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ASYMPTOTIC NEAR-TO-FAR-ZONE TRANSFORMATION
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CONFORMAL ANTENNAS EMBEDDED IN CANONICAL
STRUCTURES

presented by

JORGE M VILLA-GIRON

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ASYMPTOTIC NEAR-TO-FAR-ZONE
TRANSFORMATION FOR PERIODIC CONFORMAL
ANTENNAS EMBEDDED IN CANONICAL
STRUCTURES

By

Jorge M. Villa-Giron

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Electrical and Computer Engineering

2008

ABSTRACT

ASYMPTOTIC NEAR-TO—FAR-ZONE TRANSFORMATION FOR
PERIODIC CONFORMAL ANTENNAS EMBEDDED IN
CANONICAL STRUCTURES
By
Jorge M. Villa-Giron

Conformal antennas are important to the aerospace community because of their
aerodynamic characteristics and their versatility for electronic scanning. Computa-
tional electromagnetic methods such as the Finite Element-Boundary Integral method
have been used extensively to obtain estimations of radiation and scattering perfor-
mance of antennas on planar, elliptical and prolate spheroid surfaces. Typically, in
formulating these methods, either an infinite structure approximation or reciprocity
has been used to accomplish the near-to-far-zone transformation. At times, the need
for such transformation has been ignored all-together. In cases where a Green’s
function-that enforced cylinder boundary conditions—was used, calculations of the
far-zone field in the paraxial region were inaccurate. Several researchers have been
working in obtaining integral solutions that overcome the problems in the paraxial
and shadow zone using GTD and UTD techniques.

In this dissertation, an asymptotic periodic dyadic Green’s function will be de—
rived. A different asymptotic approximation for the periodic Green’s function will be
used to accomplish the near-to—far-zone transformation. This results will be validated

by testing expression for large radii against similar results for planar structures.

To My Inspirations, Sebastian and Nivia

iii

ACKNOWLEDGMENTS

I would like to thank several people who have helped me through the development
of my PhD. First, I would like to mention three key people and acknowledge them
for helping me during my doctoral work and were integral part of my development.
Special thanks to my three advisers: Dr. B. Shanker, Dr. Lee Kempel, and Dr.
Edward Rothwell. Thank you so much Shanker for teaching me how to think critically,
for encouraging me when I needed it the most, for teaching me electromagnetic theory;
you do not imagine how excited and pleased I was while sitting in your class listening
to you. Many thanks to you because not only you shared your knowledge, but also
your friendship; thank you for sharing the secrets of the Russian beer and the spicy
food, thank you for inviting me into your house, and for helping me to discover how
good I am in sports, jogging and basketball. These are memories I will never forget.
I do not know how to say thank you to you, Lee, I have never seen a person as
pro—student as you are. My family and I appreciate everything you have done for us.
You are one of the great role models I want to follow everyday and you are a great
example of becoming a well balanced human being that can be not only a successful
researcher, but also a manager. Dr Rothwell thanks for being there for me; being a lab
instructor for you was an invaluable experience for me and it allowed me to develop
a different way to view the electromagnetic theory; your comments were extremely
helpful to understand deeply my research topic.

Second, I thank the opportunity of being part of MSU’s Computational Elec-

tromagnetic Group during my degree; sharing time and space with great persons is

iv

invaluable. Many thanks to Dr. Kobidze and Dr. Gao were always there to answer
basic electromagnetic questions and wiling to help. Thanks to my friend Dr. Pedro
Barba for dedicating time explaining the principles of computational electromagnet-
ics, I enjoyed the time we spent at Beaners discussing finite elements topics. Thanks
to Dr. Chuan for helping me debug my codes and for showing what is in reality a
hardworker. N aveen and Vikram, thanks for sharing your culture and knowledge with
me, I admire you very much. Thanks Andrew Baczewski and Ozgur Tuncer for your
sincere friendship and for sharing great times at RCE B-102.

I am extremely grateful for the assistance of Dr. Barbara O’Kelly and Dr. Pierce
Pierre for providing me an opportunity to be part of this great institution. You helped
me begin to realize my biggest dream, become an expert in a technical area and earn
a PhD degree from a national recognized institution. Thank you so much for all the
tools you provided me for my success and thanks to you I have never worried about
money during my PhD. That helped a lot. Thank you Barbara for you friendship
and always looking after me along with Dr. Pierre.

My dissertation would have not been possible without the unconditional support
of my family. I can not find words to express my infinite gratitude to my parents:
Dumar Villa and Leonor Giron, my son Sebastian and my wife Nivia for supporting
and understanding how important this was for me. You all helped me accomplish my

biggest dream.

TABLE OF CONTENTS

LIST OF FIGURES ................................ vii

KEY TO SYMBOLS AND ABBREVIATIONS ................. ix

CHAPTER 1

Introduction ..................................... 1
1.1 Literature Review ............................. 2

1.2 Asymptotic Solution for the Electromagnetic Fields due to an Aperture

on a Circular Cylinder using UTD .................... 6
1.3 Electromagnetic Fields in the Shadow and Transition Region ..... 8
CHAPTER 2
Electromagnetic fields in terms of Green’s Functions .............. 17
2.1 Derivation of Green’s Functions ..................... 20
2.2 Approximation of Green’s Function On—Surface ............ 33
2.3 Approximation of Green’s Function in the Far Zone .......... 40
CHAPTER 3
Modified Modal Solution for the Dyadic Green’s Function ........... 63
3.1 Exact Modal and Asymptotic Solutions ................. 64
3.2 Numerical Results and Discussions ................... 70
3.3 Singly Surface Periodic Structure in 2 .................. 80
CHAPTER 4
Conclusions and Future Work ........................... 93
BIBLIOGRAPHY ................................. 96

vi

Figure 1.1
Figure 1.2
Figure 1.3

Figure 1.4
Figure 1.5
Figure 1.6
Figure 1.7
Figure 2.1
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

LIST OF FIGURES

Diffraction of surface rays ......................
Infinite circular cylinder with an aperture source M ........

Shadow, transition and illuminated regions adjacent to a magnetic
dipole on a perfectly conducting , convex surface .........

Strip formed by surface rays .....................
Coordinate system of a surface diffracted ray. ...........
Diffracted Wave front ........................
Spread of a surface diffracted ray ..................
Watson transform integration contour c formed by paths c1 and 02
Complex Icz plane ...........................
Contour of integration dictated by the radiation condition

Definition of the Branch Cut ....................
Integration around the Branch Cut .................

Magnitude convergence for exact modal solution and modified
modal solution ............................

Phase convergence for exact modal solution and modified modal
solution ................................

(a) Magnitude for exact. modal solution and modified modal solu-
tion; (b) Relative Error; 77. = 3 and R = 50A ............

(a) Magnitude for exact modal solution and modified modal solu—
tion; (b) Relative Error; 77. = 3 and R = 100A ...........

(a) Phase for exact modal solution and modified modal solution;

(b) Relative Error; 71 = 3 and R = 50A ...............

(a) Phase for exact modal solution and modified modal solution;
(b) Relative Error; 77. = 3 and R = 100A ..............

Magnitude convergence for modified modal solution and steepest
descent solution ............................

Phase convergence for modified modal solution and steepest de-
scent solution .............................

(a) Magnitude for modified modal solution and steepest descent
solution; (b) Relative error; 72. = 10, R = 50A ............

(a) Magnitude for modified modal solution and steepest descent
solution; (1)) Relative error; it = 10, R = 100A ...........

vii

8
11
13
14
35
65
66
67
68

72

74

75

76

77

79

85

87

88

89

Figure 3.15 (a) Phase for modified modal solution and steepest descent solu-

tion; (b) Relative error; n = 10, R = 50A .............. 90
Figure 3.16 (a) Phase for modified modal solution and steepest descent solu-

tion; (b) Relative error; 72. = 10, R = 100A ............. 91
Figure 3.17 Singly surface periodic structure in 2 ................ 92

viii

KEY TO SYMBOLS AND ABBREVIATIONS

CEM: Computational Electromagnetics

FE—BI: Finite Element-Boundary Integral

GTD: Geometric Theory of Diffraction

UTD: Uniform Theory of Diffraction

EMS: Exact Modal Solution

MMS: Modified Modal Solution

ix

CHAPTER 1

INTRODUCTION

The interest in developing antennas that can be mounted on structures with a convex
surface such as airplanes, automobiles, cell phones, or laptops in mobile communica-
tion systems, has increased significantly over the past few decades. These types of
antennas are known as conformal antennas. Consequently, studies to understand ra-
diation of these antennas in the presence of mounting platform has increased as well.
The analysis of these antennas is critical as the performance “in situ” is very differ-
ent from that of a stand alone structure. These antennas offer several advantages; a
potential listing is given next. Conformal antennas can help the aerodynamics of the
structure on which it is mounted and reduces its cost and weight [1]. Arrays of these
antennas permit faster electronic scan than a traditional mechanical scan. Compared
to mechanically scanned arrays, small aerodynamic drag and less space occupancy
are advantages of conformal arrays [2, 3].

Canonical shapes such as cylinders and prolate spheroids are acceptable approxi-
mations to surface that the antennas are typically mounted up on, and can be used
to model structures with singly and doubly surfaces, respectively [4, 5]. One analytic
solution to fields due to sources radiating in the presence of canonical shapes exist.
Computational electromagnetic methods have also been used to obtain an approxi-
mate solution to the problem [6, 7]. These techniques have only been efficient when
analyzing electrically small structures, or arrays with small number of elements [8, 9].

This thesis proceeds along the following lines: Different methods for analyzing
high frequencies methods will be presented in the reminder of this chapter. The
Geometric Theory of Diffraction (GTD) is a technique that finds a solution for high

frequencies [10]. A complete derivation of the GTD method for a circular cylinder,

developed by Keller [10], will be expanded in Section 1.2.

The canonical and asymptotic Green’s function for a circular cylinder was obtained
and its procedure is described in Chapter 2. After this, the function was modified to
increase efficiency at high frequencies on the surface of the cylinder and also in the
far field zone using Watson’s technique [11]. Finally the resultant integral equation
is evaluated by the method of steepest descent. It is important to mention the final
expression is not valid in the paraxial zone [12, 13, 14, 15], when the observation angle
is close to the surface but far away from the source (6 = 0 or 6 = 7r and R > 10A).

Chapter 3 contains the derivation of a modified modal solution. Validation of this
solution against the exact modal solution for an infinite circular cylinder of a small
radius (a = 0.0”) is shown. A comparison between the steepest descent solution and
the modified modal solution for the infinite circular cylinder of big radius (a = 10A)
is presented. A new asymptotic solution for an elliptical cylinder is derived, based
in Keller’s method and using the modified modal solution for a circular cylinder
matching the GTD approximation constants. Conclusions and future work will be

presented in Chapter 4.

1 . 1 Literature Review

Modeling conformal antennas is a challenge. Not only it is necessary to model the
antenna “per se” but the platform that it is mounted upon as well. Several approx-
imate methods have been proposed to analyze these antennas, and these depend of
its platform and the behavior of the antenna. The first studies assumed, that the
antenna was mounted on an infinite circular cylinder [16]. The fields were expressed
in terms of infinite Fourier series of the form 2m Cm cos(m¢), with Cm as a func-
tion of the radius, p (in cylindrical coordinates). Wait and Kahana [17] obtained the
radiation pattern for circumferential half-wave slots with ka 2 2, 3, 5, and for differ—

ent elevation angles. Similar results were obtained by Bailin [18] for large circular

cylinders. The solutions are obtained in terms of harmonic series using integer order
Bessel functions. This solution is poorly convergent for large arguments of the Bessel
functions (when ka >> 1).

Watson [19] developed a method for large values of ka, where he transformed the
poor converging harmonic series into a contour integral, deforming the integration
path for capturing individual terms poles which represent creeping waves. This new
contour integral is expressed in the form of infinite but rapid convergent series. The
problem with Watson’s method is that it can be only used for some canonical shapes.
For a slotted cylinder antenna, the residue series proposed by Watson is highly con-
vergent in the direction away from the slot; known as the deep or shadow region.
However, better results were obtained for the region forward to the slot, or the illu—
minated region, when using physical and geometrical optics [20, 21]. These methods,
by Watson, assumes that there are no surface currents in the deep or shadow region
and also approximate the induced current in the illuminated region by the current
that would be induced on the local tangent plane. While good results were obtained
at the illuminated region, incorrect results were obtained from the shadow and the
transition region, which is the boundary between the shadow and illuminated region.
With the assumption that fields propagate along rays Keller developed the Geomet-
rical Theory of Diffraction (GTD) [10]. This theory includes the effects of diffraction
which are not considered in the geometrical optics.

Diffraction happens when an incident ray is tangential to a convex surface, or
when it hits edges, or vertices of boundary surfaces, creating new rays called diffracted
rays, see Figure 1.1. The total field, defined in the geometrical optics, is a sum of
the incident and reflected rays. A diffraction coefficient can also be obtained and
used to obtain the diffracted ray by multiplying the incident ray with the diffraction
coefficient [10, 22]. This coefficient depends on the type of structure or material the

ray is hitting upon or traveling on, for instance, in Figure 1.1, r’ is the surface ray

S)

        
 

Shed Ray 1 Shed Ray 2

)r' r” /"

Diffracted Rays r3

 
   

Surface

Figure 1.1. Diffraction of surface rays

vector launched at the source point, r,- for i = 1, 2, 3... are the shed rays and ft is the
unit vector normal to the surface at the diffraction points. Laws for geometrical optics
were modified and extended at the new GTD. The generalized Fermat’s principle is
one of them. The modified principle says the least time path is the same minimum
distance path. It emphasizes that a surface-diffracted ray between two points is
a curve where the optical length is stationary among all curves between the same
points having an are on the boundary surface [10]. The optical length then defined as
the product of the geometrical distance and the refractive index (assuming a constant
refractive index).

An extension in the GTD was made by Pathak et al. [23] and he called this
new method Uniform GTD (UTD) [24]. This method introduces a new term, dyadic
torsion factor [25]. This includes the torsional surface rays that may be excited by
apertures and monopoles. All the ray fields are expressed in terms of Fock func-
tions improving the solution in the transition region including the boundary with the
shadow region, where the GT D failed [26], and reducing it geometric optics in the lit
0r illuminated region.

The incremental fields, dEm(r|r’) and de(r]r') are excited by the aperture.
These surface fields travel along the surface. The path they follow is known as the
geodesic or shortest path of propagation. It is described by a partial differential

equation theory for rays on surfaces for homogeneous medium, fulfilling the extended

Fermat’s principle [10]. The dyadic torsion factor, TF(r|r’), proposed by Pathak,
relationes the incremental fields, alEm(r|r’) and de(r|r'), where F is either E or H
depending of the problem being solved, and the differential magnetic current M by

[23]
- . — "1:3
15mm -TF(r|r')Dfe J

dF ’=
(rlr) 47, S

 

(1.1)

where dF(r|r’) is either dEm(r|r’) or de(r|r'), k is the wave number, the geodesic
path between the source and the observation point is described by s. The surface ray
divergence factor, Df, represents the change in the width of the surface ray strip, and

is given by
84140
PC Cit/1

 

Df= (1.2)

The term dilio is the angle between the surface rays adjacent to the central surface
ray from r’ to r. dw is the angle between the backward tangent rays to the adjacent
rays at the observation point, r. The distance between re and r is called the tangent
(or geodesic) radius of curvature of the geodesic circle at r [25].

