Role of bianisotropy in negative, permeability and left-handed metamaterials

Role of bianisotropy in negative permeability and left-handed metamaterials Ricardo Marque ´s,*Francisco Medina,†and Rachid Rafii-El-Idrissi Dpto. Electro ´nica y Electromagnetismo, Facultad de Fı ´sica, Univ. de Sevilla, Avda. Reina Mercedes s/n, 41012-Sevilla, Spain 共Received 9 November 2001; published 4 April 2002兲 The recently proposed artificial media with negative magnetic permeability and left-handed metamaterials are revisited at the light of the theory of artificial bi共iso/aniso兲tropic media. In particular, the existence of bianisotropic effects in those materials is investigated, making use of an approximate model. Some unexplained properties of the electromagnetic-wave propagation through these media, revealed by closer inspection of previous numerical simulations and experimental work, are highlighted. It is shown that these peculiarities are properly explained if the bianisotropy is explicitly accounted for. The bianisotropy is related to the existence of magnetoelectric coupling in the artificial constituents 共artificial atoms兲of the medium. A simple modification of the artificial atom that precludes the bianisotropy is also proposed. DOI: 10.1103/PhysRevB.65.144440 PACS number共s兲: 41.20.Jb, 42.70.Qs, 78.20.Ek I. INTRODUCTION A material medium consisting of metallic inclusions randomly or periodically distributed inside a host dielectric behaves, at least within a certain range of frequencies 共typically in the microwave region兲, as an effective continuous medium whose electromagnetic constitutive parameters may have values well outside of the range covered by ordinary materials. Thus, for instance, an artificial negative electric permittivity medium 共NEPM兲can be obtained by using long metallic wires as inclusions,1,2 which simulate the plasma behavior at microwave frequencies. Since free magnetic charges are not present in nature, this method cannot be used for manufacturing negative magnetic permeability media 共NMPM兲. Such media, however, can be built up by using small resonant metallic particles with very high magnetic polarizability. Recently, a particle having this property, the so-called split ring resonator 共SRR兲, has been proposed for this purpose.3An artificial medium consisting of an aggregate of these particles shows a negative permeability region near and above the resonance frequency. In this region, magnetic susceptibilities below ⫺1 are possible. A combination of the artificial media proposed in Refs. 2 and 3 has been experimentally demonstrated to be a left-handed artificial medium, i.e., a medium having, simultaneously, negative electric permittivity and negative magnetic permeability.4–6 On the other hand, embedding metallic resonant particles showing cross polarization effects 共i.e., an electric polarization as a response to an applied magnetic field and vice versa兲in a host dielectric medium is the usual technology for obtaining bi-isotropic and/or bianisotropic artificial media 共i.e., media which replicate optical activity at microwave frequencies兲.7–10 Indeed, all the aforementioned artificial materials 共bi-isotropic, bianisotropic, NMPM, and left-handed materials兲turn out to be very similar in many aspects. Thus, the presence of resonances in the commonly used bianisotropic and bi-isotropic inclusions suggests the existence of regions with negative permeability and/or permittivity, at least if losses are very low or, simply, ignored. Conversely, cross polarization effects could also be expected in some of the proposed resonant particles used to manufacture NMPM and left-handed materials. Our aim in this paper is to discuss the presence and effects of bianisotropy in some of the artificial NMPM and left-handed materials already proposed by other authors. We have developed for this purpose an analytical approximate model of the media accounting for bianisotropic effects. The model has been used to evaluate the magnitude of the cross polarization effects. The qualitative and quantitative results obtained from this model corroborate the main conclusions reported in Refs. 3 and 4. Moreover, the bianisotropic effects incorporated in the model explain some effects observed in the electromagnetic-wave propagation through