Bi-frequency correlation properties of the scattered intensity from dielectric rough surfaces

Geng Zhang; Zhensen Wu

doi:10.1364/OE.20.014833

Optics Express
Vol. 20,
Issue 14,
pp. 14833-14847
(2012)
•https://doi.org/10.1364/OE.20.014833

Bi-frequency correlation properties of the scattered intensity from dielectric rough surfaces

Geng Zhang and Zhensen Wu

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Geng Zhang and Zhensen Wu, "Bi-frequency correlation properties of the scattered intensity from dielectric rough surfaces," Opt. Express 20, 14833-14847 (2012)

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- Scattering
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: May 10, 2012
- Revised Manuscript: June 1, 2012
- Manuscript Accepted: June 2, 2012
- Published: June 18, 2012

Abstract

The analytical expression for the Bi-frequency correlation function of the intensity scattered from two-dimensional dielectric randomly rough surfaces obeying Gaussian distribution are presented based on the scalar Kirchhoff approximation theory with the root-mean-square (rms) slope of the surface less than 0.25 and the Gaussian moment theorem. The results show that the bi-frequency correlation properties of the scattered intensity closely depend on the incident and scattered conditions as well as on the statistical parameters and complex refractive index of the surface. Especially, the correlation function mainly comes from the specular direction, and the coherence bandwidth and the function decrease with the increase of the roughness of the rough surface. In addition, comparing with the real part, the imagery of the complex refractive index has a greater impact on the bi-frequency correlation function.

1. Introduction

The problem of the scatterings of electromagnetic waves and optics from randomly rough surfaces has always been a very important topic, and has very widely scientific and technical applications, such as in radar, SAR remote sensing, surface detection and target recognition [1–5]. In many radio and optical measurements, regardless of the radar waves or the lasers emitting pulses with some bandwidth, it is necessary to study the correlation of the scattered fields with different frequency. Ishimaru employed the two-frequency mutual coherence function to study pulse scattering from rough surfaces and discussed the pulse broadening and the enhanced backscattering effect [6]. Schertler and George derived the formulas of the two-frequency mutual correlation function of the backscattering from roughened sphere and roughened disk [7, 8]. Chen et al derived the two-frequency mutual coherent function to investigate the pulse scattering properties of the pulse plane wave and pulse beam from randomly rough surfaces [9]. The two-frequency mutual coherence function of the scattering from arbitrarily shaped rough objects were obtained and the numerical results for rough spheres and cylinders were given to analyze the dependence of the function on the shape and the size of the objects and on the roughness of the surface [10].

In practice, the intensity not the scattered field is the directly detected parameter by experiments; therefore, the correlation properties of the scattering intensity are of more considerable interest. The solution for the fourth moment equation of waves in random media was given by Xu et al [11]. The frequency spectrum of the intensity fluctuations of the scattered field was used to measure the parameters of vibrations and the surface roughness, and the experimental results were presented [12]. 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 [13]. 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 [14]. 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 [15]. Most of the discussions above simplified the correlation function of the intensity as the square of the module of the correlation function of the scattered field, since the surface considered was rougher than the incident wavelength. With Gaussian moment theorem, the correlation properties of the intensity scattered from one-dimensional and two-dimensional randomly conducting rough surfaces with RMS roughness smaller than the wavelength were discussed [16, 17] in detail. However, during the investigations the influences of the slopes and the dielectric property of the surface on the intensity correlation function were not considered for simplicity.

Based on the scalar Kirchhoff approximation theory, the bi-frequency correlation function of the scattered intensity from two-dimensional dielectric randomly rough surface with a root-mean -square slope smaller than 0.25 is investigated in this paper. Assuming the distribution of the rough surface to be Gaussian, the expression for the bi-frequency correlation function of the scattered intensity is obtained, and then the results and the influence factors are presented. It is noted that the bi-frequency correlation function of the scattered intensity closely dependent on the roughness of the surface mainly comes from the specular direction when the surface is weakly fluctuating.

2. Bi-frequency correlation function of the scattered intensity from dielectric rough surfaces

When the surface discussed is weakly rough that the roughness is smaller than the incident wavelength, the multiple scattering from different area of the rough surface and the shadowing effect can be omitted. The rms slope of the surface smaller than 0.25, according to the scalar Kirchhoff approximation theory, the scattered field from two-dimensional randomly dielectric rough surface can be expressed as [18]

E_{p q}^{s} (f) = E_{0} K (f) \int_{- \infty}^{\infty} p (x, y) U_{p q} \exp [i (k_{s} - k_{i}) \cdot r] d S

where

K (f) = - i k \exp (- i k R) / 4 π R

, and

k = 2 π / λ = 2 π f / c

is the wave-number in free space,

λ

is the incident wavelength,

f

is the incident frequency,

c

is the light velocity. The 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.

U_{p q} = a_{0} + a_{1} Z_{x} + a_{2} Z_{y}

, where

a_{0}, a_{1}, a_{2}

are polarization coefficients whose expressions are in reference [18], and

Z_{x}, Z_{y}

represent the slope of the surface along

{\hat{e}}_{x}

{\hat{e}}_{y}

directions at the point

r

, respectively.

r = x {\hat{e}}_{x} + y {\hat{e}}_{y} + h (x, y) {\hat{e}}_{z}

h (x, y)

is the fluctuating function of the surface, and

k_{s} = k (\sin θ_{s} \cos φ_{s}, \sin θ_{s} \cos φ_{s}, \cos θ_{s})

k_{i} = k (\sin θ_{i} \cos φ_{i}, \sin θ_{i} \cos φ_{i}, - \cos θ_{i})

are the scattered and incident wave vectors, where

(θ_{s}, φ_{s})

and

(θ_{i}, φ_{i})

are the scattered and incident directions, respectively.

The bi-frequency correlation function of the intensity scattered from randomly rough surfaces is defined as

$C_{12} = 〈 I (f_{1}) I (f_{2}) 〉 - 〈 I (f_{1}) 〉〈 I (f_{2}) 〉 = 〈 {| E_{p q}^{s} (f_{1}) |}^{2} {| E_{p q}^{s} (f_{2}) |}^{2} 〉 - 〈 {| E_{p q}^{s} (f_{1}) |}^{2} 〉〈 {| E_{p q}^{s} (f_{2}) |}^{2} 〉$ (2)

The rms slope of the surface discussed in this paper is smaller than 0.25 and its rms roughness is less than the incident wavelength, the surface is weak rough surface, therefore, the scattering composes coherent and incoherent parts, that is

E_{p q}^{s} (f) = Δ E_{p q}^{s} (f) + 〈 E_{p q}^{s} (f) 〉

The bi-frequency correlation function of the scattered intensity can be written as follow [17]

