High Power Microwave Sources and Technologies Using Metamaterials E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: IEEE Press Series on RF and Microwave Technology
- Sprache: Englisch
Explore the latest research avenues in the field of high-power microwave sources and metamaterials A stand-alone follow-up to the highly successful High Power Microwave Sources and Technologies, the new High Power Microwave Sources and Technologies Using Metamaterials, demonstrates how metamaterials have impacted the field of high-power microwave sources and the new directions revealed by the latest research. It's written by a distinguished team of researchers in the area who explore a new paradigm within which to consider the interaction of microwaves with material media. Providing contributions from multiple institutions that discuss theoretical concepts as well as experimental results in slow wave structure design, this edited volume also discusses how traditional periodic structures used since the 1940s and 1950s can have properties that, until recently, were attributed to double negative metamaterial structures. The book also includes: * A thorough introduction to high power microwave oscillators and amplifiers, as well as how metamaterials can be introduced as slow wave structures and other components * Comprehensive explorations of theoretical concepts in dispersion engineering for slow wave structure design, including multi-transmission line models and particle-in-cell code virtual prototyping models * Practical discussions of experimental measurements in dispersion engineering for slow wave structure design * In-depth examinations of passive and active components, as well as the temporal evolution of electromagnetic fields High Power Microwave Sources and Technologies Using Metamaterials is a perfect resource for graduate students and researchers in the areas of nuclear and plasma sciences, microwaves, and antennas.
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Veröffentlichungsjahr: 2021
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Table of Contents
Cover
Series Page
Title Page
Copyright
Editor Biographies
List of Contributors
Foreword
Preface
1 Introduction and Overview of the Book
1.1 Introduction
1.2 Electromagnetic Materials
1.3 Effective‐Media Theory
1.4 History of Effective Materials
1.5 Double Negative Media
1.6 Backward Wave Propagation
1.7 Dispersion
1.8 Parameter Retrieval
1.9 Loss
1.10 Summary
References
2 Multitransmission Line Model for Slow Wave Structures Interacting with Electron Beams and Multimode Synchronization
2.1 Introduction
2.2 Transmission Lines: A Preview
2.3 Modeling of Waveguide Propagation Using the Equivalent Transmission Line Model
2.4 Pierce Theory and the Importance of Transmission Line Model
2.5 Generalized Pierce Model for Multimodal Slow Wave Structures
2.6 Periodic Slow‐Wave Structure and Transfer Matrix Method
2.7 Multiple Degenerate Modes Synchronized with the Electron Beam
2.8 Giant Amplification Associated to Multimode Synchronization
2.9 Low Starting Electron Beam Current in Multimode Synchronization‐Based Oscillators
2.10 SWS Made by Dual Nonidentical Coupled Transmission Lines Inside a Waveguide
2.11 Three‐Eigenmode Super Synchronization: Applications in Amplifiers
2.12 Summary
References
3 Generalized Pierce Model from the Lagrangian
3.1 Introduction
3.2 Main Results
3.3 Pierce's Model
3.4 Lagrangian Formulation of Pierce's Model
3.5 Hamiltonian Structure of the MTLB System
3.6 The Beam as a Source of Amplification: The Role of Instability
3.7 Amplification for the Homogeneous Case
3.8 Energy Conservation and Transfer
3.9 The Pierce Model Revisited
3.10 Mathematical Subjects
3.11 Summary
References
4 Dispersion Engineering for Slow‐Wave Structure Design
4.1 Introduction
4.2 Metamaterial Complementary Split Ring Resonator‐Based Slow‐Wave Structure
4.3 Broadside Coupled Split Ring Resonator‐Based Metamaterial Slow‐Wave Structure
4.4 Iris Ring‐Loaded Waveguide Slow‐Wave Structure with a Degenerate Band Edge
4.5 Two‐Dimensional Periodic Surface Lattice‐Based Slow‐Wave Structure
4.6 Curved Ring‐Bar Slow‐Wave Structure for High‐Power Traveling Wave Tube Amplifiers
4.7 A Corrugated Cylindrical Slow‐Wave Structure with Cavity Recessions and Metallic Ring Insertions
4.8 Summary
References
5 Perturbation Analysis of Maxwell's Equations
5.1 Introduction
5.2 Gain from Floating Interaction Structures
5.3 Gain from Grounded Interaction Structures
5.4 Electrodynamics Inside a Finite‐Length TWT: Transmission Line Model
5.5 Corrugated Oscillators
5.6 Summary
References
6 Similarity of the Properties of Conventional Periodic Structures with Metamaterial Slow Wave Structures
6.1 Introduction
6.2 Motivation
6.3 Observations
6.4 Analysis of Metamaterial Surfaces from Perfectly Conducting Subwavelength Corrugations
References
7 Group Theory Approach for Designing MTM Structures for High‐Power Microwave Devices
7.1 Group Theory Background
7.2 MTM Analysis Using Group Theory
7.3 Inverse Problem‐Solving Using Group Theory
7.4 Designing an Ideal MTM
7.5 Proposed New Structure Using Group Theory
7.6 Design of Isotropic Negative Index Material
7.7 Multibeam Backward Wave Oscillator Design using MTM and Group Theory
7.8 Particle‐in‐Cell Simulations
7.9 Efficiency
7.10 Summary
References
8 Time‐Domain Behavior of the Evolution of Electromagnetic Fields in Metamaterial Structures
8.1 Introduction
8.2 Experimental Observations
8.3 Numerical Simulations
8.4 Attempts at a Linear Circuit Model
References
9 Metamaterial Survivability in the High‐Power Microwave Environment
9.1 Introduction
9.2 Split Ring Resonator Loss
9.3 CSRR Loss
9.4 Artificial Material Loss
9.5 Disorder
9.6 Summary
References
10 Experimental Hot Test of Beam/Wave Interactions with Metamaterial Slow Wave Structures
10.1 First‐Stage Experiment at MIT
10.2 Second‐Stage Experiment at MIT
10.3 Metamaterial Structure with Reverse Symmetry
10.4 Experimental Results on High‐Power Generation
10.5 Frequency Measurement in Hot Test
10.6 Steering Coil Control
10.7 University of New Mexico/University of California Irvine Collaboration on a High Power Metamaterial Cherenkov Oscillator
10.8 Summary
References
11 Conclusions and Future Directions
References
Index
Series Page
End User License Agreement
List of Tables
Chapter 4
Table 4.1 Dimensions of the SWS shown in Figure 4.10.
Table 4.2 Dimensions of the SWS shown in Figure 4.15 for an alignment angle,
Chapter 7
Table 7.1 Character table for
point group.
Table 7.2 Character table for the
point group.
