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Target configuration effect on wavescattering in random media withhorizontal polarizationHosam El-Ocla aa Department of Computer Science , Lakehead University , 955Oliver Road, Thunder Bay, Ontario, Canada , P7B 5E1Published online: 30 Jun 2009.

To cite this article: Hosam El-Ocla (2009) Target configuration effect on wave scattering in randommedia with horizontal polarization, Waves in Random and Complex Media, 19:2, 305-320, DOI:10.1080/17455030802503004

To link to this article: http://dx.doi.org/10.1080/17455030802503004

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Waves in Random and Complex MediaVol. 19, No. 2, May 2009, 305–320

Target configuration effect on wave scattering in random media withhorizontal polarization

Hosam El-Ocla∗

Department of Computer Science, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario,Canada P7B 5E1

(Received 4 May 2008; final version received 23 September 2008)

In a previous study, the radar cross-section (RCS) was shown to be influenced largely by thecurvature of the illumination region of the target. Scattering data from smooth concave–convexcontours reveals the obvious impact of specular reflections on the behavior of RCS. This effectneeds to be investigated more especially with relatively complex surfaces. Here, we workon a numerical calculation of the RCS and analyze its characteristics with different targetconfigurations including complexity and size. We postulate a concave illumination region andconsider targets that are taking large sizes of about five wavelengths. In this communication,we assume wave propagation and scattering from targets in free space and random mediumwith consideration of horizontal incident wave polarization.

1. Introduction

The subject of evaluating fields in regions of smooth caustics is not new; some useful solutionshave been presented, such as in [1–4], where many other publications handling the problem canbe found. In [2,4], the authors considered strictly a convex and curved wedge scatterer, whichhas limited engineering applications. High-frequency scattering from, or propagation along, aperfectly conducting boundary with a smooth concave–convex surface profile is of interest fora variety of applications. In [1], a uniform field solution is valid across a smooth caustic ofrays reflected by smoothly indented boundaries. In that analysis, inflection surfaces formed byconcave–convex boundaries were considered. However, it neglected the effects of edge diffractionand those surface rays that creep around the convex portion of the boundary past the inflectionpoint. The latter effects, which have been ignored, may result in inaccurate scattering data.These early studies were followed up by a series of investigations performed by El-Ocla and hiscollaborators. In our investigations, the problem of scattering waves from conducting targets wassolved efficiently via an exact method [5] based on Yasuura’s method [6]. That method assumesa current generator operator to calculate the electromagnetic field on the whole surface of thetarget; actually there are many articles that use this method (see e.g. [7–9]), where other referencesare available. In those studies, numerical results were presented for the RCS and backscatteringenhancement. We considered a perfectly conducting target with an analytic concave–convexboundary shape. The generated numerical results revealed characteristics that exist with partiallyconvex cross-sections, and are absent with a typical convex surface such as circular and ellipticshapes [10].

∗Email: hosam@lakeheadu.ca

ISSN: 1745-5030 print / 1745-5049 onlineC© 2009 Taylor & Francis

DOI: 10.1080/17455030802503004http://www.informaworld.com

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306 H. El-Ocla

In addition to the double passage effect [11], the backscattering enhancement was found tovary remarkably with many parameters including curvature of the illumination region, incidentwave polarization, and spatial coherence length. When a concave surface is illuminated by planewave incidence, the scattering wave may undergo focusing, which is absent when the surfaceand/or illuminated area of the surface is convex. Focusing and defocusing of scattering wavesplay a role when the scatterer has a smoothly deformed contour comprising concave and convexportions. Detailed studies of these phenomena were undertaken for partially convex targets withinflection points in [8]. However, those studies were limited to the case where the normalized targetsize is in the range of one wavelength of incident waves in free space. These features appeared tobe related to the contributions from the specular area as the scattering data were different betweenconvex and concave illumination regions. Some studies were presented showing the spread ofstationary points on the illumination regions and creeping wave effects on the scattering waves asin [12,13]. In this work, we probe the impact of normalized target size together with the prescribedeffects of curvature of the illumination region on the scattering waves from targets in randommedia in the far-field response. In this regard,we consider targets that are extended to take largesizes up to five wavelengths with a concave illumination region. Horizontal polarization (E-waveincidence) is assumed. The time factor exp(−iwt) is assumed and suppressed in the followingsection.

