Towards a systematic design of isotropic bulk magnetic metamaterials using the cubic point groups of symmetry

Towards a systematic design of isotropic bulk magnetic metamaterials using the cubic point groups of symmetry J. D. Baena,*L. Jelinek,†and R. Marqués‡ Departamento de Electrónica y Eletromagnetismo, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain 共Received 8 May 2007; revised manuscript received 27 September 2007; published 17 December 2007兲 In this paper, a systematic approach to the design of bulk isotropic magnetic metamaterials is presented. The roles of the symmetries of both the constitutive element and the lattice are analyzed. For this purpose, it is assumed that the metamaterial is composed of cubic split ring resonators 共SRRs兲arranged in a cubic lattice. The minimum symmetries needed to ensure an isotropic behavior are analyzed, and some particular configurations are proposed. Besides, an equivalent circuit model is proposed for the considered cubic SRRs. Experiments are carried out in order to validate the proposed theory. We hope that this analysis will pave the way to the design of bulk metamaterials with strong isotropic magnetic response, including negative permeability and left-handed metamaterials. DOI: 10.1103/PhysRevB.76.245115 PACS number共s兲: 42.70.Qs, 41.20.Jb, 42.25.Bs I. INTRODUCTION Metamaterials are artificial media exhibiting exotic electromagnetic properties not previously found in nature. Among them, media showing simultaneously negative electric permittivity and magnetic permeability in some frequency range, or “left-handed” metamaterials, are of particular interest. The striking properties of left-handed metamaterials, including backward-wave propagation, negative refraction, and inverse Cerenkov and Doppler effects were first reported by Veselago1in 1968. However, the realistic implementations of left-handed metamaterials came several decades later, as a combination of split ring resonators 共SRRs兲and metallic wires.2SRRs are small planar resonators exhibiting a strong magnetic response, which were proposed in 1999 by Pendry et al.3as suitable “atoms” for the development of negative magnetic permeability metamaterials. One year later, Smith et al. demonstrated the possibility of making up a left-handed medium by periodically combining metallic wires—which provide an effective negative permittivity at microwaves4—and SRRs.2In subsequent works, other SRR designs were proposed,5–8in order to reduce electrical size and/or cancel the bianisotropic behavior of the original Pendry’s design. However, all the aforementioned implementations of negative permeability and lefthanded metamaterials are highly anisotropic—or even bianisotropic5—providing only a uniaxial resonant magnetization, while isotropy is needed for many interesting applications of metamaterials, as, for instance, the “perfect lens” proposed by Pendry.9 The aforementioned implementations are, in fact, a combination of two separate systems, one providing the negative magnetic permeability 共the SRR system兲and the other providing the negative electric permittivity 共the wire system兲. How both subsystems can be combined in order to obtain a new system, whose electromagnetic properties were mainly the superposition of the magnetic and the electric properties of each subsystem, is an interesting and controversial issue10,11 that is, however, beyond the scope of this paper. In what follows, we will assume that it is possible to find some combination of two isotropic subsystems, one made of metallic wires 共or other elements providing a negative electric permittivity兲and the other made of SRRs, whose superposition gives a left-handed metamaterial, and will focus our attention on the design of isotropic systems of SRRs. Actually, since isotropic media with negative magnetic permeability are not found in nature, an isotropic system of SRRs providing such property in some frequency range will be an interesting metamaterial by itself. These metamaterials could provide the dual of negative electric permittivity media, with similar applications 共in imaging,9for instance兲. They would be also of interest for magnetic shielding and other practical applications. A first attempt to design an isotropic magnetic metamaterial was carried out by Gay-Balmaz and Martin,12 who designed a spherical magnetic resonator—formed by two SRRs crossed in right angle—which is isotropic in two dimensions. This result was later generalized in Ref. 