Electromagnetic Behaviour of Metallic Wire Structures

143 Pages·2013·2.979 MB·English

by  S. T. Chui, Lei Zhou (auth.)

Electromagnetic Behaviour of Metallic
Wire Structures
S. T. Chui Lei Zhou
•
Electromagnetic Behaviour
of Metallic
Wire Structures
123
S. T.Chui LeiZhou
Department of Physicsand Astronomy Department of Physics
Universityof Delaware FudanUniversity
Newark, NJ Shanghai
USA China
ISBN 978-1-4471-4158-7 ISBN 978-1-4471-4159-4 (eBook)
DOI 10.1007/978-1-4471-4159-4
SpringerLondonHeidelbergNewYorkDordrecht
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(cid:2)Springer-VerlagLondon2013
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Preface
Overthelast10yearstherehasbeenalotofexcitingdevelopmentinthephotonics
ofmetallicwirestructures.Theconceptandthephysicsinvolvedarequitesimple.
Yet this message is usually obscured and not explained and emphasized. In this
book we try to present our point of view on how these phenomena can be
understood in a simple way without carrying out detailed numerical calculations
every time. It turns out that the method presented here also provides for a very
efficientalgorithmfordealingwithelectromagneticwavesinthesestructures.We
have tried to make the book simple enough so that it will be useful for non-
specialistsinotherfieldstoenterintothisarea.Thisbookisintendedforageneral
audience in both physics and engineering. To include the possibility of using this
as a textbook, we have provided detailed derivations of different formulae. We
have tried to make the book self-contained by providing enough background
information so that it is not necessary to chase through different sources to
understand and follow the development of the ideas and the derivation of the
formulae.
STCwouldliketothankhiscolleaguesforhelpfuldiscussionandwhoprovided
keyideaspresentedinthisbook.Inparticular,hewouldliketothankProf.Zhifang
Lin, Prof. John Xiao, and Prof. Weiyi Zhang. He thanks the US DOE and NASA
forfinancialsupport.ZLthanksMr.CheQuforhelpinnumericalsimulationsand
the NSFC for financial support.
v
Contents
1 Electromagnetic Waves in Metallic Wire Structures . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Circuit Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Resonance Properties of Metallic Ring Systems: A Single Ring . . . 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Simple Physical Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Circuit Parameters for a Single Ring . . . . . . . . . . . . . . . . . . . . 9
2.4 Implementation of the Boundary Conditions. . . . . . . . . . . . . . . 13
2.4.1 The Gap Resistance Approach . . . . . . . . . . . . . . . . . . . 13
2.4.2 The Local Field Approach. . . . . . . . . . . . . . . . . . . . . . 14
2.5 Analytical Results for a Single Ring SRR . . . . . . . . . . . . . . . . 16
2.6 Numerical Results for a Single Ring SRR . . . . . . . . . . . . . . . . 19
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Resonance Properties of Metallic Ring Systems:
More Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Double-Ring SRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Coplanar Double-Ring SRR . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 BC–SRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Helical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Circuit Parameters for an Arbitrary-Shaped
Metallic Thin Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
vii
viii Contents
4.3 Formalism for a Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Circuit Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Helical Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6.1 End-Fire Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.6.2 Broadside Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.7 Helical Circular Polarizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.8 Faraday Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.8.1 Numerical Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 General Multiply Connected Metallic Wire Networks:
T and H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 The ‘‘T’’ Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 Classifying the Eigenmodes by Network Symmetry . . . . 72
5.3.2 Physical Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4 The H Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4.1 Physical Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.5 Comparison Between Results in Previous Sections
and Results from FDTD Simulations. . . . . . . . . . . . . . . . . . . . 86
5.5.1 Descriptions of the FDTD Simulations . . . . . . . . . . . . . 86
5.5.2 T-Shape Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5.3 H-Shape Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 Jerusalem Cross. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7 Metallic Wire Structures Under a Moderate
Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8 Electromagnetic Waves in Wire Composites I: Plasmonics . . . . . . 105
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.2 Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.3 Effect of the Variation of the Current Along the Wire. . . . . . . . 108
8.4 Variation Perpendicular to the Wire:
The ‘‘Pseudopotential’’ Idea . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.5 Photonic Band Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Contents ix
9 Electromagnetic Waves in Wire Composites II: Anisotropic,
Off-Diagonal Magnetoelectric Wire Composites. . . . . . . . . . . . . . . 121
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.2 Propagation of Plane Waves. . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.2.1 One Type of Split Rings . . . . . . . . . . . . . . . . . . . . . . . 125
9.2.2 Medium with Two Types of Rings . . . . . . . . . . . . . . . . 127
9.2.3 3-Ring Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.3 Reflection and Refraction from Composites of Split Rings. . . . . 134
9.4 Propagation of Electromagnetic Waves in Helixes. . . . . . . . . . . 136
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Chapter 1
Electromagnetic Waves in Metallic
Wire Structures
1.1 Introduction
With the advance of nanofabrication technology, more and more artificial
materials have been proposed and developed in the past 20 years. Two repre-
sentativeexamplesarephotonicandphononiccrystals[1–4].Theexistenceofthe
forbidden gap makes it possible to manipulate and control the flow of electro-
magnetic (EM) and acoustic waves. More recently, the focus has switched to
metallic wire structures. Macroscopic metallic wire structures have been much
studied in the past, partly connected with antenna applications. Recently, EM
compositematerialswithmetallicwirecomponentshavereceivedmuchattention.
