IEEE MTT-V053-I04B (2005-04) [53, 04B ed.]

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APRIL 2005

VOLUME 53

NUMBER 4

IETMAB

(ISSN 0018-9480)

PART II OF TWO PARTS

SPECIAL ISSUE ON METAMATERIAL STRUCTURES, PHENOMENA, AND APPLICATIONS Guest Editorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Itoh and A. A. Oliner

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PAPERS

Analysis and Modeling Homogenization of 3-D-Connected and Nonconnected Wire Metamaterials . . . . . M. G. Silveirinha and C. A. Fernandes Fundamental Modal Properties of Surface Waves on Metamaterial Grounded Slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, G. Lovat, and S. Paulotto Refraction Laws for Anisotropic Media and Their Application to Left-Handed Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. M. Grzegorczyk, M. Nikku, X. Chen, B.-I. Wu, and J. A. Kong Equivalent-Circuit Models for Split-Ring Resonators and Complementary Split-Ring Resonators Coupled to Planar Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. D. Baena, J. Bonache, F. Martín, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García–García, I. Gil, M. Flores Portillo, and M. Sorolla Efficient Modeling of Novel Uniplanar Left-Handed Metamaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Y. Guo, G. Goussetis, A. P. Feresidis, and J. C. Vardaxoglou Numerical Analysis Propagation Property Analysis of Metamaterial Constructed by Conductive SRRs and Wires Using the MGS-Based Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-Y. Yao, W. Xu, L.-W. Li, Q. Wu, and T.-S. Yeo FDTD Study of Resonance Processes in Metamaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E. A. Semouchkina, G. B. Semouchkin, M. Lanagan, and C. A. Randall Periodic Finite-Difference Time-Domain Analysis of Loaded Transmission-Line Negative-Refractive-Index Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Kokkinos, C. D. Sarris, and G. V. Eleftheriades Modeling of Metamaterials With Negative Refractive Index Using 2-D Shunt and 3-D SCN TLM Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. P. M. So, H. Du, and W. J. R. Hoefer

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Experimental Simulation of Negative Permittivity and Negative Permeability by Means of Evanescent Waveguide Modes—Theory and Experiment . . . . . . . . .J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura Effective Electromagnetic Parameters of Novel Distributed Left-Handed Microstrip Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.-G. Mao, S.-L. Chen, and C.-W. Huang Experimental Realization of a One-Dimensional LHM–RHM Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong Super-Compact Multilayered Left-Handed Transmission Line and Diplexer Application . . . Y. Horii, C. Caloz, and T. Itoh

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A Positive Future for Double-Negative Metamaterials (Invited Paper) . . . . . . . . . . . . . N. Engheta and R. W. Ziolkowski

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Information for Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Digital Object Identifier 10.1109/TMTT.2005.847875

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

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Guest Editorial I. INTRODUCTION

II. TERMINOLOGY

ETAMATERIALS, the topic of this TRANSACTIONS’ Special Issue, is a vague term. Broadly, it means something beyond natural materials, and implies something artificial, not ordinarily found in nature. Originally, the word metamaterials included a wider range of artificial materials, but in this TRANSACTIONS’ Special Issue, the word applies to those materials or structures in which the phase velocity and the group (power transmission) velocity point in opposite directions. This property leads to a series of fascinating performance features, such as a backward electromagnetic wave in the material, a negative index of refraction, a reversed Doppler effect, etc. These performance features in themselves are stimulating to contemplate, but we are learning that novel and practical microwave components can also be devised. Among these are forward or backward directional couplers with improved features like arbitrary coupling ratios, shorter coupling lengths, broader bandwidths, etc. Other coupling components with improved features include branch-line couplers and hybrid rings. A different type of novel component that followed directly from the backward-wave property is a leakywave antenna that can be frequency scanned or electronically scanned from backward endfire through broadside to forward endfire. Previously, such scanning could be obtained by employing the first higher space harmonic, but here, the radiation is supplied by the dominant space harmonic. Yet another novel device is a perfect lens, consisting of a planar slab of metamaterial, which has the startling property that the sharpness of the image is not limited by the wavelength of light. This truly unusual feature follows from its negative-refractive-index property and the fact that evanescent waves are amplified when traversing the metamaterial slab. All of these structures have been built, and their performances have been verified by measurements at microwave frequencies. This topic is rather new, about a half-dozen years old, but it has expanded very rapidly, particularly in the physics community. In the microwave community, the contributions began only about three years ago, but progress has also been rapid, although the goals and stress have usually been somewhat different from those in the physics community. The nature of these contributions and their potential for novel applications, however, are not yet widely understood or appreciated by most microwave engineers. It is hoped that this TRANSACTIONS’ Special Issue will help to clarify the state-of-the-art for our community and illustrate some of the opportunities for future novel applications in the microwave field. Since we believe that many people in the microwave field are not very familiar with this topic, we are devoting much of this Guest Editorial to some background information for those who may fit into this category.

The class of materials or structures that are referred to as metamaterials are characterized by various terms such as: 1) left-handed (LH); 2) BW; 3) negative-index (NI) or negative-refractive index (NRI); or 4) double-negative (DNG), and perhaps others. The term “left-handed” was used in the original paper by Veselago, and many people use it for that reason. It signifies a contrast with the well-known “right-hand rule” for the direction of the Poynting vector relative to the directions of the electric and magnetic fields. An objection to its use is that “LH” is used in classifying chiral media and types of circularly polarized radiation. The term BW is not as widely used because backward waves can be produced in several other ways. The NRI property was originally thought to be obtainable only from a metamaterial medium, but it is now known that it is possible to obtain this property from a special periodic arrangement of ordinary materials. Nevertheless, the NRI term is a very appropriate description when dealing with two- or three-dimensional structures. The terms above represent in each case a property that results from a wave that propagates within the metamaterial structure. The remaining term, double negative, follows from the fundamental properties of the metamaterial itself. Double-negative signifies that the permittivity and permeability of the material (or the effective values of and of the structure comprising the metamaterial) are both negative. For ordinary materials, these values are both positive. The letters DNG have been found to be easily generalizable to double-positive (DPS) or to single-negative materials, as only-epsilon-negative (ENG) and only-mu-negative (MNG). Unfortunately, there is no commonly agreed-upon term, so we have permitted in this TRANSACTIONS’ Special Issue each paper to employ its own preference.

M

Digital Object Identifier 10.1109/TMTT.2005.845126

III. BRIEF HISTORICAL BACKGROUND In 1968, Veselago [1] authored a paper that asked what electrodynamic behavior could be found if the and of a material are simultaneously negative rather than positive. He found that the results would be very interesting and unusual, and he pointed out that, among these new effects, one would find backward waves, a negative refractive index, a reversed Doppler effect, a reversed ˇ Cerenkov effect, a planar slab lens, etc. He then asked if any materials in nature could be found with simultaneously negative and values, and he concluded that there are none. As a result of this conclusion, no further progress along these lines was made until the recent turn of the century. It had been known for some time that a medium consisting of an array of conducting rods can serve as a medium with a negative value of , but what was missing was how to achieve a negative value of . The breakthrough was provided in 1999 by Pendry et al. in [2], which presented several examples of how one can obtain negative values of from conductors. The most interesting of

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

them was a split-ring resonator, an array of which provided a negative value of over a portion of its resonance frequency range. The combination of the rods and split-ring resonators yielded an artificial medium possessing both a negative and over a narrow frequency range. Encouraged by the implications of these results, a team at the University of California at San Diego, La Jolla, built a composite medium along these lines at microwave frequencies, and made measurements that clearly demonstrated that this medium exhibited a negative index of refraction. These measurements [3] proved that the physical effect is real, and they triggered a flood of papers, especially in the physics community. The microwave community was also excited by the potential for novel effects and novel components, but it realized that the guiding structure and constituent elements that were available at that time were not suitable for application to microwave integrated circuits. Since the structure on which the measurements were made included a split-ring resonator for the effective negative , a widespread perception developed that the array of elements must contain a resonant constituent, which is obviously undesirable because the resonant element is narrow-band and also lossy in precisely the metamaterial range of operation. Another key problem is that the physics community preferred their structures to be three-dimensional (3-D), or at least two-dimensional (2-D), whereas for microwave integrated circuits, the guiding structure must be one-dimensional (1-D). During June 2002, at the IEEE Microwave Theory and Techniques (MTT) and Antennas and Propagation (AP)/International Scientific Radio Union (URSI) Symposia, solutions were presented by three different groups that successfully addressed the above-mentioned problems. All three used existing transmission lines as the supporting guiding structure; Oliner [4] chose strip line, while Iyer and Eleftheriades [5] and Caloz et al. [6] selected microstrip line. All three also chose basically similar discontinuity elements (shunt inductances and series capacitances) to provide the required negative and values. The big difference between [4] and both [5] and [6] is that [4] employed analytical expressions for the discontinuities valid over a wide frequency range, whereas [5] and [6] used L and C terms and stressed the low-frequency features and the design simplicity associated with the L, C transmission-line approach. These papers provided the foundation for both the approach and methodology regarding how to proceed when designing for 1-D transmission lines. This is especially true for the approaches that use L and C. The authors of [5] and [6], together with various additional later contributors, then proceeded to develop a series of novel microwave components for use with microstrip line. IV. ADDITIONAL OBSERVATIONS Veselago’s original paper [1] is a classic “what-if” paper. It asks, “What if a medium were to have negative values of and instead of positive values? In what ways would the electrodynamic behaviors be different?” If the behaviors are not much different, or the results otherwise uninteresting, the study would serve to tell us that the structure is not worth investigating further. Or, if the results are interesting, but there is no way in which measurements can be made, the structure must be set aside in

the meantime. That was the situation for Veselago’s paper until recently. Now that we know how to create an artificial medium with such properties, all of the interesting performance predictions are being verified experimentally. “What-if” papers can, therefore, be very valuable under the right circumstances. In this TRANSACTIONS’ Special Issue, Engheta and Ziolkowski have submitted an excellent invited review paper with the intriguing title “A Positive Future for Double Negative Metamaterials.” Many portions of their paper inquire into structures that may lead to potential applications. In that sense, these inquiries are “what-if” papers that may be tested experimentally in the near future. One example that is discussed in their paper was initially a purely “what-if” paper (in 2002), but it has now been experimentally verified in one of the papers in this TRANSACTIONS’ Special Issue. The physical structure is a bilayer in which one layer is DNG and the other layer is DPS (an ordinary medium), forming a two-layer resonant cavity that can have a sub-wavelength arbitrarily small thickness. This experimental paper by Li et al. is entitled “Experimental Realization of a One-Dimensional LHM–RHM Resonator.” We were hoping for numerous papers with novel applications, but the microwave community is not yet ready for many such papers. Instead, the hardware-oriented creativity is directed more toward novel ways to obtain negative values of and . One such paper in this TRANSACTIONS’ Special Issue is entitled “Simulation of Negative Permittivity and Negative Permeability by Means of Evanescent Waveguide Modes—Theory and Experiment,” by Esteban et al. The key concept involves the recognition that an empty metallic waveguide below cutoff supports a decaying wave comparable to a wave in a stop band or to “propagation” through an over-dense plasma. The authors extend the earlier paper by Marqués et al., which relates a medium with a negative to evanescent TE modes, to a new equivalence, between evanescent TM modes and a negative . A second novel structure involves split-ring resonators. The first steps are to miniaturize the size of the split rings and place them in an array along the guide direction. As is well known from earlier studies, such arrays will produce a negative value of permeability. A newer contribution is a dual of such split-ring resonators, called a complementary split-ring resonator, which is fabricated by etching the negative image of the split ring in the ground plane of a microstrip line underneath the conductor strip. The complementary resonator produces a negative value of , rather than . The properties of the original split-ring structure and its complementary version are discussed, and their equivalent circuits are developed, together with how they give rise to bandpass and bandstop filters. The narrow-band and lossy nature of these resonators are still concerns. It is not our intention to cover all the novel features in this TRANSACTIONS’ Table of Contents, but we have listed these few examples as illustrative of their flavor. The field has been moving ahead nicely, and we hope that this TRANSACTIONS’ Special Issue will serve to stimulate further creativity, particularly in the direction of novel microwave components. In these introductory remarks, we have presented much in the way of background information in an attempt to be of help to those who are less well informed on this topic and are in need of some guidance.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

We wish to express our sincere thanks to our many reviewers who conscientiously contributed their time and effort in order to help us produce a quality Special Issue. We received almost 40 submitted manuscripts and, in many cases, we needed to call on the same reviewer to evaluate more than one manuscript. We also wish to express our appreciation to those authors who patiently cooperated with the rigors of

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the review process, and diligently made the modifications requested by the reviewers. We are particularly indebted to this TRANSACTIONS’ Editor-in-Chief, Prof. Michael B. Steer, for providing the necessary guidance in both matters of principle and in the many procedural details. We also greatly appreciate the many ways in which he graciously offered help and kept us on track.

List of Reviewers Alkim Akyurtlu Ikuo Awai Mario Sorolla Ayza Maurizio Bozzi Paolo Burghignoli Christophe Caloz Steve Cammer Filippo Capolino Kai Chang Weng Cho Chew Peter de Maagt Michael Dydyk George Eleftheriades Nader Engheta Fabrizio Frezza Spartak Gevorgian Thomasz Gregorczyk Wojciech Gwarek Wolfgang Hoefer Yasushi Horii Akira Ishimaru

David Jackson Dieter Jaeger Rolf Jakoby Vikram Jandhyala Mikko Kärkkäinen Shigeo Kawasaki Edward Kuester Yasuo Kuga Paolo Lampariello Ralph Levy Le-Wei Li Didier Lippens Wen-tao Lu Ricardo Marqués Wolfgang Menzel Francisco Mesa Eric Michielssen Hossein Mosallaei Michal Mrozowski Michal Okoniewski Atsushi Sanada

Hiroshi Shigesawa Dan Sievenpiper Ari Sihvola Richard Snyder Alexsander Sochava Roberto Sorrentino Costas Soukoulis Sergei Tretyakov Mikio Tsuji Ching-Kuang C. Tzuang Tetsuya Ueda Yuanxun Wang Dylan Williams Bae-Ian Wu Ke Wu Fan Yang Junho Yeo Tsukasa Yoneyama Jan Zehentner Richard Ziolkowski

TATSUO ITOH, Guest Editor University of California at Los Angeles (UCLA) Department of Electrical Engineering Los Angeles, CA 90095-1594 USA

ARTHUR A. OLINER, Guest Editor Polytechnic University Department of Electrical Engineering Lexington, MA 02421 USA

REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 10, pp. 509–514, Jan.–Feb. 1968. [2] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [3] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001.

[4] A. A. Oliner, “A periodic-structure negative-refractive-index medium without resonant elements,” in IEEE AP-S/URSI Int. Symp. Dig., San Antonio, TX, Jun. 16–21, 2002, p. 41. [5] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2D waves,” in IEEE MTT-S Int. Microwave Symp. Dig., Seattle, WA, Jun. 2–7, 2002, pp. 1067–1070. [6] C. Caloz, H. Okabe, T. Iwai, and T. Itoh, “Transmission line approach of left-handed materials,” in IEEE AP-S/URSI Int. Symp. Dig., San Antonio, TX, Jun. 16–21, 2002, p. 39.

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Tatsuo Itoh (S’69–M’69–SM’74–F’82) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1969. From September 1966 to April 1976, he was with the Electrical Engineering Department, University of Illinois at Urbana-Champaign. From April 1976 to August 1977, he was a Senior Research Engineer with the Radio Physics Laboratory, SRI International, Menlo Park, CA. From August 1977 to June 1978, he was an Associate Professor with the University of Kentucky, Lexington. In July 1978, he joined the faculty at The University of Texas at Austin, where he became a Professor of Electrical Engineering in 1981 and Director of the Electrical Engineering Research Laboratory in 1984. During the summer of 1979, he was a Guest Researcher with AEG-Telefunken, Ulm, Germany. In September 1983, he was selected to hold the Hayden Head Centennial Professorship of Engineering at The University of Texas at Austin. In September 1984, he was appointed Associate Chairman for Research and Planning of the Electrical and Computer Engineering Department, The University of Texas at Austin. In January 1991, he joined the University of California at Los Angeles (UCLA) as Professor of Electrical Engineering and Holder of the TRW Endowed Chair in Microwave and Millimeter Wave Electronics. He was an Honorary Visiting Professor with the Nanjing Institute of Technology, Nanjing, China, and at the Japan Defense Academy. In April 1994, he was appointed an Adjunct Research Officer with the Communications Research Laboratory, Ministry of Post and Telecommunication, Japan. He currently holds a Visiting Professorship with The University of Leeds, Leeds, U.K. He has authored or coauthored 350 journal publications, 650 refereed conference presentations, and has written 30 books/book chapters in the area of microwaves, millimeter waves, antennas, and numerical electromagnetics. He has generated 64 Ph.D. students. Dr. Itoh is a member of the Institute of Electronics and Communication Engineers of Japan, and Commissions B and D of USNC/URSI. He served as the editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (1983–1985). He serves on the Administrative Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He was vice president of the IEEE MTT-S in 1989 and president in 1990. He was the editor-in-chief of IEEE MICROWAVE AND GUIDED WAVE LETTERS (1991–1994). He was elected an Honorary Life Member of the IEEE MTT-S in 1994. He was elected a member of the National Academy of Engineering in 2003. He was the chairman of the USNC/URSI Commission D (1988–1990) and chairman of Commission D of the International URSI (1993–1996). He is chair of the Long Range Planning Committee of the URSI. He serves on advisory boards and committees for numerous organizations. He has been the recipient of numerous awards including the 1998 Shida Award presented by the Japanese Ministry of Post and Telecommunications, the 1998 Japan Microwave Prize, the 2000 IEEE Third Millennium Medal, and the 2000 IEEE MTT-S Distinguished Educator Award.

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Arthur A. Oliner (M’47–SM’52–F’61–LF’87) was born on March 5, 1921, in Shanghai, China. He received the B.A. degree from Brooklyn College, Brooklyn, NY, in 1941, and the Ph.D. degree from Cornell University, Ithaca, NY, in 1946, both in physics. He joined the Polytechnic Institute of Brooklyn (now Polytechnic University) in 1946, and became Professor in 1957. He then served as Department Head from 1966 to 1974, and was Director of its Microwave Research Institute from 1967 to 1982. He was a Walker-Ames Visiting Professor with the University of Washington, in 1964. He has also been a Visiting Professor with the Catholic University, Rio de Janeiro, Brazil, the Tokyo Institute of Technology, Tokyo, Japan, the Central China Institute of Science and Technology, Wuhan, China, and the University of Rome, Rome, Italy. In 2003, the University of Rome (“La Sapienza”) granted him an Honorary Doctorate, and organized an associated special symposium in his honor. He is a member of the Board of Directors of Merrimac Industries. He has authored over 300 papers, various book chapters, and has coauthored or coedited three books. His research has covered a wide variety of topics in the microwave field including network representations of microwave structures, guided-wave theory with stress on surface waves and leaky waves, waves in plasmas, periodic structure theory, and phased-array antennas. He has made pioneering and fundamental contributions in several of these areas. His interests have also included waveguides for surface acoustic waves and integrated optics, novel leaky-wave antennas for millimeter waves, and leakage effects in microwave integrated circuits. Lately, he has contributed to the topics of metamaterials, and to enhanced propagation through subwavelength holes. Dr. Oliner is a Fellow of the American Association for the Advancement of Science (AAAS) and the Institution of Electrical Engineers (IEE), U.K. He was a Guggenheim Fellow. He was elected a member of the National Academy of Engineering in 1991. He has been the recipient of prizes for two of his papers: the IEEE Microwave Prize in 1967 for his work on strip-line discontinuities, and the Institution Premium of the IEE in 1964 for his comprehensive studies of complex wave types guided by interfaces and layers. He was President of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), its first Distinguished Lecturer, and a member of the IEEE Publication Board. He is an Honorary Life Member of IEEE MTT-S (one of only six such persons). In 1982, he was the recipient of its highest recognition, the Microwave Career Award. A special retrospective session was held in his honor at the IEEE MTT-S International Microwave Symposium (reported in detail in the December 1988 issue of this TRANSACTIONS, pp. 1578-1581). In 1993, he was the first recipient of the Distinguished Educator Award of the IEEE MTT-S. He was also a recipient of the IEEE Centennial and Millennium Medals. He is also a past U.S. chairman of Commissions A and D of the International Union of Radio Science (URSI), a long-time member of and active contributor to Commission B, and a former member of the U.S. National Committee of URSI. In 1990, he was the recipient of the URSI van der Pol Gold Medal, which is given triennially, for his contributions to leaky waves. In 2000, the IEEE awarded him a second gold medal, the Heinrich Hertz Medal, which is its highest award in the area of electromagnetic waves.

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Homogenization of 3-D-Connected and Nonconnected Wire Metamaterials Mário G. Silveirinha, Member, IEEE, and Carlos A. Fernandes, Member, IEEE

Abstract—The homogenization of composite structures made of long thin metallic wires is an important problem in electromagnetics because they are one of the basic components of the doublenegative medium. In this paper, we propose a new analytical model to characterize the effective permittivity of the three-dimensionalwire medium in the long wavelength limit. We study two different topologies for the wire medium. The first structure consists of a lattice of connected wires, whereas the second one consists of a lattice in which the wires are not connected. Our results show that the propagation of electromagnetic waves in the two metamaterials is very different. While one of the structures exhibits strong spatial dispersion, the other one seems to be a good candidate for important metamaterial applications. We also found that, for extremely low frequencies, one of the structures supports modes with hyperbolic wave normal contours, originating negative refraction at an interface with air. We validated our theoretical results with numerical simulations. Index Terms—Double-negative (DNG) medium, homogenization theory, metamaterials, negative refraction, wire medium.

I. INTRODUCTION

I

N [1], Smith et al. proposed an original structure that consists of a lattice of long metallic wires and split-ring resonators (SRRs). Theoretical and experimental studies show that the composite structure behaves as a double negative (DNG) medium (also known as left-handed medium) with a negative index of refraction [1]–[3]. This remarkable result motivated much research on metamaterials and potential applications. For example, numerous studies try to explore the focusing property of a DNG slab surrounded by conventional media [4]–[6]. It was even suggested, with considerable controversy, that metamaterials could be used to fabricate a super lens with no limit of resolution [6]. Other studies show that metamaterials may favor the miniaturization of some devices, and the realization of subwavelength cavity resonators [7]. The negative refraction of the DNG metamaterial is attributed to the combined effect of the arrays of wires and SRRs. As is well known, the lattice of wires is modeled as a medium with negative permittivity, whereas the lattice of SRRs is characterized by a negative permeability. To a first-order approximation, there is no coupling between the two basic inclusions.

Manuscript received May 20, 2004; revised September 20, 2004. This work was supported by the Fundação para Ciência e a Tecnologia under Project POSI 34860/99. M. G. Silveirinha is with the Electrical Engineering Department–Instituto de Telecomunicações, Polo II da Universidade de Coimbra, 3030 Coimbra, Portugal (e-mail: [email protected]). C. A. Fernandes is with the Instituto Superior Técnico–Instituto de Telecomunicações, Technical University of Lisbon, 1049-001 Lisbon, Portugal (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845128

Fig. 1. (a) Fragment of a metamaterial formed by a lattice of connected wires. (b) Fragment of a metamaterial formed by a lattice of nonconnected wires (adjacent orthogonal wires are spaced of half-lattice constant).

The configuration originally proposed in [1] is characterized by DNG parameters only for a unique polarization, and for propagation along a specific direction of space. The fabrication of an isotropic DNG medium clearly compels for a basic cell with extra symmetry. Apparently, apart from practical difficulties related to technological limitations, the generalization seems to be straightforward. However, the situation may not be so plain. For example, one might expect that a medium formed by wires that are parallel to the coordinate axes would interact with the radiation as an isotropic medium with negative permittivity. However, numerical results reported by Silveirinha and Fernandes [8] support that, at least for the nonconnected topology studied in [8], that is not the case. Indeed, we found that near the “plasma frequency” the referred metamaterial is not isotropic. Moreover, we verified that the numerical results are consistent with the hypothesis that, analogous with the one-dimensional (1-D)-wire medium formed by an array of parallel wires [9] and other related structures [10], the considered geometry for the three-dimensional (3-D)-wire medium has strong spatial dispersion in the long wavelength limit, i.e., the permittivity depends not only on the frequency, but also on the wave vector. The results reported in [8] raise the obvious question: “Is it possible to fabricate an isotropic metamaterial with negative permittivity?” This is a fundamental subject not only to understand the possible limitations of DNG metamaterials, but also because metamaterials with negative permittivity, when paired with metamaterials with negative permeability, may have interesting properties and applications, as described in [11]. In this paper, we study two different topologies for the 3-Dwire medium. The first structure consists of a simple cubic lattice of connected wires, and the second one consists of a simple cubic lattice in which the wires are not connected (i.e., the geometry studied in [8] using numerical methods). In Fig. 1, we depict fragments of both metamaterials.

0018-9480/$20.00 © 2005 IEEE

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A. Connected Topology In this case, the metallic region in the unit cell consists of three cylindrical wires with radius and length . We have (1)

Fig. 2. Geometry of the unit cell. (a) Connected wire geometry. (b) Nonconnected wire geometry.

We propose a new analytical procedure to homogenize the periodic metamaterials. We will prove that the two topologies for the 3-D-wire medium are not equivalent, and that the interaction of electromagnetic waves with the two structures is very different. We will derive an approximate analytical formula for the effective permittivity of the composite structures. To our best knowledge, no results were reported in the literature that concern the homogenization of related structures with the exception of the numerical analysis described in [8], and the experimental results reported in [12] (the geometry considered in [12] is slightly more complex than the ones considered here). This paper is organized as follows.1 Sections II–IV concern the characterization of the Floquet modes that propagate in the periodic material. In Section II, we describe the geometry of the wire medium. In Section III, we formulate the modal problem, and we obtain the characteristic system using an integral-equation-based approach. In Section IV, we propose a simplified characteristic system that can be solved using analytical methods. Sections V and VI concern the homogenization of the metamaterials. In Section V, we explain how the effective permittivity dyadic can be obtained directly from the characteristic system. In Section VI, we discuss the physical phenomena implied by our formulas, and we compare the developed theory with full-wave numerical results. Finally, in Section VII, we present conclusions. II. GEOMETRY We denote a generic point of space by , and the unit vector directed along the -direction by . The wire medium is obtained by the periodic repetition of the unit cell shown in Fig. 2. The unit cell is centered at the origin. The wires are arranged into a simple cubic lattice with lattice constant . The boundary of the metallic region in the , and the outward unit normal vector unit cell is denoted by is denoted by . For simplicity, we admit that the wires are embedded in air. We consider two different topologies for the wire medium. The geometry shown in Fig. 2(a) corresponds to the connected case, and the geometry of Fig. 2(b) corresponds to the nonconnected case. In Sections II-A and –B, we describe each configuration in detail. 1While

this paper was being prepared, Simovski and Belov studied the homogenization of the nonconnected wire medium using a local field approach [23].

In the above, represents the surface of the wire section -direction. Note that the wires intersect oriented along the mutually, forming a junction near the origin. Our objective is to propose an analytical treatment for the homogenization problem. Thus, it is desirable to have a model as simple as possible for the current that flows along the wires. We , where is the wavelength of will assume that radiation in the dielectric region. Therefore, the thin-wire approximation can be used. Within the thin-wire approximation, the density of current over each wire flows along the direction of the axis, and has approximately circular symmetry. We will also admit that is a traveling wave characterized by the wave vector (the justification for this assumption will be given in Section III). Therefore, we admit that (2) where is a periodic function that stands for the unknown current. In order to take into account the coupling at the junction, we admit the possibility of the current induced over the th wire being discontinuous at . In general, we have . The conservation of current requires that (see also [13]) (3) Note that, within the described model, we completely neglect the shape of the wire junctions. Essentially, in our model, the junctions are replaced by infinitesimal gaps located at (i.e., the points where we allow the current to be discontinuous) with a circular transverse section. Nevertheless, our simplified model is expected to yield good results since each junction occupies only a very small fraction of the unit cell. B. Nonconnected Topology The nonconnected geometry is depicted in Fig. 2(b). As in the previous case, the metallic region in the unit cell consists and length . Now the of three cylindrical wires with radius , i.e., wires do not intersect, and the wire axes are spaced by a half-lattice constant. The wire oriented along the -direction is centered at (4) As before, we admit that the density of current over the th is a continuous periodic function wire is given by (2). Now because there are no junctions.

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III. MODAL PROBLEM In order to homogenize the 3-D-wire medium, we will first characterize the Floquet modes propagating in the periodic metamaterial. Here, we will prove that the density of current associated with a generic electromagnetic mode satisfies a homogeneous characteristic system. Later we will show that the effective permittivity dyadic of the metamaterial can be obtained directly from the characteristic system. asBy definition, an electromagnetic Floquet mode satisfies the folsociated with the wave vector lowing equations: dielectric region

(5a)

dielectric region

(5b)

on the metallic surfaces is periodic

(5c) (5d)

is the free-space where is the impedance of free space, wavenumber, is the angular frequency, and is the velocity of light in vacuum. In the above, (5a) and (5b) are the frequencydependent Maxwell equations, (5c) is the boundary condition at the metallic interfaces, and (5d) is the Floquet wave condition. As is well known, for each wave vector , the system of (5) represents an eigenvalue problem. It has nontrivial solutions only for a countable set of wavenumbers . The eigenvalues form the band structure of the periodic medium [14]. In what follows, we obtain an integral-equation-based formulation for the eigenvalue problem. To begin with, we note that an electromagnetic mode associated with the eigenvalue and the wave vector has the following integral representation: (6) In the above, the surface integral is over the primed coordinates (the integration is performed over the metallic surface in the unit is the cell), is in the dielectric region, and lattice Green function [15]–[17]. The lattice Green function is the Floquet solution of the following equation: (7) where is a multiindex of integers, is the observation point, is a source point, is a lattice point, and is Dirac’s distribution. We note that the Green function depends on both and . The lattice Green function can be efficiently evaluated as explained in [15]–[17]. Here, we instead consider the so-called spectral representation of the Green function, which is obtained into a Fourier series. The result is the following by expanding slowly convergent series:

(8)

is the volume of the unit cell, where is a multiindex of integers, and . The spectral representation of the Green function has the important advantage that, due to its simplicity, many relevant integrals can be calculated in a closed analytical form. is a Floquet wave It is clear that current density associated with the same wave vector as the electromagnetic fields. This justifies our earlier assumption that is a traveling wave [see (2)]. Now the idea is to obtain an integral equation for the current density. To this end, we impose that the tangential component of the electric field given by (6) is zero over the metallic surface. We test (6) with a generic tangential density , and then we . Using the standard integrate the resulting equation over method of moments (MoM) procedure, and also assuming that is a traveling wave associated with the same wave vector as the electromagnetic mode, we are able to prove that (9) where

is the Hermitian form

(10) In the above, the symbol denotes the conjugate of a complex denotes the surface divergence. The previous number, and result shows that for an electromagnetic mode associated with the eigenvalue and the wave vector , the corresponding curis such that (9) holds for an arbitrary test denrent density sity . In analogy with (2), we assume in this paper that the test functions are of the form (11) where uous at

is a periodic function of , possibly discontin, but such that , where . The expression for is consistent with the thin-wire approximation described in Section II. The function can be discontinuous only if the geometry of the wire medium corresponds to the connected case. It can be shown that is a consequence of the current conservation law our simplified model for the wire junctions. For a given wave vector , we can compute the eigenvalues using the standard approach deinto a complete set of basis scribed below. First, we expand functions (with unknown coefficients), and substitute it in (9). We then test the resulting equation with a basis of test functions . In this way, we obtain a homogeneous linear system for the unknown coefficients. A nontrivial solution exists only if the determinant of the linear system vanishes. This occurs only if is coincident with an eigenvalue.

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We assume that the test functions are the same as the expansion functions. The test/expansion functions are denoted by . The density of current is written as (12)

where are the unknown coefficients of the expansion. , we assume that evaluates to Over the wire section Fig. 3.

“Saw” function.

IV. LONG WAVELENGTH LIMIT (13)

where, as in (11), is a periodic function, possibly (only if the wire medium is condiscontinuous at nected). The surface divergence of a generic test/expansion function satisfies

(14) In the above, stands for the usual derivative of (i.e., the Dirac impulses that eventually arise must be discarded; indeed, the discontinuities are already taken into account by the current conservation law). Using the approach delineated before, we obtain the homogeneous linear system (15a) We will refer to the above system as “the characteristic system.” Straightforward calculations show that

Apart from the thin-wire approximation, the formulation presented in Section III is completely general. That formulation is mainly appropriate for the numerical calculation of the electromagnetic modes. However, numerical methods are computationally demanding and give no insight into the physics of the problem. The objective here is to propose a simplified formulation that may allow characterizing the wire medium using analytical methods. The scope of application of our results is the long wavelength limit. The long wavelength limit approximation holds when and

(16)

The idea is to reduce the size of the characteristic system (which theoretically has infinite dimension) so that an analytof ical description is possible for the first few bands the periodic medium (i.e., for the eigenvalues with smaller amplitude). To begin with, it is appropriate to discuss some basic properties of the expansion functions. A. Basis for the Test/Expansion Functions Comparing (2) with (12), we see that the current over the wire oriented along the th direction is given by (17)

Since the to take the

(15b) For a given , the eigenvalues are the solutions of .

current is periodic, it seems appropriate functions equal to the Fourier harmonics , This set of expansion functions is clearly sufficient to model the electric current in the nonconnected topology case. However, in the connected topology case, the set of Fourier harmonics is not enough to describe the electric current that flows along the wires (even though the Fourier basis is complete). In fact, the set of Fourier harmonics is continuous at and, thus, fails to model the possible discontinuous behavior of the current at the junctions (also note that the derivative of the current must be defined in the usual sense, and not in the distributional sense, as referred earlier). Hence, we need to incorporate some additional expansion functions in the expansion set. The step discontinuity of the current can be cancelled out with another expansion function that has similar characteristics. This

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extra expansion function may be rather arbitrary. For simplicity, , depicted in Fig. 3. we choose it to be the “saw” function , the “saw” function is given by For for

For convenience, we introduce the following vectors:

(18)

Obviously, the current can be written as , where and is a continis continuous, it can uous periodic function. Note that since be expanded into a Fourier series with a good convergence rate. must follow the conIt is also important to remember that servation law (3). The previous discussion shows that, in the connected topology case, an appropriate basis for the test/expansion functions is formed by the two subsets described below. The vectors of type first subset (subset I) is formed by the (13) such that the projection over a specific wire is a Fourier harmonic, and over the remaining wires is zero. The elements of this infinite set are continuous functions. Therefore, the is automatically observed. conservation law The second subset (subset II) is formed by two elements only, and describes the discontinuous component of the currents. The , elements of subset II are such that where are constants, which ensure that the current conseris satisfied (there exist precisely two vation law independent vectors in such conditions). The eigenvalues of the propagation problem can be rigorously calculated (within the scope of our model) with the complete basis formed by subsets I and II. In the nonconnected topology case, subset I is sufficient to characterize the wire medium, as explained earlier. B. Simplified Characteristic System The dimension of the characteristic system (15) is infinite. This seems to preclude the use of analytical techniques, as is our objective. However, in the long wavelength limit, the propagation in the wire medium can hopefully be described adequately using only the elements of the basis corresponding to the lowest order Fourier harmonics. Thus, we propose to truncate subset I, discarding all its elements, except the three vectors associated Fourier harmonics . We do not with the discard any element from subset II. Within this approximation, the dimension of linear system (15) is either drastically reduced in the to 3 in the nonconnected topology case or to connected topology case. This is equivalent to assume that, apart from the propagation factor, the amplitude of the current over each wire section is either constant (nonconnected topology) or a linear function (connected topology). with The elements of the truncated subset I are denoted by , and are such that (19) On the other hand, the elements of subset II are denoted by with and, as explained above, they satisfy

(20)

(21) , We do not consider any specific choice for coefficients . We only assume that vectors , , and (defined as above) form an orthonormal basis of the space, and are real valued. Thus, we have (22) Note that the orthogonality condition ensures that is verified. Within this approximation, we can rewrite the current expansion (12) as connected topology (23a) nonconnected topology and where the unknowns of the expansion are that since the “saw” function is odd, we have

(23b) . We note

(24) is the average current over the th wire. Therefore, The characteristic system (15) is now rewritten using dyadic notation. In the connected topology case, we obtain connected topology

(25a)

(25b)

(25c)

(25d) In the above, denotes the conjugate of the transpose of the and dyadic , denotes the form (10), and denote the vectors that represent the unknowns. Similarly, in the nonconnected topology case, the character. Note that the dyadic istic system becomes depends on the specific topology of the wire medium. Indeed, in the connected case, the wires in the unit cell are centered at the origin, whereas in the nonconnected case, they are not and, will be different in the two situations. thus,

SILVEIRINHA AND FERNANDES: HOMOGENIZATION OF 3-D-CONNECTED AND NONCONNECTED WIRE METAMATERIALS

It is convenient to rewrite the characteristic system in a unified is invertible. manner. As proven in Appendix A, the dyadic and Therefore, we can calculate vector as a function of considerably simplify (25). After straightforward calculations, we find that

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A. Homogenization Basics be an electromagnetic Floquet mode in a generic Let metallic crystal, i.e., a solution of (5) with , where is the magnetic induction. We define the average fields and as follows:

where (30) connected topology nonconnected topology

(26)

Using (5), it can be verified that the following equations hold:

For a given , the eigenvalues associated with the first few modes can be obtained by solving the characteristic equation for .

(31a)

C. Dyadic In order to keep the readability of this paper, the calculation of the dyadic is addressed in Appendix A. In the long-wavelength limit, we obtain

(31b) denotes the surface of the metallic region in the unit where is the surface current over the metallic cell and interfaces. Using (31) and (24), after straightforward manipulations, we obtain

(27a) (32)

(27b) connect. nonconnect. (27c) where is the identity dyadic, , and . Constants and are defined by (B5) and (B6), and depend exclusively on the wire radius and lattice constant. For future reference, we note that the inverse dyadics are given by (28)

Provided that magnetization and higher dipole moments can be neglected, the right-hand side of the above equation is approx, where is the polarization vector (i.e., imately equal to the spatial average electric dipole moment in a unit cell). That is the case in this paper because, at least within the scope of our thin-wire model, the magnetization is exactly zero. For future reference, we note that (32) can be rewritten as (33) where

is defined by (28).

B. Effective Permittivity Dyadic Here, we prove that the effective medium can be characterized with a permittivity dyadic. From (26) and (33), we have (34)

connected topology

(29a)

We claim that the characteristic system alent to the characteristic system both systems yield the same eigenvalues

nonconnected top (29b)

V. HOMOGENIZATION OF THE WIRE MEDIUM Here, we will explain how the effective permittivity of the wire medium can be directly obtained from the characteristic system derived in Section IV. To begin, it is appropriate to introduce some auxiliary results and definitions that will be important later.

problem). Indeed, although

is equiv(i.e., for the propagation for

have for because pole and there is the following pole-zero cancellation:

, we has a

(35) Next, we obtain an explicit formula for the effective permittivity of the metamaterial. Inserting (35) into (34), after simple manipulations, we obtain (36)

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Using (28), the above formula can be rewritten as follows:

VI. DISCUSSION

where (37) Comparing the above expression with the characteristic equation for the average electric field in an anisotropic medium [18, p. 202], we recognize that is necessarily the (relative) effective permittivity dyadic. Indeed, from (32) and (37) (and noting, as ), we have before, that the right-hand side of (32) is (38) From (29), the effective permittivity of the wire medium is given by

connected topology

(39a)

nonconnected topology

Here, we will characterize the electromagnetic modes that can propagate in the homogenized medium, and discuss the physical implications of the results. We also compare our analytical model with full-wave numerical simulations. To begin with, we note that, from (37), the dispersion characteristic of the modes can be calculated by solving the characteristic equation (41) The solutions of the characteristic equation depend on the topology of the wire medium, and are described in Sections VI-A and B. A. Connected Geometry Here, we admit that the wires are connected and, thus, the permittivity dyadic is given by (39a). This model for the permittivity is exactly coincident with the one described in [19] for the effective permittivity of a nonmagnetized plasma considering the effect of pressure forces. Unlike the classic “cold” plasma model [19], which applies when the pressure forces are neglected, this model is characterized by spatial dispersion in the long wavelength limit. A “cold” plasma is characterized by . the permittivity It is easy to verify that (39a) predicts that there are two degen) with the dispersion erate TEM modes (i.e., such that characteristic

(39b) modes Note that the effective permittivity depends explicitly on the wave vector and, thus, the wire medium suffers from spatial dispersion. We could homogenize the 1-D-wire medium (array of parallel wires) and the two-dimensional (2-D)-wire medium (two arrays of orthogonal wires) using exactly the same technique (see Appendix C). For the 1-D-wire medium case, we would find that the permittivity dyadic is given by (39b) with the index re(assuming that the wires are oriented along stricted to the -direction). Comparing that formula with the results demust be equal scribed in [9], we conclude that the constant to , where is the plasma (angular) frequency, and is the velocity of light in vacuum. Indeed, numerical simulations and analytical results show that, to a good approximation, given by (B5) is coincident with the result of [9] (40)

The above formula can be used to compute quickly and with good accuracy. On the other hand, (B5) is a slowly convergent double series and, thus, numerous terms need to be summed. To conclude, we note that the homogenization method described here can be generalized to arbitrary periodic structures, and used to extract the permittivity, permeability, and magnetoelectric terms of the effective medium.

(42)

Since when , we con. clude that the TEM waves “see” the effective permittivity Therefore, in the long wavelength limit, TEM propagation in the 3-D-wire medium is equivalent to propagation in “cold” plasma. Apart from the TEM waves discussed above, the 3-D-wire medium supports a longitudinal wave. The longitudinal wave is such that the polarization is parallel to the wave vector. It starts propagating near the plasma frequency and has the dispersion characteristic

longitudinal mode

(43)

The “cold” plasma model also predicts a similar mode [19]. However, since the “cold” plasma model neglects the “pressure . forces,” the corresponding mode is dispersionless, i.e., If the “pressure forces” are considered, the electrodynamics is exactly the same as in the wire medium. The existence of a longitudinal mode was also conjectured in [12] based on experimental results. Next, we present full-wave numerical simulations that validate the above-described results. We numerically implemented the full-wave method described in [8] also using the thin-wire approximation. We expanded the unknown currents with

SILVEIRINHA AND FERNANDES: HOMOGENIZATION OF 3-D-CONNECTED AND NONCONNECTED WIRE METAMATERIALS

Fig. 4. Dispersion characteristic of the modes as a function of the  angle for ' = 0 (full line: numerical results; dashed line: theory).

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medium with magnetic particles, it seems reasonable to assume (provided there is low interaction between the wires and magnetic particle) that the new composite medium is described by the permittivity dyadic (39a) and a permeability due to the magnetic effects. We assume that is a scalar and that in some frequency band below the plasma frequency. We claim that this new composite material behaves as an ideal isotropic DNG material in the referred frequency band and that no spatial dispersion occurs in this band. Indeed, it is easy to verify that only two TEM modes will propagate in this regime. The longitudinal mode of the wire medium will remain in cutoff even . The reason is very simple: the magnetic field though associated with the longitudinal mode is exactly zero and, thus, this mode does not interact with the magnetic particles. Thus, we conclude that this topology of the wire medium may be adequate for DNG material applications. B. Nonconnected Geometry

Fig. 5. Dispersion characteristic of the modes as a function of the  angle for ' = 45 (full line: numerical results; dashed line: theory).

expansion functions (the continuous component of each current was expanded with five Fourier harmonics; the coupling between the wires is modeled with two expansion functions; the expansion basis for the currents is the same as the one discussed in Section IV). and, thus, the plasma We admit that and . We put wavenumber is with , and we computed numerically for the relevant modes as a function (Fig. 4) and (Fig. 5). of for As can be seen in Figs. 4 and 5, the agreement between the full-wave results and the analytical model is good. The results show that the connected medium is a good candidate to synthesize a metamaterial with negative permittivity, at least if the longitudinal mode is not significantly excited. To study this subject with greater detail, one would need to know the boundary conditions for the electromagnetic fields at an interface with a dielectric. As is well known, the usual boundary conditions (i.e., the continuity of the tangential components) are insufficient to describe the reflection of plane waves at an interface. Indeed, we need an additional boundary condition [20], [21] as a consequence of the medium response being spatially nonlocal (i.e., the electric displacement at one point depends on the electric field in the whole material). Unfortunately, no general method to obtain the additional boundary conditions is available [20]. A detailed discussion of this topic is outside the scope of this paper. We will instead discuss the use of the connected geometry of the wire medium in DNG material applications. Let us suppose that an ideal isotropic magnetic particle is available (e.g., a generalization of the split ring resonator [22]). If we load the wire

Here, we consider that the wires are not connected. Unlike the previous case, the dispersion characteristic and polarization vector cannot be calculated in closed analytical form. To circumvent this problem, we proceed as follows. First, we write the wave vector in polar coordinates, i.e., . We then admit that the dispersion characteristic has the Taylor expansion , where , , etc. are unknown coefficients that, in general, depend on and . Inserting these formulas into (41), we obtain, after some simplifications, an equation of the form , where is a polynomial function. In order to calculate recursively the unknown coeficients , , etc., we impose that the successive derivatives of the function (in ) at the origin vanish. For simplicity, we will admit in the , i.e., the modes propagate remainder of this section that plane. in the Using the approach delineated above, we verified that there exist five modes in the long wavelength limit. The dispersion characteristic of the modes is

(44a)

(44b) (44c) (44d) represents an expression that vanishes at the In the above, same rate as . Equation (44c) represents the dispersion characteristic of two different modes: one associated with the “ ” sign and the other one with the “ ” sign. The polarization of the modes can be calculated using [18, p. 202 (with a different notation)] (45)

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+

Fig. 6. Angle between the electric field and the x -axis for the “ ” mode defined by (44c) (full line: numerical results; dashed line: theory).

The modes defined by (44c) and (44d) were discussed in [8]. For completeness, we will briefly review their main properties. The mode defined by (44d) is a TEM mode with polarization along the -direction (i.e., normal to the plane of propagation). On the other hand, the two modes defined by (44c) have elliptic wave normal contours (the principal axes are defined by ). The polarization of these modes is on the plane and, to a first-order approximation, is independent of the wave . This result is the leading term of vector the formula that is obtained by substituting (44c) into (45). In Fig. 6, we compare the formula for the electric field with full-wave numerical simulations obtained using the method proposed in [8] (we computed the electromagnetic modes, and then we averaged the fields over the unit cell to calculate the polarization of the field). The wire radius is . We put , where , and we computed the angle that the average electric field associated with the “ ” mode makes with the -axis as a function of . We compared . As seen in Fig. 6, the result with the expected value the agreement is good, especially for directions not too close to the axes. The discrepancy along the wire axes is not relevant because the modes defined by (44c) are degenerate along those directions. The described results show that the propagation in the nonconnected wire medium is very different from the propagation in the connected wire medium. Indeed, the modes that propagate in the two materials have distinct polarizations and dispersion characteristics. The nonconnected medium does not support TEM waves and, thus, it is not suitable for applications in which the metamaterial is supposed to mimic the properties of ideal cold plasma. In what follows, we discuss the properties of the electromagnetic modes corresponding to (44a) and (44b). The mode associated with (44a) is longitudinal and nearly dispersionless (for ). Thus, it is not relevant for propagation on the plane. On the other hand, the mode associated with (44b) has a remarkable feature: the wave normal contours are hyperbolic near the static limit. This situation is rather peculiar and does not occur in standard (nonartificial) dielectric materials, which invariably have elliptic wave normal surfaces. In Fig. 7(a), we plot a generic contour. We also depict a generic wave vector and the corresponding Poynting vector . As is well known, the Poynting vector is perpendicular to the wave normal surface [14, p. 95].

Fig. 7. (a) Generic wave normal contour of the mode defined by (44b). (b) Theoretical geometrical relation between the associated average electric field and wave vector.

Fig. 8. Negative refraction occurs at an interface between air and the nonconnected wire medium (the inset shows the orientation of the metamaterial).

For

, the polarization is to a first approximation . This expression is the leading term of the formula that is obtained by inserting (44b) into (45). In particular, if the wave vector is along a coordinate axis, the mode is longitudinal. The geometrical relation between the wave vector and electric field is depicted in Fig. 7(b). The electric field and wave vector make the same angle with the coordinates axes. To our best knowledge, no one has ever reported an artificial material with similar properties near the static limit. This mode propagates only at very large wavelengths before it enters in cutoff. The interesting thing is that the hyperbolic contours originate negative refraction at an interface with air. Indeed, since the component of parallel to the interface is preserved and the rays are parallel to the Poynting vector, we have the ray picture illustrated in Fig. 8. Note that we have negative refraction, but not a backward wave. In Fig. 8, the vector represents the wave vector of the ray is the wave vector associated that impinges on the interface, with the reflected ray, and is the wave vector associated with contour. the transmitted ray. We also depicted the The transmitted wave vector lies on the contour and its projection onto the interface is equal to the projection of and onto the interface. The transmitted ray propagates . It is clear from along the direction of the Poynting vector Fig. 8 that the negative refraction phenomenon occurs. Note that the interface with air is normal to the direction , where is measured relatively to the -axis (as usual, the wires are directed along the coordinate axes).

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Fig. 9. Calculated (full line) and theoretical (dashed line) along a hyperbolic contour in the k -plane (i.e., as a function of the '-angle).

in closed analytical form. We have proven that the permittivity dyadic can be extracted unambiguously from the characteristic system. We obtained an analytic model for the permittivity dyadic of the wire medium. The permittivity dyadic depends on the wave vector and, thus, there is always spatial dispersion. However, we found that the connected wire medium supports two TEM degenerate waves. These waves propagate as in an ideal isotropic medium with negative permittivity. Therefore, as discussed in the text, this topology of the wire medium is a good candidate for many relevant metamaterial applications (e.g., DNG materials). On the other hand, in the nonconnected medium, the waves are, in general, neither TEM, nor degenerate. Moreover, near the plasma frequency, the polarization of the fields is almost independent of the wave vector. Thus, we conclude that this topology of the wire medium is not appropriate for metamaterial applications in which the material must supposedly mimic the properties of an ideal plasma. Our results also show that, near the static limit, the dispersion characteristic of the modes is intrinsically hyperbolic. This result is remarkable because the materials usually available in nature have elliptic wave normal surfaces. A consequence of this unusual property is that negative refraction may occur at an interface between air and the nonconnected wire medium.

Fig. 10. Angle between the electric field and wave vector as a function of the '-angle (mode with hyperbolic contour; full line: numerical results; dashed line: theory).

APPENDIX A

In Fig. 9, the theoretical dispersion characteristic of the mode (44b) is compared with full-wave numerical results obtained using the method proposed in [8]. We put , where is calculated in order that (the free-space the wave vector satisfies (44b) for ). We then computed numerically as wavelength is a function of , and we compared it with the theoretical value (i.e., ). As seen in Fig. 9, the computed value compares well with the theoretical value, and is practically constant (even near the coordinate axes where the long wavelength limit is not approximation fails; indeed the condition fulfilled for ). In addition, we calculated the polarization of the electromagnetic mode numerically and we compared it with . As before, we put the formula , and we computed the angle between the average electric field and wave vector as a function of . The result is depicted in Fig. 10. The agreement between the analytical model and simulations is excellent.

Here, we calculate the dyadics defined by (25) and (26). To begin, we note that, in the long wavelength limit (16), we can replace the term by in the spectral representation of the Green function (8), except in the parcel . We obtain the approximate associated with the index formula (A1) is a multiindex of integers and . In the following, we evaluate the desired dyadics using always the above approximation for the Green function. where

A. Connected Topology Here, we assume that the topology of the wire medium is connected. We first calculate the dyadic defined by (25b). Inserting (19) into (15b), we easily obtain (A2)

VII. CONCLUSIONS In this paper, we discussed the electrodynamics of the 3-D-wire medium in the long wavelength limit. We considered two distinct topologies for the wire inclusions: the connected topology and nonconnected topology. We found that the properties of the effective medium are surprisingly dependent on the topology of the metamaterial. Based on simple physical considerations and using an integral-equation-based formulation, we were able to reduce the modal problem to the calculation of the zeros of the determinant of a characteristic system known

where

is defined by (B1). Using (B4) and the identity , we find that

(A3)

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where is the identity dyadic. The constants and are defined in Appendix B. We now calculate the dyadic defined by (25d). Inserting (19) and (20) into (15b), we obtain

is defined by (B10b). In the long wavelength The constant limit, the first term on the right-hand side can be neglected. Indeed, if we would decide to keep that term, we would find that the correction in the final result would be comparable with the error introduced by the approximation (A1). Hence, using (22), we conclude that (A10)

(A4) is defined by (B1b) and, as proven in ApThe coefficient pendix B, is identically zero. Using (B4) and the continuity , we find that equation

We are now ready to calculate the dyadic Straightforward calculations show that

defined by (26).

(A5) Therefore, we have (A11) We note that (22) implies that (A6) The dyadic

(A12)

is given by However, we then necessarily have (A13) (A7)

We next calculate the dyadic (20) into (15b), we obtain

and, in particular, we find that (A14)

defined by (25c). Inserting

Inserting the above result into (A11) and using (21), we obtain

(A15) Substituting the previous formula and (A3) into (26), we obtain (27). (A8) B. Nonconnected Topology coefficients are defined by (B1). Using In the above, the (B4), (B8), (B10), and the continuity condition, we obtain

(A9)

Here, we calculate the dyadic for the case in which the topology of the wire medium is nonconnected. From (26), we . Since the results of the above section remain have valid, the dyadic is still given by (A3). However, we note that the dyadic is not the same in the two cases because the constant depends on the topology of the wire medium [see(B6)]. We also note that because is given by an oscilis given by a double nonoscillating lating series, whereas

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series. Therefore, in the long wavelength limit, we can clearly in (A3). This approximation yields (27). neglect the term

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where and are integers. On the other hand, the constant depends on the topology of the wire medium, and can be negative. It can be written as

APPENDIX B Here, we calculate the following auxiliary integrals assuming that (A1) holds exactly:

(B1a)

connected topology nonconnected topology

(B6)

where is an integer different from zero. In the remainder of this Appendix, we will always assume that . We obtain the wires are connected. Next, we calculate

(B1b) (B7a) (B1c) Using (A1), we easily find that

is equal to (B2)

where

is the constant (independent of

(B7b)

and ) defined by

Taking into account the symmetries of the summation range and the fact of the “saw” function being odd, it in the index is easy to verify that (B8) (B3a)

. We have

Finally, we calculate

(B9) (B3b) stands for the Bessel function of the first kind In the above, is the center of the wire directed in the and order 0, and -direction. For the connected topology, we have , is given by (4). We while for the nonconnected topology, note that has only two different values, more specifically, if and if . Therefore, we can rewrite (B2) as follows:

If , the series vanishes for reasons similar to those discussed during the calculation of . Therefore, we have (B10a)

(B10b) (B4)

is independent of the wire medium being conThe constant nected or not, and is always positive. It is defined by

(B5)

Note that the definition of is a constant. and that

is independent of the

index,

APPENDIX C Here, we generalize without proof the results of the paper to the 2-D-wire medium. In this metamaterial, the wires are oriented exclusively along the - and - directions. In the case of the nonconnected topology, the permittivity dyadic is given

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by (39b), except that the index is restricted to . On the other hand, in case of the connected topology, the permittivity dyadic is given by (39a) with the symbol replaced by , and the dyadic inside brackets replaced by . The constant is defined as shown in (39a) with the symbol “3” replaced by “2.” REFERENCES [1] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [2] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [3] D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B, Condens. Matter, vol. 65, pp. 195 1041–195 104 5, Apr. 2002. [4] V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys.—Usp., vol. 10, pp. 509–514, Jan.–Feb. 1968. [5] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [6] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, no. 18, pp. 3966–3969, Oct. 2000. [7] N. Engheta, “An idea for thin, subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propag. Lett., vol. 1, no. Dec., pp. 10–13, 2002. [8] M. Silveirinha and C. A. Fernandes, “A hybrid method for the efficient calculation of the band structure of 3-D-metallic crystals,” IEEE Trans. Microw. Theory Tech., no. 3, pp. 889–902, Mar. 2004. [9] P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B, Condens. Matter, vol. 67, pp. 113 1031–113 1034, 2003. [10] C. A. Moses and N. Engheta, “Electromagnetic wave propagation in the wire medium: A complex medium with long thin inclusions,” Wave Motion, vol. 34, pp. 301–317, 2001. [11] A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2558–2571, Oct. 2003. [12] D. F. Sievenpiper, M. E. Sickmiller, and E. Yablanovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett., vol. 76, pp. 2480–2483, 1996. [13] E. Sayre, “Junction discontinuities in wire antenna and scattering problems,” IEEE Trans. Antennas Propag., vol. 21, no. 2, pp. 216–217, Mar. 1973. [14] K. Sakoda, Optical Properties of Photonic Crystals, ser. Opt. Sci.. Berlin, Germany: Springer-Verlag, 2001, vol. 80. [15] P. P. Ewald, “Die berechnung optischer und elektrostatischer gitterpotentiale,” Ann. Phys. (Germany), vol. 64, pp. 253–287, 1921. [16] M. Silveirinha and C. A. Fernandes, “A new acceleration technique with exponential convergence rate to evaluate periodic Green’s functions,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 347–355, Jan. 2005.

[17] M. Silveirinha, “Electromagnetic waves in artificial media with application to lens antennas,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Univ. Técnica de Lisboa, Lisbon, Portugal, 2003. [18] R. E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEE Press, 1991. [19] C. Dougherty, Electrodynamics of Particles and Plasmas. Reading, MA: Addison-Wesley, 1969, pp. 175–178. [20] S. I. Maslovski and S. A. Tretyakov, “Additional boundary conditions for spatially dispersive media,” in Proc. 8th Bianisotropics Conf., Lisbon, Portugal, Sep. 2000, pp. 7–10. [21] W. A. Davis and C. M. Krowne, “The effects of drift and diffusion in semiconductors on plane wave interaction at interfaces,” IEEE Trans. Antennas Propag., vol. 36, no. 1, pp. 97–103, Jan. 1998. [22] J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [23] C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 70, p. 046 616, Oct. 2004.

Mário G. Silveirinha (S’99–M’03) received the Licenciado degree in electrical engineering from the University of Coimbra, Coimbra, Portugal, in 1998, and the Ph.D. degree in electrical and computer engineering from the Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal, in 2003. His research interests include propagation in photonic crystals and homogenization and modeling of metamaterials.

Carlos A. Fernandes (S’86–M’89) received the Licenciado, M.Sc., and Ph.D. degrees in electrical and computer engineering from the Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal, in 1980, 1985, and 1990, respectively. In 1980, he joined the IST, where, since 1993, he has been an Associate Professor with the Department of Electrical and Computer Engineering. He is involved in the areas of microwaves, radio-wave propagation, and antennas. Since 1993, he has been a Senior Researcher with the Instituto de Telecomunicações, where he is currently the Coordinator of the wireless communications scientific area. He coauthored a book, a book chapter, and several technical papers in international journals and conference proceedings in the areas of antennas and radio-wave propagation modeling. His current research interests include artificial dielectrics, dielectric antennas for millimeter-wave applications, and propagation modeling for mobile communication systems. Dr. Fernandes has been the leader of antenna activity with National and European Projects such as RACE 2067—Mobile Broadband System (MBS) and ACTS AC230—System for Advanced Mobile Broadband Applications (SAMBA).

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Fundamental Modal Properties of Surface Waves on Metamaterial Grounded Slabs Paolo Baccarelli, Member, IEEE, Paolo Burghignoli, Member, IEEE, Fabrizio Frezza, Senior Member, IEEE, Alessandro Galli, Member, IEEE, Paolo Lampariello, Fellow, IEEE, Giampiero Lovat, Student Member, IEEE, and Simone Paulotto, Student Member, IEEE

Abstract—This paper deals with the analysis of surface waves supported by a metamaterial layer on a ground plane, and investigates the potentiality of these grounded slabs as substrates for planar antennas. Both double- and single-negative media, either epsilon- or mu-negative, are considered. As is known, such structures may support two kinds of surface waves, i.e., ordinary (transversely attenuating only in air) and evanescent (transversely attenuating also inside the slab) surface waves. A graphical analysis is performed for proper real solutions of the dispersion equation for TE and TM modes, and conditions are presented that ensure the suppression of a guided-wave regime for both polarizations and kinds of wave. In order to demonstrate the feasibility of substrates with such desirable properties, numerical simulations based on experimentally tested dispersion models for the permittivity and permeability of the considered metamaterial media are reported. Moreover, the effects of slab truncation on the field radiated by a dipole source are illustrated by comparing the radiation patterns at different frequencies both in the presence and in the absence of surface waves. The reported results make the considered structures promising candidates as substrates for planar antennas and arrays with reduced edge-diffraction effects and mutual coupling between elements. Index Terms—Grounded slab, guided waves, metamaterial media, surface-wave suppression.

I. INTRODUCTION

F

ABRICATION and experimental verification of materials with negative values of magnetic permeability and/or dielectric permittivity have been demonstrated in recent years by several research groups (see, e.g., [1]–[8]). The proposed designs are based on one-, two-, or three-dimensional periodic structures that may be modeled, at least in certain frequency ranges, as homogeneous materials (metamaterials) exhibiting scalar negative and dispersive constitutive parameters. Various peculiar features of electromagnetic-wave propagation exist when such metamaterials are involved, some of which were examined by Veselago in his seminal paper published in the late 1960s [9], [10]. In this connection, during the last few years, different fundamental issues have been addressed. Properties of plane waves in a double-negative (DNG) medium have been considered, e.g., in [11] and [12], where the “backward” or “left-handed” nature of these waves and the negative refraction index of the medium are discussed. Reflection and refraction of Manuscript received May 31, 2004; revised September 13, 2004. The authors are with the Department of Electronic Engineering, “La Sapienza” University of Rome, 00184 Rome, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845208

plane waves at planar interfaces are of special interest in view of the proposed application of planar configurations to obtain focusing effects at optical or microwave frequencies [13], [14]. Guided-wave propagation in structures containing metamaterial media has also been considered (see, e.g., [15]–[19]). In particular, modes supported by DNG slabs surrounded by an ordinary medium are studied in [17]–[19], where the existence of both ordinary (transversely attenuating only in air) and evanescent (transversely attenuating also inside the slab) surface waves is pointed out, and a number of modal features differing from those found in double-positive (DPS) slabs are reported, including dispersion and energy-flux properties. In this paper, we aim at studying the modal properties of a basic open structure for microwave circuits and antennas, namely, a slab placed on a perfectly conducting ground plane, when the medium of the slab is assumed to be a metamaterial, either DNG or single-negative (SNG). On the basis of simple graphical discussions of the involved dispersion equations, conditions are derived for the existence of real modes either ordinary or evanescent. As a result of our analysis, sufficient conditions are obtained, which ensure the absence of proper surface waves of both TE and TM polarizations in the considered structure (some preliminary results of this kind have recently been published in [20] and [21] for the case of a DNG grounded slab). These conditions of surface-wave suppression are very attractive in view of a possible employ of a metamaterial slab as a substrate for planar antennas and arrays with reduced edge-diffraction effects and mutual coupling between elements. This paper is organized as follows. In Section II, we describe the considered structure and we derive the dispersion equations for both TE and TM modes, either ordinary or evanescent, via transverse resonance. In Section III, we present results for DNG grounded slabs, discussing in particular the possibility of achieving surface-wave suppression. In Section IV, a similar analysis is presented for SNG grounded slabs, both mu-negative (MNG) and epsilon-negative (ENG). In Section V, numerical results are presented for the dispersion properties of modes supported by DNG slabs, on the basis of simple experimentally tested dispersion models for the permittivity and permeability of the considered metamaterial media, demonstrating the possibility to obtain surface-wave suppression in certain frequency ranges. In Section VI, the effects of surface-wave suppression on the radiated field excited by a magnetic dipole placed on the ground plane of truncated DNG slabs are illustrated. Finally, in Section VII, conclusions are drawn on the various results of this study.