13, where a fully isotropic spherical magnetic resonator was proposed. However, from a practical standpoint, it is usually easier to work with cubic designs. A first attempt on such direction was made by Simovski and co-workers in Refs. 14–16, where cubic arrangements of planar SRRs and omega particles were proposed 关see Figs. 1共a兲and 1共b兲兴. If only the magnetic and/or electric dipole representations of the SRRs and/or omega particles are considered, these arrangements are invariant under cubic symmetries. However, it has been shown13,17 that this invariance is not enough to guarantee an isotropic behavior since couplings between the planar resonators forming the cubic arrangement can give rise to an anisotropic behavior, even if its dipole representations suggest an isotropic design. The first isotropic metamaterial design fully invariant under the whole group of symmetry of the cube was proposed and simulated in Ref. 18. It is formed by volumetric square SRRs with four gaps, in order to provide 90° rotation symmetries about any of the cube axes. However, this design is unfortunately very difficult to implement in practice because it cannot be manufactured by using standard photoetching techniques, as previous SRR designs,2,3,5–8,13–17 and the gaps of the SRR have to be filled with a high relative permittivity dielectric 共about 100兲. The idea of using spatial symmetries to design isotropic metama- PHYSICAL REVIEW B 76, 245115 共2007兲 1098-0121/2007/76共24兲/245115共14兲©2007 The American Physical Society245115-1 terials was further developed in Refs. 13,17, and 19 leading to the structures depicted in Figs. 1共c兲and 1共d兲. A second group of attempts to design isotropic metamaterials is developed in Ref. 20 and 21. In these works, lattices of dielectric and/or paramagnetic spheres with very high refractive index are proposed. If the refractive index of the spheres is high enough, the internal wavelength becomes small with regard to the macroscopic wavelength, and Mie resonances of the spheres can be used to produce the negative effective permittivity and/or permeability. Since the metamaterial “atoms” are spheres, the isotropy is ensured by simply placing them in a cubic lattice. However, practical difficulties to implement such proposals are not easy to overcome. First of all, lossless media with the very high refractive index needed for the spheres are difficult to obtain. Secondly, the system has a very narrow band.21 All the previously reported proposals for isotropic magnetic metamaterial design use a “crystal-like” approach. That is, they are based on the homogenization of a system of magnetic resonators which, according to causality laws, exhibit a strong diamagnetic response above resonance. There is, however, another approach widely used in the microwave community which is based on the transmission line analogy to effective media. Initially proposed for two-dimensional metamaterial design,22 it was recently generalized to threedimensional isotropic structures.23–26 The main advantage of this approach is its broadband operation, since no resonators are necessary for the design. However, it also presents disadvantages with regard to crystal-like approaches. The transmission line approach to metamaterials does not seem to be applicable beyond the microwave range, whereas a significant magnetic response of the SRR has been shown in the terahertz range and beyond.27,28 In addition, the coupling to free space of the reported transmission line metamaterials seems to be difficult and sometimes needs an additional specific interface 共e.g., an antenna array25兲, whereas this coupling is direct in crystal-like metamaterials. Finally, regarding isotropic left-handed metamaterial design, it should be mentioned that some recent proposals based on random arrangements of chiral particles29,30 have the advantage of providing simultaneously both electric and magnetic negative polarizabilities. This approach can be straightforwardly extended to the design of SRR magnetic metamaterials, by simply considering random arrangements of such elements. There is, however, a major difficulty with this approach: the constitutive elements in a random composite have to be very small in comparison with the macroscopic wavelength to show a true statistical behavior, but it is not easy to design a SRR much smaller than one-tenth of the wavelength. Due to this fact, periodic arrangements will be considered in what follows. The main aim of this paper is