C_{12} = {| I_{p q 1} (f_{1}, f_{2}) |}^{2} + {| I_{p q 2} (f_{1}, f_{2}) |}^{2} - 2 {〈 E_{p q 1}^{s} 〉}^{2} {〈 E_{p q 2}^{s} 〉}^{2}

where

I_{p q 1} (f_{1}, f_{2}) = 〈 E_{p q 1}^{s} E_{p q 2}^{s *} 〉

and

I_{p q 2} (f_{1}, f_{2}) = 〈 E_{p q 1}^{s} E_{p q 2}^{s} 〉

2.1 Solution for the term $I_{p q 1} (f_{1}, f_{2})$

According to Eq. (1), the term $I_{p q 1} (f_{1}, f_{2})$ is

I_{p q 1} = \int \int 〈 U_{p q} (f_{1}) U_{p q}^{*} (f_{2}) \exp [i (V_{1} \cdot r_{1} - V_{2} \cdot r_{2})] 〉 d S_{1} d S_{2}

neglecting the unnecessary factor before the integral, and

V_{1, 2} = {(k_{s} - k_{i})}_{1, 2}

. For convenience, let the vectors

V

and

r

V = V_{⊥} + V_{z} {\hat{e}}_{z}

r = r_{⊥} + h (r_{⊥}) {\hat{e}}_{z}

. And the term

U_{p q} (f_{1}) U_{p q}^{*} (f_{2})

can be written as

U_{p q} (f_{1}) U_{p q}^{*} (f_{2}) = a_{01} a_{02}^{*} + a_{11} a_{02}^{*} Z_{x 1} + a_{01} a_{12}^{*} Z_{x 2} + a_{21} a_{02}^{*} Z_{y 1} + a_{01} a_{22}^{*} Z_{y 2}

The second number in the subscript of the polarization coefficient represents the frequency. And here the second terms of the slopes are omitted, which because when the maximum radius of curvature of the surface is much greater than the incident wavelength, the second terms of the slope will be much smaller than the first terms.

From Eq. (6), it can be seen that $I_{p q 1} (f_{1}, f_{2})$ can be decomposed into two terms: non-slope term and slope term,

I_{p q 1} (f_{1}, f_{2}) = I_{p q 10} (f_{1}, f_{2}) + I_{p q 1 s} (f_{1}, f_{2})

The non-slope term is

I_{p q 10} = a_{01} a_{02}^{*} \int \int 〈 \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} - V_{⊥ 2} \cdot r_{⊥}_{2})] d S_{1} d S_{2}

The slope term includes $x$ -slope term and $y$ -slope term

I_{p q 1 s} (f_{1}, f_{2}) = I_{p q 1 x} (f_{1}, f_{2}) + I_{p q 1 y} (f_{1}, f_{2})

where

I_{p q 1 x} = a_{11} a_{02}^{*} I_{p q 1 x 1} + a_{01} a_{12}^{*} I_{p q 1 x 2}

I_{p q 1 y} = a_{21} a_{02}^{*} I_{p q 1 y 1} + a_{01} a_{22}^{*} I_{p q 1 y 2}

with

I_{p q 1 x j} = \int \int 〈 Z_{x j} \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} - V_{⊥ 2} \cdot r_{⊥}_{2})] d S_{1} d S_{2}

I_{p q 1 y j} = \int \int 〈 Z_{y j} \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} - V_{⊥ 2} \cdot r_{⊥}_{2})] d S_{1} d S_{2}

where

j = 1, 2

The rough surface is assumed to be Gaussian, then the joint-characteristic function has the form

F (V_{z 1}, - V_{z 2}, ρ) = 〈 \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 = \exp [- (β_{11} + β_{22}) / 2] \exp [β_{12} ρ (r_{⊥ 1} - r_{⊥ 2})]

where

β_{i j} = V_{z i} V_{z j} δ^{2} (i, j = 1, 2)

, and

ρ (r_{⊥ 1} - r_{⊥ 2}) = \exp [- {| r_{⊥ 1} - r_{⊥ 2} |}^{2} / l_{c}^{2}]

is the correlation coefficient of the rough surface,

δ

and

l_{c}

are the rms roughness and the correlation length, respectively. For a Gaussian randomly rough surface, there are the relationships as follow [18]

〈 Z_{x j} \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 = - i V_{z (3 - j)} δ^{2} \frac{\partial ρ}{\partial x_{d}} F (V_{z 1}, - V_{z 2}, ρ)

〈 Z_{y j} \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 = - i V_{z (3 - j)} δ^{2} \frac{\partial ρ}{\partial y_{d}} F (V_{z 1}, - V_{z 2}, ρ)

where

x_{d} = x_{1} - x_{2}, y_{d} = y_{1} - y_{2}

Making the following change of variables

\begin{array}{l} r_{c} = (r_{⊥ 1} + r_{⊥ 2}) / 2 r_{d} = r_{⊥ 1} - r_{⊥ 2} \\ V_{c} = (V_{⊥ 1} + V_{⊥ 2}) / 2 V_{d} = V_{⊥ 1} - V_{⊥ 2} \end{array}

Substituting Eqs. (14)and(17) into Eq. (8) gives the non-slope terms as

I_{p q 10} = a_{01} a_{02}^{*} β \iint_{\infty} d r_{c} d r_{d} \exp (- \frac{2 {| r_{c} |}^{2}}{D^{2}}) \exp (- \frac{{| r_{d} |}^{2}}{2 D^{2}}) \exp (β_{12} ρ) \exp [i (V_{d} \cdot r_{c} + V_{c} \cdot r_{d})]

with

β = \exp [- (β_{11} + β_{22}) / 2]

To evaluate the integral, we must expand the term $\exp (β_{12} ρ)$ into Taylor series

\exp [β_{12} ρ (r_{d})] = \sum_{n = 0}^{\infty} \frac{{(β_{12})}^{n}}{n!} ρ^{n} (r_{d})

Then the solution is

I_{p q 10} = a_{01} a_{02}^{*} π^{2} D^{4} l_{c}^{2} β \exp [- \frac{D^{2} {| V_{d} |}^{2}}{8}] \sum_{n = 0}^{\infty} \frac{{(β_{12})}^{n}}{n! (l_{c}^{2} + 2 n D^{2})} \exp (- \frac{D^{2} l_{c}^{2} {| V_{c} |}^{2}}{2 l_{c}^{2} + 4 n D^{2}})

To solve the integrals in Eqs. (12) and (13), some new variables are introduced

x_{d} = ξ \cos α y_{d} = ξ \sin α

The partial derivatives of the function $ρ$ become

\frac{\partial ρ}{\partial x_{d}} = \frac{\partial ρ}{\partial ξ} \cos α \frac{\partial ρ}{\partial y_{d}} = \frac{\partial ρ}{\partial ξ} \sin α

Inserting Eqs. (15), (17), (21) and (22) in Eq. (12) and carrying out the integration over $r_{c}$ , yields

