Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects

Geng Zhang; Zhensen Wu

doi:10.1364/OE.19.007007

1. Introduction

When a laser pulse illuminates a rough object, the scattering return contains some very important information about the location of the object, physical dimension and its profile and so on which is of great significance to the target recognition, tracking and positioning and the inversion of the optical characteristics of rough surfaces. The theory of the pulse scattering and its experimental research provide vigorous support for the radar system design, the feature extraction of rough objects and the culture remote sensing, its study is of considerable interest at all times [1–5]. According to the pulse wave scattering theory presented by Ishimaru [6], the time domain scattering field is the Fourier transformation of the frequency domain scattering field, and the correlation function of the time domain scattering field or the pulse scattering power is closely related to the two-frequency mutual coherent function, the kernel problem of the time domain scattering is to solve the two-frequency mutual coherent function of all kinds of the scattering model. Ishimaru investigated the pulse scattering from random rough surface and discussed the pulse broadening and the enhanced backscattering effect [7]. Chen et al [8] and Guo and Kim [9] also studied the pulse scattering from rough surfaces using the two-frequency coherent function. Actually, the studied object has been often in three dimensional size, and its scattering problem is of more importance. Bahar and his associates analyzed the scattering cross sections of spheres and infinite cylinders with full wave approach [10,11]. Berlasso and his associates have researched the scattering from cylinders with rough surface [14,15]. Wu [16] and Wu and Cui [17] have studied the backscattering and bistatic scattering cross sections of the infrared laser scattering from arbitrarily shaped objects with rough surfaces by using the Kirchhoff approximation, the results can readily be reduced to the cases of smooth perfectly conducting objects with simple shapes. Schertler and George derived the formulas of the two-frequency backscattering mutual correlation function from roughened sphere and roughened disk and given the results of the backscattering cross section but did not give the final numerical results of the two-frequency mutual correlation function [18,19]. Combing with the backscattering cross sections of three dimensional targets, Li et al have researched the laser range profile of the targets [4,5].

In this paper, the two-frequency mutual coherent function is obtained to investigate the scattering from arbitrarily shaped rough objects, providing some theoretical basis for the further study on the laser pulse scattering from 3-D rough objects. From scalar Helmholtz integral relation, the scattering formula in the far field from an arbitrarily shaped object is derived detailed, and then the two-frequency mutual coherence function is obtained. At last, some brief numerical results and analysis are given taking rough spheres and cylinders for examples.

2. Two frequency mutual coherence function of the scattering from arbitrarily shaped rough objects

A plane wave $E_{i} (\vec{r}) = \exp (i k \hat{k} \cdot \vec{r})$ illuminates a roughened convex object, the scattering geometry is as illustrated in Fig. 1 . The surface S is the unperturbed surface, $\hat{n}$ is the corresponding external normal, ${\vec{r}}_{c}$ is its vector distance and $θ_{i}$ is the local incident angle at ${\vec{r}}_{c}$ while $S^{'}$ is the roughened surface which is the surface S plus a random fluctuation $ξ ({\vec{r}}_{c})$ , $\hat{N}$ is its corresponding normal, and ${\vec{r}}^{'}$ is the vector distance, $θ_{i}^{'}$ is the incident angle at ${\vec{r}}^{'}$ . $\hat{k}$ and ${\hat{k}}_{s}$ are the incident unit vector and the scattering unit vector, respectively. $k = 2 π / λ = ω / c$ is the wavenumber, λ is the wavelength and ω is its angular frequency. The time harmonic factor $\exp (- i ω t)$ is omitted for convenience.

Fig. 1 scattering geometry for a roughened object.

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According to the scalar Helmholtz integral relation, the scattered field from a rough object at a receiver point P in the far field can be expressed as [6]

E_{s} ({\vec{r}}_{s}) = \int_{S^{'}} [E ({\vec{r}}^{'}) \frac{\partial G ({\vec{r}}_{s}, {\vec{r}}^{'})}{\partial \hat{N}} - G ({\vec{r}}_{s}, {\vec{r}}^{'}) \frac{\partial E ({\vec{r}}^{'})}{\partial \hat{N}}] d S^{'}

where

{\vec{r}}_{s}

is the vector distance between the observation point P and the origin of coordinate,

