.

DIFFRACTION OF ELECTROMAGNETIC PLANE WAVES

FROM STRIPS AND SLITS USING THE METHOD OF

KOBAYASHI POTENTIAL

M. AMJAD IMRAN

In Partial Fulfilment of the Requirements

for the Degree of

Doctor of Philosophy

Department of Electronics

Quaid-i-Azam University

Islamabad, Pakistan

2010

– ii –

.

CERTIFICATE

It is to certify that Mr. M. Amjad Imran carried out the work contained in this dissertation under my

supervision.

Dr. Qaisar Abbas Naqvi

Associate Professor,

Department of Electronics,

Quaid-i-Azam University

Islamabad, Pakistan

Submitted through

Dr. Qaisar Abbas Naqvi

Chairman,

Department of Electronics,

Quaid-i-Azam University

Islamabad, Pakistan

– iii –

.

Acknowledgments

I humbly thank Almighty Allah, the most Gracious and the Most Merciful, who blessed me much than I

wished. I also pay the tributes to the Holy Prophet Muhammad (PBUH), the most perfect and exalted

First and foremost, I feel great pleasure and honor to express my sincere gratitude and heartfelt thanks

to my supervisor Dr. Qaisar Abbas Naqvi, associate professor, Department of Electronics Quaid-i-

Azam University Islamabad for his dynamic supervision, propitious guidance, continuous encouragement,

constructive criticism and consolatory behavior that he rendered during my stay with him.

Special thanks are due to the dearest Prof. K. Hongo from Japan who introduced me the method of

“Kobayashi Potential” during his tour to Pakistan in 2004. Whenever I approached him in this regard for

any kind of support, guidance or favor, always found him kind, very humane and professional. He is the

person who helped me out in refining my thoughts particularly in the research.

I am exceedingly lucky to have wonderful friends Fazli Mannan, Akhtar Hussain, Maj. Naveed,

Shakeel Ahmed and Abdul Ghaffar who shared all the good and bad times of my Ph.D research tenure

and have always boosted my morale especially in times of difficulty. Some very special and cheerful thanks

are extended to Ahsan Allahi for his sincere attitude towards me.

I do not have words at my command to express my heartiest thanks, gratitude and profound admiration to

my affectionate parents and family who are the source of encouragement for me. In fact this work became

possible only because of their love, moral support and prayers for success. My wife and kids deserve special

admiration for patiently enduring the false optimism of my often repeated claim that “ It would be finished

by next month”

(M. Amjad Imran)

– iv –

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Abstract

Kobayashi potential method has successfully been applied to potential as well as scattering geometries

containing perfect electrically conducting (PEC) objects by many investigators. The purpose of present

study is to extend Kobayashi potential method to study the scattering from non-PEC objects hence, to

enhance the applicability of the method. First, geometries containing strip are considered and diffracted

fields have been determined from an impedance strip, from a strip placed at dielectric slab and from a

perfectly electromagnetic conducting (PEMC) strip. Then slit geometries are included. And studies are

conducted to analyze the diffraction from an impedance slit placed at the interface of two different media,

from two parallel slits in an impedance plane and from a slit in PEMC plane.

While applying this method to above type of problems, diffracted fields are considered in terms of unknown

weighting functions. Imposition of boundary conditions give dual integral equations. These dual integral

equations are then used to decide the nature of weighting functions by using the discontinuous properties of

Weber-Schafheitlin’s integral. Edge conditions are also taken into account at this moment. Finally, matrix

equations are obtained to evaluate the expansion coefficients. The elements of these matrix equations are the

infinite integrals and are usually, very complex in nature and hard to solve analytically. So these integrals

and the matrix equations are then solved numerically for unknown expansion coefficients.

Diffracted fields are presented for each geometry. Their dependence on different parameters like angle

of incidence, slit/ strip size, impedance of plane, relative permittivity of the surrounding media has been

discussed and analyzed. Comparison with physical optics is also presented in some problems to validate the

presented results.

– v –

.

List of Publications

This manuscript is based on the following research articles

1 A. Imran, Q. A. Naqvi and K. Hongo, “Diffraction of Plane Wave by two Parallel Slits in an Infinitely Long

Impedance Plane using the Method of Kobyashi Potential” Progress in Electromagnetic Research, PIER

63, 107-123, 2006

2 A. Imran, Q. A. Naqvi and K. Hongo, “Diffraction of Electromagnetic Plane Wave by an Impedance Strip”

Progress in Electromagnetic Research, PIER 75, 303-318, 2007

3 A. Imran, Q. A. Naqvi and K. Hongo, “Diffraction from a Slit in an Impedance Plane placed at the Interface

of Two Semi Infinite Half Spaces of Different Media ” Progress in Electromagnetic Research B, PIER B

vol. 10, 191-209, 2008

4 A. Imran, Q. A. Navi and K. Hongo, ”Diffraction of Electromagnetic Plane Wave by an Infinitely Long

Conducting Strip placed on Dielectric Slab” Optics Communications, vol. 282, 443-450, 2009

5 A. Imran, Q. A. Naqvi and K. Hongo, “Diffraction of Electromagnetic Plane Wave from PEMC Slit”

Progress in Electromagnetic Research M, PIER M, vol. 8, 67-77, 2009

– vii –

.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 General Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Impedance Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 PEMC Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter 2: The Method and Prerequisites of Kobayashi Potential . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 The Method of Kobayashi Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Mathematical Prerequisites of Kobayashi Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Hypergeometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

A. Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

B. Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

C. Differential Equation of the Hypergeometric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Weber-Schafheitlin’s Discontinuous Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

A. The Values of W (µ, ν, λ; r) in the Limit r → 0 and r →∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

B. Special Cases of Weber-Schafheitlin’s Discontinuous Integrals . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 Jacobi’s Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

A. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

B. Differential Equation of the Jacobi’s Polynomial and Its Orthogonal Properties . . . . . . . 15

C. Integral Representation of the Jacobi’s Polynomials G(α, γ, x) . . . . . . . . . . . . . . . . . . . . . . . 15

D. Special Cases of Jacobi’s Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.4 Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

– viii –

Chapter 3: Electromagnetic Diffraction from Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 General Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Diffraction of Electromagnetic Plane Wave by an Impedance Strip . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Mathematical Formulation and Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

A. E-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

B. H-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 Physical Optics Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

A. E-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

B. H-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.3 Computations and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

A. Computation of the Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

B. Diffracted Far Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

C. Current Distribution on the Strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Diffraction of Electromagnetic Plane Wave from a Conducting Strip placed on Dielectric Slab . 34

3.3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A. E-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

B. H-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.2 Computations and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A. Computation of Integrals KK(ν, µ) and GG(ν, µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

B. Field Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Diffraction of Electromagnetic Plane Wave from a PEMC Strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.1 Formulation and Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

A. E-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

B. H-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Chapter 4: Electromagnetic Diffraction from Slit(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1 General Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Diffraction from a Slit in an Impedance Plane placed at the interface of Two Different Media . 53

4.2.1 Formulation and solution of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A. E-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

B. H-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 Computations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A. Evaluation of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

B. Diffracted Far Field Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Diffraction from Two Parallel Slits in an Impedance Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.2 Approximate Values of the Expansion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

– ix –

4.3.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Diffraction of Electromagnetic Plane Wave from a Slit In a PEMC Plane . . . . . . . . . . . . . . . . . . . . . . 80

4.4.1 Formulation and Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A. E-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B. H-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Introduction:

.

CHAPTER 1

Introduction

1.1 General ConsiderationIn some fields of investigation particularly in terrain modelling, electromagnetic interface, electromagnetic

compatibility, the role of material composition is very important. In the latest years, an increasing interest

has been devoted to the design and fabrication of composite materials [1],[2] and these materials has found

many applications in electromagnetics. As, for instance, metamaterial slabs, dichroic screens, frequency/

polarization selective reflectors, absorbers, or multi-layer and periodic surfaces are being more and more often

applied in antenna and microwave devices technology. Therefore understanding the effect of the properties

of non perfect electrically conducting (non-PEC) materials on diffraction phenomenon is an important and

interesting investigation.

Therefore in last few decades, the research trends in eletromagnetics has changed. An urgent need was

felt to systematically investigate canonical problems with arbitrary surface impedances. Hence the solutions

were obtained by asymptotic evaluation of the pertinent diffraction integrals involved in canonical cases

investigated through uniform theory of diffraction (UTD) [3] and uniform asymptotic theory of diffraction

(UAT) [4] approaches. The problem of scattering by a half plane with two face impedances was formulated by

Maliuzhinets [5] in integral form. A UTD diffraction coefficient was extracted by Tiberio et al [6] in 1985 for

the straight wedge with arbitrary impedances on the two faces. Volakis [7] reported a UTD coefficient for an

imperfectly conducting half plane based on exact Wiener-Hofp solution. Tiberio and Pelosi [8] also formulated

the problem of scattering from a surface impedance discontinuity on a flat surface in UTD formate. Earlier,

considerable investigations were reported by Senior [9]-[11] on the integral equation formulations involving

surfaces with imperfect conductivity and presented what they called diffraction tensor. Also Senior [12],[13]

presented several investigations on imperfect wedges at skew incidence. Several investigators have dealt

with the problem of the strips and half planes with resistive tapers. Resistive tapers can be used to control

induced current and hence radar cross section. Senior [14] treated back scattering from resistive strips.

Haupt and Liepa [15] systematically carried out synthesis of resistive tapers. Exact solutions were obtained

1

Introduction:

by Young and Senior [16] for E-polarized scattering from resistive half planes with linearly varying resistivity.

Solutions in the UAT format for arbitrary incidence for scattering by a half plane with two face impedances

had been obtained by Sanyal and Bhattacharyya [17], and for an imperfect right-angled wedge with the

one face having imperfect impedance and one face PEC by Senior and Volakis [18]. The former solution

[17] also included estimation of surface wave contributions on both sides of half plane. GTD solutions for

wave interactions with thick PEC and imperfect half planes were treated by Volakis and Ricoy [19] and

Volakis [20]. A generalized version of Maliuzhinets method was used by Volakis and Senior [21] to apply the

generalized boundary conditions to the case of scattering by a metal-backed dielectric half plane. Osipov et

al [22] found out the diffracted fields from an arbitrary angled impedance wedge and approximated them

in the region near the wedge. In [23] Osipov et al determined currents in the shadow boundary region for

the case of circular impedance cylinder. Naqvi et al used Maliuzhinets method to investigate the surface

fields scattered by an isotropic and anisotropic [24], [25] impedance wedge. Simultaneously, solutions for

dielectric half planes and wedges were also progressing. Scattering by dielectric half planes were presented

by Anderson [26], Chakarvorty [27] and Senior and Volakis [28],[29]. Dielectric wedge problems have been

treated by using PO [30], by corrections to PO solutions [31] and by hybrid techniques [32]-[38]. And a

general double diffraction coefficient for impedance wedges [39] based on extended spectral rays and for PEC

wedges [40] was derived in terms of double Fresnel integrals.

If we summarize our discussions we can say that the principal advances in electromagnetics over the last

few decades have been along the following lines and are still growing.

1: Electromagnetic wave interactions with a half plane with arbitrary impedances on two sides.

2: Wave interactions with a straight wedge with arbitrary impedance on two sides.

3: Scattering from the edges of impedances discontinuities on a plane surface

4: Scattering by a half plane with various resistance/ impedance tapers.

5: Scattering by dielectric half planes and wedges

6: Scattering by double and multiple edges.

7: Improved models of curved surface diffraction: radiation, scattering and coupling.

8: Improved methods of fast sampling and efficient evaluation of otherwise time consuming PO integrals

for large surface, and fast algorithms for ray tracing in complicated objects.

9: Scattering by electrically thick PEC and dielectric half planes.

10: Development of hybrid methods combining the advantages of two or more methods to widen the scop

of the HF technique in resonance regions and for efficient solutions of objects of intermediate electrical

size.

11: Extensive applications in antenna scattering, wave propagation, coupling, EMC and EMI, terrain

propagation, clutter modelling and so on.

12: Generation of variously user friendly computer codes for users with different requirement.

A large number of analytical and numerical techniques have been developed over the decades to obtain

accurate solutions to handle the ever increasing complexity of practical problems. The most prominent

among these are listed below

i: Geometric optics (GO)

ii: Physical optics (PO)

iii: Geometrical theory of diffraction (GTD)

iv: Uniform asymptotic theory (UAT)

2

Introduction:

v: Uniform theory of diffraction (UTD)

vi: Physical theory of diffraction (PTD)

vii: Spectral theory of diffraction (STD)

viii: Method of equivalent current (MEC)

ix: Hybrid methods

Each approach has its own advantages, limitation and difficulties [41]-[45]. The above techniques have

three principal advantages. The memory requirements for even electrically large problems are sometime

not very high; recent advances allow such techniques to handle many new classes of problems; they provide

adequate physical insight the problems under investigation. The chief disadvantages are that ray tracing may

be involved for complex objects, and they are not recommended for electrically small objects and objects

without canonical problems.

Hybrid methods were introduced in the early 1970s for solving some difficulties encountered with asymp-

totic methods. A typical situation, for example, is the diffraction from an object which is large compared

to the wavelength and/or which have locally, in some area, complexities in the geometry or in the boundary

conditions which fall outside the domain of applicability of asymptotic methods. To this category, singulari-

ties of the surface for which no explicit expressions for the coefficient of diffraction exist, can also be added.

In hybrid techniques, two or more methods are combined to make use of the advantages of each approach

in solving the complex problems effectively. The main advantage of hybrid approaches is that they reduces

the memory requirements as well as the computer process time if we compare to conventional numerical

techniques for example the method of moments (MoM) and the methods based on the finite element method

or finite difference techniques- which become more expensive at high frequency. Therefore hybrid methods

provide an efficient and cost-effective solutions, at least, in these problems.

The Kobayashi potential (KP) method is an analytico-numerical technique used for solving mixed bound-

ary problems. It may be used as an alternative to hybrid techniques. The formulation through this method is

a bit different and rather simple. The same objectives that can be obtained by using hybrid techniques may

be achieved using this method. Therefore through KP method, the complicated problems may be amenable

to efficient solutions.

In this thesis, KP method is applied to study the diffracting properties of impedance and perfectly

electromagnetic conducting (PEMC) strips/ slit(s). Prior to this, this method was successfully applied to the

problems which involved the perfectly conducting objects. Therefore the purpose of present investigations

is manifold. Firstly, it is required to study the diffracting behavior of strips/slit(s) in non-PEC planes.

Secondly, to determine the nature of the difficulties one come across while applying this method to such

problems and thirdly, the effectiveness of the method in the study of non-conducting objects.

The purpose of selecting the strips/slit(s) to study the diffraction through KP method is that these

are the basic structures in the field of diffraction and so, is of paramount importance in electromagnetics.

Strips/slit(s) in different geometries have found many applications in the passive microwave devices such as

filters, reflectors and antenna covers. These are also popular in designing the frequency-selective surface.

1.2 Impedance Boundary Conditions

When the infinite planes are composed of imperfectly conductors or the materials whose properties vary

slowly from point to point, these may be approximated by the impedance surfaces. Other structures like

metal backed dielectric slabs or rough surfaces may also be approximated by the same. The surfaces whose

radii curvature are large compared with the penetration depth also fall in the same category. Recently

3

Introduction:

such surfaces attracted the attention of many investigators because these are more realistic for practical

applications.

The surface impedance concept was introduced in early 1940’s [46]-[48]. The basic concept in the derivation

of these conditions is that if the skin depth in the conducting body is so short that the variation of the

field in the direction tangential to the body’s surface is much less than the field variations in the normal

direction, then the original 3-D equation of the electromagnetic field diffusion into the body can be replaced

by a 1-D equation in the direction normal to the surface of the body. Analytical or numerical solutions

of the reduced equation can be then used to derive impedance boundary conditions on the body’s surface.

Impedance boundary conditions was first introduced by Leontovich [49] in an attempt to solve the problems

dealing with the propagation of radio waves over the earth. Use of the exact boundary conditions to

describe the earth’s surface, which is both imperfectly conducting and inhomogeneous, result in mathematical

equations too complicated to perform analytically or numerically. Leontovich showed that on the surface

of a nearly conductor, the boundary conditions reduce to impedance boundary conditions, thus relating

the tangential components of the electric to the magnetic surface fields via a surface impedance. Since this

boundary condition relates only the fields outside the scatter, the scattered fields can be evaluated without the

involvement of the internal fields. Thus the analysis of the scattering problem become considerably simplified.

Impedance boundary conditions are also very helpful in analyzing the problems involving magnetic materials.

In such problems, even two dimensional problem can place unreasonable demands on the computational time

and storage. The use of these conditions offers an approximate but efficient formulation. At present, various

impedance boundary conditions of different approximations are being used in combination with the BE,

FE and FDTD methods for analysis of wide range practical applications such as inductive heating devices,

microstrip lines, HF power applications, transmission lines, plasma and magnetic leviation devices, non-

destructive analysis, electromagnetic scattering and geophysical problems.

Approximate boundary conditions on an impedance surface may be written as

E− (n̂.E)n̂ = ηn̂×H

where η is the impedance of the surface and n̂ is the unit normal to the surface. E, and H are the electric

and magnetic fields. If surface is plane with normal along z-axis then the above conditions may be written

as

Ex = −ηHy, Ey = ηHx

Detailed discussions may be seen in [49]-[53]

1.3 PEMC Boundary Conditions

The idea of perfectly electromagnetic conducting medium (PEMC) was introduced by Lindell and Sihvola

[54] as a generalization of the well-known and used PEC and PMC media in which certain linear combination

of electromagnetic fields become extinct. The definition of a PEMC medium through differential-form

formalism [55] corresponds to the simplest possible electromagnetic medium defined by a single real-valued

parameter known as the admittance parameter M , which can vary from zero to infinity. A null admittance

corresponds to a PEC and an admittance of infinity to a PMC medium, when the field magnitudes are finite.

It is an isotropic medium and like PEC and PMC, PEMC do not let any electromagnetic energy to pass

through it, therefore it can serve as boundary material. Although no power can through it, the fields are not

yet zero as shown by Jancewicz [56]. The most notable property of this medium which distinguishes it from

4

Introduction:

the ordinary media is its non-reciprocity when M has a finite non zero value. This has been theoretically

demonstrated by many authors, for example for the case of PEMC slab , air/PEMC interface [56], PEMC

sphere [57], PEMC cylinder [58], PEMC spheroid [59], that after scattering from these geometries, the

electromagnetic wave also have cross-polarized component along with the co-polarized component. The

possible realization of such medium has been discussed in [60]. In which they pointed out that a layer

of bi-isotropic Tellegen medium or gyrotropic anisotropic medium added with a guiding structure may act

as a PEMC boundary when the parameters of the structure are chosen in an appropriate manner. Several

interesting applications can be invented which rely on PEMC operation, for example polarization transformer,

since a linearly polarized wave suffers a rotation of the polarization while reflecting from a PEMC surface.

The rotation angle is in a close relation with admittance parameter M .

The boundary conditions to be satisfied on a PEMC surface can be written by using the PEC and PMC

boundary conditions and the fact that PEMC is the generalization of PEC and PMC as follow [54]

n̂× (H + ME) = 0, n̂.(D−MB) = 0

as the PEC and PMC boundaries conditions are

n̂×E = 0, n̂.B = 0 (PEC)

n̂×H = 0, n̂.D = 0 (PMC)

where M is defined as the PEMC admittance and n̂ is the unit normal to the boundary. Recently many

investigators showed interest in these materials [61]-[68].

1.4 Organization of the ThesisThis thesis is organized into five chapters. In the next chapter (Chapter 2), method of Kobayashi po-

tential is introduced, its basics and preliminaries has been discussed in detail. The most important part

of this chapter includes the discussions on the Weber-Schafheitlin’s integral and Jacobi’s Polynomials. The

remaining part of this thesis is devoted to the applications of this method to two dimensional geometries.

Chapter 3 includes three problems all of which deal with the diffraction from strip but in different geome-

tries. In the first problem, diffraction from an impedance strip is discussed. The thickness of the strip is

assumed negligible. Both E- and H-polarization are considered. Discussions on the dependence of diffracted

fields on different parameters, for example angle of incidence, strip width and impedance of the strip etc.

are included. The current distributions on the strip are also given.

The second problem deals with the diffraction from a strip placed on the dielectric slab of finite thickness.

This problem may be regarded as a simplified model of microstrip antenna. The purpose of this is to

study how the diffracted fields depend upon the material as well as the thickness of the slab. Current

distributions on the strip are again included for the two reasons. Firstly, one can estimate the correctness of

the formulation and secondly, it gives the behavior of current density near the edges of the strip. Physical

optics (PO) method is used as an alternative approach to verify our results. The third problem, included

in this chapter, deals with diffraction from PEMC strip. These materials may be regarded as the high

impedance materials [69]. The discussions are included that how the diffracting behavior changes as the

admittance parameter is changed which describe the impedance of the strip.

Chapter 4 is on the slit(s) problems. It includes three problems. In the first problem, KP method is used to

study the diffraction from a slit in an impedance plane placed at the interface of two semi-infinite half spaces.

5

Introduction:

Special algorithms are developed for the evaluation of different integrals and illustrative computations are

carried out for the parameters of interests.

The second problem in this chapter deals with diffraction from two parallel slits in an impedance plane.

In the literature such problems find very few references. The references which are available mostly deal

with the slits in the conducting plane. The third problem deals with the diffraction from the slit in PEMC

plane. It has been discussed in this problem that how the co-polarized and cross-polarized components of

the diffracted fields depend upon the admittance parameter, angle of incidence, slit width etc. Finally, the

chapter 5 contains the concluding remarks on the problems included in this manuscript.

6

The Method and Prerequisites of Kobayashi Potential:

.

CHAPTER 2

The Method and Prerequisites of Kobayashi Potential

This chapter contains the discussion that what this method is about and what type of problems it is

suitable for? Some of the topics are reviewed which are perquisite to this method and the formulas are

summarized which will be used in subsequent work.

2.1 The method of Kobayashi PotentialThe method of Kobayashi potential (KP Method) is an analytico-numerical technique developed to solve

the mixed boundary value problems. It was developed by Iwao Kobayashi in the beginning of 1930’s. More

than thirty years later, this method was named Kobayashi potential by Snedden in his book [70]. In its

primitive stage, this method was applied to analyze the electrostatic potential of circular disk and was then

extended to derive the expression for circular capacitor. In his original work, Kobayashi firstly introduced

the idea that the solutions of potential problems associated with conducting disc and strip can be effectively

constructed by using the discontinuous properties of Weber-Schafheitlin’s integrals. He also discussed the

properties of Jacobi’s polynomials which were used as the basis of functional space in this method. He

applied this method to potential problems of circular, semi-circular and quadrant discs, coupling of two discs

facing each other, capacitance of a disk on the grounded dielectric substrate and coupling between two disks

located on a common plane in free space or on the grounded dielectric substrate [71]-[74].

Following the publication of Kobayashi work, many of the investigators working in the field applied this

method to many physical problems. Namura applied this method to fluid dynamics as well as electrostatic

problems [75]-[78]. He also attempted the acoustic problems and electromagnetic diffraction problems which

involved the circular and rectangular apertures [79]-[85]. In his formulation, he obtained the matrix equations

for determining the expansion coefficients.The matrix elements contained double infinite integrals. Therefore,

he could not give the numerical results. Hongo came up with the numerical solutions of these integrals and

evaluated these integrals with high accuracy. Using this and similar algorithms, he and his student Serizawa

studied extensively the problems which included the diffraction of acoustic and electromagnetic waves from

7

The Method and Prerequisites of Kobayashi Potential:

rectangular aperture in a thin and thick plate, electromagnetic radiation from flanged rectangular waveguide

and coupling between two wave guides in a common flange [86]-[90].

Hongo worked extensively to make this technique get the position where it is now. He solved variety of

difficulties that must be overcome in applying the KP method to different canonical problems. He applied

this method to many problems that have many engineering applications such as thin slit, thick slit, slit

embedded in an isotropic plasma, two parallel slits, microstrip disks, flanged parallel plate wave guides,

circular disk and circular hole to name a few and obtained numerous results [91]-[104].

It can be observed in the above sited literature that the problems, which are solved through this method,

included the objects which were conducting. Application of this method to the non-PEC objects are not

tried so far. Therefore to check the validity and hence, enhance the scop of the method, this method is being

extended to the problem which involve non-PEC ( Impedance/PEMC ) objects.

2.1.1 Methodology

In the wave diffraction problems, the governing equation is the Helmholtz equation. Diffracted or scattered

fields are the general solutions of the above equation which contain unknown weighting functions. Enforce-

ment of boundary conditions yield dual or triple integral equations for the weighting functions. These

equations are solved using the discontinuous properties of the Weber-Schafheitlin’s Integrals (discussed in

the coming sections). At this stage, edge conditions are usually, incorporated in the solution. The resulting

equations can be converted into matrix equations by applying the projection method with a functional space

that consists of a set of Jacobi’s polynomials as basis functions. These matrix equations are then solved

for infinitely large no of unknowns. But in practical situations these large no of unknowns are truncated to

finite numbers, the quantity of which of course vary from problem to problem. The elements of the matrix

equations are usually infinite single or double integrals. In our cases, these are single infinite integrals having

branch points as well as poles. These integrals are usually hard to solve analytically and are therefore, solved

numerically. The convergence of these integrals is rather very slow. Therefore, algorithms are developed to

compute these integrals successfully.

Following advantages of this technique can be cited over the others

1: The formulation through this method is comparatively simple for both 2-D and 3-D problems ( as

compared with Wiener-Hofp or Maliuzhinets methods).

2: This method has wide range of applications i.e. from electrostatic [71]-[74] to wave motion [75]-[104].

Therefore it is a more versatile method.

3: This method does not pose any considerable difficulty while working in any coordinate system i.e

rectangular, cylindrical or spherical [105].

4: This method is equally applicable for both PEC/non PEC boundary value problems.

5: KP method may be applied to more complex problems with related geometries. These problems may

be formulated in a manner similar to the eigenfunction expansions in cylindrical and spherical geometries

[106].

6: This method is very efficient in studding the problems which involve multiple diffractions or high

order interactions [93],[99].

The disadvantage may be

i: Tractable geometries of this method are limited to special shapes like rectangular and circular plates

and their related geometries. However, a similar situation may be seen for other conventional eigenfunction

expansions [88].

8

The Method and Prerequisites of Kobayashi Potential:

ii: Sometimes, one get such expressions in the way of solving a problem that their numerical simulations,

sometimes, become very cumbersome. Therefore, special type of algorithms are to developed to solve the

problem [106].

2.2 Mathematical Prerequisites of Kobayashi PotentialIn this section, mathematical preliminaries of the KP method are presented and some of the details of

the special functions are given on which this method depend upon. This include Hypergeometric Functions,

Weber-Schafheitlin’s Integrals and the Jacob’s Polynomials.

2.2.1 Hypergeometric SeriesA: Definition and Properties

A hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a

rational function of n. The series, if convergent, will define a hypergeometric function, which may then be

defined over a wider domain of the argument by analytical continuation. Hypergeometric functions generalize

many special functions. These functions have very rich theory and are being introduced here briefly. Detailed

discussions can be viewed elsewhere [105]-[108]. Mathematically, these series are defined as

F (a, b, c; x) = 1 +a · b1 · cx +

a(a + 1) · b(b + 1)1 · 2 · c(c + 1)

x2 + · · · (2.1)

This series converges for |x| < 1 and diverges |x| > 1 except for c = 0 or c = −n where n is a positive integer.