Table 7.3 The character of the chosen basis currents on the SRR.
Table 7.4
character table.
Table 7.5 Characters of the basis currents for complementary MTM.
Table 7.6 Dimensions of the designed MTM SWS.
Table 7.7 Cutoff frequencies of coaxial waveguide with
mm.
Chapter 8
Table 8.1 Delay and risetimes of test cases of Figure 8.18.
List of Illustrations
Chapter 1
Figure 1.1 Broad categorization of materials based on the real components of...
Figure 1.2 Kock's artificial dielectric lens, consisting of conducting spher...
Figure 1.3 (a) Double SRR geometry building block, and (b) An array of SRRs....
Figure 1.4 (a) Cerenkov effect in a DPM, (b) The same effect in a DNG materi...
Figure 1.5 (a) Electric field lines in the SRR at resonance. (b) Magnetic fi...
Figure 1.6 The dispersion relation for various waveguide loaded artificial d...
Figure 1.7 (a) The real components of
and
calculated via Eq. (1.11), usi...
Chapter 2
Figure 2.1 Transmission lines are an equivalent and convenient way to descri...
Figure 2.2 Schematic diagram of an MTL made of
TLs coupled among each othe...
Figure 2.3 Schematic diagram of a periodic waveguide of period
, where each...
Figure 2.4 (a) Example of circular waveguide SWS interacting with an electro...
Figure 2.5 (a) Schematic of an electron beam interacting with a periodic wav...
Figure 2.6 A system made of a SWS supporting two modes in each direction (mo...
Figure 2.7 A representative dispersion diagram of a “cold” (i.e. in absence ...
Figure 2.8 Schematic of a periodic metallic SWSs that supports several Bloch...
Figure 2.9 Supersynchronism condition: four modes with DBE interact with the...
Figure 2.10 Conceptual example of a periodic SWS (modeled as a periodic MTL ...
Figure 2.11 (a) Gain,
, of the MTL system in Figure 2.10, assuming an avera...
Figure 2.12 (a) Schematic of the equivalent MTL‐electron beam interaction mo...
Figure 2.13 Starting electron beam current
. (oscillation threshold current...
Figure 2.14 Comparison between a degenerate band edge oscillator (DBEO) obta...
Figure 2.15 (a) Example of coupled nonidentical TLs to realize higher‐order ...
Figure 2.16 The butterfly SWS is placed within a circular waveguide for real...
Figure 2.17 Comparison of the dispersion diagrams with (a) and without (b) t...
Figure 2.18 (a)
dispersion diagram of the first
modes of the butterfly S...
Figure 2.19 (a) We consider an amplifying regime based on the synchronizatio...
Figure 2.20 Dispersion diagrams for modes in the SWS‐electron beam interacti...
Chapter 3
Figure 3.1 Discrete element of the TL‐beam system in Pierce's model. The arr...
Figure 3.2 (a)
: the parabola
intersects each branch of
(where
is as ...
Figure 3.3 Pierce's dispersion relation for
: for large
the parabola
is...
Chapter 4
Figure 4.1 Geometries of the (a) SRR and (b) CSRR. Source: Falcone et al. [6...
Figure 4.2 (a) MTM plates composed of CSRR, (b) a single CSRR design, and (c...
Figure 4.3 Design of the MTM CSRR for backward wave oscillator application. ...
Figure 4.4 A negative index MTM waveguide featuring plates made using CSRRs ...
Figure 4.5 Dispersion curve for the lowest‐order modes that can propagate in...
Figure 4.6 (a) Schematic of the shortened SWS section to be fabricated for c...
Figure 4.7 Simulated (light gray) and measured (dark gray)
measurement fro...
Figure 4.8 (a) Waveguide components for cold test assembly, (b) assembled MT...
Figure 4.9 Simulated (light gray) and measured (dark gray)
measurement fro...
Figure 4.10 A MTM SWS. Design consists of a cylindrical waveguide‐loaded wit...
Figure 4.11 Dispersion curve of the first eight modes in the SWS shown in Fi...
Figure 4.12 Fabricated SWS from Figure 4.10 along with the
mode launchers....
Figure 4.13 Measured
data for SWS shown in Figure 4.12. Source: Yurt et al...
Figure 4.14 Extracted effective permittivity and permeability from the measu...
Figure 4.15 Iris‐loaded cylindrical waveguide with DBE dispersion. The angle...
Figure 4.16 The first four derivatives of angular frequency
with respect t...
Figure 4.17 Dispersion relation of the four modes that coalesce at the DBE f...
Figure 4.18 Fabricated components for the iris‐loaded DBE SWS (a) Elliptical...
Figure 4.19 Measured and simulated dispersion curves for the iris‐loaded DBE...
Figure 4.20 (a) A two‐dimensional PSL SWS. (b) Longitudinal view of SWS, and...
Figure 4.21 Formation of an eigenmode using a volumetric, transversely symme...
Figure 4.22 (a) Low‐coupling condition (small
value) dispersion curve for ...
Figure 4.23 Fabricated cylindrical waveguide with a two‐dimensional periodic...
Figure 4.24 Measured
response of a W‐band PSL waveguide shown in Figure 4....
Figure 4.25 (a) CRB Structure with dimensions: pitch,
mm, radius,
mm, wi...
Figure 4.26 Equivalent circuit model of a ring‐bar SWS. This design is a sim...
Figure 4.27 (a) Geometrical overview of curved section of a CRB/Ring‐loop. (...
Figure 4.28 Simulated and calculated dispersion relation of a CRB SWS loaded...
Figure 4.29 (a) A CRB SWS:
mm,
mm,
mm,
mm,
mm, and
mm. (b) Waveg...
Figure 4.30 (a) Measurement setup showing the loaded cavity connected to the...
Figure 4.31 (a) Measurement setup showing the alumina rod inserted in the SW...
Figure 4.32 (a) Conventional corrugated SWS design, and (b) Dispersion curve...
Figure 4.33 (a) Proposed deeply corrugated SWS with cavity recessions and me...
Figure 4.34
group velocity plots for different corrugation depths given th...
Figure 4.35 (a) Dispersion curve of a conventional SWS with
and
modes al...
Figure 4.36 SWS fabrication process: (1)
plane shifting to expose recessio...
Figure 4.37 Fabricated SWS components and assembly for cold test measurement...
Figure 4.38 Full
mode dispersion curve. Simulated vs. analytical dispersio...
Figure 4.39 Measured
response of the SWS cavity used to generate the dispe...
Figure 4.40 Fabricated SWS components: (a) Single SWS cell, (b) Three‐sectio...
Figure 4.41 Measured
mode dispersion curves for homogeneous and inhomogene...
