Effect of the illumination region of targets on waves scattering in random media with H-polarization

This article was downloaded by: [University of North Texas]On: 27 November 2014, At: 19:03Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Waves in Random and Complex MediaPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/twrm20

Effect of the illumination region oftargets on waves scattering in randommedia with H-polarizationHosam El-Ocla aa Department of Computer Science , Lakehead University , 955Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1Published online: 22 Oct 2009.

To cite this article: Hosam El-Ocla (2009) Effect of the illumination region of targets on wavesscattering in random media with H-polarization, Waves in Random and Complex Media, 19:4,637-653, DOI: 10.1080/17455030903108166

To link to this article: http://dx.doi.org/10.1080/17455030903108166

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/twrm20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/17455030903108166

http://dx.doi.org/10.1080/17455030903108166

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Waves in Random and Complex MediaVol. 19, No. 4, November 2009, 637–653

Effect of the illumination region of targets on waves scattering in

random media with H-polarization

Hosam El-Ocla*

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

(Received 24 October 2008; final version received 9 June 2009)

This paper addresses some parameters that have a significant effect onwave scattering in random media. These parameters are: target configura-tion, including size and curvature; random media strength, represented inthe spatial coherence length; and incident wave polarization. Here, Ipresent numerical calculations for the radar cross-section (RCS) ofconducting targets and analyze the backscattering enhancement withdifferent configurations. I postulate a concave illumination region andconsider targets taking large sizes of about five wavelengths. In this aspect,waves scattering from targets are assumed to propagate in free space and arandom medium with H-polarization. This polarization produces what iswell known as creeping waves which in turn have an additional effect on thescattering waves that is absent in the case of E-polarization.

1. Introduction

The calculation of the radar cross-section (RCS) is a vital problem in radar detectionapplications to the extent that several techniques have been proposed over the years[1–6]. This problem becomes more interesting when RCS is calculated for targetswith the consideration of different wave polarizations, the wave nature, and thesurrounding media. In these methods, some of the effects were ignored, such as thecreeping rays around the target. Also, having strictly a typical convex object andusing the physical optics method that considers the scattering from the illuminationregion only, are some of the constraints of these methods. The lack of these effectsmay result in inaccurate scattering data. Scattering of electromagnetic waves fromtargets embedded in random media has received great attention in the fields of radarengineering and remote sensing, in particular, under the condition that back-scattering enhancement occurs [7]. On the other hand, the backscattering enhance-ment produced due to the double passage effect on waves propagating in randommedia has also attracted considerable attention among condensed-matter physicists,resulting in a large number of publications [8–10].

Over the past years, we have conducted numerous studies solving the scatteringproblem using our exact method [11,12] based on Yasuura’s method [13].

*Email: [email protected]

ISSN 1745–5030 print/ISSN 1745–5049 online

� 2009 Taylor & Francis

DOI: 10.1080/17455030903108166

http://www.informaworld.com

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

That method assumes a current generator operator to calculate the electromagneticfield on the whole surface of the target. Using this method, many studies were doneand numerical results were presented for different cases as described in [14–16],where additional references can be found. Our results are in excellent agreement withthose conducted for a circular cylinder in free space in [17]. The generated numericalresults revealed characteristics that exist with partially convex cross-sections and areabsent with a typical convex surface such as circular and elliptic shapes [18]. High-frequency scattering from, or propagation along, a perfectly conducting boundarywith a smooth concave–convex surface profile is of interest for a variety ofapplications. When a concave surface is illuminated by an incident plane wave, thescattering wave may undergo focusing, which is absent when the surface and/orilluminated area of the surface is convex. Focusing and defocusing of scatteringwaves play a role when the scatterer has a smoothly deformed contour comprisingconcave and convex portions [19]. A detailed study of these phenomena wasundertaken for partially convex targets with inflection points in [15]. However, thatstudy was limited to the case where the normalized target size is in the range of onewavelength of incident waves in free space. These features appeared to be related tothe contributions from the specular area as the scattering data were different betweenconvex and concave illumination regions. Some investigations showed the effects ofcreeping waves and the stationary points, that spread on the illumination region, onthe scattered waves as in [19,20].

In this work, we investigate the impact of target configuration including size andcurvature complexity together with the creeping waves produced by H-polarization(H-wave incidence) on the scattering waves from targets in random media in the far-field response. In this regard, we consider concave illumination region of targets thatare extended to take large sizes up to five wavelengths with the concave illuminationregion. The time factor exp(�iwt) is assumed and suppressed in the following section.

