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1716 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Hosam El-Ocla
Effect of H-wave polarization on laser radardetection of partially convex targets in
random media
Hosam El-Ocla1,2
1Department of Computer Science, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario P7B 5E1, Canada2Electrical Engineering Department, King Saud University, Box 2454, Riyadh, Saudi Arabia
Received March 16, 2010; revised May 18, 2010; accepted May 20, 2010;posted May 25, 2010 (Doc. ID 125550); published June 25, 2010
A study on the performance of laser radar cross section (LRCS) of conducting targets with large sizes is inves-tigated numerically in free space and random media. The LRCS is calculated using a boundary value methodwith beam wave incidence and H-wave polarization. Considered are those elements that contribute to theLRCS problem including random medium strength, target configuration, and beam width. The effect of thecreeping waves, stimulated by H-polarization, on the LRCS behavior is manifested. Targets taking large sizesof up to five wavelengths are sufficiently larger than the beam width and are sufficient for considering fairlycomplex targets. Scatterers are assumed to have analytical partially convex contours with inflection points.© 2010 Optical Society of America
OCIS codes: 290.1350, 290.5855.
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. INTRODUCTIONave scattering from targets has been the subject of re-
earch for many decades. This research is vital especiallyn military applications based on electromagnetic mea-urements for radar detection [1]. Moreover, civilian ap-lications have been found in remote sensing in differentedia such as the atmosphere and sea [2]. In these appli-
ations, the calculation of radar cross section (RCS) is anmportant problem to the extent that several techniquesere proposed over the years [3–10]. Proposed solutionsf these scattering problems are generally complicatednd are usually restricted to only simple canonical ob-ects.
One of the well-known phenomena in the electromag-etics is the backscattering enhancement that motivatedeveral researchers as in [11–14]. The double passage ef-ect is produced as a result of the wave propagation in aandom medium where the RCS is enhanced to be twicehat in free space.
Over the past years, a method has been presented forolving the scattering problem as a boundary value prob-em [15]. Numerical experimental data have been shownor the RCS of conducting convex bodies such as circularnd elliptic cylinders [16]. Later, the effects of the targetonfiguration, the random media, and the polarization onhe RCS for the plane wave incidence were analyzed (e.g.,17–21], where other references can be found). It has beenhown that these parameters have a primary influence onhe RCS together with the double passage effect. Our re-ults as in [18] are in excellent agreement with those con-ucted for a circular cylinder in free space in [22].In general the nature of the incident waves is antici-
ated to have implications on the behavior of the scat-ered waves. The plane wave is the ideal type that results
1084-7529/10/071716-7/$15.00 © 2
n an authentic RCS calculation reflecting the character-stics of the target. This is because the target is alwaysovered by waves and therefore enough surface currentould be generated to enable the reflection of the featuresf the target. The generation of the plane wave that stayside enough around a target of great size is quite hard inractical applications especially in the far field. A pro-osal to generate a plane wave as a collection of Gaussianeam waves was formulated in [23]. Some solutions ofave scattering problems with Gaussian beam incidencesave been presented in [24–28]. Effects of wave propaga-ion and polarization in turbulence on the spectral inten-ity of light detection and ranging systems attracted re-earchers in optical imaging and radar applications29–33].
In this work, a study is presented on the laser radarross section (LRCS) performance with a relatively com-lex cross section. In this regard, the LRCS will be calcu-ated and analyzed numerically where targets are as-umed to have a partially convex contour with beam wavencidence of a limited width. We investigate the cases ofave scattering in free space and in a random medium
uch as turbulence. Targets take large sizes of up to fiveavelengths in free space. This size would be enoughreater than the beam width to suit the need of practicaladar applications. This scattering problem is solved two-imensionally assuming vertical polarization (H-wave in-idence). Creeping waves [34] that are generated with the-polarization are expected to have an impact on the
cattered waves.In [17], the obvious difference in the behavior of the
CS with the illumination curvature was clarified. Con-equently, the focus here is on wave backscattering fromhe convex illumination portion only. Scalar waves are as-
010 Optical Society of America
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Hosam El-Ocla Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1717
umed based on the conditions of a quite small randomedium strength and a great local scale size of the ran-
om medium particles. This assumption implies that noepolarization would occur. The time factor exp�−i�t� isssumed and suppressed in the following section.
