Electromagnetic Diffraction Modeling and Simulation using MATLAB 9781630817794

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Electromagnetic Diffraction Modeling and Simulation with MATLAB®

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For a complete listing of titles in the Artech House Electromagnetics Series turn to the back of this book.

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Electromagnetic Diffraction Modeling and Simulation with MATLAB® Gökhan Apaydin Levent Sevgi

artechhouse.com

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Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Cover design by by Andy Meaden meadencreative.com ISBN 13: 978-1-63081-779-4 © 2021 ARTECH HOUSE 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.   All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. 10 9 8 7 6 5 4 3 2 1

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Contents Preface ix Acknowledgments xi Useful MATLAB Scripts xiii

1

Introduction

1

1.1  Maxwell Equations 1 1.1.1  Two-Dimensional Maxwell Equations in a Cartesian Coordinate System 3 1.2  Wave Equation 6 1.3  Boundary Conditions 7 1.4  Green’s Function Problem 8 1.5  Scattered Fields, Diffracted Fields, Fringe Fields 8 1.6  EM Wave-Object Interaction: Radar Cross Section 9 1.7  Problem Modeling 10 1.8 Summary 11 References 12

2

Two-Dimensional Canonical Wedge Problem

13

2.1  Introduction and 2-D Wedge Structure 2.1.1  Source Placement 2.2  Exact Solution and High-Frequency Asymptotics 2.2.1  Geometric Optics 2.2.2  Physical Optics 2.2.3  Physical Theory of Diffraction 2.2.4  Geometric Theory of Diffraction 2.2.5  Uniform Theory of Diffraction 2.2.6  Brief Summary and Conclusions 2.3  Numerical Methods 2.3.1  Method of Moments

13 14 16 21 24 26 30 31 31 32 32

v

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vi Contents

2.3.2  Finite Difference Time Domain 56 63 2.4 Summary References 64

3

Two-Dimensional Canonical Strip Problem

67

3.1 Introduction 67 68 3.1.1  The Strip Problem 70 3.2  Physical Theory of Diffraction 70 3.2.1  Backscattering by a Soft Strip (SBC) 3.2.2  Backscattering by a Hard Strip (HBC) 70 70 3.2.3  Backscattering by a Soft-Hard Strip (SHBC) 3.3  Method of Moments 71 72 3.3.1  Backscattering by a Soft Strip (SBC) 3.3.2  Backscattering by a Hard Strip (HBC) 72 73 3.3.3  Backscattering by a Soft-Hard Strip (SHBC) 3.4  Finite Difference Time Domain 81 86 3.5 Summary References 86

4

Two-Dimensional Canonical Triangular Cylinder ­Problem

89

4.1 Introduction 89 91 4.2  Physical Theory of Diffraction 4.3 Summary 98 References 99

5

Diffraction at a Rectangular Plate

101

5.1 Introduction 101 5.2  Scattering at Leading and Trailing Edges 102 5.3  Scattering at Side Edges 110 5.4  Backscattering in the Direction Normal to the Plate 112 5.5  Numerical Results 113 5.6 Summary 119 References 120 Appendix 5A: Functions B1,2,3 121

6

Diffraction with Rounded Edges

123

6.1 Introduction 123 6.2  Wedge with Rounded Edges 123 6.3  Trilateral Cylinder with Rounded Edges 136 6.4 Summary 147 References 147

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Contents vii

7

Double-Tip Diffraction Modeling

151

7.1 Introduction 151 152 7.2  Double-Tip Diffraction Structure 154 7.3  FDTD-Based Diffraction Modeling 7.4  MoM-Based Diffraction Modeling 156 7.5  Examples and Comparisons 159 164 7.6 Summary References 165

8

WedgeGUI Virtual Package

169

8.1 Introduction 169 170 8.2  WedgeGUI Software 8.3  Characteristic Examples 171 176 8.4 Summary References 176

9

FringeGUI Virtual Package

177

9.1 Introduction 177 178 9.2  FRINGEGUI Software 9.3  Characteristic Examples 181 189 9.4 Summary References 189

10

WedgeTOOL Virtual Package

191

10.1 Introduction 191 191 10.1.1  Wedge Diffraction Modeling 10.1.2  Finite Difference Time Domain Modeling 192 193 10.1.3  Method of Moments Modeling 10.2  WedgeTOOL Software 193 195 10.3  Characteristic Examples 10.4 Summary 197 References 197

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Selected Bibliography

199

List of Acronyms

201

About the Authors

203

Index

205

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Preface

W

aves, either electromagnetic (EM) or acoustic (AC), interact with objects and scatter. The addition of incident and scattered fields form the total fields. Scattered fields include (but are not limited to) reflections, refractions, and diffractions. Scattering has gained much attention because of the increase of wireless communication in our environment. For example, recent iPhone devices deliver gigabit-class long-term evolution for superfast download speeds. The next 5G wireless network technologies are serving more devices than ever and the number of antennas has increased to satisfy the connections; therefore, the behavior of electromagnetic propagation in our world has to be considered. As well, the Internet of Things (IoT) has gained great significance in our daily life, and 23 billion IoT devices are expected to be used in the world by 2023. The sensors that operate with low power and need to send their data within a limited time frame make up a vast majority of them and the success of transmission is crucial to ensure reliable operation. Hence, a proper understanding of electromagnetic propagation is required to set up indoor and outdoor low-power wireless networks. Scattering problems are complex, and therefore modeling and simulation have played a significant role in the design of new products before production. The solution strategies should be grouped as measurements, analytical modeling, and numerical simulations. Measurement is not easy to make in electromagnetics, and in most cases, it is also time-consuming and expensive. On the other hand, there is a limited number of analytical solutions for highly idealized engineering problems and therefore numerical simulations are mostly preferred for many real-life engineering electromagnetic problems. Diffraction has also attracted considerable attention in antennas and scattering problems. Its critical usage in the design of the F-117 and B-2 stealth aircraft demonstrates its significance. ix

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x Preface

This book is written for electrical, electronics, communication, and computer engineers in industry (e.g., antenna, radar, signal processing, GSM) as well as for university students, researchers, and professors, and for anyone who has to deal with real-life complex physical problems. The goal is to discuss recent publications of the authors about the fundamentals of diffraction of two-dimensional canonical structures especially on the wedge, strip, and triangular cylinder with different boundary conditions. Analytical and different numerical methods are used to show diffraction. The book also introduces some simple MATLAB scripts for several well-known electromagnetic diffraction problems. This book is a companion to and extension of the book Fundamentals of the Physical Theory of Diffraction by Pyotr Ya Ufimtsev in 2014 and presents the application of diffraction with canonical structures using analytical and numerical methods. The powerful numerical methods are given with several scenarios. Chapter 1 introduces some fundamental concepts of electromagnetic problems, identities, and definitions for diffraction modeling. Basic coordinate systems, Maxwell equations, boundary conditions, wave equation, and Green’s function problem are given. Scattered fields, diffracted fields, and fringe fields, and radar cross section for diffraction modeling are presented. Chapter 2 presents the behavior of electromagnetic waves around the two-dimensional canonical wedge. Chapter 3 presents the behavior of electromagnetic waves around the two-dimensional canonical strip. The behavior of electromagnetic waves around the two-dimensional canonical triangular cylinder is presented in Chapter 4. Chapter 5 presents the diffraction at a rectangular plate. The diffraction of trilateral cylinders and wedges with rounded edges is investigated in Chapter 6. In Chapter 7, the double-tip diffraction using the finite difference time domain and method of moments is discussed. Chapter 8 presents a MATLAB®-based virtual tool of diffraction from a perfectly reflecting wedge and Chapter 9 presents a MATLABbased virtual tool developed in MATLAB with a graphical user interface (GUI) for the visualization of both fringe currents and fringe waves. Chapter 10 presents a MATLAB-based electromagnetic wedge diffraction virtual tool. The virtual tool uses numerical FDTD and MoM algorithms and a high-frequency asymptotics approach. This book uses a MATLAB license under the MathWorks Book Program in developing book materials.

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Acknowledgments

W

e express our gratitude to Professor Pyotr Ya Ufimtsev. Most of the material used in this book were derived from studies carried out in collaboration with him. We would like to thank Feray Hacivelioglu and M. Alper Uslu for their valuable contributions in the preparation of some of the virtual tools.

xi

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Useful MATLAB Scripts Name: Wedge_GO_GTD.m Purpose: To compute GO and GTD diffracted and total fields around a 2D PEC Wedge. Name: Wedge_PO_PTD.m Purpose: To compute PO and PTD diffracted and total fields around a 2D PEC Wedge. Name: Wedge_GTD_UTD.m Purpose: To compute GTD and UTD diffracted and total fields around a 2D PEC Wedge. Name: RCS_Strip_SBC.m Purpose: To compute backscatter RCS of a PEC strip diffracted and total fields around a 2D PEC strip with Soft BC. Name: RCS_Strip_HBC.m Purpose: To compute backscatter RCS of a PEC strip of a 2D PEC strip with Hard BC. Name: RCS_Strip_SHBC.m Purpose: To compute backscatter RCS of a PEC strip of a 2D PEC strip with one face Soft other Hard BC. Name: RCS_Strip_MOM.m Purpose: To compute backscatter RCS of a PEC strip of a 2D PEC strip with Soft BC computed via Method of Moments. Name: RCS_TRIANGLE_PTD.m Purpose: To compute backscatter RCS of a 2D PEC Triangular Cylinder with Soft BC via PTD. Name: RCS_TRIANGLE_MOM.m Purpose: To compute backscatter RCS of a 2D PEC Triangular Cylinder with Soft BC via MOM. Name: RCS_PLATE_PTD.m Purpose: To compute backscatter RCS of a 2D PEC Plate with Soft BC via PTD. Name: RCS_PLATE_PO.m Purpose: To compute backscatter RCS of a 2D PEC Plate with Soft BC via PO.

xiii

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xiv

Useful MATLAB Scripts

Name: RoundWedge_Fringe.m Purpose: To compute diffracted and total fields around a 2D PEC rounded wedge with Soft BC via PTD. Name: BiRCS_RoundTRIANGLE_PTD.m Purpose: To compute bi-static RCS of a 2D triangular cylinder with Soft BC via PTD. Name: BiRCS_RoundTRIANGLE_MOM.m Purpose: To compute bi-static RCS of a 2D triangular cylinder with Soft BC via MOM. Name: DoubleTip_Diff_FDTD.m Purpose: To compute double tip diffraction using FDTD method. Name: DoubleTip_Diff_MOM.m Purpose: To compute double tip diffraction using MOM.

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CHAPTER

1 Contents 1.1  Maxwell Equations 1.2  Wave Equation 1.3 Boundary Conditions 1.4  Green’s Function Problem 1.5  Scattered Fields, Diffracted Fields, Fringe Fields

Introduction

T

his chapter presents fundamental electromagnetic (EM) concepts, identities, and definitions for diffraction modeling. Basic coordinate systems, Maxwell equations, boundary conditions, wave equation, and the Green’s function problem are given. The scattered fields, diffracted fields and fringe fields, and radar cross section for diffraction modeling are presented.

1.1  Maxwell Equations

1.7  Problem Modeling

The Maxwell equations are four differential equations that show the classical properties of electromagnetic fields by using electric and magnetic fields. The equations in the differential form are related to [1]

1.8 Summary

Gauss’s law:

1.6  EM Wave-Object Interaction: Radar Cross Section



! ∇ ⋅ D = r (1.1)



! ∇ ⋅ B = 0 (1.2)

where ρ is the electric volume charge density ! ! in C/m3, and D and B show the electric and magnetic flux density vectors in C/m 2 and Wb/ m 2, respectively. Faraday’s law:



! ! ∂B ∇×E = − (1.3) ∂t 1

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2 Introduction

Maxwell-Ampere’s law: ! ! ! ∂D (1.4) ∇×H = J + ∂t



! ! where E and H show the electric and magnetic field intensity vectors in ! V/m and A/m, respectively, and J is the electric current density vector in A/m 2. The continuity relation:



! ∂r ∇⋅J = (1.5) ∂t

Here, the vectors using arrows represent the fields in the time domain. Equations (1.1) and (1.2) of Gauss’s law and (1.5) are related to the divergence of vectors, and the others (1.3) and (1.4) are related to the curl operation of vectors. The divergence of a vector is the net flow out per unit volume for ! A = Ax xˆ + Ay yˆ + Az zˆ   in Cartesian coordinates,



! ∂Ax ∂Ay ∂Az (1.6) ∇⋅A = + + ∂x ∂y ∂z

! A = Ar rˆ + Aj jˆ + Az zˆ in cylindrical coordinates,

( )



! 1 ∂(rAr ) 1 ∂ Aj ∂A ∇⋅A = + + z (1.7) ∂z r ∂r r ∂j

! A = AR Rˆ + Aq qˆ + Aj jˆ in spherical coordinates,

( )



2 ! 1 ∂( R AR ) 1 ∂( Aq sin q ) 1 ∂ Aj ∇⋅A = 2 + + (1.8) ∂R ∂q R sin q R sin q ∂j R

The curl of a vector is the degree of its rotation for ! A = Ax xˆ + Ay yˆ + Az zˆ in Cartesian coordinates,



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⎛ ∂Ay ∂Ax ⎞ ∂Ay ⎞ ! ⎛ ∂A ⎛ ∂A ∂A ⎞ ∇×A =⎜ z − − ⎟ xˆ + ⎜ x − z ⎟ yˆ + ⎜ ⎟ zˆ (1.9) ∂z ⎠ ∂x ⎠ ∂y ⎠ ⎝ ∂z ⎝ ∂y ⎝ ∂x

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1.1  Maxwell Equations 3

! A = Ar rˆ + Aj jˆ + Az zˆ in cylindrical coordinates,

( )



⎛ ! ⎛ 1 ∂Az ∂Aj ⎞ ⎛ ∂A ∂A ⎞ ∂A ⎞ 1 ∂ rAj ∇×A =⎜ − − r ⎟ zˆ (1.10) ⎟ rˆ + ⎜ r − z ⎟ jˆ + ⎜⎜ ∂z ⎠ ∂r ⎠ ∂r ∂j ⎟ r ⎝ ∂z ⎝ r ∂j ⎝ ⎠

! A = AR Rˆ + Aq qˆ + Aj jˆ in spherical coordinates, ! ∇×A =

(

)

(

⎛ ⎛ ∂A ⎞ 1 ⎜ ∂ sin qAj 1 1 ∂AR ∂ RAj − q ⎟ Rˆ + ⎜ − ∂q ∂j ⎟ ∂R R sin q ⎜ R ⎜ sin q ∂j ⎝ ⎠ ⎝

) ⎞⎟ qˆ + 1 ⎛⎜ ∂(RA ) − ∂A ⎟ ⎠

R ⎜⎝ ∂R

R

⎞ ⎟ jˆ ∂q ⎟⎠ R

(1.11) The relations between the field vectors and the flux density vectors in a simple medium are

! ! D = eE (1.12)



! ! B = mH (1.13)



! ! J = sE (1.14)

where the constitutive parameters ε , μ , and σ denote the permittivity (dielectric constant) (F/m), the permeability (magnetic constant) (H/m), and the electric conductivity (S/m) of the medium, respectively. The permittivity and permeability model the electric and magnetic field storage capability. These parameters are independent of the magnitude and direction of the field and therefore the medium is linear and isotropic in a simple medium. The constitutive parameters are also independent of the position of the medium and do not vary with respect to the frequency that is homogeneous and nondispersive medium, respectively. These parameters for free space are



e0 ≈

10−9 (F/m), m0 = 4p ⋅ 10−7 (H/m), s0 = 0 (S/m) (1.15) 36p

1.1.1  Two-Dimensional Maxwell Equations in a Cartesian Coordinate System Two-dimensional diffraction problems can be solved by using simplified equations obtained from the longitudinal and transverse decomposition of Maxwell equations. These are considered as the transverse electric (TE)

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4 Introduction

and transverse magnetic (TM) representations under perpendicular/parallel polarizations or horizontal/vertical polarizations [1−3]. ! Using the curl operation of electric E = E x xˆ + E y yˆ + Ez zˆ and magnet! ic H = H x xˆ + H y yˆ + H z zˆ fields in (1.3) and (1.4) in a Cartesian coordinate system



⎛ ∂Ez ∂E y ⎞ ⎛ ∂E y ∂E x ⎞ ⎛ ∂E ∂E ⎞ − − ⎜ ⎟ xˆ + ⎜ x − z ⎟ yˆ + ⎜ ⎟ zˆ ∂z ⎠ ∂x ⎠ ∂y ⎠ ⎝ ∂z ⎝ ∂y ⎝ ∂x (1.16) ∂H y ∂H z ∂H x ˆ ˆ ˆ = −m x−m y−m z ∂t ∂t ∂t



⎛ ∂H z ∂H y ⎞ ⎛ ∂H y ∂H x ⎞ ⎛ ∂H ∂H z ⎞ − − ⎜ ⎟ xˆ + ⎜ x − ⎟ zˆ ⎟ yˆ + ⎜ ∂z ⎠ ∂x ⎠ ∂y ⎠ ⎝ ∂z ⎝ ∂y ⎝ ∂x (1.17) ⎛ ∂E y ⎞ ⎛ ∂Ez ⎞ ⎛ ∂E x ⎞ = ⎜ J x + e ⎟ xˆ + ⎜ J y + e ⎟ yˆ + ⎜ J z + e ⎟ ∂t ⎠ ∂t ⎠ ∂t ⎠ ⎝ ⎝ ⎝

TE/TM polarizations are defined by assuming no electric/magnetic field component in the direction of propagation. Using the plane of incidence as the plane containing the normal to the boundary surface, the electric field is either perpendicular to the plane of incidence for perpendicular polarization or parallel to the plane of incidence for parallel polarization. If x-y is taken as the plane of incidence (see Figure 1.1), the governing equations are TE polarization:



∂Ez ∂H = −m x , ∂y ∂t

∂H y ∂Ez = m , ∂x ∂t

∂H y ∂H x ∂E − = J z + e z (1.18) ∂x ∂y ∂t

TM polarization:



∂H z ∂E = Jx + e x , ∂y ∂t

∂E ∂H z = −J y − e y , ∂x ∂t

∂E y ∂E x ∂H − = −m z (1.19) ∂x ∂y ∂t

If y-z is taken as the plane of incidence, the governing equations are TE polarization:



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∂H y ∂E x = −m , ∂z ∂t

∂H ∂E x = m z, ∂y ∂t

∂H z ∂H y ∂E − = J x + e x (1.20) ∂y ∂z ∂t

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1.1  Maxwell Equations 5

TM polarization:



∂E ∂H x = Jy + e y , ∂z ∂t

∂E ∂H x = −J z − e z , ∂y ∂t

∂Ez ∂E y ∂H − = −m x (1.21) ∂y ∂z ∂t

If x-z is taken as the plane of incidence, the governing equations are TE polarization:



∂E y ∂H = m x, ∂z ∂t

∂E y ∂H = −m z , ∂x ∂t

∂E ∂H x ∂H z − = J y + e y (1.22) ∂z ∂x ∂t

Figure 1.1  Perpendicular (TE) polarization and parallel (TM) polarization on the x-y plane.