0018-9480/$20.00 © 2005 IEEE

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Fig. 1. (a) Reference metamaterial grounded-slab structure with the relevant physical and geometrical parameters and coordinate system. (b) Transverse equivalent network for TE and TM modes of the grounded slab.

II. PROBLEM STATEMENT AND BACKGROUND The reference structure considered here is a grounded slab of height made of an ideal linear stationary isotropic homogeneous lossless metamaterial medium with permeability and permittivity , both of which may be negative (see Fig. 1(a), where the relevant coordinate system is also shown). The problem addressed here is the study of guided modes supported by this structure, propagating along the longitudinal -direction with a real propagation constant . A is assumed and suppressed time–harmonic dependence throughout. No variation of the electromagnetic field is assumed along the -direction, thus, the two-dimensional nature of the problem allows us to separately study TE and TM modes. As is well known, a transverse equivalent network in the -direction can be associated to each TE or TM mode, as shown in Fig. 1(b). The expressions of the relevant characteristic impedances for the two polarizations in the air and slab regions (subscripts 0 and , respectively) are as follows:

(1) where (2) with and . The dispersion equation for TE and TM modes can be obtained by enforcing the transverse-resonance condition, e.g., at , on the relevant equivalent network. The result is [22] (3) This equation that is, of course, the same as the ones reported in [17]–[19] and [22] will be studied in Sections III–VI for

Fig. 2. Transverse profile of the absolute value of the E (TE case) and E (TM case) component of the electric field of ordinary and evanescent surface waves propagating along the grounded slab of Fig. 1.

TE and TM modes by considering DNG and SNG slabs separately. Since the grounded slab is a transversely open structure, its modes can be either proper, i.e., attenuating at infinity , or improper, i.e., in the transverse -direction [23]. In particular, surface diverging at infinity waves have a purely imaginary transverse wavenumber in air for proper surface waves and with for improper surface waves . In addition to this, when one (or both) of the constitutive parameters is negative, two kinds of real solutions, corresponding to surface waves supported by the structure, have to be considered, i.e., ordinary surface waves with a real transverse inside the slab, and evanescent surface wavenumber inside the waves with an imaginary wavenumber slab. A sketch of the transverse profile of the modal field for ordinary and evanescent waves on a grounded metamaterial slab is depicted in Fig. 2: the field of ordinary modes is transversely attenuating only in air, whereas the field of evanescent modes is transversely attenuating also inside the slab. It is known that ordinary surface waves cannot exist in SNG slabs, while evanescent surface waves cannot exist in a DPS isotropic grounded slab, although they are known to be present in SNG and DNG slabs [18], [19], [24], and in other specific structures, e.g., ferrite slabs [25], [26]. The issue of the location of surface-wave poles on the Riemann surface of the spectral Green’s function for the grounded slab will not be addressed here for the sake of brevity. However, we would emphasize that, in the case of a grounded slab, a guided mode is constrained to have a longitudinal phase congreater than : this implies that the relevant pole in stant the complex plane is always captured in a steepest descent deformation of the integral path. On the other hand, the graphical method presented in Section III will be applied to functions that give rise to proper modes only (i.e., modes corresponding to poles located on the top Riemann sheet); improper modes (i.e., modes corresponding to poles located on the bottom Riemann sheet) could be found only by considering different functions. III. MODAL PROPERTIES OF DNG GROUNDED SLABS Since in a DNG medium both the permeability and permitis positive. tivity are negative, the squared wavenumber ) or Therefore, surface waves may be either ordinary (if evanescent (if ). A graphical analysis of the dispersion equation for surface waves is provided in Section III-A for TE

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waves and in Section III-B for TM waves. A discussion on the possibility to achieve proper surface-wave suppression in DNG slabs is provided in Section III-C. A. TE Surface Waves 1) Ordinary TE Surface Waves: In this case, the dispersion equation (3) for proper real modes can be written as (4) By taking into account that

, (4) becomes (5)

The condition waves cannot exist if Therefore, only the case can be written as

implies that ordinary TE surface (which would imply ). will be studied, for which (5) (6)

in terms of the adimensional variables (7) Since we aim at studying surface waves, positive real solutions for of (6) are sought. These can be obtained graphically by finding the intersections between the tangent function at the at the right-hand left-hand side of (6) and the function side of the same equation. A straightforward analysis shows that is defined for , is concave and monothe function , is zero at , and tonically increasing in the interval tends to infinity for (see Fig. 3). We seek conditions that inhibit propagation of guided modes: this corresponds to determine the conditions under which no and . By observing intersection occurs between Fig. 3, it can be seen that sufficient conditions to avoid such intersection are either: 1) there exists a straight line such that lies below it and lies above it, for with [see Fig. 3(a)] or 2) there exists a straight line such that lies above it and lies with [see Fig. 3(b)]. below it, for is monotonically increasing, Since the derivative of , we can choose equal to the value of such derivative at i.e., equal to : in order to satisfy condition 1), the derivahas to be less than the derivative of at tive of and has to be less than . After some algebra, this can analytically be expressed as

(8)

On the other hand, since the derivative of is mono, we can choose equal to the tonically increasing in , i.e., equal to 1: in order to value of such derivative at has to be less than with satisfy condition 2),

Fig. 3. Graphical representation of the functions occurring in (6) for ordinary proper TE surface waves. Two cases of surface-wave suppression, corresponding to the curves labeled f ( ), are shown in (a) and (b).

. After some algebra, this can analytically be expressed as

(9) 2) Evanescent TE Surface Waves: In this case, and the dispersion equation (3) for proper real modes can be written as

(10) Both the cases

now have to be considered.

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Fig. 4. Graphical representation of the functions occurring in (11) and (14) for evanescent proper TE surface waves. Two cases of surface-wave suppression when  " > 1 are shown, corresponding to the curves labeled f ( ). The case of surface-wave suppression when  " < 1 corresponds to the curve labeled g ( ).

If

, (10) can be written as (11)

Fig. 5. Graphical representation of the functions occurring in (18) for ordinary proper TM surface waves.

condition 1) requires . Moreover, since the derivais monotonically decreasing, we can choose tive of equal to the value of such derivative at , i.e., equal to 1. After some algebra, condition 1) can then analytically be expressed as

in terms of the adimensional variables

(15) (12)

The function at the right-hand side of (11) is defined for , is concave and monotonically decreasing in ( , ), , and tends to for tends to infinity for (see Fig. 4). Therefore, by observing Fig. 4, it can easily be seen that a necessary and sufficient condition to avoid intersecand is that the horizontal asymption between lies below the horizontal asymptote of . tote of The necessary and sufficient condition to avoid propagation of is then proper evanescent TE surface waves when

On the other hand, the first part of condition 2) requires that . Moreover, since the derivative of is monotonically decreasing, we can choose equal to the value of such , i.e., equal to . After some algebra, derivative at condition 2) can then analytically be expressed as (16)

(13) If

B. TM Surface Waves

, (10) can be written as (14)

The function at the right-hand side of (14) is defined for every , is convex and monotonically increasing in the interval , is zero at , and tends to for (see Fig. 4). By observing Fig. 4, it can be seen that sufficient conditions and are that: to avoid intersection between lies above the hori1) the horizontal asymptote of , and there exists a straight line zontal asymptote of such that lies below it and lies lies below the horizontal asymptote of above it, as far as ; alternatively: 2) the horizontal asymptote of lies below the horizontal asymptote of , and there exists a straight line such that lies below lies above it, as far as lies below the it, and . By taking into account that horizontal asymptote of is equal to and the the horizontal asymptote of horizontal asymptote of is equal to 1, the first part of

1) Ordinary TM Surface Waves: In this case, the dispersion equation (3) for proper real modes can be written as

(17) As for the TE case, the condition implies that . The case ordinary surface waves cannot exist if will then be examined, and (17) can be written as (18) on the right-hand side of (18) is defined The function , tends to plus infinity for and monotonically decreasing in , and is zero at (see Fig. 5). By observing Fig. 5, it can easily be seen that a sufficient and condition to avoid intersection between is (19)

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TABLE I SUMMARY OF CONDITIONS FOR SUPPRESSION OF PROPER SURFACE WAVES OF DIFFERENT KINDS ON DNG GROUNDED SLABS WITH  " < 1. THE CONDITION FOR EVANESCENT TE WAVES IS NECESSARY AND SUFFICIENT, THE CONDITION FOR EVANESCENT TM WAVES IS ONLY SUFFICIENT. ORDINARY WAVES CANNOT EXIST IN THIS CASE

Fig. 6. Graphical representation of the functions occurring in (21) and (23) for ( ) corresponds evanescent proper TM surface waves. The curve labeled f to  " > 1, while the curve labeled g ( ) corresponds to  " < 1.

2) Evanescent TM Surface Waves: In this case, and the dispersion equation (3) for proper real modes can be written as

(20) have to be considered. Both the cases , (20) can be expressed as If (21) The function on the right-hand side of (21) is defined , is convex and monotonically increasing in , for is zero at , and tends to for (see Fig. 6). Therefore, by observing Fig. 6, a sufficient condition to avoid and is that the horizontal intersection between lies below the horizontal asymptote of asymptote of and that the point lies on the right of the point . A sufficient condition to avoid propagation is then of proper evanescent TM surface waves when (22) Finally, if

, (20) can be written as

C. Discussion on Surface-Wave Suppression in DNG Grounded Slabs The analysis in Sections III-A and III-B has shown that, for each kind of proper surface wave supported by a DNG grounded slab, conditions may be found that inhibit its propagation. However, it is necessary to ascertain if simultaneous suppression of all kinds of such waves can be obtained. In this connection, the will be considered separately. cases of Sufficient conditions to avoid proper surface-wave propaare summarized in Table I. In gation in the case particular, the condition reported for TE modes is also a necessary condition, whereas that for TM modes is only a sufficient one. By examining Table I, it can be concluded that, in order to inhibit the propagation of every kind of surface wave (both or, a sufficient condition is dinary and evanescent) when that the following set of inequalities is satisfied: (25) provided that (26) , the last inequality can In particular, since be satisfied, at a fixed frequency , by choosing the slab height sufficiently large; in fact, the condition on in (26) can also be expressed as (27)

(23) The function on the right-hand side of (23) is defined , is concave and monotonically decreasing in the infor , tends to infinity for , and tends to terval for (see Fig. 6). Therefore, by observing Fig. 6, a necessary and sufficient condition to avoid intersection between and is that the horizontal asymptote of lies above the horizontal asymptote of . The necessary and sufficient condition to avoid propagation of proper evanesis then cent TM surface waves when (24)

where is the speed of light in a vacuum. Sufficient conditions to avoid proper surface-wave propagaare collected in Table II. In particular, tion in the case the condition reported for TM evanescent waves is also a necessary condition, while the other conditions are only sufficient ones. By examining the alternative conditions for TE-mode suppression, it can be deduced that the only consistent pairs are (o1)–(e1) or (o2)–(e2). However, it can easily be seen that the only pair compatible with the reported conditions for TM-mode suppression is the (o1)–(e1) pair. Therefore, in order to inhibit the propagation of every kind of surface wave (both ordinary and evanescent) when

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TABLE II SUMMARY OF CONDITIONS FOR SUPPRESSION OF PROPER SURFACE WAVES OF DIFFERENT KINDS ON DNG GROUNDED SLABS WITH  " > 1. THE CONDITION FOR EVANESCENT TM WAVES IS NECESSARY AND SUFFICIENT, THE OTHER CONDITIONS ARE ONLY SUFFICIENT

In the following, we separately consider proper real waves on MNG and ENG metamaterials in Sections IV-A and IV-B, respectively; a discussion on surface-wave suppression on SNG grounded slabs is provided in Section IV-C. A. MNG Grounded Slabs In the case of MNG grounded slabs, the dispersion equation for proper real TE waves is from (3) (32) in terms of the adimensional variables and of (12). Since (32) is formally equal to (11) for the case of evanescent TE proper , on the basis real waves on a DNG grounded slab with of the analysis carried out in Section III-A, it can be concluded that a necessary and sufficient condition to avoid proper TE sur. face waves on an MNG grounded slab is that The dispersion equation for proper real TM waves on a MNG grounded slab is from (3) (33)

, a sufficient condition is given by the following set of inequalities:

Since the left- and right-hand sides of (33) have opposite signs, it can be concluded that proper real TM waves cannot exist on an MNG grounded slab. B. ENG Grounded Slabs

(28)

In the case of ENG grounded slabs, the dispersion equation for proper real TE waves is from (3) (34)

that can also be written as (29) provided that

Since the left- and right-hand sides of (34) have opposite signs, it can be concluded that proper real TE waves cannot exist on an ENG grounded slab. The dispersion equation for proper real TM waves on an ENG grounded slab is from (3)

(30)

(35)

In this case, since , the condition on can be achieved, at a fixed frequency , by choosing the slab height sufficiently small; in fact, the inequality in (30) can be expressed as

Since (35) is formally equal to (21) for the case of evanescent , TM proper real waves on a DNG grounded slab with on the basis of the analysis carried out in Section III-B, it can be concluded that a sufficient condition to avoid proper TM surface waves on a MNG grounded slab is

(31)

IV. MODAL PROPERTIES OF SNG GROUNDED SLABS For the sake of completeness, propagation of surface waves on SNG grounded slabs is briefly addressed here. In SNG inside the slab is grounded slabs, the squared wavenumber negative: therefore, for surface waves with a real propagation , the transverse wavenumber inside the constant slab is always purely imaginary, i.e., only evanescent surface waves may exist.

(36)

C. Discussion on Surface-Wave Suppression in SNG Grounded Slabs On the basis of the results of Sections IV-A and IV-B, conditions for surface-wave suppression on SNG grounded slabs can be summarized as shown in Table III. For the case of MNG grounded slabs, surface-wave suppreswithout any additional sion is achieved if and only if condition on the slab height. For the case of ENG grounded

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TABLE III SUMMARY OF CONDITIONS FOR SUPPRESSION OF PROPER SURFACE WAVES ON MNG AND ENG GROUNDED SLABS

slabs, a sufficient condition to achieve surface-wave suppresprovided that the slab height satisfies the sion is that inequality (37)

V. NUMERICAL RESULTS

Fig. 7. Relative permeability  (solid line with circles) and relative permittivity " (solid line with diamonds) as a function of frequency f for the medium model described by (38). The product  " is shown with a light-gray solid line. The frequency range in which the conditions in (29) are satisfied is represented as a shaded area. The maximum slab height h to have surface-wave suppression, calculated according to (31), is reported with a dashed line. Adimensional units (a.u.) are reported on the left vertical axis.

In order to verify the possibility to achieve proper surfacewave suppression in DNG grounded slabs, numerical results are presented here for the dispersion properties of TE and TM modes of different kinds supported by two different structures, indicated in the following as Case 1 and Case 2. Results on improper and complex modes will also be shown in the dispersion diagrams for the sake of completeness. Case 1 The first considered case consists of a DNG grounded slab made of a metamaterial medium modeled as in [2] and [3] with slab height and constitutive parameters chosen as in [19], i.e., mm and relative permeability and permittivity given by (38) , GHz, and GHz. The where region of simultaneously negative permeability and permittivity GHz to GHz. in this case ranges from In Fig. 7, the values of the relative permeability (solid line with circles) and permittivity (solid line with diamonds) are reGHz to GHz, ported in a frequency range from (light-gray solid together with the values of the product , the line). For the considered medium model, when conditions expressed in (25) are never satisfied, while when the conditions in (29) hold in the range of frequencies GHz to GHz, represented from approximately as a shaded area in this figure. In order to have surface-wave suppression inside this range of frequencies, the additional condition in (31), which fixes an upper limit for the slab height, has to be satisfied. Such a limit is also reported in Fig. 7 as a function of frequency (dashed line). Since the slab height has mm, by inspection of this figure, it can been chosen as be concluded that the range of surface-wave suppression is apGHz to GHz, as delimited by the proximately from vertical dashed lines (more exactly, between 5.103–5.2 GHz). The condition of surface-wave suppression will now be illustrated by means of the dispersion diagrams of the TE and TM modes supported by the considered structure. In Fig. 8, the

Fig. 8. Dispersion curves of the TE and TE modes supported by a grounded metamaterial slab with slab medium as in Fig. 7 and slab height h = 20 mm. The shaded area represents the predicted range of surface-wave suppression for both TE and TM modes. Legend: normalized phase constants =k : Solid lines: proper real ordinary waves, dotted lines: improper real ordinary waves, light-gray solid line: proper real evanescent wave, black dashed–dotted lines: complex waves. Normalized attenuation constants  =k . Gray dashed lines: complex waves. Thin solid line: = k .

dispersion curves for the two TE modes with cutoff frequency nearest to the predicted range of surface-wave suppression (represented as a shaded area) are reported. The modes are convenand without any reference to the stantionally labeled dard mode labeling of DPS slabs; moreover, we will show frequency ranges in which the reported modes have a leaky regime . With with a complex propagation constant , proper real orreference to normalized phase constants dinary waves are represented with solid lines, improper real ordinary waves are represented with dotted lines, proper real evanescent waves are represented with a light-gray solid line, and complex waves are represented with dashed–dotted lines. are represented with Normalized attenuation constants gray dashed lines. The thin solid line represents the line where , and divides the region corresponding to evanescent surface waves above it from the region of ordinary surface waves below it. For both the reported modes, at the cutoff frequency,

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TM TM

TM

Fig. 9. Dispersion curves of the , , and modes supported by a grounded metamaterial slab, as in Fig. 8. The shaded area represents the predicted range of surface-wave suppression for both TE and TM modes. Legend: as in Fig. 8.

where , one improper real solution turns into a real proper ordinary one. By increasing the frequency, this merges with another proper real ordinary solution of backward type at a splitting-point frequency, giving rise to a proper complex solution [27], [28]. It can be observed that the upper real proper mode is evanescent at lower frequencies and branch for the ordinary at higher frequencies. Turning now to TM modes, in Fig. 9, three modes are re, , and (without ported, conventionally labeled any reference to the standard mode labeling of DPS slabs) with cutoff frequencies inside the same frequency range, as in Fig. 8; the predicted range of surface-wave suppression is represented and modes behave as the again as a shaded area. The mode in Fig. 8, their real proper branches being always of the ordinary kind. The mode instead behaves differently, being proper, real, and evanescent above cutoff, and improper, real, and ordinary below cutoff. It can be observed that the predicted region of surface-wave GHz, i.e., slightly above the suppression starts at mode, and ends exactly at splitting-point frequency for the the cutoff frequency of the mode. Inside such region, no proper real waves exist, as expected. Instead, proper complex , , , and branches exist (for instance, of the modes), together with improper real branches (for instance, the mode). It can also be observed that the attenuation constants of the complex modes inside the predicted range of surface-wave suppression are very high, thus, they are not expected to give rise to directive radiation if excited by a finite source, as will be shown in Section VI. It can also be pointed out that, on the basis of the dispersion results presented in Figs. 8 and 9, the actual frequency range GHz where no proper surface waves exist is from to GHz, thus, it contains, but is larger than, the predicted range of surface-wave suppression. This is consistent with the fact that the latter was obtained by using a set of sufficient conditions for surface-wave suppression. In particular, with reference to Table II, the condition for TE-wave suppression is only a sufficient one, whereas that for TM evanescent-wave suppression is also necessary. Therefore, the reason why the region of surface-wave suppression is exactly predicted at its upper bound,

Fig. 10. Relative permeability  (solid line with circles) and relative permittivity " (solid line with diamonds) as a function of frequency f for the medium model described by (39). The product  " is shown with a gray solid line. The frequency range in which the conditions in (25) are satisfied is represented as a shaded area. The minimum slab height h to have surface-wave suppression in the DNG and ENG ranges, calculated according to (27) and (37), respectively, is reported with a dashed line. Adimensional units (a.u.) are reported on the left vertical axis.

whereas it is not exactly predicted at its lower bound, is that it is limited on the left by the splitting-point frequency of a TE mode, whereas on the right, it is limited by the cutoff frequency of an evanescent TM mode. Case 2 The second considered case consists of a DNG grounded slab with height mm and a metamaterial medium modeled as in [4] with relative permeabilities and permittivities chosen as in [29] and given by (39) GHz, GHz, (i.e., where losses have been neglected), and GHz. The region of simultaneously negative permeability and permittivity in this GHz to GHz. case ranges from In Fig. 10, the values of the relative permeability (solid line with circles) and permittivity (solid line with diamonds) are reGHz to GHz, ported in a frequency range from (light-gray solid together with the values of the product , the line). For the considered medium model, when conditions expressed in (29) are never satisfied, while when , the conditions in (25) hold in the range of frequenGHz to GHz, represented as a shaded cies from area in the figure. In order to have surface-wave suppression inside this range of frequencies, the additional condition in (27), which fixes a lower limit for the slab height, has to be satisfied. Such a limit is also reported in Fig. 10 as a function of frequency (dashed line). Since the slab height has been chosen mm, by inspection of Fig. 10, it can be concluded that, as when the medium is DNG, the range of surface-wave suppresGHz to GHz. sion is the whole interval from GHz to GHz In Fig. 10, the shown range from is a part of the frequency interval where the medium is ENG. According to the results reported in Table III, in order to achieve surface-wave suppression, a sufficient condition is that , provided that the slab height is higher than the lower limit

BACCARELLI et al.: FUNDAMENTAL MODAL PROPERTIES OF SURFACE WAVES ON METAMATERIAL GROUNDED SLABS

Fig. 11. Dispersion curves of the TE , TE , and TE modes supported by a grounded metamaterial slab with slab medium as in Fig. 10 and slab height h = 5 mm. The shaded area represents the predicted range of surface-wave suppression for both TE and TM modes. Legend: normalized phase constants =ko: Gray dashed–dotted lines: improper complex waves. Normalized attenuation constants  =ko: Black dashed lines: improper complex waves. Other lines: as in Fig. 8.

expressed in (37). Such a lower limit has also been reported in Fig. 10, and it is seen to be the continuation of the lower limit mm and the valid for the DNG case. Therefore, since relative permittivity is less than one in absolute value in all the ENG range (although it is not completely shown in Fig. 10), it can be concluded that no surface waves may also exist in the ENG range. As in Case 1, the condition of surface-wave suppression will be illustrated by means of the dispersion diagrams of the relevant TE and TM modes. In Fig. 11, the dispersion curves of , , and , three TE modes, conventionally labeled are reported in a frequency range between GHz and GHz, together with the line ; the shaded area again represents the predicted range of surface-wave suppresand behave sion for the DNG range. Modes labeled as mode in Fig. 8, while mode is improper real below cutoff and proper real above cutoff. Moreover, the proper real mode is ordinary at lower frequencies and branch of the evanescent at upper frequencies; its normalized phase constant GHz since there becomes tends to infinity at curve does not intersect greater than one and, thus, the curve any more (see Fig. 4). the It is interesting to note that, as the structure becomes ENG at GHz, the phase constants of the proper complex and modes vanish, and these modes turn into improper is repcomplex modes (whose normalized phase constant resented with gray dashed–dotted lines and whose normalized is represented with black dashed attenuation constant lines) [27], [28]. In Fig. 12, the dispersion curves of three TM modes, conven, , and are reported in the same tionally labeled and frequency range as in Fig. 11, again, with the line the shaded area of predicted surface-wave suppression in the DNG range. The three modes have a similar behavior, the only mode difference being that the proper real branch of the and is evanescent, while the proper real branches of the modes are ordinary; in this case, the complex branches of the three modes also remain proper in the shown ENG range above GHz [27], [28].

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Fig. 12. Dispersion curves of the TM , TM , and TM modes supported by a grounded metamaterial slab as in Fig. 11. The shaded area represents the predicted range of surface-wave suppression for both TE and TM modes. Legend: as in Fig. 8.

Fig. 13. Truncated metamaterial grounded slab with relevant physical and geometrical parameters, as in Fig. 11. The finite-size slab is assumed to be circular with radius R = 10 .

As already observed in Case 1, also in Case 2, inside the predicted range of surface-wave suppression, no proper surface waves, but only proper complex or improper real waves, exist. Moreover, also in this case, the attenuation constants of the complex modes are very high in almost the entire suppression range. Finally, it can be observed that, in Case 2, the region of surface-wave suppression, in the DNG range, is exactly predicted, GHz at being limited on the left by the frequency which the phase constant of one evanescent TE mode tends to infinity, whereas on the right being limited by the frequency GHz, at which the material ceases to be DNG, becoming SNG. In fact, with reference to Table I, the condition for TE proper evanescent surface-wave suppression is a necessary and sufficient one. VI. RADIATION PATTERNS IN THE PRESENCE OF TRUNCATED DNG SUBSTRATES Here, the far field radiated by a dipole source in the presence of a finite-size DNG slab is considered in order to verify surface-wave suppression and to show its effects on radiation patterns. A magnetic dipole source is assumed to be placed on the ground plane of a DNG slab along the -axis to model radiation from a short and narrow slot; the finite-size slab is assumed to be circular with radius (see Fig. 13). Comparisons will be presented between the far fields radiated in the presence of an infinite and a finite-size DNG slab. In the infinite case, the radiated field can easily be calculated by means of the dyadic Green’s function of the infinite grounded slab, which is known

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Fig. 14. Normalized radiation patterns (in decibels) of a magnetic dipole placed on the ground plane of a DNG slab along the x-axis in the elevation planes: (a)  = 0 and (b)  = 90 . Physical parameters: as in Fig. 11 at f = 22:5 GHz. Legend: infinite structure: black line with diamonds; circular finite structure (radius R = 10 ): gray line.

in a simple closed form [22]. In the finite case, a perfect absorber is assumed to be present beyond the edges of the slab so that the radiated field may be calculated through a physical-optics approximation of the aperture field on the air–slab interface; in particular, the convenient expressions reported in [30] have been used. In Fig. 14, a DNG slab with physical parameters, as in Fig. 11, GHz; the radius of the is considered at the frequency finite structure, as in Fig. 13, is , where is the free-space wavelength. From Figs. 11 and 12, it can be seen that, at this frequency, one TE surface wave of evanescent type and no TM surface waves are present. In Fig. 14(a), the radiation patterns of the infinite (black line with diamonds) and of the finite (gray line) structures are presented in the elevation plane ; in Fig. 14(b), the same are shown in the elevation plane . The effect of TE surface-wave diffraction at the edges of the finite structure is clearly evident in the plane, where the far field is mostly due to TE waves. On the other hand, plane is mainly due to TM waves, the pattern in the thus, the TE-wave diffraction deteriorates the pattern only close are due to the excito broadside. The broad maxima at mode in a physical leaky regime with a nortation of the (see Fig. 12, where malized attenuation constant and modes are seen to have a much higher attenthe uation constant, thus giving a negligible contribution to the radiation pattern); however it may be noticed that, due to its rapid leaky mode does not contribute appreciably to decay, this edge diffraction. GHz. In Fig. 15, the same structure is considered at From Figs. 11 and 12, it can be seen that now neither TE, nor TM surface waves (of any kind) are present. As a consequence, no diffraction effects are found in the radiation patterns, neither

Fig. 15. Same as in Fig. 14 at f = 23:5 GHz.

plane [see Fig. 15(a)], nor in the plane in the plane, an almost circular pattern [see Fig. 15(b)]. In the can be observed since the TE leaky waves have an extremely high attenuation constant that does not give rise to any visible plane, the maxima at are beam. Instead, in the leaky mode, now with a again due to the excitation of the higher attenuation constant than at GHz, which results in an even broader beam. VII. CONCLUSION In this paper, an investigation has been done on modal properties of metamaterial grounded slabs with particular reference to surface waves. By means of a simple graphical approach to the dispersion equations of the involved TE and TM modes on DNG grounded slabs, it has been shown that conditions may be found that ensure the absence (suppression) of any proper surface wave propagating along the considered structures. These conditions involve the constitutive parameters of the medium, slab height, and operating frequency. SNG media have also been considered by deriving analogous conditions for proper surfacewave suppression both in the MNG and ENG cases. Two different specific structures have been simulated with experimentally tested models for the dispersion behavior of the relevant constitutive parameters. The results on the dispersion properties of the TE and TM modes supported by the considered structures have fully validated the predictions made on the basis of the theoretical analysis concerning such novel modal features. Moreover, the radiation patterns of a magnetic dipole placed on the ground plane of both infinite and truncated metamaterial slabs have been presented in order to demonstrate the reduction of edge-diffraction effects when surface waves are absent. Even though the considered structures are idealized, and the realization of practical metamaterials with predictable constitutive relations may be difficult, the results of this study show that

BACCARELLI et al.: FUNDAMENTAL MODAL PROPERTIES OF SURFACE WAVES ON METAMATERIAL GROUNDED SLABS

metamaterial grounded slabs have a potential as enhanced substrates for planar antennas and arrays with reduced edge-diffraction and mutual coupling between elements.

[24] [25] [26]

REFERENCES [1] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [2] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett., vol. 76, pp. 4773–4776, Jun. 1996. [3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [4] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [5] C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett., vol. 90, no. 10, pp. 107 401–107 401, Mar. 2003. [6] A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys., vol. 92, pp. 5930–5935, Nov. 2002. [7] R. W. Ziolkowski, “Design, fabrication and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1516–1529, Jul. 2003. [8] C. Caloz, C.-C. Chang, and T. Itoh, “Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,” J. Appl. Phys., vol. 90, pp. 5483–5486, Dec. 2001. [9] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan.–Feb. 2001. , “The electrodynamics of substances with simultaneously negative [10] values of " and ” (in Russian), Usp. Fiz. Nauk., vol. 92, pp. 517–526, 1967. [11] R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 64, no. 5, pp. 056 625–056 625, Oct. 2001. [12] I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—Media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett., vol. 31, pp. 129–133, Oct. 2001. [13] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966–3969, Oct. 2000. [14] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [15] I. S. Nefedov and S. A. Tretyakov, “Waveguide containing a backwardwave slab,” Radio Sci., vol. 38, pp. 1101–1109, Dec. 2003. [16] A. Alú and N. Engheta, “Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or doublepositive (DPS) layers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 199–210, Jan. 2004. [17] H. Cory and A. Barger, “Surface-wave propagation along a metamaterial slab,” Microwave Opt. Technol. Lett., vol. 38, pp. 392–395, Sep. 2003. [18] B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys., vol. 93, pp. 9386–9388, Jun. 2003. [19] I. W. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 67, no. 5, pp. 057 602–057 602, May 2003. [20] P. Baccarelli, P. Burghignoli, G. Lovat, and S. Paulotto, “Surface-wave suppression in a double-negative metamaterial grounded slab,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 19, pp. 269–272, 2003. [21] P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, G. Lovat, and S. Paulotto, “New dispersion characteristics and surface-wave suppression in double-negative metamaterial grounded slabs,” in URSI Int. Electromagnetic Theory Symp., vol. 1, Pisa, Italy, May 23–27, 2004, pp. 379–381. [22] R. E. Collin, Field Theory of Guided Waves, 2nd, Ed. Piscataway, NJ: IEEE Press, 1991. [23] T. Tamir and A. A. Oliner, “Guided complex waves. Part I: Fields at an interface. Part II: Relation to radiation patterns,” Proc. Inst. Elect. Eng., vol. 110, no. 2, pp. 310–334, Feb. 1963.

[27]

[28] [29] [30]

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, “The spectrum of electromagnetic waves guided by a plasma layer,” Proc. IEEE, vol. 51, no. 2, pp. 317–332, Feb. 1963. R. W. Damon and J. R. Eshbach, “Magnetostatic modes of a ferromagnetic slab,” J. Phys. Chem. Solids, vol. 19, no. 3/4, pp. 308–320, 1961. P. Baccarelli, C. Di Nallo, F. Frezza, A. Galli, and P. Lampariello, “Novel behaviors of guided and leaky waves in microwave ferrite devices,” in Proc. 8th Mediterranean Electrotechnical Conf., vol. 1, Bari, Italy, May 13–16, 1996, pp. 587–590. P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, G. Lovat, and S. Paulotto, “The nature of radiation from leaky waves on single- and double-negative metamaterial grounded slabs,” in IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, Fort Worth, TX, Jun. 8–13, 2004, pp. 309–312. , “Effects of leaky-wave propagation in metamaterial grounded slabs excited by a dipole source,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 32–44, Jan. 2005. M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, “Abnormal wave propagation in passive media,” IEEE J. Sel. Topics Quantum Electron., vol. 9, no. 1, pp. 30–39, Jan.–Feb. 2003. H. Ostner, E. Schmidhammer, J. Detlefsen, and D. R. Jackson, “Radiation from dielectric leaky-wave antennas with circular and rectangular apertures,” Electromagnetics, vol. 17, pp. 505–535, May 1997.

Paolo Baccarelli (S’96–M’01) received the Laurea degree in electronic engineering and Ph.D. degree in applied electromagnetics from “La Sapienza” University of Rome, Rome, Italy, in 1996 and 2000, respectively. In 1996, he joined the Department of Electronic Engineering, “La Sapienza” University of Rome, where he is a Contract Researcher since 2000. From April 1999 to October 1999, he was a Visiting Scholar with the University of Houston, Houston, TX. His research interests concern analysis and design of planar leaky-wave (LW) antennas, numerical methods, periodic structures, and propagation and radiation in metamaterials and anisotropic media.

Paolo Burghignoli (S’97–M’01) was born in Rome, Italy, on February 18, 1973. He received the Laurea degree (cum laude) in electronic engineering and Ph.D. degree in applied electromagnetics from “La Sapienza” University of Rome, Rome, Italy, in 1997 and 2001, respectively. In 1997, he joined the Electronic Engineering Department, “La Sapienza” University of Rome, where he is currently a Contract Researcher. From January 2004 to July 2004, he was a Visiting Research Assistant Professor with the University of Houston, Houston, TX. His scientific interests include analysis and design of planar leaky-wave (LW) antennas, numerical methods for the analysis of passive guiding and radiating microwave structures, periodic structures, and propagation and radiation in metamaterials. Dr. Burghignoli was the recipient of the 2003 Graduate Fellowship Award presented by the IEEE Microwave Theory and Techniques Society (IEEE MTT-S).

Fabrizio Frezza (S’87–M’90–SM’95) received the Laurea degree (cum laude) in electronic engineering and Doctorate degree in applied electromagnetics from the “La Sapienza” University of Rome, Rome, Italy, in 1986 and 1991, respectively. In 1986, he joined the Electronic Engineering Department, “La Sapienza” University of Rome, where, from 1990 to 1998, he was a Researcher, from 1994 to 1998, a Temporary Professor of electromagnetics, and since 1998, an Associate Professor of electromagnetics. His main research activity concerns guiding structures, antennas and resonators for microwaves and millimeter waves, numerical methods, scattering, optical propagation, plasma heating, and anisotropic media. Dr. Frezza is a member of Sigma Xi, the Associazione Elettrotecnica ed Elettronica Italiana (AEI), the Italian Society of Optics and Photonics (SIOF), the Italian Society for Industrial and Applied Mathematics (SIMAI), and the Italian Society of Aeronautics and Astronautics (AIDAA).

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Alessandro Galli (S’91–M’96) received the Laurea degree in electronic engineering and Ph.D. degree in applied electromagnetics from the “La Sapienza” University of Rome, Rome, Italy, in 1990 and 1994, respectively. In 1990, he joined the Electronic Engineering Department, “La Sapienza” University of Rome. In 2000, he became and Assistant Professor and, in 2002, an Associate Professor with “La Sapienza” University of Rome, where he currently teaches electromagnetic fields for telecommunications engineering. His scientific interests mainly involve electromagnetic theory and applications, particularly regarding analysis and design of passive devices and antennas (dielectric and anisotropic waveguides and resonators, leaky-wave (LW) antennas, etc.) for microwaves and millimeter waves. He is also active in bioelectromagnetics (modeling of interaction mechanisms with living matter, health safety problems for low-frequency applications and mobile communications, etc.). Dr. Galli was the recipient of the 1994 and 1995 Quality Presentation Recognition Award presented by the IEEE Microwave Theory and Techniques Society (IEEE MTT-S).

Paolo Lampariello (M’73–SM’82–F’96) was born in Rome, Italy, on May 17, 1944. He received the Laurea degree (cum laude) in electronic engineering from the University of Rome, Rome, Italy, in 1971. In 1971, he joined the Institute of Electronics, University of Rome, where he was an Assistant Professor of electromagnetic fields. Since 1976, he has been engaged in educational activities involving electromagnetic-field theory. In 1986, he became a Professor of electromagnetic fields. From November 1988 to October 1994, he was Head of the Department of Electronic Engineering, “La Sapienza” University of Rome, Rome, Italy. Since November 1993, he has been the President of the Council for Electronic Engineering Curriculum of “La Sapienza” University of Rome. Since September 1995 he has been the President of the Center Interdepartmental for Scientific Computing, “La Sapienza” University of Rome. From September 1980 to August 1981, he was a North Atlantic Treaty Organization (NATO) PostDoctoral Research Fellow with the Polytechnic Institute of New York, Brooklyn. He has been engaged in research in a wide variety of topics in the microwave field including electromagnetic and elastic wave propagation in anisotropic media, thermal effects of electromagnetic waves, network representations of microwave structures, guided-wave theory with stress on surface waves and leaky waves, traveling-wave antennas, phased arrays, and, more recently, guiding and radiating structures for the millimeter- and near-millimeter-wave ranges. Prof. Lampariello is a member of the Associazione Elettrotecnica ed Elettronica Italiana (AEI). He is a past chairman of the Central and South Italy Section of the IEEE and a past chairman of the IEEE Microwave Theory and Techniques (MTT)/Antennas Propagation (AP) Societies Joint Chapter of the same section. He is currently the President of the Specialist Group “Electromagnetism” of the AEI.

Giampiero Lovat (S’02) was born in Rome, Italy, on May 31, 1975. He received the Laurea degree (cum laude) in electronic engineering from “La Sapienza” University of Rome, Rome, Italy, in 2001, and is currently working toward the Ph.D. degree in applied electromagnetics at “La Sapienza” University of Rome. From January 2004 to July 2004, he was a Visiting Scholar with the University of Houston, Houston, TX. His scientific interests include theoretical and numerical studies on leakage phenomena in planar structures and guidance and radiation in metamaterials and general periodic structures.

Simone Paulotto (S’97) received the Laurea degree (cum laude and honorable mention) in electronic engineering from “La Sapienza” University of Rome, Rome, Italy, in 2002, and is currently working toward the Ph.D. degree in applied electromagnetics at “La Sapienza” University of Rome. In 2002, he joined the Electronic Engineering Department, “La Sapienza” University of Rome. His scientific interests focus on electromagnetic propagation/radiation in planar structures and scattering theory.

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Refraction Laws for Anisotropic Media and Their Application to Left-Handed Metamaterials Tomasz M. Grzegorczyk, Member, IEEE, Madhusudhan Nikku, Xudong Chen, Bae-Ian Wu, and Jin Au Kong, Fellow, IEEE

Abstract—The refraction of phase and power at the interface between free space and anisotropic media is studied. Under a TE in= 0 incidence (where is cidence (electric field along ^) and the ^ component of the wave vector), a closed-form generalization of Snell’s laws is proposed. The two relations (one for the phase and one for the power) are expressed in terms of the incident angle, the constitutive parameters of the anisotropic medium (which can take negative values), and the tilting angle of the dispersion relation. In addition, both positive and negative dispersions are discussed, making the formulas directly usable for the design of left-handed metamaterials. The validation is done by two methods. First, the electromagnetic fields in layered anisotropic media are computed analytically. The angle of refraction of the wave vector is thus directly obtained, while that of the power is obtained by computing the time-average Poynting power. Second, numerical simulations are performed using HFSS to record the deflection of the power by a prism characterized by positive or negative constitutive parameters. For the specific prism experiment, in which the incident angle is equal to the tilting angle of the dispersion relation, we show that the refraction laws reduce to a simple Snell’s law form. Index Terms—Anisotropic media, left-handed metamaterials (LHMs), refraction laws.

I. INTRODUCTION

M

ETAMATERIALS composed of a succession of rods and ring resonators (of various possible shapes) are currently extensively studied since they are probably the only engineered structures that allow to have a certain control on both the effective permittivity tensor and the effective permeability tensor of a material. In particular, they have significantly broadened the range of achievable permittivities and permeabilities by encompassing negative values as well. Hence, it is currently possible to realize metamaterials with negative permittivity, negative permeability, or both. We refer to the later metamaterials as “left-handed” metamaterial (LHM), following the research of [1]. One of the well-known property of LHM is to bend the waves negatively, i.e., in the direction opposite to what is expected from standard materials. This property has been initially verified with a prism experiment [2], and later with the deflection Manuscript received May 31, 2004; revised October 8, 2004. This work was supported by the Defence Advanced Research Projects Agency under Contract N00014-03-1-0716, by the Department of the Air Force under Air Force Contract F19628-00-C-0002, and by the Office of Naval Research under Contract N00014-01-1-0713. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the U.S. Government. The authors are with the Research Laboratory of Electronics, Center for Electromagnetic Theory and Applications, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845206

of a Gaussian beam [3]–[6], as well as in a T-junction [7], [8]. Yet, in general, this negative refraction cannot be explained on the sole account of an isotropic negative index of refraction and governed by Snell’s law, but must be viewed in light of the specific dispersion relation exhibited by these metamaterials, which are intrinsically anisotropic. Indeed, a negative refraction is not specific to LHM, but can be found in standard anisotropic media or moving media as well [9], while some anisotropic LHM exhibit positive refraction. The study of waves in anisotropic LHMs has been first exposed in [10], where the general theory has been presented along with some discussions on positive and negative dispersions. More specifically, the concepts of refraction of the phase (or vector, where the bar indicates a vector) and of the Poynting power in anisotropic metamaterials have been exposed in [11]. Yet no systematic laws were given, which would yield the two refraction angles (of the wave vector and power) as functions of the incident angles. A generalization of Snell’s law to anisotropic media was proposed in [12], although the refraction angles were ultimately obtained numerically. In this paper, closed-form analytical formulas are proposed to calculate the refraction of both the phase and the time-average Poynting power for a wave incident from an isotropic medium (in our case, free space) onto an anisotropic medium. We study the two cases of elliptic and hyperbolic dispersion relations, which are inherent to LHM and, more generally, are exhibited by anisotropic media with diagonal permittivity and permeability tensors along the principal axes of the metamaterial. Two necessary cases are distinguished: when the interface is aligned with the principal axes of the metamaterial, and when it is not. The second case is necessary to account for waves impinging on slanted interfaces like in the well-known prism experiment [2], [13]. Finally, since dispersion relations can be either positive or negative, we specifically mention how the refraction angles should be computed in these two cases. The refraction laws are derived on simple mathematical grounds for an electric field polarized in the -direction (TE incidence) at , and are verified by both an analytical method that computes the electromagnetic fields in layered anisotropic media, as well as by numerical simulations using the commercial package HFSS. In the analytical method, we calculate the electric and magnetic fields due to a plane-wave incidence onto a medium of arbitrary anisotropic constitutive parameters. The problem is formulated as an eigenvalue problem, in which the eigenvalues yield the components of the wave vector ( being the direction of propagation), from which the refraction angle of is obtained. The eigenvectors of the system yield the transverse

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polarization states of the electric and magnetic fields, from which the longitudinal components can be obtained and, finally, the time-average Poynting power. From its and components, we deduce the refraction angle of the power. The refraction laws are also verified by numerical simulations. Using the retrieved parameters from [14, Fig. 8] of the unit cell presented in [15], we simulate a prism using HFSS and measure the angle of deflection of the power. Three prism angles are selected, i.e., 18 , 26.57 , and 45 , which correspond to ratios of approximately 1 : 3, 1 : 2, and 1 : 1 of the unit cells of [15], and the deflected angle is shown to be in agreement with the theoretical predictions. Finally, we show that, in the specific case of a prism experiment, of which its particularity is to have an incident angle exactly opposite to the tilting angle of the dispersion relation, the refracted angles of the wave vector and of the power are identical and can be predicted by Snell’s law, provided the index of refraction is properly defined. II. DISPERSION RELATIONS AND REFRACTION LAWS Although LHM were initially characterize with isotropic parameters and, in particular, isotropic permittivities and permeabilities, it is currently well accepted that a better model is to consider anisotropic constitutive parameters, which can be diagonalized in the coordinate system collinear with the principal axes of the metamaterial (note that we neglect bianisotropic effects despite the fact that they might be important for the characterization of some rings [16]). Thus, metamaterials composed of an arrangement of rings (either split rings, -rings [17], [18], -rings [19], or others) and rods are characterized by (1a) (1b) for which the dispersion relation for a TE incidence (electric field polarized in the -direction) at is

where the first conditions [i.e., (3a) and (3c)] correspond to a positive dispersion, while the other two [i.e., (3b) and (3d)] correspond to a negative dispersion. Although the distinction between these two cases is important to deduce if positive or negative refraction occurs, the angles obtained in each case can be deduced from one another. In order to obtain the refraction laws for the phase and for the Poynting power in these two types of media, we study the general cases of ellipses and hyperbolae defined by Ellipse:

(4a)

Hyperbola:

(4b)

where (5) where is the free-space wavenumber. Note that and are aligned with and , which correspond to the situation where the interface of the metamaterial coincides with the direction of its lattice. The rotated case will be treated hereafter. For a given incident angle , the angle of refraction of the phase ( ) is simply obtained by phase matching, while the angle of refraction of the Poynting power ( ) is obtained by computing the gradient of the dispersion relation [9], [20]. In addition, since the two dispersion relations can be obtained from , the refraction one another by the transformation laws are identical (but obviously different for the wave vector and power). After some simple algebra, the generalization of Snell’s law reads (6a)

(6b)

(2) A general study on the shape of the dispersion relation as function of the sign of these parameters has already been offered in [11]. In this paper, we are interested in the two cases referred to as cutoff and never cutoff, i.e., the two cases when the dispersion relations are either elliptic or hyperbolic, as also studied in [10]. In terms of constitutive parameters, this implies the following. • For the elliptic dispersion relation (3a) or (3b) • For the hyperbolic dispersion relation (3c) or (3d)

Note that (6) are valid for both ellipses and hyperbolae upon using the corresponding values of , , and . The distinction between a positive and negative dispersion depends on the convention chosen. In this paper, we define all angles with -axis (it will also be the case for tilted disperrespect to the sion relations) in a range of , where and both refer to a backward propagation. In addition, angles are counted positive if the component of the wave vector is positive, and reversely. Under these conventions, the following exists. • Equation (6a) is the refraction law of the wave vector for positive elliptic and hyperbolic dispersions. The angles for negative dispersions are obtained by the transformation . • Equation (6b) is the refraction law for the time-average Poynting power for a positive elliptic dispersion and a negative hyperbolic dispersion. The angles for the dual cases are obtained by the transformation .

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It should be mentioned, however, that the laws of (6) are not easily experimentally verified since they require an incident wave from free space and a measurement inside the metamaterial (or reversely). A better experimental setup would be to use a prism configuration like the one in [2], where both incidence and measurement are located in free space. The difficulty generated in this case comes from the second interface of the prism, which is not aligned with one of the principal axes of the medium. The refracted angles for both the wave vector and power, therefore, need be predicted from a tilted dispersion relation in the plane . In the new reference frame (denoted by primed coof ordinates), the permittivity and permeability tensors can be expressed as (7a) (7b) where is the rotation angle and is the rotation matrix about the -axis. It is seen that under this rotation, , but that loses its diagonal property if . If they are equal, however, the dispersion relation is circular and any rotation has no effect on the angles. After some tedious, but simple algebra (details are given in the Appendix ), the refraction laws for tilted dispersion relations by an angle are obtained as (8a) (8b) where

has been defined as (9)

It should be noted that (8) are the unique laws needed to characterize tilted elliptic and hyperbolic dispersion relations, as well as positive and negative dispersion relations. Yet in order to conform to the previous convention, special attention must be paid to the way the arctangent function is computed. In particular, if the MATLAB or Fortran function is used, both numerator and denominator might have to be simultaneously multiplied , , or . However, these manipulations are only cosby metic, and we can conclude that (8a) and (8b) are unique and characterize the refraction of phase and power in tilted diagonal anisotropic media completely (it can be verified that (8) reduce ). to (6) for Finally, the relations between the angles obtained for positive and negative dispersions can be generalized to tilted cases as well. Denoting positive and negative dispersions by the superand , respectively, it can be shown that scripts (10a) (10b) for ellipses when

(10c)

Fig. 1. Elliptic dispersion relation with  = 0:5k , = 1:5k , and  = 45 . All angles are defined with respect to the +^z -axis (coinciding with k ). The continuous lines have been obtained from (8), while the superposed pentagrams and hexagrams have been obtained from (16). (a) Dispersion relation and definition of the refraction angles. Case (1) and case (2) represent positive and negative dispersions, respectively. The dispersion relation is  is the normal vector at point P (i = 1; 2). (b) rotated by an angle . n Positive dispersion ( and  ). (c) Negative dispersion ( and  ).

0

for hyperbolae, when

(10d)

Examples of relations between , , and are given in Figs. 1 and 2 for some values of , , and chosen arbitrarily for

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III. ANALYTICAL AND NUMERICAL VALIDATIONS A. Analytical Computation of

and

Various methods are available in the literature to compute the electromagnetic fields in layered bianisotropic media where each layer is described by four fully populated constitutive tensors [22], [23]. One of the simplest ones is based on a decomposition of Maxwell’s equations into transverse ( , ) and longitudinal ( ) components: the transverse components of the fields are solved from an eigenvalue problem, and are then used to compute the remaining longitudinal components. This method has been presented for anisotropic layered media in [24], and has been generalized to bianisotropic layered media in [21]. In the current problem, the medium is only anisotropic such that the governing equations are substantially simplified. Without further detail, we write [21]

(11) and a dual equation can be written for . Equation (11) is written as a succession of 3 3 matrix multiplies, (the subscript denotes where , and where the transpose operator) and similarly for

(12) In addition, the permittivity tensor has been split as (13)

Hyperbolic dispersion relation with  = 0:5k , = 0:5k , and  = 30 . All angles are defined with respect to the +^z -axis (coinciding with k ). The continuous lines have been obtained from (8), while the superposed Fig. 2.

pentagrams and hexagrams have been obtained from (16). (a) Dispersion relation and definition of the refraction angles. Case (1) and case (2) represent positive and negative dispersions, respectively. The dispersion relation is  rotated by an angle  (approximately 20 for the purpose of illustration). n is the normal vector at point P (i = 1; 2). (b) Positive dispersion ( and  ). (c) Negative dispersion ( and  ).

the sake of illustration. These curves have also been validated by an eigenvalue solver able to compute the electric and magnetic fields in multilayer bianisotropic media [21]. In Section III, we outline the method and particularize it to anisotropic media.

with further zero padding to the individual tensors in order to reach a dimension of 3 3. The permeability tensor has been split in a similar fashion. Equation (11) and its dual are then reduced to a 2 2 system and combined to yield the 4 4 eigenvalue system (14) where the matrix is directly computed from (11) and its dual equation [24]. Equation (14) is a simple partial differential equation with weighted exponentials in the -direction as a solution. Therefore, the eigenvalues of (14) yield the propagating components of the wave vector ( ), and the eigenvectors yield the polarization states of the electric and magnetic fields in the

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transverse direction. The longitudinal components of the fields are obtained from [21], [24] (15a) (15b) Finally, knowing completely the polarization of the fields and their propagation vectors, we can compute the refraction angles of both the phase and power as (16a) (16b) with and being the and components, respectively, of the time-average Poynting power , where denotes the real part operator and the star denotes the complex conjugate. Since both (8) and the current method are analytical and without approximations, they should yield exactly the same refraction angles for identical inputs. Figs. 1 and 2 depict the and for a tilted elliptic relation and a tilted evolution of hyperbolic relation, respectively (the specific parameters are given in the caption of these figures). In both cases, the connected lines have been obtained from (8), while the pentagrams have been obtained from (16a) and the hexagrams have been obtained from (16b). As expected from two analytical methods, the results match perfectly. B. Numerical Simulations of Prisms A further validation of (8) has been performed using the commercial electromagnetic simulation software HFSS, with which we have simulated the deflection of an incident TEM wave by a prism having homogeneous -, -, and -parameters. The were evaluated by the standard Drude and parameters and was either taken to be equal to Lorentz models [2], while or equal to one. The specific plasma and resonant frequencies were obtained by first performing method of moments (MoM) simulations with periodic boundary conditions [25] of the unit cell presented in [15] to obtain the Fresnel reflection and transmission coefficients at normal incidence. Secondly, these coefficients were used to retrieve the electric and magnetic resonant and plasma frequencies [14]. Since the retrieved loss terms were very small, we have neglected them here. Using the same noGHz, tation as [2], the retrieved parameters are [14] GHz, GHz, and GHz. These analytical models were then used to define the constitutive parameters of the prism in HFSS. The setup of the simulation is shown in Fig. 3. A TEM mode is fed to a parallel-plate waveguide in which the prism is located. The prism is bounded on two sides by absorbing layers, as shown in this figure, and the far-field radiation is measured. For each case of , three different prism angles are simulated and the angle of refraction in each situation is determined from the direction of maximum radiation. The operating frequency is chosen to be 13.8 GHz in

Fig. 3. Simulation setup in HFSS. A TEM wave is incident from the left onto a prism of angle  and the far-field radiated power is recorded as function of frequency. The parameters are a = 16 mm, b = 10 mm, and c = 72 mm. TABLE I REFRACTION ANGLE OF THE POWER ( ) OBTAINED FROM THE HFSS SIMULATION OF FIG. 3. THE OPERATING FREQUENCY IS f = 13:8 GHz, FOR WHICH  = 0:46 AND  = 1:29

0

0

TABLE II REFRACTION ANGLE OF THE POWER ( ) OBTAINED FROM BOTH (8) AND (16) = 0:46 AND  = 1:29. FOR f = 13:8 GHz, FOR WHICH  NOTE THAT THE VALUE OF  NEED NOT BE SPECIFIED FOR THE CASE OF A PRISM EXPERIMENT IN WHICH THE INCIDENT ANGLE IS EQUAL TO THE TILTING ANGLE 

0

0

all cases, which corresponds to negative values of and . The results are summarized in Table I. Using the analytical models at the frequency of 13.8 GHz yields and (where we have neglected the imaginary parts). Using these values into (8) and (16), the angles of refraction for the various cases are calculated directly, and the results are summarized in Table II. Comparing these results to those in Table I, it can be seen that the numerical simulations agree satisfactorily with the theory. It should be noted that (8) and (16) have been derived for an incidence from free space onto the metamaterial. In the prism experiment, the incidence is from the metamaterial onto free and are known and we need to solve for space such that to obtain the values reported in Table II. In addition, it can be seen that Table II does not make the distinction between and . This is due to the very specific inciand dence imposed by the prism setup, for which , i.e., the incidence is exactly opposite to the tilting angle of the dispersion relation. Under this condition, it is obvious that or , but also, by inverting (8), we find that in both cases (17) where the sign refers to positive and negative dispersions. Equation (17) is seen to be independent of , and is identical to . Snell’s law with an index of refraction equal to This property can be understood physically by realizing that the lower point of the hyperbola (the argument for the ellipse is simand only. ilar) depends on only, which is a function of

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APPENDIX REFRACTION LAWS FOR A TILTED HYPERBOLIC DISPERSION RELATION

Fig. 4. Comparison between HFSS numerical results and analytical predictions from (8) and (16) of the refraction angle of the power ( ) as function of frequency for a prism of 26.57 . Case 1 corresponds to  =  and Case 2 corresponds to  = 1. The permittivity follows a Drude model = 16:6 GHz and the permeability follows a Lorentz model with with f = 13:2 GHz and f = 14:54 GHz. f

Therefore, this type of Snell’s law can be used to interpret the prism experiments. Other experiments such as the deflection of a Gaussian beam by an anisotropic metamaterial [6] should be interpreted using the refraction laws of (8) if the incidence is different from the opposite of the tilting angle (and if the metamaterial is not isotropic). Finally, the same setup as shown in Fig. 3 has been used to span frequencies and record the angle of maximum radiation for a prism of 26.57 . The simulations were again performed for and . Fig. 4 shows the numerical results compared to the analytical ones. Although the results are in fairly good agreement, some discrepancies can still be noticed. We attribute those to the fact that the numerical simulations may need a very fine mesh to reach convergence, as well as to the fact that the analytical results are obtained under the assumption of an infinite interface, which is different from a prism configuration with absorbers. IV. CONCLUSION A generalization of Snell’s refraction law to the case of an interface between an isotropic medium and an anisotropic medium at incidence has been proposed. The refraction laws of both the phase and power have been expressed in closed form, and are valid for arbitrary tilting angles between the material principal axes and interface. The validation has been done both analytically and numerically via entirely solving for the electromagnetic fields in layered anisotropic media and HFSS simulations, respectively. The analytical method agrees exactly with the predictions, while the numerical results are in reasonable agreement. Finally, we have shown that, for the specific case of a prism experiment, where the incident angle is exactly opposite of the . tilting angle, Snell’s law can still be applied with For any other incidences, however, (8) should be used.

The tilted dispersion relations in the reference frame for an ellipse and a hyperbola are depicted in Figs. 1(a) and 2(a), respectively. In both cases, the circle represents the dispersion relation of free space, and the wave is propagating in the -direction. Upon an incidence at (point on the circle), phase and , which have to be dismatching yields the points tinguished by examining whether the dispersion is positive or negative. For the sake of illustration, we detail here the method to obtain the refraction law for a tilted hyperbolic dispersion relation at an angle . Due to the similarity between ellipses and hyperbolae, the laws for a tilted ellipse are obtained in the same fashion and yield similar relations. We can define the natural reference frame of the hyperbola in which the equation is simply by (18) The primed reference frame is related to the original (unprimed) one by a rotation about the -axis (19) where is the rotation matrix. Thus, from (18), we obtain the coordinate system as equation in the (20a) where (20b)

Since the propagation is in the -direction, the component at any given incidence is directly obtained from phase matching and the component is obtained by solving (20). From this, the primed components are directly obtained and are via (19) and the normal vector at the points and , respectively. Finally, the normal vectors in the original reference frame are computed from (21) Thus, knowing the coordinates of and in , as well as the normal vectors, we compute the refracted angles of the wave vector ( ) and of the power ( ) from (22a) (22b)

GRZEGORCZYK et al.: REFRACTION LAWS FOR ANISOTROPIC MEDIA AND THEIR APPLICATION TO LHMs

in which the subscript denotes either the point or and yields and , respectively. After some manipulations, the relations can be cast in a unique form as (23a) (23b) where (23c) In (23), the upper sign refers to normal a dispersion and the lower sign refers to a negative dispersion. Also, the transformatransforms a positive elliptic dispersion into tion a negative hyperbolic dispersion, and a negative elliptic dispersion into a positive hyperbolic dispersion. Finally, (23) can be directly related to the permittivity and permeability of the metamaterial via (5) to eventually yield (8). REFERENCES [1] V. Veselago, “The electrodynamics of substances with simultaneously negative values of  and ,” Sov. Phys.—Usp., vol. 10, pp. 509–514, Jan.–Feb. 1968. [2] R. Shelby, D. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [3] J. A. Kong, “Electromagnetic wave interaction with stratified negative isotropic media,” Prog. Electromagn. Res., vol. 35, pp. 1–52, 2002. [4] J. A. Kong, B.-I. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett., vol. 80, pp. 2084–2086, Mar. 2002. , “A unique lateral displacement of a Gaussian beam transmitted [5] through a slab with negative permittivity and permeability,” Microwave Opt. Technol. Lett., vol. 33, pp. 136–139, Apr. 2002. [6] L. Ran, J. Huangfu, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Beam shifting experiment for the characterization of lefthanded properties,” J. Appl. Phys., vol. 95, pp. 2238–2241, Mar. 2004. [7] C. Caloz, C.-C. Chang, and T. Itoh, “Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,” J. Appl. Phys., vol. 90, pp. 5483–5486, Dec. 2001. [8] H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “T-junction waveguide experiment to characterize left-handed properties of metamaterials,” J. Appl. Phys., vol. 94, pp. 3712–3718, Sep. 2003. [9] J. A. Kong, Electromagnetic Wave Theory. New York: EMW, 2000. [10] I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—Media with negative parameters, capable of supporting backward waves,” Microwave Opt. Technol. Lett., vol. 31, pp. 129–133, Oct. 2001. [11] D. R. Smith and D. Shurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors materials,” Phys. Rev. Lett., vol. 90, pp. 077 405–1–4, Feb. 2003. [12] M. A. Slawinski, R. A. Slawinski, and R. J. Brown, “A generalized form of Snell’s law in anisotropic media,” Geophysics, vol. 65, pp. 632–637, Mar.–Apr. 2000. [13] D. R. Smith, P. Kolinko, and D. Shurig, “Negative refraction in indefinite media,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 21, pp. 1032–1043, May 2004. [14] X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, “Improved method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 70, no. 016 608, pp. 1–7, 2004. [15] T. M. Grzegorczyk, C. D. Moss, J. Lu, and J. A. Kong, “New ring resonator for the design of left-handed metamaterials at microwave frequencies,” in PIERS, Honolulu, HI, Oct. 13–16, 2003, p. 286. [16] R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B, Condens. Matter, vol. 65, pp. 144 440–??? ???, 2002.

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[17] M. M. I. Saadoum and N. Engheta, “A reciprocal phase shifter using novel pseudochiral or medium,” Microwave Opt. Technol. Lett., vol. 5, pp. 184–188, Apr. 1992. [18] J. Huangfu, L. Ran, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of -like metallic patterns,” Appl. Phys. Lett., vol. 84, pp. 1537–1539, Mar. 2004. [19] H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Left-handed metamaterials composed of only S-shaped resonators,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., to be published. [20] L. Landau and E. Lifshitz, The Theory of Classical Fields. New York: Pergamon, 1975, vol. 2. [21] T. M. Grzegorczyk, X. Chen, J. Pacheco, Jr., J. Chen, B.-I. Wu, and J. A. Kong, “Reflection coefficients and Goos–Hänchen shifts in anisotropic and bianisotropic left-handed metamaterials,” Progr. Electromagn. Res. (Special Issue), 2004, to be published. [22] J. L. Tsalamengas, “Interaction of electromagnetic waves with general bianisotropic slabs,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 10, pp. 1870–1878, Oct. 1992. [23] W. Y. Yin, B. Guo, and X. T. Dong, “Comparative study on the interaction of electromagnetic waves with multi-layer omega(chiro)ferrite slabs,” J. Electromagn. Waves Applicat., vol. 17, no. 1, pp. 15–29, 2003. [24] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Van Nostrand, 1990. [25] T. M. Grzegorczyk, L. Ran, X. Zhang, K. Chen, X. Chen, and J. A. Kong, “Two dimensional periodic approach for the study of left-handed metamaterials,” in Wave Propagation. Scattering and Emission in Complex Media, Y.-Q. Jin, Ed. Beijing, Singapore: World Sci., 2003, pp. 175–186.