The dyadic torsion factor, T(r|r’), was introduced because the field on the surface
is not just the surface ray field between the source and the observation point. It is
a function of the launching of the surface ray field at the source, the variation of the
surface ray field from r’ to r, and its attachment to the surface. All these additional

effects are modeled by the dyadic torsion factor, T(r|r’) and is defined as:
T(r|r’) = T1 ff, + Tgff), + T3ffi/ + T4fif" + T560, + T6011, + T7711? + Tgfzf)’ + Tg'ftfil (1.3)

where 73. is unit vector normal to the surface at any point, f a tangential unit vector
to the surface, and f) as the cross product between f and 73., named the binormal unit

vector. For a differential magnetic current that is tangential to the convex surface,

(1.3) reduces to
fimh=nfi+nW+nw+QW+RMHrma (M)
as dM(r) - 7’7. 2 0.

1.2 Asymptotic Solution for the Electromagnetic Fields due to an Aper-

ture on a Circular Cylinder using UTD

Next, a method to find the field produced by a magnetic current due to an aperture
on an uniformly circular cylinder is presented. The problem that we will attempt to
solve is to find the electromagnetic field produced by an aperture in the convex surface
of an infinite circular cylinder. It is to assumed the external medium is free space.

The problem is shown in Figure 1.2. It is known that the tangential component of the

 

 

\ ffic __
V1
au=axdE V
C»E§ [afi
Surface
VWV

Figure 1.2. Infinite circular cylinder with an aperture source M

electric field E, is zero everywhere on the cylinder surface except at the aperture. The
tangential component of E on the aperture S, can be expressed in terms of differential

magnetic current M as in (1.5). It is assumed that this current is known and is given

dM(r’) = Ea(r’) x mm (1.5)

Here, (IA is an element of area in the aperture, Ea(r’) represents the electric field at a
point r’ inside the aperture. Electric and magnetic fields can be found at some point r
outside the surface then integrating the incremental fields dEm(r|r’) and de(r|r’),

over the aperture as in 1.7.

Em(r)=//dEm(r]r') (1.6)
Sa

Hm(r) =//de(r]r’) (1.7)
So.

This equations is evaluated using UTD to find the field. Figure 1.3 shows how
the differential magnetic current dM, together with the tangent plane at this point,
divides the area in two regions. These regions are the shadow and the illuminated
regions, with a transition region separating them. UTD’s expression for electric field
is obtained for both, shadow and illuminated regions, and also provides a smooth
transition between the shadow and the illuminated region. In the next three subsec-

tions, the procedure to obtain expressions for the fields in each region is explained.

Illuminated Region

Transition
Region

Shadow
Region

 

Figure 1.3. Shadow, transition and illuminated regions adjacent to a magnetic dipole
on a perfectly conducting , convex surface

1.3 Electromagnetic Fields in the Shadow and Transition Region

The electric fields in the shadow region are expressed in terms of rays. These rays
are excited by the magnetic current source (1M, and they propagate along the surface
from the lit or illuminated region into the deep or shadow region. The surface ray
has associated an amplitude A(s) and a phase ¢(8) which varies with space, and can

be expressed as

a(s) = A(s) exp[j(q§0 — 155)] exp(jwt) (1.8)

The principle of conservation of energy, applied to a narrow band is used to de-
termine A(s) along the surface [27]. Due to the diffraction, the surface ray sheds
energy as they move, meaning that surface ray diffracts tangentially, decaying as it
travels in the direction of propagation [23]. Figure 1.1 depicts this effect. The energy
is proportional to A(s) and to the cross section area at s. For this case it would be
just the width 10(5) of the band, as in Figure 1.4. The energy between two points

inside the strip is
As

/m

Figure 1.4. Strip formed by surface rays

A2(s + As)w(s + As) — A2(s)w(s) = —2a(s)A2(s)w(s)As (1.9)

The energy lost, due to shedding is described by the factor 2a(s). It can be shown

that the derivative of A(s) is

d

dslz42(s)w(s)l ——- —2a<s)[.42(s)w(s)] (1.10)

Integrating with respect to s from 30 to s

 

 

MW) dl42( '> (’)I 8

x S If) 8 = _2 3, d3,

/ IAQ<s'>w<s'>J / a‘ ’

A2(80)w(80) SO

A2(s)w(s> 3 , ,

:> In A2(80)1U(80)] —280 0(5 MS

———> A(s) = A(30) %:%)-exp -—/a(s’)d3’ (1.11)
30

Suppressing the time dependence and solving it at 3 equal to 50 results in

14(50) = a(80) emf-flee - ksoll (11?)

Substituting (1.12) in (1.11)

 

A(s) = 0(80) eXI)(-j(¢o — a,» 1(3)) exp — / a<s’>ds’ (1.13)
30

Using (1.13) and (1.8), it can be shown that

 

a(s)=a<so)exp<-1(¢o—kso)) ’“(So’exp -/a<s'>ds’ expWO—ks»
111(3) 30
111(30) 3

=> a(s) = a(sO)

 

exp(—jk(s—so))exp -—/a(s')ds’
30

111(5)
(1.14)

The width of the strip can be written in terms of arc length, Aw(s) = (177(3). The
arc length is determined by the distance 3 and the angle, (11/10, formed by the surface

rays as in

8
d)"
a(s)=a(sO)\/%. fiexp —jks—/a(s’)ds’ (1.15)
30

The dependence of the above expression on 30 is assumed by writing lim0 (1(30), /s =
80—)

C’. Using C" in (1.15) it is obtained

(1.16)

 

The proportional constant CI represents the strength of the source. The next step is
to associate a direction to the surface ray and to relate it directly with the current
source. Because this method should work for any shape, the canonical coordinate
systems already defined cannot be used at this moment. The coordinate system that
is used here is composed by an unit vector ft that is normal to the surface at any point,
a tangential unit vector to the surface f that points to the direction of propagation

of the surface ray, as it is shown in Figure 1.5. To complete the set of our coordinate

10

system, we must define an unit vector perpendicular both to ft and also to f. This
unit vector is known as the binormal unit vector, and it is defined as b = A x it
As before, 6, 15,71 represent the observation point and f)’, f’,ft’ the source point. The

source (1M is expressed in terms of the new coordinate system as follows

dM =13’(i/-d1x1)+£’(£’.d1v1) (1.17)

From the analysis of canonical problems, it is found that the source current excites an

 

Figure 1.5. Coordinate system of a surface diffracted ray.

infinite number of modes for the normal component of field and another set of infinite
modes for the tangential component of the field. Those sets are independent of each
other, and they satisfy the different boundary condition. This definition allows us to

define C" in terms of the boundary condition for a single mode, as shown in (1.18).

(3;, = CLS‘S(30) [13’ - M + i’ - M] (1.18)

Here, Lg’s denotes the launching coefficient, and it depends on the boundary condi-

tion it satisfies and on mode. The constant C does not depend of the mode or the
boundary condition. This constant is used to fix the final expression with the one
obtained from the canonical problem.

The normal component of the electric field, ft - dE, has to satisfy the N ewmann

boundary condition, and is

S

- . d.)
it. dEp = CLg(so) [b’ - dM + tI - leI] II—cffy—O exp —jks — /ag(s’)ds’ (1.19)
0
where Ep is the contribution of a single p mode of the total electric field E. The
tangential component of the electric field (3 - dE, that is also perpendicular to the
surface ray trajectory, f, is obtained by satisfying the Dirichlet boundary condition

as in (1.20).
d s
f)- dEp = CLI‘SXSO) 13,-(1M +13, - dM] “Bigexp —jks —/01]3,(s')dsI (1.20)
0

It is obvious that this component vanishes on the surface; however, this term is
important to calculate the electric field outside the surface. The total electric field

on any point r on the surface due to a source point r’ is written as
alE(r|r’) = fidEn(r|r') +13dEd(r|r’) (1.21)

With (1.21), dB is calculated for any given point on the surface, and using GTD
dB is calculated for points outside the surface.

Keller [10] developed the GTD and shows that the wavefront of a surface diffracted
field in any point r3 outside the surface, can be represented in terms of other wavefront

diffracted field at some point r0. This is presented in Figure 1.6. The relation between

12

wavefronts is given by

 

pdpd -
dE(r3]r’) ~dE(ro|r’) d 1 2d e—Jkso (1.22)
(p1 + sax/22 + so)

 

    

Diffracted
Wavefront
“ / 4
/”fi./ _.
a. / _ I;
’2
Diffracted Ray
Strips
—i'
I”

Figure 1.6. Diffracted Wave front

By moving the reference point ro to the diffraction point r, Pathak [2] relates
dE(rs|r’) directly to the source dIVI. As shown in Figure 1.6 and Figure 1.7, when

ro —> r, p611 —-> 0, p51 —> pc and so —> s, the following expression is obtained
lim (/p‘11dE(ro|r’) ~ dE(r|r')D(r) (1.23)
p‘f->0

where D(r) is the attachment coefficient when the field is shedding or escaping from
the surface object or it is defined as the diffraction coefficient when the field is tan-
gentially hitting the surface object. This is due to the reciprocity, which states a
source M located at 1" produces a field in ro equal to the field at ro when the source

is place at ro. The coefficient D(r) depends on two factors: the nature of the field

13

      
   

l'c

Source

\l' _,.

Surface Ray
Strip

Figure 1.7. Spread of a surface diffracted ray

and the infinitesimal area around the object. Equation (1.23) can be rewritten as
lim (/p‘ide(ro|rl) ~ dM(r’) -T(r|r') (1.24)
[ff—+0
replacing (1.24) into (1.22)
dE(r3]r’) ~ L(r|r’) __pc_ 18—ij (1.25)
3(90 + 3)

where L(r|r’) is a linear transfer function, and it is defined as
L(r|r’) ~ dM(r') .T'(r|r’) (1.26)

T(r]r’) is the dyadic transfer function and using (1.19), (1.20), (1.21), (1.23), (1.24),

14

and (1.26) can be written as
d /'
TiJ-(rlr’) ~ CZLp(rI)Dp(r) —’*’—0exp —jksD — / ap(s’)ds’ (1.27)
1) d7)

where i and j are the f, f). and 6 components defined before for source and observation
points. Equation (1.25) can be expressed in terms of the source point leI and the
dyadic transfer function that relates the physical phenomenons from the diffracted

point to the source point as

dE(r3|r’) ~ dM(r’)-T(r|r’) Eff?) 63—ij (1.28)

The attachment coefficient Dp(r), the attenuation constant ap(s’), and the con—
stant C are obtained by comparing (1.28) to the asymptotic solution of canonical
problems. The asymptotic series expansion demonstrates a highly convergence when
the observation point is in the shadow region, and just few terms are normally needed
to obtain an acceptable accuracy. However, as the observation point moves to the
boundary between the transition and the shadow region more terms will be needed
to obtain the same accuracy. This problem is solved by changing the series repre-
sentation using Fock integral representation [28], which describes creeping waves, as
the observation point moves from the shadow to the transition region. The dyadic

transfer function on ( 1.28) can be written for the TEt case as

T(r]r’) = C [Ta(r’)H5’fz + Tb(r’)Sf'f) + Tc(r')Hl3’I3 + Td(r')Sf’f1]

1
(11110 p9(r) 6 —jks
V d0 ng(r’)I e D (129)

The torsion factor Tk(r’) for k = a,b,c,d, H and S are all related to the hard

 

and soft boundary conditions and satisfy the Robin boundary condition for the TEt

15

case. Pathak et al. [23] found that a circular cylinder Ta(r’) and Tb(r’) are equal
to 1, Td(r’) is equal to zero and T C(r’ ) = sin 2a’/a sin2 0/. Pathak also obtained the
surface radius of curvature pg(r’) = a/ sin2 0/ for a circular cylinder.

In this Chapter, we have obtained semianalytic methods that can be used for
analyzing sources near canonical prefect electrical conducting objects. In the next
chapters, we derive dyadic Green’s functions that may be integrated within differential

equation solvers

16

CHAPTER 2

ELECTROMAGNETIC FIELDS IN TERMS OF GREEN ’8
FUNCTIONS

Unique electric or magnetic fields at any observation point are obtained by solving a
second-order differential equation subject to specific boundary conditions. As said in
Chapter 1, the usual solution for this type of problem is an infinite series, but these
ones usually converge slowly. For that reason, a closed form solution would be useful.

A Green’s function is a solution of the partial differential equation when using a
unit source as the driving function subject to appropriate boundary condition [29].
The solution for the partial differential equation with the actual forcing function, is
then given by the convolution of the Green’s function with the actual forcing function.
Hence, the Green’s function serves as the transfer function of the system.