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6 Introduction

TM polarization: ∂H y ∂E = −J x − e x , ∂z ∂t



∂H y ∂E = Jz + e z , ∂x ∂t

∂H y ∂E x ∂Ez (1.23) − = −m ∂z ∂x ∂t

1.2  Wave Equation The wave equations for the electric and magnetic fields can be obtained by using Maxwell equations. Table 1.1 shows the steps to obtain wave equations in a linear, homogeneous, isotropic, source-free, lossless medium. If the electric and magnetic fields are to be time-harmonic with the time-dependence e−iωt, the wave equations !in a linear, homogeneous, isotropic medium for electric/magnetic fields y = ψ e−iωt can be written as



! ! ∂2 y ∇ y − me 2 = 0 (in the time domain) (1.24) ∂t



∇2y + w 2 mey = 0 (in the frequency domain) (1.25)



∇2y + k 2y = 0 (1.26)

2

where k = w me = 2p/l is the wavenumber, λ is the wavelength, ω is the angular frequency, ψ shows the time-harmonic electric or magnetic field in

Table 1.1 Steps to Obtain Homogenous Wave Equations ! The Wave Equation for E ! Take the curl of ∇ × E and use (1.3) ! ! ∂ ∇ × ∇ × E = −m ( ∇ × H ) ∂t

! The Wave Equation for H ! Take the curl of ∇ × H and use (1.4) ! ! ∂ ∇ × ∇ × H = e (∇ × E ) ∂t

Use vector identity, Ampere’s law, and Gauss’s law ! ! ∂ ⎛ ∂E ⎞ ∇ × ∇ × E = −m ⎜ e ⎟ ∂t ⎝ ∂t ⎠ ! ! ! ∂2 E ∇ ( ∇ ⋅ E ) − ∇2E = −me 2 ∂t ! ! ∂2 E 2 ∇ E − me 2 = 0 ∂t

Use vector identity, Faraday’s law, and Gauss’s law ! ! ∂ ⎛ ∂H ⎞ ∇ × ∇ × H = e ⎜−m ⎟ ∂t ⎠ ∂t ⎝ ! 2 ! ! ∂ H ∇ ( ∇ ⋅ H ) − ∇2H = −me 2 ∂t ! ! ∂2 H 2 ∇ H − me 2 = 0 ∂t

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1.3  Boundary Conditions 7

the frequency domain, and the Laplace operator ∇2 is the divergence of the gradient of the field. Note that (1.26) becomes an inhomogeneous equation by adding source term I on the right side as ∇2y + k 2y = I (1.27)



Equation (1.27) can be also written in Cartesian coordinates:



∂2 u ∂2 u ∂2 u + + + k 2u = I (1.28) ∂x 2 ∂y 2 ∂z 2

in cylindrical coordinates:



1 ∂ ⎛ ∂u ⎞ 1 ∂2 u ∂2 u + + k 2u = I (1.29) ⎜r ⎟ + r ∂r ⎝ ∂r ⎠ r 2 ∂j 2 ∂z 2

and in spherical coordinates: 1 ∂ ⎛ 2 ∂u ⎞ ∂ ⎛ ∂u ⎞ ∂2 u 1 1 + k 2u = I (1.30) ⎜R ⎟+ 2 ⎜sin q ⎟ + 2 2 2 ∂R ⎝ ∂R ⎠ R sin q ∂q ⎝ ∂q ⎠ R sin q ∂j 2 R    where u shows the component of electric/magnetic field vectors. For example: u = Ez for the z-component of electric fields in cartesian coordinate system; u = Hr for the r-component of magnetic fields in cylindrical coordinate system.

1.3  Boundary Conditions Let u be the total field that is the sum of incident (uinc) and scattered (usc) waves, nˆ be the outward unit vector normal to the surface; two types of boundary conditions are used on the perfectly conducting surface. The boundary conditions are either Dirichlet (u = 0) or Neumann ( nˆ ⋅ ∇u = ∂u/∂n = 0) boundary conditions. In the case of acoustic waves, Dirichlet conditions describe the soft (pressure-release) surfaces and Neumann conditions describe the hard (rigid) surfaces. In the case of electromagnetic waves, Dirichlet (Neumann) conditions relate to the waves with electric (magnetic) vector parallel to the edge (z-axis).

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8 Introduction

The Sommerfeld’s radiation condition satisfies



⎛ ∂u ⎞ lim kr ⎜ − iku ⎟ = 0 (1.31) r→∞ ⎝ ∂r ⎠

as the distance (r) from the scattering object goes to infinity.

1.4  Green’s Function Problem The electric/magnetic fields satisfy the homogeneous wave equation as ∇2u ( r ) + k 2u ( r )  = 0 (1.32)



The scalar Green’s function represents the field radiated from a source point using the Dirac delta function as [4−9]

∇2G ( r , r ′) + k 2G ( r , r ′) = −d ( r − r ′) (1.33)

where G(r,r′) is the scalar Green’s function and the source is represented with the Dirac delta δ (⋅); r′ and r show the position of the source and observer, respectively. The total field is obtained for r′ ∈ C and r ∈ V as u (r ) = uinc ( r ) +

⎛

∫ ⎜⎝u (r ′)

C

⎞ ∂G ( r , r ′) ∂u ( r ′) − G ( r , r ′) ⎟ dr ′ (1.34) ∂n ∂n ⎠

1.5  Scattered Fields, Diffracted Fields, Fringe Fields Electromagnetic waves interact with objects and scatter. The word scattering includes reflection, refraction, and diffraction. For the sake of clear understanding and completeness, we include the definitions of the total, scattered, diffracted, and fringe fields. The total field is the addition of the incident and scattered fields. In other words, to obtain the scattered field, one needs to extract/subtract the incident field from the total field. The subtraction of the incident and reflected fields from the total field yields the diffracted field. In other words, the diffracted field is equal to the scattered field minus the reflected field. The fringe field is the part of the diffracted field generated by the source-induced fringe (nonuniform) currents. These currents exist because of any deviations of a scattering surface from a tangential plane [10].

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1.6  EM Wave-Object Interaction: Radar Cross Section 9

The scattered field over the scattering surface S can be represented [10] as: usc =

1 4p

⎛ ∂ ⎛ e ikr ⎞ ∂u e ikr ⎞ ⎟ ds (1.35) ⎟− r ⎠ ∂n r ⎠

∫ ⎜⎝u ∂n ⎜⎝ S

Here, the currents for the soft and hard surface in terms of total fields are js =



∂u (1.36) ∂n

jh = u (1.37)



1.6  EM Wave-Object Interaction: Radar Cross Section The scattering of an interacted object is usually represented by its radar cross section (RCS) for a two-dimensional target [1]: 2



s2D

usc = lim 2pr inc (1.38) r→∞ u

and for a three-dimensional target: 2



s3D = lim 4pR2 R→∞

usc (1.39) uinc

The normalized scattering cross section is calculated Φ (j,j0 ) s = 2 = (1.40) kl kl 2



s norm

with a target of length l and directivity patterns Φ(φ ,φ 0) obtained from



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u = Φ (j,j0 )

ei(kr+p/4 ) (1.41) 2pkr

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10 Introduction

1.7  Problem Modeling The word scattering is used to represent all components produced from the interaction of electromagnetic waves with objects and it includes mainly reflection, refraction, and diffraction. One reason for diffraction is a sharp boundary discontinuity such as an edge and/or a tip. In this book, the goal is to discuss the fundamentals of diffraction for two-dimensional canonical structures especially on wedge, strip, rectangular, and triangular cylinder with different boundary conditions. Figure 1.2 shows the geometry of the problem on the xy-plane. The structure is infinite along the z-axis. Using polar coordinates, a line source at (r0,φ 0) excites the structure. Note that the interacted structure can be illuminated by a plane wave when r0 → ∞. Analytical high-frequency techniques, such as geometric optics (GO), physical optics (PO), geometric theory of diffraction (GTD), uniform theory of diffraction (UTD), and physical theory of diffraction (PTD) can be used when the wavelength is small compared with the interacted object size. Various numerical methods, such as the method of moments (MoM), finite element method (FEM), and finite difference time domain (FDTD), are also used to show the surface currents, diffracted fields, scattered fields, fringe fields, and total fields. Figure 1.3 shows the total and diffracted magnetic fields at r = 5λ around the hard wedge using a line source at r0 = 10λ and (φ 0 = 50°). The polar fields using exact series summation and MoM are compared (see Chapter 2 for details). Figure 1.4 shows the total and scattered electric fields around the soft strip. A plane wave is coming from the −x-direction (φ 0 = 180°). Here, bright and dark regions represent the field variation around the strip (see Chapter 3 for details).

Figure 1.2  Problem geometry.

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1.8 Summary 11

Figure 1.3  Wedge scattering: (left) total fields versus angle, and (right) diffracted fields versus angle around the hard wedge.

Figure 1.4  Strip scattering: (left) total electric fields, and (right) scattered electric fields around the soft strip.

1.8 Summary Diffraction is critical in understanding EM wave phenomena under different geometrical (i.e., shape and boundary) conditions. It is necessary in different engineering problems such as antenna and radiation, wave propagation, radar and radar cross section modelling, electromagnetic compatibility, etc. EM diffraction modelling has long been investigated in EM community and methods/techniques such as geometric optics (GO), geometric theory of diffraction (GTD), uniform theory of diffraction (UTD), physical optics (PO),

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12 Introduction

physical theory of diffraction (PTD), parabolic equation method (PEM), etc., have been introduced. These are also called high frequency asymptotics (HFA) because all are valid under the frequency [Hz] goes to infinity (i.e., the wavelength [m] tends to zero). In addition, as will be shown in the following chapters, time and frequency domain numerical methods such as finite difference time domain (FDTD), method of moments (MoM), etc., can also be used in diffraction modelling.

References [1]

Balanis, C., Advanced Engineering Electromagnetics, Second Edition, New York: Wiley Press, 2012. [2] Sevgi, L., Complex Electromagnetic Problems and Numerical Simulation Approaches, Piscataway, NJ: IEEE Press/Wiley, 2003. [3] Sevgi, L., “Guided Waves and Transverse Fields: Transverse to What?” IEEE Antennas and Propagation Magazine, Vol. 50, No. 6, 2008, pp. 221−225. [4] Fock, V. A., Electromagnetic Diffraction and Propagation Problems, Oxford, United Kingdom: Pergamon,1965. [5] Garg, R., Analytical and Computational Methods in Electromagnetics, Norwood, MA: Artech House, 2008. [6] Sadiku, M. N. O., Numerical Techniques in Electromagnetics with MATLAB, Third Edition, Boca Raton, FL: CRC Press, 2009. [7] Jin, J. M., Theory and Computation of Electromagnetic Fields, Hoboken, NJ: John Wiley, 2011. [8] Tsang, L., J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, John Wiley and Sons, 2001. [9] Bourlier, C., N. Pinel, and G. Kubiche, Method of Moments for 2D Scattering Problems, John Wiley and Sons, 2013. [10] Ufimtsev, P. Ya., Fundamentals of the Physical Theory of Diffraction, Second Edition, Hoboken, NJ: John Wiley & Sons, 2014.

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CHAPTER

2 Contents 2.1  Introduction and 2-D Wedge Structure 2.2  Exact Solution and High-Frequency Asymptotics 2.3  Numerical Methods 2.4 Summary

Two-Dimensional Canonical Wedge Problem This chapter presents the behavior of electromagnetic waves around the two-dimensional (2-D) canonical wedge [1−4].

2.1  Introduction and 2-D Wedge Structure The word scattering is used to represent all wave components produced from the interaction of electromagnetic waves with objects and include incidence, reflection, refraction, and diffraction. The analytical methods are used when the frequency is high (i.e., when the electromagnetic signal wavelength is quite low compared with the interacted object size) and these are GO, PO, GTD, UTD, and PTD [5−10]. Two-dimensional nonpenetrable wedge structure is shown in Figure 2.1. The tip of the wedge is located at the origin on the xy-plane. The semi-infinite wedge is located in free space. The polar coordinates r, φ , z are used. The z-axis is aligned along the edge of the wedge. The exterior angle of the wedge equals α . The top face of the wedge is located along the positive x-axis (φ = 0). The bottom face is located along φ = α . The structure is infinite along the z-axis. The angle φ is measured from the top face of the wedge.

13

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14

Two-Dimensional Canonical Wedge Problem

Figure 2.1  Two-dimensional wedge diffraction scenario.

The wedge is illuminated by a line source at a distance r0 from a direction φ = φ 0 in Figure 2.1. In other words, the source and observation points are given by (r0, φ 0) and (r, φ ), respectively. The receivers are located on a circle with a specified radius r around the tip of the wedge in Figure 2.1. Then, the total, scattered, diffracted, and fringe waves are calculated. 2.1.1  Source Placement According to the position of the source, either single-side illumination (SSI) or double-side illumination (DSI) can be considered in Figures 2.2 and 2.3, respectively. The first scenario (Case I) in Figure 2.2 belongs to SSI for the illumination of the top face only (0 < φ 0 < α − π ). In this case, the two-dimensional scattering plane around the wedge may be divided into three regions in terms of critical wave phenomena occurring there. The region (0 < φ < π − φ 0) includes all the field components (incident field, reflected field, and diffracted field). The critical angle (φ = π − φ 0) is called reflection shadow boundary (RSB). The region (π − φ 0 < φ < π + φ 0) contains only incident and diffracted fields. The critical angle (φ = π + φ 0) is the limiting boundary of the incident field and called incident shadow boundary (ISB). The third region (π + φ 0 < φ < α ) is the shadow region where only diffracted fields exist. The second scenario (Case II) in Figure 2.3 belongs to DSI for the illumination of both faces (α − π < φ 0 < π ). The regions (0 < φ < π − φ 0) and (2α − π − φ 0 < φ < α ) contain all the field components. The region between these two (π − φ 0 < φ < 2α − π − φ 0) contains no reflected fields and only incident and diffracted fields exist.

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2.1  Introduction and 2-D Wedge Structure 15

Figure 2.2  Single-side illumination.

Figure 2.3  Double-side illumination.

The third scenario (Case III) for the illumination of the bottom face only (π < φ 0 < α ) belongs to SSI, too. For this case, RSB at (φ = 2α − π − φ 0) and ISB at (φ = φ 0 − π ) divide into three regions. The region (2α − π − φ 0 < φ < α ) includes all field components (incident field, reflected field, and diffracted field). The second region (φ 0 − π < φ < 2α − π − φ 0) contains only incident and diffracted fields. The third region (0 < φ < φ 0 − π) is the shadow region where only diffracted fields exist.

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16

Two-Dimensional Canonical Wedge Problem

2.2  Exact Solution and High-Frequency Asymptotics The nonpenetrable wedge diffraction problem is canonical and plays a fundamental role in understanding and construction of high-frequency asymptotics (HFA) techniques as well as for the numerical tests. The exact solution to the scattering problem was first obtained by Sommerfeld [11] in the particular case of a half-plane. For a wedge with an arbitrary angle between its faces, the solution was obtained by Macdonald [12] and later by Sommerfeld who developed the method of branched wave functions. The perfectly electrical conducting (PEC) wedge in Figure 2.1 is located in a homogenous medium and illuminated by a cylindrical wave diverging from the line source at (r0, φ 0). The time dependence exp(−iω t) is considered. The field outside the wedge (0 ≤ φ ≤ α ) satisfies Helmholtz’s equation [5]:



⎛ ∂2 1 ∂ ⎞ I 1 ∂2 + 2 2 + k 2 ⎟ u = 0 d (r − r0 ) d (j − j0 ) (2.1) ⎜ 2+ ∂r r r ∂r ∂j r ⎝ ⎠

where k = 2p/l = w e0 m0 is the free-space wave number, I0 is the line current amplitude, and δ (·) is the Dirac delta functions. The wave function u represents either electric or magnetic field components according to its polarization. The case when u = Ez satisfies TM polarization and the magnetic field has x- and y-components (H x and Hy). In this case, regardless of the angle of incidence, the electric field is always parallel to the faces and edge of the wedge. The case when u = Hz satisfies TE polarization and the electric field has x- and y-components (Ex and Ey). In acoustic applications, either the field or its normal derivative should be zero on the surface and these conditions refer to acoustically soft boundary condition (SBC) and hard boundary condition (HBC) wedges, respectively. The boundary conditions (BC) on the faces (φ = 0) and (φ = α ) are

us = 0 for the soft face(s) of the wedge (2.2)



∂uh = 0 for the hard face(s) of the wedge (2.3) ∂n

and Sommerfeld’s radiation condition (SRC) at infinity is



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⎛ ∂u ⎞ lim kr ⎜ − iku ⎟ = 0 (2.4) r→∞ ⎝ ∂r ⎠

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2.2  Exact Solution and High-Frequency Asymptotics 17

For the source placement, either a line source or a plane wave excitation can be used. The Green’s function solution is exact in polar coordinates but requires an infinite number of mode summation. The exact total fields in polar coordinates using Green’s function with totally SBC and HBC are [6]

ustot

p I = ia 0

l =1

l =1

uhtot

p = I ia 0

l =0

l =0

J v ( kr ) Hv(1) ( kr0 ) sin (v l j0 ) sin (v l j ) with r l

l

J v ( kr0 ) Hv(1) ( kr ) sin (v l j0 ) sin (v l j ) with r l

l

( ) (

)

el J v ( kr ) Hv(1) kr0 cos v l j0 cos (v l j ) with r l

l

( )

(

)

el J v kr0 Hv(1) ( kr ) cos v l j0 cos (v l j ) with r l

l

r0 r0

(2.5)



r0

(2.6)



r0

Here, J v and H v(1) are Bessel and Hankel functions, respectively; l l vl = lπ /α , and ε 0 = 0.5, ε 1 = ε 2 = ε 3 = ⋯ = 1. The critical issue here is the specification of the number of terms included that increases with the frequency and/or the distance. Table 2.1 gives the MATLAB® code to calculate the total fields using (2.5) and (2.6). Figure 2.4 shows the total fields of wedges with soft and hard faces. A line source is located at (r0,  φ 0) = (100, 45°). The frequency is 15 MHz. The line source radiates the field uinc =



1 I H(1) (kr1 ) (2.7) 4i 0 0

r 2 + r02 − 2rr0 cos (j − j0 ) (2.8)

r1 =

The wedge structure can be illuminated by a plane wave when r0 → ∞, then the source with an incidence angle (φ = φ 0) illuminates the wedge as

−ikr cos(j−j0 )

uinc = u0e

−ik x cosj0 + y sin j0 ) = u0e ( (2.9)

The total field solutions of the Helmholtz’s equation using plane-wave excitation with SBC and HBC for both SSI and DSI are [6]



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tot ush = u0 ⎡⎣u (j − j0 ) ∓ u (j + j0 )⎤⎦ (2.10)