the aforementioned media, which cannot be fully understood at the light of the theory proposed in Refs. 3 and 4. II. APPROXIMATE ANALYSIS OF THE SRR PARTICLE The SRR particle used for designing the NMPM 共Ref. 3兲 and the left-handed material4under study is shown in Fig. 1. It is formed by two coupled conducting rings printed on a dielectric slab of thickness t. As far as the size of the particle is much smaller than the free space wavelength at resonance 共in our case the particle size is about one-tenth of that FIG. 1. The split ring resonator 共SRR兲. For numerical calculations we have chosen the same dimensions as in Refs. 4 and 5, i.e., c⫽0.8 mm, d⫽0.2 mm, r0⫽2.3 mm, and t⫽0.216 mm. The relative permittivity of the dielectric layer is ⑀ r⫽3.4. PHYSICAL REVIEW B, VOLUME 65, 144440 0163-1829/2002/65共14兲/144440共6兲/$20.00 ©2002 The American Physical Society65 144440-1 wavelength4兲, a quasistatic analysis is plausible. Under such an assumption, we will analyze the behavior of the particle when an external magnetic field B⫽Bz extexp(i ␻ t)z ˆis applied. In that case an electromotive force E⯝⫺i ␻␲ r0 2Bzis induced along the rings which is responsible for creating a current flow which produces a total magnetic moment in the particle. The slot between the rings acts as a distributed capacitance, which stores the same amount of charge 共but of opposite sign兲at both sides of the slot. From charge conservation, dI d ␾ ⯝⫺i ␻ r0共␭i⫹␭0兲⫽0, 共1兲 where Iis the total current flowing on both rings and ␭iand ␭oare the per unit length 共p.u.l.兲charge at the inner and outer rings, respectively. Equation 共1兲shows that the total current on the SRR does not depend on ␾ . However, the currents supported by the inner, Ii, and the outer, Io, rings are, of course, functions of ␾ satisfying the following equation: dIi,o d ␾ ⯝⫺i ␻ r0␭i,o⫽⫺i ␻ r0C共Vi⫺Vo兲,共2兲 where Cis the p.u.l. capacitance between the rings and Vi,o are the quasistatic voltages along the inner and outer rings respectively. The quasistatic voltages at the points marked A, B,C, and Din Fig. 1 can be obtained using Faraday’s law: VA⫽VD⫽⫺VB⫽⫺VC⯝i ␻ 2共LI⫹ ␲ r0 2Bz ext兲,共3兲 where Lis the total inductance of the SRR. The variation of the quasistatic voltage drop along the slot could be obtained by applying transmission line theory. However, as far as the size of the particle is small with respect to the free space wavelength, a first-order approach that considers a linear variation with ␾ of Viand Vois justified. Considering lossless rings and using this approximation, together with Eqs. 共1兲–共3兲, the analysis shows that there is an approximately constant voltage drop, Vi( ␾ )⫺Vo( ␾ )⫽⌬V, across the upper half of the slot (0⬍ ␾ ⬍ ␲ ) whereas, from the symmetry of the particle, the voltage drop across the lower half of the slot is just the opposite 关i.e., Vi( ␾ )⫺Vo( ␾ )⫽⫺⌬Vfor ␲ ⬍ ␾ ⬍2 ␲ 兴. The analysis also shows that the particle has a resonance at the angular frequency ␻ 0 2⯝2 ␲ r0CL;共4兲 i.e., the SRR essentially behaves as a resonant LC circuit. A more involved analysis using transmission line theory without approximations yields the same results. This result is also consistent with those obtained in Ref. 3 if we assume L ⬃ ␮ 0r0. This assumption will be justified in the following. The quasistatic analysis also accounts for the magnetic moment mz⫽ ␲ r0 2Iof the particle, which is given by mz⫽ ␣ zz mmBz ext , ␣ zz mm⯝ ␲ 2ro 4 L 冉 ␻ 0 2 ␻ 2⫺1 冊 ⫺1 .共5兲 Equation 共5兲shows that the particle is diamagnetic above the resonance frequency. Until this point our analysis does not essentially differ from that in 共Ref. 3兲. However, a careful consideration of the behavior of the particle shows that the SRR should act not only as a magnetic dipole, but also as an electric dipole. This behavior is expected from the fact that quasistatic potentials and charges for ␲ ⬍ ␾ ⬍2 ␲ are the electrical images of these quantities for 0⬍ ␾ ⬍ ␲ ,asis sketched in Fig. 1, so that the particle has a nonzero dipolar electric moment along the ydirection. After some calculations, this electric dipole is found to be py⫽i ␣ yz emBz ext , ␣ yz em⫽2 ␻ 0 ␲ r0 3C0deff ␻ 0 ␻ 冉 ␻ 0 2 ␻ 2⫺1 冊 ⫺1 , 共6兲 where C0is the p.u.l. capacitance between the rings when the dielectric is removed, and the effective distance deff gives the p.u.l. polarization Pralong the slot as Pr⫽deff␭i ⬇deffC0(Vi⫺Vo) for 0⬍ ␾ ⬍ ␲ . Here C0is used instead of Cbecause it is the total 共free plus polarization兲charge, which contributes to the particle polarizability. From application of the well-known Onsager symmetry principle to Eqs. 