Muhammad Naveed - Higher Education...

SCATTERING OF ELECTROMAGNETIC WAVES FROM

A PEMC CIRCULAR CYLINDER PLACED

UNDER WIDE DOUBLE WEDGE

Muhammad Naveed

Department of Electronics

Quaid-i-Azam University

Islamabad, Pakistan

2011

SCATTERING OF ELECTROMAGNETIC WAVES FROM

A PEMC CIRCULAR CYLINDER PLACED

UNDER WIDE DOUBLE WEDGE

by

Muhammad Naveed

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy



Islamabad, Pakistan

2011

– ii –

CERTIFICATE

It is to certify that Mr. Muhammad Naveed carried out the work contained in

this dissertation under my supervision.

Dr. Qaisar Abbas Naqvi

Associate Professor



Islamabad, Pakistan

Submitted through

Dr. Qaisar Abbas Naqvi

Chairman



Islamabad, Pakistan

– iii –

Acknowledgments

I am thankful to Allah Almighty, the Most Beneficent the Most Merciful, Who’s

Blessings have always given me strength and wisdom. I offer my heartiest praises to

the Prophet Muhammad (Peace Be Upon Him), whose life is a true picture of Quran

and is a glorious model for the whole humanity.

I express my heartiest gratitude and pay my sincere regards to honorable advisor

Dr. Qaisar Abbas Naqvi, who has been a continuous source of inspiration and en-

couragement throughout my research work. I am indeed indebted to him for all his

support and cooperation without which it wont have been possible for me to complete

my PhD research work. It is only because of his insight, enthusiasm, and continuous

encouragement which helped me to complete this uphill task. I would also like to pay

my heartfelt wishes and special thanks to Prof. Dr. Kohei Hongo, Toho University,

Japan, whose thought provoking ideas and research techniques guided me through my

research work and made me capable of achieving such a highest goal of my life for

which I am extremely grateful to him.

I have my special and warm feelings for all my friends who have always been with

me in the difficult times of research work. Particularly, I would like to mention the

names of Dr. Shakeel Ahmed and Dr. Ahsan Ilahi for their extra ordinary support in

the crucial times and in the time of need. Besides I would like to say my thanks to

my friends Dr. Abdul Ghaffar, Dr. Amjad Imran, Dr. Akhtar Hussain, Fazli Manan,

Abdul Aziz, Khalid Nasir, Muhammad Ayub, Shahid Iqbal, Yamin, Naeem Iqbal, and

Anjum for their company and a very good time together.

I am also grateful to my brothers, Arshad Akhund, Atif Riaz, Muhammad Shafiq,

Muhammad Ateeq, and my only sister Dr. Shagufta Yousaf for their moral support

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and inspiration. It was their continuous encouragement and motivation which kept

me moving towards my goal.

I have my most special and deepest gratitude to my wife, Faiqa Naveed, who was

always their to support me whenever I felt depressed. She has been supporting me

greatly by her love and encouragement during this difficult period. It was because

of the support of all my family members and specially my wife that I managed to

achieve my objective. Their support and prayers were the major source of inspiration

in completion of this work. I am also grateful to my cousins Dr. Anwar Hussain,

Taimoor Khurshid, Shahzad Afzal, Amir Afzal, Yasir Afzal, Faisal Afzal, Mehtab

Afzal, Muhammad Sohail Malik and all those who kept praying for my success.

At the end, I would like to dedicate this work to my parents, Mrs. Shamim Akhtar

and Mr. Muhammad Yousaf. There is no way that I could reciprocate their love and

affection in the manner they did in my childhood. I would particularly mention my

mother who’s prayers have been a real source of inspiration for me.

Muhammad Naveed

– v –

To

My Parents and Family

– vi –

Abstract

A new wedge diffraction function, called as Naveed-Naqvi-Hongo (NNH) wedge

diffraction function, is derived and evaluated asymptotically by applying the steepest

descent method. It is found that the total field with NNH solution is continuous at the

shadow boundaries and gives the well known non-uniform expression for the observa-

tion point far from the shadow boundaries. Numerical comparison of NNH solution

is made with exact series solution and Pauli-Kouyumjian-Pathak (PKP) result. It

is found that the agreement among these three is fairly well. In contrast to the ex-

pressions proposed by Kouyoumjian and Pathak, the NNH solution does not need the

parameters to switch for the region of validity, hence it is easier to make a numerical

code. The validity of NNH wedge diffraction function is further checked by evaluating

the diffracted field from a geometry which contains an infinite slit in a perfect electric

conducting (PEC) plane. It is further extended to a more complex geometry consisting

of two parallel PEC wedges and a perfect electromagnetic conductor (PEMC) circular

cylinder which is placed under the PEC wide double wedge. It is found that by using

the NNH solution, the results evaluated for some special cases, including the trans-

mission coefficient of PEC slit, are in fairly good agreement with the published work.

The transmission coefficient and the diffraction pattern of PEC wide double wedge is

studied and elaborated by considering a geometry consisting of a coated PEMC cylin-

der placed under the two parallel wedges. The cylinder is coated with double positive

(DPS) or double negative (DNG) materials. Finally, a comparison of the transmission

coefficient of PEC geometry, as evaluated by using NNH solution, is made with the

geometry of more practical nature, that is, an impedance slit.

– vii –

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter I: Introduction 1

Chapter II: Evaluation of Uniform Wedge Diffraction Function 12

2.1. Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1. Exact Series Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2. Pauli-Kouyoumjian-Pathak (PKP) Representation . . . . . . . . . . 14

2.2. Naveed-Naqvi-Hongo (NNH) Wedge Diffraction Function . . . . . . . . . . . 16

2.2.1. Small Argument Approximation and Behavior . . . . . . . . . . . . . . .

at the Shadow Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2. Surface Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter III: PEMC Cylinder Placed Under PEC Wide Double Wedge 24

3.1. PEC Slit Excited by Uniform Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1. Field Diffracted From a Wide PEC Slit . . . . . . . . . . . . . . . . . . . . . 29

3.2. PEMC Cylinder Placed Under PEC Wide Double Wedge . . . . . . . . . 31

3.2.1. A PEC Wedge Excited by Plane Wave . . . . . . . . . . . . . . . . . . . . . . .

and Cylindrical Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.2. A PEMC Cylinder Excited by Plane Wave . . . . . . . . . . . . . . . . . . .


3.2.3. PEMC Cylinder Below PEC Wide Double Wedge . . . . . . . . . . 35

3.2.3. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

– viii –

3.3. Coated PEMC Cylinder Placed Under PEC Wide Double Wedge . 46

3.2.1. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter IV: PEMC Cylinder Placed Under an Impedance Slit 56

4.1. Impedance Slit Excited by Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1.1. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2. PEMC Cylinder Placed Under an Impedance Slit . . . . . . . . . . . . . . . . . . 65

4.2.1. Isolated Impedance Half Plane Excited by . . . . . . . . . . . . . . . . . . .

Plane Wave and Cylindrical Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.2. A PEMC Cylinder Excited by Plane Wave . . . . . . . . . . . . . . . . . . .


4.3. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 5: Summary and Conclusion 74

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

– ix –

List of publications

List of Publications

[1] M. Naveed, Evaluation of uniform wedge diffraction function, presented in All

Pakistan Mathematical Conference held in Islamabad on 7 September 2007.

[2] M. Naveed, and Q. A. Naqvi., ”Scattering of electromagnetic plane wave by a per-

fectly conducting slit and a PEMC parallel cylinder, Progress In Electromagnetics

Research M, Vol. 1, 45-58, 2008.

[3] M. Naveed, Q. A. Naqvi and K. Hongo, Diffraction of em plane wave by a slit

in an impedance plane using Maliuzhinets function, Progress In Electromagnetics

Research B, Vol. 5, 265-273, 2008.

[4] M. Naveed, S. Ahmed and Q. A. Naqvi, Scattering of electromagnetic plane waves

from a coated PEMC circular cylinder placed under PEC wide double wedge,

Mathematical Problems in Engineering, Volume 2010, Article ID 254025, 26 pages

doi : 10.1155/2010/254025

[5] S. Ahmed, A. Ghaffar, Q. A. Naqvi, M. Naveed., ”Effect of dissipative and disper-

sive DNG material coating on the scattering behavior of parallel Nihility circular

cylinders, Accepted for publication, Mathematical problems in engineering, 2011.

– 1 –

CHAPTER I

Introduction

Light, being the most phenomenal display of electricity and magnetism, has been

attracting the attention of many great philosophers and scientists. Francesco Maria

Grimaldi was the first one who observed the process of bending of light through a

narrow slit [1] and named the phenomenon as diffringere, a latin word meaning to

break in different directions [2]. Although, he identified diffraction but was unable to

formulate a theoretical explanation [2]. Sir Isaac Newton in the 17th century presented

the corpuscular theory of light [3] which explained the geometrical optics phenomena

such as rectilinear propagation, reflection, and refraction but was insufficient to explain

the diffraction and interference phenomena. Huygen, in 1690, introduced the concept

of secondary wavefronts which could explain the phenomena of diffraction [4]. Almost

over 100 years did not witness any substantial progress in the explanation of diffraction.

In 1802, Thomas Young presented the interference pattern of light produced when light

passed through a narrow slit [5]. Fresnel, in 1815, combined the Huygen’s principle

with interference and suggested that the phase of the elementary wave be taken into

account while calculating the secondary wavefront [6]. After about fifty years Kirchhoff

devised his scalar theory of diffraction [7] that confirmed Fresnel’s diffraction theory.

The term Diffracted Ray was first introduced by Kalashnikov, in 1911, who also

suggested an objective proof of their existence by recording them on photographic

plates [8]. Now it is believed that any deviation of light rays from rectilinear path

which cannot be interpreted as reflection or refraction is called diffraction.

Most common display of diffraction phenomena in every day life is observed on a

CD or DVD which act as a diffraction grating to form the familiar rainbow pattern on

the disk. This principle can be extended to engineer a grating such that it can produce

– 2 –

any desired diffraction pattern. Diffraction is important in many practical applications,

such as the resolution of a camera, telescope, microscope, x-ray diffraction studies of

crystals, holography, microwave remote sensing [9-11] etc. The transmitted signal in

a radar system is also diffracted. The larger the aperture (antenna) the narrower the

transmitted wave pattern. Moreover, many of the objects of interest on the earth’s

surface including tree branches, wheat stalks, wind induced ripples over water and

ocean waves are all examples of features within the scale of the diffracting objects. It

is, therefore, of practical interest to have the analytical as well as numerical analysis

of the diffracted field patterns through various geometries.

In electromagnetics, the diffraction through edges, corners or tips may be studied

by considering the objects having same local geometries, called the canonical objects.

The ray structure of diffracted fields was established theoretically, in 1924, by Rubi-

nowicz [12], and later by other authors [13-14]. Finally the concept of diffracted rays

was formulated, in the most general form, by Keller in 1956, and from his formulation

the famous geometrical theory of diffraction (GTD) was born [15-18]. Keller extended

laws of geometrical optics (GO) so that it includes diffraction by introducing diffracted

rays in addition to the usual rays. Thus the diffracted field in a framework of GO can

be computed. He started from the eigenfunction (modal) form of the related Green’s

function, transformed it into a contour integral in the complex plane, chose a suit-

able path and evaluated the integral asymptotically by means of a steepest-descend

method. This way he could separate from the total field the incident, reflected and

transmitted GO terms and obtained a closed form result of the far field caused by the

edge for an incident plane wave.

GTD allows the extension of usual GO to efficiently treat electrically large scat-

tering objects even if they include sharp edges and are hence not treatable by usual

ray optics. Within GTD, the field diffracted by these sharp edges is calculated based

on the solution of the pertinent canonical problem (the wedge, half plane etc), and is

– 3 –

given in a GO adapted form, that is, the GTD edge diffraction co-efficient of the wedge.

GTD provides a simple and physical approach to the description of the diffraction of an

electromagnetic wave by an object, as it contains only trigonometric functions. This is

a great advantage of the GTD over other conventional methods. Among the canonical

problems in electromagnetics and acoustic scattering theory [19], the solutions for per-

fectly conducting wedge and its special case, the half plane, served as starting point

for the GTD. A particular method to derive the GTD diffraction co-efficient for the

wedge has been proposed by Blume and Wittich [20]. They investigated the eigenfunc-

tion expansion of the Green’s function of the wedge for the case of an incident plane

wave and used a suitable distributional analysis to decompose the total field into the

incoming, the transmitted, the reflected, and the edge diffracted parts.