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18

Two-Dimensional Canonical Wedge Problem Table 2.1 MATLAB Code for the Total Fields of Wedge Structure Using Line Source Excitation (2.5) and (2.6)

function u=Exact_Series(k,alfa,r0,phi0,r,phi,pol) % k wave number (rad/m) % alfa exterior angle (rad) % (r0,phi0) source position % (r,phi) observer position % pol polarization 0 for SBC, 1 for HBC kr=k*r; kr0=k*r0; err=1e-200; er=10; if pol==0, % for Soft BC n=1; vn=n*pi/alfa; =(rr0)*besselj(vn,kr0)*besselh(vn,1,kr)*sin(vn*phi0)*sin  (vn*phi); while er>err, n=n+1; vn=n*pi/alfa;  un=(rr0)*besselj(vn,kr0)*besselh(vn,1,kr)*sin(vn*phi0)  *sin(vn*phi); er=abs(un)/abs(u); if isnan(un), er=0; un=0; end u=u+un; end elseif pol==1 % for Hard BC n=0; vn=n*pi/alfa; =0.5*((rr0)*besselj(vn,kr0)*besselh(vn,1,kr)*cos(vn*phi0)*cos  (vn*phi)); while er>err, n=n+1; vn=n*pi/alfa;  un=(rr0)*besselj(vn,kr0)*besselh(vn,1,kr)*cos(vn*phi0)  *cos(vn*phi); er=abs(un)/abs(u); if isnan(un), er=0; un=0; end u=u+un; end end u=pi/(1i*alfa)*u;

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2.2  Exact Solution and High-Frequency Asymptotics 19

where (−) and (+) are for SBC and HBC, respectively, and ⎛p ⎞

∞

−i⎜ ⎟v l 2p u (y ) = el e ⎝ 2 ⎠ J v (kr ) cos ( v l y ) (2.11) ∑ l a l=0



u0 =



1 I H(1) (kr0 ) (2.12) 4i 0 0

The diffracted fields can be calculated by using the Sommerfeld ray ­asymptotics for kr ≫ 1: diff us,h = u0 ⎡⎣v (j − j0 ) ∓ v (j + j0 )⎤⎦ (2.13)



where (−) and (+) are for SBC and HBC, respectively ⎛p ⎞ sin ⎜ ⎟ i⎛⎜kr+ p ⎞⎟ ⎝ n ⎠ ⎝ 4⎠ v (y ) = e np 2

∞

2

e−krs ds (2.14) ⎡ ⎛p ⎞ ⎛ y + z ⎞⎤ z −∞ ⎢cos ⎜ ⎟ − cos ⎜ ⎟⎥ cos 2 ⎝ n ⎠⎦ ⎣ ⎝n⎠

∫

with n = α /π and s=



p i z 2e 4 sin (2.15) 2

Retaining the first term in (2.14) under

kr cos (y/2) ≫ 1



⎛p ⎞ ⎛ p⎞ i⎜kr+ ⎟ sin ⎜ ⎟ e ⎝ 4⎠ ⎝n⎠ v (y ) ≈ (2.16) ⎡ ⎛p ⎞ ⎛ y ⎞⎤ 2pkr n ⎢cos ⎜ ⎟ − cos ⎜ ⎟⎥ ⎝ n ⎠⎦ ⎣ ⎝n⎠



e ⎝ 4⎠ = u0 f s,h (j,j0 ) (2.17) 2pkr

⎛ p⎞ i⎜kr+ ⎟

diff us,h

fs,h (j,j0 )

⎞ ⎛p ⎞⎛ sin ⎜ ⎟ ⎜ ⎟ 1 1 ⎝ n ⎠⎜ ⎟ (2.18) = ∓ ⎛ j − j0 ⎞ ⎛ j + j0 ⎞ ⎟ ⎛p ⎞ ⎛p ⎞ n ⎜ ⎟ cos ⎜ ⎟ − cos ⎜ ⎟⎟ ⎜ cos ⎜ ⎟ − cos ⎜ ⎝n⎠ ⎝n⎠ ⎝ n ⎠ ⎝ n ⎠⎠ ⎝

where (−) and (+) are for SBC and HBC, respectively.

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20

Two-Dimensional Canonical Wedge Problem

The functions can be also determined by [6]:



⎧ −ikr cos(j−j0 ) u (j − j0 ) = v (j − j0 ) + ⎨ e 0 ⎪⎩

0 ≤ j < p + j0 p + j0 ≤ j ≤ a





⎧ −ikr cos(j+j0 ) u (j + j0 ) = v (j + j0 ) + ⎨ e 0 ⎪⎩

0 ≤ j < p − j0 p − j0 ≤ j ≤ a



(2.19)

(2.20)

for the illumination of the top face only (SSI Case I) −ikr cos(j−j0 )

u (j − j0 ) = v (j − j0 ) + e



u (j + j0 )

⎧ −ikr cos(j+j0 ) ⎪ e = v (j + j0 ) + ⎨ 0 ⎪ −ikr cos(2a−j−j0 ) ⎩ e

(2.21)

0 ≤ j < p − j0 p − j0 ≤ j < 2a − p − j0 (2.22) 2a − p − j0 ≤ j ≤ a

for the illumination of both faces (DSI-Case II). Arbitrary impedance conditions should be also considered. For example, the impedance on one face (φ = 0) can be zero, while another face (φ = α ) is infinity [3]. In electromagnetic terms, one face is electric (with infinite electric conductivity), while another face is magnetic (with infinite magnetic conductivity). In acoustics, one face is soft, while the other is hard (as SHBC). The exact total fields in polar coordinates using Green’s function with SHBC:

tot ush

p = I ia 0

l =0

l =0

J v ( kr ) Hv(1) ( kr0 ) sin (v l j0 ) sin (v l j ) with r l

Jv

l

l

( )

kr0 Hv(1) l

( kr ) sin (v l j0 ) sin (v l j ) with r

r0 r0

(2.23)

where vl = (2l + 1)π /(2α ) and ε 0 = 0.5, ε 1 = ε 2 = ε 3 = ⋯ = 1. The total field solution with SHBC is

tot ush = u0 ⎡⎣u (j − j0 ) − u (j + j0 )⎤⎦ (2.24)

u (y ) =

∞

⎛p ⎞

−i⎜ ⎟v l 2p e ⎝ 2 ⎠ J v (kr ) cos ( v l y ) (2.25) ∑ l a l=0

where v l = (2l + 1)p/2a .

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2.2  Exact Solution and High-Frequency Asymptotics 21

Figure 2.4  The three-dimensional gray-scale plots of total fields outside the wedge: (top) TM/SBC, (bottom) TE/HBC (α = 300, λ = 20m, r0 = 100m, φ 0 = 45°).

2.2.1  Geometric Optics GO is a ray-based approach, which models incident, reflected, and refracted fields between the source and the receiver. The GO describes a wave field in the limiting case when a wavelength tends to zero. It deals with reflection and refraction on the illuminated side of the object and the diffracted field cannot be taken into account. According to GO, the field in the shadow region behind the opaque objects equals zero, including the points on the object surface. On the illuminated side of the object, the GO field is the sum of the incident and reflected waves.

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22

Two-Dimensional Canonical Wedge Problem

The GO fields for a line source under SSI (0 < φ 0 < α − π ):

GO us,h



⎧ H(1) (kR ) ∓ H(1) (kR ) 0 ≤ j < p − j0 1 0 2 I0 ⎪ 0 (1) = ⎨ H0 (kR1 ) p − j0 ≤ j < p + j0 (2.26) 4i ⎪ 0 p + j0 ≤ j ≤ a ⎩

and under DSI (a − p < j0 < p) :

GO us,h



⎧ H(1) (kR ) ∓ H(1) (kR ) 0 ≤ j < p − j0 1 0 2 I0 ⎪ 0 = ⎨ H0(1) (kR1 ) p − j0 ≤ j < 2a − π − j0 (2.27) 4i ⎪ (1) (1) 2a − p − j0 ≤ j ≤ a ⎩ H0 (kR1 ) ∓ H0 (kR3 )

Here, signs (−) and (+) are for SBC and HBC, respectively, and

R1 =

r 2 + r02 − 2rr0 cos (j − j0 ) (2.28)



R2 =

r 2 + r02 − 2rr0 cos (j + j0 ) (2.29)

R3 =



r 2 + r02 − 2rr0 cos (2a − j − j0 ) (2.30)

H0(1)(kR1) is the incident plane wave in the illuminated region, and the terms H0(1)(kR 2) and H0(1)(kR3) are the reflected plane waves from face (φ = 0) and face (φ = α ), respectively. The GO fields for a plane wave under SSI (0 < j0 < a − p) :

GO us,h



⎧ e−ikr cos(j−j0 ) ∓ e−ikr cos(j+j0 ) ⎪⎪ −ikr cos(j−j0 ) = u0 ⎨ e ⎪ 0 ⎪⎩

0 ≤ j < p − j0 p − j0 ≤ j < p + j0 (2.31) p + j0 ≤ j ≤ a

and under DSI (a − p < j0 < p) :

GO us,h

⎧ e−ikr cos(j−j0 ) ∓ e−ikr cos(j+j0 ) ⎪⎪ −ikr cos(j−j0 ) = u0 ⎨ e ⎪ e−ikr cos(j−j0 ) ∓ e−ikr cos(2a−j−j0 ) ⎪⎩

0 ≤ j < p − j0 p − j0 ≤ j < 2a − p − j0 (2.32) 2a − p − j0 ≤ j ≤ a

Here, signs (−) and (+) are for SBC and HBC, respectively, e−ikrcos(φ−φ0) is the incident plane wave in the illuminated region, and the terms e−ikrcos(φ+φ0)

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2.2  Exact Solution and High-Frequency Asymptotics 23 Table 2.2 MATLAB Code for the GO Fields of Wedge Structure Using Plane Wave Excitation (2.31) and (2.32) if phi00 =

+

i ∂u ( r′ )x π [6,27]. Because jhfr is finite for sharp wedges

Figure 6.7  Total, PO, and PTD (fringe) surface currents of the 30° SBC wedge.

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6.2  Wedge with Rounded Edges 135

Figure 6.8  PTD (fringe) surface currents of the 30° SBC and HBC wedges.

it is not surprising that it has only slight changes in the vicinity of the junction points where the surface L is smooth and only its second derivative undergoes discontinuities. Also remember that away from the edge (kr ≫ 1) on the sharp wedge (with β = 15°) the SBC current jsfr drops as (1/kr)3/2 while the HBC current jhfr attenuates as (1/kr)1/2 [6]. The curves in Figure 6.8 for ⎪r⎪ > 3λ relate to large values (kr > 18) and qualitatively agree with those for a sharp wedge. Notice as well that at point ψ = 0 (r = 0 in Figure 6.8) on the circular cylinder alone (without L1, L2), the SBC and HBC fringe currents are determined by (14.53) and (14.54) of [6] where one should set γ = π /2 and ψ = −π /2. According to these equations, ⎪jsfr⎪ = k⎪jhfr⎪. For λ = 10m (with f = 30 MHz) taken in our calculations, this relationship means that ⎪jsfr⎪ ≈ 0.6⎪jhfr⎪ while for the rounded wedge according to Figure 6.8 we have ⎪jsfr⎪ ≈ 0.7⎪jhfr⎪. Finally, in Figure 6.9 for comparison purposes, we plot both the soft and hard fringe waves for rounded wedges with a = a2 = λ /5 at the distance ρ = a2 + 2λ . They relate to the scenario in Figure 6.2(b). Note that the curves for the wedge with β = 30° are similar to those for the sharp wedge given in Figure 4.5 of [6]. Fringe integral equations suitable for analysis of the field scattered by rounded soft and hard wedges are developed. Numeric results are obtained via a regular MoM procedure. A comparison with scattering from sharp wedges is illustrated and confirms that the rounded wedge can be considered, approximately, as the sharp wedge when the radius of rounding does not exceed one-tenth of the wavelength. The results are also important from the theoretical/methodical point of view because they demonstrate the direct extension of PTD for objects with rounded edges.

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136

Diffraction with Rounded Edges

Figure 6.9  Fringe fields around different SBC/HBC wedges (for the scenario in Figure 6.2(b)).

6.3  Trilateral Cylinder with Rounded Edges This part presents the example of the extension of the physical theory of diffraction for a trilateral cylinder with rounded edges. This approximation contains only the primary (single-diffracted) edge waves and it is validated by comparison with the solution which includes multiple diffracted edge waves. A very good agreement has been already found for objects with faces about three wavelengths long. For soft cylinders with large faces (L ≫ λ ), this PTD approach allows one to decrease the computer time by a factor L/3λ compared to the numeric solution of the surface integral equations for the total currents.

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6.3  Trilateral Cylinder with Rounded Edges 137

This part also demonstrates scattering at objects with rounded edges in which radii of curvature are less than the wavelength. However, the developed theory is also applicable for large curvature radii. We investigate diffraction at an equilateral cylinder with L1 = L2 = L3 under the boundary condition u = 0. The problem scenario is pictured in Figure 6.10. In acoustics, this condition relates to a soft object; in electromagnetics, to a perfectly conducting object and electromagnetic waves with Ez -polarization. Mathematically, the solutions of these acoustic and electromagnetic problems are identical, u(x,y) = Ez(x,y). For simplicity, we consider the symmetric scattering of the incident wave

uinc = Ezinc = exp(ikx) (6.22)

when u(x,y) = u(x,-y). Sections L01,L02,L03 are parts of the circular cylinders with radius a (see Figure 6.10). They are smoothly conjugated with the faces of the tangential wedges. Points 1, 2, and 3 denote the tips of these wedges. The angle between faces equals to 2β = π /3 = 60°; that is, β = π /6 = 30°. In a high-frequency situation, when the wavelength is very small compared to the distance between edges, one can neglect the multiple diffracted edge waves. In this case, the fringe currents jfr in the vicinity of the object edges are asymptotically identical to those on the rounded tangential wedges with the same shape of the edge and with infinite faces. We refer to them as primary (single-diffracted) currents and focus on their

Figure 6.10  Cross section of the scattering object. d = 2a cot β + L, sections L01,L02,L03 are parts of the circular cylinders with radius a, and the length of sections L1,L2,L3 is L, β = π /6.

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138

Diffraction with Rounded Edges

calculation. Denote these currents as j1fr, j2fr, j3fr for the wedges whose faces intersect each other at points 1, 2, and 3, respectively. These currents are calculated by using i p.v. ∫ j fr( x ′ , y ′)H0(1) ( kr ) dl ′ = uinc(x, y) 4 L

(6.23) i − p.v. ∫ j PO( x ′ , y ′)H0(1) ( kr ) dl ′ 4 L



Here, always L > 0 and dl′ ≥ 0. The symbol p.v.∫ means the Cauchy principal value of the integral. The second term on the right side of (6.23) represents the PO field. Equation (6.23) can be solved by the classical MoM [28]. Actually, we need to solve three integral equations associated with three tangential wedges:  The wedge with L = L01 + L1 + L2;  The wedge with L = L02 + L1 + L3;  The wedge with L = L03 + L3 + L2. where L1, L2, L3 formally are infinitely long, but during practical calculations, it was established that they can be taken finite and equal to L1 = L2 = L3 = L in the case when L ≥ 3λ . Denote the fringe currents on the tangential wedges as j1fr, j2fr, j3fr, respectively. The total fringe current on the actual object surface is the sum j fr,tot = j1fr + j2fr + j3fr (6.24)



Solving (6.23), we find the fringe currents (6.24). To test/validate this approximation we find the total surface current on the object solving the integral equation uinc ( x, y ) −

i p.v. ∫ j tot ( x ′ , y ′ ) H0(1) ( kr ) dl ′ = 0 (6.25) 4 L tot

and subtract the PO currents. We also calculate the field generated by the total currents in the far zone and consider it as the exact scattered field since it contains all scattering components including multiple diffracted waves. The comparison demonstrated below in figures shows very good agreement with the asymptotic PTD approximation already for objects with D = 3λ .

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6.3  Trilateral Cylinder with Rounded Edges 139

Note that in these figures we plot the normalized scattering cross-section σ norm defined by (5.45) in [6] with l = 2a + L. The current and field data related to the first-order PTD are labeled as PTD. The data related to the solution of (6.25), which we consider as exact ones, are labeled as MoM. Equations (6.23) and (6.25) have been solved numerically by MoM. Using MoM, the object under investigation is discretized and replaced with a number of neighboring segments. The segment lengths are specified according to the wave frequency. As a rough criterion, the length of each segment should be equal to one-tenth of the wavelength for discretization in almost all frequency and time domain models (this is a rough discretization; depending on the problem at hand as many as several dozen segments may be required). Fringe currents, fringe fields, and scattered fields are comparatively given in this section. Figure 6.11 shows fringe currents induced on the upper surface (y ≥ 0) of the soft rounded trilateral cylinder. Here, solid and dashed curves belong to ⎪jfr,MoM⎪ and ⎪jfr,PTD⎪, respectively. The discretization parameters used in the calculations are mentioned in the figure captions. As observed, the agreement is very good. Notice that picks here are associated with the curvature discontinuities at the conjugated points and demonstrate that fringe currents concentrate in the vicinity of these points.

Figure 6.11  Nonuniform/fringe component of the currents induced on the upper surface (y ≥ 0) of the soft rounded trilateral cylinder (jzfr component of the electric currents). Here, 0 ≤ l ≤ L01/2 + L1 + L02 + L3/2, NL01 = NL02 = NL03 = 84, NL1 = NL2 = NL3 = 462,a = λ /5, D = 3λ , dl = λ /200.

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Diffraction with Rounded Edges

Figure 6.12 displays bistatic scattering at a soft rounded trilateral cylinder (i.e., 2D RCS vs. angle) for a = λ /5. Here, solid, dashed, and dashedMoM PTD PO dotted curves belong to s norm , s norm , and s norm , respectively. As observed, MoM and PTD solutions agree very well, while PO is insufficient in representing the diffraction contributions. Two important observations follow from this figure. Maximum scattering happens in the vicinity of the shadow direction. It is a well-known phenomenon of forward scattering. This phenomenon is the result of cophase interference of elementary PO waves in the forward direction that represents the focal line for these waves. The second maximum is observed in the vicinity of the direction ϕ = 60° and relates to the specular reflection from the face L1. Figure 6.13 demonstrates the field generated by fringe currents. Again, it is seen that main maxima relate to the shadow radiation and specular reflection. According to Figures 6.11, 6.12, and 6.13, the multiple diffracted fringe waves actually do not reveal themselves. Therefore, the first-order PTD is totally applicable for objects about 3 wavelengths long, and certainly for longer objects. Figure 6.14 compares the scattering from a soft trilateral cylinder with sharp and rounded edges. The data for the sharp cylinder are based on the

Figure 6.12  Bistatic scattering at a soft rounded trilateral cylinder (Ez-polarization, a = λ /5, D = 3λ , dl = λ /200).