to present a systematic approach to the design of metamaterial structures based on periodic arrangements of SRRs. The first section is focused on the spatial symmetries which are necessary to ensure an isotropic behavior in the metamaterial. Cubic arrangements of SRRs placed on cubic lattices are considered, and the minimum symmetry requirements for both the individual resonators and the lattices are investigated. The second section is devoted to a deeper analysis of the isotropic cubic SRRs forming the basis of the crystal structure. In the third section, an equivalent circuit model for such cubic SRRs is developed and applied to some specific examples. The fourth section is focused on the experimental verification of the analysis developed in the previous ones. Finally, the main conclusions of the work are presented. II. ROLE OF CUBIC SYMMETRIES Let us assume that constitutive elements and the unit cell of the material are much smaller than the operating wavelength. In such a case, the interaction of electromagnetic field with the material is described by means of constitutive relations. Besides, the material is supposed to be linear, so the most general way to express those relations between electromagnetic intensities and electromagnetic flux densities is31 D= ␧ ·E+ ␰ ·H, B= ␨ ·E+ ␮ ·H,共1兲 where ␧ , ␮ are second rank constitutive tensors and ␰ , ␨ are second rank constitutive pseudotensors. In order to get a macroscopic isotropic behavior, all constitutive tensors and pseudotensors ␧ , ␮ , ␰ , and ␨ must become scalars or pseudoscalars. Let us now address the problem of forcing the tensors 共or pseudotensors兲in Eq. 共1兲to be scalars 共or pseudoscalars兲for the specific case of a periodic structure. It is well known32,33 that there are 32 symmetry point groups for periodic crystals which can be classified in 7 crystallographic systems. It is also known that the cubic system is the only one that forces any second rank tensor 共or pseudotensor兲to be a scalar 共or a pseudoscalar兲.33 Since any material satisfying the linear constitutive relations 关Eq. 共1兲兴 and being invariant under the cu- FIG. 1. Cubic constitutive elements for isotropic metamaterial design. Cubes 共a兲and 共b兲were studied in Refs. 14–16. Their hidden faces are arranged in such a way that the cube satisfies the central symmetry to avoid magnetoelectric coupling. Cubes 共c兲and 共d兲 were proposed in Ref. 13 as truly three-dimensional 共3D兲isotropic cubic resonators. BAENA, JELINEK, AND MARQUÉS PHYSICAL REVIEW B 76, 245115 共2007兲 245115-2 bic symmetries exhibits an isotropic macroscopic behavior, this section will be focused on the analysis of such cubic symmetries. It is clear that any structure invariant under all the symmetry transformations of the cube must be isotropic, as already proposed by Koschny et al.18 Furthermore, the full symmetry group of the cube contains four different subgroups also belonging to the cubic system and, thus, providing an isotropic macroscopic behavior. Since a less symmetric design is subjected to less structural constraints, it may be guessed that using these subgroups—instead of the whole symmetry group of the cube—may have practical advantages. Keeping this in mind, we will first give a short overview on the five cubic point groups. Next, we shall connect these point groups with some real structures made of planar resonators commonly used in metamaterials. This will be done in two parts: the study of the symmetries of the constitutive element, or the basis, and the analysis of the suitable periodic arrangements, or the lattice. At the end of the section some practical isotropic structures will be specifically analyzed. A. Cubic point groups The five cubic point groups are schematically represented in Fig. 2. Following Schöenflies’ notation and ordering by degree of symmetry, these groups and their generators are as follows: 共1兲T=具兵1,4x·4y,4y·4x其典=proper rotations of the regular tetrahedron 共12 operations兲; 共2兲Th=具兵1,−1,4x·4y,4y·4x其典=Texpanded by the inversion 共24 operations兲; 共3兲Td=具兵1,−2x,4x·4y,4y·4x其典=proper and improper rotations of the regular tetrahedron 共24 operations兲; 共4兲O=具兵1,4x,4y其典=proper rotations of the cube 共24 operations兲; 共5兲Oh=具兵1,−1,4x,4y其典=full symmetry group of the cube 共48 operations兲. We have used a widely used notation for symmetry transformations, 1being the identity operator, −1the inversion, npan n-fold rotation axis about the paxis, and −npthe n-fold axis about the paxis followed by the inversion. For