E_{s} ({\overset{⇀}{k}}_{i}, {\overset{⇀}{k}}_{s}) = Δ E_{s} ({\overset{⇀}{k}}_{i}, {\overset{⇀}{k}}_{s}) + 〈 E_{s} ({\overset{⇀}{k}}_{i}, {\overset{⇀}{k}}_{s}) 〉

Δ E_{s} ({\overset{⇀}{k}}_{i}, {\overset{⇀}{k}}_{s})

is the zero-mean Gaussian fluctuating component, and

〈 E_{s} ({\overset{⇀}{k}}_{i}, {\overset{⇀}{k}}_{s}) 〉

is its mean value, namely, the specular component.

Since the scattered intensity is $I = E E^{*}$ , and the fluctuation correlation of the scattered intensity is $C_{12} = 〈 I_{1} I_{2} 〉 - 〈 I_{1} 〉〈 I_{2} 〉$ , using Eq. (2) yields

\begin{array}{l} C_{12} = 〈 Δ E_{s 1} Δ E_{s 1}^{*} Δ E_{s 2} Δ E_{s 2}^{*} 〉 - 〈 Δ E_{s 1} Δ E_{s 1}^{*} 〉 〈 Δ E_{s 2} Δ E_{s 2}^{*} 〉 + 〈 E_{s 1} 〉 〈 E_{s 2} 〉 \\ \times (〈 Δ E_{s 1} Δ E_{s 2} 〉 + 〈 Δ E_{s 1} Δ E_{s 2}^{*} 〉 + 〈 Δ E_{s 1}^{*} Δ E_{s 2} 〉 + 〈 Δ E_{s 1}^{*} Δ E_{s 2}^{*} 〉) \end{array}

I_{n}

(n=1,2) indicates the intensity for different wavelengths or different angles.

Based on Gaussian-moment theorem [26], the fourth-order moment in Eq. (3) can be simplified as:

\begin{array}{l} 〈 Δ E_{s 1} Δ E_{s 1}^{*} Δ E_{s 2} Δ E_{s 2}^{*} 〉 = 〈 Δ E_{s 1} Δ E_{s 1}^{*} 〉 〈 Δ E_{s 2} Δ E_{s 2}^{*} 〉 \\ + 〈 Δ E_{s 1} Δ E_{s 2} 〉 〈 Δ E_{s 1}^{*} Δ E_{s 2}^{*} 〉 + 〈 Δ E_{s 1} Δ E_{s 2}^{*} 〉 〈 Δ E_{s 1}^{*} Δ E_{s 2} 〉 \end{array}

Inserting Eq. (4) and Eq. (2) into Eq. (3), we can obtain the fluctuation correlation of the scattered intensity which be expressed by two kinds of second-order moments and first-order moments of the scattered field

C_{12} = {| 〈 E_{s 1} E_{s 2}^{*} 〉 |}^{2} + {| 〈 E_{s 1} E_{s 2} 〉 |}^{2} - 2 {〈 E_{s 1} 〉}^{2} {〈 E_{s 2} 〉}^{2}

The normalized fluctuation correlation can be defined as

γ_{12} = \frac{〈 I_{1} I_{2} 〉 - 〈 I_{1} 〉 〈 I_{2} 〉}{\sqrt{(〈 I_{1}^{2} 〉 - {〈 I_{1} 〉}^{2}) (〈 I_{2}^{2} 〉 - 〈 I_{2}^{2} 〉)}} = \frac{C_{12}}{\sqrt{C_{11} C_{22}}}

By now, if we solve each term in Eq. (5), the result for the normalized intensity fluctuation correlation can be given.

3. Solution of the moments of the scattered field

In order to solve the normalized intensity fluctuation correlation $γ_{12}$ , we will derive in detail the moments of the scattered field in Eq. (5) in the following.

By Eq. (1), we can easily get the first term in Eq. (5)

〈 E_{s 1} E_{s 2}^{*} 〉 = C_{1} ∭ \int_{- \infty}^{\infty} p (x_{1}, y_{1}) p (x_{2}, y_{2}) 〈 \exp (i {\vec{υ}}_{1} \cdot {\vec{r}}_{1} - i {\vec{υ}}_{2} {\vec{r}}_{2}) 〉 d x_{1} d y_{1} d x_{2} d y_{2}

where

C_{1} = K_{1} K_{2}^{*} F_{1} F_{2}^{*}

.