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6.3  Trilateral Cylinder with Rounded Edges 141

Figure 6.13  The field generated by the fringe currents induced on a soft rounded trilateral cylinder (Ez-polarization, a = λ /5, D = 3λ , dl = λ /200). MoM is applied to the whole object with Ltot.

solution of the same equations (6.23) and (6.25) where the curvature radius a is set to zero. Note that here the MoM and PTD lines coincide within the graphical accuracy. For the sharp cylinder, they also coincide with the PTD curve in Figure 5.11 of [6] where it was calculated by the PTD analytic form in terms of simple trigonometric functions. This figure also admits clear physical interpretation. Shadow radiation and specular reflection for both objects are nearly the same due to their close dimensions. However, in the vicinity of backscattering (120° ≤ ϕ ≤ 180°), the rounded object creates higher scattering due to the specular reflection from the front rounded edge. This part provides an extension of PTD for objects with rounded edges. This extension represents a fruitful combination of the PTD fundamental concept of fringe currents with the MoM modeling of the currents. The rounding shape was chosen as a circular cylinder. However, it is clear that the developed approach can be also applied to objects with other rounding shapes, as well as for other more complex objects. Note that this first-order PTD approximation describes only single-diffracted fringe waves. The exact data (based on the direct application of MoM to the whole object) include the contribution of multiple diffracted fringe waves. A comparison of the asymptotic PTD and exact data reveals a very good agreement already in the case when the size of the object is about three wavelengths. Hence, for

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Figure 6.14  Comparison of scattering from a soft trilateral cylinder with sharp and rounded edges (Ez-polarization, a = λ /5, D = 3λ , dl = λ /200).

these and longer objects the multiple diffracted fringe waves are already negligible and the first-order PTD is totally acceptable. This comparison also shows that for soft cylinders with large faces (L ≫ λ ) this PTD approach allows one to decrease the computer time by a factor L/3λ compared to the numeric solution of the surface integral equations for the total currents. For the hard rounded trilateral cylinder, the fringe currents jfr in the vicinity of the object edges are asymptotically identical to those on the rounded tangential wedges with the same shape of the edge and with infinite faces. We refer to these as primary (single-diffracted) currents and focus on their calculation. Denote these currents as j1fr, j2fr, j3fr for the wedges whose extended faces intersect each other at points 1, 2, 3, respectively. To calculate them, we apply the fringe integral equations introduced in [25]. They are reproduced here briefly: i 1 fr ∂ j ( x, y ) − p.v. ∫ j fr ( x ′ , y ′ ) H0(1) ( kr ) dl ′ 4 2 ∂n L =

i ∂ p.v. ∫ j PO ( x ′ , y ′ ) H0(1) ( kr ) dl ′ 4 ∂n L

(6.26)

for the illuminated side Lill and

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6.3  Trilateral Cylinder with Rounded Edges 143

i 1 fr ∂ j ( x, y ) − p.v. ∫ j fr ( x ′ , y ′ ) H0(1) ( kr ) dl ′ 4 2 ∂n L = uinc ( x, y ) +

(6.27) i ∂ p.v. ∫ j PO ( x ′ , y ′ ) H0(1) ( kr ) dl ′ 4 ∂n L

for the shadowed side Lsh. Here, L > 0 and dl′ ≥ 0. The integral terms on the right-hand side of (6.26) and (6.27) represent the PO field. Equations (6.26) and (6.27) can be solved by the classic MoM [28]. Actually, we need to solve three sets of integral equations associated with three tangential wedges:  The wedge with L = L01 + L1 + L2;  The wedge with L = L02 + L1 + L3;  The wedge with L = L03 + L3 + L2. where L1, L2, L3 formally are infinitely long, but during the practical calculation, they have been set to equal L1 = L2 = L3 = L and finite. It was established that this assumption is effective already in the case when L ≥ 3λ [6]. Denote the fringe currents on the rounded tangential wedges as j1fr, fr j2 , j3fr, respectively. The total fringe current on the actual object is the sum

j fr,PTD = j1fr + j2fr + j3fr (6.28)

Solving (6.26) and (6.27), the fringe current (6.28) is obtained. To test/validate this PTD approximation, the exact fringe current has been calculated as the difference

j fr,exact = j tot,exact − j PO (6.29)

Here jtot,exact is found by solving the integral equation for the actual total current on the object



i 1 tot ∂ j ( x, y ) − p.v. ∫ j tot ( x ′ , y ′ ) H0(1) ( kr ) dl ′ = uinc ( x, y ) (6.30) 4 2 ∂n L

that includes all multiple fringe waves/currents. The fields generated by the surface currents are calculated as u ( x, y ) =

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i ∂ j ( x ′ , y ′ ) H0(1) ( kr ) dl ′ (6.31) 4 ∫L ∂n

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The currents in (6.28) and (6.29) and the fields generated by them are indicated in the following figures by the labels PTD and MoM, respectively. The MoM data is considered as the exact ones. Note that in the following figures, we plot the normalized scattering cross section σ norm defined by (5.45) in [6] with l = 2a + L. Equations (6.26), (6.27), and (6.30) have been solved numerically by MoM. Examples of its applications to fringe integral equations are given in [26, 29]. MoM has been used in scattering and diffraction modeling [21, 31]. In this method, the object under investigation is discretized and replaced with some neighboring segments. The segment lengths are specified according to the wave frequency. As a rough criterion, the length of each segment should be equal to one-hundredth of the wavelength for discretization in almost all frequency and time domain models (this is a rough discretization; depending on the problem at hand as many as several dozen segments may be required). Fringe currents, fringe fields, and scattered fields are comparatively given in this section. Figure 6.15 shows fringe currents induced on the upper surface (y ≥ 0) of the hard rounded trilateral cylinder. Here, solid and dashed curves belong to ⎪jfr,MoM⎪ and ⎪jfr,PTD⎪, respectively. The discretization parameters used in the calculations are mentioned in the figure captions. As observed, the agreement is good. The MoM and PTD curves here totally coincide with the graphical resolution. Note here the high peak between two conjugation points of the section L02 with L1 and L3. It is clearly associated with the curvature discontinuities at these points. In the next figures, we plot the normalized scattering cross section σ norm. Figure 6.16 displays the bistatic scattering of waves at the hard rounded trilateral cylinder for a = λ /5. Here, dashed, solid, and dashed-dotted PO MoM PTD curves belong to s norm , s norm , and s norm , respectively. As observed, MoM and PTD solutions agree quite well, while PO is insufficient in representing the diffraction phenomena. Two important observations follow from this figure. Maximum scattering happens in the vicinity of the shadow direction φ = 0. It is a well-known phenomenon of forward scattering. Its physical nature is considered in Section 1.5 of [6] where it is interpreted as the shadow radiation. This phenomenon is the result of cophase interference of PO waves in the forward direction that represents the focal line for these waves. The second maximum is observed in the vicinity of the direction φ = 60° and relates to the specular reflection from the face L1. The field oscillations are due to the interference of three edge waves. The difference between MoM and PTD observed in Figure 6.17 is caused by the multiple fringe waves that are absent in this first-order PTD

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6.3  Trilateral Cylinder with Rounded Edges 145

Figure 6.15  Fringe currents induced on the upper surface (y ≥ 0) of the hard rounded trilateral cylinder (a = λ /5, D = 5λ , dl ≅ λ /100, NL01 = NL02 = NL03 = 44, NL1 = NL2 = NL3 = 432).

Figure 6.16  Bistatic scattering of waves at the hard rounded trilateral cylinder (a = λ /5, D = 5λ , dl ≅ λ /100, NL01 = NL02 = NL03 = 44, NL1 = NL2 = NL3 = 432).

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Figure 6.17  Fringe waves generated in the far zone by the fringe currents induced on the hard rounded trilateral cylinder (a = λ /5, D = 3λ , dl ≅ λ /100, NL01 = NL02 = NL03 = 44, NL1 = NL2 = NL3 = 232).

approximation. Their influence becomes noticeably smaller in Figure 6.18 for the large cylinder with its size D = 5λ . Figure 6.19 demonstrates the scattering of waves by the hard and soft cylinders where the PTD data for the soft cylinder were reproduced from [30]. For large objects, the main lobes generated by the PO currents are

Figure 6.18  Fringe waves generated in the far zone by the fringe currents induced on the hard rounded trilateral cylinder (a = λ /5, D = 5λ , dl ≅ λ /100, NL01 = NL02 = NL03 = 44, NL1 = NL2 = NL3 = 432).

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References 147

Figure 6.19  Bistatic scattering of waves at the hard and soft rounded trilateral cylinder (a = λ /5, D = 5λ , dl ≅ λ /100, NL01 = NL02 = NL03 = 44, NL1 = NL2 = NL3 = 432.

actually the same for both cylinders. The difference is observed only in side lobes due to different fringe waves.

6.4 Summary This chapter provides the extension of PTD for diffraction of waves at hard finite objects with rounded edges. This extension consists of a combination of the PTD fundamental concept of fringe currents with the MoM modeling of the currents. The rounding shape is chosen as a circular cylinder. It is clear that the developed approach can also be applied to objects with other rounding shapes, as well as for other more complex objects. A comparison of PTD with the exact data reveals a good agreement already in the case when the size of the object is about 3−5 wavelengths. Hence, for these and longer objects the extended first-order PTD is fully acceptable. Note also that compared to the exact data found by MoM the developed PTD approach allows one to reduce the computer time approximately by a factor L/(5λ ). This is the essential advantage of PTD over direct numeric solutions.

References [1]

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Kalashnikov, A., “The Gouy-Sommerfeld Diffraction” (in Russian), J. Russian Physical-Chem. Soc., Vol. 44, Physical Section, No. 3, St. Petersburg, Russia, 1912, pp. 137−144.

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[2] [3] [4] [5] [6] [7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

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Hansen, C. (ed.), Geometrical Theory of Diffraction, IEEE Press, 1981. Bowman, J. J., T. B. A. Senior, and P. L. E. Uslenghi (eds.), Electromagnetic and Acoustic Scattering by Simple Shapes, Hemisphere Publishing Corporation, 1987. Kravtsov, Yu. A., and N. Y. Zhu, Theory of Diffraction: Heuristic Approaches, Alpha Science Series on Wave Phenomena, 2010. Ufimtsev, P. Ya., Fundamentals of the Physical Theory of Diffraction, Hoboken, NJ: John Wiley & Sons, 2007. Ufimtsev, P. Ya., Fundamentals of the Physical Theory of Diffraction, Second Edition, Hoboken, NJ: John Wiley & Sons, 2014. Hamid, M. A. K., “Diffraction Coefficient of a Conducting Wedge Loaded with A Cylindrical Dielectric Slab at the Apex,” IEEE Transactions on Antennas and Propagation, Vol. 21, May 1973, pp. 398−399. Elsherbeni, A. and M. Hamid, “Diffraction by a Wide Double Wedge with Rounded Edges,” IEEE Transactions on Antennas and Propagation, Vol. 33, September 1985, pp. 1012−1015. Hallidy, W., “On Uniform Asymptotic Green’s Function for the Perfectly Conducting Cylinder Tipped Wedge,” IEEE Transactions on Antennas and Propagation, Vol. 33, No. 9, September 1985, pp. 1020−1025. Mitzner, K. M., Kaplin, K. J., and Cashen, J. F., “How Scattering Increases As An Edge Is Blunted,” In the book: H. N. Kriticos, D. L. Jaggard (Editors) Recent Advances in Electromagnetic Theory, Springer, 1990. Vasiliev, E. N., V. V. Solodukhov, and A. I. Fedorenko, “The Integral Equation Method in the Problem of Electromagnetic Waves Diffraction by Complex Bodies,” Electromagnetics, Vol. 11, No. 2, October 2007, pp. 161−182. Yarmakhov, I. G., “Investigation of Diffraction of Electromagnetic Waves at Edges of Perfectly Conducting and Impedance Wedges with a Rounded Edge” (in Russian), Radiotekhnika I Elektronika, Vol. 36, No. 10, 1991, pp. 1887−1895), J. Commun. Technol. Electron., Vol. 49, No. 4, 2004, p. 379. Christou, M. A., Polycarpou, A. C., and Papanicolau, N. C., “Soft Polarization Diffraction Coefficient for A Conducting Cylinder-Tipped Wedge,” IEEE Transactions on Antennas and Propagation, Vol. 58, No. 12, pp. 4082−4085, Dec. 2010. Balanis, C. A., L. Sevgi, and P. Ya. Ufimtsev, “Fifty Years of High Frequency Diffraction,” International Journal on RF and Microwave Computer-Aided Engineering, Vol. 23, No 4, July 2013, pp. 394−402. Ufimtsev, P. Ya., “The 50-Year Anniversary of the PTD: Comments on the PTD’s Origination and Development,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp. 18−28. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries: Analysis of Asymptotic Techniques,” IEEE Antennas and Propagation Magazine, Vol. 53, June 2011, pp. 232−253. Cakir, G., L. Sevgi, and P. Ya. Ufimtsev, “FDTD Modeling of Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries: Comparisons Against Analytical Models and Calibration,” IEEE Transactions on Antennas and Propagation, Vol. 60, No. 7, July 2012, pp. 3336−3342.

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References 149 [18] Hacivelioglu, F., M. A. Uslu, and L. Sevgi, “A MATLAB-Based Simulator for the Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 6, December 2011, pp. 234−243. [19] Uslu, M. A., and L. Sevgi, “MATLAB-Based Virtual Wedge Scattering Tool for the Comparison of High Frequency Asymptotics and FDTD Method,” Applied Computational Electromagnetics Society Journal, Vol. 27, No. 9, September 2012, pp. 697−705. [20] Apaydin, G., and L. Sevgi, “A Novel Wedge Diffraction Modeling Using Method of Moments (MoM),” Applied Computational Electromagnetics Society Journal, Vol. 30, No. 10, October 2015, pp. 1053−1058. [21] Uslu, M. A., G. Apaydin, and L. Sevgi, “Double Tip Diffraction Modeling: Finite Difference Time Domain vs. Method of Moments,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 12, December 2014, pp. 6337−6343. [22] Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Wedge Diffracted Waves Excited by a Line Source: Exact and Asymptotic Forms of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 61, No. 9, September 2013, pp. 4705−4712. [23] Bhattacharyya, A. K., “Scattering by a Right-Angled Penetrable Wedge: A Stable Hybrid Solution (TM Case),” Applied Computational Electromagnetics Society Journal, Vol. 5, No. 3, 1990. [24] Ikiz, T., and M. K. Zateroglu, “Diffraction of Obliquely Incident Plane Waves by an Impedance Wedge with Surface Impedances Being Equal to The Intrinsic Impedance of the Medium,” Applied Computational Electromagnetics Society Journal, Vol. 26, No. 3, March 2011, pp. 199−205. [25] Apaydin, G., L. Sevgi, and P. Ya. Ufimtsev, “Fringe Integral Equations for the 2-D Wedge with Soft and Hard Boundaries,” Radio Science, Vol. 61, No. 9, September 2016, pp. 4705−4712. [26] Apaydin, G., and L. Sevgi, “Two Dimensional Non-Penetrable Wedge Scattering Problem and a MATLAB-Based Fringe Wave Calculator,” IEEE Antennas and Propagation Magazine, Vol. 58, No. 2, April 2016, pp. 86−93. [27] Ufimtsev, P. Ya., “Fast Convergent Integrals for Nonuniform Currents on Wedge Faces,” Electromagnetics, Vol. 18, No. 3, May-June 1998, pp. 289−313. Corrections in Electromagnetics, Vol. 19, No. 5, 1999, p. 473. [28] Harrington, R. F., Field Computation by Moment Methods, Oxford University Press, 1996. [29] Apaydin, G., F. Hacivelioglu, L. Sevgi, and P. Ya. Ufimtsev, “Fringe Waves from a Wedge with One Face Electric and the Other Face Magnetic,” IEEE Transactions on Antennas and Propagation, Vol. 64, No. 3, 2016, pp. 1125−1130. [30] Apaydin, G., L. Sevgi, and P. Ya. Ufimtsev, “Extension of Ptd for Finite Objects with Rounded Edges: Diffraction at a Soft Trilateral Cylinder,” IEEE Antennas and Wireless Propagation Letters, Vol. 16, 2017, pp. 2590−2593. [31] Apaydin, G., F. Hacivelioglu, L. Sevgi, and P. Ya. Ufimtsev, “Wedge Diffracted Waves Excited by a Line Source: Method of Moments (MoM) Modeling of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 8, 2014, pp. 4368−4371.

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CHAPTER

7 Contents

Double-Tip Diffraction Modeling

7.1 Introduction

This chapter presents the double-tip diffraction using FDTD and MoM.

7.2 Double-Tip Diffraction Structure

7.1 Introduction

7.3 FDTD-Based Diffraction Modeling 7.4 MoM-Based Diffraction Modeling 7.5  Examples and Comparisons 7.6 Summary

Electromagnetic waves interact with objects and scatter. Major scattered field components are reflected, refracted, and diffracted fields. HFA methods, such as GO, PO, GTD, UTD, and PTD have long been used to analyze scattered fields when the wavelength is small compared to object size [1−6]. A useful MATLAB-based virtual tool has also been introduced for the use of HFA modeling in the classical wedge problem [7]. Fringe waves excited by a finite-distance line source are extracted in [8]. Backscattering from a wedge with different boundary conditions is also modeled analytically [9]. An interesting discussion on the diffraction modeling can be found in [10]. Diffraction modeling has also been investigated numerically [11−15]. Diffracted waves and diffraction coefficients are extracted with the FDTD method using time-gating in [11]. A more general multistep FDTD approach is also used in diffraction modeling [12,13]. Similarly, it is shown in [14,15] that MoM is also successful in diffraction modeling. Double diffraction has also been investigated analytically and numerically [16−25]. The double wedge or double tip is a canonical 151

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geometry that arises in many practical structures. The analysis via a spectral extension of the UTD has been described, which yields closed-form expressions for the field doubly diffracted in the far zone by the edges of two interacting wedges illuminated by a plane wave in [17,18]. The UTD has been extended to include double diffraction by an arbitrary configuration of two wedges and a scalar double diffraction coefficient is defined in [19]. A HFA analysis of the scattering by a double impedance wedge via the extended spectral ray method and diffraction coefficients were derived for up to and including the triple diffraction mechanism in [23]. A time-domain single-diffraction solution of a wedge-type obstacle is extended to double diffraction and the resulting waveform is compared with the corresponding solution in the frequency domain by applying the inverse Fourier transform of the waveform in [25]. In this chapter, novel double-tip diffraction modeling approaches are introduced by using both FDTD and MoM. An analytical solution using a spectral approach for the problem of scattering by 2-D semi-infinite or finite polygonal objects with an imperfectly reflective surface, illuminated by a plane wave, can be found in [26].

7.2  Double-Tip Diffraction Structure The structure shown in Figure 7.1 is used in double-tip diffraction modeling. The polar coordinates ρ ,φ ,z are used. It is a nonpenetrable rectangular object infinite along the z-axis excited with a line source. The problem has a translational symmetry along z and therefore can be investigated in 2-D on the xy-plane. The origin is chosen at the midpoint of the top edge with length L and therefore the top boundary extends from (−L/2,0) to (+L/2,0) on the x-axis. The lengths of left and right boundaries are infinite (dL = dR → ∞). The line source is assumed only in the first quadrant (0 ≤ φ 0 ≤ π /2) because of the structural symmetry. The dashed circle with radius ρ shows the locations of receivers, and 360 receivers are located on this circle, which yields Δφ = 1° angular resolution. Under these assumptions, top-boundary reflected fields exist for the receivers located between the lines L1 and L 2, while side-boundary (specular) reflections occur only for the receivers between the lines L3 and L 4. Incident fields do not exist in the shadow region bounded by the left side of the structure and the line SB. The two tips are responsible for the creation of the diffracted fields that exist everywhere. Note that Figure 7.1 shows the reflection and shadow regions for ρ 0cosφ 0 > L/2. Mathematically, the problem is postulated via the 2-D wave equation in polar coordinates

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7.2  Double-Tip Diffraction Structure 153

Figure 7.1  Double-tip structure (Structure-1).



