Science in China Series G: Physics, Mechanics & Astronomy

© 2008 SCIENCE IN CHINA PRESS

Springer-Verlag

Received March 7, 2006; accepted March 24, 2007 doi: 10.1007/s11433-007-0021-2 †Corresponding author (email: ychrenmail@126.com) Supported by the National Natural Science Foundation of China (Grant No. 60571058) and the National Defense Foundation of China (Grant No. 51403020505DZ0111)

Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 157-164

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Two-frequency mutual coherence function and its applications to pulse scattering by random rough surface

REN YuChao† & GUO LiXin School of Science, Xidian University, Xi’an 710071, China

An analytic expression of the two-frequency mutual coherence function (MCF) was derived for a two-dimensional random rough surface. The scattered field was cal-culated by the Kirchhoff approximation, which is valid in the case that the radius of curvature of the surface is much larger than the incident wave length. The scatter-ing problem of narrowband pulse was investigated to simplify the analytic expres-sion of the two-frequency MCF. Numerical simulations show that the two-frequency MCF is greatly dependent on the root-mean-square (RMS) height, while less de-pendent on the correlation length. The analytic solutions were compared with the results of Monte Carlo simulation to assess the accuracy and computational effi-ciency.

mutual coherence function, pulse scattering, random rough surface

The problem of millimeter wave or optics scattering from a random rough surface has been in-vestigated extensively for several decades, because of its broad applications in science and tech-nology. The frequency characteristics of the conventional scattering cross section for rough sur-faces cannot give the pulse characteristics, so it is necessary to study the correlation of the scattered field at two different frequencies. The “two-frequency mutual coherence function (MCF)”[1] is the correlation of the scattering waves at two different frequencies for given incident and scattering angles, and it represents the coherence bandwidth of the scattering waves. The inverse Fourier transform of the two-frequency MCF is the scattering pulse shape in the time domain.

In the literature, Galdi et al.[2] simulated the time-dependent scattering characteristics of elec-tromagnetic wave by rough interfaces based on the physical optics method. Phu[3] studied the relation between the surface roughness and the angular or frequency correlations of the scattering wave with optical and millimeter wave experiments. Up to date, Ishimaru et al. have used the Monte Carlo technique to obtain the numerical solutions of the two-frequency MCF by one-di-

158 REN YuChao et al. Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 157-164

mensional random rough surface, and their work was reported in a series of papers[4－7]. Monte Carlo technique is an excellent approach to the study of rough surface scattering, when the illu-minated region is small. For an extremely large illuminated region of two-dimensional surface, Monte Carlo technique becomes computationally expensive. Guo[8] has recently derived analytic formulas of the two-frequency MCF for the fractal rough surface. It is noted that the analytic method is more efficient than the Monte Carlo technique as it saves computation time significantly. However, Guo’s work was only devoted to the one-dimensional rough surface.

In this paper, we focus on deriving an explicitly analytic expression of the two-frequency MCF in the case of two-dimensional rough surface. The root-mean-square (RMS) slope of the rough surfaces is assumed to be less than 0.5, so that multiple scattering is negligible and only the first-order Kirchhoff approximation needs to be included to calculate the scattering from random rough surfaces.

1 Two-frequency MCF The Kirchhoff approximation is known to be reliable as the incident wave number becomes much larger than the maximum surface wave number, provided the local incidence angle remains low. It relies on the physical optics approximation, which amounts to identifying the surface with its local tangent plane. According to the Kirchhoff approximation, the scattered fields are given by[9]

0( , ) ( ) ( )exp[i( ) ] ,spq pq s iE E K U dSω ω ω ′ ′= − ⋅∫r k k r (1)

where 0 0 0 0( ) i exp( i ) / 4 ,K k k R Rω π= − − 0k cω= is the free-space wave number, ω is the radial frequency of incident wave (with c denoting the light velocity in free space). The remaining term in eq. (1) is 0 1 2( ) ( ) ( ) ( ) ,pq pq pq x pq yU a a Z a Zω ω ω ω= + + 0 ( ),pqa ω 1( )pqa ω and 2 ( )pqa ω are po-

larization coefficients described in ref. [9], xZ and yZ are the surface slopes in x and y di-

rection for a point ′r on the rough surface. ik and sk in eq. (1) are the respective incident and scattered wave vectors, and dS ′ denotes the differential surface element.