Let ${\vec{υ}}_{⊥} = (υ_{x}, υ_{y}), {\vec{r}}_{⊥} = (x, y)$ , the term $〈 \exp (i {\vec{υ}}_{1} \cdot {\vec{r}}_{1} - i {\vec{υ}}_{2} {\vec{r}}_{2}) 〉$ can reduce to

〈 \exp (i {\vec{υ}}_{1} \cdot {\vec{r}}_{1} - i {\vec{υ}}_{2} {\vec{r}}_{2}) 〉 = \exp [i ({\vec{υ}}_{⊥ 1} \cdot {\vec{r}}_{⊥ 1} - {\vec{υ}}_{⊥ 2} \cdot {\vec{r}}_{⊥ 2})] 〈 \exp [i (υ_{z 1} ζ_{1} - υ_{z 2} ζ_{2})] 〉

where

〈 \exp [i (υ_{z 1} ζ_{1} - υ_{z 2} ζ_{2})] 〉

is the joint-characteristic function of the Gaussian random height variation

ζ (x, y)

which is given by

〈 \exp [i (υ_{z 1} ζ_{1} - υ_{z 2} ζ_{2})] 〉 = \exp {- \frac{1}{2} δ^{2} [υ_{z 1}^{2} - 2 υ_{z 1} υ_{z 2} ρ ({\vec{r}}_{⊥ 1} - {\vec{r}}_{⊥ 2}) + υ_{z 2}^{2}]}

δ

is the surface roughness, and

ρ ({\vec{r}}_{⊥ 1} - {\vec{r}}_{⊥ 2})

is the normalized autocorrelation function of the rough surface,

ρ ({\vec{r}}_{⊥ 1} - {\vec{r}}_{⊥ 2}) = \exp (- {| {\vec{r}}_{⊥ 1} - {\vec{r}}_{⊥ 2} |}^{2} / l_{c}^{2})

l_{c}

is the correlation length of the rough surface.

Making the following change of variables

\begin{array}{l} {\vec{r}}_{⊥ d} = {\vec{r}}_{⊥ 1} - {\vec{r}}_{⊥ 2} {\vec{r}}_{⊥ c} = ({\vec{r}}_{⊥ 1} + {\vec{r}}_{⊥ 2}) / 2 \\ {\vec{υ}}_{⊥ d} = {\vec{υ}}_{⊥ 1} - {\vec{υ}}_{⊥ 2} {\vec{υ}}_{⊥ c} = ({\vec{υ}}_{⊥ 1} + {\vec{υ}}_{⊥ 2}) / 2 \end{array}

and applying Eq. (8) and (9) to Eq. (7), Eq. (7) can be reduced to

\begin{array}{l} 〈 E_{s 1} E_{s 2}^{*} 〉 = C_{1} \exp [- \frac{1}{2} δ^{2} (υ_{z 1}^{2} + υ_{z 2}^{2})] \int \int_{- \infty}^{\infty} d {\vec{r}}_{⊥ c} d {\vec{r}}_{⊥ d} \exp (- \frac{2 {| {\vec{r}}_{⊥ c} |}^{2}}{D^{2}}) \\ \times \exp (- \frac{{| {\vec{r}}_{⊥ d} |}^{2}}{2 D^{2}}) \exp [i ({\vec{υ}}_{⊥ d} \cdot {\vec{r}}_{⊥ c} + {\vec{υ}}_{⊥ c} \cdot {\vec{r}}_{⊥ d})] \exp [υ_{z 1} υ_{z 2} δ^{2} ρ ({\vec{r}}_{⊥ d})] \end{array}

Now we can see that to solve the integral, the power function $\exp [υ_{z 1} υ_{z 2} δ^{2} ρ ({\vec{r}}_{⊥ d})]$ must be expanded into Taylor series expansion