In this chapter, we explain the manner in which a closed form integral equation
that relates the magnetic field due to an aperture on a circular cylinder may be
obtained, using a dyadic Green’s function. The electromagnetic field, is governed by
the vector wave equation (2.1), the boundary conditions at the surface of the cylinder

and the radiation condition.
VxVxF—%F=0 on

where the vector field F can be either E or H. The dyadic Green’s function G must
solve the same differential equation, but using a unit source as a driving function, as

shown in (2.2)

v x V x E — 1.36 = I6(r — r’) (2.2)

multiplying (2.1) by G and (2.2) by F, subtracting the results and grouping the terms,

17

it gives the differential equation (2.3)
F-(VXVxG)—(VxVxF)-G=F-I6(r—r’) (2.3)
using the identity (2.4)

Vx(AxB)=B-(VXA)—A-(VXB)

B-(VxA)=Vx(A><B)-A~(V><B) (2.4)

with A = V x G and B = F for the first term in the left hand side of (2.3) and using

A = V x F and B = G for the second term, produces
V-VxGxF—V-VXFxG=F~I6(r—r’) (2.5)

Using A x B = —B X A and grouping terms, (2.5) becomes
—V-[FxV><G+VxeG]=F-I6(r—r’) (2.6)

(2.6) is the integrated over the volume exterior to the surface (the region in which

fields are sought)

—/J V-[FxVxG+VxFxG]dV=[V//F-I6(r—r’)dV (2-7)

and using the divergence theorem, (2.7) is converted in (2.8)

—flfi-[FXVXG+VxeG]dS=—F(r’) (2.8)
S

If F is H, using Faraday’s Law for a source free region (V x H = jweoE), (2.8) can

18

be written as

H(r’)=fl'ftxH-VdeS+jweo#fiXE-Gd5 (2.9)

S S
Because the cylinder is a perfect electric conductor (PEC), the Green’s function is
chosen such that the normal derivative V X G is equal to zero (satisfying the Newman
boundary condition) on the surface of the cylinder. Then, the first integral is zero.
Also, as the tangential component of the electric field, ft x E, on the PEG surface
is zero, the only region the integral can be non zero is at the aperture. For these

reasons, (2.9) reduce to
H(r’) = jwco #71 x E - G (15' (2.10)
Sa

where Se is the aperture on the cylinder surface. Due to symmetry of the Green’s

function, (2.10) can be written as in
H(r) = j'wcoflfi x E(r') -G(r’ I r) dS’ (2.11)
Sa

interchanging source and observation points. To relate the dyadic Green’s function
to the torsion factor of Chapter 1, it is recognized that by replacing dF(r) by dH(r)
in (1.1) and with M = ft x E as the magnetic source on the surface, the relation

between the dyadic torsion factor and the dyadic Green’s function is defined as

<T(r, I r) = C(r’ | r) (2.12)

19

2.1 Derivation of Green’s Functions

To relate and derive the relations of the dyadic Green’s function, we should start in-
troducing the concept of dyadic electric and magnetic field and rewriting the Maxwell
equation in terms of these fields. First, there are three orthogonal current distribu-
tions, Ji, with (2' = 1, 2, 3) which produce three sets of harmonically oscillating fields
in the same environment and at the same frequency. In this way, we would have fields
such as E, and H,- [29]. However, if the coordinates :13, y, z are replaced by 231,222,133,

the magnetic and electric fields could be written in the dyadic form

3 3
F: 2:17,]- (2.13)
3:1

' i=1

where F could be either E or H. The dyadic electric field can be represented or called

the electric dyadic Green’s function
E = 68 (2.14)
whereas the magnetic dyadic Green’s function is given as
jwuoH = Gm (2.15)
Normalizing the current moment, it is found that
jwpoj = I6(R — R’) (2.16)
Maxwell’s equations can be rewritten in terms of the new notation as:

V x GeaRlR’) = EmomIR’) (2.17)

20

v x E,,,0(R|R’) = I6(R — R’) + k2—éeO(RIRl) (2.18)
_ 1 _

v . Geo(R|R’) = 32V - [16(R — R’)] (2.19)

V EmaRlR’) = 0 (2.20)

The subscript zero is used to indicate that only the free-space radiation boundary

condition is enforced. The dyadic magnetic Green’s function, Gmo, may be decom-

mo, such that

posed into two terms, €3an and G

_+.. —_
GmO = G'rn.OU(p _ pl) "I" G'mOUmI ‘— p) (2'21)

This is because GmO has a singularity at p = p, where again p’ denotes the source

radius. The unit step functions for cylindrical coordinates are defined as

1, if p > p’
U<p — p’) = (2.22)
0, if p < ,0,
1, if p’ > p
U 01' - p) = (223)
0, if p’ < p
Taking curl is applied to (2.21), results in
_ ——+ —_
V X Gmfl = V X IGmOU(p _ pl) + GmOUU), _ 10)] (2'24)
Applying the identity (2.25)
Vx(aB)=anB—BXVa (2.25)

21

on (2.24) and using A x B = --B x A, yields

__ —-+ “—
VXG-mo = (VxG...0>U<p—p’>+VU(p—p')x6250

+ (v x 57716)U(p’— p) + VUlp’ — p) x 6.7.0 (2.26)

Evaluating the divergence of the step functions, we have

 

_,_.QL_7 “BK ~8U_~ _/
VU(p [0—100 pa¢+za, p5(p p) (53-27)
. —1 .
VUlp’-p)=p5(-1(p—p))3p( p) l—Tlpflp—p') (228)

Substituting (2.27) and (2.28) on (2.26), results in

__ ._.__+_ A _
v x on,0 -_= (v x Gmo)U(p — p’) + pd(p — p’) x 9.7;,
+ (V X C;10)U(PI - p) - r360) - p') X @7710
= (v x stomp — p’) + (v x mat/(12’ — p)

+ 13662 - p’) x (53.0 — 5.7.0) (2.29)

Using the boundary condition for a magnetic field )6 x (Hg-,0 — E1210) = E, in terms

of the dyadic functions, substituting (2.15) and (2.16), we have

72 X (Gfi-IO — 6;;2‘0) = i36(l‘ — I") (2.30)

where I; is the surface idem factor and 6 (r — r’) is the surface delta function. Con-
verting (2.30) to cylindrical coordinates, 75. = p“ and 6(r — r') = 6(6) — (b’)6(z — 2').

Substituting Is by I — {if}, yields
. —+ -—— — . .
p X (GINO — GmO) = (I — PP)6(¢ - ¢,)6(3 ‘ 2,) (2-31)

22

h-‘Iultiplying both sides of (2.31) by 6(p -— p’)

- - —+ —
{Riff} ‘— 10,) X (GmO '— Gm())= (1 "— pp)6(p_ plé) (¢_ Cblé) (Z _ 2,) (2'32)
and substituting (2.32) into (2.29), yields

v x mm = (v x agar/(p — p’) + (v x 6,7,0)U(p’ — ,0) +

(I-p/3)6(p- #5) ((P— W (3- Z') (2-33)

Using (2.18) in (2.33), we have

Mp - p’)5(¢ — 2W»? — 2’) + legit—60 = (V x 6;,0)U(p — p’) +

(I - (3x3)6(p- 9’5) (45- W (Z - 2') + (V X Gall/(H - p) (2-34)

and simplifying (2.34), the expression for the electric dyadic Green’s function for free

space can be written as

GeaRIR’): [(V x Gimp — p’) + (v x G;.0>U<p’ —p> -226<R— R’)l (2.35)

>-
cioll] H

An expression for Geo(R|R') in terms of eigenfunctions must be found, but we will
not do this directly. First, Gmo(R|R')i will be written in terms of eigenfunctions,
and then, the expression will replace Gmo(R]R')i in (2.35). To do this, the curl is

applied to (2.18), obtaining:

v x v x G,,,o(R|R’) = v x [I6(R -— R’)] + 1.3V x 6,0(Rla’) (2.36)

23

and using (2.17) in (2.36), we obtain
V x V x Emoann’) — kgfimomjn’) = V x [I6(R — R’)] (2.37)

Equation (2.37) shows that Gmo(R]R’) is also a solution for the dyadic wave equa-
tion. For this reason, a solution for Gmo(R]R’) could be found in terms of eigen-
functions. The eigenfunctions will be formed on vector wave functions [30]. These
functions are a set of eigenfunctions that can be found using scalar wave functions, cpl
and (p2 as the generation functions. In this work, two kinds of vector wave functions
will be used: M and N. These two, in addition to L, were introduced by Hansen [30].

To construct M, it is assumed that (pl is a solution for the scalar Helmholtz equation
V2e1+ (62901 = 0 (2.38)
and that there is a function F such that

F=wami am)
where 13 is the pilot vector. It can be verified that F satisfies the vector wave equation.
VxVxF—k2F=0 (2.40)

Using (2.39) in (2.40), and factorizing the curl in the equation:
Vxwaxeam—HVerFAi an)

VXWxwam%t%mkfl an)

24

using the identity (2.43)
VxVxA=V(V-A)—V2A

in (2.42) results in

V x IV<V - m) — V299115 — We] = 0
Using V - (aB) = aVB + BVa in (2.44) simplified it as

V X [WWV '13 +15 A $01) — V2<PII3 - 16290113] = 0

Because V - 15 = 0, (2.45) reduces to

V x [Va A m) — V2212 - 627212] = 0

and again to

V x l—ng — 624912] = 0

Factorizing —p from (2.47), it is obtained

V x [(V2m — 6221231 = 0

(2.43)

(2.44)

(2.45)

(2.46)

(2.47)

(2.48)

If (,01 is a solution for the Helmholtz equation (2.38), then (2.48) is zero, which

means that F is a solution for the vector wave equation (2.39). One set of vector

eigenfunctions is given by

M: V X (90113)

25

(2.49)

The other set of vector eigenfunctions is given by

NlexVXWfi) am)

Q

Using (2.50) in the vector wave equation (2.40) and factorizing, it is obtained

1 . 1 -
V x V x [EV x V x (8219)] — k2[EV x V x (8211)] = 0

1
=> V x V x [EV x V x (99213) — 16,9213] = 0 (2.51)
Using identity (2.43) with (2.51), it becomes

1 . . .
V X V X I;(V(V - 9021)) - V2902P) - W214 = 0 (2.52)

applying V - (aB) = aVB + BVa to (2.52), it becomes
1 . . . .
V X V X IE(V(992V ' P + PV¢2I - V2902?) - (69021)] = 0 (53-53)

with V ~13 = 0, (2.53) reduces to

l

k.
1

=> V X V X I—EV2902I3 - W213i = 0

V X V X I (WW2 - V2902?) - W273] = 0

:Vxprgw%s—Hmn==o QM)

As before, if 992 is a solution for the Helmholtz equation (2.38), then (2.54) is zero,

which means that N is also a solution for the vector wave equation. If 991 = 4,22, M

can be written in terms of N by placing (2.49) in (2.50)

1

N=k

V x M (2.55)

26

This can be shown by a rather simple development. The vector wave equation in

terms of M is expressed as
VxVxM—k2M=O (2.56)
Finally, substituting (2.55) into (2.56)
IN x N — k2M = 0 (2.57)

and so

V x N = kM (2.58)

To find an expression for Gmo(R|R’) in terms of vector wave equations, only the
first two vector wave functions, M and N, called the solenoidal vector wave functions
are necessary. Conversely, if we want to find an expression for Geo(R]R'), the three
vector wave functions, M, N and L will be necessary. For convenience, Gmo(RIR’)
is first derived, then Geo(R]R') is represented in terms of Gmo(R|R’). To expand
Gmo in terms of the solenoidal vector wave functions, it is convenient to follow the
Ohm—Rayleigh method. The method is explained in [31] on page 179, or in a more
applied way by [29], renamed as the Gm method. Basically, it says that a source
function, V x [I6(R — R’)], can be written in an expansion by simply finding the

appropriate vector wave functions that enforces the boundary condition.

V x [I6(R — R’)] = /dl.~,, [de Z [N(lcz)A(kz) + M(lcz)B(kz)] (2.59)
0 —oc 71:0

27

Integrating the scalar product between N’(—k.z) and (2.59) over the entire volume,

// N’(—kz) - V x [I6(R — R’)]dV =

[41:12:46.2 // N’< <—-Icz> 1N<kz>A (a M(z)B(kz)ldV (2.60)

7120

and using the orthogonal properties for M(kz) and N (kz) proposed by [29] in page
150, A(kz) and B063) are obtained as

 

A k _ M’ —k,. 2.61

( 2) 47,2; ( ) ( )
2 50 I

B I.» - —-——N —k 2.62

( .) 47121 (- z) ( )

The source function can this be represented as

 

kalef" ’kz) + MszlNl("kz))l

00 00

V x [15( (R— R’ 1:07:1kp /dch :24;
_w 2:0

(2.63)

Then Gmo is written as a function of the terms obtained in (2.63). Furthermore, the

integral with respect to kp can be eliminated. This is because our problem involves

an infinite cylinder [29]. Then function, Gmo is expressed as

I

'1» 0‘? 00

_:t _]a

GmO(RIR,) = Er— / dkz 2
_m :—

[N(2)(kz)M’(—kz) + M(2)(kz)N’(—k;)] , ifp > p’
[N(A~.z)M’<2>(—kz) + MeaN’l’k—ka] . ifp < p’
(2.64)

H

where the superscript “(2) means that the vector wave equation is in terms of second

28

kind Hankel functions. Taking the curl of 67:10 and 5,7,0 we obtain

—jk 00 0° 1
V X G77+10(RIR’) = 87 / dkz Z 3
—oo nz—oo P

{\7 x N(2 )(k )M’—( 712) +V x M(2)(kZ)N'(—kz)} , p > p’ (2.65)

VxG (R|R’)=Zd1~’—" i0: 1

7710 8—71“ 2 (5—2

n=— 00 P
{VxNikaM’W—kawxM<kz)N’( let-.1} , p<p’ (2.66)

replacing (2.55) and (2.58) in (2.65), the following is obtained

—-8jk2
V x G,,_’:_o(R|R)=

 

   

.12
—oo n=—oo kl)

{M<’>(kz)M’<—kz)+N<2><62)N’(—kz>}dkz. p>p’ (2.67)

Using identical steps as before, replacing (2.55) and (2.58) in (2.66), it can be ex-

pressed as

V X G7770(RIR’_) _ -—:28j/ Z k—le

—OO 71:

{MeaMW—ka+N<kz>N’<2> (—k.>}dkz , p<p’ (2.68)

29

Using (2.67) and (2.68) in (2.35), the field expression is obtained

-1' 00 00 1
GeomlR’) = _ki,_,s,33(11 R’) +8— / dkz W: 7.3
—— 7)

"OO

[N(2)(kz)M’(— k )+
+

[NaaM’l’M—k.)
(2.69)

Finally, to obtain the second-kind dyadic Green’s function, Geo, the scattering su-
perposition method can be used [29]. A scattered wave term has to be added to
the free-space dyadic Green’s function previously obtained, as we see in the following

equation

Ee2(R|R’) = a720(RIR') + E23(RIR') (2-70)

This scattered wave term, G83, must satisfy the boundary condition for the specific
problem, where p x V x G62 = 0 at p = a on the cylinder surface. The Neumann
boundary condition can only be satisfied if the vector wave functions from the observa-
tion point are the same as those used in Geo for p < p’, M’I2)(—kz) and N'(2)(—kz).
The radiation condition of outgoing waves can be satisfied also by M(2)(kz) and
N (2)073) from the observation point. The expression for CBS is then deduced from

the expression of Geo as in

-j DC 00 1
G2szIR/)=8—7r / dkz Z F2
7)

{77,114 2~><e>M ’W— r.)+b.,N<2><k.)N’<2>(—k.)} , p>p’ (2.71)

The coefficients (1,, and b" are found by making G82 satisfying the Neumann boundary

30

condition at the cylinder surface
,3 x V x [M + a,-,M(2) + N + b.,,N(2)]p=a = 0’ (2.72)
Next, to obtain the vector wave function, the scalar wave function
WP: 95. Z) = Jnfkalejn¢€jkzz (273)

is used [32]. Using (2.73) into (2.49), which defines one of the two groups of solenoidal

vector wave functions, with pilot vector 13 = 2, the vector wave function M is obtained

M __: V x (;J7)(kpp)ejӢejkzz)

 

_. livfifiga
p04” 0/)

= —— —k (75 2.74
(1,, p33” ” (kpp) ( ’

For convenience, we define .17 = kpp, and rewrite (2.74) as

 

 

 

 

 