E ({\vec{r}}^{'})

and

\partial E ({\vec{r}}^{'}) / \partial \hat{N}

are the total electric field and its normal derivative on the scattering surface

S^{'}

,

G ({\vec{r}}_{s}, {\vec{r}}^{'})

and

\partial G ({\vec{r}}_{s}, {\vec{r}}^{'}) / \partial \hat{N}

are Green’s function and its normal derivative, respectively. The Green’s function

G ({\vec{r}}_{s}, {\vec{r}}^{'})

is given by

G ({\vec{r}}_{s}, {\vec{r}}^{'}) = \frac{\exp [i k | {\vec{r}}_{s} - {\vec{r}}^{'} |]}{4 π | {\vec{r}}_{s} - {\vec{r}}^{'} |}

The total electric field

E ({\vec{r}}^{'})

is a sum of the incident field

E_{i} ({\vec{r}}^{'})

and the scattered

E^{'} ({\vec{r}}^{'})

on the surface

S^{'}

, that is

E ({\vec{r}}^{'}) = E_{i} ({\vec{r}}^{'}) + E^{'} ({\vec{r}}^{'})

Since the incident field

E_{i} ({\vec{r}}^{'})

satisfies

\int_{S^{'}} [E_{i} ({\vec{r}}^{'}) \frac{\partial G ({\vec{r}}_{s}, {\vec{r}}^{'})}{\partial \hat{N}} - G ({\vec{r}}_{s}, {\vec{r}}^{'}) \frac{\partial E_{i} ({\vec{r}}^{'})}{\partial \hat{N}}] d S^{'} = 0

The scattered field

E_{s} ({\vec{r}}_{s})

at point P can be obtained

E_{s} ({\vec{r}}_{s}) = \int_{S^{'}} [E^{'} ({\vec{r}}^{'}) \frac{\partial G ({\vec{r}}_{s}, {\vec{r}}^{'})}{\partial \hat{N}} - G ({\vec{r}}_{s}, {\vec{r}}^{'}) \frac{\partial E^{'} ({\vec{r}}^{'})}{\partial \hat{N}}] d S^{'}

The radius of principal curvature at any point of the surface is assumed to be much larger than the incident wavelength, and then the tangent-plane approximation can be applied. The scattered field and its normal derivative at

{\vec{r}}^{'}

on the rough surface are written as following, respectively

E^{'} ({\vec{r}}^{'}) = (1 + R_{i}) E_{i} ({\vec{r}}^{'})

\frac{\partial E^{'} ({\vec{r}}^{'})}{\partial \hat{N}} = i (1 - R_{i}) k \hat{k} \cdot \hat{N} E_{i} ({\vec{r}}^{'})

where

R_{i}

is the Fresnel reflection coefficient.

Because the point P is in the far field, the following approximation can be used

\frac{\partial G ({\vec{r}}_{s}, {\vec{r}}^{'})}{\partial \hat{N}} = - i k {\hat{k}}_{s} \cdot \hat{N} G ({\vec{r}}_{s}, {\vec{r}}^{'})

Inserting Eqs. (6), (7) and (8) into Eq. (5), we can get

E_{s} ({\vec{r}}_{s}) = \frac{i k}{4 π} \int_{S^{'}} (R_{i} \vec{V} - \vec{W}) \cdot \hat{N} \frac{\exp [i k (| {\vec{r}}_{s} - {\vec{r}}^{'} | + \hat{k} \cdot {\vec{r}}^{'})]}{| {\vec{r}}_{s} - {\vec{r}}^{'} |} d S^{'}

where

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

,

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

,

(θ_{i}, φ_{i})

is the incident direction, and

(θ_{s}, φ_{s})

is the scattering direction, and

\vec{V} = \hat{k} - {\hat{k}}_{s}

,

\vec{W} = \hat{k} + {\hat{k}}_{s}

.

As illustrated in Fig. 1, the vector distance of the point on the rough surface $S^{'}$ can be approximated as the vector distance of the point on the unperturbed surface S plus its fluctuation along the normal direction, i.e.