Metallicwirestructuressuchasfractalshavebeenshowntobeefficientwideband
filters and ultra-compact reflectors [5–7]. Man-made metamaterials with artificial
magnetism [8, 9] and negative refraction index [10, 11] were designed and
experimentally verified. The original negative-index material [10, 11] consists of
straightmetallicwiresandmetallicsplit-ringresonators(SRRs),andthepurposeis
toadjusttheelectricdipoleeigenmodeinthewireandthe‘‘magnetic’’eigenmode
in the SRR, so that their out-of-phase responses have an overlapping frequency
region.Sincethen,near-fieldsubwavelengthimageshavebeenrealizedbyseveral
groups with different types of metamaterials [12–15], based on early theoretical
proposals[16,17].Thepossibilityofmakingmetallicwirestructuresofsizesless
than microns opens the door to explore new physical phenomena. Micron size
helixes may exhibit a giant (orders of magnitude larger than current materials)
Faradayrotation at infraredfrequencies[18].The possibility ofmakinghelixes at
micron scales enables the recent experimental study of these as efficient circular
polarizers at the far infrared [19]. Some metallic wire structures possess rather
intriguing EM properties. While the magnetization (electric polarization) in
ordinarymaterialsisgenerallyinducedbyanexternalmagnetic(electric)field,in
some particular metallic wire structures, an external electric (magnetic) field can
induceamagnetic(electric)polarization.Materialsthatarebothferromagneticand
ferroelectricatthesametime(multiferroics)exhibitthistypeofphenomenawhich
S.T.ChuiandL.Zhou,ElectromagneticBehaviourofMetallicWireStructures, 1
DOI:10.1007/978-1-4471-4159-4_1,(cid:2)Springer-VerlagLondon2013
2 1 ElectromagneticWavesinMetallicWireStructures
is called magnetoelectric effect. Interests in this type of materials have recently
revived due to improved sophistication in creating multiphase nanostructure
material. While the magnetoelectric effect in multiferroic materials decreases
drasticallyabovethespinwavefrequency,ifthewirestructuresaresmallenough
the magnetoelectric effect can persists up to much higher frequencies. These
magnetoelectric materials can exhibit novel EM properties. For example, the
Poyntingvectorcanbeperpendiculartothewavevector,andtheEMeigenmodes
inside such a medium can be quire nontrivial [20].
The fascinating phenomena mentioned above occur in composite materials
involving metallic wire structures. Knowledge of the EM eigenmodes in metallic
wirestructuresiscrucialtounderstandthesephysicaleffects.Full-wavenumerical
simulations based on finite-difference time-domain (FDTD) and finite-element
method were frequently adopted to reproduce the observed phenomena. Such
calculations shed very little light on the inherent physics behind the physical
phenomena.Wehaverecentlydevelopedarigorousandgeneralcircuittheorythat
provides a modular, simple, and physical way to understand wire structures with
metallic components. Furthermore, we find that quite often this approach is
numericallymuchmoreefficientthancurrentalgorithms.Inthisbook,weexplain
this theoretical framework to study the EM eigenmodes in arbitrary metallic wire
structures, and apply the theory to various examples to illustrate our understand-
ings of the inherent physics behind the physical phenomena occurring in such
medium.