example, the operator −2xis the rotation through 180° about the xaxis followed by the inversion. B. Cubic basis In order to simplify the problem, the symmetries can be separately imposed on the basis and the lattice of the structure. For the sake of simple fabrication, we will assume that the basis is formed by six planar resonators placed over the faces of an inert rigid cube, as in Fig. 1. If the crystal was diluted enough, then the coupling between two neighboring cubes would be much weaker than the coupling between the six SRRs of the same cube and thus each cube could be seen as a single cubic resonator 共CR兲electromagnetically coupled to others. Such consideration implies that the interaction between the CRs forming the material can be described by dipole-dipole interactions, higher order multipole interactions being negligible. In such approximation, all the CRs are properly described by second rank polarizability tensors connecting the external field, Eext and Bext, with the dipolar moments, pand m, induced in the CRs,31,34 p= ␣ ee ·Eext + ␣ em ·Bext, m= ␣ mm ·Bext − ␣ em t·Eext,共2兲 where ␣ ee, ␣ mm, and ␣ em are the electric, magnetic, and magnetoelectric polarizability tensors, and the superscript t means transpose operation. The constitutive tensors in Eq. 共1兲can be derived from these polarizabilities and from the lattice structure by applying a homogenization technique. In what follows, different kinds of CRs will be named by its cubic group symmetry followed by the acronym CR 共group-CR兲. In order to design an isotropic CR, we have to find suitable planar resonators and place them correctly over the cube so as to fulfill the necessary symmetries. Obviously, the planar resonators have to be invariant under certain symmetry transformations of the square. To classify all different possibilities, a list of the symmetry subgroups of the square is shown in Table I, as well as their geometrical representations, and some examples of planar resonators commonly used in metamaterial design and obeying these symmetries. This table also provides a systematic terminology for planar resonators by using the symbol of the symmetry group followed by the term SRR 共group-SRR兲. In what follows, we will use the term SRR in a general sense covering any type of geometry derived from the SRR and the omega particle. By direct inspection on Fig. 2, it can be seen that any of the five cubic point groups contains three twofold rotation axes 共180° rotations兲parallel to the edges of the cube. Thus, only resonators belonging to the last five rows of Table Iare appropriate for designing isotropic CRs. At this point, it may be worth mentioning that Pendry’s SRRs3as well as Omega particles35 are not appropriate for such purpose because they correspond to the C1-SRR and D1-SRR topologies. In summary, in order to get an isotropic CR, we have to choose six identical SRRs pertaining to the classes C2-, D2-, C4-, or D4-SRR and arrange them according to one of the cubic point groups T,Td,Th,O,orOhshown in Fig. 2. Although all five cubic point groups mentioned above are equally useful to achieve isotropic CRs, a specific choice may strongly affect the properties of an isotropic metamaterial. For instance, using isotropic CRs of low symmetry may FIG. 2. Objects with the symmetries of the five cubic point groups. TOWARDS A SYSTEMATIC DESIGN OF ISOTROPIC BULK…PHYSICAL REVIEW B 76, 245115 共2007兲 245115-3 be quite advantageous since the electrical size of the CRs can be made smaller. This fact can be justified in terms of the LC circuit models for the SRRs5–8because the effective capacitances of low symmetry SRRs are usually higher than those of high symmetry SRRs,8thus providing a smaller resonance frequency. Following these considerations, the best choice of basis would be a T-CR made of six planar resonators of the C2-SRR type. A good candidate among all possibilities is the cube shown in Fig. 1共c兲made of six nonbianisotropic SRRs 共NB-SRRs兲,8,36 a configuration already proposed in Refs. 13 and 17. Furthermore, it was shown in Ref. 13 that this configuration shows a bi-isotropic behavior, due to the lack of inversion symmetry of the cubic arrangement. However, sometimes, an effective isotropic medium without bianisotropy 共 ␰ , ␨ =0兲is desired. Since ␰ and ␨ are pseudotensors, the invariance of the CR under inversion is required in order to avoid such property. In this case, the lowest symmetry group is the Thgroup. A CR invariant under the last group