_ kp 8(Jn(:r)ej"9’ejk~”z) . ,. 6(J7-L(:r)ej"¢ejkzz) ~
M — — , , P—fvp Cb
:1: 06’) 0:17
2 J'nkpJn(x)€jn¢ejk;zfi_ A: paJn.($)ejn¢ejkzz(/3
CL‘ 81:
-. , , J (:13) . 8.171(1)
= k 3,145 sz/v ___" 2.75
pe e [ ]71 :7: p— 01: ———-¢ ] ( )
Using (2.58) with N = 115V x M, we obtain M, as
1 - nJ (- ). 3_J_n(1‘ )
N = __ k. jmpjjk I: '1
kOV x [ pe ( [j .7: —p— ('31: —qb ]
BJ (.r]
_ kP jnq) ejkz z «0171(33 )~ 15 71"»(1’) A 712.1,),(1‘); 182: {117 ;
— A—e A; 0 ——p—nz———¢+— 2————a——..
A: ..
_____ A_P€jn<be ejk; z [:A,zazl‘+(~1)fi_nkzxilnfl7)¢_ kanfl’fi] (2.76)
,0 .L

31

We define M’(—kz), M(2)(I\iz), N’(—kz), and N(2)(kz) from 2.75 and 2.76 as

 

 

 

 

. . j I 7 .
M’(—k;) = kpe—J'M’Ie—szz’ —j71£’f(—jc—)p'—a—Jfl(—$2(pf (2.77)
J: 8.1:
, , <2) <2) .
M(2)(kz) : kpcjng’oejkzz j‘an (5E)/3_aH77 (xlqb (2.78)
2: 83:
I kp _- C), __ A~ I
N(—k,,) — —e 3” c J ~Z
k0
, BJ 17’ A J,:7:’ ~ A
[—sz S], )p’—n7.~z "2E, )¢-kan(LL‘,)Z, (2.79)
(2) kP Yup 'k z
N (k) —eJ e7 7'
A70
. (2) (2)
. 0H. 7, H ~ 2 .
[ij—Ja—Eflp-nkz ”I(x)¢—-ka,(l)(:r)z (2.80)

Finally, substituting (2.77), (2.78), (2.79) and (2.78) into (2.72), we can find an and

 

bn
a _ —J,.,
,7 _ 2
HRH?)
_af
1777 = _— 87 (2.81)
(2)
6H7} ('7

where ”y = kpa. The complete expression for G25 is obtained by substituting (2.81)

32

into (2.71)

—8J
M<2>(1-)MI<2>(1,)+_LYN<2>(7.,)N'<2>(_1~,)] . p>p’

(2.82)

Evaluating (2.82) and (2.69) in (2.70), and expressing 3H? (, )/8’)’ as Hn( 2),(7) yields
the expression for the electric dyadic Green’s function of the second kind for a perfect

conducting circular cylinder

 

 

 

 

1 0° - 00
_ 'n' , —.k;.-Z—
G62(p,¢,2]pl,¢l,2’) = W 71—2006] (,0 / (111726 J
' —oo
—j-n.H.‘.”(x> in 1:. 2H’é”<a- . kzkaffz)($) ,
(2) , +‘(Zfi) H12) p" " ’ -2 1(2) Z
7$Hn (’7) (7:) VKOHn (’7)
(<2) ,. 2 -. .- <2) .
+ Hn(2)(:r)_ (2::zi2in 7122(1) (15¢; + nkzkpfizzzf-T) 9(:27
V'Hn (A) 07 $Hf1(2))('7_] 7:143an ('1’)
772A78Hff2’h) 124))

where5=<p-q§’and2=z—z.

2.2 Approximation of Green’s Function On-Surface

As it was said in Section 2.1, (2.83) represents the dyadic Green’s function as a series
of eigenfunction. In this section, G28 will be specialized for the case where both the
observation and source points are on the surface. Then, :17 would also be equal to

kpp and p = a, where a is the cylinder radius. Also, using the boundary condition,

33

the radial components of €62 will not be necessary and can be suppressed in (2.83).

Using those conditions, the dyadic Green’s function on the surface can be written as

 

   

 

00
Ge2(p.¢ zla 7’ 2’) 262:: e222 / d7. 72—3222
noo=— —oo
1H____(_7)H',,(2_(n__kz)2_11;1__,nkzka,(,2)() <52,
7 H596) ’“02 7 2+73H35’27’H
71k 21(12)ka . __ 117(3)”) MI
7276371,?) ((3)) 245’ 7 VIZ—DY ————H,(2)(7) zz (2.84)

Equation (2.84) is exact for any cylinder radius. However, for cylinder of large
radius, (2.84) is very slowly convergent. The Hankel function of large order has to
be computed to obtain a reasonable approximation. This order is approximately 2ka
series terms to obtain a good accuracy. To solve this, and to avoid the time consump-
tion by evaluating computationally the Hankel function, the Watson transform is used
[33, 34]. Watson showed that an infinite series of the form E? f(71)e'jmr could be
represented by a complex integral of the form fc( f (17) / sin(v7f)o)(dv. If we assume that
f(v) vanishes at -00 and co the integral can be rewritten as f01+C2 (f(v)/ sin(v7r))dv.
He also proposed that this integral may be evaluated if the residues at each pole are
taken. The surface of the cylinder is part of the shadow region. Watson showed
that in this zone, few terms are needed to obtain an accurate and highly conver-
gent expression [34]. Physically, this method of pole residues represents the creeping

waves, which are launched by the surface ray when it is travelling in the direction of

propagation. The original Watson transform is the first part of (2.85), but it can be

34

modified to be used with the series that we have in (2.84) as follows,

 

2j Z “me—fin” = fo1+c2 sifiEZLflU

0° . -— . J '7‘)
gj Z f(n)ejn¢e-er = / Mm,
—oo c

1+02 sin(v7r)

 

m .4‘.‘ .
. —. . . , JL¢ Jvfl'
2j E N708]we_Jnflejmr = / f(L)€ e dv
--00 61+62

sin(v7r)

oc ' _
, 7745+") _ _1_ f(v)€JUI¢+") ,
_ZOO f(n.)e 3 — 23, c1+C2 sin(v7r) dv (2.80)

 

where Cl and 02 comprise the closed path of the integration contour, see Figure 2.1
[34]. The 22’ component of (2.84) represents the 2 component of the surface field

produced by the :2 component of the magnetic dipole source. Taking this component

 

 

 

 

lm(v)
L AC1 jo A LC
[,7 V V r\\
K s s s s s e e > 7' Re(v)
7% : -jo : Jr
02

 

Figure 2.1. Watson transform integration contour 0 formed by paths c1 and c2

35

 

.,, 1 x — °° 7.. . 7-,. 77.9%
G25 = -— 2 Z €JI1'¢/dk26_] :9.ij (2.86)
(27") n=—oo -00 akO Hn (’7)

and applying the Watson transform obtained in (2.85) with f(v) = H52)(7)/H1/;(2)(7)

and cl from —00 -—j0 to 00 -j0 and 62 from oo+ja to 00 —ja, we have the following

 

 

 

 

 

result.
00 _
G22 _ 1 l P *jkzzdk ejv(¢+7r)H2(J )(A/)
2 " (2)22 (77-26 Z 7(2)
-00 'o W, sm<mr>Hv (7)
oo oo—ja 2
= 1 l / p e—Jl‘zzdl, / €3v<¢+fllH§ )(7)d
,, 2 .2 7 2
(2 ) 2—oo “Ito —oc—j0 sm(v7r)Hv( )(7)
‘00“ .-,- 2
/ 6JL(¢+W)H'(’)I7)dv (2 87)
.- , ’(2) A '
(”+30 s1n(t7r)H ( )

Mapping V to —z/ in the third integral of (2.87), and inverting the integration limits,

with H(_2,L),("/) = e—jmrngz) and Hfiiny) = e—jmrHLQ), (2.87) may be rewritten as

 

1 ' 00 I»
Z J 5p —k~2
G Z = — / ———6 J ~ d7.-
62 (27f)22 aka Z
—-00

oo— '0 oo—ja

/] er(5+W)HI(J2) (,7) dv _ / e—jv(a+7r)e_jv7er(}2)(7)

_ a 3111(mr)H,IJI2)(~) —sin(v7r)e—jv7rH,,,(2)(’y)

 

 

do

I

 

 

—oc ] —-OO—j0
oo- o ' , —' _ ' . " 2
1 j 00 kp -1“: J [631((D+7r) +6 ]L(¢+7r)] H18 )(7)
= — ——8 J ’~' dk (it) (2.88)
(27f)2 2 (11:2 - ’(2)
-00 'o -00-]. sm<mr>Hv ('7)

To simplify (2.88), the Poisson’s sum formula (2.89) can be used, where l is the
number of times that the creeping wave encircles the cylinder. This equation varies

depending on which path is chosen. In this case, the path Cl was chosen, when a < 0.

36

The procedure to obtain (2.89) is presented in [34] on p. 334. Using

ejmr

 

OC
— axe—12’” (2.89)
M)

sin(mr) _

the term ejv7r can be factorized in (2.88) as

 

 

 

 

 

 

~~ 1 j lip _‘k
G“ = - —— J zzdk
C2 (27r)22 (11.786
—00
00— eij [63v(5+7r)+e ]v((.b+7r)] H(2) A
/ sin(v7r) V H/(2)( ) dU (2.90)
—oo—ja v '7
Replacing (2.89) with (2.90), it is express as
1 ' 00 7.
(:32: l / ie—jkzidk
82 (27r)22 (7/98 2
—00
OH“ 00 _ [ejv(5+vr>+e—jv<7¢+vr)] 11.32)”)
/ 2j 25mm 7(2) d”
—OO—jo’ (=0 H'v (AI)
00
= —1 fie-jkzzdk
(2702—00 (1kg
00 oo—ja (e—jv(2l7r+$+27r)+ejv($—217r)) H52)(7)
Z/ . [(2) (1v
[:0 _OO—JO Hv (’7')
(2.91)

is obtained for large radius cylinders. The orders of 1 larger than zero are negligible

as the magnitude of the creeping wave, after a complete encirclement, has decayed

37

enough to ignore them. Taking the lowest term of l, the (2.91) is simplified as

 

 

—1 0C ‘ mi)!» foo—j” Iefiwm)”IDEWPI”)
e ”’ C

(117 (2.92)
(27?)2 aka —oo—j0' H1],(2)(“/)

For the last Hankel functions, we can use the expansion for Hankel function with a

large argument, in terms of Fock-type Airy functions as [8],

H3297) ~ 7mg”) (2.93)

 

705(7)

m2 (f7?

where 1772(7) as the Fock-type Airy functions of the second kind. Using the notation

1152(7) ~ —7

 

(2.94)

of Fock [28], the quantity T is related to m as

T = 1(1/ — '7) (2.95)

m

Substituting (2.93) and (2.94) into (2.92), and with v = 'ITLT + ”y [33], we find that
dv = de and (2.92) can be expressed as

}

oo oo—ja —jv(<p+27r) jug) 492(7)
G22 _ —1 / kp—e—jkzzdk / (6 +6 Jmfi
62 (2702 (71% I“ . 10“")
—oo

-oc—7a 2%

dv

 

 

 

 

OO— '0. _ . —’ . —,
1 00 k0 _‘jk-7'E 1k J (6 ]v(¢+27r) + GJUCD) Ill/12(7) d
: (2W)2 We “ ( z u), (7') m U
-OC I '0 —oo——ja 2
(2.96)

38

Considering only the short path term, ej 11¢, (2.96) can be written as

 

 

 

 

oc . —
1 k , . _ -_7(m‘r+kpa)d>
0:; = 2 / ie—szzdkz/e , w2(T)m(m)dT
(21r) aka 102(7)
1 00 2k -jmr'c3 —jkpa6
= 2 / T-Tpe—jkzzdkz/ e 6, w2(T)dT (2.97)
(27r) 00 also 1‘ 102(7)
_ 1

Expressing above equation in polar coordinates,

k2 = k0 sina

kp = 1:0 cosa

a5 = scosé

E = ssiné

[3 = m5 (2.98)

Simplifying this resulting expressions. Here, 3 is the geodesic distance, and 5 is the

angle subtended by the geodesic curve from the azimuthal plane of the cylinder at

39

the source point. Substituting (2.98) in (2.97) the final expression is given by

 

 

 

1 00 2k jflT

~-y TN. 7 _,‘ . 7 _'. 7 8— ’l.’ 7'

G3 = 2 / 2% fire Jkpa‘l’dkz/——, L2( )dT

’ (27f) akO 102(7')
00
= 1 f) m 0/90 C050 e—jLOSina(ssin(5) e—jkocosa(scosd)dk
(27r)2 01:2

—00

If;— e—Jfi‘ru W2
d7.
11/2 (T)

I‘1
2
: Fifi/We e—jA.Osiria(ssin6) e—jkocosa(9cos6)k0cosada
Ca
/€_JBTU'2(T)
——7—————d7’
w2(7)
1
1 ,‘2 :2 , .‘ ‘ —jfiT )
z _7 / me—Ji.oscos<a—6>da/ 14292,, (299)
(27r) 0. “12(7)
01 1

/jB e jfiqum
4—; F/———— dT (2.100)

 

~ 1 m2 cos2a 471' “k .
G2“ : __ __ 3 —_7 ‘03 cos(a—6)d 2101
62 (2,7,2 / a «fly/(we a < >

9505

The remaining dyadic terms are given by G62 , GS; and G; z<b

2.3 Approximation of Green’s Function in the Far Zone

We start again with the expression for the expanded form of the electric dyadic

Green’s function of the second kind for an infinite PEC circular cylinder (2.83). In

40

this case, only the source point will be on the cylinder surface at p = a. In contrast
with the on-surface case presented in Section 2.2, the observation point can be in any
point outside the surface including at an infinite distance from the cylinder. Because

of this, x = kpp where a < p < 00. For this case, (2.83) must be written as

 

G'p,( ¢,z|a,(b, z) 1)2 Z ejmfi Z: dkz 63—sz 2

712—00

 

 

 

 

—jnH(2ar)< )+j_n (2)2Hfl:l<:) -Q, _j kzkaé’lem
$511792) ('7) 72 k0 H52)(7) 7k8H52)(7)_
H32)(x) _ 1kg 2 115,2)(19 M, nkzka£2)(:r) ,A,
+ (2) k «2) f + «2) q”
7H7; (7) 07 an (7) 7$k3Hn Ml
. (2) 2 (2)
n-kzkan (~73) 293/__1_ (fig) Hn (517) 221 (2.102)
2k2H'(2)( ) 7 k0 nglh)

As noted in Section 2.2, this expression slowly converges for the case where the
parameter kpp is large. Later, asymptotic expression will be derived similar to the
surface case. To evaluated this Greens’s function in the far—zone the Hankel function

and its spectral derivative are approximated as

 

 

. 2
pgmooH’QM’u ~—2H£)<x> (2.103)
(2)
lim H” 1: ~ lim . (2.104)
p—+oc a: p—m erpf=