From the definition of the two-frequency MCF[1], we have

* 2 *

1 2 1 2 0 1 2

*1 2 1 2 1 1 2 2

( , ) ( ) ( ) | | ( ) ( )

( ) ( )exp[i( )] ,

s spq pq pq

pq pq

E E E K K

dS dS U U

ω ω ω ω ω ω

ω ω

Γ = =

′ ′ ′ ′× ⋅ − ⋅∫ ∫ V r V r

(2)

where 1,2 1,2( )s i= −V k k , the brackets ⋅ denote an ensemble average.

We make use of the relation of ˆzV z⊥= +V V and ˆ( )f z⊥ ⊥′ ′ ′= +r r r , then expand the

two-frequency MCF pqΓ in terms of the zero-order slope and the first-order slope, i.e.

1 2 0 1 2 1 2( , ) ( , ) ( , ),pq pq pqsω ω ω ω ω ωΓ ≈ Γ + Γ (3)

where

2 * *

0 1 2 0 1 2 1 2 0 1 0 2

1 1 2 2 1 1 2 2

( , ) | | ( ) ( ) ( ) ( )

exp[i( )] exp[i( )],pq pq pq

z z

E K K d d a a

V f V f

ω ω ω ω ω ω⊥ ⊥

⊥ ⊥ ⊥ ⊥

′ ′Γ =

′ ′× − ⋅ − ⋅∫ ∫r r

V r V r

(4)

and 2 * *

1 2 0 1 2 1 2 1 1 0 2 1( , ) | | ( ) ( ) [ ( ) ( )pqs pq pq xE K K d d a a Zω ω ω ω ω ω⊥ ⊥′ ′Γ = ∫ ∫r r

REN YuChao et al. Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 157-164 159

* * *

0 1 1 2 2 2 1 0 2 1 0 1 2 2 2

1 1 2 2 1 1 2 2

( ) ( ) ( ) ( ) ( ) ( ) ]

exp[i( )] exp[i( )].pq pq x pq pq y pq pq y

z z

a a Z a a Z a a Z

V f V f

ω ω ω ω ω ω

⊥ ⊥ ⊥ ⊥

+ + +

′ ′⋅ − ⋅ − ⋅V r V r

(5)

In eq. (5), the terms of the second-order slope are ignored.

1.1 Derivation of Γpq0

It is expected that the integrations in eqs. (4) and (5) can be evaluated using asymptotic approxi-mations. Let 1 2( ) / 2,c ⊥ ⊥= +V V V 1 2 ,d ⊥ ⊥= −V V V (6) and 1 2( ) / 2,c ⊥ ⊥′ ′ ′= +r r r 1 2.d ⊥ ⊥′ ′ ′= −r r r (7) To derive an analytic expression of the two-frequency MCF, it is assumed that the rough surface is a stationary Gaussian process. In this case, the characteristic function is

2 2 2 21 1 2 2 1 2 1 2exp[i( )] exp{ [( ) / 2 ]},z z z z z zV f V f V V V Vσ σ ρ− = − + − (8)

where 2 21 2 exp( | | / )df f lρ ′= = − r is the correlation function of Gaussian correlated surface. σ

is the RMS height, and l is the correlation length. Substituting eqs. (6), (7) and (8) into eq. (4) yields

2 * *

0 1 2 0 1 2 0 1 0 2

2 2 2 21 2 1 2

( , ) | | ( ) ( ) ( ) ( ) exp(i )

exp(i )exp{ [( ) / 2 ]}.

pq pq pq c d c

d c d z z z z

E K K a a d

d V V V V

ω ω ω ω ω ω

σ σ ρ

′ ′Γ = ⋅

′ ′× ⋅ − + −

∫∫

r V r

r V r

(9)

By making use of the integral identity

1 sinexp(i ) sin ( ),2

L

L

VLVx dx c VLL VL−

= =∫ (10)

we arrive at the integration over cd ′r in eq. (9), i.e.

exp(i ) sin ( )sin ( ),c d c dx dyd A c V L c V L′ ′ ′⋅ =∫ r r r (11)

where 2 2 ,A L L= × and 2L is the length of the illuminated region in x and y directions.

To evaluate the integration over dd ′r in eq. (9), we make Taylor expansions:

2

2 1 21 2

0

( )exp( ) .!

nnz z

z zn

V VV Vn

σσ ρ ρ

∞

=

= ∑ (12)

Then the integration over dd ′r is simplified as

0 00 0 ,nI I I= + (13) where 2 2 2

00 1 2exp[ ( ) / 2] sin ( )sin ( ),z z cx cyI V V A c V L c V Lσ= − + (14)

2 2 22

2 2 2 2 1 20 1 2

1

( )( )exp[ ( ) / 2] exp ,! 4

ncx cyz z

n z zn

V V lV VI V V ln n n

σσ π

∞

=

⎡ ⎤+= − + −⎢ ⎥

⎣ ⎦∑ (15)

in which 0n = denotes the case of coherent scattering, and 1n≥ denotes the case of incoherent scattering. Eq. (15) is evaluated under the assumption that the linear dimension of the rough surface is much larger compared with the surface height correlation length.