For x = 1, the series converge for c > a + b and, when x = −1, the series converges for c > a + b− 1. Using

the following notation

(a)r = a(a + 1)(a + 2) · · · (a + r − 1) =Γ (a + r)

Γ (a)(a)0 = 1 (2.2)

The last expression may be expressed by

2F1(a, b, c; x) = F (a, b, c;x) =∞∑

n=0

(a)n(b)n

(c)n

xn

n!=

Γ (c)Γ (a)Γ (b)

∞∑n=0

Γ (a + n)Γ (b + n)Γ (c + n)Γ (n + 1)

xn (2.3)

In the above equation subscript 2 and 1 in 2F1 means that two parameters (a)n and (b)n are in the

numerator and one parameter (c)n is in the denominator. One property of these functions which can readily

be proved that

F (a, b, c; x) = F (b, a, c;x) (2.4)

Using the definition of the Gamma function, recurrence relations may be derived

d

dxF (a, b, c; x) =

∞∑n=1

(a)n(b)n

(c)n(n− 1)!xn−1

=ab

c

∞∑n=0

(a + 1)n(b + 1)n

(c + 1)nn!xn =

ab

cF (a + 1, b + 1, c + 1; x)

(2.5)

andd

dxF (a, b, c;x) =

b

x[F (a, b + 1, c; x)− F (a, b, c; x)] (2.6)

Using (2.5) and (2.6) it can be written as

d

dx

[xbF (a, b, c; x)

]= bxb−1F (a, b + 1, c; x) (2.7)

9

The Method and Prerequisites of Kobayashi Potential:

Some of the elementary functions may be expressed in terms of hypergeometric functions. For example

(1 + x)n = F (−n, b, b;−x), ln(1 + x) = xF (1, 1, 2;−x), exp(x) = limb→∞

F(1, b, 1;

x

b

)(2.8)

where b is arbitrary constant.

B: Integral Representation

Series expression of the Hypergeometric function can be used to derive its integral representation. Using

the integral representation of Beta function

(a)n

(c)n=

Γ (a + n)Γ (c)Γ (a)Γ (c + n)

=Γ (a + n)Γ (c− a)Γ (c)Γ (a)Γ (c + n)Γ (c− a)

=B(a + n, c− a)

B(a, c− a)

=1

B(a, c− a)

∫ 1

0

(1− t)c−a−1ta+n−1dt

(2.9a)

Therefore

F (a, b, c; x) =1

B(a, c− a)

∞∑n=0

(b)nxn

n!

∫ 1

0

(1− t)c−a−1ta+n−1dt

=1

B(a, c− a)

∫ 1

0

(1− t)c−a−1ta−1dt

∞∑n=0

(b)n(xt)n

n!

(2.9b)

Taking into account the relation∞∑

n=0

(b)n(xt)n

n!= (1− xt)−b

Finally, the following expression is obtained

F (a, b, c; x) =1

B(a, c− a)

∫ 1

0

(1− t)c−a−1ta−1(1− xt)−bdt (2.10)

It is noted that the above equation is valid only for |x| < 1 and if the condition c > a > 0 is satisfied. For

x = 1, the above equation gives

F (a, b, c; 1) =1

B(a, c− a)

∫ 1

0

(1− t)c−a−b−1ta−1dt

=B(a, c− a− b)

B(a, c− a)=

Γ (c)Γ (c− a− b)Γ (c− a)Γ (c− b)

(2.11)

In (2.10) we let s = 1− t and use the relation

[1− x(1− s)]−b = (1− x)−b

[1− xs

x− 1

]−b

then, another integral representation of F (a, b, c; x) can be obtained as

F (a, b, c; x) =(1− x)−b

B(a, c− a)

∫ 1

0

(1− s)a−1sc−a−1

[1− xs

x− 1

]−b

dt

= (1− x)−bF

(b, c− a, c;

x

x− 1

)(2.12)

C: Differential Equation of the Hypergeometric Function

The differential equation of Hypergeometric function is given by

x(1− x)d2y

dx2+ [c− (a + b + 1)x]

dy

dx− aby = 0 (2.13)

10

The Method and Prerequisites of Kobayashi Potential:

which has regular singularities at x = 0, x = 1 and x → ∞. The solutions can be obtained by considering

series expansion of the above equation around each singularity.

C1: Expansion around x=0

Substituting

y = xp∞∑

n=0

anxn (2.14)

into (2.13) and let the coefficient of each power xp+n be zero, following is obtained

(p + n + 1)(p + n + c)an+1 = (p + n + a)(p + n + b)an (n = 0, 1, 2, · · ·)

or

an =(p + a)n(p + b)n

(p + 1)n(p + c)na0 (2.15)

Indicial equation is given by

p(p− 1) + cp = 0, p = 0, or p = 1− c (2.16)

When 1− c is not negative integer or zero, independent solutions are given by

y1 = F (a, b, c; x)

y2 = x1−cF (a + 1− c, b + 1− c, 2− c; x)(2.17)

C2: Expansion Around x=1

Putting q = 1− x in (2.13)

q(1− q)d2y

dq2+ [a + b− c + 1− (a + b + 1)q]

dy

dq− aby = 0 (2.18)

and its solution is given by

y1 = F (a, b, a + b− c + 1; 1− x)

y2 = (1− x)c−a−bF (c− a, c− b, c− a− b + 1; 1− x)(2.19)

C3: Expansion around x →∞Putting x =

1s

and y = saw in (2.13),

d2w

ds2+

1 + a− b− (2− c + 2a)ss(1− s)

dw

ds− a(1− c + a)

s(1− s)w = 0 (2.20)

The independent solutions of this equation can readily be found which are

y1 = x−aF

(a, 1− c + a, 1 + a− b;

1x

)

y2 = x−bF

(b, 1− c + b, 1 + b− a;

1x

) (2.21)

It is noted that, since the expressions in (C1) are valid in the range (−1, 1), (C2)’s expressions are valid in

(0,2) and (C3)’s expressions are valid in (1,∞) and (−∞,−1), there overlap the regions of validity. Hence

11

The Method and Prerequisites of Kobayashi Potential:

each expression can be expressed in terms of linear combination of the solutions in the other region. For

example

F (a, b, c; x) = AF (a, b, a + b− c + 1; 1− x) + B(1− x)c−a−bF (c− a, c− b, c− a− b + 1; 1− x) (2.22)

where (a + b < c < 1). The above equation holds in the range (0,1). Letting x=0 and x=1, the coefficients

A and B can be determined. The final results are given by

F (a, b, c; x) =Γ (c)Γ (c− a− b)Γ (c− a)Γ (c− b)

F (a, b, a + b− c + 1; 1− x)

+Γ (c)Γ (a + b− c)

Γ (a)Γ (b)(1− x)c−a−bF (c− a, c− b, c− a− b + 1; 1− x)

(2.23)

2.2.2 Weber-Schafheitlin’s Discontinuous Integrals [105],[106],[109]

The integral of the form

W (µ, ν, λ; r) =∫ ∞

0

Jµ(ru)Jν(u)uλ

du (2.24)

is known as the Weber-Schafheitlin’s Integral in the literature. It was first investigated by Weber for some

special cases, namely

(i) λ = µ = 0, ν = 1 (ii) λ =−12

, µ = 0, ν = ±12

The integral was then investigated, for all values of λ, µ, ν for which it is convergent, by Sonine (Math. Ann.

XVI. pp. 51-52 1880 ). But he did not examined in great detail nor did he lay any stress on the discontinuity

which occure when r = 1. The special case in which λ = 0 was discussed by Gubler who used a very elegant

transformation of contour integral but unfortunately the same analysis can not be adopted for the more

general case in which λ 6= 0. Some year latter this integral was investigated very thoroughly by Schafheitlin

(Math. Ann. XXX. pp. 990-991 1887).

This integral can be expanded by the series

W (µ, ν, λ; r)

=rµΓ [ 12 (µ + ν − λ + 1)]

2λΓ [ 12 (−µ + ν + λ + 1)]Γ (µ + 1)F

[µ + ν − λ + 1

2,µ− ν − λ + 1

2, µ + 1; r2

]0 ≤ r < 1

=Γ [ 12 (µ + ν − λ + 1)]

2λrν−λ+1Γ [ 12 (µ− ν + λ + 1)]Γ (ν + 1)F

[µ + ν − λ + 1

2,−µ + ν − λ + 1

2, ν + 1;

1r2

], r ≥ 1

(2.25)

where F (a, b, c; x) are the hypergeometric functions. A distinguished properties of W (µ, ν, λ; r) which we

will refer through out in our work and on which KP method base, is that If there exists a relation µ−ν +λ =

−2m − 1(m = 0, 1, 2, · · ·) among the parameters, the function becomes W (µ, ν, λ; r) = 0 for r > 1 because

of the properties of the gamma function |Γ (−n)| → ∞ ( where n is a positive integer). For r in the range

0 ≤ r < 1, W (µ, ν, λ; r) takes finite value. Now we describe some of the the properties of the function

W (µ, ν, λ; r). The detailed treatment can be consulted elsewhere [2-34],[2-38].

12

The Method and Prerequisites of Kobayashi Potential:

A: The Values of W (µ, ν, λ; r) in the Limit r → 0 And r →∞The convergence of the integral is determined from the integrand around the limits of the integration. In

the limit u → 0, the integrand of (2.24) becomes

Jµ(u) = [Γ (µ + 1)]−1(u

2

)µ

,Jµ(ru)Jν(u)

uλ= Auµ+ν−λ, A =

rµ

Γ (µ + 1)Γ (ν + 1)2µ+ν(2.26a)

Hence, the integral converges in the lower limit u → 0 for <(µ + ν − λ) > −1. In the limit u → ∞, using

the asymptotic approximation of the Bessel function

Jµ(u) '√

2πu

cos[u− (2µ + 1)π

4

](2.26b)

the behavior of the integrand for large u becomes

u−λJµ(ru)Jν(u) =1

πuλ+1√

r

[cos

{(r + 1)u− (µ + ν + 1)π

2

}+ cos

{(r − 1)u− (µ− ν)π

2

}](2.26c)

Taking into account the conditions of convergence for the integral∫ ∞ 1

umdu (m > 1),

∫ ∞ cos u

umdu (m > 0) (2.27a)

the integral (2.24) converges for λ + 1 > 0 when r 6= 1 . When r = 1, it converges for λ > 0 and µ − ν =

odd integer and for λ + 1 > 0 when µ− ν 6= odd integer. Summarizing the results we find the conditions

µ + ν + 1 > λ > −1 for r 6= 1 µ + ν + 1 > λ > 0 for r = 1 (2.27b)

are needed for the integral (2.24) to have the definite values. Limit of W (µ, ν, λ; r) can be evaluated for

r → 0 and r →∞.

Using the relation Jµ(ru) ' ( 12ru)µ

Γ (µ + 1)for small r, W (µ, ν, λ; r) becomes

W (µ, ν, λ; r) =( 12r)µ

Γ (µ + 1)

∫ ∞

0

uµ−λJν(u)du (2.28a)

which may be written as [103]

W (µ, ν, λ; r) =Γ [ 12 (µ + ν − λ + 1)]rµ

2λΓ (µ + 1)Γ [ 12 (ν − µ + λ + 1)](2.28b)

Therefore, in the limit r → 0,

limr→0

r−µW (µ, ν, λ; r) =1

2λΓ (µ + 1)· Γ [ 12 (µ + ν − λ + 1)]Γ [ 12 (ν − µ + λ + 1)]

(2.29)

provided −(ν + 1) < µ− λ < 12 .

In the limit r →∞, the integral (2.24) may be changed to

W (µ, ν, λ; r) = rλ−1

∫ ∞

0

Jµ(y)Jν(y/r)yλ

dy

by the transformation ru = y. Since r is very large, Jν(y/r) is approximated by Jν(y/r) ' (y/2r)ν

Γ (ν + 1). Then

the integration with respect to y can be carried out and the value of W in the limit r →∞ becomes

limr→∞

rν+1−λW (µ, ν, λ; r) =−Γ [ 12 (µ + ν + 1− λ)]

2λΓ (ν + 1)Γ [ 12 (µ + 1− ν + λ)],

(λ− µ− 1 < ν <

12

+ λ

)(2.30)

13

The Method and Prerequisites of Kobayashi Potential:

B: Special Cases of Weber-Schafheitlin’s Discontinuous Integrals

Following are the most special cases of the Weber-Schafheitlin’s Discontinuous Integrals. Numerous other

cases are given by Nielsen ( Ann. di. Mat.(3) XIV. pp. 82-90 1908)

∫ ∞

0

Jµ(au)Jν(au)uλ

du =( 12a)λ−1Γ (λ)Γ

[12 (µ + ν − λ + 1)

]

2Γ[12 (ν + λ− µ + 1)

]Γ

[12 (µ + ν + λ + 1)

]Γ

[12 (µ + λ− ν + 1)

] (2.31a)

∫ ∞

0

Jµ+p(au)Jµ−p−1(bu)du =

bµ−p−1Γ (µ)aµ−pΓ (µ− p)

F

(µ,−p, µ− p;

b2

a2

)for b < a

(−1)p

2afor b = a

0 for b > a

(2.31b)

∫ ∞

0

Jµ(au)Jν(au)u

du =2π

sin[ 12 (ν − µ)π]ν2 − µ2

(2.31c)

∫ ∞

0

Jν(au)Jν(bu)u

du =

12ν

(b

a

)ν

for b < a

12ν

(a

b

)ν

for b > a, <(µ) > 0(2.31d)

∫ ∞

0

Jν(au) sin(bu)u

du =

1ν

sin[ν sin−1 b

a

]for b ≤ a

aν sin(νπ/2)ν[b + (b2 − a2)

12 ]ν

for b ≥ a(2.31e)

∫ ∞

0

Jν(au) cos(bu)u

du =

1ν

cos[ν sin−1 b

a

]for b ≤ a

aν cos(νπ/2)ν[b + (b2 − a2)

12 ]ν

for b ≥ a(2.31f)

∫ ∞

0

Jν(au) sin(bu)du =

1√a2 − b2

sin[ν sin−1 b

a

]for b ≤ a

∞ or 0 for a = baν cos(νπ/2)√

b2 − a2[b +√

b2 − a2]νfor b > a

[<(ν) > 2]

(2.31g)

∫ ∞

0

Jν(au) cos(bu)du =

cos[ν sin−1(b/a)

]√

a2 − b2for b ≤ a

∞ or 0 for a = b

− aν sin(νπ/2)√b2 − a2[b +

√b2 − a2]ν

for b > a

[Re(ν) > 2] (2.31h)

∫ ∞

0

J0(au) sin(bu)du =

0 for b < a∞ for a = b

1√b2 − a2

for b > a(2.31i)

∫ ∞

0

J0(au) cos(bu)du =

1√a2 − b2

for b < a

∞ for a = b0 for b > a

(2.31j)

∫ ∞

0

uµ−1Jν(au)du = 2µ−1a−µ Γ ( 12ν + 1

2µ)Γ (1 + 1

2ν − 12µ)

−<ν < <µ <32

(2.31k)

14

The Method and Prerequisites of Kobayashi Potential:

2.2.3 Jacobi’s Polynomials [105],[112]In mathematics, Jacobi polynomials are a class of orthogonal polynomials. They can be obtained from

hypergeometric series in cases where the series is in fact finite. They are named after Carl Jacobi. We will

give some of the properties concerning Jacobi’s polynomials which are closely related to Kobayashi potential.

A: Definition

Jacobi’s polynomials are defined by the hypergeometric functions as follow

G(α, γ, x) = F (−n, α + n, γ;x)

= 1 +n∑

k=1

(−1)k

(nk

)(α + n)(α + n + 1) · · · (α + n + k − 1)

γ(γ + 1) · · · (γ + k − 1)xk

(γ 6= 0,−1,−2, · · · ,−n + 1)

(2.32a)

The explicit expressions of the polynomials with lower order are

G0(α, γ, x) = 1

G1(α, γ, x) = 1− α + 1γ

x

G2(α, γ, x) = 1− 2α + 2

γx +

(α + 2)(α + 3)γ(γ + 1)

x2

G3(α, γ, x) = 1− 3α + 3

γx + 3

(α + 3)(α + 4)γ(γ + 1)

x2 − (α + 3)(α + 4)(α + 5)γ(γ + 1)(γ + 2)

x3 (2.32b)

For γ > 0 and α > γ − 1, all the zeros of Gn(α, γ, x) are located in the range 0 ≤ x ≤ 1.

B. Differential Equation of the Jacobi’s Polynomial and Its Orthogonal Properties

Jacobi’s polynomial satisfies the differential equation

x(1− x)d2y

dx2+ [γ − (α + 1)x]

dy

dx+ n(α + n)y = 0 (2.33a)

This equation can be written as

d

dx

{xγ(1− x)α+1−γ dy

dx

}+ n(α + n)xγ−1(1− x)α−γy = 0 (2.33b)

and from which orthogonality can be derived, which is given by∫ 1

0

xγ−1(1− x)α−γGn(α, γ, x)Gm(α, γ, x)dx =1

α + 2n

Γ (n + 1)Γ 2(γ)Γ (α− γ + n + 1)Γ (γ + n)Γ (α + n)

δnm (2.33c)

This relation can be proved by using the Rodrigues’s representation for Gn(α, γ, x) which is given by

Gn(α, γ, x) =Γ (γ)

Γ (γ + n)x1−γ(1− x)γ−α dn

dxn

{xn+γ−1(1− x)α+n−γ

}(2.33d)

C: Integral Representation of the Jacobi’s Polynomials G(α, γ, x)

By using the integral representation of the Weber-Schfheitlin’s discontinuous integral given by

W (λ, µ, ν, x) =∫ ∞

0

Jµ(√

xξ)Jν(ξ)ξλ

dξ

=Γ

[12 (µ + ν + 1− λ)

]xµ/2

2λΓ (µ + 1)Γ[

λ+ν+1−µ2

] F

[µ + ν + 1− λ

2,µ + 1− λ− ν

2, µ + 1, x

] (2.34a)

15

The Method and Prerequisites of Kobayashi Potential:

By comparing with parameters of hypergeometric function with that of (2.32a) we have the integral repre-

sentation of the Jacobi’ polynomials given by

Gn(α, γ, x) =2γ−αΓ (n + 1)Γ (γ)

Γ (n + α)x

12 (1−γ)

∫ ∞

0

Jγ−1(√

xξ)J2n+α(ξ)ξγ−α

dξ (2.34b)

Using the orthogonality and Rodrigues’s representation of G(α, γ, x), expansion formulas for the Bessel

functions in terms of Jacobi’s polynomials can be obtained, which will frequently appear in the subsequent

analysis and is given by.

x−µ/2Jµ(p√

x) =∞∑

n=0

(α + 2n)Γ (α + n)Γ (n + 1)Γ (γ)

(p

2

)γ−α−1

J2n+α(p)Gn(α, γ, x) (γ = µ + 1) (2.34c)

D: Special Cases of Jacobi’s Polynomials

We will give the definition and orthogonal properties for the Jacobi’s polynomials which are frequently

used in our analysis .

umn (x) = G

(m +

12,m + 1, x

)= F

(n + m +

12,−n,m + 1; x

)

=√

2Γ (n + 1)Γ (m + 1)Γ (n + m + 1

2 )x−m/2

∫ ∞

0

Jm(√

xξ)J2n+m+ 12(ξ)√

ξdξ

=Γ (m + 1)

Γ (n + m + 1)x−m(1− x)

12

dn

dxn

{xn+m(1− x)n− 1

2

}(2.35a)

vmn (x) = G

(m +

32,m + 1, x

)= F

(n + m +

32,−n,m + 1; x

)

=Γ (n + 1)Γ (m + 1)√

2Γ (n + m + 32 )

x−m/2

∫ ∞

0

√ξJm(

√xξ)J2n+m+ 3

2(ξ)dξ

=Γ (m + 1)

Γ (n + m + 1)x−m(1− x)−

12

dn

dxn

{xn+m(1− x)n+ 1

2

}(2.35b)

pmn (x) = G (m + 1,m + 1, x) = F (n + m + 1,−n,m + 1; x)

=Γ (n + 1)Γ (m + 1)

Γ (n + m + 1)x−m/2

∫ ∞

0

Jm(√

xξ)J2n+m+1(ξ)dξ

=Γ (m + 1)

Γ (n + m + 1)x−m dn

dxn

{xn+m(1− x)n

}(2.35c)

wmn (x) = (1− x)−

12 G

(m +

12,m + 1, x

)= (1− x)−

12 F

(n + m +

12,−n,m + 1; x

)

=Γ (n + 1

2 )Γ (m + 1)√2Γ (n + m + 1)

x−m/2

∫ ∞

0

Jm(√

xξ)J2n+m+ 12(ξ)

√ξdξ

=Γ (m + 1)

Γ (n + m + 1)x−m dn

dxn

{xn+m(1− x)n− 1

2

}(2.35d)

qmn (x) = G

(m +

13,m + 1, x

)= F

(n + m +

13,−n,m + 1; x

)

= 223Γ (n + 1)Γ (m + 1)

Γ (n + m + 13 )

x−m/2

∫ ∞

0

Jm(√

xξ)J2n+m+ 13(ξ)

ξ23

dξ

=Γ (m + 1)

Γ (n + m + 1)x−m(1− x)

23

dn

dxn

{xn+m(1− x)n− 2

3

}(2.35e)

rmn (x) = G

(m +

43,m + 1, x

)= F

(n + m +

43,−n,m + 1; x

)

= 2−13Γ (n + 1)Γ (m + 1)

Γ (n + m + 43 )

x−m/2

∫ ∞

0

Jm(√

xξ)J2n+m+ 43(ξ)ξ

13 dξ

=Γ (m + 1)

Γ (n + m + 1)x−m(1− x)−

13

dn

dxn

{xn+m(1− x)n+ 1

3

}(2.35f)

16

The Method and Prerequisites of Kobayashi Potential:

The expansion formulas of the Bessel function by the Jacobi’s polynomials are given by

x−m/2Jm(ξ√

x) =∞∑

n=0

√2(2n + m + 1

2 )Γ (n + m + 12 )

Γ (n + 1)Γ (m + 1)

J2n+m+ 12(ξ)√

ξum

n (x)

=∞∑

n=0

√8(2n + m + 3

2 )Γ (n + m + 32 )

Γ (n + 1)Γ (m + 1)

J2n+m+ 32(ξ)

ξ32

vmn (x)

=∞∑

n=0

2(2n + m + 1)Γ (n + m + 1)Γ (n + 1)Γ (m + 1)

J2n+m+1(ξ)ξ

pmn (x)

=∞∑

n=0

√2(2n + m + 1

2 )Γ (n + m + 12 )

Γ (n + 1)Γ (m + 1)

J2n+m+ 12(ξ)√

ξ(1− x)

12 wm

n (x)

=∞∑

n=0

213(2n + m + 1

3 )Γ (n + m + 13 )

Γ (n + 1)Γ (m + 1)

J2n+m+ 13(ξ)

ξ13

qmn (x)

=∞∑

n=0

243(2n + m + 4

3 )Γ (n + m + 43 )

Γ (n + 1)Γ (m + 1)

J2n+m+ 43(ξ)

ξ43

rmn (x)

(2.36)

The orthogonal properties for the Jacobi’s polynomial are given by∫ 1

0

xm(1− x)−12 um

n (x)umn′(x)dx =

Γ (n + 1)Γ 2(m + 1)Γ (n + 12 )

(2n + m + 12 )Γ (n + m + 1)Γ (n + m + 1

2 )δn,n′ (2.37a)

∫ 1

0

xm(1− x)12 vm

n (x)vmn′(x)dx =

Γ (n + 1)Γ 2(m + 1)Γ (n + 32 )

(2n + m + 32 )Γ (n + m + 1)Γ (n + m + 3

2 )δn,n′ (2.37b)

∫ 1

0

xmpmn (x)pm

n′(x)dx =Γ 2(m + 1)Γ 2(n + 1)

(2n + m + 1)Γ 2(n + m + 1)δn,n′ (2.37c)

∫ 1

0

xm(1− x)12 wm

n (x)wmn′(x)dx =

∫ 1

0

xm(1− x)−12 um

n (x)umn′(x)dx (2.37d)

∫ 1

0

xm(1− x)−23 qm

n (x)qmn′(x)dx =

Γ (n + 1)Γ 2(m + 1)Γ (n + 13 )

(2n + m + 13 )Γ (n + m + 1)Γ (n + m + 1

3 )δn,n′ (2.37e)

∫ 1

0

xm(1− x)13 rm

n (x)rmn′(x)dx =

Γ (n + 1)Γ 2(m + 1)Γ (n + 43 )

(2n + m + 43 )Γ (n + m + 1)Γ (n + m + 4

3 )δn,n′ (2.37f)

2.2.4 Spherical Bessel Function

Spherical Bessel’s functions jn(z), nn(z) are the solutions of the differential equation

d2w

dz2+

2dw

zdz+

[1− l(l + 1)

z2w

]= 0 (2.38)

These functions may be defined in terms of Bessel’s function of first and second kind as follow

jn(z) =√

π

2zJn+ 1

2(z), nn(z) =

√π

2zNn+ 1

2(z) (2.39a)

The Spherical Bessel’s function of third kind have the relation

h(1)n (z) =

√π

2zH

(1)

n+ 12(z), h(2)

n (z) =√

π

2zH

(2)

n+ 12(z) (2.39b)

The spherical Bessel functions can be expanded in terms of trigonometric function as

h(1)n (z) =

1z

exp[j

{z − (n + 1)π

2

}] n∑m=0

(−1)m (n + 12 ,m)

j2zm= h(2)∗

n (z) (2.40a)

17

The Method and Prerequisites of Kobayashi Potential:

jn(z) =1z

sin[z − nπ

2

] [n/2]∑m=0

(−1)m (n + 12 , 2m)

(2z)m

+1z

cos[z − nπ

2

] [(n−1)/2]∑m=0

(−1)m (n + 12 , 2m + 1)

(2z)2m+1

(2.40b)

nn(z) = −1z

cos[z − nπ

2

] [n/2]∑m=0

(−1)m (n + 12 , 2m)

(2z)m

+1z

sin[z − nπ

2

] [(n−1)/2]∑m=0

(−1)m (n + 12 , 2m + 1)

(2z)2m+1

(2.40c)

where [x] is Gauss’s symbol which extracts the integer part of x. The lower order of these functions are:

j0(x) =sin(x)

x, j1(x) =

1x2

(sin x− x cos x), j2(x) =1x3{(3− x2) sin x− 3x cosx} (2.41a)

n0(x) =cos(x)

x, n1(x) = − 1

x2(cos x + x sin x), n2(x) = − 1

x3{(3− x2) cos x + 3x sin x} (2.41b)

h(1),(2)0 (x) = ∓ j

xexp(±jx), h

(1),(2)1 (x) = − 1

x2(x± j) exp(±jx),

h(1),(2)2 (x) = − 1

x3(3x± j3− jx2) exp(±jx) (2.41c)

From the series solution, with conventional normalization[] it can be shown that

fl−1(z) + fl+1(z) = (2l + 1)z−1fl(z), lfl−1(z)− (l + 1)fl+1(z) = (2l + 1)dfl(z)

dz(2.42a)

d

dz

[zl+1fl(z)

]= zl+1fl−1(z),

d

dz

[z−lfl(z)

]= z−lfl+1(z) (2.42b)

where fl(z) may be any of the jn(z), nn(z), h1n(z) and h2

n(z). These two recurrence relations give back the

differential equation and induction on l leads to the Rayleigh formulas

jn(z) = (−1)nzn[z−1 d

dz

]n

jo(z), nn(z) = (−1)nzn[z−1 d

dz

]n

no(z) (2.43)

From the above expressions it is easy to extract the limiting behaviors of the spherical Bessel’s functions.