I_{p q 1 x j} = - i V_{z (3 - j)} \frac{π D^{2} δ^{2} β}{2} \exp [- \frac{D^{2} {| V_{d} |}^{2}}{8}] \int_{0}^{\infty} ξ d ξ \exp (β_{12} ρ) \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} Γ (ξ)

where

Γ (ξ) = \int_{0}^{2 π} d α \cos α \exp [i ξ | V_{c} | \cos (α - χ)]

and

χ = arc \tan (V_{c y} / V_{c x})

With the complex exponential term in Eq. (23) which can be expressed as [19]

\exp [i b \cos (α - χ)] = \sum_{n = - \infty}^{\infty} i^{n} J_{n} (b) \exp [i n (α - χ)]

and the integral relationships [20]

\int_{0}^{2 π} d α \cos α \exp (i n α) = {\begin{cases} π n = \pm 1 \\ 0 n \neq \pm 1 \end{cases}

\int_{0}^{2 π} d α \sin α \exp (i n α) = {\begin{cases} \pm i π n = \pm 1 \\ 0 n \neq \pm 1 \end{cases}

and

J_{- n} = {(- 1)}^{n} J_{n}

, then the function

Γ (ξ)

has the solution

Γ (ξ) = i 2 π J_{1} (ξ | V_{c} |) \cos χ

where

J_{n} (...)

is the nth-order Bessel function.

Inserting Eq. (28) into Eq. (23) and utilizing Eq. (19), the final solution for Eq. (12) can be attained

\begin{array}{l} I_{p q 1 x j} = - 2 π^{2} D^{6} V_{z (3 - j)} δ^{2} β V_{c x} \exp (- D^{2} {| V_{d} |}^{2} / 8) \\ \times \sum_{n = 0}^{\infty} \frac{l_{c}^{2} β_{12}^{n}}{n! {[l_{c}^{2} + 2 (1 + n) D^{2}]}^{2}} \exp {- \frac{D^{2} l_{c}^{2} {| V_{c} |}^{2}}{2 [l_{c}^{2} + 2 (1 + n) D^{2}]}} \end{array}

In the same way, the solution of Eq. (13) can be obtained as

\begin{array}{l} I_{p q 1 y j} = - 2 π^{2} D^{6} V_{z (3 - j)} δ^{2} β V_{c y} \exp (- D^{2} {| V_{d} |}^{2} / 8) \\ \times \sum_{n = 0}^{\infty} \frac{l_{c}^{2} β_{12}^{n}}{n! {[l_{c}^{2} + 2 (1 + n) D^{2}]}^{2}} \exp {- \frac{D^{2} l_{c}^{2} {| V_{c} |}^{2}}{2 [l_{c}^{2} + 2 (1 + n) D^{2}]}} \end{array}

Then substituting Eq. (29) and Eq. (30) into Eq. (10) and Eq. (11), respectively, the solution for the slope term $I_{p q 1 s}$ is

\begin{array}{l} I_{p q 1 s} = - [(a_{11} a_{02}^{*} V_{z 2} + a_{01} a_{12}^{*} V_{z 1}) V_{c x} + (a_{21} a_{02}^{*} V_{z 2} + a_{01} a_{22}^{*} V_{z 1}) V_{c y}] 2 π^{2} D^{6} δ^{2} β \\ \times \exp [- \frac{D^{2} {| V_{d} |}^{2}}{8}] \sum_{n = 0}^{\infty} \frac{l_{c}^{2} β_{12}^{n}}{n! {[l_{c}^{2} + 2 (1 + n) D^{2}]}^{2}} \exp {- \frac{D^{2} l_{c}^{2} {| V_{c} |}^{2}}{2 [l_{c}^{2} + 2 (1 + n) D^{2}]}} \end{array}

With Eqs. (20)and(31), the analytical expression for the term

I_{p q 1} (f_{1}, f_{2})

can be achieved.

2.2 Solution for the term $I_{p q 2} (f_{1}, f_{2})$

Similarly, from Eq. (1) the term $I_{p q 2} (f_{1}, f_{2})$ can be written as

I_{p q 2} = \int \int 〈 U_{p q 1} U_{p q 2} \exp [i (V_{1} \cdot r_{1} + V_{2} \cdot r_{2})] 〉 d S_{1} d S_{2}

where

U_{p q 1} U_{p q 2} = a_{01} a_{02} + a_{11} a_{02} Z_{x 1} + a_{01} a_{12} Z_{x 2} + a_{21} a_{02} Z_{y 1} + a_{01} a_{22} Z_{y 2}

Also $I_{p q 2} (f_{1}, f_{2})$ can be expressed by the non-slope term and the slope term

I_{p q 2} (f_{1}, f_{2}) = I_{p q 20} (f_{1}, f_{2}) + I_{p q 2 s} (f_{1}, f_{2})

and

I_{p q 20} = a_{01} a_{02} \int \int \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} + V_{⊥ 2} \cdot r_{⊥ 2})] 〈 \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 d r_{⊥ 1} d r_{⊥ 2}

The slope term $I_{p q 2 s} (f_{1}, f_{2})$ is then

I_{p q 2 s} (f_{1}, f_{2}) = I_{p q 2 x} (f_{1}, f_{2}) + I_{p q 2 y} (f_{1}, f_{2})

where

I_{p q 2 x} = a_{11} a_{02} I_{p q 2 x 1} + a_{01} a_{12} I_{p q 2 x 2}

I_{p q 2 y} = a_{21} a_{02} I_{p q 2 y 1} + a_{01} a_{22} I_{p q 2 y 2}

and

I_{p q 2 x j} = \int \int 〈 Z_{x j} \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} + V_{⊥ 2} \cdot r_{⊥ 2})] d r_{⊥ 1} d r_{⊥ 2}

I_{p q 2 y j} = \int \int 〈 Z_{y j} \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} + V_{⊥ 2} \cdot r_{⊥ 2})] d r_{⊥ 1} d r_{⊥ 2}

Here the joint-characteristic function is

F (V_{z 1}, V_{z 2}, ρ) = 〈 \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 = β \exp (- β_{12} ρ)

and

〈 Z_{x j} \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 = {(- 1)}^{3 - j} i V_{z (3 - j)} δ^{2} \frac{\partial ρ}{\partial ξ} \cos α F (V_{z 1}, V_{z 2}, ρ)

〈 Z_{y j} \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 = {(- 1)}^{3 - j} i V_{z (3 - j)} δ^{2} \frac{\partial ρ}{\partial ξ} \sin α F (V_{z 1}, V_{z 2}, ρ)

By Eqs. (17)and(21), and finishing the integration over $r_{c}$ , Eqs. (35) and (39)-(40) can be transformed as follow

I_{p q 20} = a_{01} a_{02} \frac{π β D^{2}}{2} \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \int_{0}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \exp (- β_{12} ρ) Η (ξ)

I_{p q 2 x j} = {(- 1)}^{3 - j} i δ^{2} V_{z (3 - j)} \frac{π β D^{2}}{2} \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \int_{0}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} \exp (- β_{12} ρ) Λ (ξ)