Figure 4.42 Measured normalized group velocity
versus operational frequenc...
Chapter 5
Figure 5.1 Section of infinitely long TWT consisting of electron beam surrou...
Figure 5.2 Cross‐section of infinitely long TWT showing electron beam surrou...
Figure 5.3 Cross‐section of infinitely long TWT showing electron beam surrou...
Figure 5.4 (a) Cut‐away view of corrugated waveguide with
‐periodic corruga...
Figure 5.5 Corrugation geometries considered: (a) truncated sinusoidal corru...
Figure 5.6 Unit‐periodic geometry with unit‐periodic corrugation and profile...
Figure 5.7 Plane view of domains
and
.
Figure 5.8 Admittance
as a function of frequency for the rectangular corru...
Figure 5.9 The finite length TWT.
Figure 5.10 Transmission line model for finite length device. Transmission c...
Figure 5.11 Lozenge‐shaped (above) and ellipsoidal (below) cross‐sections wi...
Figure 5.12 Transmission coefficient as a function of frequency for an isotr...
Figure 5.13 The effect of aspect ratio
on gain for lozenge‐shaped inclusio...
Figure 5.14 The effect of filling fraction on gain for symmetric lozenge‐sha...
Figure 5.15 Axial symmetry square rib corrugations in domain
of cylindrica...
Figure 5.16 Square rib corrugated waveguide with
‐period.
Figure 5.17 Corrugation geometries considered: (a) square rib corrugations, ...
Figure 5.18 Domains
,
, and
.
Figure 5.19 Azimuthal symmetry.
Figure 5.20 T‐shaped ribbed standing wave magnetic field
, for the cold str...
Figure 5.21 Larger T‐shaped ribbed structure standing wave magnetic field
,...
Figure 5.22 Simple ribbed standing wave magnetic field
, for the cold struc...
Chapter 6
Figure 6.1 (a) SWS comprising coupled cavities and (b) Its corresponding dis...
Figure 6.2 (a) Anode block of a magnetron showing the cavities; (b) Split ri...
Figure 6.3 Cylindrical SWS with a sinusoidal profile.
Figure 6.4 Dispersion curves for the two lowest modes with their correspondi...
Figure 6.5 Dispersion diagram for the lowest‐order modes with different corr...
Figure 6.6 Dispersion diagram for the lowest modes with different amplitude ...
Figure 6.7 Dispersion diagram for the lowest modes with different amplitude ...
Figure 6.8Figure 6.8 Dispersion diagram for the
mode in an SWS with a prof...
Figure 6.9 Structure of the hybrid
mode in an all‐metallic SWS with sinuso...
Figure 6.10 Dependence of cutoff frequencies of low‐order modes on corrugati...
Figure 6.11 Cylindrical SWS with
cm and rectangular (meander) profile of p...
Figure 6.12 Dispersion of lowest modes in rectangular profile systems with p...
Figure 6.13 Dispersion of lowest modes in rectangular profile systems with p...
Figure 6.14 Dispersion of lowest modes in rectangular profile systems with p...
Figure 6.15 Dispersion of lowest modes in rectangular profile systems with p...
Figure 6.16 Transverse structure of the lowest modes in the SWS (Figure 6.9)...
Figure 6.17 Dispersion diagrams for lowest mode in an SWS with small period
Figure 6.18 Truncated sinusoidal corrugations in annular domain
of cylindr...
Figure 6.19 (a) Cut‐away view of corrugated waveguide with
‐periodic corrug...
Figure 6.20 Corrugation geometries considered: (a) truncated sinusoidal corr...
Figure 6.21 Unit‐periodic geometry with unit‐periodic corrugation and profil...
Figure 6.22 Plane view of domains
and
.
Figure 6.23 Admittance
as a function of corrugation depth ratio
for (a) ...
Figure 6.24 (a) Admittance
as a function of frequency for fixed ratio
wi...
Figure 6.25 (a) Admittance
as a function of frequency for fixed ratio
wi...
Figure 6.28 Admittance
as a function of frequency for the rectangular corr...
Figure 6.26 (a) Admittance
as a function of frequency for fixed ratio
wi...
Figure 6.27 (a) Admittance
as a function of frequency for fixed ratio
wi...
Figure 6.29 (a) Dispersion curves for rectangular corrugations with
for co...
Figure 6.30 (a) Dispersion curves for truncated sawtooth corrugations. The p...
Figure 6.31 (a) Dispersion relation for the rectangular corrugations conside...
Figure 6.32 (a) Normalized group velocity
and (b) integrated Poynting vect...
Chapter 7
Figure 7.1 Twofold rotation
Figure 7.2 Reflection mirror plane.
Figure 7.3 Inversion center and symmetry of a molecule.
Figure 7.4 An
rotation (
plus
mirror plane).
Figure 7.5 Decision chart to find the symmetry group of an MTM.
Figure 7.6 SRR geometry.
Figure 7.7 SRR and its symmetry operations. (a) Basic SRR unit cell (b) A un...
Figure 7.8 (a) Four quadrants of the EM constitutive tensor and (b) their co...
Figure 7.9 Behavior of SRR basis currents under symmetry elements of the
g...
Figure 7.10 SRR permittivity and permeability for different excitations.
Figure 7.11 Flowchart outlining the inverse‐problem steps for using group th...
Figure 7.12 (a) Complementary SRR design with dimensions
mm,
mm,
mm,
Figure 7.13 Behavior of MTM unit cell basis currents under symmetry elements...
Figure 7.14 MTM negative permeability and permittivity for different frequen...
Figure 7.15 One sheet of an isotropic negative index MTM.
Figure 7.16 EM constitutive parameters of the proposed MTM.
Figure 7.17 Two isotropic negative index MTM unit cells that allow negative ...
Figure 7.18 Geometry of the designed MTM: (a) one plate dimensions, and (b) ...
Figure 7.19 (a) One unit of an MTM‐loaded circular waveguide, and (b) dimens...
Figure 7.20 Dispersion diagram for the designed MTM for the first two modes ...
Figure 7.21 Electric field distribution between MTM plates for a phase advan...
Figure 7.22 Schematic of one period of the MTM‐BWO and schematic of the BWO ...
Figure 7.23 Output power for a 1 kA proposed MTM BWO.
Figure 7.24 Fourier transform of the output electric field for the
mode.
Figure 7.25 Output port signal modes.
Figure 7.26 Electric field distribution along the proposed BWO and the outpu...
Figure 7.27 PIC phase space plot of electrons at a)
ns, b)
ns, c)
ns....
Chapter 8
Figure 8.1 Schematic of the experimental arrangement to measure time respons...
Figure 8.2 Bandstop filter (BSF) experimental arrangement. The test waveguid...