Tomasz M. Grzegorczyk (M’00) received the Ph.D. degree from the Laboratoire d’Electromagnetisme et d’Acoustique (LEMA), École Polytechnique Federale de Lausanne (Swiss Federal Institute of Technology, Lausanne), Lausanne, Switzerland, in 2000. His doctoral research concerned the modeling of millimeter and submillimeter structures using numerical methods, as well as their technological realizations with the use of micromachining techniques. In January 2001, he joined the Research Laboratory of Electronics (RLE), Massachusetts Institute of Technology (MIT), Cambridge, where he is currently a Research Scientist. He has been a Visiting Scientist with the Institute of Mathematical Studies, National University of Singapore. In July 2004, he became an Adjunct Professor with The Electromagnetics Academy, Zheijiang University, Hangzhou, China. His research interests include the study of wave propagation in complex media including LHMs, the polarimetric study of oceans and forests, electromagnetic induction from spheroidal objects for unexploded ordnance modeling, waveguide and antenna design, and wave propagation over rough terrains. He is on the Editorial Board of the Journal of Electromagnetic Waves and Applications. Dr. Grzegorczyk has been part of the Technical Program Committee of the Progress in Electromagnetics Research Symposium since 2001.

Madhusudhan Nikku received the B.Tech. degree in mining engineering from the Institute of Technology, Banaras Hindu University, Varanasi, India, in 2002, the M.S. degree from the Massachusetts Institute of Technology (MIT), Cambridge, in 2004, and is currently working toward the Ph.D. degree at MIT. From 2003 to 2004, he was a Research Assistant with the Center for Electromagnetic Theory and Applications, Research Laboratory of Electronics, MIT. His research interests include electrodynamics of anisotropic media, metamaterials and bioengineering.

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Xudong Chen received the B.S. and M.S. degrees in electrical engineering from Zhejiang University, Hangzhou, China, in 1999 and 2001, respectively, and is currently working toward the Ph.D. degree at the Massachusetts Institute of Technology (MIT), Cambridge. He is currently with the Center for Electromagnetic Theory and Applications, Research Laboratory of Electronics, MIT. His research interests are mainly metamaterial and electromagnetic inverse problems. Mr. Chen was the recipient of the 1997 Chinese National Mathematical Contest in Modeling First Prize.

Bae-Ian Wu was born in Hong Kong, on November 3, 1975. He received the B.Eng. degree in electronic engineering from the Chinese University of Hong Kong, in 1997, and the M.S. and Ph.D. degrees in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge, in 1999 and 2003, respectively. Since 1997, he has been a Research Assistant with the Department of Electrical Engineering and Computer Science, Research Laboratory of Electronics, MIT, where he has concentrated on electromagneticwave theory and applications. From 1998 to 2002, he was a Teaching Assistant with the Department of Electrical Engineering and Computer Science, MIT, for several undergraduate and graduate electromagnetic (EM) courses. He is a currently a Post-Doctoral Associate with the Center for Electromagnetic Theory and Applications, Research Laboratory of Electronics, MIT.

Jin Au Kong (S’65–M’69–SM’74–F’85) is the President of The Electromagnetics Academy and a Professor of Electrical Engineering with the Massachusetts Institute of Technology (MIT), Cambridge. His research interest is in the area of electromagnetic-wave theory and applications. He has authored or coauthored over 30 books including Electromagnetic Wave Theory (New York: Wiley–Interscience, 1975, 1986, 1990; EMW Publishing since 1998) and over 600 refereed journal papers, book chapters, and conference papers. He is Editor for the Wiley Series in Remote Sensing, Editor-in-Chief of the Journal of Electromagnetic Waves and Applications (JEMWA), and Chief Editor for the book series Progress in Electromagnetics Research (PIER).

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Equivalent-Circuit Models for Split-Ring Resonators and Complementary Split-Ring Resonators Coupled to Planar Transmission Lines Juan Domingo Baena, Jordi Bonache, Ferran Martín, Ricardo Marqués Sillero, Member, IEEE, Francisco Falcone, Txema Lopetegi, Member, IEEE, Miguel A. G. Laso, Member, IEEE, Joan García–García, Ignacio Gil, Maria Flores Portillo, and Mario Sorolla, Senior Member, IEEE

Abstract—In this paper, a new approach for the development of planar metamaterial structures is developed. For this purpose, split-ring resonators (SRRs) and complementary split-ring resonators (CSRRs) coupled to planar transmission lines are investigated. The electromagnetic behavior of these elements, as well as their coupling to the host transmission line, are studied, and analytical equivalent-circuit models are proposed for the isolated and coupled SRRs/CSRRs. From these models, the stopband/passband characteristics of the analyzed SRR/CSRR loaded transmission lines are derived. It is shown that, in the long wavelength limit, these stopbands/passbands can be interpreted as due to the presence of negative/positive values for the effective and of the line. The proposed analysis is of interest in the design of compact microwave devices based on the metamaterial concept. Index Terms—Duality, metamaterials, microwave filters, splitring resonators (SRRs).

I. INTRODUCTION

I

N RECENT years, there has been a growing interest for the design of one-, two-, and three-dimensional artificial structures (also called metamaterials) with electromagnetic properties generally not found in nature. Among them, special attention has been devoted to double-negative media. These are artificial periodic structures composed of sub-wavelength constituent elements that make the structure behave as an effective medium with negative values of permittivity ( ) and permeability ( ) at the frequencies of interest. The properties of such media were already studied by Veselago [1] over 30 years ago. Due to the simultaneous negative values of and , the wave vector and the vectors and (the electric- and magnetic-field intensity) form a left-handed triplet, with the result of antiparallel phase and group velocities, or backward-wave propagation. Due to left-handedness, exotic electromagnetic properties are Manuscript received June 1, 2004; revised October 20, 2004. This work was supported by the Dirección General de Investigación and the Comisión Interministerial de Ciencia y Tecnología under Contract TIC2002-04528-C02-01, Contract TEC2004-04249-C02-01, Contract TEC2004-04249-C02-02, and Contract PROFIT-070000-2003-933. J. D. Baena and R. Marqués Sillero are with the Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla, 41012 Seville, Spain. J. Bonache, F. Martín, J. García–García, and I. Gil are with the Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain F. Falcone, T. Lopetegi, M. A. G. Laso, M. Flores Portillo, and M. Sorolla are with the Electrical and Electronic Engineering Department, Public University of Navarre, E-31006 Pamplona, Spain. Digital Object Identifier 10.1109/TMTT.2005.845211

expected for left-handed metamaterials (LHMs); namely, inversion of the Snell law, inversion of the Doppler effect, and backward Cherenkov radiation. It is also worth mentioning the controversy originated four years ago from the paper published by Pendry [2], where amplification of evanescent waves in LHMs is pointed out [3]–[6]. In spite of these interesting properties, it was not until 2000 that the first experimental evidence of left-handedness was demonstrated [7]. Following this seminal paper, other artificially fabricated structures exhibiting a left-handed behavior were reported [8]–[11] including the experimental demonstration of negative refraction [12]–[14] and backward wave radiation [15]. The original medium proposed in [7] consists of a bulky combination of metal wires and split-ring resonators (SRRs) [16] disposed in alternating arrows. However, SRRs are actually planar structures, and wires can be easily substituted by metallic strips [8]. Therefore, the extension of these designs to planar configurations can be envisaged [17], [18], thus, opening the way to new planar microwave devices. In fact, in coplanar waveguide (CPW) technology, miniaturized stopband [19] and bandpass filters [20] have been recently reported by some of the authors. In these implementations, SRRs are etched in the back substrate side, underneath the slots, to achieve high magnetic coupling between line and rings at resonance. The presence of the rings leads to an effective negative-valued permeability in a narrow band above resonance, where signal propagation is inhibited. By simply adding shunt metallic strips between the central strip and ground planes, the authors have demonstrated the switch to a bandpass characteristic [18], [20]. This effect has been interpreted as due to the coexistence of effective negative permeability and permittivity (the latter introduced by the additional strips) [18]. In microstrip technology, SRRs etched in the upper substrate side, in proximity to the conductor strip, have been found to provide similar effects [21]. Broad-band negative- media in microstrip technology can also be fabricated by periodically etching series gaps in the conductor strip [11], [22]–[26]. However, the implementation of an associated effective negative requires the use of shunt inductances, which are associated to metallic vias to the ground [11], [22]–[26]. Now, a key question arises: is it possible to conceive the dual counterpart of SRRs? If so, an effective negative permittivity could be introduced in microstrip devices by using this concept. In a recent paper [27], it was demonstrated by some of the authors that by periodically

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numerous workers in the field, at least as a first approach to the design (of course, conventional commercial simulation tools will also take a place in this approach as a second-order approximation and fine-tuning tools). Comparison between the frequency responses provided by the proposed analytical circuit models and those experimentally obtained from fabricated SRR- and CSRR-loaded planar transmission lines are also provided in this paper. A satisfactory agreement between the proposed analytical models and reported experimental results is shown in all cases. II. ELECTROMAGNETIC BEHAVIOR OF SRRs AND CSRRs A. Physics of SRRs and CSRRs and Its Equivalent-Circuit Models

Fig. 1. Topologies of the: (a) SRR and (b) CSRR, and their equivalent-circuit models (ohmic losses can be taken into account by including a series resistance in the model). Grey zones represent the metallization.

etching the negative image of SRRs in the ground plane of a microstrip line underneath the conductor strip, a narrow stopband appeared at approximately the resonant frequency of a conventional SRR of identical dimensions etched on the same substrate. The stopband characteristic obtained in the above-cited structure was interpreted as due to a negative effective permittivity introduced by these new elements, electrically coupled to the host transmission line. For the reasons that will be explained in Section II, these new elements have been termed as complementary split-ring resonators (CSRRs) (see Fig. 1). More recently, it has been shown that, by periodically etching capacitive gaps along the aforementioned CSRR-loaded microstrip line, the reported stopband switches to a passband [28], [29]. This effect has been interpreted as due to a left-handed behavior of the line. In summary, by properly coupling SRRs and/or CSRRs to a host planar transmission line (CPW or microstrip), planar structures with effective negative constituent parameters can be obtained. By adding shunt strips and/or capacitive gaps, a left-handed behavior is achieved. These structures are fully planar (i.e., without vias or other no-planar objects) and can be easily fabricated by using standard photo-etching techniques. The main purpose of this paper is to provide a simple and analytical technique for the design of these structures. This technique is based on lumped-element circuit models, able to describe the elements and their coupling to the host transmission lines, as well as on analytical formulas to determine the main circuit parameters for these models. As a consequence of this analytical approach, the proposed circuit models can be directly programmed and run in a PC station with negligible computation time. Therefore, the proposed approach can provide useful ab initio calculations on the physical behavior of the analyzed structures. Although today almost all microwave designers are equipped with useful simulation tools, able to analyze the studied structures, such an approach to the design could be time consuming and blind. Therefore, we feel that the analytical tools presented in this paper will be useful for

The electromagnetic properties of SRRs have been already analyzed in [30] and [31]. This analysis shows that SRRs beresonator that can be excited by an external maghave as an netic flux, exhibiting a strong diamagnetism above their first resonance. SRRs also exhibit cross-polarization effects (magnetoelectric coupling) [31] so that excitation by a properly polarized time-varying external electric field is also possible. Fig. 1 shows the basic topology of the SRR, as well as the equivalent-circuit stands for the total camodel proposed in [30]. In this figure, pacitance between the rings, i.e., , where is the per unit length capacitance between the rings. The resonance frequency of the SRR is given by , where is the series capacitance of the upper and lower halves of the SRR, i.e., . The inductance can be approximated by that of a single ring with averaged radius and width [30]. If the effects of the metal thickness and losses, as well as those of the dielectric substrate are neglected, a perfectly dual behavior is expected for the complementary screen of the SRR [28]. Thus, whereas the SRR can be mainly considered as a resonant magnetic dipole that can be excited by an axial magnetic field [30], the CSRR (Fig. 1) essentially behaves as an electric dipole (with the same frequency of resonance) that can be excited by an axial electric field. In a more rigorous analysis, the cross-polarization effects in the SRR [30], [31] should be considered and also extended to the CSRR. Thus, this last element will also exhibit a resonant magnetic polarizability along its -axis (see Fig. 1) and, therefore, its main resonance can also be excited by an external magnetic field applied along this direction [28]. These features do not affect the intrinsic circuit model of the elements, although they may affect its excitation model. The intrinsic circuit model for the CSRR (dual of the SRR model) is also shown in Fig. 1. In this circuit [32], the inductance of the SRR model is substituted by the capacitance of a disk of radius surrounded by a ground plane at a distance of its edge. Conversely, the series connection of the two capacitances in the SRR model is substituted by the parallel combination of the two inductances connecting the inner disk to the ground. Each inductance is given by , where and is the per unit length inductance of the CPWs connecting the inner disk to the ground. For infinitely thin perfect conducting screens, and in the absence of any dielectric substrate, it directly follows from duality that the parameters of the circuit models for the SRRs and CSRRs are related by and . The factor

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Fig. 3. Sketch of the: (a) electric- and (c) magnetic-field lines of an SRR on a dielectric substrate. (b) Magnetic- and (d) electric-field lines of a similar CSRR on the same dielectric substrate are also sketched.

Fig. 2. Sketch of the electric- and magnetic-field lines in the SRR (left-hand side) and the CSRR (right-hand side). (a) Electric-field lines in the SRR at resonance. (b) Magnetic-field lines in the dual CSRR. (c) and (d) Magnetic- and electric-field lines in the SRR and CSRR, respectively. (e) Magnetic induction field in the equivalent ring inductance used for the computation of L in the SRR [27]. (f) Electric field in the dual equivalent capacitor proposed for the computation of C for the CSRR.

of 4 appearing in these relations is deduced from the different symmetry properties of the electric and magnetic fields of both elements, as is sketched in Fig. 2. From the above relations, it is easily deduced that the frequency of resonance of both structures is the same, as is expected from duality. The proposed analysis can be easily extended to other planar topologies derived from the basic geometry of the SRR [33]. Some examples are shown in Fig. 2. It is worth noting that some of these topologies do not exhibit cross-polarization effects and, hence, these effects are also absent in their complementary counterparts. The proposed equivalent circuits for these topologies, as well as for their complementary configurations, are also shown in this figure. The nonbianisotropic split-ring resonator (NB SRR) is a slight modification of the basic SRR topology, which shows a 180 rotation symmetry in the plane of the element. As a consequence of this symmetry, cross-polarization effects are not possible in the NB SRR. However, the equivalent-circuit model and resonant frequency of the NB SRR are identical to those of the SRR. The double-slit SRR (D SSR) also presents the aforementioned symmetry, thus avoiding cross polarization. However, the D-SSR equivalent circuit differs from that of the SRR, being the frequency of resonance twice than that of the SRR (of identical size). Finally, the spiral resonator (SR) [34], as well as the double spiral resonator (DSR) [32] allows for a reduction of the resonant frequency with respect to the SRR, as can be seen from its proposed equivalent circuits. It has been already mentioned that the behavior of SRRs and CSRRs (as well as their derived geometries) are strictly dual for perfectly conducting and infinitely thin metallic screens placed in vacuum. However, deviations from duality—which may give rise to a shift in the frequencies of resonance—arise from losses,

Fig. 4. Topologies corresponding to: (a) the NB SRR, (b) the D SRR, (c) the SR, and (d) the DSR. The equivalent circuits for these topologies are depicted in the second column, while the circuits models for the complementary counterparts are represented in the third column.

finite width of metallizations, and the presence of a dielectric substrate. The latter is expected to be the main cause of deviations from duality. This fact is due to the variations of the elements of the CSRR circuit model, and , from the values extracted from the SRR circuit model parameters and by duality ( , and ). As is sketched in Fig. 3, these variations arise directly from the presence of a dielectric substrate, which affects and , but leave and unaltered. Similar deviations from duality arise in the derived topologies shown in Fig. 2. Analytical expressions for and in the SRR when a dielectric substrate is present were provided in [30]. As we have already mentioned, the capacitance in Figs. 1 and 4 is that corresponding to a metallic disk of radius surrounded by a ground plane at a distance (see Fig. 1). An analytical approximate expression for when a dielectric substrate is

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Fig. 5. Capacitance of the CSRR is approximately equal to that corresponding c=2 surrounded by a ground plane at a to a metallic disk of radius a = r distance b a = c, being r , the averaged radius of the CSRR, and c, the width of the slots in this. The dielectric substrate is characterized by its permittivity " and thickness h.

0

0

present (see Fig. 5) is derived in the Appendix . The final expression is

(1) where the meaning of the different symbols is explained in the Appendix. The inductance in Figs. 1 and 4 is that corresponding to a circular CPW structure of length , strip width , and slot width . For the present purposes, the design formulas given in [35] for the per unit length CPW inductance provide enough accuracy and have been used in all numerical computations throughout this study. B. Numerical Calculations and Experimental Validation The effect of the dielectric substrate in the frequency of resonance of the CSRR and SRR is shown in Fig. 6. As is expected, there is no difference for the two limiting values of a zero and an infinite substrate thickness. However, significant differences in the values of the frequency of resonance for both elements can be observed for intermediate thicknesses. The accuracy of the circuit models for the SRR and its derived geometries (see Figs. 1 and 4) has been already experimentally checked in some previous papers [30], [33], [34]. In order to experimentally verify the accuracy of the proposed circuit models for the CSRR and derived geometries, a set of these resonators with different topologies were etched on a metallized microwave substrate and its frequencies of resonance were measured. Their dual counterparts were also manufactured and measured for completeness. The resonant frequencies were obtained from the transmission coefficient , measured in a rectangular waveguide, properly loaded with the corresponding element [32]. The waveguide was excited in the fundamental mode and connected to an Agilent 8510 network analyzer. The SRRs or derived geometries were placed in the central -plane so that they were excited by the magnetic field perpendicular to the element plane. Their dual counterparts were etched in the top wall of the waveguide, being excited by the electric field perpendicular to the element plane. Fig. 7 shows the transmission coefficients for an SRR and an NB SRR with identical geometrical parameters, as well as the same coefficients for its duals [CSRR and complimentary NB SRR (C-NB SRR)]. It can be

Fig. 6. Numerical calculations showing the dependence of the resonant frequency of SRRs (solid lines) and CSRRs (dashed lines) on the substrate parameters. (a) Dependence on the dielectric thickness for different values of the relative permittivity of the substrate (shown at right). (b) Dependence on the value of the relative dielectric constant for different substrate thickness (in millimeters).

easily seen that the SRR and NB SRR have the same frequency of resonance (the small shift can be attributed to tolerances in the manufacturing process). The same can be said for its complementary elements. Fig. 8 illustrates the cross-polarization effects in the SRR, as well as the absence of these effects in the NB SRR (this last element is not excited in positions 3 and 4), as is predicted by the theory [33]. This figure also shows that the magnetic excitation is by far the most efficient for the SRR. From duality, it can be deduced that the electric excitation will be the dominant one for the CSRR. Finally, the frequencies of resonance for different configurations, measured following the method illustrated in Fig. 7, are shown in Table I [32]. The theoretical values shown in this table were obtained from the proposed circuit models (see Figs. 1 and 4). As can be seen, a reasonable agreement between theory and experiment was obtained. It is remarkable that the CSRRs always resonate at frequencies slightly higher than those of the SRRs. This effect is sharper for the higher dielectric constants. III. LUMPED-ELEMENT CIRCUIT MODELS FOR SRRs AND CSRRs COUPLED TRANSMISSION LINES Let us now focus on finding the equivalent-circuit models corresponding to transmission-line structures periodically loaded with SRRs or CSRRs. These models should describe the host transmission line, resonators (SRR or CSRRs), and their

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TABLE I MEASURED AND THEORETICAL VALUES FOR THE FREQUENCY OF RESONANCE. THE RESONATORS ARE PRINTED ON A SUBSTRATE WITH THICKNESS t = 0:49 mm AND RELATIVE PERMITTIVITY " = 2:43. THE PARAMETERS OF r = 1:7 mm, c = d = 0:2 mm; RINGS, NAMED JUST AS IN FIG. 1, ARE r = 3:55 mm, c = d = 0:3 mm

Fig. 7. Frequency response obtained in a rectangular waveguide loaded with SRRs and NB SRRs, as well as its dual counterparts (CSRR and C-NB SRR). The method of excitation is sketched in the inset of this figure. The elements parameters are those of Table I.

Fig. 8. (a) Experimental demonstration of the cross-polarization effects in the SRR. Position 1: electric and magnetic excitation. Position 2: only magnetic excitation. Position 3: only electric excitation. Position 4: no excitation. The behavior of the NB SRR under the same excitations is shown in (b). The absence of electric excitation (3 and 4) shows the absence of cross-polarization effects at resonance.

coupling. It has been indicated that the basic SRR and CSRR topologies (Fig. 1) exhibit cross-polarization effects. This means that both SRRs and CSRRs can be magnetically and/or

electrically excited if the rings are properly oriented. However, it has been verified (see Fig. 8) that magnetic/electric coupling are the dominant coupling mechanisms in SRRs/CSRRs. Therefore, cross-polarizations effects can be ignored in a first-order approximation (this assumption will be strictly valid for NB SRRs and other nonbianisotropic configurations). As discussed in Section II, to properly excite SRRs by means of a time-varying magnetic field, a significant component in the axial direction is required. This makes the CPW structure the preferred host transmission line for SRRs excitation. It was previously shown by the authors [18]–[20] that by etching the SRRs in the back substrate side, underneath the slots, high magnetic coupling is achieved. Alternatively, SRRs can be etched in the upper substrate side, between signal and ground, but this requires very wide slots to accommodate the rings and produces significant mismatch [36]. In contrast, since CSRRs require electric coupling, with a significant component of the electric field perpendicular to the CSRRs surface, microstrip lines with rings etched in the ground plane (below the conductor strip) are more convenient [27], [28]. This do not mean that CPW structures should be ruled out, although electric coupling to CSRRs is softer (in comparison to microstrip) and rejection in the vicinity of the resonant frequency is degraded. Due to the small electrical dimensions of SRRs and CSRRs at resonance, the structures (CPW or microstrip loaded lines) can be described by means of lumped-element equivalent circuits. For the SRR loaded transmission line, the proposed equivalent-circuit model is shown in Fig. 9(a) [18], [19]. and are the per-section inductance and capacitance of the line, while the and SRRs are modeled as a resonant tank (with inductance capacitance ) magnetically coupled to the line through a mutual inductance, . Due to the symmetry of the structure, the magnetic wall concept has been used and the circuit shown in Fig. 9(a) actually corresponds to one-half of the basic cell. The equivalent impedance of the series branch can be simplified to that shown in the circuit of Fig. 9(b) [18], which is formally identical to the series impedance corresponding to a left-handed transmission line [23]–[26] (in that region where the total series impedance is capacitive). From the circuit of Fig. 9(b), the dispersion relation can be easily obtained as follows:

(2)

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Fig. 9. (a) Lumped-element equivalent circuit for the basic cell of the SRR loaded transmission line. (b) Simplified circuit with the series branch replaced by its equivalent impedance.

Fig. 11. Theoretical (solid lines) and simulated (dashed lines) dispersion diagram for the infinite periodic SRR–CPW structures with unit cells identical to those shown in Figs. 10 and 12.

Fig. 10. Layout of the fabricated SRR loaded CPW structure drawn to scale. = 1:9 mm. Adjacent ring pairs are SRR dimensions: c = d = 0:2 mm and r separated 5 mm. The strip and slots widths (W = 5:4 mm and G = 0:3 mm) have been determined to achieve a 50- line. The structure has been fabricated on an Arlon 250-LX-0193-43-11 substrate with thickness h = 0:49 mm and dielectric constant " = 2:43. Actual device length (including access lines) is 35 mm.

with

,

, and . is the propagation constant for Bloch waves and is the period of the structure. For an SRR-loaded CPW structure, line parameters ( and ) can and be determined from a transmission-line calculator, from the aforementioned SRR circuit model, and can be inferred from the fraction of the slot area occupied by the rings according to (3) These circuit elements have been calculated for the structure shown in Fig. 10 (a CPW with pairs of SRRs etched in the back substrate side). The dispersion relation for the corresponding infinite periodic structure is depicted in an – diagram in Fig. 11. A frequency gap around the theoretical frequency of resonance GHz) is observed. The explanation is the of the rings ( following: in a narrow region starting at , the series impedance in Fig. 9(b) becomes negative and signal propagation is inhibited. In contrast, just below the resonance, the series impedance is highly inductive, and makes the second term in (2) positive with and higher than unity. The result is a stopband around a level of rejection that depends on the number of SRR pairs etched in the line. It was previously reported [18] that the aforementioned stopband can be switched to a passband by periodically inserting metallic strips between the central CPW strip and ground planes (Fig. 12). These additional strips make the structure to behave as a microwave plasma with a negative effective permittivity below the plasma frequency [15]. If this frequency is above the

Fig. 12. Layout of the fabricated SRR loaded CPW structure with shunt metal strips, drawn to scale. Dimensions are identical to those of Fig. 10 and the width of the shunt strips is 0.2 mm.

resonant frequency of the SRRs, a narrow passband with backward wave propagation is expected in that region where negative effective permittivity and permeability coexist (i.e., above the resonant frequency of the rings). The strips can be modeled that should be added to the by shunt connected inductances shunt impedance of the circuit of Fig. 9. From this circuit, the dispersion relation can be calculated [18], i.e.,

(4)

and represented in an – diagram (Fig. 12). In practical computations, can be estimated from the simulated frequency response of the strip-loaded CPW (SRRs removed), where the plasma frequency is given by the resonator composed by and . A narrow passband is present above the resonant frequency of the SRRs. The propagation constant for Bloch waves ( ) decreases with frequency, which is indicative of antiparallel phase and group velocities, and is, therefore, in agreement with the theory. It is worth noting that (4) can be deduced from (2) by simply changing the line capacitance in (2) by the effective capacitance associated to the parallel connection of and the strip inductance , which is negative below the aforementioned plasma frequency. Thus, in the long wavelength limit

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Fig. 13. Lumped-element equivalent circuit for the basic cell of the CSRR loaded microstrip line.

(when is very small, i.e., at the upper frequencies), the passband can be associated to the simultaneous presence of a negative series impedance and a negative shunt admittance along the line, i.e., to the simultaneous presence of negative effective and [24], [26]. It is worth noting that the upper frequency limit of the passband for the structure of Fig. 12 coincides with the lower frequency limit of the stopband for the structure in Fig. 10. This fact is consistent with the aforementioned interpretation: in the long wavelength limit, the structure of Fig. 10 presents an effective negative and an effective positive . The dispersion diagrams, computed from electromagnetic simulations (using the Agilent Momentum commercial software) are also shown in Fig. 11. In agreement with the theory, both the theoretical and simulated passbands for the structure of Fig. 12 are located inside the corresponding stopbands for the structure of Fig. 10, and the upper limit of the passband for the structure in Fig. 12 coincides with the lower limit of the upper stopband for the structure in Fig. 10. There is a small shift in frequency between theory and simulations, which is usual in this kind of resonant structures [7]. Let us now analyze the CSRR loaded transmission lines. Since CSRRs are etched in the ground plane, and they are mainly excited by the electric field induced by the line, this coupling can be modeled by series connecting the line capacitance to the CSRRs. According to this, the proposed lumped-element equivalent circuit for the CSRR loaded transmission line is that depicted in Fig. 13. Again, and are the per-section inductance and capaciand model the CSRR, as has been tance of the line, while previously shown. From the circuit of Fig. 13, the dispersion relation can be obtained by simple calculation as follows:

Fig. 14. Theoretical (solid lines) and simulated (dashed lines) dispersion diagram for the infinite CSRR- microstrip structures with unit cells identical to those shown in Figs. 15 and 16. The mismatch between the stopbands and passbands is due to the different dimensions of the CSRRs in Figs. 15 and 16.

Fig. 15. Layout of a 50- microstrip line with CSRRs etched on the back substrate side. Dielectric substrate is Rogers RO3010 (h = 1:27 mm, " = 10:2). CSRR dimensions are c = d = 0:3 mm, r = 3:0 mm and the periodicity is 7 mm. The conductor strip has a width of W = 1:2 mm corresponding to a characteristic impedance of 50 .

(5)

Fig. 16. Layout of a 50- microstrip line with CSRRs etched on the back substrate side and series gaps etched in the conductor strip. Dimensions and substrate are as those reported in Fig. 15, except for the external radius of the CSRRs, which has been set to 2.5 mm and periodicity, which is now 6 mm.

is the angular resonant frewhere quency of the CSRRs. Inspection of (5) points out the presence of a frequency gap in the vicinity of . This is confirmed by the theoretical dispersion relation (see Fig. 14) corresponding to the structure depicted in Fig. 15, a 50- microstrip line with CSRRs etched in the back side metal (ground plane) [25]. In and have been inferred from a theoretical calculations, and have been obtransmission-line calculator, while tained according to the model described in Section II. In the frequency interval delimited by (upper limit) and

(lower limit), the shunt impedance is dominated by the tank inductance, and the structure behaves as a one-dimensional effective medium with negative permittivity. Therefore, propagating modes are precluded in this frequency band. When the discrete nature of the structure is explicitly taken into account, it is realized that the rejection band should extend slightly below due to the extreme values of the shunt admittance in a narrow band below this frequency. In order to obtain a left-handed transmission line based on CSRRs, it is now necessary to introduce an effective negativevalued permeability to the structure. This can be achieved by periodically etching capacitive gaps in the conductor strip at periodic positions (see Fig. 16) [22]–[26]. These gaps provide a

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Fig. 17. Insertion loss measured on the fabricated prototypes (solid lines) and results obtained from the equivalent-circuit models (dashed lines). (a) SRR loaded CPW without shunt strips. (b) SRR loaded CPW with shunt strips. (c) CSRR loaded microstrip lines without series gaps. (d) CSRR loaded microstrip lines with series gaps.

negative effective permeability up to a frequency that can be tailored by properly designing the gap dimensions. If this frequency is set above , a narrow left-handed transmission band is expected. The dispersion relation of the CSRR transmission line with series gaps is deduced from the equivalent circuit as

(6)

is the gap capacitance. Again, the – representawhere tion is indicative of left-handed wave propagation in the allowed can be inferred from band (see Fig. 14). The gap capacitance the cutoff frequency of the structure without CSRRs since this cutoff frequency is given by the frequency of resonance of the resonator formed by the line inductance and the gap capac. The simulated dispersion diagrams for the infinite itance periodic structures with unit cells identical to those of Figs. 15 and 16 are also shown in Fig. 14. The mismatch between the stopbands and passbands is due to the different periodicity and dimensions of the CSRRs in both structures. It is worth noting that all the parameters of the proposed equivalent circuits, used in our theoretical computations, have been either inferred from the analytical SRR or CSRR models ( , , , and ) or estimated from independent physical arguments ( , , , and ). Thus, the reported models are

self-consistent and do not rely on any kind of parameter adjustment external to the model. IV. COMPARISON WITH EXPERIMENTAL DATA AND DISCUSSION The measured frequency responses for the previous finite size structures (Figs. 10, 12, 15, and 16, respectively) have been measured and the results are shown in Fig. 17 (the Agilent 8722ES vector network analyzer has been used for the measurements). The SRR loaded CPW structures were manufactured on an Arlon 250-LX-0193-43-11 thin dielectric substrate in order to obtain high inductive coupling between line and rings. For the CSRR loaded microstrip lines, a high-permittivity Rogers RO3010 dielectric substrate was used in order to enhance capacitive couplings. The comparison between theoretical and experimental results (see Fig. 17) shows that the proposed circuit models predict the stopbands/passbands with a reasonable accuracy. The bandwidth seems to be better predicted for the microstrip CSRR devices than for the CPW SRR ones. However, the location of the stopbands/passbands is better evaluated in the SRR loaded CPWs than in the CSRR loaded microstrip lines. These results are consistent with the discrepancies and correspondences between theory and simulations reported in Section III. More theoretical and experimental research is needed to explain these discrepancies. However, the agreement between theory and experiments shown in Fig. 17

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is actually noticeable. If we mainly take into account that the effects of the coupling between adjacent SRRs or CSRRs, as well as the eventual modification of the SRR/CSRR behavior by the proximity of the line, are not taken into account in the models. This fact seems to confirm that the SRRs and CSRRs behave as almost closed structures, only tightly coupled to the external lines. Finally, in order to evaluate the effect of cross-polarization [31] (not taken into account in the circuit model for the coupling between SRRs/CSRRs and lines), we have simulated the different measured structures, but with the SRRs/CSRRs rotated by 90 . The obtained results (not shown) do not substantially differ from the results shown here. This fact seems to demonstrate that cross-polarization effects can be actually neglected in the analyzed structures, although they are crucial for other devices such as frequency-selective surfaces [28].

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with a suitable trial function for the electrostatic potential . For this purpose, we first express the spectral-domain charge density in terms of a suitable Green’s function [35]

or

where

is the trial function, which is chosen as if if if

V. CONCLUSIONS An analytical procedure for the study of a family of planar structures with negative effective parameters, including left-handed behavior, has been presented in a unified way. The analyzed structures are based on the coupling of SRRs and CSRRs to conventional planar lines. They are fully planar, i.e., they neither incorporate vias, nor other nonplanar inserts, and can be implemented in both CPW and microstrip technology. They can also incorporate modifications of the basic SRR/CSRR geometry. This research has been specifically devoted to obtain analytical tools for the ab initio analysis of these structures. To this end, we have first studied the physics of the isolated SRR and CSRR and we have inferred their equivalent-circuit models. The frequencies of resonance obtained from these circuit models have been compared to those obtained experimentally, and a satisfactory agreement has been found. The coupling between planar transmission lines and SRRs/CSRRs has been modeled by means of a mutual inductance and a shunt capacitance, respectively. From the resulting equivalent circuits, the behavior of periodic and finite structures has been inferred. Four structures were considered: two CPW structures coupled to SRRs (with and without shunt metal strips) and two microstrip lines loaded with CSRRs (with and without series gaps). The qualitative behavior of these structures was shown to be in agreement with the previously reported theory of effective media with negative parameters. In addition, the measured frequency responses of these structures was in reasonable quantitative agreement with the theoretical predictions, thus showing the validity of the lumped-element circuit models. This agreement is indicative of the usefulness of the reported circuit models as practical design tools. APPENDIX VARIATIONAL CALCULATION OF THE CAPACITIVE FOR THE CSRR The capacitance pression

can be obtained from the variational ex-

where and and function

are the geometrical parameters shown in Fig. 5 is defined as

with and being the th-order Struve and Bessel functions. These expressions directly follow (1), which can be easily computed in few steps by using standard integration routines. ACKNOWLEDGMENT The authors extend their thanks to Conatel s.l. and Omicron Circuits. The fabricated prototypes presented in this paper are patent pending. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 10, pp. 509–514, 1968. [2] J. B. Pendry, “Negative refraction makes perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966–3969, 2000. [3] N. García and M. Nieto-Vesperinas, “Left handed materials do not make perfect lens,” Phys. Rev. Lett., vol. 88, pp. 207 403(1)–07 403(4), 2002. [4] G. Gómez–Santos, “Universal features of the time evolution of evanescent moded in a left handed perfect lens,” Phys. Rev. Lett., vol. 90, pp. 077 401(1)–077 401(4), 2003. [5] D. R. Smith, D. Schurig, M. Rosenbluth, S. Shultz, S. Annanta–Ramakrishna, and J. B. Pendry, “Limitations on subdiffraction imaging with a negative refractive index slab,” Appl. Phys. Lett., vol. 82, pp. 1506–1508, 2003. [6] R. Marqués and J. Baena, “Effect of losses and dispersion on the focusing properties of left handed media,” Microwave Opt. Technol. Lett., vol. 41, pp. 290–294, 2004. [7] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, 2000. [8] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional isotropic left handed metamaterial,” Appl. Phys. Lett., vol. 78, pp. 489–491, 2001. [9] R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left handed media simulation and transmission of EM waves in sub-wavelength SRR-loaded metallic waveguides,” Phys. Rev. Lett., vol. 89, pp. 183 901(1)–183 901(4), 2002. , “A new 2D isotropic left handed metamaterial design: Theory and [10] experiment,” Microwave Opt. Technol. Lett., vol. 35, pp. 405–408, 2002.

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[11] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [12] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, 2001. [13] C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett., vol. 90, pp. 107 401(1)–07 401(4), 2003. [14] A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left handed material that obeys Snell’s law,” Phys. Rev. Lett., vol. 90, pp. 137 401(1)–137 401(4), 2003. [15] A. Grbic and G. V. Eleftheriades, “Experimental verification of backward wave radiation from a negative refractive index metamaterial,” J. Appl. Phys., vol. 92, pp. 5930–5935, 2002. [16] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [17] M. Schβüler, A. Fleckenstein, J. Freese, and R. Jakoby, “Lef-handed metamaterials based on split ring resonators for microstrip applications,” in Proc. 33rd Eur. Microwave Conf., Munich, 2003, pp. 1119–1122. [18] F. Martín, F. Falcone, J. Bonache, R. Marqués, and M. Sorolla, “A new split ring resonator based left handed coplanar waveguide,” Appl. Phys. Lett., vol. 83, pp. 4652–4654, 2003. [19] F. Martín, F. Falcone, J. Bonache, T. Lopetegi, R. Marqués, and M. Sorolla, “Miniaturized coplanar waveguide stopband filters based on multiple tuned split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 12, pp. 511–513, Dec. 2003. [20] F. Falcone, F. Martín, J. Bonache, R. Marqués, T. Lopetegi, and M. Sorolla, “Left handed coplanar waveguide band pass filters based on bi-layer split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, pp. 10–12, Jan. 2004. [21] J. García–García, F. Martín, F. Falcone, J. Bonache, I. Gil, T. Lopetegi, M. A. G. Laso, M. Sorolla, and R. Marqués, “Spurious passband suppression in microstrip coupled line band pass filters by means of split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 9, pp. 416–418, Sep. 2004. [22] I. H. Lin, C. Caloz, and T. Itoh, “Transmission characteristics of left handed non uniform transmission lines,” in Proc. Asia–Pacific Microwave Conf., vol. 3, 19–22, 2002, pp. 1501–1504. [23] C. Caloz, H. Okabe, H. Iwai, and T. Itoh, “Transmission line approach of left-handed metamaterials,” in Proc. USNC/URSI Nat. Radio Sci. Meeting, San Antonio, TX, 2002, p. 39. [24] A. A. Oliner, “A periodic-structure negative-refractive index medium without resonant elements,” in Proc. USNC/URSI Nat. Radio Sci. Meeting, San Antonio, TX, 2002, p. 41. [25] O. F. Siddiqui, M. Mojahedi, and G. V. Eleftheriades, “Periodically loaded transmission line with effective negative refractive index and negative group velocity,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2619–2625, Oct. 2003. [26] G. V. Eleftheriades, O. Siddiqui, and A. Iyer, “Transmission line models for negative refractive index media and associated implementations without excess resonators,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 2, pp. 53–55, Feb. 2003. [27] F. Falcone, T. Lopetegi, J. D. Baena, R. Marqués, F. Martín, and M. Sorolla, “Effective negative-" stopband microstrip lines based on complementary split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 280–282, Jun. 2004. [28] F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marqués, F. Martín, and M. Sorolla, “Babinet principle applied to metasurface and metamaterial design,” Phys. Rev. Lett., vol. 93, pp. 197 401(1)–197 401(4), 2004. [29] R. Marqués, J. D. Baena, F. Martín, J. Bonache, F. J. Falcone, T. Lopetegi, M. Beruete, and M. Sorolla, “Left-handed metamaterial based on dual split ring resonators in microstrip technology,” in Proc. Int. URSI Electromagnetic Theory Symp., Pisa, Italy, May 23–27, 2004, pp. 1188–1190. [30] R. Marqués, F. Mesa, J. Martel, and F. Medina, “Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—Theory and experiment,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2572–2581, Oct. 2003. [31] R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left handed metamaterials,” Phys. Rev. B, Condens. Matter, vol. 65, pp. 144 441(1)–144 441(6), 2002.

[32] J. D. Baena, J. Bonache, F. Martín, R. Marqués, F. Falcone, T. Lopetegi, M. Beruete, M. A. G. Laso, J. García–García, F. Medina, and M. Sorolla, “Modified and complementary split ring resonators for metasurface and metamaterial design,” in Proc. 10th Bianisotropics Conf., Ghent, Belgium, 2004, pp. 168–171. [33] R. Marqués, J. D. Baena, J. Martel, F. Medina, F. Falcone, M. Sorolla, and F. Martín, “Novel small resonant electromagnetic particles for metamaterial and filter design,” in Proc. Electromagnetics in Advanced Applications Int. Conf., Turin, Italy, Sep. 2003, pp. 439–442. [34] J. D. Baena, R. Marqués, F. Medina, and J. Martel, “Artificial magnetic metamaterial design by using spiral resonators,” Phys. Rev. B, Condens. Matter, vol. 69, pp. 014 402(1)–014 402(5), 2004. [35] I. Bahl and P. Bhartia, Microwave Solid State Circuit Design. Toronto, ON, Canada: Wiley, 1988. [36] F. Falcone, F. Martin, J. Bonache, R. Marqués, and M. Sorolla, “Coplanar waveguide structures loaded with split ring resonators,” Microwave Opt. Technol. Lett., vol. 40, pp. 3–6, 2004. [37] Y. Chang and L.-T. Chang, “Simple method for the variational analysis of a generalized N -dielectric-layer transmission line,” Electron. Lett., vol. 6, pp. 49–50, Feb. 1970.

Juan Domingo Baena was born in El Puerto de Santa María, Cádiz, Spain, in August 1976. He received the Licenciado degree in physics from the Universidad de Sevilla, Seville, Spain, in 2001, and is currently working toward the Ph.D. degree at the Universidad de Sevilla. In 1999, he was a Software Programmer with Endesa (providing company of electricity in Spain). In September 2002, he joined the Electronic and Electromagnetism Department, Universidad de Sevilla. His current research interests include analysis, design, and measurement of artificial media with exotic electromagnetic properties (metamaterials). Mr. Baena was the recipient of a Spanish Ministry of Science and Technology Scholarship.

Jordi Bonache was born in Barcelona, Spain, in 1976. He received the Physics and Electronics Engineering degrees from the Universitat Autònoma de Barcelona, Barcelona, Spain, in 1999 and 2001, respectively, and is currently working toward the Ph.D. degree at the Universitat Autònoma de Barcelona. In 2000, he joined the High Energy Physics Institute of Barcelona (IFAE), where he was involved in the design and implementation of the control and monitoring system of the MAGIC telescope. In 2001, he joined the Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, where he is currently an Assistant Professor. His research interests include active and passive microwave devices and metamaterials.

Ferran Martín was born in Barakaldo (Vizcaya), Spain, in 1965. He received the B.S. degree in physics and Ph.D. degree from the Universitat Autònoma de Barcelona, Barcelona, Spain, in 1988 and 1992, respectively. Since 1994, he has been an Associate Professor of electronics with the Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona. He has recently been involved in different research activities including modeling and simulation of electron devices for high-frequency applications, millimeterwave and terahertz generation systems, and the application of electromagnetic bandgaps to microwave and millimeter-wave circuits. He is also currently very active in the field of metamaterials and their application to the miniaturization and optimization of microwave circuits and antennas.

BAENA et al.: EQUIVALENT-CIRCUIT MODELS FOR SRRs AND CSRRs COUPLED TO PLANAR TRANSMISSION LINES

Ricardo Marqués Sillero (M’95) was born in San Fernando (Cádiz), Spain, in 1954. He received the Ph.D. degree from the Universidad de Sevilla, Seville, Spain, in 1987. He is currently an Associate Professor with the Universidad de Sevilla. Since 1984, he has been with the Microwave Group, Department of Electronics and Electromagnetism, Universidad de Sevilla. His main fields of interest include computer-aided design (CAD) for microwave integrated circuit (MIC) devices, wave propagation in ferrites, and other complex and anisotropic media and field theory. His recent research interest is focused on the analysis and design of artificial media with exotic electromagnetic properties (metametarials), including negative refraction, sub-wavelength focusing, and their applications in microwave technology. Francisco Falcone was born in Caracas, Venezuela, in 1974. He received the M.Sc. degree in telecommunication engineering from the Public University of Navarre, Navarre, Spain, in 1999, and is currently working toward the Ph.D. degree in telecommunication engineering from the Public University of Navarre. From 1999 to 2000, he was with the Microwave Implementation Department, Siemens-Italtel, where he was involved with the layout of the Amena mobile operator. Since 2000, he has been a Radio Network Engineer with Telefónica Móviles España. Since the beginning of 2003, he has also been an Associate Lecturer with the Electrical and Electronic Engineering Department, Public University of Navarre. His main research interests include electromagnetic-bandgap devices, periodic structures, and metamaterials. Txema Lopetegi (S’99–M’03) was born in Pamplona, Navarre, Spain, in 1973. He received the M.Sc. and Ph.D. degrees in telecommunication engineering from the Public University of Navarre, Navarre, Spain, in 1997 and 2002, respectively. Since 1997, he has been with the Electrical and Electronic Engineering Department, Public University of Navarre, as an Academic Associate from 1997 to 1999, and as an Assistant Professor since 2000. During 2002 and 2003, he was a Post-Doctoral Researcher with the Payload Systems Division, European Space Research and Technology Center (ESTEC), European Space Agency (ESA), Noordwijk, The Netherlands. His current research interests include metamaterials and their applications in microwave and millimeter-wave technologies (electromagnetic-bandgap structures, left-handed media, and SRRs), as well as coupled-mode theory and synthesis techniques using inverse scattering. Dr. Lopetegi was the recipient of a 1999 and 2000 grant from the Spanish Ministry of Education to support the research of his doctoral thesis. Miguel A. G. Laso (S’99–M’03) was born in Pamplona, Spain, in 1973. He received the M.Sc. and Ph.D. degrees in telecommunication engineering from the Public University of Navarre, Navarre, Spain, in 1997 and 2002, respectively. Since 2001, he has been an Assistant Lecturer with the Electrical and Electronic Engineering Department, Public University of Navarre. He has been involved in several projects funded by the Spanish Government and the European Union. He was a Post-Doctoral Researcher supported by the Spanish Ministry of Science and Technology with the Payload System Division, European Space Research and Technology Center (ESTEC), European Space Agency (ESA), Noordwijk, The Netherlands, where he was involved with satellite applications of electromagnetic crystals in the microwave range. His current interests include electromagnetic crystals, metamaterials, and periodic structures in planar microwave and millimeter-wave technologies and in the optical wavelength range. Dr. Laso was the recipient of a grant from the Spanish Ministry of Education to support the research of his doctoral thesis from 1998 to 2002.

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Joan Garcia–Garcia was born in Barcelona, Spain in 1971. He received the Physics degree and Ph.D. degree in electrical engineering from the Universitat Autònoma de Barcelona, Barcelona, Spain, in 1994 and 2001, respectively. He then became a Post-Doctoral Research Fellow with the Institute of Microwaves and Photonics, The University of Leeds, Leeds, U.K., working under the INTERACT European project. In 2002, he was a Post-Doctoral Research Fellow with the Universitat Autònoma de Barcelona, working under the Ramon y Cajal project of the Spanish Government. In November 2003, he become an Associate Professor of electronics with the Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona.

Ignacio Gil was born in Barcelona, Spain, in 1978. He received the Physics and Electronics Engineering degrees from the Universitat Autònoma de Barcelona, Barcelona, Spain, in 2000 and 2003, respectively, and is currently working toward the Ph.D. degree at the Universitat Autònoma de Barcelona. He is also an Assistant Professor with the Universitat Autònoma de Barcelona. His research interests include active and passive microwave devices and metamaterials.

Maria Flores Portillo was born in Pamplona, Spain, in 1980. She received the Telecommunication Engineering degree from the Universidad Pública de Navarra, Navarra, Spain, in 2004. Her research interests include passive microwave devices and metamaterials.

Mario Sorolla (S’82–M’83–SM’01) was born in Vinaròs, Spain, in 1958. He received the M.Sc. degree from the Polytechnic University of Catalonia, Catalonia, Spain, in 1984, and the Ph.D. degree from the Polytechnic University of Madrid, Madrid, Spain, in 1991, both in telecommunication engineering. From 1986 to 1990, he designed very high-power millimeter waveguides for plasma heating for the Euratom-Ciemat Spanish Nuclear Fusion Experiment. From 1987 to 1988, he was an Invited Scientist with the Institute of Plasma Research, Stuttgart University, Stuttgart, Germany. He has been involved with MICs and monolithic microwave integrated circuits for satellite communications with Tagra, Les Franqueses del Vallés, Spain, and Mier Communications, Barcelona, Spain. From 1984 to 1986, he was an Assistant Lecturer with the Polytechnic University of Catalonia, Vilanova i la Geltrú, Spain. From 1991 to 1993, he was an Assistant Lecturer with the Ramon Llull University, Barcelona, Spain. From 1993 to 2002, he was an Assistant Professor with the Public University of Navarre, Navarre, Spain, where he is currently a Full Professor with the Electrical and Electronic Engineering Department. His research interest include high-power millimeter waveguide components and antennas, coupled-wave theory, quasi-optical systems in the millimeter and terahertz range, and applications of metamaterials and enhanced transmission phenomena to microwave circuits and antennas.

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Efficient Modeling of Novel Uniplanar Left-Handed Metamaterials Yunchuan Guo, Student Member, IEEE, George Goussetis, Member, IEEE, Alexandros P. Feresidis, Member, IEEE, and John C. Vardaxoglou, Member, IEEE

Abstract—This paper presents an efficient modeling technique for the derivation of the dispersion characteristics of novel uniplanar metallodielectric periodic structures. The analysis is based on the method of moments and an interpolation scheme, which significantly accelerates the computations. Triangular basis functions are used that allow for modeling of arbitrary shaped metallic elements. Based on this method, novel uniplanar left-handed (LH) metamaterials are proposed. Variations of the split rectangularloop element printed on grounded dielectric substrate are demonstrated to possess LH propagation properties. Full-wave dispersion curves are presented. Based on the dual transmission-line concept, we study the distribution of the modal fields and the variation of series capacitance and shunt inductance for all the proposed elements. A verification of the left-handedness is presented by means of full-wave simulation of finite uniplanar arrays using commercial software (HFSS). The cell dimensions are a small fraction of 24) so that the structures can the wavelength (approximately be considered as a homogeneous effective medium. The structures are simple, readily scalable to higher frequencies, and compatible with low-cost fabrication techniques. Index Terms—Left-handed (LH) materials, metamaterials, method of moments (MoM), Rao–Wilton–Glisson (RWG) basis function.

I. INTRODUCTION

T

HERE HAS been increased interest over the past few years on composite media (metamaterials) that support left-handed (LH) propagation. LH describes the fact that form an LH triplet, instead of a the vectors , , and right-handed (RH) triplet, as is the case in conventional RH media [1]. Thus, in LH media, the Poynting vector (that points at the direction of energy propagation and group velocity) is antiparallel to the wave vector (that points at the direction of phase velocity). LH materials possess an effective negative refractive index (NRI), , which is related to the phase velocity according to

where is the phase velocity, is the frequency, and is the propagation constant. For NRI metamaterials, the dispersion relation of the first-order mode has a negative gradient. This indiManuscript received June 16, 2004; revised October 29, 2004. This work was supported by the U.K. Engineering and Physical Sciences Research Council under Research Grant EP/C510607/1. The authors are with the Wireless Communications Research Group, Department of Electronic and Electrical Engineering, Loughborough University, Loughborough LE11 3TU, U.K. (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845204

) and group velocity ( ) cates that the phase velocity ( are antiparallel. Recently, artificial LH media have been synthesized and their unusual properties such as negative refraction and subwavelength focusing have been experimentally demonstrated [2]–[4]. A transmission-line approach for the analysis and design of planar LH media has been proposed [5]–[7]. According to this approach, LH propagation occurs in a dual transmission line (TL) in one and two dimensions. Instead of the shunt capacitance and series inductance of the standard circuit representation of RH TLs, an LH TL can be produced introducing series capacitance and shunt inductance. Based on this analysis, many properties of NRI media have been predicted and the performance of finite structures has been demonstrated. Distributed LH structures, which are compatible with photolithographic techniques, have also been presented [4], [6]–[8]. The distributed planar NRI structures proposed thus far require either grounding vias [4], [6], [7] or planar elements embedded vertically in a planar grounded dielectric substrate [8]. While equivalent circuits have been proposed for the dispersion characterization of LH metamaterials, accurate analysis and derivation of modal field distributions require full-wave modeling. This can be computationally costly, particularly for complex geometries in the unit cell. In this paper, we propose an efficient modeling technique for the analysis of novel uniplanar LH metamaterials with arbitrary element geometries. The analysis is based on the method of moments (MoM) and an interpolation scheme, which significantly accelerates the computations. The interpolation technique has been initially proposed in [9] and later employed in order to analyze plane-wave scattering problems [10] and general planar microstrip structures [11]. Recently it has been employed for deriving the dispersion characteristics of free-standing periodic metallodielectric fractal arrays [12]. Here, we employ this technique for efficient dispersion characterization and derivation of modal field distribution of planar LH metamaterials on grounded dielectric substrates without vias (also referred to as uniplanar metamaterials throughout this paper). A thorough study of the accuracy of the technique when applied to the derivation of dispersion properties is presented, and the method is validated by comparison to the commercial software High Frequency Structure Simulator (HFSS).1 Based on this method, novel uniplanar LH metamaterials are proposed. Variations of the split rectangular-loop element printed on grounded dielectric substrate are demonstrated to exhibit LH 1High Frequency Structure Simulator (HFSS), ver. 9.0, Ansoft Corporation, Palo Alto, CA, 2003.

0018-9480/$20.00 © 2005 IEEE

GUO et al.: EFFICIENT MODELING OF NOVEL UNIPLANAR LH METAMATERIALS

propagation properties. In order to explain their performance, the modal field distribution of the unit cell is studied in conjunction with the emerging dispersion relation. In conducting this study, emphasis is given to the distributions of the - and -fields and how they satisfy the requirement of LH media for shunt inductance and series capacitance. The proposed structures are simple to fabricate and are scalable to higher frequencies [13]. A verification of the left-handedness of the proposed structures is presented by means of full-wave simulation of finite uniplanar arrays using commercial software (HFSS). II. THEORY To model the fields and derive the eigenvalue equations in periodic structures with complex elements, the mixed-potential integral equation (MPIE) is formed using the Rao–Wilton– Glisson (RWG) triangular patch subdomain basis functions [14]. The MoM is employed to solve the MPIE in conjunction with an interpolation technique, which accelerates the computations of the dispersion properties of the proposed structures. A brief outline of this method is produced here. can be expressed as a function of the The electric field surface current by (1) where

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is the th testing function and is the th basis where function. In order to derive the generic solution for allowed modes, nontrivial solutions of the homogeneous equation (4) are sought for propagation constant values along the contour of the Brillouin zone. These correspond to the roots of the determinant of the -matrix in (4). Therefore, the pairs of frequency and propagation constants values (real) for which the determinant becomes zero correspond to propagating modes (surface waves). Once a pair of and that satisfies (4) has been found, we can solve backward to (3) and (1) in order to derive the current and field distribution of each mode. Typically, the dispersion characterization of a periodic array and its determiinvolves a large number of evaluations of nant, particularly when the zeros are sharp. Furthermore, complex array elements require a large number of basis functions and the procedure for a single calculation of the -matrix can be very time consuming. This makes the derivation of the dispersion diagram a cumbersome procedure due to the excessive computational time required. However, unlike the resonant behaviors of the currents and fields, the -matrix elements usually have a regular and smooth variation versus frequency. That enables us to calculate the -matrix at only a few frequency points and obtain the -matrix at other frequency points by using interpolation. We initially chose the functions

is the magnetic vector potential

(6) (2)

(7)

is the Green’s function and is the integral domain of the metal surface. A closed-form periodic Green function can be used for an infinite array on a grounded dielectric substrate [15]. According to the MoM, the unknown surface current is expanded as a summation of products of known basis functions and unknown coefficients as follows:

as the interpolation functions for the real and imaginary parts of the -matrix elements for our structures [10]. For each interpolation interval, the -matrix needs to be calculated at only three frequency points in order to estimate , , and . The interpolation technique is more accurate for small unit cells with respect to wavelength since the phase variations between the basis and test functions are small [9]. A thorough study of the effect of the interpolation step on the accuracy of the derived dispersion properties is presented in Section III for evaluation and validation of the method.

(3) For treating elements with complex or arbitrary shapes, the triangular patch subdomain basis functions, first introduced by RWG, are found to be versatile. on Applying the boundary condition and the test procedure, the matrix equation can be derived as follows:

(4)

The elements are defined as

(5)

III. UNIPLANAR DESIGNS Here, we evaluate and apply the technique described in Section II in order to model the dispersion characteristics and modal fields of uniplanar LH metamaterials. Our initial investigation has shown that moving from the square split-loop element to the rectangular split loop, the first resonant mode changes from RH to LH. We, therefore, focus our study on a rectangular unit cell with an edge ratio of 2 : 1. The investigation presented here includes a closed rectangular loop, a split rectangular loop, and two variations of the split loop that increase magnetic and electric effects according to the dual TL concept for LH media. The arrays are placed on a grounded dielectric substrate. The dispersion properties of all four uniplanar arrays are presented. The dispersion results presented have been verified with HFSS throughout this section. Finally, we used HFSS in order to simulate finite LH arrays and, thus, verify the left-handedness of the proposed structures.

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Fig. 3. Error (%) in calculated roots of the determinant of the Z -matrix due to the application of the interpolation technique with different interpolation steps and interpolation functions.

Fig. 1. Unit cells of the: (a) rectangular loop, (b) split loop, (c) loaded loop, and (d) spiral loop. (e) Reciprocal lattice and irreducible Brillouin zone.

Fig. 2. Variation of Z versus frequency (real and imaginary part) for the spiral-loaded element [see Fig. 1(d)].

A. Calculation of Dispersion Characteristics The unit cells of the arrays discussed hereafter are shown in Fig. 1. Fig. 1 shows the unit element of a split rectangular loop [see Fig. 1(b)], a capacitive-loaded split rectangular loop [see Fig. 1(c)], and a spiral-loaded rectangular loop arrays [see Fig. 1(d)]. For completeness, the closed rectangular loop [see Fig. 1(a)] is also discussed. The unit cell is kept constant throughout (3 mm 6 mm). All metallic line widths are 0.2 mm and the gaps in the split-loop variations are mm and mm in order to be compatible with conventional photolithographic techniques. The element dimensions are mm and mm. The dielectric substrate has a dielectric constant of 2.2 and a thickness of 1.13 mm. In order to validate the application of the interpolation technique, the elements of the -matrix have been calculated for a range of frequency points and element geometries. As an exelement are shown in Fig. 2 ample, the values of a random for the array with unit element depicted in Fig. 1(d). It is evident that the -matrix element values, both real and imaginary, vary smoothly and regularly with frequency. This allows for the implementation of the interpolation technique described in Section II. Our simulations showed that large interpolation steps have not produced successful results for the dispersion characteriza-

tion of metallodielectric arrays. We, therefore, performed a detailed study of the effect of the interpolation step on the accuracy of the dispersion results. The range of frequencies was broken down to smaller sections where interpolation was applied. An error value defined as the difference between the spectral posiwith and without interpolation has tions of a minimum of been calculated for different interpolation steps. Initially, we have used the quadratic (6) and inverse (7) functions for the real and imaginary parts of the -matrix elements, respectively (re). Subsequently, we used (7) as the interpolation ferred to as function for both real and imaginary parts (referred to as Inv). Fig. 3 shows the percentage error value for different frequencies and interpolation steps for the element depicted in Fig. 1(d). It is evident that the inverse function approximates both real and imaginary parts better. For an interpolation step equal to 1 GHz, the error becomes substantially small, well below 1%, which corresponds to a relatively good accuracy of the final result. Based on this study, an interpolation step of 1 GHz and only the inverse interpolation function has been used to produce the dispersion diagrams. For higher values of dielectric constant, corresponding to typical commercially available printed circuit board (PCB) substrates, similar accuracy can be achieved with the same interpolation step. In order to demonstrate the acceleration achieved by the proposed technique for bandgap characterization, Fig. 4 shows a typical variation of the determinant of the impedance matrix with frequency. The zero of the determinant, which is the solution that we want to identify, corresponds to the sharp minimum at 4.31 GHz. As shown in the inset graph, in order to have 1% accuracy in determining this value, the step for producing a graph such as the one shown in Fig. 4 would require approximately 30 frequency points. Employing the interpolation technique, we can produce the same number of points with the computational cost of only three frequency points. The time required for each matrix generation using interpolation together with the calculation of the determinant is negligible compared to the time required to generate the matrix without interpolation. Hence, the interpolation technique accelerates the computations by approximately ten times.

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Fig. 6. Refractive index of the loaded and spiral-loaded split-loop structures calculated from the dispersion diagram. Fig. 4. Determinant of impedance matrix with frequency. A close-up of the root (sharp minimum) is shown in the inset.

Fig. 5. Dispersion relation for the grounded periodic arrays with elements shown in Fig. 1. Results obtained with the theoretical technique presented here (Int) and also with HFSS.

Following the evaluation of the interpolation scheme, the dispersion characteristics of the elements depicted in Fig. 1 can now be calculated by locating the zeros of the impedance matrix in the – space, as discussed in Section II. By varying and along the boundary of the irreducible Brillouin zone, a number of frequency sweeps are carried out. All the corresponding determinants of are calculated and plotted versus and . From the determifrequency for different values of nant plots, zeros are identified as sharp minima. The dispersion relation of the first resonant mode of these arrays is shown in Fig. 5, where the light line is also shown. From the dispersion relation, we can readily calculate the equivalent refractive index according to (8) where is the speed of light and is the phase velocity of the mode. Fig. 6 shows the refractive index for the loaded split loop and spiral split loop, as calculated from the dispersion relation of Fig. 5. B. Dispersion Properties and Field Distributions Following Fig. 5, one can observe that, as we move from the rectangular loop to the spiral-loaded loop, the frequency of the first resonant mode drops significantly. It is worth noting

that there is an essential difference between the closed rectangular loop [see Fig. 1(a)] and the variations of the split loop [see Fig. 1(b)–(d)]. In the first case, the current flows co-directionally along the two longer parallel sides, with zeros at the centers of the short sides. However, in all the variations of the split loop, the current flows co-directionally along the metallization of the elements from end to end, resulting in longer electrical length. This, in turn, results in lowering the frequency of the first resonant mode. Among the elements of Fig. 1(b)–(d), we can attribute the reduction of the resonant frequency to the increase of the resonant length of the element. Alternatively, in an equivalent-circuit model of the structure, the capacitance between end-loaded elements and the inductance (in the case of the spiral) increase and, hence, the resonant frequency decreases. The spiral element array has a unit cell of approximately in the -direction, where is the center wavelength of the first resonant mode. This allows an effective medium description of the structure. The gradient of the – curve (Fig. 5) changes from positive in the case of Fig. 1(a) to negative in all the remaining cases of Fig. 1 for . As discussed in Section I, this indicates an LH medium. The first resonant mode corresponds to electric-field polarization parallel to the larger dimension of the element and aligned with the gap in the split-loop variations. Therefore, in -direction, the first resonant mode is predominantly TE, the -direction, the first mode is predominantly TM. In and in the and sections of the Brillouin contour, the modes the are hybrid. The bandwidth of the LH mode can be determined from the dispersion diagram. It is worth noting, however, that as we move away from the light line toward an increasing propagation constant and decreasing frequency, the effective refractive index increases significantly. This makes it harder to couple energy in the negative mode from free space (or an RH medium with a small value of ). Following the dual TL concept for LH media, it is instructive to study the distribution of the electric and magnetic field in the unit cell, taking into account the requirement for shunt inductance and series capacitance. Shunt inductance is related to the magnetic field that would be produced by an ideal shunt inductor, i.e., the -component of the magnetic field ( ). Similarly, a series capacitor would produce an electric field along the

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Fig. 8. FEM simulation of the transmission magnitude for arrays of loaded split-loop elements with four, five, and six elements.

Fig. 7. Normalized values of the: (a)–(d) E component and the (e)–(h) H component of the loop, split-loop, loaded split-loop, and spiral split-loop element arrays at the first resonant mode in 0X .

