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

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

Geng Zhang and Zhensen Wu*

School of Science, Xidian University, xi’an 710071, China *[email protected]

Abstract: Based on the Physical optics approximation, the scattering field in the far zone by arbitrarily shaped objects with slightly rough surface which obeys Gaussian distribution and its two-frequency mutual coherence function are derived theoretically, and the numerical results for rough spheres and rough cylinders are given and analyzed. The results show that the function has closely relationship with the roughness and the dimension of the rough objects. The roughness and the curvature of the object influence both the amplitude and the profile of the two-frequency mutual coherence function. Also, the smaller the radius of the object, the larger the coherent bandwidth. The two-frequency mutual coherence function can be used to investigate the laser pulse scattering characteristics of arbitrarily shaped rough objects, provide theoretical basis for target recognition.

©2011 Optical Society of America

OCIS codes: (290.1350) Scattering, rough surface; (140.3538) Lasers, pulsed; (260.0260) Physical optics.

References and links

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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).

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).

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

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). 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). 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). 10. E. Bahar, and M. A. Fizwater, “Scattering and depolarization by large conducting spheres with rough surface,”

Appl. Opt. 24(12), 1820–1825 (1985). 11. E. Bahar, and M. A. Fitzwater, “Scattering and depolarization by conducting cylinders with rough surfaces,”

Appl. Opt. 25(11), 1826–1832 (1986). 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). 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).

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).

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).

#134720 - $15.00 USD Received 13 Sep 2010; revised 17 Oct 2010; accepted 18 Oct 2010; published 29 Mar 2011(C) 2011 OSA 11 April 2011 / Vol. 19, No. 8 / OPTICS EXPRESS 7007

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).

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

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).

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 ˆ( ) exp( )iE r ikk r illuminates a roughened convex object, the scattering

geometry is as illustrated in Fig. 1. The surface S is the unperturbed surface, n is the

corresponding external normal, cr is its vector distance and

i is the local incident angle at cr

while S is the roughened surface which is the surface S plus a random fluctuation ( )cr , N is

its corresponding normal, and r is the vector distance, i is the incident angle at r . k

and ˆsk are the incident unit vector and the scattering unit vector, respectively.


2 / /k c is the wavenumber, is the wavelength and is its angular frequency. The

time harmonic factor exp( )i t is omitted for convenience.

S

sk

S

Nnk

i i

k

cr

cr r

X

Y

Z

O

sr

r

P

Fig. 1. scattering geometry for a roughened object.

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]

,

,ˆ ˆs

s s sS

G r r E rE r E r G r r dS

N N

(1)

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

E r and ˆE r N are the total electric field and its normal derivative on the scattering

surface S , ,sG r r and ˆ,sG r r N are Green’s function and its normal derivative,

respectively. The Green’s function ,sG r r is given by

exp

,4

s

s

s

ik r rG r r

r r

(2)

The total electric field E r is a sum of the incident field iE r and the scattered E r on

the surface S , that is

iE r E r E r (3)

Since the incident field iE r satisfies

,

, 0ˆ ˆs i

i sS

G r r E rE r G r r dS

N N

(4)

The scattered field s sE r at point P can be obtained

,

,ˆ ˆs

s s sS

G r r E rE r E r G r r dS

N N

(5)

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 ron the rough surface are written as following, respectively

1 i iE r R E r (6)


ˆ ˆ1ˆ i i

E ri R kk NE r

N

(7)

whereiR is the Fresnel reflection coefficient.

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

, ˆ ˆ ,

ˆs

s s

G r rikk NG r r

N

(8)

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

êxp

ˆ4

s

s s iS

s

ik r r k rikE r RV W N dS

r r

(9)

where ˆ sin cos ,sin sin , cosi i i i ik , ˆ sin cos ,sin sin ,coss s s s s sk , ,i i is

the incident direction, and ,s s is the scattering direction, and ˆ ˆsV k k , ˆ ˆ

sW k k .

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.