. SCATTERING PROBLEMhe geometry of the problem is shown in Fig. 1. A randomedium is assumed as a sphere of radius L around a tar-
et of mean size a�L, and is further described by the di-lectric constant ��r�, the magnetic permeability �, andhe electric conductivity �. For simplicity ��r� is ex-ressed as
��r� = �0�1 + ���r��, �1�
here �0 is assumed to be constant and equal to the freepace permittivity and ���r� is a random function with
����r�� = 0, ����r����r��� = B�r,r��, �2�
B�r,r�� � 1, kl�r� � 1. �3�
ere, the angular brackets denote the ensemble average;�r ,r��, l�r� are the local intensity and local scale size of
he random medium fluctuation, respectively; and k���0�0 is the wavenumber in free space. Also � and �re assumed to be constant; �=�0, �=0.Consider the case where a directly incident wave is pro-
uced by a line source f�r�� distributed uniformly alonghe y axis. Here, let us designate the incident wave byin�r�, the scattered wave by us�r�, and the total wave by�r�=uin�r�+us�r�. The target is assumed to be a conduct-
ng cylinder whose cross section is expressed by
r = a�1 − � cos 3� − ��, �4�
here is the rotation index and � is the concavity index.e can deal with this scattering problem two-
imensionally using Eq. (3); therefore, we represent r as= �x ,z�. Assuming an H-polarization of the incidentaves (H-wave incidence), we can impose the Neumannoundary condition for the wave field u�r� on the cylinderurface S. That is, �� /�n�u�r�=0, where u�r� representsx. We consider uin�r1 �rt� to be represented as
q
f
r =(x,z)
L
Target
Randommedium
x
z
z
Incidentwave
Scatteredwave
0
B(r,r)
B0
S
: Normalizedfluctuation intensityof random medium
a
y
L >>a
ig. 1. Geometry of the problem of wave scattering from a con-ucting cylinder in a random medium.
uin�r1�rt� = G�r1�rt�exp− kx1
kW�2� , �5�
here W is the beam width. The beam expression is use-ul approximately only around the cylinder.
For practical turbulent media, condition (3) may be sat-sfied. As a result, depolarization of electromagneticaves can be neglected. Therefore, we can assume the for-ard scattering approximation and the scalar approxima-
ion [35], and re-incident waves are then negligible [15].ased on these assumptions and using the current gen-rator YH and the Green’s function in a random medium�r �r��, we can express the scattered wave as [21]
us�r� = − S
dr1 S
dr2 �
�n2G�r�r2��YH�r2�r1�uin�r1�rt�� .
�6�
ere, YH is the operator that transforms incident wavesnto surface currents on S and depends only on the scat-ering body. The current generator can be expressed inerms of wave functions that satisfy the Helmholtz equa-ion and the radiation condition. That is, the surface cur-ent is obtained as
− S
YH�r2�r1�uin�r1�rt�dr1
� −��M
� �r2�
�nAH
−1 S
���MT �r1�,uin�r1�rt���dr1, �7�
here
S
���MT �r1�,uin�r1�rt���dr1 �
Sm�r1�
�uin�r1�rt�
�n
−�m�r1�
�nuin�r1�rt��dr1. �8�
he above equation is sometimes called “reaction” namedy Rumsey [36]. In Eq. (8), the basis functions �M arealled the modal functions and constitute the complete setf wave functions satisfying the Helmholtz equation inree space and the radiation condition; �M�−N ,−N+1, . . . ,m , . . . ,N�, �M
� and �MT denote the com-
lex conjugate and the transposed vectors of �M, respec-ively; M=2N+1 is the total mode number, m�r�=Hm
�1�
�kr�exp�im�; and AH is a positive definite Hermitianatrix given by
AH =��−N
�n,�−N
�n � . . . �−N
�n,�N
�n �] � ]
�N
�n,�−N
�n � . . . �N
�n,�N
�n � � , �9�
n which its m ,n element is the inner product of m and,
n
Yt
H
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W
wm
w
T
M
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O
wa
B
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1718 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Hosam El-Ocla
�m
�n,�n
�n � � S
�m
�n
�n�
�ndr. �10�
H is proved to converge in the sense of the mean to therue operator when M→ .
