Effect of Medium Parameters on RCS of Conducting Targets for Horizontal Polarization

Hosam El-Ocla

Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada hosam @lakeheadu.ca

Abstract— This work is to study effects of the medium config­uration on radar cross section of conducting targets. Medium parameters to be investigated include correlation function of the medium and the correlation distance around the object. Numerical results are analyzed for partially convex objects.

I. INTRODUCTION

Backscattering enhancement phenomenon occurs when radar cross section (RCS) of targets in random medium is twice that in free space and this is known as double passage effect [1],[2]. It was presented a method to solve the scattering problems as a boundary value problem and it shows the medium effects on the RCS of targets [3]. In earlier studies, we used this method on concave-convex targets in random media and showed numerical results for exact RCS calculation and backscattering enhancement with considering incident wave polarization [4]-[6]. These studies prove that the spatial coherence length (SCL) of waves around the target in random medium is a key parameter together with the target configuration characteristics. Using our method and besides the free-space consideration, we investigate here the effects of random medium parameters on the RCS and backscattering enhancement. These parameters include the fluctuations intensity and SCL. The target size is assumed to be greater than one wavelength with convex illumination region of concave-convex target. Horizontal polarization (E-wave incidence) is considered with a plane wave incidence. The time factor exp(—iwt) is assumed and suppressed in the following sections.

II. FORMULATION

Scattering problem geometry of the problem is shown in Fig. 1. A random medium is assumed as a sphere of radius L around a target of the mean size a <C L, and also to be described by the dielectric constant e(r), the magnetic permeability /i, and the electric conductivity v. For simplicity e(r) is expressed as

e(r) = e0[l + 5e(r) (1)

where EQ is assumed to be constant and equal to free space permittivity and 5e(r) is a random function with

<fe(r))=0, (6e(r)6e(r')) = B(r,r'), (2)

B ( r , r ' ) < l , fcZ(r)»l. (3)

Here, the angular brackets denote the ensemble average,

Random £ medium

Incident wave

Scattered wave

Target

B(r,r): Normalized fluctuation intensity of random medium

Fig. 1. Geometry of the problem of wave scattering from a conducting cylinder in random media.

£>(r, r') and l(r) are the local intensity and local scale-size of the random medium fluctuation, respectively, and k = UO^/EQJIQ is the wavenumber in free space. Also /i and v are assumed to be constants; /i = //Q, V = 0. For a practical turbulent atmosphere, the condition (3) may be satisfied. Therefore, we can assume the forward-scattering approximation and the scalar approximation [7]. Consider the case where a directly incident beam wave is produced by a line source / ( r ' ) along the y axis. The beam wave incidence maintains a specific width around the target. The line source is located at rt beyond the turbulence in the far field. An electromagnetic wave radiated from the field source propagates in the random medium illuminating a conducting target and it induces a current on its surface. A scattered wave from the target is produced by the surface current and propagates back to the observation point that coincides with the source point. Here, let us designate the incident wave by Uin(r), the scattered wave by us(r), and the total wave by u(r) = uin(r) + us(r). The target is assumed to be a conducting cylinder of which cross-section is expressed by

a[l - 6 cos 3(6 - </))], (4)

where 6 is the incident point on the target, (j) is the rotation index and S is the concavity index. We can deal with this

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scattering problem two dimensionally under the condition (3); therefore, we represent r as r = (x,z). Then these waves satisfy Helmholtz equation, i.e.,

where

[V2 + fc2(l + <fe(r))Mr) = 0. (5) Also all waves satisfy the Dirichlet boundary condition stated above and the radiation condition (6), i.e.,

/ <C ^ ( r i ) , uin(r1 | r t) > drx Js

= / Um(ri) Js L

giXm(r i | r t ) dn ^ m ( r i | r t ) d n .