The diffraction integral for perfectly conducting wedge is used as the canonical

problem in many high frequency techniques. The history of wedge diffraction function

dates back to 1896, when Sommerfeld developed a rigorous solution for diffraction by

a perfectly conducting half plane [21]. He showed that the wave in the shadow region

is a cylindrical wave that originates at the edge of the half plane. In the lit region,

Sommerfeld showed that the wave could be expressed as the summation of a cylindri-

cal wave and the incident plane wave. The asymptotic expansion of the Sommerfeld

solution was derived by many researchers [22-26]. In these, Pauli’s work [24] is note-

worthy. Pauli derived the asymptotic expansion for the Sommerfeld’s formulation and

his result is finite at one of the two shadow boundaries, and the singular boundary can

be changed into the regular one by transforming the argument of the derived result.

Kouyumjian and Pathak [25] introduced a parameter to the Pauli’s solution which en-

ables to switch the variable in accordance with the location of the shadow boundaries

and it gives the uniform solution. Another improved formula for the diffraction by

a perfectly conducting wedge was presented by Liu and Ciric [26]. They employed a

– 4 –

new function which resulted into an expression to give a uniform field at all shadow

boundaries.

Wedge forms an integral part of the solution of a large class of high frequency

diffraction problems dealing with even more complex bodies [27-58] (slit, two bodies

scattering problems etc). It has found its many practical applications in the field of

microwave devices [40-47] such as in filters, reflectors and antenna covers etc. The

problem of diffraction by an infinite conducting slit has been studied extensively [30-

39]. Morse and Rubenstein [30] treated the problem of diffraction of acoustic waves

by using the method of separation of variables. Clemmow [31] derived dual integral

equations for the diffracted field by a slit using plane wave spectrum representation

of electromagnetic waves. Hongo [33] studied diffraction from two parallel slits in a

conducting plane using method of Kobayashi potential. Karp and Russek [34] used the

technique of fictitious line sources located according to the geometry of each scatterer.

Elsherbeni and Hamid [37-38] used the technique of Karp and Russek [34] to deal with

the diffraction from wide double wedge.

The problem of diffraction through slit has been extended to the double body

and/or multiple objects problems by many researchers [47-57]. In all these problems

the results were restricted to two or more perfectly conducting bodies of the same

type. Elsherbeni and Hamid [58] showed that the technique used by Karp and Russek

can be extended to two or more different geometries/scatterers provided that all of

them are infinite along one of the coordinate axes. It may be noted that scattering

from slits/half planes and two body problems are still considered the topics of current

interest [59-64].

PEMC has been recently introduced meta-materials by by Lindell and Sihvola [65-

66]. It is a very fundamental type of medium, at the same time extremely simple and

very complex. It is a generalization of both perfect electric conductor (PEC) and

– 5 –

perfect magnetic conductor (PMC) media for which the medium is labeled as PEMC.

Due to the cross-components in addition to the co-components in the scattered field,

it is bi-isotropic. The possible applications of this material include ground planes for

low-profile antennas, field pattern purifiers for aperture antennas, polarization trans-

formers, radar reflectors, and generalized high-impedance surfaces. Many researchers

have worked on this material [67-91]. It is well known that PEC boundary may be

defined by the boundary conditions

n×E = 0, n.B = 0

while PMC boundary may be defined by the conditions

n×H = 0, n.D = 0

The PEMC boundary conditions are of the more general form

n× (H + ME) = 0, n.(D−MB) = 0

where M denotes the admittance of the PEMC boundary. It may be noted that,

PEMC corresponds to PMC for M = 0, while it corresponds to PEC for M → ±∞.

In order to fulfill the boundary conditions, co-polarized as well as cross-polarized

field components are required in the field representation, which earned PEMC a non-

reciprocal attribute. It is due to these unusual and impressive properties of the material

that PEMC geometries are of active area of research nowadays.

Recent years have also witnessed an increased interest in materials, such as double-

negative (DNG) and single-negative (SNG), epsilon-negative (ENG) and mu-negative

(MNG), as well as combinations of these with conventional double-positive (DPS)

materials [80-91]. Veselago [80] mentioned the unusual properties of DNG materials,

which are characterized by a negative real part of the permittivity as well as the

– 6 –

permeability. Pendry [81] gave the concept of the so called perfect lens consisting of

a specific DNG slab which has attracted much of attention. Shelby et al. [82] gave

the experimental verification of a negative index of refraction. Lakhtakia discussed an

electromagnetic trinity from negative permittivity and negative permeability [83-84].

Moreover, combinations of DNG and DPS materials lead to a new paradigm in the

miniaturization of devices such as cavity resonators [85]. Ziolkowski and Kipple [86],

used the double negative materials to increase the power radiated by electrically small

antennas. Alu and Engheta [87] used different combinations of these materials in

the waveguides to study the guided modes. Eleftheriades and Balmain [88] discussed

the fundamental principles and applications of the negative refraction materials. Li

and Shen [89] found that an isolated conducting cylinder coated with metamaterials

has anomalous scattering cross section compared to that coated with conventional

materials like dielectric. By comparing the back scattered cross section of a cylinder

coated with a metamaterial and the same cylinder coated with a conventional material,

it was found that the back scattered cross sections of TM incidence are of very similar

behavior: they both have large forward scattering. However, a cylinder coated with

a conventional material has smaller forward scattering for TE incidence compared to

a cylinder coated with metamaterial. Ahmed and Naqvi [90] gave a comparison of

normalized bistatic echo width of a coated PEMC cylinder. It was found by these

authors that, for the case of TE polarization, co-polarized component of the bistatic

echo width of the coated PEMC cylinder for admittance parameter Mη1 → ±∞ is in

agreement with that of a coated PEC cylinder, while the cross-polarized component

disappears as Mη1 → ±∞. Moreover, the co-polarized components of normalized

bistatic echo width show relatively different behaviors for the same configurations

when Mη1 = ±1 while the cross-polarized components show similar behavior for the

two polarizations for different coating layers when Mη1 = ±1. By interchanging the

values of εr and µr, TM case reduces to TE case and vice versa.

– 7 –

Problems of more practical nature, such as radar absorbing materials etc, were dis-

cussed by many scientists by introducing the concept of impedance/non-conducting

materials. It was a fundamental step in the study of diffraction of electromagnetic

waves by objects which are not perfectly conducting. Surface impedance boundary

conditions, introduced by Leontovich [92], may provide a useful model for several

practical configurations. The 2-D problem of plane wave diffraction by a wedge with

impedance boundary conditions was independently solved by Maliuzhinets [93-95], Se-

nior [96] and Williams [97]. These authors used techniques, different in detail but

similar in essence, employing the Sommerfeld representation of wave fields that re-

duced the diffraction problem to a scalar Hilbert problem of conjugation, or one of

its equivalents such as Wiener-Hopf or difference equations, which admit conventional

closed-form analytic solutions. The Maliuzhinets method consists of expressing the

total field as a spectrum of plane waves which can be written as an integral with an

unknown spectral function. The key step in the Maliuzhinets method is the transfor-

mation of an integral equation into a first order functional difference equation whose

solution yields the unknown spectral function. Main steps of the method are

(1) Expressing the unknown solution of the Helmholtz equation in Sommerfeld

integrals, that is, as a linear superposition of plane waves, the simplest solutions to

the wave equation, but with unknown amplitudes, the spectra.

(2) Inserting the Sommerfeld integrals into the boundary conditions at wedge

faces, inverting the resultant integral equations for the spectra and obtaining in this

way a matrix difference equation for the spectra.

(3) Solving the matrix difference equation, and lastly.

(4) Evaluating the Sommerfeld integrals with the saddle point method and de-

ducing a first-order uniform asymptotic expression for the far field.

– 8 –

Wedge-shaped non-conducting objects, and their special case, impedance half

plane, received a lot of attention by many researchers [98-116]. Further contributions

to the Maliuzhinets theory were made by Tuzhilin who developed a theory of related

functional equations [117-120] and demonstrated the possibility of extending the Mal-

iuzhinets approach to more sophisticated boundary conditions [120]. Depending upon

the value of the vertex angle, the model of an impedance wedge uniformly includes a

variety of canonical geometries, including an imperfect half-plane, a flat surface with

an impedance step, an impedance horn and an impedance slit. Many papers have

appeared dealing with both electromagnetic and acoustic applications in these config-

urations. For instance: plane wave scattering from an impedance strip [121], radiation

of a line source at the tip of an absorbing wedge [122-124], Greens functions [125-126],

diffraction of plane, surface and cylindrical waves [127-138] by an impedance wedge of

arbitrary angle. The corresponding mathematical solution for the impedance wedge

can therefore serve as a universal basis for treating scattering and diffraction problems

of all these geometries.

The versatility of the phenomena of diffraction, its multidimensional aspects and

its display in a wide variety of practical applications, as highlighted in the preceding

paragraphs, became the major source of inspiration of the research work carried out in

this thesis. A new wedge diffraction function (called as Naveed-Naqvi-Hongo (NNH)

wedge diffraction function) is evaluated which could give a uniform field around the

wedge, even at the shadow boundaries. After achieving this objective, the NNH so-

lution is then further extended to solve complex problems involving more than one

geometries. Comparison of the results obtained by using NNH wedge diffraction func-

tion with the known solutions for some special cases of these geometries is also made,

which further confirmed its credibility. Both the transmission coefficient of the slit

and the diffraction pattern of the field is studied in all such cases. The term ’trans-

mission coefficient’ is generally used in electrical engineering when wave propagation

– 9 –

in a medium containing discontinuities is considered. It is the ratio of transmitted to

incident wave at a discontinuity ( may be a slit in a plane) in a transmission medium.

It may also be called as penetration probability which is a measure of how much of an

electromagnetic wave passes through a surface. Diffraction pattern is infact the pattern

which the electromagnetic waves will form when diffracted from an obstacle/aperture.

A comparison of some of these results (as obtained by using the NNH wedge diffraction

function) for special cases of the PEC wedge (the half plane), is made with that of a

more practical geometry, that is, an impedance slit. Comparison of the transmission

coefficient of PEC geometries (the slit) in the presence of PEMC cylinder and that of

an impedance slit (the geometry of more practical nature) is evaluated and analyzed.

It is further observed that these problems could be further extended to more complex

geometries of practical nature.

In Chapter II of the thesis a new wedge diffraction function called as Naveed-

Naqvi-Hongo (NNH) wedge diffraction function is presented. The method used to

derive NNH solution does not require the employment of any new function as was pre-

sented by Liu and Ciric [26]. The results derived are simple in numerical computation.

It can be seen that by using simple mathematical transformation, the solution is valid

at all shadow boundaries without switching to any parameter. Therefore, the total

field with the NNH solution is continuous at the shadow boundaries and gives the

well known non-uniform expression for the observation far from the shadow bound-

aries. The procedure of the analysis is an application of the standard steepest descent

method and it is readily applied to derive the derivative of the wedge diffraction func-

tion which is required to get the slope diffraction coefficient. The result is of slightly

different form from that of [24] but it gives same asymptotic expression far from the

shadow boundaries as those derived by others. To verify the validity and precision of

the NNH method, its comparison has also been made with the exact solution based on

the eigen function expansion and Pauli-Kouyumjian-Pathak (PKP). The results are

– 10 –

also presented in the form of a table which shows that the three results give almost

the same values. The comparison of the plots shows more clearly the authenticity of

the method . It can be seen that the comparison among the three methods is fairly

well. The expression of the current density induced on the surface of the wedge is

also derived for the grazing incidence for both E and H polarizations by using the

expression derived by NNH method.

In chapter III, the results of chapter II are utilized. In the first section of the

chapter, the validity and authenticity of NNH solution is established by dealing with

the problem of a PEC slit. A comparison is also made with published work. In the

second section of the chapter, a more complicated problem is dealt with by placing a

PEMC cylinder under the PEC wide double wedge. The transmission coefficient and

the diffraction pattern of the wide double wedge, in the presence of PEMC cylinder,

is determined and the results of special cases of PEMC cylinder are compared with

the published work. The practical applications of the geometry is generally found in

the remote sensing and detection of objects under the cracks/crevices etc. In the third

section, PEMC cylinder is replaced with a DNG/DPS coated PEMC cylinder and both

the transmission coefficient and diffraction pattern of the PEC wide double wedge are

found. It may be noted that the increase or decrease in the transmission coefficient of

PEC slit in the presence of coated cylinder (both DNG and DPS), is because of the

anomalous behavior of the coatings [90].

In chapter IV, a comparison of some of the results obtained in chapter III, by us-

ing the NNH wedge diffraction function, for special cases of the PEC wedge (the half

plane), is made with that of an impedance slit, which is a more practical geometry.