2. Scattering problem

The geometry of the problem is shown in Figure 1. A random medium is assumed as a sphere ofradius L around a target of mean size a � L, and also to be described by the dielectric constantε(r), the magnetic permeability µ, and the electric conductivity σ . For simplicity ε(r) is expressedas

ε(r) = ε0[1 + δε(r)] (1)

where ε0 is assumed to be constant and equal to the permittivity of free space and δε(r) is arandom function with

〈δε(r)〉 = 0, 〈δε(r) δε(r′)〉 = B(r, r′) (2)

and

B(r, r) � 1, kl(r) � 1. (3)

Figure 1. Geometry of the problem of wave scattering from a conducting cylinder in a random medium.

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Waves in Random and Complex Media 307

Here, the angular brackets denote the ensemble average and B(r, r), l(r) are the local intensityand local scale-size of the random medium fluctuation, respectively, and k = ω

√ε0µ0 is the

wavenumber in free space. Also µ and σ are assumed to be constant; µ = µ0, σ = 0. Forpractical turbulent media the condition (3) may be satisfied. Therefore, we can assume theforward scattering approximation and the scalar approximation [14].

Consider the case where a directly incident wave is produced by a line source f (r′) distributeduniformly along the y-axis. The incident wave is cylindrical and becomes plane approximatelyaround the target because the line source, this is located at rt beyond the random medium, isvery far from the target. An electromagnetic wave radiated from the source propagates in therandom medium illuminating the target and induces a current on its surface. A scattered wavefrom the target is produced by the surface current and propagates back to the observation pointthat coincides with the source point. That is, the monostatic scattering is considered underthe condition that the backscattering enhancement occurs [15]. The target is assumed to be aconducting cylinder of which cross-section is expressed by

r = a[1 − δ cos 3(θ − φ)] (4)

where φ is the rotation index and δ is the concavity index. We can deal with this scattering problemtwo dimensionally under the condition (3); therefore, we represent r as r = (x, z). Assuming a hor-izontal polarization of incident waves (E-wave incidence), we can impose the Dirichlet boundarycondition for wave field u(r) on the cylinder surface S. That is, u(r) = 0, where u(r) represents Ey .

Here, let us designate the incident wave by uin(r), the scattered wave by us(r), and the totalwave by u(r) = uin(r) + us(r). Then these waves satisfy Helmholtz equation:

[∇2 + k2(1 + δε(r))]u(r) = 0. (5)

Also all waves satisfy the Dirichlet boundary condition stated above and the radiation condition(6):

limr→∞ r

[∂u(r)

∂r+ jku(r)

]= 0 (6)

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

[∇2 + k2(1 + δε(r))]G(r|r′) = −δ(r − r′) (7)

limr→∞ rg

[∂G(r|r0)

∂rg

+ jkG(r|r0)

]= 0 (8)

where rg = |r − r0|. Therefore, the solution of Helmholtz equation for the total wave u is obtainedas

u(r) = uin(r) +∫

S

[G(r|r0)

∂u(r0)

∂n0− u(r0)

∂G(r|r0)

∂n0

]dr0. (9)

By applying the boundary condition, us(r) can be expressed as

us(r) =∫

S

G(r|r0)∂u(r0)

∂n0dr0. (10)

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308 H. El-Ocla

Greenís Functionin R.M

Incident wave

CurrentGeneratoron Body

SourceSurfaceCurrent

GreenísFunctionin R.M

Scattered Wave

Observation

Re-incident Wave

Figure 2. Schematic diagram for solving the scattering problem where a conducting body is surroundedwith a random medium.