{\vec{r}}^{'} = {\vec{r}}_{c} + \hat{n} ({\vec{r}}_{c}) ξ ({\vec{r}}_{c})

Also we have

| {\vec{r}}_{s} - {\vec{r}}^{'} | \approx | {\vec{r}}_{s} - {\vec{r}}_{c} | - ξ ({\vec{r}}_{c}) \hat{n} ({\vec{r}}_{c}) \cdot {\hat{k}}_{s}

Since

d S^{'} = \hat{n} \cdot \hat{N} d S

Assuming the mean square slope is much smaller than unit,

\hat{n} \cdot \hat{N} \approx 1 (R_{i} \vec{V} - \vec{W}) \cdot \hat{N} \approx (R_{i} \vec{V} - \vec{W}) \cdot \hat{n}

Using Eqs. (10) to (13) in Eq. (9) yields

E_{s} ({\vec{r}}_{s}) = \frac{i k}{4 π} \int_{S^{'}} (R_{i} \vec{V} - \vec{W}) \cdot \hat{n} \exp (i k \vec{V} \cdot \hat{n} ξ) \exp [i k (| {\vec{r}}_{s} - {\vec{r}}_{c} | + \hat{k} \cdot {\vec{r}}_{c})] / | {\vec{r}}_{s} - {\vec{r}}_{c} | d S

Assuming the object is to be conducting, Eq. (14) can be further approximated as following

E_{s} ({\vec{r}}_{s}) = \frac{- i k}{2 π} \int_{S} \hat{k} \cdot \hat{n} \exp (i k \vec{V} \cdot \hat{n} ξ) \frac{\exp [i k (| {\vec{r}}_{s} - {\vec{r}}_{c} | + \hat{k} \cdot {\vec{r}}_{c})]}{| {\vec{r}}_{s} - {\vec{r}}_{c} |} d S

Making use of the far-zone approximation, the exponential term above can be rewritten as

| {\vec{r}}_{s} - {\vec{r}}_{c} | \approx R - {\vec{r}}_{c} \cdot {\hat{k}}_{s}

and the denominator in Eq. (15) can be approximated as R which is the distance between the point P and the surface S. Then the far-field scattered field from a rough object can be given by

E_{s} = \frac{- i k \exp (i k R)}{2 π R} \int_{S} \hat{k} \cdot \hat{n} \exp (i k \vec{V} \cdot \hat{n} ξ) \exp (i k \vec{V} \cdot {\vec{r}}_{c}) d S

Equation (17) is the tangent-plane approximation solution of the scattered field in the far field by an arbitrarily shaped rough object. From the equation we can see that comparing with the scattering from a rough surface [8], the scattering from a rough object has a factor

\exp (i k \vec{V} \cdot {\vec{r}}_{c})

which is introduced by the curvature of the object.

According to the reference [7], the two-frequency mutual coherence function of the scattering from an arbitrarily shaped object with rough surface can be easily written as

〈 E_{s f 1} E_{s f 2}^{*} 〉 = K \int d S_{1} \int d S_{2} (\hat{k} \cdot {\hat{n}}_{1}) (\hat{k} \cdot {\hat{n}}_{2}) \exp [i \vec{V} \cdot (k_{1} {\vec{r}}_{c 1} - k_{2} {\vec{r}}_{c 2})] (χ_{t} - χ_{1} χ_{2})

where

E_{s f 1}

is the incoherent part of the scattering field,

K = k_{1} k_{2} \exp (i ω_{d} R / c) / {(2 π R)}^{2}

, and

χ_{1, 2} = < \exp (\pm i k_{1, 2} \vec{V} \cdot {\hat{n}}_{1, 2} ξ_{1, 2} >

,

χ_{t} = < \exp [i \vec{V} \cdot (k_{1} ξ_{1} {\hat{n}}_{1} - k_{2} ξ_{2} {\hat{n}}_{2})] >

are the first- and second-order characteristic function of the random variables, respectively.Since the mean curvature radius of the object is much larger than the incident wavelength and the correlation length of the rough surface, the tangent-plane approximation can be applied to simplify the Eq. (18)

〈 E_{s f 1} E_{s f 2}^{*} 〉 = K \int d S \int d {\vec{R}}_{⊥} {(\hat{k} \cdot \hat{n})}^{2} \exp (i ω_{d} \vec{V} \cdot {\vec{r}}_{c} / c) \exp (- i k_{2} \vec{V} \cdot {\vec{R}}_{⊥}) (χ_{t} - χ_{1} χ_{2})

where

{\vec{R}}_{⊥}

is the tangent plane at

{\vec{r}}_{c}

.The fluctuation

ξ ({\vec{r}}_{c})

obeys Gaussian distribution with rms δ and correlation length

l_{c}

χ_{1, 2} = \exp (- k_{1, 2}^{2} V_{z}^{2} δ^{2} / 2)

