[ieee 2010 42nd southeastern symposium on system theory (ssst 2010) - tyler, tx, usa...

SCATTERING OF ELECTROMAGNETIC

RADIATION FOR A PERFECT ELECTRIC

CONDUCTING CYLINDER BY USING

MULTIPLE ANGLES OF POLARIZATION

Sandeep Narkimelli Department of Electrical Engineering

The University of Texas

Tyler, United States

[email protected]

Hector A. Ochoa Assistant Professor of Electrical Engineering

The University of Texas

Tyler, United States

[email protected], [email protected]

Abstract--- Radar is a device which detects distant or non

visible objects by means of reflected radio waves [1]. The

quality of the reflected signal depends on the shape and

orientation of the target with respect to the type of polarization

used. Most of the antennas currently used on radar systems

employ one type of polarization at a time for target detection.

This paper describes the importance of employing

polarizations of multiple angles on targets, approximately at

the same instant of time. To implement this idea a generalized

set of equations have been derived which represent the

backscatter generated by a cylindrical object. Using different

angles in these equations would give the backscatters of

different polarizations employed on the same target, at small

time intervals. The logic behind this is that there would be at

least one angle that would have the maximum signal strength

in the backscatter. The backscatter for this angle would satisfy

the best quality criterion as compared with the rest of the

polarization angles.

Keywords: Polarimetry, Types of Polarizations, Multiple

polarizations, Transverse electric, Transverse magnetic

I.INTRODUCTION

Materials contain charged particles, and when these

materials are subjected to electromagnetic fields, their

charged particles interact with the electromagnetic field,

producing currents and modifying the electromagnetic wave

propagation in those media compared to that in free space

[2]. This has been the operational point for all

electromagnetic wave based systems and radar is one of

them. Polarization is an underlying concept, in all of the

electromagnetics oriented applications, that explains the

different ways in which electromagnetic waves can be made

to interact with materials. Polarization forms one of the base

points for radar system applications. The emphasis of this

paper has being laid on deriving a set of equations using

linear polarization that would characterize the case for the

simultaneous use of different angles of polarizations rather

than just horizontal and vertical. To implement this,

polarizations of all possible angles are being tested on the

target because maximum signal strength occurs only when

there is a proper alignment between the transmitter and the

target. For the analysis, a stationary cylindrical target has

being considered. The reason for this consideration was that

most of the real world objects tend to be cylindrical;

therefore, the consideration for using a cylindrical test target

would yield to more realistic results. A generalized set of

equations describing the electromagnetic parameters for a

cylindrical target are derived, these equations can be used to

derive the parameters for every possible angle. Moreover,

dual polarization radars that are being currently used require

two antennas for simultaneous transmission of polarizations,

whereas, the application of this concept practically can be

achieved by a single antenna system using short time gaps

between each transmission.

II.THEORETICAL BACKGROUND

This section gives an overview of the background

concepts that are useful in understanding the steps.

Polarization is defined as “that part of a radiated

electromagnetic wave describing the time-varying direction

and relative magnitude of the electric field vector” [2].

There are different types of polarizations that are currently

being used and they are linear, circular and elliptical

polarizations [6]. Linear polarization is divided into

horizontal and vertical polarizations.

Horizontal Polarization:

When an electric field is perpendicular to the plane of

incidence the polarization is referred to as perpendicular

polarization; in other words, if the electric field is parallel to

the interface or parallel to the earth’s surface, then it is

known as horizontal polarization. An example of a system

that employs horizontal polarization would be weather

radars.

Vertical Polarization:

When the electric field is parallel to the plane of

incidence, the polarization is referred to as parallel

polarization and because in this case the component of

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42nd South Eastern Symposium on System TheoryUniversity of Texas at TylerTyler, TX, USA, March 7-9, 2010

M1C.2

electric field is also perpendicular to the interface when the

magnetic field is parallel to the interface, it is also known as

vertical polarization.

Currently, there are different types of antennas being

used and each of these antennas is efficient with respect to

the application. However, polarization plays a major role in

the efficiency of these antennas by being the common stand

point for all of them. Dual-polarization is a concept that

makes use of more than one polarization at a time and it

finds its use in various applications. Weather radar would be

one that comes into picture mostly when it comes to dual

polarization. The term “dual polarization” is also referred as

polarimetry, but it is termed for more than just two

polarizations. Radars employing polarimetry are called

polarimetric radars. Weather radars come into the class of

polarimetric radars by the use of dual polarizations,

horizontal and vertical. These radars transmit on one

polarization and receive on other. Polarimetric radar is

relatively a new discipline with great potential, primarily

through its impact on remote sensing of the environment

from aircraft, satellites or space shuttles. It provides

important information associated with the management of

natural resources as well as on numerous other scientific and

industrial applications like radar meteorology, mine

detection technology, SAR interferometry for ground

mapping and air/ground traffic surveillance and control [3].