Putting eqs. (11) and (13) into eq. (9) gives

160 REN YuChao et al. Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 157-164

0 1 2 1 2 1 2( , ) ( , ) ( , ),pq pqc pqnω ω ω ω ω ωΓ = Γ + Γ (16)

where

2* 20 1 2 1 2 0

1 2 0 1 0 220

2 2 21 2

| | exp[ i( ) ]( , ) ( ) ( ) sin ( )

(4 )

sin ( )sin ( )sin ( )exp[ ( ) / 2],

pqc pq pq dx

dy cx cy z z

E k k k k Ra a A c V L

R

c V L c V L c V L V V

ω ω ω ωπ

σ

− −Γ =

× − +

(17)

2*0 1 2 1 2 0

1 2 0 1 0 220

2 2 2222 2 2 1 21 2

1

| | exp[ i( ) ]( , ) ( ) ( ) sin ( )

(4 )

( )( ) sin ( )exp[ ( ) / 2] exp .

! 4

pqn pq pq dx

ncx cyz z

dy z zn

E k k k k Ra a A c V L

R

V V lV Vlc V L V Vn n n

ω ω ω ωπ

σπσ∞

=

− −Γ =

⎡ ⎤+× − + −⎢ ⎥

⎣ ⎦∑

(18)

1.2 Derivation of Γpqs

Let us examine the expression of two-frequency MCF pqsΓ given in eq. (5). We hope to evaluate

the analytic expression of the ensemble average, so that eq. (5) may also be integrated in a close form. For the Gaussian random rough surfaces, we have[9]

2 2 2 2 21 1 2 2 (3 ) 1 2 1 2exp[i( )] i exp{ [( ) / 2 ( , )]},pj z z z j z z z z d d

dZ V f V f V V V V V x y

pρσ σ σ ρ−

∂− = − − + −

∂

, ,p x y= 1, 2.j = (19) Noting eq. (19), the integration for the case (p=x) in eq. (5) is given by

* 2 * 20 1 1 2 1 0 2 1 1 2

2 2 2 21 2 1 2

i[ ( ) ( ) ( ) ( ) ] exp(i )

exp(i )exp{ [( ) / 2 ]}.

sx pq pq z pq pq z c d c

d c d z z z zd

I a a V a a V d

d V V V Vx

ω ω σ ω ω σ

ρ σ σ ρ

′ ′= − + ⋅

∂′ ′× ⋅ − + −∂

∫

∫

r V r

r V r

(20)

To simplify the integrals, we assume that the rough surface is isotropic. Using the following variable transformations: cos ,dx ξ α= sin ,dy ξ α= (21) the partial derivates of correlation function take the following form:

( , ) ( ) cos ,d d

d

x yx

ρ ρ ξ αξ

∂ ∂=

∂ ∂

( , ) ( ) sin .d d

d

x yy

ρ ρ ξ αξ

∂ ∂=

∂ ∂ (22)

The integration over dd ′r in eq. (20) may be evaluated as

2 2 22 1

2 2 2 2 1 21 1 2

1

( )( )i exp[ ( ) / 2] exp ,! 4

ncx cyz z

sx cx z zn

V V lV VI V l V Vn n n

σπ σ

−∞

=

⎡ ⎤+= − − + −⎢ ⎥

⎣ ⎦∑ (23)

in which we have expanded 1 2exp( )z zV V ρ into Taylor series in terms of Gaussian correlation function. Substituting eqs. (11), (23) into eq. (20) yields

* *0 1 1 2 1 0 2 1 1 2

2 2 22 12 2 2 2 1 2

1 21

[ ( ) ( ) ( ) ( ) ] sin ( )sin ( )

( )( ) ( ) exp[ ( ) / 2] exp .! 4

sx pq pq cx z pq pq cx z dx dy

ncx cyz z

z zn

I a a V V a a V V A c V L c V L

V V lV Vl V Vn n n

ω ω ω ω

σπ σ σ

−∞

=

= − +

⎡ ⎤+× − + −⎢ ⎥

⎣ ⎦∑

(24)