Figure 8.3 Bandpass filter (BPF) experimental arrangement. Elements other th...
Figure 8.4
SRR array card for the bandstop system. Upper scale in inches; ...
Figure 8.5 Eccostcok PP
foam block to hold SRR array cards in place within ...
Figure 8.6
SRR array card for the bandpass system. Upper scale in inches; ...
Figure 8.7 Transmission,
, (dark gray) and reflection,
, (light gray) of t...
Figure 8.8 Measured transmission,
, (a) and reflection,
, (b) parameters v...
Figure 8.9 Measured transmission and reflection versus frequency with sample...
Figure 8.10 Time response as measured by the output detector voltage for var...
Figure 8.11 Time response as measured by the output detector voltage of the ...
Figure 8.12 Measured transmission,
, (dark gray) and reflection,
, (light ...
Figure 8.13 Measured transmission,
, (a) and reflection,
, (b) parameters ...
Figure 8.14 Slotted waveguide arrangement for movable RF probing of the SRR‐...
Figure 8.15 Upstream and downstream slotline RF probe measurements in the SR...
Figure 8.16 Measured reflection and transmission versus frequency of the WR
Figure 8.17 Time response as indicated by output detector voltage for variou...
Figure 8.18 Transmitted signals for cases of no test WR
waveguide section a...
Figure 8.19 Log plot of SRR‐loaded waveguide at resonance showing a response...
Figure 8.20 SRR unit cell.
Figure 8.21 SRR cards for the bandstop filter (BSF) system. WR
waveguide is...
Figure 8.22 SRR cards for the bandpass filter (BPF) system. WR
waveguide is...
Figure 8.23 Simulated
‐parameters for the bandstop (BSF) system. (a): trans...
Figure 8.24 Normalized signals versus time for the bandstop system at resona...
Figure 8.25 Simulated
‐parameters for the bandpass system at resonance. (a)...
Figure 8.26 Vertical electric field (
) maps showing negative phase velocity...
Figure 8.27 Normalized voltage signals versus time for the pass band system ...
Chapter 9
Figure 9.1 (a)Individual SRR on dielectric substrate with wire array on oppo...
Figure 9.2 Simulation of an MTM loaded X‐band waveguide. (a) HFSS simulation...
Figure 9.3 (a) One section of X‐band waveguide loaded with an array of meta‐...
Figure 9.4 (a) One section of X‐band waveguide loaded with an array of meta‐...
Figure 9.5 (a) CSRR unit cell. (b) simulated permittivity and permeability b...
Figure 9.6 HFSS and ANSYS simulations of thermal loading of a waveguide load...
Figure 9.7 (a) CSRR unit‐cell geometry optimized to minimize loss. (b) Effec...
Figure 9.8 Steady‐state temperature versus power. Source: Image provided by ...
Figure 9.9 Single element of the SRR array. Each ring is
thick, with the d...
Figure 9.10 Experimental configuration in cross‐section. 6 by 6 mm waveguide...
Figure 9.11 Numerical results of
versus frequency for an open defect on in...
Figure 9.12 Numerical results of
versus frequency for a short defect on in...
Chapter 10
Figure 10.1 A geometry of the MTM plate (a) and a photograph of the fully as...
Figure 10.2 Dispersion relation for the symmetric (light gray) and antisymme...
Figure 10.3 Schematic of the experiment with a HPM source.
Figure 10.4 PIC simulation of the MTM structure. Beam energy is 480 keV, cur...
Figure 10.5 Microwave power (thin dark gray trace), electron gun voltage (th...
Figure 10.6 Measurement results: measured (black dots) and simulated (light ...
Figure 10.7 Design of MTM with reverse symmetry (MTM‐R). The period of the M...
Figure 10.8 Dispersion curves of the MTM‐R structure. The Cherenkov and the ...
Figure 10.9 Electric field distribution of Modes 1 and 2. (a) E field on the...
Figure 10.10 Particle‐in‐cell (PIC) simulation results: (a) helical beam tra...
Figure 10.11 High‐power operation map: (a) High‐power region in the solenoid...
Figure 10.12 Magnetic field profiles and beam radius of the 490 keV electron...
Figure 10.13 Sample pulse in the first category, with HPM radiation in both ...
Figure 10.14 Sample pulse in the second category, with HPM radiation in both...
Figure 10.15 Sample pulse in the third category, with HPM generation at mult...
Figure 10.16 Tuning of the output microwave frequency with the beam. Data we...
Figure 10.17 Frequency tuning of the output microwave pulse with the solenoi...
Figure 10.18 Illustration of the position of the steering coils. One pair of...
Figure 10.19 Sample pulse with 0.2 A of current applied in the steering coil...
Figure 10.20 High‐power operation map with the steering coils applied. (a) H...
Guide
Cover
Table of Contents
Series Page
Title Page
Copyright
Editor Biographies
List of Contributors
Foreword
Preface
Begin Reading
Index
Series Page
WILEY END USER LICENSE AGREEMENT
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IEEE Press Editorial Board
Ekram Hossain,
Editor in Chief
Jón Atli Benediktsson
Xiaoou Li
Jeffrey Reed
Anjan Bose
Lian Yong
Diomidis Spinellis
David Alan Grier
Andreas Molisch
Sarah Spurgeon
Elya B. Joffe
Saeid Nahavandi
Ahmet Murat Tekalp
High Power Microwave Sources and Technologies Using Metamaterials
Edited by
John LuginslandConfluent Sciences, LLCIthaca, New York
Jason A. MarshallNaval Research LaboratoryWashington, DC
Arje NachmanAir Force Office of Scientific ResearchArlington, VA
Edl SchamilogluUniversity of New MexicoAlbuquerque, NM
IEEE Press Series on RF and Microwave Technology
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Editor Biographies
John Luginsland – Dr. John Luginsland is a senior scientist at Confluent Sciences, LLC and an adjunct professor at Michigan State University. Previously, he worked at AFOSR serving as the plasma physics and lasers and optics program officer, as well as in various technical leadership roles. Additionally, he worked for SAIC and NumerEx, as well as the directed energy directorate of the Air Force Research Laboratory (AFRL). He is a fellow of the IEEE and AFRL. He received his BSE, MSE., and PhD in nuclear engineering from the University of Michigan in 1992, 1994, and 1996, respectively.
Jason A. Marshall – Dr. Jason A. Marshall is the associate superintendent, Plasma Physics Division, Naval Research Laboratory. Prior to this, he was a principal scientist with the Air Force Office of Scientific Research responsible for management and execution of the Air Force basic research investments in Plasma and Electro‐energetic Physics. He received BS degrees in anthropology and chemistry from Eastern New Mexico University in 1994 and 1995, respectively; an MS degree in chemistry from Washington State University in 1998; and a PhD in chemical physics from Washington State University in 2002.