The average intensity of the backscattering wave for-wave incidence is given by
��us�r��2� = S
dr01 S
dr02 S
dr1� S
dr2�
�YH�r01�r1��YH� �r02�r2��
�exp− kx1�
kW�2�exp− kx2�
kW�2� �
�n01
�
�n02
��G�r�r01�G�r�r02�G��r�r1��G��r�r2���. �11�
n our representation of ��us�r��2�, we use an approximateolution for the fourth moment of the Green’s function inrandom medium M22 [19],
M22 = �G�r�r1��G�r�r01�G��r�r2��G��r�r02��
� �G�r�r1��G��r�r2����G�r�r01�G��r�r02��
+ �G�r�r1��G��r�r02���G�r�r01�G��r�r2���. �12�
In wave propagation through a strong continuous ran-om medium, we may assume that the Green’s functionecomes approximately complex Gaussian random [37].ere, we obtain an analytical form for the second moment
f the Green’s function M11 that has been given in20,38–40], for instance,
�G�r�r01�G��rt�r02�� = �G��,z��01,z01�G���t,z��02,z02��
= M11��,�t,z��01,�02,z01,z02�. �13�
e solve M11 using the Helmholtz equation as in [37,41],
�
�z− j
1
2k��2 − �t
2��M11��,�t,z��01,�02,z01,z02�
=�−k2
4 0
z−z0
D� − �t,z −z�
2,z��dz���
�M11in ��,�t,z��01,�02,z01,z02�, �14�
here z0 is defined in Eq. (15) and M11in is the second mo-
ent of the Green’s function in free space,
z0 = a, �15�
here a is the mean size of the target,
M11in ��,�t,z��01,�02,z01,z02� = G0��,z��01,z01�G0
���t,z��02,z02�.
�16�
herefore, we can represent Eq. (14) as
11��,�t,z��01,�02,z01,z02� = M11in ��d,�s,z�m��d,�0d�, �17�
here �d=�−�t , �s= ��+�t� /2 , �0d=�1−�2 , �0s= ��1�2� /2. For a two-dimensional problem, M11
in can be for-ulated as follows:
G0��,z��1,z1�G0���,z��1�,z1�� �
1
8�kzexp�ik�z − z0�−
z1 − z1�
z − z0
+1
2
�� − �1�2 − �� − �1��2
�z − z0�2 �� .
�18�
n the other hand, m��d ,�0d� can be expressed as
m��d,�0d� = exp−k2
4 z0
z
Dt z� − z0
z − z0�d
+z� − z0
z − z0�0d,z��z0�dz�� , �19�
Dt��,z�z0� = 0
z−z0
D�,z −z�
2,z��dz�, �20�
D�,z −z�
2,z�� = 2B0,z −
z�
2,z�� − B�,z −
z�
2,z��� ,
�21�
here D is called the structure function of turbulence [35]nd can be expressed as [15]
D��,z+,z−� = B�z+� �
l�z+��2
exp− z−
l�z+��2� . �22�
y solving Eq. (19), we get
m��d,�0d� = exp−k2
4����z − z0��d
2 + ��z − z0��d�0d
+ ��z − z0��0d2 �� , �23�
here � ,� ,� ,� are given in [20]. Let us assume that theoherence of waves is kept almost complete in propaga-ion of a distance 2a equal to the mean diameter of theylinder. This assumption is acceptable in practical casesnder condition (3). On the basis of the assumption, it is
mportant here to point out that we are going to present auantitative discussion for the numerical results in Sec-ion 3. We can obtain the LRCS by using Eq. (11),
� = ��us�r��2�k�4�z�2. �24�
The calculation of scattering data has been restricted tohe interval 0.1�ka�30. It is quite difficult to exceedhis ka limit since a larger ka requires a large M whichonsequently increases the calculation time dramatically.
. NUMERICAL RESULTSlthough the incident wave becomes sufficiently incoher-nt, we should pay attention to the spatial coherenceength (SCL) of the incident wave around the target asas been pointed out in earlier papers. The degree of spa-ial coherence is defined by
wf=aktStucuf
gF�Nct
otabwnag
Fa
Ff
Hosam El-Ocla Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1719
���,z� =�G�r1�rt�G��r2�rt��
��G�r0�rt��2�, �25�
here r1= �� ,0�, r2= �−� ,0�, r0= �0,0�, and rt= �0,z�. In theollowing calculation, we assume B�r ,r�=B0 and kB0L3�; therefore, the coherence attenuation index � defineds k2B0Ll /4 given in [15] is 15�2, 44�2, and 150�2 forl=20�, 58�, and 200�, respectively, which means thathe incident wave becomes sufficiently incoherent. TheCL is defined as 2k� at which ���=e−1�0.37. It is notedhat D in Eq. (22) has the following relation: D��2. Fig-re 2 shows a relation between the SCL and kl in thisase and that the SCL is equal to 3, 5.2, and 9.7. We willse the SCL to represent one of the random medium ef-ects on the LRCS.