-axis ( -field is parallel to the -axis and, therefore, “voltage is applied” along ). The -component of the electric field ( ) is, therefore, a measure of the equivalent series capacitance. The and for the first resonant mode of the ardistributions of rays studied in this section are shown in Fig. 7. As mentioned earlier, the current in the closed rectangular loop flows in parallel on the two long edges with a zero at the middle of each shorter edge. Therefore, opposite charges accumulate on closely spaced short sides of successive elements, resulting in an equivalent capacitance between the edges of successive elements [see Fig. 7(a)]. The currents on the two long edges create magnetic-field components along the -axis, which corresponds to shunt inductance [see Fig. 7(e)]. However, both series capacitance and shunt inductance are weak and the structure supports an RH mode. The situation is different in the case of the split loop and its variations. Additional capacitance emerges in the gap, where opposite charges gather on either side [see Fig. 7(b)]. Furthermore, the current now circulates along the open loop, resulting in stronger magnetic fields that also appear along the short edges [see Fig. 7(f)]. The structure at the first resonant mode is, therefore, LH for the TE mode propagating in the -direction. However, it is still positive for the TM modes that propagate in . This is attributed to the fact that the polarization of the electric- and magnetic-field components of the TM modes result in a weaker excitation of currents on the loops, as has been

readily observed in corresponding field plots obtained from our simulations. Further enhancement of the series capacitance is obtained for the loaded split loop [see Fig. 1(c)]. In this topology, we have introduced larger reentrant faces where opposite charges accumulate and the capacitance values are increased. This is indeed confirmed by the field distribution of Fig. 7(c). Furthermore, current now also flows along the reentering loads, with results [see Fig. 7(g)]. The structure is more in further increase of miniature and supports LH propagation in both directions. In order to further enhance the LH effect, we extend the capacitive loads to spirals. It is worth noting that the double spiral geometry that is formed resembles that of [16], where it has been initially proposed for small antenna applications. The flow of current on the spirals acts as a planar inductor and, thus, . This is indeed confirmed in increases the magnetic field Fig. 7(h). In [16], it is also noted that the mutual inductance between the two spirals is positive, resulting in an increase of the total inductance. This is advantageous for our application. In addition, stronger electric fields in the gap and, hence, higher series capacitance are also obtained [see Fig. 7(d)]. The structure supports LH propagation throughout the Brillouin zone, and the bandwidth of the LH mode is now wider as compared to the previous geometries. C. Validation of Uniplanar LH Propagation by Simulation of Finite Structure In order to validate LH propagation in the proposed structure, we carried out full-wave finite-element method (FEM) simulations (HFSS) of finite structures. To ease of the computational effort required, the simulations were made for the loaded split loop rather than the spiral split loop. One dimensional arrays with four, five, and six elements have been simulated, respectively. The elements are aligned in a single column along the -direction, where LH propagation occurs for the first TE mode. A perfect electric conductor (PEC) boundary condition has been imposed on either side in order to ensure that TE polarization is maintained. The structure is fed with a waveguide mode of the same polarization, and the transmission coefficients (magnitude and phase) are obtained for the three arrays. Fig. 8 shows the magnitude of the transmission coefficient for the three arrays. The passband is identified as the LH mode and extends between 4.2–5.8 GHz. The lower edge is in good agreement with our simulations (Fig. 5). The upper edge is higher

GUO et al.: EFFICIENT MODELING OF NOVEL UNIPLANAR LH METAMATERIALS

Fig. 9. HFSS simulation of the transmission phase for arrays of loaded split-loop elements with four, five, and six elements.

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of the interpolation step on the accuracy of the results. The dispersion diagrams of the proposed structures have been produced with an interpolation step of 1 GHz, which resulted in good accuracy. The method was validated by comparison with commercial software (HFSS). The refraction index of the structures has been derived. A thorough investigation of the modal field distribution provided with a detailed explanation of the LH properties of the structures based on the dual TL concept. A full-wave-based validation of the left-handedness has been , which allows for an presented. The unit cell is as small as equivalent effective medium model. The structures are simple, compatible with low-cost fabrication techniques, and scalable to higher frequencies. ACKNOWLEDGMENT

than what is presented in Fig. 5 due to the leaky part of the mode, which can carry energy, but is not shown there. The passband shown in Fig. 8 is a very good indication of the existence of the predicted LH mode. In order to validate the LH properties of the mode, Fig. 9 shows the calculated phase of the transmission coefficient for the arrays with four, five, and six elements. In general, the difference of the transmission phase between two TLs of lengths and is [17] (9) and RH media with positive refractive index For ), the difference in the transmission phase is negative. ( However, in the case of LH medium ( ), the phase difference is expected to be positive. The transmission phases in Fig. 9 are plotted with every complete cycle added to the total phase shift so that an actual comparison of the phase is possible. In agreement to the predicted LH nature of the mode, it is observed that, within the bandwidth of the LH mode, the transmission phase of the six-element array is higher than that of the five-element array, which, in turn, is higher than that of the four-element array. In the bandgap between the modes, where a standing wave is formed, these phases are all equal to zero. At higher frequencies, the transmission phases then interchange, corresponding to a positive second-order mode that exists for this array. Note that the transmission phase of the finite structure is in very good agreement to the phase predicted from the refractive index (Fig. 6) using (9). Figs. 8 and 9 are, therefore, a good verification of the LH nature of the first-order mode of the presented arrays. IV. CONCLUSION An efficient modeling technique for producing the dispersion diagrams of uniplanar metamaterial arrays of arbitrary shaped metallic elements on grounded dielectric substrate has been presented. The method was based on interpolation of the values of the -matrix elements for varying frequency. Based on this method, novel uniplanar LH metamaterials have been designed. A study was carried out with regards to the effect

The authors would like to thank two of the reviewers for their most constructive comments. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [2] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [3] G. V. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2711, Dec. 2002. [4] A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1252–1263, Apr. 2004. [5] A. Grbic and G. V. Eleftheriades, “Periodic analysis of a 2-D negative refractive index transmission line structure,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2604–2611, Oct. 2003. [6] C. Caloz and T. Itoh, “Transmission line approach of left-handed (LH) materials and microstrip implementation of an artificial LH transmission line,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1159–1166, May 2004. [7] A. Grbic and G. V. Eleftheriades, “Dispersion analysis of a microstripbased negative refractive index periodic structure,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 4, pp. 155–157, Apr. 2003. [8] C.-Y. Cheng and R. W. Ziolkowski, “Tailoring double-negative metamaterial responses to achieve anomalous propagation effects along microstrip transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2306–2314, Dec. 2003. [9] E. H. Newman, “Generation of wide-band data from the method of moments by interpolating the impedance matrix,” IEEE Trans. Antennas Propag., vol. 36, no. 12, pp. 1820–1824, Dec. 1988. [10] A. S. Barlevy and Y. Rahmat-Samii, “Characterization of electromagnetic band-gaps composed of multiple periodic tripods with interconnecting vias: Concept, analysis and design,” IEEE Trans. Antennas Propag., vol. 49, no. 3, pp. 343–353, Mar. 2001. [11] J. Yeo and R. Mittra, “An algorithm for interpolating the frequency variations of method-of-moments matrices arising in the analysis of planar microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 1018–1025, Mar. 2003. [12] Y. Guo, A. P. Feresidis, G. Goussetis, and J. C. Vardaxoglou, “Efficient modeling of novel fractal loaded electromagnetic band gap arrays,” in Proc. IEE Int. Computation in Electromagnetics Conf., Apr. 2004, pp. 147–148. [13] T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science, pp. 1494–1496, Mar. 2004. [14] S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, no. 5, pp. 409–418, May 1982.

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[15] M. J. Tsai, F. DeFlaviis, O. Fordham, and N. G. Alexopoulos, “Modeling planar arbitrarily shaped microstrip elements in multilayered media,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 3, pp. 330–337, Mar. 1997. [16] R. C. Fenwick, “A new class of electrically small antennas,” IEEE Trans. Antennas Propag., vol. AP-13, no. 5, pp. 379–383, May 1965. [17] O. F. Siddiqui, M. Mojahedi, and G. V. Eleftheriades, “Periodically loaded transmission line with effective negative refractive index and negative group velocity,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2619–2625, Oct. 2003.

Yunchuan Guo (S’03) was born in Sichuan, China, in 1980. He received the B.Eng. degree in electrical engineering from Tianjin University, Tianjin, China, in 2002, and is currently working toward the Ph.D. degree in electronic and electrical engineering at Loughborough University, Loughborough, U.K. His doctoral research concerns efficient computational techniques of metamaterials. His research interests include metamaterials, antennas, and computational electromagnetics.

George Goussetis (M’01) was born in Athens, Greece, in 1976. He received the Electrical and Computer Engineering degree from the National Technical University of Athens, Athens, Greece, in 1998, the B.Sc. degree in physics from University College London, London, U.K., in 2002, and the Ph.D. degree in the area of waveguide filters from the University of Westminster, Westminster, U.K., in 2002. In 1998, he joined the Space Engineering Spa, Roma, Italy, as Trainee RF Engineer. In 1999, he joined the Wireless Communications Research Group, University of Westminster, as a Research Assistant. In 2002, he joined the Wireless Communications Research Group, Loughborough University, Loughborough, U.K., where he is currently a Senior Research Fellow. He has authored or coauthored over 50 peer reviewed journals and conference papers. His research interests include the modeling and design of microwave filters and passive components and electromagnetic bandgap (EBG) and periodic structures.

Alexandros P. Feresidis (M’98) was born in Thessaloniki, Greece, in 1975. He received the B.Sc. degree in physics from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1997, the M.Sc. (Eng.) degree in radio communications and high-frequency engineering from The University of Leeds, Leeds, U.K., in 1998, and the Ph.D. degree in electronic and electrical engineering from Loughborough University, Loughborough, U.K., in 2002. During the first half of 2002, he was a Research Associate with the Wireless Communications Research Group, Department of Electronic and Electrical Engineering, Loughborough University. He is currently a Lecturer in the same department. He has authored or coauthored over 40 refereed journals and conference proceeding papers. His research interests include analysis and design of EBG materials and frequency-selective surfaces (FSSs), metamaterial structures, array antennas, base-station antennas, computational electromagnetics, and microwave circuits.

John C. Vardaxoglou (M’87) received the B.Sc. degree in mathematics (mathematical physics) and Ph.D. degree from the University of Kent at Canterbury, Canterbury, U.K., in 1981 and 1985, respectively. In January 1988, he became a Lecturer in communications with the Department of Electronic and Electrical Engineering, Loughborough University of Technology, Loughborough, U.K. In January 1992, he became a Senior Lecturer. In 1998, he became a Professor of wireless communications. He holds the Chair of Wireless Communications with Loughborough University and is the founder of the Centre for Mobile Communications Research (CMCR). He established the Antennas and Microwaves Research Group, Loughborough University and heads the CMCR. He has been active in the area of electromagnetic modeling and applications of FSSs. His contribution to the CMCR is in the analysis and design of small loaded antennas for mobile telephony. He is the Technical Director of Antrum Ltd. He has served as a consultant to various industries in the U.K. and abroad. He holds three patents. He has authored or coauthored over 120 refereed journals and conference proceeding papers and has written a research monograph on FSSs. His current research interests include wireless communication networks, array antennas, FSSs, radomes, leaky-wave resonant antennas, optical control of microwaves and devices, periodic surfaces and EBG/photonic-bandgap (PBG) materials, and mobile telephone antennas. Dr. Vardaxoglou is a member of the Executive Committee of the Institution of Electrical Engineers (IEE), U.K. and the Professional Network in Antennas and Propagation. He chaired the 1st IEE Antenna Measurements and SAR (AMS’02) Conference and has been on the Organizing Committee of the 2001 and 2003 IEE International Conferences on Antennas and Propagation.

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Propagation Property Analysis of Metamaterial Constructed by Conductive SRRs and Wires Using the MGS-Based Algorithm Hai-Ying Yao, Wei Xu, Senior Member, IEEE, Le-Wei Li, Fellow, IEEE, Qun Wu, Senior Member, IEEE, and Tat-Soon Yeo, Fellow, IEEE

Abstract—An efficient numerical algorithm based on the modified Gram–Schmidt (MGS) procedure is developed in this paper to solve the electrical-field integral equation for characterizing metamaterials. The method of moments (MoM) with rooftop basis functions is implemented in the integral-equation solver. The MGSbased algorithm is implemented into the solver for decomposing the local MoM dense matrix without prior knowledge of the matrix elements. Although each element of the metamaterials is electrically small in size, the number of elements is very large so that the number of unknowns for characterizing the metamaterials is very large. Even so, this algorithm in the MoM solver has demonstrated via examples to be efficient and accurate. Numerical results of unknowns show that the CPU time per iteraon the number tion and the memory requirements are both reduced from ( 2 ) to ( 1 5 ). After implementing this algorithm in the solver, propagation characteristics are finally presented when electromagnetic waves pass through a metamaterial prism that is synthesized using square split-ring resonators and wires in free space. Index Terms—Electrical-field integral equation (EFIE), lefthanded material, metamaterial, modified Gram–Schmidt (MGS) procedure.

I. INTRODUCTION

M

ETAMATERIALS represent a class of artificial electromagnetic materials that exhibit some special electromagnetic properties, e.g., the materials in which the phase and group velocities of electromagnetic waves are opposite to form the left-handed rule (the materials are also referred to as left-handed materials) [1]. Since 2001 [2], wave theory, medium characterization, optimum or purpose-driven specific design, systematic synthesis, and potential engineering applications of metamaterials have led to a fast growing area and attracted more and more attention and interests of scientists and engineers [3]–[6]. The artificial metamaterials possess some specific abnormal properties, which the natural materials usually do not exhibit, e.g., both negative permittivity and Manuscript received June 1, 2004; revised October 22, 2004. This work was supported by the Singapore–MIT Alliance. H.-Y. Yao, W. Xu, and T.-S. Yeo are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260. L.-W. Li is with the Department of Electrical and Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 and also with the Department of Electronics and Communications Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]). Q. Wu is with the Department of Electronics and Communications Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845210

negative permeability, negative index of refraction, and reversal Doppler shift [1]. Properties of negative permittivity and negative permeability of artificial metamaterials can be realized from different physical or mechanical synthesis and fabrication methodologies. Although general periodic structures can be fabricated to realize the special properties, the primary material synthesis is still based on the individual inclusions [7], [2], [8]–[11] or the L–C microwave striplines/devices [12]–[14]. To design such artificial materials for different specific engineering applications, simulation tools are essential for saving efforts and costs. The numerical methods for characterizing these structures are still those fundamental numerical simulation techniques, i.e., the differential-equation (DE) method [5], [15], the integral-equation (IE) method [16], [17], and the equivalent-circuit models [18]–[20]. Among the three methods usually utilized for designing artificial synthetic materials, the electromagnetic solver developed based on the method of moments (MoM) for solving the electrical-field integral equation (EFIE) is rigorous and widely adopted in various electromagnetic simulations. It is certainly a very good option for analyzing electromagnetic characteristics of metamaterials. However, the MoM will lead to a full matrix whose numerical operations and iterative solution requires memory to store the matrix elements, where and denote the number of unknowns and number of iterations, respectively. The computational cost and memory requirement will increase much faster with the larger number of unknowns. On the one hand, the operating frequency of electromagnetic waves in a medium of metamaterials falls usually within the microwave frequency band or even higher. On the other hand, the spacings and dimensions of conducting inclusions that compose the artificial media are required to be much smaller than the wavelength in order to synthesize a practical continuous material so that the number of inclusions is numerous and the resultant total number of unknowns is very large, even though the dimensions of such synthesized metamaterials are electrically small in this frequency band. To achieve this objective, a fast algorithm has to be integrated into the MoM procedure to accelerate the matrix–vector multiplications. For this problem, some techniques such as the improved multipole-based fast algorithm [21], [22] and the QR factorizationbased algorithms [23]–[25] are usually employed to speed up the matrix-vector multiplications. The QR-based algorithms utilize the low rank compression to the local MoM matrix, which

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expresses the interaction of two well-separated geometrical regions. Recently, the multilevel QR-based technique is developed to extract the parasitic capacitance, which adopts the binarytree structure of the fast multipole method [26]. In this paper, this kind of algorithms for efficiently reducing the MoM matrix order is chosen. The current algorithm is developed based on the modified Gram–Schmidt (MGS) procedure in an iterative procedure. It can reduce both the computational complexity and the , which will be evident from memory requirement to later discussions and numerical examples. II. FORMULATIONS For a perfectly electric conducting (PEC) object illuminated by a plane wave , the EFIE can be obtained by enforcing the boundary conditions of the tangential electric-field components on its surface, and is given by

groups. The First, we divide the basis functions into number of basis functions in each group is, at most, . By evaluating the interactions of two groups numbered by and using (4), the corresponding local matrix can be attained, in which and are the numbers of basis functions in groups and , respectively. The objective of this algorithm is to factorize matrix as follows: (6) is constructed by the most linearly inwhere the matrix dependent orthonormal column vectors (which span the column within a prescribed tolerance of ) and the maspace of represents the expansion coefficients, which form the trix corresponding matrix with respect to the column vectors of matrix when constructing . To find orthonormal sets, which compose the matrix , the MGS procedure is applied. Conveniently, the elements of the matrix can be rewritten by a series of column vectors (7)

(1) where is the induced current density, denotes the unit tangenstands for the wave impedance tial vectors of the surface , in the free space, represents identifies the free-space scalar Green’s function, and the propagation constant. The MoM technique is used to subsequently solve the equation. The unknown currents, denoted by , are expanded in as follows: terms of basis functions (2)

where ( ) denotes the -dimensional column vector. In the MGS-based algorithm (MGSA), we mark with and subsequently refer to it as . In the current algorithm, we choose as the criterion in which denotes the ( )th orthonormalized matrix. If is less than a specified tolerance , then the MGS orthonormal operation stops. Moreover, according to the procedure of MGS orthonormality, an orthonormal basis must be found to make the module of the orthonormalized masmallest, which ensures the efficient matrix comtrix pression. The module of the orthonormalized matrix can be expressed by

Proper testing functions are then applied to obtain an algebraic matrix equation as follows: (3) impedance matrix and idenin which denotes an tifies an unknown column vector, while represents a coefficient column vector, both of the length . The elements of the impedance matrix are defined as

(4) while the elements of the coefficient column vector are calculated by (5) demands a storage of Evidently, the impedance matrix . The linear system in (3) can be solved by either a direct approach or an iterative method. A direct solver requires computational cost and an iterative solver requires per iteration. Once the number of unknowns becomes large, these requirements could easily run out of the given memory and the efficiency will become horribly poor. To resolve this problem, an algorithm based on the MGS procedure is subsequently presented.

(8) in which is the unit vector of an orthonormal basis , denotes the MGS orthonormality to every column vector in the matrix based on , and the superscript denotes the complex conjugate. In order to make the module smallest, an orthonormal basis is needed to make the term of in (8) as big as possible. If we make use of this term to seek for the basis directly, an linear equation system is requested to be solved, which will lead to operations. Hence, we simplify this term to to construct an orthonormal basis. In this paper, we present an circulatory operation to seek for this orthonormal basis. The procedure of this algorithm is briefly described as follows. 1) Grouping: Divide the discretized currents into groups of size at most, in which is the number of current basis functions. Let Choose a tolerance For

. .

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2)

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Constructing an orthonormal basis by an circulatory scheme Initialization: Let flag

Construction: While flag , Begin: , Let flag For Begin: Fig. 1. Bistatic RCSs versus  for a conducting sphere with an electric radius of ka = 4:7.

If If

let

,

, flag

; Else If . flag End. End.

let

,

,

III. NUMERICAL RESULTS

Then do

3)

computational complexity and to save the memory. The compu, and the same is tational cost of each local matrix is for the required memory. Assume that the average number of . The computational basis functions in each group is . For all the local maresource requirements are both in trix–vector multiplications, the computational complexity and . memory requirement are both in

An orthonormal basis is then found. Producing the matrices and Perform the MGS orthogonalization procedure to based on the vector , while a expansion coefficient row can be obtained as follows: vector

If , then end this procedure; else continue doing 2) and 3). and can be given by Finally, the matrix (9) and (10) , then If the size of the orthonormal basis is less than is stored as two matrices and instead of a dense matrix. Thus, and are used in the iterative so as to reduce the solver of (3) to replace the local matrix

Here, several examples are provided to validate the accuracy and efficiency of this algorithm. The rooftop basis function defined on two adjacent cells is employed in the MoM. The matrix equation is iteratively evaluated by the biconjugate gradient (BCG) method and its numerical results are obtained. The tolerance of in the MGSA is chosen. To validate the current method and analysis procedure and to gain more insight into the applicability and accuracy, we have considered a couple of subsequent examples. First, we consider a conducting sphere with an electrical dimension of . The object is discretized into ten patches. The number of unknowns is 2112, which are divided into 46 groups. The -plane bistatic radar cross sections (RCSs) as a function of are obtained using the MGSA and the exact solution from the Mie series is also obtained. Both results are shown, respectively, in Fig. 1. An excellent agreement is observed in the comparison, which demonstrates the fairly good accuracy of the current algorithm. To show the efficiency of the current algorithm when used to characterize the metamaterial structure, we consider some cubeshaped metamaterial samples synthesized by a square split-ring resonator (SRR) and a wire. The geometry and dimensions of this kind of inclusions are shown in Fig. 2. The space distances of the inclusions denoted, respectively, by , , and in the -, -, and -directions, are all 3.3 mm. Five samples are considered in the simulation. The number of the inclusions of these samples in these three directions are the same and are set as 5–8, and 10, which correspond to 2850, 4932, 7840, 11 712, and 22 900 unknowns. The CPU time and memory requirements

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Fig. 2. Geometry of a single inclusion composed by a square SRR and wire. The dimensions are assumed to be L = 3:3 mm, L = 2:63 mm, w = 0:25 mm, d = 0:3 mm, g = 0:46 mm, and t = 0:254 mm. (a) Front view (b) Top view.

Fig. 5. Rank map for the cube-shaped metamaterial sample with 11 712 unknowns.

Fig. 3. CPU time values of each iteration consumed in the MGSA and the conventional MoM, respectively.

Fig. 6.

Fig. 4. Memory requirements of the MGSA and conventional MoM, respectively.

on a Pentium third-generation (3G) PC requested by the conventional MoM and MGSA for these examples are presented in Figs. 3 and 4. The lines of for the CPU time and memory cost are also provided for the reference, in which represents the number of unknowns. As we expect, the computational cost of the MGSA is indeed , whereas the

Top view of a metamaterial prism.

MoM takes much larger memory and longer CPU time to obtain the solution. The savings in memory and CPU time becomes especially significant when the number of unknowns becomes larger. The rank map for the metamaterial sample with 11 712 unknowns is also presented in Fig. 5 to show the efficient compression to every local matrix. Next, we employ this algorithm to analyze the propagation characteristics of electromagnetic wave in the metamaterial sample, which is placed in the free space. A metamaterial prism (wedge-shaped sample) synthesized by the inclusions shown in Fig. 2 is considered and depicted in Fig. 6. The total number of inclusions is 234. The numbers of inclusions along the shortest side and the longest side are 3 and 15, respectively. The incline of this prism is approximately 63.43 . In this case, the angle incident angle is 25.64 . Firstly, the scattering cross section (SCS) of a single inclusion is evaluated to find the resonant

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Fig. 7. SCS versus frequency for a single inclusion.

Fig. 9. Distribution of electric-field component E (r ; t) in polar plot due to the metamaterial prism at f = 16:21 GHz.

Fig. 10. Distribution of electric-field component E (r ; t) in rectangular linear plot around the metamaterial prism at f = 16:21 GHz.

Fig. 8. Electric-field component E (r ; f ) distribution due to the metamaterial prism. (a) Larger spectrum from 15.2 to 17.0 GHz plotted at a frequency step of 0.1 GHz. (b) Zoomed-in smaller spectrum from 16.1 to 16.3 GHz plotted at a frequency step of 0.01 GHz.

frequency, which is shown in Fig. 7. It can be found that the resonant frequency is approximately 16.25 GHz. In order to gain more insight into the effects or influences of the prism edge of the metamaterials, we consider a beamed plane wave as the incidence, which is displayed as the gray part in Fig. 6 and it, thus, illuminates normally on the first interface. Due to the existence of a small passband near the res-

onant frequency of the inclusion [27], we calculate the field GHz distribution in the frequency range with a step of 0.1 GHz. This structure resulted in 5148 unknowns, which are divided into 72 groups. The MGSA needs 119.5 MB of memory and takes 0.13 s/iterations. Fig. 8(a) deversus picts the amplitude of the electric-field component is obtained at a distance of the the frequency, in which prism. We can observe a special phenomena around 16.2 GHz. At this frequency, the electric field refracted by the metamaterial wedge peaks at a negative refractive angle. To obtain more details, we also evaluate the field distribution in the frequency GHz with a step of 0.01 GHz, which range of is shown in Fig. 8(b). As can be seen, the negative refraction GHz, which is near the resonance occurred around of the inclusion. In Fig. 9, we plot the amplitude of the elecat GHz. From this figure, tric-field component the refractive angle is found to be approximately 5.36 and the corresponding refractive index is 0.2159. The time–harat 16.21 GHz is shown in Fig. 10 in which monic field . the phase angle

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IV. DISCUSSION

Case 3)

From the macroscope view, the metamaterial sample we analyze in this paper is a one-dimensional left-handed medium. It posses negative permittivity and negative permeability simultaneously in some frequency range near the resonance of the inclusion [17]. The effective permittivity and permeability tensors of this metamaterial are given by

(11)

and

(15c)

Case 3) requires , which is not easy to realize physically, thus, we will not subsequently consider it. Here, only the field components in Case 1) and 2) can be discussed and expressed. Case 1)

Here, we analyze this structure in the system [28]. The constitutive relations in this system can be expressed as

(16a) (16b)

If the wave propagation constant is in the ), then words,

(12a)

(16c)

(12b)

(16d)

-plane (in other

Case 2)

(17a) (13a)

(17b)

(17c)

(13b) Maxwell equations in the

system lead to

(14)

where the speed of waves is and the propagation . Equation (14) must be satisfied when constant waves are propagating in the medium. Then, we have the following three cases. Case 1)

and

(15a)

Case 2)

and (15b)

(17d) where , , and denote three unit vectors of the system, while , , and denote the displacement compocoordinate system. For , nents projected onto the . we have In our model, the polarization of the incident electric field is parallel to the -axis. Thus, Case 1) is not applicable. Wave propagates by means of Case 2). As shown in Fig. 6, the propagation vector is pre-set along the -axis. However, due to the prism’s edge or tip effect and microscopic diffraction of the inclusions, we can find from Fig. 10 that the wave inside the metamaterial prism does not actually propagate along the -axis, but has an angle with the -axis. That is the reason we retain the element in the above deduction. When , the wave is an extraordinary wave whose Poynting’s vector is not in the direction parallel to . In addition, the effective permittivity and permeability of the SRR-constructed artificial material have a very large imaginary part [17], which occurs in the same frequency range with the large real part. As a result, the extraordinary wave is drastically attenuated after transmission [28] so it causes only a small refractive angle as observed. However, if the metamaterials are two-dimensional isotropic, the extraordinary wave will not exist no matter how the propagation vector is diffracted in the -plane.

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V. CONCLUSIONS In this paper, a hybrid technique combining the MGS-based matrix compression and an iterative solver has been presented to accelerate the IE solution. By using this algorithm, the CPU . time per iteration and memory requirements are both Moreover, this current algorithm is independent of the electrical size and integral kernels. Thereby, it is especially suitable to analyze the metamaterials, which usually have a large number of unknowns, but electrically small inclusions. Herein, several cubic-shape metamaterial samples with different numbers of unknowns have been modeled and their physical quantities have been calculated to demonstrate the efficiency of this algorithm. Moreover, a metamaterial prism has been synthesized to analyze propagation properties of electromagnetic waves. The negative refraction phenomena has been observed. Finally, the detailed explanation of the propagation properties of the waves in the arsystem. tificial media has been made by using the REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Soviet Phys—Usp., vol. 10, no. 4, pp. 509–514, 1968. [2] D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett., vol. 90, no. 7, p. 077 405, Feb. 2003. [3] A. Alu and N. Engheta, “Guided modes in a waveguide filled with a pair of single-negative (SNG), double-negative (DNG), and/or doublepositive (DPS) layers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 199–210, Jan. 2004. [4] J. A. Kong, “Electromagnetic wave interaction with stratified negative isotropic media,” Progr. Electromagn. Res., vol. PIER35, pp. 1–52, 2002. [5] H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “T-junction waveguide experiment to characterize lefthanded properties of metamaterials,” J. Appl. Phys., vol. 94, no. 6, pp. 3712–3716, Sep. 2003. [6] A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2550–2557, Oct. 2003. [7] F. Falcone, F. Martin, J. Bonache, R. Marques, T. Lopetegi, and M. Sorolla, “Left handed coplanar waveguide bandpass filters based on bi-layer split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, pp. 10–12, Jan. 2004. [8] R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1516–1529, Jul. 2003. [9] N. Engheta, “Metamaterials with negative permittivity and permeability: Background, salient features, and new trends,” in IEEE MTT-S International Microwave Symp. Dig., vol. 1, 2003, pp. 187–190. [10] P. Gay-Balmaz and O. J. F. Martin, “Efficient isotropic magnetic resonators,” Appl. Phys. Lett., vol. 81, no. 5, pp. 939–941, Jul. 2001. [11] L.-W. Li, H.-X. Zhang, and Z.-N. Chen, “Representation of constitutive relation tensors of metamaterials: An approximation for FFB media,” in Proc. Progress in Electromagnetics Research, Waikiki, HI, Oct. 13–16, 2003, p. 617. [12] C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 980–992, Mar. 2004. [13] A. A. Oliner, “A planar negative-refractive-index medium without resonant elements,” in IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, 2003, pp. 191–194. [14] A. Grbic and G. V. Eleftheriades, “Dispersion analysis of a microstripbased negative refractive index periodic structure,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 4, pp. 155–157, Apr. 2003. [15] T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R. Smith, D. C. Vier, and S. Schultz, “Ab initio numerical simulation of left-handed metamaterials: Comparison of calculations and experiments,” J. Appl. Phys., vol. 90, no. 10, pp. 5419–5424, Nov. 2001.

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[16] P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys., vol. 92, no. 5, pp. 2929–2936, Sep. 2002. [17] H.-Y. Yao, L.-W. Li, Q. Wu, and J. A. Kong, Macroscopic Performance Analysis of Metamaterials Synthesized from Microscopic 2-D Isotropic Cross Split-Ring Resonator Array. Boston, MA: EMW Publishing, 2004, vol. 84. [18] A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 2, pp. 68–70, Feb. 2004. [19] W. Xu, L.-W. Li, and Q. Wu, “Design of left-handed materials with broad bandwidth and low loss using double resonant frequency structure,” in IEEE AP-S Int. Symp. Dig., vol. 4, Monterey, CA, Jun. 20–26, 2004, pp. 3792–3795. [20] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [21] K. F. Warnick, G. Kang, and W. C. Chew, “Regulated kernel for the electric field integral equation,” in IEEE AP-S Int. Symp., vol. 4, Jul. 2000, pp. 2310–2313. [22] E. Darve and P. Have, “Efficient fast multipole method for low-frequency scattering,” J. Comput. Phys., vol. 197, pp. 341–363, 2004. [23] S. Kapur and D. E. Long, “IES : A fast integral equation solver for efficient 3-dimensional extraction,” in IEEE Int. Computer-Aided Design Conf., Nov. 1997, pp. 448–455. [24] N. A. Ozdemir and J. F. Lee, “Single level dual rank SVD algorithm for volume integral equations of electromagnetic scattering,” in IEEE AP-S Int. Symp., Jun. 2003, pp. 302–305. [25] K. Zhao and J.-F. Lee, “A novel 2-level IE–SVD algorithm to model large microstrip antenna arrays,” in 19th Annu. Review of Progress in Applied Computational Electromagnetics, Monterey, CA, Mar. 24–28, 2003, pp. 826–829. [26] D. Gope and V. Jandhyala, “PILOT: A fast algorithm for enhanced 3D parasitic capacitance extraction efficiency,” Microwave Opt. Technol. Lett., vol. 41, no. 3, pp. 169–173, May 2004. [27] C. D. Moss, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Numerical studies of left handed metamaterials,” Progress Electromagn. Res., vol. PIER35, pp. 315–334, 2002. [28] J. A. Kong, Electromagnetic Wave Theory. Boston, MA: EMW Publishing, 1999, ch. 3.

Hai-Ying Yao received the B.Eng. degree in electronic engineering, M.Eng. degree in electromagnetic field and microwave technology, and Ph.D. degree in electromagnetic field and microwave technology from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1996, 1999, and 2002, respectively. From April 2000 to October 2001, she was a Research Assistant with the City University of Hong Kong, Hong Kong. Since 2002, she has been with the High Performance Computations of Engineered Systems (HPCES) Programme, Singapore–MIT Alliance (SMA), National University of Singapore, Singapore, as a Research Fellow. Her current research interests include fast algorithms and hybrid methods in computational electromagnetics, and radio-wave propagation and scattering in various media.

Wei Xu (SM’03) received the B.Eng. degree in electronic engineering and information science from the University of Science and Technology of China (USTC), Hefei, China, in 2003, and is currently working toward the M.Eng. degree in electrical engineering (with a specialization in electromagnetics) from the National University of Singapore, Singapore. His current research interests include analysis and design of metamaterials and composite material structures.

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Le-Wei Li (S’91–M’92–SM’96–F’05) received the B.Sc. degree in physics from Xuzhou Normal University, Xuzhou, China, in 1984, the M.Eng.Sc. in electrical engineering from China Research Institute of Radiowave Propagation (CRIRP), Xinxiang, China, in 1987, and the Ph.D. degrees in electrical engineering from Monash University, Melbourne, Australia, in 1992. In 1992, he was a Research Fellow with La Trobe University (jointly with Monash University), Melbourne, Australia. Since 1992, he has been with the Department of Electrical and Computer Engineering, National University of Singapore (NUS), Singapore, where he is currently Director of the NUS Centre for Microwave and RF and a Centre for Microwave and Radio Frequency (CMRF) Chair Professor of Electromagnetics. From 1999 to 2004, he was also with the High Performance Computations on Engineered Systems (HPCES) Programme, Singapore–MIT Alliance (SMA), as an SMA Fellow. His current research interests include electromagnetic theory (e.g., dyadic Green’s functions), computational electromagnetics (fast algorithms of the adaptive integral method and pre-corrected fast Fourier transform method), radio wave propagation and scattering in various media (metamaterials, chiral media, and bi-anisotropic media), microwave propagation and scattering in tropical environments (forest and rainfall attenuation), and analysis and design of various antennas. In these areas, he coauthored Spheroidal Wave Functions in Electromagnetic Theory (New York: Wiley, 2001), 42 book chapters, over 220 international refereed journal papers, 25 regional refereed journal papers, and over 230 international conference papers. He is an Associate Editor for Radio Science and the Journal of Electromagnetic Waves and Applications, and an Overseas Editorial Board Member of the Chinese Journal of Radio Science. Dr. Li was a recipient of the Best Paper Award presented by the Chinese Institute of Communications for his 1990 Journal of China Institute of Communications paper, and the Prize Paper Award presented by the Chinese Institute of Electronics for his 1991Chinese Journal of Radio Science paper. He was also the recipient of a 1995 Ministerial Science and Technology Advancement Award presented by the Ministry of Electronic Industries, China and a 1996 National Science and Technology Advancement Award with a medal presented by the National Science and Technology Committee, China. He is a member of The Electromagnetics Academy based at the Massachusetts Institute of Technology (MIT).

Qun Wu (SM’03) received the B.Sc. degree in radio engineering, the M.Eng. degree in electromagnetic fields and microwave technology, and Ph.D. degree in communication and information systems engineering from the Harbin Institute of Technology (HIT), Harbin, China in 1977, 1988, and 1999, respectively. From 1998 to 1999, he was a Visiting Professor with the Seoul National University (SNU), Seoul, Korea. From 1999 to 2000, he was a Visiting Professor with the Pohang University of Science and Technology, Pohang, Korea. Since 1990, he has been with the Department of Electronics and Communications Engineering, HIT, China, where he is currently a Professor. He has authored or coauthored over 30 international and regional refereed journal papers. His recent research interests are mainly in microwave active circuits, electromagnetic compatibility, monolithic microwave integrated circuits (MMICs), and millimeter-wave microelectromechanical systems (MEMS) devices. Dr. Wu was the recipient of two Third-Class Prizes and one Second-Class Prize of Scientific Progress Awards presented by the Ministry of Aerospace of China in 1989 and 1992, respectively.

Tat-Soon Yeo (M’79–SM’93–F’03) received the B.Eng. (Hons I) degree from the University of Singapore, Singapore, in 1979, the M.Eng. degree from the National University of Singapore (NUS), Singapore, in 1981, and the Ph.D. degree from the University of Canterbury, Christchurch, New Zealand, in 1985. He is currently a Professor with the Electrical and Computer Engineering Department and a Vice Dean of the Faculty of Engineering, NUS. He is also concurrently the Director of the Radar and Signal Processing Laboratory, and Antennas and Propagation Laboratory, Department of Electrical Engineering, NUS. He is also the Director of the Temasek Defence Systems Institute, a teaching institute jointly established by NUS and the U.S. Naval Postgraduate School (NPS). His current research interests are scattering analysis, synthetic aperture radar, antenna and propagation study, numerical methods in electromagnetics, and electromagnetic compatibility. Dr. Yeo is the past-chairman and Executive Committee member of the Microwave Theory and Techniques (MTT)/Antennas and Propagation (AP) and Electromagnetic Compatibility (EMC) Chapters, IEEE Singapore Section, and the chairman of the Singapore EMC Technical Committee. He was the recipient of a Colombo Plan Scholarship, a 1997 Singapore Ministry of Defence–NSU Joint Research and Development Award, the 2000 IEEE Millennium Medal, and the 2002 Singapore Standard Council’s Distinguish Award.

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FDTD Study of Resonance Processes in Metamaterials Elena A. Semouchkina, Member, IEEE, George B. Semouchkin, Michael Lanagan, Member, IEEE, and Clive A. Randall

Abstract—The finite-difference time-domain simulations in the frequency domain are used to study resonance phenomena in left-handed metamaterials consisting of the arrays of split-ring resonators (SRRs) and metal rods. It is demonstrated that, at frequencies corresponding to the band of enhanced transmission of the metamaterial, the half-wavelength resonances occur in both the SRRs and rods. The observed resonances in rods make questionable the applicability of the plasma concept to the analysis of the metamaterial performance. We also show that overlapping of electric or magnetic fields at resonance causes coupling between resonators and assembling them in three-dimensional groups, which rearrange in dependence on frequency inside the transmission band. As the result, the resonance phenomena in the metamaterial proceed essentially nonuniform, although the size of the metamaterial units is less than the wavelength. We suggest that coupling between resonators is capable of providing the electromagnetic response, similar to that observed at the backward-wave propagation in double-negative media. The latter is demonstrated on the example of the all-dielectric metamaterial composed from an array of dielectric resonators. Index Terms—Composite medium, electromagnetic (EM) coupling, finite-difference time-domain (FDTD) methods, metamaterials, microwave resonators, permeability, permittivity.

I. INTRODUCTION

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ICROWAVE metamaterials with a negative index of refraction have recently attracted much attention within the electromagnetic (EM) community. The idea of “left-handed” or “double-negative substances” was forwarded by Veselago in 1968 [1]; however, the boom of interest to metamaterials has started approximately five years ago since Pendry et al. [2] proposed an artificial material consisting of the arrays of split-ring resonators (SRRs) and metal rods. It is suggested that the SRRs, which exhibit magnetic response to the EM wave at resonant frequencies, could provide for the effective negative permeability of the medium, while the array of rods, which acts as a high-pass filter for the EM wave at plasma frequency, provides for the effective negative permittivity of the medium. At frequencies when both parameters are negative, the fascinating properties of materials are expected; in particular, high transmission for the backward wave within the stopband of the SRR array. Manuscript received June 2, 2004; revised November 11, 2004. This work was supported in part by the National Science Foundation under Award DMI0339535 and by the Center for Dielectric Studies under Grant 0120812. E. A. Semouchkina, G. B. Semouchkin, and M. Lanagan are with the Materials Research Institute, The Pennsylvania State University, University Park, PA 16802 USA (e-mail: [email protected]). C. A. Randall is with the Interdisciplinary Research Collaboration, Cambridge University, Cambridge CB3 0HE, U.K., on leave from the Materials Research Institute, The Pennsylvania State University, University Park, PA 16802 USA. Digital Object Identifier 10.1109/TMTT.2005.845203

Smith et al. [3] and Shelby et al. [4] confirmed experimentally that metamaterials constructed from SRRs and rods really demonstrate enhanced transmission within the SRR stopband. Later, negative index of refraction for a lens of that metamaterial was also verified [5]. However, several other observations for these periodic structures caused questions. In particular, Smith et al. [3] have mentioned that resonances in the rods, which occurred because of the lack of electric connectivity between the strips and the bounding metal surfaces, produced serious complication of the observed phenomena. It is desirable to clear up the question as to whether the rods well connected to the bounding metal plates in metamaterial could resonate or not. The nature of the SRR’s response is also not completely understood. Shelby et al. [4] suggested that there were no half-wavelength requirements for resonance in the SRRs. This point-of-view seems to originate from the concepts applied at designing two concentric SRRs by Pendry et al. [2], who supposed that the smaller ring could load the bigger ring by introducing an additional capacitance, and that such loading could change the resonance wavelength. However, Katsarakis et al. [6] have pointed out that concentric SRRs can basically be represented by the outer (bigger) ring alone, while it is well known that, in such SRRs, the magnetic moment along the normal axis can only be provided by the resonance mode, for which the half-wavelength is equal to the perimeter of the SRR. At last, a common approach in explaining the EM response of metamaterials is based on the assumption that the properties of two arrays (the SRRs and rods) are simply superimposed so that the interaction between these arrays could be omitted. It looks contradicting to another often-used assumption about strong coupling between the SRRs because, in such a case, coupling between the SRRs and rods should also be accounted for. In this paper, we study the resonance phenomena in metamaterials constructed from the SRRs and rods, by means of numerical modeling using the finite-difference time-domain (FDTD) method. First, we introduce our metamaterial model, which is close to the one described by Moss et al. [7], but is slightly modified, in order to produce more symmetric geometry, which makes understanding the formation of resonant modes easier. We describe the simulated transmission spectra for our model and demonstrate that they are similar to the ones presented by Moss et al. [7] and that the band of enhanced transmission appears in our spectra at the same frequencies, at which Moss et al. have demonstrated left-handed behavior for EM waves propagating through the metamaterial slab. We then present the analysis of the amplitude and phase distributions for oscillations of electric and magnetic fields in metamaterial sample at different frequencies. These distribu-

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Fig. 1. (a) View from the source on the column of unit cells presented in (c). The SRRs are shown in black and the microstrip are shown in a dark gray. Dashed lines show the boundaries of units located behind the front dielectric plate. Shaded areas are used to show positions of side dielectric cross sections of the metamaterial unit cells constructed plates. (b) and (c) from the SRRs and microstrips. (b) Conventional design. (c) Design with an -plane of symmetry. Grey blocks represent dielectric plates. Wide black strips show the locations of the SRRs and narrow strips show the locations of the microstrips.

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tions were obtained by using the technique for frequency-domain data representation that enabled us to visualize EM-field standing waves in the structures [8]. We show that, although the size of metamaterial particles is smaller than the wavelength, the metamaterial could not be strictly considered as uniform media since SRRs and rods form coupled groups rearranging with frequency. We suggest that coupling between resonators could affect EM response of the metamaterial in a way similar to that observed at the backward-wave propagation in double-negative media. In order to make our concept more clear, in Section IV, we present a new all-dielectric metamaterial composed from an array of dielectric resonators (DRs), which demonstrates, in particular, enhanced transmission and negative refraction due to cooperative phenomena caused by coupling between the resonators. II. MODEL DESCRIPTION A. Basic Model The metamaterial model used in our simulations was similar to the one described by Moss et al. [7] and based on the experimental setup presented in [5]. The metamaterial was placed in a parallel-plate waveguide, i.e., between the two perfectly conducting sheets separated by 13.5 mm. Perfectly matched layers were used to terminate other boundaries of the computational domain. The SRRs were constructed from two concentric square rings with the gap in the smaller ring located opposite to the gap in the bigger ring [see Fig. 1(a)]. Both gaps were 0.25-mm wide. The

width of the strips forming the rings was also equal to 0.25 mm. Instead of rods, we used microstrips with the same width of 0.25 mm. The SRRs were placed on one side, while the microstrips were placed on the opposite side of dielectric plates with a permittivity of 3.4 and a thickness of 1 mm. Location of the SRRs and microstrips on the plates are shown in Fig. 1(c). The metamaterial structure was excited by an electric wall -plane, which created an electric field source located in along the -axis and a horizontal magnetic field along the -axis. The distance between the source and front surface of the metamaterial sample was usually 1.5 mm, the distance between the source and PML boundary was 5 mm, while the back surface of the sample was separated from opposite PML boundary by 7.5 mm. When the cell size was chosen properly, these distances allowed for preventing the voltage signatures from instabilities during long simulation runs. Most often, the cell size of 0.125 mm was used for all three dimensions. As the perimeter of the large rings in the SRRs was 11.5 mm, the half-wavelength resonance in such rings could be expected at a frequency of 13 GHz for rings located in the air and at approximately 7 GHz for rings placed in dielectric media with a permittivity of 3.4. Since EM response of microstrips located on the surface of dielectric substrates corresponds to an effective dielectric permittivity, which is less than that of the substrate, we could expect the half-wavelength resonance in the large rings to occur at approximately 10 GHz. The half-wavelength resonance in the microstrips with the length of 13.5 mm used in our model instead of rods could be expected at frequencies of 11 GHz in air, 6 GHz in dielectric media with the permittivity of 3.4, and at approximately 9 GHz in metamaterial, respectively. It is worth noting that Moss et al. [7] used normalized transmitted power to characterize interaction between the incident EM wave and the SRRs; however, they did not specify the technique for calculations. In this study, we simulated voltage between the metal sheets of the parallel-plate waveguide at the positions in front and behind the sample. We used the fast Fourier transformation of the time-domain data for the voltage to obtain the spectrum characterizing EM response of the metamaterial. The spectra sampled at the distance 1.5 mm behind the structure, demonstrated deep drops at the resonances that shorted the circuit. Such drops could be considered as the stopbands for transmission. B. Design of Metamaterial Unit Cells Fig. 1(b) presents the top view of the typical unit cell of the metamaterial, constructed from the SRRs and rods, which was used by numerous authors. This design emerged from the initial one-dimensional and strongly anisotropic design [9] in which the SRRs were located only in the -plane, i.e., with their axis parallel to the direction of magnetic field ( ) because it was supposed that only such orientation could provide for effective negative permeability of the metamaterial above the resonance -plane were added frequency of the SRRs. Resonators in the to decrease anisotropy, however, it was not clarified whether the additional resonators were active or not at excitation by the EM wave traveling along the -axis. The design presented in Fig. 1(b) had only a single diagonal plane of symmetry. Meanwhile, we have found that, in metamaterials with low symmetry

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Fig. 2. Spectra of the voltage (in volts versus hertz) between the metal sheets behind the unit cell for the two cells shown in Fig. 1. (a) For conventional design. (b) For modified design.

of the unit cell, the resonance patterns are often distorted, especially for samples consisting from a relatively small quantity of resonators. Fig. 1(c) presents a modified model of the unit cell with fourfold rotational symmetry with respect to the -axis, which was used in our simulations for a resonant mode study. The shortest distance between the SRRs in the proposed model is the same for all resonators related to this cell and it equal to 2.12 mm, while the shortest distance between the microstrips representing the rods is 2.475 mm. The estimation of the plasma frequency for such a distance between the “rods” gives the frequency higher than 20 GHz, thus, we could expect the effective permittivity of the media to be negative at all frequencies in the range under study, i.e., from 8 to 19 GHz. To make sure that the modification of the unit cell design does not significantly change the EM response of the cell, we calculated and compared frequency spectra for the voltage between the parallel waveguide plates for two types of unit cells. Fig. 2 demonstrates that the main features of the spectra are very similar, and this justifies the employment of the modified design in our study. To build a sample of metamaterial from the cells of a type shown in Fig. 1(b), Moss et al. [7] have used the translation of two dielectric plates so that the array representing the metamaterial sample included 24 SRRs and eight rods (microstrips) [see Fig. 3(a)]. In order to build a sample of metamaterial from the modified cells, we had to include in the array not only the cells of the type shown in Fig. 1(c), but also the cells of a “reverse” type [see Fig. 3(b)]. Our samples consisted of 36 SRRs and 12 rods. As seen from Fig. 3, the samples of both types have absolutely the same sets of distances between the SRRs and between the rods. III. SIMULATION RESULTS FOR METAMATERIAL MODEL

Fig. 3. (a) Array of SRRs and rods similar to the one simulated in [7]. -planes of symmetry; dashed line shows (b) Array of modified cells with the position of the source (electric wall), and two points behind the sample mark the positions, at which the voltage between the two metal sheets was sampled.

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Fig. 4. Spectrum of the normalized voltage (in decibels versus hertz) between the metal sheets sampled at the position 1 [see Fig. 3(b))]. The arrow points at the passband, which, according to [7], is related to the backward-wave propagation.

A. Transmission Spectra of Metamaterial Sample The spectra of the voltage sampled at two positions, shown by points 1 and 2 in Fig. 3(b), were slightly different in magnitude; however, their basic features were similar and comparable to the features of the spectra for unit cells shown in Fig. 2.

Fig. 4 presents the normalized voltage spectrum sampled at the position 1 [see Fig. 3(b)]. Comparison of this spectrum with the transmission power spectrum presented by Moss et al. [7, Fig. 5] for metamaterial with close parameters shows that both spectra

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have identical sets of passbands and stopbands. The band of enhanced transmission at approximately 10 GHz related by Moss et al. to the backward-wave propagation has the same shape and values in decibels. However, the main passband in our case is wider and the shallow stopband is located between 18.5–19.5 GHz, while in [7], near 16.0 GHz, the sharp transmission peak occurs at 20.0 GHz instead of 16.5 GHz, and there is a weak peak of transmission at 7.5 GHz instead of a well-shaped band observed in [7]. These differences could be partly related to the modifications introduced to the unit cells. As seen in Fig. 4, the metamaterial sample blocks power transmission at frequencies below 9 GHz and between 19.0–22.5 GHz since, at these frequencies, the voltage values behind the sample are close to zero. Moss et al. [7] observed a similar lack of transmission at low and high frequencies, particularly the drops deeper than 20 dB below 9.5 GHz and between 16–18 GHz. It was suggested that low transmission is related to the EM response of the rods because the samples containing the rods only demonstrated even deeper drops of transmission at all frequencies below the plasma frequency estimated to be equal to 18 GHz in [7]. Transmission through the metamaterial samples containing the SRRs [7] was found to be many orders of magnitude higher in a wide frequency range, and the only deep drop was observed at frequency of approximately 9.0 GHz. The authors related this drop to the SRR resonance. According to existing views, the permeability of the metamaterial at the high-frequency side of the SRR resonance is negative. Correspondingly, the passband for the backward-wave propagation in the samples containing both the SRRs and rods should be observed right above 9 GHz. The spectrum presented in Fig. 4 corresponds to these expectations because the band with the enhanced transmission is located between 9–10.5 GHz. However, a similar transmission band in [7] was shifted to higher frequencies by more than 1 GHz with respect to the SRR resonance. Such frequency shift between the SRRs resonance and the passband related to the backward-wave propagation is not quite understandable since negative permeability is not expected to exist over a wide frequency range. In order to get deeper insight into the passband formation, we began our study of resonance processes in the metamaterial by examining frequencies in the band of enhanced transmission. B. EM Response of the SRRs in the Unit Cell Numerical experiments with the sample consisting of one unit cell [see Fig. 1(c)] have shown that, at frequencies close to 10 GHz, the SRRs and the “rods” (microstrips) located on front and back dielectric plates, and those located on side dielectric plates of the unit cell, demonstrate essentially different EM response. The amplitude distributions of the field oscillations in the SRRs located on front and back plates, i.e., perpendicular to the direction of wave propagation, have the features of full-wavelength resonance in bigger split rings, while the amplitude field distributions in the SRRs located on side dielectric plates, i.e., parallel to the direction of wave propagation, correspond to the half-wavelength resonance in these rings. Let us examine the type of field distribution, which could be expected at half-wavelength and full-wavelength resonances in split rings. Schematics of two resonances are shown in Fig. 5,

Fig. 5. Charges, currents, and magnetic fields in split rings at: (a) half-wavelength resonance and (b) full-wavelength resonance. The locations of the maximum charge density are shown by circles, while the areas of maximum current flows are shown by straight arrows. Markers point at the areas of maximum electric fields, while markers 0 point at the areas of maximum magnetic fields.

+

and the consideration is based on well-known publications related to microstrip ring resonators (see, e.g., [10]) and to openloop resonators (see, e.g., [11]). At half-wavelength resonance, the charge densities should be maximal at the slot edges, while current flow should have maximum density in the ring rib opposite to the rib with the slot [see Fig. 5(a)]. The charges in two halves of the ring divided by the axis shown in Fig. 5(a) should be of opposite sign and, correspondingly, the oscillations of electric fields around these halves should be opposite in phase. In difference from the charges, the current in the ring should have the same direction along the whole length of the ring during each half period and, correspondingly, the oscillations of magnetic fields everywhere inside the ring should be in-phase. The antinodes of the electric-field standing wave along with the nodes of the magnetic-field standing wave should be observed near the slot edges, while the antinode of the magnetic-field standing wave along with the node of the electricfield standing wave should be observed near the middle of the strip opposite to the slot. At full-wavelength resonance, the charges of opposite signs should be located mostly on the ring rib with a slot and on the opposite rib, while the currents should be maximal in the side ring strips [see Fig. 5(b)]. In this case, the charges in the two halves of the ring divided by a horizontal axis, shown in Fig. 5, should be opposite in sign, and the oscillations of electric fields in these halves should be opposite in phase. Currents in both side strips should be in phase; however, the oscillations of magnetic fields created by these currents inside the ring should be opposite in-phase and should terminate each other in the center of the ring. The antinodes of the electric-field standing wave along with the nodes of the magnetic-field standing wave should be observed near the slot and near the middle of the opposite side of the ring, while the antinodes of the magnetic-field standing wave along with the nodes of the electric-field standing wave should be observed near the middles of the side ring strips. Resonances in the SRRs similar in design to those used in metamaterials have been studied by Gay-Balmaz and Martin [12] and Katsarakis et al. [6]. Gay-Balmaz and Martin studied transmission of the waveguide with inserted individual or coupled SRRs and have observed deep stopband, when the SRRs

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Fig. 7. 3-D patterns of the amplitude distributions of current density in vertical (Jz ) and horizontal (Jx) strips of the SRRs located on the side dielectric plate of the unit cell shown in Fig. 1(c).

Fig. 6. (a) and (b) Three-dimensional (3-D) patterns and (c) and (d) top views (2-D images) of the distributions of (a), (b), and left columns in (c) and (d) amplitude and [right columns in (c) and (d)] phase of oscillations of (a) and (c) electric- and (b) and (d) magnetic-field components normal to the XZ -plane in which the side SRRs are located at frequency f = 9:925 GHz. The numbers in phase patterns illustrate the phase values and show that phase difference of the fields in differently shaded areas is close to  .

were oriented in parallel to the wave vector, i.e., when the external magnetic field penetrated through the SRRs and generated circular currents in the rings. Current distributions in the SRRs for this case are calculated by using the Green’s tensor technique [12], exactly corresponded to the above description for the half-wavelength resonance in simple rings. When the SRRs were placed perpendicular to the wave vector, the “magnetic” resonance was observed only if the SRRs were turned by 90 with respect to the direction of external electric field so that the slot was located in vertical ribs. Katsarakis et al. [6] measured and calculated transmission spectra of two-dimensional (2-D) lattices combined from SRRs with one of four different orientations with respect to the direction of the external electric field and to the direction of wave propagation. They confirmed the results of Gay-Balmaz and Martin, and also found the electric resonance in the SRRs located perpendicular to the wave vector. The description of this resonance completely corresponds to the above discussed case of the full-wavelength resonance. However, both magnetic and electric resonances in [6] were suggested to occur at close frequencies, while the full-wavelength resonance is expected at approximately twice higher frequency than the half-wavelength resonance. Fig. 6 shows amplitude distributions for standing waves of the oscillations of electric- and magnetic-field components in the column of the SRRs located on the side dielectric plate of the unit cell shown in Fig. 1(c). The presented data correspond to the resonance frequency, at which the field amplitudes reach their maximum values, which exceed several times the values

Fig. 8. (a) and (b) 3-D patterns, and (c) and (d) top views (2-D images) of the distributions of (a) and (b) and left columns in (c) and (d) amplitude and [right columns in (c) and (d)] phase of oscillations of (a) and (c) electric- and (b) and (d) magnetic-field components normal to the Y Z -plane in which the front SRRs are located at frequency f = 10:4 GHz. The numbers on the phase patterns illustrate the phase values and show that phase difference of the fields in differently shaded areas is close to  .

at frequencies far from resonance. As seen from this figure, the character of field distributions in bigger rings corresponds to the half-wavelength resonance. Fig. 7 demonstrates that the distributions of current density in the SRRs located on the side dielectric plate of the unit cell at the same frequency also correspond to this type of resonance. The obtained results agree with the data presented in [6] and [12] since the SRRs located on the side dielectric plates of our unit cell correspond to the SRRs oriented parallel to the wave vector in [6] and [12]. It is worth noting that ring resonance in the unit cell of metamaterial occurred at 9.925 GHz and not at 9.0 GHz, as was suggested by Moss et al. [7] based on transmission data for samples without rods. This difference allows for omitting the problem of the frequency shift between the resonance in the SRRs and the enhanced transmission above 10 GHz related by Moss et al. [7] to the backward-wave propagation. Fig. 8 presents the amplitude and phase distributions of electric- and magnetic-field components in the column of the SRRs

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Fig. 9. 3-D patterns of the amplitude distributions of current density in vertical (J z ) and horizontal (J y ) strips of the SRRs located on the front dielectric plate of the unit cell shown in Fig. 1(c).

located on the front (or back) dielectric plates of the unit cell shown in Fig. 1(c). As seen from this figure, the distributions possess with basic features expected for standing waves of electric and magnetic fields at full-wavelength resonance. The distributions of current density presented in Fig. 9 also corresponded to this type of resonance. However, in difference from the case with SRRs located parallel to the wave vector, no resonance increase of the amplitudes of the field and current oscillations was found at frequencies around 10 GHz and up to 20 GHz, where full-wavelength resonance could be expected. Instead, we observed gradual growth of the maximal amplitudes of the field and current oscillations with frequency until 17.5 GHz and gradual decrease at higher frequencies, while the type of field and current distributions was more or less similar to the ones presented in Figs. 8 and 9. In the vicinity of 10 GHz, the values of the field amplitudes and the current density were about ten times less than the ones in the SRRs located parallel to the wave vector. The latter result agrees with the results of calculations current values in individual SRRs located perpendicularly to the wave vector [12] when no transmission drops, i.e., no resonance response was also observed. We conjecture that the field and current distributions in the SRRs located perpendicular to the wave vector, despite their resemblance in the distributions at the full-wavelength resonance in slot rings, could not be related to the resonance. Instead, they reflect oscillations of external fields between the metal sheets used to guide the EM wave in the metamaterial sample. The electric field between the sheets could induce opposite charges on the horizontal strips of each SRR and cause current oscillations in vertical strips of the SRRs. The external magnetic fields could also affect the SRRs located perpendicularly to the wave component of the mode could interact vector since the with the magnetic fields induced by current oscillations in vertical strips of these SRRs. Katsarakis et al. [6] and Gay-Balmaz and Martin [12] excluded any coupling of the external magnetic field with the SRRs located perpendicular to the wave vector. However, it seems to be true only for the case of half-wavelength resonance. C. EM Response of the Rods in the Unit Cell Our simulations have shown that the EM response of the SRRs located on the front and back dielectric plates of the unit cell of the metamaterial is strongly affected by the rod behavior. We have found that, at 9.9 GHz, the half-wavelength resonances occur in the front and back rods (microstrips), which is well seen from the amplitude and phase distributions of electric-field

Fig. 10. Top views on 3-D patterns of (a) phase and (b) amplitude distributions of the electric-field E x component in the X Z -plane, which crosses the front and back rods of the unit cell, as shown in Fig. 1(c) by a dashed–dotted line, at frequency f = 9:9 GHz. Arrows on the phase distribution mark the areas where fields are opposite in phase. Arrows on the amplitude distribution point at the areas of maximal coupling between the rods and SRRs. (c) Side view on the unit cell.

oscillations in the -plane crossing both front and back rods (Fig. 10). Essentially increased in amplitude oscillations of electric field form standing waves with the antinodes located near the middle of the rods, and the nodes observed at the connection points between the rods and metal sheets [see Fig. 10(b)]. The charges on the two rods responsible for the electric fields around the rods are always opposite in sign since the field oscillations in all space between the rods have the same phase [see Fig. 10(a)]. The obtained data add some questions about the mechanisms responsible for the enhanced transmission of metamaterial samples at frequencies close to 10.0 GHz. It is particularly doubtful that resonating rods can generate the response expected from the rod array and provide for effective negative permittivity of the media at frequencies below the plasma frequency. To find out whether or not the resonant rod behavior depends on the quantity of unit cells in the metamaterial sample, we repeated the study of resonances for the sample consisting of four cells [see Fig. 3(b)]. D. EM Response of the Four-Cell Sample In order to study the resonance processes in the four-cell sample, we have simulated the amplitude field distributions in the - and -planes located as shown in Fig. 11. It was found that, in difference from the one-cell sample, half-wavelength resonances in four-cell sample at frequencies around 10 GHz are observed not only in the SRRs located on the -plane, but also in the SRRs dielectric plates parallel to the located on the dielectric plates parallel to the -plane. However, the EM response of the SRRs belonging to the columns of both orientations was found to be essentially nonuniform. Fig. 12 describes the resonance behavior of the five columns marked by white color in Fig. 11. As seen from this figure, for most of the columns of the SRRs, there are three frequencies,

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Fig. 13. Amplitude distribution of oscillations of electric field in the -plane A–A (Fig. 11), at a frequency of 10.2 GHz.

XZ YZ

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Fig. 11. Locations of the - and -planes used to study EM-field standing waves in the SRRs (dashed lines 1–5) and in the rods (dashed–dotted lines A–A and B–B) in the four-cell sample. White rectangles are drawn instead of black rectangles to mark the SRR columns, for which field distributions were simulated.

Fig. 14. Frequency spectra of the electric-field amplitude (relative units) for the rods crossed by the A–A (1–3) and B–B (4–6) planes (see Fig. 11). Numbers for the rods in each plane are given from the left- to right-hand side (Fig. 11). Numbered arrows at the frequency axis mark the resonant frequencies of the SRRs in the columns nearest to the rods with the same number.