The following sections describe the concepts involved

in the derivation of the scatter equation.

III. SCATTERING OF NORMALLY INCIDENT PLANE

WAVE BY CONDUCTING CIRCULAR CYLINDER

This section deals with the description of methods

employed and procedures involved in deriving the

generalized set of equations for the scattered

electromagnetic waves polarized at different angles. A

stationary circular cylinder is considered as the target,

because of the simplicity of the solution and it can be

represented in terms of well known tabulated functions such

as Bessel and Hankel functions. The scattering of plane

waves by circular conducting cylinders of infinite length at

normal incidence are considered. The practical case of

scattering would be for finite length cylinders, which will be

obtained by transforming the scattered fields of infinite

lengths using approximate relationships.

The assumption here is that a uniform plane wave

represented by iE is incident normally on a perfectly

conducting cylinder of radius ‘a’. Assuming the wave is

polarized with some angle θ with respect to the plane of reference. This means that the angle θ is the angle between the electric field of the incident wave and the reference

plane. This is shown on Figure 1.

z

y

x

θ

cosiE θ

iE

iE sinθ

y

z

Figure 1. Uniform plane wave incident on a conducting circular cylinder

If a wave is at an angle of 0° ,180° , or 360° with respect to the plane of reference, then it can be considered to

be horizontally polarized and if it’s at an angle of 90°or 270° with respect to the plane of reference, it can be considered to be vertically polarized. In order to perform the

analysis, a waveform with an arbitrary polarization angle is

considered; therefore, first this waveform will need to be

decomposed into horizontal and vertical polarized

waveforms.

The decomposed waveforms will be processed using

the transverse electric and transverse magnetic polarization

methods, which are the horizontal and vertical polarizations

respectively. Based on Figure 1, the incident electric field

can be written as

ˆ ˆi i i

y y z zE a E a= +E (1)

Where equation (1) is a linear combination of horizontal and

vertical polarizations using the triangle law of vector

addition [4]

Using the properties of right angled triangle,i

yE and

i

zE can be derived as

ˆcosi i

y yE E aθ=

ˆsini i

z zE E aθ=

Where the electric field iE can be written as 0

j xE e

β−[2]

Substituting these values in equation(1), the following

expression for the incident electric field is obtained

0 0ˆ ˆcos sin

i j x j x

y zE e a E e a

β βθ θ− −= +E (2)

Equation (2) is the incident wave with an arbitrary

angle of polarization θ with the reference plane. In the general derivation of a transverse electric polarization the

magnetic components of the waveform are considered, this

is shown in equation (3). cos

0ˆ j

za H eβ φ−=iH (3)

Equation (3) is represented in cylindrical coordinates

where cosx ρ φ= , ρ - radial distance, φ - azimuth angle and β - propagation constant.

In order to derive the electric fields with magnetic

components, there needs to be some form of transformation

from the magnetic to electric components. This has been

achieved by using the relationship between the electric and

61

magnetic energy densities [5] shown by the equations (4)

through (8) 2

0

2

EU

ε= (4)

2

0

2

HU

µ= (5)

0 0cosH E

εθ

µ= (6)

cos

0ˆcos j

z zE e aβρ φε

θµ

−=H (7)

cos

0ˆj

z zH e aβρ φ−=H (8)

Using the equations above, the derivation follows by

deriving the incident magnetic and electric fields using the

transverse electric and transverse magnetic polarizations

respectively.

IV. TRANSVERSE ELECTRIC POLARIZATION

Let us assume a transverse electric uniform plane wave

travelling in the positive x-direction is normally incident on

the cylindrical target of radius a shown in the Figure 2.