\exp [υ_{z 1} υ_{z 2} δ^{2} ρ ({\vec{r}}_{⊥ d})] = \sum_{n = 0}^{\infty} \frac{{(υ_{z 1} υ_{z 2} δ^{2})}^{n}}{n!} ρ^{n} ({\vec{r}}_{⊥ d})

Finally the result of the Eq. (7) can be obtained

\begin{array}{l} 〈 E_{s 1} E_{s 2}^{*} 〉 = C_{1} π^{2} D^{4} \exp [- \frac{1}{2} δ^{2} (υ_{z 1}^{2} + υ_{z 2}^{2})] \exp [- \frac{D^{2} {| {\vec{υ}}_{⊥ d} |}^{2}}{8}] \\ \times \sum_{n = 0}^{\infty} \frac{l_{c}^{2} {(υ_{z 1} υ_{z 2} δ^{2})}^{n}}{n! (l_{c}^{2} + 2 n D^{2})} \exp (- \frac{D^{2} l_{c}^{2} {| {\vec{υ}}_{⊥ c} |}^{2}}{2 l_{c}^{2} + 4 n D^{2}}) \end{array}

With Eq. (1), the second kind of correlation function of the scattered field can be easily written as

\begin{array}{l} 〈 E_{s 1} E_{s 2} 〉 = C_{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d {\vec{r}}_{⊥ 1} d {\vec{r}}_{⊥ 2} p ({\vec{r}}_{⊥ 1}) p ({\vec{r}}_{⊥ 2}) \\ \times \exp [i ({\vec{υ}}_{⊥ 1} \cdot {\vec{r}}_{⊥ 1} + {\vec{υ}}_{⊥ 2} \cdot {\vec{r}}_{⊥ 2})] 〈 \exp [i (υ_{z 1} ζ_{1} + υ_{z 2} ζ_{2})] 〉 \end{array}

where

C_{2} = K_{1} K_{2} F_{1} F_{2}

and the term

〈 \exp [i (υ_{z 1} ζ_{1} + υ_{z 2} ζ_{2})] 〉

is given by

〈 \exp [i (υ_{z 1} ζ_{1} + υ_{z 2} ζ_{2})] 〉 = \exp {- \frac{1}{2} δ^{2} [υ_{z 1}^{2} + 2 υ_{z 1} υ_{z 2} ρ ({\vec{r}}_{⊥ d}) + υ_{z 2}^{2}]}

Utilizing Eq. (11) and also making a integral of ${\vec{r}}_{⊥ c}$ , Eq. (15) can be reduce to

\begin{array}{l} 〈 E_{s 1} E_{s 2} 〉 = \frac{C_{2} π D^{2}}{2} \exp [- \frac{1}{2} δ^{2} (υ_{z 1}^{2} + υ_{z 2}^{2})] \exp (- \frac{D^{2} {| {\vec{υ}}_{⊥ c} |}^{2}}{2}) \\ \times \int_{- \infty}^{\infty} d {\vec{r}}_{⊥ d} \exp (- \frac{{| {\vec{r}}_{⊥ d} |}^{2}}{2 D^{2}}) \exp (i {\vec{υ}}_{⊥ c d} \cdot {\vec{r}}_{⊥ d} / 2) \exp [- υ_{z 1} υ_{z 2} δ^{2} ρ ({\vec{r}}_{⊥ d})] \end{array}

Let $x_{d} = r \cos θ, y_{d} = r \sin θ$ , then

\begin{array}{l} 〈 E_{s 1} E_{s 2} 〉 = \frac{C_{2} π D^{2}}{2} \exp [- \frac{1}{2} δ^{2} (υ_{z 1}^{2} + υ_{z 2}^{2})] \exp (- \frac{D^{2} {| {\vec{υ}}_{⊥ c} |}^{2}}{2}) \int_{0}^{\infty} r d r \exp (- \frac{r^{2}}{2 D^{2}}) \\ \times \exp [- υ_{z 1} υ_{z 2} δ^{2} ρ (r)] {\int_{0}^{2 π} d θ \exp [\frac{i r}{2} (υ_{x d} \cos θ + υ_{y d} \sin θ)]} \end{array}

Using the integral relation [27]