I0 ⎧1 ∂ ⎛ ∂ ⎞ 1 ∂2 2⎫ ⎨ r ∂r ⎜⎝ r ∂r ⎟⎠ + 2 2 + k ⎬ u = r d r − r0 d j − j0 (7.1) r ∂j ⎭ ⎩

(

) (

)

where k is the wave-number, I0 is the line current amplitude, (ρ 0,φ 0) and (ρ ,φ ) specify the source and the observation points, respectively, δ (⋅) is the Dirac delta functions. The related nonpenetrable BCs are u = 0 (TM case) or ∂u/∂n = 0 (TE case) on the structure. The radiation condition also applies. In the limit when the lengths of left and right boundaries go to zero (dL = dR → 0) the structure yields another canonical scattering problem: the infinite strip problem. This is pictured in Figure 7.2. The tips marked with number 1 and number 2 are responsible for the creation of the diffracted fields. In this case, reflected fields only exist for the receivers located between the lines L1 and L 2. The region between the lines SB1 and SB2 is the shadow region where no incident field exists. The incident, scattered, and diffracted field components all exist elsewhere. Note also that the structure in Figure 7.1 also reduces to (vertical) half-plane problem when L → 0 meaning that three important canonical problems can be investigated at the same time once numerical models are established for the Structure-1 in the figure.

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Figure 7.2  Infinite strip problem (Structure-2).

The electromagnetic line source may be the z-component of either electric field intensity (u = Ez, TM case) or magnetic field intensity (u = Hz, TE case). In the case of acoustic waves, these conditions refer to acoustically soft (TM → SBC) and hard (TE → HBC) boundary conditions, respectively. Note that the word scattering represents all types of waves generated from EM (incident) wave−object interaction (reflections, refractions, diffractions, creeping waves, whispering gallery waves, etc.). The addition of scattered and incident fields yields total fields. Diffractions occur from the edge and/or tip type discontinuities.

7.3  FDTD-Based Diffraction Modeling The FDTD method is one of the most popular and widely used models in electromagnetics [27]. It has been widely used in a variety of electromagnetic problems, from radiation, propagation, and scattering to microstrip circuit design, from subsurface imaging to antenna design, and so on. The FDTD method has also been used in the calculation of diffraction coefficients and there are many studies in modeling diffraction from various wedges [11−13]. Note that the incident field is a pulse in the time therefore broadband diffraction characteristics can be obtained via a single FDTD simulation. Once

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7.3  FDTD-Based Diffraction Modeling 155

incident and diffracted pulses are recorded, discrete/fast Fourier transform (DFT/FFT) can be applied and diffraction coefficient versus frequency and/ or diffraction coefficients versus angle variations can be obtained. The 2-D FDTD models used for TM and TE polarizations on the xy-plane contain sets of H x,Hy,Ez, and Ex,Ey,Hz components, respectively. The update equations for TM z polarization are 1 H x 21 i ,j + 2 n+



1 H y 12 i + ,j 2 n+



Ez

n+1 i ,j

= Ez

n i ,j



=

1 H x 21 i ,j + 2

n n ⎤ ⎡ Δt ⎢ Ez i ,j +1 − Ez i ,j ⎥ −     (7.2a) ⎥ m 1⎢ Δy i ,j + ⎢ ⎥⎦ 2 ⎣

=

1 H y 12 i + ,j 2

n n ⎡ ⎤ Δt ⎢  Ez i +1,j − Ez i ,j ⎥ +   (7.2b) ⎥ m 1 ⎢ Δx i + ,j ⎢ ⎥⎦ 2 ⎣

n−

n−

1 1 1 1 n+ ⎡ ⎤ n+ n+ n+ 2 2 2 2 − H  H − H  H ⎢ y i + 1 ,j x i ,j + 1 y i ,j − 1 ⎥ y i − 1 ,j Δt ⎢ 2 ⎥ (7.2c) 2 2 2 +   −   ⎥ mi ,j ⎢ Δx Δy ⎢ ⎥ ⎢⎣ ⎥⎦

Likewise, the update equations for TE z mode are



1 1 n+ n+ ⎤ ⎡ 2 2 H − H ⎢ z i + 1 ,j + 1 z i + 1 ,j − 1 ⎥ Δt ⎢ 2 2 2 2 ⎥ +     (7.3a) m 1  ⎢ Δy ⎥ i + ,j ⎢ ⎥ 2 ⎣ ⎦



1 1 n+ n+ ⎤ ⎡ 2 2 H − H ⎢ z i + 1 ,j + 1 z i − 1 ,j + 1 ⎥ Δt ⎢ 2 2 2 2 ⎥ −     (7.3b) m 1  ⎢ Δx ⎥ i ,j + ⎥ 2 ⎢ ⎣ ⎦

n+1

n

E x i + 1 ,j = E x i + 1 ,j 2

Ey

1 n+ H z 12 1 i + ,j + 2 2

=

2

n+1 i ,j +

1 2

= Ey

1 n− H z 12 1 i + ,j + 2 2

+

n i ,j +

m

1 2

Δt

1 1 i + ,j + 2 2

n n n n ⎡ ⎤ − E E E − E 1 1 1 1 y y x i + ,j ⎢ x i + 2 ,j +1 i +1,j + i ,j + ⎥ 2 2 2 − ⎢  ⎥ Δy Δx ⎢ ⎥ ⎢⎣ ⎥⎦

(7.3c)

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The multistep FDTD approach introduced in [13] is extended here for the calculation of double-tip diffractions. The incident, reflected, and diffracted fields are separated in the time, and then total, scattered, and/or diffracted fields versus angle are obtained by the application of FFT. For the structure in Figure 7.1, diffracted fields are extracted with the following four steps: 1. Run the FDTD simulation with the structure and record transient responses at the specified number of receivers on the observation circle. This will yield total fields. 2. Remove the structure, rerun the FDTD simulation in free space, and record transient responses at the same receivers. This will yield incident fields. 3. Replace the structure with infinite plane (in other words, extend the top edge of the structure infinitely on the horizontal axis) and rerun the FDTD simulations. Recorded fields will include only incident and reflected fields on the upper half-plane. Use them only for the receivers located between the lines L1 and L2. 4. Extend the right side of the structure infinitely on the vertical direction and repeat step 3. Recorded fields will include only incident and reflected fields on the right half-plane. Use them only for the receivers located between the lines L3 and L4. Once time variations of the fields for the four steps are obtained, incident and reflected fields are extracted in regions where they exist and only time variations of diffracted fields are left at the receivers on the observation circle. Diffracted fields at a specified frequency can then be extracted by the application of FFT on all receivers’ data. Note that only the first three steps are enough to extract diffracted field data for the infinite strip shown in Figure 7.2. Moreover, only the first two steps are enough to obtain a scattered field.

7.4  MoM-Based Diffraction Modeling A similar multistep MoM is also used in diffraction modeling as described in [14,15]. Here, the method is extended to the double-tip diffraction problem. In this model, the three boundaries of Structure-1 in Figure 7.1 are divided into small segments (where segment lengths are much smaller than the wavelength). Although dL = dR → ∞, they have to be finite in numerical algorithms. Side-boundary lengths between 10λ − 100λ are enough depending on the parameters of the problem at hand. The length of the

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7.4  MoM-Based Diffraction Modeling 157

top-boundary is finite. The currents on each segment are assumed to be constant. In the standard MoM, the source-excited segment fields are calculated, the matrix system is built, and the segment currents are calculated numerically from the solution of the derived system of equations [14]. The segment-scattered fields at the observer are then accumulated. Necessary MoM equations (with the exp(−iω t) time dependence) in this procedure are

( )

(

)

( )

(

)

Vn = −Ezinc rn = −e0 H0(1) k rn − r0 (SBC) (7.4a)



Vn = −H zinc rn = −h0 H0(1) k rn − r0 (HBC) (7.4b)



where Vn denotes the incident field at matching points (ρ n) and the impedance matrix is obtained



kh0 Δ (1) ⎧ ⎪⎪ − 4 H0 k rm − rn , m ≠ n (SBC) (7.5a) ≅ ⎨ kh Δ ⎡ 2 ⎛ g kΔ ⎞ ⎤ , m = n ⎪ − 0 ⎢1 + i log ⎜ ⎟ ⎝ 4e ⎠ ⎥⎦ p 4 ⎣ ⎪⎩



⎧⎪ ikΔ (1) Zmn ≅ ⎨ − 4 H1 k rm − rn nˆ n ⋅ rˆ nm , m ≠ n (HBC) (7.5b) 0.5, m = n ⎩⎪

Zmn

(

)

(

)(

)

where Δ is the segment length, η 0 ≈ 120 π is the intrinsic impedance of free space, H0(1) and H1(1) are the first kind Hankel function with order zero and one, respectively, γ = 1.781 is the exponential of the Euler constant, nˆ n denotes the unit normal vector of the segment at ρ n , and rˆnm is the unit vector in the direction from source ρ n to the receiving element ρ m . While considering the effects of segment currents, the scattered fields are obtained from





Ezsc(r r) ≅ −

H zsc(r r) ≅ −

kh0 Δ 2N + M I n H0(1) k r − rn (SBC) (7.6a) 4 ∑ n=1

(

ikΔ 2N + M I n H1(1) k r − rn 4 ∑ n=1

(

)

)( nˆ

n

)

⋅ rˆn (HBC) (7.6b)

The MoM procedure is implemented as follows: The incident fields upon segments in (7.4) are calculated by using the free-space Green’s function.

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The impedance matrix in (7.5) is formed. Then, the source-induced segment currents are obtained. Finally, scattered fields in (7.6) on the chosen observation points are calculated from the superposition of segment radiations using the Green’s function. The direct wave from the source to the receiver and scattered waves from all segments to the receiver are added and total wave at the receiver is obtained. For the structure in Figure 7.1, MoM-computed diffracted fields are extracted with the following three steps: 1. Run the MoM simulation with the structure and record the scattered fields at the specified number of receivers on the observation circle. 2. Replace the structure with infinite plane (in other words, extend the top edge of the structure infinitely on the horizontal axis) and rerun the MoM simulations. Recorded scattered fields will include only reflected fields on the upper half-plane. Use them only for the receivers located between the lines L1 and L2. 3. Extend the right side of the structure infinitely in the vertical direction and repeat step 2. Recorded scattered fields will include only reflected fields on the right half-plane. Use them only for the receivers located between the lines L3 and L4. Only the first two steps are enough to extract diffracted field data for the infinite strip (Structure-2) shown in Figure 7.2. MoM directly yields scattered fields; therefore, only the first step is enough for the extraction of scattered fields. MATLAB algorithms are developed for both FDTD- and MoM-based diffraction modeling. The next section presents several comparisons. Note that diffracted fields presented in the following examples contain singleand double-tip diffractions from both tips. Figure 7.3 illustrates single- and double-diffracted waves from the left tip. The right tip also contributes to the same single- and double-diffracted waves. Note that standard free-space FDTD and MoM algorithms are used here. The FDTD space is terminated with absorbing boundary layers and edges that are directly extended into these layers. This is how infinite length structure (dL = dR → ∞) is simulated. On the other hand, edges are truncated so that dL and dR are finite, but the lengths of the truncated edges are long enough to simulate dL = dR → ∞. One needs to check if the first segment beyond the truncation has negligible induced current. Beyond the truncation, this (i.e., the simulation of the infinite edges) is achieved if the scattered field at the nearest receiver, caused by the segment currents is less

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7.5  Examples and Comparisons 159

Figure 7.3  Single- and double-tip diffractions: (a) single diffraction from the right tip, (b) double diffractions from the first right tip, then the left tip, (c) single diffraction from the left tip, and (d) double diffractions from the first left tip, then the right tip.

than a specified value corresponding to the stated accuracy and/or error. Relative accuracy of 1% or less is used to generate all the examples presented in the next section.

7.5  Examples and Comparisons The examples given in this section, present total, diffracted, and scattered fields around Structure-1, Structure-2, and for the vertical half-plane for a given line source at 30 MHz. Source and observer radial distances are 100m (ρ 0 = 10λ ) and 80m (ρ = 8λ ), respectively. The polarizations and angle of incidences are mentioned in Figures 7.4–7.10. In Figure 7.4, total, diffracted, and scattered fields around the tip of a vertical half-plane, simulated with both FDTD and MoM approaches, are given. Here, the angle of incidence is φ 0 = 30°. The top side of Structure-1 is taken as L = λ /10. As observed in the total field plot, the ripples in the angular region −30° ≤ φ ≤ 210° correspond to the interference of incident

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Double-Tip Diffraction Modeling

Figure 7.4  Total, diffracted, and scattered fields around Structure-1 at 30 MHz; (L = λ /10, ρ 0 = 10λ , ρ = 8λ , φ 0 = 30°, TM/SBC case, solid line = MoM, dashed line = FDTD).

and diffracted fields, and the ripples in the angular region 270° ≤ φ ≤ 330° correspond to the interference of incident, diffracted, and reflected fields. The dominant diffraction occurs along the two critical boundaries, ISB and RSB. On the other hand, forward scattering and specular reflections dominate the scattered fields. Figures 7.5, 7.6, and 7.7 belong to scattering from Structure-1. Total, diffracted, and scattered fields simulated with both the FDTD and MoM approaches for different illumination angles and top surface lengths are shown in Figures 7.6 and 7.7. Only total and diffracted fields are given in Figure 7.8. Note that there are two tips and four critical boundaries. Observe the diffracted fields along these boundaries and their interference for different top surface lengths. Figures 7.8, 7.9, and 7.10 belong to the FDTD and MoM simulation results for Structure-2 (infinite strip). Scattered fields are also included in Figures 7.9 and 7.10. As observed angular variations of the total, scattered, and diffracted fields for different angles of illumination with different top surface lengths, the forward scattering, and specular reflections dominate

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7.5  Examples and Comparisons 161

Figure 7.5  Total, diffracted, and scattered fields around Structure-1 at 30 MHz; (L = λ , ρ 0 = 10λ , ρ = 8λ , φ 0 = 45°, TE/HBC case, solid line = MoM, dashed line = FDTD).

the scattering fields, but, as mentioned above, dominant diffractions are observed along critical boundaries. On the other hand, interference of the double diffractions may change the picture significantly depending on the angle of illumination and top surface lengths. Note that different discretizations are required in MoM simulations for the TM (SBC) and TE (HBC) polarizations. Infinite sides of Structure-1 are approximated by 10λ -long finite sides for the TM polarization. On the other hand, up to 100λ -long side-lengths may be required for the TE polarization (because ill-conditioned matrices may be obtained in this polarization). The segment lengths are chosen to be λ /20. The FDTD cell sizes are taken as λ /20. The source is above the horizontal plane in these examples, but it can be located arbitrarily anywhere in the angular domain. In this case, one has to pay attention to the infinite boundaries in both FDTD and MoM procedures. In other words, when a plane (or cylindrical) wave of incidence below the horizontal plane is considered, the infinite boundaries must extend well beyond the source.

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Figure 7.6  Total, diffracted, and scattered fields around Structure-1 at 30 MHz; (L = λ , ρ 0 = 10λ , ρ = 8λ , φ 0 = 30°, TM/SBC case, solid line = MoM, dashed line = FDTD).

Figure 7.7  Total (left) and diffracted (right) fields around Structure-1 at 30 MHz; (L = 4λ , ρ 0 = 10λ , ρ = 8λ , φ 0 = 60°, TM/SBC case, solid line = MoM, dashed line = FDTD).

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7.5  Examples and Comparisons 163

Figure 7.8  Total (left) and diffracted (right) fields around Structure-2 at 30 MHz; (L = 2λ , ρ 0 = 10λ , ρ = 8λ , φ 0 = 45°, TM/SBC case, solid line = MoM, dashed line = FDTD).

Figure 7.9  Total, diffracted, and scattered fields around Structure-2 at 30 MHz; (L = 5λ , ρ 0 = 10λ , ρ = 8λ , φ 0 = 30°, TM/SBC case, solid line = MoM, dashed line = FDTD).

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Double-Tip Diffraction Modeling

Figure 7.10  Total, diffracted, and scattered fields around Structure-2 at 30 MHz; (L = 6λ , ρ 0 = 10λ , ρ = 8λ , φ 0 = 60°, TM/SBC case, solid line = MoM, dashed line = FDTD).

Note also that there are highly effective commercial FDTD and MoM packages that can be used in numerical simulation of a broad range of EM problems. Unfortunately, they cannot be used in solving the problems discussed here. By using a commercial package, total fields can be reproduced, but scattered and/or diffracted fields cannot be discriminated using the multistep approach introduced here.

7.6 Summary Double-tip diffraction modeling using multistep FDTD and MoM approaches in two-dimension were discussed, MATLAB-based diffraction algorithms were developed, and numerical results presented. Very good agreement between the results shows that both FDTD and MoM can be used effectively in diffraction modeling. The novel multistep multitip diffraction modeling introduced here is highly effective for the numerical models such as FDTD and MoM. Its extension to 3-D is straightforward. Since the power and beauty of these numerical models is their application capability directly in 3-D, distinguishing

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References 165

and discriminating scattered and diffracted fields for the real objects in 3-D would be very helpful in understanding and designing low-visible objects. The reader is referred to [27−31] for indoor, anechoic chamber measurement results that belong to 2-D propagation above the flat, perfectly reflecting surface with single- and double diffractive obstacles.

References [1] [2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

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Ufimtsev, P. Ya., Fundamentals of the Physical Theory of Diffraction, Wiley & Sons, 2007. Balanis, C., L. Sevgi, and P. Ya. Ufimtsev, “Fifty Years of High Frequency Asymptotics,” International Journal on RF and Microwave Computer-Aided Engineering, Vol. 23, No. 4, July 2013 pp. 394−402. Pelosi, G., Y. Rahmat-Samii, and J. Volakis, “High Frequency Techniques in Diffraction Theory: 50 Years of Achievements in GTD, PTD, and Related Approaches,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp. 16−17. Ufimtsev, P. Ya., “The 50-year Anniversary of PTD: Comments on the PTD’s Origin and Development,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp.18−28. Rahmat-Samii, Y., “GTD, UTD, UAT, and STD: A Historical Revisit and Personal Observations,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp. 29−40. Pelosi, G., and S. Selleri, “The Wedge-Type Problem: The Building Brick in High-Frequency Scattering from Complex Objects,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp. 41−60. Hacivelioglu, F., M. A. Uslu, and L. Sevgi, “A MATLAB-based Virtual Tool for The Electromagnetic Wave Scattering from a Perfectly Reflecting Wedge,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 6, 2011, pp. 234−243. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Wedge Diffracted Waves Excited by a Line source: Exact and Asymptotic Forms of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 61, No. 9, 2013, pp.4705−4712. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Backscattering from a SoftHard Strip: Primary Edge Waves Approximation,” IEEE Antennas and Wireless Propagation Letters, Vol. 12, 2013, pp. 249−252. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “On the Modified Theory of Physical Optics,” IEEE Transactions on Antennas and Propagation, Vol. 61, No. 12, December 2013, pp. 6115−6119. Stratis, G., V. Anantha, and A. Taflove, “Numerical Calculation of Diffraction Coefficients of Generic Conducting and Dielectric Wedges Using FDTD,” IEEE Transactions on Antennas and Propagation, Vol. 45, No. 10, 1997, pp. 1525−1529. Cakir, G., L. Sevgi, and P. Ya. Ufimtsev, “FDTD Modeling of Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries: Comparisons Against Analytical Models and Calibration,” IEEE Transactions on Antennas and Propagation, Vol. 60, No. 7, July 2012, pp. 3336−3342.