of symmetry can be made by using planar resonators of the D2-SRR type as, for instance, the symmetric SRR37 or the modified double-slit broadside coupled SSR 共BC-SRR兲 shown in Fig. 1共d兲.13 However, as will be shown in the following, such symmetry requirements can be relaxed if the lattice symmetries are properly chosen. C. Cubic lattices Above findings give precise instructions for choosing suitable geometries for isotropic metamaterial constitutive elements. The next step is to create an isotropic metamaterial with these elements. The cubic shape of the considered constitutive elements suggests that the best periodical arrangements are the simple cubic 共sc兲, body centered cubic 共bcc兲, and face centered cubic 共fcc兲lattices shown in Fig. 3. All these lattices obey the full symmetry group of the cube, Oh. Therefore, the whole metamaterial 共lattice plus basis兲retains the cubic point group symmetries and the macroscopic isotropic behavior. Although all previously mentioned lattices can provide isotropic metamaterials, it is convenient to look deeply into the possible structures because some particular choices may offer interesting advantages. Regarding Fig. 3,ais the edge size of the CR and bis the edge size of the cubic unit cell. In order to describe CR interactions as dipole-dipole interactions, bmust be chosen much larger than a, so that the metamaterial properties can be deduced from Eq. 共2兲and the appropriate homogenization procedure. However, usually, we are also interested in a high density of dipoles in order to get a strong electromagnetic response. Therefore, bshould TABLE I. Classification of SRR types based on the symmetry subgroups of the square. The second column shows Schöenflies’ notation and the generator of groups. The symbols of transformations are 1 =identity; 4=90° rotation; 2=180° rotation; −4=−90° rotation; mx,my=line reflections respect to the xand yaxes, respectively; mx,y,mx−y=line reflections respect to both diagonals of the square. Each group is schematically represented by the objects in the second column which can be replaced by the planar resonators shown in the third column. SRR types Symmetry subgroups of the square Geometrical representation Examples of resonators C1-SRR C1={1} D1-SRR D1x={1,mx} D1y={1,my} D1,x,y ={1,mx,y} D1,x,-y ={1,mx,-y} C2-SRR C2={1,2} D2-SRR D2x=D2y={1,mx,my,2} D2xy =2xy D={1,mx,y,mx,-y,2} C4-SRR C4={1,4,2,-4} D4-SRR D4={1,4,2,-4,mx,my,mx,y,mx,-y} BAENA, JELINEK, AND MARQUÉS PHYSICAL REVIEW B 76, 245115 共2007兲 245115-4 be as small as possible. However, decreasing bmay lead to a failure of the aforementioned homogenization procedure. However, in any case, the combination of a basis and a lattice with the appropriate symmetries will provide an isotropic metamaterial, regardless of the homogenization procedure. Finally, there are some practical limitations to the values that aand bcan reach as, for instance, the obvious inequality b艌a, derived from the fact that CRs are supposed to be impenetrable. Additional limitations appear for each specific structure. In the case of a sc lattice with T-, Td-, or O-CRs, the lack of inversion symmetry implies that opposite sides of a CR are not oriented in the same way. Thus, the constrain b⬎ais necessary in order to avoid a mutual short circuit between the SRRs of neighboring CRs. To allow the minimum distance b=a, the noncentrosymmetric CRs in the sc lattice must be replaced by Th-orOh-CRs, so that the SRRs on contacting sides of neighboring CRs exactly overlap. In the case of a bcc lattice, the contact between corners implies that the inequality b艌amust be fulfilled for any type of CR. Finally, for the fcc lattice, the contact between edges of neighboring CRs establishes the harder condition b艌2a. The particular case of an fcc lattice with the minimum cell size, b=2a, deserves a specific analysis. When Th-or Oh-CRs are used as the basis of the fcc lattice, the structure turns into an sc lattice with the highest possible compactness, i.e., b=a, because the holes between each eight neighboring CRs have the same shape as the CRs forming the basis. The case of an fcc lattice with a T-, Td-, or O-CR basis is even more special and interesting because each hole exactly corresponds with the inversion of the CR of the basis. Therefore, although the basis of the structure is not invariant under inversion, the fcc structure is brought into coincidence with itself by inversion centered at the center of a CR, followed by a translation