I_{p q 2 y j} = {(- 1)}^{3 - j} i δ^{2} V_{z (3 - j)} \frac{π β D^{2}}{2} \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \int_{0}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} \exp (- β_{12} ρ) Φ (ξ)

where

Η (ξ) = \int_{0}^{2 π} d α \exp [\frac{i ξ | V_{d} |}{2} \cos (α - ε)]

Λ (ξ) = \int_{0}^{2 π} d α \cos α \exp [\frac{i ξ | V_{d} |}{2} \cos (α - ε)]

Φ (ξ) = \int_{0}^{2 π} d α \sin α \exp [\frac{i ξ | V_{d} |}{2} \cos (α - ε)]

with

ε = arc \tan (V_{y d} / V_{x d})

By Eqs. (25)-(27), The functions $Η (ξ)$ , $Λ (ξ)$ and $Φ (ξ)$ have the solutions

Η (ξ) = 2 π J_{0} [\frac{ξ}{2} | V_{d} |]

Λ (ξ) = i 2 π J_{1} (\frac{ξ}{2} | V_{d} |) \cos ε

Φ (ξ) = i 2 π J_{1} (\frac{ξ}{2} | V_{d} |) \sin ε

Noting that in Eqs. (44)-(46), because of the term $\exp [- β_{12} ρ (ξ)]$ , it is difficult to get the solutions analytically, a more simple approximation should be taken which we have discussed in our paper [16] before. Using the Eq. (16) in the reference [16], and inserting Eqs. (50)-(52)into Eqs. (44)-(46), respectively, we can get

\begin{array}{l} I_{p q 20} = π^{2} D^{2} a_{01} a_{02} β \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) {\exp (- β_{12}) \int_{0}^{ξ_{0}} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \\ \times J_{0} [\frac{ξ}{2} | V_{d} |] + \int_{ξ_{0}}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) J_{0} [\frac{ξ}{2} | V_{d} |]} \end{array}

\begin{array}{l} I_{p q 2 x j} = {(- 1)}^{4 - j} V_{z (3 - j)} π^{2} D^{2} δ^{2} β \cos ε \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) {\exp (- β_{12}) \int_{0}^{ξ_{0}} ξ d ξ \\ \times \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} J_{1} [\frac{ξ}{2} | V_{d} |] + \int_{ξ_{0}}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} J_{1} [\frac{ξ}{2} | V_{d} |]} \end{array}

\begin{array}{l} I_{p q 2 y j} = {(- 1)}^{4 - j} V_{z (3 - j)} π^{2} D^{2} δ^{2} β \sin ε \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) {\exp (- β_{12}) \int_{0}^{ξ_{0}} ξ d ξ \\ \times \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} J_{1} [\frac{ξ}{2} | V_{d} |] + \int_{ξ_{0}}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} J_{1} [\frac{ξ}{2} | V_{d} |]} \end{array}

where

ξ_{0} = l_{c} {\ln (\frac{β_{12}}{\ln 2 - \ln [1 + \exp (- β_{12})]})}^{1 / 2}

Since the Bessel functions of integer order $J_{n} (z)$ can be expressed explicitly as [21]

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

And there are the relationships [20]

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 solutions for $I_{p q 20}$ and $I_{p q 2 s}$ can be given by

\begin{array}{l} I_{p q 20} = π^{2} D^{4} a_{01} a_{02} β \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} {(| V_{d} | D)}^{2 k}}{2^{3 k} {(k!)}^{2}} \\ \times [\exp (- β_{12}) γ (k + 1, \frac{ξ_{0}^{2}}{2 D^{2}}) + Γ (k + 1, \frac{ξ_{0}^{2}}{2 D^{2}})] \end{array}

\begin{array}{l} I_{p q 2 s} = [(a_{11} a_{02} V_{z 2} - a_{01} a_{12} V_{z 1}) V_{d x} + (a_{21} a_{02} V_{z 2} - a_{01} a_{22} V_{z 1}) V_{d y}] π^{2} D^{6} δ^{2} β \\ \times \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} {| V_{d} |}^{2 k} D^{2 k} l_{c}^{2 k + 2}}{2^{3 k} k! (k + 1)! {(l_{c}^{2} + 2 D^{2})}^{k + 2}} [\exp (- β_{12}) γ (k + 2, x_{0}) + Γ (k + 2, x_{0})] \end{array}

with

x_{0} = ξ_{0}^{2} / 2 D^{2} + ξ_{0}^{2} / l_{c}^{2}

2.3 Mean scattered field $〈 E_{p q j}^{s} (f_{j}) 〉$

At last, the mean scattered field $〈 E_{p q j}^{s} (f_{j}) 〉$ should be derived. By Eq. (1), $〈 E_{p q j}^{s} (f_{j}) 〉$ can be written as

〈 E_{p q j}^{s} 〉 = \int 〈 (a_{0 j} + a_{1 j} Z_{x} + a_{2 j} Z_{y}) \exp (i V_{z j} h) 〉 \exp (i V_{⊥ j} \cdot r_{⊥}) d S

The term $〈 \exp (i V_{z j} h) 〉$ in the equation above is the characteristic function of the Gaussian random variable

〈 \exp [i V_{z j} h (r_{⊥})] 〉 = \exp (- V_{z j}^{2} δ^{2} / 2) = \exp (- β_{j j} / 2)

and by reference [18]

〈 Z_{x} \exp [i V_{z j} h (r_{⊥})] 〉 = 〈 Z_{y} \exp [i V_{z j} h (r_{⊥})] 〉 = 0

The mean scattered field is then

〈 E_{p q j}^{s} 〉 = π D^{2} a_{0 j} \exp (- β_{j j} / 2) \exp (- D^{2} {| V_{⊥ j} |}^{2} / 4)

Thus every term needed has been derived, inserting these solutions in Eq. (4), the analytical expression for the bi-frequency correlation function of the scattered intensity from two- dimensional randomly dielectric weak rough surface can then be obtained.

3. Numerical results and analyses

In this section, the bi-frequency correlation function of the scattered intensity will be calculated numerically to analyze its bi-frequency correlation properties and the influencing factors. In the calculation, let the center frequency be $f_{0} = (f_{1} + f_{2}) / 2$ and the frequency difference be $f_{d} = f_{1} - f_{2}$ , and the medium be assumed non-dispersive or in the extent of the bandwidth the refractive index have hardly change, thus for the frequency difference $f_{d}$ , the polarization coefficients $a_{0 j}, a_{1 j}, a_{2 j}$ are constant. From the formulas derived above, it can be seen that the bi-frequency correlation function $C_{12}$ of the scattered intensity from two-dimensional dielectric rough surface with a root-mean-square slope smaller than 0.25 and rms roughness less than the incident wavelength is closely dependent on the incident and scattered conditions as well as on the statistical parameters and refractive index of the surface.