χ_{t} = \exp {- [(k_{1}^{2} + k_{2}^{2}) V_{z}^{2} δ^{2} / 2 - k_{1} k_{2} V_{z}^{2} δ^{2} < ξ_{1} ξ_{2} >]}

where

V_{z} = \vec{V} \cdot \hat{n}

. Let

ω_{c} = (ω_{1} + ω_{2}) / 2

,

ω_{d} = ω_{1} - ω_{2}

, and for a narrow incident pulse wave,

ω \approx | ω_{c} | > > ω_{d}

, the two-frequency mutual coherence function can be simplified as

\begin{array}{l} 〈 E_{s f 1} E_{s f 2}^{*} 〉 = K \int d S {(\hat{k} \cdot \hat{n})}^{2} \exp (i ω_{d} \vec{V} \cdot {\vec{r}}_{c} / c) \exp (- δ^{2} V_{z}^{2} ω_{d}^{2} / 2 c^{2}) \\ \times \int d {\vec{R}}_{⊥} \exp (- i k \vec{q} \cdot {\vec{R}}_{⊥}) (χ_{t} - χ^{2}) \end{array}

χ_{t} = \exp [- k^{2} δ^{2} V_{z}^{2} (1 - < ξ_{1} ξ_{2} >)], χ = \exp (- k^{2} δ^{2} V_{z}^{2} / 2)

According to the definition of the scattering cross section per unit area when a plane wave illuminates a rough surface [9]

σ_{p}^{0} = \frac{V_{z}^{2}}{4 π} \int d {\vec{R}}_{⊥}^{'} \exp (- i k \vec{V} \cdot {\vec{R}}_{⊥}^{'}) (χ_{t} - χ^{2})

Then the Eq. (20) can be rewritten as

〈 E_{s f 1} E_{s f 2}^{*} 〉 = 4 π K γ_{12}

and

γ_{12} = \int d S σ_{p}^{0} \exp (i ω_{d} \vec{V} \cdot {\vec{r}}_{c} / c) \exp (- δ^{2} V_{z}^{2} ω_{d}^{2} / 2 c^{2})

which is the main factor in the two-frequency mutual coherence function.

So far, we get the two-frequency mutual coherence function by arbitrarily shaped objects with Gaussian fluctuating rough surface.

3. Numerical results and analysis

For simplicity, we define $γ_{12}$ to be the two frequency scattering function, and in the following, the numerical results and analysis are for $γ_{12}$ . As an example, the functions $γ_{12}$ of rough conducting spheres and cylinders with Gaussian slightly rough surface will be computed and analyzed in detail. The incident wavelength is 1.06μm, the roughness of the rough surface is characterized by rms δ and correlation length $l_{c}$ . In the following we will give the numerical results to illustrate the effect of the roughness and the dimension of the object on the two-frequency mutual coherence function. The correlation length $l_{c}$ in all the numerical results keeps invariable.

3.1 Rough spheres

As illustrated in Fig. 2 , the center of the sphere is located at the origin of coordinate, its radius is a, the incident wave illuminates the sphere along the direction $- \hat{Z}$ , the observation plane is in the plane of $X O Z$ , therefore, the angles $θ_{i} = φ_{i} = φ_{s} = 0^{°}$ , the function $γ_{12}$ changes with the scattering angle $θ_{s}$ .

Fig. 2 scattering geometry for rough spheres.

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According to Eq. (22), the variations of the function $γ_{12}$ with the scattering angle and the frequency difference under different conditions are illustrated in Figs. 3 –5 .

Fig. 3 Function $γ_{12}$ of spheres with $δ = 0.03 μ m, l_{c} = 5 δ, a = 5 c m .$

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Fig. 5 Function $γ_{12}$ of spheres with $δ = 0.03 μ m, l_{c} = 5 δ, a = 2 c m .$

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From the figures above, we can see that the function $γ_{12}$ decrease rapidly with the increases of the scattering angle and the frequency difference. From Fig. 3 and Fig. 4 , it illustrates the effect of the roughness on the function $γ_{12}$ . With the increase of the roughness, the peak value of $γ_{12}$ increases, and the decrease of $γ_{12}$ with the frequency difference becomes smoother. From Fig. 4 and Fig. 5, we can see the effect of the radius a on the function $γ_{12}$ . With same roughness, the smaller the radius, the smaller the peak value of $γ_{12}$ , the slower the decrease of $γ_{12}$ with the frequency difference.