In the first part of the chapter the transmission coefficient and the diffraction of an

electromagnetic plane wave from a slit in an impedance plane is studied. The method

is based on Maliuzhinets technique for impedance surfaces. The formulation of the

problem is done in GTD regime. A comparison of the results for different values of

– 11 –

face impedance and for different incident angles is made. Moreover, the comparison

of both E- and H- polarized fields is also made. In the second section of the chapter,

a comparison of the transmission coefficient of PEC the slit in the presence of PEMC

cylinder (as obtained in chapter III), is made with that of an impedance slit. Further-

more, the results by changing the different parameters such as admittance parameter

of the cylinder, slit width and by changing the incident angles etc are also presented

and studied.

In the entire thesis time dependence is assumed to be exp(jωt) and it is suppressed

throughout the analysis.

Chapter 5 contains the conclusions of the work done in the thesis.

– 12 –

CHAPTER II

Evaluation of Uniform Wedge Diffraction Function

Diffraction integral for perfectly conducting wedge is important in diffraction the-

ory since wedge is used as a canonical problem in high frequency techniques. In this

chapter, a new expression for the uniform wedge diffraction function, called as Naveed-

Naqvi-Hongo wedge diffraction function(referred to as NNH wedge diffraction function

hereafter), is evaluated. It is found that the total field obtained by using NNH wedge

diffraction function is continuous at the shadow boundaries and gives the well known

non-uniform expression for the observation far from the shadow boundaries. A com-

parison of the numerical results computed by three different methods, the exact series

solution based on the eigen function expansion, PKP method, and the NNH method

is also made. It is found that the three results give almost the same values.

2.1. Previous Work

In this section, the expression of exact series solution and PKP method are given

for the comparison with the NNH solution as presented in next section.

2.1.1. Exact Series Solution

Consider the geometry, which contains a PEC wedge of infinite extent, as shown

in Fig. 2.1. Faces of the wedge are located at φ = 0 and φ = φw. The edge of the

wedge coincides with z-axis of the coordinate system. For a plane wave incidence given

below(

Eiz

Hiz

)=

(E0

H0

)exp[jk(x cosφ0 + y sin φ0)] (2.1.1)

where k = 2πλ = ω

√µε is the wave number of isotropic medium. The z-components

of total electromagnetic field around the PEC wedge are obtained as [21].

– 13 –

�

��

Fig. 2.1. Diffraction from a PEC wedge - Incident and reflected shadow boundaries.

Ez =4π

φωE0

∞∑m=1

exp[jpmπ

2

]Jm(kρ) sin(pmφ) sin(pmφ0)

=E0[u(ρ, φ− φ0)− u(ρ, φ + φ0)] (2.1.2)

Hz =2π

φωH0

∞∑m=1

εm exp[jpmπ

2

]Jm(kρ) cos(pmφ) cos(pmφ0)

=H0[u(ρ, φ− φ0) + u(ρ, φ + φ0)] (2.1.3)

where

u(ρ, ψ) =π

φω

∞∑m=0

εm exp[jpmπ

2

]Jm(kρ) cos(pmψ), ψ = φ∓ φ0 (2.1.4)

In above equations (ρ, φ) are the cylindrical coordinates, Jm(.) is the Bessel function

of the first kind and order m, φω is the wedge angle, φ0 is the incident angle, and

p = 1n = π

φω. The Neumann number εm = 1 for m=0 and 2 for m > 0. The diffracted

wave may be obtained by subtracting the incident and reflected wave from the total

field.

– 14 –

2.1.2. Pauli-Kouyoumjian-Pathak (PKP) Representation

The series solution may be transformed into contour integral representation by

using the integral representation of Bessel function as

Jm(kρ) =12π

∫

C1

exp j[kρ cos β + pm

(β − π

2

)]dβ

C1 :[−π

2+ j∞,−3π

2+ j∞

](2.1.5)

By using (2.1.5) in (2.1.4), the residues of the poles contained in the closed contour

C1 + C2−D1−D2 represent the GO field (along C1 and C2), that is the incident and

reflected field and the contribution along the contour D1 +D2 gives the diffracted field

as shown in Fig 2.2a.

�

�

�

�

�

Fig. 2.2a. The contours C1, C2, D1, D2 in complex β plane.

The equation becomes

v(ρ, ψ) =− j

2φωsin(pπ)

∫

D0

exp[−jkρ cos t]cos (pπ)− cos p(t + ψ)

dt, ψ = φ± φ0 (2.1.6)

where the contour D0 is given by[−π

2 + ε− j∞, π2 − ε + j∞]

with 0 < ε < π2 and

φω=πp as shown in Fig. 2.2b. The contours D1 and D2 has been further transformed

to D0 by using the relations t = β − π and t = β + π, respectively.

– 15 –

� � � �

� � � ��

Fig. 2.2b. The contour D0 in complex β plane.

The final expression for the diffracted field can be obtained as

(Ed

z

Hdz

)=

(v(ρ, φ− φ0)− v(ρ, φ + φ0)v(ρ, φ− φ0) + v(ρ, φ + φ0)

)

=− p√2

{Q1+F

[|a1+|

√kρ

]+ Q1−F

[|a1−|

√kρ

]

∓Q2+F[|a2+|

√kρ

]∓Q2−F

[|a2−|

√kρ

]}exp [−jkρ] (2.1.7)

where F (x) is the Fresnel integral defined as

F (x) =1√π

exp[j(x2 +

π

4

)] ∫ ∞

x

exp[−jt2]dt (2.1.8)

and

Q1+ =|a1+| cot(p

2(π + φ− φ0)

)

Q1− =|a1−| cot(p

2(π − φ + φ0)

)

Q2+ =|a2+| cot(p

2(π + φ + φ0)

)

Q2− =|a2−| cot(p

2(π − φ− φ0)

)(2.1.9)

– 16 –

a1+ =√

2 cos(

12

(φ− φ0 − 2N+

1 π))

a1− =√

2 cos(

12

(φ− φ0 − 2N−

1 π))

a2+ =√

2 cos(

12

(φ− φ0 − 2N+

1 π))

a2− =√

2 cos(

12

(φ− φ0 − 2N−

1 π))

(2.1.10)

In equation (2.1.10), the factors N+ and N− are positive or negative integers or zero

which most closely satisfy the equations

2π

pN+ − (φ∓ φ0) = π,

2π

pN− − (φ∓ φ0) = −π (2.1.11)

When observations are far away from each of the shadow boundaries, the argument

of the Fresnel integral becomes large and the value of the integral can be given by its

asymptotic approximation F (x) ' 12√

πxexp[−j π

4 ]. Hence, equation (2.1.7) reduces to

a well known non-uniform solution

v(ρ,φ− φ0)∓ v(ρ, φ + φ0) ' p sin (pπ)√2πkρ

exp[−jkρ− j

π

4

]

×{

1cos (pπ)− cos p(φ− φ0)

∓ 1cos (pπ)− cos p(φ + φ0)

}(2.1.12)

which becomes singular at the shadow boundaries.

2.2. Naveed-Naqvi-Hongo (NNH) Wedge Diffraction Function

In this section, it is shown that the results of integral evaluated by new method

does not need factor as introduced by PKP method and is valid for all observation

points. Using the identity

sin (pπ)cos (pπ)− cos p(t + ψ)

=− 12

{cot

(p

2(π + t + ψ)

)+ cot

(p

2(π − t− ψ)

)}(2.2.1)

Then (2.1.6), that is, v(ρ, ψ) is decomposed into two terms as

v(ρ, ψ) = f(ρ, ψ) + f(ρ,−ψ) (2.2.2)

– 17 –

where

f(ρ, ψ) =j

4φw

∫

D0

cos(

p2 (π + ψ + t)

)

sin(

p2 (π + ψ + t)

) exp [−jkρ cos t] dt

=j

4φw

∫

D0

cos(

p2 (π + ψ + t)

)

sin(

p2 (π + ψ + t)

) exp[−jkρ + j

kρ

2t2

]dt (2.2.3)

In above equations

cos t = 1 +t2

2+ · · ·

has been used. This integral is evaluated by applying the stationary phase method

of integration. It is readily seen that the stationary point is located at t = 0. When

the observation point is far from the shadow boundary, following simple solution is

obtained

f(ρ, ψ) = − p√8πkρ

cot(

p(π + ψ)2

)exp

[−jkρ− j

π

4

](2.2.4)

Near the shadow boundaries the factor cot p2 (π±ψ) becomes singular. In this case the

saddle point and the pole are very close. By using the transformation

t =√

2 exp[jπ

4

]s

equation (2.2.3) can be rewritten as

f(ρ, ψ) =j√

24φw

exp[−jkρ + j

π

4

]cos

(p(π + ψ)

2

)

×∫ ∞

−∞

exp[−kρs2

]

sin(

p(π+ψ)2

)cos

(pt2

)+ cos

(p(π+ψ)

2

)sin

(pt2

)dt (2.2.5)

The integral in (2.2.5) can be transformed into the following form

P =j4p2

sin(

p(π + ψ)2

) ∫ ∞

0

exp[−kρs2]s2 + ju2

ds (2.2.6)

– 18 –

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} ~ � � � � � � � � � � � � � � � � � � � � �� ¡ ¢ £ ¤ ¥

¦ § ¨ © ª « ¬ ¬ ® ¯ ° ® ± ° ± ² ³ ´ ² µ ¶ µ· ¸ ¹ º » ¼ ½ ¹ ½ ¾ ¿ À Á Â Á Ã Ä Å Æ Ç È Ç É

Ê Ë Ì Í Î Ï Ð Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ú ÜÝ Þ ß à á â á á á ã ä å æ ç è æ é ê ë ì í î ï

ð ñ ò ó ô õ ö ô ÷ ø ù ú û ü ø ý þ ÿ � � � � �� ! " # $ % & ' ( ) * + ,- . / 0 1 2 3 4 2 5 6 7 8 9 8 : ; < = > = ? >

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7 8 8 9 : ; ; 9 < = > ? @ @ > A @ B C D D B E FG H I J K L M N O L P Q R S T S U V W X Y Z Z Y[ \ ] ^ _ ` a b c d e f g h i i j k l m m m n op q r s t u v u u w x y z { z | z } ~ � � � � �� ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Á ÃÄ Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × × ØÙ Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ðñ ò ó ô õ ô ö ÷ ö ø ù ø ú û ü ü ý þ ý ÿ � � �� ! " ! # $ %& ' ( ) * ) + , - + . / . 0 1 2 3 4 5 4 6 7 4 89 : ; < = < > ? > @ A B A C D C E F G F H I J KL M N O P Q R O S Q T U V W X Y Z Y W [ \ ] ^ _ ` a b b c d e f g

Fig. 2.3 Comparison among the three different methods.

– 19 –

where

u =√

2p

sin(

p(π + ψ)2

)

Using above in (2.2.5) yields

f(ρ, ψ) =− u

πexp

[−jkρ + j

π

4

]cos

(p

2(π + ψ)

)

× sin(p

2(π + ψ)

) ∫ ∞

0

exp[−kρs2]s2 + ju2

ds

=− sgn(u) cos(p

2(π + ψ)

)exp[−jkρ]F

[√kρ|u|

](2.2.7)

where sgn(u) = 1, the signum function, for u ≥ 0 and sgn(u) = −1 for u < 0, and

F (x) is the Fresnel integral defined by (2.1.8). By summarizing the results

v(ρ, φ− φ0)∓ v(ρ, φ + φ0) ={−sgn

(sin

(p

2(π + φ− φ0)

))cos

(p

2(π + φ− φ0)

)

× F

[√2kρ

p

∣∣∣sin(p

2(π + φ− φ0)

)∣∣∣]

− sgn(sin

(p

2(π − φ + φ0)

))cos

(p

2(π − φ + φ0)

)

× F

[√2kρ

p

∣∣∣sin(p

2(π − φ + φ0)

)∣∣∣]

± sgn(sin

(p

2(π + φ + φ0)

))cos

(p

2(π + φ + φ0)

)

×F

[√2kρ

p

∣∣∣sin(p

2(π + φ + φ0)

)∣∣∣]

±sgn(sin

(p

2(π − φ− φ0)

))cos

(p

2(π − φ− φ0)

)

×F

[√2kρ

p

∣∣∣sin(p

2(π − φ− φ0)

)∣∣∣]}

exp(−jkρ)

(2.2.8)

In Fig. 2.3, the comparison among the results by exact series solution, PKP solution

and NNH solution are shown for the case φω = 2700, φ0 = 1500 and kρ = 10. The

agreement is seen to be very good.