Also, the incident wave can be generally expressed as

uin(r) =∫

VT

G(r|r′)f (r′)dr′ = G(r|rt ). (11)

By referring to Figure 2 and according to the current generator method [5] which uses the currentgenerator YE and Green’s function in random medium G(r|r′), we can express the surface currentwave as

∂u(r0)

∂n0= JE(r2) =

∫S

YE(r2|r1) uin(r1|rt ) dr1. (12)

Accordingly, the scattered wave is given as

us(r) =∫

S

JE(r2) G(r|r2) dr2 (13)

which can be represented as

us(r) =∫

S

dr1

∫S

dr2 [G(r|r2)YE(r2|r1)uin(r1|rt)] . (14)

Here, YE is the operator that transforms incident waves into surface currents on S and dependsonly on the scattering body. The current generator can be expressed in terms of wave functionsthat satisfy Helmholtz equation and the radiation condition. According to Yasuura’s method, thesurface current under the Dirichlet condition can be approximated by means of a truncated modalexpansion as follows:

∂u(r)

∂n�

M∑m=1

bm(M)φ∗m(r) = bM�∗T

M = �∗MbT

M (15)

where the basis functions φm are called the modal functions and constitute the complete setof wave functions satisfying the Helmholtz equation in free space and the radiation condi-tion (6). Here, the basis functions �M are called the modal functions and constitute the com-plete set of wave functions satisfying the Helmholtz equation in free space and the radiation

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Waves in Random and Complex Media 309

condition; �M = [φ−N, φ−N+1, . . . , φm, . . . , φN ], �∗M and �T

M denote the complex conjugateand the transposed vectors of �M , respectively, M = 2N + 1 is the total mode number, andφm(r) = H

(1)m (kr) exp(imθ ). The coefficient vector bM , defined as {b1, b2, . . . , bM}, can be ob-

tained by the ordinary mode-matching method as shown below. Let us minimize the mean squareerror by the method of least squares:

E(M) =∫

S

∣∣∣∣∣M∑

m=1

bm(M)φ∗m(r) − ∂u(r)

∂n

∣∣∣∣∣2

dr. (16)

That is, we partially differentiate (16) with respect to b∗m and obtain the algebraic equation

M∑m=1

bm(M)∫

S

φn(r)φ∗m(r)dr =

∫S

φn(r)∂u(r)

∂ndr, n = 1 ∼ M. (17)

Because of u = uin + us = 0 on S, the right-hand side of (17) can be written as

∫S

(φm

∂u

∂n− ∂φm

∂nu

)dr =

∫S

(φm

∂uin

∂n− ∂φm

∂nuin

)dr +

∫S

(φm

∂us

∂n− ∂φm

∂nus

)dr.

(18)

Using Green’s theorem for φm, us in the region surrounded by S and infinity, and using theradiation condition for φm, us , we obtain

∫S

(φm

∂us

∂n− ∂φm

∂nus

)dr = 0 (19)

and hence the right-hand side of (17) can be given as the reaction of φm and uin:

∫S

φm

∂u

∂ndr =

∫S

� φm(r), uin(r) � dr. (20)

Where �,� means

� φm(r), uin(r) �≡ φm(r)∂uin(r)

∂n− ∂φm(r)

∂nuin(r). (21)

We can therefore write (17) as

AEbTM =

∫S

� �m(r), uin(r) � dr (22)

where AE is a positive definite M × M Hermitian matrix except for the internal resonancefrequencies, and is given by

AE =

⎛⎜⎝

(φ−N, φ−N ) · · · (φ−N, φN )...

. . ....

(φN, φ−N ) · · · (φN, φN )

⎞⎟⎠ (23)

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310 H. El-Ocla

in which its m, n elements are the inner products of φm and φn:

(φm, φn) ≡∫

S

φm(r)φ∗n(r)dr. (24)

From (22), the bTM is given by

bTM = A−1

E

∫S

� �M (r′), uin(r′) � dr. (25)

Substituting (25) into (15) and comparing it with (12), we can approximately express the currentgenerator as follows:

YE(r|r′) � �∗M (r)A−1

E � �TM (r′). (26)

Therefore, the surface current is obtained as

∫S

YE(r2|r1) uin(r1|rt ) dr1 � �∗M (r2)A−1

E

∫S

� �TM (r1), uin(r1|rt ) � dr1 (27)

where

∫S

� �TM (r1), uin(r1|rt ) � dr1 ≡

∫S

[φm(r1)

∂uin(r1|rt )

∂n− ∂φm(r1)

∂nuin(r1|rt )

]dr1.