Fig. 4 Function $γ_{12}$ of spheres with $δ = 0.05 μ m, l_{c} = 5 δ, a = 5 c m .$

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In order to demonstrate the effect of the roughness and the radius of the sphere on the function $γ_{12}$ more concretely, in the following we give the normalized numerical results of $γ_{12}$ in the backscattering direction under different conditions.

From Fig. 6 below, we can see that with the increase of the roughness, the backscattering $γ_{12}$ becomes smoother, but its fluctuation is very obvious. If we define that the coherent bandwidth is the frequency difference $ω_{d}$ when $γ_{12}$ reaches its first minimize, the coherent bandwidth of $γ_{12}$ is approximately invariant with different roughness.

Fig. 6 Normalized $γ_{12}$ of spheres with different roughness.

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However, in Fig. 7 , we can see that with a smaller radius, the coherent bandwidth increases and $γ_{12}$ becomes smoother. In Fig. 8 , $ω_{d} = 0, θ_{s} \neq θ_{i}$ , the normalized function $γ_{12}$ changes into the normalized bistatic scattering cross section. The figure illustrates that the bistatic scattering cross section of the sphere is depend on the roughness not the radius which is consistent with the reduced scale theory proposed in reference [20].

Fig. 7 Normalized $γ_{12}$ of spheres with different radius.

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Fig. 8 Normalized $γ_{12}$ of rough spheres versus scattering angle under different condition.

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3.2 rough cylinders

As illustrated in Fig. 9 , the center of the cylinder is located at the origin of coordinate, its radius and length are a and L, respectively. The incident wave illuminates the cylinders along the direction $- \hat{X}$ , the observation plane is in the plane of $X O Y$ , the incident direction $(θ_{i}, φ_{i}) = (90^{°}, 180^{°})$ and the scattering angle $θ_{s}$ is $90^{°}$ , the function $γ_{12}$ changes with the scattering azimuth angle $φ_{s}$ .

Fig. 9 scattering geometry for rough cylinders.

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The variations of the function $γ_{12}$ with the scattering angle and the frequency difference under different conditions are illustrated in Figs. 10 –12 .

Fig. 10 Function $γ_{12}$ with $δ = 0.03 μ m, l_{c} = 5 δ, a = L = 5 c m .$

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Fig. 12 Function $γ_{12}$ of cylinders with $δ = 0.03 μ m, l_{c} = 5 δ, a = 2 c m, L = 5 c m .$

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Fig. 11 Function $γ_{12}$ of cylinders with $δ = 0.05 μ m, l_{c} = 5 δ, a = L = 5 c m .$

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From Figs. 10–12 above, we can see that the function $γ_{12}$ of rough cylinders decreases rapidly with the increases of the scattering azimuth angle and the frequency difference. With the increase of the roughness, the peak value of $γ_{12}$ increases, the decrease of $γ_{12}$ with the frequency difference becomes smoother; With same roughness, the smaller the radius, the smaller the peak value of $γ_{12}$ , the slower the decrease of the function $γ_{12}$ against the frequency difference.

Also, in order to demonstrate the effect of the roughness and the dimension of the cylinder on the function $γ_{12}$ more concretely, we give the normalized numerical results of $γ_{12}$ in the backscattering direction under different conditions.

From Fig. 13 below, we can see that the decrease of the backscattering $γ_{12}$ against the frequency difference is also fluctuating, with the increase of the roughness, this fluctuation becomes weakened, the profile becomes smoother, and the decrease of the function $γ_{12}$ becomes slower.

Fig. 13 Normalized $γ_{12}$ of cylinders with various roughness.

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In Fig. 14 , we only can see the effect of the radius not the length of the cylinders on the function $γ_{12}$ . Combing with Fig. 14 and Fig. 15 , we can see that the smaller the radius of the cylinder, the larger the coherent bandwidth, the slower the decrease of the normalized function $γ_{12}$ while the larger the length of the cylinder, the bigger the value of the function $γ_{12}$ . That is, the radius mainly influents the profile of the function $γ_{12}$ while the length only influents the value of $γ_{12}$ . The coherent bandwidth is closely dependent on the radius of the object, but has no relationship with the length and the roughness.

Fig. 14 Normalized $γ_{12}$ of cylinders with various sizes.

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Fig. 15 Backscattering $γ_{12}$ of rough cylinders with different dimensions.

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In Fig. 16 , $ω_{d} = 0$ , the normalized two-frequency scattering function changes into the normalized bistatic scattering cross section. The figure illustrates that the bistatic scattering cross section of the cylinder only depend on the roughness not the dimension.