2. Scattering problem

The geometry of the problem is shown in Figure 1. A random medium is assumed asa sphere of radius L around a target of mean size a�L. The random medium isfurther described by the dielectric constant "(r), the magnetic permeability �, and theelectric conductivity �. For simplicity "(r) is expressed as

"ðrÞ ¼ "0½1þ D"ðrÞ� ð1Þ

where "0 is assumed to be constant and equal to the free space permittivity and D"(r)is a random function with

hD"ðrÞi ¼ 0, hD"ðrÞD"ðr0Þi ¼ B ðr, r0Þ ð2Þ

and

B ðr, rÞ � 1, klðrÞ�1: ð3Þ

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

638 H. El-Ocla

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

k ¼ !ffiffiffiffiffiffiffiffiffiffi"0�0p

is the wavenumber in free space. Also � and � are assumed to be

constant; �¼�0, �¼ 0. For practical turbulent media the condition (3) may be

satisfied. Therefore, we can assume the forward scattering approximation and the

scalar approximation [21].We can approximately express (2) under the condition (3) as follows:

B ðr, rÞ ¼ Bð�1 � �2, zþ, z�Þ

¼ BðzþÞ exp �ð�1 � �2Þ

2þ z2�

l 2ðzþÞ

� �ð4Þ

where r¼ (�, z), �¼ ixþ iy, zþ¼ (z1þ z2)/2, and z�¼ z1� z2, and where

BðzþÞ ¼B0, z0 � zþ � L

B0ðzþ=LÞ�n, zþ � L

�ð5Þ

l ðzþÞ ¼ l0 ð6Þ

z0 ¼ a: ð7Þ

An actual target surface may be regarded as a rough surface or coated with a

material where particles stick partly on the surface. To avoid this we assume that an

infinitesimal thin layer of free space wraps the surface of the target where the

thickness of this layer tends to zero. In (5), n denotes the normalized thickness of the

transition layer shielding the target where n41 and n 6¼ 2, 3 [11]. Consider the case

where a directly incident wave is produced by a line source f(r0) distributed uniformly

along the y axis. Then, the incident wave is cylindrical and becomes plane

θ

φ

r = (x,z)

L

Target

Randommedium

x

z

z

Incidentwave

Scatteredwave

0

B0

S

B(r,r): Normalizedfluctuation intensityof random medium

a

y

L >> a

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

Waves in Random and Complex Media 639

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

approximately around the target because the line source is very far from the target.

The target is assumed to be a conducting cylinder with cross-section expressed by

r ¼ a½1� � cos 3ð� � �Þ� ð8Þ

where � is the rotation index and � is the concavity index. We can deal with this

scattering problem two dimensionally under the condition (3); therefore, we

represent r as r¼ (x, z). Assuming an H-polarization of incident waves (H-wave

incidence), we can impose the Neumann boundary condition for the wave field u(r)

on the cylinder surface S. That is,

@

@nuðrÞ ¼ 0 ð9Þ

where u(r) represents Ex. Here, let us designate the incident wave by uin(r), the

scattered wave by us(r), and the total wave by u(r)¼ uin(r)þ us(r). Then these waves

satisfy the Helmholtz equation:

½r2 þ k2ð1þ �"ðrÞÞ�uðrÞ ¼ 0: ð10Þ

Also all waves satisfy the Neumann boundary condition (9), and the radiation

condition (11):

limr!1

r@uðrÞ

@rþ jkuðrÞ

� �¼ 0 ð11Þ

where r¼ jrj. Also the Green’s function in a random medium satisfies the boundary

condition, Helmholtz equation and the radiation condition as

½r2 þ k2ð1þ �"ðrÞÞ�Gðrjr0Þ ¼ ��ðr� r0Þ ð12Þ

limr!1

rg@Gðrjr0Þ

@rgþ jkGðrjr0Þ

� �¼ 0 ð13Þ

where rg¼ jr� r0j. Therefore, the solution of the Helmholtz equation for the total

wave u is obtained by

uðrÞ ¼ uinðrÞ þ

ZS

Gðrjr0Þ@uðr0Þ

@n0� uðr0Þ

@Gðrjr0Þ

@n0

� �dr0: ð14Þ

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

usðrÞ ¼ �

ZS

@Gðrjr0Þ

@n0uðr0Þ dr0: ð15Þ

Also, the incident wave whose dimension coefficient is understood can be generally