(28)

The above equation is sometimes called a ‘reaction’, named by Rumsey [16]. The YE is provedto converge in the sense of the mean on the true operator when M → ∞.The average intensity of the backscattering wave for E-wave incidence is given by

〈|us(r)|2〉 =∫

S

dr01

∫S

dr02

∫S

dr′1

∫S

dr′2 YE(r01|r′

1)Y ∗E(r02|r′

2)〈G(r|r′1)G(r|r01)

G∗(r|r′2)G∗(r|r02)〉. (29)

We can obtain the RCS by using equation (29)

σ = 〈|us(r)|2〉 · k(4πz)2. (30)

The calculation of scattering data has been restricted to the interval 0.1 < ka < 30. It is quitedifficult to exceed this ka’s limit since larger ka requires a big M which consequently enlargesthe calculation time dramatically.

3. Numerical results

Although the incident wave becomes sufficiently incoherent, we should pay attention to the spatialcoherence length (SCL) of the incident wave around the target [7]. The degree of spatial coherence

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Waves in Random and Complex Media 311

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

kl = 200

=20 π

π=58 π

e-1

Γ(kρ

)

kρ

SCL=9.7

SCL=5.2SCL=3

Figure 3. The degree of spatial coherence of an incident wave about the cylinder.

is defined by

�(ρ, z) = 〈G(r1|rt )G∗(r2|rt )〉〈|G(r0|rt )|2〉 (31)

where r1 = (ρ, 0), r2 = (−ρ, 0), r0 = (0, 0), rt = (0, z). In the following calculation, we as-sume B(r, r) = B0 and kB0L = 3π ; therefore, the coherence attenuation index α defined ask2B0Ll/4 given in reference [10] is 15π2, 44π2, and 150π2 for kl = 20π , 58π , and 200π ,respectively, which means that the incident wave becomes sufficiently incoherent. The SCL isdefined as the 2kρ at which |�| = e−1 � 0.37. In the Appendix, we formulate a simplified formof the second moment of the Green’s function in a random medium needed to calculate (31). Weuse the SCL to represent one of the random medium effects on RCS; that is the relation betweenSCL and kl shown in Figure 3.

Here, we point out that N in (23) depends on the target parameters and polarization of theincident waves. For example, we choose N = 24 at δ = 0.1 for E-wave incidence in the range of0.1 < ka < 5; at ka = 20, we choose N = 40 at δ = 0.1. As a result, our numerical results areaccurate because these values of N lead to convergence of RCS.

Based on the assumption of wave coherence completion in the propagation of distance 2a, letus define the effective illumination region (EIR) as that surface that is illuminated by the incidentwave and restricted by the SCL. Therefore, we expect that the target configuration including δ

and ka together with SCL are going to affect the EIR. Accordingly the behavior of RCS and theenhancement factor in RCS (ERCS) are influenced by EIR in a way that will be clarified shortly.In the following, we conduct numerical results for RCS and normalized RCS (NRCS), defined asthe ratio of RCS in random media σ to RCS in free space σ0.

3.1. The RCS in free space

In this section, we discuss numerical results for RCS of concave–convex targets for a concaveillumination region shown in Figure 4. It is noted that the behavior of RCS here is differentfrom that one with a convex illumination region that was presented in [7]. As was shown, thetarget’s curvature represented by δ has a limited effect on the scattering waves from the convexportion. On the other hand, δ has more influence on the waves scattered from the concave

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312 H. El-Ocla

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25 30

δ = 0.1 = 0.15

= 0.2

= 0.18

σ0/

(2a)

ka

Figure 4. RCS vs. target size in free space with different δ.

region. This influence is attributed to the contributions from complex saddle points that lienear the concave-to-convex transitions on the physical contour as has been described in [12].As we perceive with big δ, RCS increases with ka to a certain limit and then decreases in astepwise manner. Such behavior can be explained as follows: as magnifying δ, the concavity ofthe incident area expands which in turn increases the effective illumination region. Therefore,the contributions from this region become sometimes in phase so they add up and sometimesout of phase so they cancel out with ka depending on the real and complex scattering raydirections and that leads to such up-down behaviour with δ = 0.18, 0.2. We can understandthat with having bigger ka and/or δ, EIR would be stretched in a way leading to have inflectionpoints in RCS at certain ka. Specifically, the ka inflection points, for example, are: ka ∼= 14 withδ = 0.2 and ka ∼= 19 with δ = 0.18. It is notable that as having bigger δ, as getting smallerka’s inflection point and that reflects the impact of the illumination curvature on the RCS.On the contrary, with smaller δ, the specular reflections reduce as a result of lowering theconcavity slope of the EIR. Accordingly, the scattering waves progress in same directions andbe in phase so they add up resulting in that gradual increase in RCS with ka as the case withδ = 0.1, 0.15.