Fig. 16 Normalized $γ_{12}$ versus scattering angle with different roughness and different dimensions.

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4. Conclusion

Based on the Physical optics approximation, the narrow pulse plane wave scattering from slightly rough conducting object was investigated by its two-frequency mutual coherence function. The scattering field in the far zone by arbitrarily shaped objects with Gaussian rough surface and its two-frequency mutual coherence function were derived theoretically. And for simplification, the numerical results for a rough sphere and a rough cylinder were given and analyzed. The results showed that the two-frequency scattering function $γ_{12}$ had closely relationship with the roughness and the radii of the objects, and when the light was vertical incident on a cylinder, the length of the cylinder only influenced the amplitude not the profile of $γ_{12}$ . The function $γ_{12}$ decreased rapidly with the increase of the scattering angle and the frequency difference. The rougher the object, the bigger the peak value of $γ_{12}$ , the slower the decrease of $γ_{12}$ ; the coherent bandwidth of $γ_{12}$ had no obvious relationship with the roughness but was closely dependent on the curvature radius of the object, the smaller the radius, the bigger the coherent bandwidth. The work in this paper will be further applied to investigate the time domain scattering and the statistical properties of speckle from complicated rough objects and the range-Doppler imaging, then it can provide some theoretical basis for the radar system design, target detection and reorganization, tracking and positioning and feature extraction.

Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant No. 60771038.

References and links

1. S. Abrahamsson, B. Brusmark, H. C. Strifors, and G. C. Gaunaurd, “Extraction of target signature features in the combined time-frequency domain by means of impulse radar,” Proc. SPIE 1700, 102–113 (1992). [CrossRef]

2. H. C. Strifors and G. C. Gaunaurd, “Scattering of electromagnetic pulses by simple-shaped targets with radar cross section modified by a dielectric coating,” IEEE Trans. Antenn. Propag. 46(9), 1252–1262 (1998). [CrossRef]

3. L. Mees, G. Gouesbet, and G. Gréhan, “Scattering of laser pulses (plane wave and focused gaussian beam) by spheres,” Appl. Opt. 40(15), 2546–2550 (2001). [CrossRef]

4. Y. H. Li and Z. S. Wu, “Targets recognition using subnanosecond pulse laser range profiles,” Opt. Express 18(16), 16788–16796 (2010). [CrossRef] [PubMed]

5. Y. H. Li, Z. S. Wu, Y. J. Gong, G. Zhang, and M. J. Wang, “Laser one-dimensional range profile,” Acta Phys. Sin. 59, 6985–6990 (2010).

6. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).

7. 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]

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

9. L. Guo and C. Kim, “Study on the two-frequency scattering cross section and pulse broadening of the one-dimensional fractal sea surface at millimeter wave frequency,” Prog. Electromagn. Res. 37, 221–234 (2002). [CrossRef]

10. E. Bahar and M. A. Fizwater, “Scattering and depolarization by large conducting spheres with rough surface,” Appl. Opt. 24(12), 1820–1825 (1985). [CrossRef] [PubMed]

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

12. R. Schiffer, “The Coherent scattering cross-section of a slightly rough sphere,” J. Mod. Opt. 33(8), 959–980 (1986).

13. M. K. Abdelazeez, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antenn. Propag. 31(2), 375–377 (1983). [CrossRef]

14. R. G. Berlasso, F. P. Quintián, M. A. Rebollo, N. G. Gaggioli, B. L. Sánchez Brea, and M. E. Bernabeu Martínez, “Speckle size of light scattered from slightly rough cylindrical surfaces,” Appl. Opt. 41(10), 2020–2027 (2002). [CrossRef] [PubMed]

15. R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, and N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39(31), 5811–5819 (2000). [CrossRef]

16. Z. S. Wu, “IR Laser Backscattering by arbitrarily shaped dielectric object with rough surface,” SPIE's International Symposium on Optical Science and Engineering in San Diego, California, (21–26, July,1991).

17. 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]

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

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

20. S. Zhendong and L. Hongwei, “Model-measurement and reverse evaluation for RCS of stealthy targets,” J. Univ. Electron. Sci. Technol, China 24, 13–17 (1995).

Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects

Abstract

1. Introduction

2. Two frequency mutual coherence function of the scattering from arbitrarily shaped rough objects

3. Numerical results and analysis

3.1 Rough spheres

3.2 rough cylinders

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (16)

Equations (26)

Optics Express