\int_{0}^{2 π} \exp (i A \cos θ) d θ = 2 π J_{0} (| A |)

where

J_{0} (\cdot \cdot \cdot)

is the zero-order Bessel function, leads to a simpler result for Eq. (18)

\begin{array}{l} 〈 E_{s 1} E_{s 2} 〉 = C_{2} π^{2} D^{2} \exp [- \frac{1}{2} δ^{2} (υ_{z 1}^{2} + υ_{z 2}^{2})] \exp (- \frac{D^{2} {| {\vec{υ}}_{⊥ c} |}^{2}}{2}) \\ \times \int_{0}^{\infty} r d r \exp (- \frac{r^{2}}{2 D^{2}}) \exp [- υ_{z 1} υ_{z 2} δ^{2} ρ (r)] J_{0} (\frac{r | {\vec{υ}}_{⊥ d} |}{2}) \end{array}

Noting that in Eq. (19), because of the term $\exp [- υ_{z 1} υ_{z 2} δ^{2} ρ (r)]$ , it is difficult to get the solution analytically, therefore, a more simple approximation should be taken for it which we have discussed in our paper [25]. Using the Eq. (16) in the paper [25], Eq. (19) can be transformed into

\begin{array}{l} 〈 E_{s 1} E_{s 2} 〉 = C_{2} π^{2} D^{2} \exp [- \frac{1}{2} δ^{2} (υ_{z 1}^{2} + υ_{z 2}^{2})] \exp (- \frac{D^{2} {| {\vec{υ}}_{⊥ c} |}^{2}}{2}) {\exp (- υ_{z 1} υ_{z 2} δ^{2}) \\ \times \int_{0}^{r_{d}} r d r \exp (- \frac{r^{2}}{2 D^{2}}) J_{0} (\frac{r | {\vec{υ}}_{⊥ d} |}{2}) + \int_{r_{d}}^{\infty} r d r \exp (- \frac{r^{2}}{2 D^{2}}) J_{0} (\frac{r | {\vec{υ}}_{⊥ d} |}{2})} \end{array}

The value of $r_{d}$ is obtained from

r_{d} = l_{c} {\ln (\frac{υ_{z 1} υ_{z 2} δ^{2}}{\ln 2 - \ln [1 + \exp (- υ_{z 1} υ_{z 2} δ^{2})]})}^{1 / 2}

For further calculation we need to expand the zero-order Bessel function into series [27],

J_{0} (z) = \sum_{i = 0}^{\infty} {(- 1)}^{k} \frac{z^{2 k}}{2^{2 k} {(k!)}^{2}}

and using the integral relation,

γ (a, x) = \int_{0}^{x} e^{- t} t^{a - 1} d t Γ (a, x) = \int_{x}^{\infty} e^{- t} t^{a - 1} d t [Re a > 0]

The final result of Eq. (15) can be given by

\begin{array}{l} 〈 E_{s 1} E_{s 2} 〉 = C_{2} π^{2} D^{4} \exp [- \frac{1}{2} δ^{2} (υ_{z 1}^{2} + υ_{z 2}^{2})] \exp (- \frac{D^{2} {| {\vec{υ}}_{⊥ c} |}^{2}}{2}) \\ \times \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} {| {\vec{υ}}_{⊥ d} |}^{2 k} D^{2 k}}{2^{3 k} {(k!)}^{2}} [\exp (- υ_{z 1} υ_{z 2} δ^{2}) γ (k + 1, \frac{r_{d}^{2}}{2 D^{2}}) + Γ (k + 1, \frac{r_{d}^{2}}{2 D^{2}})] \end{array}

For the mean value of the scattered field in Eq. (5)

〈 E_{s} 〉 = K F \int_{- \infty}^{\infty} p ({\vec{r}}_{⊥}) \exp (i {\vec{υ}}_{⊥} \cdot {\vec{r}}_{⊥}) 〈 \exp [i υ_{z} ζ ({\vec{r}}_{⊥})] 〉 d {\vec{r}}_{⊥}

The term $〈 \exp [i υ_{z} ζ ({\vec{r}}_{⊥})] 〉$ in the equation above is the characteristic function of the Gaussian random variable

〈 \exp [i υ_{z} ζ ({\vec{r}}_{⊥})] 〉 = \exp (- \frac{1}{2} υ_{z}^{2} δ^{2})