– 20 –

Fig. 2.4. Comparison of three methods at φ0 = 300.



– 21 –


The comparison has also been made among the three methods for different incident

angles as shown in Figs. 2.4-2.7. The comparison among the plots is fairly well.

2.2.1 Small Argument Approximation and Behaviour at the Shadow Bound-

aries

For very small argument of the Fresnel integral, the Fresnel integral given in

(2.1.8) may be approximated by

F (x) ' 12− x√

πexp

(jπ

4

)(2.2.9)

Then, near the shadow boundaries, 0 < ε < π2 , relation (2.2.9) is simplified as follows

(I) φ− φ0 = π − ε

v(ρ, φ− φ0)∓ v(ρ, φ + φ0) = −12

exp(−jkρ)sgn(ε) (2.2.10)

(II) φ− φ0 = −π + ε

v(ρ, φ− φ0)∓ v(ρ, φ + φ0) = −12


– 22 –

(III) φ + φ0 = π − ε

v(ρ, φ− φ0)∓ v(ρ, φ + φ0) = ±12


(IV) φ + φ0 = 2πp − π − ε

v(ρ, φ− φ0)∓ v(ρ, φ + φ0) = ±12


The total field consisting of the GO field and the diffracted field is continuous when

the observation point passes the shadow boundaries.

2.2.2 Surface Field

In this sub-section, the surface current induced on the surface of the wedge is

derived. For the case of H-polarization, the result is given by (2.1.2), (2.1.3), (2.1.7),

(2.1.13) and (2.2.9) directly. For the case of E-polarization, the series solution and its

asymptotic solution are derived as

Hρ =j4πY0E0

φw

∞∑m=1

exp(jpmπ

2

) pm

kρJm(kρ) cos(pmφ) sin(pmφ0)

' jY0E0√2πkρ

exp(−jkρ + j

π

4

) {Fs

[√2kρ

p

∣∣∣sin p

2[π + φ− φ0]

∣∣∣]

−Fs

[√2kρ

p

∣∣∣sin p

2[π − φ + φ0]

∣∣∣]− Fs

[√2kρ

p

∣∣∣sin p

2[π + φ + φ0]

∣∣∣]

+Fs

[√2kρ

p

∣∣∣sin p

2[π − φ− φ0]

∣∣∣]}

(2.2.14)

where

Fs(x) = 1− 2√

π x exp(jπ

4

)F (x) (2.2.15)

The non-uniform asymptotic solution is given by

Hρ ' jY0√2π(kρ)

32

exp(−jkρ + j

π

4

)p2 sin(pπ)

×{

sin [p(φ− φ0)][cos(pπ)− cos [p(φ− φ0)]]2

− sin [p(φ + φ0)][cos(pπ)− cos [p(φ + φ0)]]2

}(2.2.16)

– 23 –

For grazing incidence, the electric current density on the surface is given by

Hρ(π, 0, p) = Hρ(φw − π, φw, p)

'√

2πkρ

exp(−jkρ + j

π

4

)(2.2.17)

Hz(π, 0, p) = Hz(φw − π, φw, p)

' exp(−jkρ)− p√2πkρ

exp(−jkρ + j

π

4

)cos pπ (2.2.18)

2.3. Conclusion

A uniform asymptotic expression for the wedge diffraction function has been de-

rived by applying the steepest decent method of integration. In contrast to the widely

used expression given by Kouyoumjian and Pathak, the present expression gives a

uniform solution without switching of parameter in the argument depending on the

observation point. To verify the validity and precision of the present solution, nu-

merical comparison is made for the exact series solution, PKP expression and NNH

solution. The agreement among them is fairly well.

– 24 –

CHAPTER III

PEMC Cylinder Placed Under PEC Wide Double Wedge

In chapter III, scattering of electromagnetic waves from multiple objects is studied

using the NNH uniform wedge diffraction function. For this purpose, a coated PEMC

circular cylinder placed under the PEC wide double wedge is considered. The NNH

uniform wedge diffraction function, evaluated in chapter II, is employed to calculate the

diffracted field from the wedge. First section of the chapter deals with the diffraction of

plane wave from a geometry which contains an infinite slit in PEC plane by employing

the NNH wedge diffraction function.

In second section of the chapter, the transmission coefficient and diffraction pat-

tern of the PEC wide double wedge are studied in the presence of PEMC circular

cylinder. Results of special cases are compared with the published work. The method

used to incorporate interaction between wedge and cylinder is based on the work by

Karp and Russek [35].

In last section, analysis of the field scattered by a PEMC circular cylinder coated

with double-positive (DPS) or double-negative (DNG) materials and placed under

PEC wide double wedge is presented. Transmission coefficient and diffraction pattern

of PEC wide double wedge in the presence of the coated PEMC circular cylinder are

obtained.

3.1. PEC Wide Slit Excited by Uniform Plane Wave

Scattering of uniform electromagnetic plane wave from a PEC slit is studied. The

geometry and co-ordinates of the problem are shown in Fig. 3.1a. A slit may be viewed

as composed of two coplanar half-planes separated by certain distance. The problem

– 25 –

is two dimensional since the incident field and property of the slit are uniform in z-

direction. It is assumed that the slit is wide, that is, the wavelength of incident plane

wave is smaller than the width of the slit. Therefore, field diffracted by the slit may

be considered as the sum of field diffracted by each isolated half plane. That is field

�

�

�

�

Fig. 3.1a. Slit in a PEC plane.

��

� �

�

�

��

�

�

Fig. 3.1b. Angles of incident and scattered waves with PEC slit.

– 26 –

diffracted by one half plane has no interaction with other half plane. The angle between

the incoming plane wave and the normal to the plane of the screen (measured from the

positive y axis) is θ0, whereas, the angle between the observation point and normal to

the screen (measured from the negative y axis) is θ. All angles are considered positive

if measured counterclockwise with respect to the normal and negative if clockwise.

Moreover, the angles which the incident wave make with the right and left half planes

are φ01 and φ02, respectively as shown in Fig. 3.1b. The incident plane wave is given

as(

Eiz

Hiz

)=

(E0

H0


where φ0 is the angle of incidence with respect to x-axis. Uniform expression for the

field diffracted from wedge, as derived in equation (2.2.8), has the form(

Edz

Hdz

)=

exp[−j(kρ)]√ρ

D sh(ρ, φ, φ0;n)Ei

z (3.1.2)

where

D sh(ρ, φ, φ0, n) =

−√ρ

{−sgn

(sin

(π + φ− φ0

2n

))cos

(π + φ− φ0

2n

)

× F

[√2kρn

∣∣∣sin(

π + φ− φ0

2n

)∣∣∣]

− sgn

(sin

(π − (φ− φ0)

2n

))cos

(π − (φ− φ0)

2n

)

× F

[√2kρn

∣∣∣sin(

π − (φ− φ0)2n

)∣∣∣]

± sgn

(sin

(π + φ + φ0

2n

))cos

(π + φ + φ0

2n

)

× F

[√2kρn

∣∣∣sin(

π + φ + φ0

2n

)∣∣∣]

± sgn

(sin

(π − (φ + φ0)

2n

))cos

(π − (φ + φ0)

2n

)

×F

[√2kρn

∣∣∣sin(

π − (φ + φ0)2n

)∣∣∣]}

(3.1.3)

– 27 –

Ds and Dh are the diffraction coefficients of E- and H- polarization respectively, p =

1n = π

φωand sgn is the signum function. Function F (x) is the Fresnel integral defined

as

F (x) =1π

exp(jx2 + j

π

4

) ∫ ∞

x

exp(−jµ2)dµ (3.1.4)

For n = 2, wedge angle φω is equal to 2π and wedge becomes half plane. By setting

n=2, the diffraction co-efficient for the half plane is obtained as

D sh(ρ, φ, φ0) =

−√ρ

{−sgn

(sin

(π + φ− φ0

4

))cos

(π + φ− φ0

4

)

× F

[√8kρ

∣∣∣sin(

π + φ− φ0

4

)∣∣∣]

− sgn

(sin

(π − (φ− φ0)

4

))cos

(π − (φ− φ0)

4

)

× F

[√8kρ

∣∣∣sin(

π − (φ− φ0)4

)∣∣∣]

± sgn

(sin

(π + φ + φ0

4

))cos

(π + φ + φ0

4

)

× F

[√8kρ

∣∣∣sin(

π + φ + φ0

4

)∣∣∣]

± sgn

(sin

(π − (φ + φ0)

4

))cos

(π − (φ + φ0)

4

)

×F

[√8kρ

∣∣∣sin(

π − (φ + φ0)4

)∣∣∣]}

(3.1.5)

It is assumed that point of observation is far from the slit. For large argument ap-

proximation, Fresnel integral simplifies to

F (x) ≈ 12√

πxexp

[−j

π

4

](3.1.6)

and (3.1.5) becomes

D(φ, φ0) ≈ − 1√8πk

exp[−j

π

4

] [sec

(φ− φ0

2

)∓ sec

(φ + φ0

2

)](3.1.7)

– 28 –

Using the equations (3.1.2) and (3.1.7), diffracted field from right half plane is given

as

Edr (ρ1, φ1) =− 1√

8πkρ− 1

21 exp[−jkρ1] exp

[−j

π

4

]

×[sec

(φ1 − φ01

2

)∓ sec

(φ1 + φ01

2

)]exp[jks cos φ01] (3.1.8)

In the far field from the slit, ρ À d, following relations hold

ρ1 = ρ− s sin θ, φ1 =3π

2+ θ, φ01 =

π

2+ θ0

Above far-field relations are used in the exponential term whereas in the amplitude

term ρ1 = ρ is used. Equation (3.1.8) takes the following form

Edr =

exp[−j(kρ + π

4 )]

√8πkρ

exp[jks(sin θ − sin θ0)]

×[csc

(θ − θ0

2

)∓ sec

(θ + θ0

2

)], ρ À d (3.1.9)

Similarly field diffracted from left half plane can be obtained and is given below

Edl (ρ2, φ2) =− 1√

8πkρ− 1

22 exp[−jkρ2] exp

[−j

π

4

]

×[sec

(φ2 − φ02

2

)∓ sec

(φ2 + φ02

2

)]exp[jks cos φ02](3.1.10)

In the far field of the slit, ρ À d, following relations hold

ρ2 = ρ + s sin θ, φ2 =3π

2− θ, φ02 =

π

2− θ0


term ρ2 = ρ is used

Edl =

exp[−j(kρ + π

4 )]

√8πkρ

exp[−jks(sin θ − sin θ0)]

×[− csc

(θ − θ0

2

)∓ sec

(θ + θ0

2

)], ρ À d (3.1.11)

– 29 –

3.1.1. Field Diffracted From a Wide PEC Slit

Diffracted field from a PEC slit, at an observation point (ρ, φ), may be calculated

by simply adding the results given in (3.1.9) and (3.1.11) as

Ed = Edr + Ed

l =

√k

2πρexp

[−jkρ− j

π

4

]f (1) (3.1.12)

where

f (1) = jsin ks(sin θ − sin θ0)

k sin(

θ−θ02

) ∓ cos ks(sin θ − sin θ0)k cos

(θ+θ0

2

)

� � � � � � � � � � � � � � � � ��

� �

� �

� �

� �

� �

� �

θ

� � ��

Fig. 3.2a. Comparison of diffracted field with Elsherbeni’s result for ks=3, θ0 = 00.

Fig. 3.2b. Comparison of diffracted field with Elsherbeni’s result for ks=8, θ0 = 00.

– 30 –

In the limit θ → θ0, i.e., as the incident angle approaches the observation angle in the

far field, function f (1) simplifies to

f (1) = j2s cos(

θ + θ0

2

)∓ 1

k cos θ0(3.1.13)

Diffracted field patterns are plotted as function of observation angle in Figs. 3.2 and

3.3. Numerical results shown in Fig. 3.2 are in good agreement with the Elsherbeni’s

results [37]. Fig. 3.3 shows the diffraction pattern of the slit at various incident angles

for different values of slit width. It may be noted that the results are valid for all

incident angles.

Diffracted field from PEC slit for ks=4, θ0 = 00.

Fig. 3.3a. Diffracted field from PEC slit for ks=4, θ0 = 300.

– 31 –

Diffracted field from PEC slit for ks=8, θ0 = 00.

� � � � � ��

� � �

� � �

� � �

� � �

� � �

��

θ

Fig. 3.3b. Diffracted field from PEC slit for ks=8, θ0 = 300.