共6兲, it can be inferred that the analyzed SRR also will show an additional cross polarizability, resulting in an induced magnetic moment when an external electric field, with the appropriate polarization, is applied. This cross polarizability could be obtained by direct calculations. However, since the SRR is made with reciprocal media, reciprocity should also hold for the whole structure. In that case, from the symmetry relations for the generalized susceptances11,12 and Eqs. 共6兲, the aforementioned cross polarizability can be shown to be given by mz⫽i ␣ zy meEy ext⫽⫺i ␣ yz emEy ext ,共7兲 where Ey ext is an applied external electric field directed along the yaxis of the particle and ␣ yz em is given by Eqs. 共6兲. The particle also has an ␣ xx ee polarizability, which can be approximately evaluated as the polarizability of a disk,12 ␣ xx ee⯝ ⑀ 0 16 3rext 3;rext⫽r0⫹c⫹d/2 共8兲 as well as a polarizability ␣ yy ee which is given by ␣ yy ee⫽ ␣ xx ee⫹4 ␻ 0 2r0 2C0 2Ldeff 2 冉 ␻ 0 2 ␻ 2⫺1 冊 ⫺1 .共9兲 The second term on the right-hand side of Eq. 共9兲is calculated from Eq. 共7兲, which implies that any external field Ey ext will induce a current on the ring which, in turn, will create an electric polarization, which cannot be neglected around the resonance frequency. Finally, Eqs. 共5兲–共9兲are summarized as follows: px⫽ ␣ xx eeEx ext ,共10兲 py⫽ ␣ yy eeEy ext⫹i ␣ yz emBz ext ,共11兲 MARQUE ´S, MEDINA, AND RAFII-EL-IDRISSI PHYSICAL REVIEW B 65 144440 144440-2 mz⫽⫺i ␣ yz emEy ext⫹ ␣ zz mmBz ext ,共12兲 which clearly shows the bianisotropic behavior of the particle. From Eqs. 共10兲–共12兲it should be possible, after application of an appropriate homogenization procedure, to obtain the macroscopic susceptances of an effective continuous medium consisting of a random or periodic arrangement of these particles. The suitability of such a homogenization procedure will be mainly limited by the electrical size of the unit cell. There exists wide experimental evidence of an appropriate homogenization procedure that provides a good description of the main features of the electromagnetic behavior of left-handed and/or bi共iso/aniso兲tropic metamaterials, provided that the size of the unit cell is smaller than approximately one-tenth of the free space wavelength—as is the case—or even more.4,6,9,10 Losses can also play an important role in the homogenization procedures, but the numerical simulations reported in Ref. 4 show that, for the particular structure analyzed here, the main experimental results can be accounted for by neglecting losses in the analysis of the artificial atoms. Furthermore, although losses are systematically neglected along this paper, they could be easily incorporated in the proposed model by simply adding a frequency-dependent imaginary part ⫺iR/ ␻ to the inductance L共accounting for the series resistance Rof the metallic strips兲and an imaginary part i ␻ Gto the p.u.l. capacitance C 共accounting for the p.u.l. shunt conductance Gacross the slot between the rings兲. III. CONSEQUENCES AND QUALITATIVE EVIDENCE OF BIANISOTROPY In order to evaluate the physical implications of the bianisotropic nature of the SRR particle, we have analyzed both the anisotropic NMPM proposed in Ref. 3 and the twodimensional left-handed medium proposed in Ref. 4. The first medium consists of a number of identical SRR particles printed on a dielectric slab 共relative dielectric permittivity ⑀ r and thickness t) and arranged in a cubic lattice with spacing a. The left-handed material4is formed by placing between the SRR particles of the above medium wires of 共equivalent兲 infinite length, which are parallel to the yaxis of these particles 关see the inset in Fig. 2共c兲in Ref. 4兴. The wire medium behaves as an anisotropic plasma, with ⑀ yy⫽ ⑀ 0(1 ⫺ ␻ p 2/ ␻ 2), ␻ pbeing the plasma frequency.1,4 Neglecting losses and taking into account the constraints imposed by the reciprocity theorem,13 the constitutive relations for these media can be written as D⫽ ⑀ 0共1⫹ ␹ ញ e兲•E⫺i 冑 ⑀ 0 ␮ 0 ␬ ញ •H,共13兲 B⫽i 冑 ⑀ 0 ␮ 0 ␬ ញ T•E⫹ ␮ 0共1⫹ ␹ ញ m兲•H,共14兲 where, accordingly to Eqs. 