Taking the wavelength is $1.06 μ m$ and the aperture dimension is $13.3 l_{c}$ as well as the refractive index is $n = 2.43 + i 10.7$ , Figs. 1 -4 are the variation of he bi-frequency correlation function $C_{12}$ of the scattered intensity with the scattering angle $θ_{s}$ and the frequency difference $f_{d}$ . In Figs. 1-3 , $θ_{i} = φ_{i} = φ_{s} = 0^{°}$ , $l_{c} = 5.89 μ m$ , the rms roughness $δ$ increases sequentially. Comparing with Fig. 3, in Fig. 4 the incident angle is $10^{°}$ , $δ$ is invariable, $δ = 0.9 μ m$ ,the correlation length is increased to be $7.5 μ m$ .

Fig. 1 Bi-frequency correlation function $C_{12}$ versus scattering angle and frequency difference with the parameters $δ = 0.6 μ m$ and $l_{c} = 5.89 μ m$

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Fig. 4 Bi-frequency correlation function $C_{12}$ versus scattering angle and frequency difference with the parameters $θ_{i} = 10^{°}$ , $δ = 0.9 μ m$ and $l_{c} = 7.5 μ m$

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Fig. 3 Bi-frequency correlation function $C_{12}$ versus scattering angle and frequency difference with the parameters $δ = 0.9 μ m$ and $l_{c} = 5.89 μ m$

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Fig. 2 Bi-frequency correlation function $C_{12}$ versus scattering angle and frequency difference with the parameters $δ = 0.8 μ m$ and $l_{c} = 5.89 μ m$

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It is shown that the contribution for the function $C_{12}$ mainly comes from the specular direction while the values in other directions can be neglected. With a smaller rms slope $s = \sqrt{2} δ / l_{c}$ , the decrease of the function $C_{12}$ versus the frequency difference $f_{d}$ is much more slowly that the correlation bandwidth is bigger and the profile near the specular is sharper. While with an increase $s$ , the values of the function $C_{12}$ in specular direction decreases dramatically and others increase relatively. It is explained that when the rms slope is small, the scattering mode is similar to the plane scattering which scattered power only occurs in specular direction, while the surface becomes more roughness, the non-specular scattering enhances and the specular scattering weakens. In addition, the increase of rms height results in a narrower correlation bandwidth.

$θ_{i} = θ_{s} = φ_{i} = 0^{°}$ and $l_{c} = 5.89 μ m$ , with different roughness $δ$ , Figs. 5 -6 and Figs. 7 -8 are the variation of the bi-frequency correlation function $C_{12}$ against the scattering azimuth angle $φ_{s}$ and the frequency difference $f_{d}$ under HH- and VH-polarization, respectively. Under both the polarizations, the function $C_{12}$ decreases monotonously with the increase of the frequency difference $f_{d}$ and the roughness $δ$ . A bigger roughness $δ$ results in a smaller correlation bandwidth. However, in HH-polarization the variable tendency of the function $C_{12}$ with the increase of the azimuth angle $φ_{s}$ is completely different to that in VH-polarization.

Fig. 5 Bi-frequency correlation function $C_{12}$ versus scattering azimuth angle and frequency difference with the parameters $δ = 0.6 μ m$ and $l_{c} = 5.89 μ m$ , HH-polarization

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Fig. 6 Bi-frequency correlation function $C_{12}$ versus scattering azimuth angle and frequency difference with the parameters $δ = 0.8 μ m$ and $l_{c} = 5.89 μ m$ , HH-polarization

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Fig. 7 Bi-frequency correlation function $C_{12}$ versus scattering azimuth angle and frequency difference with the parameters $δ = 0.6 μ m$ and $l_{c} = 5.89 μ m$ , VH-polarization

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Fig. 8 Bi-frequency correlation function $C_{12}$ versus scattering azimuth angle and frequency difference with $δ = 0.8 μ m$ and $l_{c} = 5.89 μ m$ , VH-polarization

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To illustrate the influence of the dielectric property of the rough surface on the bi-frequency function $C_{12}$ , Fig. 9 gives the profiles of the function $C_{12}$ versus the frequency difference $f_{d}$ with different roughness and refractive indexes. It is can be seen easily that with a complex refractive index, the smaller the roughness, the bigger the function $C_{12}$ , while with a same roughness, the refractive index has influence on the value of the function $C_{12}$ not on the correlation bandwidth. Especially, the imagery of the refractive index is the main influencing factor that a smaller difference of the absolute values of the imagery of the refractive indexes results in a smaller difference of the values of the function $C_{12}$ .

Fig. 9 Bi-frequency correlation function $C_{12}$ versus frequency difference with different refractive indexes

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4. Conclusion

Based on the scalar Kirchhoff approximation theory, the Bi-frequency correlation properties of the intensity scattered from two-dimensional dielectric randomly rough Gaussian surfaces assumed to be Gaussian with a root-mean-square (rms) slope of the surface less than 0.25 are investigated in this paper. The rms roughness of the surface is assumed to be smaller than the incident wavelength. By the Gaussian moment theorem, expanding the exponential and complex exponential function into series and with some mathematical simplification, the Bi-frequency correlation function of the scattered intensity considering the influences of the scattering from the slope of the surface has been derived in detail and calculated. The numerical results show that the incident and scattered conditions as well as on the statistical parameters and complex refractive index of the surface have great impacts on the bi-frequency correlation properties of the scattered intensity. Especially, the correlation function mainly comes from the specular direction, and the coherence bandwidth decreases with the increase of the roughness of the rough surface. In HH-polarization and VH-polarization, the function $C_{12}$ versus with the scattering azimuth angle have different varying tendency. In addition, the refractive index can also affect the value of the function $C_{12}$ but not on the correlation bandwidth. Comparing with the real part, the imagery of the complex refractive index has a greater influence on the bi-frequency correlation function that the larger difference of the absolute values of the imagery of the refractive indexes, the larger difference of the values of the function $C_{12}$ . The work performed in this paper provide a much better knowledge about the scattering properties of dielectric rough surfaces to investigate the scattering problem and the identification of three-dimensional rough dielectric objects which we will discuss in future.

Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant No. 61172031 and from the Fundamental Research Funds for the Central Universities under Grant No. K50510070009.