Arje Nachman – Dr. Arje Nachman is the program officer for electromagnetics at AFOSR. He has worked at AFOSR since 1985. Before that, he was on the mathematics faculty of Texas A&M and Old Dominion University, and a senior scientist at Southwest Research Institute (SwRI). Dr. Nachman received a BS in computer science and applied mathematics in 1968 from Washington University (St. Louis) and a PhD in Mathematics in 1973 from NYU.
Edl Schamiloglu – Dr. Edl Schamiloglu is a distinguished professor of electrical and computer engineering at the University of New Mexico, where he also serves as associate dean for research and innovation in the School of Engineering, and special assistant to the Provost for Laboratory Relations. He is a fellow of the IEEE and the American Physical Society. He received his BS and MS from Columbia University in 1979 and 1981, respectively, and his PhD from Cornell University in 1988.
List of Contributors
Ahmed F. Abdelshafy
Department of Electrical Engineering and Computer Science, The University of California at Irvine
Irvine, CA
USA
Filippo Capolino
Department of Electrical Engineering and Computer Science, The University of California at Irvine
Irvine, CA
USA
Ushe Chipengo
Ansys Inc.
Canonsburg, PA
USA
Christos Christodoulou
Department of Electrical and Computer Engineering, University of New Mexico
Albuquerque, NM
USA
Adrian W. Cross
Department of Physics
University of Strathclyde
Glasgow, Lanarkshire
UK
Alexander Figotin
Department of Mathematics, The University of California at Irvine
Irvine, CA
USA
Mark Gilmore
Department of Electrical and Computer Engineering, University of New Mexico
Albuquerque, NM
USA
Mohamed Aziz Hmaidi
Luxoft
Farmington Hills, MI
USA
Jason S. Hummelt
Diamond Foundry
Santa Clara, CA
USA
Robert Lipton
Department of Mathematics, Louisiana State University
Baton Rouge, LA
USA
Xueying Lu
Northern Illinois University
DeKalb, IL
USA
John Luginsland
Confluent Sciences, LLC
Albuquerque, NM
USA
Jason A. Marshall
Naval Research Laboratory
Washington, DC
USA
Arje Nachman
Air Force Office of Scientific Research
Arlington, VA
USA
Niru K. Nahar
Electroscience Laboratory, The Ohio State University
Columbus, OH
USA
Mohamed A. K. Othman
SLAC National Accelerator Laboratory, Stanford University
Menlo Park, CA
USA
Alan D.R. Phelps
Department of Physics
University of Strathclyde
Glasgow, Lanarkshire
UK
Anthony Polizzi
Synovus Financial
Columbus, GA
USA
Guillermo Reyes
Department of Mathematics, University of Southern California
Los Angeles, CA
USA
Edl Schamiloglu
Department of Electrical and Computer Engineering, University of New Mexico
Albuquerque, NM
USA
Hamide Seidfaraji
Microsoft Corporation
Kirkland, WA
USA
Rebecca Seviour
School of Computing and Engineering, University of Huddersfield
Huddersfield
UK
Michael A. Shapiro
Plasma Science and Fusion Center, MIT
Cambridge, MA
USA
Richard J. Temkin
Plasma Science and Fusion Center, MIT
Cambridge, MA
USA
Lokendra Thakur
MIT‐Harvard Broad Institute
Cambridge, MA
USA
John L. Volakis
College of Engineering and Computing, Florida International University
Miami, FL
USA
Tyler Wynkoop
BAE Systems, Inc.
Minneapolis, MN
USA
Sabahattin Yurt
Qualcomm Technologies, Inc.
San Diego, CA
USA
Foreword
Since its inception in 1985, the Department of Defense's Multidisciplinary University Research Initiative (MURI) program has convened teams of investigators with the hope that collective insights drawn from research across multiple disciplines could facilitate the advancement of newly emerging technologies and address the department's unique problem sets. Developed in collaboration between the military services and the Office of the Secretary of Defense, MURI topics and the teams chosen to execute the research represent a dedicated source of innovation for science and technology solutions to hard national security problems. These highly competitive awards complement and augment the traditional basic research initiatives that support single‐investigator grants with research programs that can draw on a wide range of researchers and disciplines. Furthermore, longer periods of performance allow these MURIs to start new research areas at the intersection of multiple fields of study. The combination of significant and sustained support in areas critical to National Security and the Department of Defense's mission provide the potential for game changing advancement in science and technology.
This volume, edited by Drs. John Luginsland, Jason A. Marshall, Arje Nachman, and Edl Schamiloglu, summarizes the accomplishments of the FY12 MURI consortium, which was awarded an AFOSR grant on Transformational Electromagnetics. Drs. Luginsland, Marshall, and Nachman (AFOSR) were program officers for this MURI, and Dr. Schamiloglu (University of New Mexico) was the consortium PI. The other PIs on this MURI were Dr. Richard Temkin (MIT), Dr. John Volakis (The Ohio State University and toward the end Florida International University), Dr. Alexander Figotin (UC Irvine), and Dr. Robert Lipton (Louisiana State University). The contributors to this volume were the faculty, staff, and graduate students involved in performing the research.
The success of this MURI is a result of the hard work and internationally recognized expertise of the sponsored researchers. As a plasma physicist myself, I certainly appreciate the challenges in advancing the state‐of‐the‐art in directed energy microwave sources. The five universities, guided by the MURI's Advisory Board with members from the Air Force Research Laboratory, Los Alamos National Laboratory, and industry, have advanced the understanding of a new generation of directed energy microwave capability that introduces metamaterials into their beam‐wave interaction structures. Conventional microwave vacuum electronics has advanced enormously from continuous research for nearly a century. Metamaterial‐based devices have been explored for less than a decade, so one can only imagine what advances will be realized in the future.
I commend the AFOSR program officers for successfully creating such a MURI topic, and I commend the PI and his team for successfully executing this award. This is an example of how multidisciplinary teams accelerate research through cross‐fertilization of ideas. Such efforts also hasten the transition of basic research findings to practical applications and, importantly, train the next generation of the science and engineering workforce in areas of particular importance to the U.S. DoD.
In summary, I am very pleased to have this volume as an archival record of this successful five‐year effort. The editors have done a masterful job of working with the researchers to collate this huge mass of valuable information into a consistent whole. This volume is a wonderful way of disseminating the advances from this MURI to new students, and also to practitioners, in the field seeking to understand how metamaterials can be exploited to design a new generation of intense microwave sources.