共10兲–共12兲, only ␹ eyy , ␹ exx , ␬ yz , and ␹ mzz are different from zero. When losses are neglected all these quantities are real numbers. It can be easily realized that plane transverse electromagnetic 共TEM兲waves can propagate along the xaxis in an homogeneous medium described by the constitutive relations 共13兲and 共14兲共with the aforementioned restrictions兲, provided that Eand Hare polarized along the yand zaxes of the SRR’s, respectively. The wave number of these TEM plane waves is given by kx⫽ ␻ 冑 ␮ zz ⑀ yy⫺ ␮ 0 ⑀ 0 ␬ yz 2,共15兲 which differs from the expressions in Refs. 3 and 4 because of the presence of the bianisotropic term ␬ yz . Note that a very important consequence of Eq. 共15兲is the existence of a forbidden band at those frequencies satisfying ␮ zz ⑀ yy⫺ ␮ 0 ⑀ 0 ␬ yz 2⬍0, 共16兲 while transmission is possible for ␮ zz ⑀ yy⫺ ␮ 0 ⑀ 0 ␬ yz 2⬎0. 共17兲 Using Eq. 共15兲for the TEM wave number instead of the simplified equation kx⫽ ␻ 冑 ␮ zz ⑀ yy 共Refs. 3 and 4兲leads, of course, to quantitatively different results, but also to a meaningfully different qualitative behavior. Assuming that the NMPM has a positive dielectric constant ( ⑀ yy NMPM⬎0) in the whole frequency range of interest, assuming that the composite SRR and wire medium 共i.e., the left-handed material兲 has a negative dielectric constant at the same frequencies ( ⑀ yy LH⬍0), and assuming that the magnetic properties of both media are identical3,4 ( ␮ zz NMPM⫽ ␮ zz LH and ␬ yz NMPM⫽ ␬ yz LH), the simplified relation kx⫽ ␻ 冑 ␮ zz ⑀ yy, which neglects bianisotropy—i.e., the magnetoelectric coupling in the SRR—predicts a forbidden band for the NMPM which exactly coincides with the transmission band for the lefthanded material. The use of Eqs. 共15兲–共17兲, however, predicts a mismatch between the aforementioned frequency bands as far as ␬ yz⫽0. This mismatch would be located at the upper limit of both bands, where ␮ zz approaches to zero 共the lower limit is at the resonance, where ␮ zz→⫺⬁). This mismatch, although apparently ignored in the discussion by the authors of the previously cited papers, is in fact, clearly perceivable in the numerical simulations reported in Fig. 2共c兲 of Ref. 4 and in the experimental curves in Fig. 3. in the same paper. In our opinion, such a mismatch cannot be explained if ␬ yz is neglected in the dispersion relation 共15兲; i.e., it cannot be explained if bianisotropic effects are neglected. Moreover, when numerical simulations are carried out for the same structure and for plane waves propagating in the same direction, but with the electric field polarized along the xaxis of the SRR particle, the rejection band of the NMPM and the transmission band of the left-handed material exactly coincide.5This result can be interpreted by taking into account that the cross polarization ␣ xz em vanishes in the SRR particle and, therefore, the coupling parameter ␬ xz must vanish in the corresponding effective medium. We can thus conclude that the presence of bianisotropy provides an explanation of some, in other way unexplained, qualitative results of the numerical simulations and experiments presented in Refs. 4 and 5. ROLEOFBIANISOTROPYINNEGATIVE... PHYSICAL REVIEW B 65 144440 144440-3 IV. NUMERICAL AND EXPERIMENTAL EVIDENCE OF BIANISOTROPY Once it has been shown that the consideration of the magneto electric coupling 共i.e., the bianisotropy nature of the artificial medium兲can account for some significant qualitative features of the propagation of electromagnetic waves through the NMPM and left-handed materials under study, quantitative agreement between our model and the numerical simulations and experiments in Ref. 4 will be discussed in this section. For this purpose, the polarizabilities 共10兲–共12兲 must be calculated for the SRR particles used in Ref. 4. The p.u.l. capacitances Cand C0have been calculated using the routines reported in Ref. 14. It is not a simple task to develop an analytical model for the total inductance L. Due to this reason, we have deduced the value of Lfrom the experimental value of the resonance frequency for a single particle reported in Ref. 4 (f⫽4.845 GHz) and from Eq. 