3.2. PEMC Circular Cylinder Placed Under PEC Wide Double Wedge

Scattering of electromagnetic plane wave from a PEMC circular cylinder is con-

sidered. Fig. 3.4a shows that the two faces of the wedge are located at φ = 0 and

φ = 2Φ. Geometry of the problem consists of two parallel conducting wedges sepa-

rated by a distance 2s, where 2ks À 1 and a PEMC circular cylinder of radius a whose

axis is parallel to the edges of two parallel wedges. The angles which the incident and

diffracted rays make with the normal to the screen, that is, along y-axis are θ0 and θ,

respectively. It can also be observed from Fig. 3.4b that the angles which the incident

– 32 –

� �

� �

� � � � � ��

�

�

Fig. 3.4a Geometry of the problem.

� � ��

��

��

�

�

�

� � ��

Fig. 3.4b Angles of incident and scattered waves with PEC wedges.

wave make with the right and left wedges of the geometry, that is, along the x-axis

are φ01 and φ02, respectively. In order to find the solution of the above problem,

it is required to determine scattering from isolated PEC wedge and isolated PEMC

cylinder due to plane wave and cylindrical wave excitation. It may be noted that study

– 33 –

of cylindrical wave excitations is required to incorporate the interaction between two

wedges and between wedge and cylinder.

3.2.1. A PEC Wedge Excited by Plane Wave and Cylindrical Wave

In this sub-section scattering of plane wave and cylindrical wave from an isolated

PEC wedge are presented. For the plane wave excitation on the edge of the wedge at

an angle φ0 with respect to the x−axis, the incident field is given below

(Ei

z

Hiz

)=

(E0

H0


From equation (3.1.2), the uniform expression for the field diffracted from a PEC

wedge has the form

(Ed

z

Hdz

)=

exp[−j(kρ)]√ρ

D sh(ρ, φ, φ0;n)Ei (3.2.2)

where D sh(ρ, φ, φ0; n) for the wedge is defined as

D sh(ρ, φ, φ0, n) =

−√ρ

{−sgn

(sin

(π + φ− φ0

2n

))cos

(π + φ− φ0

2n

)

× F

[√2kρn

∣∣∣sin(

π + φ− φ0

2n

)∣∣∣]

− sgn

(sin

(π − (φ− φ0)

2n

))cos

(π − (φ− φ0)

2n

)

× F

[√2kρn

∣∣∣sin(

π − (φ− φ0)2n

)∣∣∣]

± sgn

(sin

(π + φ + φ0

2n

))cos

(π + φ + φ0

2n

)

× F

[√2kρn

∣∣∣sin(

π + φ + φ0

2n

)∣∣∣]

± sgn

(sin

(π − (φ + φ0)

2n

))cos

(π − (φ + φ0)

2n

)

×F

[√2kρn

∣∣∣sin(

π − (φ + φ0)2n

)∣∣∣]}

(3.2.3)

– 34 –

The scattering of cylindrical wave from an isolated PEC wedge can be determined

by solving problem of a line source in the presence of a conducting wedge whose edge is

parallel to the line source. If the source is of unit amplitude and is located at (ρ0, φ0)

parallel to the z-axis, its field in the absence of the wedge is given as [138]

Eiz =

π

2jH

(2)0 (kR) (3.2.4)

where R is the distance between the line source and the field point and H(2)0 (.) is

the Hankel function of the second kind of order zero. The asymptotic expression for

diffracted field in the presence of the wedge is given below [138].

Ez =π

2jH

(2)0 (kρ)F (φ, ρ0, φ0, n) (3.2.5)

where

F (φ, ρ0, φ0, n) ≈ H(2)0 (kρ0) exp

[−j

π

2

] sin(πn )

πn

×{[

cos(π

n

)− cos

(φ− φ0

n

)]−1

−[cos

(π

n

)− cos

(φ + φ0

n

)]−1}

(3.2.6)

3.2.2. A PEMC Cylinder Excited by Plane Wave and Cylindrical Wave

Scattering of plane wave and cylindrical wave from an isolated PEMC cylinder are

derived. A circular cylinder is defined by the surface ρ = a, while its axis coincides with

the z-axis. The scattered field due to plane wave incidence on circular cylinder [65] is

ECp =

π

2jH0(kρ)Gp(φ, φ0, a) (3.2.7)

where

Gp(φ, φ0, a) = −2j

π

∞∑n=0

εn(−1)nTn cos[n(φ− φ0)] (3.2.8)

– 35 –

Tn is the transmission co-efficient. The subscript p in (3.2.7) indicates expression for

plane wave. Similarly the scattered field due to cylindrical wave incident on circular

cylinder [65] is

ECl =

π

2jH0(kρ)Gl(φ, φ0, a) (3.2.9)

where

Gl(φ, φ0, a) =−∞∑

n=0

εnjnTnHn(kρ0) cos[n(φ− φ0)] (3.2.10)

Values of transmission coefficient for both co- and cross-polarized components of

PEMC cylinder [68] are given as

Tn =

H(2)n (ka)J/

n(ka)+M2η20Jn(ka)H(2)/

n (ka)

(1+M2η20)H

(2)n (ka)H

(2)/

n (ka)Co− polarized

2Mη0

πka(1+M2η20)H

(2)n (ka)H

(2)/

n (ka)Cross polarized

(3.2.11)

In above equations the Neumann number εn = 1 for n=0 and εn = 2 for n > 0, Jn(x)

is the Bessel function of argument x and order n and Hn(x) is the Hankel function

of the second kind of order n and argument x. Primes indicate the derivative with

respect to the whole argument.

3.2.3. PEMC Cylinder Below PEC Wide Double Wedge

The geometry is illuminated by a plane wave of unit amplitude. The field at any

point is the sum of incident field and response field. The response field consists of

non-interaction field and interaction field. The non-interaction field is the scattered

field by each of the two wedges and the cylinder due to the incident plane wave. The

interaction field is due to the three fictitious line sources located at the edge of each

wedge and at the cylinder. This is how the multiple interaction among the three

objects (two wedges and cylinder) is incorporated by adding the contributions of both

the interaction and non-interaction fields.

– 36 –

The total field is mathematically given by [58]

Et = Ei + Es (3.2.12)

where

Es = Es1 + Es2 + Es3 (3.2.13)

and the scattered fields from edges of each wedge and the cylinder are [58]

Es1 =π

2jH0(kρ1)[exp(−jks sin θ0)]D(φ1, φ01, n1)

+ c3F (φ1, s1, φ31, n1) + c2F (φ1, 2s, φ21, n1) (3.2.14)

Es2 =π

2jH0(kρ2)[exp(+jks sin θ0)]D(φ2, φ02, n2)

+ c3F (φ2, s2, φ32, n2) + c1F (φ2, 2s, φ12, n2) (3.2.15)

Es3 =π

2jH0(kρ3)[exp(−jkd cos θ0)]D(φ3, φ03, a)

+ c1G(φ3, s1, φ13, a) + c2G(φ3, s2, φ23, a) (3.2.16)

where n1 = 2π−απ and n2 = 2π−β

π . Each of the above equations contain a non-

interaction term and two interaction terms, that is, equation (3.2.14) is the field

scattered from wedge A in which D(φ1, φ01, n1) represents the non-interaction field,

whereas F (φ1, s1, φ31, n1) and F (φ1, 2s, φ21, n1) are the interaction terms due to the

fictitious line sources located at the cylinder axis and edge of wedge B, respectively.

Similarly the other two equations represent the scattered field from wedge B and the

cylinder. The strength of the fictitious lines sources located at edge of wedge and cylin-

der axis are represented by c1, c2, and c3, respectively. When the observation point

is far from the edges as compared to the width of double wedge kρ2s À 1, approximate

relations can be written. Therefore, using the following far field approximation [58,

64]

φ0 = φ01 = φ03 =π

2+ θ0, φ02 =

π

2− θ0

φ1 = φ3 ' 3π

2+ θ, φ2 ' 3π

2− θ

φ12 = φ21 ' π, φ13 = ψ ' tan−1

(d

s

)

φ31 = φ32 ' π + ψ, φ23 ' π − ψ

– 37 –

and

ρ1 ' ρ− s sin θ, ρ2 ' ρ+ s sin θ, ρ3 ' ρ− d cos θ. Also the distances between the edges

of the two wedges and the cylinder are considered as s1 and s2, respectively.

To determine c1, c2 and c3, the treatment of Karp and Russek [34] has been

followed

2c1 − c2[F (φ31, 2s, φ21, n1) + F (φ21, 2s, φ21, n1)]

− c3[F (φ31, s1, φ31, n1) + F (φ21, s1, φ31, n1)]

= exp(−jks sin θ0)[D(φ31, φ01, n1) + D(φ21, φ01, n1)] (3.2.17)

2c2 − c1[F (φ32, 2s, φ12, n2) + F (φ12, 2s, φ12, n2)]

− c3[F (φ32, s2, φ32, n2) + F (φ12, s2, φ32, n2)]

= exp(jks sin θ0)[D(φ32, φ02, n2) + D(φ12, φ02, n2)] (3.2.18)

2c3 − c1[G(φ13, s1, φ13, a) + G(φ23, s1, φ13, a)]

− c2[G(φ13, s2, φ23, a) + G(φ23, s2, φ23, a)]

= exp(−jkd cos θ0)[D(φ13, φ03, a) + D(φ23, φ03, a)] (3.2.19)

Solving (3.2.17) ∼ (3.2.19) for c1, c2 and c3, the expression for scattered field is given

as

Es =exp(−jkρ)√

πkρE(θ, s, d, n1, n2, a) (3.2.20)

where the scattered field pattern E(θ, s, d, n1, n2, a) is obtained from (3.2.13).

Finally the transmission coefficient T for plane wave incidence is calculated by

using the following expression [34]

T = Re[(1− j)E]/2ks (3.2.21)

where E is E(θ, s, d, n1, n2, a) in the limit as θ approaches θ0.

– 38 –

3.2.4 Results and Discussion

The discussion is divided into two parts. First part includes the analysis of trans-

mission coefficient of PEC wide double wedge in the presence of PEMC cylinder,

whereas, second part comprises discussion related with the diffraction pattern of PEC

wide double wedge.

A comparison of transmission coefficient of PEC slit loaded with PEC cylinder,

(Tc), is made with the transmission coefficient of the slit loaded with a PEMC cylinder.

Both the co-polarized (Tco) and cross-polarized (Tcross) components of transmission

coefficient for PEMC cylinder are studied and their comparison is made with Tc. In all

the cases cylinder radius is taken as ka=0.5. In order to check the validity of code a

comparison of Tc with Tco is made for kd = 0 and kd = 5 and corresponding results are

compared with the Elsherbeni’s work [58], as shown in Figs 3.5a and 3.5b, respectively.

It can be observed that Tco, in both the cases, shows exactly the same behavior as

that of Tc when Mη0 → ∞. In Fig. 3.6, a comparison of Tcross has been made with

the transmission coefficient of an unloaded slit (T ), by making Mη0 = 0. It can be

Fig. 3.5a. Slit transmission coefficient for θ0 = 00, kd=0, ka=0.5, Mη0 →∞.

– 39 –

Fig. 3.5b. Slit transmission coefficient for θ0 = 00, kd=5, ka=0.5, Mη0 →∞.

Fig. 3.6. Slit transmission coefficient for θ0 = 00, kd=0, Mη0 = 0.

observed that the two coefficients have the similar behavior. It is because the cross

polarized component is zero at Mη0 = 0. In Figs. 3.7 and 3.8, a comparison of Tco and

Tcross for Mη0 = 0 and Mη0 = ±1 at kd = 0 and kd = 5 are presented, respectively. In

Fig. 3.7a, it can be seen that when the cylinder, with Mη0 = 0, is at kd = 0 then Tco is

less as compared to Tcross, but when it is shifted below the center of the aperture plane,

say at kd = 5, Tco becomes larger than Tcross which is obvious from Fig. 3.7b. More-

over, the transmission coefficients oscillate with decreasing amplitude as expected and

tend to unity as the slit width ks tends to infinity. But, contrary to this effect, Fig. 3.8b

shows that Tcross is larger at kd = 5, Mη0 = ±1, whereas at kd = 0, Tcross becomes less

– 40 –

Fig. 3.7a. Slit transmission coefficient for θ0 = 00, kd=0, ka=0.5, Mη0 = 0.

Fig. 3.7b. Slit transmission coefficient for θ0 = 00, kd=5, ka=0.5, Mη0 = 0.

Fig. 3.8a. Slit transmission coefficient for θ0 = 00, kd=0, ka=0.5, Mη0 = ±1.