Similarly, the evaluation of the integration for the case ( p=y) in eq. (5) then yields

REN YuChao et al. Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 157-164 161

* *0 1 2 2 1 0 2 2 1 2

2 2 22 12 2 2 2 1 2

1 21

[ ( ) ( ) ( ) ( ) ] sin ( )sin ( )

( )( ) ( ) exp[ ( ) / 2] exp .! 4

sy pq pq cy z pq pq cy z dx dy

ncx cyz z

z zn

I a a V V a a V V A c V L c V L

V V lV Vl V Vn n n

ω ω ω ω

σπ σ σ

−∞

=

= − +

⎡ ⎤+× − + −⎢ ⎥

⎣ ⎦∑ (25)

Putting eqs. (24) and (25) into eq. (5) gives

20 1 2 1 2 0

1 2 0 1 120

* * *1 2 2 2 0 2 2 1 1 2 1

2 12 2 2 2 1 2

1 21

| | exp[ ( ) ]( , ) sin ( )sin ( ) { ( )

(4 )

[ ( ) ( ) ] ( ) [ ( ) ( ) ]}

( ) ( ) exp[ ( ) / 2] exp!

pqs dx dy pq z

pq cx pq cy pq z pq cx pq cy

nz z

z zn

E k k i k k RA c V L c V L a V

R

a V a V a V a V a V

V Vl V Vn n

ω ω ωπ

ω ω ω ω ω

σπ σ σ

−∞

=

− −Γ = −

⋅ + + +

× − + −∑2 2 2( )

.4

cx cyV V ln

⎡ ⎤+⎢ ⎥⎣ ⎦

(26)

2 Formulas for narrowband pulse

Applying the variable transformations 1 2dω ω ω= − and 0 1 2( ) 2,ω ω ω= + the arguments of the

two-frequency MCF are changed from 1,ω 2ω to ,dω 0.ω In many measurement systems, the

antenna emits a narrowband pulse, in which the center radial frequency 0ω satisfies 0 dω ω . This case is called the “wide-sense stationary uncorrelated scattering channel” (WSSUS)[1], and

0( , )dω ωΓ is a slowly varying function of 0.ω With the assumption of WSSUS, the two-fre-quency MCF reduces to 0 0 0 0( , ) ( , ) ( , ) ( , ),pq d pqc d pqn d pqs dω ω ω ω ω ω ω ωΓ = Γ + Γ + Γ (27)

where

2 2 2 20 0 00 0 02

02 2 2 2 2 2

| | exp( i / )( , ) ( ) sin ( )sin ( )

(4 )

sin ( / )sin ( / )exp( )exp( / 2 ),

dpqc d pq x y

d x d y z d z

E k R ca A c V L c V L

R

c q L c c q L c V q c

ωω ω ω

π

ω ω σ ω σ

−Γ =

× − −

(28)

2 2 20 0 00 0 02

0

2 2 22 222 2 2 2 2 2

1

| | exp( i / )( , ) ( ) sin ( / )sin ( / )

(4 )

( )( ) exp( )exp( / 2 ) exp ,

! 4

dpqn d pq d x d y

nx yz

z d zn

E k R ca A c q L c c q L c

R

V V lVlV q cn n n

ωω ω ω ω ω

π

σπσ ω σ∞

=

−Γ =

⎡ ⎤+× − − −⎢ ⎥

⎣ ⎦∑

(29)

2 20 0 0

0 20

2 2 2

2 2 22 2 12 2 2 2

1

| | exp( i / )( , ) sin ( / )sin ( / )

(4 )

2 Re{ } i Im{ }/ ( ) exp( )

( )( ) exp( / 2 ) exp .! 4

dpqs d d x d y

z pq d z pq z

nx yz

d zn

E k R cA c q L c c q L c

R

V q c l V

V V lVq cn n n

ωω ω ω ω

π

χ ω χ π σ σ

σω σ

−∞

=

−Γ = −

⎡ ⎤× + −⎣ ⎦⎡ ⎤+

× − −⎢ ⎥⎣ ⎦∑

(30)

In eqs. (28)－(30), * *0 1 2( ),pq pq pq x pq ya a V a Vχ = + 0 / ,x xV q cω= 0 / ,y yV q cω= 0 / .z zV q cω= ,xq

yq and zq are the coefficients related to incidence direction and scattering direction, defined by

sin cos sin cos ,x s s i iq θ φ θ φ= − sin sin sin sin ,y s s i iq θ φ θ φ= − cos cos .z s iq θ θ= + (31)