共4兲. This leads, for the particular case treated in Ref. 4, to the value L⫽3.03 ␮ 0r0„which is of the same order than the value obtained for the inductance of a ring of radius r0made with a wire of radius c/2 共Ref. 12兲:L⫽ ␮ 0r0关ln(16r0/c)⫺2兴 ⫽1.87 ␮ 0r0…. Finally, the effective distance deff has been approximated as deff⯝c⫹d. Using these values in Eqs. 共5兲– 共9兲, the polarizabilities 共10兲and 共11兲have been calculated. Once these polarizabilities are known, the constitutive parameters of the effective medium can be obtained by means of a homogenization procedure. A rough approach is to simply take for the nonzero macroscopic susceptibilities ␹ eyy ⫽ ␣ yy ee/a3, ␹ exx⫽ ␣ xx ee/a3, ␬ yz⫽ 冑 ␮ 0/ ⑀ 0 ␣ zy me/a3, and ␹ mzz ⫽ ␮ 0 ␣ zz mm/a3, where a3is the volume of the unit cell. This approach, which neglects the electromagnetic coupling between the individual atoms of the metamaterial, only qualitatively accounts for the behavior of the analyzed medium 共for which a⫽8 mm). Just approximate qualitative results can be expected from this method, which we will call procedure No. 1. A second scheme, which we will call procedure No. 2, is based on the well-known Lorentz approach for the local fields on the particle: El⫽E⫹P/(3 ⑀ 0) and Hl⫽H ⫹M/3. Results obtained by using this latter approach, whose main guidelines are summarized in Ref. 10, are expected to be closer to the results of the numerical simulation. Nevertheless, it is known that this approach should fail near the resonance frequencies and when applied to dense media, as is the case. Therefore results obtained following any of the procedures above are expected to agree just roughly with the experiments, but procedure No. 2 is expected to be better than procedure No. 1. Figure 2 shows the results obtained following the procedure Nos. 1 and 2 in this paper as well as those obtained from the numerical simulation in Ref. 4. Note that the numerical simulations presented in Ref. 4 were carried out by using a commercial electromagnetic mode solver which, obviously, considers the discrete nature of the metamaterials. Therefore, although bianisotropy is not explicitly incorporated in the macroscopic constitutive relations proposed in Ref. 4, the numerical simulations 共which ignore any a priori hypothesis about the nature of the metameterial兲implicitly will account for its effect, provided it is actually present. Although qualitative and quantitative agreement between our analytical model and the numerical simulations is just as good as can be expected from the rough approximations we have used, it is physically meaningful to predict a gap of frequencies, ␦ ␻ , for which propagation is not possible neither in the NMPM nor in the left-handed metamaterial. The gap appears in both analytical ( ␦ ␻ 1and ␦ ␻ 2) and numerical results ( ␦ ␻ 关4兴兲 and is highlighted in Fig. 2. This gap is clearly perceivable too in the experimental results depicted in Fig. 3 of Ref. 4兲. For comparison purposes, the measured 3 dB passband of the left-handed metamaterial, ⌬ ␻ expt , and the 3 dB forbidden frequency gap, ␦ ␻ expt , for both the NMPM and left-handed metamaterial have been depicted in Fig. 2. Our rough analytical models reasonably predict the position and order of magnitude of the forbidden frequency gap. This detail of the phenomenon, which is present in both experiment and numerical simulations, cannot be explained by just characterizing the NMPM by means of a negative permeability and/or the left-handed material by means of a simultaneously negative dielectric permittivity and magnetic permeability. However, it is perfectly accounted for by means of the hypothesis of the bianisotropic nature of the material. Moreover, it was already noted that, for the case of plane waves propagating along the yaxis of the SRR’s, with the E and Hfields polarized along the xand zaxes of the SRR’s, respectively, numerical simulations do not predict any mismatch between the NMPM forbidden band and the lefthanded medium passband.5Since the cross polarizations ␣ xz em FIG. 2. Phase advance (kxa) as a function of frequency for the NMPM 共solid lines兲and the left-handed 共dotted lines兲metamaterials reported in Refs. 4 and 5 for a TEM wave polarized with the electric field along the xaxis and the magnetic field along the zaxis of Fig. 1. The dimensions of the SRR’s are as in Fig. 1. a ⫽8 mm. The characteristics of the wire medium can