Fig. 12. Amplitude distributions of oscillations of the electric field in planes 1–5 located as shown in Fig. 11 at frequencies corresponding to maximal standing-wave amplitudes for at least one of the SRRs in each column. The numbers below the patterns represent the amplitude values in relative units.

at which field amplitudes in one, two, or all of the SRRs in the column demonstrate features typical for half-wavelength resonance in the SRRs. These frequencies are different for different columns and there are no obvious regularities in resonating of the SRRs in each of the columns. We relate this resonance behavior of the SRRs to the mode-splitting phenomena well known for systems of coupled resonators. Due to coupling between resonators and resonant mode splitting, each of the resonators could resonate at more than one frequency. As we have previously shown [13], even in a single microstrip square

patch, the resonant oscillations in each of the two orthogonal directions could be observed at two different frequencies if strong coupling between the oscillations in orthogonal directions is provided. The obtained results make doubtful the consideration of the metamaterial as a uniform media, which departs from the fact that the diameters of the SRRs and the distances between them are much smaller than the wavelength. As seen from our data, these small dimensions cannot provide for a uniform EM response of the SRRs. Simulation of the amplitude distributions of the electric field -planes A–A and B–B (Fig. 11) confirmed that the in the half-wavelength resonances in the rods (microstrips) occurred in the same frequency range in which the resonances in the SRRs were observed (see typical field distributions in Fig. 13). However, as seen from Fig. 13, when the field amplitude for one of the rods reaches its maximal resonant value, the resonances in other rods are still forming so that the resonances in different rods take place at different frequencies. Frequency spectra of the maximal electric field amplitudes simulated for six rods are presented in Fig. 14, which, in Fig. 11, -planes A–A and B–B, provide for a are crossed by the better insight into the process of resonance formation in the rods. As seen from Fig. 14, a sharp increase in the amplitude of electric-field standing waves occurs for most of the rods at two,

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Fig. 15. (a) Amplitude distribution of oscillations of the magnetic field Hy component in the XY -plane, which crosses the sample at the level, where the lower row of the SRRs is located, at a frequency of 10.05 GHz. (b) Top view on a four-cell sample.

three, or even four different frequencies, which are different for different rods. However, the minimums in the spectra of the field amplitudes for the rods are observed exactly at the resonant frequencies of the SRRs in the nearest columns. It means that the resonance formation in the rods and SRRs are interdependent. In other words, the resonating rods and SRRs are coupled. Fig. 15 presents the amplitude distribution for the oscillations component of magnetic field, which illustrates coupling of the between resonators in the four-cell sample. The field pattern is simulated in the -plane located below the midheight of the sample, where magnetic fields near the rods are maximal at the resonance. In Fig. 15, the rods with the strongest resonant fields are observed in the lower part of the pattern, where the bright spots are seen. The coupled fields between the SRRs and rods are seen all over the pattern. These fields produce an integrated EM response of the SRRs and rods. The data presented in Fig. 15 confirm that the EM response of the sample is nonuniform. It is worth noting that, while multiple resonances in the rods and coupling between the SRRs and resonating rods make questionable the applicability of the plasma concept to the analysis of the EM response of the metamaterial, there are studies indicating that the negative permeability could be provided by other means. In particular, Ziolkowski [14] and Mosallaei and RahmatSamii [15] have reported the study on metamaterials composed from two types of resonators: the SRRs and either capacitively loaded strips (CLSs) [14] or short straight wires [15]. The wires or strips were not connected with any metal parts of the waveguiding system. The authors observed passbands for their metamaterials within the frequency gap forbidden for propagation in the samples composed from resonators of only one type and related these passbands to both negative permittivity and permeability of the metamaterial. The plasma concept was not used to justify negative permittivity. Instead, Ziolkowski [14] demonstrated the Loretz-type behavior for the permittivity extracted from scattering parameters spectra for the metamaterial built from SRRs and CLSs. Although the response of such type was barely discernible in the metamaterial built from the CLSs only, i.e., without the SRRs, the author related the Loretz type of permittivity behavior to the CLSs. It follows from these considerations that the CLSs, in the presence of SRRs, were able to support the resonance modes analogous to electric dipoles. Mosallaei and Rahmat-Samii [15] suggested the formation of similar modes in short wires. Referring to these papers, we could also suppose that Loretz-type behavior of the rods in our metamaterial cause negative permittivity, and, correspondingly, negative

Fig. 16. (a) Schematic of cylindrical DR embedded into substrate under TEM-wave excitation, (b) 3-D field configuration, and (c) amplitude distribution of oscillations of the magnetic field in the XY -plane for the lowest resonant mode at 10.1 GHz. Parameters of the DR: " = 7:8, " = 77. DR diameter: 2.64 mm. DR’s height and substrate thickness: 1.5 mm.

Fig. 17. (a) Schematic of the all-dielectric metamaterial sample excitation. Two horn-shaped microstrips serve as the source and receiver to study transmission properties. (b) Amplitude distributions of oscillations of the magnetic field Hz component normal to the XY -plane at frequencies of 9.75 and 10.50 GHz.

refraction index at frequencies on the higher frequency side of the SRR resonance. However, much of our data also point at the significance of coupling between resonators for the left-handedness. The fact that, in both of the above cited papers, no signs of negative permittivity were found in the samples composed from the strip-like resonators alone also indicates at the importance of coupling effects. IV. EM RESPONSE OF ALL-DIELECTRIC METAMATERIAL To provide a better understanding of the impact of coupling between the resonators on the metamaterial transmission properties, we investigated another type of metamaterial constructed of the array of cylindrical DRs. The first resonant mode in these mode [16] and could be conresonators resembles the sidered as an equivalent to a magnetic dipole located in the -plane (Fig. 16). In this attitude, the DRs at the first resonance are comparable to the SRRs; however, the rotational symmetry of the cylinders provides magnetic dipoles with an additional opportunity to change their direction that creates better conditions for coupling between the DRs even in the absence of extended rods in the structure. Fig. 17(a) shows the schematic of a metamaterial sample constructed of DRs embedded in the low-permittivity matrix. The

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Fig. 19. Beam propagation through the prism of metamaterial designed as a low-permittivity matrix (" = 7:8) with embedded DRs (" = 62) organized in the rhombohedral lattice at frequencies of: (a) 16.0 and (b) 16.9 GHz.

Fig. 18. Transmission spectrum of the metamaterial sample (curve 1) in comparison with the spectrum of a dielectric block with the same dimensions and with the permittivity equal to averaged permittivity of the metamaterial sample (curve 2). Frequency is given in ( 10 GHz).

2

DRs formed a square lattice with a period of 3.6 mm comparable with the diameter of the DRs of 2.64 mm. Fig. 17(b) shows the field distributions in the structure at different frequencies. As seen from this figure, at frequencies when resonant conditions for individual DRs are still not achieved, the magnitudes of the field oscillations inside the DRs are small, and the magnetic dipoles are oriented perpendicularly to the wave vector of incident EM radiation. In the vicinity of the resonance, the horizontal magnetic dipoles turn to couple increased electric or magnetic fields inside the DRs and form geometric patterns of coupled fields, which are specific for each frequency [an example of the pattern at 10.5 GHz is given in Fig. 17(b)]. Fig. 18 presents the transmission spectrum of the metamaterial sample in comparison with the transmission spectrum of a piece of dielectric with the same dimensions and with the permittivity equal to averaged permittivity of the metamaterial. As seen from this figure, at frequencies in the vicinity of the first resonant frequency of a single DR, a band of enhanced transmission is observed. We also observed transmission bands in the vicinities of higher frequency resonances of individual DRs. One of them, seen in Fig. 18 at 13.5 GHz, is related to the resonant mode equivalent to a vertical electric dipole. The ability of all-dielectric metamaterial to provide negative refraction has been demonstrated by simulation EM wave propagation through the prism of metamaterial with a rhombus lattice at the frequencies close to the first higher order resonance of the DRs (Fig. 19). By examining the phase fronts, visualized by using reflections from spherically located receivers, we observed a negative beam refraction at 16.0 GHz, while at 16.9 GHz, this beam weakened and positive refraction prevailed. We should point out that the observed anomalies in refraction are not related to the Bragg-scattering phenomena described for photonic-bandgap crystals [17]. We have verified this by calculated the band structure of the DR array. The results of this study, as well as the description of higher frequency transmission bands, will be published elsewhere.

Fig. 20. Simulated and measured reflection spectra of the prototype with DRs made of BZT (" = 62, diameter: 3.06 mm, height: 1.53 mm) within glass bonded alumina matrix (" = 7:8, thickness: 1.53 mm).

The performed analysis lead us to suggest that, in the metamaterial consisting of the SRRs and rods, one of the mechanisms providing for the enhanced transmission and the left-handed behavior could also be related to the coupling between resonators. We have fabricated the prototype of all-dielectric metamaterial by using the low-temperature co-fired ceramic (LTCC) technology for co-processing bismuth zinc tantalite (BZT) with permittivity of 62 and a glass-bonded alumina matrix with a permittivity of 7.8. This process was previously developed for fabrication of miniature microstrip filters with a substrate containing high-permittivity dielectric inclusions [18]. As seen from Fig. 20, the reflection spectrum of the prototype demonstrated a good agreement with the simulated spectrum of the metamaterial with the same parameters. Both spectra have deep drops at the frequencies close to the resonance frequency of the DRs that points at increased transmission caused by coupling between the DRs. The first minimum in the spectra is related to the wave reflections between the transmitter and receiver and is observed in the uniform media as well. The second minimum corresponds to the first resonance mode of DRs, i.e., to the magnetic dipoles formation, while the third one corresponds to the second resonance mode of DRs, i.e., to the formation of electric dipoles. The location and bandwidth of these two minimums were defined by the DRs parameters and were not observed in uniform substrate structures.

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V. CONCLUSION We have examined resonance processes in metamaterials consisting of SRRs and rods by using the FDTD method for visualization EM-field standing waves in the frequency range where enhanced transmission was observed. It has been demonstrated that besides the half-wavelength resonances in the SRRs, similar resonances occur in the rods at close frequencies, and that both types of resonators interact with each other. It makes questionable the applicability of the concepts developed for the rod array to the analysis of the metamaterial properties, thus, the suggestion of simple superposition of the SRR and rod responses. In fact, the SRRs and rods form 3-D coupled groups, rearranging with frequency. Coupling between resonators causes resonant mode splitting and promotes channeling of EM energy by coupled fields that, possibly, contributes to formation of the bands with enhanced transmission. Simulation of all-dielectric metamaterials, as well as measurements of the fabricated prototypes constructed from arrays of DRs embedded in a low-permittivity matrix, confirmed that coupling between the resonators could cause an enhanced transmission, thus, negative refraction of the metamaterial. REFERENCES [1] V. G. Veselago, “The elctrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [2] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Young, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [3] D. R. Smith, W. J. Padfilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 128, pp. 4184–4187, May 2000. [4] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett., vol. 78, no. 4, pp. 489–491, Jan. 2001. [5] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [6] N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett., vol. 84, no. 15, pp. 2943–2945, Apr. 2004. [7] C. D. Moss, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Numerical studies of left-handed metamaterials,” Progr. Electromagn. Res., vol. PIER 35, pp. 315–334, 2002. [8] E. Semouchkina, W. Cao, R. Mittra, and W. Yu, “Analysis of resonance processes in microstrip ring resonators by the FDTD method,” Microwave Opt. Technol. Lett., vol. 28, no. 5, pp. 312–321, Mar. 2001. [9] P. Markos and C. M. Soukoulis, “Left-handed metamaterials,” arXiv: Cond-Mat/0 212 136, vol. 1, pp. 1–11, Dec. 2002. [10] K. Chang, Microwave Ring Circuits and Antennas. New York: Wiley, 1996. [11] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001, Microwave Opt. Eng.. [12] P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys., vol. 92, no. 5, pp. 2929–2936, Sep. 2002. [13] E. Semouchkina, G. Semouchkin, and M. Lanagan, “FDTD analysis of dual-mode microstrip antenna,” in IEEE AP-S Int. Symp. Dig., vol. 3, Jun. 2003, pp. 772–776. [14] R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1516–1529, Jul. 2003. [15] H. Mosallaei and Y. Rahmat-Samii, “Composite materials with negative permittivity and permeability properties: Concept, analysis, and characterization,” in IEEE AP-S Int Symp. Dig., vol. 4, 2001, pp. 378–381. [16] D. Kaifez and P. Guillon, Dielectric Resonators. Atlanta, GA: Noble, 1998.

[17] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: Negative refraction by photonic crystals,” Nature, vol. 423, no. 69-40, pp. 604–605, 2003. [18] E. Semouchkina, A. Baker, G. Semouchkin, M. Lanagan, and R. Mittra, “New approaches for designing microstrip filters utilizing mixed dielectrics,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 644–652, Feb. 2005, to be published.

Elena A. Semouchkina (M’04) received the M.S. degree in electrical engineering and Candidate of Science degree in physics and mathematics from Tomsk State University, Tomsk, Russia, in 1978 and 1986, respectively, and the Ph.D. degree in materials from The Pennsylvania State University, University Park, in 2001. She was a Scientist with Russian academic centers such as the Siberian Physics–Technical Institute, St. Petersburg State Technical University, and Ioffe Physics–Technical Institute, where she was involved with the investigation of metal–oxide–semiconductor devices and the development of infrared photodetectors. Since 1997, she has been with the Materials Research Institute, The Pennsylvania State University, initially as a Graduate Research Assistant, then as a Post-Doctoral Scholar and, since 2004, as a Research Associate. She has authored or coauthored approximately 50 publications in scientific journals. Her current research interests are focused on computational analysis of EM processes in microwave materials, metamaterials, and devices. Dr. Semouchkina was a recipient of the Xerox 2001 Research Award at The Pennsylvania State University for the best Ph.D. thesis and the National Science Foundation 2004 Advance Fellows Award.

George B. Semouchkin received the M.S. degree in electrical engineering, Ph.D. degree in materials, and Doctor of Science degree in physics and mathematics from the Leningrad Polytechnic Institute (now St. Petersburg State Technical University), St. Petersburg, Russia, in 1962, 1970, and 1990, respectively. Prior to joining The Pennsylvania State University, University Park, in 1999, he was with the St. Petersburg State Technical University, as a Professor, a Leading Scientist, a Head of the Laboratory, and earlier as a Senior Scientist, where he studied ionic crystals, ceramic materials, inorganic dielectrics, and developed microelectronic devices. He is currently a Visiting Professor of materials with the Materials Research Institute, The Pennsylvania State University. He has authored over 130 technical publications. His current research interests include designing LTCC-based microwave devices and all-dielectric metamaterials.

Michael Lanagan (M’99) received the B.S. degree in ceramic engineering from the University of Illinois at Urbana-Champaign, in 1982, and the Ph.D. degree in ceramic science and engineering from The Pennsylvania State University, University Park, in 1987. He is currently Professor of materials science and engineering, Associate Director of the Materials Research Institute, and Associate Director of the Center for Dielectric Studies with The Pennsylvania State University. Prior to joining The Pennsylvania State University, he was a Staff Scientist for 12 years with the Argonne National Laboratory, where he studied materials for superconductors, molten carbonate fuel cells, and high-energy density capacitors. He has authored over 150 technical publications. He holds eight patents. His current research interests include the development of new dielectric materials for high-energy density capacitors and microwave metamaterials. Dr. Lanagan is a member of The American Ceramic Society and The International Microelectronics and Packaging Societies. He was an invited participant to the National Academy of Engineering’s “Frontiers of Engineering,” which recognizes promising young scientists in all areas of research.

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Clive A. Randall received the B.S. degree in physics from the University of East Anglia, Norwich, U.K., in 1983, and the Ph.D. degree in experimental physics from the University of Essex, Essex, U.K., in 1987. In 1987, he joined The Pennsylvania State University, University Park, as a Research Associate, where he became a Senior Research Associate in 1992, Associate Professor of materials science and engineering in 1994, and Professor of materials science and engineering in 1999. In 1997, he became the Director of the Center for Dielectric Studies, Materials Research Institute, The Pennsylvania State University. He has been a Visiting Scientist with the Oak Ridge National Laboratory (1990), the Shonan Institute of Technology, Kanagawa, Japan (1994), and the Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland (1997). He is currently on sabbatical leave with the Interdisciplinary Research Collaboration (IRC), Cambridge University, Cambridge, U.K., where he is involved with superconductivity. He possesses over 15 years experience in electroceramics. He is an internationally recognized expert on microstructural effects in ferroelectric materials. He has authored or coauthored over 170 publications. He holds eight patents. His specific expertise includes transmission electron microscopy, size effects on ferroelectricity, defect chemistry, fine ceramic particle assemblage, and electroceramic characterization. Dr. Randall is a member of the American Ceramic Society, Materials Research Society, and the Pennsylvania Ceramics Association. He was the recipient of the 2002 Fulrath Award.

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Periodic Finite-Difference Time-Domain Analysis of Loaded Transmission-Line Negative-Refractive-Index Metamaterials Titos Kokkinos, Costas D. Sarris, Member, IEEE, and George V. Eleftheriades, Senior Member, IEEE

Abstract—In this paper, a systematic methodology is proposed for the full-wave time-domain analysis of planar loaded transmission-line (TL)-based negative-refractive-index (NRI) media. Lumped inductors and capacitors are incorporated in a finite-difference time-domain (FDTD) mesh via an “extended FDTD” approach, combining Maxwell’s equations with lumped-element voltage–current characteristics. The analysis is facilitated by periodic boundary conditions that restrict the simulated domain to a single unit cell of the periodic loaded TL grid. Thus, fast numerical characterization of the NRI structures under study is attained. The proposed technique is successfully compared to measured results and is proven to consist of an effective tool for the rigorous explanation of experimental observations through the dispersion analysis and computation of modal patterns and relative strengths of the types of waves supported by an NRI medium. Index Terms—Dispersion, finite difference time domain (FDTD), negative refractive index (NRI).

I. INTRODUCTION

I

N A seminal paper [1], Veselago predicted the possibility of media exhibiting simultaneously negative dielectric permittivity and magnetic permeability . Analysis of the definition of the index of refraction, as a square root of the product of and , shows that, in such media, the index assumes negative values. Therefore, they can be referred to as negative-refractive-index (NRI) media, as opposed to conventional positive-refractive-index (PRI) ones. NRI implies the support of backward waves in such media. As a result, a number of unconventional properties, also pointed out by Veselago, are enabled, such as negative refraction, inverted Doppler shift, Cerenkov radiation, and furthermore, Pendry’s concept of a “perfect lens” [2]. Recently, Veselago’s theoretical predictions were experimentally validated through the engineering of composite media of periodic inclusions, macroscopically exhibiting an NRI. Examples of those are the three-dimensional split-ring resonator (SRR) medium of [3] and the planar loaded transmission-line (TL) grid media of [4] and [5], which will be further investigated in this paper. On the front of modeling, earlier studies in the frequency domain (via the finite-element method and commercial packages) Manuscript received June 16, 2004; revised October 27, 2004. This work was supported by the Natural Sciences and Engineering Research Council of Canada under a Discovery Grant and a Strategic Grant. The authors are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada M5S 3G4 (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845197

Fig. 1. Unit cell of the loaded TL NRI medium of [5].

have verified the properties that were analytically predicted by Veselago [6], the growth of evanescent waves in the loaded TL NRI media [7], and the dispersion properties of the latter [8]. Yet the time-domain modeling of metamaterials, via the finite-difference time-domain (FDTD) method, is motivated by the richness of the transient behavior that it captures. In fact, recent FDTD analyses of negative refraction at an interface between a positive and negative index medium [9], [10] have contributed to the resolution of questions regarding the causal evolution of negatively refracted wavefronts. Both [10] and [11] and the widely referenced work by Ziolkowski and Heyman [12] pursue the FDTD implementation of the macroscopic dispersive index of of NRI media. Hence, the FDTD cell size does refraction not necessarily follow the size of the (electrically small) inclusions that practically give rise to the negative index. On the other hand, the existence of backward waves requires a modified formulation of the numerically involved perfectly matched layer (PML) absorber for NRI media [13], while resistive terminations can only offer a narrow-band alternative. Obviously, such models are useful for the qualitative evaluation of generic NRI structures, while they offer limited insight to the operation of particular implementations of them. More recently, the TL-matrix method was successfully modified to model NRI media, as reported in [14]. This paper extends upon the research previously reported by the authors in [15] and focuses on the modeling of the loaded two-dimensional TL-NRI metamaterial of [5]. The latter is a periodic structure of a unit cell shown in Fig. 1. This is a dual two-dimensional TL in the sense that it is derived by replacing the shunt capacitors and series inductors of the conventional TL with shunt inductors and series capacitors, respectively. Continuous one-dimensional dual TLs have been fairly well known [16], and were recently related to NRI materials via periodic implementations of those in [4] and [17]–[19]. The modeling of

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the lumped elements, shunt inductors, and series capacitors, included in the unit cell of Fig. 1, is readily accomplished by the “extended FDTD” method [20], [21], which couples lumpedelement voltage–current characteristics with Maxwell’s equations. Since the unit cell is electrically small (typically of the order of a tenth of a wavelength), its discretization imposes a severe limitation on the choice of the FDTD cell size. Consequently, the simulation of practical grids consisting of moderate-to-large numbers of unit cells is bound to present a computationally intensive problem. To address this issue, periodic boundary conditions (PBCs) are implemented at the boundaries of the unit cell, enabling the fast modeling of metamaterial structures. For this purpose, the so-called sine–cosine method [24], [25] is employed for the translation of the PBCs to the time do. In the context of main for any lattice wave vector the proposed method, the use of well-known FDTD absorbing boundary conditions, including the PML, is possible since the mesh is truncated along an isotropic nondispersive boundary. In fact, this approach is far more general since circuit models can be associated to other NRI implementations, notably the SRR NRI [22], [23]. These models can be readily expressed in terms of state equations in the sense of [21]. Hence, they can be incorporated in an FDTD simulation domain, terminated with periodic/absorbing boundary conditions. It is noted, however, that since PBCs are enforced, phenomena associated with truncated periodic structures cannot be accounted for. These can be readily modeled by applying the conventional FDTD technique to the truncated structure of interest using the update equations presented in this paper for the working volume and any kind of absorbing boundary conditions for mesh termination. The validity of this approach was demonstrated by the authors in [15] and is not further negotiated here. The structure of this paper is as follows. First, the proposed modeling approach is explained and the possibilities that it provides (as a full-wave method) for the insightful physical characterization of the simulated structures are mentioned. Subsequently, the Brillouin diagrams for different implementations of the NRI medium of [5] are deduced and successfully compared with available measured and simulated data. Finally, modal field plots for forward, backward, and surface waves excited at distinct frequencies are presented. Their analysis and discussion illuminates the physics of the dual TL NRI medium and supports several published experimental observations. II. PERIODIC/EXTENDED FDTD ANALYSIS OF LOADED TL-NRI METAMATERIALS Floquet’s boundary conditions for two-dimensional periand along the – and – odic media of spatial period axes of a rectangular coordinate system state that phasor field components one period away in either direction differ only , by a constant attenuation and phase-shift term , respectively ( and being complex propagation constants). Several ways have been proposed for the translation of this frequency-domain relationship to the time domain. In this study, the propagation constants are defined to be real, while the sine–cosine method of [25] is implemented. According to it, two separate grids are considered,

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and corresponding to field components with time dependence and . For example, if is the lattice vector and is the lattice wave vector, Floquet’s conditions on electric-field phasors [denoted by a tilde ( )] assume the form (1) If a frequency-to-time domain transform is considered for and , the following set of time-domain conditions is then derived:

(2) The propagation constants and in these conditions are chosen to represent a point on the – -, – -, – -axis of the Brillouin diagram [26]. For each such point, the timedomain waveforms of fields are sampled within the unit cell. Subsequently, their Fourier transform (from the time domain to frequency domain) reveals the resonances that represent the . modal frequencies The three-dimensional FDTD domain for the NRI medium of [5] is depicted in Fig. 2. The open boundary in the -direction is terminated in Mur’s first-order absorbing boundary condition, which is deemed sufficiently accurate for a distance of a few wavelengths from the interface. Periodicity in the - and -directions is enforced by the PBCs. The unit cell of the simulated NRI medium (Fig. 1) includes lumped capacitive and inductive elements. These can be practically realized by components such as capacitive gaps or vias provided that the latter are electrically small at the operating frequency of the NRI medium. This typically being the case, the use of lumped-element equivalent representations of these components facilitates both their analytical and numerical modeling. An example of how lumped-element equations are coupled to FDTD updates is given for the case of a capacitor along a parallel to the -axis edge of Yee’s cell with a voltage–current characteristic . Assuming that is the -component of the electric field sampled on a node at the center of the capacitor, its update equation can be written as (3) Similarly, electric-field components along –oriented inductors are updated as

(4) As mentioned in [25], a point source can excite all the modes implicitly introduced by the boundary confor a set of ditions. Line sources can be used as well, properly oriented in or modes. This technique also proorder to excite the vides the possibility to study one of the modes supported by

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Fig. 3. Brillouin diagram for loaded TL-NRI structure of [27], computed by the periodic/extended FDTD method of this paper.

III. DISPERSION ANALYSIS

Fig. 2. Computational unit cell for the simulation of a loaded TL NRI medium.

the structure for a fixed wave vector . The pattern of that mode is extracted by the Fourier transform of the sampled . Contime-domain signal, evaluated at the frequency versely, modes of the same frequency and different wave vector can be studied and directly compared, their patterns being extracted from separate simulations of domains terminated with their respective boundary conditions. In the results that follow, a line source has been used for the excitation of the TM with respect to the -axis waves.

Implementing the aforementioned concepts, the dispersion analysis of the loaded TL NRI medium of [27] is performed. mm, the height The periodicity in both and is mm, and its relative permittivity is of the substrate is . The width of the TL is 0.4 mm and the characteristic . The values impedance of the microstrip line is of the lumped elements loading the TL mesh are nH pF, respectively. The unit cell is discretized in and cells with , while the substrate occupies three cells. The time step is set to ps (0.9 of the Courant limit). As an excitation, a Gabor function with and is GHz used in the sine–cosine grid, respectively, with and GHz. Fast convergence was obtained in all cases with the necessary number of time steps being limited to approximately 7000. The results of this analysis are shown in Fig. 3, where the wavenumber varies within the irreducible Brillouin zone [26]. Three lower order TM modes appear to be present, which are: 1) a surface TM wave, which follows the light cone and is spatially confined close to the TL metal; 2) a TM forward wave similar to the one supported by an unloaded TL mesh; and 3) a TM backward wave. The latter is readily characterized by its time-domain field waveforms, as discussed in [15]. In addition, Fig. 4 presents a comparison between our results and relevant experimental data of [27], along with results obtained by Ansoft’s HFSS. While an overall satisfactory agreement between experimental and theoretical results is observed, the following notes are in order. First, the full-wave analysis indicates the existence of a lower stopband formed at around 2.4 GHz. As mentioned in [8], this stopband stems from the contra-directional coupling of the backward wave with the surface wave. This effect is not captured by the TL theory-based periodic analysis of the two-dimensional TL-NRI medium [31] since the latter does not include surface-wave modes. Second, for frequencies where the backward wave and surface wave

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Fig. 4. Comparison of the dispersion diagram of Fig. 3 to measured results from [27]. Results from Ansoft’s HFSS and TL theory are appended.

are not phase matched, the measured phase follows closely the backward-wave dispersion curve. Good agreement is also observed for the frequencies where the forward wave is supported. On the other hand, the measurements deviate from the simulated phase at frequencies where the two waves tend to become phase matched. This effect is further investigated and explained in Section IV. Recently, loaded TL NRI structures, designed so that the edges of the backward-wave and forward-wave dispersion , thus closing the upper stopband curves meet at existing between the two, have attracted a particular interest, among other reasons, due to their significance for the implementation of Pendry’s planar “perfect lens” [28], [29]. The condition for the closure of the stopband assumes the form [5, eq. (29)], [30]

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Fig. 5. Brillouin diagram for a closed stopband loaded TL-NRI structure.

computational time that this technique requires for the extraction of the whole Brillouin diagram is less than 10 min. These conclusions are relatively insensitive to the values of the loading capacitors and inductors. IV. MODAL FIELD PATTERNS IN THE LOADED TL-NRI MEDIUM A. Modal Pattern Determination From an FDTD Simulation The field patterns of the three types of modes indicated in the dispersion diagram of Fig. 3 are further investigated here. These are derived from the Fourier transform of field components within the FDTD computational domain, iteratively applied on their time samples. In particular, the Fourier transform of the electric field is evaluated at a frequency as

(6) (5) where is the characteristic impedance of the TL and and are the loading inductors and capacitors. Fig. 5 displays the dispersion diagram of the previously analyzed structure, where the 5.6-nH inductor has been replaced by a 10-nH inductor so that condition (5) is met. The simulation , result confirms that the stopband is closed at GHz, within the accuracy, in the frequency domain, provided by the finite time step of the FDTD. For the calculations MHz. presented in Fig. 5, the frequency resolution is The lower stopband is formed between 2.15–2.35 GHz. In terms of computational efficiency, an exceptional performance of the algorithm was observed. The results of a convergence study for several frequency points of the Brillouin diagram were presented in [15]. In that study, a uniform discretization rate of 16 grid points per period proved to be optimal in terms of accuracy and execution time. Approximately 15 s were sufficient for the determination of the resonant frequencies for each lattice wave vector on a Pentium 4 2.6-GHz PC (without any code optimization). Hence, a rough estimate for the total

where is the position vector of is the number of time steps used an FDTD grid point and for the Fourier transform. The presented case studies refer to the following three points on the Brillouin diagram (indicated as A–C in Fig. 3, respectively), all three being within the – region of the Brillouin zone. • Point A corresponds to a surface wave excited at a frequency GHz propagating with . • Point B corresponds to a backward wave excited at the previous frequency GHz propagating with . • Point C corresponds to a forward wave propagating with the previous phase difference per unit cell ( ) at a frequency GHz. Field data for these three points are selectively presented in Figs. 6–11. All fields have been sampled in the middle of the dielectric substrate. In Fig. 6, the magnitude of the -compowithin nent of the backward-wave electric field (point B) the unit cell is plotted. It is noted that this component is mainly

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Fig. 6. Magnitude of the x-component of the backward-wave electric field for propagation in the x-direction at 0.99 GHz. The field is supported at the capacitors, where it contributes to a displacement current.

Fig. 8. Magnitude of the x-component of the surface-wave electric field for propagation in the x-direction at 0.99 GHz. It assumes negligible values throughout the unit cell.

Fig. 7. Magnitude of the x-component of the forward-wave electric field for propagation in the x-direction at 12.83 GHz. It assumes negligible values since this TL mode is predominantly z -directed.

Fig. 9. Magnitude of the z -component of the backward-wave electric field for propagation in the x-direction at 0.99 GHz. The field cannot propagate along the gap of the capacitors in the x-direction (where the E supports the displacement current) and, hence, it drops to zero there.

supported within the capacitors at the ends of the cell in the direction of propagation. Indeed, this mode is dominantly -polarized. However, as the TL along the -axis is interrupted by -component is developed, supporting a the capacitors, the displacement current at the two gaps. On the other hand, the -components of the forward wave and surface wave assume negligible values throughout the cell, as shown in Figs. 7 and 8. Indeed, the forward wave corresponds to an unloaded TL mode and, hence, it is expected to be vertically polarized. Figs. 9–11 support these comments by depicting the -components of the three modal fields. First, Fig. 9 shows the backward-wave being reduced to zero at the capacitors along the propagation gives rise to a displacement current. The foraxis, where ward wave of point C is plotted in Fig. 10. The field is maximized at the capacitors along the -direction. This stems from

the fact that the symmetry of the unit cell, combined with the chosen direction of wave propagation (along the -axis) imply the existence of two -directed magnetic walls passing through the centers of the gaps of these capacitors. Finally, Fig. 11 provides the surface-wave (point A) component , which follows closely the metal of the TL mesh. B. Analysis of Coupling Between Backward-Wave and Surface-Wave Modes In the following, the results of Fig. 4 are further discussed based on the extracted modal patterns. The fields of Figs. 9 and GHz and they have been 11 are at a common frequency excited by the same Gabor pulse provided by a line source. Thus, the comparison of their magnitudes reveals the relative coupling coefficients of the source to these modes. Careful inspection of

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Fig. 10. Magnitude of the z -component of the forward-wave electric field for propagation in the x-direction at 12.83 GHz. The field assumes a maximum at the magnetic walls at the end capacitors of the domain in the y -direction.

Fig. 12. Magnitude of the z -component of the backward-wave electric field for propagation in the x-direction at 1.94 GHz. The field becomes phase matched to the surface-wave mode and follows the TL metal.

Fig. 11. Magnitude of the z -component of the surface-wave electric field for propagation in the x-direction at 0.99 GHz. The field follows closely the TL metal, while it is weakly excited compared to the backward wave.

Fig. 13. Magnitude of the z -component of the surface-wave electric field for propagation in the x-direction at 1.94 GHz. As surface and backward waves become phase matched, their amplitudes assume similar values.

those establishes the fact that the surface wave is excited two orders of magnitude below the backward wave. This result offers a numerical validation to the dispersion measurements of [27], where the phase seems to follow the backward-wave dispersion curve without being affected by the surface-wave line prediced by both HFSS and this method. Evidently, this is due to the dominance of the backward wave in these frequencies. A question that naturally arises is how these coupling coefficients vary as the two waves become phase matched. This is the subject of the numerical experiment that follows, in which backward-wave and surface-wave vertical fields are computed , GHz) and ( , for ( GHz), respectively. These wave vector-frequency pairs are close to the point where the lower branch of the dispersion diagram is separated into the backward wave and surface wave

(where the two waves are phase matched). The results of this experiment are shown in Figs. 12 and 13. Indeed, the two waves appear similar in both shape and magnitude, which is in agreement with the intuitive expectation that this would be the case as they move toward their phase-matching point. Hence, the measured phase close to the phase-matching point deviates from the backward-wave dispersion curve, reflecting the strong excitation of the surface wave and its strong coupling to the backward wave. V. CONCLUSION This paper has presented a rigorous full-wave time-domain technique for the modeling of loaded TL based NRI media. The modeling of the lumped elements loading a TL mesh was allowed for by the “extended FDTD” approach of [20] and [21].

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PBCs were applied to reduce the modeling of the whole structure to the simulation of a single unit cell, thus rendering the necessary execution time for each of the aforementioned studies (computation of Brillouin diagrams and modal field patterns) to the order of a few minutes. The proposed method is absolutely stable, straightforward to implement, and requires no special absorbing boundary conditions. The presented results were supported by successful comparisons to existing experimental and theoretical data and agreed with fundamental physical arguments. They were shown to provide critical insights to the physical properties of the simulated medium and the types of waves that it supports. Specifically, this technique enabled the examination of the relative excitation magnitudes of surfacewave and backward-wave modes, which led to the conclusion that the latter is dominant compared to the former, which in agreement with experimental results reported in [5] and [27]. In consequence, the proposed technique constitutes an efficient tool for the in-depth understanding of the physics of the loaded TL-based NRI structures and the computer-aided design of applications based on those. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of  and ,” Soviet Phys.—Usp., vol. 10, pp. 509–514, 1968. [2] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966–3969, Oct. 2000. [3] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 78, pp. 2933–2936, Oct. 2000. [4] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D waves,” in IEEE MTT-S Int. Microwave Symp. Dig., Seattle, WA, Jun. 2–7, 2002, pp. 1067–1070. [5] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [6] C. Caloz, C.-C. Chang, and T. Itoh, “Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,” J. Appl. Phys., vol. 90, no. 11, pp. 5483–5486, Dec. 2001. [7] A. Grbic and G. V. Eleftheriades, “Growing evanescent waves in negative-refractive-index transmission-line media,” Appl. Phys. Lett., vol. 82, no. 12, pp. 1815–1817, Mar. 2003. , “Dispersion analysis of a microstrip based negative refractive [8] index periodic structure,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 4, pp. 155–157, Apr. 2003. [9] S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett., vol. 90, no. 10, p. 107 402, Mar. 2003. [10] S. A. Cummer, “Dynamics of causal beam refraction in negative refractive index materials,” Appl. Phys. Lett., vol. 82, 2003. , “Simulated causal subwavelength focusing by a negative refrac[11] tive index slab,” Appl. Phys. Lett., vol. 82, p. 1503, 2003. [12] R. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 64, p. 056 625, 2001. [13] S. A. Cummer, “Perfectly matched layer behavior in negative refractive index materials,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 172–175, 2004. [14] P. M. So and W. J. R. Hoefer, “Time domain modeling of metamaterials with negative refractive index,” in IEEE MTT-S Int. Microwave Symp. Dig., Ft. Worth, TX, Jun. 6–11, 2004, pp. 1779–1782. [15] T. Kokkinos, R. Islam, C. D. Sarris, and G. V. Eleftheriades, “Rigorous analysis of negative refractive index metamaterials using FDTD with embedded lumped elements,” in Proc. IEEE MTT-S Int. Microwave Symp. Dig., Ft. Worth, TX, Jun. 6–12, 2004, pp. 1783–1786.

[16] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. New York: Wiley, 1994. [17] A. Grbic and G. V. Eleftheriades, “A backward-wave antenna based on negative refractive index L–C networks,” in IEEE Int. AP-S Symp., vol. 4, San Antonio, TX, Jun. 16–21, 2002, pp. 340–343. [18] C. Caloz, H. Okabe, H. Iwai, and T. Itoh, “Transmission line approach of left-handed materials,” in USNC/URSI Nat. Radio Science Meeting, San Antonio, TX, Jun. 16–21, 2002, p. 39. [19] A. A. Oliner, “A periodic-structure negative-refractive-index medium without resonant elements,” in USNC/URSI Nat. Radio Science Meeting, San Antonio, TX, Jun. 16–21, 2002, p. 41. [20] W. Sui, D. A. Christensen, and C. H. Durney, “Extending the two-dimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elements,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 4, pp. 724–730, Apr. 1992. [21] M. Picket-May, B. Houshmand, and T. Itoh, “High speed electronic circuits with active and passive lumped elements,” in Computational Electrodynamics : The Finite Difference Time Domain Method, A. Taflove and S. Hagness, Eds. Norwood, MA: Artech House, 1995, ch. 15. [22] G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, “Transmission line models for negative refractive index media and associated implementations without excess resonators,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 2, pp. 51–53, Feb. 2003. [23] K. Sarabandi and H. Mosallaei, “Embedded-circuit meta-materials for novel design of tunable electro-ferromagnetic permeability, band-gap and bi-anisotropic media,” in Proc. IEEE AP-S Symp., 2003, pp. 355–358. [24] A. Taflove and S. Hagness, “Analysis of periodic structures,” in Computational Electrodynamics: The Finite Difference Time Domain Method. Norwood, MA: Artech House, 1995, ch. 13. [25] P. Harms, R. Mittra, and W. Ko, “Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures,” IEEE Trans. Antennas Propag., vol. 42, no. 9, pp. 1317–1324, Sep. 1994. [26] L. Brillouin, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, 1st ed. New York: McGraw-Hill, 1946. [27] A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades, “Experimental and theoretical verification of focusing in a large periodically loaded transmission line negative refractive index metamaterial,” Opt. Express, vol. 11, pp. 696–708, Apr. 2003. [28] A. Grbic and G. V. Eleftheriades, “Negative refraction, growing evanescent waves and sub-diffraction imaging in loaded transmission line metamaterials,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 12, pp. 2297–2305, Dec. 2003. , “Overcoming the diffraction limit with a planar left-handed trans[29] mission line lens,” Phys. Rev. Lett., vol. 92, no. 11, pp. 117 403–??? ???, Mar. 2004. [30] A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 2, pp. 68–70, Feb. 2004. [31] A. Grbic and G. V. Eleftheriades, “Periodic analysis of a 2-D negative refractive index transmission line structure,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2604–2611, Oct. 2003.

Titos Kokkinos received the Diploma degree in electrical and computer engineering from the National Technical University of Athens (NTUA), Athens, Greece, in 2003, and is currently working toward the M.A.Sc. degree in electrical and computer engineering at the University of Toronto, Toronto, ON, Canada. In 2001, he joined Siemens AG, Zürich, Switzerland. From May 2002 to June 2003, he was with the Optical Networking Group, NTUA. Since September 2003, he has been with the Electromagnetics Group, University of Toronto. His research interests include computational electromagnetics, NRI metamaterials, antennas design, microwave circuits design, and optical networks. Mr. Kokkinos was nominated for the Fulbright Fellowship. He was the recipient of the University of Toronto Open Fellowship and the Onassis Foundation Fellowship for graduate studies.

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Costas D. Sarris (M’03) received the Diploma degree (with distinction) in electrical and computer engineering from the National Technical University of Athens (NTUA), Athens, Greece in 1997, and the M.Sc. degree in electrical engineering, M.Sc. degree in applied mathematics, and Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 1998, 2002, and 2002, respectively. In November 2002, he joined the Edward S. Rogers Sr. Department of Electrical and Computer Engineering (ECE), University of Toronto, Toronto, ON, Canada, where he is currently an Assistant Professor. His research interests are in the area of computational electromagnetics with emphasis on time-domain techniques, including wavelet-based methods, wireless channel characterization and stochastic ray-tracing, wave propagation modeling in linear/nonlinear media and periodic structures, and electromagnetic compatibility/interference problems (EMC/EMI). Dr. Sarris was the recipient of a Student Paper Award (honorable mention) presented at the 2001 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium, Phoenix, AZ.

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George V. Eleftheriades (S’86–M’88–SM’02) received the Diploma degree (with distinction) in electrical engineering from the National Technical University of Athens, Athens, Greece in 1988, and the Ph.D. and M.S.E.E. degrees in electrical engineering from The University of Michigan at Ann Arbor, in 1993 and 1989, respectively. From 1994 to 1997, he was with the Swiss Federal Institute of Technology, Lausanne, Switzerland, where he was engaged in the design of millimeter and sub-millimeter-wave receivers and in the creation of fast computer-aided design (CAD) tools for planar packaged microwave circuits. He is currently a Professor with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada. He has authored or coauthored over 80 papers in refereed journals and conference proceedings. His current research interests include negative-refraction metamaterials, integrated-circuit (IC) antennas and components for broadband wireless communications, novel beam-steering techniques, low-loss silicon micromachined components, millimeter-wave radiometric receivers, and electromagnetic design for high-speed digital circuits. Dr. Eleftheriades was a corecipient of the 1990 Best Paper Award presented at the 6th International Symposium on Antennas (JINA) and the Ontario Premier’s 2001 Research Excellence Award. His graduate students were the recipients of Student Paper Awards presented at the 2000 Antenna Technology and Applied Electromagnetics Symposium, the 2002 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS), and the 2002 IEEE International Symposium on Antennas and Propagation. Moreover, he was the recipient of a Steacie Memorial Fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC) in 2004.

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Modeling of Metamaterials With Negative Refractive Index Using 2-D Shunt and 3-D SCN TLM Networks Poman P. M. So, Senior Member, IEEE, Huilian Du, and Wolfgang J. R. Hoefer, Fellow, IEEE

Abstract—We present two novel techniques for modeling twoand three-dimensional metamaterials with negative refractive index by transmission-line matrix (TLM) networks. The TLM networks are numerical models of reactively loaded periodic networks that support backward waves and constitute artificial media with negative refractive index. The TLM models, their implementations, and results computed with the models will be presented. Index Terms—Artificial dielectrics, backward waves, focusing, inter-cell scattering procedure, left-handed media (LHM), metamaterials, negative permeability, negative permittivity, negative refractive index, periodic structures, transmission-line matrix (TLM) method.

I. INTRODUCTION

M

ATERIALS with negative refractive index, first discussed by Veselago [1] in the late 1960s, have recently attracted much attention. Veselago predicted that such media would support backward waves, focus electromagnetic radiation emitted by a point source situated outside the medium, and reverse the Doppler and Vavilov–Cerenkov effects. These theoretical predictions were based on the formal solutions of Maxwell’s equations in which both and were assumed to be negative. Veselago also concluded that and can both be negative only in the presence of frequency dispersion so one could not envisage the existence of substances with constant negative constitutive parameters over a wide range of frequencies. It took approximately 30 years before such media—now referred to as metamaterials—were first realized by researchers such as Pendry et al. [2], Smith et al. [3], and Shelby et al. [4] in the form of periodic structures composed of thin wire cylinders and split-ring resonators. Due to their resonant nature, such structures exhibit the negative refractive index property over a rather narrow bandwidth. In 2002, Iyer and Eleftheriades [5], Eleftheriades et al. [6], Caloz and Itoh [7], and Oliner [8] independently proposed and described alternative realizations of metamaterials that were structurally less complex and less dispersive than those based on thin wires and split-ring resonators. They consisted of host transmission lines with embedded lumped series capacitors and shunt inductors. Observing that these periodic structures

Manuscript received April 21, 2004; revised July 5, 2004. This work was supported by the Natural Science and Engineering Council of Canada. The authors are with the Computational Electromagnetics Research Laboratory, Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada V8W 3P6 (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845196

can support backward waves as discussed by Ramo et al. [9], Caloz and Itoh [7] periodically loaded a microstrip line with series capacitors and shunt inductors; Iyer and Eleftheriades [5] and Eleftheriades et al. [6] realized a two-dimensional (2-D) mesh of microstrip transmission lines with embedded discrete series capacitors and shunt inductors, and modeled them in the frequency domain using a circuit simulator and a thin-wire moment-method simulator. Analytical, numerical, and experimental studies confirmed that these loaded transmission-line models exhibit indeed the properties predicted by Veselago. The modeling of electromagnetic fields by means of transmission-line networks harks back to the 1940s when Whinnery and Ramo [10] and Whinnery et al. [11] employed them to analyze waveguide resonators and discontinuities. This work had inspired Johns and Beurle [12] to create, in 1971, the transmission-line matrix (TLM) method, which is a numerical network model of Maxwell’s field equations. It was thus obvious to the authors that we could create a numerical network model of metamaterials by embedding reactive elements into Johns and Beurle’s TLM mesh in the same way as Iyer and Eleftheriades and Caloz and Itoh had realized physical models with such wave properties. The advantages of having a numerical TLM model of metamaterials are considerable. Not only is such a model many times faster than an analog implementation of the transmission-line networks on a circuit simulator, but it allows us to solve much larger structures with millions of cells, as compared to hundreds or thousands used in previous simulations. Furthermore, the TLM being a time-domain method, we can perform both time–harmonic and transient simulations with arbitrary excitation functions that demonstrate the dispersive behavior of metamaterials. By integrating a TLM metamaterial model into an existing TLM simulator that handles arbitrary boundary and material geometries, extracts scattering parameters, and dynamically visualizes fields and energy flow, we obtain a powerful computer-aided design (CAD) tool and virtual test-bed for research, teaching, and engineering design involving metamaterials. In this paper, we present two different TLM models of metamaterials. The first is a modified 2-D shunt node model that emulates the one-dimensional (1-D) and 2-D transmission-line structures proposed by Caloz and Itoh, [7] and Iyer and Eleftheriades [5] and Eleftheriades et al. [6]. The second is based on the 3-D symmetrical condensed node (SCN) proposed by Johns in 1987 [13] and employs an inter-cell connection algorithm that models the reactive periodic elements embedded in the network. The 2-D shunt model has a larger stable time step and is computationally faster than the 3-D SCN model. However, the latter is capable of modeling metamaterials in three-dimensional (3-D)

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SO et al.: MODELING OF METAMATERIALS WITH NEGATIVE REFRACTIVE INDEX USING 2-D SHUNT AND 3-D SCN TLM NETWORKS

space. No 3-D loaded transmission-line model of metamaterials has been realized to date. The two TLM models also have their advantages and disadvantages in regard to their implementation. The first approach requires a significant modification of the impulse scattering procedure at the nodes, but does not affect the connection between cells or the implementation of boundaries and interfaces between different media. To implement a metamaterial in 2-D shunt TLM with air as the host medium, a 9 9 impulse scattering matrix per unit cell is needed. A corresponding scattering matrix for the 3-D SCN would be of size 27 27; such a big scattering matrix is not easy to derive or implement. The intercell approach, on the other hand, preserves the original TLM node scattering matrix and introduces interface scattering matrices at cell boundaries. As we will show later, at least five types of inter-cell scattering matrices are needed to handle practical modeling problems. In the following sections, we derive the two TLM models by building upon the reactively loaded transmission-line network models of metamaterials described by Iyer and Eleftheriades and Caloz and Itoh. We reformulate them in terms of transmission-line modeling concepts. In particular, we represent the lumped elements in these models by reactive transmission-line stubs and develop new TLM algorithms that describe the propagation of short voltage impulses in such structures. The resulting 2-D shunt and 3-D SCN TLM scattering and connection algorithms are then coded and validated by test simulations. In particular, the dispersion error of the numerical models is verified against analytically accurate eigenfrequencies of a 3-D cavity filled uniformly with metamaterial. Furthermore, the salient features of the benchmark refraction problem analyzed by Pendry [17] (Pendry’s perfect lens) are confirmed by inspection of high-resolution field maps generated with the numerical TLM model and compared with experimental and simulated field patterns published by Grbic and Eleftheriades [18]. II. THEORY

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Fig. 1. Unit cell of a 2-D distributed network that models a 2-D continuous medium at low frequencies. The elements are labeled in units of impedance and susceptance per unit length, respectively (after Eleftheriades et al. [6]).

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Fig. 2. Unit square cell of size ` ` of a host transmission-line network loaded with lumped series capacitors and a shunt inductor to model negative refractive index materials.

realized by making the series elements capacitive and shunt element inductive [9]. In this case, the phase velocity on the network will be opposite to the group velocity (the latter is positive in the direction of , defined energy flow), resulting in a negative refractive index as (3)

The generic network model for 2-D electromagnetic-wave and permeability propagation in a medium of permittivity is shown in Fig. 1. and , specified in units The series and shunt elements of complex impedance ( /m) and complex shunt admittance per unit length (S/m) are related to the constitutive parameters of the modeled continuous medium by the following equivalence: (1) For infinitesimal cell size, the network equations and 2-D Maxwell equations for the TM-to- case are isomorphic when the following identities are introduced: (2) Iyer and Eleftheriades [5], Eleftheriades et al. [6], and Caloz and Itoh [7] have exploited the ability of the model shown in Fig. 1 to support backward waves when both the series and shunt elements are made negative imaginary, causing the permittivity and permeability values in (1) to become negative. This can be

To build such a network, lumped series capacitors and shunt inductors are embedded in a 2-D host network of transmission lines [5]–[8], as shown in Fig. 2. The equivalent series inducand the shunt capacitance tances of the host transmission-line network must be compensated by modifying the embedded lumped elements such that the hybrid and at the design frequency. network has the desired and shunt inductance emHence, the series capacitance bedded in each cell (see Fig. 2) must be (4) where and are the intrinsic effective constitutive parameters of the host network without the embedded lumped elements. The host network parameters can be expressed in terms of the link transmission-line parameters as follows: (5)

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where and are the characteristic impedance and the phase velocity of the link transmission lines of the host network, and is the mesh parameter or cell size. III. TLM MODELS OF METAMATERIALS We have used both 2-D shunt and 3-D SCN TLM meshes as host networks to realize computational models for metamaterials. In the following, we will first describe the design of the TLM models, starting with the specifications of the material properties, and derive numerical algorithms for implementation. A number of numerical experiments will then be performed to validate the TLM models. A. Design of TLM Models for Metamaterials The design of a metamaterial begins with the choice of its intrinsic wave impedance and refractive index . Since and , we can express the constitutive parameters of the metamaterial as (6) Next we must choose the design frequency at which these properties are to be realized. Finally, we must select of the discrete network that has these properties the cell size at the design frequency. The cell size should be much smaller than the guided wavelength for the discrete model to emulate a continuous medium. The choice should satisfy the condition (7) Using (4) and (5), we can now determine the lumped elements that must be embedded into the TLM mesh to realize the square . unit cell with the chosen cell size Unlike physical host networks for metamaterials, the TLM network needs not to be realizable. In particular, the TLM host network can have the wave properties of free space, which requires link transmission lines of characteristic impedance and phase velocity in the 2-D TLM, and of impedance and phase velocity in the 3-D TLM. Naturally, the properties of the host TLM network may be set different from that of air using either the traditional stub loading or different link line properties.

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Fig. 3. 2-D shunt TLM cell of size ` ` equipped with four open-circuited series stubs and one short-circuited shunt stub that model the embedded reactive elements. Scattered impulses and line characteristic impedances are also defined in this figure.

B. Implementation in 2-D Shunt TLM A convenient way to realize the unit cell in Fig. 2 by means of transmission-line elements is to model the series capacitances and shunt inductance by open- and short-circuited stubs of , as shown in Fig. 3. length The characteristic impedances of these stubs are related to and by the following expressions: (8) where is the phase velocity on the stub lines. These expressions are approximations that are valid for electrically short . transmission lines with The nine-port junction of transmission lines in Fig. 3 forms . The voltage impulse scattering a square cell of size matrix of this junction is shown in (9) at the bottom of this page. The elements of this scattering matrix have been derived in [16] and, thus, will not be repeated here. C. Implementation in 3-D SCN TLM In a 2-D shunt TLM network that models a material with a and characteristic impedance , both the phase velocity are times link line velocity and link-line impedance these values (see Table I, note that an index refers to the properties of the material, while the index refers to the properties of the TLM link lines.) We can interpret this by assuming that, in the 2-D shunt TLM model, the link lines have a permeability equal to the material

(9)

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TABLE I RELATIONSHIP BETWEEN THE PROPERTIES OF A MATERIAL AND OF THE LINK LINES FORMING ITS 2-D AND 3-D TLM MODEL

Fig. 5. Unit cell of a 3-D SCN TLM mesh with inter-cell networks that model relative permittivity and permeability.

(15) Solving (14) and (15) for Fig. 4. Unit cell of a 3-D SCN TLM network that models a 3-D continuous medium. The elements are labeled in units of impedance and susceptance per unit length, respectively.

permeability , and a permittivity half the material permit. Hence, tivity (10) In the 3-D TLM model of this material, the link lines have a permeability of and a permittivity of . Hence, (11) These relationships are summarized in Table I. The generic 3-D SCN TLM model for an isotropic medium of permittivity and permeability is shown in Fig. 4. There we only show the lumped-element equivalents of the link lines deployed in the -direction and polarized in the -direction. Similar circuits can be drawn for the two other coordinate directions. We have

and

yields (16) (17)

If and are positive, we can represent them by a ) and a lumped capacitor lumped inductor ( ( ), respectively [19]. Hence, (18) (19) Note that these elements are independent of frequency. However, in metamaterials with negative and , and become negative; we must then represent them by a lumped ca) and a lumped inductor ( pacitor ( ), respectively, resulting in strongly dispersive intercell network elements (20)

(12) (13) Dielectric and magnetic materials with other constitutive parameters are traditionally modeled by adding reactive stubs to the TLM nodes. Here, we present an alternative model that leaves the node unchanged and introduces relative permittivity and permeability in the links between the nodes in the form of and inter-cell networks [19], as shown in Fig. 5. are the host 3-D TLM node parameters per cell. In order to model the material with the network shown in Fig. 5, we must have

(14)

(21) In the 3-D implementation of metamaterials, we place inter-cell networks between two SCN nodes along the -, -, and -directions for both polarizations. This involves additional processing of impulses exchanged between neighboring nodes. The inter-cell network for modeling metamaterials is shown and . For a given in Fig. 6. There operating frequency and mesh size , the values of and can be calculated using (20) and (21). Since boundaries and interfaces are situated halfway between nodes, we must split the inter-cell networks up in these locations, as shown in Fig. 6. The next step is to embed these capacitors and inductors into the TLM network. Again, we model them by open- and short-

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Fig. 6. Splitting of the inter-cell network for metamaterials at boundaries and material interfaces.

Fig. 7. Equivalent stub model of the lumped-element inter-cell network for modeling negative " and  .

circuited stubs of length and derive an equivalent scattering matrix at the boundary between two cells, as shown in Fig. 7. The characteristic impedances of these stubs are related to and shunt inductance by the folthe series capacitance lowing expressions: (22) where is the phase velocity on the stub lines, as well as on the host network link lines. These expressions are approximations that are valid for electrically short transmission lines with . The inter-cell scattering matrix for the metamaterial is

These and the following expressions have been derived by means of the transmission-line theory. As explained earlier and as shown in Fig. 6, four more inter-cell scattering matrices are needed to handle interfaces between metamaterial and various types of boundaries, namely, the electric wall, magnetic wall, regular dielectric, and absorbing boundary. These matrices are given in (24)–(27). The scattering matrix of the inter-cell network inserted between a metamaterial and perfect electric wall is (24) where

The scattering matrix of the inter-cell network inserted between a metamaterial and perfect magnetic wall is (25) where

(23) where

The scattering matrix of the inter-cell network inserted between a metamaterial and an absorbing boundary is (26)

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where

To model two adjacent, but different metamaterials, a general 6 6 inter-cell scattering matrix is needed as follows:

The scattering matrix of the inter-cell network inserted at the interface between a metamaterial and a regular material is

(27)

where

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TABLE II COMPARISON BETWEEN THEORETICAL AND COMPUTED EIGENFREQUENCIES (IN GIGAHERTZ) OF A 20 15 10 mm CAVITY FILLED UNIFORMLY  : , ` WITH METAMATERIAL OF " mm

2 2

=

= 01 5 1 = 1

Fig. 8. Metamaterial slab in air. A uniform 10-GHz TEM wave is incident normally from the left-hand side.

Fig. 9. Two snapshots of the electric field propagating across the air and metamaterial sections at 10 GHz.

(28)

IV. MODELING RESULTS AND VALIDATION We have implemented the new 2-D and 3-D metamaterials models in our TLM field solver engines. We have then computed the electrodynamic responses of several structures containing metamaterials for which error-free analytically solutions exist. We have also verified that the numerical field solutions obtained with these models exhibit all the salient features predicted theoretically [1], [17] and confirmed experimentally [6], [18]. Some typical examples are presented below. A. Validation of Constitutive Parameters In the first numerical experiment, we have verified the accuracy of the effective constitutive parameters and boundary positions of the 3-D TLM model by computing several eigenfrequencies of rectangular cavities filled uniformly with metamaterial. Table II compares the analytically accurate and numer-

ically computed eigenfrequencies of the first four modes in a rectangular cavity of size 20 15 10 mm . The metamaterial has negative relative permittivity and permeability of 1.5 each, mm) and is modeled by a cubic 3-D SCN TLM mesh ( using inter-cell networks. The relative error of the TLM results lies well within the range predicted by a dispersion analysis of the TLM network [14] confirming that it models both the constitutive parameters and boundary implementation with the dispersion error margin. Analogous results have been obtained with the 2-D shunt TLM model. B. Validation of Impedance and Negative Phase Velocity To validate the characteristic impedance property of the metamaterial model, we have computed the transmission of a uniform TEM wave through a slab of metamaterial. The geometry is shown in Fig. 8. We have modeled the structure with a 1-mm 3-D TLM grid. At 10 GHz, the refractive index of the metamaterial is ( , ) and its characteristic impedance is 376.7 . Fig. 9 shows two snapshots of the electric field propagating across the metamaterial slab shown in Fig. 8. Note that the

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Fig. 11. Pendry’s perfect lens consisting of a slab of metamaterial of refractive index 1. The ray diagram predicts perfect focusing due to the negative refraction of the wave incident on the lens (TE case).

0

Fig. 10. Focusing of a cylindrical wave (TE case) emitted by a point source situated approximately a half-wavelength from the interface between air and metamaterial simulated with the modified TLM algorithm. (a) 3-D surface display. (b) 2-D contour display of the instantaneous transverse electric field after 500 time steps.

power flows in the positive -direction, while the phase velocity in the metamaterial is in the negative -direction. Quantitative analysis of the above waveforms and their displacement in the -direction shows clearly that: 1) the wavelength in the metamaterial is half that in air; 2) its phase velocity is negative and half that in air; and 3) all subsections are matched since no scattering occurs at the material interfaces. This experiment validates the accurate modeling of the characteristic impedance and the negative phase velocity within the margins predicted by dispersion analysis. C. Validation of Negative Diffraction and Focusing In the next validation experiment, we have modeled the refraction of a cylindrical monochromatic wave at an air–metamaterial interface. Fig. 10 shows the focusing effect predicted by Veselago [1] that occurs when a point source is placed in air near the material interface, and recreates the wave pattern produced by Iyer and Eleftheriades [5] and Eleftheriades et al. [6]. What is not obvious from the still pictures is the fact that the wave envelope in the metamaterial is traveling toward the interface rather than away from it, which gives the illusion that energy is actually converging from both sides toward the interface. In fact, the group velocity (and, hence, the velocity of energy transport) is opposed to the phase velocity in the metamaterial due to the embedded lumped elements that form a periodic structure supporting backward waves. This predicted behavior is indeed

clearly visible when the time-domain solution is displayed dynamically on the computer screen. In this particular simulation, a monochromatic 10-GHz sine wave was injected at a source point in the air region situated at 9.5 mm from the air–metamaterial interface. We have used a 2-D mm. Characteristics of the air TLM shunt mesh with ; region and host TLM grid: link line impedance: ; , , , link line velocity: , Characteristics of the metamaterial: cell size 1 mm 1 mm , , pF m, H m Loading elements per cell are as follows. • Series cap.: 0.0671907 pF, shunt Ind.: 9.53609 nH. • Computational domain: 200 200 cells, 500 time steps; (1.18-ns real time); CPU time: 16 s on a 900-MHz PC. D. Validation of Sub-Wavelength Focusing One of the most remarkable properties of a metamaterial slab is its ability to recreate an image with a resolution much finer than the wavelength. This property has been theoretically demonstrated by Pendry [17], Fig. 11, and experimentally verified by Grbic and Eleftheriades [18]. Fig. 12 shows several phases of the dynamic buildup of the electric field in Pendry’s perfect lens simulated with the 3-D SCN TLM model. The modeling parameters are as follows. mm, refractive index • Metamaterial Slab: thickness , wave impedance ,( ). , wave impedance , • Air: refractive index ). ( from the air–metamaterial The source point is situated at interface and emits a monochromatic cylindrical wave of frequency 16.31 GHz. The sequence of field plots in Fig. 12 also shows the progressive buildup of evanescent fields at the air–metamaterial interfaces toward the formation of two images of the source, one situated in the center of the slab (inner focus) and one situated in air on the opposite side of the slab. The transmitted wave pattern is that of a cylindrical wave emanating from the source image on the right-hand side of the lens. These results quantitatively confirm the theoretical and experimental data published in [17] and [18].

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tional efficiency, accuracy, resolution, and flexibility. When embedded in a time-domain electromagnetic-field simulator, these models provide powerful CAD capabilities and serve as a computational test-bed for researching the properties of media with a negative refractive index and for developing innovative components based on metamaterials. REFERENCES

Fig. 12. Three phases of the dynamic buildup of the electric field in Pendry’s perfect lens (Fig. 11) simulated with the 3-D SCN TLM model. t = 0 is the time at which the excitation is started at the source. (a) Electric field at t = 373 ps (b) Electric field at t = 500 ps (c) Electric field at t = 1361 ps.

