Beamwidth Effects on Backscattering Enhancement from Targets inRandom Media for E-Wave Polarization

Hosam El-Ocla

Department of Computer Science, Lakehead University

955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1

E-mail: hosam@lakeheadu.ca

Abstract: In this paper we present a study on the

effects of the beamwidth on the backscattering en-

hancement of waves from object in random medium.

Considered random medium as turbulence is isotropic

with different strengths of correlation functions. Tar-

gets take sizes more than twice of the wavelength in

free space to suit object dimensions of real world such

as aircrafts. Horizontal incident wave polarization (E-

wave incidence) is assumed.

1 Introduction

Several methods have been proposed to formulate the

scattering waves in literature (e.g. [1]–[3]). Current

generator method was developed and used over years

to solve the scattering problem as a boundary value

problem [4]–[6]. This method is characterized by the

calculation of the current on the whole surface includ-

ing the shadow region. Therefore this method gives an

accurate calculation of the scattered waves intensity.

Backscattering enhancement in random media at-

tracted researchers in the areas of radar engineering

and remote sensing as in [7, 8]. Double passage of

waves backscattering from point objects results in an

enhancement in the RCS in random medium and ac-

cordingly it is twice that in free space. Effects of inci-

dent waves are quite significant depending upon its na-

ture and polarization particularly in random medium.

To generate waves of infinitely large plane wave fronts,

an infinitely large source should be used. This can not

easily happen for plane waves at the fronts of large size

targets in the far field and, therefore, we study the ef-

fect of beam wave with a limited beamwidth compared

to the inifinte width of the plane wave. It should be

noted that the current generator method is normal-

ized to the wavenumber and, consequently, it is valid

for the radio and optical frequencies.

In this work, we investigate the impact of

beamwidth kW on the backscattering enhancement of

targets with finite size. We use the spatial coherence

length (SCL) of waves around the object to represent

the strength of the random medium. We compare

between SCL and kW to measure the capability of

the radar model we propose. We deal with the scat-

tering problem two-dimensionally assuming horizontal

polarization (E-wave incidence). Numerical results are

for the normalized LRCS of concave-convex targets of

sizes more than twice wavelengths to suit the real di-

mensions of objects such as aircrafts. The time factor

exp(-iwt) is assumed and suppressed in the following

section.

2 Formulation

Geometry of the problem is shown in Figure 1. A

random medium is assumed as a sphere of radius L

around a target of the mean size a � L, and also to be

described by the dielectric constant ε(r), the magnetic

permeability µ, and the electric conductivity ν. For

simplicity ε(r) is expressed as

ε(r) = ε0[1 + δε(r)] (1)

where ε0 is assumed to be constant and equal to free

space permittivity and δε(r) is a random function with

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

and

B(r, r) � 1, kl(r) � 1 (3)

Here, the angular brackets denote the ensemble aver-

age and B(r, r), l(r) are the local intensity and local

scale-size of the random medium fluctuation, respec-

tively, and k = ω√ε0µ0 is the wavenumber in free

space. Also µ and ν are assumed to be constants;

µ = µ0, ν = 0. For practical turbulent media the con-

dition (3) may be satisfied. Therefore, we can assume

5048978-1-4799-5775-0/14/$31.00 ©2014 IEEE IGARSS 2014

θ

φ

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

Figure 1: Geometry of the problem of wave scattering

from a conducting cylinder in a random medium.

the forward scattering approximation and the scalar

approximation. Consider the case where a directly in-

cident beam wave is produced by a line source f(r′)along the y axis. 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

r = a[1− δ cos 3(θ − φ)] (4)

where φ is the rotation index and δ is the concav-

ity index. We can deal with this scattering problem

two dimensionally under the condition (3); therefore,

we represent r as r = (x, z). Assuming a horizontal

polarization of incident waves (E-wave incidence), we

can impose the Dirichlet boundary condition for wave

field u(r) on the cylinder surface S. That is, u(r) = 0,

where u(r) represents Ey .

Using the current generator YE and Green’s func-

tion in random medium G(r | r′), we can express the

scattered wave as

us(r) =

∫S

dr1

∫S

dr2 [G(r | r2)YE(r2 | r1)uin(r1 | rt)](5)

where rt represents the source point location and it

is assumed as rt = (0, z) in section 3. We consider

uin(r1 | rt), whose dimension coefficient is understood,

to be represented as:

uin(r1 | rt) = G(r1 | rt) exp[−(kx1

kW)2] (6)

whereW is the beamwidth. The beam expression [9] is

approximately useful only around the cylinder. Here,

YE is the operator that transforms incident waves into

surface currents on S and depends only on the scatter-

ing body. The current generator can be expressed in

terms of wave functions that satisfy Helmholtz equa-

tion and the radiation condition. YE is well formulated

in [5] for E-polarization. That is, the surface current

is obtained as ∫S

YE(r2 | r1)uin(r1 | rt) dr1 � (7)

Φ∗M (r2)A

−1E

∫S

� ΦTM (r1), uin(r1 | rt) � dr1,

where ∫S

� ΦTM (r1), uin(r1 | rt) � dr1 ≡∫

S

[φm(r1)

∂uin(r1 | rt)∂n

− ∂φm(r1)

∂nuin(r1 | rt)

]dr1. (8)

The above equation is sometimes called “reaction”.