Here, we point out that N in Eq. (9) depends on the tar-et parameters and polarization of the incident waves.or example, for H-wave incidence, we choose N=28 at=0.1 in the range of 0.1�ka�5; at ka=20, we choose=46 at �=0.1. As a result, our numerical results are ac-
urate because these values of N lead to convergence ofhe LRCS.
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8
kl = 200
=20 π
π=58 π
e-1
(ρ)
ρ
Γ
k
kSCL=9.7
SCL=5.2SCL=3
ig. 2. The degree of spatial coherence of an incident wavebout the cylinder.
uin us SCL
EIRr
(b) Generic incident wave
uin us 2kw
EIRbr
(a) Beam wave incidence
2a
2a
Fig. 3. Geometry showing the EIRs in a random medium.
On the basis of the assumption of coherence completionf the waves in the propagation of distance 2a, we definehe effective illumination region (EIR) of targets. EIRbrnd EIRr in Fig. 3 are those surfaces that are illuminatedy the incident wave and restricted by 2kW for the beamave and the SCL for the generic wave regardless of itsature (plane or beam), respectively. Consequently, it isnticipated that target aspects including � and ka to-ether with the SCL and kW are going to influence the
0
1
2
3
0 5 10 15 20 25 30
free spaceSCL = 3
= 5.2= 7.5
ka
σ/(2a)
b
(a)
0
1
2
3
0 5 10 15 20 25 30
free spaceSCL = 3
= 5.2= 7.5
σ/(2a)
b
ka
(b)
0
1
2
3
0 5 10 15 20 25 30
free spaceSCL = 3
= 5.2= 7.5
σ/(2a)
b
ka
(c)ig. 4. LRCS versus target size in free space and at three dif-
erent SCLs for kW=2 where (a) �=0, (b) �=0.1, and (c) �=0.2.
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Ff
1720 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Hosam El-Ocla
IR generally. Subsequently, the LRCS and the enhance-ent in laser radar cross section (ELRCS) will be affected
n a way that will be clarified in the numerical results. Inhe following, we conduct numerical results for the LRCSnd normalized laser radar cross section (NLRCS), de-ned as the ratio of the LRCS in random media �b to theRCS in free space �0.
. Radar Cross Sectionn the following, numerical results for the LRCS �b shownn Fig. 4 are discussed. The LRCS undergoes oscillatingehavior with ka and this is attributed to three factorshich are the target complexity represented in �, the ran-om medium inhomogeneities reflected in the length ofhe SCL, and the creeping waves produced as a result ofhe H-wave polarization. These oscillations are more se-ere with a big � in the resonance region as a result of theontributions from the inflection points in addition to thepecular reflections as was explained in the E-wave inci-ence case [27]. This behavior agrees with an earliertudy handling scattering reflections from illuminatednd shadow portions of analytical targets with inflectionoints [42,43]. On the other hand, having a smaller SCL,he LRCS has a greater oscillation. This is interpreted ashe creeping waves coupling with direct waves in the ran-om medium. The former waves located inside the SCLre correlated with the latter waves leading to in-phaseddition, while waves creeping on the target’s surface out-ide the SCL are uncorrelated with the direct waves lead-ng to an out-of-phase situation. These in- and out-of-hase waves result in an oscillating RCS. This effectbviously reduces with wider SCLs since waves becomeore in-phase. More explanation about the impact of theCL on the LRCS will be analyzed with a closer lookhen the ELRCS is discussed in the next subsection.As ka becomes greater than kW, the LRCS lessens ow-
ng to the lack of the surface current resulting in aradual dwindling in the scattered wave intensity witha. The manner of this is almost similar irrespective ofoth � and the SCL. Figure 5 shows that smaller kWpeeds up the monotonic decrease behavior with ka asas shown in [28]. For ka�kW, the LRCS is almost in-ariant with � since the EIR becomes almost a flat surfaceor the small kW. At certain ka, scattered waves diminishnd the incident wave is unable to generate enough sur-ace current for the radar detection.