V. CONCLUSION Two novel extensions of the TLM method for modeling metamaterials have been derived and implemented. They represent numerical incarnations of the reactively loaded transmissionline models of negative refractive index materials proposed in the literature [5]–[8]. One model is realized by modifying the TLM scattering process, and the other by modifying the connection process between TLM cells (inter-cell network method). While both models give virtually identical results, they differ considerably in their implementation. Their respective advantages and disadvantages have been discussed. The results obtained with these TLM models confirm the theoretically predicted behavior of metamaterials and reproduce the results published by Eleftheriades et al. and Pendry with high computa-

[1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [2] J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [3] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 18, pp. 4184–4187, May 2000. [4] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [5] A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D waves,” in IEEE MTT-S Int. Microwave Symp Dig., vol. 2, Jun. 2–7, 2002, pp. 1067–1070. [6] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [7] C. Caloz and T. Itoh, “Application of the transmission line theory of lefthanded (LH) materials to the realization of a microstrip LH transmission line,” in IEEE AP-S Int. Symp. Dig., vol. 2, Jun. 16–21, 2002, pp. 412–415. [8] A. A. Oliner, “A periodic-structure negative-refractive-index medium without resonant elements,” in USNC/URSI Nat. Radio Science Meeting, San Antonio, TX, Jun. 16–21, 2002, p. 41. [9] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed. New York: Wiley, 1994, pp. 263–264. [10] J. R. Whinnery and S. Ramo, “A new approach to the solution of high frequency field problems,” Proc. IRE, vol. 32, no. 5, pp. 284–288, May 1944. [11] J. R. Whinnery, C. Concordia, W. Ridgway, and G. Kron, “Network analyzer studies of electromagnetic cavity resonators,” Proc. IRE, vol. 32, no. 6, pp. 360–367, Jun. 1944. [12] P. B. Johns and R. L. Beurle, “Numerical solution of 2-dimensional scattering problems using a transmission line matrix,” Proc. Inst. Elect. Eng., vol. 118, no. 9, pp. 1203–1208, Sep. 1971. [13] P. B. Johns, “A symmetrical condensed node for the TLM method,” IEEE Trans. Microw. Theory Tech., vol. AP-35, no. 4, pp. 370–377, Apr. 1987. [14] C. Christopoulos, The Transmission-Line Modeling Method: TLM. Piscataway, NJ: IEEE Press, 1995, p. 232. [15] P. P. M. So, Eswarappa, and W. J. R. Hoefer, “A two-dimensional transmission line matrix microwave field simulator using new concepts and procedures,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 1877–1884, Dec. 1989. [16] P. P. M. So and W. J. R. Hoefer, “Time domain TLM modeling of metamaterials with negative refractive index,” in IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2004, pp. 1779–1782. [17] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, no. 18, pp. 3966–3969, Oct. 2000. [18] A. Grbic and G. V. Eleftheriades, “Subwavelength focusing using a negative-refractive-index transmission line lens,” IEEE Antennas Wireless Propagat. Lett., vol. 2, pp. 186–189, 2003. [19] H. Du, P. P. M. So, and W. J. R. Hoefer, “Inter-cell scattering framework in TLM for modeling material properties,” in Eur. Microwave Conf. Dig., Amsterdam, The Netherlands, Oct. 12–14, 2004, pp. 865–868. [20] P. P. M. So and W. J. R. Hoefer, “Time domain TLM modeling of metamaterials with negative refractive index,” in Eur. Microwave Conf. Dig., Amsterdam, The Netherlands, Oct. 12–14, 2004, pp. 1213–1216.

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Poman P. M. So (M’87–SM’00) received the B.Sc. degree in computer science and physics from the University of Toronto, Toronto, ON, Canada, in 1985, the B.A.Sc. and M.A.Sc. degrees in electrical engineering from the University of Ottawa, Ottawa, ON, Canada, in 1985 and 1987, respectively, and the Ph.D. degree from the University of Victoria, Victoria, BC, Canada, in 1996. He is currently an Adjunct Assistant Professor and a Senior Research Engineer with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada. He possesses over 15 years of hands-on experience in object-oriented software development for microwave and millimeterwave engineering using the TLM method. From April 1997 to June 1998, he was a Senior Antenna Engineer with EMS Canada Ltd. (formerly Spar Aerospace Ltd.). His research has included high-frequency (10–40 GHz) antennas and feed components design for commercial satellite systems, as well as -band active antenna CAD software development. In October 1993, he was with the Ferdinand-Braun-Institut für Höchstfrequenztechnik Berlin, Berlin, Germany, as an invited Research Scientist. From August 1990 to February 1991, he was a Visiting Researcher with the University of Rome, Rome, Italy, and the Laboratoire d’Electronique, Sophia Antipolis, France. During his time in Europe, he developed a number of electromagnetic wave simulators for the digital MPP and Connection Machine CM2 massively parallel computers. He is co-founder of the Faustus Scientific Corporation and the Chief Software Architect of the MEFiSTo line of products of the Faustus Scientific Corporation. He is a reviewer for Wiley’s International Journal of Numerical Modeling—Electronic Networks, Devices and Fields. Dr. So is a Registered Professional Engineering in the Province of British Columbia, Canada.

Ka

Huilian Du received the Bachelor and Master degrees in control engineering (with a major in semiconductor physics and devices) from the Harbin Institute of Technology, Harbin, China, in 1988 and 1991, respectively, and is currently working toward the Ph.D. degree in computational electromagnetics at the University of Victoria, Victoria, BC, Canada. For ten years, she was involved in the areas of very large scale integration (VLSI) design, RF bipolar power transistor, amplifier, and module design and testing in China. From November 2001 to April 2002, she was a Visiting Scholar with the Microelectronics Research Laboratory, Dalhousie University, Halifax, NS, Canada. In May 2002 she joined the Department of Electrical and Computer Engineering, University of Victoria.

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Wolfgang J. R. Hoefer (M’71–SM’78–F’91) received the Dipl.-Ing. degree in electrical engineering from the Technische Hochschule Aachen, Aachen, Germany, in 1965, and the D.Ing. degree from the University of Grenoble, Grenoble, France, in 1968. From 1968 to 1969, he was a Lecturer with the Institut Universitaire de Technologie de Grenoble, Grenoble, France, and a Research Fellow with the Institut National Polytechnique de Grenoble, Grenoble, France. In 1969, he joined the Department of Electrical Engineering, University of Ottawa, Ottawa, ON, Canada, where he was a Professor until March 1992. Since April 1992, he holds the Natural Sciences and Engineering Research Council (NSERC) Industrial Research Chair in Radio Frequency Engineering with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada. During several sabbatical leaves, he has been a Visiting Scientist and Professor with the Space Division of AEG-Telefunken, Backnang, Germany (now ATN), the Electromagnetics Laboratory, Institut National Polytechnique de Grenoble, the Space Electronics Directorate, Communications Research Centre, Ottawa, ON, Canada, the University of Rome “Tor Vergata,” Rome, Italy, the University of Nice–Sophia Antipolis, France, the Technical University of Munich, Munich, Germany, the Ferdinand Braun Institute for High Frequencies, Berlin, Germany, and the Gerhard Mercator University, Duisburg, Germany. In 1989, he was an Invited Lansdowne Lecturer with the University of Victoria. His research interests include numerical techniques for modeling electromagnetic fields and waves, CAD of microwave and millimeter-wave circuits, microwave measurement techniques, and engineering education. He is the cofounder and Managing Editor of the International Journal of Numerical Modeling. He serves on the Editorial Boards of the Proceedings of the Institution of Electrical Engineers, the International Journal of Microwave and Millimeter-Wave Computer Aided Engineering, Electromagnetics, and the Microwave and Optical Technology Letters. Dr. Hoefer was an elected Fellow of the British Columbia Advanced Systems Institute (BC-ASI) in 1992 and a Fellow of the Royal Society of Canada in 2003. He serves regularly on the Technical Program Committees of IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and IEEE Antennas and Propagation Society (IEEE AP-S) Symposia. He is the co-chair and former chair of the MTT Technical Committee on Field Theory (MTT-15). He served as associate editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (1998–2000) and was co-editor of the December 2002 Special Issue of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He serves on the Editorial Boards of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the recipient of the Peter B. Johns Prize for the best paper published in the International Journal of Numerical Modeling in 1990. He is a Distinguished Microwave Lecturer of the IEEE MTT-S (2005–2007).

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Simulation of Negative Permittivity and Negative Permeability by Means of Evanescent Waveguide Modes—Theory and Experiment Jaime Esteban, Carlos Camacho-Peñalosa, Member, IEEE, Juan E. Page, Teresa M. Martín-Guerrero, and Enrique Márquez-Segura, Member, IEEE

Abstract—In this paper, the theoretical foundations of the equivalence between waveguide propagation below cutoff and artificial plasmas are carefully analyzed through the derivation of the propagation constants of normal modes in waveguides filled with anisotropic plasmas. The equivalence between waveguide and dielectric plasma proposed by Marqués et al., which is valid for evanescent TE modes, has a dual counterpart for magnetic plasmas and evanescent TM modes. This new equivalence states that a negative magnetic permeability medium can be simulated by means of TM modes below their cutoff frequencies. The need of an anisotropic filling of the waveguide for the equivalence between plasmas and evanescent modes is also highlighted. To exemplify the applicability of this new equivalence, a structure that implements a double-negative medium has been proposed. Full-wave simulations of the proposed structure and measurements from an experimental setup are presented, both of which corroborate the new equivalence’s validity. Index Terms—Backward waves, electric plasmas, evanescent modes, magnetic plasmas, metamaterials, negative permeability, negative permittivity, periodic structures.

In Section II, the equivalence between waveguide and dielectric plasma, valid for evanescent TE modes, is introduced. In order to provide a more rigorous explanation, the modes of a rectangular waveguide filled with a magnetic plasma are presented, and the role of the anisotropy of the waveguide inserts is highlighted. Section III focuses on a new equivalence, namely, that a negative magnetic permeability medium can be simulated by means of TM modes below their cutoff frequencies. This new equivalence was previously introduced at a workshop [2], but no evidence (simulated or measured results of a realizable structure), apart from a simplified theoretical analysis, was provided.1 In this paper, a practicable structure is proposed, considering that a rigorous analysis of this geometry, and a suitable experimental setup, will be conclusive to confirm the equivalence. Section IV centers on the rigorous analysis of the proposed geometry, while Section V concentrates on the results obtained with an ad hoc experimental setup. Conclusions can be found in Section VI.

I. INTRODUCTION

T

HE development of artificial materials with simultaneous negative values of electric permittivity and magnetic permeability parameters (“double-negative” or “left-handed” media) has recently become a topic of interest because of its potential applications. In this area, Marqués et al. [1] proposed simulating artificial negative electric permittivity media, also called artificial plasmas, by using an empty waveguide operating at frequencies below the cutoff frequency of the dominant mode. In their approach, the left-handed medium is achieved by placing a periodic array of split-ring resonators (SRRs) inside a rectangular empty waveguide below cutoff.

Manuscript received May 31, 2004; revised October 7, 2004. This work was supported by the Spanish Ministry of Science and Technology and by the European Regional Development Funds of the European Union under Grant TIC2003-05027. J. Esteban and J. E. Page are with the Departamento de Electromagnetismo y Teoría de Circuitos, Escuela Técnica Superior de Ingenieros de Telecomunicación, Universidad Politécnica de Madrid, 28040 Madrid, Spain (e-mail: [email protected]). C. Camacho-Peñalosa, T. M. Martín-Guerrero, and E. Márquez-Segura are with the Departamento de Ingeniería de Comunicaciones, Escuela Técnica Superior de Ingenieros de Telecomunicación, Universidad de Málaga, 29071 Málaga, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845194

II. MAGNETIC-PLASMA-FILLED RECTANGULAR WAVEGUIDE As is well known, the propagation constant of the mode of a rectangular waveguide with dimensions can be expressed as [3] , where and being and . Here, the rectangular waveguide filled with a plasma is analyzed with both an isotropic and an anisotropic magnetic plasma. A. Isotropic Magnetic Plasma As suggested by Marqués et al. [1], plasma simulation can be significantly simplified by using an empty metallic waveguide. They argue that the fundamental TE mode of the empty waveguide has a phase constant of the form (1) where is the angular frequency, is the permeability of the is an effective dielectric constant given by vacuum, and (2) 1For the sake of completeness, part of the material presented in the workshop [2] is reproduced in Sections II and III.

0018-9480/$20.00 © 2005 IEEE

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Fig. 1. Vertically placed SRRs in a rectangular waveguide.

where is the cutoff frequency for the considered mode and is the permittivity of the vacuum. As this phase constant is the same as that of a TEM mode propagating in a lossless plasma , the TE mode is said to simulate with plasma frequency the propagation of a TEM wave in a lossless plasma medium exactly. Filling the waveguide with an isotropic magnetic plasma should then reproduce double-negative media phenomena. However, for an isotropic lossless magnetic plasma (with no and electrical properties) filling the waveguide, with (3) the propagation constant becomes

Fig. 2. Propagation constant of the TE mode in an anisotropic-magneticplasma-filled waveguide. a = 12:95 mm, b = 6:48 mm, and f = 11:58 GHz. Plasma as in (5) and (9) with f = 10:05 and f = 10:95 GHz.

that correspond to two families of hybrid modes, the first one, i.e., (6), with unperturbed propagation constants (with respect to the empty waveguide modes). , as in the case of the fundamental mode of When the empty waveguide, (6) is not applicable and (7) becomes the characteristic equation of TE modes, and then

(4) , with the same behavior as that of the empty waveguide ( ), i.e., with the same forward-wave modes, but with the to . cutoff frequencies shifted from Therefore, the equivalence of the SRR-loaded waveguide with a TEM mode in a double-negative medium presented in [1] bears no relationship to an isotropic-magnetic-plasma-filled waveguide. This equivalence must be searched for in the anisotropic characteristics of the interaction between SRRs and the magnetic field. B. Anisotropic Magnetic Plasma The main response of the SRRs to the fields is a magnetic polarizability along the normal to the rings. Locating the rings, as shown in Fig. 1, the waveguide can be considered as filled with and an idealized an uniaxial anisotropic medium with relative permeability tensor given by

(5)

where the plasma behavior occurs along the -axis. Solving the Maxwell equations for this simple problem leads to a characteristic equation with two solutions (6) (7)

(8) These TE modes propagate at the frequencies at which the two parentheses have the same sign. A backward wave will be oband , and a tained for frequencies that are lower than both forward wave at frequencies higher than both values. Between and , there is a stopband, no matter which of the two is , the mode becomes all-pass. higher. When Consider now the case of a WR51 waveguide 6.48 mm) filled with the macroscopic plasma (12.95 model of permeability for the SRRs described in [4], i.e., (9) This is a more complete plasma model (of plasma frequency GHz), which includes losses MHz and a magnetic resonance frequency GHz . Therefore, the discussion of (8) in terms of the simple plasma model (3) is no longer valid. However, the existence of passbands will still be related to the sign changes of and the real part of . In Fig. 2, the propagation constant and is shown. Backward-wave of the mode with behavior occurs at the frequency range where the real part of the mode permeability is negative and the empty-waveguide is under cutoff. The mode is forward where the permeability is positive and the empty-waveguide mode is propagating.

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Fig. 3. Mode impedance of the TE mode in an anisotropic-magneticplasma-filled waveguide. a = 12:95 mm, b = 6:48 mm, and f = 11:58 GHz. Plasma as in (5) and (9) with f = 10:05 and f = 10:95 GHz.

Fig. 5. Propagation constant of the TE mode in an anisotropic-magneticplasma-filled waveguide. a = 12:95 mm and b = 6:48 mm. Plasma as in (5) and (9) with f = 10:05 and f = 10:95 GHz.

SRR-loaded waveguide in the negative permeability frequency range. modes ( odd) is The quite different behavior of the worth mentioning, which is derived from the fact that these modes have a magnetic field perpendicular to the medium’s axis -mode family that shows plasma behavior (the axis). The propagates in a narrow passband below the magnetic resonance frequency , but as a forward-wave mode instead of a backward-wave mode. As an example, the propagation constant of mode is presented in Fig. 5. the III. ELECTRIC-PLASMA-FILLED RECTANGULAR WAVEGUIDE

Fig. 4. Phase constants of the TE modes in an anisotropic-magneticplasma-filled waveguide. a = 12:95 mm, b = 6:48 mm, and f = 11:58 GHz. Plasma as in (5) and (9) with f = 10:05 and f = 10:95 GHz.

Propagating and nonpropagating frequency ranges can also be revealed by means of the mode impedance. For the TE modes, the impedance is

A similar analysis can be carried out for the case of a waveguide filled with an electric plasma. If an isotropic plasma is selected, the result is the same frequency shift of the emptywaveguide cutoff frequencies obtained for the isotropic-magnetic-plasma-filled waveguide. Significant results are obtained only with anisotropic plasmas. A. Anisotropic Electric Plasma (Case 1) Consider a rectangular waveguide filled with an anisotropic , and a relative permittivity tensor given by medium with

(10) (11) which is represented in Fig. 3. In the experimental setup presented in [1] and [5], there is no reason for modes different from the fundamental not to be excited. All modes with a uniform magnetic field in the -direction would produce the expected magnetic response of the SRRs. These are the modes when is odd. A more complete dispersion diagram (including the phase constants of modes) is shown in Fig. 4. some of the This continuum of “lower order” modes, which is distinctive from anisotropic-plasma-filled waveguides [6], contributes to the reported transmission of electromagnetic waves through the

where, analogous to (3), (12) Solving Maxwell equations leads to a characteristic equation with the following two solutions: (13) (14)

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Fig. 6. (a) “Rodded” and (b) “wired” media in rectangular and square waveguides.

which correspond to two families of hybrid modes, the first one, i.e., (13), with unperturbed propagation constants (with respect to the empty waveguide modes). , which includes the fundamental mode of the When empty waveguide, (13) is not applicable and (14) becomes the characteristic equation of TE modes

Fig. 7. Propagation constant of the TM mode in an anisotropic-electricplasma-filled waveguide. a = b = 22:86 mm and f = 9:28 GHz. Plasma as in (16) and (21) with f = 8 GHz.

(15) This is the same result as that presented in [7], where a plasmafilled waveguide is simulated with a “rodded medium,” i.e., a two-dimensional array of metallic wires, as depicted in Fig. 6(a).

at frequencies higher than both values. The mode impedance given by

(20) B. Anisotropic Electric Plasma (Case 2) More interesting, as far as double-negative media simulation is concerned, is a uniaxial plasma with the relative permittivity tensor

(16)

where the plasma behavior is now along the - and -axis. Solving this problem for normal modes, the following two solutions are obtained: (17) (18) the first one for TE modes and the second one for TM modes. For any TM mode, the propagation constant is analogous to (8), i.e.,

(19)

Therefore, the same conclusion can be made, namely, that these TM modes propagate at the frequencies at which the two parentheses have the same sign. A backward wave is expected for frequencies that are lower than both and , and a forward wave

yields the same conclusion. As an example, a square waveguide has been analyzed (square instead of rectangular for convenience, since TM modes are going to be dealt with). The waveguide is filled, as shown in Fig. 6(b), with a transverse bidimensional array of thin metallic wires, aligned with an - and -axis to obtain the desired plasma effect. Assuming some losses, the permittivity function of the wires can be modeled as [8]

(21)

which represents a low-pass negative permittivity phenomenon (which is worth mentioning), as opposed to the case of SRRs, resonant structures whose plasma model shows narrow-band negative permeability. For a 22.86 22.86 mm waveguide, and a plasma with GHz and MHz, the dispersion diagram of the mode is presented in Fig. 7 and the mode impedance is presented in Fig. 8. In both figures, the two passbands (backward and forward) are conspicuous. A more detailed dispersion diagram (with the phase constant of a number of modes) is presented in Fig. 9 as an approach to the waveguide’s continuum of “lower order” modes.

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Fig. 10.

Crossed-strip loaded square waveguide.

Fig. 8. Mode impedance of the TM mode in an anisotropic-electricplasma-filled waveguide. a = b = 22:86 mm and f = 9:28 GHz. Plasma as in (16) and (21) with f = 8 GHz.

2

Fig. 11. Dispersion behavior for the structure of Fig. 10. Waveguide 22.86 22.86 mm, T = 12 mm, s = 1 mil, t = 18 m, w = 0:1 mm, " = = 1:07(1 j 0:002).Solid lines correspond to phase 3:7(1 j 0:002), and " constants, while dashed lines are for attenuation constants. The fundamental mode curves are thicker.

0

Fig. 9. Phase constants of the TM modes in an anisotropic-electricplasma-filled waveguide. a = b = 22:86 mm and f = 9:28 GHz. Plasma as in (16) and (21) with f = 8 GHz.

These results suggest that a new equivalence can be proposed: dual counterpart of evanescent TE modes as electric plasmas when loading waveguides with anisotropic magnetic inserts. In this new equivalence, TM modes under cutoff behave as magnetic plasmas when loading the waveguide with anisotropic electric inserts of the “case 2” type. IV. RIGOROUS FULL-WAVE ANALYSIS OF A STRIP-FILLED WAVEGUIDE The structure shown in Fig. 6(b) can be slightly modified, turning the wires into thin strips, as shown in Fig. 10, where only one vertical and one horizontal strip is considered, and adding to handle the strips. This a dielectric of low permittivity geometry has two considerable advantages. On the one hand, it can be manufactured with planar-circuit technology (this is

0

the reason for the low-permittivity dielectric substrate). On the mode of the square waveguide should be other hand, the easily excited by the TEM mode of a coaxial cable. Furthermore, it can be rigorously analyzed as a periodic closed waveguide by the mode-matching technique of [9] through some suitable modifications (detailed in the Appendix). A periodically loaded square waveguide (22.86 22.86 mm) has been characterized by this method. All the results in this paper have been computed using 160 modes in the square waveguide regions, and checked with up to 240 modes to verify the absolute convergence of the results. The number of modes in other regions were determined with a modal ratio slightly lower than the optimum one in order to avoid the relative convergence phenomenon (see [10]). The dispersion diagram for the first few modes with odd–odd symmetry is shown in Fig. 11. There are forward modes, like the fundamental mode above 9 GHz and the first higher order mode, which propagates for frequencies above 9.7 GHz, complex pairs, as usually found on periodic structures, and backward waves, like the fundamental mode below 8 GHz. Our interest is focused on the fundamental mode. Its behavior shows excellent agreement with the magnetic-plasma model predictions (compare with Fig. 7). The stopband ranges

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Fig. 12. Dispersion behavior for the structure of Fig. 10 (T = 7:3 mm, all other dimensions as in Fig. 11). Solid lines correspond to phase constants, while dashed lines are for attenuation constants. The fundamental mode curves are thicker.

from 8.04 GHz (identifiable as the plasma frequency of the strips) to 8.96 GHz, which is roughly the cutoff frequency of mode of a square waveguide filled with a dielectric the . The main difference between the of relative permittivity magnetic-plasma model predictions and the rigorous full-wave analysis results lies in the fact that the backward-wave band of the periodic structure has a lower cutoff (Bragg) frequency at 6.9 GHz. This is a predictable difference between a homogeneous and a discrete periodic structure. At this frequency, the (where is the period phase constant reaches the value of the periodically loaded waveguide), which is the maximum attainable value for the phase constant, due to the periodicity in the phase constant of the periodic structures’ dispersion diagram. The plasma frequency of a “wire” array should be controllable by varying the radius and distance between wires [8]. In the crossed-strip-loaded waveguide, the control variables are the width of the strips and the structure period. This is confirmed by the dispersion diagram in Fig. 12, where the period has been and to obtain an all-pass mode. tuned to make V. EXPERIMENTAL SETUP AND MEASUREMENTS A periodic square waveguide has been built (see Fig. 13) filling a hollow square waveguide with crossed strips etched , ) onto a 1-mil polymide substrate ( with 0.5-oz/ft copper cladding. These thin sheets are spaced from each other by means of a polymethacrylmide foam ( , ). In order to make a good metallic contact with the waveguide walls, the crossed strips were etched onto a square substrate larger than the waveguide cross section, i.e., with a surrounding copper frame. The extra material was, after removing the corners, folded back on the polymethacrylmide foam. The pressure of the foam on the folded flaps ensures the contact with the four waveguide walls and helps to keep the crossed strips in place. The waveguide is excited by means of a centered coaxial whose inner conductor protrudes from a short-circuit wall into

Fig. 13. Experimental setup. On the right-hand side, a period is disassembled to show the etched crossed strips and the stacked films of polymethacrylmide foam.

the waveguide and contacts with the first crossed strips. This transition, by means of a quadruple current loop, has proven that is capable of generating the transverse magnetic field of the TM modes. The propagation constant of the first mode of the structure has been measured by an improved statistical version of the method proposed by Bianco and Parodi in [11] that can be found in [12]. While in [11], four lengths are measured to obtain a single determination of the propagation constant, in this case, a waveguide filled with a large number of periods, for an overall length of approximately 10 cm, was measured. Up to 19 different waveguide lengths (using from 5 to 23 periods) were measured to obtain up determinations of the propagation constant at to each frequency. This large number of determinations provides the statistical reduction of the measurements’ uncertainty. A sample of the measurement results is shown in Fig. 14 and is compared with the theoretical full-wave results computed as explained in [9] and the Appendix. Unfortunately, due to inherent limitations of the measurement method, it is not possible to determine the propagation constant when the phase constant value is too low or when the attenuation constant is too high. The stopband exists (around 9 GHz), but it is wider than predicted. Slight deviations of the phase constant are also visible at lower frequencies. A second example of the measurement results is given in Fig. 15. In this case, the period has been significantly reduced and, therefore, the plasma frequency has been increased, while the cutoff frequency of the empty waveguide remains at the same value. A stopband ranges from this cutoff frequency ( 9 GHz) to a new and higher plasma frequency. The lower cutoff frequency, due to the periodicity of the structure, has

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Fig. 16. Current and voltages at the discontinuities of one period of the crossed-strip-loaded waveguide.

negative propagation constant ranges well below the cutoff modes of the hollow square frequency of the waveguide (6.56 GHz). VI. CONCLUSION

6 0

Fig. 14. Theory (thick line) and measurements (thin line with error bars 2 standard deviations). Waveguide 22.86 22.86 mm, T = 7:21 mm, s = 1 mil, t = 18 m, w = 0:21 mm, " = 3:7(1 j 0:002), and " = 1:07(1 j 0:002).

2

0

The theory behind the equivalence between propagation in cutoff empty waveguides and effective permeability and permittivity parameters has been described. The theoretical analysis has shown that the equivalence between waveguide and plasma proposed by Marqués et al. [1], which is valid for TE modes, is due to the anisotropic characteristics of the inserts (in this case, SRRs). As another outcome of the performed analysis, a dual equivalence has been proposed for TM modes under cutoff, which can be considered as “one-dimensional magnetic plasmas” when suitably loaded with anisotropic electric inserts (rods, wires, and strips in the examples presented). This new equivalence has been numerically and experimentally verified by analyzing, using a full-wave modal analysis technique, and by measuring an ad hoc experimental set up. It is believed that this equivalence could suggest new realizations of double-negative media simulations and could open new possibilities for the theoretical and experimental study of wave propagation in media with both negative permittivity and permeability parameters. APPENDIX MODIFICATION OF THE METHOD OF [9] FOR THE CHARACTERIZATION OF CROSSED-STRIP LOADED WAVEGUIDES

6

Fig. 15. Theory (thick line) and measurements (thin line with error bars 2 standard deviations). T = 3:41 mm, all other dimensions are as shown in Fig. 14.

decreased since the maximum absolute value of the phase constant has been increased. As a result, the backward-wave

The geometry of one period of the strip loaded waveguide can be fitted to the general structure considered in [9, Fig. 1] by dividing region II into two homogeneous regions, i.e., IIa and IIb, as depicted in Fig. 16. Region I is made up of the four square waveguides formed by the crossed strips and the square waveguide walls. Region IIa is the square waveguide filled with the dielectric substrate, while region IIb is where the foam fills the waveguide. All the formulation presented in [9] is then applicable since the discontinuities between regions IIa and I and between IIb and III are of the reduction-in-section type and have the same frequency-independent matrix. The only differences to take into account are the definition of the mode propagation conand ) and admittances ( and ) for restants ( gions IIa and IIb, and the substitution of and matrices

ESTEBAN et al.: SIMULATION OF NEGATIVE PERMITTIVITY AND NEGATIVE PERMEABILITY BY MEANS OF EVANESCENT WAVEGUIDE MODES

for some more complicated relations. With the definitions of Fig. 16, [9, eq. (7)] must be replaced by

(22) where

,

,

, and

are diagonal matrices with

(23) From these new values, [9, eq. (10)] become

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[3] C. T. A. Johnk, Engineering Electromagnetic Fields and Waves, 2nd ed. New York: Wiley, 1988. [4] R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [5] R. Marqués, F. Medina, F. Mesa, and J. Martel, “On the electromagnetic modeling of left-handed metamaterials,” in Advances in Electromagnetics of Complex Media and Metamaterials, S. Zohudi, A. Shivola, and M. Arsalane, Eds. Dordrecht, The Netherlands: Kluwer, 2002, pp. 123–141. [6] C. C. Johnson, Field and Wave Electrodynamics. New York: McGrawHill, 1965, pp. 402–404. [7] W. Rotman, “Plasma simulation by artificial dielectrics and parallelplate media,” IRE Trans. Antennas Propag., vol. AP-10, no. 1, pp. 82–95, Jan. 1962. [8] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett., vol. 76, pp. 4773–4776, 1996. [9] J. Esteban and J. M. Rebollar, “Characterization of corrugated waveguides by modal analysis,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 6, pp. 937–943, Jun. 1991. [10] R. Sorrentino, M. Mongiardo, F. Alessandri, and G. Schiavon, “An investigation of the numerical properties of the mode-matching technique,” Int. J. Numer. Modeling, vol. 4, pp. 19–43, 1991. [11] B. Bianco and M. Parodi, “Measurement of the effective relative permittivities of microstrip,” Electron. Lett., vol. 11, pp. 71–72, 1975. [12] E. Márquez-Segura and C. Camacho-Peñalosa, “Broadband experimental characterization of the propagation constant of planar transmission lines,” in Proc. 5th Int. Recent Advances in Microwave Technology Symp., Kiev, Ukraine, Sep. 11–16, 1995, pp. 464–467.

Jaime Esteban was born in Madrid, Spain, in 1963. He received the Ingeniero de Telecomunicación and Dr.Eng. degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1987 and 1990, respectively. Since January 1988, he has been with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. In 1990, he became Profesor Interino and, in 1992, Profesor Titular de Universidad. His research topics include the analysis and characterization of waveguides, transmission lines, planar structures and periodic structures, the analysis and design of microwave and millimeter-wave passive devices, and numerical optimization techniques (genetic algorithms and evolution programs). His current research is focused on the analysis and applications of left-handed double-negative metamaterials. Dr. Esteban was the recipient of a Spanish Ministry of Education and Science scholarship (1988–1990).

(24) and from them, the derivation of the expressions corresponding to [9, eq. (11)–(18)] is straightforward, although some care must . be taken with the fact that now ACKNOWLEDGMENT The authors would like to thank Prof. R. Marqués, University of Sevilla, Seville, Spain, for introducing them to the subject of left-handed media. REFERENCES [1] R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett., vol. 89, no. 18, Oct. 2002. Paper 183901. [2] C. Camacho-Peñalosa, J. Esteban, T. M. Martín-Guerrero, and E. Márquez-Segura, “On the simulation of negative electric permittivity and magnetic negative permeability by means of evanescent waveguide modes,” in 27th ESA Antenna Technology Workshop, Santiago de Compostela, Spain, Mar. 9–11, 2004, pp. 461–468.

Carlos Camacho-Peñalosa (S’80–M’82) received the Ingeniero de Telecomunicación and Doctor Ingeniero degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1976 and 1982, respectively. From 1976 to 1989, he was with the Escuela Técnica Superior de Ingenieros (ETSI) de Telecomunicación, Universidad Politécnica de Madrid, as Research Assistant, Assistant Professor, and Associate Professor. From September 1984 to July 1985, he was a Visiting Researcher with the Department of Electronics, Chelsea College (now King’s College), University of London, London, U.K. In 1989 he became a Full Professor with the Universidad de Málaga, Málaga, Spain. He was the Director of the ETSI de Telecomunicación (1991–1993), Vice-Rector (1993–1994), and Deputy Rector (1994) of the Universidad de Málaga. From 1996 to 2004, he was the Director of the Departamento de Ingeniería de Comunicaciones, Universidad de Málaga. From 2000 to 2003, he was Co-Head of the Nokia Mobile Communications Competence Centre, Málaga, Spain. His research interests include microwave and millimeter solid-state circuits, nonlinear systems, and applied electromagnetism. He has been responsible for several research projects on nonlinear microwave circuit analysis, microwave semiconductor device modeling, and applied electromagnetics.

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Juan E. Page was born in Madrid, Spain, in 1946. He received the Ingeniero de Telecomunicación and Doctor Ingeniero degrees from the Universidad Politécnica de Madrid, Madrid, Spain, in 1971 and 1974, respectively. Since 1983, he has been a Professor with the Departamento de Electromagnetismo y Teoría de Circuitos, Universidad Politécnica de Madrid. His activity includes teaching of electromagnetic theory and research in the field of computer-aided design (CAD) of microwave devices and systems.

Teresa M. Martín-Guerrero was born in Málaga, Spain. She received the Licenciado en Ciencias Físicas degree (M.Sc. equivalent) from the Universidad de Granada, Granada, Spain, in 1990, and the Doctor Ingeniero de Telecomunicación degree (Ph.D. equivalent) from the Universidad de Málaga, Málaga, Spain, in 1995. Her doctoral dissertation focused on distributed effects and modeling of field-effect transistor (FET)-type devices. In 1991 she joined the Departamento de Ingeniería de Comunicaciones, Universidad de Málaga, as Assistant Professor, and in 1999, became an Associate Professor. Her current research activities deal with microwave and millimeter-wave device modeling, and differential techniques for positioning using global satellite systems.

Enrique Márquez-Segura (S’93–M’95) was born in Málaga, Spain, in April 1970. He received the Ingeniero de Telecomunicación and Doctor Ingeniero de Telecomunicación degrees from the Universidad de Málaga, Málaga, Spain, in 1993 and 1998, respectively. In 1994, he joined the Departamento de Ingeniería de Comunicaciones, Universidad de Málaga, where, in 2001, he became an Associate Profesor. His current research interests include electromagnetic material characterization, measurement techniques, and RF and microwave circuits design for communication applications.

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Effective Electromagnetic Parameters of Novel Distributed Left-Handed Microstrip Lines Shau-Gang Mao, Member, IEEE, Shiou-Li Chen, Student Member, IEEE, and Chen-Wei Huang

Abstract—The novel one-dimensional left-handed microstrip lines (LHMLs) consisting of the arrays of thin wires and two-layer split-ring resonators are investigated theoretically and experimentally in this paper. Unlike the conventional left-handed metamaterials for waveguides or microstrip lines, which are bulky three-dimensional constructions or require the lumped elements for high-pass configuration, this distributed structure can be directly implemented on a substrate by photolithographic techniques without soldering any chip inductors or capacitors. Moreover, it can also be easily realized at a higher frequency region by scaling the dimensions of the structure, making it highly efficient and flexible in millimeter-wave applications. To characterize the inhomogeneous LHML, the effective medium description is developed for extracting the effective electromagnetic parameters, i.e., the complex effective permittivity and permeability, as well as the refractive index. Results show that not only the simultaneously negative real permittivity and permeability, but also the antiparallel phase and group velocities may be achieved in the design passband region. In contrast to the antenna array using the conventional microstrip delay line, the LHML is incorporated in the series-fed microstrip combline array to exhibit the leading phase between the successive elements. Index Terms—Backward-wave radiation, distributed metamaterial, effective electromagnetic parameters, left-handed microstrip lines (LHMLs).

I. INTRODUCTION

I

NTEREST IN the new and bizarre class of materials for the electromagnetic community has built up over the past decades, and some impetus has been provided by the increasing focus on system performance and the advent of novel applications [1]–[3]. Over the last few years, numerous groups of researchers have emerged with a similar goal: to explore the exceptional properties of materials not really observed in nature. One of the highlights of the unprecedented metamaterials is the left-handed metamaterial (LHM). LHMs are a new class of artificially ordered composites whose real parts of the permittivity and permeability are simultaneously negative. The history of LHMs can be traced back to Veselago’s theoretical hypothesis in 1968 [4], in which he demonstrated that the LHM would result in unusual optical phenomena when light passed through it, including the anomalous refraction and reversal of both Doppler effect and Cerenkov radiation. In 2003, the controversy over

Manuscript received June 1, 2004; revised October 25, 2004. This work was supported by the National Science Council of Taiwan, R.O.C., under Grant NSC 93-2213-E-027-024 and Grant NSC 93-2752-E-002-004-PAE. The authors are with the Institute of Computer and Communication Engineering, National Taipei University of Technology, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845192

whether LHMs could be realized was resolved, thanks to the experiments performed in [5] and [6]. The research of using LHMs to provide novel applications in the practical microwave devices, such as the LHM-based antennas, phase shifters, couplers, resonators and lens, is a relatively new and exciting development [7]–[14]. Most of the recently presented LHM structures use combinations of the lumped-element inductors and capacitors on the one-dimensional (1-D) or two-dimensional (2-D) high-pass transmissionline configurations. Several sophisticated theoretical studies have currently been conducted to deal with the macroscopic constitutive relations of the LHM [15]–[17]. In this paper, the distributed 1-D left-handed microstrip line (LHML) consisting of the periodically loaded thin wires and split-ring resonators (SRRs) along the two-layer microstrip line is proposed. The similar types of left-handed coplanar waveguides and microstrip lines were presented in [18] and [19]. Here, we investigate in more detail the effective electromagnetic parameters of the proposed LHML and its application to the anand permeability tenna array. First, its effective permittivity are determined to examine how the LHML interacts with the electromagnetic wave. Our approach here inverts the scatand tering parameters of the LHML to extract the complex . Note that the technique we describe is readily applicable to both the experiment and simulation of the LHML whenever the scattering parameters are obtained. Second, the proposed effective parameter extraction method is verified by investigating not only the traditional microstrip line, but also the left-handed microstrip structure using interdigital capacitors and short-circuited stub inductors [20]. We also demonstrate that the opposite and could exist without sign of the imaginary parts of violating the entropy condition. Third, the significant features in the passband of the LHML, i.e., the simultaneously negative real permittivity and permeability, negative real refractive index, and characteristic impedance are presented. Finally, the phase-advanced characteristic, considered useful in phase-shifter design, is utilized to establish the LHML-based antenna array. II. DISTRIBUTED LHML CONFIGURATION Fig. 1 shows the physical configuration of a unit cell of the novel two-layer LHML structure. This multilayer architecture with different dielectric layers cannot only suppress the power leakage [21], but also realize the high-performance circuits in compact size [22]. The black and shaded areas represent the conductors located on the top of dielectrics 1 and 2, respectively. The two-layer split-ring resonators (SRRs) consist of two concentric square rings, whose splits are oriented opposite to each

0018-9480/$20.00 © 2005 IEEE

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Fig. 2. Signal flow graph for Fig. 1.

two-layer SRR more compact than the single-layer SRR for the same resonant frequency. Moreover, the proposed structure can avoid the bianisotropy in comparison with the single-layer SRR, which benefits the scope of the LHML applications [23]. III. EFFECTIVE ELECTROMAGNETIC PARAMETERS Fig. 1. Physical configuration of a single-cell 1-D LHML structure (h = 0:635 mm, h = 1:5748 mm, " = 10:2, " = 2:2, L = 3:15 mm, L = 0:25 mm, L = 0:75 mm, W = 1:4 mm, W = 0:4 mm, W = 0:5 mm, S = 0:2 mm).

other. The SRRs can be designed for a particular resonant frequency by careful choice of the ring inductances and gap capacitances [23]. The vertical via is placed parallel to the -axis and bounded to the strip conductor and ground plane. When the applied electric field is parallel to the thin wires, the negative real permittivity can be obtained [18], [24]. As to the negative real permeability, it can be produced when the magnetic field is parallel to the axis of SRRs [25], [26]. In the case when the quasi-TEM mode propagates along the -axis of the two-layer microstrip line, the electric field is concentrated between the strip conductor and ground plane, and the magnetic field encircles the strip conductor. Hence, the array of vertical wires parallel to the -axis, considered as artificial plasma, demonstrates negative real permittivity below its plasma frequency. The periodic SRRs nearby the strip conductor to couple the fringing magnetic field generate a negative real permeability. Therefore, the proposed two-layer microstrip line with periodically loaded SRRs and thin wires may exhibit the left-handed characteristic. These effective permittivity and permeability for the SRR-only, wire-only, and LHML structures are discussed separately in Section IV. The commercial software Ansoft HFSS based on the finite-element method is used to examine the structures in this paper, while the vector network analyzer HP 8720C is used to obtain the experimental data. The reference planes for the simulated and measured scattering parameters are indicated in Fig. 1. Several significant characteristics of the proposed LHML are shown compared with the conventional LHMs for the bulky constructed waveguides or the lumped-element microstrip lines. One is that it can be directly implemented on the substrate by photolithographic techniques without soldering any chip inductors or capacitors, and it can also be easily realized at a higher frequency region by scaling the dimensions of structure [27]. and are selected, the Additionally, because larger capacitance in the small separation region between the rings can be generated by the two-layer SRR with two concentric square rings on both sides of dielectric 1. This makes the

As shown in Fig. 1, the fields in the proposed heterogeneous LHML are highly inhomogeneous. To characterize this structure under the dominant quasi-TEM mode excitation, the effective medium theory is used in the long-wavelength approximation, i.e., when the wavelength is much larger than the dimensions of the constituent scattering elements that compose the medium. This theory is based on the idea of replacing the inhomogeneous LHML by an equivalent microstrip structure (EMS) with homoand permeability geneously filled effective permittivity so that the fields in the EMS are equal to the mean fields in the original structure. The size of the unit cell of the LHML is 3.65 mm, which indicates that the resonant frequency is approximately 5 GHz with free-space wavelength of approximately 6 cm. This indicated wavelength exceeds by a factor of 15 the structural details of the LHML. Therefore, the proposed LHML can be treated as a and . macroscopically homogeneous EMS with These complex effective medium parameters can be extracted from the transmitted and reflected waves of the EMS with the corresponding signal flow graph, as shown in Fig. 2. Consider propagating along the a wave with the time dependence of -axis, the reflection coefficient for a wave passing from the reference transmission line into the EMS is (1) and are the characteristic impedances of the where EMS and the reference transmission line, respectively. Additionally, when the wave propagates through the EMS, it will attenuate and shift in phase according to the complex propagation . The propagation factor of the EMS with constant (Fig. 1) is a physical length

The two-port scattering parameters can then be determined by using the signal flow graph [28] (2) (3)

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Solving (2) and (3) by the Niconlson–Ross–Weir (NRW) approach [29], [30], we can obtain the equations of (4) where the sign of

is determined by

The relationship between is transmission line

and

and the air-filled reference

(5) Hence, (5) as follows:

can be obtained by combining (1) and

(6) and can be determined by the analytical where formulas [31] and is expressed by scattering parameters, as shown in (4). Furthermore, the propagation constant can be determined by Bloch wave analysis [32] (7)

Fig. 3. Measured and simulated " substrate.

and 

of a microstrip line with the FR4

The effective refractive index of the proposed LHML is then and by using the above technique are not in contradiction with the law of increase of entropy, as is demonstrated in Section IV. (8) If the material is passive, the requirement of causality fixes the choice of sign in (8) [33]. With the relations expressed in (6) and (8), the complex effective permittivity and permeability can be separated as (9) (10) Moreover, and , associating with the electric and magnetic losses, respectively, are closely related to the dissipation is determined by using of energy. The dissipated energy Poynting’s power balance theorem to a dispersive medium [34]

(11)

For the lossy material for which , it indicates that at least and in (11) must be larger than zero over some one of range of positive frequency. Note that the extracted complex

IV. RESULTS AND DISCUSSION To illustrate the effective parameter extraction method, a 10-mm-long 50- microstrip line is made on an FR4 substrate with the well-known dielectric properties (relative permittivity , relative permeability , thickness mm, ). In Fig. 3, complex and and loss tangent are obtained from the measured and simulated scattering parameters of the microstrip line. The dispersion characteristic from the closed-form equation [31] is also included for of and are in close agreement with comparison. Measured the results given by the full-wave simulation and the analytical equation, which verifies the proposed effective parameterextraction method. To obtain the simultaneously negative real permittivity and permeability, the design of the artificially created LHML should be realized by treating the electric and magnetic properties separately [35]. Typical frequency dependences of the simulated effective parameters of the SRR- and wire-only structures are shown in Fig. 4. Note that the SRR-only around the 4.5-GHz resonant structure displays a negative frequency and the wire-only structure produces a negative lasting until 5 GHz.

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Fig. 4. Simulated " and  for the one-cell SRR- and wire-only structures (for Figs. 4, 5, and 7–9, the dimensions of the SRR and wire are the same as those in Fig. 1).

Fig. 5 depicts the measured and simulated complex effective permittivity and permeability for the LHML. There is a close correspondence between the measured and simulated results regardless of the slight frequency misalignment resulting from the sensitivity of the relative positions of both SRRs and wires. By combining the arrays of wires and SRRs, there is a frequency region over which both and are negative. Compared with the results of Fig. 4, the addition of the wires to the SRRs does not significantly alter the permeability properties of the SRRs. However, the composed wires and the SRR’s structure results in a negative real permittivity that is opposite to that for the and need to SRR alone. Moreover, as described in [36], be positive simultaneously to preserve causality (dissipated en). However, and have opposite signs in our ergy study, which seems to violate the requirement of causality. To demonstrate the positive for the proposed passive LHML, the in (11) is determined and presented value is still positive, in Fig. 5. It is observed that resulting in the condition even if and during the frequency range of interest. In addition to the proposed distributed LHML, the lefthanded microstrip structure using interdigital capacitors and short-circuited stub inductors presented by [20] is also investigated to further demonstrate the versatility of the proposed effective parameter-extraction method for a wide range of and are negative simultaneously applications. In Fig. 6,

Fig. 5. Measured and simulated " and  for the one-cell LHML structures consisting of SRRs and a vertical wire. The value " j j  j" j is also presented.

+

Fig. 6.

Simulated "

and 

for the other proposed LHML [20].

MAO et al.: EFFECTIVE ELECTROMAGNETIC PARAMETERS OF NOVEL DISTRIBUTED LHMLs

Fig. 7. Measured and simulated wire-only, and LHML structures.

S

and

S

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of the three-cell SRR-only,

between 1.4–3.8 GHz, which permits the passband of the balanced composite right/left-handed structure, as performed in [20, Figs. 2 and 3]. Fig. 7 shows the simulated scattering parameters for the three-cell SRR-only, wire-only, and LHML structures. Due to [see Fig. 4], the transmission through the array of negative SRRs is large for all the frequencies outside the resonant interval (4.47–4.65 GHz) and decreases to 15 dB in this range. Also, is negative [see Fig. 4], the transmission of the array since of metallic wire is very small for all frequencies below 5 GHz. The LHML structure created by the combination of an array of SRRs and wires exhibits high transmission within the resoand are negative, as shown in nance interval, where both Fig. 5. For frequencies outside the resonant interval, the product of the LHML is negative so that the transmission deand of the LHML structure cays rapidly. Measured are also presented for verification. Inherently, due to the dielectric and conductor losses of the LHM investigated in [19], the proposed three-cell LHML with the measured around 2.5 dB in the passband is reasonable. More accurate and morecostly fabrication processes should lead to larger transmission in the passband. Also note that the proposed LHML exhibits the narrow passband performance since it uses the resonator-type constituent (i.e., SRR). The magnitude and angle of measured and for the one-cell LHML are shown in Fig. 8. Within phase response is the passband region of the LHML, the positive, hence, implying that the phase velocity is oppositely directed to the power flow [12], [37]. This backward-wave propagation is one of the necessary signatures of an LHML. One of the useful applications for the phase-advanced characteristic of the proposed LHML, such as the phase shifter in antenna arrays, is presented in Section V. The measured real and imaginary parts of the effective refractive index obtained from (8) are also shown in Fig. 8. Note that is negative and almost vanishes for frequencies between 4.47–4.6 GHz, which indicates the passband of the LHML [see Fig. 7]. However, for frequenhas a positive value, showing a cies outside this passband, lossy mechanism that deteriorates the transmission magnitude of the LHML. In the passband region, the guided wavelength [33] increases from 23 to 790 mm as depicted is significantly larger than the in Fig. 8, which indicates that

Fig. 8. Measured magnitude and angle of S and S , effective refractive index, guided wavelength, and characteristic impedance for the one-cell LHML.

dimensions of the unit cell (3.65 mm). Hence, the LHML can be effectively considered as a homogeneous medium and the effective medium theory can be used successfully. The measured of the LHML obtained from (5) characteristic impedance is also depicted in Fig. 8. The real part of is close to 50 and its imaginary part is approaching to zero at 4.55 GHz, which in the passband region. results in maximum V. LHML-BASED ANTENNA ARRAY The radiation characteristic of the microstrip combline array with two open-circuited stubs is investigated here. The four-cell LHML is incorporated in the series-fed configuration for a phase-shifting element. A quarter-wavelength microstrip line

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the endfire direction for frequencies from 4.63 to 4.65 GHz, as indicated in Fig. 10. These results show that the LHML-based microstrip combline array exhibits the significant backward radiation and the frequency-scanning main beam because the LHML possesses a dispersive phase-advanced characteristic. VI. CONCLUSIONS

Fig. 9. Physical configuration of the LHML-based microstrip combline array (L = 7:5 mm, L = 9 mm, L = 7:97 mm, L = 8:3 mm, L = 8:3 mm, W = 5:5 mm, W = 1 mm, W = 2:4 mm, W = 2:4 mm, W = 0:95 mm, ' = 35 and  = 40 : the dimensions of the unit cell LHML are the same as those in Fig. 1).

The planar distributed LHMs realized by the combination of the arrays of SRRs and wires along the two-layer microstrip line have been constructed, simulated, and then measured, showing a good agreement between measurement and simulation. To characterize the proposed LHML based on the effective medium , and have been extracted from the theory, complex scattering parameters, and these results have then been used to explain the propagation characteristics of the LHML. Additionand , indicating electric and magnetic losses of the ally, LHML, have been determined without contradicting the entropy condition. Moreover, the technique we developed has been further applied to examine other LHM structures to illustrate the usefulness of the formulation. To achieve unusual backward radiation, the CMDL in the microstrip combline array is replaced by the LHML as the phase shifter. Some potential applications of the proposed LHML, such as the LHML-based antennas and filters with unique properties, are currently under investigation. ACKNOWLEDGMENT

Fig. 10. Measured co-polarized radiation pattern at 4.63 and 4.65 GHz in the x–z plane for the series-fed microstrip combline array using the LHML and CMDL as a phase shifter, respectively.

The authors would like to thank the reviewers for their constructive comments. Author S.-G. Mao, wishes to gratefully acknowledge Prof. C. H. Chen, Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C., for his continued encouragement and support throughout all phases of this research.

is used at the input of the array in order to minimize reflection at the input terminal. The illustration of two microstrip open-stub radiators along with the four-cell LHML, impedance matching network, and feed line are shown in Fig. 9. The wave traveling in the LHML exhibits the antiparallel phase and group velocities, thus incurring a positive phase shift along the radiating elements of the combline array away from the feed end. Therefore, the proposed LHML-based combline array is capable of supporting the backward-wave radiation. The measured co-polarized radiation patterns at 4.63 and 4.65 GHz in the – plane (Fig. 1) for the series-fed microstrip combline array with and without the LHML are depicted in Fig. 10. The antenna is fed from the positive -direction, and the radiation pattern is measured in an anechoic chamber with the far-field antenna measurement system. At 4.63 GHz, the main , i.e., beam of the LHML-based array is directed at 22 toward the backfire direction from the broadside; whereas at 4.65 GHz, the main beam moves closer to the broadside . This is because the direction and is approximately phase and the negative are progressively approaching zero with increase in frequency, as shown in Fig. 8. By contrast with the antenna array using the LHML phase shifter, the main beam of the array using the conventional microstrip delay line (CMDL) scans slightly away from the forward direction toward

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[11] A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1252–1263, Apr. 2004. [12] C. Caloz and T. Itoh, “Transmission line approach of left-handed (LH) materials and microstrip implementation of an artificial LH transmission line,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1159–1166, May 2004. [13] M. Anioniades and G. V. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 7, pp. 103–106, Jul. 2003. [14] A. Grbic and G. V. Eleftheriades, “Experiment verification of backwardwave radiation from a negative refractive index metamaterial,” J. Appl. Phys., vol. 92, pp. 5930–5935, Nov. 2002. [15] C.-Y. Cheng and R. W. Ziolkowski, “Tailoring double-negative metamaterial responses to achieve anomalous propagation effects along microstrip transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2306–2314, Dec. 2003. [16] C. M. Krowne, “Electromagnetic-field theory and numerically generated results for propagation in left-handed guided-wave single-microstrip structures,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2269–2283, Dec. 2003. [17] IEEE Trans. Antennas Propag. (Special Issue), vol. 51, no. 10, Oct. 2003. [18] F. Falcone, F. Martin, J. Bonache, R. Marques, T. Lopetegi, and M. Sorolla, “Left handed coplanar waveguide band pass filters based on bi-layer split ring resonators,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 1, pp. 10–12, Jan. 2004. [19] M. Schussler, A. Fleckenstein, J. Freese, and R. Jakoby, “Left-handed metamaterials based on split ring resonators for microstrip applications,” in Eur. Microwave Conf., vol. 3, Oct. 2003, pp. 1119–1122. [20] A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 2, pp. 68–70, Feb. 2004. [21] Y. Liu, K. Cha, and T. Itoh, “Non-leaky CPW with conductor backing,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 5, pp. 1067–1072, May 1995. [22] I. J. Bahl, “High- and low-loss matching network elements for RF and microwave circuits,” IEEE Microw. Mag., vol. 1, no. 9, pp. 64–73, Sep. 2000. [23] R. Marques, F. Mesa, J. Martel, and F. Medina, “Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design—Theory and experiments,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2572–2581, Oct. 2003. [24] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys., Condens. Matter, vol. 10, pp. 4785–4809, 1998. , “Magnetism from conductors and enhanced nonlinear phe[25] nomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [26] D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B, Condens. Matter, vol. 65, pp. 195 104/1–195 104/5, 2002. [27] T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science, vol. 303, pp. 1494–1496, Mar. 5, 2004. [28] G. Gonzalez, Microwave Transistor Amplifies: Analysis and Design, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1997. [29] A. M. Nicolson and G. Ross, “Measurement of intrinsic properties of materials by time domain techniques,” IEEE Trans. Instrum. Meas., vol. 19, no. 11, pp. 377–382, Nov. 1970. [30] W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE, vol. 62, no. 1, pp. 33–36, Jan. 1974. [31] T. Itoh, Ed., Planar Transmission Line Structures. New York: IEEE Press, 1987. [32] S.-G. Mao and M.-Y. Chen, “Propagation characteristics of finite-width conductor-backed coplanar waveguides with periodic electromagnetic bandgap cells,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 11, pp. 2624–2628, Nov. 2002. [33] J. O. Dimmock, “Losses in left-handed materials,” Opt. Express, vol. 11, pp. 2397–2402, Sep. 22, 2003.

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[34] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. New York: Pergamon, 1982. [35] F. J. Rachford, D. L. Smith, P. F. Loschialpo, and D. W. Forester, “Calculations and measurements of wire and/or split-ring negative index media,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 66, pp. 036 613/1–036 613/5, 2002. [36] R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microwave Opt. Technol. Lett., vol. 41, no. 4, pp. 315–316, May 2004. [37] O. F. Siddiqui, S. J. Erickson, G. V. Eleftheriades, and M. Mojahedi, “Time-domain measurement of negative group delay in negative-refractive-index transmission-line metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1449–1454, May 2004.

Shau-Gang Mao (S’97–M’98) received the B.S. degree in atmosphere science and the M.S. and Ph.D. degrees in electrical engineering from the National Taiwan University, Taipei, Taiwan, R.O.C., in 1992, 1994, and 1998, respectively. From October 1998 to July 2000, he fulfilled military service with the Department of Communication, Electronics, and Information, Coast Guard Administration, Executive Yuan, Taiwan, R.O.C., where he conducted projects on coastal surveillance and communication systems. From August 2000 to January 2002, he was with the Department of Electrical Engineering, Da-Yeh University. Since February 2002, he has been with the Institute of Computer and Communication Engineering, National Taipei University of Technology, Taipei, Taiwan, R.O.C., where he is currently an Associate Professor. His research includes microwave and millimeter-wave circuits and antennas. Dr. Mao was the secretary of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Taipei Chapter in 2001. He is the member of the Editorial Boards of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the recipient of the 2001 Best Paper Award presented at the Asia–Pacific Microwave Conference (APMC) and the 2004 International Union of Radio Science (URSI) Young Scientist Award presented at the International Symposium on Electromagnetic Theory.

Shiou-Li Chen (S’04) was born in I-Lan, Taiwan, R.O.C., on November 26, 1979. She received the B.E. degree in computer and communication engineering from Da-Yeh University, Changhua, Taiwan, R.O.C., in 2002, the M.S. degree from the Institute of Computer and Communication Engineering, National Taipei University of Technology, Taipei, Taiwan, R.O.C., in 2004, and is currently working toward the Ph.D. degree at the Institute of Computer and Communication Engineering, National Taipei University of Technology. Her research interests include design and analysis of planar microwave circuits and antennas for wide-band and wireless communication applications.

Chen-Wei Huang was born in Taichung, R.O.C., on May 28, 1980. He received the B.E. degree in electrical engineering from the National Taipei University of Technology, Taipei, Taiwan, R.O.C., in 2002, and is currently working toward the M.S. degree at the National Taipei University of Technology. His research interests include electromagnetic bandgap structures, material measurements, planer left-handed material and modeling of the LHM’s microstrip lines equivalent circuits.

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Experimental Realization of a One-Dimensional LHM–RHM Resonator Yan Li, Lixin Ran, Hongsheng Chen, Jiangtao Huangfu, Xianmin Zhang, Kangsheng Chen, Tomasz M. Grzegorczyk, Member, IEEE, and Jin Au Kong, Fellow, IEEE

Abstract—We present the experimental realization of a onedimensional left-handed/right-handed resonator. The left-handed (LH) part of the resonator is formed by a solid-state-like metamametallic structures, while the right-handed terial composed of part is just air. The resonant behavior is successfully observed value is obtained. Through phase measureand an acceptable ment, we verify that the LH metamaterial layer acts as a phase compensator. Further proof is given by varying the thicknesses of both layers simultaneously to achieve the same resonant frequency under the limitation of total phase difference less than , which is a criterion dramatically different from traditional cavity resonators.



2

Index Terms—Left-handed metamaterials (LHMs), resonator.

I. INTRODUCTION

M

ATERIALS with simultaneously negative permittivity and negative permeability over a certain range of frequencies, termed left-handed metamaterials (LHMs) by Veselago early in 1968 [1], are attracting more and more interest in many scientific disciplines. Numerous potential applications of LHM have been proposed and studied theoretically, such as flat lenses [2], filters [3], resonators [4], antennas [5], etc. Currently, several types of artificial LHMs constructed from various split-ring resonators and rods [6]–[10], -like pattern resonators [11], [12], or transmission-line-based structures [13], [14] have been reported, and some of them have yielded acceptable electromagnetic characteristics. These achievements have rendered possible the quest for real applications based on LHMs. In this paper, we perform a series of experiments to test the characteristics of a one-dimensional (1-D) LHM–right-handed material (RHM) resonator. The LHM part of the resonator is constructed from -like rings printed on low-loss substrates compressed using a hot-press technique to realize a solid-state-like material [12], while the RHM part of the resonator is simply air. The experimental results show that, compared with traditional cavity resonators, the LHM–RHM resonator indeed works in a Manuscript received June 1, 2004; revised September 16, 2004. This work was supported by the Chinese Natural Science Foundation under Contract 60201001, Contract 60271010, and Contract 60371010, by the Defence Advanced Research Projects Agency under Contract N00014-03-1-0716, and by the Office of Naval Research under Contact N00014-01-1-0713. Y. Li, L. Ran, H. Chen, J. Huangfu, X. Zhang, and K. Chen are with the Department of Information and Electronic Engineering, Zhejiang University, Hangzhou 310027, China. T. M. Grzegorczyk and J. A. Kong are with the Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA and also with The Electromagnetics Academy, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845191

Fig. 1. Illustration of a sub-wavelength cavity resonator as proposed in [4]. The left layer of thickness d is a conventional isotropic material with  > 0,  > 0(n > 0), and the right layer of thickness d is taken to be an isotropic metamaterial with  < 0 and  < 0(n < 0).

different mode and exhibits some unique properties, which are in agreement with the descriptions provided in [4] within acceptable differences due to the 1-D nature of the metamaterial used in the experiment and to the inherent losses and impedance mismatch at the boundaries. II. THEORY AND EXPERIMENTAL CONSIDERATIONS Fig. 1 shows the basic structure of a 1-D sub-wavelength cavity resonator consisting of LHM and RHM layers, as originally proposed in [4]. Assuming that both layers are lossless, the dispersion relation can be described by [15] (1) where represents the wave vector in free space. In conventional (right-handed (RH) only) materials, the dispersion relation of (1) can only be satisfied under some very specific conand . If one of the media is left-handed (LH), ditions for however, where the effective permeability and index of refraction are negative, the solution to (1) becomes much less dependent on the two thicknesses and , such that this 1-D cavity resonator can be realized with a thickness far smaller than half of the wavelength. In such a structure, the LHM layer acts as a phase compensator in its effective frequency band, and resonance occurs under the condition of zero total phase difference

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Fig. 3. Magnitude of S 21 measured when a 5-cm-long LHM sample with ten unit cells is placed into an empty waveguide. The vertical axis represents the magnitude of S 21 in dB and the horizontal axis represents the frequency in gigahertz.

Fig. 2. Experimental setup for the realization of an LHM–RHM resonator. (a) Lateral view. (b) General configuration and dimensions.

between the front and back planes of the resonator. A detailed theoretical study of this resonator can be found in [4]. In this paper, we look at verifying some of the conclusions proposed in [4] by building a similar structure. In the experimental setup, the LHM sample is constructed from a solid-state-like block consisting of multilayer low-loss microwave substrates on which -like metallic patterns are alternately printed. We have shown that such arrangement of printed circuit boards (PCBs) printed with -like metallic patterns yields a clean and wide band of negative index of refraction with reduced losses [12], and that replacing the air gaps with the same substrate and compressing all the layers into a solid-state form significantly improves the performance. Several experiments have been performed in a planar waveguide, showing that the sample exhibits LH properties in a 1-GHz-wide passband with insertion losses of less than 0.5 dB per unit cell [12]. The experiments reported in this study are based on such a sample and are performed via the following three steps. Step 1) We measure the transmission property of an LHM sample to ensure that the LH behavior still exists when the metamaterial is placed in a rectangular waveguide instead of in a parallel-plate waveguide.

Fig. 4. Comparison of the estimated data and measured data about the phase difference corresponding to a 5-mm-long LHM sample.