Brendan B. Godfrey is retired from a career of research management in government and industry, most recently as part of the Senior Executive Service (SES). He is a full‐time volunteer, not only principally for IEEE‐USA, but also for IEEE‐Nuclear and Plasma Sciences Society, the National Academies, Lawrence Berkeley National Laboratory, and Ars Lyrica Houston. He has led organizations with as many as 1500 people and budgets as large as US$ 500 million. He was director of the Air Force Office of Scientific Research from 2004 to 2010. His personal research centers on intense‐charged particle beams, high‐power microwave sources, and computational plasma methods. He is an IEEE Fellow and American Physical Society Fellow, and holds a PhD from Princeton University.
Preface
Aristotle identified a distinction between natural and artificial things. He ascribed the difference to motion and change. Natural things have a source of motion or change within them. Artificial things don't have any source of change in them, so they need an external cause. In this book, we explore the change in artificial materials caused by high‐power electromagnetic radiation.
This book presents a snapshot in time of the status of research on high‐power microwave (HPM) sources and technologies using metamaterials circa 2021. The focus of this book is on research that resulted from an FY2012 Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) award on Transformational Electromagnetics that was funded for over US$ 7.5 million and for over five years. The award was also supplemented with substantial Defense University Research Instrumentation Program (DURIP) grants. This MURI award builds on decades of AFOSR support for HPM research. The exploration of metamaterials essentially doubles the space of materials that can be exploited in the design of HPM sources, a space previously occupied by only conventional metals.
One of the editors (ES) was the lead Principal Investigator (PI) on the award and the remaining editors (JL, JAM, and AN) served as program officers for part or all of the award.
The team of university researchers was led by the University of New Mexico (ES) and included MIT (Richard Temkin, PI), the Ohio State University (John Volakis, PI), the University of California at Irvine (Alex Figotin, PI), and Louisiana State University (Robert Lipton, PI). The title of their proposal was Innovative Use of Metamaterials in Confining, Controlling, and Radiating Intense Microwave Pulses.
Supporting this MURI team were collaborators at the Air Force Research Laboratory's (AFRL's) Directed Energy (DE) Directorate (Dr. Robert E. Peterkin, Chief Scientist for AFRL's Directorate at the time). In addition, an esteemed group of scientists served as the advisory board for this MURI, providing feedback and guidance. Members of the Advisory Board were:
Dr. Dave Abe, Naval Research Laboratory, Washington, DC
Dr. Richard Albanese, ADED Co., San Antonio, TX
Dr. Carter Armstrong, L‐3 Communications EDD, San Carlos, CA
Dr. Bruce Carlsten, Los Alamos National Laboratory, Los Alamos, NM
Mr. Charles Chase, Lockheed Martin, Palmdale, CA
Mr. Chuck Gilman, SAIC, Albuquerque, NM (Retired)
Dr. John Petillo, Leidos Corp., Billerica, MA
Dr. Don Sullivan, Raytheon, Albuquerque, NM
Dr. Jeffrey P. Tate, Raytheon Space and Airborne Systems, El Segundo, CA
Dr. Pravit Tulyathan, Boeing, Huntington Beach, CA (Retired)
Chapter 1, written by Rebecca Seviour, presents an introduction to metamaterials and the scope of the book. Chapter 2, led by Ahmed F. Abdelshafy, presents a multitransmission line model for beam/wave interaction structures. Chapter 3, led by Alex Figotin, presents a generalized Pierce model from the Lagrangian. Chapter 4, led by Ushemadzoro Chipengo, reviews dispersion engineering for slow‐wave structure design. Chapter 5, led by Robert Lipton, presents a perturbation analysis of Maxwell's equations. Chapter 6, led by Sabahattin Yurt, presents a comparison of the properties of conventional periodic structures with deep corrugation with those of metamaterials. Chapter 7, led by Hamide Seidfaraji, presents a group theory approach for designing metamaterial structures for HPM devices. Chapter 8, led by Mark Gilmore, describes the temporal evolution of microwave electromagnetic fields in metamaterial structures. Chapter 9, written by Rebecca Seviour, discusses metamaterial survivability in the HPM environment. Chapter 10, led by Michael A. Shapiro, presents hot test results of beam/wave interaction with metamaterials structures. Finally, Chapter 11, written by the editors presents the conclusions and future directions.
The proceeds from the sales of this book will be directed to the SUMMA Foundation, a philanthropic organization that supports scholarships for students studying and scientific workshops on the subject of high‐power electromagnetics (http://ece‐research.unm.edu/summa/).
Finally, special thanks go to Dustin Fisher for converting original Word documents to LATEX. We also thank Dr. Brendan Godfrey for graciously agreeing to contribute the Foreword to this book. Special thanks also go to Mary Hatcher, Teresa Netzler, and Victoria Bradshaw at Wiley for supporting this project and patiently awaiting completion of the manuscript.
1Introduction and Overview of the Book
Rebecca Seviour
University of Huddersfield, School of Computing and Engineering, Queensgate, Huddersfield HD1 3DH, UK
1.1 Introduction
High‐power microwaves (HPMs), or directed energy RF, is an evolution of vacuum electron devices (VEDs) that seeks to generate the highest peak power levels in the frequency range of 100 s MHz through 100 GHz (and even higher frequencies) in short pulses (10–100 s ns in duration) that can be repetitively pulsed [1,2]. They came onto the scene in the late 1960s following the advent of pulsed power drivers that not only provided high‐energy electron beams (in the order of a MeV and higher), but concomitantly provided high currents as well (1–10's kA) [3]. Similar to VEDs, the electron beam is the power source from which the microwaves grow. Unlike VEDs, HPM sources have much less‐stringent vacuum and material requirements since their applications tend to be limited in scope with short mission times.
The state‐of‐the‐art in the practice of HPM sources has been led by intense beam‐driven oscillators whose output scale as , where is the peak output microwave power and is the operating frequency [2,4]. This is the Figure‐of‐Merit (FOM) for HPM oscillators. The equivalent FOM for HPM amplifiers is where is the bandwidth (BW). Until recently, conventional wisdom suggested that for emerging defense applications, the highest power on target (highest intensity field) was of greatest utility. However, recent advances in the understanding of the interaction of intense microwave fields with components and circuits argue that a tailored waveform synthesized at low power and amplified to very high power, might provide even superior capabilities. This is termed waveform diversity. Consider a comparison of the state‐of‐the‐art oscillator and amplifier in terms of the FOM: (i) the ITER/DIII‐D's plasma‐heating gyrotron oscillator at 110 GHz, 1 MW (10 s pulse), 1.1 MHz BW, has a FOM W‐ and essentially no BW. (ii) Haystack radar's gyrotron amplifier at 94 GHz, 55 kW output power (5.5 kW average), 1600 MHz BW yields a FOM W‐. Thus, there is a 2 order‐of‐magnitude opportunity to advance the FOM in high‐power amplifiers with considerable BW.