expressed as

uinðrÞ ¼

ZVT

Gðrjr0Þfðr0Þdr0 ¼ GðrjrtÞ: ð16Þ

640 H. El-Ocla

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

By referring to Figure 2 and according to the current generator method [11] which

uses the current generator YH and the Green’s function in the random medium

G(r jr0), we can express the surface current wave as

uðr0Þ ¼ JHðr2Þ ¼ �

ZS

YHðr2jr1Þ uinðr1jrtÞdr1: ð17Þ

Here, YH 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 as has been derived in Appendix 1 where it is given as:

YHðrjr0Þ ’ �

@(�MðrÞ

@nA�1H � (T

Mðr0Þ: ð18Þ

Therefore, the surface current in (17) is obtained as

ZS

YHðr2jr1Þ uinðr1jrtÞdr1 ’ �@(�Mðr2Þ

@nA�1H

ZS

� (TMðr1Þ, uinðr1jrtÞ �dr1 ð19Þ

where rt represents the source point location and it is assumed that rt¼ (0, z) in

Section 3. Here YH converges when M!1. That is:ZS

� (TMðr1Þ, uinðr1jrtÞ �dr1

ZS

�mðr1Þ@uinðr1jrtÞ

@n�@�mðr1Þ

@nuinðr1jrtÞ

� �dr1: ð20Þ

The above expression is sometimes called the ‘reaction’, named by Rumsey [22].The scattered wave can be divided into two parts:

us ¼ husi þ Dus ð21Þ

where husi and Dus are called the coherent and the incoherent scattered waves,

respectively. The scattered wave in (15) can be expressed using (17) as:

usðrÞ ¼ �

ZS

dr1

ZS

dr2@

@n2Gðrjr2Þ

� �YHðr2jr1Þuinðr1jrtÞ

� �: ð22Þ

Green’s Functionin R.M

CurrentGeneratoron Body

SourceSurfaceCurrent

Green’sFunctionin R.M

ScatteredWave

Observation

Re-incident Wave

Incidentwave

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


Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

Using (16) and (22), husi is formulated as

husðrÞi ¼ �

ZS

dr1

ZS

dr2 YHðr2jr1Þ@

@n2Gðrjr2ÞGðr1jrtÞ

� � �: ð23Þ

Therefore, the average (coherent) intensity of scattered waves for H-wave incidenceis given by

hjusðrÞj2i ¼

ZS

dr01

ZS

dr02

ZS

dr01

ZS

dr02 YHðr01jr01ÞY�Hðr02jr

02Þ

@

@n01

@

@n02hGðrjr01ÞGðrjr01ÞGðrjr

02ÞGðrjr02Þi: ð24Þ

In our representation of hjus(r)j2i, we use an approximated solution for the fourth

moment of the Green’s function in the random medium [14]. Final forms for (24) infree space and a random medium are derived in Appendix 2. Let us assume that thecoherence of waves is kept almost complete in propagation of distance 2a whichequals the mean diameter of the cylinder. This assumption is acceptable in practicalcases under the condition (3). On the basis of the assumption, it is important here topoint out that we are going to present a quantitative discussion for the numericalresults in Section 3. We can obtain the RCS by using Equation (24)

� ¼ hjusðrÞj2i � kð4�zÞ2: ð25Þ

The calculation of scattering data has been restricted to the interval 0.15 ka5 30.It is quite difficult to exceed this ka limit since larger ka requires a big M whichconsequently expands the calculation time dramatically.

3. Numerical results

Although the incident wave becomes sufficiently incoherent, we should pay attentionto the spatial coherence length of the incident wave around the target (SCL) as hasbeen pointed out in my papers. The degree of spatial coherence is defined by

�ð�, zÞ ¼hGðr1jrtÞG

�ðr2jrtÞi

hjGðr0jrtÞj2i

ð26Þ

where r1¼ (�, 0), r2¼ (��, 0), r0¼ (0, 0), rt¼ (0, z). In the following calculation, weassume B(r, r)¼B0 and kB0L¼ 3�; therefore, the coherence attenuation index defined as k2B0Ll/4 given in [11] is 15�2, 44�2, and 150�2 for kl¼ 20�, 58�, and 200�,respectively, which means that the incident wave becomes sufficiently incoherent.The SCL is defined as the 2k� at which j�j ¼ e�1’ 0.37. In [16], a simplified form ofthe second moment of the Green’s function in a random medium needed to calculate(26) is formulated. Figure 3 shows the relation between SCL and kl in this case forthe SCL equal to 3, 5.2, and 9.7. We will use the SCL to represent one of the randommedium effects on RCS.