In (8), the basis functions ΦM are called the

modal functions and constitute the complete set

of wave functions satisfying the Helmholtz equation

in free space and the radiation condition; ΦM =

[φ−N , φ−N+1, . . . , φm, . . . , φN ], Φ∗M and ΦT

M denote

the complex conjugate and the transposed vectors of

ΦM , respectively, M = 2N +1 is the total mode num-

ber, φm(r) = H(1)m (kr) exp(imθ), and AE is a positive

definite Hermitian matrix given by

AE =

(φ−N , φ−N ) . . . (φ−N , φN )...

. . ....

(φN , φ−N ) . . . (φN , φN )

(9)

in which its m,n element is the inner product of φm

and φn:

(φm, φn) ≡∫S

φm(r)φ∗n(r)dr (10)

The YE is proved to converge in the sense of mean on

the true operator when M → ∞.

Therefore, the average intensity of backscattering

wave for E-wave incidence is given by

〈|use(r)|2〉 =

∫S

dr01

∫S

dr02

∫S

dr′1

∫S

dr′2

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YE(r01 | r′1)Y ∗E(r02 | r′2)×

exp

[−(kx′

1

kW

)2]exp

[−(kx′

2

kW

)2]×

〈G(r | r01)G(r | r02)G∗(r | r′1)G∗(r | r′2)〉 (11)

We can obtain the LRCS σ using equation (11)

σ = 〈|us(r)|2〉 · k(4πz)2 (12)

3 Numerical Results

Although the incident wave becomes sufficiently inco-

herent, we should pay attention to the spatial coher-

ence length (SCL) of incident waves around the target.

The degree of spatial coherence is defined as [4]

Γ(ρ, z) =〈G(r1 | rt)G∗(r2 | rt)〉

〈| G(r0 | rt) |2〉 (13)

where r1 = (ρ, 0), r2 = (−ρ, 0), r0 = (0, 0), rt =

(0, z). In the following calculation, we assume

B(r, r) = B0 and kB0L = 3π; therefore the coher-

ence attenuation index α defined as k2B0Ll/4 is 15π2,

44π2, and 150π2 for kl = 20π, 58π, and 200π, respec-

tively, which means that the incident wave becomes

sufficiently incoherent. The SCL is defined as the 2kρ

at which | Γ |= e−1 � 0.37. Figure 2 shows a relation

between SCL and kl in this case and SCL, accordingly,

is equal to 3, 5.2, and 9.7. We will use the SCL to

represent one of the random medium effects on LRCS.

The integrations in (11) are calculated using the trape-

zoidal rule.

In the following, we conduct numerical results for

the normalized LRCS (NLRCS), defined as the ratio

of LRCS in random media σ to LRCS in free space σ0.

3.1 Backscattering Enhancement

Here, we discuss the numerical results for the NLRCS

in figure 3. For ka � SCL, the NLRCS is two regard-

less the concavity index δ due to the double passage

effect. In this range, beam wave acts as being a plane

wave at the small ka which tends to be of a point tar-

get.

Considering a finite size object, NLRCS deviates

from two and decreases monotonically with greater ka.

This behavior is more obvious with smaller beamwidth

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

k

SCL=9.7

SCL=5.2SCL=3

Figure 2: The degree of spatial coherence of an inci-

dent wave about the cylinder.

kW . NLRCS oscillates regularly in sinusoidal behav-

ior owing to the random medium fluctuations effect.

When SCL is wide enough, NLRCS approaches two as

well shown in figure 4. In the resonance region where

2ka is comparable with the SCL, NLRCS suffers from

an anomalous oscillated behavior. These oscillations

are stronger with more complex target cross-sections

(case of δ = 0.08 compared to δ = 0) due to the effects

of inflection points on the concave-to-convex curva-

ture region. Also these oscillations are more obvious

with narrower kW . Inflection points contributions are

more obvious in the random medium than in free space

which in turn affect NLRCS significantly with smaller

kW . When ka � kW , NLRCS would diminish with

large enough object size and the beam wave would be

incapable of remote sensing.

4 ConclusionThe behavior of the backscattering enhancement, rep-

resented in the normalized LRCS (NLRCS), depends

greatly on the object complexity, random medium cor-

relation function (represented in kl in this paper and

accordingly the SCL), and the beamwidth. For point

target, NLRCS is two because of the double passage ef-

fect. This is not the case with an object having a finite

size where NLRCS deviates from two with ka. How-

ever, NLRCS approaches two with wider SCL and/or

wider beamwidth. Objects can hide from the remote

sensing radars in the resonance region through manu-

facturing with complex cross sections having more in-

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flection points. This can be avoided and overcome by

having a beamwidth wide enough to generate needed

current on the surface of the object which, in turn,

enhances the capability of the radar system regardless

the SCL of the random medium.

References

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0.4

0.8

1.2

1.6

2

2.4

0 2 4 6 8 10 12 14 16

kW=5

kW=10

SCL=4=6

σ/σ

b

0

ka(a)

0.4

0.8

1.2

1.6

2

2.4

0 2 4 6 8 10 12 14 16

SCL=4=6

σ/σ

b0

ka

kW=10

kW=5

(b)

Figure 3: normalized LRCS vs. target size in random

medium where (a) δ = 0, (b) δ = 0.08.

0.4

0.8

1.2

1.6

2

2.4

0 2 4 6 8 10 12 14 16

σ/σ

b

0

kW=5=10

ka

Figure 4: normalized LRCS vs. target size in random

medium where at SCL=30.

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