Hi

Ei

a

σ = ∞

y

z

x

Figure 2. Uniform plane wave incident on a conducting circular

cylinder

Using equation (8) the incident magnetic field can be

written in terms of Bessel’s functions as

0

0

ˆ ( ) ( ) cos( )n

z z n n

n

H H a j J nε βρ φ∞

=

= −∑ (9)

The corresponding incident electric field can be

obtained using Maxwell’s equation jω∇× =H D

1j

jωε

ωε∇× = ⇒ = ∇×H E E H (10)

Solving the curl of equation (10) the electric field

components in ρ and φ coordinates will be obtained as shown in equations (11) and (12) below

( )101 1 1i

i n jnz

n

n

HHE nj J e

j j

φρ βρ

ωε ρ φ ωε ρ

∞− +

=−∞

∂= = ∂

∑ (11)

01( )

i

i n jnz

n

n

HHE j J e

j j

φφ

ββρ

ωε ρ ωε

∞−

=−∞

−∂−′= =

∂ ∑ (12)

Since the scattered fields travel in the outward

direction, they must be represented by cylindrical travelling

wave functions. Thus the magnetic field sH for transverse

electric polarization is represented by equation (13)

(2)

0ˆ ˆ ( )s

z z z n n

n

a H a H d H βρ∞

=−∞

= = ∑sH (13)

Where n

d represents the unknown amplitude coefficients

which can be found by applying the appropriate boundary

conditions

Before the boundary conditions on the vanishing total

tangential electric field on the surface of the cylinder can be

applied, it is necessary to first find the corresponding

electric fields. These can be accomplished by using

Maxwell’s equations (10) which for the scattered magnetic

field of (13) leads to

( )(2)01 1 1s

s nz

n

n

H dHE H

j jρ βρ

ωε ρ φ ωε ρ φ

∞

=−∞

∂∂= =

∂ ∂∑ (14)

(2)01( )

s

s z

n n

n

HHE d H

j jφ

ββρ

ωε ρ ωε

∞

=−∞

−∂− ′= =∂ ∑ (15)

Since the cylinder is a perfect electric conductor, the

tangential components of the total electric field must vanish

on its surface ( )aρ = . Thus applying the boundary

conditions on the total electric field, the unknown amplitude

coefficient n

d will be derived as equation (16)

(2)

( )

( )

n jnn

n

n

J ad j e

H a

φβ

β

− ′= −

′ (16)

Thus the scattered electric and magnetic fields can be

written using equations (13) and (16) as

0s s s

zE H Hρ φ= = = (17)

1 (2)0

(2)

( )1( )

( )

s n jnn

n

n n

H J aE nj H e

j H a

φρ

ββρ

ωε ρ β

∞− +

′=−∞

′−= ∑ (18)

(2)0

(2)

( )( )

( )

s n jnn

n

nn

H J aE j H e

j H a

φφ

β ββρ

ωε β

∞−

=−∞

′′=

′∑ (19)

(2)

0(2)

( )( )

( )

s n jnn

z n

nn

J aH H j H e

H a

φββρ

β

∞−

=−∞

′= −

′∑ (20)

The total electric field for the equations(18), (19) and

(20) can obtained using the equation t i sE E E= + . The

scattered equations above, along with the set of equations

derived with the transverse magnetic polarization will be

used to deduce the final set of generalized equations for

electromagnetic waves of different polarizations. The

derivation for the transverse magnetic polarization is based

on the same assumption as for that of the transverse electric

polarization.

62

V. TRANSVERSE MAGNETIC POLARIZATION

Let us assume that a transverse magnetic uniform plane

travelling in the positive x-direction is normally incident on

the cylindrical target as shown in the Figure 3. The incident

electric field can be written as in equation (21) below

HiEia

σ = ∞

y

z

x

Figure 3. Uniform plane wave incident on circular conducting

cylinder Transverse Magnetic

' cos

0ˆ ˆi i j

z z zE a E a E eβρ− Φ= = (21)

Equation (21) can be written in terms of Bessel’s

functions as

'

0

'

0

0

ˆ ( )

ˆ ( ) ( )cos( )

i n jn

z z n

n

n

z n n

n

E E a j J e

E a j J n

φβρ

ε βρ φ

∞−

=−∞

∞

=

=

= −

∑

∑ (22)

The term '

0E on equation (22) is

'

0 0 sinE E θ= and the

prime has been used just for representation. The

corresponding magnetic field components can be obtained

using the Maxwell’s Faraday equation E j Hω∇× = −

E j Hωµ∇× = − (23)

Solving the curl of the equation(23), the magnetic

components for the ρ and φ components will be obtained as shown in the equations (24) and (25) below

( )'

10

1 1

1

i

i z

n jn

n

n

EH

j

Enj J e

j

ρ

φ

ωµ ρ φ

βρωµ ρ

∞− +

=−∞

∂= ∂

−= ∑

(24)

'

01( )

i

i n jnz

n

n

EEH j J e

j j

φφ

ββρ

ωµ ρ ωµ

∞−

=−∞

∂′= =

∂ ∑ (25)

In a perfect electric conductor the total electric field is

given according to equation

(2)

0ˆ ˆ ( )t i s s

z z z n n

n

E E E a E a E c H βρ∞

=−∞

= + = = ∑ .