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[13] Uslu, M. A., and L. Sevgi, “MATLAB-Based Virtual Wedge Scattering Tool for the Comparison of High Frequency Asymptotics and FDTD Method,” Applied Computational Electromagnetics Society Journal, Vol. 27, No. 9, 2012, pp. 697−705. [14] Apaydin, G., and L. Sevgi, “A Novel Wedge Diffraction Modeling Using Method of Moments,” Applied Computational Electromagnetics Society Journal, Vol. 30, No. 10, 2015. [15] Apaydin, G., F. Hacivelioglu, L. Sevgi, and P. Ya. Ufimtsev, “Wedge Diffracted Waves Excited by a Line source: Method of Moments (MoM) Modeling of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 8, August 2014, pp. 4368−4371. [16] Tiberio, R., Manara, G., Pelosi, G., and Kouyoumjian, R. G., “High-frequency Electromagnetic Scattering of Plane Waves from Double Wedges,” Radio Science, Vol. 17, pp. 323−336, 1982. [17] Tiberio, R., G. Manara, G. Pelosi, and R. G. Kouyoumjian, “High Frequency Diffraction by a Double Wedge,” IEEE International Symposium on Antennas Propagation, Vancouver, Canada, June 1985. [18] Tiberio, R., G. Manara, G. Pelosi, and R. G. Kouyoumjian, “High Frequency Electromagnetic Scattering of Plane Waves from Double Wedges,” IEEE Transactions on Antennas and Propagation, Vol. 37, No. 9, September 1989, pp. 1172−1180. [19] Schneider, M., and R. Luebbers, “A General UTD Diffraction Coefficient for Two Wedges,” IEEE Transactions on Antennas and Propagation, Vol. 39, January 1991, pp. 8−14. [20] Ivrissimtzis, L. P., and R. J. Marhefka, “A Note on Double Edge Diffraction for Parallel Wedges,” IEEE Transactions on Antennas and Propagation, Vol. 39, No. 10, October 1991. [21] Albani, M., F. Capolino, S. Maci, and R. Tiberio, “Double Diffraction Coefficients for Source and Observer at Finite Distance for a Pair of Wedges,” IEEE International Symposium on Antennas and Propagation, Vol. 2, 1995, pp. 1352−1352. [22] Capolino, F., and S. Maci, “Uniform High-Frequency Description of Singly, Doubly and Vertex Diffracted Rays for a Plane Angular Sector,” J. Electromagn. Wave Applicat., Vol. 10, No. 9, October 1996, pp. 1175−1197. [23] Herman, M. I., and J. Volakis, “High Frequency Scattering by a Double Impedance Wedge,” IEEE Transactions on Antennas and Propagation, Vol. 36, No. 5, May 1988, pp. 664−678. [24] Albani, M., “A Uniform Double Diffraction Coefficient for a Pair of Wedges in Arbitrary Configuration,” IEEE Transactions on Antennas and Propagation, Vol. 53, No. 2, February 2005, pp. 702−710. [25] Karousos, A., and C. Tzaras, “Time-Domain Diffraction for a Double Wedge Obstruction,” IEEE International Symposium on Antennas Propagat., Hawaii, June 2007, pp. 4581−4584. [26] Bernard, J. M. L., “A Spectral Approach for Scattering by Impedance Polygons,” Q. Jl Mech. Appl. Math., Vol. 59, No. 4, 2006, pp. 517−550. [27] Yee, K. S., “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Transactions on Antennas and Propagation, Vol. 14, 1966, pp. 302−307.

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References 167 [28] Erricolo, D., U. G. Crovella, and P. L. E. Uslenghi, “Time-Domain Analysis of Measurements on Scaled Urban Models with Comparisons to Ray-Tracing Propagation Simulation,” IEEE Transactions on Antennas and Propagation, Vol. 50, No. 5, May 2002, pp. 736−741. [29] Erricolo, D., G. D’Elia, and P. L. E. Uslenghi, “Measurements on Scaled Models of Urban Environments and Comparisons with Ray-Tracing Propagation Simulation,” IEEE Transactions on Antennas and Propagation, Vol. 50, No. 5, May 2002, pp. 727−735. [30] Erricolo, D., “Experimental Validation of Second Order Diffraction Coefficients for Computation of Path-Loss Past Buildings,” IEEE Transactions on Electromagnetic Compatibility, Vol. 44, No. 1, February 2002, pp. 272−273. [31] Negishi, T., V. Picco, D. Spitzer, et al., “Measurements to Validate the UTD Triple Diffraction Coefficient,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 7, July 2014, pp. 3723−3730.

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CHAPTER

8 Contents 8.1 Introduction 8.2  WedgeGUI Software 8.3 Characteristic Examples 8.4 Summary

WedgeGUI Virtual Package This chapter presents a MATLAB-based virtual tool of diffraction from a perfectly reflecting wedge.

8.1 Introduction Electromagnetic wave scattering has been substantially investigated on the canonical, 2-D wedge problem for many decades. Numerical difficulties of various analytical models based on complex integration as well as series summation are also discussed in detail in [1, 2]. Comparisons against numerical techniques such as the FDTD method are given in [3]. In all these studies, wave pieces like reflection, refraction, and diffraction, which are the components of scattering are revisited through analytical exact as well as HFA methods, such as GO, GTD, UTD, PO, PTD, elementary edge waves (EEW), and parabolic equation (PE) methods [4−20]. In this chapter, a MATLAB-based virtual tool—WedgeGUI—for electromagnetic wave scattering from a PEC wedge is introduced in 2-D. There are many different forms of these models, but the ones presented here are numerically the most efficient. The virtual tool WedgeGUI is discussed together with some examples. The nonpenetrable wedge diffraction problem is canonical and plays a fundamental role in the construction of HFA techniques as well as for numerical tests. The exact solution to this 169

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scattering problem was first obtained by Sommerfeld [4] in the particular case of a half-plane. For a wedge with an arbitrary angle between its faces, the solution was obtained by Macdonald [5] and later on by Sommerfeld, who developed the method of branched wave functions [6].

8.2  WedgeGUI Software WedgeGUI, with the front panel displayed in Figure 8.1, is organized for the investigation of wedge diffraction in 2-D with various exact and HFA models. The panel is divided into three parts. The top block is reserved for the structure. The wedge figure is shown on the top right. The wedge exterior angle and incident distance/angle are supplied on the top left. The user also selects either of the soft and hard BCs and total and diffracted fields in this block. A pop-up menu allows the user to choose the type of the source: plane wave excitation or cylindrical wave excitation. For each source type, the methods used in simulations are given with check boxes. Multiple selection is possible. The flow chart of the virtual tool is given in Figure 8.2. The area below the structure block is divided into two parts. On the left, the results are presented in terms of polar plots (i.e., fields as a function of angle). On the right, the results are presented in terms of Cartesian plots (i.e., field as a function of radial range or field as a function of frequency).

Figure 8.1  The front panel of the EM virtual tool WedgeGUI.

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8.3  Characteristic Examples 171

Figure 8.2  Basic flow chart of WedgeGUI.

For the polar plots in the left block, one needs to specify the frequency and the radial distance from the tip. The rest is handled automatically once the Plot button is pressed. The fields are calculated at N observation points equidistant from the tip, located Δφ = α /N apart. The progress bar on the bottom left shows the status of the calculations.

8.3  Characteristic Examples Figures 8.3, 8.4, and 8.5 show example scenarios and simulation results. The user may edit/modify the plot in another figure window. This is possible by clicking on the polar plot. Similarly, one can plot the field as a function of range or field as a function of frequency in the right block. The Clear button clears the plot for the next simulation, and the Save Data button records the data in a text file with the name given by the user. Table 8.1 lists sample recorded data that belongs to the simulations presented in Figure 8.5. Data recording is important because WedgeGUI does not run simulations for both soft boundary conditions and hard boundary conditions, or both total and diffracted fields, at the same time. One needs to run WedgeGUI twice for the examples presented in Figures 8.6, 8.7, and 8.8.

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Figure 8.3  Example scenario and the output of WedgeGUI at 30 MHz: diffracted fields vs. angle for HBC (α = 350°, φ 0 = 45°, r = 50m, kr = 10π ). Curves belong to exact, PTD, UTD, and PE models.

Figure 8.4  Example scenario and the output of WedgeGUI at 30 MHz: Total fields vs. angle for SBC (α = 330°, φ 0 = 70°, r = 50m, kr = 10π ). Curves belong to exact, PO, and PTD models.

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8.3  Characteristic Examples 173 Table 8.1 Sample Recorded Data for Angle vs. Diffracted Fields: Amplitude vs. Angle, Created by WedgeGUI Angle (Degree)

Exact Series

PTD

UTD

0.0000e+000

0.0000e+000 2.0847e-003

0.0000e+000 2.0904e-003

0.0000e+000 2.0889e-003

1.3300e+000

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2.6600e+000

4.1726e-003

4.1850e-003

4.1823e-003

3.9900e+000

6.2679e-003

6.2882e-003

6.2845e-003

5.3200e+000

8.3748e-003

8.4045e-003

8.4001e-003

6.6500e+000

1.0497e-002

1.0538e-002

1.0533e-002

7.9800e+000

1.2641e-002

1.2694e-002

1.2689e-002

9.3100e+000

1.4810e-002

1.4878e-002

1.4873e-002

1.0640e+001

1.7008e-002

1.7094e-002

1.7089e-002

1.1970e+001

1.9241e-002

1.9348e-002

1.9342e-002

1.3300e+001

2.1515e-002

2.1645e-002

2.1640e-002

1.4630e+001

2.3833e-002

2.3991e-002

2.3986e-002

1.5960e+001

2.6197e-002

2.6393e-002

2.6388e-002

1.7290e+000

2.8619e-002

2.8856e-002

2.8852e-002

1.8620e+000

3.1102e-002

3.1389e-002

3.1385e-002

1.9950e+001

3.3652e-002

3.3998e-002

3.3995e-002

2.1280e+001

3.6275e-002

3.6693e-002

3.6690e-002

2.2610e+001

3.8978e-002

3.9482e-002

3.9480e-002

2.3940e+001

4.1768e-002

4.2375e-002

4.2373e-002

2.5270e+001

4.4651e-002

4.5382e-002

4.5382e-002

2.6600e+001

4.7636e-002

4.8516e-002

4.8517e-002

2.7930e+001

5.0729e-002

5.1789e-002

5.1791e-002

2.9260e+001

5.3938e-002

5.5216e-002

5.5219e-002

3.0590e+001

5.7273e-002

5.8812e-002

5.8816e-002

3.1920e+001

6.0741e-002

6.2595e-002

6.2600e-002

3.3250e+000

6.4352e-002

6.6584e-002

6.6590e-002

3.4580e+001

6.8116e-002

7.0801e-002

7.0808e-002

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Figure 8.5  Example scenario and the output of WedgeGUI at 30 MHz: Diffracted fields vs. angle for HBC (α = 250°, φ 0 = 150°, r = 50m, kr = 10π ). Curves belong to exact, PTD, and UTD models.

Figure 8.6  Diffracted fields vs. angle for (left) SBC, and (right) HBC computed with exact and UTD models at 30 MHz (α = 240°, φ 0 = 110°, r = 50m, kr = 10π ).

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Figure 8.7  Total fields vs. angle for (left) SBC and (right) HBC computed with exact and UTD models at 30 MHz (α = 240°, φ 0 = 110°, r = 50m, kr = 10π ).

Figure 8.8  Diffraction coefficients vs. (top) frequency and (bottom) range for the plane wave illumination at 30 MHz (SBC, α = 350°, φ 0 = 60°).

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8.4 Summary A MATLAB-based virtual diffraction tool, WedgeGUI, has been introduced. The WedgeGUI presented results of electromagnetic wave scattering from a wedge-shaped object under different structures as well as operational parameters. Various models under both line source and plane wave illuminations were included. Comparisons among UTD, PO, PTD, and PE models through many scenarios and illustrations are possible. The WedgeGUI virtual tool can be used in many graduate-level courses such as advanced electromagnetic theory, high-frequency asymptotic methods in electromagnetics, and diffraction theory.

References [1]

[2]

[3]

[4] [5] [6]

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Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries: Analysis of Asymptotic Techniques,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 3, June 2011, pp. 232−253. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “On the Numerical Evaluations of Diffraction Formulas for the Canonical Wedge Scattering Problem,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 5, 2013, pp. 257−272. Cakir, G., L. Sevgi, and P. Ya. Ufimtsev, “FDTD Modeling of Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries: Comparisons against Analytical Models and Calibration,” IEEE Transactions on Antennas and Propagation, Vol. 60, No. 7, 2012, pp. 3336−3342. Sommerfeld, A., “Mathematische Theorie der Diffraction,” Mathematische Annalen, Vol. 16, 1896, pp. 317−374. Macdonald, H. M., Electric Waves, Cambridge, England: The University Press, 1902, pp. 186−198. Sommerfeld, A., “Theorie der Beugung,” Chapter 20 in the book Die Differential- und Integralgleichungen der Mechanik und der Physik, The second (Physical) part, Ph. Frank and R. Mises (eds.), Friedr. Vieweg & Sohn, Braunschweig, Germany, 1935. American publication by Mary S. Rosenberg, New York, 1943.

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CHAPTER

9 Contents 9.1 Introduction 9.2 FRINGEGUI Software 9.3 Characteristic Examples 9.4 Summary

FringeGUI Virtual Package This chapter presents a MATLAB-based virtual tool for the visualization of both fringe currents and fringe waves.

9.1 Introduction Electromagnetic waves interact with objects and scatter. Scattered fields include (but are not limited to) reflected waves, refracted waves, and diffracted waves. One reason for diffraction is a sharp boundary discontinuity such as an edge or a tip. The two-dimensional wedge with nonpenetrable boundaries is a canonical structure where all these wave phenomena can be investigated, discriminated, and visualized. Early analytical studies can be grouped into two types: ray-based models such as GTD [1,2] and UTD [3], and source-induced-current based model such as PTD [4−6]. The paper by Balanis, Sevgi, and Ufimtsev [7] also reviews diffraction models/approaches of the last 50 years, which also includes comparisons between ray (GTD) and source-induced-current (PTD) based diffraction methods. Another review paper, with many visuals and useful MATLAB scripts, is presented by Hacivelioglu, Sevgi, and Ufimtsev [8]. Numerical techniques used in the computation of complex GTD, UTD, and PTD equations/ integrals and MATLAB codes prepared for this purpose can be found in [9]. A MATLAB-based, free diffraction virtual tool WedgeGUI, which 177

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can be used in the visualization of all wave pieces around the wedge, was discussed in Chapter 8 and [10]. Papers in [11,12] focus on the FDTD modeling of wedge diffraction. Papers in [13,14] focus on MoM modeling of wedge diffraction. Double-tip diffraction modeling using FDTD, MoM, and FEM is investigated in [15,16]. According to PTD, source-induced surface currents may be divided into two types: uniform currents (caused by planar boundaries) and nonuniform currents (caused by edges and/or tips). Uniform and nonuniform currents are also known as PO and fringe currents, respectively. Fringe currents generate fringe waves [6]. Recently, fringe waves generated around a nonpenetrable wedge by a line source are investigated in [17]. Fringe currents and fringe waves for the same nonpenetrable wedge structure are also modeled using MoM [18−20], FDTD [21], and FEM [22]. Almost perfect agreement among analytical (PTD) and numerical (MoM, FDTD, FEM) models demonstrate the validity of the fringe waves obtained via the model called modified theory of physical optics (MTPO) [23]. A new useful MATLAB package—Fringe GUI—for the visualization of both fringe currents and fringe waves, is presented in this chapter. The package uses the new fringe integrals presented in [19]. The PTD and MTPO models are also included so that a user can visualize fringe waves computed with MoM, PTD, and MTPO separately, or all at once in the same figure.

9.2  FRINGEGUI Software The two-dimensional nonpenetrable wedge structure is pictured in Figure 9.1. The exterior angle of the wedge is α . The tip of the wedge is located at the origin on the xy-plane. The top face of the wedge is located along the positive x-axis (φ = 0°). The bottom face is located along φ = α . The structure is infinite along the third (z-) axis. A plane wave (uinc) with an incidence angle φ = φ inc illuminates the wedge. The wave function u represents either of electric/magnetic field components according to its polarization. The case when u = Ez is the SBC. The magnetic field has x- and y-components (H x and Hy). In this case, regardless of the angle of incidence, the electric field is always parallel to the faces of the wedge. The electric field is also parallel to the edge of the wedge. The case when u = Hz is the HBC. In this case, the electric field has x- and ycomponents (Ex and Ey). A number of receivers are located on a circle with a specified radius around the tip of the wedge and the scattered waves are recorded. Analytical models such as PTD yield total fields, scattered fields, diffracted fields, as well as fringe fields. On the other hand, numerical models use multistep approaches as explained in [11,13]. Note that the two previously introduced

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Figure 9.1  The two dimensional wedge diffraction scenario.

free packages WedgeGUI [10] and WedgeFDTD [12] can be used in extracting and observing diffracted fields around the tip of the wedge. Here in FringeGUI, fringe waves around the tip of the wedge are extracted using the new PTD formulations derived in [17], fringe MoM integrals (presented in [19]), and MTPO formulations [23]. The flow chart of the algorithms inherited behind FringeGUI is given in Figure 9.2. The main executable file of the package is FringeGUI.EXE. Figure 9.3 displays the front panel and Table 9.1 lists input parameters. The panel has

Figure 9.2  A flowchart of FringeGUI algorithm.

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FringeGUI Virtual Package Table 9.1 User-Specified Operational Parameters of the FringeGUI Package

Operational Parameters Exterior wedge angle (deg) Wedge length (lambda) Boundary condition Frequency (MHz) Incident angle (deg) Observer radius (m) N MoM PTD MTPO 3-D fields Currents RUN CLEAR CLOSE

Explanation Exterior angle of the wedge Length of the wedge for MoM calculation Soft BC or hard BC on wedge surface Operating frequency Incidence angle of the plane wave excitation Observer radius for fringe wave calculation Number of segments per wavelength for MoM calculation Method of moments Physical theory of diffraction Modified theory of diffraction Plot three-dimensional fringe fields distribution Plot surface currents on the wedge Run the program Clear figure Close the program

two parts. The right part belongs to plot the fringe waves in polar form. The left part is divided into three vertical panels. The wedge parameters (i.e., the exterior wedge angle, wedge length, boundary type) are supplied from the top panel. The mid panel is for the source properties. A plane wave excitation is used. The bottom panel is reserved for the output. There are also three operational RUN, CLEAR, and CLOSE pushbuttons. Figures of threedimensional fields, surface currents, and the GUI panel may be saved by using the TIFF box after fringe wave calculation. The user may select both

Figure 9.3  The front panel of the FringeGUI MATLAB package.

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analytical (PTD) and numerical (MoM and MTPO) methods for comparison. The user may also plot results using the MTPO method.