3.2. The RCS in a random medium

We discuss the numerical results for the RCS shown in Figure 5. We notice from this figure thatthere are two effects on RCS. The first is the effect of target curvature and can be seen clearly withδ. With enlarging δ, the RCS obviously gets bigger with ka as a result of the EIR expansion to acertain inflection point limit and then lessens in ascending manner due to the effect of stationarypoints as has been explained above. On the contrary, the RCS reduces with δ in case of a convexillumination region that was shown in [7]. The second is the effect of SCL; as the SCL increases,the behavior of the RCS in random media becomes closer to its behaviour in free space exceptfor the magnitude. In summary, we can understand that as magnifying either δ and/or SCL, asRCS increases due to the EIR extension, and vice versa. On the other hand, it is observed thatRCS suffers from oscillating behavior especially with small SCL and we attribute this manner tothe random medium effect.

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Waves in Random and Complex Media 313

4

5

6

7

8

9

10

11

12

13

0 5 10 15 20 25 30

δ = 0.1= 0.15

= 0.18= 0.2

ka

σ/(2

a)

(a)

4

5

6

7

8

9

10

11

12

13

0 5 10 15 20 25 30

δ = 0.1= 0.15

= 0.18= 0.2

σ /(2

a)

ka(b)

4

5

6

7

8

9

10

11

12

13

0 5 10 15 20 25 30

δ = 0.1= 0.15

= 0.18= 0.2

σ /(2

a)

ka(c)

Figure 5. NRCS versus target size with (a) SCL = 3, (b) SCL = 5.2, (c) SCL = 7.5, (d) SCL = 9.7, (e)SCL = 30. (Continued)

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314 H. El-Ocla

4

5

6

7

8

9

10

11

12

13

0 5 10 15 20 25 30

δ = 0.1= 0.15

= 0.18= 0.2

σ /(2

a)

ka(d)

5

10

15

20

0 5 10 15 20 25 30

δ = 0.1= 0.15

= 0.18= 0.2

σ /(2

a)

ka(e)

Figure 5. (Continued)

3.3. Backscattering enhancement

To manifest the behavior of ERCS in random media compared to free space propagation, wepresent numerical results for NRCS in Figure 6. We analyze the NRCS with different ranges ofka with respect to SCL.

For ka � SCL, the NRCS equals two due to the double passage effect of waves in randommedia. This value of NRCS is realized, independent of the curvature of the illumination portion,i.e. independent of the concavity index δ. NRCS decreases monotonically with ka especially withgreater δ. NRCS keeps lowering to a certain inflection point of ka with big δ and then grows upin the opposite direction owing to the effect of EIR as was pointed out earlier. Such a monotonicdecrease is slower with higher SCL and be closer to two as the waves are more correlated aroundthe target and therefore the random medium has less effect.

The real data of RCS is the one that realizes, as a result of the double passage effect, thatNRCS = 2 to reflect the actual radar measurement of the target. However, as explained above andbecause of the SCL and the target’s complexity that lead to the aggressive specular contributions,this rule does not hold. When the target size approaches a certain range of the wavelength in free

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Waves in Random and Complex Media 315

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

δ = 0.1= 0.15

= 0.18= 0.2

σ/σ

0

ka(a)

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

δ = 0.1= 0.15

= 0.18= 0.2

σ/σ 0

ka

(b)

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

σ/σ 0

δ = 0.1= 0.15

= 0.18= 0.2

ka(c)

Figure 6. RCS vs. target size for targets in random media with (a) SCL = 3, (b) SCL = 5.2, (c) SCL =7.5, (d) SCL = 9.7, (e) SCL = 30. (Continued)