(16)

lim r r—Yoo

du(r) dr

+ jku(r) 0 (6)

where r = | r |. Green's function in random medium is needed to formulate the scattering waves and it satisfies the boundary condition and Eqs. (5) and (6) as

[V2 + k2(l + fe(r))] G(r | r ;) = -S(r - r')

lim dG(r | r0)

dr„ j fcG(r | r 0 )

(7)

(8)

where rn r — r0 |. Therefore the solution of Helmholtz equation for the total wave u is obtained as

U(T) =uin(r) + Js

" i ,du(r0) <9G(r|r0) dn0 dn0

dr0. (9)

Above equation is sometimes called "reaction" named by Rumsey [8]. The basis functions <3>M are called the modal functions and constitute the complete set of wave functions satisfying the Helmholtz equation in free space and the radi­ation condition; $ M = [<t>-N, <I>-N+I, • • •, </>m, • • •, </>N], $ M and ^ ^ denote the complex conjugate and the transposed vectors of <1>M, respectively, M = 2N + 1 is the total mode number, 0m(r) = Hm (kr) exp(im#). AE is a positive definite Hermitian matrix given by

( ((/>_ AT, </>-7v) . . . (0_AT,0AT) \

: •. : h (17> (0AT,0-AT) . . . (0AT,0AT) /

in which its m, n element is the inner product of (f)m and <j)n\ By applying the boundary condition, us(r) can be expressed as (07

^s(r)= [ G(r\r0)^^dr0. (10) Js vno

Also, the incident wave can be generally expressed as

uin(v) = [ G(r | r ' ) / ( r ' )dr ' = G(r | r t ) . (11)

i,<l>n) = / 0m(r )^ ( r )d r . Js

(18)

YE is proved to converge in the sense of mean on the true operator when M —> oo. Accordingly, the average intensity of backscattering wave for E-wave incidence is given by

According to the current generator method [5] that uses the current generator YE and Green's function in random medium G(r | r '), we can express the surface current wave as

du(r0) dn0 j s

Accordingly, the scattered wave is given as

JE(T2) = I yE(r2 | r i ) ^ n ( n | r t) d n . (12) Js

d wave is given as

/ J ^ ( r 2 ) G ( r | r 2 ) d r 2 (13) Js

^ s(r) =

that can be written as

us(r) = / dri / dr2 [G(r | r2)FE(r2 | r1)uin(r1 | r t)] , JS JS (14)

where r t represents the source point location and it is assumed that rt = (0, z) in Section III. Here, YE is the operator that transforms incident waves into surface currents on S and depends only on the scattering body. The current generator can be expressed in terms of wave functions that satisfy Helmholtz equation and the radiation condition as explained in [4]-[6]. That is the surface current is obtained as

/ yE(r2 I r1)uin(r1 | r t) drx Js

~ Qlt^A-1 / < *M(r i ) , Win(ri | r t) > dri , (15) Js

( M r ) | 2 ) = y d r 0 i Idr02 Idr[ Idrf2 YE(T01\T,

1)

x ^ ( r 0 2 | r ' 2 ) ( G ( r | r[)G(r | r01)G*(r | r^)G*(r | r02)>. (19)

In our representation of (|^s(r)|2), we use an approximate solution for the fourth moment of the Green's function in random medium M22 as

M22 = (G(r | r[) G(r | r0 i) G*(r | r2) G*(r | r02)) ^ ( G ( r | r ; ) G * ( r | r 2 ) ) ( G ( r | r 0 1 ) G * ( r | r 0 2 ) ) + (G(r | ri) G*(r | r02)> (G(r | r01) G*(r | r2)). (20)

In wave propagation through a continuous random medium, we may assume that the Green's function becomes approxi­mately complex Gaussian random. Here, we obtain an ana­lytical form for the second moment of Green's function Mn as

(G( r | r 0 i )G*( r | r 0 2 )> = (G(p,z | poi,z01)G*(p,z | ̂ 02,^02)) = Mu(p,Z | p01,p02,Z01,Z02). (21)

We solve M n using the Helmholtz equation as [5]. We can obtain the RCS by using (19) as

a=(\us(r)\2)k(4irz)2. (22)

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III. NUMERICAL RESULTS

Although the incident wave becomes sufficiently incoherent, we should pay attention to the spatial coherence length (SCL) of incident waves around the target. The degree of spatial coherence is defined as

r(P,*) < G ( r i | r t ) G * ( r 2 | r t ) ) (I G(r0 | r t) |2) '

(23)

where n = (p,0), r2 = (-p,0) , r0 = (0,0), andr t = (0, z). In the following calculation, we assume £>(r,r) = B0 and kBoL = 3TT; therefore the coherence attenuation index a = k2B0Ll/A is 15TT2 and 44TT2 for kl = 20TT and 58TT, respectively. Hence, the incident wave becomes sufficiently incoherent. The SCL is defined as the 2kp at which | T | = e _ 1 ~ 0.37. Fig. 2 shows a relation between SCL and kl in this case, and SCL is equal to 3 and 5.2, accordingly. We will use the SCL to represent one of the random medium effects on RCS.