References and links

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2. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

3. E. Bahar and S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt. 24(12), 1820–1825 (1985). [CrossRef] [PubMed]

4. E. Bahar and M. A. Fitzwater, “Scattering and depolarization by conducting cylinders with rough surfaces,” Appl. Opt. 25(11), 1826–1832 (1986). [CrossRef] [PubMed]

5. W. Zhensen and C. Suomin, “Bistatic scattering by arbitrarily shaped objects with rough surface at optical and infrared frequencies,” Int. J. Infrared Millim. Waves 13(4), 537–549 (1992). [CrossRef]

6. A. Ishimaru, L. Ailes-Sengers, P. Phu, and D. Winebrenner, “Pulse broadening and two-frequency mutual coherence function of the scattered wave from rough surfaces,” Waves Random Media 4(2), 139–148 (1994). [CrossRef]

7. D. J. Schertler and N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A 9(11), 2056–2066 (1992). [CrossRef]

8. D. J. Schertler and N. George, “Backscattering cross section of a roughened sphere,” J. Opt. Soc. Am. A 11(8), 2286–2297 (1994). [CrossRef]

9. C. Hui, W. Zhensen, and B. Lu, “Infrared laser pulse scattering from randomly rough surfaces,” Int. J. Infrared Millim. Waves 25(8), 1211–1219 (2004). [CrossRef]

10. G. Zhang and Z. S. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express 19(8), 7007–7019 (2011). [CrossRef] [PubMed]

11. Z. W. Xu, J. Wu, Z. S. Wu, and Q. Li, “Solution for the Fourth Moment Equation of Waves in Random Continuum Under Strong Fluctuations: General Theory and Plane Wave Solution,” IEEE Trans. Antenn. Propag. 55(6), 1613–1621 (2007). [CrossRef]

12. V. N. Bronnikov and M. M. Kalugin, “Measuring the parameters of vibrations and surface roughness, using the frequency spectrum of the intensity fluctuations of scattered radiation,” J. Opt. Tech. 76(11), 697–701 (2009). [CrossRef]

13. M. J. Wang, Z. S. Wu, Y. L. Li, X. A. Zhang, and H. Zhang, “The fourth order moment statistical characteristic of the laser pulse scattering on random rough surface,” Acta Phys. Sin-CH ED 58, 2390–2396 (2009).

14. Y. Xin, Y. J. He, Y. R. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35(23), 4000–4002 (2010). [CrossRef] [PubMed]

15. H. C. Jacks and O. Korotkova, “Intensity-intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28(6), 1139–1144 (2011). [CrossRef] [PubMed]

16. W. Zhen-Sen and Z. Geng “Intensity Correlation Function of Light Scattering from a Weakly One-Dimensional Random Rough Surface,” Chin. Phys. Lett. 26(11), 114208 (2009). [CrossRef]

17. G. Zhang and Z. Wu, “Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces,” Opt. Express 20(2), 1491–1502 (2012). [CrossRef] [PubMed]

18. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing, Vol. 2 (Addison-Wesley Publishing,1982).

19. C. Bourlier, “Azimuthal Harmonic Coefficients of the Microwave Backscattering from a Non-Gaussian Ocean Surface With the First-Order SSA Model,” IEEE Trans. Geosci. Rem. Sens. 42(11), 2600–2611 (2004). [CrossRef]

20. J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

21. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1958).

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Fig. 1 Bi-frequency correlation function

C_{12}

versus scattering angle and frequency difference with the parameters

δ = 0.6 μ m

and

l_{c} = 5.89 μ m

View in Article | Download Full Size | PDF

Fig. 4 Bi-frequency correlation function

C_{12}

versus scattering angle and frequency difference with the parameters

θ_{i} = 10^{°}

δ = 0.9 μ m

and

l_{c} = 7.5 μ m

View in Article | Download Full Size | PDF

Fig. 3 Bi-frequency correlation function

C_{12}

versus scattering angle and frequency difference with the parameters

δ = 0.9 μ m

and

l_{c} = 5.89 μ m

View in Article | Download Full Size | PDF

Fig. 2 Bi-frequency correlation function

C_{12}

versus scattering angle and frequency difference with the parameters

δ = 0.8 μ m

and

l_{c} = 5.89 μ m

View in Article | Download Full Size | PDF

Fig. 5 Bi-frequency correlation function

C_{12}

versus scattering azimuth angle and frequency difference with the parameters

δ = 0.6 μ m

and

l_{c} = 5.89 μ m

, HH-polarization

View in Article | Download Full Size | PDF

Fig. 6 Bi-frequency correlation function

C_{12}

versus scattering azimuth angle and frequency difference with the parameters

δ = 0.8 μ m

and

l_{c} = 5.89 μ m

, HH-polarization

View in Article | Download Full Size | PDF

Fig. 7 Bi-frequency correlation function

C_{12}

versus scattering azimuth angle and frequency difference with the parameters

δ = 0.6 μ m

and

l_{c} = 5.89 μ m

, VH-polarization

View in Article | Download Full Size | PDF

Fig. 8 Bi-frequency correlation function

C_{12}

versus scattering azimuth angle and frequency difference with

δ = 0.8 μ m

and

l_{c} = 5.89 μ m

, VH-polarization

View in Article | Download Full Size | PDF

Fig. 9 Bi-frequency correlation function

C_{12}

versus frequency difference with different refractive indexes

View in Article | Download Full Size | PDF

Equations (63)

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E_{p q}^{s} (f) = E_{0} K (f) \int_{- \infty}^{\infty} p (x, y) U_{p q} \exp [i (k_{s} - k_{i}) \cdot r] d S

E_{p q}^{s} (f) = Δ E_{p q}^{s} (f) + 〈 E_{p q}^{s} (f) 〉

C_{12} = {| I_{p q 1} (f_{1}, f_{2}) |}^{2} + {| I_{p q 2} (f_{1}, f_{2}) |}^{2} - 2 {〈 E_{p q 1}^{s} 〉}^{2} {〈 E_{p q 2}^{s} 〉}^{2}

I_{p q 1} = \int \int 〈 U_{p q} (f_{1}) U_{p q}^{*} (f_{2}) \exp [i (V_{1} \cdot r_{1} - V_{2} \cdot r_{2})] 〉 d S_{1} d S_{2}

U_{p q} (f_{1}) U_{p q}^{*} (f_{2}) = a_{01} a_{02}^{*} + a_{11} a_{02}^{*} Z_{x 1} + a_{01} a_{12}^{*} Z_{x 2} + a_{21} a_{02}^{*} Z_{y 1} + a_{01} a_{22}^{*} Z_{y 2}

I_{p q 1} (f_{1}, f_{2}) = I_{p q 10} (f_{1}, f_{2}) + I_{p q 1 s} (f_{1}, f_{2})

I_{p q 10} = a_{01} a_{02}^{*} \int \int 〈 \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} - V_{⊥ 2} \cdot r_{⊥}_{2})] d S_{1} d S_{2}

I_{p q 1 s} (f_{1}, f_{2}) = I_{p q 1 x} (f_{1}, f_{2}) + I_{p q 1 y} (f_{1}, f_{2})

I_{p q 1 x} = a_{11} a_{02}^{*} I_{p q 1 x 1} + a_{01} a_{12}^{*} I_{p q 1 x 2}

I_{p q 1 y} = a_{21} a_{02}^{*} I_{p q 1 y 1} + a_{01} a_{22}^{*} I_{p q 1 y 2}

I_{p q 1 x j} = \int \int 〈 Z_{x j} \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} - V_{⊥ 2} \cdot r_{⊥}_{2})] d S_{1} d S_{2}