Interest in metamaterials (MTMs) grew rapidly following the publication of Pendry [5] and its practical implementation by Smith afterwards [6]. As discussed in this chapter, the history of MTMs dates back to the nineteenth century with numerous contributors, many of whom have only recently been rediscovered. This history has been reviewed in several books [7,8] and continues to be unraveled.
While numerous books have been written on the EM properties of MTMs, all of the applications that have been described in these books to‐date are at low‐power levels. In this book, we bring together advances that have been made in studying MTMs as slow‐wave structures (SWSs) for active electron beam‐driven HPM devices. We discuss structures that satisfy Wasler's definition of a MTM (see Section 1.2), and we also describe periodic SWSs with degenerate band edges (DBEs) that do not satisfy this definition, yet do offer novel engineered dispersion relations that are relevant to our overall goal‐seeking to discover novel beam/wave interactions that can be exploited for new HPM amplifiers.
1.2 Electromagnetic Materials
In many VEDs, the particle wave interaction is mediated in part via a material, where the functionality of the material manipulates the electromagnetic (EM) wave in a controlled fashion. The creativity of engineers to construct new devices is largely limited by the EM properties of available materials and the ability to precision engineer geometries from these materials. Of course, we are not restricted to naturally occurring materials; for decades, RF engineers have used materials synthesized at the molecular level with peculiar RF properties, such as Polytetraflufoethylene (Teflon™) and . These molecular synthesized materials can be used in VEDs to modify the behavior of an EM wave in a useful manner. In a simplistic form, this behavior between wave and material is described via the constitutive relations:
Here the permittivity () and the permeability () are the complex averaged EM response functions of the molecules that make up the material due to the interaction with the electric and magnetic components of an incident wave. The molecules in the material respond to the incident EM wave by forming dipoles, and these individual responses are averaged over all molecules in a volume to yield the permittivity and permeability. This averaging process discussed further in Section 1.3 even holds for gases as the number of molecules is still large enough that the parameters and accurately describe the interaction of an EM wave well into ultraviolet frequencies.
As and are the primary parameters that define a materials response to an EM wave it is useful to categorize materials based on the real components of these two parameters, as shown in Figure 1.1. Materials in the upper right quadrant of Figure 1.1 are often termed Double Positive Media (DPM), common dielectric materials, such as Polytetraflufoethylene, . The upper‐left and lower‐right quadrants of Figure 1.1 are the single negative media, such as plasmas or metals with a negative permittivity and negative permeability materials such as “wet” ice crystals. Unlike the DPM these single‐negative media only allow evanescent wave transport. The lower‐left quadrant of Figure 1.1 represents a special case of materials where both permittivity and permeability are simultaneously negative. These Double Negative materials (DNGs) like their double‐positive counterparts support wave propagation though the media. The key difference between the DNG quadrant and the other three is that single‐negative and double‐positive media all occur naturally, whereas we are yet to find a naturally occurring DNG media.
Figure 1.1 Broad categorization of materials based on the real components of the permittivity and permeability.
Although presenting fantastic opportunities, molecular synthesized materials are limited in the range of RF properties they can produce due to the nature of the EM interaction with the molecules of the material. An interaction where the light‐mass negatively charged electrons surrounding the relatively large‐mass positively charged nucleus of the atoms move in response to an EM wave forming a dipole. This response is fixed by both the fundamental properties (charge, mass) and the chemical bonds formed in the material, limiting the available parameter range and these materials can access. These limitations have led scientists and engineers to create a range of artificial composite structures with periodic subwavelength functional inclusions. Although these inclusions are many orders of magnitude larger than the molecules of the constitutive materials, they are still much smaller that the EM wavelength of interest. In this case, to an incident EM wave, these inclusions respond no differently than giant molecules with a very large polarizability. This enables the interactions between wave and the collective structures to be described in terms of the “homogenized” abstracted bulk material parameters permittivity and permeability. Treating the collective periodic structures in this homogenized manner is called an “effective” medium or material. This approach in theory allows the engineer to fabricate artificial effective materials with specific engineered EM properties, most notable of which is the creation of the above DNG materials. There are of course restrictions on achievable physical material properties that are impossible to engineer, such as the creation of media where waves propagate with group velocities greater than the speed of light in vacuum.
Around 20 years ago, the word “MTM” entered the lexicon to refer to certain types of effective media. Even though a large number of peer‐reviewed papers using the word “MTM” have been published an agreed definition of what a MTM is remains elusive. The origin of the word “meta” from the Greek “beyond” implies in some sense that “metamaterials” are a form of material beyond conventional materials. Sources suggest the term “MTM” was first coined by Rodger Walser in 1999 [9], who defined a MTM as; “…macroscopic composites having man‐made, three‐dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to specific excitation.” Whereas the Metamorphose Network defines a metamaterial as “…an arrangement of artificial structural elements, designed to achieve advantageous and unusual electromagnetic properties” [10].
This later definition although encompassing the Walser definition could be considered too “broad,” as, for example it does not recognize the critical differences between MTMs, photonics structures, and other man‐made structures such as multi‐input, multi‐output (MIMO) antenna arrays. To quote Cai and Shalaev [8]; “Metamaterials are, above all, man‐made materials. The structural units of a metamaterial, known as meta‐atoms or metamolecules, must be substantially smaller than the wavelength being considered, and the average distance between neighboring metaatoms is also subwavelength in scale. The subwavelength scale of the inhomogeneities in a metamaterial makes the whole material macroscopically uniform, and this fact makes a metamaterial essentially a material instead of a device. The scale of the inhomogeneities also distinguishes metamaterials from many other electromagnetic media.” These last two sentences from Wei are critical in defining the underlaying physics that enables us to consider MTMs as “effective media.” For example some definitions would allow the eye of a lobster to be defined as a MTM, even though the structure of the lobster's eye works on reflection with a periodicity of 10 m [11], many times larger than the wavelength of light entering the lobster eye meaning that the system cannot really be treated as an effective media.
1.3 Effective‐Media Theory
Effective media theory builds upon the theoretical framework developed in the nineteenth century by Mossitti [12] and Clausius [13] on the homogenization of materials. For example consider a system of small, subwavelength, particles arranged into a lattice. If the particles are small enough, then the response of the system to an EM wave is the same as if the system were a collection of molecules with a large polarizability, i.e. if the scale of the inhomogeneities is small compared to the incident wavelength, then the system appears homogeneous to the wave. This homogenization approach allows us to predict the EM behavior of a heterogeneous system by evaluating the effective permittivity and permeability of a macroscopically homogeneous medium. Where the effective permittivity and permeability of the bulk material is found in terms of the permittivities, permeabilities, and geometry of the individual constituents of the system. This approach is the basis for many “effective media” theories, Lakhtakia [14] presents a comprehensive review of the early work on effective media theories and a review of more modern work can be found in the paper by Belov and Simovski [15] that also discusses the homogenization of MTMs including a radiation term.