18

Electromagnetic Diffraction from Strip:

.

CHAPTER 3

Electromagnetic Diffraction from Strips

3.1 Introduction

This chapter contains three problems. First problem deals with the diffraction from an impedance strip.

In the second problem diffraction from a conducting strip placed at the dielectric slab has been investigated.

Third problem involves the PEMC strip. Electromagnetic plane wave has been considered as a source of

excitation. Both TM and TE cases are discussed for each geometry. Algorithms are developed to compute

the field patterns and are presented graphically.

3.2 Diffraction of Electromagnetic Plane Wave by an Impedance Strip

3.2.1 Mathematical Formulation and Solution of the Problem

Consider an impedance strip of width 2a, as depicted in Fig. 3.1. The geometry is illuminated by a

uniform plane wave. The strip is considered infinite in length along z-direction, therefore it can be treated

as a two dimensional object. If Eiz and Hi

z be the incident fields for E- and H- polarization respectively, then

(

Eiz

Hiz

)

= exp[

jk0(x cos φ0 + y sin φ0)]

(3.1)

where φ0 be the angle of incidence with respect to x-axis and k0 = ω√

µ0ε0 is wave number of the medium.

It is assumed, for simplicity, that the amplitude of the incident field is unity.

Waves in two dimensional space are governed by Helmholtz equation given by

∂2%

∂x2+

∂2%

∂y2+ k2

0% = 0 (3.2)

19

Electromagnetic Diffraction from Strip:

where % represents the z−component of electric or magnetic field depending on the polarization. The

elementary solutions of (3.2) are given by

cos(ξxa) exp

[

∓√

ξ2 − κ20ya

]

, sin(ξxa) exp

[

∓√

ξ2 − κ20ya

]

where ξ is the separation variable, and xa = xa, y

a = ya, κ0 = k0a.

Using the relations

cosx =

√

πx

2J−

12(x), sinx =

√

πx

2J 1

2(x) (3.3a)

the diffracted field from the strip in the half space y > 0 as well as in the half space y < 0 can be expressed

by the general solution of the wave equation (3.2). That is

(

Ed+

z

Hd+

z

)

=

√

πxa

2

∫ ∞

0

{

g1(ξ)J−12

(xaξ) + g2(ξ)J 12

(xaξ)}

exp

[

−√

ξ2 − κ20ya

]

ξ12 dξ, y > 0 (3.3b)

(

Ed−

z

Hd−

z

)

=

√

πxa

2

∫ ∞

0

{

h1(ξ)J−12

(xaξ) + h2(ξ)J 12

(xaξ)}

exp

[

√

ξ2 − κ20ya

]

ξ12 dξ, y < 0 (3.3c)

where g1,2(ξ) and h1,2(ξ) are weighting functions to be determined from the required boundary conditions

on the plane y = 0. It is noted that (3.3b) and (3.3c) represent the Fourier sine and cosine transforms of

the wave functions since the Bessel functions in the above expressions are equivalent to the trigonometric

functions with the relation (3.3a).

Since in two dimensional problems, the waves in each polarization (E- and H-) does not couple, therefore

same symbols g1,2(ξ) and h1,2(ξ) for the unknown functions are used for both the cases.

x= ax= - a z

x

y

Figure 3.1 Geometry of the problem

( , )

0

A. E-Polarization

The required boundary conditions are given by

Etz

∣

∣

∣

y=0+

= −Z+Htx

∣

∣

∣

y=0+

, Etz

∣

∣

∣

y=0−

= Z−Htx

∣

∣

∣

y=0−

, |xa| ≤ 1 (3.4a)

Etz

∣

∣

∣

y=0+

= Etz

∣

∣

∣

y=0−

, Htx

∣

∣

∣

y=0+

= Htx

∣

∣

∣

y=0−

, |xa| ≥ 1 (3.4b)

20

Electromagnetic Diffraction from Strip:

where t in superscript means total and Z+ and Z− are assumed to be the surface impedances of the upper

and lower surfaces of the strip respectively. The conditions (3.4a) are the standard impedance boundary

conditions (SIBCs) and are discussed in detail in [50]-[53]. From the conditions (3.4b)∫ ∞

0

{[

g1(ξ) − h1(ξ)]

cos(xaξ) +[

g2(ξ) − h2(ξ)]

sin(xaξ)}

dξ = 0, |xa| ≥ 1 (3.5a)

∫ ∞

0

√

ξ2 − κ20

{[

g1(ξ) + h1(ξ)]

cos(xaξ) +[

g2(ξ) + h2(ξ)]

sin(xaξ)}

dξ = 0, |xa| ≥ 1 (3.5b)

Using discontinuous properties of the Weber-Schafheitlin’s integrals and incorporating the edge conditions

of electric and magnetic fields, the above expressions yield

g1(ξ) − h1(ξ) =

∞∑

m=0

AmJ2m+ 32(ξ)ξ−

32 , g2(ξ) − h2(ξ) =

∞∑

m=0

BmJ2m+ 52(ξ)ξ−

32 (3.6a)

g1(ξ) + h1(ξ) =

∞∑

m=0

Cm

J2m+ 12(ξ)

√

ξ2 − κ20

ξ−12 , g2(ξ) + h2(ξ) =

∞∑

m=0

Dm

J2m+ 32(ξ)

√

ξ2 − κ20

ξ−12 (3.6b)

Manipulation of the above expressions gives

g1(ξ) =

∞∑

m=0

AmJ2m+ 32(ξ)ξ−

32 + Cm

J2m+ 12(ξ)

√

ξ2 − κ20

ξ−12 (3.7a)

g2(ξ) =

∞∑

m=0

BmJ2m+ 32(ξ)ξ−

32 + Dm

J2m+ 32(ξ)

√

ξ2 − κ20

ξ−12 (3.7b)

h1(ξ) =

∞∑

m=0

AmJ2m+ 32(ξ)ξ−

32 − Cm

J2m+ 12(ξ)

√

ξ2 − κ20

ξ−12 (3.7c)

h2(ξ) =

∞∑

m=0

BmJ2m+ 32(ξ)ξ−

32 − Dm

J2m+ 32(ξ)

√

ξ2 − κ20

ξ−12 (3.7d)

where Am, Bm, Cm, Dm are the expansion coefficients. From the conditions (3.4a), it is obtained∫ ∞

0

[

1 − jζ+

κ0

√

ξ2 − κ20

]

[

g1(ξ) cos(xaξ) + g2(ξ) sin(xaξ)]

dξ

= −2[

1 − ζ+ sinφ0

]

exp[

jκ0xa cos φ0

]

(3.8a)∫ ∞

0

[

1 − jζ−κ0

√

ξ2 − κ20

]

[

h1(ξ) cos(xaξ) + h2(ξ) sin(xaξ)]

dξ

= −2[

1 + ζ− sin φ0

]

exp[

jκ0xa cos φ0

]

(3.8b)

In the above expressions ζ± = Z±/Z0 and Z0 =√

µ0

ε0be the impedance of free space. Putting the values

g1,2(ξ) and h1,2(ξ) in the last expressions and then separating even and odd functions, following expressions

are obtained

∞∑

m=0

∫ ∞

0

Ξ+(ξ)[

AmJ2m+ 32(ξ)ξ−

32 + Cm

J2m+ 12(ξ)

√

ξ2 − κ20

ξ−12

]

cos(xaξ)dξ = −Φ+(φ0) cos[

κ0xa cos φ0

]

(3.9a)

∞∑

m=0

∫ ∞

0

Ξ+(ξ)[

BmJ2m+ 52(ξ)ξ−

32 + Dm

J2m+ 32(ξ)

√

ξ2 − κ20

ξ−12

]

sin(xaξ)dξ = −Φ+(φ0)j sin[

κ0xa cos φ0

]

(3.9b)

∞∑

m=0

∫ ∞

0

Ξ−(ξ)[

AmJ2m+ 32(ξ)ξ−

32 − Cm

J2m+ 12(ξ)

√

ξ2 − κ20

ξ−12

]

cos(xaξ)dξ = Φ−(φ0) cos[

κ0xa cosφ0

]

(3.9c)

∞∑

m=0

∫ ∞

0

Ξ−(ξ)[

BmJ2m+ 52(ξ)ξ−

32 − Dm

J2m+ 32(ξ)

√

ξ2 − κ20

ξ−12

]

sin(xaξ)dξ = Φ−(φ0)j sin[

κ0xa cosφ0

]

(3.9d)

21

Electromagnetic Diffraction from Strip:

where the symbols Ξ±(ξ) and Φ±(φ0) stand for

Ξ±(ξ) =[

1 − jζ±κ0

√

ξ2 − κ20

]

Φ+(φ0) = 2[

1 − ζ+ sinφ0

]

Φ−(φ0) = 2[

1 + ζ− sinφ0

]

Expanding the trigonometric functions in the above expressions in terms of Jacboi’s polynomials pmn (section

2.2.3) and making use of (3.3a) and the following relations

x−m/2Jm(ξ√

x) =

∞∑

n=0

2(2n + m + 1)Γ (n + m + 1)

Γ (n + 1)Γ (m + 1)

J2n+m+1(ξ)

ξpm

n (x) (3.10a)

∫ 1

0

xmpmn (x)pm

n′(x)dx =Γ 2(m + 1)Γ 2(n + 1)

(2n + m + 1)Γ 2(n + m + 1)δn,n′ (3.10b)

where δn,n′ is the the delta function and Jacobi’s polynomial is given below

pmn (x) = F (n + m + 1,−n,m + 1;x)

=Γ (n + 1)Γ (m + 1)

Γ (n + m + 1)x−m/2

∫ ∞

0

Jm(√

xξ)J2n+m+1(ξ)dξ (3.10d)

The following matrix equations are obtained for the evaluation of expansion coefficients.

∞∑

m=0

[

KRE

(

2m +3

2, 2n +

1

2; ζ+

)][

Am

]

+[

GRE

(

2m +1

2, 2n +

1

2; ζ+

)][

Cm

]

= −Φ+(φ0)[

JE

]

(3.11a)

∞∑

m=0

[

KRE

(

2m +3

2, 2n +

1

2; ζ−

)][

Am

]

−[

GRE

(

2m +1

2, 2n +

1

2; ζ+

)][

Cm

]

= Φ−(φ0)[

JE

]

(3.11b)

∞∑

m=0

[

KRE

(

2m +5

2, 2n +

3

2; ζ+

)][

Bm

]

+[

GRE

(

2m +3

2, 2n +

3

2; ζ+

)][

Dm

]

= −jΦ+(φ0)[

JO

]

(3.11c)

∞∑

m=0

[

KRE

(

2m +5

2, 2n +

3

2; ζ−

)][

Bm

]

−[

GRE

(

2m +3

2, 2n +

3

2; ζ+

)][

Dm

]

= jΦ−(φ0)[

JO

]

(3.11d)

n = 0, 1, 2...

where the correspondence between the matrices and their elements are

[

JE

]

⇐⇒J2n+ 1

2(κ0 cos φ0)

(κ0 cosφ0)12

[

JO

]

⇐⇒J2n+ 3

2(κ0 cos φ0)

(κ0 cosφ0)12

(3.12a)

and

KRE(m,n; ζ±) =

∫ ∞

0

Ξ±(ξ)Jm(ξ)Jn(ξ)

ξ2dξ (3.12b)

GRE(m,n; ζ±) =

∫ ∞

0

Ξ±(ξ)Jm(ξ)Jn(ξ)

ξ√

ξ2 − κ20

dξ (3.12c)

22

Electromagnetic Diffraction from Strip:

Equations (3.11) are the simultaneous equations and can be solved for expansion coefficients Am, Bm, Cm,

Dm as below

{[

KRE

(

2m +3

2, 2n +

1

2; ζ+

)]−1[

GRE,

(

2m +1

2, 2n +

1

2; ζ−

)]

+[

KRE

(

2m +3

2, 2n +

1

2; ζ+

)]−1[

GRE

(

2m +1

2, 2n +

1

2; ζ−

)]}[

Cm

]

= −{

Φ+(φ0)[

KRE

(

2m +1

2, 2n +

1

2; ζ+

)]−1

+ [Φ−(φ0)[

KRE

(

2m +1

2, 2n +

1

2; ζ+

)]−1}[

JE

]

(3.13a)

[

Am

]

= −[

KRE

(

2m +3

2, 2n +

1

2; ζ+

)]−1[

GRE

(

2m +1

2, 2n +

1

2; ζ+

)][

Cm

]

− Φ+(φ0)[

KRE

(

2m +3

2, 2n +

1

2; ζ+

)]−1[

JE

]

(3.13b)

{[

KRE

(

2m +5

2, 2n +

3

2; ζ+

)]−1[

GRE

(

2m +3

2, 2n +

3

2; ζ+

)]

+[

KRE

(

2m +5

2, 2n +

3

2; ζ+

)]−1[

GRE

(

2m +3

2, 2n +

3

2; ζ−

)]}[

Dm

]

= −j{

Φ+(φ0)[

KRE

(

2m +5

2, 2n +

3

2; ζ+

)]−1

+ Φ−(φ0

[

KRE

(

2m +5

2, 2n +

3

2; ζ+

)]−1}[

JO

]

(3.13c)

[

Bm

]

= −[

KRE

(

2m +5

2, 2n +

3

2; ζ+

)]−1[

GRE

(

2m +3

2, 2n +

3

2; ζ+

)][

Dm

]

− jΦ+(φ0)[

KRE

(

2m +5

2, 2n +

3

2; ζ+

)]−1[

JO

]

(3.13d)

If surface impedances of the upper and lower surfaces are equal i.e ζ+ = ζ− = ζ, equations (3.11) may be

simplified to

∞∑

m=0

[

KRE

(

2m +3

2, 2n +

1

2; ζ

)][

Am

]

= 2ζ sin(φ0)[

JE

]

(3.14a)

∞∑

m=0

[

KRE

(

2m +5

2, 2n +

3

2; ζ

)][

Bm

]

= 2jζ sin(φ0)[

JO

]

(3.14b)

∞∑

m=0

[

GRE

(

2m +1

2, 2n +

1

2; ζ

)][

Cm

]

= −2[

JE

]

(3.14c)

∞∑

m=0

[

GRE

(

2m +3

2, 2n +

3

2; ζ

)][

Dm

]

= −2j[

JO

]

, n = 0, 1, 2... (3.14d)

The values of expansion coefficients Am, Bm, Cm, Dm can be computed from the above expressions. Once

these expansion coefficients are computed, these can be used to evaluate the field expressions and the other

quantities of interest.

A1. Diffracted Fields

Diffracted far field in the upper region may be evaluated by applying the saddle point method of integration.

If the values of g1,2(ξ) from (3.7a) and (3.7b) are substituted into (3.3b), following is obtained

Ed+

z =1

2

∞∑

m=0

∫ ∞

0

{[

Am

J2m+ 32(ξ)

ξ32

+ Cm

J2m+ 12(ξ)

√

ξ(ξ2 − κ20)

]

cos(xaξ)

+

[

Bm

J2m+ 52(ξ)

ξ32

+ Dm

J2m+ 32(ξ)

√

ξ(ξ2 − κ20)

]

sin(xaξ)

}

exp[

−√

ξ2 − κ20ya

]

dξ

Putting ξ = κ0 cos θ, xa = ρ cos φ, ya = ρ sin φ in the above expression

23

Electromagnetic Diffraction from Strip:

Ed+

z =1

2

∞∑

m=0

∫

c

{[

AmJ2m+ 32(κ0 cos θ) + jBmJ2m+ 5

2(κ0 cos θ)

]

tan(θ)

−j[

CmJ2m+ 12(κ0 cos θ) + jDmJ2m+ 3

2(κ0 cos θ)

]}

(κ0 cos θ)−1 exp[

−jκ0ρ cos(θ − φ)]

dθ

where c is the path from −j∞ to π + j∞ in the complex plane. The saddle point is θ = φ. So the last

expression, for far observation point, reduces to

Ed+

z '√

π

8

1√k0ρ

exp[

−jk0ρ + jπ

4

]

∞∑

m=0

{

[

AmJ2m+ 32(κ0 cosφ) + BmJ2m+ 5

2(κ0 cos φ)

]

tan φ

− j[

CmJ2m+ 12(κ0 cos φ) + DmJ2m+ 3

2(κ0 cos φ)

]

}

(κ0 cosφ)−12 , kρ � 1 (3.15)

where φ is the angle of observation with reference to positive x-axis . For computation of the fields, the

expansion coefficients Am, Bm, Cm, Dm, as described earlier, can be obtained from equations (3.13) or (3.14).

A2. Current Induced on the Strip

The current density induced on the impedance strip may be obtained as follows

Jz = −[

Htx

∣

∣

∣

y=0+

−[Htx

∣

∣

∣

y=0−

]

=jY0

κ0

∞∑

m=0

∫ ∞

0

[

CmJ2m+ 12(ξ) cos(xaξ) + DmJ2m+ 3

2(ξ) sin(xaξ)

]

ξ−12 dξ

=jY0√2κ0

∞∑

m=0

Γ (m + 1

2)

Γ (m + 1)

{

Cmp−

12

m (x2a) + xa(2m + 1)Dmp

12m(x2

a)}

(3.16)

where Y0 = 1

Z0and Γ (.) is the gamma function. While getting the expression (3.16), (3.10) has been used.

B. H-Polarization

The required boundary conditions for H-Polarization case are given by

Etx

∣

∣

∣

y=0+

= Z+Htz

∣

∣

∣

y=0+

, Etx

∣

∣

∣

y=0−

= −Z−Htz

∣

∣

∣

y=0−

, |xa| ≤ 1 (3.17a)

Etx

∣

∣

∣

y=0+

= Etx

∣

∣

∣

y=0−

, Htz

∣

∣

∣

y=0+

= Htz

∣

∣

∣

y=0−

, |xa| ≥ 1 (3.17b)

The incident wave is given by (1) and the diffracted wave is given by (3.3b). The conditions (3.17b) give the

same expressions as (3.6) and g1,2(ξ) ∼ h1,2(ξ) are given by (3.7). And the conditions (3.17a) yield

∫ ∞

0

[

√

ξ2 − κ20 + jκ0ζ+

][

g1(ξ) cos(xaξ) + g2(ξ) sin(xaξ)]

dξ

= jκ0(sin φ0 − ζ+) exp[

jκ0xa cos φ0

]

(3.18a)∫ ∞

0

[

√

ξ2 − κ20 + jκ0ζ−

][

h1(ξ) cos(xaξ) + h2(ξ) sin(xaξ)]

dξ

= −jκ0(sin φ0 + ζ−) exp[

jκ0xa cos φ0

]

(3.18b)

24

Electromagnetic Diffraction from Strip:

Separating even and odd functions and then projecting the resulting equations into the functional space

with elements p±

12

n (x2a), following matrix equations are obtained after some manipulations

[

K+

RH,E

][

Am

]

+[

G+

RH,E

][

Cm

]

= j[

sin φ0 − ζ+

][

JE

]

(3.19a)[

K−

RH,E

][

Am

]

−[

G−

RH,E

][

Cm

]

= j[

sin φ0 + ζ−

][

JE

]

(3.19b)[

K+

RH,O

][

Bm

]

+[

G+

RH,O

][

Dm

]

= −[

sin φ0 − ζ+

][

JO

]

(3.19c)[

K−

RH,O

][

Bm

]

−[

G−

RH,O

][

Dm

]

= −[

sin φ0 + ζ−

][

JO

]

(3.19d)

where the correspondence between the matrices and their elements are given by

[

K±

RH,E

]

⇐⇒ KRH

(

2n +1

2, 2m +

3

2; ζ±

)

[

K±

RH,O

]

⇐⇒ KRH

(

2n +3

2, 2m +

5

2; ζ±

)

[

G±

RH,E

]

⇐⇒ GRH

(

2n +1

2, 2m +

1

2; ζ±

)

[

G±

RH,O

]

⇐⇒ GRH

(

2n +3

2, 2m +

3

2; ζ±

)

[

JE

]

⇐⇒ 2κJ2n+ 1

2(κ cos φ0)

(κ cos φ0)12

[

JO

]

⇐⇒ 2κJ2n+ 3

2(κ cos φ0)

(κ cos φ0)12

(3.20)

and

KRH(m,n; ζ) =

∫ ∞

0

[

jζκ0 +

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ2dξ (3.21a)

GRH(m,n; ζ) =

∫ ∞

0

[

jζκ0 +

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ√

ξ2 − κ20

dξ (3.21b)

The matrix equations (3.19) may be used to get the expansion coefficients Am, Bm, Cm, Dm as follows

{[

K+

RH,E

]−1[

G+

RH,E

]

+[

K−

RH,E

]−1[

G−

RH,E

]}[

Cm

]

= j{

[sin φ0 − ζ+][

K+

RH,E

]−1

+ [sin φ0 + ζ−][

K+

RH,E

]−1}[

JE

]

(3.22a)

[

Am

]

= −[

K+

RH,E

]−1[

G+

RH,E

][

Cm

]

+ j[sin φ0 − ζ+][

K+

RH,E

]−1[

JE

]

(3.22b)

{[

K+

RH,O

]−1[

G+

RH,O

]

+[

K−

RH,O

]−1[

G−

RH,O

]}[

Dm

]

={

−[sin φ0 − ζ+][

K+

RH,O

]−1

+ [sin φ0 + ζ−][

K−

RH,O

]−1}[

JO

]

(3.22c)

[

Bm

]

= −[

K+

RH,O

]−1[

G+

RH,O

][

Dm

]

− [sin φ0 − ζ+][

K+

RH,O

]−1[

JO

]

(3.22d)

If ζ+ = ζ− = ζ, equation (3.19) reduces to

[

Am

]

= j sinφ0

[

KRH,E

]−1[

JE

]

(3.23a)

[

Bm

]

= − sin φ0

[

KRH,O

]−1[

JO

]

(3.23b)

[

Cm

]

= −jζ[

GRH,E

]−1[

JE

]

(3.23c)

[

Dm

]

= ζ[

GRH,O

]−1[

JO

]

(3.23d)

25

Electromagnetic Diffraction from Strip:

Far H-polarized scattered field in the upper half space can be evaluated by applying the saddle point method

of integration and the result has the same form as (3.15), but the expansion coefficients Am ∼ Dm are given

by (3.22) or (3.23), instead of (3.13) or (3.14).

B1. Current Induced on the Strip

The current density induced on the impedance strip is obtained as follows.

Jx = Htz

∣

∣

∣

y=0+

−Htz

∣

∣

∣

y=0−

=

∞∑

m=0

∫ ∞

0

[

Am

J2m+ 32(ξ)

ξ32

cos(xaξ) + Bm

J2m+ 52(ξ)

ξ32

sin(xaξ)

]

dξ

=1√2

∞∑

m=0

Γ (m + 1

2)

Γ (m + 1)

{

Am

4m + 3

[

p−

12

m (x2a) +

m + 1

2

m + 1p−

12

m+1(x2a)

]

+xa2m + 1

4m + 3Bm

[

p12m(x2

a) +m + 3

2

m + 1p

12

m+1(x2a)

]}

(3.24)

where (3.10) is used similar to the E-polarized case.

3.2.2 Physical Optics Approximate Solutions

The above KP solutions can be verified by solving the problem by some other method. So physical optics

(PO) method has be used as an alternative approach.

A. E- Polarization:

The total field on the surface of the strip is

Etz =

2ζ+ sin φ0

1 + ζ+ sin φ0

exp[

jk0x cos φ0

]

, Htx = − 2Y0 sin φ0

1 + ζ+ sin φ0

exp[

jk0x cos φ0

]

(3.25)

The equivalent current is

Mx = −Etz, Jz = −Ht

x

Far field expression of the vector potential is given by

Az =µY0

j2Q0C(k0ρ)

∫ a

−a

exp[

jk0(cos φ + cos φ0)x′

]

dx′ =µY0

j2Q0C(k0ρ)S0(φ),

Fx = − εζ+

j2Q0C(k0ρ)

∫ a

−a

exp[

jk0(cos φ + cos φ0)x′

]

dx′ = − εζ+

j2Q0C(k0ρ)S0(φ) (3.26)

where Az and Fx be the components of magnetic and electric vector potential respectively. And

Q0 =2 sin φ0

1 + ζ+ sinφ0

C(k0ρ) =

√

2

πk0ρexp

[

−jk0ρ + jπ

4

]

Thus far electric field is derived as

Ez = −jωAz +1

ε

∂Fx

∂y= −k0

4

(

1 − ζ+ sin φ)

Q0C(k0ρ)S0(φ)

= − 1 − ζ+ sinφ

1 + ζ+ sin φ0

k0a sin φ0

√

2

πk0ρexp

(

−jk0ρ + jπ

4

)

sinc[

k0a(

cos φ + cosφ0

)]

(3.27)

26

Electromagnetic Diffraction from Strip:

where sinc(x) = sin xx .

B. H-Polarization

The total field on the surface of the strip is

Htz =

2 sin φ0

ζ+ + sinφ0

exp[

jk0x cos φ0

]

Etx = Z0

2ζ+ sinφ0

ζ+ + sin φ0

exp[

jk0x cos φ0

]

(3.28)

The equivalent current is

Jx = −Htz, Mz = Et

x

Far field expression of the vector potential is given by

Ax =µ

j2Q1C(k0ρ)S0(φ), Fz =

εζ+

j2Z0Q1C(k0ρ)S0(φ) (3.29)

where Q1 =2 sin φ0

ζ+ + sinφ0

. Thus far magnetic field is derived as

Hz = −jωFz −1

µ

∂Fx

∂y=

k0

4

(

sin φ − ζ+

)

Q0C(k0ρ)S0(φ)

= − ζ+ − sin φ

ζ+ + sinφ0

k0a sin φ0

√

2

πk0ρexp

(

−jk0ρ + jπ

4

)

sinc[

k0a(

cos φ + cos φ0

)]

(3.30)

where the terms have the same meanings as given in the proceeding section.

3.2.3 Computations and Discussion

A. Computation of the Integrals KRE(m,n; ζ), GRE(m,n; ζ), KRH(m,n; ζ) and GRH(m,n; ζ)

First we take the integral

KRE(m,n; ζ) =

∫ ∞

0

[

1 − jζ

κ0

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ2dξ

=

∫ x

0

[

1 − jζ

κ0

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ2dξ +

∫ ∞

x

[

1 − jζ

κ0

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ2dξ

= KxRE(m,n; ζ) + K∞RE(m,n; ζ) (3.31a)

where x is a large constant and it is taken as x = 300 in present computations. The integral KxRE(m,n; ζ)

may be evaluated by any standard numerical method method. The second integral K∞RE(m,n; ζ) can be

written as follow

K∞RE(m,n; ζ) =

∫ ∞

x

[

1 − jζ

κ0

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ2dξ

=

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ2− j

ζ

κ0

∫ ∞

x

Jm(ξ)Jn(ξ)

ξdξ + j

ζκ0

2

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ3dξ (3.31b)

To perform the above numerical integrations, Hankel approximation of the Bessel function has been used.