I_{p q 1 y j} = \int \int 〈 Z_{y j} \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} - V_{⊥ 2} \cdot r_{⊥}_{2})] d S_{1} d S_{2}

F (V_{z 1}, - V_{z 2}, ρ) = 〈 \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 = \exp [- (β_{11} + β_{22}) / 2] \exp [β_{12} ρ (r_{⊥ 1} - r_{⊥ 2})]

〈 Z_{x j} \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 = - i V_{z (3 - j)} δ^{2} \frac{\partial ρ}{\partial x_{d}} F (V_{z 1}, - V_{z 2}, ρ)

〈 Z_{y j} \exp [i (V_{z 1} h_{1} - V_{z 2} h_{2})] 〉 = - i V_{z (3 - j)} δ^{2} \frac{\partial ρ}{\partial y_{d}} F (V_{z 1}, - V_{z 2}, ρ)

\begin{array}{l} r_{c} = (r_{⊥ 1} + r_{⊥ 2}) / 2 r_{d} = r_{⊥ 1} - r_{⊥ 2} \\ V_{c} = (V_{⊥ 1} + V_{⊥ 2}) / 2 V_{d} = V_{⊥ 1} - V_{⊥ 2} \end{array}

I_{p q 10} = a_{01} a_{02}^{*} β \iint_{\infty} d r_{c} d r_{d} \exp (- \frac{2 {| r_{c} |}^{2}}{D^{2}}) \exp (- \frac{{| r_{d} |}^{2}}{2 D^{2}}) \exp (β_{12} ρ) \exp [i (V_{d} \cdot r_{c} + V_{c} \cdot r_{d})]

\exp [β_{12} ρ (r_{d})] = \sum_{n = 0}^{\infty} \frac{{(β_{12})}^{n}}{n!} ρ^{n} (r_{d})

I_{p q 10} = a_{01} a_{02}^{*} π^{2} D^{4} l_{c}^{2} β \exp [- \frac{D^{2} {| V_{d} |}^{2}}{8}] \sum_{n = 0}^{\infty} \frac{{(β_{12})}^{n}}{n! (l_{c}^{2} + 2 n D^{2})} \exp (- \frac{D^{2} l_{c}^{2} {| V_{c} |}^{2}}{2 l_{c}^{2} + 4 n D^{2}})

x_{d} = ξ \cos α y_{d} = ξ \sin α

\frac{\partial ρ}{\partial x_{d}} = \frac{\partial ρ}{\partial ξ} \cos α \frac{\partial ρ}{\partial y_{d}} = \frac{\partial ρ}{\partial ξ} \sin α

I_{p q 1 x j} = - i V_{z (3 - j)} \frac{π D^{2} δ^{2} β}{2} \exp [- \frac{D^{2} {| V_{d} |}^{2}}{8}] \int_{0}^{\infty} ξ d ξ \exp (β_{12} ρ) \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} Γ (ξ)

Γ (ξ) = \int_{0}^{2 π} d α \cos α \exp [i ξ | V_{c} | \cos (α - χ)]

\exp [i b \cos (α - χ)] = \sum_{n = - \infty}^{\infty} i^{n} J_{n} (b) \exp [i n (α - χ)]

\int_{0}^{2 π} d α \cos α \exp (i n α) = {\begin{cases} π n = \pm 1 \\ 0 n \neq \pm 1 \end{cases}

\int_{0}^{2 π} d α \sin α \exp (i n α) = {\begin{cases} \pm i π n = \pm 1 \\ 0 n \neq \pm 1 \end{cases}

Γ (ξ) = i 2 π J_{1} (ξ | V_{c} |) \cos χ

\begin{array}{l} I_{p q 1 x j} = - 2 π^{2} D^{6} V_{z (3 - j)} δ^{2} β V_{c x} \exp (- D^{2} {| V_{d} |}^{2} / 8) \\ \times \sum_{n = 0}^{\infty} \frac{l_{c}^{2} β_{12}^{n}}{n! {[l_{c}^{2} + 2 (1 + n) D^{2}]}^{2}} \exp {- \frac{D^{2} l_{c}^{2} {| V_{c} |}^{2}}{2 [l_{c}^{2} + 2 (1 + n) D^{2}]}} \end{array}

\begin{array}{l} I_{p q 1 y j} = - 2 π^{2} D^{6} V_{z (3 - j)} δ^{2} β V_{c y} \exp (- D^{2} {| V_{d} |}^{2} / 8) \\ \times \sum_{n = 0}^{\infty} \frac{l_{c}^{2} β_{12}^{n}}{n! {[l_{c}^{2} + 2 (1 + n) D^{2}]}^{2}} \exp {- \frac{D^{2} l_{c}^{2} {| V_{c} |}^{2}}{2 [l_{c}^{2} + 2 (1 + n) D^{2}]}} \end{array}

\begin{array}{l} I_{p q 1 s} = - [(a_{11} a_{02}^{*} V_{z 2} + a_{01} a_{12}^{*} V_{z 1}) V_{c x} + (a_{21} a_{02}^{*} V_{z 2} + a_{01} a_{22}^{*} V_{z 1}) V_{c y}] 2 π^{2} D^{6} δ^{2} β \\ \times \exp [- \frac{D^{2} {| V_{d} |}^{2}}{8}] \sum_{n = 0}^{\infty} \frac{l_{c}^{2} β_{12}^{n}}{n! {[l_{c}^{2} + 2 (1 + n) D^{2}]}^{2}} \exp {- \frac{D^{2} l_{c}^{2} {| V_{c} |}^{2}}{2 [l_{c}^{2} + 2 (1 + n) D^{2}]}} \end{array}

I_{p q 2} = \int \int 〈 U_{p q 1} U_{p q 2} \exp [i (V_{1} \cdot r_{1} + V_{2} \cdot r_{2})] 〉 d S_{1} d S_{2}

U_{p q 1} U_{p q 2} = a_{01} a_{02} + a_{11} a_{02} Z_{x 1} + a_{01} a_{12} Z_{x 2} + a_{21} a_{02} Z_{y 1} + a_{01} a_{22} Z_{y 2}

I_{p q 2} (f_{1}, f_{2}) = I_{p q 20} (f_{1}, f_{2}) + I_{p q 2 s} (f_{1}, f_{2})

I_{p q 20} = a_{01} a_{02} \int \int \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} + V_{⊥ 2} \cdot r_{⊥ 2})] 〈 \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 d r_{⊥ 1} d r_{⊥ 2}

I_{p q 2 s} (f_{1}, f_{2}) = I_{p q 2 x} (f_{1}, f_{2}) + I_{p q 2 y} (f_{1}, f_{2})