Here, we point out that N in (35) depends on the target parameters andpolarization of incident waves. For example for H-wave incidence, we choose N¼ 28

642 H. El-Ocla

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

at �¼ 0.1 in the range of 0.15 ka5 5; at ka¼ 20, we choose N¼ 46 at �¼ 0.1. Asa result, our numerical results are accurate because these values of N lead toconvergence of RCS.

Based on the assumption of wave coherence completion in the propagation ofdistance 2a, let us define the effective illumination region (EIR) as that surface that isilluminated by the incident wave and restricted by the SCL. Therefore, we expect thatthe target configuration including � and ka together with SCL are going to affect theEIR and accordingly both RCS and the enhancement factor in RCS (ERCS) in away that will be clarified in Section 3. In the following, we conduct numerical resultsfor RCS and normalized RCS (NRCS), defined as the ratio of RCS in random media� to RCS in free space �0.

In the following sections and as an extension of our previous work in [16], weemploy our method to calculate RCS and NRCS of concave–convex targets inrandom media for H-wave incidence.

3.1. RCS in free space

In this section, we discuss numerical results for RCS of concave–convex targets forthe concave illumination region shown in Figure 4. It is noted that the behavior ofRCS here is different from that with a convex illumination region which waspresented in [14] as has been explained in [16]. On the other hand, � has moreinfluence on the waves scattered from a concave region. This influence is attributedto the contributions from complex saddle points that lie near the concave-to-convextransitions on the physical contour as has been described in [19]. It is perceived thatwith bigger �, the RCS increases with ka to a certain inflection limit and thendecreases in a stepwise manner. The ka for inflection points here are quite similar to

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8

kl = 200 π

=20 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 a cylinder.


Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

those with E-wave incidence in [16]. Also, we notice that RCS suffers from anoscillating behavior with H-wave incidence with a low ka range. The oscillatingbehaviour that is absent in the case of E-wave incidence is due, as a matter of fact, tothe effect of creeping waves [23]. At the point of tangency, each wave creeps aroundthe surface at a velocity less than the direct wave and is attenuated by tangentialradiation. The creeping wave’s attenuation reduces its effectiveness, rapidly resultingin diminishing the interference effect gradually with ka as previously shown in [14].

3.2. RCS in a random medium

We discuss the numerical results for RCS shown in Figure 5. We notice from thisfigure that there are two factors that obviously affect the behavior of RCS as well asin case of E-wave incidence. The first is the effect of SCL; as the SCL increases, thebehavior of RCS in a random medium becomes closer to its behavior in free spaceexcept for the magnitude owing to the EIR expansion. In addition to the creepingwaves, RCS undergoes obvious fluctuations due to the random medium effect.That is, the oscillations in Figure 5(a) are due to the creeping waves coupling withdirect waves in the random medium. Creeping waves located inside the SCL arecorrelated with the direct waves leading to in-phase addition, while those wavesoutside the SCL are uncorrelated with the direct waves, leading to their being out-of-phase. These in- and out-of-phase waves result in such oscillating RCS. This effectreduces with wider SCL since the waves become more in-phase. Also this effectdisappears with E-wave incidence owing to the absence of creeping waves as wasshown in [14].

Second is the effect of target curvature and can be seen clearly with changing �.As either � and/or SCL get bigger, the RCS increases due to the EIR extension, andvice versa.

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25 30

σ/(2

a)

ka

δ = 0.1

= 0.2

= 0.18

Figure 4. RCS versus target size in free space with different �.