ˆ( ) ( )c c cr r n r r (10)

Also we have

ˆˆ( ) ( )s s c c c sr r r r r n r k (11)

Since

ˆˆdS n NdS (12)

Assuming the mean square slope is much smaller than unit,

ˆ ˆˆ ˆ1 ( ) ( )i in N RV W N RV W n (13)

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

ˆˆ êxp exp4

s s i s c c s cS

ikE r RV W n ikV n ik r r k r r r dS

(14)

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

êxp

ˆ ˆ êxp2

s c c

s sS

s c

ik r r k rikE r k n ikV n dS

r r

(15)

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

ˆs c c sr r R r k (16)

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

exp ˆ ˆ êxp exp2

s cS

ik ikRE k n ikV n ikV r dS

R

(17)


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 cikV r 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

*

1 2 1 2 1 2 1 1 2 2 1 2ˆ ˆˆ ˆ( )( )expsf sf c c tE E K dS dS k n k n iV k r k r (18)

where 1sfE is the incoherent part of the scattering field, 2

1 2 exp 2dK k k i R c R , and

1,2 1,2 1,2 1,2ˆexp( ik V n , 1 1 1 2 2 2

ˆ ˆexp[ ]t iV k n k n 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)

* 2

1 2 2 1 2ˆ ˆ( ) exp / expsf sf d c tE E K dS dR k n i V r c ik V R (19)

where R is the tangent plane at cr .

The fluctuation ( )cr obeys Gaussian distribution with rms and correlation length cl

2 2 2

1,2 1,2exp( / 2)zk V

2 2 2 2 2 2

1 2 1 2 1 2exp{ [( ) 2 ]}t z zk k V k k V

where ˆzV V n . Let 1 2 2c ,

1 2d , and for a narrow incident pulse wave,

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

* 2 2 2 2 2

1 2

2

ˆ ˆ( ) exp exp 2

exp

sf sf d c z d

t

E E K dS k n i V r c V c

dR ikq R

(20)

where 2 2 2 2 2 2

1 2exp[ (1 )], exp( / 2)t z zk V k V .

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

2

0 2exp( )( )4

zp t

VdR ikV R

Then the Eq. (20) can be rewritten as

*

1 2 124sf sfE E K (21)

and

0 2 2 2 2

12 exp exp 2p d c z ddS i V r c V c (22)

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 12 to be the two frequency scattering function, and in the

following, the numerical results and analysis are for12 . As an example, the functions

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 and correlation lengthcl . 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 cl 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 Z , the observation plane is

in the plane of XOZ , therefore, the angles 0i i s , the function 12 changes with the

scattering angles .

According to Eq. (22), the variations of the function12 with the scattering angle and the

frequency difference under different conditions are illustrated in Figs. 3–5.

-Y

Z

XO

s

ˆsk

k

Fig. 2. scattering geometry for rough spheres.

015

3045

6075

90

0.0

0.2

0.4

0.6

0.8

1.0

180

150

12090

6030

0

s /

deg

d / GHZ

×10-3

Fig. 3. Function 12 of spheres with 0.03 , 5 , 5 .cm l a cm


015

3045

6075

90

0.0

0.5

1.0

1.5

2.0

2.5

3.0

180

150

12090

6030

0

s /

deg

d / GHZ

×10-3


015

3045

6075

90

0.0

0.1

0.2

0.3

0.4

0.5

180

150120

9060

300

s /

deg

d / GHZ

×10-3


From the figures above, we can see that the function12 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 function12 . With the increase of the roughness, the peak value

of12 increases, and the decrease of

12 with the frequency difference becomes smoother. From

Fig. 4 and Fig. 5, we can see the effect of the radius a on the function12 . With same

roughness, the smaller the radius, the smaller the peak value of12 , the slower the decrease of

12 with the frequency difference.

In order to demonstrate the effect of the roughness and the radius of the sphere on the

function12 more concretely, in the following we give the normalized numerical results

of12 in the backscattering direction under different conditions.

From Fig. 6 below, we can see that with the increase of the roughness, the backscattering

12 becomes smoother, but its fluctuation is very obvious. If we define that the coherent

bandwidth is the frequency differenced when

12 reaches its first minimize, the coherent

bandwidth of12 is approximately invariant with different roughness.


0 15 30 45 60 75 900.0

0.2

0.4

0.6

0.8

1.0

=0.03m

=0.04m

=0.05m

No

rm

ali

zed

d / GHZ

lc=5

a=5cm

Fig. 6. Normalized 12 of spheres with different roughness.

0 15 30 45 60 75 900.0

0.2

0.4

0.6

0.8

1.0

No

rm

ali

zed

d / GHZ

a=1cm

a=2cm

a=5cm

=0.05m

lc=5

Fig. 7. Normalized 12 of spheres with different radius.