I_{p q 2 x} = a_{11} a_{02} I_{p q 2 x 1} + a_{01} a_{12} I_{p q 2 x 2}

I_{p q 2 y} = a_{21} a_{02} I_{p q 2 y 1} + a_{01} a_{22} I_{p q 2 y 2}

I_{p q 2 x j} = \int \int 〈 Z_{x j} \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} + V_{⊥ 2} \cdot r_{⊥ 2})] d r_{⊥ 1} d r_{⊥ 2}

I_{p q 2 y j} = \int \int 〈 Z_{y j} \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 \exp [i (V_{⊥ 1} \cdot r_{⊥ 1} + V_{⊥ 2} \cdot r_{⊥ 2})] d r_{⊥ 1} d r_{⊥ 2}

F (V_{z 1}, V_{z 2}, ρ) = 〈 \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 = β \exp (- β_{12} ρ)

〈 Z_{x j} \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 = {(- 1)}^{3 - j} i V_{z (3 - j)} δ^{2} \frac{\partial ρ}{\partial ξ} \cos α F (V_{z 1}, V_{z 2}, ρ)

〈 Z_{y j} \exp [i (V_{z 1} h_{1} + V_{z 2} h_{2})] 〉 = {(- 1)}^{3 - j} i V_{z (3 - j)} δ^{2} \frac{\partial ρ}{\partial ξ} \sin α F (V_{z 1}, V_{z 2}, ρ)

I_{p q 20} = a_{01} a_{02} \frac{π β D^{2}}{2} \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \int_{0}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \exp (- β_{12} ρ) Η (ξ)

I_{p q 2 x j} = {(- 1)}^{3 - j} i δ^{2} V_{z (3 - j)} \frac{π β D^{2}}{2} \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \int_{0}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} \exp (- β_{12} ρ) Λ (ξ)

I_{p q 2 y j} = {(- 1)}^{3 - j} i δ^{2} V_{z (3 - j)} \frac{π β D^{2}}{2} \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \int_{0}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} \exp (- β_{12} ρ) Φ (ξ)

Η (ξ) = \int_{0}^{2 π} d α \exp [\frac{i ξ | V_{d} |}{2} \cos (α - ε)]

Λ (ξ) = \int_{0}^{2 π} d α \cos α \exp [\frac{i ξ | V_{d} |}{2} \cos (α - ε)]

Φ (ξ) = \int_{0}^{2 π} d α \sin α \exp [\frac{i ξ | V_{d} |}{2} \cos (α - ε)]

Η (ξ) = 2 π J_{0} [\frac{ξ}{2} | V_{d} |]

Λ (ξ) = i 2 π J_{1} (\frac{ξ}{2} | V_{d} |) \cos ε

Φ (ξ) = i 2 π J_{1} (\frac{ξ}{2} | V_{d} |) \sin ε

\begin{array}{l} I_{p q 20} = π^{2} D^{2} a_{01} a_{02} β \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) {\exp (- β_{12}) \int_{0}^{ξ_{0}} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \\ \times J_{0} [\frac{ξ}{2} | V_{d} |] + \int_{ξ_{0}}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) J_{0} [\frac{ξ}{2} | V_{d} |]} \end{array}

\begin{array}{l} I_{p q 2 x j} = {(- 1)}^{4 - j} V_{z (3 - j)} π^{2} D^{2} δ^{2} β \cos ε \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) {\exp (- β_{12}) \int_{0}^{ξ_{0}} ξ d ξ \\ \times \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} J_{1} [\frac{ξ}{2} | V_{d} |] + \int_{ξ_{0}}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} J_{1} [\frac{ξ}{2} | V_{d} |]} \end{array}

\begin{array}{l} I_{p q 2 y j} = {(- 1)}^{4 - j} V_{z (3 - j)} π^{2} D^{2} δ^{2} β \sin ε \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) {\exp (- β_{12}) \int_{0}^{ξ_{0}} ξ d ξ \\ \times \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} J_{1} [\frac{ξ}{2} | V_{d} |] + \int_{ξ_{0}}^{\infty} ξ d ξ \exp (- \frac{ξ^{2}}{2 D^{2}}) \frac{\partial ρ}{\partial ξ} J_{1} [\frac{ξ}{2} | V_{d} |]} \end{array}

ξ_{0} = l_{c} {\ln (\frac{β_{12}}{\ln 2 - \ln [1 + \exp (- β_{12})]})}^{1 / 2}

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

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]

\begin{array}{l} I_{p q 20} = π^{2} D^{4} a_{01} a_{02} β \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} {(| V_{d} | D)}^{2 k}}{2^{3 k} {(k!)}^{2}} \\ \times [\exp (- β_{12}) γ (k + 1, \frac{ξ_{0}^{2}}{2 D^{2}}) + Γ (k + 1, \frac{ξ_{0}^{2}}{2 D^{2}})] \end{array}

\begin{array}{l} I_{p q 2 s} = [(a_{11} a_{02} V_{z 2} - a_{01} a_{12} V_{z 1}) V_{d x} + (a_{21} a_{02} V_{z 2} - a_{01} a_{22} V_{z 1}) V_{d y}] π^{2} D^{6} δ^{2} β \\ \times \exp (- \frac{D^{2} {| V_{c} |}^{2}}{2}) \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} {| V_{d} |}^{2 k} D^{2 k} l_{c}^{2 k + 2}}{2^{3 k} k! (k + 1)! {(l_{c}^{2} + 2 D^{2})}^{k + 2}} [\exp (- β_{12}) γ (k + 2, x_{0}) + Γ (k + 2, x_{0})] \end{array}

〈 E_{p q j}^{s} 〉 = \int 〈 (a_{0 j} + a_{1 j} Z_{x} + a_{2 j} Z_{y}) \exp (i V_{z j} h) 〉 \exp (i V_{⊥ j} \cdot r_{⊥}) d S

〈 \exp [i V_{z j} h (r_{⊥})] 〉 = \exp (- V_{z j}^{2} δ^{2} / 2) = \exp (- β_{j j} / 2)

〈 Z_{x} \exp [i V_{z j} h (r_{⊥})] 〉 = 〈 Z_{y} \exp [i V_{z j} h (r_{⊥})] 〉 = 0

〈 E_{p q j}^{s} 〉 = π D^{2} a_{0 j} \exp (- β_{j j} / 2) \exp (- D^{2} {| V_{⊥ j} |}^{2} / 4)

Abstract

1. Introduction

2. Bi-frequency correlation function of the scattered intensity from dielectric rough surfaces

2.1 Solution for the termIpq1(f1,f2)

2.2 Solution for the termIpq2(f1,f2)

2.3 Mean scattered field〈Epqjs(fj)〉

3. Numerical results and analyses

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Equations (63)

Optics Express

2.1 Solution for the term $I_{p q 1} (f_{1}, f_{2})$

2.2 Solution for the term $I_{p q 2} (f_{1}, f_{2})$

2.3 Mean scattered field $〈 E_{p q j}^{s} (f_{j}) 〉$