41

Substituting those in (2.102), it is written as

 

 

 

I I 1 00 (3an 00 —jk E n ’92 2 H782)(37) "I
G(p,(b,zla,¢,z) = (2702 "goo _[O (1sz 7—2’ (m) Hg2)( ) 10¢)
_ kzka£2)($) -. _ JHT(12)($) 6043/
7112,1152) (7) 7Hr(12)(7) 1
72112111 (”(7) 7 k0 H530)
Using
H£2)(1:) ~ «fie—flanged (2.100)

factorizing it from the numerator on (2.105), and rewriting (2.105) grouping terms in

function of the source points, we obtain

G,(p,¢zla¢>’, z’) =(72r2——)2 Z «2an

 

 

 

n=—oo
00
/ dkze_jkzzl{ lez2 [:zp ,. _ 22]? <5,
_ 1607211790) k0
+ 1 [14.3% 1th,] ,- }9303’ejn§e—j(kpp+kzz)
— Z __ I
koaH H’mo) "0 k0 711.82% > WWW

(2.107)

Replacing [Esp — $73.3] with 0 in (2.107), and expressing it in terms of d\ adic com-
ponents resulting in

00

#71006]: ejn(¢>+g) [Gwéay + 69262’ + 0005‘;le

(2.108)

42

where

 

OO . / -7r .
szZ 6321 k "3(kpp+kzzl
0‘9 f e n 2 6 (11;; (2.109)

G =
2 ‘
e (kpa)2k0H,ll(2)(kpa) V 27rka

—00

 

62 =

dkz (2.110)

—00

 

CO . I ~7r .
_- Jk z :1 -J(kpp+kzz)
<b¢_ / ( Jle z 6] 8 (2.111)

G62 __00 kpaH£2)(kpa) [Tr/6N) dkz
Each dyadic component can be evaluated using the method of steepest descent. This
method is used to calculate complex integrals with large parameters; in our case with
kpp large. This method works due to the fact that we can deform the contour of
integration and the result will not change provided this function is analytic. We first
need to find critical points of the integral. Then for those points, a path must be
identified and finally, we must deform the contour to follow the paths [35]. We start

from the canonical steepest descent integral which is given by:

SDP
where 10' is the large parameter and SDP is the contour that contains the saddle
points. After applying the method, we will obtain a first order approximation of the

form

, /2W'F(k§)eflg(k§)ejw
1
ng”(k§)|2

 

G(kz) ~ (2.113)

with kg as the saddle-point and 112 as the angle between the saddle-point and the
intersection with the contour. For convenience, we change the parameters in (2.109)

from cylindrical coordinates to spherical coordinates as

kp = 1:0 sin(6), p = Rsin(6), z = Rcos(6) (2.114)

43

where 6 is the angle between the z axis and the observation point in the far-zone.
Taking e—j(k/’p+kzz) from (2.109), replacing p and z from (2.114), and comparing

this term with the canonical steepest descent integral, we can write it as

€—j(kpRsin(6)+k;Rcos(6)) = e—jR(kpsin(6)+kz cos(9)) = 61290:) (2,115)

with k = R. Using k2 = kg + k3, we can substitute it on

g(ky) = ——j(kp sin(6) + k2 cos(9)) (2.116)

A!

to obtain
9(a) = —j(,/I.~g — k3 5111(0) + kz cos(0)) (2.117)

The saddle—point is obtained by setting the g%(,%2|kz=k§ = 0. Thus,

 

l k~ ——k~ ' 6
(iii “) = —j __~_s1_n_(_) + cos(6) (2.118)
'Z (1% ~13)?

Therefore, when Egg?) = 0, the value found for k2 is a saddle—point, kg";

——-k3 ' 0
—j AM+COS(8) = 0
03—12.)?

1
kf 3111(6) = (1.2 — 1.2)? c0s(9)
4. 0 4

kg tan(9) = (k8 -— kg)?
(1.3)? = k8 cos(6)2
kg = k0 cos(6) (2.119)

44

then function g”(k;) and g(kz) can be evaluated at the saddle-point k2 2 kg 2

k0 cos(l9) as

 

 

ll ’63
9 (k3) = jSin(9) 2 2 3
(1.0 — k0 c052(6))2
k2
= jsin(6) O 3
13(1— c082(0))2
j
= 2.120
kg si112(6) ( )
2 2 2 1
g(kg) = —j[(k0 —k0 cos (6))?sin(9) +k0 cos(6)cos(9)]

= —- j(k3[sin2(6) + cos2(9)l

= —jk0 (2.121)

It follows that the function F(k~) would be written as

Q

 

 

 

 

jkzzl if, .
F003) = e [(233] ”I” (2.122)
with kp = ksin(19)
jk0c0s(6)z’ JET; . 6
1705;) = e [(2) 6 "CO“ ) (2.123)
(ksin(9)a)2Hn (ksin(6)a)sin(6)\/27rkli’
Substituting equations (2.123), (2.121) and (2.120) into (2.113) resulting in
. J :0
27f GJI‘O 005(0)4 6121',” 005(9) 6R(—jk0)€jw
06¢ (k sin(6)a)2H;,(2)(k sin(6)a)sin(6)\/27rkR (2 124)
82 N - 1 '
2
0mm

45

.' _ 71'
“11h T/J — II

6310 cos(9)z’ej {£82 gem-11:0,. (308(0) . MO

(k sin(6)a)2H7,1(2) (k sin(6)a) m

 

00>
662 N

je-ijO cos(0)ejk0 cos(6)z’ 71
[{(ka 5111(0))? H.152)(ksin(6)a)

 

(2.125)

We follow the same procedure for (2.110). New, g(kz)S and g”(kz)s are the same as

in (2.109), F(kz) is given by

 

 

 

 

 

jkzz’equ'
F003) = e [(2) (2.120)
(koa)2k0H.n (kpa), /2' 71"ka
and
'k cos(0)z’ 'n
170.8) — e] 0 6121 (2127)
'2 _ .
(koa)2k0H;,(2) (kpa) , /27rkpp
Replacing (2.113), we obtain:
. J -H , ,
J27? «Si/~000Mb €13 eR(-Jl’0)ew
G 2 (k()a)2H,, (ksin(6)a)sin(0)\/27rkR
62 ~ . 1
—-%— 2
leosin (6)l
- ,_ijO _jkO cos(l9)z’
3" Rck ,(2) 1 (2.128)
(a) Hn (ksin(0)a)

Finally for (2.111), g(kg) and g"(kg) are the same as in (2.109) and (2.110). F(kz) is

._ ° jkzZ/ejgf
F(kz) = ’6 (2.129)

(kpa)H,’f2) (kpa) , /27rk‘p/)

 

and .7r
_jejko cos(0)z’ej 3

(ka sin(6))H,/l(2) (ka sin(t9)) sin(6) 277/013

 

Fug) = (2.130)

46

There for, it can be verified that

 

 

- ”k 6 .~.’ 3'" , ,
«‘27. (7):] 0”“ l e 7‘ em—akmeyw
C33 (ka 5111(0))Hn‘") (k sin(9)a) 3111(0) 22ka
782 N . 1
R—12—— 2
l k0 sin (6)|
e—ijO ejko cos(0)z’ 1

 

. . (2.131)

R(ka sm(6)) Hff2)(ksin(6)a)
Summarizing, the three components of 623 after applying the steepest decent method
can be written as

r . -. . I -, . 7r
9d) 6—1R’1‘0j2kcos(6)e]k0905(9)2 0° neJn(<P+g)

 

 

 

 

 

 

G . (2.132)
62 (COR (27r)2(kasm(9))2 ”goo Hg,(2)(ka sin(0))
0 .. ejn<5+3l
Ge2~ M (270% Z [(2) . (2.133)
0 nz—oo Hn (kas1n(9))
92¢ e—ijO 211‘erk005(9)2’ 00 ejn($+g)
(2.134)

62 N RkO (2n)2(kasin(6)) 71;“, H.,’,(2)(kasin(9))

Equations (2.132), (2.133) and (2.134) can be written into separate sums using
23000 [7(7) = 2900 F(",") + 28° 17(7) -— F(0). Doing that for (2.132), substitut-

ing 72. by —n in the first summation and changing its intervals, the following equation

47

is obtained

06¢ e‘ijO j2kO cos(6)ejk COS(6)Z,
82 2013 (2702021 3111(0))?

i i 7167723833135 00 nej 1158ng

 

 

 

— 0
n=_oo Hi,(2)(kasin(0)) nzo 215221251110»

e‘ijO j2k0 cos(6)ejk C°S(6)ZI
kOR (27r)2(ka sin(6))2

0° _ —jn.5 if"; 00 3716 -n.
2 7:2) 6. + Z [(3)6 J_ (2.135)
71.20 H_n (ka 8111(0)) n=0 Hn (ka sm(6))

 

 

 

Using the transform HI? (7) = e‘jwnH.;,(2)(7) for analytic functions [36] and sub-

stituting it in (2.135), we have

G995 ~ e—ijO j2kO cos(6)ejk C°S(6)Z,
e2 kOR (27r)2(kasin(6))2
[ 00 —ne‘jn$e—jn§ 00 nejnaj" J
n

+
:0 e‘jflnH.,’1(2)(kasin(6)) 72:0 Hg,(2)(ka sin(6))

 

 

 

e—ijO 32,150 cos(l9)ejk COS(9)ZI
kOR (27r)2(ka sin(9))2

[i njn(—e_jn$) +§ 713-718an J

7120 H;1(2)(ka sin(6)) 72:0 H;z(2)(ka sin((9))

 

 

 

(2.136)

48

Factorizing the summation in (2.136) and applying the Euler’s identity, (2.136) can

be simplified as

 

 

e2 koR (271’)2(kasin(9))2 2 2j

0905 e—ijO j2k0 008(9)ejk cos(6)z’ [ 00 ”341(6an — 64715)] 3];
n=0 Hileucasinw»

 

 

~ e_ij0 (‘4)1‘30 008(9lejkCOS(6)z, [00 njn sin(n5)
[COR (27)2(ka.sin(0))2 n=0 H112) (kasin(6))
(2.137)

We can use the same procedure in (2.133) to obtain

e—ijO j2ejk cos(6)z’
1:012 (27020
i 6.771% ej"a 00 ej "'2 ejna 1

+2

n=_oo H32)(kasin(0)) ”:0 H152) (ka 5111(0)) _ H’(2)(ka 5111(0))

 

92
G62

 

 

 

 

 

 

 

0
e—ijO erjk COS(0)z’ 00 jn(ejn6 _ 8—3713)?- _ 1 1
kOR (2”)2‘1 =0 H52)(ka sin(6)) 2 H6(2)(ka sin(6))

d

(2.138)

The last term in (2.138) can be introduced to the summation by introducing the term

an (called the Neumann’s constant [37]) when 5n = 1 for n = 0 and En = 2 for n # 0

as

62:
082 N

 

 

e—ijO j2€jkcos(0)z' [ 00

enj” cos(n$)
2.139
kg}? (27r)2a ”2:0 H52)(ka sin(0))] ( )

49

The process for obtaining 032$ in (2.133) is exactly the same as the one for G25

 

 

—ij0 - jkcos(6)z’ 00 -n _—
Gq§¢ 8 211.08 [ 5n] (303014)) ] (2140)

62 "’ (COR (2w)2kasin(0) ”:0 H£2)(kasin(9))

At this point, an asymptotic approximation for the electric dyadic Green’s function
is found when the observation point is far way from the origin. However, this ap—
proximation works for circular cylinders with an electrically small diameter. For a
large radius of curvature, this approximation converges poorly. As it was mentioned
in Section 2.2, to obtain a good convergence, the order of the Hankel function has
to be bigger than kpp (77. >> 2kpp). As in Section 2.2, Watson transform has to be

derived for this specific case, and is done as follows

 

1+c2 sin(v7r)

2j Z f(n)e—jn7r _____ A f(’U) d’U

0° .— . 'v5
2jZf(n)eJ"¢e”3m = / M611,

12.02 sin(mr)

 

 

00 - — -v37r
E : 71$ —— 'mr 7137f _ f(v)e]v¢ej T
2] _Oof(n)e] 6 J e3 T — [61+c2 sin(v7r) dv
0° . — '1) 5+3”)
”(3+0 _ i f(v)eJ ( 7
200: f(n)e7 2 _ 2j €1+C2 8mm) d0 (2.141)

Simplifying (2.133) and replacing ka sin(9) by 'y, we obtain

e—ijO jk sin(9)ejk C°S(9)zl 00 Him—(5+ %)
~ 2 2
Rko 271' 7 n=—oo Hf] )(7)

 

0
G62 .3 (2.142)

 

Apply Watson transform in (2.142), with f (v) = 1

50

 

 

6:
062 N

. . ,' — 3w
e-JRkojksin(6)eJkCOS(9lzl 1 / BJMHT) (2.143)

Bk 2,. 7 . I 2
0 2” , 3W, sm<mr>HJ )m

Expanding the integrals, changing z/ to —1/ in the third integral of (2.143), it may be

rewritten as

 

 

 

 

 

 

 

 

a ksinre)ejkcos<9>z’ was)
682 4 2 / [(2) dv
7r 7 Cl+c2 sin(v7r)Hv ('7)
ksin(6)€jkcos(6l)z’ 00—30 ejv($+§27£) —OO+J0 ejv(a+2’27l)
"’ 4737 f - «2) d“ f - ,(2) d”
—oo—_7a SlIl(’U7T)Hv (7) oo+ja 5111(1)”)111) (’7')
ksin(6)ejkcos(6)z’ 00—30 ejv($+3§£) —OO—]0 e—jv(<—b+§j:)
N 4737 / . 1(2) dv+ / . [(2) d”
-—oo—]a sm(v7r)Hv (7) oo—ja s1n(—v7r)H_v (7)

(2.144)

51

Replacing H52) 7' = e—ijl/P), and sin -v7r with -—sin mr and inverting the
v

integration limits in (2144) we have

 

 

 

 

 

 

 

 

gz k sin(0)ejk (305(9):,
G62 471’2
’7
W mm) 0...... 644544)
/ ‘ [(2) d'U- / - —'v7r ’(2) dv
—OO—]0' s1n(v7r)HU (7) —oo—ja —s1n(v7r)e J HU (7)
ksin(6)ejkcos(9)zl 00-30 ejv($+§271) 00—30 e-J'v(5+§)
—oo—]0 Sln(v7r)Hv (’7) —OO—j0' Sln(U7T)Hv (7)
ksin(9)ejkcos(9)z’ 00—30 ejv(5+§2K)+B—jv($+g)
N 4 2 f [(2) do (2.145)
7r 7 _Oc_j0 sin(v7r)HU (7)

Following the same procedure as in Section 2.2, using the Poisson sum, we need to

factorize the term ejmr in (2.145). Thus results in

 

 

 

- OO— '0 _ jv(6+7r) _jv(5+37r)

9;; ksin(6)le~'COS(9)Z’ J ejmr (8 7 + e 7
052 47r27‘ / sin('v7r) Hl(2)( , (£2146)

—OO—j0’ v 7)
For path Cl when 0 < 0, the Poisson sum that it is used is
ejmr oo .