– 41 –

Fig. 3.8b. Slit transmission coefficient for θ0 = 00, kd=5, ka=0.5, Mη0 = ±1.

than Tco. It can also be observed from Fig. 3.8 that Tco and Tcross are larger than

Tc at both kd = 0 and kd = 5. To further highlight the effect of Mη0 on Tcross, it

is obvious from Fig. 3.9 that Tcross is maximum when Mη0 = ±1 and decreases for

other values of Mη0. Similarly Fig. 3.10 shows the effect of variation in ka on Tcross

at Mη0 = ±1. Obviously the value of Tcross is larger for ka = 0.5 and decreases for

smaller values of ka. Both these Figs are for kd = 0. The behavior of Tco and Tcross

for obliquely incident plane wave at θ0 = 150 and θ0 = 300 for ka = 0.1, kd = 0 and

Mη0 = ±1 is shown in Figs. 3.11a and 3.11b. It is observed that at θ0 = 150, Tcross is

higher than unity in the lower range of ks (ks ≤ 2) and is larger than Tco. For the

Fig. 3.9. Slit transmission coefficient (cross polarized) for θ0 = 00, kd=0, ka=0.5.

– 42 –

� � � � � � � � ��

� � �

� � �

� � �

� � �

� � ��

� � � � � ��

�

� �

Fig. 3.10. Slit transmission coefficient (cross polarized) for θ0 = 00, kd=0, Mη0 = ±1.

Fig. 3.11a. Slit transmission coefficient for θ0 = 150, kd=0, ka=0.1, Mη0 = ±1.

� � � � � � � ��

� � � �

� � � �

� � � �

� � � �

� � � �

� � � �

�

� �

� � � � � � � � � �� !

Fig. 3.11b. Slit transmission coefficient for θ0 = 300, kd=5, ka=0.1, Mη0 = ±1.

same cylinder parameters but with θ0 = 300, both Tco and Tcross becomes less than

unity. However, Tcross oscillates with greater amplitude as compared to Tco. Hence

– 43 –

incident angle effects the peak locations of Tco and Tcross. To see the effect of interior

wedge angle on the transmission coefficient when Mη0 = ±1, ka = 0.1, it is observed

that as the wedge angle is increased, the amplitude of oscillation in both Tco and Tcross

Fig. 3.12a. Slit transmission coefficient (co-pol component) for θ0 = 00,

kd=0, ka=0.1, Mη0 = ±1.

� � � � � � � � ��

� � �

� � �

� � �

� � �

αα == ββ == 0000

αα == ββ == 330000

�

Fig. 3.12b. Slit transmission coefficient (cross-pol component) for θ0 = 00

kd=0, ka=0.1, Mη0 = ±1.

is increased i.e., the interior wedge angle effects the levels of maxima and minima of

the oscillation in both the cases, however this effect is more dominant in Tcross as

compared to Tco as shown in Figs. 3.12a and 3.12b.

The normalized diffraction pattern of the slit loaded with PEC cylinder (Dc), com-

– 44 –

pared with the corresponding normalized diffraction patterns in the presence of PEMC

cylinder, is presented. Comparison between co-polarized (Dco) and cross-polarized

(Dcross) components for different values of Mη0 is made. Fig. 3.13a presents Dc com-

pared with Dco for kd=0 and ks=8. The solid curve in the Fig. 3.13a represents Dc. It

is observed that Dco shows similar behavior as that of Dc for Mη0 →∞. Moreover, in

Fig. 3.13b it can be observed that Dcross for both Mη0 = 0 and Mη0 →∞ gives the

same diffraction patterns as that of an unloaded slit (D) which is in good agreement

with the published work [64]. This shows that cross polarized component exists only

Fig. 3.13a. Slit diffraction pattern for θ0 = 00, kd=0, ka=0.5, ks=8.

Fig. 3.13b. Slit diffraction pattern for θ0 = 00, kd=0, ka=0.5, ks=8.

– 45 –

Fig. 3.14. Slit diffraction pattern for θ0 = 00, kd=0, ka=0.5, ks=8, Mη0 = ±1.

Fig. 3.15a. Slit diffraction pattern (co-polarized) for θ0 = 00,

kd=0, ka=0.5, ks=8, Mη0 = ±1.

Fig. 3.15b. Slit diffraction pattern (cross-polarized) for θ0 = 00,

kd=0, ka=0.5, ks=8.

– 46 –

Fig. 3.16. Slit diffraction pattern for θ0 = 00, kd=1.5, ka=0.5, ks=8, Mη0 = ±1.

for Mη0 = ±1 and becomes zero for other values of Mη0. To further investigate

the effect of Mη0 on Dco and Dcross, Fig. 3.14 shows the comparison of both these

diffraction patterns for Mη0 = ±1. It can be seen that the beam width for cross-

polarized component is less than that of co-polarized component. In order to see the

effect of slit width on Dco and Dcross, plots for different values of ks at kd = 0 and

Mη0 = ±1 are shown in Figs. 3.15a and 3.15b. It is observed that the number of

side lobes increases with the increase in slit width for both Dco and Dcross. When the

cylinder is shifted to kd = 1.5 for ks = 8 and Mη0 = ±1, comparison of both Dco and

Dcross is shown in Fig. 3.16.

3.3. Coated PEMC Cylinder Placed Under PEC Wide Double Wedge

The problem of the diffraction of electromagnetic plane wave from a geometry

which contains PEC double wedge separated by a distance 2s, where 2ks À 1, and

a coated PEMC circular cylinder, is presented. The PEMC cylinder is taken to be

infinite along its axis and has been coated with a double positive (DPS) or double

negative (DNG) material. The radius of the inner cylinder is a and the radius of

the coated cylinder is b, both coinciding with the z-axis, as shown in Fig. 3.17. The

– 47 –

geometry is excited by a plane wave. In order to determine the interaction contribution

of the geometry, as already discussed in section (3.2), the scattered field due to plane

wave and cylindrical wave incident on an isolated circular cylinder is required, which

has been given by equations (3.2.7-3.2.10).

� �

� �

� � � � � ��

�

�

�

�

Fig. 3.17. Geometry of the problem.

For coated PEMC cylinder the transmission co-efficient, Tn, for co- and cross

polarized components is given as [90]

Tco =J ′n(k0b)

η0(A)− Jn(k0b)

η1(B)

H(2)n (k0b)

η1(B)− H

(2)′n (k0b)

η0(A)

(3.3.1)

Tcross =jMη1

[H(1)

n (k1a)− H(2)n (k1a)H(1)′

n (k1a)

H(2)′n (k1a)

]

×[

Jn(k0b)H(2)′n (k0b)− J ′n(k0b)H

(2)n (k0b)

η0H(2)n (k0b)(B)−H

(2)′n (k0b)(A)

](3.3.2)

where

A =

[(ac

b+ d

)− jMη1

H(2)n (k1a)

H(2)′n (k1a)

(ea

b+ f

)][H(1)

n (k1b)− H(2)n (k1b)H

(1)′n (k1a)

H(2)′n (k1a)

]

+ jMη1H

(2)n (k1b)

H(2)′n (k1a)

(ea

b+ f

)[H(1)

n (k1a)− H(2)n (k1a)H(1)′

n (k1a)

H(2)′n (k1a)

](3.3.3)

– 48 –

B =

[H(1)′

n (k1b)− H(2)′n (k1b)H

(1)′n (k1a)

H(2)′n (k1a)

] [(ac

b+ d

)− jMη1

H(2)n (k1a)

H(2)′n (k1a)

(ea

b+ f

)]

+ jMη1H

(2)′n (k1b)

H(2)′n (k1a)

(ea

b+ f

)[H(1)

n (k1a)− H(2)n (k1a)H(1)′

n (k1a)

H(2)′n (k1a)

](3.3.4)

and

a =H

(2)n (k0b)

η0− H

(2)n (k1b)H

(2)′n (k0b)

η1H(2)′n (k1b)

(3.3.5)

b =1η1

[H(1)

n (k1b)− H(2)n (k1b)H

(2)′n (k1b)

H(2)′n (k1b)

](3.3.6)

c = H(1)n (k1a)− H

(2)n (k1a)H(1)′

n (k1b)

H(2)′n (k1b)

(3.3.7)

d =H

(2)n (k1a)H(2)′

n (k0b)

H(2)′n (k1b)

(3.3.8)

e = jMη1

[H(1)′

n (k1a)− H(1)′n (k1b)H

(2)′n (k1a)

H(2)′n (k1b)

](3.3.9)

f = jMη1H

(2)′n (k0b)H

(2)′n (k1a)

H(2)′n (k1b)

(3.3.10)

The field at an observation point is considered to be composed of the incident field

plus a response field from each of the edge of two wedges and the cylinder. The total

field is given as

Et = Ei + Es

where

Es = Es1 + Es2 + Es3 (3.3.11)

Es1, Es2, and Es3 are defined by equations (3.2.2) − (3.2.4). Far field conditions are

used, as already presented in section (3.2). The scattered field is determined by using

the equations (3.2.13-3.2.16).

– 49 –


The transmission coefficient and diffraction pattern of PEC wide double wedge

in the presence of coated PEMC circular cylinder are presented. Behavior of both co-

polarized (T cco) and cross-polarized (T c

cross) components of coated PEMC cylinder are

discussed. In all the plots radius of un-coated cylinder is taken as ka=0.15 and that of

coated cylinder as kb=0.2. The validity of the code has been checked by making the

coating equal to zero. Results are found to be in agreement with un-coated

Fig. 3.18a. Slit transmission coefficient for θ0 = 00,

Mη1 = ±1, kd=0, εr = −1.5, µr = −1.

Fig. 3.18b. Slit transmission coefficient for θ0 = 00,

Mη1 = ±1, kd=5, εr = −1.5, µr = −1.

– 50 –

PEMC cylinder. Comparison of T cco and T c

cross for Mη1 = ±1 at kd = 0 and kd = 5

taking relative permitivity εr = −1.5 and relative permeability µr = −1, are shown in

Figs. 3.18a and 3.18b, respectively. It can be seen that in both the cases, T ccross is larger

than T cco, which is contrary to un-coated PEMC cylinder for kd = 5 in which Tco is less

than Tcross at Mη0 = ±1. Furthermore, it is observed that the transmission coefficient

is large in the presence of coated PEMC cylinder as compared to PC cylinder. In both

the cases T cco and T c

cross are greater than unity whereas Tc, in general, remains less

than unity. The variation in the radius of coated cylinder also effects the behavior of

� � � � � � � � ��

� � � �

� � � �

� � � �

� � � �

� � � �

� � � � � � �

� � � � � � ��

�

� �

Fig. 3.19a. Slit transmission coefficient for θ0 = 00, kd=0.

� � � � � � ��

�

�

�

�

�

��

�

� �

Fig. 3.19b. Slit transmission coefficient for θ0 = 00, kd=0.

T cco and T c

cross as shown in Fig. 3.19. Fig. 3.19b shows that T cco oscillates with greater

amplitude as the value of b is increased. However, T ccross does not show considerable

– 51 –

Fig. 3.20a. Slit transmission coefficient for θ0 = 00, kd=0, ka=0.15, kb=0.2.

Fig. 3.20b. Slit transmission coefficient for θ0 = 00, kd=5, ka=0.15, kb=0.2.

Fig. 3.21a. Slit transmission coefficient for θ0 = 00, kd=0, ka=0.15, kb=0.2.

– 52 –

Fig. 3.21b. Slit transmission coefficient for θ0 = 00, kd=0, ka=0.15, kb=0.2.

change in behavior with the increase in radius b as hi-lighted in Fig. 3.19a. The behavior

of T cco and T c

cross for oblique incidence case with incident angles θ0 = 200 and θ0 = 300

are shown in Fig. 3.20. Fig. 3.20a shows that T ccross becomes less than unity as the

angle of incidence is increased from zero, whereas the amplitude of oscillation for T cco

decreases with the increase of incidence angle θ0 as shown in Fig. 3.20b. All the plots of

Figs. 3.19 and 3.20 are for Mη1 = ±1. Further more, it can be observed from Fig. 3.21

that interior wedge angle effects the peak-to-peak values of the oscillations both in the

case of T cco and T c

cross. In case of T ccross, as shown in Fig. 3.21a, the oscillations are

always around unity and decreases with increasing ks whereas in case of T cco as shown

in Fig. 3.21b, the oscillations are larger and are greater than unity. The plots for DPS

coated cylinder show almost similar behavior as that of DNG coated cylinder.