0
1
2
3
0 5 10 15 20 25 30
free spaceSCL = 3
= 5.2= 7.5
ka
σ/(2a)
b
ig. 5. LRCS versus target size in free space and at three dif-erent SCLs for kW=1.5, where �=0.
. Backscattering Enhancementn Fig. 6, we analyze the NLRCS with respect to the re-ationship between kW and the SCL. In Fig. 6(a), one canee that when SCL�2kW, the NLRCS is farther awayrom 2 but with less fluctuations than when SCL=2kW.his is attributed to the case where EIRr�EIRbr and
herefore some of the incident waves are incoherent lead-ng to more deviation from 2. In Fig. 6(b), where SCL
2kW and therefore EIRr�EIRbr, the NLRCS rises uploser to 2 but the fluctuations are stronger with smallerW. When SCL�2kW, as shown in Fig. 6(c), the NLRCS
1
1.2
1.4
1.6
1.8
2
2.2
0 5 10 15 20 25 30
kw =1.5= 2
σ/σ
b
0
ka(a)
1
1.2
1.4
1.6
1.8
2
2.2
0 5 10 15 20 25 30σ/σ
b
0
ka
kw =1.5= 2
(b)
1.92
1.94
1.96
1.98
2
2.02
2.04
2.06
0 5 10 15 20 25 30
σ/σ
b
0
kw =1.5= 2
ka
(c)ig. 6. Normalized LRCS versus target size with �=0 at differ-nt kW’s, where (a) SCL=3, (b) SCL=5.2, (c) SCL=30; �b , �0 areRCS in random media and in free space, respectively.
iwflb�apgSaR
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4IrflTcs(SiliHwstcwbtd
ieo(fahRsl
ATau
R
1
1
1
1
1
1
1
1
1
1
2
2
2
Hosam El-Ocla Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1721
s closer to 2 and the strength of the fluctuations is lessith wider kW. Therefore, to reduce the strength of theuctuations in the NLRCS, the following relation has toe maintained: EIRr�EIRbr, i.e., the condition SCL2kW should be valid. This relation assures that EIRr isprimary factor over the EIRbr effect. Therefore, com-
ared to the case of plane wave incidence, the EIR of tar-ets surrounded by random media is a function of theCL regardless of the incident wave. Moreover, the SCLnd kW as well should have greater lengths to have a NL-CS closer to 2.For ka�SCL, the NLRCS equals 2 as a result of the
ouble passage effect and this is realized apart from thencident wave polarization. The beam wave acts as if it is
plane wave for small ka. By enlarging ka compared tohe SCL, the NLRCS oscillates remarkably and irregu-arly deviating from 2 as a result of a combination of ef-ects including creeping waves, target inflection points,nd the randomness of the medium as discussed earlier.his behavior is different from that for E-wave incidencehere oscillations have regular sinusoidal behavior as a
esult of the absence of creeping waves.
. CONCLUSIONn this paper several parameters that influence the laseradar cross section (LRCS) of analytical targets having in-ection points with beam wave incidence were addressed.hese issues include the target’s concavity index and theontributions from the points in the neighborhood of thehadow portion beside the effective illumination regionEIR). Also, the random medium effect represented in theCL of waves around the target was proven to have a key
mpact on the LRCS. The LRCS reduces with a quitearge target size, and the beam wave of a relatively lim-ted length becomes unable to sense the target properly.aving a wider beam width allows the LRCS to dwindleith ka at a slower rate. This is because the generated
urface current, with a limited beam width, is not enougho produce the scattered waves that reflect the wholeharacteristics of the target. Creeping waves producedith H-wave incidence result in a substantial oscillatingehavior especially in the resonance region, and this ac-ivity does not exist that severely with the E-wave inci-ence.To have a better monostatic radar detection of targets
n random media, double passage is the effect that can bexclusively accepted. To realize this, the strength of thescillations in the normalized laser radar cross sectionNLRCS) should, therefore, be minimal by attaining theollowing condition: SCL�2kW. Nevertheless, the SCLnd in turn the beam size should have wider lengths toave the enhancement in laser radar cross section (EL-CS) approaching 2. These two conditions are valid irre-pective of both target configuration [27] and the wave po-arization for the laser radar detection.
CKNOWLEDGMENThis work was supported in part by the National Sciencend Engineering Research Council of Canada (NSERC)nder grant 250299-02.
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