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316 H. El-Ocla

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

δ = 0.1= 0.15

= 0.18= 0.2

σ/σ 0

ka

(d)

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 5 10 15 20 25 30

σ/σ 0

ka

δ = 0.1= 0.15

= 0.18= 0.2

(e)

Figure 6. (Continued)

space, NRCS decreases, obviously getting far away from two especially with small SCL and/orbig δ. This range varies widely with SCL which one has no control over. However, we can comeup with an approximate common range among the different values of SCL. This range could be:1λ ≤ a ≤ 4λ where target can easily hides from radars since the collected data of RCS would notreflects the real data.

4. Conclusion

The behavior of RCS of smooth targets with inflection points is influenced by the effective il-lumination region in correlation with the contributions from the vicinity of concave-to-convextransitions on the scatterer surface. In addition, random medium represented in spatial coherencelength (SCL) of waves around the target has a key effect on the scattering data. All features in refer-ence data for RCS using our exact method have been explained completely. Target’s configuration,including curvature and size, is playing a primary role on RCS especially for concave illuminationregion of partially convex targets. In radar applications, it would be recommended to manufacture

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Waves in Random and Complex Media 317

the target with the following rule: 1λ ≤ a ≤ 4λ. Moreover, the index of the target’s curvature com-plexity should be high. Under these circumstances, the enhancement in RCS deviates obviouslyfrom two. Therefore, the double passage effect, that predicts the actual image of the target, doesnot hold. As a result and as keeping these two recommended conditions in mind, therefore, targetswould be able to hide from radars since the picked radar data will not reflect the real information.

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

References[1] P. Pathak and M.-C. Liang, On a uniform asymptotic solution valid across smooth caustics of rays

reflected by smoothly indented boundaries, IEEE Trans. Antennas Propagat. 38 (1990), pp. 1192–1203.[2] E. Constantinides and R. Marhefka, Plane wave scattering from 2-D perfectly conducting superquadric

cylinders, IEEE Trans. Antennas Propagat. 39 (1991) pp. 367–376.[3] K. Goto, T. Ishihara, and L.B. Felsen, High-frequency (Whispering-Gallery Mode)-to-beam conversion

on a perfectly conducting concave–convex boundary, IEEE Trans. Antennas Propagat. 50 (2002), pp.1100–1119.

[4] G.N. Milford and J.D. Cashman, The high-frequency magnetic field at the edge of a curved face wedge:TE-polarized plane wave or creeping wave illumination, IEEE Trans. Antennas Propagat. 52 (2004),pp. 1355–1361.

[5] M. Tateiba and Z.Q. Meng, Wave scattering from conducting bodies in random media – Theory andnumerical results, in Electromagnetic Scattering by Rough Surfaces and Random Media, PIER 14, M.Tateiba and L. Tsang, eds., PMW Publishers, Cambridge, MA, 1996, pp. 317–361.

[6] M. Tateiba, A general aspect of Yasuura’s method for analyzing scattering problems, in Proceedings ofthe Sino-Japan Joint Meeting on Optical Fibre Science and Electromagnetics Theory, Wuhan, China,1997, pp. 78–83.

[7] H. El-Ocla and M. Tateiba, Backscattering enhancement for partially convex targets of large sizes incontinuous random media for E-Wave Incidence, Waves Random Media 12 (2002), pp. 387–397.

[8] H. El-Ocla, Backscattering from conducting targets in continuous random media for circular polar-ization, Waves Random Media 15 (2005), 91–99.

[9] H. El-Ocla, Laser backscattered from conducting targets of large sizes in continuous random mediafor E-Wave polarization, J. Opt. Soc. Amer. A 23 (2006), pp. 1908–1913.

[10] Z.Q. Meng and M. Tateiba, Radar cross sections of conducting elliptic cylinders embedded in strongcontinuous random media, Waves Random Media 6 (1996), pp. 335–45.

[11] Yu.A. Kravtsov and A.I. Saishev, Effects of double passage of waves in randomly inhomogeneousmedia, Sov. Phys. Usp 25 (1982), pp. 494–508.