A circuit theory includes circuit parameters such as the inductances and the
capacitances of the circuit. In general, the current distribution along the wires is
not uniform and the circuit elements cannot be approximated by a single ‘‘local’’
(self)lumpedelements;themodulationalongthecurrentpathhastobetakeninto
consideration.Twodifferentrouteshavebeenexplored[21–23,18,24,25].Oneis
the ‘‘partial equivalent element circuit’’ (PEEC) method that discretizes the
inductances and capacitances into small elements distributed in space [21] and
considers the self and mutual circuit elements between them. We have recently
developedanothermethodbyexpandingthecurrentdistributionalongthewirein
terms of a Fourier series [18, 22–25]. We found that this latter approach provides
for a simpler, more physical, and numerically very efficient way to understand
metallicwirenetworks.Insomecases,theproblembecomesanalyticallytractable.
Forexample,thereisafolklorethat,forametallicwireofarbitraryshapeoflength
L, the resonance wavelengths k is closetotwiceits length divided by aninteger,
r
k ¼2L=n; but no proof has ever been given. In our approach, we can show that
r
thisistrueinthethin-wirelimit.Waystocalculatethecorrectionsasthewiregets
thicker are demonstrated. We summarize and explain briefly here the advantages
of the nonlocal basis functions that we adopted. The details will be presented in
different chapters following this introduction.
(1) For thin wires, the ‘‘self’’ circuit impedances (inductances and capacitances)
betweenthesameFouriermodesareoftheorderoflog(radiusofwire/length
of wire) and much larger than that (‘‘mutual’’) between different Fourier
1.1 Introduction 3
modes, which are not log divergent. The circuit problem generally involves
inverting the circuit impedance matrix to relate the induced current in the
wirestotheexternallyappliedelectricfield.IntheFourierbasis,thematrixis
approximately diagonal; its inversion is easier and the results are easier to
understand. The circuit equations are simplified greatly.
(2) In this basis, the value of the impedance matrix element increases rapidly as
the square of the mode index. For low lying excitations, only a few Fourier
modes can already produce accurate results. The PEEC method [21] expands
the circuit parameters in a local basis set and thus do not have these
simplifications.
(3) The local basis works, in principle, for any metallic wire network and the
accuracy is determined by the mesh points chosen in a specific computation,
but the physics behind the eigenmodes is not very clear. The nonlocal-basis
theory is generally less computation intensive if better basis functions are
chosen; it is also more ‘‘modular’’ and offers a better physical picture on the
characteristics of the eigenmodes. Of course, for complicated structures, it is
easier to pick a local basis. The wire structures are simple enough that a
nonlocal basis can generally be used.
The expansionof the electric field interms of‘‘entire domainbasis functions’’
had been discussed by the electrical engineering community in the past. But
eventuallyitfelloutoffavor.Asthereaderwillfindout,fortheclassofproblems
discussed here, our method really works well. One of the problems in using
nonlocal basis functions is that the electric field can be very different at a few
placessuchaswireendsandjunctions.Wefindthatthisproblemcanbeovercome
byintroducingauxiliaryvariablescorrespondingtolocalizedelectricfieldsatthese
places. We summarize, next, the circuit equation that we shall study in different
chapters in this book.
1.2 Circuit Equation
As shown in Fig. 1.1, we assume that the current density at position~r inside a
metallic wire structure and time t is denoted by ~jð~r; tÞ, and the resistivity
distribution of the system is described by qð~rÞ; then the following equation holds
qð~rÞ~jð~r; tÞ¼~E ð~r; tÞþ~E ð~r; tÞþ~E ð~r; tÞ: ð1:1Þ
ext L C
This equation states that the total electric field is composed by three parts: the
external field ~E ð~r; tÞ; the inductive field ~E ð~r;tÞ¼(cid:2)o~Að~r;tÞ=cot; and the
ext L
capacitive field ~E ð~r; tÞ¼(cid:2)ruð~r; tÞ where ~Að~r; tÞ;uð~r; tÞ are the vector and
C
scalarpotentials.WerepresentaphysicalquantityXasaFourierseriesintimeand
focusonasingleFouriercomponent:~Xð~r; tÞ¼~Xð~rÞexpðixtÞ;withxdenotingthe
frequency. The inductive field can be expressed in terms of the vector potential