be found in Ref. 4. Curves labeled No. 1, No. 2, and 关4兴show the results obtained following the two analytical methods 共No. 1, No. 2兲reported in this paper and the results of the numerical simulations in Fig. 2共c兲of Ref. 4. The frequency gaps for which transmission is not possible neither in the NMPM nor in the left-handed metamaterial are labeled as ␦ ␻ 1, ␦ ␻ 2, ␦ ␻ 关4兴, and ␦ ␻ expt . The experimental 3 dB transmission band for the left-handed metamaterial is shown as ⌬ ␻ expt . MARQUE ´S, MEDINA, AND RAFII-EL-IDRISSI PHYSICAL REVIEW B 65 144440 144440-4 and ␣ zx em vanish for the SRR particle, it is obvious that our analytical model does not predict a mismatch either. This is an additional confirmation of the hypothesis of bianisotropy as the explanation for the behavior of these media. A final evidence in favor of the theory in this paper is that, for plane waves traveling along the zaxis of the SRR’s, with the fields Eand Hpolarized along the yand xaxes of the particles, numerical simulations predict a narrow transmission frequency band for the left-handed material just below the resonance 关see Fig. 2共d兲in Ref. 4兴. This result is also coherent with Eq. 共9兲. Indeed, this equation predicts a resonance in ␣ yy ee which produces very large and positive values of this polarizability just below the resonance. These high values would result in large and positive values of the dielectric permittivity of the NMPM, which, eventually, would cancel the negative values of the dielectric permittivity of the wire medium, thus giving a positive global dielectric permittivity for the left-handed material. Since the permeability of the NMPM is positive below the magnetic resonance frequency, this results in a narrow passband for the composite lefthanded material at these frequencies. Once again our hypothesis seems to be confirmed by numerical simulation. From the quantitative point of view adopted in this section some discrepancies can be observed between our model and the numerical simulations and experiments reported in Ref. 4, which deserve some comments. Thus, in numerical simulations the frequency gap is of about 100 MHz, whereas the analytical model predicts a gap of about 50–60 MHz. Moreover, the graphics in Fig. 2, although similar, show a systematic displacement in frequency. These discrepancies between the numerical simulations and the analytical model might be due to the limitations of the homogenization procedure and/or the numerical simulations or to the presence of spatial dispersion, which cannot be taken into account by the assumed local constitutive relations 共13兲and 共14兲. Some degree of inaccuracy in the numerical simulations cannot be discarded, since the location of the experimental pass and forbidden bands shown in Fig. 2 is better reproduced by the analytical models in this paper than by the numerical simulations. Note that the bandwidths calculated from the analytical models are smaller than the experimental bandwidths. This discrepancy could be attributed to Ohmic losses, which are not taken into account in our model. V. AVOIDING BIANISOTROPY Since bianisotropy might be an undesired property of the medium, a method to eliminate it could be of interest. There is a slight modification of the SRR particle which would eliminate the magnetoelectric coupling of the SRR particle 共and then the bianisotropy of the artificial medium兲while keeping all its other interesting features. This modification consists in replacing one of the rings 共the internal ring, for instance兲by another ring located just behind the external ring, at the opposite side of the dielectric substrate, with the slits still placed at opposite sides. This modification of the SRR particle is sketched in Fig. 3. The electromagnetic behavior of this modified SRR 共MSRR兲should be in many essential aspects similar to that of the SRR shown in Fig. 1. In particular, when a time-harmonic varying external magnetic field is applied along the zaxis of the particle, it will induce—by Faraday’s law—electric currents on the metallic rings with a behavior described by Eqs. 共1兲–共4兲, where now Lshould be the total inductance of the MSRR and Cthe p.u.l. capacitance between the two opposite metallic strip rings. Therefore, the MSRR shows a magnetic polarizability also given by Eq. 