[12] H. Ikuno and L.B. Felsen, Complex ray interpretation of reflection from concave–convex surface, IEEETrans. Antennas Propagat. 36 (1988), pp. 1260–1271.

[13] H. Ikuno and L.B. Felsen, Complex rays in transient scattering from smooth targets with inflectionpoints, IEEE Trans. Antennas Propagat. 36 (1988), pp. 1272–1280.

[14] A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press, New York, 1977.[15] A. Ishimaru, Backscattering enhancement: from radar cross sections to electron and light localizations

to rough surface scattering, IEEE Antennas Propagat. Mag. 33 (1991), pp. 7–11.[16] V.H. Rumsey, Reaction concept in electromagnetic theory, Phys. Rev. 94 (1954), pp. 1483–91.[17] M. Tateiba, Multiple scattering analysis of optical wave propagation through inhomogeneous random

media, Radio Sci. 17 (1982), pp. 205–210.[18] M. Tateiba, Numerical analysis of nonreciprocity for spatial coherence and spot dancing in random

media, Radio Sci. 17 (1982), pp. 1531–1535.[19] M. Tateiba, Some useful expressions for spatial coherence functions propagated through random

media, Radio Sci. 20 (1985), pp. 1019–1024.[20] M. Tateiba, Some characteristics of the second moment of waves propagated through an inhomogeneous

random medium, J. Waves–Material Interact. 1 (1986), pp. 54–65.[21] M. Tateiba, The Lorentz reciprocity in random media–The derivation from Maxwell’s equations, Radio

Sci. 26 (1991), pp. 499–503.

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318 H. El-Ocla

Appendix 1. A simplified form of the second moment of Green’s function

In wave propagation through a strong continuous random medium, we may assume that the Green’s functionbecomes approximately complex Gaussian random [17].

Here, we obtain an analytical form for the second moment of the Green’s function M11 that has beengiven in [18,20,21]: for instance,

〈G(r|r01)G∗(rt |r02)〉 = 〈G(ρ, z|ρ01, z01)G∗(ρt , z|ρ02, z02)〉= M11(ρ, ρt , z|ρ01, ρ02, z01, z02). (32)

We solve M11 using the Helmholtz equation and as in [17,19]:

[∂

∂z− j

1

2k

(∇2 − ∇2

t

)]M11(ρ, ρt , z|ρ01, ρ02, z01, z02)

={

−k2

4

[∫ z−z0

0D

(ρ − ρt , z − z′

2, z′

)dz′

]}Min

11(ρ, ρt , z|ρ01, ρ02, z01, z02) (33)

where z0 is defined in (34) and Min11 is the second moment of the Green’s function in free space

z0 = a (34)

where a is the mean size of the target.

Min11(ρ, ρt , z|ρ01, ρ02, z01, z02) = G0(ρ, z|ρ01, z01)G∗

0(ρt , z|ρ02, z02). (35)

Therefore, we can represent (33) as

M11(ρ, ρt , z|ρ01, ρ02, z01, z02) = Min11(ρd, ρs, z)m(ρd, ρ0d ) (36)

where ρd = ρ − ρt , ρs = (ρ + ρt )/2, ρ0d = ρ1 − ρ2, ρ0s = (ρ1 + ρ2)/2. For a 2-D problem, Min11 can be

formulated as follows:

G0(r|r1) = G0(ρ, z|ρ1, z1) = 1

4iH

(1)0

(k√

(ρ − ρ1)2 + (z − z1)2)

(37)

G∗0(r|r′

1) = G∗0(ρ, z|ρ ′

1, z′1) = −1

4iH

(2)0

(k

√(ρ − ρ ′

1)2 + (z − z′1)2

). (38)

Because of kz � 1 we let

R1 = k√

(ρ − ρ1)2 + (z − z1)2 (39)

R2 = k

√(ρ − ρ ′

1)2 + (z − z′1)2 (40)

Equations (37) and (38) can be expressed approximately as

G0(ρ, z|ρ1, z1) = 1

4iH

(1)0 (kR1) � 1

4i

√2

πkR1expi(kR1− π

4 ) (41)

G∗0(ρ, z|ρ1, z1) = −1

4iH

(2)0 (kR2) � −1

4i

√2

πkR2exp−i(kR2− π

4 ) . (42)