Two commonly used effective media theories that illustrate the general approach are the Maxwell–Garnett [16] and the Bruggeman [17] approach. Each approach is based upon slightly different assumptions about the topology and material properties of the constituent materials. In the Maxwell–Garnett approach it is assumed that the inclusions are well‐defined spheres sparsely scattered across the host medium. The Bruggeman approach is essentially a percolation approach, where the two mediums are equally intermingled. These examples highlight a key point about effective‐media theories. As the effective‐permittivity/permeability are averaged differently in each model, different effective‐media theories cannot be directly compared to each other even when the same subwavelength configuration is considered.
1.4 History of Effective Materials
1.4.1 Artificial Dielectrics
The realization of artificial materials began a hundred years before the term “metamaterial” was introduced, with the work of Rayleigh and Bose in the 1890s. Rayleigh proposed a system of small scatterers as an equivalent continuous medium [18], and Bose produced an artificial chiral material by twisting “jute” root [19]. This work was extended in 1914 by Lindman who considered small wire helices embedded into a host medium to create an artificial chiral material [20]. The first practical applications did not appear until the 1940s with the pioneering work of Kock [21]. Kock created Artificial Dielectrics from arrays of subwavelength metallic structures (spheres, rods, and plates) to form Dielectric Lenses [21] of the form shown in Figure 1.2, with the aim to develop light weight RF lens compared to their metal counterparts.
Figure 1.2 Kock's artificial dielectric lens, consisting of conducting spheres embedded in a low index foam, taken from [21]. Source: Kock [21] / with permission of John Wiley & Sons, Inc.
In 1953 Brown [22] extended the work of Kock, considering a lattice of thin metallic wires, showing the system could be considered to have a plasma frequency. Brown demonstrated that the system formed an artificial plasma and could be considered an effective medium with negative permittivity. In the case of lossless wires, the wire array can be modeled as an array of inductors with inductance . In this case, the effective permittivity () of the system becomes
Importantly, Kharadly and Jackson [23] generalized this work to consider effective media formed from lattices of metal ellipsoids, disks, and rods, with the assumption that the frequency of operation is low and the Rayleigh quasi‐static restriction holds. Interest in this type of effective medium grew as the possibilities for exploitation were realized, most comprehensively by Rotman [24], who explored these artificial materials as plasma analogs to investigate the effect of plasmas on antenna systems. This type of wire array media have been turned into an “active” material by the inclusion of diodes enabling the media to be actively switched from a negative to a positive permittivity medium. Progress with this type of media resulted in the material becoming commercially available in the 1970s [25]. Even today wire‐array based media are still attracting interest as subwavelength elements for epsilon negative (ENG) and DNG materials. Also, especially, in configurations that exhibit spatial dispersion (i.e. a dependence of the permittivity or permeability on the wavevector, and ) [26–28].
1.4.2 Artificial Magnetic Media
Research into artificial magnetic media dates back to the work of Schelkunoff and Friis [29] in the 1950s and the proposed Split Ring Resonator (SRR). Engineered high‐permeability materials are especially interesting as most conventional materials of the magnetic field component of the EM wave couples only weakly to the material [30]. Magnetism without magnetic materials has been known for sometime, such as “wet” ice crystals, where the water in the system causes a diamagnetic behavior, although even in these systems, the relative permeability is low. Today, the SRR remains the magnetic meta‐atom of choice for researchers, although multiple researchers have examined the SRR in depth (see for example [31,32]), the basic geometry remains the same as that originally proposed by Schelkunoff in 1950.
Figure 1.3 (a) Double SRR geometry building block, and (b) An array of SRRs. (c) The equivalent circuit diagram from the SRR shown in (a).
Due to the significance of the SRR, it is pertinent to review the key aspects of its function and behavior. Consider the geometry shown in Figure 1.3, a double SRR formed from concentric metallic tracks similar to the design of Pendry et al. [33]. We consider the case where this SRR meta‐atom is much smaller than the wavelength of interest allowing a system of multiple SRRs to be described by effective medium theory. At the level of an individual meta‐atom, the incident wave upon a SRR produces a magnetic flux to oppose the incident field. Without the split, this interaction would be purely an inductive nonresonant phenomena, resulting in a weakly diamagnetic system. The split prevents the current circulating causing a collection of charge at the split edge creating a capacitance.
A meta‐atom with a single SRR will accumulate charge at the gap creating a large electric dipole moment that in most cases dominate over the magnetic dipole moment. A second concentric SRR where the “gaps” of the SRRs are opposite each other offers control over the capacitance of the meta‐atom, allowing the electric dipole moment of the inner ring to suppress the electric dipole moment of the outer ring, allowing the magnetic moment to dominate the system.
The resulting SRR configuration can be modeled as an equivalent subwavelength quasi‐static LCR circuit, shown in Figure 1.3. This circuit although a crude first approximation can present great insights into the system's response and behavior of the artificial material over all. The inductive elements of the equivalent circuit are relatively easy to determine, estimated by . The Ohmic loss in the system can be estimated as . Determining the capacitance is tricky as in addition to the capacitive effects of the split “gaps,” there is also the capacitance from the gap that separates the two SRRs. An analysis conducted by Baena et al. [34] approximates the capacitance of the double SRR system by , where is the combined width of the rings and the separation between the rings. This enables the resonant frequency of the meta‐atom to be estimated as . Using the resonant frequency, we can estimate, to first order, the magnetic moment of an individual meta‐atom in response to an incident wave of magnetic field, [35]:
Using Eq. (1.3) one can then determine the effective permeability [35] () of an artificial material formed from a lattice of individual subwavelength SRRs:
is the unit‐cell volume for an individual meta‐atom. This approach is of course rather crude and does not take into account electric coupling or the bianisotropic nature of the material. Although it does enable us, at least to first order, to gain useful insights into how engineered changes to the unit‐cell geometry will alter the effective permeability of our artificial material.
1.5 Double Negative Media
Although several researchers have considered materials with simultaneous negative permittivity and permeability (DNG materials) [36], and materials with a negative index of refraction [37] prior to 1965. The first systematic study of the general properties of a hypothetical DNG medium with a negative refractive index is attributed to the seminal 1967 paper by Veselago [38]