In the last part of discussion, diffraction pattern of wide double wedge in the

presence of coated PEMC cylinder is presented. In Fig. 22, the effect of Mη1 on

the diffraction pattern is shown. Behavior of both co-polarized and cross-polarized

components of coated PEMC cylinder, that is, (Dcco) and (Dc

cross), taking εr = 1.5

and µr = 1 for kd = 0 and ks = 8 is studied. In both the cases, it can be seen that

Dcco and Dc

cross show slight different behavior for Mη1 = 1 as compared to other values

of Mη1. Fig. 23 shows the variation in Dcco and Dc

cross for εr = 1.5, µr = 1 and kd = 0

– 53 –

Fig. 3.22a. Slit diffraction pattern for θ0 = 00, kd=0, ks=8, ka=0.15, kb=0.2.

Fig. 3.22b. Slit diffraction pattern for θ0 = 00, kd=0, ks=8, ka=0.15, kb=0.2.

Fig. 3.23a. Slit diffraction pattern for θ0 = 00, kd=0, ka=0.15, kb=0.2.

– 54 –

� � � � � ��

� � � �

� � � �

� � � �

� � � ��

��

θθ

Fig. 3.23b. Slit diffraction pattern for θ0 = 00, kd=0, ka=0.15, kb=0.2.

with respect to the slit width. It is observed that both Dcco and Dc

cross show different

behavior for different values of slit widths.

3.4. Conclusion

The diffraction pattern of PEC slit using the NNH wedge diffraction function is

presented. It is found that the results are in fairly good agreement with the published

work which shows the validity of the newly derived NNH wedge diffraction function.

It is shown that the diffracted field remains uniform at all incident angles. Further

investigation of the wedge diffraction function is also made by considering a complex

problem of three scatterers, that is, two parallel PEC wedges and a PEMC cylinder.

The transmission coefficient and the diffraction pattern of PEC wide double wedges

loaded with a PEMC cylinder are presented and compared with published work. The

comparison shows that the transmission coefficient of PEC wide double wedge has a

high value in the presence of PEMC cylinder instead of PEC cylinder. Furthermore, it

is observed that the transmission coefficient varies under particular conditions such as

by either shifting the cylinder below the center of the aperture plane of PEC double

wedge or by coating the PEMC cylinder with DPS or DNG materials. Variations

in the transmission coefficient with respect to the admittance parameter of both un-

coated and coated PEMC cylinders is also studied. It is found that the behavior of

– 55 –

Tco and Tcross of an un-coated PEMC cylinder and T cco and T c

cross of coated PEMC

cylinder varies not only with the incident angles of the original plane wave but also a

considerable change takes place in the behavior of the transmission coefficients if the

interior wedge angles are changed.

– 56 –

CHAPTER IV

PEMC Cylinder Placed Under an Impedance Slit

The chapter deals with the comparison of transmission coefficient of PEC slit (as

evaluated by using the NNH wedge diffraction function in chapter III) with that of

impedance geometry. Being more closer to practical nature, the comparison, as made

in this chapter for some special cases of impedance wedge, can be extended to more

complex problems to give further insight of the transmission coefficient viz-a-viz PEC

geometries. First section of the chapter deals with the diffraction of plane wave from

a geometry which contains an infinite slit in an impedance plane using Maliuzhinets

function. Results for both E- and H-polarized fields are presented. The transmission

coefficient of an impedance slit is further studied in the presence of a PEMC cylinder

and results of special cases are compared with those obtained in chapter III. The

method employed for the analysis and determination of transmission coefficient is

same as that used in chapter III.

4.1. Impedance Slit Excited by Plane Wave

Scattering of uniform electromagnetic plane wave from an impedance slit is stud-

ied. The geometry and co-ordinates of the problem are shown in Fig. 4.1a. An

impedance slit may be viewed as composed of two coplanar half-planes separated by

certain distance each having the same values of face impedance. The face impedance

for each half plane may be defined as ζ±, where positive and negative signs represent

upper and lower surfaces of the half plane, respectively. The problem is two dimen-

sional since the incident field and property of the slit are uniform in z-direction. It is

assumed that the slit is wide, i.e., the wave length of incident plane waves is smaller

than the width of the slit. Therefore, field diffracted by the slit may be considered as

– 57 –

the sum of field diffracted by each isolated impedance half plane, that is, field diffracted

by one half plane has no interaction with other half plane.

�

�

�

�

Fig. 4.1a. Slit in an impedance plane

��

� �

�

�

��

�

�

Fig. 4.1b. Angles of incident and scattered waves with impedance slit.

The angles between the incident and diffracted rays with the normal to the screen,

i.e., along y-axis, are θ0 and θ, respectively. Whereas the angles which the incident

– 58 –

wave make with the right and left half planes are φ01 and φ02, respectively as shown

in Fig. 4.1b. The incident plane wave is given as(

Eiz

Hiz

)=

(E0

H0


The uniform expression for the field diffracted from wedge has the form(

Edz

Hdz

)=

exp[−jkρ)]√ρ

D sh(ρ, φ, φ0; p)Ei (4.1.2)

The diffraction coefficient for impedance wedge can be defined as [93]

D sh(ρ, φ, φ0, n) =

−√ρ

{Ψ(Φ− π − φ)

Ψ(Φ− φ0)sgn

(sin

(π + φ− φ0

2n

))

× cos(

π + φ− φ0

2n

)F

[√2kρn

∣∣∣sin(

π + φ− φ0

2n

)∣∣∣]

+Ψ(Φ + π − φ)

Ψ(Φ− φ0)sgn

(sin

(π − (φ− φ0)

2n

))

× cos(

π − (φ− φ0)2n

)F

[√2kρn

∣∣∣sin(

π − (φ− φ0)2n

)∣∣∣]

− Ψ(Φ− π − φ)Ψ(Φ− φ0)

sgn

(sin

(π + φ + φ0

2n

))

× cos(

π + φ + φ0

2n

)F

[√2kρn

∣∣∣sin(

π + φ + φ0

2n

)∣∣∣]

− Ψ(Φ + π − φ)Ψ(Φ− φ0)

sgn

(sin

(π − (φ + φ0)

2n

))

× cos(

π − (φ + φ0)2n

)F

[√2kρn

∣∣∣sin(

π − (φ + φ0)2n

)∣∣∣]}

(4.1.3)

Ds and Dh are the diffraction coefficients for E and H polarizations respectively with

the same form except the definition of sin θ± contained in the functions Ψ(α), that is,

sin θ± = 1ζ±

for E-polarization and sin θ± = ζ± for H-polarization.

Ψ(α) = ψΦ

(α + θ+ +

π

2

)ψΦ

(α− θ+ +

3π

2

)ψΦ

(α + θ− − 3π

2

)ψΦ

(α− θ− − π

2

)

(4.1.4)

– 59 –

where ψΦ(.) is the Maliuzhinets function.

For the half plane, n = 2, φω = 2π and Φ = π, therefore

D sh(ρ, φ, φ0, 2) =

−√ρ

{Ψ(−φ)

Ψ(π − φ0)sgn

(sin

(π + φ− φ0

4

))

× cos(

π + φ− φ0

4

)F

[√8kρ

∣∣∣sin(

π + φ− φ0

4

)∣∣∣]

+Ψ(2π − φ)Ψ(π − φ0)

sgn

(sin

(π − (φ− φ0)

4

))

× cos(

π − (φ− φ0)4

)F

[√8kρ

∣∣∣sin(

π − (φ− φ0)4

)∣∣∣]

− Ψ(−φ)Ψ(π − φ0)

sgn

(sin

(π + φ + φ0

4

))

× cos(

π + φ + φ0

4

)F

[√8kρ

∣∣∣sin(

π + φ + φ0

4

)∣∣∣]

− Ψ(2π − φ)Ψ(π − φ0)

sgn

(sin

(π − (φ + φ0)

4

))

× cos(

π − (φ + φ0)4

)F

[√8kρ

∣∣∣sin(

π − (φ + φ0)4

)∣∣∣]}

(4.1.5)

and

Ψ(α) = ψπ

(α + θ+ +

π

2

)ψπ

(α− θ+ +

3π

2

)ψπ

(α + θ− − 3π

2

)ψπ

(α− θ− − π

2

)

(4.1.6)

The Maliuzhinets function ψπ(α) for the half-plane is given by [93]

Ψπ(α) = exp

[− 1

8π

∫ α

0

π sin t− 2√

2π sin t2 + 2t

π cos tdt

](4.1.7)

Using the equations (4.1.2) and (4.1.5), diffracted filed from right half plane is given

as

Edr (ρ1, φ1) =Dr(ρ1, φ01, φ1)ρ

− 12

1 exp [−jkρ1] Ei (4.1.8)

– 60 –

where

Dr(ρ1, φ01, φ1) = − 14√

2πk

[Ψ(−φ1)

Ψ (π − φ01)cot

(π + φ1 − φ01

4

)

+Ψ(2π − φ1)Ψ (π − φ01)

cot(

π − φ1 + φ01

4

)

− Ψ(−φ1)Ψ (π − φ01)

cot(

π + φ1 + φ01

4

)

−Ψ(2π − φ1)Ψ (π − φ01)

cot(

π − φ1 − φ01

4

)](4.1.9)

In the far field of the slit (ρ À s)

ρ1 = ρ− s sin θ, φ1 =3π

2+ θ, φ01 =

π

2+ θ0


term ρ1 = ρ is used.

Dr

(ρ,

π

2+ θ0,

3π

2+ θ

)= − 1

4√

2πk

[Ψ

(− 3π2 − θ

)

Ψ(

π2 − φ0

) cot(

2π + θ − θ0

4

)

+Ψ

(π2 − θ

)

Ψ(

π2 − φ0

)cot(

θ0 − θ

4

)

− Ψ(− 3π

2 − θ)

Ψ(

π2 − φ0

) cot(

3π + θ + θ0

4

)

− Ψ(

π2 − θ

)

Ψ(

π2 − φ0

)cot(−π − θ − θ0

4

)](4.1.10)

The field diffracted from the left half plane is

Edl (ρ2, φ2) = Dl(ρ2, φ02, φ2)ρ

− 12

2 exp[−jkρ2]Ei (4.1.11)

where

Dl(ρ2, φ02, φ2) = − 14√

2πk

[Ψ(−φ2)

Ψ (π − φ02)cot

(π + φ2 − φ02

4

)

+Ψ(2π − φ2)Ψ (π − φ02)

cot(

π − φ1 + φ02

4

)

− Ψ(−φ2)Ψ (π − φ02)

cot(

π + φ2 + φ02

4

)

−Ψ(2π − φ2)Ψ (π − φ02)

cot(

π − φ2 − φ02

4

)](4.1.12)

– 61 –

In the far field of the slit (ρ À s)

ρ2 = ρ + s sin θ, φ2 =3π

2− θ, φ02 =

π

2− θ0


term ρ2 = ρ is used.

Dl

(ρ,

π

2− θ0,

3π

2− θ

)= − 1

4√

2πk

[Ψ

(− 3π2 + θ

)

Ψ(

π2 + φ0

) cot(

2π − θ + θ0

4

)

+Ψ

(π2 + θ

)

Ψ(

π2 + φ0

)cot(

θ − θ0

4

)

− Ψ(− 3π

2 + θ)

Ψ(

π2 + φ0

) cot(

3π − θ − θ0

4

)

− Ψ(

π2 − θ

)

Ψ(

π2 − φ0

)cot(−π + θ + θ0

4

)](4.1.13)

The field diffracted from the slit can be calculated by taking the linear combination

of the two terms (4.1.8) and (4.1.11)

E(ρ, φ) = Er(ρ1, φ1) + El(ρ2, φ2) (4.1.14)


The diffraction pattern and transmission coefficient of an impedance slit are stud-

ied. The results shown in Figs. 4.2 to 4.5 give the comparison of E-polarized and

H-polarized fields. It may be noted that the results are valid for all incident angles

and are in fairly good agreement with the published work. The plots presented in

Figs 4.6 and 4.7 give the comparison of the diffraction pattern of E-polarized plane

wave when the face impedance is changed. Also the comparison of H-polarized plane

wave for different values of face impedance is given in Figs 4.8 and 4.9. The diffrac-

tion patterns show slight variations in the fields with the change in the values of face

impedance.

– 62 –

Fig. 4.2. Face impedance 0.2-0.5j, kd=4 and θ0 = 00.



– 63 –


Fig. 4.6. E-polarized field - Comparison for different values of

face impedance at kd=4, θ0 = 00.