That is given by

Jn(ξ) =

√

2

πξ

{[

1 − (4n2 − 1)(4n2 − 9)

128ξ2+

(4n2 − 1)(4n2 − 9)(4n2 − 25)(4n2 − 49)

98304ξ4

]

cos

(

ξ − 2n + 1

4π

)

−[

(4n2 − 1)

8ξ− (4n2 − 1)(4n2 − 9)(4n2 − 25)

3072ξ3

]

sin

(

ξ − 2n + 1

4π

)}

(3.32)

27

Electromagnetic Diffraction from Strip:

Using this formula it can be written as

Jm(ξ)Js(ξ) =1

π

[

1

ξ− a2 + b2 − a1b1

ξ3+

a4 + b4 + a2b2 − a1b3 − b1a3

ξ5

]

cos(m − s)π

2

+1

π

[

a1 − b1

ξ2− a1b2 + a3 − (a2b1 + b3)

ξ4

]

sin(m − s)π

2

+1

π

[

1

ξ− a2 + b2 + a1b1

ξ3

]

cos

(

2ξ − m + s + 1

2π

)

− 1

π

[

a1 + b1

ξ2− a1b2 + a3 + (a2b1 + b3)

ξ4

]

sin

(

2ξ − m + s + 1

2π

)

a1 =4m2 − 1

8, a2 =

(4m2 − 1)(4m2 − 9)

128, a3 =

(4m2 − 1)(4m2 − 9)(4m2 − 25)

3072

a4 =(4m2 − 1)(4m2 − 9)(4m2 − 25)(4m2 − 49)

98304

b1 =4s2 − 1

8, b2 =

(4s2 − 1)(4s2 − 9)

128, b3 =

(4s2 − 1)(4s2 − 9)(4s2 − 25)

3072

b4 =(4s2 − 1)(4s2 − 9)(4s2 − 25)(4s2 − 49)

98304(3.33)

Integrating by parts and then retaining the terms up to ξ−5 with constant coefficients and ξ−4 multiplied

by trigonometric functions, following integrals may be written as

∫ ∞

x

Jm(ξ)Jn(ξ)

ξdξ =

1

π

[(

− 1

x+

A1

3x3

)

cosm − n

2π +

(

− A3

2x2+

A5

4x4

)

sinm − n

2π]

+[( 1

2x2− A2

2x4+

3A4

4x4− 3

4x4

)

sin(2x − β) +(

− 1

2x3+

3

2x5+

A2

x5+

A4

2x3− 3A4

2x5

)

× cos(2x − β)] 1

π(3.34a)

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ2dξ =

1

π

[(

− 1

2x2+

A1

4x4

)

cosm − n

2π +

( 1

2x3− 3

2x5+

A4

x5− A2

2x5

)

sin(2x − β)

+(

− 3

4x4+

A4

2x4

)

cos(2x − β) +(−A3

3x3+

A5

5x5

)

sin(m − n

2π)

]

(3.34b)∫ ∞

x

Jm(ξ)Jn(ξ)

ξ3dξ =

1

π

[(

− 1

3x3+

A1

5x5

)

cosm − n

2π +

1

2x4sin(2x − β) +

( 1

x5+

A4

2x5

)

cos(2x − β)

− A3

4x4sin(

m − n

2π)

]

(3.34c)∫ ∞

x

Jm(ξ)Jn(ξ)

ξ4dξ =

1

π

[

− 1

4x4cos(

m − n

2π) +

1

2x5sin(2x − β) − A3

5x5sin(

m − n

2π)

]

(3.34d)

where

β = (m + n + 1)π

2, A1 = a2 + b2 − a1b1, A2 = a2 + b2 + a1b1,

A3 = a1 − b1, A4 = a1 + b1, A5 = a1b2 + a3 − (a2b1 + b3)

Using the expressions (3.34), the integral KRE(m,n; ζ) can be computed as follow

KRE(m,n; ζ) = KxRE(m,n; ζ) +1

π

{[ −1

2x2+

A1

4x4− P1

(−1

x+

A1

3x3

)

+ P2

( −1

3x3+

A1

5x5

)]

cosm − n

2π

+[−A3

3x3+

A5

5x5− P1

(−A3

2x2+

−A5

4x4

)

+ P2

(−A3

4x4

)]

sinm − n

2π

+[ 1

2x3− 3

2x5+

A4

x5− A2

2x5− P1

( 1

2x2− 1

x4− A2

2x4+

3A4

4x4

)

+ P2

( 1

x4

)]

sine

+[

− 3

4x4+

A4

2x4− P1

(

− 1

2x3+

3

2x5+

A2

x5+

A4

2x3− 3A4

2x5− A6

2x5

)

+ P2

(

− 1

x5+

A4

2x5

)]

coz}

(3.35)

28

Electromagnetic Diffraction from Strip:

where P1 = jζκ0

, P2 = jκ0ζ2

, coz = cos(2x − β) sine = sin(2x − β) and A6 = a1b2 + a3 + (a2b1 + b3).

Following the same procedure, the integral GRE(m,n; ζ) can be evaluated as follow

GRE(m,n; ζ) =

∫ ∞

0

[

1 − jζ

κ0

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ√

ξ2 − κ20

dξ

=

∫ x

0

[

1 − jζ

κ0

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ√

ξ2 − κ20

dξ +

∫ ∞

x

[

1 − jζ

κ0

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ√

ξ2 − κ20

dξ

= GxRE(m,n; ζ) + G∞RE(m,n; ζ) (3.36a)

where G∞RE(m,n; ζ) may be written as

G∞RE(m,n; ζ) =

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ2dξ +

κ2

2

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ4dξ − j

κ0

ζ

∫ ∞

x

Jm(ξ)Jn(ξ)

ξdξ

1

π

{[ −1

2x2+

A1

4x4− P4

(−1

x+

A1

3x3

)

+ P6

( −1

3x3+

A1

5x5

)]

cosm − n

2π

+[−A3

3x3+

A5

5x5− P4

(−A3

2x2+

−A5

4x4

)

+ P6(−A3

4x4)]

sinm − n

2π

+[ 1

2x3− 3

2x5+

A4

x5− A2

2x5− P4

( 1

2x2− 1

x4− A2

2x4+

3A4

4x4

)

+ P6

( 1

x4

)]

sine

+[

− 3

4x4+

A4

2x4− P4

(

− 1

2x3+

3

2x5+

A2

x5+

A4

2x3− 3A4

2x5− A6

2x5

)

+ P6

(

− 1

x5+

A4

2x5

)]

coz}

(3.36b)

where P3 =κ20

2and all other notations have the same meaning as described above. The other integrals

KRH(m,n; ζ) and GRH(m,n; ζ) can also be evaluated in the similar manner. The final results, after some

manipulations, are

KRH(m,n; ζ) =

∫ ∞

0

[

jζκ0 +

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ2dξ

=

∫ x

0

[

jζκ0 +

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ2dξ +

∫ ∞

x

Jm(ξ)Jn(ξ)

ξdξ + jκ0ζ

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ2dξ

−κ20

2

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ3dξ

Therefore

KRH(m,n; ζ) = KxRH(m,n; ζ) + K∞RH(m,n; ζ)

=KxRH(m,n; ζ) +1

π

{[

P4

(

− 1

2x2+

A1

4x4

)

− κ20

2

(

− 1

3x3+

A1

5x5

)

− 1

x+

A1

3x3

]

cosm − n

2π

+[

P4

(−A3

3x3+

A5

5x5

)

+κ2

0

2

A3

4x4+

(

− A3

2x2+

A5

4x4

)]

sinm − n

2π

+[

P4

( 1

2x3− 3

2x5+

A4

x5− A2

2x5

)

− κ20

2

1

2x4+

1

2x2− A2

2x4− 1

x5+

3A4

4x4− 1

x4

]

sine

+[

P4

(

− 3

4x4+

A4

2x4

)

− 1

2x3+

3

2x5+

A2

x5+

A4

2x3− 3A4

2x5− A6

2x5− κ2

0

2

(

− 1

x5+

A4

2x5

)]

coz}

(3.37)

where P4 = jκ0ζ

GRH(m,n; ζ) =

∫ ∞

0

[

jζκ0 +

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ√

ξ2 − κ20

dξ

=

∫ x

0

[

jζκ0 +

√

ξ2 − κ20

]Jm(ξ)Jn(ξ)

ξ√

ξ2 − κ20

dξ +

∫ ∞

x

Jm(ξ)Jn(ξ)

ξdξ + jκ0ζ

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ2dξ

29

Electromagnetic Diffraction from Strip:

+jκ3

0ζ

2

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ4dξ

=GxRH(m,n; ζ) + G∞RH(m,n; ζ)

=GxRH(m,n; ζ) + +1

π

{[

P4

(

− 1

2x2+

A1

4x4

)

− P5

4x4− 1

x+

A1

3x3

]

cosm − n

2π

+[

P4

(

− A3

3x3+

A5

5x5

)

− P5A3

5x5− A3

2x2+

A5

4x4

]

sinm − n

2π

+[

P4

( 1

2x3− 3

2x5+

A4

x5− A2

2x5

)

+ P5

1

2x5+

1

2x2− A2

2x4− 1

x5+

3A4

4x4− 1

x4

]

sine

+[

P4

(

− 3

4x4+

A4

2x4

)

− 1

2x3+

3

2x5+

A2

x5+

A4

2x3− 3A4

2x5− A6

2x5

]

coz}

(3.38)

where P5 =P4κ2

0

2

B. Diffracted far Fields

In the scattering or diffraction problems, one of the most interesting thing to see is the field patterns. In

the problem discussed above, far field patterns can be computed using equation (3.15) for both the E- and

H- polarization. But the expansion coefficients Am, Bm, Cm, Dm for both the cases are different. For E-

polarization, these can be computed using equation (3.13) or (3.14). Same may be computed from equation

(3.22) or (3.23) for H-polarization. The matrix size is taken as m × n = (2κ0 + 1) × (2κ0 + 1). Diffracted

far fields are then computed using these expansion coefficients in equation (3.15) for both the cases. Fig.

3.2 and Fig. 3.3 give the far field patterns for E-polarization for different values of angle of incidence and

κ0 = 4.0, ζ± = 0.1+0.3i. The validity of these results are verified by comparing them with those obtained

using physical optics through equation (3.27). Fig. 3.4 and Fig. 3.5 give the diffracted field patterns and their

comparisons for H-polarization corresponding to φ0 = π/2 and φ0 = π/3, κ0 = 4.0, ζ± = 0.1 + 0.3i

with those obtained through physical optics using equation (3.30). Observing these graphs, it can be noted

that the comparison of the KP and physical optics results is reasonable for both the polarizations.

C. Current Distribution on the Strip

Current distributions on the strip is another important parameter. These can be computed from the

expressions (3.16) for E-polarization and (3.24) for H-polarization. Fig. 3.6 and Fig. 3.7 shows the same for

different values of angle of incidence. It can be noticed that the magnitude of current density is maximum

but finite at the edges of the strip for E-polarization and reverse is the case for H-polarization. The results

are as expected.

3.2.4 Conclusion

In this problem, electromagnetic diffraction from an impedance strip has been studied. The thickness of

the strip is taken as vanishingly small. The unknown weighting functions are expanded by taking into account

the edge conditions. Detailed discussions are presented to evaluate the integrals involved. Numerical results

are presented for far field patterns and the current densities induced on the strip. It is found that not only

the amplitude of the diffracted fields but position of the main lobe of the fields also depend upon the angle

of incidence. With the increase of the angle of incidence, the amplitude of the main lobe gradually increases

and it shifts towards the lower values of the angle of observation. The same problem is also attempted using

the physical optics technique and the solutions are presented to validate the KP method results. It can easily

be noticed that the comparison is good. Current distributions on the strip show that, for E-polarization,

current density has maximum value at the edges but not infinite as in case of conducting strip.

30

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

PO KP

E- polarization

Figure 3.2 Diffracted far field for φ0 = 60 and its comparison with PO

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

DIF

FRA

CTE

D F

IELD

OBSRVATION ANGLE

PO KP

E- polarization

Figure 3.3 Diffracted field pattern for normal incidence

31

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200

0.0

0.5

1.0

1.5

2.0

2.5

3.0

FAR

DIF

FRA

CTE

D F

IELD

OBSEVATION ANGLE

KP PO

H- polarization

Figure 3.4 Comparison between the two methods for H-polarization

-20 0 20 40 60 80 100 120 140 160 180 200-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

KP PO

H- polarization

Figure 3.5 Comparison of physical optics and KP field patterns

32

Electromagnetic Diffraction from Strip:

-1.0 -0.5 0.0 0.5 1.01.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

CU

RR

EN

T M

AG

NIT

UD

E

xa

E- polarization

Figure 3.6 Current distribution for E-polarization

-1.0 -0.5 0.0 0.5 1.0-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

CU

RR

EN

T M

AG

NIT

UD

E

x/a

H-polarization

Fig. 3.7 Current distribution ( H-Polarization)Figure 3.7 Current distribution for H-polarization

33

Electromagnetic Diffraction from Strip:

3.3 Diffraction of Electromagnetic Plane Wave from a Conducting Strip

Placed on a Dielectric Slab

3.3.1 Formulation of the Problem

A. E-Polarization

The geometry of the problem and the coordinate system are shown in Fig. 3.8. A strip of negligible

thickness and width 2a is located on a dielectric slab. The thickness of the slab is taken as d. The constitutive

parameters of medium occupying the slab (−d < y < 0) are assumed as (ε, µ0), while rest of the space has

medium with constitutive parameters (ε0, µ0). Let the electromagnetic plane wave is incident upon the

geometry. Therefore incident field Eiz

Eiz = exp

[

jk0(x cos φ0 + y sin φ0)]

(3.39a)

with φ0 as the angle of incidence. For convenience, the whole space has been divided into three regions.

Region I comprises the portion y > 0, above the strip. Region II (−d < y < 0) is the dielectric slab while

region III is considered to occupy the portion y < −d . In each region, scattered field may be assumed in

terms of unknowns. The expressions for scattered fields Esuz in region I, Esd

z in the slab, and Eslz in region III,

are supposed in the form

Esuz =

∫ ∞

0

[

g1(ξ) cos (ξxa) + g2(ξ) sin (ξxa)]

exp (−uya) dξ, y > 0 (3.39b)

Esdz =

∫ ∞

0

cos (ξxa)[

f1(ξ) exp (vya) + h1(ξ) exp (−vya)]

dξ

+

∫ ∞

0

sin (ξxa)[

f2(ξ) exp (vya) + h2(ξ) exp (−vya)]

dξ, −d < y < 0 (3.39c)

Eslz =

∫ ∞

0

[

`1(ξ) cos (ξxa) + `2(ξ) sin (ξxa)]

exp[

u(ya + da)]

dξ, y < −d (3.39d)

where k0 =√

µ0ε0 and k =√

µ0ε are the wave numbers of free space and dielectric slab respectively and

the other parameters are u =√

ξ2 − κ20, v =

√

ξ2 − κ2, κ0 = k0a, κ = ka, da = da , xa = x

a , ya = ya . The

functions f1,2(ξ), g1,2(ξ), h1,2(ξ), l1,2(ξ) are the weighting functions to be determined from the boundary

conditions. The boundary conditions of the problem are given by

(i) Ez is continuous at y = 0 for all values of x(ii) Ez andHx are continuous for y = −d(iii) Hx is continuous at y = 0 and for |xa| ≥ 1

(iv) Eiz + Er

z + Esuz = 0 at y = 0 and for |xa| ≤ 1

(3.40)

Using boundary conditions (i) and (ii) the following relations are obtained

f1(ξ) + h1(ξ) = g1(ξ) (3.41a)

f2(ξ) + h2(ξ) = g2(ξ) (3.41b)

f1(ξ) exp(−p) + h1(ξ) exp(p) = `1(ξ) (3.41c)

f2(ξ) exp(−p) + h2(ξ) exp(p) = `2(ξ) (3.41d)

f1(ξ) exp(−p) − h1(ξ) exp(p) = u`1(ξ) (3.41e)

f2(ξ) exp(−p) − h2(ξ) exp(p) = u`2(ξ) (3.41f)

34

Electromagnetic Diffraction from Strip:

where p = vda. Similarly applying the boundary conditions (iii) and (iv)

∫ ∞

0

[{

ug1(ξ) + v[

f1(ξ) − h1(ξ)]}

cos(ξxa) +{

ug2(ξ) + v[

f2(ξ) − h2(ξ)]}

sin(ξxa)]

dξ = 0 (3.42a)

∫ ∞

0

[

g1(ξ) cos(ξxa) + g2(ξ) sin(ξxa)]

dξ = −(1 + ΛE) exp[

jk0x sin φ0

]

(3.42b)

where ΛE is the reflection coefficient of the slab for E-polarization which may be evaluated from the expres-

sions

ΛE =(1 − P 2

E)(e∗ − e)

∆E, PE =

Y0 sinφ0

Y sin φr

e = exp(jkd cos φr), e∗ = exp(−jkd cos φr)

∆E =(1 + PE)2e − (1 − PE)2e∗

where Y0 =

√

ε0µ0

and Y =

√

ε

µ0

are intrinsic admittances of free space and dielectric slab respectively.

Permeability is assumed to be constant all over the space. The angle of refraction φr is related with the

angle of incident φ0 through Snell’s law

k cosφr = k0 cosφ0

x

y

Fig: 3.8 Geometry of the problem

µ0,

µ0, 0

zX= - a X= a

d

Eiz

Manipulating the expressions (3.41a) − (3.41f), following is obtained

f1(ξ) =(u + v) exp(p)

(v + u) exp(p) + (v − u) exp(−p)g1(ξ) (3.43a)

h1(ξ) =(v − u) exp(−p)

(v + u) exp(p) + (v − u) exp(−p)g1(ξ) (3.43b)

`1(ξ) =2v

(v + u) exp(p) + (v − u) exp(−p)g1(ξ) (3.43c)

f2(ξ) =(u + v) exp(p)

(v + u) exp(p) + (v − u) exp(−p)g2(ξ) (3.43d)

35

Electromagnetic Diffraction from Strip:

h2(ξ) =(v − u) exp(−p)

(v + u) exp(p) + (v − u) exp(−p)g2(ξ) (3.43e)

`2(ξ) =2v

(v + u) exp(p) + (v − u) exp(−p)g2(ξ) (3.43f)

Equations (3.42a) and (3.42b) are the dual integral equations. So making use of discontinuous properties of

Weber-Schafheitlin’s integrals and incorporating the edge, it can be written as

ug1(ξ) + v[

f1(ξ) − h1(ξ)]

=

∞∑

m=0

AmJ2m(ξ) (3.44a)

ug2(ξ) + v[

f2(ξ) − h2(ξ)]

=

∞∑

m=0

BmJ2m+1(ξ) (3.44b)

where Am and Bm are the expansion coefficients to be determined and Jm(.) is the bessel’s function. Using

equations (3.43), weighting functions g1(ξ) and g2(ξ) can be expressed as

g1(ξ) =u + v + (v − u) exp(−2p)

(v + u)2 − (v − u)2 exp(−2p)

∞∑

m=0

AmJ2m(ξ) (3.45a)

g2(ξ) =u + v + (v − u) exp(−2p)

(v + u)2 − (v − u)2 exp(−2p)

∞∑

m=0

BmJ2m+1(ξ) (3.45b)

Putting these values of g1,2(ξ) in (3.42b), comparing even and odd functions and then projecting the resulting

equations into the functional space with elements u±

12

n (x2a) (chapter 2, section 2.2.3), following is obtained

∞∑

m=0

AmGG(2m, 2n) = −(1 + ΛE)J2n(κ0 cos φ0) (3.46a)

∞∑

m=0

BmGG(2m + 1, 2n + 1) = −j(1 + ΛE)J2n+1(κ0 cos φ0) (3.46b)

where GG(ν, µ) is defined by

GG(ν, µ) =

∫ ∞

0

u + v + (v − u) exp(−2p)

(v + u)2 − (v − u)2 exp(−2p)Jν(ξ)Jµ(ξ)dξ (3.46c)

In obtaining (3.46a) and (3.46b), the following relations are used

x−m/2Jm(ξ√

x) =

∞∑

n=0

√2(2n + m + 1

2)Γ (n + m + 1

2)

Γ (n + 1)Γ (m + 1)

J2n+m+ 12(ξ)

√ξ

umn (x)

umn (x) =

√2Γ (n + 1)Γ (m + 1)

Γ (n + m + 1

2)

x−m/2

∫ ∞

0

Jm(√

xξ)J2n+m+ 12(ξ)

√ξ

dξ

where umn (.) be the Jacobi’s polynomials and F(.) be the hypergeometric function.

A1. Diffracted far Fields

Since the geometry supports surface waves. If the observation point is far from the surface of the strip

then this contribution can be neglected and diffracted field dominate. Substituting g1(ξ) and g2(ξ) from

(3.45a) and (3.45b) in expression (3.39b) and then using saddle point method, far diffracted field for region

y > 0 may be written as

Esuz ' C(k0ρ)

√

εr − cos2 φ + sinφ +(

√

εr − cos2 φ − sin φ)

exp(

−j2k0d√

εr − cos2 φ)

(

√

εr − cos2 φ + sin φ)2

−(

√

εr − cos2 φ − sin φ)2

exp(

−j2k0d√

εr − cos2 φ)

×∞∑

m=0

[

AmJ2m(κ0 cos φ) + jBmJ2m+1(κ0 cos φ)]

(3.47)

where C(k0ρ) =√

π2koρ exp

[

j π4− jkoρ

]

, φ is the angle of observation and εr = εε0

. Coefficients Am and Bm

can be computed from matrix equations (3.46a) and (3.46b).

36

Electromagnetic Diffraction from Strip:

A2. Current Distribution

Current density, induced on the strip, may be computed from the expression

Jz = −Hx

∣

∣

∣

∣

y=0+

+ Hx

∣

∣

∣

∣

y=0−

=Y0

jκ0

∞∑

m=0

∫ ∞

0

[

AmJ2m(ξ) cos(ξxa) + BmJ2m+1(ξ) sin(ξxa)]

dξ

=Y0

jκ0

(

1 − x2a

)−12

∞∑

m=0

{

Am cos[

2m sin−1 xa

]

+ Bm sin[

(2m + 1) sin−1 xa

]}

(3.48)

B. H-Polarization

The formulation for H-polarization can be conducted in a manner similar to E-polarization. The expres-

sions for the scattered fields corresponding to (3.39a)-(3.39d) for H-polarization may be written as

Hiz = exp

[

jk0(x cos φ0 + y sinφ0)]

(3.49a)

Hsuz =

∫ ∞

0

[

g1(ξ) cos (ξxa) + g2(ξ) sin (ξxa)]

exp (−uya) dξ, y > 0 (3.49b)

Hsdz =

∫ ∞

0

cos (ξxa)[

f1(ξ) exp (vya) + h1(ξ) exp (−vya)]

dξ

+

∫ ∞

0

sin (ξxa)[

f2(ξ) exp (vya) + h2(ξ) exp (−vya)]

dξ, −d < y < 0 (3.49c)

Hslz =

∫ ∞

0

[

`1(ξ) cos (ξxa) + `2(ξ) sin (ξxa)]

exp[

u(ya + da)]

dξ, y < −d (3.49d)

The required boundary conditions are given by

(i) Ex is continuous at y = 0 for all values of x(ii) Hz andEx are continuous for y = −d(iii) Hz is continuous at y = 0 and for |xa| ≥ 1

(iv) Eix + Er

x + Esux = 0 at y = 0 and for |xa| ≤ 1

(3.50)

Using these boundary conditions and proceeding in the similar manner as last section, following dual integral

equations are obtained∫ ∞

0

[{

g1(ξ) − f1(ξ) − h1(ξ)}

cos(ξxa) +{

g2(ξ) − f2(ξ) − h2(ξ)}

sin(ξxa)]

dξ = 0, |xa| ≥ 1 (3.51a)

∫ ∞

0

u[

g1(ξ) cos(ξxa) + g2(ξ) sin(ξxa)]

dξ = −j cos φ0κ0(1 − ΛH) exp[

jk0x sin φ0

]

, |xa| ≤ 1(3.51b)

and unknown coefficients

f1(ξ) = − εru

v

(v + εru)

(εru + v) + (εru − v) exp(−2p)g1(ξ) (3.52a)

h1(ξ) =εru

v

(εru − v) exp(−2p)

(εru + v) + (εru − v) exp(−2p)g1(ξ) (3.52b)

`1(ξ) = − 2εru exp(−p)

(εru + v) + (εru − v) exp(−2p)g1(ξ) (3.52c)

f2(ξ) = − εru

v

(v + εru)

(εru + v) + (εru − v) exp(−2p)g2(ξ) (3.52d)

h2(ξ) =εru

v

(εru − v) exp(−2p)

(εru + v) + (εru − v) exp(−2p)g2(ξ) (3.52e)

`2(ξ) = − 2εru exp(−p)

(εru + v) + (εru − v) exp(−2p)g2(ξ) (3.52f)

37

Electromagnetic Diffraction from Strip:

where ΛH be the reflection coefficient of the slab for H-polarization and it can be calculated as

ΛH =(1 − P 2

H)(e∗ − e)

∆H, PH =

Z0 sinφ0

Z sin φr,

e =exp(jkd cos φr), e∗ = exp(−jkd cos φr)

∆H =(1 + PH)2e − (1 − PH)2e∗

where Z0 =

√

µ0

ε0and Z =

√

µ0

εare intrinsic impedance of free space and dielectric slab respectively. The

angle of refraction φr is related with the angle of incident φ0 by

k cosφr = k0 cosφ0

The equations (3.51a) and (3.51b) are the dual integral equations. Proceeding in the similar manner as

in last section, it can be written as

g1(ξ) − f1(ξ) − h1(ξ) =

∞∑

m=0

AmJ2m+1(ξ)

ξ(3.53a)

g2(ξ) − f2(ξ) − h2(ξ) =

∞∑

m=0

BmJ2m+2(ξ)

ξ(3.53b)

Substituting f1,2(ξ) and h1,2(ξ) given by (3.52a)-(3.52b) and (3.52d)-(3.52e) into (3.53), weighting functions

g1(ξ) and g2(ξ) can be determined as given below

g1(ξ) = v(εru + v) + (εru − v) exp(−2p)

(εru + v)2 − (εru − v)2 exp(−2p)

∞∑

m=0

AmJ2m+1(ξ)

ξ(3.54a)

g2(ξ) = v(εru + v) + (εru − v) exp(−2p)

(εru + v)2 − (εru − v)2 exp(−2p)

∞∑

m=0

BmJ2m+2(ξ)

ξ(3.54b)

Substituting equations (3.54) into (3.51b), and projecting the resulting equations into the functional space

with elements v±

12

n (x2a) (chapter 2), following is obtained

∞∑

m=0

AmKK(2m + 1, 2n + 1) = −jcos φ0

sin φ0

(1 − ΛH)J2n+1(κ cos φ0) (3.55a)

∞∑

m=0

BmKK(2m + 2, 2n + 2) =cosφ0

sin φ0

(1 − ΛH)J2n+2(κ cos φ0) (3.55b)

where KK(ν, µ) is defined by

KK(ν, µ) =

∫ ∞

0

uv(εru + v) + (εru − v) exp(−2p)

(εru + v)2 − (εru − v)2 exp(−2p)

Jν(ξ)Jµ(ξ)

ξ2dξ (3.55c)

B1. Diffracted far Fields

Diffracted far fields in region y > 0 can be calculated using saddle point method. The final results are

Hsuz ' C ′(k0ρ)

(

εr sin φ +√

εr − cos2 φ)

+(

εr sinφ +√

εr − sin2 φ)

exp(

−j2k0d√

εr − cos2 φ)

(

√

εr − cos2 φ + εr sin φ)2

−(

εr sin φ −√

εr − cos2 φ)2

exp(

−j2k0d√

εr − cos2 φ)

×∞∑

m=0

[

AmJ2m+1(κ0 cos φ) + jBmJ2m+2(κ0 cosφ)] sin φ

cosφ(3.56)

38

Electromagnetic Diffraction from Strip:

where C ′(k0ρ) =√

π2koρ exp

[

−j π4− jkoρ

]

, φ be the angle of observation and εr = εε0

.