共5兲with the aforementioned new interpretation for Land C. However, as can be easily seen by inspection of Fig. 3, the electric charge is distributed on both rings of the MSRR in such a way that no net electric polarization is produced. In fact, as is sketched in Fig. 3, the electric polarization of the upper half side (y⬎0) of the MSRR must be just the opposite of the polarization of its lower half side (y⬍0), in order to allow the field displacement current to close the ohmic current lines. That is, the MSSR is not a bianisotropic particle. Apart from the absence of magnetoelectric coupling, the MSRR presents an interesting additional advantage. Since the p.u.l. capacitance between the broadside coupled metallic strips of the MSRR can be made considerably higher than the p.u.l. capacitance between the edge coupled split rings 共by simply widening the strips and/or by using a very thin dielectric slab and/or increasing its dielectric permittivity兲,it is expected that the resonant frequency 共4兲could be meaningfully reduced 共the total size of the particle remaining unchanged兲with respect to the SRR configuration. This will result in a smaller overall electrical size of the particle at the frequencies of operation of the resulting NMPM and/or lefthanded metamaterials. This is a very important aspect if the artificial discrete medium has to be described as an effective continuous medium at microwave frequencies. FIG. 3. The modified split ring resonator 共MSRR兲proposed as an alternative to the conventional SRR in order to avoid bianisotropic effects. ROLEOFBIANISOTROPYINNEGATIVE... PHYSICAL REVIEW B 65 144440 144440-5 VI. CONCLUSIONS The recently reported artificial negative magnetic permeability media and left-handed metamaterials have been revisited at the light of the theory of bi共iso/aniso兲tropic artificial media. The conclusions obtained from this analysis complement the results of the aforementioned previous works, explaining some relevant features of the electromagnetic behavior of such media. It has been highlighted that the electromagnetic behavior of artificial bianisotropic media, NMPM, and left-handed metamaterials, made with resonant metallic inclusions in a host uniform medium, present noticeable similarities. In particular, the bianisotropic characteristics of recently reported NMPM and left-handed metamaterials have been investigated. An analytical model, accounting for magnetoelectric coupling, has been proposed for the split ring resonator 共SRR兲, which is the elementary atom of the aforementioned NMPM and left-handed metamaterials. That coupling is responsible for the bianisotropic behavior of the equivalent continous medium consisting of an aggregate of those particles. Considering bianisotropy, some up-to-date unexplained features of the electromagnetic waves propagating through those media can be explained. In particular, it has been shown that the transmission and forbidden frequency bands for those materials can be more adequately described accounting for bianisotropy. Finally, a new modified split ring resonator, which does not present bianisotropic effects, has been proposed. This new MSRR could be useful in the design of new nonbianisotropic NMPM and lefthanded metamaterials. ACKNOWLEDGMENTS The authors thank Professor D.R. Smith for kindly providing all required information about his calculations and experiments. This work was supported by the Comisio ´n Interministerial de Ciencia y Tecnologı ´a, Spain, under Project No. TIC2001-3163 and by Junta de Andalucı ´a. *Electronic address: [email protected] †Electronic address: [email protected] 1J. Pendry, A. Holden, W. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 共1996兲. 2J. Pitarke, F.G. a Vidal, and J. Pendry, Phys. Rev. B 57,15261 共1998兲. 3J. Pendry, A. Holden, D. Robbins, and W. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 共1999兲. 4D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 共2000兲. 5D. Smith, W. Padilla, D. Vier, R. Shelby, S. Nemat-Nasser, N. Kroll, and S. Schultz, in Photonic Crystals and Light Localization in the 21st Century, Proceedings of the NATO–ASI Conference on Photonic Crystals and Light Localization, Crete 共Greece兲, June 18–30, 2000, edited by Costas M. 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