However, in Fig. 7, we can see that with a smaller radius, the coherent bandwidth

increases and12 becomes smoother. In Fig. 8, 0,d s i , the normalized function

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].


0 30 60 90 120 150 180

0.0

0.2

0.4

0.6

0.8

1.0

No

rm

ali

zed

s / deg

=0.05m, a=5cm

=0.05m, a=2cm

=0.03m, a=5cm

=0.03m, a=2cm

lc=5

d=0

Fig. 8. Normalized 12 of rough spheres versus scattering angle under different condition.

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 X , the observation plane is in the plane of XOY , the incident direction

, 90 ,180i i and the scattering angle s is 90 , the function

12 changes with the

scattering azimuth angles .

The variations of the function12 with the scattering angle and the frequency difference

under different conditions are illustrated in Figs. 10–12.

Yk ˆ

sk

Z

X

s

Fig. 9. scattering geometry for rough cylinders.


015

3045

6075

90

0.0

0.1

0.2

0.3

0.4

0.5

0.6

180

150120

9060

300

1

2

s /

deg

d / GHZ

×10-3

Fig. 10. Function 12 with 0.03 , 5 , 5 .cm l a L cm

015

3045

6075

90

0.0

0.5

1.0

1.5

2.0

2.5

180

150120

9060

300

12

s /

deg

d / GHZ

×10-3

Fig. 11. Function 12 of cylinders with 0.05 , 5 , 5 .cm l a L cm

015

3045

6075

90

0.00

0.03

0.06

0.09

0.12

0.15

0.18

0.21

180

150

12090

6030

0

1

2

s /

deg

d / GHZ

×10-3

Fig. 12. Function 12 of cylinders with 0.03 , 5 , 2 , 5 .cm l a cm L cm


From Figs. 10–12 above, we can see that the function12 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 of12 increases, the decrease of

12 with the

frequency difference becomes smoother; With same roughness, the smaller the radius, the

smaller the peak value of12 , the slower the decrease of the function

12 against the frequency

difference. Also, in order to demonstrate the effect of the roughness and the dimension of the cylinder

on the function12 more concretely, we give the normalized numerical results of

12 in the

backscattering direction under different conditions.

From Fig. 13 below, we can see that the decrease of the backscattering12 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 12

becomes slower.

0 15 30 45 60 75 900.0

0.2

0.4

0.6

0.8

1.0

=0.03m

=0.04m

=0.05m

No

rm

ali

zed

d / GHZ

lc=5

a=5cm

L=5cm

Fig. 13. Normalized 12 of cylinders with various roughness.

0 15 30 45 60 75 900.0

0.2

0.4

0.6

0.8

1.0

No

rm

ali

zed

d / GHZ

a=1cm,L=5cm

a=2cm,L=5cm

a=5cm,L=5cm

a=5cm,L=2cm

a=5cm,L=8cm

=0.05m

lc=5

Fig. 14. Normalized 12 of cylinders with various sizes.


In Fig. 14, we only can see the effect of the radius not the length of the cylinders on the

function12 . 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

function12 while the larger the length of the cylinder, the bigger the value of the function

12 .

That is, the radius mainly influents the profile of the function 12 while the length only

influents the value of12 . The coherent bandwidth is closely dependent on the radius of the

object, but has no relationship with the length and the roughness.

0 15 30 45 60 75 900.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

d / GHZ

a=1cm,L=5cm

a=2cm,L=5cm

a=5cm,L=5cm

a=5cm,L=2cm

a=5cm,L=8cm

=0.05m

lc=5

×10-3

Fig. 15. Backscattering 12 of rough cylinders with different dimensions.

0 30 60 90 120 150 180

0.0

0.2

0.4

0.6

0.8

1.0

No

rm

ali

zed

s / deg

=0.03m, a=5cm,L=5cm

=0.03m, a=3cm,L=5cm

=0.03m, a=5cm,L=3cm

=0.05m, a=3cm,L=5cm

=0.05m, a=5cm,L=5cm

=0.05m, a=5cm,L=3cm

lc=5

d=0

Fig. 16. Normalized 12 versus scattering angle with different roughness and different

dimensions.

In Fig. 16, 0d , 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.

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 function12 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 12 . The function

12 decreased rapidly with the increase of the scattering angle and the

frequency difference. The rougher the object, the bigger the peak value of12 , the slower the

decrease of12 ; the coherent bandwidth of

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


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

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