B1. Current Induced on the Strip

Current density may be computed from the expression

Jx = Hz

∣

∣

∣

∣

y=0+

− Hz

∣

∣

∣

∣

y=0−

=

∞∑

m=0

∫ ∞

0

[

AmJ2m+1(ξ)

ξcos ξxa + Bm

J2m+1(ξ)

ξsin ξxa

]

dξ

=

∞∑

m=0

{

Am

2m + 1cos

[

2m + 1 sin−1 xa

]

+Bm

2m + 2sin

[

(2m + 2) sin−1 xa

]

}

(3.57)

3.3.2 Computations and Discussion

A. Computation of integrals KK(ν, µ) and GG(ν, µ)

Integrals GG(ν, µ) and KK(ν, µ) can be represented in the form

GG(ν, µ) =

∫ ∞

0

1

2

Jν(ξ)Jµ(ξ)√

ξ2 − κ2dξ +

∫ ∞

0

FG(ξ, ν, µ)dξ =G(ν, µ;κ)

2+

∫ ∞

0

FG(ξ, ν, µ)dξ

KK(ν, µ) =1

1 + εr

∫ ∞

0

√

ξ2 − κ2

ξ2Jν(ξ)Jµ(ξ)dξ +

∫ ∞

0

FK(ξ, ν, µ)dξ =K(ν, µ;κ)

1 + εr+

∫ ∞

0

FK(ξ, ν, µ)dξ

where

K(ν, µ;κ) =

∫ ∞

0

√

ξ2 − κ2

ξ2Jν(ξ)Jµ(ξ)dξ

G(ν, µ;κ) =

∫ ∞

0

Jν(ξ)Jµ(ξ)√

ξ2 − κ2dξ

FG(ξ, ν, µ) =v2 − u2 + (v − u)(3v − u) exp(−2p)

2v [(v + u)2 − (v − u)2 exp(−2p)]Jν(ξ)Jµ(ξ)dξ

FK(ξ, ν, µ) = v(εru + v)

[

(1 + εr)ξ − (εru + v)]

+ (εru − v)[

(1 + εr)ξ + (εru − v)]

exp(−2p)

(1 + εr)[

(v + εru)2 − (εru − v)2 exp(−2p)]

Jν(ξ)Jµ(ξ)

ξ2dξ

The integrals FG(ξ, ν, µ) and FK(ξ, ν, µ) can be regarded as the correction integrals. These integrals can

be computed by the standard methods. However the integrands of these correction integrals have branch

points at ξ = κ0 and ξ = κ and poles between κ0 ≤ ξ ≤ κ. These integrals can be managed to computed

by deforming the path of integration. The integrals K(ν, µ;κ) and G(ν, µ;κ) can be computed by changing

these into summation form. This is discussed in detail by Hongo [91] and [105]

B. Field Patterns

Far field patterns as a function of angle of observation, have been computed using equation (3.47) for

E-polarization and equation (3.56) for H-polarization. These expressions contain expansion coefficients Am

and Bm. These expansion coefficients have been computed using equations (3.46) for E-polarization and

39

Electromagnetic Diffraction from Strip:

equations (3.55) for H-polarization. KK(n,m) and GG(n,m) are the matrices, the elements of which are in

terms of infinite integrals and have been computed as discussed in above section. These integrals have been

computed for finite values of m and n values. Fig. 3.9 gives the field patterns for different values of angle of

incidence for E-polarization. It is evident from the patterns that as the angle of incidence is increased, the

corresponding main lobe shifts towards the lower value of angle of observation φ. To check the validity of the

results KP patterns are compared with those of obtained through physical optics. Fig. 3.10 and Fig. 3.11

give the comparisons for E- and H-polarized field respectively for φ0 = 60◦, κ0 = 4.0 κ = 6.0 da = 2.0.

The comparison seems good. Current distributions on the strip are also obtained. Fig. 3.12 and Fig. 3.13

give the same. It is evident from the graphs, that the current density is maximum at the strip edges for

E-polarization and reverse is true for H-polarization. Fig. 3.14 and Fig. 3.15 give the effects of strip widths

on the field patterns. Fig. 3.15a shows the effects of the slab width on the field patterns. It can be deduced

from the patterns that as da is decreased, the field intensifies in the upper half space

3.3.3 Conclusion

The problem of diffraction from a strip placed on a dielectric slab is rigorously solved using KP method.

The formulation involved large number of unknown weighting functions (eight). But the boundary conditions

made it possible to write six of them in terms of remaining two. This greatly simplified the calculations.

The integrals which emerged during manipulations are very complex in nature and have singularities. So

effective methods were devised to compute them. Far fields are computed and presented to observe their

dependence on different parameters of interest i.e size of strip, angle of incidence, width of the slab. Physical

optics solutions of the problem are also developed for comparison. And it is seen that the two results are

in good agreement. It can be noticed from the graphs that as the size of the strip is increased the width of

main lobe diminishes, the side lobes become prominent and at the same time amplitude of the main lobe

increases. Current distributions on the strip are also given which testify the incorporation of correct edge

conditions in the solution.

40

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

6

7

8

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

0=60

0=80

0=90

da=2.0

E-polarization

Figure 3.9 Variation of diffracted fields with angle of incidence

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

KP PO

0= 60

da= 2.0

E- polarization

Fig. 3.10 Comparison of field patterns obtained through KP and PO methodsFigure 3.10 Diffracted field pattern and its comparison

41

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

PO KP

da= 2.0

H- polarization

Fig. 3.11 Comparision of diffracted field patterns (H-Polarization)Figure 3.11 Comparison of the field patterns for H-polarization

-1.0 -0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

2.5

CU

RR

EN

T M

AG

NIT

UD

E

xa

da= 2.0

H- polarization

Fig. 3.13 Current distribution on the strip (H-Polarization)Figure 3.12 Current distribution on the strip

42

Electromagnetic Diffraction from Strip:

-1.0 -0.5 0.0 0.5 1.01

2

3

4

5

6

7

8

9

CU

RR

EN

T M

AG

NIT

UD

E

xa

0=45

0=90

da= 2.0

E- polarization

Fig. 3.12 Current distribution on the strip (E-polarization)Figure 3.13 Current distribution on the strip

-20 0 20 40 60 80 100 120 140 160 180 200

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

da= 2.0

E- polarization

Fig. 3.14 Variation of diffracted field as a function of strip widthFigure 3.14 Variations of diffracted fields as a function of strip width

43

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200

-10123456789

101112131415161718

DIF

FR

AC

TE

D F

IELD

OBSERVATION ANGLE

0=4.0

0=6.0

0=8.0

da=2.0

H- polarization

Fig. 3.15 Variation of field patterns with width of stripFigure 3.15 Variations of diffracted fields as a function of strip width

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

da= 2.0

da= 3.0

da= 4.0

H-polarization

Figure 3.15a Variations of diffracted fields as a function of da

44

Electromagnetic Diffraction from Strip:

3.4 Diffraction of Electromagnetic Plane Wave from a PEMC Strip

3.4.1 Formulation and solution of the problem

A. E-Polarization

The geometry of the problem is similar as shown in Fig. 3.1. This time, it contains a PEMC strip of

negligible thickness and of width 2a. If φ0 is the angle of incidence, then the incident field Eiz, co-polarized

component Edz and cross-polarized component Hd

z can be written as

Eiz = exp

[

jk0(x cos φ0 + y sinφ0)]

(3.58a)

Edz =

∫ ∞

0

[

fe(ξ) cos (xaξ) + ge(ξ) sin (xaξ)]

exp

[

−√

ξ2 − κ20ya

]

dξ y > 0 (3.58b)

Hdz =

∫ ∞

0

[

fh(ξ) cos (xaξ) + gh(ξ) sin (xaξ)]

exp

[

−√

ξ2 − κ20ya

]

dξ y > 0 (3.58c)

All the notations have the same meaning as described in the previous sections. The fe,h(ξ) and ge,h(ξ) are

the weighting functions to be determined from the boundary conditions.

The required boundary conditions are given by

(i) The fields are continuous at |xa| ≥ 1 and y = 0

(ii) Htx + MEt

x = 0 and Htz + MEt

z = 0 for |xa| ≤ 1 and y = 0

where superscript t stands for total. From (i)

∫ ∞

0

√

ξ2 − κ20

[

fe(ξ) cos(xaξ) + ge(ξ) sin(xaξ)]

dξ = 0, |xa| ≥ 1 (3.59a)

∫ ∞

0

√

ξ2 − κ20

[

hh(ξ) cos(xaξ) + gh(ξ) sin(xaξ)]

dξ = 0, |xa| ≥ 1 (3.59b)

Boundary condition (ii) gives

∫ ∞

0

{[

fh(ξ) + Mfe(ξ)]

cos(xaξ) +[

gh(ξ) + Mge(ξ)]

sin(xaξ)}

dξ

= −M exp[

jκ0xa cosφ0

]

(3.60a)∫ ∞

0

√

ξ2 − κ20

{[

fe(ξ) − MZ20fh(ξ)

]

cos(xaξ) +[

ge(ξ) − MZ20gh(ξ)

]

sin(xaξ)}

dξ

= jκ0 sin φ0 exp[

jκ0xa cosφ0

]

(3.60b)

The above expressions are the dual integral equations. Equation (3.59a) and (3.59b) can be used to decide

the nature of weighting functions fe,h(ξ) and ge,h(ξ) by making use of the discontinuous properties of Weber-

Schafheitlin’s integrals, as follow

fe(ξ) =1

√

ξ2 − κ20

∞∑

m=0

AmJ2m(ξ) (3.61a)

ge(ξ) =1

√

ξ2 − κ20

∞∑

m=0

BmJ2m+1(ξ) (3.61b)

fh(ξ) =1

√

ξ2 − κ20

∞∑

m=0

CmJ2m(ξ) (3.61c)

gh(ξ) =1

√

ξ2 − κ20

∞∑

m=0

DmJ2m+1(ξ) (3.61d)

45

Electromagnetic Diffraction from Strip:

where Jm(.) be the Bessel’s function of order m and Am, Bm, Cm and Dm are the expansion coefficients.

Separating even and odd functions of the expressions (3.60a) and (3.60b) and then expanding the trigono-

metric functions in terms of Jacobi’s polynomials u±

12

n (x2a) and v

±12

n (x2a) (section 2.2.3), following matrix

equations are obtained for the expansion coefficients∞∑

m=0

[

G(2m, 2n;κ0)][

Cm + MAm

]

= −M[

J2n(κ0 cosφ0)]

(3.62a)

∞∑

m=0

[

G(2m + 1, 2n + 1;κ0)][

Dm + MBm

]

= −Mj[

J2n+1(κ0 cos φ0)]

(3.62b)

∞∑

m=0

[

H(2m, 2n + 1;κ0)][

Am − MCmZ20

]

= j tan(φ0)[

J2n+1(κ0 cos φ0)]

(3.62c)

∞∑

m=0

[

H(2m + 1, 2n + 2;κ0)][

Bm − MDmZ20

]

= − tan(φ0)[

J2n+2(κ0 cos φ0)]

(3.62d)

n = 0, 1, 2, · · ·

where

G(α, β;κ0) =

∫ ∞

0

Jα(ξ)Jβ(ξ)√

ξ2 − κ20

dξ (3.63a)

H(α, β;κ0) =

∫ ∞

0

Jα(ξ)Jβ(ξ)

ξdξ (3.63b)

In writing the equation (3.62), following relations are used

x−m/2Jm(ξ√

x) =

∞∑

n=0

√2(2n + m + 1

2)Γ (n + m + 1

2)

Γ (n + 1)Γ (m + 1)

J2n+m+ 12(ξ)

ξ12

umn (x) (3.64a)

=

∞∑

n=0

√2(2n + m + 1

2)Γ (n + m + 1

2)

Γ (n + 1)Γ (m + 1)

J2n+m+ 12(ξ)

ξ12

umn (x) (3.64b)

umn (x) = F (n + m +

1

2,−n,m + 1;x), vm

n (x) = F (n + m +3

2,−n,m + 1;x) (3.64c)

where F (m,n, l;x) is the hypergeometric series [112].

The equations (3.62) may be solved to evaluate the expansion coefficients Am, Bm, Cm, Dm. The co-

polarized component Edz and cross-polarized component Hd

z may be computed from the equations (3.58b)-

(3.58c) using the saddle point method. The final results are

Edz '

√

π

2k0ρexp

[

−jk0ρ − jπ

4

]

∞∑

m=0

[

AmJ2m(κ0 cos φ) + jBmJ2m+1(κ0 cos φ)]

tan φ (3.65a)

Hdz '

√

π

2k0ρexp

[

−jk0ρ − jπ

4

]

∞∑

m=0

[

CmJ2m(κ0 cos φ) + jDmJ2m+1(κ0 cosφ)]

tan φ (3.65b)

where (ρ, φ) are the cylindrical coordinates of the observation point. Far field in the lower region can also

be derived similarly.

B. H-Polarization

The field expressions corresponding to expressions (3.58) for H-polarization may be written as

Hiz = exp

[

jk0(x cos φ0 + y sin φ0)]

(3.66a)

Edz =

∫ ∞

0

[

fe(ξ) cos (xaξ) + ge(ξ) sin (xaξ)]

exp[

−√

ξ2 − κ20ya

]

dξ (3.66b)

Hdz =

∫ ∞

0

[

fh(ξ) cos (xaξ) + gh(ξ) sin (xaξ)]

exp[

−√

ξ2 − κ20ya

]

dξ (3.66c)

46

Electromagnetic Diffraction from Strip:

All the notations used in the above expressions have the same meaning as described in last section. The

boundary conditions remain the same.

Imposition of boundary conditions give∫ ∞

0

[

fe(ξ) cos(xaξ) + ge(ξ) sin(xaξ)]

dξ = 0, |xa| ≥ 1 (3.67a)

∫ ∞

0

[

hh(ξ) cos(xaξ) + gh(ξ) sin(xaξ)]

dξ = 0, |xa| ≥ 1 (3.67b)

∫ ∞

0

{[

fh(ξ) + Mfe(ξ)]

cos(xaξ) +[

gh(ξ) + Mge(ξ)]

sin(xaξ)}

dξ = − exp[

jκ0xa cosφ0

]

(3.68a)

∫ ∞

0

√

ξ2 − κ20

{[

Y 20 fe(ξ) − Mfh(ξ)

]

cos(xaξ) +[

Y 20 ge(ξ) − Mgh(ξ)

]

sin(xaξ)}

dξ

= −jκ0 sin φ0 exp[

jκ0xa cosφ0

]

(3.68b)

Using the discontinuous properties of Weber-Schafheitlin’s integrals and incorporating the edge conditions

for H-field, following is obtained

fe(ξ) =

∞∑

m=0

AmJ2m+1(ξ)

ξ, ge(ξ) =

∞∑

m=0

BmJ2m+2(ξ)

ξ(3.69a)

fh(ξ) =

∞∑

m=0

CmJ2m+1(ξ)

ξ, gh(ξ) =

∞∑

m=0

DmJ2m+2(ξ)

ξ(3.69b)

Proceeding in a similar manner as in last section, matrix equations takes the form

∞∑

m=0

[

K(

2m + 2, 2n + 2;κ0

)][

Y 20 Bm − MDm

]

= M tan φ0

[

J2n+2(κ0 cos φ0)]

(3.70a)

∞∑

m=0

[

K(

2m + 1, 2n + 1;κ0

)][

Y 20 Am − MCm

]

= −Mj tan φ0

[

J2n+1(κ0 cosφ0)]

(3.70b)

∞∑

m=0

[

H(

2m + 1, 2n;κ0

)][

Cm + MAm

]

= −[

J2n(κ0 cos φ0)]

(3.70c)

∞∑

m=0

[

H(

2m + 2, 2n + 1;κ0

)][

Dm + MBm

]

= −j[

J2n+1(κ0 cos φ0)]

(3.70d)

n = 0, 1, 2, · · ·

where

K(α, β;κ0) =

∫ ∞

0

√

ξ2 − κ20

ξ2Jα(ξ)Jβ(ξ)dξ (3.71a)

H(α, β;κ0) =

∫ ∞

0

Jα(ξ)Jβ(ξ)

ξdξ (3.71b)

The above expressions are the matrix equations and can be solved for the expansion coefficients Am, Bm,

Cm, Dm by any standard method.

Diffracted far fields for the co-polarized Hdz and cross-polarized Ed

z components may be obtained from

(3.66b),(3.66c) by using the saddle point method. The final results are

Edz '

√

π

2k0ρexp

[

−jk0ρ + jπ

4

]

∞∑

m=0

[

AmJ2m+1(κ0 cos φ) + jBmJ2m+2(κ0 cosφ)]

tan φ (3.72a)

Hdz '

√

π

2k0ρexp

[

−jk0ρ + jπ

4

]

∞∑

m=0

[

CmJ2m+1(κ0 cos φ) + jDmJ2m+2(κ0 cos φ)]

tan φ (3.72b)

47

Electromagnetic Diffraction from Strip:

where (ρ, φ) are the cylindrical coordinates of the observation point.

3.4.2 Results and Discussions

The far field patterns for co-polarized and cross-polarized components can be computed from equations

(3.65) for E-Polarized incident field and equations (3.72) for H-polarization case. These equations contain

the unknowns expansion coefficients Am, Bm, Cm and Dm. The values of these expansion coefficients

can be computed from equations (3.62) for E-polarization and (3.70) for H-polarization. How to compute

the integrals G(α, β;κ0) and K(α, β;κ0) are discussed in [91] and [105] in detail. Since M , the admittance

parameter is interesting in our work, therefore the dependance of field patterns on this parameter is explored.

Fig. 3.16 and Fig. 3.17 show the variations in the field patterns as a function of M taking φ0 = 70◦, κ0 = 4

for E-polarization. It turns out that there exist no cross-polarized component Hdz for PMC and PEC case

and it dominates for M = 1.0. The amplitude of co-polarized component Ez slightly increases due to increase

in M . Fig. 3.19 and Fig. 3.20 show the field patterns for H-Polarization. Cross-polarized component Edz

follow the same trend as in case of E-polarization. The dependance of co- and cross-polarized components

on the strip width can be seen in Fig. 3.20 and Fig. 3.21. From these figures, it can be conclude that as the

strip width κ0 is increased, side lobes start to appear for both the components. Similar trends can also be

seen for H-polarization. Another important parameter is the angle of incidence φ0. Fig. 3.22 and Fig.3.23

show how this parameter affects the field patterns.

3.4.3 Conclusion

To see the versatility of the method, KP formulation is developed to study the diffraction from PEMC

strip. Both the principal polarization are considered. Numerical computations are carried out to explore

the dependence of the co- and cross-polarized components on admittance parameter. It can be seen that

cross-component does not exist for both PEC and PMC cases and it dominates for unity value of admittance

parameter for both E- and H-polarization cases. Effects of strip width and angle of incidence on both

components are also investigated. As the strip width is increases, both the co- and cross- components

behave in similar manner, i.e side lobes in the field patterns start to appear.

48

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

M= 0.0 M= 1.0 M= 3.0 M= 100.0

E- polarization

Figure 3.16 Variations of co-component as a function of M (E-Polarization)

-20 0 20 40 60 80 100 120 140 160 180 200-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

M= 0.0 M=1.0 M= 3.0 M=10.0 M=100.0

E- polarization

Figure 3.17 Variations of cross-component as a function of M (E-Polarization)

49

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

M= 0.0 M= 1.0 M= 3.0 M= 100.0

H- polarization

Figure 3.18 Dependance of co-component as a function of M (H-Polarization)

-20 0 20 40 60 80 100 120 140 160 180 200

0.0

0.5

1.0

1.5

2.0

2.5

3.0

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

M= 0.0 M= 1.0 M= 3.0 M= 100.0

H- polarization

Figure 3.19 Dependance of cross-component on M (H-Polarization)

50

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

DIF

FR

AC

TE

D F

IELD

OBSERVATION ANGLE

H-POLARIZATION

Fig. 7 Variation of co-polarized component as a function of strip widthFigure 3.20 Variation of co-component as a function of strip width

-20 0 20 40 60 80 100 120 140 160 180 200

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

DIF

FR

AC

TE

D F

IELD

OBSERVATION ANGLE

H-POLARIZATION

Fig. 6 variation of cross-polarized component as a function of strip widthFigure 3.21 Variation of cross-component as a function of strip width

51

Electromagnetic Diffraction from Strip:

-20 0 20 40 60 80 100 120 140 160 180 200-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

DIF

FR

AC

TE

D F

IELD

OBSERVATION ANGLE

0= 70

0= 80

0= 90

M=1.0

Fig. 9 Dependence of co-polarized component on the angle of incidence

E-POLARIZATION

Figure 3.22 Variation of co-component as a function of angle of incidence

-20 0 20 40 60 80 100 120 140 160 180 200

-0.20.00.20.40.60.81.01.21.41.61.82.02.22.42.62.83.03.2

DIF

FR

AC

TE

D F

IELD

OBSERVATION ANGLE

E-POLARIZATION

Figure 3.23 Variation of cross-component as a function of angle of incidence

52

Electromagnetic Diffraction from Slit(s):

.

CHAPTER 4

Electromagnetic Diffraction from Slit(s)

4.1 General Consideration

In this chapter KP formulation is employed to study the diffraction from slit(s) in an imperfectly con-

ducting/PEMC plane In the first problem, diffraction has been studied from an impedance slit placed at

the interface of two different media. Second problem include the diffraction from two parallel slits in an

impedance pane. Special emphasis has been given to study the interactions between the slits. And the final

problem deals with the PEMC slit. It can be found that the Kobayashi Potential method may be generalized

to these problems in relatively straight forward manner.

4.2 Diffraction from a Slit in an Impedance Plane placed at the Interface

of Two Different Media

4.2.1 Formulation and solution of the problem

A. E-Polarization

The configuration of the problem is shown in Fig. 4.1. An impedance plane of negligible thickness

contains a slit of width 2a. The surface impedances of the upper and lower surfaces of the impedance plane

are assumed as Z+ and Z− respectively. If E-polarized field, Eiz is incident upon the slit with angle of

incidence φ0 and Erz be the reflected field from the impedance plane, then

Eiz = exp

[

jk0(x cos φ0 + y sinφ0)]

(4.1a)

Erz = −Z0 − Z+ sinφ0

Z0 + Z+ sinφ0

exp[

jk0(x cos φ0 − y sin φ0)]

(4.1b)

53

Electromagnetic Diffraction from Slit(s):

where k0 and Z0 be the wave number and intrinsic impedance of the upper half space (y > 0). Let the

scattered fields Ed+

z in the upper half space (y > 0) and Ed−

z in the lower half space (y < 0) can be written

in terms of unknowns as follow

Ed+

z =

∫ ∞

0

[

g1(ξ) cos (xaξ) + g2(ξ) sin (xaξ)]

exp

[

−√

ξ2 − κ20ya

]

dξ, y > 0 (4.1c)

Ed−

z =

∫ ∞

0

[

h1(ξ) cos (xaξ) + h2(ξ) sin (xaξ)]

exp[

√

ξ2 − κ2ya

]

dξ, y < 0 (4.1d)

where all the parameters have the same meaning as described in the proceeding sections. g1,2(ξ), h1,2(ξ) are

the unknown weighting functions.

µ0, 0

µ0,

Fig. 4.1 Geometry of the problem

y

x

z x= - ax=a

Related boundary conditions are

Etz

∣

∣

∣

y=0+

= −Z+Htx

∣

∣

∣

y=0+

, Etz

∣

∣

∣

y=0−

= Z−Htx

∣

∣

∣

y=0−

, |xa| ≥ 1 (4.2a)

Etz

∣

∣

∣

y=0+

= Etz

∣

∣

∣

y=0−

, Htx

∣

∣

∣

y=0+

= Htx

∣

∣

∣

y=0−

, |xa| ≤ 1 (4.2b)

From the conditions (4.2a)

∫ ∞

0

[

1 − j

√

ξ2 − κ20

κ0

ζ+

]

[

g1(ξ) cos(xaξ) + g2(ξ) sin(xaξ)]

dξ = 0, |xa| ≥ 1 (4.3a)

∫ ∞

0

[

1 − j

√

ξ2 − κ2

κζ−

]

[

h1(ξ) cos(xaξ) + h2(ξ) sin(xaξ)]

dξ = 0, |xa| ≥ 1 (4.3b)

where ζ± are the normalized surface impedances. From the boundary conditions (4.2b), it can be written as

∫ ∞

0

{[

h1(ξ) − g1(ξ)]

cos(xaξ) +[

h2(ξ) − g2(ξ)]

sin(xaξ)}

dξ

54

Electromagnetic Diffraction from Slit(s):

=2ζ+ sinφ0

1 + ζ+ sin φ0

exp[

jκ0xa cos φ0

]

(4.3c)

∫ ∞

0

{[

√

ξ2 − κ2h1(ξ) +

√

ξ2 − κ20g1(ξ)

]

cos(xaξ) +[

√

ξ2 − κ2h2(ξ) +

√

ξ2 − κ20g2(ξ)

]

sin(xaξ)}

dξ

=j2κ0 sinφ0

1 + ζ+ sin φ0

exp[

jκ0xa cos φ0

]

(4.3d)

The above expressions are the dual integral equations (DIEs). Expressions (4.3a) and (4.3b) permit us to

write the unknown weighting functions in the terms of unknown expansion coefficient as follow

g1(ξ) =1

jκ0η+ +√

ξ2 − κ20

∞∑

m=0

AmJ2m+ 32(ξ)ξ−

32 (4.4a)

g2(ξ) =1

jκ0η+ +√

ξ2 − κ20

∞∑

m=0

BmJ2m+ 52(ξ)ξ−

32 (4.4b)

h1(ξ) =1

jκη− +√

ξ2 − κ2

∞∑

m=0

CmJ2m+ 32(ξ)ξ−

32 (4.4c)

h2(ξ) =1

jκη− +√

ξ2 − κ2

∞∑

m=0

DmJ2m+ 52(ξ)ξ−

32 (4.4d)

where η± = ζ−1± and Am, Bm, Cm, Dm are the expansion coefficients. Separating even and odd functions of

the expressions (4.3c) and (4.3d) and then projecting the resulting equations into the functional space with

elements p±

12

n (x2a) (chapter 2 section 2.2.3), the following matrix equations are obtained

∞∑

m=0

[

−AmGSE

(

2m +3

2, 2n +

1

2; ζ+

)

+ CmGSE

(

2m +3

2, 2n +

1

2; ζ−

)]

=2ζ+ sin φ0

1 + ζ+ sinφ0

J2n+ 12(κ0 cos φ0)

(κ0 cos φ0)12

(4.5a)

∞∑

m=0

[

−BmGSE

(

2m +5

2, 2n +

3

2; ζ+

)

+ DmGSE

(

2m +5

2, 2n +

3

2; ζ−

)]

=j2ζ+ sinφ0

1 + ζ+ sinφ0

J2n+ 32(κ0 cos φ0)

(κ0 cos φ0)12

(4.5b)

∞∑

m=0

[

AmKSE

(

2m +3

2, 2n +

1

2; ζ+

)

+ CmKSE

(

2m +3

2, 2n +

1

2; ζ−

)]

=j2κ0 sin φ0

1 + ζ+ sinφ0

J2n+ 12(κ0 cos φ0)

(κ0 cos φ0)12

(4.5c)∞∑

m=0

[

BmKSE

(

2m +5

2, 2n +

3

2; ζ+

)

+ DmKSE

(

2m +5

2, 2n +

3

2; ζ−

)]

= − 2κ0 sinφ0

1 + ζ+ sinφ0

J2n+ 32(κ0 cos φ0)

(κ0 cos φ0)12

n = 0, 1, 2, · · · (4.5d)

where

GSE(m,n; ζ±) =

∫ ∞

0

1

jκη± +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξ2dξ (4.5e)

KSE(m,n; ζ±) =

∫ ∞

0

√

ξ2 − κ2

jκη± +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξ2dξ (4.5f)

55

Electromagnetic Diffraction from Slit(s):

The above matrix equations can be solved for the computation of expansion coefficients Am, Bm, Cm, Dm.