Step 2) We measure and calculate the total phase difference of the two-layer resonator to see whether it is approaching zero, which is an indication that the structure resonates. Step 3) We vary the thicknesses of both layers simultaneously and measure the corresponding resonant frequencies. The main expectation is to have an invariant resonant frequency for various thicknesses, which is obviously impossible to achieve with a resonator composed of two RH materials under (yielding the limitation of total phase difference less than the minimum length of the resonator). Fig. 2 shows the experimental setup. It is composed of a 13-mm-long -band rectangular waveguide with a cross sec10.16 mm. The right end of this wavetion of 22.86 mm guide is connected with a short-circuit piston of which the position within the waveguide provides an accurate control on the total length of the cavity. The left end of the waveguide is short circuited by a copper sheet with the same cross section, but with a rectangular slot in the center, from where the microwave power is fed into the resonator. The LHM sample is placed into

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TABLE I PHASE ANALYSIS OF THE LHM–RHM RESONATOR. THE NEGATIVE SIGN IN THE FOURTH COLUMN MEANS A PHASE ADVANCE

the waveguide, and the RHM medium is realized by the remaining air. Different lengths of LHM samples are adopted, and screwing the piston in or out allows to vary the length of the air layer. The experimental data are recorded using a PC from a general-purpose interface bus (GPIB) port of an Agilent 8722ES -parameter network analyzer. III. EXPERIMENTAL RESULTS Before performing the resonance experiment, we first measure the transmission property of a 5-cm-long LHM sample in a rectangular waveguide to ensure that the LH properties are still measured at the expected frequencies. The results are shown in with [12, Fig. 2(a)], we see Fig. 3. Comparing the curve of that the transmission band located between 8.5–9.7 GHz still exists, suggesting that the sample still works properly inside the rectangular waveguide. In addition, by observing the negative -parameter when different lengths of phase changes of the samples are measured, we can also conclude that the transmission band indeed corresponds to an LH band. The results of these measurements are shown in Fig. 4, where the solid curve represents the phase shift corresponding to a 5-mm-long LHM sample. For comparison, the estimated values (dashed line) are also shown, which are calculated from , where represents the width of represents the refractive index obtained the waveguide and from the prism refraction experiment [12]. Our experiments for this specific metamaterial indicate that the values of range essentially between and in the LH band. The results reported hereafter have been obtained with , knowing that this introduces a the average value of phase uncertainty of approximately 10 . It can be seen that the measured phase advance oscillates around the theoretical estimation. The difference may be explained by the fact that we have neglected the impedance mismatch between the media and that the index of refraction has been estimated using a refraction experiment. Yet the results are sufficiently close to allow us to conclude that the sample of metamaterial still exhibits its LH properties within a similar frequency range when it is located inside a rectangular waveguide versus a parallel-plate waveguide. An important difference between this resonator and standard resonators is that, in this case, the total phase difference between the front and back planes can become zero if we as-

Fig. 5. Magnitude of S 11 measured when the resonators resonate at 9.33 GHz.

sume that the -component of the wave vector in the LHM is opposite to the one in air. A phase analysis is, therefore, performed in the second step. First, an empty waveguide is mea-parameter is obtained. A 5-mmsured and the phase of the long sample is then inserted into the empty waveguide and the -parameter is measured such that the phase difphase of the ference caused by the LHM layer can be estimated. The same sample is then placed into the resonator, the piston is adjusted to make the resonance occur at some different frequencies in -parameter, the resthe LH band, and the magnitude of the are onant frequency, and the resonator’s total length recorded. The same procedure is also repeated using a 10-mmlong sample. The corresponding experimental data are shown in Table I and Fig. 5. In Table I, the last column shows the total phase difference between the front and back planes of the two-layer structure. As can be seen, they are all relatively close to zero, which verifies our previous assumption on the reversal of the -component of the wave vector. We attribute the deviation in the results to the material losses and the difficulty of accurately measuring the length of the sample. curves for two different lengths of resFig. 5 shows the onators: the dashed curve is obtained for a resonator with an LH sample of 5 mm and a 16-mm-long air layer, while the solid curve is obtained for a resonator with an LH sample of

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Fig. 6.

Relationship between the resonator length and resonant frequency.

10 mm and an 18-mm-long air layer (the parameters and resonance condition for both cases are included in Table I). Fig. 5 clearly shows that the two resonators resonate at the same frequency, namely, 9.33 GHz, which suggests that we can achieve the same resonant frequency by simultaneously shortening the length of the LHM layer and of the air layer. This would be impossible with a conventional resonator under the limitation of total phase difference less than . Fig. 5 also gives us confidence that a high loaded value (244.8 for a 5-mm sample and 94.69 for a 10-mm sample) can be obtained with such resonators. The aforementioned fundamental property of this resonator is further illustrated in Fig. 6, which depicts the resonant frequency as function of the resonator length for two cases of LHM samples. The dashed line represents the resonator with a 5-mm-long LHM sample and the solid line represents the resonator with a 10-mm-long LHM sample. It is seen that the dashed line lies to the left of the solid line, indicating that a shorter resonator can be realized for an identical resonant frequency. Fig. 6, therefore, convinces us that in order to satisfy the resonant condition (total phase difference equal to zero), the same resonant frequency can be achieved by shortening the lengths of the two layers of materials (RHM and LHM), which marks the uniqueness of such a resonator. IV. CONCLUSION In this paper, a 1-D two-layer LHM–RHM resonator has been realized and measured to verify the prototype proposed in [4]. Our results have shown that the LHM layer indeed acts as a phase compensator and that the total phase difference of the two-layer resonator can reach zero at resonance, which is a completely different operating mode compared with traditional cavity resonators. Using this property, the LHM/RHM resonator can be realized with a size much smaller than the one of standard resonators, and is only limited by the size of the unit cell of the metamaterial. In addition, a relatively high- value can also be obtained. These results indicate that LHM can indeed be used to designs new types of resonators, which, with the reduction of the sizes of the unit cells, could be used in real applications.

[1] V. Veselago, “The electrodynamics of substances with simultaneously negative values of  and ,” Sov. Phys.—Usp., vol. 10, pp. 509–514, Jan.–Feb. 1968. [2] J. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966–3969, Oct. 30, 2000. [3] J. Lu and S. He, “Numerical study of a Gaussian beam propagating in media with negative permittivity and permeability by using a bidirectional beam propagation method,” Microwave Opt. Technol. Lett., vol. 37, pp. 292–296, Mar. 2003. [4] N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propag. Lett., vol. 1, no. 1, pp. 10–13, Dec. 2002. [5] S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vinvcent, “A metamaterial for directive emission,” Phys. Rev. Lett., vol. 89, pp. 213 9021–213 902-4, Nov. 2002. [6] J. Pendry, A. J. Holten, D. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [7] R. Shelby, D. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [8] R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B, Condens. Matter, vol. 65, pp. 144 440-1–144 440-6, 2002. [9] S. O’Brien and J. Pendry, “Magnetic activity at infrared frequencies in structured metallic photonic crystals,” J. Phys., Condens. Matter, vol. 14, pp. 6383–6394, 2002. [10] T. M. Grzegorczyk, C. D. Moss, J. Lu, and J. A. Kong, “New ring resonator for the design of left-handed metamaterials at microwave frequencies,” in PIERS 2003, Honolulu, HI, Oct. 13–16, 2003, p. 286. [11] J. Huangfu, L. Ran, H. Chen, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of -like metallic patterns,” Appl. Phys. Lett., vol. 84, pp. 1537–1539, Mar. 1, 2004. [12] L. Ran, J. Huangfu, H. Chen, Y. Li, X. Zhang, K. Chen, and J. A. Kong, “Microwave solid-state left-handed material with a broad bandwidth and a ultra-low loss,” Phys. Rev. B, Condens. Matter, vol. 70, pp. 073 1021–073 102-3, 2004. [13] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [14] C. Caloz and T. Itoh, “Transmission line approach for left-handed (LH) materials and microstrip implementation of an artificial LH transmission line,” IEEE Trans. Antennas Propagat., vol. 52, no. 5, pp. 1159–1166, May 2002. [15] J. A. Kong, Electromagnetic Wave Theory. New York: EMW, 2000.

Yan Li was born in Haerbin, China, in 1980. She received the B.S. degree in information and electronic engineering from Zhejiang University, Hangzhou, China, in 2003, and is currently working toward the M.S. degree at Zhejiang University. Her current research interests include LHM applications and embedded systems.

Lixin Ran was born in Jiangsu, China, in 1968. He received the Bachelor, Master, and Ph.D. degrees in information and electronic engineering from Zhejiang University, Hangzhou, China, in 1991, 1994 and 1997, respectively. He is currently an Associate Professor with Zhejiang University. His research interests involve microwave circuits and chaos, LHMs, and high-speed digital circuits.

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Hongsheng Chen was born in Zhejiang, China, in 1979. He received the B.S. degree in information and electronic engineering from Zhejiang University, Hangzhou, China, in 2000, and is currently working toward the Ph.D. degree at Zhejiang University. His current research interests include the application of the LHM, design of wide-band low-loss metamaterial structures, and numerical studies on electromagnetic properties of metamaterials.

Jiangtao Huangfu was born in Henan, China, in 1978. He received the B.S. and Ph.D. degrees in information and electronic engineering from Zhejiang University, Hangzhou, China, in 1999 and 2004, respectively. His current research interests include LHM and wireless communication.

Tomasz M. Grzegorczyk (M’00) received the Ph.D. degree from the Laboratoire d’Electromagnetisme et d’Acoustique (LEMA), École Polytechnique Federale de Lausanne (Swiss Federal Institute of Technology, Lausanne), Lausanne, Switzerland, in 2000. His doctoral research concerned the modeling of millimeter and submillimeter structures using numerical methods, as well as their technological realizations with the use of micromachining techniques. In January 2001, he joined the Research Laboratory of Electronics (RLE), Massachusetts Institute of Technology (MIT), Cambridge, where he is currently a Research Scientist. He has been a Visiting Scientist with the Institute of Mathematical Studies, National University of Singapore. In July 2004, he became an Adjunct Professor with The Electromagnetics Academy, Zheijiang University, Hangzhou, China. His research interests include the study of wave propagation in complex media including LHMs, the polarimetric study of oceans and forests, electromagnetic induction from spheroidal objects for unexploded ordnance modeling, waveguide and antenna design, and wave propagation over rough terrains. He is on the Editorial Board of the Journal of Electromagnetic Waves and Applications. Dr. Grzegorczyk has been part of the Technical Program Committee of the Progress in Electromagnetics Research Symposium since 2001.

Xianmin Zhang was born in Zhejiang, China, in 1965. He received the B.S. and Ph.D. degrees in physical electronics and optoelectronics from Zhejiang University, Hangzhou, China, in 1987 and 1992, respectively. In 1994, he became an Associate Professor of information and electronic engineering with Zhejiang University, and a Full Professor in 1999. He is currently the Director of the Institute of Electronic Information Technology and System, Zhejiang University. His research interests include LHM, fiber optics, and microwave photonics.

Kangsheng Chen is a Professor of information and electronic engineering with Zhejiang University, Hangzhou, China. From 1992 to 1996, he was the Director of the Information and Electronic Engineering Department, Zhejiang University. From 1996 to 1999, he was the Director of the Personnel Office, Zhejiang University. His research interests are in the area of microwave and optical waveguide technology, signal integrity, and LHM. He has authored or coauthored three books and over 100 refereed papers.

Jin Au Kong (S’65–M’69–SM’74–F’85) is the President of The Electromagnetics Academy and a Professor of Electrical Engineering with the Massachusetts Institute of Technology (MIT), Cambridge. His research interest is in the area of electromagnetic-wave theory and applications. He has authored or coauthored over 30 books including Electromagnetic Wave Theory (New York: Wiley–Interscience, 1975, 1986, 1990; EMW Publishing since 1998) and over 600 refereed journal papers, book chapters, and conference papers. He is Editor for the Wiley Series in Remote Sensing, Editor-in-Chief of the Journal of Electromagnetic Waves and Applications (JEMWA), and Chief Editor for the book series Progress in Electromagnetics Research (PIER).

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Super-Compact Multilayered Left-Handed Transmission Line and Diplexer Application Yasushi Horii, Member, IEEE, Christophe Caloz, Member, IEEE, and Tatsuo Itoh, Fellow, IEEE

Abstract—A novel super-compact multilayered (ML) composite-right/left-handed (CRLH) transmission line (TL) is proposed. This ML architecture consists of the periodic repetition of pairs of U-shaped parallel plates connected to a ground enclosure by meander lines. The parallel plates provide the left-handed (LH) series capacitance, and the meander lines provide the LH shunt inductance, while the right-handed parasitic series inductance and shunt capacitance are generated by the metallic connections in the direction of propagation and by the voltage gradient from the TL to the ground enclosure, respectively. In contrast to previously reported planar LH or CRLH TLs, the ML TL has its direction of propagation along the vertical direction, perpendicular to the plane of the substrates. This presents the distinct advantage that large electrical length can be achieved over an extremely short TL length and small transverse footprint. The LH-range characteristics of the multilayer CRLH TL are analyzed by the finite-element method and finite-difference time-domain (FDTD) full-wave simulations. In addition, the CRLH equivalent-circuit model is applied to gain simple insight into the behavior of the structure. Finally, the theoretical results are confirmed by experiments using the initial prototype with the very small length (thickness) of 0 016 and footprint of 0 06 0 08 ( = 0 = 235 mm at the center of the LH band, 0.4 GHz). The proposed miniaturized ML line can find applications in bandpass filters, delay lines, and numerous phase-advance components. As an example of application, a 1-GHz/2-GHz diplexer, composed of two ML CRLH TLs, is demonstrated. The ML CRLH TL proposed here presents a great potential for the development of novel microwave components taking profit of new multilayer technologies such as low-temperature co-fired ceramic technology. Index Terms—Composite right/left-handed (CRLH) transmission lines (TLs), delay/advance lines, diplexers, multilayered (ML) structures.

I. INTRODUCTION

O

VER THE last decade, left-handed (LH) materials, characterized by antiparallel phase and group velocity, have drawn considerable interest in the microwave community due to their potential for novel types of devices and components. The concept of left-handedness was initially introduced by Veselago in 1968, who theorized that a material having simultaneously negative permittivity and permeability would support

backward-wave propagation and exhibit negative refractive index [1]. Recently, Smith et al. succeeded in experimentally demonstrating an LH structure made of negative- thin wires [2] and negative- split-ring resonators [3]. An equivalent circuit approach was developed by Caloz and Itoh [4] and Eleftheriades et al. [5]–[7] and this approach has lead to the extended concept of composite right/left-handed (CRLH) materials, which fully take into account the parasitic right-handed (RH) effects naturally occurring in a practical LH structure [8], [9]. Several practical applications of CRLH structures have been demonstrated such as backfire-to-endfire leaky-wave antennas [10], zeroth-order resonators and antennas [11], broadband 0–3-dB directional couplers [12], branch-line couplers [13], and compact hybrid rings [14]. These applications are basically implemented in a microstrip technology and do not exhibit particularly small dimensions compared with the latest miniaturized components used in mobile communication systems. In order to reduce the dimensions of these microwave devices toward miniaturization, a novel multilayered (ML) CRLH transmission line (TL) with vertically stacked LH unit cells, perpendicular to the plane of the substrates, was theoretically proposed in [15]. In this paper, this structure is fully characterized and demonstrated experimentally, and a super-compact diplexer application of it is presented. The ML TL also presents the advantage of providing a broader LH bandwidth due to the very large series capacitance of the parallel-plate capacitors. Moreover, such ML structures can be easily implemented using modern low-temperature co-fired ceramic (LTCC) processes. This paper is organized as follows. Section II describes the ML CRLH architecture and its fundamental characteristics based on a TL model. A practical ML CRLH TL is demonstrated in Section III by finite-element method (FEM)/finite-difference time-domain (FDTD) simulations, circuit analysis, and measurements. In Section IV, a compact 1-GHz/2-GHz diplexer, integrating two ML CRLH bandpass filters, is presented as an application. II. ML CRLH TL ARCHITECTURE A. Motivation

Manuscript received June 26, 2004. Editorial process for this paper has been carried out by Prof. M. Steer, Editor-in-Chief, IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. Y. Horii is with the Faculty of Informatics, Kansai University, Osaka 569-1095 Japan (e-mail: [email protected]). C. Caloz is with the Microwave Research Group, Poly-Grames, École Polytechnique Montréal, Montréal, QC, Canada H3T 2B1. T. Itoh is with the Electrical Engineering Department and Microwave Electronics Laboratory, University of California at Los Angeles, Los Angeles, CA 90095 USA. Digital Object Identifier 10.1109/TMTT.2005.845189

Achieving a small-size LH structure seems a priori a challenging task. An LH structure requires a series capacitance and a shunt inductance , whereas natural materials provide a and a shunt capacitance , which globseries inductance ally corresponds to the CRLH equivalent circuit shown in Fig. 1. and must be produced As a consequence, the reactances artificially under the form of structured components such as interdigital capacitors and stub inductors [8]. Unfortunately, attempts to reduce the size of such planar components unavoid-

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Fig. 1. Equivalent circuit of an ML CRLH TL composed of four circuit parameters: LH-shunt inductance L , LH-series capacitance C , RH-series inductance L , and RH-shunt capacitance C .

ably results into reduction of the desired LH contributions, es. Eventually, left-handpecially that of the series capacitance edness can completely disappear if the size of the unit cell becomes too small. by parallel-plate caBy replacing the planar capacitor pacitors perpendicular to the direction of propagation, large series capacitance values can be obtained over a very small physical length, and miniaturized LH structures can thereby be obtained. The LH shunt inductance can then still be generated by a stub inductor. In addition, the RH series- /shunt- can be reduced to extremely small values if necessary, which improves the design flexibility of the TL. An immediate benefit of such an architecture for metamaterials is that the reduced size of the unit cell provides improved homogeneity in comparison with that achieved in planar configurations. These considerations suggest an ML CRLH TL structure of the type shown in Fig. 2. Such a structure can be easily fabricated by today’s LTCC processes, where distance between the plates can be accurately controlled with the resolution of up to 1 m. B. Description of the Structure The ML CRLH TL is constituted of periodically stacked LH unit cells, each of which is constituted by a pairs of U-shaped parallel plates connected to a ground enclosure by meander lines. The ports of this TL are located on either side of the structure, one at the top of it and the other one at the bottom of it. The gap between the U-shaped metallizations provides the , while the meander line provides the LH series capacitance LH shunt inductance . The vertical conductors linking to two plates of the U-shaped structure introduce a small RH series and the spacing between these conductors and inductance the metallic enclosure introduce an RH shunt capacitance , which can be made extremely small with a large spacing. The 50- termination strip lines, lying at the top or bottom of the ground enclosure, are connected to the output/input patches, which couple to the body of LH unit cells with the capacitance value of for matching, according to Fig. 1. Fig. 2 also shows the design parameters of the TL, which will be demonstrated in Section III, where the overall structure and strip-line ports will be embedded in a substrate with a permittivity and where total dimension of the TL will be ( mm at the center of the LH band, 0.4 GHz).

Fig. 2. Geometry of the ML CRLH TL. (a) Three-dimensional (3-D) view. (b) yz -plane cross section a–a at x = 400 mil, and locations of equivalent-circuit parameters. (c) xy -plane cross section b–b at z = 80 mil. The dimensions are in mil.

C. Equivalent Circuit and Useful Relations [9], [12] The equivalent circuit of a CRLH unit cell, as shown in Fig. 1, , an LH shunt inis composed of an LH series capacitance , an RH series inductance , and an RH shunt ductance . The matrix of this unit cell is given capacitance by

(1)

where (2) (3)

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The matrix of a finite-size -cell TL is simply obtained by taking the th power of the unit cell matrix (4) and the corresponding scattering parameters are then computed by the usual formulas (5)

(6)

On the other hand, the dispersion relation of an infinitely periodic CRLH TL can be derived by applying Bloch–Floquet thematrix of the unit cell (1) [9] orem to the

Fig. 3. Simulated frequency characteristics of the transmission parameter jS j and the reflection parameter jS j for the ML CRLH TL of Fig. 2. For the equivalent-circuit model, the extracted parameters corresponding to the circuit model of Fig. 1 are C = 0:1 pF, L = 4:7 nH, C = 9:6 pF, and = 9:2 nH. L

(7) where , , and represent the propagation constant of the TL, the period or size of the unit cell, and the angular frequency, respectively. This relation has two branches. The lower frequency branch exhibits antiparallel phase and group velocities (LH branch) and the higher frequency branch exhibits parallel phase and group velocities (RH branch). The eigenfrequenin (7) as cies at the spectral origin are obtained by setting (8) (9) and are correspond to the higher edge of the where LH branch and the lower edge of the RH branch, respectively, delimiting a frequency bandgap, in which propagation is prohibited. On the other hand, the lower edge of the LH branch is determined by (10) We should note, however, that in the balanced condition , these frequencies become identical, so that the gap closes up, yielding a seamless LH to RH transition. Finally, the phase velocity , group velocity , and group delay can be derived from (7) as

(11)

(12)

and (13)

III. DEMONSTRATION OF THE ML CRLH TL Fig. 3 shows the simulation results for the magnitude of the and refraction coefficient of the transmission coefficient ML CRLH TL. These curves are calculated by both Ansoft’s FEM commercial software HFSS 9.0 and an in-house FDTD code. All the metallizations are treated as a perfect electric conductor with zero thickness. These results show good agreement with each other, and show that the CRLH TL exhibits a 10-dB passband from 0.26 to 0.66 GHz. Though the passband includes large ripples, it will be improved by choosing appropriate structural parameters. To better characterize the behavior of the TL, the equivalent-circuit parameters of Fig. 1 are extracted. The extraction is estimated approximately procedure is the following. First, , where by the classical electrostatic expression is the permittivity of free space F/m , is the relative permittivity of the medium, is the area of each metal plate, and is the spacing between the plates. Next, and are estimated from the simulated the frequencies , and a first estimate of and is obtained magnitude of by (8) and (10). If can be read from the simulation, it will be by (9). Finally, these parameters are easier to decide and obtained by (5) adjusted by curve fitting between and (6) and their full-wave simulated counterparts. The paramepF, nH, ters extracted by this procedure are

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Fig. 6. Time variation of the electric-field distribution in the ML CRLH TL observed at xy -plane cross section c–c at y = 315 mil in Fig. 2(a). Calculation is carried out at 0.4 GHz (period T = 2:5 10 s) by FDTD. The solid arrow shows the direction of power flow, and the dashed arrow shows the direction of the field variation in the LH section. (a) t = 0. (b) t = 1=4 T. (c) t = 1=2 T. (d) t = 3=4 T. (e) t = T .

2

Fig. 4. Dispersion diagram of the ML CRLH TL of Fig. 2 obtained by FEM simulation and circuit analysis.

Fig. 5. (a) Phase velocity v and group velocity v . (b) Group delay t of the ML CRLH TL obtained by FEM simulation and circuit analysis.

pF, and nH. and calculated by the circuit model are also shown in Fig. 3. Excellent agreement between circuit model theory and the full-wave analysis can be observed. The dispersion diagram is shown in Fig. 4. The equivalent-circuit curve is obtained by closed-form expression (7). The FEM curve is obtained by applying the periodic boundary conditions to the edges of the unit cell, and by solving the relation between the phase difference at these edges and the eigenfrequency of the structure. These results verify that this structure operates as an LH TL.

Fig. 7. Prototype of the ML CRLH TL depicted in Fig. 2. (a) Assembled prototype. (b) Layer parts before assembling. Parts #1 and #12: patch with a port. Parts #2 and #11: separator for generation of 2C (h = 10 mil). Parts #3 and #4, #6 and #7, and #9 and #10: assembled U-shaped metallization. The meander line is sandwiched inside the U-shape, Parts #5 and #8: separator for generation of C (h = 20 mil).

Fig. 5(a) shows the phase velocity and group velocity , and Fig. 5(b) shows the group delay computed with (11)–(13) for the circuit model and from the computed dispersion relation for the FEM. In both results, the phase velocity is negative and has a resonance pole corresponding to the cutoff around 0.8 GHz. The group velocity is zero in the stopbands (LH high-pass stopband from 0 to 0.26 GHz and unbalanced CRLH stopband above 0.8 GHz). The group delay, although strongly varying at the edge of the passband, becomes relatively

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Fig. 8. Experimental results for the ML CRLH TL compared with full-wave FEM results. In the lossless model, all of conductors are treated as perfect 10 S/m electric conductors. In the lossy model, a conductivity of 5.8 (copper) is introduced for all the metals. The frequency shift between the simulation and the measurement is explained by the presence of unavoidable air gaps between the layers in the handmade prototype fabrication.

2

Fig. 9. Block diagram of the diplexer.

flat around the center frequency of the LH passband, which means that group-delay dispersion is moderate and that a modulated signal can, therefore, be transmitted along this line without significant distortion. In order to illustrate the backward-wave propagation phenomenon along the structure, Fig. 6 shows FDTD-simulated field distributions at different times along the structure, computed at the center of the LH band (0.4 GHz). The electric field is observed at the cross section of the – plane in Fig. 2(a). It can be read from the results that the wave comes in from the left side of the bottom and goes out from the top of the structure with the time progression, while the field in the stacked LH section goes down from the top to the bottom. This result clearly validates the backward wave propagation. To confirm the theoretical predictions, experiments have been carried out. Due to the unavailability of LTCC technology to the authors, the handmade prototype was built for the proof of concept. This prototype is shown in Fig. 7. The measured scattering parameters of this prototype are shown in Fig. 8 in comparison with the FEM results in the case of lossless metals (reported from Fig. 3) and of lossy metals with a conductivity of 5.8 10 S/m. The measured characteristics are shifted to higher frequency as a consequence of the presence of unavoidable thin air gaps in the prototype, which decrease the effective permittivity of the structure. The

Fig. 10. Geometry of the diplexer composed of two ML CRLH TLs. (a) 3-D view. (b) yz -plane cross section a–a at x = 275 mil. (c) xy -plane cross section b–b at z = 80 mil. The dimensions are in mil.

Fig. 11. Prototype of the diplexer. Port 1 is the common input. Port 2 is the output of structure “A.” Port 3 is the output of structure “B.”

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Fig. 12. FEM-simulated and measured scattering characteristics of the diplexer. (a) Reflection coefficient jS j observed at input port 1. (b) Transmission coefficient jS j from port 1 to port 2. (c) Transmission coefficient jS j from port 1 to port 3. (d) Isolation jS j between port 2 and port 3.

measured minimum insertion loss is 2.2 dB at 0.63 GHz and the maximum ripple is nearly 10 dB in the LH passband (evaluated from 0.38 to 0.69 GHz). This error is probably due to misalignment in the different 12 layers of the prototype and should be avoidable by using the modern LTCC fabrication technology. Even though the errors between theory and experiment are not negligible, the concept of the ML CRLH TL is demonstrated by this experiment. IV. DIPLEXER Conventional diplexers [17], [18] often use a low-pass and a high-pass filters for design simplicity, low insertion loss, and low cost. However, in applications where higher selectivity is required, the utilization of two narrow bandpass filters is necessary. This is typically done at the expense of increased size in a monopackaged device to avoid spurious coupling effects between the integrated capacitors and inductors of the two channels. Thus, there is a tradeoff between the size of a diplexer and its isolation in conventional RH implementations. The miniature ML CRLH TL presented in Section III presents an obvious potential interest in highly selective compact diplexers. This TL can be designed as a narrow-band filter by setting the frequencies and , given by (8) and (10), respectively, close to each other. Very high selectivity with a very small footprint is achievable in this structure by increasing the number of the vertically stacked unit cells. In addition, by using opposite directions of propagation through the two filters, excellent isolation can be obtained due to cancellation of mutual coupling. For the proof of concept, here we present a CRLH diplexer for 1 GHz/2 GHz. Fig. 9 shows a simple block diagram of the diplexer, consisting of two bandpass filters tuned for 1 GHz (called structure “A”) and 2 GHz (called structure “B”), respectively. Structures “A” and “B” are both ML CRLH TLs

differing only by their parameters to provide the two desired distinct passbands. The dimensions of “A” are larger than those of “B” since “A” is operating at a lower frequency than “B.” As shown in Fig. 10, “A” and “B” are arranged back-to-back with an inter-spacing of 80 mil. Vertical metals connected to the ground enclosure are added at the center of “A” and “B” to provide some and RH series inductance amount of RH shunt capacitance . The lines sections interconnecting structures “A” and “B” are much shorter than the guided wavelength at the designed frequencies of 1 and 2 GHz and, therefore, they exhibit negligible distributed effects. The 1-GHz wave originating from the input port is guided mostly to “A” through the three-port connection because “B” is designed to block the 1-GHz wave and, therefore, the line to “B” is seen as an open circuit at this frequency, while the 2-GHz wave is guided to “B” for the same reason. The overall , and the structure is embedded in a substrate with total dimension of the diplexer is ( mm at 1.0 GHz). Fig. 11 shows a picture of the corresponding fabricated prototype. In the prototype, the strip line connected to structure “B” is bent by 90 in order to avoid conflicting with the strip line connected to structure “A” in the fixation of the two subminiature A (SMA) connectors. The FEM-simulated and measured scattering characteristics are shown in Fig. 12. In the FEM results, the bandpass centered and at around 1 GHz in the transmission coefficient the bandpass centered at around 2 GHz in the transmission are clearly apparent. The excellent isolation coefficient of 38.8 dB at 1 GHz and 48.9 dB at 2 GHz is obtained between . As in the case of the simple TL, the two output ports prototype characteristics are shifted to higher frequencies, but otherwise good agreement with simulation can be observed, which demonstrates the validity of this miniaturized ML CRLH diplexer.

HORII et al.: SUPER-COMPACT ML LH TL AND DIPLEXER APPLICATION

V. CONCLUSION A novel super-compact ML CRLH TL architecture, consisting of the periodic repetition of pairs of U-shaped parallel plates connected to a ground enclosure by meander lines, has been proposed. This architecture, which supports propagation in the vertical direction perpendicular to the planes of the substrates, presents the advantage of providing large electrical length over an extremely short TL length and small transverse footprint, which can be exploited in diverse miniaturized components. An ML CRLH TL has been characterized by FEM and FDTD full-wave simulations, compared with the results of the CRLH equivalent circuit model and confirmed by measurements, using the initial prototype with the very small and footprint of length (thickness) of ( mm at the center of the LH band, 0.4 GHz). A 1-GHz/2-GHz diplexer, composed of two ML CRLH TLs, has been demonstrated as an example of application. The proposed ML architecture, in addition to reduced size, offers increased flexibility in the design of CRLH TLs. Whereas in previously reported structures, the parasitic RH contributions are difficult to control, they can be easily reduced if necessary in ML configurations. Moreover, by replacing the vertical metallizations to the metallic posts, as demonstrated in the diplexer application, the design flexibility can be highly extended. Furthermore, by engaging in the latest LTCC technology, much better performances can be obtained than those shown in this paper with handmade prototypes. For instance, significantly enhanced selectivity and isolation can be achieved in the diplexer by using more layers with suppressed air gaps. The proposed miniaturized ML structure may be applied to bandpass filters, couplers, delay/advance lines, and several other CRLH devices where miniaturization is a concern. ACKNOWLEDGMENT The authors would like to thank Dr. S.-M. Han, University of California at Los Angeles, for his assistance with prototype fabrication. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 10, no. 4, pp. 509–514, Jan.–Feb. 1968. [2] D. R. Smith, D. C. Vier, W. Padilla, S. C. Nemat-Nasser, and S. Schultz, “Loop-wire for investigating plasmons at microwave frequencies,” Appl. Phys. Lett., vol. 75, no. 10, pp. 1425–1427, Sep. 1999. [3] D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, no. 18, pp. 4184–4187, May 2000. [4] C. Caloz and T. Itoh, “Application of the transmission line theory of lefthanded (LH) materials to the realization of a microstrip ‘LH line’,” in IEEE AP-S Int. Symp., vol. 2, San Antonio, TX, Jun. 2002, pp. 412–415. [5] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L–C loaded transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec. 2002. [6] A. Grbic and G. V. Eleftheriades, “Periodic analysis of a 2-D negative refractive index transmission line structure,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2604–2611, Oct. 2003. [7] O. F. Siddiqui, M. Mojahedi, and G. V. Eleftheriades, “Periodically loaded transmission line with effective negative refractive index and negative group velocity,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2619–2625, Oct. 2003. [8] C. Caloz and T. Itoh, “Novel microwave devices and structures based on the transmission line approach of meta-materials,” in IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, Philadelphia, PA, Jun. 2003, pp. 195–198.

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[9] A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 2, pp. 68–70, Feb. 2004. [10] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backfire-to-endfire scanning capability,” Electron. Lett., vol. 38, no. 23, pp. 1414–1416, Nov. 2002. [11] A. Sanada, M. Kimura, I. Awai, H. Kubo, C. Caloz, and T. Itoh, “A planar zeroth-order resonator antenna using a left-handed transmission line,” in Eur. Microwave Conf., Amsterdam, The Netherlands, Oct. 2004, pp. 1341–1344. [12] C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 980–992, Mar. 2004. [13] H. Lin, M. De Vincentis, C. Caloz, and T. Itoh, “Arbitrary dual band components using composite right/left-handed transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 4, pp. 1142–1149, Apr. 2004. [14] H. Okabe, C. Caloz, and T. Itoh, “A compact enhanced-bandwidth hybrid ring using a left-handed transmission line,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 798–804, Mar. 2004. [15] Y. Horii, C. Caloz, and T. Itoh, “Vertical multi-layered implementation of a purely left-handed transmission line for super-compact and dualband devices,” in Eur. Microwave Conf., Amsterdam, The Netherlands, Oct. 2004, pp. 471–474. [16] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley, 2003. [17] A. R. Brown and G. M. Rebeiz, “A high-performance integrated K -band diplexer,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1477–1481, Aug. 1999. [18] K. L. Wu, Y. J. Zhao, J. Wang, and M. K. K. Cheng, “An effective dynamic coarse model for optimization design of LTCC RF circuits with aggressive space mapping,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 393–402, Jan. 2004.

Yasushi Horii (M’95) received the B.E., M.E., and D.Eng. degrees in communication engineering from Osaka University, Osaka, Japan, in 1988, 1990, and 1994, respectively. In April 1994, he joined the Faculty of Informatics, Kansai University, Osaka, Japan, as a Research Associate, and became a Lecturer in 1997 and an Associate Professor in 2000. From September 2003 to January 2004 and April 2004 to September 2004, he was a Visiting Research Scientist with the University of Victoria, Victoria, BC, Canada. From February 2004 to Match 2004, he was a Visiting Research Associate with the University of California at Los Angeles (UCLA). He is currently with Kansai University, Osaka, Japan. His research interests include design of microwave devices and circuits, ML fabrication technology, photonic-bandgap (PBG) structures and LH metamaterials. Dr. Horii is a member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan.

Christophe Caloz (S’99–M’03) was born in Sierre, Switzerland, in 1969. He received the Diplôme d’Ingénieur en Électricité and Ph.D. degree from the École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, in 1995 and 2000, respectively. From 2001 to 2004, he was a Research Engineer with the Microwave Electronics Laboratory, University of California at Los Angeles (UCLA), where he conducted research on PBG structures and microwave applications on metamaterials. In June 2004, he joined the École Polytechnique of Montréal, Montréal, QC, Canada, where he is currently an Associate Professor and a member of the Microwave Research Group, Poly-Grames. He has authored and coauthored over 90 technical conference, letter, and journal papers. He is currently authoring Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications—The Engineering Approach. He has filed several patents. His current interests include electromagnetic theory, numerical methods, planar circuits and antennas, nonlinear and active devices, monolithic microwave integrated circuit (MMIC) technology, ferroelectrics, nanoferrites, ultrawide-band systems, and LH metamaterials. Dr. Caloz was the recipient of the 2004 UCLA Chancellor’s Award for postdoctoral research.

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Tatsuo Itoh (S’69–M’69–SM’74–F’82) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1969. From September 1966 to April 1976, he was with the Electrical Engineering Department, University of Illinois at Urbana-Champaign. From April 1976 to August 1977, he was a Senior Research Engineer with the Radio Physics Laboratory, SRI International, Menlo Park, CA. From August 1977 to June 1978, he was an Associate Professor with the University of Kentucky, Lexington. In July 1978, he joined the faculty at The University of Texas at Austin, where he became a Professor of Electrical Engineering in 1981 and Director of the Electrical Engineering Research Laboratory in 1984. During the summer of 1979, he was a Guest Researcher with AEG-Telefunken, Ulm, Germany. In September 1983, he was selected to hold the Hayden Head Centennial Professorship of Engineering at The University of Texas at Austin. In September 1984, he was appointed Associate Chairman for Research and Planning of the Electrical and Computer Engineering Department, The University of Texas at Austin. In January 1991, he joined the University of California at Los Angeles (UCLA) as Professor of Electrical Engineering and Holder of the TRW Endowed Chair in Microwave and Millimeter Wave Electronics. He was an Honorary Visiting Professor with the Nanjing Institute of Technology, Nanjing, China, and with the Japan Defense Academy. In April 1994, he was appointed an Adjunct Research Officer with the Communications Research Laboratory, Ministry of Post and Telecommunication, Japan. He currently holds a Visiting Professorship with The University of Leeds, Leeds, U.K. He has authored or coauthored 350 journal publications, 650 refereed conference presentations, and has written 30 books/book chapters in the area of microwaves, millimeter waves, antennas, and numerical electromagnetics. He has generated 64 Ph.D. students. Dr. Itoh is a member of the Institute of Electronics and Communication Engineers of Japan, and Commissions B and D of USNC/URSI. He served as the editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (1983–1985). He serves on the Administrative Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He was vice president of the IEEE MTT-S in 1989 and president in 1990. He was the editor-in-chief of IEEE MICROWAVE AND GUIDED WAVE LETTERS (1991–1994). He was elected an Honorary Life Member of the IEEE MTT-S in 1994. He was elected a member of the National Academy of Engineering in 2003. He was the chairman of the USNC/URSI Commission D (1988–1990) and chairman of Commission D of the International URSI (1993–1996). He is chair of the Long Range Planning Committee of the URSI. He serves on advisory boards and committees for numerous organizations. He has been the recipient of numerous awards including the 1998 Shida Award presented by the Japanese Ministry of Post and Telecommunications, the 1998 Japan Microwave Prize, the 2000 IEEE Third Millennium Medal, and the 2000 IEEE MTT-S Distinguished Educator Award.

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A Positive Future for Double-Negative Metamaterials Nader Engheta, Fellow, IEEE, and Richard W. Ziolkowski, Fellow, IEEE

Invited Paper

Abstract—Metamaterials (MTMs), which are formed by embedding inclusions and material components in host media to achieve composite media that may be engineered to have qualitatively new physically realizable response functions that do not occur or may not be easily available in nature, have raised a great deal of interest in recent years. In this paper, we highlight a large variety of the physical effects associated with double- and single-negative MTMs and some of their very interesting potential applications. The potential ability to engineer materials with desired electric and magnetic properties to achieve unusual physical effects offers a great deal of excitement and promise to the scientific and engineering community. While some of the applications we will discuss have already come to fruition, there are many more yet to be explored. Index Terms—Antennas, metamaterials (MTMs), negative index material, negative refraction, resonators, waveguides.

I. INTRODUCTION

O

VER 30 years ago, Veselago theoretically considered a homogeneous isotropic electromagnetic material in which both permittivity and permeability were assumed to have negative real values, and he studied uniform plane-wave propagation in such a material, which he referred to as “left-handed (LH)” medium [1], [2]. In such a medium, he concluded, the direction of the Poynting vector of a monochromatic plane wave is opposite to that of its phase velocity, suggesting that this isotropic medium supports backward-wave propagation and its refractive index can be regarded negative. Since such materials were not available until recently, the interesting concept of negative refraction, and its various electromagnetic and optical consequences, suggested by Veselago had received little attention. This was until Smith et al. [5], University of California at San Diego, La Jolla, inspired by the work of Pendry et al. [3], [4] constructed a composite “medium” in the microwave regime by arranging periodic arrays of small metallic wires and split-ring resonators [5]–[8] and demonstrated the anomalous refraction at the boundary of this medium, which is the result of negative refraction in this artificial medium [8]. Since then, many aspects of this class and other related types of artificial materials, now termed metamaterials (MTMs), are being investigated by several groups worldwide, and various ideas and suggestions for potential applications of these media have been mentioned (e.g., [9]–[115]). This has led

Manuscript received July 11, 2004; revised November 8, 2004. N. Engheta is with the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104-6314 USA (e-mail: [email protected]). R. W. Ziolkowski is with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2005.845188

to a renewed interest in using fabricated structures to develop composite MTMs that have new physically realizable response functions that do not occur, or may not be readily available, in nature. Among these recent examples of engineered materials, one can mention double-negative (DNG) materials [12], [13] (also known as LH medium [1], negative-index materials (NIMs) [14], [15], backward-wave media (BW) [16], and negative-phase-velocity (NPV) media [17], [18] to name a few); electromagnetic bandgap (EBG) structured materials, and complex surfaces such as high-impedance ground planes and artificial magnetic conductors (AMCs). The new response functions of these MTMs are often generated by artificially fabricated inhomogeneities embedded in host media (volumetric or three-dimensional (3-D) MTMs) or connected to or embedded on host surfaces (planar or two-dimensional (2-D) MTMs). It is important to point out that the history of artificial materials appears to date back to the late part of the 19th Century when Bose published his work in 1898 on the rotation of the plane of polarization by man-made twisted structures, which were indeed artificial chiral structures by today’s definition [19]. Lindman in 1914 studied artificial chiral media formed by a collection of randomly oriented small wire helices [20]. Afterwards, there were several other investigators in the first half of the 20th Century who studied various man-made materials. In the 1950s and 1960s, artificial dielectrics were explored for lightweight microwave antenna lenses, such as the work of Kock [21]. The ‘bed of nails’ wire grid medium was used in the early 1960s to simulate wave propagation in plasmas [22]. The interest in artificial chiral materials was resurrected in the 1980s and 1990s (see, e.g., [23]) and they were investigated for various potential device and component applications such as microwave radar absorbers. Although the majority of the research related to MTMs reported in the recent literature has been concentrated on electromagnetic properties of DNG (LH, NIM. BW, NPV) media, it is worth noting that single-negative (SNG) materials in which only one of the material parameters, not both, has a negative real value may also possess interesting properties when they are juxtaposed in a complementary manner. These media include the only epsilon-negative (ENG) media, such as plasmonic materials like noble metals (silver, gold, etc.) in the visible and infrared (IR) regimes, and the only mu-negative (MNG) media. It has been shown that suitably arranged SNG media may exhibit exciting properties, which may lead to the design of interesting future devices and components (see, e.g., [24]–[27]). In this paper, we provide an overview of some of the unusual characteristics of DNG MTMs and review some of their exciting potential applications. This paper attempts to address some of the “what-if” questions, namely, if one is able to easily construct

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such DNG media (and there have been ample experimental evidence pointing to construction and engineering aspects of such media), what can one do with them? As such, some of the ideas reviewed in this manuscript are speculative in nature, although they are based on mathematical foundations and are consistent with physical realizability conditions. It is important to point out that currently there are several active research directions in this field, one of which is the research efforts of various groups aimed at the construction and fabrication of 3-D volumetric DNG MTMs by embedding in host media various classes of small inclusions such as wires and split-ring resonators [8], [28]–[34], broadside coupled split-ring resonators [35], capacitively loaded strips and split-ring resonators [13], omega structures [36], [37], and space-filling elements [38] to name a few. Another direction of research effort by several groups is focused on 2-D planar DNG MTMs that utilize circuit and transmission-line implementations. These lumped and distributed circuit element realizations have been used to construct negative-index structures and transmission lines for a variety of applications [39]–[48]. Engineering bandgap structures to control the wave’s phase front in order to effectively achieve negative refraction is yet another active area of research relevant to MTMs [49]–[53]. Not all of these topics will be reviewed here since the focus of this paper is on the characteristics and potential applications of DNG MTMs. Furthermore, some of these topics are the subjects of other papers in this TRANSACTIONS. Although in this paper, we focus on the DNG media, occasionally we will also make remarks on some aspects of SNG media (e.g., plasmonic media) since some of the speculated potential devices that can be formed by DNG media can also be envisioned using SNG materials. This paper cannot obviously include all the potential applications studied by all the groups active in this field, and it only addresses a selected sample of ideas. Thus, we apologize in advance for any omission and oversight in this regard. A comment about the terminology and notations: among the various possible terminologies for this class of MTMs currently used by various communities, we favor the descriptor DNG for the isotropic case because, in our opinion, it emphasizes the fundamental description of the material. It will be used throughout even though many of the other terms have been equally popular. We use the time–harmonic convention for monochromatic time variations. It is also assumed that the DNG MTMs are lossless at the frequency of interest unless specified otherwise. However, when dissipation is considered, the complex paand are used where rameters and are nonnegative quantities for passive media. We will also consider any losses to be relatively small, i.e., and . Furthermore, we also simplify the discussion by assuming that the MTMs under discussion are isotropic. Almost all of the realizations of DNG or SNG MTMs to date are by their nature anisotropic or bianisotropic. There is, however, a strong motivation to achieve isotropic properties and this too is under investigation. For instance, the role of anisotropy in the sign of the permittivity and permeability of materials has been investigated in order to achieve certain unconventional features in wave propagation [116]. II. NEGATIVE REFRACTION AND CAUSALITY IN DNG MEDIA The index of refraction of a DNG MTM has been shown to be negative (e.g., [6], [12], and [39]), and there have now been

several theoretical and experimental studies that have been reported confirming this negative index of refraction (NIR) property and applications derived from it such as phase compensation and electrically small resonators [54], negative angles of refraction (e.g., [8], [54]–[59]), sub-wavelength waveguides with lateral dimension below diffraction limits (e.g., [26], [27], [60]–[63]) enhanced focusing (see [46] and [64]), backward ˇ wave antennas [44], Cerenkov radiation [65], photon tunneling [66], [67], and enhanced electrically small antennas [68]. These studies rely heavily on the concept that a continuous wave (CW) excitation of a DNG medium leads to a negative refractive index and, hence, to negative or compensated phase terms. Ziolkowski and Heyman thoroughly analyzed this concept mathematically using detailed steps, and have shown that, in DNG media, the refractive index can be negative [12]. One must exercise some care with the definitions of the electromagnetic properties in a DNG and in a lossless DNG medium, medium. When with the branch-cut choices, as shown in [12], one should write and . This leads to the following expressions for the definitions of the wavenumber and the wave impedance, respectively: (1) which are needed to properly describe the interaction of a wave with a DNG medium. If the index of refraction of a medium is negative, then the refracted angle, according to Snell’s law, should also become “negative.” This suggests that the refraction is anomalous, and the refracted angle is on the same side of the interface normal as the incident angle is. This will be clearly shown here later. Veselago in his 1968 paper mentions certain temporal dispersions for negative permittivity and permeability [1]. As for the causality, we note that if one totally ignores the temporal dispersion in a DNG medium and carefully consider the ramifications of a homogeneous nondispersive DNG medium and the resulting NIR, one will immediately encounter a causality paradox in the time domain, i.e., a nondispersive DNG medium is noncausal. However, a resolution of this issue was uncovered in [69] by taking the dispersion into account in a time-domain study of wave propagation in DNG media. The causality of waves propagating in a dispersive DNG MTM was investigated there both analytically and numerically using the one-dimensional (1-D) electromagnetic plane-wave radiation from a current sheet source in a dispersive DNG medium. In that study, a lossy Drude model of the DNG medium was used, and the solution was generated numerically with the finite-difference time-domain (FDTD) method. The analogous problem in a nondispersive DNG medium was also considered, and it was shown that the solution to this problem is not causal in agreement with similar observations given in [6]. An approximate solution was constructed that combined a causal envelope with a sinusoid, which has the nondispersive NIR properties; it compared favorably with the FDTD results for the dispersive DNG case. It was thus demonstrated that causal results do indeed require the presence of dispersion in DNG media and that the dispersion is responsible for a dynamic reshaping of the pulse to maintain causality. The CW portions of a modulated pulse (i.e., excluding its leading and trailing edges) do obey all of the NIR effects expected from a

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time–harmonic analysis in a band-limited “nondispersive” DNG medium. Therefore, one can conclude that the CW analyses of DNG media are credible as long as very narrow bandwidth pulse trains are considered for any practical realizations. This has been the case in all of the experimental results reported to date that we are aware of. Moreover, time delays for the realization of the NIR effects are inherent in the processes dictated by the dispersive nature of the physics governing these media. Since, in this paper, in addition to analytical descriptions, we present several FDTD simulation results for wave interactions with DNG media, particularly the FDTD representation of negative refraction, here we need to briefly discuss some of the features of the FDTD simulator specific to the DNG structures. It should be emphasized that the use of this purely numerical simulation approach does not involve any choices in defining derived quantities to explain the wave physics, e.g., neither wave vector directions, nor wave speeds are stipulated a priori. In this manner, it has provided a useful approach to studying the wave physics associated with DNG MTMs. As in [12], [57], [58], and [104], lossy Drude polarization and magnetization models are used to simulate the DNG medium. In the frequency domain, this means the permittivity and permeability are described as

(2) where and denote the corresponding plasma and damping frequencies, respectively. Although in some of the analytical and numerical studies, as well as experiments considered by other groups (e.g., [5]–[8], [36], [113], and [114]), the Lorentz model and its derivatives have been used, here the Drude model is preferred for the FDTD simulations for both the permeability and permittivity functions because it provides a much wider bandwidth over which the negative values of the permittivity and permeability can be obtained. This choice is only for numerical convenience, and it does not alter any conclusions derived from such simulations since the negative refraction is observed in either choice. However, choosing the Drude model for the FDTD simulation also implies that the overall simulation time can be significantly shorter, particularly for low-loss media. In other words, the FDTD simulation will take longer to reach a steady state in the corresponding Lorentz model because the resonance region where the permittivity and permeability acquire their negative values would be very narrow in this model. Choice of electric and magnetic currents, polarization and magnetization fields, and the discretization of the computational space for the FDTD simulation are done self-consistently following the conventional FDTD method [119]. The simulation space was truncated with an MTM-based absorbing boundary condition [120], [121]. to The FDTD cell size in all cases presented here was minimize the impact of any numerical dispersion on the results. As an example of numerical simulation of negative refraction, Fig. 1 presents two cases of CW Gaussian wave interaction with DNG slabs when the CW frequency is chosen to be GHz. (Needless to say, this choice is arbitrary; the numerical results presented below can be obtained at any frequency with

Fig. 1. FDTD predicted electric-field intensity distribution for the interaction of the CW Gaussian beam that is incident at 20 to a matched DNG slab having: 1 and (B) n (! ) 6 with ! = 2 3 10 rad/s. (A) n (! ) A negative angle of refraction equal and opposite to the angle of incidence is clearly observed.

0

0

2 2

a proper scaling of the parameters.) In order to reduce the effect of reflection and, thus, to observe the negative refraction more clearly, the parameters of these slabs are chosen such that the slab are impedance matched to the free space. Therefore, the electric and magnetic Drude models were selected to be identical, i.e., and . In all cases, only low-loss . This values were considered by setting means that the index of refraction has the form

(3)

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In Fig. 1(A), the matched DNG slab has , rad/s and, hence, when . For the other matched DNG slab, shown in , for which the model parameters Fig. 1(B), rad/s and were selected to be . As can be seen in Fig. 1, the negative angle of refraction is , clearly seen in both cases. In panel (A), where the refracted angle is equal and opposite to the angle of inci, the negative dence, while in panel (B), with angle of refraction is negative, but less than the incident angle. Due to the change in wavelength in the DNG slab, the beam becomes highly compressed along the beam axis. The discontinuities in the derivatives of the fields at the double-positive (DPS)–DNG interfaces (i.e., the so-called “V-shaped” patterns at both interfaces) are clearly seen. These cases clearly show the presence and effects of the negative angle of refraction realized when an obliquely incident Gaussian beam interacts with a DNG slab. Power flow at the negative angles predicted by Snell’s law is confirmed. Fine resolution-in-time movies (not shown here) of the behavior of the electric field in the interaction cases demonstrate that the phase propagation is indeed in the opposite direction to the power flow shown in the figures given here. The FDTD simulation results such as these have numerically confirmed many of the fundamental properties of beam interactions with a DNG medium. We note that there had been some controversy about this negative angle of refraction [14] despite initial experimental verification [8] with 3-D MTM constructs. However, this has already been resolved through the subsequent explanation [15]. More recent planar negative-index transmission line [40], [43] and related planar refractive cone experiments [70], [71] have also more clearly verified this effect. It is also worth mentioning that Foster’s reactance theorem has been shown to be satisfied for the lossless DNG MTMs [72], just as it is for the conventional DPS media. III. PHASE COMPENSATION IN DNG MEDIA One of the interesting features of DNG media is their ability to provide phase compensation or phase conjugation due to their negative index. Consider a slab of conventional lossless DPS material with positive index of refraction and thickness and a slab of lossless DNG MTM with negative refractive index and thickness . Although not necessary, but for the sake of simplicity in the argument, we assume that each of these slabs is impedance matched to the outside region (e.g., free space). Let us take a monochromatic uniform plane wave normally incident on this pair of slabs. As this wave propagates through the slab, the phase difference between the exit and entrance faces of the first slab is obviously , where , while the total phase difference between the front and back faces of this two-layer structure is , implying that whatever phase difference is developed by traversing the first slab, it can be decreased and even compensated by traversing the second slab. If the ratio of and is chosen to be at the given frequency, then the total phase difference between the front and back faces of this two-layer structure will become zero. This means that the DNG slab acts as a phase compensator

Fig. 2. FDTD predicted electric-field intensity distribution for the phase compensator/beam translator system of a DPS–DNG stacked pair. The Gaussian beam is normally incident on a stack of two slabs, the first being a DPS slab with n (!) = +3 and the second being a DNG slab with n (! ) 3. The initial beam expansion in the DPS slab is compensated by its refocusing in the DNG slab. The Gaussian beam is translated from the front face of the system to its back face with only a 0.323-dB attenuation over the 4 distance.

0

0

in this structure [54]. We should note that such phase compensation/conjugation does not depend on the sum of thicknesses , rather it depends on the ratio of and . Thus, in principle, can be any value as long as satisfies the above condition. Therefore, even though this two-layer structure is present, the wave traversing this structure would not experience the phase difference. This feature can lead to several interesting ideas in device and component designs, as will be discussed later. Such phase compensation can also be verified using the FDTD simulation, as shown in Fig. 2. A Gaussian beam is launched toward a pair of DPS and DNG layers, each layer having a thickness of . The DPS slab has , . As is evident from while the DNG layer has Fig. 2, the beam expands in the DPS slab and then refocuses in the DNG slab, and the waist of the intensity of the beam is recovered at the back face. The electric-field intensity could, in . There principle, be maintained over the total thickness of is only a 0.323 dB (7.17%) reduction in the peak value of the intensity of the beam when it reaches the back face. Moreover, the phase of the beam at the output face of the stack is the same as its value at the entrance face. Therefore, the phase compensator thus translates the Gaussian beam from its front face to its back face with low loss.

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Using multiple matched DPS–DNG stacks, one could produce a phase-compensated time-delayed waveguiding system. Each pair in the stack would act as shown in Fig. 2. Thus, the phase compensation–beam translation effects would occur throughout the entire system. Moreover, by changing the index of any of the DPS–DNG pairs, one changes the speed at which the beam traverses that slab pair. Consequently, one can change the time for the beam to propagate from the entrance face to the exit face of the entire DPS–DNG stack. In this manner, one could realize a volumetric low-loss time-delay line for a Gaussian beam system. This phase compensation can lead to a wide variety of potential applications that could have a large impact on a number of engineering systems. One such set of applications offers the possibility of having sub-wavelength electrically small cavity resonators and waveguides with lateral dimension below diffraction limits. These ideas are briefly reviewed here. The interested reader is referred to [26], [27], and [54] for further details. IV. SUB-WAVELENGTH COMPACT CAVITY RESONATOR USING DNG MEDIA Let us take the pair of DPS and DNG layers discussed above and put two perfect reflectors (e.g., two perfectly conducting plates) at the two open surfaces of this bilayer structure, forming a 1-D cavity resonator [see Fig. 3(A)]. It has been shown by Engheta [54] that such a cavity resonator may possess a nonzero mode even when the thickness of the cavity is, in principle, . In that analysis, the much smaller than the conventional dispersion relation was found to be

(4) and since , can be rewritten as

,

, and

, it

(5) (The choice of sign for and is irrelevant here since either sign does not change this relation.) This relation does not show any direct constraint on the sum of thicknesses of and . Instead, it deals with the ratio of tangent of these thicknesses (with multiplicative constants). This implies that, contrary to a conventional DPS–DPS cavity resonator, here, as is reduced, the value of can also become smaller in order to satisfy the above dispersion relation (5), and the layers can conceptually be as thin or as thick as otherwise needed as long as relation (5) is satisfied. The total thickness of such a may turn out to be much smaller than thin cavity , which, for low-frequency applications, may the standard provide significant miniaturization in cavity resonator designs. The electric- and magnetic-field expressions for the mode in such a DPS–DNG 1-D cavity have been derived in [54]. Fig. 3(B) presents a sketch of these field distributions in this cavity. As noted in Figs. 1 and 2, and as shown in Fig. 3(B), the tangential electric field possesses a discontinuous first derivative at the boundary between the two

Fig. 3. (A) Sub-wavelength compact cavity resonator formed by a pair of DPS and DNG layers sandwiched between two reflectors. The cavity mode can exist in this structure even when the total thickness may be less than the conventional =2. (B) Sketch of the normalized magnitude of electric (E ) and magnetic field (H ) distribution (with respect to  H ) as a function of the z coordinate. = = 0:1, d = = 0:05, " =" , = , Here, we take d " = 2" , and  = 2 . In this case, the DPS slab occupies the region 0 < z= < 0:1, while the DNG slab occupies 0:05 < z= < 0. From [60].

0

layers, i.e.,

0

0

Interface Interface with and . A similar argument can be used for the magnetic-field behavior at this interface. This idea has also been extended to the cases of cylindrical and spherical cavities filled with a pair of coaxial DPS–DNG layers and concentric DPS–DNG shells, respectively [73], [74]. The corresponding dispersion relations and field distributions have been obtained. As in the case of a 1-D cavity resonator, it has been found that it is possible to have these 2-D and 3-D cavity geometries with the dimension below the conventional cavity size when they are filled with a pair of DPS and DNG layers [73], [74]. An experimental confirmation of the 1-D cavity resonator concept has been recently shown by Hrabar et al. at a recent symposium [75]. It is worth noting that it is also possible to have such compact cavity resonators using SNG media, such as pairs of ENG and

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MNG layers [26], [27]. In the case of a 1-D ENG–MNG cavity resonator, the dispersion relation takes the form

and . In this case, the dispersion relations in (7) may be approximated, respectively, as case

(6) Such a compact cavity of an ENG–MNG pair can support only one mode at a given frequency. Since this paper deals with the concept of DNG materials and their potential applications, here we do not include the various aspects of the SNG materials and their relevant applications. Further details for the case of ENG–MNG cavities can be found in [26] and [27]. V. SUB-WAVELENGTH GUIDED-WAVE STRUCTURES WITH LATERAL DIMENSIONS BELOW DIFFRACTION LIMITS The concept of sub-wavelength cavity resonators using DPS–DNG or ENG–MNG layers, reviewed above, has been extended to the case of guided-wave structures that can have lateral dimensions smaller than the conventional diffraction . [26], [27] When the structure shown in Fig. 3(A) limit is used as a parallel-plate waveguide with the DPS–DNG pair, the dispersion relation for guided mode with longitudinal waveguide wavenumber has been shown to be modes modes

(7)

for the TE mode and a correwhere sponding expression for the TM mode [26]. For DPS and DNG slabs, and , respectively, and the transverse wavenumber and may be real or imaginary, depending on the value of . In standard metallic waveguides filled with DPS materials, it is known that guided modes cannot be supported when the lateral dimensions fall below a certain limit, i.e., below the cutoff thickness. In other words, in a closed waveguide with a metallic wall, reducing the cross-sectional size of the guide results in cutting off the modes one after the other until the dominant mode is also cut off. However, in the DPS–DNG or ENG–MNG waveguides, it may be possible to overcome this limitation and to devise a waveguide capable of supporting guided modes without any cutoff thickness [26], [27]. By properly choosing the material parameters and thicknesses of the DPS and DNG layers in the waveguide, one can obtain various interesting features for such a waveguide. Some of these properties for the case of parallel-plate waveguides are summarized below. Fig. 4(A) presents a sample of the dispersion diagram for the TE mode in the DPS–DNG parallel-plate waveguide, showing the relationship among the normalized total thickness , , and . For comparison, the corresponding TE-mode dispersion diagram for the DPS–DPS parallel-plate waveguide is shown in Fig. 4(B). Referring to Fig. 3(A) as the geometry of the waveguide and Fig. 4 for its dispersion diagram, let us first consider the case where the DPS and DNG layers are very thin, i.e.,

case

(8)

where is shorthand for and obviously should always be a positive quantity. If instead of a pair of DPS–DNG layers, this thin waveguide were filled with a pair of DPS–DPS layers (and similarly with a pair of ENG–ENG, DPS–ENG, MNG–MNG, DNG–MNG, or DNG–DNG layers), the first equation in (8) would not be satisfied because, for these pairs, the right-hand side of that equation would not be positive and, thus, no TE mode could propagate in such a thin DPS–DPS waveguide, as expected. This can be seen from Fig. 4(B) around the region of the diagram where thickness of one of the layers (e.g., the first DPS layer) is taken to be very small, and we note that, in this region, the total thickness of the guide cannot be lower than a certain limit. This indeed would represent the diffraction limit mentioned above for standard waveguides. However, if the thin waveguide is filled with a pair of DPS–DNG slabs (or a pair of DNG–ENG, DPS–MNG, or ENG–MNG slabs), the right-hand side of the first equation in (8) would be positive and, thus, that equation may be approximately satisfied for a certain value of . In this case, the exact expression for should be obtained by solving the TE dispersion relation in (7). As we can see from that equation and also from Fig. 4(A), and the region where both is allowed and, hence, the total thickness can be very small, contrary to the case of the DPS–DPS waveguide. In such a limit and as long as the TE dispersion relation in (7) is satisfied, one (and only one) propagating TE mode may exist in principle, as Fig. 4(A) shows, no matter how thin these layers are. This means a parallel-plate waveguide filled with DPS–DNG layers does not have a cutoff thickness for the TE modes; thus, it can support a TE guided mode, even though the lateral dimension can be well below the diffraction limit. As for the TM mode, the second equation in (8) provides the of the dominant TM mode when approximate value for is a real quantity for a given set of and material parameters. For a DPS–DPS or DNG–DNG thin waveguide, this TM mode exists lies between the wavenumbers of the for any ratio , and its two layers. The allowable ranges of variation of in (8) in terms of have been discussed in detail in [26] and [27]. It has been shown that, for the DPS–DNG parallel-plate waveguide, indeed differs from the ones in the the range of variation of standard DPS–DPS waveguides in that may attain values and (effectively only outside the interval between “complementary” to the standard DPS–DPS case where is in this interval). The fact that thin waveguides loaded with complimentary pairs of MTMs (e.g., DPS–DNG or ENG–MNG) , may offer interesting possibilmay support nonlimited ities in the design of very thin resonant cavities and very thin , the waveguides having guided modes with high . With waveguide wavelength will be very small, which

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Fig. 4.

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Dispersion plots for: (A) the DPS–DNG parallel-plate waveguide and (B) the DPS–DPS parallel-plate waveguide for comparison. Minimum total thickness )= for the waveguide is shown in terms of = k and d = . Here, we have " = 3" ,  = 3 , " = 3" , and  =  where the upper signs are for (A) with DPS–DNG and the lower signs are for (B) with DPS–DPS. From [76]. (d

+d

j

j

may give rise to compact resonators and filters. A similar observation regarding the possibility of being very large has also been made by Nefedov and Tretyakov [61]. Other salient features of DPS–DNG waveguides such as unusual properties in mode excitation, backward flow of power in the DPS–DNG waveguides, and the effect of geometric discontinuity in the DPS–DNG slabs in such waveguides have been studied and reported in the literature [27], [76]. Moreover, some of the unconventional characteristics of such DPS–DNG waveguides and cavities can be explained and justified using the distributed-circuit-element approach with appropriate choice of elements [77]. This “circuit-element” approach may also be applied to ENG–MNG, or DPS–SNG waveguides [77]. It is worth noting that there are other techniques to construct subwavelength waveguides by embedding split-ring resonators in a waveguide below cutoff [117], [118]. These split-ring resonators can be considered as lumped elements within such subwavelength waveguides, whereas the structures considered above are subwavelength due to the resonant pairing of the DPS and DNG layers. The case of cylindrical waveguide with a metallic wall filled with coaxial layers of DPS–DNG or ENG–MNG materials have also been studied by Alù and Engheta [73], and analogous results have been obtained. Guided-wave structures involving SNG materials have also been analyzed in detail by Alù and Engheta [26], [27],

7

7

and similarities and differences between the ENG–MNG and DPS–DNG parallel-plate waveguides have been discussed and reported [26]. For thin waveguides, these two sets of waveguide have many similarities. However, as the parallel-plate waveguides gets thicker, the differences between these two classes of waveguide become evident. For example, we have shown that it is possible to have a mono-modal thick parallel-plate waveguide filled with a pair of ENG and MNG layers, whereas the same size DPS–DNG waveguide may support more than one mode. Other characteristics of the SNG waveguides can be found in [26] and [27]. The case of surface-wave propagation along the open ungrounded DNG slab waveguides has also been studied [27]. The geometry of such a DNG waveguide is shown in Fig. 5(A). The dispersion relation is given as

(9) and is half of the slab thickness. Fig. 5(B) where shows the dispersion diagrams for the odd TE mode in the open DNG slab waveguide (solid line) and the corresponding DPS . One slab waveguide (dashed line), where

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Fig. 5. (A) Open ungrounded DNG slab waveguide. (B) Its dispersion diagram = 4 and " = 2" . (solid line) for the odd TE mode with  The dispersion diagram for the corresponding open DPS slab waveguide with  = 4 and " = 2" is shown with the dashed line.