Since the scattered fields travel in the outward

direction, they must be represented by cylindrical travelling

wave functions. Thus sE is represented is

(2)

0ˆ ˆ ( )s s

z z z n n

n

E a E a E c H βρ∞

=−∞

= = ∑ (26)

Where n

C represents the unknown amplitude coefficients

which are found by applying the boundary conditions to the

total electric field, similar to that applied for transverse

electric polarization. The amplitude coefficients nc are

given by the equation (27) below

(2)

( )

( )

n jnn

n

n

J ac j e

H a

φββ

−= − (27)

Thus the scattered field of equation (26) reduces to

(2)

0 (2)

( )( )

( )

s n jnn

z n

n n

J aE E j H e

H a

φββρ

β

∞−

=−∞

= − ∑

(2)

0 (2)0

( )( ) ( ) cos

( )

n n

n n

n n

J aE j H n

H a

βε βρ φ

β

∞

=

= − −∑ (28)

Similarly the corresponding scattered magnetic field

components can be obtained by using Maxwell’s equations

(23) which leads to

1 (2)0

(2)

1 1

( )1( )

( )

s

s z

n jnn

n

n n

EH

j

E J anj H e

j H a

ρ

φ

ωµ ρ φβ

βρωµ ρ β

∞− +

=−∞

∂−=

∂

= ∑ (29)

(2)0

(2)

1

( )( )

( )

s

s z

n jnn

n

n n

EH

j

E J aj H e

j H a

φ

φ

ωµ ρβ β

βρωµ β

∞−

=−∞

∂=

∂

−′= ∑

(30)

0 0H E

εµ

= (31)

Substituting the term 0

H from equation (31) the final set

of scattered electric and magnetic field components gathered

from the transverse electric and magnetic polarizations are

given by the equations from (32)through (37)

1 (2)0

(2)

cos ( )1( )

( )

s n jnn

n

n n

E J aE nj H e

j H a

φρ

θ βεβρ

ωε µ ρ β

∞− +

′=−∞

′−= ∑

(32)

(2)0

(2)

cos ( )( )

( )

s n jnn

n

nn

E J aE j H e

j H a

φθ

β θ βεβρ

ωε µ β

∞−

=−∞

′′=

′∑ (33)

(2)

0 (2)0

( )( ) ( ) cos

( )

s n n

z n n

n n

J aE E j H n

H a

βε βρ φ

β

∞

=

= − −∑ (34)

1 (2)0

(2)

( )1( )

( )

s n jnn

n

n n

E J aH nj H e

j H a

φρ

ββρ

ωµ ρ β

∞− +

=−∞

= ∑ (35)

(2)0

(2)

( )( )

( )

s n jnn

n

n n

E J aH j H e

j H a

φφ

β ββρ

ωµ β

∞−

=−∞

−′= ∑ (36)

(2)

0(2)

( )( )

( )

s n jnn

z n

nn

J aH H j H e

H a

φββρ

β

∞−

=−∞

′= −

′∑ (37)

The magnitude of the total scattered electric field

sE can be obtained by using the following equation

63

( ) ( ) ( )2 2 2s s s

zE E Eρ φ= + +s

E (38)

The magnitude of the total scattered electric field for

different angles of polarization are shown in Figures 4-7.

VI. RESULTS

For simulation purposes the target was assumed to be a

cylinder standing on its base. For a maximum reflection a

vertical polarized waveform should yield better results as

compared with horizontal polarization. This means that the

simulations should show an increment in the signal strength

while going from zero to ninety degrees and the inverse

should be observed from ninety to zero degrees. Graphs for

a normalized scattered electric field have been plotted for

different angles starting from 0 degrees.

The plots for the angles of polarization 0, 45, 90 and 150

degrees are shown in Figures 4-7. From the graphs it can be

observed that the amplitude increases with the increase in

the angle from 0 to 90 and the maximum occurs at 90

degrees. As the cylinder for this simulation has been

considered to be standing on its base oriented in the vertical

direction with reference to its axis, the plots should yield

more signal strength for any angle closer to 090 or it odd

multiples. This proves that the equations derived satisfy the

condition, that when a target is oriented in a particular

direction, the corresponding polarization would yield

optimum results.