Since the equations (4.5) are the matrix equations so these may be written in standard matrix notation as

under

[

G+

SE,E

][

Am

]

−[

G−

SE,E

][

Cm

]

= −ζ+

[

JE

]

(4.6a)[

K+

SE,E

][

Am

]

+[

K−

SE,E

][

Cm

]

= jκ0

[

JE

]

(4.6b)[

G+

SE,O

][

Bm

]

−[

G−

SE,O

][

Dm

]

= −jζ+

[

JO

]

(4.6c)[

K+

SE,O

][

Bm

]

+[

K−

SE,O

][

Dm

]

= −κ0

[

JO

]

(4.6d)

where the correspondence between the matrices and their elements are given by

[

G±

SE,E

]

⇐⇒ GSE

(

2n +1

2, 2m +

3

2; ζ±

)

[

G±

SE,O

]

⇐⇒ GSE

(

2n +3

2, 2m +

5

2; ζ±

)

[

K±

SE,E

]

⇐⇒ KSE

(

2n +1

2, 2m +

3

2; ζ±

)

[

K±

SE,O

]

⇐⇒ KSE

(

2n +3

2, 2m +

5

2; ζ±

)

[

JE

]

⇐⇒ 2 sin φ0

1 + ζ+ sin φ0

J2n+ 12(κ cos φ0)

(κ cos φ0)12

[

JO

]

⇐⇒ 2 sin φ0

1 + ζ+ sin φ0

J2n+ 32(κ cos φ0)

(κ cos φ0)12

Expressions (4.6a)-(4.6d) can be solved for the expansion coefficients as follow

{[

G+

SE,E

]−1[

G−

SE,E

]

+[

K+

SE,E

]−1[

K−

SE,E

]}[

Cm

]

={

ζ+

[

G+

SE,E

]−1

+ jκ0

[

K+

SE,E

]−1}[

JE

]

(4.7a)

[

Am

]

=[

G+

SE,E

]−1[

G−

SE,E

][

Cm

]

− ζ+

[

G+

SE,E

]−1[

JE

]

(4.7b)

{[

G+

SE,O

]−1[

G−

SE,O

]

+[

K+

SE,O

]−1[

K−

SE,O

]}[

Dm

]

={

jζ+

[

G+

SE,O

]−1

− κ0

[

K+

SE,O

]−1}[

JO

]

(4.7c)

[

Bm

]

=[

G+

SE,O

]−1[

G−

SE,O

][

Dm

]

− jζ+

[

G+

SE,O

]−1[

JO

]

(4.7d)

The geometry supports the surface wave but when the observation point is far from the surface, these waves

can be neglected and diffracted waves dominates. Diffracted far fields in the upper region can be evaluated

from (4.1c). So after applying the saddle point method of integration, the final expression are as follow

Ed+

z =

∞∑

m=0

∫ ∞

0

jκ0

jκ0 +√

ξ2 − κ20ζ+

{

Am

J2m+ 32(ξ)

ξ3/2cos(xaξ) + Bm

J2m+ 52(ξ)

ξ3/2sin(xaξ)

}

exp[

−√

ξ2 − κ20ya

]

dξ

=

√

π

2

tan φ

1 + ζ+ sin φ

1√k0ρ

exp[

−jk0ρ + jπ

4

]

∞∑

m=0

[

Am

J2m+ 32(κ0 cos φ)

√κ0 cosφ

+ jBm

J2m+ 52(κ0 cos φ)

√κ0 cosφ

]

(4.8)

where (ρ, φ) are the cylindrical coordinates of the observation point. A far field in the lower half plane can

also be derived similarly.

56

Electromagnetic Diffraction from Slit(s):

B. H-Polarization

The field expressions corresponding to equation (4.1) for H-polarization may be written as

Hiz = exp

[

jk0(x cos φ0 + y sin φ0)]

(4.9a)

Hrz =

−Z+ + Z0 sin φ0

Z+ + Z0 sinφ0

exp[

jk0(x cos φ0 − y sinφ0)]

(4.9b)

Hd+

z =

∫ ∞

0

[

g1(ξ) cos (xaξ) + g2(ξ) sin (xaξ)]

exp[

−√

ξ2 − κ20ya

]

dξ y > 0 (4.9c)

Hd−

z =

∫ ∞

0

[

h1(ξ) cos (xaξ) + h2(ξ) sin (xaξ)]

exp[

√

ξ2 − κ2ya

]

dξ y < 0 (4.9d)

And the boundary conditions are

Etx

∣

∣

∣

y=0+

= Z+Htz

∣

∣

∣

y=0+

, Etx

∣

∣

∣

y=0−

= −Z−Htz

∣

∣

∣

y=0−

; |xa| ≥ 1 (4.10a)

Etx

∣

∣

∣

y=0+

= Etx

∣

∣

∣

y=0−

, Htz

∣

∣

∣

y=0+

= Htz

∣

∣

∣

y=0−

; |xa| ≤ 1 (4.10b)

Using (4.10a), following is obtained

∫ ∞

0

[

u + jκ0ζ+

][

g1(ξ) cos(xaξ) + g2(ξ) sin(xaξ)]

dξ = 0, |xa| ≥ 1 (4.11a)

∫ ∞

0

[

v + jκζ−

][

h1(ξ) cos(xaξ) + h2(ξ) sin(xaξ)]

dξ = 0, |xa| ≥ 1 (4.11b)

where ζ± have the same meaning as given in the last section and u =√

ξ2 − κ20, v =

√

ξ2 − κ2. The last

expressions are only possible if

g1(ξ) =1

jκ0ζ+ + u

∞∑

m=0

AmJ2m+ 12(ξ)ξ−

12 (4.12a)

g2(ξ) =1

jκ0ζ+ + u

∞∑

m=0

BmJ2m+ 32(ξ)ξ−

12 (4.12b)

h1(ξ) =1

jκζ− + v

∞∑

m=0

CmJ2m+ 12(ξ)ξ−

12 (4.12c)

h2(ξ) =1

jκζ− + v

∞∑

m=0

DmJ2m+ 32(ξ)ξ−

12 (4.12d)

From equation (4.10b)

∫ ∞

0

{[

g1(ξ) − h1(ξ)]

cos(xaξ) +[

g2(ξ) − h2(ξ)]

sin(xaξ)}

dξ

=2 sin φ0

ζ+ + sinφ0

exp[

jκ0xa cos φ0

]

, |xa| ≤ 1 (4.13a)

∫ ∞

0

{[

vh1(ξ) + εrug1(ξ)]

cos(xaξ) +[

vh2(ξ) + εrug2(ξ)]

sin(xaξ)}

dξ

=j2κ0ζ+ sin φ0

ζ+ + sin φ0

exp[

jκ0xa cosφ0

]

, |xa| ≤ 1 (4.13b)

57

Electromagnetic Diffraction from Slit(s):

Proceeding in a similar manner as in last section, following matrix equations are obtained

∞∑

m=0

[

AmGSH

(

2m +1

2, 2n +

1

2; ζ+

)

− CmGSH

(

2m +1

2, 2n +

1

2; ζ−

)]

=2 sin φ0

ζ+ + sin φ0

J2n+ 12(κ0 cos φ0)

(κ0 cosφ0)12

(4.14a)

∞∑

m=0

[

BmGSH

(

2m +3

2, 2n +

3

2; ζ+

)

− DmGSH

(

2m +3

2, 2n +

3

2; ζ−

)]

=j2 sin φ0

ζ+ + sin φ0

J2n+ 32(κ0 cos φ0)

(κ0 cosφ0)12

(4.14b)

∞∑

m=0

[

AmKSH

(

2m +1

2, 2n +

1

2; ζ+

)

+ CmKSH

(

2m +1

2, 2n +

1

2; ζ−

)]

=j2κ0ζ+ sinφ0

ζ+ + sin φ0

J2n+ 12(κ0 cos φ0)

(κ0 cos φ0)12

(4.14c)

∞∑

m=0

[

BmKSH

(

2m +3

2, 2n +

3

2; ζ+

)

+ DmKSH

(

2m +3

2, 2n +

3

2; ζ−

)]

= −2κ0ζ+ sin φ0

ζ+ + sinφ0

J2n+ 32(κ0 cosφ0)

(κ0 cos φ0)12

n = 0, 1, 2, · · · (4.14d)

where

GSH(m,n; ζ±) =

∫ ∞

0

1

jκζ± +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξdξ (4.15a)

KSH(m,n; ζ±) =

∫ ∞

0

εr

√

ξ2 − κ2

jκζ± +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξdξ (4.15b)

The above expressions are the matrix equations and can be solved for the expansion coefficients Am, Bm,

Cm and Dm as described in the preceding section.

Diffracted far fields in the upper space may be obtained by applying the saddle point method on expression

(4.9c). The final result is given by

Hd+

z =

√

π

2

sinφ

ζ+ + sinφ

1√k0ρ

exp[

−jk0ρ − jπ

4

]

×∞∑

m=0

[

Am

J2m+ 12(κ0 cos φ)

√κ0 cos φ

+ jBm

J2m+ 32(κ0 cos φ)

√κ0 cosφ

]

(4.16)

where (ρ, φ) are the coordinates of observation point.

4.2.2 Computations and Discussions

A. Evaluation of the Integrals KSE(m,n;κ), GSE(m,n;κ), KSH(m,n;κ) and GSH(m,n;κ)

This section includes the discussions that how the integrals involved in the matrix equations may be

computed. Discussion is initiated with the integral GSE(m,n;κ). The path of integration may be divided

as bellow

GSE(m,n; ζ) =

∫ ∞

0

1

jκη +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξ2dξ

58

Electromagnetic Diffraction from Slit(s):

=

∫ x

0

1

jκη +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξ2dξ +

∫ ∞

x

1

jκη +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξ2dξ

= GxSE(m,n; ζ) + G∞SE(m,n; ζ) (4.17a)

where x is a fairly large constant as described in section 3.2.3 . The integral GxSE(m,n; ζ) may be evaluated

by any standard method of numerical integration. The integral G∞SE(m,n; ζ) may be written as

G∞SE(m,n; ζ) =

∫ ∞

x

1

jκη +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξ2dξ

'∫ ∞

x

Jm(ξ)Jn(ξ)

ξ3− jκη

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ4dξ +

κ2

2

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ5dξ (4.17b)

Using the expressions (3.34) in the last chapter, the above expressions may be written as

G∞SE(m,n; ζ) ' 1

π

{[ −1

3x3+

A1

5x5− P6

4x5− κ2

10x5

]

cosm − n

2π +

[ A3

4x4− P6

A3

5x5

]

sinm − n

2π

+[ 1

2x4+

P6

2x5

]

sin(2x − β) +[

− 1

x5+

A4

2x5

]

cos(2x − β)}

(4.18)

where P6 = jκη, Similarly

KSE(m,n; ζ) =

∫ ∞

0

√

ξ2 − κ2

jκη +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξ2dξ

= −P6GSE(m,n; ζ) +

∫ ∞

0

Jm(ξ)Jn(ξ)

ξ2dξ (4.19)

Adopting the same procedure as above, the integral GSH(m,n;κ) may be written as

GSH(m,n; ζ) =

∫ ∞

0

1

jκζ +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξdξ

'∫ x

0

1

jκζ +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξdξ

+

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ2dξ − P1

∫ x

0

Jm(ξ)Jn(ξ)

ξ3dξ + P7

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ4dξ (4.20)

where P7 = ( 1

2− ζ2)κ2. If

G∞SH(m,n; ζ) =

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ2dξ − P1

∫ x

0

Jm(ξ)Jn(ξ)

ξ3dξ + P7

∫ ∞

x

Jm(ξ)Jn(ξ)

ξ4dξ

then it may be computed, using expressions (3.34), as under

G∞SH(m,n; ζ) ' 1

π

{[

− 1

2x2+

A1

4x4+ P1

(

− 1

3x3+

A1

5x5

)

− P7

4x4

]

cosm − n

2π

+[

− A3

3x3+

A5

5x5− P1A3

4x4− P7A3

5x5

]

sinm − n

2π

+[

− 3

4x4+

A4

2x5− P1

x5+ P1

( A4

2x5

)]

sin(2x − β)

+[ 1

2x3− 3

2x5+

A4

x5− A2

2x5+

P1

2x4+

P7

2x5

]

cos(2x − β)}

(4.21)

59

Electromagnetic Diffraction from Slit(s):

Similarly

KSH(m,n; ζ) =

∫ ∞

0

√

ξ2 − κ2

jκζ +√

ξ2 − κ2

Jm(ξ)Jn(ξ)

ξdξ

= −jκζGSH(m,n; ζ) +1

m + nδm,n (4.22)

B. Diffracted far field Patterns

To study the scattering properties of the slit, far field patterns are obtained. These Field patterns are eval-

uated using equation (4.8) for E-polarization and equation (4.16) for H-polarization. The values of unknowns

expansion coefficients Am, Bm, Cm, Dm can be computed from equations (4.7) for E-polarization and (4.14)

for H-polarization. These expressions contain the integrals GSE(m,m; ζ±), KSE(m,m; ζ±), GSH(m,m; ζ±)

and KSH(m,m; ζ±). How to compute these integrals, is discussed in the above section. In the simulation of

field patterns, the matrix size is takes as (2κ0 +3)× (2κ0 +3) in the computations. The far field patterns are

shown in Fig. 4.2 and Fig. 4.6 for E- and H-polarizations respectively for different angle of incidence. To

verify the validity of KP computations, comparison is given with the physical optics. Fig. 4.3 gives the same

for E-polarization. The angle of incidence φ0 is π2

and the surface impedances are chosen as ζ± = 0.2− 0.5i.

Similarly Fig. 4.7 shows the comparison for H-polarization. All the parameters are same except φ0, which is

π3

in this case. The comparison is fairly good for both the cases. Fig. 4.4 and Fig. 4.8 give the variations in

the field patterns with the slit width. It is evident that with the increase in the width of the slit, the width

of the main lobe somewhat decreases but at the same time minor lobes start to emerge in the pattern. Fig.

4.5 is intended to show the effects of material properties of the plane on the diffracted fields. It gives, with

the decrease in the impedance values of the plane, the strength of the field patterns intensifies. How the

medium parameters of lower space affect the scattered field is given in Fig. 4.9. The field patterns are given

for εr = 1.0, εr = 2.25, and εr = 4.0. The other parameters are as φ0 = π3, ζ± = 0.2 − 0.5i.

4.2.3 Conclusion

In this problem, KP solutions are obtained for the diffracted fields from a slit in an impedance plane placed

at the interface of two semi infinite half spaces of different media. After applying the boundary conditions and

following the standard procedure, the problem finally reduced to matrix equations for unknown expansion

coefficients. The elements of these equations are the infinite integrals. These integrals are evaluated by

making use of Hankel approximation of the Bessel functions which expedited the convergence of the integrals.

Numerical results are computed and presented for the diffracted fields as a function of angle of incidence, slit

width etc. These results are compared with those obtained through physical optics and found them in good

agreement. Effects of surrounding media of the slit and the material of the plane itself on the field patterns

are also shown. And it is observed that as the impedance of the plane is decreased, the fields intensify in

the upper half space. It is also seen that the relative permittivity of surrounding media has no effects on the

position of the main lobe nor on the general trends of the patterns but it changes the amplitude of the field

only.

60

Electromagnetic Diffraction from Slit(s):

-20 0 20 40 60 80 100 120 140 160 180 200-1

0

1

2

3

4

5

6

7

8

9

10

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

E- polarization

Figure 4.2 Variation of field patterns with angle of incidence

-20 0 20 40 60 80 100 120 140 160 180 200

0

2

4

6

8

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

KP PO

E- polarization

Figure 4.3 Comparison of KP and PO patterns

61

Electromagnetic Diffraction from Slit(s):

-20 0 20 40 60 80 100 120 140 160 180 200-2

0

2

4

6

8

10

12

14

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

0= 6.0

0= 7.0

0= 8.0

E- polarization

Figure 4.4 Dependance of field patterns on slit width

-20 0 20 40 60 80 100 120 140 160 180 200

0

2

4

6

8

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

E-polarization

Figure 4.5 ς (Impedance of plane) dependance of the patterns

62

Electromagnetic Diffraction from Slit(s):

-20 0 20 40 60 80 100 120 140 160 180 200-1

0

1

2

3

4

5

6

7

8

9

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

H- polarization

Fig.6 Scattering patterns for different values of angle of incidenceFigure 4.6 Field patterns as a function of angle of incidence (H-polarization)

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

KP PO

H- polarization

Figure 4.7 Comparison of field pattern obtained by the two methods

63

Electromagnetic Diffraction from Slit(s):

-20 0 20 40 60 80 100 120 140 160 180 200-1

0

1

2

3

4

5

6

7

8

9

10

11

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

H- polarization

Figure 4.8 Effects of the slit width on field patterns (H-polarization)

-20 0 20 40 60 80 100 120 140 160 180 200-1

0

1

2

3

4

5

6

7

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

r

r

r

H- polarization

Figure 4.9 Field patterns as a function of εr (H-polarization)

64

Electromagnetic Diffraction from Slit(s):

4.3 Diffraction from Two Parallel Slits in an Impedance Plane

4.3.1 Statement of the Problem

The geometry of the problem is shown in Fig. 4.10. The slits of width 2a and 2b in an impedance plane

are separated by the distance d. The slit having width 2a is termed as slit 1 while slit 2 has the width of

2b. For simplicity, E-polarized electromagnetic plane wave is considered as the incident wave . Therefore

the incident field Eiz may be written as

Eiz = exp

[

jk(x cos φ0 + y sinφ0)]

(4.23a)

where φ0 is the angle of incidence and k is the propagation constant of the space. The reflected field from

the impedance plane may be written as

Erz = −Z0 − Z+ sinφ0

Z0 + Z+ sinφ0

exp[

jk(x cos φ0 − y sin φ0)]

(4.23b)

where Z0 be the impedance of the free space and Z+ be the surface impedance of upper side of the plane.

The diffracted fields may be considered in the form

Ed+z =

∫ ∞

0

[

g1(ξ) cos (xaξ) + g2(ξ) sin (xaξ)]

exp[

−√

ξ2 − κ2aya

]

dξ

+

∫ ∞

0

[

g3(ξ) cos (xbξ) + g4(ξ) sin (xbξ)]

exp

[

−√

ξ2 − κ2byb

]

dξ, y > 0 (4.23c)

Ed−z =

∫ ∞

0

[

h1(ξ) cos (xaξ) + h2(ξ) sin (xaξ)]

exp[

√

ξ2 − κ2aya

]

dξ

+

∫ ∞

0

[

h3(ξ) cos (xbξ) + h4(ξ) sin (xbξ)]

exp

[

√

ξ2 − κ2byb

]

dξ, y < 0 (4.23d)

where g1,2,3,4(ξ) h1,2,3,4(ξ) are the weighting functions to be determined from the boundary conditions,

and κa = ka, κb = kb, ya = ya , yb = y

b , xa = xa , xb = x

b .

x2

y2

x1

y1

zx

y

z2z1

d

x1= - a x1= -a x2= - b x2= b

Fig. 4.10 Geometry of the problem

d2d1

Using the discontinuous properties of Weber-Schafheitlin’ integral, Weighting functions for slit 1 may be

written as follow

g1(ξ) =jκa

jκa +√

ξ2 − κ2aζ+

∞∑

m=0

AmJ2m+ 32(ξ)ξ−

32 (4.24a)

65

Electromagnetic Diffraction from Slit(s):

g2(ξ) =jκa

jκa +√

ξ2 − κ2aζ+

∞∑

m=0

BmJ2m+ 52(ξ)ξ−

32 (4.24b)

h1(ξ) =jκa

jκa +√

ξ2 − κ2aζ−

∞∑

m=0

CmJ2m+ 32(ξ)ξ−

32 (4.24c)

h2(ξ) =jκa

jκ +√

ξ2 − κ2aζ−

∞∑

m=0

DmJ2m+ 52(ξ)ξ−

32 (4.24d)

Similarly, weighting functions for slit 2 are

g3(ξ) =jκb

jκb +√

ξ2 − κ2bζ+

∞∑

m=0

EmJ2m+ 32(ξ)ξ−

32 (4.25a)

g4(ξ) =jκb

jκb +√

ξ2 − κ2bζ+

∞∑

m=0

FmJ2m+ 52(ξ)ξ−

32 (4.25b)

h3(ξ) =jκb

jκb +√

ξ2 − κ2bζ−

∞∑

m=0

GmJ2m+ 32(ξ)ξ−

32 (4.25c)

h4(ξ) =jκb

jκb +√

ξ2 − κ2bζ−

∞∑

m=0

HmJ2m+ 52(ξ)ξ−

32 (4.25d)

where ζ+ and ζ− are the normalized impedance of upper and lower surfaces and Am, Bm, Cm, Dm, Em,

Fm, Gm and Hm are the expansion coefficients to be determined.

The remaining boundary conditions are

Htx

∣

∣

∣

y=0+

= Htx

∣

∣

∣

y=0−

, |xa| ≤ 1 (4.26a)

Htx

∣

∣

∣

y=0+

= Htx

∣

∣

∣

y=0−

, |xb| ≤ 1 (4.26b)

Etz

∣

∣

∣

y=0+

= Etz

∣

∣

∣

y=0−

, |xa| ≤ 1 (4.26c)

Etz

∣

∣

∣

y=0+

= Etz

∣

∣

∣

y=0−

, |xb| ≤ 1 (4.26d)

where superscript t means total.

Applying (4.26a), using the relation x1 − d = x2 between the two local coordinates and then comparing

even and odd functions, following is obtained∫ ∞

0

√

ξ2 − κ2b

[{

g3(ξ) + h3(ξ)}

cos (xaξr) cos (daξr) −{

g4(ξ) + h4(ξ)}

cos (xaξr) sin (daξr)]

dξ

+

∫ ∞

0

√

ξ2 − κ2a

[

g1(ξ) + h1(ξ)]

cos (xaξ) dξ

=2jκa sinφ0

1 + ζ+ sinφ0

cos(κaxa cos φ0) (4.27a)

and∫ ∞

0

√

ξ2 − κ2b

[{

g3(ξ) + h3(ξ)}

sin (xaξr) sin (daξr) +{

g4(ξ) + h4(ξ)}

sin (xaξr) cos (daξr)]

dξ

+

∫ ∞

0

√

ξ2 − κ2a

{

g2(ξ) + h2(ξ)}

sin (xaξ) dξ

= − 2κa sin φ0

1 + ζ+ sinφ0

sin(κaxa cos φ0) (4.27b)

66

Electromagnetic Diffraction from Slit(s):

where da = da and r = a

b . While obtaining the above expressions, addition theorem for the trigonometric

functions is used. Now expanding the trigonometric in terms of Jacobi’s polynomials (chapter 2), it can be

written as

∫ ∞

0

√

ξ2 − κ2a

{

g1(ξ) + h1(ξ)}

J2n+ 12

(ξ)1√ξdξ

+

∫ ∞

0

√

ξ2 − κ2b

[{

g3(ξ) + h3(ξ)}

J2n+ 12

(ξr) cos (daξr)]

√

r

ξdξ

−∫ ∞

0

√

ξ2 − κ2b

[{

g4(ξ) + h4(ξ)}

J2n+ 12

(ξr) sin (daξr)]

√

r

ξdξ

=2jκa sinφ0

1 + ζ+ sinφ0

J2n+ 12(κa cosφ0)

(κa cos φ0)1/2(4.28a)

∫ ∞

0

[

√

ξ2 − κ2a

a

]{

g2(ξ) + h2(ξ)}

J2n+ 32

(ξ)1√ξdξ

+

∫ ∞

0

[

√

ξ2 − κ2b

b

] [{

g3(ξ) + h3(ξ)}

J2n+ 32

(ξr) sin (daξr)]

√

r

ξdξ

+

∫ ∞

0

[

√

ξ2 − κ2b

b

][{

g4(ξ) + h4(ξ)}

J2n+ 32

(ξr) cos (daξr)]

√

r

ξdξ

= − 2κa sinφ0

1 + ζ+ sin φ0

J2n+ 32(κa cos φ0)

(κa cosφ0)1/2(4.28b)

Let ζ+ = ζ− = ζ and putting the values of g1,2,3,4(ξ) and h1,2,3,4(ξ) in the above expressions

∞∑

m=0

∫ ∞

0

[

jκa

√

ξ2 − κ2a

jκa +√

ξ2 − κ2aζ

]

J2m+ 32

(ξ) J2n+ 12

(ξ)[

Am + Cm

]

ξ−2dξ

+

∫ ∞

0

[

jκb

√

ξ2 − κ2b

jκb +√

ξ2 − κ2bζ

]

J2m+ 32

(ξ) J2n+ 12

(ξr) cos (daξr)[

Em + Gm

]

r12 ξ−2dξ

−∫ ∞

0

[

jκb

√

ξ2 − κ2b

jκb +√

ξ2 − κ2bζ

]

J2m+ 52

(ξ) J2n+ 12

(ξr) sin (daξr)[

Fm + Hm

]

r12 ξ−2dξ

=2jκa sinφ0

1 + ζ sin φ0

J2n+ 12(κa cos φ0)

(κa cosφ0)12

(4.29a)

and

∞∑

m=0

∫ ∞

0

[

jκa

√

ξ2 − κ2a

jκa +√

ξ2 − κ2aζ

]

J2m+ 52

(ξ) J2n+ 32

(ξ)[

Bm + Dm

]

ξ−2dξ

+

∫ ∞

0

[

jκb

√

ξ2 − κ2b

jκb +√

ξ2 − κ2bζ

]

J2m+ 32

(ξ) J2n+ 32

(ξr) sin (daξr)[

Em + Gm

]

r12 ξ−2dξ

+

∫ ∞

0

[

jκb

√

ξ2 − κ2b

jκb +√

ξ2 − κ2bζ

]

J2m+ 52

(ξ) J2n+ 32

(ξr) cos (daξr)[

Fm + Hm

]

r12 ξ−2dξ

= − 2κa sinφ0

1 + ζ sinφ0

J2n+ 32(κa cosφ0)