0

0

for a sub-wavelength guided-wave structure with lateral dimension below the diffraction limit and that it will be tightly confined . Several other details about the to this structure since dispersion properties of the open ungrounded DNG slab waveguides can be found in [27], among those one can mention the fact that such a waveguide supports guided modes in which the portion of the power flowing in the surrounding vacuum is antiparallel with respect to the portion flowing inside the DNG slab. This feature can offer an interesting possibility for the “backward” coupling between such DNG open waveguides and standard DPS open waveguide placed in their proximity [27], [78]. This will be briefly reviewed in Section VI. It is also important to note that the properties of the grounded DNG slab has been studied by Baccarelli et al. with an emphasis toward conditions for the absence of surface waves in such grounded structures with potential application to printed antenna systems [79]. can The possibility of having guided modes with provide a possible solution for the transport of RF and optical energy in structures with small lateral dimensions, below the diffraction limit with possible applications to miniaturization of optical interconnects and nanophotonics. Ideally, in the lossless case, there is, in principle, no limitation on the compactness of such waveguides and the confinement of the guided mode. However, in practice, loss is present and may limit the performance and, thus, should be taken into consideration. Similar characteristics have also been obtained for the thin open cylindrical waveguides formed by a DNG or a SNG cylinder or coaxial layers of DNG–DPS or DPS–SNG materials [80]. VI. BACKWARD COUPLERS USING DNG SLAB WAVEGUIDES

striking difference between the two dispersion diagrams is the presence of a TE odd mode in the DNG slab as the expression tends to small values below . We also note that, for this solution of the dispersion relation (i.e., the lowest solid line in increases as . In fact, Fig. 5(B)), the value of can become even greater than . In order to highlight the importance of this solution, let us first consider the case of a standard DPS slab waveguide in which propagating guided modes are possible only when . In particular, the first even mode has no cut off, i.e., even when , a solution for still exists and is . However, this DPS behavior also implies that the lateral distribution of the field of such a guided mode is widespread in the region surrounding the slab, and essentially the mode is only weakly guided. Therefore, when the guiding structure with the DPS material becomes very thin, the effective cross section of the guided mode becomes very large, i.e., in the limit of zero slab thickness, the guided mode is simply a uniform plane wave. Consequently, if we consider reducing the slab thickness, the guided mode will travel with a transverse section much larger than the slab lateral dimension. This issue is indeed another manifestation of the diffraction limitation, which does not traditionally allow the signal transport in a guided structure thinner than a given dimension determined by the wavelength of operation. For the DNG slab waveguide, the situation is quite different. In this case, since the first odd mode has no cutoff thickness, the fields are more concentrated and confined near the slab surface. This is an important advantage of such DNG open-slab waveguides. It implies that the guided surface wave can be present even

As mentioned above, owing to the peculiar behavior of guided modes in open DNG slab waveguides, they can exhibit backward coupling properties when they are in proximity of open DPS slab waveguides, i.e., if one of the two waveguides is excited to carry power in one direction, the second waveguide, through the coupling, might “redirect” back some of this power in the opposite direction. We should note that an analogous phenomenon for the planar circuits has been observed and studied in the negative-index transmission-line couplers by Islam et al. [45] and Caloz and Itoh [42]. The geometry of the problem is shown in Fig. 6, in which the two open-slab waveguides are separated and surrounded by a simple medium (e.g., free space). In this figure, a simple physical description of the backward coupling of the power flows in the two slabs is also given in terms of the phase and Poynting vectors. Using a rigorous modal analysis of the structure, the following dispersion relation for the supported TE modes can be obtained: (10) where

(with

) with

(11) is the modal dispersion relation of each slab alone (i.e., without coupling), represented by the product of the dispersion rela-

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“redirected” back into the other waveguide continuously and exponentially with a factor that increases as decreases. In the exact approach, this behavior is due to the fact that the two modes share the same real part, but start to have two oppositely valued imaginary parts. The backward coupler with a strong coupling, therefore, acts similarly to a periodically corrugated waveguide (grating reflector [122]) in its stopband, but with the unusual feature that the “reflected” power is effectively flowing in a separate channel and is isolated from the “incident” one. This implies that the incident and reflected power flows are spatially localized in the two different waveguides.

Fig. 6. Geometry of the backward coupler with the DNG and DPS slab waveguides with a sketch of the power flows for a given supported mode. From [78].

tions of even and odd TE modes in the isolated slab, is the (with ), and waveguide wavenumber, . The coefficients on the right-hand side of (10) take into account the coupling effects and are given by

(12) The TM mode dispersion relation may be straightforwardly obtained using the duality principle. When the two waveguides are far apart (i.e., is sufficiently large), the coupling term on the right-hand side of the dispersion relation vanish and, as expected, (10) reduces into the dispersion relations for the two “decoupled” open waveguides. The modes in each waveguide are unperturbed and, thus, there is no coupling present. When is reduced, however, the modes supported by each one of the two waveguides have field distributions that extend into the region occupied by the other waveguide. Thus, a new set of modes should be found to satisfy the exact dispersion relation with a field distribution obtainable by solving the boundary value problem. The properties of these modes can be obtained by using perturbation techniques or the exact formulation. Some of the details can be found in [27]. Here, we briefly describe an interesting distinction between a DPS–DNG waveguide coupler and a corresponding standard DPS–DPS waveguide coupler. In a standard waveguide directional coupler, when we fix the geometry of the two waveguides separately and, therefore, and without coupling, fix the values of the exact solutions for the wavenumbers and move farther from each other as is reduced. Moreover, their interference spatial period consequently decreases. When instead we consider a DPS–DNG backward coupler, the two solutions and move closer as is reduced, thus increasing the spatial period of their coupling. One can then get to a point at which the two supported modes have the same ; and the interference wavenumber, i.e., is no longer present (i.e., its period is infinite). By decreasing the distance further and, hence, increasing the coupling coefficient, an exponential variation for the power exchange (rather than a sinusoidal variation) results. Consequently, the power is

VII. SUB-WAVELENGTH FOCUSING WITH A FLAT LENS OR A CONCAVE LENS MADE OF DNG MATERIALS Another one of the interesting potential applications of DNG media, which was first theoretically suggested by Pendry [64], is the idea of a “perfect lens” or focusing beyond the diffraction limit. In his analysis of the image formation process in a flat slab of lossless DNG material, Pendry showed that, in addition to the faithful reconstruction of all the propagating spatial Fourier components, the evanescent spatial Fourier components can also ideally be reconstructed. The evanescent wave reconstruction is due to the presence of a “growing exponential effect” in the DNG slab, leading to the formation of an image with a resolution higher than the conventional limit. His idea has motivated much interest in studying wave interaction with DNG media. Various theoretical and experimental works by several groups have explored this possibility; they have shown the possibility and limitations of sub-wavelength focusing using a slab of DNG or negative-index MTMs [81], [82]. The sub-wavelength focusing in the planar 2-D structures made of negative-index transmission lines has also been investigated [46]. The presence of the growing exponential in the DNG slab has also been explained and justified using the equivalent distributed circuit elements in a transmission-line model [83]. It has also been shown that “growing evanescent envelopes” for the field distributions can be achieved in a suitably designed, periodically layered stacks of frequency selective surfaces (FSSs) [84]. In Fig. 7(A), we present an FDTD simulation of the focusing effects for a planar slab of DNG materials with . A diverging CW-modulated Gaussian beam is assumed to be normally incident on such a planar DNG slab in order to determine whether it can focus such a diverging Gaussian beam or not. The focal plane (or waist) of the beam was taken to be in front of the slab. Since it was expected that the DNG slab would have an NIR and would focus the beam, i.e., it would bend the wave vectors of a diverging beam back toward the beam axis, a strongly divergent beam was considered in this simulation. A , diffraction limited beam, whose waist was approximately away from was used. The location of the waist was set at the DNG interface so that there would be sufficient distance for the beam to diverge before it hit the interface. The DNG slab . Therefore, if the DNG slab refocuses also had a depth of the beam, the waist of the beam at the back face of the DNG slab should be approximately the same as its initial value (the Drude medium has some small losses). A snapshot of the FDTD predicted electric-field intensity distribution is shown in Fig. 7(A). This result clearly shows that the planar DNG medium turns the

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Fig. 7. (A) FDTD predicted electric-field intensity distributions illustrate the focusing of the Gaussian beam as it propagates in the n (! ) 1 DNG slab. Focusing at the back surface is observed. (B) Similar FDTD simulation for a DNG slab having n (! ) 6. Channeling of the beam in the DNG slab is observed; the wings of the beam are seen to feed the center of the beam.

0

0

diverging wave vectors toward the beam axis and, hence, acts as a lens to focus the beam. Since all angles of refraction are the negative of their angles of incidence for the slab, the initial beam distribution is essentially recovered at the back face of the slab, i.e., as designed, the focal plane of the beam in the DNG medium is located at the back face of the DNG slab. From the electric-field intensity obtained from the FDTD simulation, we note that the peak intensity is approximately 18% lower than its value at the original waist of the Gaussian beam.

This variance stems from the presence of additional wave processes, such as surface-wave generation, and from dispersion and loss in the actual Drude model used to define the DNG slab in the FDTD simulation. We note that if, as shown by Ziolkowski and Heyman in [12], from the front face of an a point source is at a distance planar slab of thickness , then the first focus of the source is found in the center of the slab a distance from the front face and the second focus is located beyond the from the back face. The slab effectively slab at a distance translates the source to the exterior focus position. If the reconstruction by the growing evanescent fields in the slab is to achieve sub-wavelength focusing at the image point and if the slab has even a small amount of loss, the slab will have to be thin. The source and its image will then have to be very near, respectively, to the front and back faces of the slab. The configuration will then essentially become a near-field one, and the “perfect lens” situation is thus lost to only near-field configurations when real media with losses are involved. Nonetheless, since there are numerous near-field imaging situations, such as breast tumor sensing, the slab translator may still provide interesting imaging possibilities. We note that if the source is moved further away from the front face of the slab, the foci inside and outside the slab will move closer to the back face of the slab. Consequently, even a normal lens configuration in which the source is far from the lens, the “perfect lens” only becomes useful in the near field of the output face of the system. The corresponding results for the Gaussian beam interacting , shown in with the matched DNG slab with Fig. 7(B), reveals related, but different results. In contrast to the case, when the beam interacts with the matched DNG slab with , there is little focusing observed. The negative angles of refraction dictated by Snell’s Law are shallower for this higher magnitude of the refractive index, i.e., . Rather than a strong focusing, the medium channels power from the wings of the beam toward its axis, hence, maintaining its amplitude as it propagates into the DNG medium. The width of the beam at the back face is only slightly narrower, yielding only a slightly higher peak value there in comparison to its values at the front face. The strong axial compression of the beam caused by the (factor of six) decrease in the slab also occurs. We note that, wavelength in the in these DNG cases, the beam appears to diverge significantly once it leaves the DNG slab. The properties of the DNG medium hold the beam together as it propagates through the slab. Once it leaves the DNG slab, the beam must begin diverging, i.e., if the DNG slab focuses the beam as it enters, the same physics will cause the beam to diverge as it exits. Moreover, there will be no focusing of the power from the wings to maintain the center portion of the beam. The rate of divergence of the exiting beam will be determined by its original value and the properties and size of the DNG medium. We also point out that a beam focused into a DNG slab will generate a diverging beam within the slab and a converging beam upon exit from the slab. This behavior has also been confirmed with the FDTD simulator. One potential application for these results is clearly the use of a matched flat DNG slab with an index of refraction as a lens/translator, as originally suggested by Pendry [64].

ENGHETA AND ZIOLKOWSKI: POSITIVE FUTURE FOR DNG MTMs

This can have various applications, for instance, in a variety of near-field microwave or optical systems. Another potential application is to channel the field into a particular location, e.g., to ) as use a large negative index DNG slab (e.g., a superstrate (over-layer) on a detector so that the beam energy would be channeled efficiently onto the detector’s face. Most superstrates, being simple dielectrics, defocus the beam. Often one includes a curved DPS lens over the detector face to achieve the focusing effect. The flatness of the DNG slab has further advantages in packaging the detectors into an array or a system. Yet another potential application is to combine the negative index properties of the DNG slab with its negative refraction features to realize a low-loss phase compensator, as discussed earlier. It should be mentioned that a planar DNG slab is unable to focus a collimated beam (i.e., flat beam) or a plane wave since the negative angle of refraction can occur only if there is oblique incidence. This is the reason why only FDTD simulations for expanding Gaussian beams have been shown up to this point. In order to focus a flat Gaussian beam (one with nearly an infinite radius of curvature), one must resort to a curved lens. However, in contrast to focusing (diverging), a plane wave with a convex (concave) lens composed of a DPS medium, here one must consider focusing (diverging) a plane wave with a concave (convex) lens composed of a DNG medium. Such a plano-concave DNG is shown in Fig. 8(A) for the FDTD lens with simulation. It was formed by removing a parabolic section from deep and wide. The the backside of a slab that was focal length was chosen to be , and the location of the focus was chosen to be at the center of the back face of the slab. The full width of the removed parabolic section at the back face was . A Gaussian beam with a waist of was launched distance away from the planar side of this lens and was normally incident on it. with It is known that a DPS plano-convex lens of index a similar radius of curvature [the dark gray region in Fig. 8(A)] would have a focus located a distance from its back face. Thus, to have the focal point within the very near field, as it is in the DNG case, the index of refraction would have to be very large. In fact, to have it located at the back face would require . This would also mean that very little of the incident beam would be transmitted through such a high index lens because the magnitude of the reflection coefficient would approach one. In contrast, the DNG lens achieves a greater bending of the incident waves with only moderate absolute values of the refractive index and is impedance matched to the incident medium. Moreover, since the incident beam waist occurs at the lens, the expected waist of the focused beam would be [123]. Hence, for a , the transverse waist at the focus normal glass lens would be and the corresponding intensity . The longitudinal size half-max waist would be of the focus is the depth of focus, which, for the normal glass . Again, to achieve a focus that lens would be is significantly sub-wavelength using a DPS lens, a very large index value would be required and would lead to similar disadvantages in comparison to the DNG lens. However, for the DNG plano-concave lens, one obtains more favorable results.

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Fig. 8. (A) Plano-concave DNG lens geometry for the FDTD simulation. The launching plane and beam axis are shown by the black lines. The electric-field sampling points were located at the intersections of the beam axis and horizontal lines. The location of the DNG lens region is shown in light gray. The dark gray region is air, as is the entire region surrounding the lens. (B) Snapshot of the FDTD-predicted electric-field intensity distributions for the DNG concave lens. The peak of the intensity occurs at the predicted focal point. This sub-wavelength focal region is significantly smaller than would be expected from the corresponding traditional DPS lens.

Fig. 8(B) shows a snapshot of the FDTD-predicted electric-field intensity distribution when the intensity is peaked at the focal point. The radius of the focus along the beam axis (half intenand along the sity radius) is measured to be approximately . This sub-wavetransverse direction it is approximately length focal region is significantly smaller than would be expected from the corresponding traditional DPS lens. Moreover, even though the focal point is in the extreme near field of the lens, the focal region is nearly symmetrical and has a resolution that is much smaller than a wavelength. Such a sub-wavelength source has a variety of desirable features that may have applications for high-resolution imaging in near-field scanning optical microscopy (NSOM) systems. In particular, the field intensity has been concentrated into a sub-wavelength region without a

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Fig. 9. Spherical scatterer composed of two concentric shells of DPS and DNG isotropic materials.

guiding structure. It could thus act as a much smaller aperture NSOM source than is available with a typical tapered optical fiber probe and without the associated aperture effects. VIII. SUB-WAVELENGTH SPHERICAL AND CYLINDRICAL RESONANT STRUCTURES CONTAINING DNG METAMATERIALS Earlier we reviewed the possibility of designing a sub-wavelength 1-D cavity resonator by pairing DPS and DNG layers. This resonance phenomenon can be intuitively explained by noting that if the DPS layer acts as an equivalent reactive impedance, the corresponding DNG layer may be considered as another reactive impedance, but with an opposite sign (since the permittivity and permeability of the DNG layer have opposite sign to those of the DPS layer) [54]. Therefore, pairing these two reactive “impedances” with opposite signs may produce the conditions for a resonance. Furthermore, reducing the thicknesses of both layers can affect the values of these equivalent impedances, but their signs stay opposite. As a result, the DPS–DNG pair can still remain resonant, as we reduce their thickness while maintaining a certain ratio of thicknesses (although the bandwidth of this resonance is affected as the size is reduced). This feature is not present for a pair of conventional DPS–DPS thin layers, and it is due to juxtaposing DPS and DNG layers (and, similarly, also for ENG and MNG layers). Such pairing of complementary materials can provide us with the possibility of having sub-wavelength “compact resonant structures” in the form of sub-wavelength cavities, waveguides, and scatterers. This can be extended to the cylindrical and spherical geometries formed by pairs of complementary MTMs, as studied by Alù and Engheta [73], [74], [85], who have theoretically shown how suitable pairs of two concentric spherical shells made of DPS–DNG, ENG–MNG, or even of ENG–DPS or MNG–DPS materials may lead to significant enhancement of wave scattering when compared with scattering from structures with the same shape and dimensions, but made of standard DPS media only. In other words, these structures are indeed electrically tiny scatterers with much larger scattering cross section than ordinary scatterers of the same size. With no loss, this enhancement can be of several orders of magnitude when compared with scatterers of comparable dimensions made by standard dielectrics. This result is consistent with the phenomenon of surface plasmon resonance in nanoparticles made of noble metals [124]. These resonant structures are briefly reviewed here. Consider a spherical structure composed of two concentric shells made of DPS and DNG materials with radii , , surand permeability rounded by free space (with permittivity ) (Fig. 9). A monochromatic incident plane wave illuminates

this structure. As is well known, the peaks in the scattering coefficients for such a structure are due to the excitation of the natural modes (i.e., material polaritons) supported by the structure. In other words, if at a given frequency the scatterer supports a natural mode, its scattering cross section will show a resonant peak. At a fixed frequency, this can be explored by varying the total dimension of the scatterer until we achieve the case for which a natural mode may be excited. For spherical structures made of conventional media, the presence of such modes has a cutoff dimension for the outer radius (usually comparable with the wavelength of operation) below which no natural mode is supported and, thus, the scattering coefficients are low. On the other hand, when DNG or SNG MTMs are combined with complementary materials, this “cutoff” limitation may be removed. Specifically, the dispersion relation for the material natural modes supported by the concentric-shell structure in Fig. 9 has been obtained in general [85], and in the limit of small radii, it can be simplified as

(13) where is the spherical modal order of the natural mode (referring to the variation along ). In order to have a physical solusatisfying the above equation, one or both of the tion for expressions above containing the material permittivities or permeabilities should attain a value between zero and unity. This cannot be achieved if the concentric shells are all made of conventional DPS materials, implying that no natural mode is supported for electrically tiny spherical shells with conventional materials. However, if we use a DNG or SNG layer combined with a DPS or another properly chosen complementary MTM concentric layer, it will become possible to have a solution for . As a result, for a specific ratio of radii fulfilling the above equation, the condition for the presence of a natural mode for the tiny DPS–DNG (or ENG–MNG, DPS–ENG, or DPS–MNG) concentric shells exists, which depends only on the ratio of shell radii without any direct constraint on the outer dimension of the scatterer. This behavior gives rise to the possibility of having a very high scattered field from a very tiny concentric shell particle [73], [74], [85]. Fig. 10 shows, as an example, the behavior of the scattering for the mode, i.e., the dipolar scattered coefficient spherical TM wave, for a case where a DPS material is covered with an ENG layer. Comparison with the DPS–DPS case is also shown in Fig. 10, which reveals a major enhancement of the scattering phenomenon. We note from Fig. 10(A) that the

ENGHETA AND ZIOLKOWSKI: POSITIVE FUTURE FOR DNG MTMs

Fig. 10. Behavior of the scattering coefficient c versus a =a for a DPS–ENG combination with  =  =  , " = 10" , " = [ 1:5 j Im(" )]" . In (A), Im(" ) is assumed zero, and the plot shows the behavior of c in terms of a =a for several values of a. In (B), the effect of loss, i.e., nonzero Im(" ), is shown for a =  =20, where  is the wavelength in region 1.

0 0

j

j

scattering coefficient can reach its maximum, even though the outer radius is much smaller than the wavelength. As the outer radius gets smaller, the maximum scattering can still occur, al. When though it becomes much more sensitive to the ratio the loss is included, the peak of the scattering coefficient is decreased as expected; however, it may still be greater than the scattering from the corresponding DPS case, as shown in Fig. 10(B). One can speculate that by embedding these highly polarizable DPS–DNG spherical particles in a host medium, a bulk composite material with effective permittivities and/or permeabilities exhibiting resonances can be formed [85]. The resulting effective material parameters can be strongly affected by the and can attain negative values for a cerchoice of the ratio tain range of that ratio [85]. There are two other interesting features about the scattering from these resonant structures worth noting. First, the large scattering amplitude from the spherical DPS–DNG shells shown above in Fig. 10 was for the dipolar term. As is well known, higher order multipoles begin to contribute more as the size of the sphere increases. However, as studied by Alù and Engheta [86], (13) can be satisfied for different ratios of , e.g., for quadrupolar or the radii for cases with

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for octopolar scattering. This implies that one can obtain a strong resonant scattering amplitude for a higher order multipole (e.g., quadrupole or octopole) while keeping the electrical size of the object small and the scattering amplitudes of the lower and higher order multipoles weak. Therefore, if the ratio of the radii is chosen judiciously, a very small spherical object made from a combination of ENG, MNG, DNG, and/or DPS layers can, in principle, strongly scatter quadrupolar or octopolar fields. It is interesting to notice that ordinarily electrically small scatterers re-radiate like small dipoles because the phase retardation within them is negligible. However, these electrically and physically small DPS–DNG two-shell spheres may scatter like a quadrupole. This can offer interesting potential applications for realizing optical nanotransmission lines by arranging these nanoshells into linear arrays to transport optical energy below diffraction limits and to act as nanoantennas with quadrupole and higher order multipole radiation patterns. The second interesting feature for such DNG–DPS scatterers involves the opposite effect, the possibility of reducing the total scattering cross section of a structure. It has been suggested by Alù and Engheta [87] that, for a different ratio of radii, a “transparency” condition may be achieved, which results in the reduction of the total scattering cross section of these scatterers. In the case of thin spherical scatterers, their total cross section is generally dominated by the dipolar term in the multipole expansion. This dipolar scattering may vanish with a proper choice of the two-shell radii, and the total scattering cross section can thus be reduced. When larger scatterers are considered, this overall reduction is less effective because the multiple contributions from different multipole terms contribute more to the overall scattering cross section and cannot all be reduced simultaneously. Nonetheless, a noticeable effect is still present even in this scenario for certain proposed geometries. Similar features have been obtained for the case of thin cylindrical structures formed by coaxial DPS and DNG (or ENG–MNG) layers [73]. For the small radii approximation, the dispersion relation for the natural modes in such cylindrical DPS–DNG structures can be expressed as

(14)

satisfies this equation, the Again, when the ratio of radii scattering amplitude reaches its maximum. Other characteristics of these cylindrical structures are discussed in [73]. IX. DNG METAMATERIALS AND ANTENNA APPLICATIONS We mentioned that pairing DNG and DPS materials (or pairing complementary SNG materials) can be regarded as joining two reactive impedances with opposite signs, resulting in a resonance phenomenon. One can then ask the following question: Can a DNG (or an SNG) layer be used to modify

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Fig. 11. (A) Electrically small electric dipole surrounded by a DNG shell. mode into a (B) Lumped-element circuit for such an antenna radiating a free-space sphere of radius a that is surrounded by a lossless DNG spherical shell of thickness ` a a , which is then terminated with a free-space region.

TM

= 0

the input impedance of an antenna, providing the possibility to improve the antenna performance? This problem has been studied analytically and numerically by Ziolkowski and Kipple [68]. They have considered the possibility of matching an electrically small electric dipole to free space by surrounding it with a DNG shell, as shown in Fig. 11(A), and have successfully demonstrated that the dipole-DNG shell system produces large radiated power for an electrically small antenna [68]. The complex power generated by an electrically small electric centered in a spherical region of dipole with current moment radius filled with a homogeneous DPS medium is well known [125] (15) When it is compared to those generated by that antenna when it is embedded in a DNG medium, it is observed that, while the radiated power in both cases was equal, the reactive power in the DNG case was equal in magnitude, but opposite in sign to the free space (DPS) case, i.e., the complex powers . This property stimulated the investigation into the effect of placing a DNG shell around an electrically small electric dipole in an attempt to “match” the dipole to free space. An approximate lumped-element circuit model of the dipole-DNG shell system confirmed that it was possible to treat the presence of the DNG shell as a matching network. This spatial model is shown in Fig. 11(B). The fundamental mode radiated by the dipole into free space sees a high-pass filter and, thus, is basically in a cutoff situation. Normally, this mode sees a potential barrier and tunnels through it to the receiver, leaving a large amount of reactive power behind. The DNG shell produces a low-pass system that can compensate for the high-pass behavior. The DNG shell provides a way to match the corresponding CL and LC resonances to produce a system that is impedance matched to free space, i.e., by

Fig. 12. Radiated power gain for the dipole-DNG shell system is considerable even in the presence of losses.

joining two regions with reactive impedances of opposite signs (capacitive for the DPS sphere and inductive for the DNG shell), the total reactance can be reduced to zero. The potential barrier and, consequently, the reactive power near the to propagate antenna are reduced to zero allowing the freely into free space. This matched-resonance source-DNG shell description is reciprocal to Alù and Engheta’s resonant scattering arguments and is consistent with the 2-D planar MTM realizations considered by Eleftheriades et al. [43] and Caloz and Itoh [40]. The problem of an infinitesimal electric dipole enclosed within a DNG shell was solved analytically, and numerical evaluations of a variety of DNG cases were provided [68]. These results confirmed that the dipole-DNG shell system caused the radiated power to be increased by orders of magnitude with a simultaneous decrease in the reactance and to values below a corresponding decrease in the radiation the Chu limit. Moreover, this behavior does not disappear in the presence of losses. Rather, since it is a resonance-based effect, losses broaden the resonance and decrease the peak of the response. The results for the “super-gain” case in [68] for which the DNG shell has and an inner radius m are shown in Fig. 12. The dipole length was assumed to be 100 m. The radiated power gain, the power radiated by the dipole-DNG shell system relative to the power radiated in free space by a dipole whose half-length equals the outer radius of the DNG shell, is plotted against the outer radius of the shell. The peak of this radiated m. power gain occurs for an outer radius of The lossless case is compared to the lossy cases with the of the DNG MTM loss tangent being 0.0001 and 0.001. It illustrates the expected behavior. Despite the presence of losses, the radiated power gain is substantial. The natural question arose as to the relationship between the enhanced source and scattering results. If reciprocity holds, there should be a one-to-one correspondence between a particular enhanced DPS–DNG scattering configuration and the corresponding enhanced DNG dipole-shell radiating system. To

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stored energy in the inner sphere associated with the real part of the electric field goes to zero and with the magnetic field goes to a maximum. This behavior is correlated to an inductive reactive power; it cancels the capacitive reactive electric power associated with the dipole formed by the scattering from the electrically small shell. As a consequence of the demonstrated reciprocity between a pair of DPS–DNG and ENG–MNG shells, Kipple and Ziolkowski have also considered using only an ENG shell to produce the necessary inductive reactive power needed to cancel the capacitive reactive power of the electrically small dipole. This dipole-ENG shell system has been shown to produce a radiated power gain similar to the dipole-DNG shell system. It may be much simpler to realize physically than a dipole-DNG system.

For a DNG shell of inner radius 1.0 mm and (" ; ) = ; 03 ), the gain in the energy stored in the inner DPS sphere when a T M (n = 1; m = 1) wave is scattered from the shell and the gain the power Fig. 13.

0

( 3"

radiated from an electrically small dipole antenna center in the shell are given as a function of the outer radius of the shell. The correlation between these two quantities is clearly noticeable.

verify that the reciprocity holds for these DNG and ENG–MNG systems, Kipple and Ziolkowski [88] have also considered several very general sphere scattering problems both analytically and numerically. In particular, plane-wave scattering from a sphere that was coated with two concentric spherical shells, which could be DPS, DNG, ENG or MNG materials, was analyzed. The coated spheres were considered to be located in a DPS medium, i.e., free space. This allowed a direct comparison with the concentric ENG–MNG and DPS–DNG sphere results [85], a direct comparison of the free-space sphere embedded in a DNG shell with the reciprocal DNG dipole-shell system, and other interesting combinations of DPS, DNG, ENG, and MNG shells. It was verified that reciprocity holds for all of the configurations studied. To demonstrate this behavior, consider the “manufacturable” case of [68] in which an electrically small (1.2-mm length) electric dipole is embedded in a small sphere of free space with a radius of 1.0 mm, that is, in turn, surrounded by a DNG shell having permittivity and perme. The external ability values region is assumed to be free space, and the radiation frequency is 10 GHz. The radiated power gain, here, the ratio of the power radiated by the dipole in the presence of the DNG shell and in free space, is plotted in Fig. 13 as a function of the DNG shell’s outer radius. The peak of this radiated power gain occurs for an mm. The energy stored gain, when a outer radius of wave is incident on these nested spheres, is also shown in Fig. 13. The gain is the ratio of the energy stored in the inner DPS (free space) sphere in the presence of the DNG shell and in the same sphere when all the regions are free space. The peak in the stored energy gain occurs at the same outer DNG shell radius, as does the peak in the radiated power gain. A strong correlation is observed between all of the corresponding radiation and scattering results. It has also been demonstrated that the scattering resonance occurs where the

X. DISPERSION COMPENSATION IN A TRANSMISSION LINE USING DNG MTMs Another interesting potential application of DNG MTM is in its possible use for dispersion compensation. Cheng and Ziolkowski have considered the use of volumetric DNG MTMs for the modification of the propagation of signals along a microstrip transmission line [89]. If one could compensate for the dispersion along such transmission lines, signals propagating along them would not become distorted. This could lead to a simplification of the components in many systems. Microstrip dispersion can be eliminated by correcting for the frequency dependence of the effective permittivity associated with this type of transmission line. As shown in [126] and [127], for a microstrip transmission line of width and a conventional dielectric substrate height , one has the approximate result for the effective relative permittivity of the air–substrate–microstrip system (16)

where

the

with and the

constants ,

electrostatic

the

characteristic

relative

impedance

permittivity with

, and . The goal is to design an MTM that can be included with the microstrip line in some manner to make it dispersionless, i.e., we want to produce a dispersion-compensated segment of transmission line. This means we want to introduce an MTM with and permeability so that the relative permittivity overall relative permittivity and permeability of the system is

(17)

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in such a manner that the wave impedance in the MTM remains the same as it is in the original substrate, i.e.,

(18) and the index of refraction in the medium compensates for the dispersion effects associated with the microstrip geometry itself, i.e., the effective index becomes that of free space

(19) These conditions are satisfied if so that

Fig. 14. Real part of the index of refraction of the microstrip-only, of the MTM-only, and the total MTM-dispersion-compensated transmission line.

(20) We note that the effective permittivity and permeability of such an MTM should be negative, implying that a DNG material must be utilized for this purpose. [The range of validity of condition (20) should be consistent with that of the effective medium approximation (16).] A plot of the index of refraction of the uncompensated line, of the MTM compensator, and of the dispersion-compensated line is shown in Fig. 14 for a microstrip transmission line at 10 GHz using Roger’s Duroid 5880 substrate. The substrate had the relative permittivity and its height was mil mm The width of the mm mil to achieve transmission line was a 50- impedance. As shown in Fig. 14, in principle, complete dispersion compensation is theoretically possible. XI. MTMs OTHER THAN DNG MEDIA Although the focus of this paper is on the DNG MTMs, there are other classes of MTMs that can exhibit equally exciting and interesting features. Here, we briefly mention some of these media. A. MTMs With Near-Zero Refractive Index MTMs, in which the permittivity and/or permeability are near zero and, thus, the refractive index is much smaller than unity, can offer exciting potential applications. Planar MTMs that exhibit both positive and negative values of the index of refraction near zero have been realized experimentally by several research groups [39]–[48]. Within these studies, there have also been several demonstrations, both theoretically and experimentally, of planar MTMs that exhibit a zero index of refraction within a specified frequency band. In particular, by matching the resonances in a series-parallel lumped-element circuit realization of a DNG MTM at a specified frequency, the propa-

gation constant as a function of frequency continuously passes through zero (giving a zero index) with a nonzero slope (giving a nonzero group speed) in its transition from a DNG region of its operational behavior to a DPS region [40], [43], [115]. Several applications of these series-parallel MTMs have been proposed and realized, e.g., phase shifters, couplers, and compact resonators. Several investigations have also presented volumetric MTMs that exhibit near-zero-index medium properties, e.g., [49]–[53]. These zero-index EBG structure studies include working in a passband. By introducing a source into a zero-index EBG with an excitation frequency that lies within the EBG’s passband, Enoch et al. [51], [52] and Tayeb et al. [53] produced an extremely narrow antenna pattern. Alù et al. have also shown theoretically that by covering a sub-wavelength tiny aperture in a flat perfectly conducting , one can sigscreen with a slab of materials with nificantly increase the power transmitted through such a hole due to the coupling of the incident wave into the “leaky” wave supported by such a layer [90]. By covering both sides of the hole, not only can one increase the transmitted power through the hole, but this power can be directed as a sharp beam in a given direction [90], [91]. These results stimulated a study by Ziolkowski [92] that details the propagation and scattering properties of a passive dispersive MTM that is matched to free space and has an index of refraction equal to zero. 1-D, 2-D, and 3-D problems corresponding to source and scattering configurations have been treated analytically. The 1-D and 2-D results have been confirmed numerically with FDTD simulations. It has been shown that the electromagnetic fields in a matched zero-index medium , so that and (i.e., ) take on a static character in space, yet remain dynamic in time in such a manner that the underlying physics remains associated with propagating fields. Zero phase variation at various points in the zero-index medium has been demon-

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tailored wave speed. Complete spatio-temporal wavefront engineering could then be realized. Many of these issues are currently being investigated. B. MTMs as Artificial Magentic Conductors

Fig. 15. (A) Line source is located at the center of a matched zero-index square cylinder of Drude medium with side length 1:2 . (B) Electric-field intensity produced by such a line source driven at 30 GHz.

strated once steady-state conditions are obtained. These behaviors have been used to illustrate why a zero-index MTM, such as a zero-index electromagnetic bandgap structured medium, significantly narrows the far-field pattern associated with an antenna located within it. The geometry and FDTD results for a line source centered in a square matched zero-index cylinder of and driven at 30 GHz are shown in Fig. 15. side length The uniformity of the constant electric field over the entire interior of the square cylinder is clearly seen. Moreover, one can see that the fields radiated into free space arise locally as though they are driven by uniform fields across apertures corresponding to the sides of the square. There may be a variety of potential applications for matched zero-index media beyond their already demonstrated use for compact resonators and highly directive sources and apertures. These include delay lines with no phase differences between their inputs and outputs and wavefront transformers, i.e., a transformer that converts wavefronts with small curvature into output beams with large curvature (planar) wavefronts. Other MTMs could be designed by engineering the permittivity and permeability models to yield a matched zero-index medium with a

If an MTM can be engineered to possess a large permeability, it will behave as a “magnetic conductor.” Several planar and volumetric MTM structures have been investigated that act as AMCs, i.e., slabs that produce reflection coefficient with zero phase, i.e., an in-phase reflection [93]–[99]. It has been shown by Erentok et al. [100] that a volumetric MTM constructed from a periodic arrangement of capacitively loaded loops (CLLs) acts as an AMC when the incident wave first interacts with the capacitor side of the CLLs and as a perfect electric conductor (PEC) from the opposite direction. The CLL MTM has effective material properties that exhibit a two-time-derivative Lorentz material (2TDLM) behavior for the permeability and a Drude behavior for the permittivity. The resonance of the real part of the 2TDLM model and the zero crossing of the real part of the Drude model occur at the same frequency at which the in-phase reflection occurs. This concurrence of the critical frequencies of both models produces an MTM slab with a high-impedance . Nustate at that frequency, i.e., merical simulation and experimental results for the CLL-based MTM slab have shown good agreement. The use of the two-CLL-deep MTM AMC block for antennas was also considered [100]. Numerical simulations of the interaction of a dipole antenna with such an MTM block have shown the expected AMC enhancements of the radiated fields. The dipole–AMC block configuration is shown in Fig. 16(A). The behavior of this system as a function of the antenna length and the distance of the antenna from the block have been studied. As shown in Fig. 16(B), it was found that resonant responses are obtained when the distance between the dipole and MTM block was optimized. Significantly enhanced electric-field values in the reflected field region and front-to-back ratios have been demonstrated. The - and -plane patterns of the dipole–AMC block system and of the free-space dipole are ancompared in Fig. 17 for the optimized case of a mm near a CLL-based AMC tenna driven at 10 GHz block with dimensions 7.1 mm 6.6 mm 25.4 mm. The broadside power is more than doubled in the presence of the AMC block. The realized front-to-back ratio, as shown in Fig. 16(B), for this case, is 164.25. C. Single-Negative MTMs and Plasmonic Media We mentioned earlier that some of the exciting features and interesting potential applications of DNG MTMs may also be developed using SNG materials such as plasmonic media. This is particularly the case where the electrical and physical dimensions of devices and components involving these materials are small. Furthermore, as shown above, when complementary SNG materials are paired, e.g., when an ENG layer (e.g., a plasmonic layer such as silver or gold in the visible or IR regimes) is juxtaposed with an MNG one, some interesting features, which are specific to pairing of these layers and which are not present for each single layer alone, may appear [25]–[27]. One of these features, a counterpart to the lensing effect of

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Fig. 16. (A). Dipole antenna and CLL-MTM block configuration. (B) Resonant interaction between the dipole antenna and CLL-based MTM block produces very large front-to-back ratios.

DNG slabs, is the virtual image formation, wave tunneling, growing evanescent fields, and evanescent wave displacement in a pair of ENG–MNG slabs [25]. Consider a pair of lossless ENG–MNG slabs, in front of which a point (or a line) source of monochromatic wave is placed (Fig. 18). As is usually done, the field distribution at the object plane can, in general, be expanded in terms of all spatial Fourier components (propagating and evanescent parts). Alù and Engheta have theoretically found that by judiciously selecting the ENG and MNG material , , , and and the thicknesses parameters and , one can achieve a situation in which all spatial Fourier components can “tunnel” through this paired ENG–MNG structure and, thus, the pair effectively becomes “transparent” [25]. In other words, the propagating, as well as evanescent waves at the “entrance” face of this pair of slabs, can, under certain conditions, tunnel through the pair, and show up at the exit face with the same corresponding values (in both magnitude and phase) as their values at the entrance face. As a result, an observer on the other side of this “conjugate” matched pair of ENG–MNG slabs will see a “virtual” image of the point (or line) source as though it were seated closer to the observer providing near-field observation by the amount of the objects with ideally all spatial Fourier components present [25]. This effect may conceptually provide an interesting future application in image reconstruction, resolution

Fig. 17. Far-field patterns of the dipole antenna and CLL-based MTM block system (solid line), shown in Fig. 16, are compared to those produced by a free-space dipole antenna (dotted line). (A) E -plane pattern. (B) H -plane pattern.

Fig. 18. Pair of ENG–MNG layers under certain conditions may provide image displacement and virtual image reconstruction. From [25].

enhancement, near-field subwavelength imaging, and NSOM. It is important to point out that an analogous matched pairing of DNG and DPS slabs would also “preserve” and allow “tunneling” of the evanescent waves, analogous to what Pendry has found for his DNG lens surrounded by a conventional medium. However, Pendry’s lens ideally forms a “real” image of the

ENGHETA AND ZIOLKOWSKI: POSITIVE FUTURE FOR DNG MTMs

object, whereas this ENG–MNG bilayer may conceptually displace a “virtual” image [25]. The field distributions inside such an ENG–MNG bilayer have been thoroughly analyzed, and using equivalent transmission-line models with appropriate distributed series and shunt reactive elements, various effects such as growing evanescent fields at the ENG–MNG interface, tunneling and transparency have been explained and physically justified [25]. Numerical simulations for some of these features are currently being conducted [101]. XII. EPILOGUE We have tried to share with you a wide variety of physical effects associated with DNG and SNG MTMs and their potential applications. While the physics of MTMs appears to be much better understood now through analysis and numerical simulations, there are significant challenges ahead in the areas of fabrication and measurements. There have been several successful microwave realizations of the volumetric MTMs that have demonstrated the unusual properties discussed here. However, since the required inclusion size is much smaller than a wavelength for these MTMs, the move to millimeter, terahertz, IR, and visible frequencies will require the development of a host of innovative structures and fabrication processes. Nonetheless, the ability to tailor material properties to achieve physical effects not thought to be possible only a few years ago is motivating a large number of activities in these directions. The future is indeed very positive for DNG MTMs. ACKNOWLEDGMENT The authors would like to thank their graduate students at the University of Pennsylvania, Philadelphia, and the University of Arizona, Tucson, respectively, for all their contributions to the MTM research activities in their groups. For the materials presented here, the authors particularly thank A. Alù, University of Pennsylvania and Universita di Roma Tre, Rome, Italy, Allison Kipple, University of Arizona and U. S. Army Electronic Proving Ground, Ft. Huachuca, AZ, and Aycan Erentok, University of Arizona. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.—Usp., vol. 47, pp. 509–514, Jan.–Feb. 1968. [2] , “The electrodynamics of substances with simultaneously negative values of " and ” (in Russian), Usp. Fiz. Nauk, vol. 92, pp. 517–526, 1967. [3] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2081, Nov. 1999. [4] , “Low-frequency plasmons in thin wire structures,” J. Phys., Condens. Matter, vol. 10, pp. 4785–4809, 1998. [5] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett., vol. 84, pp. 4184–4187, May 2000. [6] D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett., vol. 85, pp. 2933–2936, Oct. 2000. [7] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett., vol. 78, pp. 489–491, Jan. 2001. [8] A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 77–79, Apr. 2001. [9] IEEE Trans. Antennas Propag. (Special Issue), vol. 51, no. 10, Oct. 2003.

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[102] A. Alù and N. Engheta, “Radiation from a traveling-wave current sheet at the interface between a conventional material and a material with negative permittivity and permeability,” Microwave Opt. Technol. Lett., vol. 35, no. 6, pp. 460–463, Dec. 2002. [103] J. Pacheco, Jr., T. M. Grzegorczyk, B.-L. Wu, Y. Zhang, and J. A. Kong, “Power propagation in homogeneous isotropic frequencydispersive left-handed media,” Phys. Rev. Lett., vol. 89, Dec. 2002. Paper 257 401. [104] R. W. Ziolkowski, “Gaussian beam interactions with double negative (DNG) metamaterials,” in Negative Refraction Metamaterials: Fundamental Properties and Applications, G. V. Eleftheriades and K. G. Balmain, Eds., to be published. [105] S. A. Cummer, “Dynamics of causal beam refraction in negative refractive index materials,” Appl. Phys. Lett., vol. 82, pp. 2008–2010, Mar. 2003. [106] C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2596–2603, Oct. 2003. [107] A. N. Lagarkov and V. N. Kisel, “Electrodynamics properties of simple bodies made of materials with negative permeability and negative permittivity,” Dokl. Phys., vol. 46, no. 3, pp. 163–165, 2001. [108] , “Electrodynamics properties of simple bodies made of materials with negative permeability and negative permittivity” (in Russian), Dokl. Akad. Nauk SSSR, vol. 377, no. 1, pp. 40–43, 2001. [109] R. A. Silin and I. P. Chepurnykh, “On media with negative dispersion,” J. Commun. Technol. Electron., vol. 46, no. 10, pp. 1121–1125, 2001. , “On media with negative dispersion” (in Russian), Radiotekhnika, [110] vol. 46, no. 10, pp. 1212–1217, 2001. [111] M. W. Feise, P. J. Bevelacqua, and J. B. Schneider, “Effects of surface waves on behavior of perfect lenses,” Phys. Rev. B, Condens. Matter, vol. 66, 2002. Paper 035 113. [112] A. Ishimaru and J. Thomas, “Transmission and focusing of a slab of negative refractive index,” in Proc. URSI Nat. Radio Science Meeting, San Antonio, TX, Jul. 2002, p. 43. [113] P. F. Loschialpo, D. L. Smith, D. W. Forester, and F. J. Rachford, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 67, 2003. Paper 025 602(R). [114] M. K. Kärkkäinen, “Numerical study of wave propagation in uniaxially anisotropic Lorentzian backward-wave slabs,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 68, 2003. Paper 026 602. [115] R. W. Ziolkowski and C.-Y. Cheng, “Lumped element models of double negative metamaterial-based transmission lines,” Radio Sci., vol. 39, Apr. 2004. Paper RS2017. [116] D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett., vol. 90, no. 7, Feb. 2003. Paper 077 405. [117] S. Hrabar and J. Bartolic, “Backward waveguide based on uniaxial anisotropic negative permeability metamaterials,” in Proc. 17th Int. Applied Electromagnetics and Communications Conf., Oct. 2003, pp. 251–254. [118] R. Marques, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength slit-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett., vol. 89, no. 18, Oct. 28, 2002. Paper 183 901. [119] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 1995. [120] D. C. Wittwer and R. W. Ziolkowski, “Two time-derivative Lorentz material (2TDLM) formulation of a Maxwellian absorbing layer matched to a lossy media,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 192–199, Feb. 2000. [121] D. C. Wittwer and R. W. Ziolkowski, “Maxwellian material based absorbing boundary conditions for lossy media in 3D,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 200–213, Feb. 2000. [122] D. L. Lee, Electromagnetic Principles of Integrated Optics. New York: Wiley, 1986. [123] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics. New York: Wiley, 1991, pp. 94–95. [124] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983. [125] C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [126] F. Gardiol, Microstrip Circuits. New York: Wiley, 1994, pp. 48–50. [127] W. J. Getsinger, “Microstrip dispersion model,” IEEE Trans. Microw. Theory Tech., vol. MTT-21, no. 1, pp. 34–39, Jan. 1973.

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Nader Engheta (S’80–M’82–SM’89–F’96) received the B.S. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 1978, and the M.S. degree in electrical engineering and Ph.D. degree in electrical engineering (with a minor in physics) from the California Institute of Technology (Caltech), Pasadena, in 1979 and 1982, respectively. From June 1982 to June 1983, he was a Post-Doctoral Research Fellow with Caltech. From June 1983 to June 1987, he was a Senior Research Scientist with the Dikewood Division, Kaman Sciences Corporation, Santa Monica, CA. In July 1987, he joined the faculty of the University of Pennsylvania, Philadelphia, where he is currently the H. Nedwill Ramsey Professor of Electrical and Systems Engineering. He is also a member of the David Mahoney Institute of Neurological Sciences, University of Pennsylvania, and a member of the Bioengineering Graduate Group, University of Pennsylvania. He was the graduate group chair of electrical engineering from July 1993 to June 1997. He was an Associate Editor for Radio Science (1991–1996). He was on the Editorial Board of the Journal of Electromagnetic Waves and Applications. He has guest edited/co-edited several special issues, namely, the “Special Issue of Wave Interaction with Chiral and Complex Media” of the Journal of Electromagnetic Waves and Applications (1992), Special Issue of Antennas and Microwaves (from the 13th Annual Benjamin Franklin Symposium) of the Journal of the Franklin Institute (1995), the Special of Issue of Electrodynamics in Complex Environments of Wave Motion (2001). His research interests and activities are in the areas of fields and waves phenomena, MTMs, theory of nanooptics and nanophotonics, nanoelectromagnetism, miniaturized antennas, through-wall microwave imaging, electromagnetics/electrophysics of event-related brain cortical potentials [e.g., electroencephalography (EEG)], physics of information contents in polarization vision, bio-inspired/biomimetic sensing, processing, and displaying polarization information, reverse-engineering of polarization vision and information sensing in nature, bio-inspired hyperspectral imaging, mathematics of fractional operators, and fractal domains. Dr. Engheta is a Guggenheim Fellow and a Fellow of the Optical Society of America. He is a member of the American Physical Society (APS), the American Association for the Advancement of Science (AAAS), Sigma Xi, Commissions B and D of the U.S. National Committee (USNC) of the International Union of Radio Science (URSI), and a member of the Electromagnetics Academy. He was the chair (1989–1991) and vice-chair (1988–1989) of the joint chapter of the IEEE Antennas and Propagation (AP)/Microwave Theory and Techniques (MTT) Philadelphia Section. He is an elected member of the Administrative Committee (AdCom) of the IEEE Antennas and Propagation Society (IEEE AP-S) since January 2003. He has organized and chaired various special sessions in international symposia. He is an associate editor for the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS (2002–present) and was an associate editor for the IEEE TRANSACTIONS ON ANTENNA AND PROPAGATION (1996–2001). He coedited the Special Issue on Metamaterials of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (2003). He was a recipient of the IEEE Third Millennium Medal. He has also been the recipient of various awards and distinctions for his scholarly research contributions and teaching activities including the UPS Foundation Distinguished Educator Term Chair (July 1999–June 2000), the Fulbright Naples Chair Award for Naples, Italy (1998), a 1989 National Science Foundation (NSF) Presidential Young Investigator (PYI) Award, two-time recipient of the S. Reid Warren, Jr. Award for distinguished teaching from the School of Engineering and Applied Science, University of Pennsylvania (1993 and 2002) , the 1994 Christian F. and Mary R. Lindback Foundation Award, and the W. M. Keck Foundation’s 1995 Engineering Teaching Excellence Award. He served as an IEEE Antennas and Propagation Society Distinguished Lecturer from 1997 to 1999.

Richard W. Ziolkowski (M’87–SM’91–F’94) received the Sc.B. degree in physics (magna cum laude) (with honors) from Brown University, Providence, RI, in 1974, and the M.S. and Ph.D. degrees in physics from the University of Illinois at Urbana-Champaign, in 1975 and 1980, respectively. From 1981 to 1990, he was a member of the Engineering Research Division, Lawrence Livermore National Laboratory, and served as the leader of the Computational Electronics and Electromagnetics Thrust Area for the Engineering Directorate from 1984 to 1990. In 1990, he joined the Department of Electrical and Computer Engineering, University of Arizona, Tucson, as an Associate Professor, and became a Full Professor in 1996. He currently serves as the Kenneth Von Behren Chaired Professor. His research interests include the application of new mathematical and numerical methods to linear and nonlinear problems dealing with the interaction of acoustic and electromagnetic waves with complex media, MTMs, and realistic structures. For the Optical Society of America (OSA), he was a co-guest editor of the 1998 special issue of Journal of Optical Society of American A, Optical Image Science featuring mathematics and modeling in modern optics. Prof. Ziolkowski is a member of Tau Beta Pi, Sigma Xi, Phi Kappa Phi, the American Physical Society, the Optical Society of America, the Acoustical Society of America, and Commissions B (Fields and Waves) and D (Electronics and Photonics) of URSI (International Union of Radio Science). He served as the vice chairman of the 1989 IEEE Antennas and Propagation Society (IEEE AP-S) and URSI Symposium, San Jose, CA, and as the Technical Program chairperson for the 1998 IEEE Conference on Electromagnetic Field Computation, Tucson, AZ. He served as a member of the IEEE Antennas and Propagation Society Administrative Committee (AdCom) (2000–2002). He was an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (1993–1998). He was a co-guest editor ( for the October 2003 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION “Special Issue on Metamaterials.” He served as the IEEE AP-S vice president in 2004. He is currently serving as the IEEE AP-S president. For the U.S. URSI Society, he has served as secretary for Commission B (Fields and Waves) (1993–1996) and as chairperson of the Technical Activities Committee (1997–1999), and as secretary for Commission D (Electronics and Photonics) (2001–2002). He served as a member-at-large of the U.S. National Committee (USNC), URSI (2000–2002) and currently serves as a member of the International Commission B Technical Activities Board. He was a co-organizer of the Photonics Nanostructures Special Symposia at the 1998, 1999, and 2000 OSA Integrated Photonics Research (IPR) Topical Meetings. He served as the chair of the IPR Sub-Committee IV, Nanostructure Photonics (2001). He was a Steering Committee member for the 27th European Space Agency (ESA) Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left-handed Materials, Fractal and Frequency Selective Surfaces, Santiago de Compostela, Spain (March 2004). He was the recipient of the 1993 Tau Beta Pi Professor of the Year Award and the 1993 and 1998 IEEE and Eta Kappa Nu Outstanding Teaching Award.

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Digital Object Identifier 10.1109/TMTT.2005.845214

Digital Object Identifier 10.1109/TMTT.2005.847877

Digital Object Identifier 10.1109/TMTT.2005.847878

Digital Object Identifier 10.1109/TMTT.2005.847879

EDITORIAL BOARD Editor: M. STEER Associate Editors:A. CANGELLARIS, A. CIDRONALI, K. ITOH, B. KIM, S. MARSH, W. MENZEL, A. MORTAZAWI, Y. NIKAWA, Z. POPOVIC, A. RÄISÄNEN, V. RIZZOLI, P. RUSSER, D. WILLIAMS REVIEWERS M. Abdul-Gaffoor M. Abe R. Abou-Jaoude M. Abouzahra A. Abramowicz L. Accatino R. Achar D. Adam E. Adler M. Adlerstein K. Agarwal D. Ahn H.-R Ahn M. Aikawa C. Aitchison M. Akaike C. Akyel A. Akyurtlu B. Albinsson F. Alessandri A. Alexanian C. Algani W. Ali-Ahmad F. Alimenti B. Allen D. Allsopp D. Allstot R. Alm B. Alpert A. Alphones A. Altintas A. Alvarez-Melcom M. Alzona S. Amari L. Andersen B. Anderson Y. Ando O. Anegawa K.-S. Ang I. Angelov R. Anholt Y. Antar G. Antonini D. Antsos K. Anwar I. Aoki R. Aparicio K. Araki J. Archer P. Arcioni F. Arndt R. Arora U. Arz M. Asai P. Asbeck K. Ashby H. Ashok J. Atherton A. Atia I. Awai K. Aygun S. Ayuz Y. Baeyens T. Bagwell Z. Baharav I. Bahl D. Baillargeat S. Bajpai J. Baker-Jarvis E. Balboni S. Banba J. Bandler I. Bandurkin R. Bansal D. Barataud I. Barba F. Bardati I. Bardi S. Barker D. Barlage J. Barr D. Batchelor B. Bates H. Baudrand S. Beaussart R. Beck D. Becker K. Beilenhoff B. Beker V. Belitsky D. Belot H. Bell T. Benson M. Berroth G. Bertin S. Best W. Beyenne A. Beyer S. Bharj K. Bhasin P. Bhattacharya Q. Bi M. Bialkowski E. Biebl P. Bienstman R. Bierig R. Biernacki S. Bila L. Billonnet T. Bird B. Bishop G. Bit-Babik D. Blackham B. Blalock M. Blank P. Blondy P. Blount D. Boccoli B. Boeck F. Bögelsack L. Boglione R. Boix J. Booske N. Borges de Carvalho V. Boria V. Borich O. Boric-Lubecke E. Borie J. Bornemann R. Bosisio H. Boss S. Bousnina P. Bouysse M. Bozzi E. Bracken P. Bradley R. Bradley T. Brazil G. Brehm K. Breuer B. Bridges L. Briones T. Brookes S. Broschat E. Brown G. Brown R. Brown S. Brozovich S. Bruce

S. Bryan H. Bu D. Budimir T. Budka M. Bujatti C. Buntschuh J. Burghartz P. Burghignoli O. Buric-Lubecke D. Butler Q. Cai M. Calcatera C. Caloz E. Camargo R. Cameron N. Camilleri R. Camisa S. Cammer C. Campbell R. Campbell M. Campovecchio F. Canavero A. Cangellaris F. Capolino A. Cappy J.-L. Carbonero G. Carchon J. Carlin G. Carrer R. Carter F. Casas A. Cassinese J. Catala R. Caverly M. Celik M. Celuch-Marcysiak Z. Cendes B. Cetiner J. Cha N. Chaing H. Chaloupka M. Chamberlain C.-H. Chan C.-Y. Chang C. Chang F. Chang H.-C. Chang K. Chang H. Chapell W. Chappell W. Charczenko K. Chatterjee G. Chattopadhyay S. Chaudhuri S. Chebolu C.-C. Chen C.-H. Chen D. Chen H.-S. Chen J. Chen J.-I. Chen J. Chen K. Chen S. Chen W.-K. Chen Y.-J. Chen Y.-K. Chen Z. Chen K.-K. Cheng S. Cherepko W. Chew W.-C. Chew C.-Y. Chi Y.-C. Chiang T. Cho D. Choi J. Choi C.-K. Chou D. Choudhury Y. Chow C. Christopoulos S. Chung R. Cicchetti A. Cidronali T. Cisco J. Citerne D. Citrin R. Clarke J. Cloete E. Cohen L. Cohen A. Coleman R. Collin F. Colomb B. Colpitts G. Conciauro A. Connelly D. Consonni H. Contopanagos F. Cooray I. Corbella J. Costa E. Costamagna A. Costanzo C. Courtney J. Cowles I. Craddock G. Creech J. Crescenzi S. Cripps D. Cros T. Crowe M. Crya R. Culbertson C. Curry W. Curtice Z. Czyz S. D’Agostino C. Dalle G. Dambrine K. Dandekar A. Daryoush B. Das N. Das M. Davidovich M. Davidovitz B. Davis I. Davis L. Davis G. Dawe H. Dayal F. De Flaviis H. De Los Santos P. De Maagt D. De Zutter B. Deal A. Dec J. Deen J. Dees J. DeFalco D. Degroot C. Deibele J. Del Alamo A. Deleniv M. DeLisio S. Demir J. DeNatale E. Denlinger N. Deo

A. Deutsch Y. Deval T. Dhaene A. Diaz-Morcillo G. D’Inzeo C. Diskus B. Dixon T. Djordjevic M. A. Do J. Doane J. Dobrowolski W. Domino S. Dow C. Dozier P. Draxler R. Drayton A. Dreher F. Drewniak S. Dudorov S. Duffy L. Dunleavy V. Dunn J. Dunsmore A. Dutta D. Duvanaud A. Duzdar S. Dvorak L. Dworsky M. Dydyk L. Eastman J. Ebel R. Egri R. Ehlers T. Eibert H. Eisele B. Eisenstadt G. Eisenstein G. Eleftheriades I. Elfadel S. El-Ghazaly F. Ellinger T. Ellis B. Elsharawy R. Emrick N. Engheta B. Engst Y. Eo H. Eom N. Erickson J. Eriksson C. Ernst M. Eron L. Escotte M. Essaaidi J. Everard G. Ewell A. Ezzeddine M. Faber C. Fager D.-G. Fang N. Farhat M. Farina W. Fathelbab A. Fathy A. Fazal E. Fear R. Feinaugle M. Feldman P. Feldman A. Ferendeci C. Fernandes A. Fernandez A. Ferrero I. Fianovsky J. Fiedziuszko I. Filanovsky P. Filicori D. Filipovic A. Fliflet P. Focardi B. Fornberg K. Foster P. Foster G. Franceschetti A. Franchois M. Freire R. Freund A. Freundorfer F. Frezza R. Fujimoto V. Fusco G. Gabriel T. Gaier Z. Galani I. Galin D. Gamble B.-Q. Gao M. Garcia K. Gard R. Garver G. Gauthier B. Geller V. Gelnovatch P. Genderen G. Gentili N. Georgieva W. Geppert J. Gerber F. Gerecht F. German S. Gevorgian R. Geyer O. Ghandi F. Ghannouchi K. Gharaibeh G. Ghione D. Ghodgaonkar F. Giannini A. Gibson S. Gierkink J. Gilb B. Gilbert B.Gimeno E.Glass A. Glisson M. Goano E. Godshalk J. Goel M. Goldfarb C. Goldsmith P. Goldsmith M. Golio R. Gómez R. Gonzalo S. Goodnick S. Gopalsami A. Gopinath R. Gordon P. Gould K. Goverdhanam J. Graffeuil L. Gragnani B. Grant G. Grau A. Grebennikov B. Green T. Gregorzyk I. Gresham E. Griffin

J. Griffith A. Griol G. Groskopf C. Grossman T. Grzegorczyk M. Guglielmi P. Guillon K.-H. Gundlach A. Gupta K. Gupta R. Gupta F. Gustrau R. Gutmann W. Gwarek R. Haas J. Hacker G. Haddad S. Hadjiloucas C. Hafner M. Hagmann S. Hagness H.-K. Hahn A. Hajimiri D. Halchin A. Hallac B. Hallford K. Halonen R. Ham K. Hamaguchi M. Hamid J.-H. Han A. Hanke V. Hanna V. Hansen G. Hanson Y. Hao L. Harle M. Harris L. Hartin H. Hartnagel J. Harvey H. Hasegawa K.-Y. Hashimoto K. Hashimoto J. Haslett G. Hau S. Hay H. Hayashi J. Hayashi L. Hayden B. Haydl S. He T. Heath J. Heaton I. Hecht G. Hegazi P. Heide E. Heilweil W. Heinrich G. Heiter M. Helier R. Henderson R. Henning D. Heo J. Herren K. Herrick N. Herscovici J. Hesler J. Heston M. Heutmaker C. Hicks R. Hicks A. Higgins M. Hikita D. Hill G. Hiller W. Hioe J. Hirokawa T. Hirvonen V. Ho W. Hoefer R. Hoffmann M. Hoft J. Hong S. Hong W. Hong K. Honjo G. Hopkins Y. Horii D. Hornbuckle J. Horng J. Horton K. Hosoya R. Howald H. Howe J.-P. Hsu Q. Hu C.-C. Huang C. Huang F. Huang H.-C. Huang J. Huang P. Huang T.-W. Huang A. Huber D. Huebner H.-T. Hui A. Hung C. Hung H. Hung I. Hunter J. Hurrell M. Hussein B. Huyart I. Huynen H.-Y. Hwang J. Hwang K.-P. Hwang J. Hwu C. Icheln T. Idehara S. Iezekiel P. Ikonen K. Ikossi K. Inagaki A. Ishimaru T. Ishizaki Y. Ismail K. Itoh T. Itoh F. Ivanek A. Ivanov T. Ivanov C. Iversen D. Iverson D. Jablonski D. Jachowski C. Jackson D. Jackson R. Jackson A. Jacob M. Jacob H. Jacobsson D. Jaeger N. Jaeger N. Jain R. Jakoby G. James R. Janaswamy

Digital Object Identifier 10.1109/TMTT.2005.847876

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