Inserting Eq. (26) in Eq. (25), and with the integral relation

\int_{- \infty}^{\infty} \exp (- A^{2} x^{2}) \exp (i B x) d x = \frac{\sqrt{π}}{A} \exp (- \frac{B^{2}}{4 A^{2}})

Then the result is given

〈 E_{s} 〉 = K F π D^{2} \exp (- \frac{1}{2} υ_{z}^{2} δ^{2}) \exp (- \frac{D^{2} {| {\vec{υ}}_{⊥} |}^{2}}{4})

Substituting Eq. (14), (24) and (28) into Eq. (5), the analytical expression $C_{12}$ of the fluctuation correlation of the scattered intensity from two-dimensional weak Gaussian random rough surface can be obtained. Let the index ‘12’ be ‘11’ or ‘22’, we can get the correlation functions $C_{11}$ and $C_{22}$ , then the normalized fluctuation correlation function of the scattered intensity $γ_{12}$ .

4. Numerical results and analyses

In this paper, the correlation between the fluctuations of two scattered intensities under different conditions is investigated to analyze its influencing factors.

First the normalized two-frequency and two-angle correlation function of the scattered intensity fluctuation is discussed, as illustrated by Fig. 2 and Fig. 3 . The central wavelength is $0.6328 μ m$ and the surface roughness $δ$ and the correlation length $l_{c}$ are 0.6 $μ m$ and $4 μ m$ , respectively. The dimension of the aperture D is 2 mm. The angles are $θ_{s 1} = θ_{i 1} = θ_{1} = 20^{°}$ , $φ_{s 1} = φ_{s 2} = 0^{°}$ . Figure 2 gives the profile of the function $γ_{12}$ against the second angles $θ_{s 2} = θ_{i 2} = θ_{2}$ with different incident wavelength difference $Δ λ$ , while Fig. 3 shows the profile of the function $γ_{12}$ versus the wavelength difference $Δ λ$ when the second angle $θ_{2}$ is different.

Fig. 2 Correlation function $γ_{12}$ versus second angle $θ_{2}$ .

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Fig. 3 Correlation function $γ_{12}$ versus wavelength difference $Δ λ$ .

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In Fig. 2 there are two peaks in the profile of the function $γ_{12}$ when the wavelength difference is zero, one of the two peaks is at the specular reflection direction and the other is at its contrary direction, which is the so-called memory effect. Increasing the wavelength difference, the two peaks reduce to one gradually, also the values of the peaks decreases. In Fig. 3, the situation of the peak value of the profile of the function $γ_{12}$ versus the wavelength difference $Δ λ$ with an increasing second angle $θ_{2}$ moves to smaller wavelength differences step by step. Especially, when the two angles are equal, $θ_{2} = θ_{1}$ , the peak is at the center.

In order to explain the influence of the parameters on the function $γ_{12}$ more clearly, we decompose the function into two kinds: the two-frequency correlation, i.e., $λ_{1} \neq λ_{2}$ , $θ_{1} = θ_{2}$ , as shown in Fig. 4 - Fig. 7 ; the angular correlation, i.e., $λ_{1} = λ_{2}$ , $θ_{1} \neq θ_{2}$ , as shown in Fig. 8 –Fig. 11 .

Fig. 4 TFCF $γ_{12}$ versus RMS $δ$ with different wavelength-difference $Δ λ$ .

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Fig. 7 TFCF $γ_{12}$ versus wavelength difference $Δ λ$ with different $κ$ .

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Fig. 8 ACF $γ_{12}$ versus RMS $δ$ with different angles.

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Fig. 11 ACF $γ_{12}$ versus angle $θ_{2}$ with different $κ$ and RMS $δ$ .