(κa cos φ0)1/2(4.29b)

which may be written as follow

∞∑

m=0

{

Kea

(

2m +3

2, 2n +

1

2;κa

)

Am + Kea

(

2m +3

2, 2n +

1

2;κa

)

Cm

67

Electromagnetic Diffraction from Slit(s):

+ Kecb

(

2m +3

2, 2n +

1

2; da

)

Em + Kecb

(

2m +3

2, 2n +

1

2; da

)

Gm

− Kesb

(

2m +5

2, 2n +

1

2; da

)

Fm − Kesb

(

2m +5

2, 2n +

1

2; da

)

Hm

}

=2jκa sinφ0

1 + ζ sin φ0

J2n+ 12(κa cos φ0)

(κa cosφ0)1/2n = 0, 1, 2, 3 · · · (4.30a)

∞∑

m=0

{

Kea

(

2m +5

2, 2n +

3

2;κa

)

Bm + Kea

(

2m +5

2, 2n +

3

2;κa

)

Dm

+ Kesb

(

2m +3

2, 2n +

3

2; da

)

Em + Kesb

(

2m +3

2, 2n +

3

2; da

)

Gm

+ Kecb

(

2m +5

2, 2n +

3

2; da

)

Fm + Kecb

(

2m +5

2, 2n +

3

2; da

)

Hm

}

= − 2κa sin φ0

1 + ζ sin φ0

J2n+ 32(κa cos φ0)

(κa cosφ0)1/2n = 0, 1, 2, 3 · · · (4.30b)

where

Kea(m,n, κa) =

∫ ∞

0

[

jκa

√

ξ2 − κ2a

jκa +√

ξ2 − κ2aζ

]

Jm(ξ)Jn(rξ)

ξ2dξ (4.31a)

Kecb(m,n, da) =

∫ ∞

0

[

jκb

√

ξ2 − κ2b

jκb +√

ξ2 − κ2bζ

]

Jm(ξ)Jn(rξ) cos(rdaξ)r12 ξ−2dξ (4.31b)

Kesb(m,n, da) =

∫ ∞

0

[

jκb

√

ξ2 − κ2b

jκb +√

ξ2 − κ2bζ

]

Jm(ξ)Jn(rξ) sin(rdaξ)r12 ξ−2dξ (4.31c)

Using boundary conditions (4.26b), (4.26c) and (4.26d) one by one and proceeding in the similar manner,

following is obtained

∞∑

m=0

{

Keb

(

2m +3

2, 2n +

1

2;κb

)

Em + Keb

(

2m +3

2, 2n +

1

2;κb

)

Gm

+ Keca

(

2m +3

2, 2n +

1

2; db

)

Am + Keca

(

2m +3

2, 2n +

1

2; db

)

Cm

+ Kesa

(

2m +5

2, 2n +

1

2; db

)

Bm + Kesa

(

2m +5

2, 2n +

1

2; db

)

Dm

}

=2jκb sin φ0

1 + ζ sinφ0

J2n+ 12(κb cos φ0)

(κb cosφ0)12

n = 0, 1, 2, 3 · · · (4.32a)

∞∑

m=0

{

Keb

(

2m +5

2, 2n +

3

2;κb

)

Fm + Keb

(

2m +3

2, 2n +

3

2;κb

)

Hm

− Kesa

(

2m +3

2, 2n +

3

2; db

)

Am − Kesa

(

2m +3

2, 2n +

3

2; db

)

Cm

+ Keca

(

2m +5

2, 2n +

3

2; db

)

Dm + Keca

(

2m +5

2, 2n +

3

2; da

)

Bm

}

= − 2κb sinφ0

1 + ζ sinφ0

J2n+ 32(κb cos φ0)

(κb cos φ0)12

n = 0, 1, 2, 3 · · · (4.32b)

∞∑

m=0

{

Gea

(

2m +3

2, 2n +

1

2;κa

)

Am − Gea

(

2m +3

2, 2n +

1

2;κa

)

Cm

+ Gecb

(

2m +3

2, 2n +

1

2; da

)

Em − Gecb

(

2m +3

2, 2n +

1

2; da

)

Gm

68

Electromagnetic Diffraction from Slit(s):

− Gesb

(

2m +5

2, 2n +

1

2; da

)

Fm + Gesb

(

2m +5

2, 2n +

1

2; da

)

Hm

}

= − 2ζ sin φ0

1 + ζ sinφ0

J2n+ 12(κa cos φ0)

(κa cos φ0)12

n = 0, 1, 2, 3 · · · (4.33a)

∞∑

m=0

{

Gea

(

2m +5

2, 2n +

3

2;κa

)

Bm − Gea

(

2m +5

2, 2n +

3

2;κa

)

Dm

+ Gesb

(

2m +3

2, 2n +

3

2; da

)

Em − Gesb

(

2m +3

2, 2n +

3

2; da

)

Gm

+ Gecb

(

2m +5

2, 2n +

3

2; da

)

Fm − Gecb

(

2m +5

2, 2n +

3

2; da

)

Hm

}

= − 2jζ sin φ0

1 + ζ sinφ0

J2n+ 32(κa cos φ0)

(κa cos φ0)12

n = 0, 1, 2, 3 · · · (4.33b)

∞∑

m=0

{

Geb

(

2m +3

2, 2n +

1

2;κb

)

Em − Geb

(

2m +3

2, 2n +

1

2;κb

)

Gm

− Geca

(

2m +3

2, 2n +

1

2; db

)

Cm + Geca

(

2m +3

2, 2n +

1

2; db

)

Am

− Gesa

(

2m +5

2, 2n +

1

2; db

)

Dm + Gesa

(

2m +5

2, 2n +

1

2; da

)

Bm

}

= − 2ζ sinφ0

1 + ζ sin φ0

J2n+1/2(κb cos φ0)

(κb cosφ0)12

n = 0, 1, 2, 3 · · · (4.34a)

∞∑

m=0

{

Geb

(

2m +5

2, 2n +

3

2;κb

)

Fm − Geb

(

2m +5

2, 2n +

3

2;κb

)

Hm

+ Gesa

(

2m +3

2, 2n +

3

2; db

)

Cm − Gesa

(

2m +3

2, 2n +

3

2; db

)

Am

− Geca

(

2m +5

2, 2n +

3

2; db

)

Dm + Geca

(

2m +3

2, 2n +

5

2; db

)

Bm

}

= − 2jζ sin φ0

1 + sinφ0

J2n+ 32(κb cos φ0)

(κb cosφ0)12

n = 0, 1, 2, 3 · · · (4.34b)

where

Keb (m,n, κb) =

∫ ∞

0

[

jκb

√

ξ2 − κ2b

jκb +√

ξ2 − κ2bζ

]

Jm(ξ)Jn(qξ)ξ−2dξ (4.35a)

Keca(m,n, db) =

∫ ∞

0

[

jκa

√

ξ2 − κ2a

jκa +√

ξ2 − κ2aζ

]

Jm(ξ)Jn(qξ) cos(qdbξ)q12 ξ−2dξ (4.35b)

Kesa(m,n, db) =

∫ ∞

0

[

jκa

√

ξ2 − κ2a

jκa +√

ξ2 − κ2aζ

]

Jm(ξ)Jn(qξ) sin(qdbξ)q12 ξ−2dξ (4.35c)

Gea(m,n, κa) =

∫ ∞

0

[

jκa

jκa +√

ξ2 − κ2aζ

]

Jm(ξ)Jn(rξ)ξ−2dξ (4.35d)

Gecb(m,n, da) =

∫ ∞

0

[

jκb

jκb +√

ξ2 − κ2bζ

]

Jm(ξ)Jn(rξ) cos(rdaξ)r−12 ξ−2dξ (4.35e)

Gesb(m,n, da) =

∫ ∞

0

[

jκb

jκb +√

ξ2 − κ2bζ

]

Jm(ξ)Jn(rξ) sin(rdaξ)r−12 ξ−2dξ (4.35f)

69

Electromagnetic Diffraction from Slit(s):

and

db = db , q = 1

r

4.3.2 Approximate Values of the Expansion Coefficients

The expressions (4.30) and (4.32)-(4.34) are the matrix equations and these can be used to evaluate the

expansion coefficients Am to Hm. In these expressions Keca, Ke

cb, Kesb, Ke

sa, Geca, Ge

cb, Gesb and Ge

sa are the

interaction integrals and may be approximated by using saddle point method for larger values of r or q

and the integrals Kea, Ge

a, Keb , Ge

b may be computed using the standard methods of integration. For our

convenience, the above expressions can be written in standard matrix notation as follow

[

Kea

(

2m +3

2, 2n +

1

2;κa

)][

Am + Cm

]

+[

Kecb

(

2m +3

2, 2n +

1

2; da

)][

Em + Gm

]

−[

Kesb

(

2m +5

2, 2n +

1

2; da

)][

Fm + Hm

]

=2jκa sin φ0

1 + ζ sinφ0

[

J2n+ 12(κa cos φ0)

(κa cos φ0)12

]

[

Kea

(

2m +5

2, 2n +

3

2;κa

)][

Bm + Dm

]

+[

Kecb

(

2m +5

2, 2n +

3

2;κa

)][

Fm + Hm

]

+[

Kesb

(

2m +3

2, 2n +

3

2; da

)][

Em + Gm

]

= − 2κa sin φ0

1 + ζ sin φ0

[

J2n+ 32(κa cosφ0)

(κa cos φ0)12

]

[

Keb

(

2m +3

2, 2n +

1

2;κb

)][

Em + Gm

]

+[

Keca

(

2m +3

2, 2n +

1

2; db

)][

Am + Cm

]

+[

Kesa

(

2m +5

2, 2n +

1

2; db

)][

Bm + Dm

]

=2jκb sin φ0

1 + ζ sin φ0

[

J2n+ 12(κb cosφ0)

(κb cos φ0)12

]

[

Keb

(

2m +5

2, 2n +

3

2;κb

)][

Fm + Hm

]

+[

Keca

(

2m +5

2, 2n +

3

2; db

)][

Bm + Dm

]

−[

Kesa

(

2m +3

2, 2n +

3

2; db

)][

Am + Cm

]

= − 2κb sin φ0

1 + ζ sin φ0

[

J2n+ 32(κb cos φ0)

(κb cos φ0)12

]

[

Gea

(

2m +3

2, 2n +

1

2;κa

)][

Am − Cm

]

+[

Gecb

(

2m +3

2, 2n +

1

2; da

)][

Em − Gm

]

+[

Gesb

(

2m +5

2, 2n +

1

2; da

)][

−Fm + Hm

]

= − 2ζ sin φ0

1 + ζ sinφ0

[

J2n+ 12(κa cos φ0)

(κa cosφ0)12

]

[

Gea

(

2m +5

2, 2n +

3

2;κa

)][

Bm − Dm

]

+[

Gecb

(

2m +5

2, 2n +

3

2; da

)][

Fm − Hm

]

+[

Gesb

(

2m +3

2, 2n +

3

2; da

)][

Em − Gm

]

= − 2jζ sin φ0

1 + ζ sin φ0

[

J2n+ 32(κa cosφ0)

(κa cos φ0)12

]

70

Electromagnetic Diffraction from Slit(s):

[

Geb

(

2m +3

2, 2n +

1

2;κb

)][

Em − Gm

]

+[

Geca

(

2m +3

2, 2n +

1

2; db

)][

Am − Cm

]

+[

Gesa

(

2m +5

2, 2n +

1

2; db

)][

Bm − Dm

]

= − 2ζ sin φ0

1 + ζ sin φ0

[

J2n+ 12(κb cos φ0)

(κb cos φ0)12

]

[

Geb

(

2m +5

2, 2n +

3

2;κb

)][

Fm − Hm

]

+[

Geca

(

2m +5

2, 2n +

3

2; db

)][

Bm − Dm

]

−[

Gesa

(

2m +3

2, 2n +

3

2; db

)][

Am − Cm

]

= − 2jζ sin φ0

1 + ζ sinφ0

[

J2n+ 32(κb cos φ0)

(κb cosφ0)12

]

The above equations may be solved by using block Gauss-Seidel procedure. The contribution from the

coupling terms Keca, Ke

cb, Kesb, Ke

sa, Geca, Ge

cb, Gesb and Gsa may be neglected for sufficiently large separation

between the slits as compared to the wave length. Therefore the zeroth order solutions may be written as

[

Am + Cm

]0

=2jκa sin φ0

1 + ζ sin φ0

[

Kea

(

2m +3

2, 2n +

1

2;κa

)

]−1[

J2n+ 12(κa cosφ0)

(κa cos φ0)12

]

(4.36a)

[

Bm + Dm

]0

= − 2κa sin φ0

1 + ζ sinφ0

[

Kea

(

2m +5

2, 2n +

3

2;κa

)

]−1[

J2n+ 32(κa cos φ0)

(κa cosφ0)12

]

(4.36b)

[

Em + Gm

]0

=2jκb sinφ0

1 + ζ sin φ0

[

Keb

(

2m +3

2, 2n +

1

2;κb

)

]−1[

J2n+ 12(κb cos φ0)

(κb cos φ0)12

]

(4.36c)

[

Fm + Hm

]0

= − 2κb sin φ0

1 + ζ sinφ0

[

Keb

(

2m +5

2, 2n +

3

2;κb

)

]−1[

J2n+ 32(κb cos φ0)

(κb cosφ0)12

]

(4.36d)

[

Am − Cm

]0

= − 2ζ sinφ0

1 + ζ sin φ0

[

Gea

(

2m +3

2, 2n +

1

2;κa

)

]−1[

J2n+ 12(κa cosφ0)

(κa cos φ0)12

]

(4.36e)

[

Bm − Dm

]0

= − 2jζ sin φ0

1 + ζ sinφ0

[

Gea

(

2m +5

2, 2n +

3

2;κa

)

]−1[

J2n+ 32(κa cosφ0)

(κa cosφ0)12

]

(4.36f)

[

Em − Gm

]0

= − 2ζ sinφ0

1 + ζ sin φ0

[

Geb

(

2m +3

2, 2n +

1

2;κb

)

]−1[

J2n+ 12(κb cos φ0)

(κb cos φ0)12

]

(4.36g)

[

Fm − Hm

]0

= − 2jζ sinφ0

1 + ζ sinφ0

[

Geb

(

2m +5

2, 2n +

3

2;κb

)

]−1[

J2n+ 32(κb cos φ0)

(κb cosφ0)12

]

(4.36h)

And the first order solutions are

[

Am + Cm

]1

=[

Am + Cm

]0

−[

Kea

(

2m +3

2, 2n +

1

2;κa

)

]−1 [

Kecb

(

2m +3

2, 2n +

1

2; da

)

]

[

Em + Gm

]0

+

[

Kea

(

2m +3

2, 2n +

1

2;κa

)

]−1 [

Kesb

(

2m +5

2, 2n +

1

2; da

)

]

[

Fm + Hm

]0

(4.37a)

71

Electromagnetic Diffraction from Slit(s):

[

Bm + Dm

]1

=[

Bm + Dm

]0

−[

Kea

(

2m +5

2, 2n +

3

2;κa

)

]−1 [

Kecb

(

2m +5

2, 2n +

3

2; da

)

]

[

Fm + Hm

]0

−[

Kea

(

2m +5

2, 2n +

3

2;κa

)

]−1 [

Kesb

(

2m +3

2, 2n +

3

2; da

)

]

[

Em + Gm

]0

(4.37b)

[

Em + Gm

]1

=[

Em + Gm

]0

−[

Keb

(

2m +3

2, 2n +

1

2;κb

)

]−1 [

Keca

(

2m +3

2, 2n +

1

2; db

)

]

[

Am + Cm

]0

−[

Keb

(

2m +3

2, 2n +

1

2;κb

)

]−1 [

Kesa

(

2m +5

2, 2n +

1

2; db

)

]

[

Bm + Dm

]0

(4.37c)

[

Fm + Hm

]1

=[

Fm + Hm

]0

−[

Keb

(

2m +5

2, 2n +

3

2;κb

)

] [

Keca

(

2m +5

2, 2n +

3

2; db

)

]

[

Bm + Dm

]0

+

[

Keb

(

2m +5

2, 2n +

3

2;κb

)

] [

Kesa

(

2m +3

2, 2n +

3

2; db

)

]

[

Am + Cm

]0

(4.37d)

[

Am − Cm

]1

= [Am − Cm]0

−[

Gea

(

2m +3

2, 2n +

1

2;κa

)

]−1 [

Gecb

(

2m +3

2, 2n +

1

2; da

)

]

[

Em − Gm

]0

+

[

Gea

(

2m +3

2, 2n +

1

2;κa

)

]−1 [

Gesb

(

2m +5

2, 2n +

1

2; da

)

]

[

Fm − Hm

]0

(4.37e)

[

Bm − Dm

]1

=[

Bm − Dm

]0

−[

Gea

(

2m +5

2, 2n +

3

2;κa

)

]−1 [

Gecb

(

2m +5

2, 2n +

3

2; da

)

]

[

Fm − Hm

]0

−[

Gea

(

2m +5

2, 2n +

3

2;κa

)

]−1 [

Gesb

(

2m +3

2, 2n +

3

2; da

)

]

[

Em − Gm

]0

(4.37f)

[

Em − Gm

]1

=[

Em − Gm

]0

−[

Geb

(

2m +3

2, 2n +

1

2;κb

)

]−1 [

Gecba

(

2m +3

2, 2n +

1

2; db

)

]

[

Am − Cm

]0

−[

Geb

(

2m +3

2, 2n +

1

2;κb

)

]−1 [

Gesa

(

2m +5

2, 2n +

1

2; db

)

]

[

Bm − Dm

]0

(4.37g)

[

Fm − Hm

]1

=[

Fm − Hm

]0

−[

Geb

(

2m +5

2, 2n +

3

2;κb

)

]−1 [

Geca

(

2m +5

2, 2n +

3

2; db

)

]

[

Bm − Dm

]0

+

[

Geb

(

2m +5

2, 2n +

3

2;κb

)

]−1 [

Gesa

(

2m +5

2, 2n +

3

2; db

)

]

[

Am − Cm

]0

(4.37h)

The zeroth order coefficients give the fields by single slit 1 or slit 2 as if they are isolated and the first order

72

Electromagnetic Diffraction from Slit(s):

coefficients yield the field when the first order interaction between the slits are taken into account. Similarly

higher order interaction terms may also be introduced by iteration. Diffracted far field in the upper half

space can be evaluated by applying the saddle point method. The result is given

Ed+

z =

√

π

2

sin φ

1 + ζ+ sin φ

1√kρ

exp[

−jkρ + jπ

4

]

×∞∑

m=0

{

Am

J2m+ 32(κa cosφ)

√κa cos φ

+ jBm

J2m+ 52(κa cosφ)

√κa cos φ

}

exp[

−jkd1 cos φ]

+

{

Em

J2m+ 32(κb cosφ)

√κb cos φ

+ jFm

J2m+ 52(κb cos φ)

√κb cosφ

}

exp[

jkd2 cosφ]

(4.38)

where (ρ, φ) is the coordinates of the observation point as in the previous cases and d1 and d2 are the

distances shown in geometry of the problem. The values of unknowns Am, Bm, Em, Fm can be had from the

above expressions. The corresponding expression in the lower half space may be derived similarly.

This problem may also be solved for H-polarized field by proceeding in the similar manner as above. We

are giving here the expressions which are worth to mention. The expressions for g1,2,3,4(ξ) and h1,2,3,4(ξ) for

H-polarized incident wave upon the geometry are

g1(ξ) =1

jκaζ+ +√

ξ2 − κ2a

∞∑

m=0

AmJ2m+ 12(ξ)ξ−

12 (4.39a)

g2(ξ) =1

jκaζ+ +√

ξ2 − κ2a

∞∑

m=0

BmJ2m+ 32(ξ)ξ−

12 (4.39b)

h1(ξ) =1

jκaζ− +√

ξ2 − κ2a

∞∑

m=0

CmJ2m+ 12(ξ)ξ−

12 (4.39c)

h2(ξ) =1

jκζ− +√

ξ2 − κ2a

∞∑

m=0

DmJ2m+ 32(ξ)ξ−

12 (4.39d)

g3(ξ) =1

jκbζ+ +√

ξ2 − κ2b

∞∑

m=0

EmJ2m+ 12(ξ)ξ−

12 (4.40a)

g4(ξ) =1

jκbζ+ +√

ξ2 − κ2b

∞∑

m=0

FmJ2m+ 32(ξ)ξ−

12 (4.40b)

h3(ξ) =1

jκbζ− +√

ξ2 − κ2b

∞∑

m=0

GmJ2m+ 12(ξ)ξ−

12 (4.40c)

h4(ξ) =1

jκbζ− +√

ξ2 − κ2b

∞∑

m=0

HmJ2m+ 32(ξ)ξ−

12 (4.40d)

Adopting the same procedure, we get the expressions corresponding to (4.30)-(4.35) as follow

∞∑

m=0

{

Kha

(

2m +1

2, 2n +

1

2;κa

)

Am + Kha

(

2m +1

2, 2n +

1

2;κa

)

Cm

+ Khcb

(

2m +1

2, 2n +

1

2; da

)

Em + Khcb

(

2m +1

2, 2n +

1

2; da

)

Gm

73

Electromagnetic Diffraction from Slit(s):

− Khsb

(

2m +3

2, 2n +

1

2; da

)

Fm − Khsb

(

2m +3

2, 2n +

1

2; da

)

Hm

}

=2ζ sinφ0

ζ + sin φ0

J2n+ 12(κa cos φ0)

(κa cos φ0)1/2n = 0, 1, 2, 3 · · · (4.41a)

∞∑

m=0

{

Kha

(

2m +3

2, 2n +

3

2;κa

)

Bm + Kha

(

2m +3

2, 2n +

3

2;κa

)

Dm

+ Khsb

(

2m +1

2, 2n +

3

2; da

)

Em + Khsb

(

2m +1

2, 2n +

3

2; da

)

Gm

+ Khcb

(

2m +3

2, 2n +

3

2; da

)

Fm + Khcb

(

2m +3

2, 2n +

3

2; da

)

Hm

}

= − 2jζ sin φ0

ζ + sin φ0

J2n+ 32(κa cos φ0)

(κa cos φ0)1/2n = 0, 1, 2, 3 · · · (4.41b)

∞∑

m=0

{

Khb

(

2m +1

2, 2n +

1

2;κb

)

Em + Khb

(

2m +1

2, 2n +

1

2;κb

)

Gm

+ Khca

(

2m +1

2, 2n +

1

2; db

)

Am + Khca

(

2m +1

2, 2n +

1

2; db

)

Cm

+ Khsa

(

2m +3

2, 2n +

1

2; db

)

Bm + Khsa

(

2m +3

2, 2n +

1

2; db

)

Dm

}

= − 2ζ sin φ0

ζ + sin φ0

J2n+ 12(κb cosφ0)

(κb cos φ0)12

n = 0, 1, 2, 3 · · · (4.42a)

∞∑

m=0

{

Khb

(

2m +3

2, 2n +

3

2;κb

)

Fm + Khb

(

2m +3

2, 2n +

3

2;κb

)

Hm

− Khsb

(

2m +1

2, 2n +

3

2; db

)

Am − Khsb

(

2m +1

2, 2n +

3

2; db

)

Cm

+ Khcb

(

2m +3

2, 2n +

3

2; db

)

Dm + Khcb

(

2m +3

2, 2n +

3

2; da

)

Bm

}

= − 2jζ sinφ0

ζ + sin φ0

J2n+ 32(κb cosφ0)

(κb cos φ0)12

n = 0, 1, 2, 3 · · · (4.42b)

∞∑

m=0

{

Gha

(

2m +1

2, 2n +

1

2;κa

)

Am − Gha

(

2m +1

2, 2n +

1

2;κa

)

Cm

+ Ghcb

(

2m +1

2, 2n +

1

2; da

)

Em − Ghcb

(

2m +1

2, 2n +

1

2; da

)

Gm

− Ghsb

(

2m +3

2, 2n +

1

2; da

)

Fm + Ghsb

(

2m +3

2, 2n +

1

2; da

)

Hm

}

= − 2 sin φ0

ζ + sin φ0

J2n+ 12(κa cos φ0)

(κa cosφ0)12

n = 0, 1, 2, 3 · · · (4.43a)

∞∑

m=0

{

Gha

(

2m +3

2, 2n +

3

2;κa

)

Bm − Gha

(

2m +3

2, 2n +

3

2;κa

)

Dm

+ Ghsb

(

2m +1

2, 2n +

3

2; da

)

Em − Ghsb

(

2m +1

2, 2n +

3

2; da

)

Gm

+ Ghcb

(

2m +3

2, 2n +

3

2; da

)

Fm − Ghcb

(

2m +3

2, 2n +

3

2; da

)

Hm

}

= − 2j sin φ0

ζ + sin φ0

J2n+ 32(κa cos φ0)

(κa cosφ0)12

n = 0, 1, 2, 3 · · · (4.43b)

74

Electromagnetic Diffraction from Slit(s):

∞∑

m=0

{

Ghb

(

2m +1

2, 2n +

1

2;κb

)

Em − Ghb

(

2m +1

2, 2n +

1

2;κb

)

Gm

− Ghca

(

2m +1

2, 2n +

1

2; db

)

Cm + Ghca

(

2m +1

2, 2n +

1

2; db

)

Am

− Ghsa

(

2m +3

2, 2n +

1

2; db

)

Dm + Ghsa

(

2m +3

2, 2n +

1

2; da

)

Bm

}

= − 2 sin φ0

ζ + sin φ0

J2n+1/2(κb cosφ0)

(κb cos φ0)12

n = 0, 1, 2, 3 · · · (4.44a)

∞∑

m=0

{

Ghb

(

2m +3

2, 2n +

3

2;κb

)

Fm − Ghb

(

2m +3

2, 2n +

3

2;κb

)

Hm

+ Ghsa

(

2m +1

2, 2n +

3

2; db

)

Cm − Ghsa

(

2m +1

2, 2n +

3

2; db

)

Am

− Ghca

(

2m +3

2, 2n +

3

2; db

)

Dm + Ghca

(

2m +3

2, 2n +

3

2; db

)

Bm

}

= − 2j sinφ0

ζ + sin φ0

J2n+ 32(κb cosφ0)

(κb cos φ0)12

n = 0, 1, 2, 3 · · · (4.44b)

where

Kha (m,n, κa) =

∫ ∞

0

[

√

ξ2 − κ2a

jκaζ +√

ξ2 − κ2a

]

Jm(ξ)Jn(ξ)ξ−1dξ (4.45a)

Khb (m,n, κb) =

∫ ∞

0

[

√

ξ2 − κ2b

jκbζ +√

ξ2 − κ2b

]

Jm(ξ)Jn(ξ)ξ−1dξ (4.45b)

Khca(m,n, db) =

∫ ∞

0

[

√

ξ2 − κ2a

jκaζ +√

ξ2 − κ2a

]

Jm(ξ)Jn(qξ) cos(qdbξ)q−

12 ξ−1dξ (4.45c)

Khcb(m,n, da) =

∫ ∞

0

[

√

ξ2 − κ2b

jκbζ +√

ξ2 − κ2b

]

Jm(ξ)Jn(rξ) cos(rdbξ)r−

12 ξ−1dξ (4.45d)

Khsa(m,n, db) =

∫ ∞

0

[

√

ξ2 − κ2a

jκaζ +√

ξ2 − κ2a

]