Fig. 4.7. E-polarized field - Comparison for different values of


– 64 –

Fig. 4.8. H-polarized field - Comparison for different values of


Fig. 4.9. H-polarized field - Comparison for different values of


Fig. 4.10. Comparison of transmission coefficient of PEC slit and impedance slit.

– 65 –

Fig. 4.10 shows the comparison of transmission coefficient of PEC slit (T ) and

impedance slit (T i). It is observed that both the transmission coefficients show almost

similar behavior except that T is slightly larger than T i. Both oscillate with decreasing

amplitudes for increasing ks and tends to unity as ks tends to infinity.

4.2. PEMC Cylinder Placed Under an Impedance Slit

Scattering of electromagnetic plane wave from a PEMC cylinder placed under

impedance slit is considered. The geometry of the problem is shown in Fig. 4.11.

�

�

�

�

�

�

Fig. 4.11. Impedance slit loaded with PEMC cylinder.

Radius of PEMC cylinder is a. Parameter d represents the distance of the PEMC

cylinder from the edge of the slit. The angles between the incident and diffracted rays

with the normal to the screen, i.e., along y-axis are θ0 and θ, respectively. Whereas,

the angles which the incident waves make with the right and left half planes are φ01 and

φ02, respectively. In order to find the solution of the above problem, it is required to

determine scattering from isolated impedance half plane and isolated PEMC cylinder

due to plane wave and cylindrical wave excitation. It may be noted that study of

– 66 –

cylindrical wave excitations is required to incorporate the interaction between two half

planes and between half plane and cylinder.

4.2.1. Isolated Impedance Half Plane Excited by Plane Wave and Cylin-

drical Wave

In this sub-section scattering of plane wave and cylindrical wave from an isolated

impedance half plane are presented. For the plane wave excitation on the edge of the

half plane at an angle φ0 with respect to the x−axis, the incident field is given below(

Eiz

Hiz

)=

(E0

H0


The uniform expression for the field diffracted from impedance wedge has the form

as given in equation (4.1.3). When the observation point is far from the edge of the

wedge, the Fresnel function in this equation can be approximated by its asymptotic

expansion (4.1.9). By considering only the dominant term, equation (4.1.3) reduces to

the following equation

D sh(ρ, φ, φ0, n) ≈− 1

n√

2πk

(−j

π

4

)sin

φ0

n

{Ψ(Φ− π − φ)

Ψ(Φ− φ0)

[cos

φ0

n− cos

π + φ

n

]−1

−Ψ(Φ + π − φ)Ψ(Φ− φ0)

[cos

φ0

n− cos

π − φ

n

]−1}

(4.2.2)

The scattering of cylindrical wave from an isolated PEC wedge can be determined

by solving problem of a line source in the presence of a conducting wedge whose edge

is parallel to the source. If the source is of unit amplitude and is located at (ρ0, φ0)

parallel to the z-axis, its field in the absence of the wedge is given as [139]

Eiz =

π

2jH

(2)0 (kR) (4.2.3)

where R is the distance between the line source and the field point, k is the wave num-

ber, and H(2)0 (.) is the Hankel function of the second kind of order zero. The asymptotic

expression for diffracted field in the presence of the wedge is given below [139].

E =π

2jH

(2)0 (kρ)F (φ, ρ0, φ0, n) (4.2.4)

– 67 –

where

F (φ, ρ0, φ0, n) ≈H(2)0 (kρ0) sin(φ0

n )n

{Ψ(Φ− π − φ)

Ψ(Φ− φ0)

[cos

φ0

n− cos

π + φ

n

]−1

−Ψ(Φ + π − φ)Ψ(Φ− φ0)

[cos

φ0

n− cos

π − φ

n

]−1}

(4.2.5)

4.2.2. A PEMC Cylinder Excited by Plane Wave and Cylindrical Wave

Scattering of plane and cylindrical waves from an isolated PEMC cylinder are

derived. A circular cylinder is defined by the surface ρ = a, while its axis coincides

with the z-axis. The expressions of scattered field due to plane and cylindrical waves

incident on circular cylinder, as derived in chapter III, are given by equations (3.2.7-

3.2.10). The transmission coefficients of both co- and cross-polarized components of

un-coated PEMC cylinder has also been defined by equation (3.2.11) and (3.2.12) as

Tn =

H(2)n (ka)J/

n(ka)+M2η20Jn(ka)H(2)/

n (ka)

(1+M2η20)H

(2)n (ka)H

(2)/

n (ka)Co− polarized

2Mη0

πka(1+M2η20)H

(2)n (ka)H

(2)/

n (ka)Cross polarized

(4.2.6)

In above equations Jn(.) is the Bessel function of order n and Hn(.) is the Hankel

function of second kind of order n. Primes indicate the derivative with respect to the

whole argument.

The field at an observation point is considered to be composed of the incident

field plus a response field from each of the two half planes and the cylinder. The total

field in the forward direction is given by

Et = Ei + Es

where

Es = Es1 + Es2 + Es3 (4.2.7)

Es1, Es2 and Es3 are defined by equations (3.2.2) − (3.2.4). Well-known far field

conditions are used as already presented in section (3.2). The analysis of Karp and

Russek [34] has been followed. The scattered field has the same expressions as given

by equations (3.2.13-3.2.16).

– 68 –

4.3. Results and Discussion

In this chapter the diffraction pattern and transmission coefficient of an impedance

slit and the slit loaded with PEMC cylinder is studied. In all the cases considered

for evaluation, cylinder radius (ka) is taken as 0.5 and its location from the slit is

taken as kd = 0. A comparison between the transmission coefficients of PEC and

impedance slits loaded with a PEMC cylinder for co-polarized components (Tco and

T ico) at Mη0 → ∞ is shown in Fig. 4.12. It can be observed that both Tco and T i

co

show almost similar behavior, however, T ico is slightly less than unity. In Fig. 4.13, a

comparison of the transmission coefficient of impedance slit loaded with PEMC

Fig. 4.12. Slit transmission coefficient at θ0 = 00, ka=0.5, Mη0 →∞.

Fig. 4.13. Slit transmission coefficient at θ0 = 00, ka=0.5, Mη0 = 0.

– 69 –

Fig. 4.14. Slit transmission coefficient at θ0 = 00, ka=0.5, Mη0 = 0.

cylinder for cross polarized component (T icross) at Mη0 = 0 is made with the trans-

mission coefficient of an unloaded impedance slit (T i). It can be observed that the

two coefficients have the same behavior for the obvious reason that the cross polarized

component is zero at Mη0 = 0. Fig. 4.14 gives the comparison of T ico and T i

cross at

Mη0 = 0. It can be seen that T ico is less than unity but oscillates with decreasing

amplitude and approaches unity when ks approaches infinity. A comparison of both

T ico and T i

cross at Mη0 = ±1 is also made in Fig. 4.15. T icross is much larger than T i

co

Fig. 4.15. Slit transmission coefficient at θ0 = 00, ka=0.5, Mη0 = ±1.

– 70 –

Fig. 4.16. Slit transmission coefficient at θ0 = 00, ka=0.5.



– 71 –

Fig. 4.19. Slit transmission coefficient (cross-pol) at ka=0.1, Mη0 = ±1.

as expected. To further investigate the effect of Mη0 on T icross, it is observed from

Fig. 4.16 that T icross is maximum when Mη0 = ±1 and decreases for other values of

Mη0. Similarly Figs. 4.17 and 4.18 show the effect of variation in ka on T icross and T i

co

at Mη0 = ±1, respectively. Obviously the value of T icross is larger for ka = 0.5 and

decreases for smaller values of ka whereas T ico does not show considerable variation

for different values of ka. The behavior of T ico and T i

cross for obliquely incident plane

wave at θ0 = 150, θ0 = 200 and θ0 = 300 for ka = 0.1, kd = 0 and Mη0 = ±1 is

shown in Figs. 4.19 and 4.20. It is observed that both T ico and T i

cross become less than

unity at angles other than θ0 = 00. In Figs. 4.21 and 4.22, the effect of different values

of surface impedance on transmission coefficient is studied. A visible change in T ico

is observed in the lower range of ks (ks ≤ 4) as the surface impedance is changed.

Similar variation can also be seen in T icross but for the range ks ≤ 2. However, in this

case the amount of the variation is larger as compared to T ico.

– 72 –

Fig. 4.20. Slit transmission coefficient (co-pol) at ka=0.1, Mη0 = ±1.

Fig. 4.21. Slit transmission coefficient (co-pol) for different values of

surface impedance at ka=0.1, Mη0 = ±1.

Fig. 4.22. Slit transmission coefficient (cross-pol) for different values of

surface impedance at ka=0.1, Mη0 = ±1.

– 73 –

4.4. Conclusion

In the first part of the chapter a simple and convenient expression for the field

diffracted by an infinite slit in an impedance plane has been derived for the wavelength

greater than or equal to the slit width. The principal result is that this field can be

accurately calculated everywhere by considering each half plane composing the screen,

to be excited by the incident plane waves. Comparison of E and H polarizations is

made and the effect of surface impedance on E and H polarized fields is also presented.

In second part of the chapter, transmission coefficient of impedance slit loaded

with PEMC cylinder is discussed and compared with that of PEC geometry as eval-

uated in chapter III. Both co- and cross polarized components of PEMC cylinder are

analyzed. The effects of ka, Mη0 and φ0 on both T ico and T i

cross are presented and

discussed.

– 74 –

CHAPTER V

Conclusion

This chapter contains conclusions based on the research work carried out in this

thesis.

A new uniform expression for the wedge diffraction integral, called as NNH wedge

diffraction function, is evaluated by applying the steepest decent method. In contrast

to the widely used expression given by Kouyoumjian and Pathak, the NNH wedge

diffraction function gives a uniform solution without switching of parameter in the

argument depending on the observation point. Therefore, it is easy to make the nu-

merical code. To verify the validity and precision of the NNH solution, its numerical

comparison is made with the exact series solution and PKP function. The results are

presented in the form of a table. The agreement among the three solutions is fairly

well. Moreover, the results of the three methods are also plotted and compared for

various incident angles. It is found that the total field with the NNH result is continu-

ous at all the incident angles and shows a fairly good agreement with the exact series

solution and PKP method.

To check the validity of the NNH wedge diffraction function, the solution for the

diffraction of an incident plane wave by a slit in PEC plane using NNH solution is

studied. It is found that the results of the slit are uniform at all incident angles.

Moreover, the results compared with the published work are in fairly good agreement.

Further investigation of the NNH wedge diffraction function is made by consid-

ering a complex problem of three scatterers, that is, two parallel PEC wedges and a

PEMC cylinder. The transmission coefficient and the diffraction pattern of PEC wide

double wedge loaded with a PEMC cylinder are presented and some of the special

– 75 –

cases are compared with published work. Comparison shows that the transmission co-

efficient of PEC wide double wedge has a high value in the presence of PEMC cylinder

instead of PEC cylinder. Furthermore, it is observed that the transmission coefficient

varies under particular conditions such as by either shifting the cylinder below the

center of the aperture plane of PEC double wedge or by coating the PEMC cylinder

with DPS or DNG materials. Variations in the transmission coefficient with respect

to the admittance parameter of both un-coated and coated PEMC cylinders is also

studied. It is found that the behavior of Tco and Tcross of an un-coated PEMC cylinder

and T cco and T c

cross of coated PEMC cylinder varies not only with the incident angles

of the original plane wave but also a considerable change takes place in the behavior

of the transmission coefficients if the interior wedge angles are changed.

A comparison of the transmission coefficient of PEC slit, as evaluated by using

NNH solution are also compared with that of an impedance slit. The problem a slit

in an impedance slit is solved by the Maliuzhinets function. The comparison of the

transmission coefficient of impedance slit, made with that of PEC slit, shows that the

behavior of both the transmission coefficients is similar, however, the transmission co-

efficient of impedance slit is slightly less than that of PEC slit. Both the transmission

coefficients tends to unity as the slit width tends to infinity. It is also shown that the

diffracted field of an impedance slit can be accurately calculated everywhere and the

results are valid for all incident angles. Comparison of E and H polarized fields at

various incident angles as well as with different values of the face impedance is made.

In the final analysis, a PEMC cylinder is placed under an impedance slit. The trans-

mission coefficient obtained in the presence of PEMC cylinder is compared with that

of PEC geometry which shows slight difference in the two results. The transmission

coefficient of impedance slit in the presence of PEMC cylinder is further studied by

changing the various parameters of PEMC cylinder and the slit like admittance pa-

rameter, slit width etc. Both co-and cross-polarized components are analyzed and the

comparison of the two is also made by changing these parameters.

– 76 –

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