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Let $θ_{1} = θ_{2} = θ$ and $κ = D / l_{c}$ , Fig. 4 and Fig. 5 show the two-frequency correlation function (TFCF) $γ_{12}$ changes the surface roughness $δ$ under different conditions. There are obvious local minimums in the profile. In Fig. 4, with the increase of the wavelength difference, there is a huge change in the profile. The smaller of the magnitude of the wavelength difference, the better of the correlation of the scattered intensity fluctuation. In the same magnitude of the wavelength difference, the profile of the correlation function seems similar; the difference is the situation and the value of the local minimum that a bigger wavelength difference results in a larger situation and a smaller value of the local minimum. In Fig. 5, the two wavelengths are 0.4765 $μ m$ and 0.5145 $μ m$ .With same angles, for example, $θ = 30^{°}$ , the profiles of the two-frequency correlation functions are same on the whole except the local minimum which is a smaller value and at a bigger roughness with a smaller the ratio $κ$ . While with a same ratio, $κ = 10^{3}$ and different angles, the local minimums with same values occur at different situations.

Fig. 5 TFCF $γ_{12}$ versus RMS $δ$ with different $κ$ and angle $θ$ .

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Figure 6 shows the influence of the roughness and the incident angle and the ratio $κ$ on the two-frequency correlation function $γ_{12}$ against the wavelength difference $Δ λ$ while the incident wavelength $λ$ is 0.5143 $μ m$ . We can see that the profile of the function versus $Δ λ$ becomes smoother than that versus $δ$ . In Fig. 6, with the increase of the roughness, $θ = 45^{°}$ the profile decreases dramatically. Also the incident angle influences the correlation. Figure 7 shows that the profile of the two-frequency correlation decreases slower with the increase of the ratio $κ$ which is similar to that in Fig. 5.

Fig. 6 TFCF $γ_{12}$ versus wavelength difference $Δ λ$ with RMS $δ$ and angle $θ$ .

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We also discuss angular correlation of the scattering intensity fluctuation from rough surfaces. Figure 8 and Fig. 9 above show the angular correlation versus the roughness. In Fig. 8, with the angular difference increasing, the angular correlation function decreases more quickly and with a much smaller local minimum, however, the local minimums appear at the same roughness. The difference of the incident angle results in the different situations and values of the local minimums. The influence of the ratio $κ$ on the angular correlation function versus roughness is very little, as shown in Fig. 9. It only makes a very small change in the situation and value of the local minimum. While with same angular differences, the correlation varies dramatically.

Fig. 9 ACF $γ_{12}$ versus RMS $δ$ with different $Δ θ$ and $κ$ .

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Figures 10 and 11 above show the tendency of the angular correlation function versus the second angle. Obviously, there is the memory effect. Increasing the roughness and with same first angles results in a quicker variation in the profile except its peak values, also the incident angle gives a huge influence on the profile, as illustrated in Fig. 10; however, the ratio $κ$ does not influence the profile on the whole, as shown in Fig. 11.

Fig. 10 ACF $γ_{12}$ versus angle $θ_{2}$ with different angle $θ_{1}$ and RMS $δ$ .

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5. Conclusion

In this paper, we have studied a kind of fourth-order moment, i.e., the intensity correlation scattered from two-dimensional random weak rough surfaces based on Beckmann theory and Gaussian moment theorem. The scattered intensity correlation function is derived which can be expressed by two kinds of second-order moment and first-order moment of the scattered field. The derivation is applied to slightly fluctuating surfaces with Gaussian field statistics, an assumption that can easily be fulfilled in practice. With some mathematical approximation, analytical expressions are presented and the numerical results are given which show the dependence of the correlation function of the scattered intensity fluctuation on the incident conditions and the rough surface characteristic parameters. The numerical results show that the wavelength difference and the angular difference, as well as the roughness parameters have a great impact on the fourth-order moment. When the fourth-order moment decomposes into two kinds of correlation function, two-frequency correlation function and angular correlation, the parameters have different degree of the influence on them. Especially, in the profiles versus the roughness, there are obvious local minimums while in the profile of the angular correlation there is memory effect. The work performed in this paper gives us a much better knowledge about the scattering properties of roughness and also provides theoretical basis to investigate the more meaningful scattering problem of three-dimensional rough object which we will discuss in future.

Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant No. 61172031.

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Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces

Abstract

1. Introduction

2. Normalized fluctuation correlation of the scattered intensity from two-dimensional weakly rough surfaces

3. Solution of the moments of the scattered field

4. Numerical results and analyses

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (31)

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