= 2' ~92sz 2.147
Sin(v7r) J 1;) e ( l

52

Replacing (2.147) in (2.146), and just retaining the lowest term of 1, (2.146) is sim-

plified as

 

 

 

 

 

 

._' - 7r _ —+37r
9 ksin(9)ejkC05(9)Z, 00 J0 00 ‘ (611)“) ?)+8 JLW 7))
a 5 23-26-2211”r a
6 4n27 42)
-OO—j0' [=0 H1) (7)
jksin(9)ejkcos(6)zl 00 oc—Ja ( -Jv(2l7r <15 gl+e (21w+¢+ ))
N l 2712) Z ’(2) dv
[ZO—OO—ja Ht) (7)
(2.148)
_. _ , __7r ’_+_37r
6 jksi11(9)ejl‘7005(6)zl 00 JO" (6 JL( (15 2) +6 JUN? 7))
08:2: ~ 27r2 [(2) dv (2.149)
7 —OO—j0 H?) (7)

With (D1 = —5— 35 and (1)2 2 15+ 3?- and substituting H1} 2)7( ) bv (2. 94) [8], we have

. OO-jO’ _. _ .
jksin(6)echos(0)2’ (6 ”(1’1 + e ”32)

 

 

 

 

2: d,
062 27:27 . IL” (7') 1,
—-k Si11(9)ejkcos(9)3, 00—30 7712 (e_jvq)1 + 6—.70‘I’2)
’” 7/ , ch) (2150)
27"” . fiwgtr)
-oo-]0

(IA

With 7' = $0) — 7) )and m— — ()f3 [33], we can find that d1) = 771117, replacing it in
(2.150) yields

 

 

. .‘ . . ,l 3 —jv<I>1 —jv<I>2
—kSlIl 6 (3]A’C05(9)~ m (e + e
025 ~ ( ) / dT

 

 

271'”) 1" filaéfr)
1
-—k Si11(9)ejk (305(9):, (€_jvq)1 + €_jv(p2)
~ ‘ 4 f , I dT (2.151)
471' \/7_rw2(T)
1“1

This can be manipulated further to yield

 

 

 

 

 

 

G ~ ——ksin(6)ejkcos(0)zl (e—j(m'r+7')<I>1 + e—j(mT+7)<I>2)
" ~ (17'
62 47r / fill/2(7)
I‘1
. .- 'k cos 9 z, — 'mflI) — 'mfll’
N 4‘ b111(9)€] ( ) 6—J7‘I’1L ——e J, 1dr + e‘jlq’Q— _6 J, 2dr
4. «Fl 152(7) fin 42(7)
(2.152)
In (2.98), it is defined that ,13 _—.—. mo, substituting m<I>L2 by 512, we have
_ . - jkcos(0)z’ —jfil'r —jfi T
6'25 N 1‘81“”)? e—J7‘1’1_1_ e d, + e—J'7<P2_1_ ‘3 I 2 d,
471' fin (7') J7? “12(7)
I‘1
(2.153)

We can express (2.153) in terms of the complex conjugate far-zone hard Fock function

g<ul(;3)* that is given by

-J'fird
g<ul(3 )* _ _]T /u,—(—— Tu 6 (2.154)

Where “ =1: ” denotes complex conjugation, noting that 1L”1(7')* 2 105(7), and u is the

function order, which for our case is zero. Equation (2.153) can be expressed as

I

 

 

~ —k{." 6 ljkCOS(9)Z ., 'A
c‘éé ~ in“ if. [6"V‘plg‘0)('m<1>1>* + 6‘] "1’29(°)<m‘1’2)*l
_ ‘7 ‘. 9 ijcos(9)z’ 2 .H
N As1n( 1: Z 6—3 (@I’g(0)(ni(1>p)* (2.155)
7r
p=1

54

Now, we will follow the same procedure with the GS; component. We need to simplify

(2.134) and replace ka sin(9) by 7 to yield

e—jRA?()k06jkcos(6)z’ 00 (yd—4+3)

 

 

 

 

 

 

0‘” ~ (2.156)
62 4 2 2
R10 2W 7 'n=—00 H51 )(7)
Applying the Watson transform results in
. . - — 37r
—]RL70 k ’]A7COS(9)Z, 1 e]U(¢+—2—)
Gig) ~ ‘3 Rk ()6 2 2 T f (2) (2.157)
0 7T 7 Jc1+02 sin(v7r)HU (7)
Expanding the integrals, changing 1/ to —V, (2.157) may be rewritten as:
464) kg 3'1: eos(9)z' pug—5+?)
G62 ~ j47r27 / - (2) (11)
01+02 s1n(er)Hv (7)
kejkeos(6)zl (DO—'70 6.774543%!) —OO+JU ejv(5+§271)
-00-” sm<mr>Hv ('1) OCH, sm<mr>Hv (7')

kejkcos(6)z’ oc—ja 3”) —oo—j0 e’jUfC—b+§27:)
J W 7 oo—ja sin(mr)Hv (7') sin(—v7r)H_v(7)

oc—jor

 

 

(2.158)

55

o ,- -
We replace Hg.) 7' = 6—1 NH?) and sin —v7r with —sin mr , and invert the inte-
U

gration limits in (2.158) to yield

oo—ja

 

 

 

 

 

W9 kejk cos(19)z’ ewe—M3275) 00-]0 e—jv(5+§275)
Ge2 "’ “*7— / (2) d”— / . (2) d“
J4” 7 -OO-]O' sin(mr)Hv (7) —oo—ja —sin(v7r)e_Jv7THv (7)
kejkeos(6)z’ oc—ja eJ'v(?¢3+3§£) oo—ja e—jv(5+§)
~ T / _. (2) dv+ / . (2) dv
J —C>O-_]0 s1n('er)Hv (7) —oo-ja sm(mr)Hv (7)
kejkcos(6)z' 00—” ”($333) + e-Jv($+%)
J ’ 7 sm(v7r)Hv (7)

—oc—j0

Following the same procedure as in Section 2.2, applying the Poisson sum, we need
to factorize the term ejmT in (2.159) to get
, 00.35 (6.145%) + e—Jv(?6+34£))

G546 kejkcos(6)z / ejmr

~ _ dv (2.160)
62 ' 2 ‘ '} 2
347 7 s1n(17r) H1) )(7)

—oo—j0

Replacing (2.147) in (2.160), and just leaving the lowest term ofl (2.160) is simplified

 

as
__l ' _+7T _a’“+37r
4' keik008<9lz' 00 N 00 7 (63M ”+12 ”M 7))
G(,; N T / QjZe—fl WT (2) div

(e-jv(2lvr—5—g) + e—jv(217r+—Q§+3275))

 

 

kejkcos(0)z' 00 foo—jg
~ —‘————— d1)
2 - 2
27f 7 1:0 "'00-'30 Hz) )(“I’)
(2.161)
.. ‘ y, 00—30 (3 Jvf 5—? + e 3145+?)
m.) kejkcos(6)2
G(,? N 2’2 (2) dv (2.162)
A 7 --OC-j0‘ HI) (’7')

56

With (131: —¢- 72r_ and (D2— — (5+3; and substituting H(2 )( 7) by (2.93)[8], we have:

kejkcos(6)z' 00—10 (ca—jmpl +e—jv<I>2)

 

 

(45¢
6'82 2727 jw (T: dv
—oo-ja Jmfi
kejkcos(0)z’ oo—ja m (6"ij1 + e—jv‘b2)
~ ——,—— / d5 (2.163)
J2??? _ 751020)
—OO—jd

1
With 7' = %(v - 7) and m = ($3 [33] we find that do = de, and replace it in
(2.163) to give

 

 

 

 

 

 

. , I 2 —jv<I>1 —jv<I>2)
G¢¢ kegkcos(0)z / m (e + e d»,-
82 .1247 7771020)
1‘ 1
m2kejk cos(6)z’ (e—jmpl + e_jvq)2)
~ , / d7" (2.164)
32717 771020")
F 1
This equation can then be simplified as
¢¢ m2k€jk COS(0)Z, (e_j(m7-+7)(p1 + e—j(m'T+fiY)q)2)
G d
6? 12m / fiwze) T
F 1
2 . 75003 0 z’ mT<I> m'7'<I>
N We? 9 4741; e , ___1d,+e—m>21 /e___ , 2
J27W fin W20 ) \/— 102(Tld
(2.165)

57

In (2.98), we defined that )8 = mg), substituting m<I>1,2 by 7312, we have

 

 

2. 'kco:02’ 3T —" T
a? N m (”8], SHe--J'7<1’1__F/e__2__1;(flfl1 dT+e-J7<P2-1_ ‘3 332 0),
32717 [111 102(7)
(2.166)

We can express (2.166) in terms of the complex conjugate far-zone soft Fock function

f(u) (,3)* given by

BTdT
(u) = _j_ Tu e 3
f (3)* f /'EF_ (2167)

Having w1(T)* = w2(T) and u is the function order, which for our case is zero, (2.166)

can be expressed as:

 

 

 

 

 

 

2 jk cos(9)z’ . ,
06532 m (“3. [e—J’Y‘I’lf(0)(mq)1)* + e—JW‘I’2f(0)(mq)2)*]
e J2T7
m2kejk cos(O
~ —j7‘I’p( * 2.1
flm )2}: e f0)(m<1>p) ( 68)
Next, Watson transform can be used to accelerate the computation of Ge 995. Replacing
\ ka sin(6’) by 7 simplifies (2.132) as
(M 6—312“) jkcos(t9)ejk cos(0)z’ 00 nejn(a+2r)
G ~ 2 2 ———-——————— (2.169)
2. v N... Hm
Applying, Watson transform but with f (n) = [(3) (2.169) can be written as
Hn (’7) .
. . - _ 3,
005 6—3Rk0 jk coS(6)CJk “35(9):, 1 veflw'f—Qi)
Ge2 N k 2 2 f ,(2) (2.170)
R 0 2” "r J sin<mr>Hv (7')

C1+C2

58

Expanding the integrals, changing l/ to —1/, (2.170) yields

 

 

' ' ' _ 31
Orb e—ijO kcos(9)e~7kcos(6)z, ver(¢+_2£)
G62 N d'U

 

 

 

. 2-2 . I 2
RAD 47f / 01+02 s1n(mr)Hv( )(7)
kc08(9)ejkcos(9)zl OO-JU vejv(5+§27£) —OO+JU veijl'QQE)
"’ 42272 / H<2> d“ f . 1(2) d”
_OO_,.,, sin<mr>H <1) OOH, sm<vvr>Hv (1)

oo—jo —oo—ja

kcos(9)ejkcos(6)z, / vej“(5+§2’l) d / _ve—Jv('d3+§2"£)
Sln )

4,2,2 . (W) 1,52)” sin(—UJT)H’_(12))(7)

dv

 

 

 

—oo—ja oo—ja

(2.171)

We replace Hfii)(7) = e’jMHLQ) and sin(—v7r) with —sin(mr), and invert the

integration limits in (2.171)

I

06¢ kcos(0)ejk 005(6)}:

 

 

 

 

 

 

 

 

e2 ~ 47,272
00—]0' vejv(g+§27_r_) 00—30 _ve—jv(5+§275)
/ ' [(2) dv_ ./ —'v7r ’(2) dv
—oo—jor Sin(er)Hv ('7) -—oo—ja —sm(v7r)e 3 H0 (7)
kcos(9)e jkcos(6 —] 22er (45+?) oo—flo ve_jv($+2)
"’ 4W2 2:1” (2) 0“” f . 2(2) 0’”
ja sin (v7r)Hv ('7) —oo—]a SlIl(U7T)Hv (7)
kCOS(6)ejkcos(0)z’ 00—30 vejv(5+§275) _ ve—jv(5+1§)
N 2 2 f [(2) dv (2.172)
47f 7 _Oo_j0 sin(v7r)Hv (7)

59

As in the two other components, we factorize the term 63mr to apply the Poisson sum

and obtain

 

 

— . ' —+7T _ ' —+ :3.”
606 —kcos(6)ejl\'-COS(9)Z, 00 J0 ejmr v (ejvw 12) " e va 7))
G62 N 4W2»)? / sin(er) ’(2) do
—oo—j0 Hv (’7')
(2.173)

Replacing (2.147) in (2.173), and just leaving lowest term of I, (2.173) is simplified as

I oo—ja

 

 

- —+7r _- _+31r
6d) -kcos(9)ejkCOS(9)3 . 00 —j2lmrv(ejv(¢ 2) —8 WW 7))
082 N 4N2AQ / 2.7 :5 dv
’ - [=0
—oo—]0
v (e—jv(217r—$—g) _ e—jv(217r+$+§271))
d‘v

 

 

—jktcOS(9)ejkC05(6)Z, 0C foo—j“
2 ,2 . I 2
2” 2 1:0 20-20 H‘ ’m
(2.174)

—' —'v—_—7T _' "+37r
(71¢ —jkcos(6)ejk‘305(9)2’ 0C 30 v(e 3“ it 7)—e 30W 7
062 N

 

 

2W272 H/(2) (7)

—oo—j0

With (121 = _5 — g and (D2 = 5+ 327: and substituting H;(2)(7) by (2.94)[8], we have

I 00—.70 ”U (6—1].qu _ e—jv‘bl)

 

 

49¢) jk cos(0)ejkcos(0)z
G 2 2 2 I d”
6 2w , .wqm
_OO-JU mE 7r

 

 

,-. . oo—ja 2,, —jv<I>2 _ —jv<I)1
k. . . 6 Jltcos(9)z’ m L (e e )
N “M )8 / dv (2.176)

27772 fiwéfi)

—oo—ja

60

Using '0 = 7717' + 7 and do = de, (2.176) simplifies to

 

 

 

6¢ k cos(fi)ejk cos(6)z’ 7713(7717' + ’7) («E—j‘bfimT'l’l) + e—j<I>1(m7'+7))
G82 N 2 / I (17
27w «5702(7)
P 1
(2.177)
Expanding (2.177), we have
6d) k C08(9)2:jk2cos(6)z,
Ge N
—— 'mT‘D — ' <I>
e—j7@2_m:§_1___2d7 _ e-J’7‘I’1_1_ LEW—1d,
I I
w2(7) «1‘1 w2(7’)
+ m3 ‘7 e j7q’2_ QWT— e jm‘r‘I’z ————dT — e_j7q’1_/ejmTq)1dT
fi fi
(2.178)

Equation (2.178) can be expressed in terms of the first and second order, 11 = 1 and
u = 2, complex conjugate far-zone soft Fock function 9(1‘)(B)*. Replacing m3 by it,

results in

 

- I
g kcos 6 ejkcos(())z my _ - , _ -N ,
06? N ( )2“? {_2_ [e J’V‘I’2]g(1)(m(p2)* _6 J Iq)1]g(1)(Tn(I)1)*]

+ 127. [6-2722g(0)(m<1>2)* —e—J'2‘1’1 9(0)(m<1)1)*]} (2.179)

Simplifying (2.179), we have

61

k , (9 jkcos(6)z’ 2 . , "
Gig) N C05< >84” Z(_1)Pe—J7(DP [9(0)(7n<1>p)* + ‘Z£EQ(1)(771<I>I))*
72:1

 

(2.180)

This is the remaining component of our asymptotic solution. The expressions for
G25, G2; and Gig work very well at azimuth. Although, as the elevation angle
6 changes from g to O the accuracy and convergence decreases significatively. An

alternative solution for this problem is proposed in the next chapter.