VI. CONCLUSIONS

This paper analyzes how different angles of polarization

affect the scattered electric field for a perfect cylindrical

conductor. The analysis employs different techniques from

electromagnetic theory. A mathematical model for the

scattered electric field was obtained by using Maxwell’s

equations and the transverse electric and the transverse

magnetic modes. The scattered electric field was simulated

and analyzed for different angles of polarization using this

mathematical model in MATLAB.

As stated previously, it was assumed that the target

cylinder was standing on its base with its main axis oriented

in the vertical direction. As the angle of polarization was

increased from 0 degrees to 90 degrees it was evident how

the amplitude of the scattered electric field increases. On the

other hand, as the polarization angle was increased from 90

degrees to 180 degrees a decrease in the amplitude of the

scattered electric field was observed. This shows that in

order to receive maximum amplitude of the scattered

electric field there needs to be a proper alignment between

the target’s orientation and the angle of polarization.

This work can be directly applied to the design of radar

systems. However, more research is required on how this

system is going to affect the processing of the reflected

signal. First, the polarization angle of an electric field

completely depends on the physical orientation of the

antenna. Therefore, if the antenna is allowed to rotate on the

horizontal direction this will create electric fields with

different angles of polarization. As a consequence, if the

radar system transmits with more than one angle of

polarization it will ensure that at least one of those angles

will provide the maximum amplitude for the scattered

electric field. An issue in this case of rotating the antenna

accurately, in a small time frame between the successive

transmissions, can be solved by using a microcontroller and

stepper motor combination. Second, the type of image

processing technique that could be employed to create a

composite image from different signals needs to be

investigated. Third, the performance of the system for

moving targets needs to be analyzed. All these are

challenges that could be addressed by further analysis of this

system, by conducting physical experiments.

VII. REFERENCES

[1] John B. McKinney, “Radar: A Case History of an Innovation,” IEEE

A&E Systems Magazine, vol. 21, no. 8, pp. 13-15, August 2006.

[2] C. A. Balanis, Advanced Engineering Electromagnetics, New York: John Wiley & Sons, 1989.

[3] Eric Pottier, “Radar Polarimetry towards a future standardization,” Annals of Telecommunications, vol. 54, no. 1-2, pp. 137-141, January

1999.

[4] George B. Arfken, Hans J. Weber, Mathematical Methods for

Physicists. New York: Harcourt Academic Press, Fifth Edition, 2001.

[5] Changhui Li, George W. Kattawar, and Peng-Wang Zhai, “Electric and

magnetic energy density distributions inside and outside dielectric

particles illuminated by a plane electromagnetic wave,” Optical Society of America, vol. 13, no. 12, pp. 4554-4559, 13 June 2005.

[6] Constantine A. Balanis, Antenna Theory: Analysis and Design. New Jersey: Wiley-Interscience, Third Edition, 2005.

-4 -3 -2 -1 0 1 2 3 40

1

2

3

4

5

6

7

8

9x 10

-5

The Angle Phi The Angle Phi The Angle Phi The Angle Phi ΦΦΦΦ

Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A

ngle θθ θθ=0 degre

es

=0 degre

es

=0 degre

es

=0 degre

es

THE MAGNITUDE OF THE TOTAL SCATTERED ELECTRIC FIELD FOR ANGLE OF 0 DEGREESTHE MAGNITUDE OF THE TOTAL SCATTERED ELECTRIC FIELD FOR ANGLE OF 0 DEGREESTHE MAGNITUDE OF THE TOTAL SCATTERED ELECTRIC FIELD FOR ANGLE OF 0 DEGREESTHE MAGNITUDE OF THE TOTAL SCATTERED ELECTRIC FIELD FOR ANGLE OF 0 DEGREES

0Deg0Deg0Deg0Deg

Figure 4. Plot of scattered electric field for an angle of 0 degrees

64

-4 -3 -2 -1 0 1 2 3 40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09


The N

orm

alized B

ackscatter For Differe

nt Angles

The N

orm

alized B


nt Angles

The N

orm

alized B


nt Angles

The N

orm

alized B


nt Angles θθ θθ


45 Deg45 Deg45 Deg45 Deg


-4 -3 -2 -1 0 1 2 3 40

0.02

0.04

0.06

0.08

0.1

0.12


Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A


es

=90 degre

es

=90 degre

es

=90 degre

es


THE MAGNITUDE OF THE TOTAL SCATTERED ELECTRIC FIELD FOR ANGLE OF 90 DEGREES


-4 -3 -2 -1 0 1 2 3 40

0.01

0.02

0.03

0.04

0.05

0.06


Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A

ngle

Amplitude of the B

ackscatter for A


es

=150 degre

es

=150 degre

es

=150 degre

es




65

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