Jm(ξ)Jn(qξ) sin(qdbξ)q−

12 ξ−1dξ (4.45e)

Khsb(m,n, da) =

∫ ∞

0

[

√

ξ2 − κ2b

jκbζ +√

ξ2 − κ2b

]

Jm(ξ)Jn(rξ) sin(rdbξ)r−

12 ξ−1dξ (4.45f)

Gha(m,n, κa) =

∫ ∞

0

[

1

jκaζ +√

ξ2 − κ2a

]

Jm(ξ)Jn(ξ)ξ−1dξ (4.45g)

Ghb (m,n, κb) =

∫ ∞

0

[

1

jκbζ +√

ξ2 − κ2b

]

Jm(ξ)Jn(ξ)ξ−1dξ (4.45h)

Ghca(m,n, db) =

∫ ∞

0

[

1

jκaζ +√

ξ2 − κ2a

]

Jm(ξ)Jn(qξ) cos(qdbξ)q−

12 ξ−1dξ (4.45i)

Ghcb(m,n, da) =

∫ ∞

0

[

1

jκbζ +√

ξ2 − κ2b

]

Jm(ξ)Jn(rξ) cos(rdaξ)r−12 ξ−1dξ (4.45j)

Ghsa(m,n, db) =

∫ ∞

0

[

1

jκaζ +√

ξ2 − κ2a

]

Jm(ξ)Jn(qξ) sin(qdbξ)r−

12 ξ−1dξ (4.45k)

75

Electromagnetic Diffraction from Slit(s):

Ghsb(m,n, da) =

∫ ∞

0

[

1

jκbζ +√

ξ2 − κ2a

]

Jm(ξ)Jn(rξ) sin(rdaξ)r−12 ξ−1dξ (4.45l)

The expansion coefficients for the zeroth order solution can be obtained from the expressions

[

Am + Cm

]0

= − 2ζ sinφ0

ζ + sin φ0

[

Kha

(

2m +1

2, 2n +

1

2;κa

)

]−1[

J2n+ 12(κh

a cosφ0)

(κa cos φ0)12

]

(4.46a)

[

Bm + Dm

]0

= − 2jζ sinφ0

ζ + sinφ0

[

Kha

(

2m +3

2, 2n +

3

2;κa

)

]−1[

J2n+ 32(κa cos φ0)

(κa cosφ0)12

]

(4.46b)

[

Em + Gm

]0

=2ζ sinφ0

ζ + sin φ0

[

Khb

(

2m +1

2, 2n +

1

2;κb

)

]−1[

J2n+ 12(κb cos φ0)

(κb cosφ0)12

]

(4.46c)

[

Fm + Hm

]0

= − 2jζ sinφ0

ζ + sinφ0

[

Khb

(

2m +3

2, 2n +

3

2;κb

)

]−1[

J2n+ 32(κb cosφ0)

(κb cos φ0)12

]

(4.46d)

[

Am − Cm

]0

= − 2 sin φ0

ζ + sin φ0

[

Gha

(

2m +1

2, 2n +

1

2;κa

)

]−1[

J2n+ 12(κa cosφ0)

(κa cos φ0)12

]

(4.46e)

[

Bm − Dm

]0

= − 2j sinφ0

ζ + sinφ0

[

Gha

(

2m +3

2, 2n +

3

2;κa

)

]−1[

J2n+ 32(κa cos φ0)

(κa cosφ0)12

]

(4.46f)

[

Em − Gm

]0

= − 2 sin φ0

ζ + sin φ0

[

Ghb

(

2m +1

2, 2n +

1

2;κb

)

]−1[

J2n+ 12(κb cos φ0)

(κb cos φ0)12

]

(4.46g)

[

Fm − Hm

]0

= − 2j sinφ0

ζ + sinφ0

[

Ghb

(

2m +3

2, 2n +

3

2;κb

)

]−1[

J2n+ 32(κb cosφ0)

(κb cos φ0)12

]

(4.46h)

And the expansion coefficients for first order interaction may be computed from the expressions

[

Am + Cm

]1

=[

Am + Cm

]0

−[

Kha

(

2m +1

2, 2n +

1

2;κa

)

]−1 [

Khcb

(

2m +1

2, 2n +

1

2; da

)

]

[

Em + Gm

]0

+

[

Kha

(

2m +1

2, 2n +

1

2;κa

)

]−1 [

Khsb

(

2m +3

2, 2n +

1

2; da

)

]

[

Fm + Hm

]0

(4.47a)

[

Bm + Dm

]1

=[

Bm + Dm

]0

−[

Kha

(

2m +3

2, 2n +

3

2;κa

)

]−1 [

Khcb

(

2m +3

2, 2n +

3

2;κa

)

]

[

Fm + Hm

]0

−[

Kha

(

2m +3

2, 2n +

3

2;κa

)

]−1 [

Khsb

(

2m +1

2, 2n +

3

2; da

)

]

[

Em + Gm

]0

(4.47b)

[

Em + Gm

]1

=[

Em + Gm

]0

−[

Khb

(

2m +1

2, 2n +

1

2;κb

)

]−1 [

Khcb

(

2m +1

2, 2n +

1

2; db

)

]

[

Am + Cm

]0

−[

Khb

(

2m +1

2, 2n +

1

2;κb

)

]−1 [

Khsb

(

2m +3

2, 2n +

1

2; db

)

]

[

Bm + Dm

]0

(4.47c)

76

Electromagnetic Diffraction from Slit(s):

[

Fm + Hm

]1

=[

Fm + Hm

]0

−[

Khb

(

2m +3

2, 2n +

3

2;κb

)

] [

Khca

(

2m +3

2, 2n +

3

2; db

)

]

[

Bm + Dm

]0

+

[

Khb

(

2m +3

2, 2n +

3

2;κb

)

] [

Khsa(2m +

1

2, 2n +

3

2; db)

]

[

Am + Cm

]0

(4.47d)

[

Am − Cm

]1

=[

Am − Cm

]0

−[

Gha

(

2m +1

2, 2n +

1

2;κa

)

]−1 [

Ghcb

(

2m +1

2, 2n +

1

2; da

)

]

[

Em − Gm

]0

+

[

Gha

(

2m +1

2, 2n +

1

2;κa

)

]−1 [

Ghsb

(

2m +3

2, 2n +

1

2; da

)

]

[

Fm − Hm

]0

(4.47e)

[

Bm − Dm

]1

=[

Bm − Dm

]0

−[

Gha

(

2m +3

2, 2n +

3

2;κa

)

]−1 [

Ghcb

(

2m +3

2, 2n +

3

2; da

)

]

[

Fm − Hm

]0

−[

Gha

(

2m +3

2, 2n +

3

2;κa

)

]−1 [

Ghsb

(

2m +1

2, 2n +

3

2; da

)

]

[

Em − Gm

]0

(4.47f)

[

Em − Gm

]1

=[

Em − Gm

]0

−[

Ghb

(

2m +1

2, 2n +

1

2;κb

)

]−1 [

Ghcb

(

2m +1

2, 2n +

1

2; db

)

]

[

Am − Cm

]0

−[

Ghb

(

2m +1

2, 2n +

1

2;κb

)

]−1 [

Ghsb(2m +

3

2, 2n +

1

2; db)

]

[

Bm − Dm

]0

(4.47g)

[

Fm − Hm

]1

=[

Fm − Hm

]0

−[

Ghb

(

2m +3

2, 2n +

3

2;κb

)

]−1 [

Ghca

(

2m +3

2, 2n +

3

2; db

)

]

[

Bm − Dm

]0

+

[

Ghb

(

2m +3

2, 2n +

3

2;κb

)

]−1 [

Ghsa

(

2m +1

2, 2n +

3

2; db

)

]

[

Am − Cm

]0

(4.47h)

These solutions have the same physical interpretations as in the case of E-polarized field.

4.3.3 Numerical Results and Discussions

In order to obtain the diffracted fields using equation (4.38), one must determine the expansion coefficients

Am, Bm, Em and Fm by solving the simultaneous equations (4.36) and (4.37). In the numerical computations,

the values of expansion coefficients are computed such that interactions up to third order are included but

interactions up to any order may be considered by iteration. It is seen that the contribution from next

higher order interactions are not appreciable. The diffracted field are determined for the upper half space

for some interesting parameters e.g angle of incidence φ0, kd and surface impedance (material properties)

of the plane. Fig. 4.11 and fig. 4.12 show the same. It is noted that as the angle of incidence is varied,

the amplitude of the diffracted fields as well as the position of the corresponding main lobe changes. The

increase in angle of incidence increases the amplitude of the diffracted field. Similarly with the increase of

77

Electromagnetic Diffraction from Slit(s):

spacing between the slits the positions of the main lobes remains approximately the same but the side lobes

start to appear. Fig. 4.13 gives the dependency of diffracted field on the material properties of the plane. It

is easily observable that the amplitude of the field decreases as the value of surface impedance ζ is increased.

4.3.4 Conclusion

In this problem, mathematical expressions are derived for the fields diffracted from two parallel slits in an

impedance plane. The selection of the problem aims at to how much extent KP method facilitates to study

the interactions between the two objects. During the formulation, two local coordinate systems, with origin

at the center of each slit, were introduced. The coupling terms, in matrix equations, were involved by making

use of addition theorem of trigonometric functions. Matrix equations were written in a form which made

it possible to study multiple diffractions between the slits. Numerical results are given by including up to

third order interactions because next higher interactions contribute negligibly. Dependence of the diffracted

fields on distance between the slits, angle of incidence and the material of the plane is presented.

0 20 40 60 80 100 120 140 160 1800.0

0.2

0.4

0.6

0.8

1.0

1.2

Diff

ract

ed fi

eld

Observation angle

0= 60

0= 70

0= 80

0= 90

a b

kd

Figure 4.11 Field patterns as a function of angle of incidence

78

Electromagnetic Diffraction from Slit(s):

0 20 40 60 80 100 120 140 160 1800.0

0.2

0.4

0.6

0.8

1.0

1.2

DIF

FRA

CTE

D F

IELD

S

OBSERVATION ANGLE

kd= 3.0 kd= 4.0 kd= 5.0

a=

b

Figure 4.12 Variation of field patterns with the spacing between the slits

0 20 40 60 80 100 120 140 160 1800.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

FAR

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

a=

b

kd=4.0

Fig. 4.13 Variations of diffracted field with the impedance of planeFigure 4.13 Field patterns as a function of ζ (impedance of plane)

79

Electromagnetic Diffraction from Slit(s):

4.4 Diffraction of Electromagnetic Plane Wave from a Slit In a PEMC

Plane

4.4.1 Formulation and solution of the problem

A. E-Polarization

The geometry of the problem is shown in Fig. 4.14. Electromagnetic plane wave is incident upon the slit

in PEMC plane of negligible thickness. The width of the slit is 2a. If φ0 is the angle of incidence, then the

incident field Eiz, co-polarized component Es

z and cross-polarized component Hsz of the scattered field can

be written as

Eiz = exp

[

jk(x cos φ0 + y sinφ0)]

(4.48a)

Esz =

∫ ∞

0

{

fe(ξ) cos (xaξ) + ge(ξ) sin (xaξ)}

exp[

−√

ξ2 − κ2ya

]

dξ, y > 0 (4.48b)

Hsz =

∫ ∞

0

{

fh(ξ) cos (xaξ) + gh(ξ) sin (xaξ)}

exp[

−√

ξ2 − κ2ya

]

dξ, y > 0 (4.48c)

where κ, xa, ya and k have the same meaning as in proceeding sections. The fe,h(ξ) and ge,h(ξ) are the

weighting functions to be determined from the boundary conditions.

x= ax= - a zx

y

Fig. 4.14 Geometry of the problem

PEMC plane PEMC plane

The required boundary conditions are given by

(i) The fields are continuous at |xa| ≤ 1 and y = 0

(ii) Hsx + MEs

x = 0 and Hsz + MEs

z = 0 for |xa| ≥ 1 and y = 0

Applying boundary condition given in (ii)

∫ ∞

0

[

fh(ξ) + Mfe(ξ)]

cos(xaξ) +[

gh(ξ) + Mge(ξ)]

sin(xaξ)dξ = 0 (4.49a)

∫ ∞

0

[

√

ξ2 − κ2

] [

fe(ξ) − MZ2fh(ξ)]

cos(xaξ) +[

ge(ξ) − MZ2gh(ξ)]

sin(xaξ)dξ = 0 (4.49b)

The above expressions can be used to decide the nature of weighting functions fe,h(ξ) and ge,h(ξ). Therefore

making using of the discontinuous properties of Weber-Schafheitlin’s integrals, as follow

fh(ξ) + Mfe(ξ) =

∞∑

m=0

AmJ2m+1(ξ)ξ−1 (4.50a)

80

Electromagnetic Diffraction from Slit(s):

gh(ξ) + Mge(ξ) =

∞∑

m=0

BmJ2m+2(ξ)ξ−1 (4.50b)

fe(ξ) − MZ2fh(ξ) =

∞∑

m=0

CmJ2m(ξ)

√

ξ2 − κ2(4.50c)

ge(ξ) − MZ2gh(ξ) =

∞∑

m=0

DmJ2m+1(ξ)√

ξ2 − κ2(4.50d)

where Jm(.) be the Bessel’s function of order m and Am, Bm, Cm and Dm are the expansion coefficients.

Manipulating the above expressions

fe(ξ) =1

1 + M2Z2

∞∑

m=0

CmJ2m(ξ)

√

ξ2 − κ2+ MZ2Am

J2m+1(ξ)

ξ(4.51a)

fh(ξ) =1

1 + M2Z2

∞∑

m=0

−MCmJ2m(ξ)

√

ξ2 − κ2+ Am

J2m+1(ξ)

ξ(4.51b)

ge(ξ) =1

1 + M2Z2

∞∑

m=0

DmJ2m+1(ξ)√

ξ2 − κ2+ MZ2Bm

J2m+2(ξ)

ξ(4.51c)

gh(ξ) =1

1 + M2Z2

∞∑

m=0

−MDmJ2m+1(ξ)√

ξ2 − κ2+ Bm

J2m+2(ξ)

ξ(4.51d)

where Z be the impedance of free space.

Using boundary condition (i), separating even and odd functions of the resultant expressions and then

expanding the trigonometric functions in terms of Jacobi’s polynomials u±

12

n (x2a) and v

±12

n (x2a) (chapter 2,

section 2.2.3), following matrix equations for the expansion coefficients are obtained

∞∑

m=0

H(

2m, 2n + 1;κ)

Cm + M2Z2K(

2m + 1, 2n + 1;κ)

Am

= jκ sin φ0Ψ(MZ)J2n+1(κ cos φ0)

(κ cos φ0)(4.52a)

∞∑

m=0

H(

2m + 1, 2n + 2;κ)

Dm + M2Z2K(

2m + 2, 2n + 2;κ)

Bm

= −κ sin φ0Ψ(MZ)J2n+2(κ cos φ0)

(κ cos φ0)(4.52b)

∞∑

m=0

M2Z2G(

2m, 2n;κ)

Cm − H(

2m + 1, 2n;κ)

Am

= −MZ2J2n(κ cos φ0) (4.52c)∞∑

m=0

M2Z2G(

2m + 1, 2n + 1;κ)

Dm − H(

2m + 2, 2n + 1;κ)

Bm

= jMZ2J2n+1(κ cos φ0) (4.52d)

n = 0, 1, 2, · · ·

where

Ψ(MZ) = 1 + M2Z2, G(α, β;κ) =

∫ ∞

0

Jα(ξ)Jβ(ξ)√

ξ2 − κ2dξ (4.53a)

H(α, β;κ) =

∫ ∞

0

Jα(ξ)Jβ(ξ)

ξdξ, K(α, β;κ) =

∫ ∞

0

√

ξ2 − κ2

ξ2Jα(ξ)Jβ(ξ)dξ (4.53b)

81

Electromagnetic Diffraction from Slit(s):

Equations (4.52) may be solved to evaluate the expansion coefficients Am, Bm, Cm, Dm. The co-polarized

component Esz and cross-polarized component Hs

z may be computed from the expressions (4.48b),(4.48c)

using the saddle point method. The final results are

Esz = C(kρ)

1

Ψ(MZ)

[

∞∑

m=0

MZ2

[

AmJ2m+1(κ cos φ) + BmJ2m+2(κ cos φ)]

tan φ

− j[

CmJ2m(κ cos φ) + DmJ2m+1(κ cos φ)]]

(4.54a)

Hsz = C(kρ)

1

Ψ(MZ)

[

∞∑

m=0

[

AmJ2m+1(κ cos φ) + BmJ2m+2(κ cos φ)]

tan φ

+ jM[

CmJ2m(κ cos φ) + DmJ2m+1(κ cos φ)]]

(4.54b)

where C(kρ) =√

π2kρ exp

[

−jkρ − j π4

]

and (ρ, φ) are the cylindrical coordinates of the observation point. A

far field in the lower region can also be derived similarly.

B. H-Polarization

The field expressions corresponding to (4.48) for H-polarization may be written as

Hiz = exp

[

jk(x cos φ0 + y sinφ0)]

(4.55a)

Esz =

∫ ∞

0

{fe(ξ) cos (xaξ) + ge(ξ) sin (xaξ)} exp[

−√

ξ2 − κ2ya

]

dξ (4.55b)

Hsz =

∫ ∞

0

{fh(ξ) cos (xaξ) + gh(ξ) sin (xaξ)} exp[

−√

ξ2 − κ2ya

]

dξ (4.55c)

where Esz is the cross component and Hs

z the co component of the scattered field for H-polarized incident

field. All the notations used in the above expressions have the same meaning as described in last section.

So applying the boundary conditions (i), (ii) and following the same procedure as in the above case, finally

following matrix equations are obtained

∞∑

m=0

∫ ∞

0

[

AmJ2m+1(ξ)J2n(ξ)

ξ− M2Cm

J2m(ξ)J2n(ξ)√

ξ2 − κ2

]

dξ = −Ψ(MZ)J2n(κ cos φ0) (4.56a)

∞∑

m=0

∫ ∞

0

[

BmJ2m+2(ξ)J2n+1(ξ)

ξ− M2Dm

J2m+1(ξ)J2n+1(ξ)√

ξ2 − κ2

]

dξ

= −jΨ(MZ)J2n+1(κ cos φ0) (4.56b)∞∑

m=0

∫ ∞

0

[

M2AmJ2m+1(ξ)J2n+1(ξ)

√

ξ2 − κ2

ξ2+ Cm

J2m(ξ)J2n+1(ξ)

ξ

]

dξ

= jκ sin φ0(MZ2)J2n+1(κ cos φ0)

(κ cos φ0)(4.56c)

∞∑

m=0

∫ ∞

0

[

M2BmJ2m+2(ξ)J2n+2(ξ)

√

ξ2 − κ2

ξ2+ Dm

J2m+1(ξ)J2n+2(ξ)

ξ

]

dξ

= −κ sin φ0(MZ2)J2n+2(κ cos φ0)

(κ cos φ0)(4.56d)

The above expressions may be expressed in more precise form as follow

∞∑

m=0

H(

2m + 1, 2n;κ)

Am − M2G(

2m, 2n;κ)

Cm = −Ψ(MZ)J2n(κ cos φ0) (4.57a)

82

Electromagnetic Diffraction from Slit(s):

∞∑

m=0

H(

2m + 2, 2n + 1;κ)

Bm − M2G(

2m + 1, 2n + 1;κ)

Dm

= −jΨ(MZ)J2n+1(κ cos φ0) (4.57b)∞∑

m=0

M2K(

2m + 1, 2n + 1;κ)

Am + H(

2m, 2n + 1;κ)

Cm

= jκ sin φ0(MZ2)J2n+1(κ cos φ0)

(κ cos φ0)(4.57c)

∞∑

m=0

M2K(

2m + 2, 2n + 2;κ)

Bm + H(

2m + 1, 2n + 2;κ)

Dm

= −κ sin φ0(MZ2)J2n+2(κ cos φ0)

(κ cos φ0)(4.57d)

n = 0, 1, 2, · · ·

The above expressions are the matrix equations and can be solved for the expansion coefficients Am, Bm,

Cm, Dm by any standard method.

Far diffracted fields for the co- and cross-polarized components are the same as that of (4.54a) and (4.54b)

but the expansion coefficients are given by (4.57) instead of (4.51)

4.4.2 Results and Discussions

The equation (4.52) and (4.57) are the matrix equations and are derived to compute the unknown coeffi-

cients Am, Bm, Cm and Dm for E- and H-polarization respectively. These equations contain the integrals

G(α, β;κ), K(α, β;κ) and H(α, β;κ) and how to compute these integrals are discussed in detail in [91] and

[105]. Once the expansion coefficients are calculated, these can be used to compute the far field patterns for

co-polarized and cross-polarized components from (4.54a) and (4.54b). Since M , the admittance parameter

is most important in our work, therefore due attention is focused on this parameter. Fig. 4.15 and Fig.

4.16 show the variations in the field patterns as a function of M for φ0 = 60◦, κ = 4 for E-polarization.

It turns out that there exist no cross-polarized component Hsz for PMC and PEC case and it dominates

for M = 1.0 and as the value of M is increased, the amplitude of the cross-component gradually decrease.

Contrary to cross-component, co-polarized component Esz increases as the value of M increases. Fig. 4.17

and Fig. 4.18 give the dependance of the field patterns on angle of incidence for E-polarization case. For

H-polarization, the dependance of field patterns on M is slightly different as is shown in Fig. 4.19 and Fig.

4.20. As the value of M increases, the amplitude of the co-component, Hsz , decreases and the behavior of

cross-component Ez is similar to that of E-polarization case.

4.4.3 Conclusion

This problem deals with the diffraction from a slit in PEMC plane. In the formulation, co- and cross-

component are supposed in terms of unknown weighting functions. These weighting functions were then

obtained from the matrix equations which developed in the course of formulation. Saddle point method is

used to find out the far fields. Variations of co- and cross-component in far region are given as a function of

admittance parameter and angle of incidence. Both E- and H-polarization are considered. It is found that

there exist no cross-component for null value of admittance parameter and as the admittance approaches

infinity for both the polarization. Co-component of both E- and H-polarization is sensitive to admittance

parameter but not much.

83

Electromagnetic Diffraction from Slit(s):

-20 0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

M= 0.0 M= 1.0 M= 3.0 M= 10.0 M= 100.0

E- polarization

Figure 4.15 Variation of co-component with admittance parameter M

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

6

7

8

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

M= 0.0 M= 1.0 M= 3.0 M= 10.0 M= 100.0

Fig. 3 variation of cross-component as a function of M

E- polarization

Figure 4.16 Variation of cross-component with admittance parameter M

84

Electromagnetic Diffraction from Slit(s):

-20 0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

8

9

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

0=90

0=80

0=70

0=60

E- polarization

Fig. 4 Effect of angle of incidence on field patterns Figure 4.17 Dependence of co-component on angle of incidence

-20 0 20 40 60 80 100 120 140 160 180 2000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

E- polarization

Fig. 5 Effect of angle of incidence on cross-componentFigure 4.18 Dependence of cross-component on angle of incidence

85

Electromagnetic Diffraction from Slit(s):

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

6

7

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

M= 0.0 M=1.0 M=10.0 M=100.0

H- polarization

Fig. 6 Dependence of co-component on M (H-polarization)Figure 4.19 Variation of co-component with admittance parameter M

-20 0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

6

7

DIF

FRA

CTE

D F

IELD

OBSERVATION ANGLE

M= 0.0 M=1.0 M= 3.0 M=10.0 M=100.0

H- polarization

Fig. 7 Dependence of cross-component on M (H-polarization)Figure 4.20 Variation of cross-component with admittance parameter M

86

Concluding Remarks:

.

CHAPTER 5

Concluding Remarks

The method of Kobayashi potential has been applied to different geometries involving impedance /PEMC

slit(s) and strips. It is found that this method gives the opportunity to investigator to incorporate the

material properties of the scatterer in the formulation. Therefore it is manageable, through this method,

to study the dependence of diffracted field on the materials of the object and the media surrounding the

geometry as well. This method also has the capability to handle the multiple diffractions from two or more

slits/ strips. Matrix equations, emerged in the problem, can be rewritten in a form which is suitable for

iterations.

It is noted that the formulation involve considerably large number of unknown weighting functions and

hence matrix equations for the problems involving non PEC objects as compare to the corresponding prob-

lems involving PEC objects, which make the analysis/ numerical work somewhat cumbersome. The integrals

involved are very complex in nature. The evaluation of these integrals through usual methods is very difficult.

Therefore, one has to develop an effective methods to compute these integrals. But above sited problems are

manageable through an experience in programming/ computational work. Therefore, this technique may be

regarded as an alternative approach to the existing analytic as well as numerical techniques in such problems.

The first problem deals with diffraction from impedance strip. The analysis involves approximately double

number of weighting functions as compared to corresponding conducting strip problem. The convergence

of the integral involved is rather slow so Hankel representation of Bessel’s function is used to overcome the

problem. If impedances of upper and lower surfaces are considered similar, then final matrix equations can

be obtained in more simplified form, which can help in economizing the computer running time and storage.

Computations are carried out for diffracted fields. The results are then compared with those of obtained

through physical optics (PO). The comparison seems appreciable. Current distributions induced on the strip

are also presented as a function of angle of incidence for both E- and H-polarization cases. The point to

note is that these are finite at the edges for both the cases.

In the second problem, diffraction has been studied from a conducting strip placed at the dielectric slab.

The problem involves the integrals which have large number of poles and branch points. Therefore, effective

88

Concluding Remarks:

methods were developed to numerically evaluate these integrals. Diffracted fields are computed and presented

as a function of angle of incidence, strip size, width of slab etc . Current distributions on the strip are also

given. It is clear from the graphs that as the strip size is increased, the position of the main lobe remains

the same but side lobes start to emerge. Similarly, with the decrease in slab width, the fields in the upper

half space intensifies.

Next, Diffraction from a PEMC strip has been investigated. It has been concluded that variation of

admittance parameter, M , do not have appreciable effect on co-component. But the dependence of the

cross-component on M is noticeable. There exist no cross-component for PEC and PMC cases and has

maximum value for M = 1.0.

Remaining three problems deal with slit(s). In the fourth problem, diffraction from a slit in an impedance

placed at the interface of two semi-infinite half spaces is analyzed. In this problem effects of the material

of the plane itself and the media surrounding the slit has been investigated on the diffracting behavior of

the slit. The results are also compared with Physical optics which shows the similar tendencies. The other

parameters of interest are the slit width and angle of incidence for which the computations are performed.

In addition, diffraction has been studied from two parallel slits in an impedance plane. It has been pointed

out there that when separation between the slits is large compared to wavelength then coupling integrals may

be replaced by their corresponding asymptotic expressions. Multiple interactions are studied and results are

given by including up to third order interactions terms. Inclusion of any order of interaction is no problem,

it is the matter of iteration only.

Finally a problem is included which deals with the diffraction from PEMC slit. Results inferred are not

very much different from those of obtained for the case of PEMC strip.

Knowing the successful application of KP method to non-PEC objects, the following research directions

may by recommended

(i) Study of diffraction from N-slits in impedance plane

(ii) Scattering from thick impedance slit

(iii) Diffraction from 3D objects like impedance rectangular plate and hole

(iv) Radiation from rectangular waveguide with non-PEC walls

(v) Extension of this method to non-PEC geometries in other co-ordinate systems like cylindrical or

spherical

89

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