62

CHAPTER 3

MODIFIED MODAL SOLUTION FOR THE DYADIC GREEN’S
FUNCTION

In this Chapter, the analytic modal solution, and the steepest descent solution are
studied and discussed in detail for a magnetic source on 3. PEG circular cylinder for
observation points far away from the source point (7" —- 7'") >> 10A. The exact dyadic

)

Green’s function of the second kind, Egg , for the infinite PEC circular cylinder will
be the starting point of the analysis. The $43, component of 5:22) would be the
focus of this work, because the axial singularities that are presented when using the
steepest descent method becoming indeterminate when evaluated at the vertical axis.
In Section 3.1, an efficient way of implementing the modal solution is obtained using
the integration around the branch cuts. These will be related to using an steepest
descent path approximation to evaluate this integral. The convergence of the exact

and the modified modal solution will be studied in this Chapter. Plots of the relative

error between the solutions will be also shown.

63

3.1 Exact Modal and Asymptotic Solutions

The dyadic Green’s function of the second kind for a infinite PEC circular cylinder

of radius a, was obtained for a magnetic source was obtained as

 

 

 

 

 

 

 

00 _ 00
G(2 2)(p,qb,z|a, (15, z) — ——2—n;oo ejnd) / dkze—jsz
-J'nH(2(2))(kpp) + In (1.22)}; 11% p) M kzka’(2)(kpp> ,2,
kpakppHn2 )puc (1) (WV k0 H;,:—2_—)(Icpa ka akgH H’Q) (kpa)_
Hn2 )(kp/J) "’9 H512)(kpp "kzkanm (kfp) ~ ‘al
+ (a ‘62) m< m
kpaHn (kpa) 0 pa (kpp)Hn (kpa)¢ k pa(kpp)k2Hn(2) (kpa a_)

 

(2) H(2)
nkzkan (lip/9) z __ (ka) zz
+l< >l i ’“P” fig—0)” H’flwl }

kpa)2k(2)Hn(2)k (k pa
(3.1)

As stated earlier this exact modal solution is very difficult to evaluate when the
radius of the cylinder is electrically large (ka >> 1) and when the observation pomt

is far away from the source point. The (7395’ component of the Green’s functlon 1S

 

Z (2W5 Z dk 6 3k Zf(kz,kp) (3.2)

n=—oo

G¢¢(p,¢,z|a,¢’,zl)= (271' 1)2

where f (k2, kp) is written as

 

 

’2( ) 2 (2)
H” k kz H k A A
f(kZa 1910): (2 2()k pp p) _ (kn—kg) n limpp) $925, (33)
kpaHn (kp a) 0 p (kplen (kpa)

As shown if Figure 3.1, the integral on the real axis of the (26’ component is verv
difficult to evaluate, because as the integral being evaluated the complex argument

(j kzé) of the exponential part become very oscillatory. As shown in Chapter 2 the

64

A jkzi

‘kO

 

 

 

Figure 3.1. Complex 15;; plane

steepest descent method could be implemented, but it was shown, this method has
convergence problems near to the paraxial zone due to the branch point singularities in
15;. Instead of fixing the steepest descent problem, this solution will try to deform the
contour of integration. The first problem is to close the correct contour of integration.
The radiation condition in the 2 and the p” direction must be satisfied. Expressing
k; as a complex number, 19;; = kzr + jkzi, with kzr as the real part and km; the
imaginary part, the plane wave with propagation constant Is; in the 2 direction can

be expressed as

8—].sz = e—j(kzr+jkzi)5

= e—jkz'rlz—Zl)ekzi(Z—z’) (3.4)

As seen in Figure 3.2, if (z — z’) > 0 the contour of integration must be closed in the

lower half plane, represented by a dashed line. In this way, kzi will take only negative

I
values and attenuation of ekzz’(z_z ) will be guaranteed. For (2 — 2') < 0 the upper

half plane is chosen. The two branch points of the integrand (kp = :t‘flcg — kg) at

65

 

 

 

 

 

 

 

 

 

s ’

Figure 3.2. Contour of integration dictated by the radiation condition

It; 2 :tlco, indicated us the two possible Riemann surfaces kz could be at, during the
contour of integration. Therefore, the contour of integration must be chosen in such
a manner that the inverse transform of (3.2) exists. According to Cauchy theorem,
closing the contour of integration as in Figure 3.2, the original integration in the k;
plane could be changed by two integrals, one around the branch cut, and the other

one closing the contour at infinity as

oo
/ €_jkzzf(kz,kp)dkz : _ / e—jkfiflkzakpmkz _ [fa—jkzzfljkz’ kp)dkz
—OO Coo Cb
(3.5)

As it is shown in Figure 3.3, we have different options for setting the branch cut
of A3,) = iflkg — kg; however, the proper Riemman sheet is the one that leads to
satisfaction of the radiation condition for large values of p for each part of the contour
of integration. Also, the correct topology of the branch cut chosen guarantees the

contribution of f goes to zero [34]. In order to chose the appropriated Riemman
Coo

66

j kzi

 

,
\a b i
\
\ _ I
g Cl
‘ §
4‘0 1.1.: ~:J_ d’ kzr‘
d Ffil‘flko 7
I \
l \
|
l C, b’ \

 

Figure 3.3. Definition of the Branch Cut

sheet, the kp component can be expressed in polar coordinates

. _ 2 .2
Ip _ sad/to — A,

= ij\/kz—kO\/kz+k0
.¢++¢—
—_— :tj\/r+r“ng—T_2
+ . _
. (,9 +99
2 :tVr‘W‘efl Q +%] (3.6)

 

Looking at Figure 3.4, it seem that r+ and 1“ represent the magnitude contribution
of the branch cut singularities —k0 and k0, respectively, as well as, 90+ and cp—
contribute to the phase for a given point on the contour of integration around the
branch cut. Taking the first case, when the contour of integration is coming from
j 00 down to a value close to zero, and evaluating the angle contribution for different
points using (3.6), as shown in Figure 3.4(a), the phase for kp is calculated as 27r
which means the points are located at the first Riemman sheet. Looking at Figure

3.4(b), and following the same procedure, it is found out the lap phase for this case is

67

7r selecting the second Riemman sheet.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘F ijl a s ijl
l’ f— a 36/ "l-
| I 'H_
+ l V ‘ + if .
r+ “I r If, 9 .J I"
-k (p _J - -k z’gg \J """x _
0 E {—b (P ‘ 0 - —. K'x (p ‘
"7"}. ..... I - JET" a
f "° f "°
(a) (b)
.szi «W
l
I
11%|
4- + U
'kO 0—---" - - «r .......
i T T I: __. - >
(—
b l ko ‘9
(c) m

Figure 3.4. Integration around the Branch Cut

ekziEkazivfiljdkzz‘ + / e‘jkzI'EIIHzr,—a)dkzr
0

\O

/ e—jkfifwz, 19,0)de =

0,, oo
0 00
+/ e—szr3f(kzr,a)dkzr + [ekzi§f(jkz,,—H)jdkz,+ e"9k25f(kz,kp)dkz
—k0 0 o

(3.7)

68

Vthre a = “1:3 - k3,, ,3 = ”198+ 15%, f is the integral around the branch cut
(9

singularity. When evaluating the integral around the branch out singularity, the
small argument approximation for the Hankel function could be used, and it can
be shown that at the limit closed to the singularity, the contribution of the integral

vanishes. Thus,

0C
/ e—J'WHII, IpIdIz = few —HI - f(jkziIB)lIdkzi
Cb 0
-’\‘0
+ / e‘jkz-r-Z— [f(kzr, Cl) _ f(kzr, —Q)] (1sz (3.8)
0
o, - - - (2) —j7r _ej'mr ’2( ) —j7r _ jmr ’(1)
Usmb the identlties H7, (Ce )2 Hn1 )(g) and H7, ((6 ) — e Hn (g)

from [38] we obtain

ayw

OC

/e_1sz(kzakpdkz —e*zz/ wUkzia midkzi

Cb 0
—’~‘0
0

+ C—jkerl/I(kzr ,a')dkzr (3.9)

I I
, . __ -Hn(1)(w) Hn(2)(7p)
’I’aH-n. (Ia) I'aHn (Ia)

 

. 2 (1) Hf?)
(nu) MIFJI'I—IZML (3.10)

I; , ’1
ma“ ”I'PHn( )(Ia) "I’pH‘nQ )(Ia)

Substituting H.,(,.1)(§)- — In(§ ) +an(<) and H783“) = Jn(§) — an(§) over (3.3),

evaluation the contour integral, and after some mathematical manipulation, the 09,305

69

can be written as in (3.11), with III(kz, 7) shown in 3.12.

—II30
G¢2(p,I,zIa,I’,z’I = ——1)—2 Z eW / dkzr2j€_jkzr2¢(kzraa)

Tl——OO 0

00
+ fdkzi2ek2iz'w(jkziafi) (3-11)
0

J’(Ip)Yn(Ia)- HUI/mafia)
.I, kg, =
M 7) 70 (Jn('I’)20 + Yn(70)2)
_ (_ nkz )2 Jn(7p)YI€(Ia) - Yn(’Ip)JI'I(Ia)
kma 70(J1’z(70)2 + YI’I(Ia)2)

 

 

(3.12)

The infinite and finite integrals in expression (3.11) converge more rapidly for
increasing 2 as well as the index n. However, for cylinder with large radius, the mod—
ified modal solution converges relatively slower with respect to the steepest descent

solution.

3.2 Numerical Results and Discussions

In this Subsection, the convergence of the exact modal solution (EMS) and the modi-
fied modal solution (MMS) will be studied for a circular cylinder of radius a = 0.05A.
After proving a good agreement between the EMS and the MMS, we will compare
the steepest decent solution with the MMS for an infinite circular cylinder with large
radius (a = 10A). Values of the magnitude for different indexes n, from 1 — 7, for
EMS and MMS are shown in Figure 3.5, showing the convergence of the magnitude
for the (Mo’ components represented by (3.2) and (3.11).

Three observation points where analyzed for R = 100A. The first case when
6 = 900 is seen in Figure 3.5(a) where EMS and MMS start converging at n = 3.

Figure 3.5(b) with 6 = 450 and Figure 3.5(c) with 9 = 00 have a similar behavior to

70

 

 

 

 

 

 

 

 

 

 

 

0.70996 A I; A
0.70994- »---.__-.._--_.___+___,
0.70992 - .
OJ
'3 0.70990 - .
t:
C
00.70988 I
(U
2 0.70986
070904
070932: .
0 70990 ‘ . - . .
1 2 3 4 5 6 7
Index n
(a) 0 = 900
.3
x 10
G)
"U
3
r:
C
U)
(U
2
7. I ' 1 l l
1 2 3 4 5 6 7
Index n
(b) 9 = 450
Figure 3.5

71

-3
x 10

 

 

 

 

2.” . . a
l
\ -—o—— EMS
2.58- \ —*— MMS I
a)
g 2.56» .
a:
c
3’ 2.54»
E
2.52-
D
2.: L A 1 A .
1 2 3 4 5 6 7
Index n
(c) I9 = 0°

Figure 3.5. Magnitude convergence for exact modal solution and modified modal
solution

the previous Figure 3.5(a) where the solution converges for indexes of n greater than
2, which was expected because the size of the cylinder and wave number 77. >> 219a.
The small error on Figure 3.5(a) is explained by the following: The second integral
term in (3.11)that goes form 0 to 00 and involves an attenuation factor represented
by 619223. As 6 —I 900, 2 gets closer to z’, E ——> 0, and the attenuation factor goes to
1, therefore the integral will not be attenuated fast enough and errors introduced by
$01922, B) will have more impact in the final solution. Figure 3.6 shows phase values
for different index of EMS and MMS that converges when n > 2 and different values
of I9 = 00,450, 900. As seen MMS and EMS show good agreement in magnitude as
well in phase.

Figure 3.7 shows values of magnitude and relative error when n = 3, R = 50A and
00 < 6 < 900. Figure 3.7(b) presents the relative error. As seen the error is very small

in order of 10—6 and does not show a big variation along the scan. This is because

72

 

-1 .39745

 

 

 

 

 

 

 

{
—0— EMS
4.39750 MMS l
4.39755
3
cu 4.39760-
.c
0.
4.39765 5
4.39770»
4.39775 . 1 1 .
1 3 4 5 6
Index n
(a) 6 = 90°
-2.63 « T
-2.64 “ 7 v 7* ‘ “ 7‘
-2.65~
3 -2.66~
g .
g_ -2.67~ l
-2.68~ I l
p —0—— EMS l
‘2'69l ——#—MMS .
_27l 1 E—_,i
1 3 4 5 6 7
Index n
(b) 0 = 45°
Figure 3.6

73

 

 

 

-024r——~——..~~——7 . a..—~*4_.m—hj

~ ., ~ a; a“: .55.

 

 

 

026‘» i
-028} ‘
m 1
a:
2
3* «
n. '0 .
—0— EMS
-0326. __+__MMS i
i .
-634: . . - . I 1 _J
1 2 3 4 5 6 7
Index n
(c) 6 = 0°

Figure 3.6. Phase convergence for exact modal solution and modified modal solution

the radius of the circular cylinder is small enough to obtain a reasonable answer with
both solutions and the observation point is not far away from the source. Figure 3.8
shows the same results as Figure 3.7 but when R = 100A. Comparing Figure 3.7(b)
with Figure 3.8(b) it is noticed a maximum error occurs when 6 = 90°; this is because
1/1( j kzi, fl) is composed by Bessel functions, which arguments are proportional to kpp
and at 6 = 90° p 2 R = 100A thus making it a large argument.