9.3  Characteristic Examples The fringe wave calculations around the wedge using analytical (PTD) and numerical (MoM) methods are presented for different scenarios. A frequency of 30 MHz is chosen. Infinite wedge faces are truncated in 100 wavelengths and segment lengths are taken as λ /40 for MoM calculations. First, the total and fringe currents on both faces of the wedge up to 10 wavelengths from the tip are compared for SBC and HBC cases. Next, fringe waves around the wedge at a circle having a 2-wavelength radius are plotted using FringeGUI. Then, the fringe field distribution around the wedge is shown. Results in Figures 9.4, 9.5, and 9.6 belong to the SSI when only the top face is illuminated by a plane wave. As observed in Figure 9.5, a good

Figure 9.4  Total and nonuniform surface currents using MoM from a wedge at 30 MHz with different boundary conditions (SSI): α = 300°, φ inc = 45°, ds = λ /40, r = 2λ , N = 8000; (top) SBC, (bottom) HBC.

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Figure 9.5  Fringe waves using PTD, MoM, and MTPO from a wedge at 30 MHz with different boundary conditions (SSI): α = 300°, φ inc = 45°, ds = λ /40, r = 2λ , N = 8000; (top) SBC, (bottom) HBC.

Figure 9.6  Fringe field distribution using PTD from a wedge at 30 MHz with different boundary conditions (SSI): α = 300°, φ inc = 45°, r = 2λ ; (top) SBC, (bottom) HBC.

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agreement between the analytical (PTD) and numerical (MoM) results is obtained. A comparison of different polarizations shows the difference of the fringe waves with respect to the observation angles. Second, the examples in Figures 9.7, 9.8, and 9.9 belongs to DSI when both faces are illuminated. Finally, the same comparisons are given for the narrow interior angle of the wedge in Figures 9.10, 9.11, and 9.12. By increasing the exterior wedge angle, MoM results require more discretization to have better accuracy and therefore the computation is expensive.

Figure 9.7  Total and nonuniform surface currents using MoM from a wedge at 30 MHz with different boundary conditions (DSI): α = 300°, φ inc = 150°, r = 2λ , ds = λ /40, N = 8000; (top) SBC, (bottom) HBC.

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Figure 9.8  Fringe waves using PTD, MoM, and MTPO from a wedge at 30 MHz (DSI) with α = 300°, φ inc = 150°, ds = λ /40, r = 2λ , N = 8000; (top) SBC, (bottom) HBC.

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Figure 9.9  Fringe field distribution using PTD from a wedge at 30 MHz (DSI) with α = 300°, φ inc = 150°, r = 2λ ; (top) SBC, (bottom) HBC.

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Figure 9.10  Total and nonuniform surface currents using MoM from a wedge at 30 MHz (SSI) with α = 340°, φ inc = 135°, ds = λ /40, r = 2λ , N = 8000, (top) SBC, (bottom) HBC.

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Figure 9.11  Fringe waves using PTD, MoM, and MTPO from a wedge at 30 MHz (SSI) with α = 340°, φ inc = 135°, ds = λ /40, r = 2λ , N = 8000; (top) SBC, (bottom) HBC.

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Figure 9.12  Fringe field distribution using PTD from a wedge at 30 MHz (SSI) with α = 340°, φ inc = 135°, r = 2λ ; (top) SBC, (bottom) HBC.

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9.4 Summary A new MATLAB-based fringe wave simulator, FringeGUI, for a nonpenetrable wedge has been developed. FringeGUI uses two different models: one is analytical (PTD), and the other is numerical (MoM).

References [1] Keller, J. B., “Diffraction by an Aperture,” J. Appl. Phys., Vol. 28, 1957, pp. 426−444. [2] Keller, J. B., “Geometrical Theory of Diffraction,” J. Opt. Soc. Am., Vol. 52, 1962, pp.116−130. [3] Kouyoumjian, R. G., and P. H. Pathak, “A Uniform Geometrical Theory of Diffraction for an Edge in a Perfectly Conducting Surface,” Proc IEEE, Vol. 62, 1974, pp. 1448−1461. [4] Ufimtsev, P. Ya., “Diffraction at a Wedge and a Strip,” Part I of “Approximate Computation of the Diffraction of Plane Electromagnetic Waves at Certain Metallic Objects,” Zhurnal Teknichheskoi Fiziki, Vol. 27, No. 8, 1957, pp. 1840−1849 (English translation published by: Soviet Physics-Technical Physics). [5] Ufimtsev, P. Ya., Theory of Edge Diffraction in Electromagnetics: Origination and Validation of the Physical Theory of Diffraction, Raleigh, NC: SciTech Publishing, 2009. [6] Ufimtsev, P. Ya., Fundamentals of the Physical Theory of Diffraction, Wiley and Sons (first edition 2007, second edition 2014). [7] Balanis, C. A., L. Sevgi, and P. Ya. Ufimtsev, “Fifty Years of High Frequency Diffraction,” International Journal of RF and Microwave Computer-Aided Engineering, Vol. 23, No. 4, July 2013, pp. 394−402. [8] Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries: Analysis of Asymptotic Techniques,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 3, June 2011, pp. 232−253. [9] Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “On The Numerical Evaluation of Diffraction Formulas for the Canonical Wedge Scattering Problem,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 5, October 2013, pp. 257−272. [10] Hacivelioglu, F., M. A. Uslu, and L. Sevgi, “A MATLAB-based Virtual Tool for the Electromagnetic Wave Scattering from a Perfectly Reflecting Wedge,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 6, December 2011, pp. 234−243. [11] Cakir, G., L. Sevgi, and P. Ya. Ufimtsev, “FDTD Modeling of Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries: Comparisons Against Analytical Models and Calibration,” IEEE Transactions on Antennas and Propagation, Vol. 60, No. 7, July 2012, pp. 3336−3342. [12] Uslu, M. A., and L. Sevgi, “MATLAB-based Virtual Wedge Scattering Tool for the Comparison of High Frequency Asymptotics and FDTD Method,” Applied

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[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

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Computational Electromagnetics Society Journal, Vol. 27, No. 9, September 2012, pp. 697−705. Apaydin, G., and L. Sevgi, “A Novel Wedge Diffraction Modeling Using Method of Moments,” Applied Computational Electromagnetics Society Journal, Vol. 30, No. 10, October 2015, pp. 1053−1058. Apaydin, G., and L. Sevgi, “Method of Moments (MoM) Simulation of Backscattering by a Soft-Hard Strip,” IEEE Transactions on Antennas and Propagation, Vol. 63, No. 12, December 2015, pp. 5822−5826. Uslu, M. A., G. Apaydin, and L. Sevgi, “Double Tip Diffraction Modeling: Finite Difference Time Domain vs. Method of Moments,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 12, December 2014, pp. 6337−6343. Ozgun, O., and L. Sevgi, “Double-Tip Diffraction Modeling: Finite Element Modeling (FEM) vs. Uniform Theory of Diffraction (UTD),” IEEE Transactions on Antennas and Propagation, Vol. 63, No. 6, June 2015, pp. 2686−2693. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Wedge Diffracted Waves Excited by a Line Source: Exact and Asymptotic Forms of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 61, No. 9, September 2013, pp. 4705−4712. Apaydin, G., Hacivelioglu, F., Sevgi, L., and Ufimtsev, P. Ya., “Wedge Diffracted Waves Excited by A Line Source: Method of Moments (MoM) Modeling of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 8, pp. 4368−4371, Aug. 2014. Apaydin, G., F. Hacivelioglu, L. Sevgi, and P. Ya. Ufimtsev, “Fringe Waves from a Wedge with One Face Electric and the Other Face Magnetic” IEEE Transactions on Antennas and Propagation, Vol. 64, No. 3, 2016, pp. 1125−1130. Apaydin, G., Sevgi, L., and Ufimtsev, P. Ya., “Fringe Integral Equations for The 2‐D Wedges with Soft and Hard Boundaries.” Radio Science, Vol. 51, No. 9, 2016, pp. 1570−1578. Uslu, A., G. Apaydin, and L. Sevgi, “Finite Difference Time Domain Modeling of Fringe Waves,” Applied Computational Electromagnetics Society Journal, Vol. 32, No. 7, 2017, pp. 575−580. Ozgun, O., and L. Sevgi, “Finite Element Modeling of Fringe Waves in Wedge Diffraction Problem,” IEEE Antennas and Wireless Propagation Letters, Vol. 15, 2016. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “On the Modified Theory of Physical Optics,” IEEE Transactions on Antennas and Propagation, Vol. 61, No. 12, December 2013, pp. 6115−6119.

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CHAPTER

10

WedgeTOOL Virtual Package

Contents 10.1 Introduction 10.2 WedgeTOOL Software 10.3 Characteristic Examples 10.4 Summary

10.1 Introduction The word scattering is used to represent all components produced from the interaction of EM waves with objects and includes incidence, reflection, refraction, and diffraction. Mathematical methods, such as GO, PO, GTD, UTD, and PTD can be used when the wavelength is small compared with the interacted object size [1−10]. Diffraction modeling has also been investigated by numerical methods such as FDTD [11−14] in the time domain and MoM) [15−19] in the frequency-domain, which are effective methods for wedges. In this chapter, the MATLAB-based WedgeTOOL, which uses two-step FDTD [13] and MoM [19] approaches, is developed for the extraction and visualization of diffracted fields on the canonical nonpenetrable wedge scattering problem, and the results are validated against HFA approaches. 10.1.1  Wedge Diffraction Modeling The nonpenetrable wedge diffraction problem is canonical and plays a fundamental role in the understanding and construction of HFA techniques as well as for numerical tests. The nonpenetrable wedge scattering problem in polar coordinates is specified with the 191

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line source S(r0,φ 0), the receiver point R(r,φ ), and the exterior angle of the wedge (α ). The wedge has symmetry along z and therefore 2-D on the xy plane can be considered. According to the position of the line source, either SSI for the illumination of the top face (0 < φ 0 < α − π ) or DSI for the illumination of both faces (α − π < φ 0 < π ) are taken into account. The field outside the wedge satisfies the wave equation, the BC, and SRC at infinity. In the case of acoustic waves, either the field or its normal derivative is zero on the surface and these conditions refer to acoustically soft (SBC) and hard (HBC) wedges, respectively. In the case of EM waves, SBC (HBC) corresponds to the z-component of electric (magnetic) field intensity Ez − TM (Hz − TE) [2]. The total field solution of the wave equation based on the series summation for both SSI and DSI are given in [2,3]. Then, the diffracted fields can be calculated by subtracting the GO fields from the total field in different regions by considering shadow and reflected region. 10.1.2  Finite Difference Time Domain Modeling The diffraction modeling is introduced by using the FDTD-based MATLAB package WedgeFDTD in the visualization of diffracted fields using this multistep FDTD procedure as follows [3,14]: The FDTD simulation is run separately for each of the three scenarios:  The first scenario is modeled with a wedge structure and the total fields (incident, reflected, and diffracted) are obtained. The total fields have incident, reflected, and diffracted fields for 0 ≤ φ < π − φ 0; incident and diffracted fields for π − φ 0 ≤ φ < π + φ 0; and the diffracted fields for π + φ 0 ≤ φ < α for SSI.  For the second scenario, FDTD is run for the infinite plane for α = 180°, which yields the incident and reflected fields on the upper half-plane. Since there is no edge or tip, the total fields do not contain diffracted fields.  The third scenario is modeled in free-space by removing the wedge. The incident fields are recorded.  All transient responses are recorded. Subtracting the time data of the second scenario from the first scenario for (0 ≤ φ < π − φ 0); and the time data of the third scenario from the first scenario for (π − φ 0 ≤ φ < π + φ 0) yields diffracted-only fields all around the wedge. Note that the three-step procedure is enough to obtain diffracted fields under the SSI condition, but another infinite plane consisting of two halfplanes oriented along with the angles φ = α − π and φ = α is required for DSI.

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10.1.3  Method of Moments Modeling MoM is one of the earliest frequency-domain numerical techniques used in EM [15−18]. Wedge scattering may also be modeled with MoM. In this model, the faces of the wedge are divided into small segments compared to the wavelength. The source-excited segment fields are calculated, the matrix system is numerically solved, and the unknown segment currents are derived. Then, the segment-scattered fields at the observer are accumulated. Finally, the incident field at the observer is added and the total fields are obtained [19]. The MoM-computed diffracted fields can be extracted with the following steps [19]:  The MoM simulation is run and the incident fields and total fields are obtained for the wedge;  The reflected-only fields can be obtained from an infinite-plane scenario (i.e., by taking α = π and repeating the MoM procedure);  Subtracting the reflected-only fields from the total fields for 0 ≤ φ < π − φ 0; and subtracting the incident fields from the total fields for 0 ≤ φ < π + φ 0 will yield diffracted-only fields all around the wedge.

10.2  WedgeTOOL Software The main executable file of the package is WedgeTOOL.EXE. The front panel is displayed in Figure 10.1 and Table 10.1 shows the input parameters of the package. The panel is mainly divided into two parts. The right part belongs to plot total and diffracted fields. The left part is for choosing the input parameters and the controls. The WedgeTOOL package has been developed in MATLAB-GUI for the analysis and visualization of diffraction of the wedge. WedgeTOOL has been designed in a way that the user can:  Visualize, load, and save various wedge scenarios;  Supply input parameters;  Plot the total and diffracted fields (MoM, FDTD, HFA);  Specify the boundary condition type;  Specify the operating frequency;  Specify the wedge angle, the source position, and the observer radius.

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WedgeTOOL Virtual Package Table 10.1 User-Specified Operational Parameters of the WedgeTOOL GUI Package

Operational Parameters Number of cells in x dimension Number of cells in y dimension PML thickness Frequency (MHz) Maximum time step Segment size in wavelength Wedge angle (deg) Incident distance (m) and angle (deg) Observation distance (m) Frequency (MHz) Angle step TE/TM

Explanation FDTD cell FDTD cell FDTD analysis Operating frequency FDTD analysis MoM analysis Exterior wedge angle Line source position in polar coordinates Radial distance from the tip EM wave signal Angular resolution Wave polarization

The left part is divided into four panels. The source frequency in MHz, and the wedge angle, the FDTD, and MoM parameters are supplied from the first three panels. The two operational pushbuttons (RUN and CLEAR) are reserved for normal operations. The FDTD visualization, diffracted, and total fields versus angle results are displayed on the right plot.

Figure 10.1  The front panel of the WedgeTOOL GUI MATLAB package.

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10.3  Characteristic Examples Numerical simulation results obtained with the WedgeTOOL package are presented in this section. Examples given in Figures 10.2, 10.3, 10.4, and 10.5 belong to total and diffracted fields from the nonpenetrable wedge. Figure 10.2 shows the scenario and the parameters used in the simulations. As seen, good agreement between the analytical HFA model and FDTD- and MoM-based numerical models is observed. A wedge with 30° interior angle is taken into account. Soft BC (TM z case) is assumed. Total and diffracted fields versus angle around the tip of the wedge on a circle with 50m radius are plotted in Figures 10.2 and 10.3, respectively. The wedge is illuminated by a line source located at 70m distance with φ 0 = 60°. MoM results are compared with exact series representation and FDTD model. As observed, very good agreement is obtained. The second example in Figures 10.4 and 10.5 belongs to a nonpenetrable wedge with the same interior angle and illumination angle as the previous example, but HBC is used. As observed, the agreement among the models is also very good.

Figure 10.2  Total fields around PEC wedge as a function of angles at 30 MHz using exact series, MoM, and FDTD solutions: α = 330°, r = 50m, r0 = 70m, φ 0 = 60° (TM/SBC case).

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Figure 10.3  Diffracted fields around the PEC wedge as a function of angles at 30 MHz using exact series, UTD, MoM, and FDTD solutions: α = 330°, r = 50m, r0 = 70m, φ 0 = 60° (TM/ SBC case).

Figure 10.4  Total fields around PEC wedge as a function of angles at 30 MHz using exact series, MoM, and FDTD solutions: α = 330°, r = 50m, r0 = 70m, φ 0 = 60° (TE/HBC case).

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Figure 10.5  Diffracted fields around PEC wedge as a function of angles at 30 MHz using exact series, UTD, MoM, and FDTD solutions: α = 330°, r = 50m, r0 = 70m, φ 0 = 60° (TE/HBC case).

10.4 Summary In this chapter, a MATLAB-based electromagnetic wedge diffraction virtual tool was developed. The virtual tool uses numerical FDTD and MoM algorithms and the HFA approach.

References [1] [2] [3]

[4]

[5]

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Balanis, C. A., Advanced Engineering Electromagnetics, John Wiley & Sons, 2012. Ufimtsev, P. Ya., Fundamentals of the Physical Theory of Diffraction, John Wiley & Sons, 2007. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Electromagnetic Wave Scattering from A Wedge with Perfectly Reflecting Boundaries: Analysis of Asymptotic Techniques,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 3, June 2011, pp. 232−253. Hacivelioglu, F., L. Sevgi, and P. Ya. Ufimtsev, “Wedge Diffracted Waves Excited by a Line Source: Exact and Asymptotic Forms of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 61, No. 9, September 2013, pp. 4705−4712. Balanis, C., L. Sevgi, and P. Ya. Ufimtsev, “Fifty Years of High Frequency Asymptotics,” International Journal on RF and Microwave Computer-Aided Engineering, Vol. 23, No. 4, July 2013, pp. 394−402.

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WedgeTOOL Virtual Package

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Pelosi, G., Y. Rahmat-Samii, and J. Volakis, “High Frequency Techniques in Diffraction Theory: 50 Years of Achievements in GTD, PTD, and Related Approaches,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp. 16−17. Ufimtsev, P. Ya., “The 50-year Anniversary of PTD: Comments on the PTD’s Origin and Development,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp.18−28. Rahmat-Samii, Y., “GTD, UTD, UAT, and STD: A Historical Revisit and Personal Observations,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp. 29−40. Pelosi, G. and S. Selleri, “The Wedge-type Problem: The Building Brick in High-Frequency Scattering from Complex Objects,” IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013, pp. 41−60. Hacivelioglu, F., M. A. Uslu, and L. Sevgi, “A MATLAB-based Virtual Tool for the Electromagnetic Wave Scattering from a Perfectly Reflecting Wedge,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 6, December 2011, pp. 234−243. Yee, K., “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Transactions on Antennas and Propagation, Vol. 14, No. 3, May 1966, pp. 302−307. Stratis, G., V. Anantha, and A. Taflove, “Numerical Calculation of Diffraction Coefficients of Generic Conducting and Dielectric Wedges Using FDTD,” IEEE Transactions on Antennas and Propagation, Vol. 45, No. 10, October 1997, pp. 1525−1529. Cakir, G., L. Sevgi, and P. Ya. Ufimtsev, “FDTD Modeling of Electromagnetic Wave Scattering from a Wedge with Perfectly Reflecting Boundaries: Comparisons against Analytical Models and Calibration,” IEEE Transactions on Antennas and Propagation, Vol. 60, No. 7, July 2012, pp. 3336−3342. Uslu, M. A., and L. Sevgi, “MATLAB-based Virtual Wedge Scattering Tool for the Comparison of High Frequency Asymptotics and FDTD Method,” Applied Computational Electromagnetics Society Journal, Vol. 27, No. 9, 2012, pp. 697−705. Harrington, R. F., Field Computation by Moment Method, New York: IEEE Press, (first edition 1968), 1993. Arvas, E., and L. Sevgi, “A Tutorial on the Method of Moments,” IEEE Antennas and Propagation Magazine, Vol. 54, No. 3, June 2012, pp. 260−275. Apaydin, G., and L. Sevgi, “A Canonical Test Problem for Computational Electromagnetics (CEM): Propagation in A Parallel-Plate Waveguide,” IEEE Antennas and Propagation Magazine, Vol. 54, No. 4, August 2012, pp. 290−315. Apaydin, G., and L. Sevgi, “Method of Moments (MoM) Modeling for Resonating Structures: Propagation Inside a Parallel Plate Waveguide,” Applied Computational Electromagnetics Society Journal, Vol. 27, No. 10, October 2012, pp. 842−849. Apaydin, G., F. Hacivelioglu, L. Sevgi, and P. Ya. Ufimtsev, “Wedge Diffracted Waves Excited by a Line Source: Method of Moments (MoM) Modeling of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 8, August 2014, pp. 4368−4371.