644 H. El-Ocla

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

(e)

0

5

10

15

20

0 5 10 15 20 25 30

δ = 0.1= 0.18

= 0.2

σ/(2

a)

ka

0

2

4

6

8

10

12

0 5 10 15 20 25 30

δ = 0.1= 0.18

= 0.2σ/(2

a)

ka

(d)(c)

δ = 0.1

= 0.18

= 0.2

0

2

4

6

8

10

12

0 5 10 15 20 25 30

σ/(2

a)

ka

(b)

0

2

4

6

8

10

12

0 5 10 15 20 25 30

δ = 0.1= 0.18

= 0.2

σ/(2

a)

ka

(a)

0

2

4

6

8

10

12

0 5 10 15 20 25 30

δ = 0.1

= 0.18

= 0.2σ/

(2a)

ka

Figure 5. RCS versus target size for targets in random media with (a) SCL¼ 3, (b) SCL¼ 5.2,(c) SCL¼ 7.5, (d) SCL¼ 9.7, (e) SCL¼ 30.


Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

3.3. Backscattering enhancement

To analyze the behavior of ERCS in random media compared to free spacepropagation, we present numerical results for NRCS in Figure 6. We discuss thebehaviour of NRCS in different ranges of ka with respect to SCL.

(a)

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

δ = 0.1= 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

ka

δ = 0.1= 0.18

= 0.2

(c)

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

σ/σ 0

ka

δ = 0.1= 0.18

= 0.2

(e)

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= 0.18

= 0.2

(d)

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30

σ/σ 0

ka

δ = 0.1= 0.18

= 0.2

Figure 6. NRCS versus target size with (a) SCL¼ 3, (b) SCL¼ 5.2, (c) SCL¼ 7.5,(d) SCL¼ 9.7, (e) SCL¼ 30.

646 H. El-Ocla

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

For ka�SCL, the NRCS equals two, due to the double passage effect of wavesin random media. This value of NRCS is realized, independent of the illuminationportion curvature, i.e. independent of the concavity index �. NRCS decreasescontinuously with ka especially with greater �. NRCS keeps lowering to thoseinflection points at pre-mentioned ka with big � and then grows in the oppositedirection owing to the effect of EIR. Such a monotonic decrease is slower with largerSCL and closer to two as the waves are more correlated around the target andtherefore the random medium has less effect.

The difference in the creeping wave effect on RCS in free space and the randommedium leads to such longer infection than having each individually. For example,RCS in Figure 4, continues to suffer from heavy oscillations up to ka’ 6 solely dueto the creeping wave effect. In Figure 5(a) it continues to suffer from such obviousoscillating behavior up to ka¼ 9 as a result of the effect of creeping waves in therandom medium. In Figure 6, ERCS undergoes such fluctuation up to ka’ 13 as aresult of the difference in creeping waves propagating in free space and the randommedium.

As in the case of E-wave incidence, as the mean size of the target closes to acertain range of the wavelength in free space, NRCS lessens, obviously deviatingfrom two, particularly with small SCL and/or big �. This range differs widely withSCL which one has no control over. However, we can come up with an approximatecommon range among the different values of SCL. This range could be: 1� a� 4where the target can easily hide from radars since the collected data of RCS wouldnot reflect the real data. Yet, in this range ERCS is infected remarkably withcreeping waves.

4. Conclusion

The illumination region of the target has a primary effect on the RCS especially forthose objects with inflection points. In addition, the random medium together withthe target configuration, including size and curvature complexity, have a significantimpact on the scattering data. Creeping waves that are generated with H-polarizationinfect the behavior of RCS by having obvious oscillations especially with a target’ssmall size. In radar applications, it was pointed out in previous work forE-polarization that it would be recommended to manufacture the target with thefollowing rule: 1� a� 4. Moreover, the index of a target’s curvature complexityshould be big. Keeping these two recommended conditions in mind therefore, targetswould be able to hide from radars since the picked radar data will not reflect the realinformation. The above rule is valid with H-polarization with some deviations due tothe creeping wave effect.

Acknowledgments

This work was supported in part by National Science and Engineering Research Council ofCanada (NSERC) under Grant 250299-02.


Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

References

[1] C.L. Bennett and H. Mieras, Time domain scattering from open thin conductivity surfaces,

Radio Sci. 16 (1981), pp. 1231–1239.[2] 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.

[3] E. Constantinides and R. Marhefka, Plane wave scattering from 2-D perfectly conducting

superquadric cylinders, IEEE Trans. Antennas Propagat. 39 (1991), pp. 367–376.

[4] J.M. Rius, M. Ferrando, and L. Jofre, High-frequency RCS of complex radar targets in

real-time, IEEE Trans. Antennas Propagat. 41 (1993), pp. 1308–1319.