Figure 3.9 presents values of the phase for EMS and MMS and relative errors
between them when n = 3 and R = 50A. As seen in Figure 3.9(b) the behavior of
the relative error for the phase is similar to the relative error of the magnitude. It is
an error of a very small order, 10—5, and it remains constant though 0° < 6 < 90°.
The results when R = 100A can be seen in Figure 3.10. For Figure 3.10(b) the error
increases near 6 = 0° and 6 = 90°. The error when 6 = 0° is produced as a result
of high oscillations within the first integral of (3.11). Those oscillations exist because

when 6 = 0°, 2' becomes a big oscillatory argument for e'ijTE. The second error,

74

 

——<x—-EMS
——4e—JWMS

F’
or

Magnitude

F’
k)

F’
—A

 

 

 

 

O
O
N
O
A V
O
a)
O

80 100

 

 

Error (Mag)

 

 

 

0 in :10 60 80
angle 9°
(b)

Figure 3.7. (a) Magnitude for exact modal solution and modified modal solution; (b)
Relative Error; n = 3 and R = 50A

75

 

 

 

 

 

 

 

0.03l-s —0— EMS
1 —al1—MMS l
a) 0.025l '
,3 0.02
8;
(0 0.015 1

2 0.01 I

0.005 .~. "' *“r -~ 1.. r l
0~-~- . . . , !_l
0 20 40 60 30 100
angle 9°
(80
10°h ,
-2

a 10 r /

(U /

2 /

17103 //

O /

t ,/

LIJ '/
10° \\“‘\~~-— “de— —-~———/ 1
10.8i r L i 4

0 20 40 60 80
angle 9°
0))

Figure 3.8. (a) Magnitude for exact modal solution and modified modal solution; (b)
Relative Error; n = 3 and R = 100A

76

 

.6
_ N

'5
1_.1. LJ

 

 

_1 2. —~<}— EMS
i —Il+—MMS

 

0 " 210 ’ 40 60 80
angle 9°
(3)

 

 

Error (Phase)

 

 

 

0 20 210 60 60
angle 9°
0))

Figure 3.9. (a) Phase for exact modal solution and modified modal solution; (b)
Relative Error; n = 3 and R = 50A

77

when 6 = 90° is caused by the same issues as the magnitude relative error, which was
discussed for Figure 3.8(b).

Numerical calculations, for different values of R and 6, have been executed to
obtain the convergence of the magnitude and phase for both, the steepest descent
solution, (SPD), and the modified modal solution and only the most important re-
sults have been shown here. The 0303’ component of the dyadic Green’s function was
computed for 6 = 005729380, 6 == 46.10219° and 6 = 89.42706° for a cylinder of ra-
dio a = /\ and R = 100.0049A. Comparing Figure 3.11 and Figure 3.12, it is noticed
the fast convergence and likeliness of the magnitude and phase when the observation
angle 6 9: 90°. For 6 2 45° we can see a good agrement of the magnitude and phase.
As shown in Figure 3.11(a) and Figure 3.12(a), SPD values do not agree with values
obtained using the modal solution (6 ~ 0°). Near the paraxial zone, the axial sin-
gularities show up, the difference in the magnitudes is bigger and the convergence in
the SPD method is not as good as the convergence using the modified modal solu-
tion. The lack of convergence of the SPD and the disagreement between the solutions
increases because as we get closer to the 6 = 0 or 6 = 7r in the 10,; plane, saddle
point in the a plane is getting near to or = g or (Ir 2 :23, we have the branch cut
singularities represented at these points.

Figure 3.13 and Figure 3.14 contain values of magnitude and its relative error for
R = A and R = 100A respectively. Both figures were obtained with index n = 10
and 0° < 6 < 90°. Comparing Figure 3.13(b) and Figure 3.14(b) it can be seen that
both figures present very high errors when 6 = 0°. The order of the error decreases
as the distance from the observation point to the paraxial zone increases. The same
behavior is observed on Figure 3.15 and Figure 3.16 where the values of the phase

and its relative error for both SDP and MMS are shown.

78

 

 

 

 

l
0.5, “ l
4 l '0 i
- l ,
.C l a“, i
° ‘21 “l
-25. —0— EMS l‘ l
l —I‘rc—MMS 6'1
'30 R — —20 _ 4'0 6‘0 nab—J
angle 9°
(81)
10° T___#*__ _rflm
, W
A 1021 . i
(D I l
a) l ;’ l
2 4g l
h l i), l
O __ !
t -6’ l , / “ ‘ ~ 9- - ‘
LU 10 l’ \ R 1
l
10 1 1 --—- .
0 20 40 60 80 100
angle 9°
0))

Figure 3.10. (a) Phase for exact modal solution and modified modal solution; (b)
Relative Error; n = 3 and R = 100/\

79

3.3 Singly Surface Periodic Structure in :2

Many applications require the analysis of periodic antennas over structures such as
infinite plates [39] and circular cylinders [40]. Looking at Figure 3.17, it can be stated
that computationally, it is more efficient to analyze a single unit cell containing infor-
mation about the periodicity of the structure than simulating the complete periodic
structure. The periodic Green’s functions have been the focus of many studies similar
to the previous [41, 42, 43].

In this section, the development of a periodic Green’s function that relates the
field produced by magnetic currents due to periodic antennas in 2 over the surface of
an infinite circular cylinder will be explained. The periodic phased array of antennas
must be expressed in terms of one unit cell and the fields produced by it will be
expressed in terms of Floquet modes. The infinite array of magnetic current sources

is written in terms of a singular magnetic current term as

, 27717!’ ‘jkz 277m

L ) = M(p~=a,¢',2’)e T (3-13)

 

M(p=a,¢’,

where L is the distance between array elements in the 2 direction, and the field due

to this infinite periodic magnetic currents is expressed as

me = 11060 m; fiw M0- G(re2 r’elrl ‘Jk’T ds’ (3.14)
o‘lnSa
Periodic Green’s functions are a common approach to attack these problems, the

periodic infinite array can be translated as part of the Green’s function such that

CO
— — . 2772.71' _ 7‘, 277m
GP62(p1 05,2la, $22,) = Z G62(Pa¢32la,¢l,zl- L )6 3 2T

TTlZ—OO

(3.15)

80

in this way (3.14) is re-written in the following known way

H(r) = jweo flM(r’)-§P82(r’|r)ds’ (3.16)
Sa

and the periodic dyadic Green’s function of the second kind for a infinite PEC circular

cylinder of radius a is

00
_2 —'~_-'k 2m1‘r
$320.3 zla a 9) _,_Z Z W / a... We 3 2T

2m=—oo nz—oo “‘00

—jnHl.°)(kpp> jn k_z ))2H,',<2(k,.p) ,-,_, kzkallzleppfl

(2) + 1. 2 k [(2) °° J [(2) p2
kpakppHn (kpa) (pa) 0 Hn (kpa) kpakan (kpar
«0

H7. (ka) nk 2 H°l<kp p) nkzkaflkpp)
+ 0) kka He) °°+ 249 m
kpaHn (kpa) 0 p (kp p)Hn (kpa) kpa(kpp)k0Hn (kpa)‘

nkzkatzkkpp) L k_p 2H£°lwpp>
m) 2 "k k @) zz
(kpa)2kan (kpa) pa 0 Hn (kpa)

 

 

 

 

 

(3.17)
Using Poisson summation formula [34]
oo oo
1 ' 293
2 7e” T = 2 5(1— pT) (3.18)
with t = kz and T = L we have
00 oo
1 _ - .~2m7r
X |__ Lle WT : Z 50;; +pL) (3.19)
m=— 192—00

81

using (3.19) into (3.17)

G(2)2,,’(p,¢z|a¢, 2) —Z ejn¢dez e—jkzz E 6(kz+mL)

2n=—oo m=—oo

-J'nH(2 )(kpp) + 2'72 (2)(§3)2nH Hln.(2)(kp10) -955. kzkarl2(2)(k/)P) -,/
. (2). (ka)2 1501,42) 2 2 . .2 (<2). 2"
2)

+ H152)(kpp) _anz>2 H73 (kpp) 06+ nkzka£2)(kpp) ,3,
kpaH,(,2)(kpa) ‘Okpa (kpp)H,’,(2)(kpa) kpa(kpp)k2Hl(2)(kpa)d

+ nkzka£2l<kpp> 2 &,__1_ (a) 10.26pm 2,,
(kpa)2kgH,’,(2)(kpa) kpa ’90 Hfilmapa)

 

 

 

 

(3.20)

Solving the integrand of (3.20), the periodic Green’s function is not a spectral solution
anymore and it becomes a modal solution with eigenvalues dictated by the periodicity

of the array elements as seen

 

Gg32(pa¢1zla Q5]: ZI) [1)2 ”200 ej7l¢ 2 eij2

m=-oo

{[ -—jnHl.2)(kmpp) + jn (:19) 11,52 )(kmpp>[fiq3,

 

 

kmpakmppHfi2)(kmpa) (kmp (1)2 k0 H32)(kmpa)

 

 

2 2
, mLkmpHfif )(kmpp) [3;]- nmLkmpH£)(kmpp) (2);,
2 n2 "

+ H32)(kmpp> (ML)? H520 mp0) (5,3,
kmpaHfimepa) .Okmpa (kmpplflri2 )(kmpa)

2 2
2 ' 2
(kmpa)2k8H,’,( )(kmpa) imp“ ’20 Hf] )(kmpa)

 

 

 

 

(3.21)
where kmp = 138 - (mL)2. The next step is the efficient implementation of this

82

periodic Green’s function. This is a very Changeling problem because the summation

of F loquet modes usually converges very slowly, and this is left for future works.

83

 

 

 

 

 

Magnitude

 

 

 

 

o ‘5 1'0 15
Index n
(a) 6 = 0"

 

 

Magnitude

 

 

 

 

 

 

 

O

o 5 1o 15
Index n
(b) 0 = 450

Figure 3.11

84

x10

1D fir T

 

14-

12-

 

10-

 

 

 

Magnitude

 

 

 

 

o 5 1o 15
Index n
(c) 0 = 90"

Figure 3.11. Magnitude convergence for modified modal solution and steepest descent
solution

85

1
ll mu l.0l l0 l
4 6 Aw

 

:00

(a) 9

 

 

 

Index n

(b) 9

45"

Figure 3.12

86

 

 

 

 

-30 i— '4: kt t
.f. E
-35» . ___ i
F 1 MM :
0’ ‘ .. SPD
3 -40~ L—" l
_c .1
0. , 4'
-455 ' l l
1 1
-5o '——- — . *1
o 5 15
Index n
(c) 9 = 90"

Figure 3.12. Phase convergence for modified modal solution and steepest descent
solution

87

 

——0—r SDS 4
12*\ —a.i—— MMS

Magnitude

 

 

 

 

 

 

"'0 £0 40 so 80
angle 9°
(a)
10° ;— a 1
i l
A '1' l
:3,10 é i
(U l \\x- 1
g f \ J
a i
:- 10'2.» l
“J i \ ;
-3 ‘
10 o 20 4o 60 80
angle 9°
(1))

Figure 3.13. (a) Magnitude for modified modal solution and steepest descent solution;
(b) Relative error; n = 10, R = 50A

88

 

 

 

 

 

 

 

 

 

 

1o—-——— ., . ,, aw— —fi—~
—o—SDS
8* +MMS
o)
‘0
a 6.
C 1
a)
(U
2
4 .
2l_,_1 1 ,_ i 333333333 ~ ‘
o 20 4o 60 80
angle 9°
(a)
0
10 . 2
10'1 “ \ \ \
A f \\ \\\~~-\\\
g 10.23 F “‘x
12', * ‘\
. \\ i
§ 103; \\ 1
u‘] 43 \_ 3
10 i \\.?i
5 l.

 

1° 0 210 4‘0 6?) 810
angle 9°
0))

Figure 3.14. (a) Magnitude for modified modal solution and steepest descent solution;
(b) Relative error; 71. = 10, R = 100A

89

 

 

 

 

 

 

 

 

1 _ /
8 0’
(U
.C ‘t
l , F:
-1 4
——~O— SDS
—-Il*-—MMS
‘2” ' 20 4b 6‘0 at)
angle 9°
(a)
1
10 l—_ . a ]
A —/’”f I 4/ A\ J
3 10° \\
g \\ l
e, . \ l
5 10'1 I» \\ a
t \ .
m 1
10'21 __ . 1 i ._ _ _#__l
o 20 4o 60 80
angle 9°

(b)

Figure 3.15. (a) Phase for modified modal solution and steepest descent solution; (b)
Relative error; n = 10, R = 50A

90

 

 

 

 

 

 

 

2!. l
<12 1
g l
& 1F 4'
—0— SDS l
0' +MMS “ l
4 . . . - _J
'10 20 4o 60 80
angle 9°
(a)
1
10 1F —# l
.\\ l
100 \\\ )
g -1': \ - 3
g 10 * \» ”Ix \ l
n. \ l
v -2 \
h 10 \
e . l
L \
”J 10°: \;
‘l
10;. . » _______ .___l
o 20 4o 60 80
angle 9°
0))

Figure 3.16. (a) Phase for modified modal solution and steepest descent solution; (b)
Relative error; n = 10, R = 100A

91

 

 

Figure 3.17. Singly surface periodic structure in 2

92

CHAPTER 4

CONCLUSIONS AND FUTURE WORK

A modified modal solution for obtaining the field due to magnetic currents as an
aperture on an infinite circular cylinder was obtained. Validation of this solution was
made by comparing the MMS with the exact modal solution for the same problem.
The agreement between both methods, as expected, was very good for cylinders with
small radius (a < 0.01/\) and the distance between the source and the observation
points no larger than 50A, obtaining a relative error between the solution around order
of 10-6. For distances grater than 50)\ the relative error between them increases due
to convergence problems with the exact modal solution, specifically at 6 = 90°.

A deep analysis of the steepest descent solution was made concluding that the
solution was not accurate for the paraxial zone due to axial singularities in the .152
plane, omitted when the saddle point technique was implemented. Those axial sin-
gularities where detected to Show up when the angle 0 of the observation point was
close to 00 and 1800.

The MMS was found to be a good solution to over come the lack of accuracy around
the paraxial zone for circular cylinders of larger radius (a. = A). It was demonstrated
this solution provides the same accuracy as the steepest descent solution near the
azimuth region 350 < 9 < 1350. For angles smaller or greater than those, the
steepest descent method fails converging to an accurate solution. Although steepest
descent remains faster converging at the azimuth zone, the MMS provides a smooth
transition between the paraxial region and the azimuth region.

Although MMS was found for an infinite circular cylinder, this solution can be
expanded for an infinite elliptical cylinder by multiplying the MMS with the specific

torsion factor and radius of curvature for an elliptical cylinder. These coefficients

93

could be found by using UTD.

As a future work, a FE—BI code could be implemented using the lV'IMS as bounci-
ary integral for truncating the computational domain, this code can be validated by
using analytic solutions for the radar cross section of an antenna embedded in an
infinite circular or elliptical cylinder. Diffraction and attenuation coefficients for pro-
late spheroids can be found to expand this solution for arbitrary shapes with doubly
surface.

Additionally, a periodic Green’s function that relates the field produced by mag-
netic currents due to periodic array antennas on 2 and over the surface of an infinite
circular cylinder was obtained using F loquet’s theorem and the Poisson summation

equation. The efficient implementation of this solution is left for future works.

94

BIBLIOGRAPHY

95

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[2]

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