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Waves in Random and Complex Media 319

Then we have

G0(ρ, z|ρ1, z1) G∗0(ρ, z|ρ ′

1, z′1) = 1

8πk

1√R1R2

exp−ik(R1−R2). (43)

As the following condition satisfies the far field observation point

z � z0, z1, z′1, |ρ|, |ρ1|, |ρ ′

1|. (44)

Then

1√R1R2

= 1

z(45)

ikR1 = ik√

(z − z0 + z0 − z1)2 + (ρ − ρ1)2

= ik√

(z − z0)2 + 2(z − z1)(z − z0) + (z0 − z1)2 + (ρ − ρ1)2

� ik(z − z0)

{1 + z0 − z1

z − z0+ 1

2

(ρ − ρ1)2

(z − z0)2

}(46)

ikR2 = ik

√(z − z0 + z0 − z′

1)2 + (ρ − ρ ′1)2

= ik

√(z − z0)2 + 2(z − z′

1)(z − z0) + (z0 − z′1)2 + (ρ − ρ ′

1)2

� ik(z − z0)

{1 + z0 − z′

1

z − z0+ 1

2

(ρ − ρ ′1)2

(z − z0)2

}. (47)

Then we can obtain

G0(ρ, z|ρ1, z1) G∗0(ρ, z|ρ ′

1, z′1)

� 1

8πkzexp

{ik(z − z0)

[−z1 − z′

1

z − z0+ 1

2

(ρ − ρ1)2 − (ρ − ρ ′1)2

(z − z0)2

]}

= 1

8πkz

O(ρ, z|ρ1, z1)

O(ρ, z|ρ ′1, z

′1)

(48)

O(ρ, z|ρ1, z1) = exp

{ik

[−z′ + (ρ − ρ1)2

2(z − z1)

]}(49)

On the other hand, m(ρd, ρ0d ) can be expressed as follows:

m(ρd, ρ0d ) = exp

[−k2

4

∫ z

z0

Dt

(z′ − z0

z − z0ρd + z′ − z0

z − z0ρ0d , z

′|z0

)dz′

](50)

Dt (ρ, z|z)) =∫ z−z0

0D

(ρ, z − z′

2, z′

)dz′ (51)

D

(ρ, z − z′

2, z′

)= 2

[B

(0, z − z′

2, z′

)− B

(0, z − z′

2, z′

)](52)

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320 H. El-Ocla

where D is called the structure function of the random medium. By solving (50), we get

m(ρd, ρ0d ) = exp

[−k2

4µ{α(z − z0)ρ2

d + β(z − z0)ρdρ0d + γ (z − z0)}ρ20d

](53)

where

µ = √π

B L3

l(z − z0)2(54)

α(z − z0) = 1

3 − n

(z

L

)3−n

− 2

2 − n

(z0

L

)(z

L

)2−n

+ 1

1 − n

(z0

L

)2(z

L

)1−n

− n

3(3 − n)+ 2

2 − n

(z0

L

)− 1

1 − n

(z0

L

)2

− 1

3

(z0

L

)3

(55)

β(z − z0) = 2

(3 − n)(2 − n)

(z

L

)3−n

− 2

(2 − n)(1 − n)

(z0

L

)(z

L

)2−n

+⎡⎣(

z0

L

)2

+ 2n

1 − n

(z0

L

)2

− n

2 − n

⎤⎦(

z

L

)+ 2

3

n

3 − n

− n

2 − n

(z0

L

)− 1

3

(z0

L

)3

(56)

γ (z − z0) = 2

(3 − n)(2 − n)(1 − n)

(z

L

)3−n

−[

n

(1 − n)+

(z0

L

)] (z

L

)2

+⎡⎣ n

2 − n+

(z0

L

)2⎤⎦ (

z

L

)− 1

3

n

3 − n− 1

3

(z0

L

)3

. (57)

Here l is defined in (3) and assumed in our numerical results as shown in Figure 3, B is a constant as definedin (2), L is the rough size of the range of the random medium (see Figure 1), the positive index n denotes thethickness of the transition layer from the random medium to free space, and n = 8

3 is assumed in Section 3as in our previous work.

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