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Permeability by Means of Evanescent Waveguide. Modes—Theory and Experiment. Jaime Esteban, Carlos Camacho-Peñalosa, Member, IEEE, Juan E. Page, ...
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

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.

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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

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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

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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.