[5] 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.[6] 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.[7] A. Ishimaru, Backscattering enhancement: from radar cross sections to electron and

light localizations to rough surface scattering, IEEE Antennas Propagat. 33 (1991),

pp. 7–11.[8] Yu.A. Kravtsov and A.I. Saishev, Effects of double passage of waves in randomly

inhomogeneous media, Sov. Phys. Usp 25 (1982), pp. 494–508.[9] E. Jakeman, Enhanced backscattering through a deep random phase screen, J. Opt. Soc.

Am. 5 (1988), pp. 1638–1648.[10] Yu.A. Kravtsov, New effects in wave propagation and scattering in random media (a mini

review), Appl. Opt. 32 (1993), pp. 2681–2691.[11] M. Tateiba and Z.Q. Meng, Wave scattering from conducting bodies in random media –

theory and numerical results, in Electromagnetic Scattering by Rough Surfaces and Random

Media, M. Tateiba and L. Tsang, eds., PIER 14, PMW Publications, Cambridge, MA,

1996, pp. 317–361.[12] M. Tateiba and Z.Q. Meng, Infinite-series expressions of current generators in wave

scattering from a conducting body, Res. Rep. Informat. Sci. Elect. Eng. Kyushu

University, 4 (1999), pp. 1–6.[13] M. Tateiba, A general aspect of Yasuura’s method for analyzing scattering problems, in

Proceedings of Sino-Japan Joint Meeting on Optical Fibre Science and Electromagnetics

Theory, 1997.

[14] H. El-Ocla and M. Tateiba, Effect of H-polarization on backscattering enhancement for

partially convex targets of large sizes in continuous random media, Waves Random Media

13 (2003), pp. 125–136.[15] H. El-Ocla, Backscattering from conducting targets in continuous random media for circular

polarization, Waves Random Complex 15 (2005), pp. 91–99.[16] H. El-Ocla, Target configuration effect on wave scattering in random media with horizontal

polarization, Waves Random Complex 19 (2009), pp. 305–320.[17] M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, Academic Press,

New York, 1969.[18] Z.Q. Meng and M. Tateiba, Radar cross sections of conducting elliptic cylinders embedded

in strong continuous random media, Waves Random Media (1996), pp. 335–345.[19] H. Ikuno and L.B. Felsen, Complex ray interpretation of reflection from concave-convex

surface, IEEE Trans. Antennas Propagat. 36 (1988), pp. 1260–1271.

648 H. El-Ocla

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

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

[21] A. Ishimaru, Wave Propagation and Scattering in Random Media, IEEE Press,

Piscataway, NJ, 1977.[22] V.H. Rumsey, Reaction concept in electromagnetic theory, Phys. Rev. 94 (1954),

pp. 1483–1491.[23] M.L. Harbold and B.N. Steinberg, Direct experimental verification of creeping waves,

J. Acoust. Soc. Am. 45 (1969), pp. 592–603.

[24] R. Dashen, Path integrals for waves in random media, J. Math. Phys. 20 (1979),pp. 894–920.

Appendix 1. Current generator form

According to Yasuura’s method, the surface current under the Neumann condition can beapproximated by means of a truncated modal expression as follows:

uðrÞ ’XMm¼1

bmðMÞ@��mðrÞ

@n¼ bM

@(�TM@n¼@(�M@n

bTM: ð27Þ

The basis functions (M are called the modal functions and constitute the complete set of wavefunctions satisfying the Helmholtz equation in free space and the radiation condition;(M¼ [��N, ��Nþ1, . . . ,�m, . . . ,�N], ��M and �T

M denote the complex conjugate and thetransposed vectors of �M, respectively, M¼ 2Nþ 1 is the total mode number,�mðrÞ ¼ Hð1Þm ðkrÞ expðim�Þ. The coefficient vector bM, defined as {b1, b2, . . . , bM}, can beobtained by the ordinary mode-matching method as shown below. Let us minimize the meansquare error by the method of least squares:

�HðM Þ ¼

ZS

XMm¼1

bmðM Þ@��mðrÞ

@n� uðrÞ

2

dr: ð28Þ

That is, we partially differentiate (28) with respect to b�m and obtain the algebraic equation

XMm¼1

bmðM Þ

ZS

@�n@n

@��m@n

dr ¼

ZS

@�n@n

uðrÞ dr, n ¼ 1 �M: ð29Þ

Based on (9), the normal derivative of the total wave on S is zero, that is:

@u

@n¼@uin@nþ@us@n¼ 0: ð30Þ

Therefore, the right-hand side of (29) can be written asZS

�m@u

@n�@�m@n

u

� �dr ¼

ZS

�m@uin@n�@�m@n

uin

� �drþ

ZS

�m@us@n�@�m@n

us

� �dr ð31Þ

where uin and us are the incident and scattered waves. Using Green’s theorem for �m, us in theregion surrounded by S and infinity, and using the radiation condition (11) for �m, us,we obtain Z

S

�m@us@n�@�m@n

us

� �dr ¼ 0

hence, the right-hand side of (29) can be given as the reaction of �m and uin:ZS

@�m@n

udr ¼ �

ZS

� �mðrÞ, uinðrÞ �dr ð32Þ


Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

where� ,�means

� �mðrÞ, uinðrÞ � �mðrÞ@uinðrÞ

@n�@�mðrÞ

@nuinðrÞ: ð33Þ

We can therefore write (29) as

AHbTM ¼ �

ZS

� �mðrÞ, uinðrÞ �dr ð34Þ

where AH is a positive definite MM Hermitian matrix except for the internal resonancefrequencies, and is given by

AH ¼

�@��N@n

,@��N@n

�. . .

�@��N@n

,@�N@n

�

..

. . .. ..

.�@�N@n

,@��N@n

�. . .

�@�N@n

,@�N@n

�

0BBBBBB@

1CCCCCCA

ð35Þ

in which its m, n element is the inner product of �m and �n:

@�m@n

,@�n@n

� �

ZS

@�m@n

@��n@n

dr: ð36Þ

From (34), bTM is given by

bTM ¼ �A�1H

ZS

� (Mðr0Þ, uinðr

0Þ �dr: ð37Þ

Substituting (37) into (27) and comparing it with (17), we can approximately obtain thecurrent generator YH for the Neumann condition as

YHðrjr0Þ ’ �

@(�MðrÞ

@nA�1H � (T

Mðr0Þ ð38Þ

where

�@(�MðrÞ

@nA�1H cMðrÞ ¼ fc�NðrÞ, c�Nþ1ðrÞ, . . . cNðrÞg: ð39Þ

Appendix 2. Final form of the scattering intensity for H-wave incidence

The current generator for H-wave incidence in (38) can be represented as

YHðrjr0Þ ’ cMðrÞ � (T

Mðr0Þ ¼

XNm¼�N

cmðrÞ �mðr0Þ@

@n0�@�mðr

0Þ

@n0

� �: ð40Þ

Therefore, the product of current generators is obtained as

YHðr01jr01ÞY�Hðr02jr

02Þ ¼

XNm¼�N

cmðr01Þ �mðr01Þ@

@n01�@�mðr

01Þ

@n01

� �( )

XNn¼�N

c�nðr02Þ ��nðr02Þ@

@n02�@��nðr

02Þ

@n02

� �( )

¼XN

m¼�N

XNn¼�N

cmðr01Þc�nðr02Þ

650 H. El-Ocla

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

��mðr

01Þ��nðr02Þ

@2

@n01@n02

� �mðr01Þ@��nðr

02Þ

@n02

@

@n01

�@�mðr

01Þ

@n01��nðr

02Þ@

@n02þ@�mðr

01Þ

@n01

@��nðr02Þ

@n02

�ð41Þ

hence, the integrand of (24) can be expressed as

YHðr01jr01ÞY�Hðr02jr

02Þ

@

@n01

@

@n02hGðr01jrÞGðrjr01ÞG

�ðr02jrÞG�ðrjr02Þi

¼

" XNm¼�N

XNn¼�N


�mðr

01Þ��nðr02Þ

@2

@n01@n02

@

@n01

@




02Þ

@n02

@

@n01

@

@n01

@



�@�mðr

01Þ

@n01��nðr

02Þ@

@n02

@

@n01

@



þ@�mðr

01Þ

@n01

@��nðr02Þ

@n02

@

@n01

@



!#: ð42Þ

As in two-dimensional problems we need to calculate the surface integral which can betransformed into a line integration as