1

0.8

0.6

0.4 e1

0.2

\ \ \ \

\ \ ■ \ \

SCL=3\

>t/= 5 8 7T - ° 0 7T

-\SCL=5.2 \

\ \

0 2 4 6 8 kp

Fig. 2. The degree of spatial coherence of an incident wave about the cylinder.

Here, we point out that N in (17) depends on the target parameters and polarization of incident waves. For example, we choose N = 24 at S = 0.1 for E-wave incidence in the range of 0.1 < ka < 5; for larger ka at more complex target where S is bigger, we choose greater iV. As a result, our numerical results are accurate because these values of N lead to convergence of RCS.

Numerical results for the RCS are shown in Fig. 3 and discussed as follows. In Fig. 3(a), where the medium is a free space, RCS decreases with ka to a certain small limit and then keeps a constant value. When ka —» 0, RCS has a large value as a result of the contribution from one stationary point. As ka increases, the contributions from the illumination region rays cancel out resulting such monotonic decrease of RCS. When ka is a bit less than one, the illumination region acts as being a flat surface facing the plane wave incidence and therefore RCS does not change with ka. Cross sections with larger curvature reduce the RCS since contributions from illumination regions are also reduced.

(a)

(b)

Fig. 3. (c)B0

(C)

RCS versus target size for (a) free space (b) Bo = 5 x 1 0 - 5 .

5 x 10" and

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In Figs. 3(b) and 3(c), targets are embedded in random media with SCL equal to 2.4 and 4, respectively, as calculated above where S = 0.05. The behavior of the RCS in free space and random medium are similar apart from the magnitude; this difference in magnitude is due to the double passage effect as it is well known. RCS suffers from oscillating behavior with small SCL due to the effect of random medium explained in [6]. Waves in the forward propagation with stronger random medium, where Bo is small, lose some of their concentration power in hitting the target owing to the multiple forward scattering by medium's particles and, therefore, RCS changes slightly with different SCL size. However, RCS with relatively greater Bo undergoes more oscillating behavior as the coupling between incident waves and scattering waves increases. As a result, normalized RCS deviates more away from two due to the double passage effect. This leads to inaccurate target de­tection as RCS does not represent preciously the configuration of the object.

IV. CONCLUSION

RCS is calculated with different medium nature includ­ing free space and random medium. Parameters of random medium including the fluctuations intensity Bo and spatial coherence length around the target have been shown to affect obviously the RCS and the backscattering enhancement with plane wave incident. Illumination region varies with target complexity and therefore scattering waves amount changes resulting different RCS. RCS with bigger B0 undergoes more oscillating behavior as the coupling between incident and scattering waves increases.

ACKNOWLEDGMENT

This work was supported in part by the National Science and Engineering Research Council of Canada (NSERC) under grant 250299-02.

REFERENCES

[1] Yu. A. Kravtsov and A. I. Saishev, "Effects of double passage of waves in randomly inhomogeneous media," Sov. Phys. Usp., Vol. 25, pp. 494-508, 1982.

[2] E. Jakeman, "Enhanced backscattering through a deep random phase screen," J. Opt. Soc. Am., Vol. 5, No. 10, pp. 1638-1648, 1988.

[3] M. Tateiba, and E. Tomita, "Theory of scalar wave scattering from a conducting target in random media," IEICE Trans. Electron., Vol. E75-C, No.l, pp. 101-106, 1992.

[4] H. El-Ocla and M. Tateiba, "Backscattering enhancement for partially convex targets of large sizes in continuous random media for E-wave incidence," Waves in Random Media, Vol. 12, No. 3, pp. 387-397, 2002.

[5] H. El-Ocla, "Target configuration effect on waves scattering in random media with horizontal polarization," Waves in Random and Complex Media, Vol. 19, No. 2, pp. 305-320, 2009.

[6] H. El-Ocla, "Effect of illumination region of targets on waves scattering in random media with H-polarization," Waves in Random and Complex Media, Vol. 19, No. 4, pp. 637-653, 2009.

[7] A. Ishimaru, Wave Propagation and Scattering in Random Media. Prentice-Hall: New Jersey, 1991.

[8] V. H. Rumsey, "Reaction concept in electromagnetic theory," Physical Review, Vol. 94, pp. 1483-91, 1954.

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