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15] [16] [17]

[18]

[19]

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Selected Bibliography Apaydin, G., and L. Sevgi, “A Novel Wedge Diffraction Modeling Using Method of Moments (MoM),” Applied Computational Electromagnetics Society Journal, Vol. 30, No. 10, October 2015, pp. 1053−1058. Apaydin, G., and L. Sevgi, “Method of Moments Modeling of Backscattering by a Soft−Hard Strip,” IEEE Transactions on Antennas and Propagation, Vol. 63, No. 12, December 2015, pp. 5822−5826. Apaydin, G., F. Hacivelioglu, L. Sevgi, and P. Ya. Ufimtsev, “Fringe Waves from a Wedge with One Face Electric and the Other Face Magnetic,” IEEE Transactions on Antennas and Propagation, Vol. 64, No. 3, March 2016, pp. 1125−1130. Apaydin, G., and L. Sevgi, “The Two-Dimensional Nonpenetrable Wedge Scattering Problem and a MATLAB-Based Fringe Wave Calculator,” IEEE Antennas and Propagation Magazine, Vol. 58, No. 2, April 2016, pp. 86−93. Apaydin, G., F. Hacivelioglu, L. Sevgi, W. B. Gordon, and P. Ya. Ufimtsev, “Diffraction at a Rectangular Plate: First-Order PTD Approximation,” IEEE Transactions on Antennas and Propagation, Vol. 64, No. 5, May 2016, pp. 1891−1899. Apaydin, G., Hacivelioglu, L. Sevgi, and P. Ya. Ufimtsev, “Wedge Diffracted Waves Excited by a Line Source: Method of Moments (MoM) Modeling of Fringe Waves,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 8, August 2014, pp. 4368−4371. Apaydin, G., L. Sevgi, and P. Ya. Ufimtsev, “Fringe Integral Equations for the 2-D Wedges with Soft and Hard Boundaries,” Radio Science, Vol. 51, No. 9, September 2016, pp. 1570−1578. Apaydin, G., L. Sevgi, and Ufimtsev, P. Ya., “Diffraction at Rounded Wedges: MoM Modeling of PTD Fringe Waves,” Applied Computational Electromagnetics Society Journal, Vol. 32, No. 7, July 2017, pp. 600−607. Apaydin, G., L. Sevgi, and Ufimtsev, P. Ya., “Extension of PTD for Finite Objects with Rounded Edges: Diffraction at a Soft Trilateral Cylinder,” IEEE Antennas and Wireless Propagation Letters, Vol. 16, 2017, pp. 2590−2593. Apaydin, G., L. Sevgi, and P. Ya. Ufimtsev, “Diffraction of Acoustic Waves at TwoDimensional Hard Trilateral Cylinders with Rounded Edges: First-Order Physical Theory of Diffraction Approximation,” The Journal of the Acoustical Society of America, Vol. 143, No. 5, May 2018, pp. 2792−2795.

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Selected Bibliography

Hacivelioglu, F., M. A. Uslu, and L. Sevgi, “A MATLAB-Based Virtual Tool for the Electromagnetic Wave Scattering from a Perfectly Reflecting Wedge,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 6, December 2011, pp. 234−243. Hacivelioglu, F., G. Apaydin, L. Sevgi, and P. Ya. Ufimtsev, “Diffraction at Trilateral Cylinders with Combinations of Soft and Hard Faces: First-Order PTD Approximation,” Electromagnetics, Vol. 38, No. 4, March 2018, pp. 217−225. Sevgi, L., “Electromagnetic Modeling and Simulation: Challenges in Validation, Verification and Calibration,” IEEE Transactions on Electromagnetic Compatibility, Vol. 56, No. 4, August 2014, pp. 750−758. Sevgi, L., “Electromagnetic Diffraction Modeling: High Frequency Asymptotics vs. Numerical Techniques,” Applied Computational Electromagnetics Society Journal, Vol. 32, No. 7, July 2017, pp. 555−561. Uslu, M. A., G. Apaydin, and L. Sevgi, “Double Tip Diffraction Modeling: Finite Difference Time Domain vs. Method of Moments,” IEEE Transactions on Antennas and Propagation, Vol. 62, No. 12, December 2014, pp. 6337−6343. Uslu, M. A., G. Apaydin, and L. Sevgi, “Diffraction Modeling by a Soft-Hard Strip using Finite-Difference Time-Domain Method,” IEEE Antennas and Wireless Propagation Letters, Vol. 16, 2017, pp. 306−309. Uslu, M. A., G. Apaydin, and L. Sevgi, “Finite Difference Time Domain Modeling of Fringe Waves,” Applied Computational Electromagnetics Society Journal, Vol. 32, No. 7, July 2017, pp. 575−580.

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List of Acronyms 2-D

two-dimensional

3-D

three-dimensional

BC

boundary condition

DF

diffracted field

DBC

Dirichlet boundary condition

DSI

double-side illumination

EM

electromagnetic

FDTD

finite-difference time-domain

FEM

finite element method

FFT

fast Fourier transform

GO

geometric optics

GTD

geometric theory of diffraction

GUI

graphical user interface

HBC

hard boundary condition

HFA

high-frequency asymptotics

I

incident

ISB

incident shadow boundary

LS

line source

MoM

method of moments

NBC

Neumann boundary condition 201

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202

List of Acronyms

PEC

perfectly electrical conducting

PMC

perfect magnetic conducting

PO

physical optics

PTD

physical theory of diffraction

PW

plane wave

RCS

radar cross section

RSB

reflection shadow boundary

SB

shadow boundary

SBC

soft boundary condition

SF

scattered field

SHBC

soft-hard boundary condition

SRC

Sommerfeld’s radiation condition

SSI

single-side illumination

TE

transverse electric

TED

theory of edge diffraction

TM

transverse magnetic

UTD

uniform theory of diffraction

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About the Authors Gökhan APAYDIN is a professor in the Department of Electrical-Electronics Engineering, Uskudar University, Istanbul, Turkey. He received B.S., M.S., and Ph.D. degrees in electrical and electronics engineering from Bogazici University, Istanbul, Turkey, in 2001, 2003, and 2007, respectively. He was a teaching and research assistant with Bogazici University from 2001 to 2005, a project engineer with the University of Technology Zurich, Zurich, Switzerland from 2005 to 2010, and a visiting associate professor in the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Illinois, in 2015. He has been involved with complex electromagnetic problems and systems and his research interests include analytical and numerical methods in electromagnetics (especially on computational electromagnetics, wave propagation, diffraction modeling, scattering, and related areas). He is also interested in novel approaches in engineering education and teaching electromagnetics via virtual tools. He has authored two books, over 50 scientific journal articles, and presented over 30 conference papers. His book entitled Radio Wave Propagation and Parabolic Equation Modeling was published by IEEE Press−John Wiley in 2017. He is a Senior Member of the IEEE and associate editor at IEEE Access. Levent SEVGI is a Fellow of the IEEE, distinguished lecturer of the IEEE AP Society (2020−2022), associate editor of the IEEE AP Magazine, and the writer/editor of the “Testing Ourselves” column (since 2007), a member of the IEEE AP Society AdCom (2013−2015) and Education Committee (since 2006), an EB member of IEEE Access (2017−2019 and 2020−2022). He has been involved with complex electromagnetic problems and systems for 203

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About the Authors

more than three decades. He is also interested in novel approaches in engineering education and popular science topics such as science, technology and society, and public understanding of science. He has published many books/book chapters in English and Turkish and over 180 journal and magazine articles, papers, and tutorials, including three books, Complex Electromagnetic Problems and Numerical Simulation Approaches, Electromagnetic Modeling and Simulation, and Radiowave Propagation and Parabolic Equation Modeling, published by IEEE Press−WILEY in 2003, 2014, and 2017, respectively, and A Practical Guide to EMC Engineering, published by Artech House in September 2017.

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Index A Ampere’s law, 2, 6 Analytical high-frequency techniques, 10 Angular frequency, 6, 68 Artificial reflections, 85 B Backscattering, 69–73, 109–114 Backscattering RCS, 109, 113–115 Bistatic RCS, 115–119 Bistatic scattering, 69, 93–95, 116, 119, 140, 144–145, 147 Boundary conditions, 7, 16, 27, 69, 91, 124, 154, 171 C Cartesian coordinates, 2, 7 Cauchy principal, 52, 126, 138 Continuity relation, 2 Constitutive parameters, 3 Convolutional perfectly matched layer (CPML), 81–82 Critical angle, 14, 26, 37, 111 Curl, 2–6 Cylindrical coordinates, 2–3, 7 D Differential form, 1 Diffracted field, 10–11, 14–15, 19, 21, 23, 26–28, 35, 37, 38, 40–42,

81–86, 89, 91, 124–125, 130, 153, 156, 160 Diffraction modeling, 151–159 Dirac delta, 8, 16, 71, 153 Directivity patterns, 9, 69 Divergence, 2, 7 Double-tip diffraction, 152–156, 159–162, 164 Double-side illumination (DSI), 14–15, 22, 192 Dirichlet, 7, 68 E Edge waves, 68–69, 91, 93, 95, 101, 116, 136–137, 144, 169 Electric conductivity, 3, 20 Electric current density, 2 Exterior angle, 13, 18, 47, 170, 178, 180, 192 Euler constant, 33, 72, 157 F Faraday’s law, 1, 6 Fast Fourier transform (FFT), 81, 155–156 Finite difference time domain (FDTD), 56, 81–86, 192 Flux density vectors, 1, 3 Forward scattering, 74, 84, 89, 91, 108, 116, 119, 140, 144, 160

205

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206 Index

Frequency domain, 6–7, 12, 32, 5–60, 152, 191, 193 Fresnel function, 30, 112 Fringe current, 26–27, 37, 56, 58–59, 63, 106–107, 111, 127, 129, 131–147, 178, 181 Fringe field, 8, 55, 59–62, 112, 116, 119, 125, 128–133, 136, 139, 144, 178, 180–182, 185, 188 FringeGUI, 177–188 Fringe wave, 27–28, 34–39, 42–51, 91, 95, 98, 101–108, 111–119, 141–143, 146–147, 178–184 G Gauss’s law, 1–2, 6 Geometric optics (GO), 21–24, Geometric theory of diffraction (GTD), 30–31 Grazing incidence, 26, 108, 110, 112, 114–117, 122 Green’s function, 8, 17, 20, 32, 34, 37–39, 42–43, 59, 71, 157–158 H Hankel function, 17, 33, 43, 72–73, 157 Hard boundary conditions (HBC), 16, 43–44, 70, 72 Hard strip, 70, 72, 79 High frequency asymptotics (HFA), 12, 16–31, 82, 152 High frequency techniques, 10 Homogeneous wave equation, 7, 71 I Illumination angles, 37, 47, 74, 160 Impedance matrix, 33–34, 38, 72–74, 157–158 Incidence angle, 17, 69, 83, 111, 178, 180 Incident fields, 37–38, 74, 83–84, 86, 152, 154, 156–157, 192–193 Incident shadow boundary (ISB), 14, 31, 85, 160

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Integral equations, 45, 55, 128, 135–136, 138, 142–144, Interior angle, 40, 123, 130, 134, 183, 195 Intrinsic impedance, 33, 157 Isotropic, 3, 6 L Laplace operator, 7 Leading edge, 103–104, 106–107 Line current amplitude, 16, 153 Line source, 10, 14, 16–18, 22, 33–25, 37, 40, 56, 59, 69, 81, 152, 154, 192, 194 M Magnetic conductivity, 20 Maxwell-Ampere’s law, 2 Maxwell Equations, 1–5, 56 Method of moments (MoM), 32, 71–79, 193 Mode summation, 17 N Neumann, 7, 68, 90 Nondispersive medium, 3 Nonuniform current, 8, 26–27, 35–36, 47, 56, 58–62, 125, 178 Normalized backscattering, 69–70 Normalized scattering cross section, 9, 92, 139, 144 P Parallel polarization, 4, 89 Perfectly electrical conducting (PEC), 16, 24, 56, 58–59, 63, 81, 102, 124, 169, 195–197 perfect magnetic conducting (PMC), 81 Permittivity, 3 Perpendicular polarization, 4, 89 Physical optics (PO), 10–11, 24, 27 Physical theory of diffraction (PTD), Plane of incidence, 4–5

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Index Plane wave, 10, 17, 22–25, 45, 47, 50, 56, 69, 83, 89–91, 126, 130, 133, 152, 170, 175, 178, 180–181 Polar coordinates, 10, 13, 17, 20, 68, 152, 191, 194 Polarization, 4–5, 16, 33–34, 37, 40– 41, 59–63, 81, 89, 108, 111–119, 140–142, 161 R Radar cross section (RCS), 9, 56, 81 Radiation condition, 8, 16, 69, 153 Ray asymptotics, 19, 26, 101, 108 Rectangular plate, 101–118, Reflected field, 8, 14–15, 35, 38–39, 81, 83, 124–125, 152–153, 156, 158, 160, 192, Reflection shadow boundary (RSB), 14, 31, 85, 125, 160 Reflected rays, 28 Relative L2-error norm, 49 Rounded edges, 123–147 Rounded wedge, 124–135, S Scattered field, 8–9, 25–27, 32–35, 37, 39, 42, 69–70, 72–76, 91, 98, 108, 125, 132, 157–164 Segment currents, 33–39, 157–158, 193 Segment lengths, 32–34, 37, 40, 45, 47, 139, 144, 156, 161 Shadow boundary, 14, 28, 93, 125 Sharp wedge, 124–130, 134–135 Side edge, 112 Simple medium, 3 Single-side illumination (SSI), 14–17, 20, 22–26, 58–59, 62–63, 181– 181, 186–188, 192 Soft boundary conditions (SBC), 16–19, 22, 33, 43, 69, 70, 72, Soft-hard boundary conditions (SHBC), 70, 73–74, 76–78, 80–86 Soft cylinder, 93, 136, 142, 146 Soft strip, 10–11, 70, 72, 79

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207 Soft-hard strip, 68, 70, 73, 80 Specular direction, 70, 119, 121 Specular reflection, 74, 84, 91, 98, 109, 131, 140–141, 144, 152, 160 Spherical coordinates, 2, 3, 7 Strip, 10–11, 67–86, 111–112, 114, 116, 154–160 Surface currents, 10, 25–27, 31, 35–37, 50, 56–62, 101, 126, 134–135, 143, 180 T TE polarization, 4–5, 16, 34, 41, 63, 155, 161 TM polarization, 4–6, 16, 34, 41, 155, 161 Theory of edge diffraction (TED), 70, 78 Time domain, 2, 6, 32, 56, 58–60, 63, 81–86, 139, 144, 152 Total field, 7–11, 17–21, 39–42, 51, 69, 71–78, 82, 91, 95, 103, 124–126, 156, 193–196 Trailing edge, 102–109, 112–115 Transverse electric (TE), 4–6 Transverse magnetic (TM), 4–6 Triangular cylinder, 89–98, Trilateral cylinder, 90–96, 136–147, U Uniform current, 24–25, 36, 58, 178 Uniform theory of diffraction (UTD), 31, 40, 152, 172–176, 196–197 Unit normal vector, 34, 73, 157 V Volume charge density, 1 W Wave equation, 6–8, 71, 152–153 Wave-number, 6, 16, 102, 69, 153 Wedge, 13–14, 123, 169, 191 WedgeGUI, 169–176 WedgeTOOL, 191–197

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Recent Titles in the Artech House Electromagnetics Series Tapan K. Sarkar, Series Editor

Advanced FDTD Methods: Parallelization, Acceleration, and Engineering Applications, Wenhua Yu, et al. Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, Allen Taflove, editor Analysis Methods for Electromagnetic Wave Problems, Volume 2, Eikichi Yamashita, editor Analytical and Computational Methods in Electromagnetics, Ramesh Garg Analytical Modeling in Applied Electromagnetics, Sergei Tretyakov Anechoic Range Design for Electromagnetic Measurements, Vince Rodriguez Applications of Neural Networks in Electromagnetics, Christos Christodoulou and Michael Georgiopoulos CFDTD: Conformal Finite-Difference Time-Domain Maxwell’s Equations Solver, Software and User’s Guide, Wenhua Yu and Raj Mittra The CG-FFT Method: Application of Signal Processing Techniques to Electromagnetics, Manuel F. Cátedra, et al. Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second Edition, Allen Taflove and Susan C. Hagness Electromagnetic Waves in Chiral and Bi-Isotropic Media, I. V. Lindell, et al. Electromagnetic Diffraction Modeling and Simulation with MATLAB®, Gökhan Apaydin and Levent Sevgi Engineering Applications of the Modulated Scatterer Technique, Jean-Charles Bolomey and Fred E. Gardiol Fast and Efficient Algorithms in Computational Electromagnetics, Weng Cho Chew, et al., editors Fresnel Zones in Wireless Links, Zone Plate Lenses and Antennas, Hristo D. Hristov

Grid Computing for Electromagnetics, Luciano Tarricone and Alessandra Esposito High Frequency Electromagnetic Dosimetry, David A. Sánchez-Hernández, editor High-Power Electromagnetic Effects on Electronic Systems, D. V. Giri, Richard Hoad, and Frank Sabath Intersystem EMC Analysis, Interference, and Solutions, Uri Vered Iterative and Self-Adaptive Finite-Elements in Electromagnetic Modeling, Magdalena Salazar-Palma, et al. Numerical Analysis for Electromagnetic Integral Equations, Karl F. Warnick Parallel Finite-Difference Time-Domain Method, Wenhua Yu, et al. Practical Applications of Asymptotic Techniques in Electromagnetics, Francisco Saez de Adana, et al. A Practical Guide to EMC Engineering, Levent Sevgi Quick Finite Elements for Electromagnetic Waves, Giuseppe Pelosi, Roberto Coccioli, and Stefano Selleri Understanding Electromagnetic Scattering Using the Moment Method: A Practical Approach, Randy Bancroft Wavelet Applications in Engineering Electromagnetics, Tapan K. Sarkar, Magdalena Salazar-Palma, and Michael C. Wicks For further information on these and other Artech House titles, including previously considered out-of-print books now available through our In-Print-Forever® (IPF®) program, contact: Artech House Publishers 685 Canton Street Norwood, MA 02062 Phone: 781-769-9750 Fax: 781-769-6334 e-mail: [email protected]

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