162 REN YuChao et al. Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 157-164

The polarization coefficients 0 ,pqa 1pqa and 2pqa are independent on ,dω but determined by the

center radial frequency 0.ω With the assumption of narrowband pulse, the relation between scattering power and incident

power is simplified as

i1( ) ( ) ( )e .2

d ts d i d dP t P dωω ω ω

π−= Γ∫ (32)

The analytic theory for the scattering pulse shape allows different incident pulse shapes of ( ).i dP ω

If the incident pulse is an impulse signal ( ),tδ i( ) ( )e ,d ti d iP P t dtωω = ∫ then ( ) 1.i dP ω = There-

fore the scattering power ( )sP t is the inverse Fourier transform of the two-frequency MCF

( ).dωΓ In this case, ( )sP t is equivalent to the impulse response function of the rough surface.

3 Numerical simulation and discussion

As an example, the two-frequency MCF is calculated by eq. (27) with the central wave length of 1.06 μm,λ = the incidence angle of 0iθ = ° and the azimuthal angle of 0i sφ φ= = ° . The rough

surface is characterized by the refractive index of n=2.43+i10.7, with the area of the illuminated region of 13.3 13.3 .λ λ× Figures 1 and 2 show the two-frequency MCF as a function of fd =

(2 )dω π and .sθ It is shown that the two-frequency MCF is a maximum at the center frequency and drops rapidly towards zero as the frequency moves away from the center frequency. This re-sults in a finite coherence bandwidth, and the inverse of this bandwidth is the pulse broadening. It is also shown that the two-frequency MCF is a function of RMS height and correlation length. As shown in Figure 1, when the RMS height is small, the two-frequency MCF is less dependent on the variety of fd. With the RMS height increasing, the two-frequency MCF appears as a very narrow peak in the normal direction. It is shown in Figure 2 that the two-frequency MCF concentrates more in the specular direction for a larger correlation length.

To examine the accuracy of the analytic two-frequency MCF presented in this paper, we com-pare the analytic results with Monte Carlo simulations[10] in Figure 3. The scattering model of a perfectly conducting two-dimensional rough surface is under consideration, with 3 mm,λ = θ i = 20° , σ λ= and 3 .l λ= Figure 3(a) shows the distribution of the two-frequency MCF against the

Figure 1 Two-frequency mutual coherence function against scattering angle and frequency difference for the random rough surface. (a) σ = 0.2 μm, l = 5.89 μm; (b) σ = 0.6 μm, l = 5.89 μm.

REN YuChao et al. Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 157-164 163

Figure 2 Two-frequency mutual coherence function against scattering angle and frequency difference for the random rough surface. (a) σ = 0.8 μm, l = 5.89 μm; (b) σ = 0.8 μm, l = 7.5 μm.

Figure 3 Comparison of analytical solutions and Monte Carlo simulations. (a) fd = 0; (b) θ s = θ i = 20°.

scattering angle for the frequency difference of fd = 0. It is observed that the analytic solution is not very different from the result of the Monte Carlo simulation. The distribution of the two-frequency MCF against the frequency difference is illustrated in Figure 3(b), with 20s iθ θ= = ° . As can be seen, the analytic solution shows reasonable agreement with the result of Monte Carlo simulation throughout most of the region. The small errors between the analytic solutions and the Monte Carlo solutions are primarily due to the fact that we have made approximation operations to derive the analytic expression of the two-frequency MCF. For example, in eq. (5), only the terms of the zero-order slope and the first-order slope are retained, and the terms of the second-order slope are ignored. Significant speed-up can be expected through the use of an analytic expression to calcu-late the two-frequency MCF. The calculation of the two-frequency MCF by eq. (27) is achieved within 1 s on a 1.2 GHz PC. However, it takes several hours to perform the Monte Carlo simula-tions for 100 surface realizations in the case of millimeter scattering. Generally speaking, when the problem becomes too computationally expensive, it is recommended to rely on the analytic methods whereby it is possible to derive analytic expressions that are more efficient.

4 Conclusions

An analytic formulation of the two-frequency MCF has been derived for millimeter wave or light scattering from the random rough surface, based on the Kirchhoff scattering theory. With the help of the two-frequency MCF, it is easy to analyze the time-dependent pulse scattering power by using

164 REN YuChao et al. Sci China Ser G-Phys Mech Astron | Feb. 2008 | vol. 51 | no. 2 | 157-164

its inverse Fourier transform. The dependence of MCF on RMS height and correlation length is also analyzed in numerical simulation. The analytic solutions show good agreement with the re-sults of Monte Carlo simulations.

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