ZS

� dr ¼

Z 2�

0

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ

dr

d�

� �2s

d�: ð43Þ

Therefore, we can represent the wave intensity in (24) as

hjusðrÞj2i ¼

Z 2�

0

d�01

Z 2�

0

d�02

Z 2�

0

d�01

Z 2�

0

d�02

( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir201 þ

dr01d�01

� �2s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r202 þdr02d�02

� �2s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r021 þdr01d�01


r022 þdr02d�02

� �2s

� XNm¼�N

XNn¼�N


��mðr

01Þ��nðr02Þ

@2

@n01@n02

@

@n01

@




02Þ

@n02

@

@n01

@

@n01

@



�@�mðr

01Þ

@n01��nðr

02Þ@

@n02

@

@n01

@



þ@�mðr

01Þ

@n01

@��nðr02Þ

@n02

@

@n01

@



��): ð44Þ


Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

In our representation of hjus(r)j2i, we use an approximate solution for the fourth moment of

the Green’s function in a random medium, M22. M22 can be obtained as the sum of the secondmoment products [18,24] on the assumptions that a single point source coincides with a singlepoint observation. That is,

M22 ¼ hGðrjr01ÞGðrjr01ÞG

�ðrjr02ÞG�ðrjr02Þi

’ hGðrjr01ÞG�ðrjr02Þi hGðrjr01ÞG

�ðrjr02Þi

þ hGðrjr01ÞG�ðrjr02Þi hGðrjr01ÞG

�ðrjr02Þi

¼M0 ðM þM�Þ ð45Þ

M0 ¼ G0ðrjr01ÞG0ðrjr01ÞG

�0ðrjr

02ÞG�0ðrjr02Þ

¼ U expðX Þ ð46Þ

M ¼ expðY1Þ ð47Þ

M� ¼ expðY2Þ ð48Þ

where U,X,Y1,Y2 are defined in [14]. We express @/@n01 and @/@n02 of the moment of theGreen’s function as follows:

@


�ðr02jrÞG�ðrjr02Þi ¼

@M0

@n01ðM þM�Þ ¼

@X

@n01ðM þM�Þ ð49Þ

@


�ðr02jrÞG�ðrjr02Þi ¼

@M0

@n02ðM þM�Þ ¼

@X

@n02ðM þM�Þ ð50Þ

@2

@n01@n02

hGðr01jrÞGðrjr01ÞG�ðr02jrÞG

�ðrjr02Þi

¼@2M0

@n01@n02

ðM þM�Þ ¼@2X

@n01@n02

þ@X

@n01

@X

@n02

� �M0ðM þM�Þ: ð51Þ

The normal derivative of X can be obtained as

@X

@n01¼ �jk

@z01@n01þ

jk

z� z0�01

@�01@n01

ð52Þ

@X

@n02¼ jk

@z02@n02�

jk

z� z0�01

@�01@n02

ð53Þ

@2X

@n01@n02

¼ 0: ð54Þ

652 H. El-Ocla

Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

Therefore, the wave intensity hjus(r)j2i in (24) can be expressed in final form as

hjusðrÞj2i ¼

Z 2�

0

d�01

Z 2�

0

d�02

Z 2�

0

d�01

Z 2�

0

d�02

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir201 þ

dr01d�01


r202 þdr02d�02


r021 þdr01d�01


r022 þdr02d�02

� �2s

XN

m¼�N

cmðr01Þ@X

@n01�mðr

01Þ@X

@n01�@�mðr

01Þ

@n01

� �" #

XNn¼�N

c�nðr02Þ@X

@n02��nðr

02Þ@X

@n02�@��nðr

02Þ

@n02

� �" #M0ðM þM�Þ

�ð55Þ

and the wave intensity in free space can be expressed as

jusðrÞj2 ¼

Z 2�

0

d�01

Z 2�

0

d�02

Z 2�

0

d�01

Z 2�

0

d�02

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir201 þ

dr01d�01


r202 þdr02d�02


r021 þdr01d�01


r022 þdr02d�02

� �2s

XN

m¼�N

cmðr01Þ@X

@n01�mðr

01Þ@X

@n01�@�mðr

01Þ

@n01

� �" #

XNn¼�N

c�nðr02Þ@X

@n02��nðr

02Þ@X

@n02�@��nðr

02Þ

@n02

� �" #M0

�: ð56Þ


Dow

nloa

ded

by [

Uni

vers

ity o

f N

orth

Tex

as]

at 1

9:03

27

Nov

embe

r 20

14

Documents

Effect of the illumination region of targets on waves scattering in random media with H-polarization