of length athrough any of the cube axes. Since the wavelength of the signal illuminating the structure is supposed to be much larger than a, the system can be considered as macroscopically invariant under inversion and, therefore, any bi-isotropic behavior must disappear. Thus, we conclude that a very interesting choice in order to obtain an isotropic metamaterial is the fcc lattice with b=2aand with a basis formed by T-CRs 关example in Fig. 1共c兲兴because of its high compactness, non-bi-isotropic macroscopic behavior, and low degree of symmetry. It is worth recalling here that T-CRs have the lowest symmetry among all the possibilities shown in Fig. 2, which helps to reduce the electrical size of the unit cell, as explained above. III. RESONANCES AND POLARIZABILITIES OF CUBIC RESONATORS Until now, only the symmetry of CRs and cubic lattices useful for isotropic periodic metamaterials were analyzed. However, in order to have a complete characterization of the metamaterial, polarizabilities and couplings between individual SRRs must be considered. In dilute crystals, the approach of weak coupling between CRs, but strong coupling between the SRRs of each CR, is valid. Then, the metamaterial characterization involves two separate problems: obtaining the polarizability tensors in Eq. 共2兲for a single CR and applying the appropriate homogenization procedure to obtain the constitutive parameters for the whole structure. For dense packages, the aforementioned approach is not valid since couplings between SRRs of different CRs can be stronger than SRR couplings inside each individual CR. However, even in these cases, the analysis of the isolated CR resonances and polarizabilities still provides useful information on the behavior of the metamaterial. For instance, it allows to elucidate if the coupling between SRRs in a practical low symmetry CR can be neglected or not. In case they could be neglected, all the analysis in Sec. II B would become irrelevant because the SRRs could be substituted by its equivalent dipoles 共as it was assumed in Refs. 14–16兲, without more considerations on the SRR structure. Therefore, the analysis in this section is necessary in order to justify the practical relevance of the analysis developed in Sec. II. Further, in Sec. IV, an experimental validation of this analysis will be provided. Let us assume that the CR size is much smaller than the operating wavelength. Thus, an RLC circuit model is valid for describing the behavior of single Pendry’s SRRs,3as well as for any type of modified SRRs5–8or omega particles.38 Furthermore, if the resonators are not too close 共so that the interaction energies are small with regard to the self-energy FIG. 3. Cubic Bravais’ lattices. Their top views are also depicted for the particular case of b=2a. Black and gray small cubes represent cubic resonators on successive planes. TOWARDS A SYSTEMATIC DESIGN OF ISOTROPIC BULK…PHYSICAL REVIEW B 76, 245115 共2007兲 245115-5 of each SRR兲, then the CR can be considered as six RLC circuits coupled through mutual impedances. The positive directions of the electric currents on each loop are arbitrarily defined in Fig. 4. The relation between currents and electromotive forces exciting the CR can be written as Z·I=F,共3兲 where Zisa6⫻6 square impedance matrix, Iis a column matrix whose ith component is the current flowing over the ith SRR, and Fis a column matrix whose ith component is the external electromotive force acting on the ith SRR. The diagonal components of the impedance matrix are the selfimpedances of each SRR, i.e., Zii=R+j ␻ L+1/共j ␻ C兲, with R,L, and Cbeing the resistance, self-inductance, and selfcapacitance of a single SRR.8The nondiagonal components Zij are the mutual impedances between the ith and jth SRRs. From the reciprocity theorem,39 we know that the impedance matrix must be symmetric, i.e., Zij=Zji. This reduces the number of independent elements of Zto 21. This number can be further reduced by applying the geometrical symmetries of the CR, as shown in the next paragraph. The application of any symmetry operation changes the components of Iaccording to the rule I⬘=S·I, where Sis the corresponding operator of symmetry. It is well known that any symmetry operation Sof the cubic point groups can be expressed as some combination of the three orthogonal fourfold rotations and the inversion, whose matrix representations, in the six-dimensional space defined by I, are 4x⬅ 冤 10 01 00 00 0−1 −1 0 010 01 0 冥 , 4y⬅ 冤 00 10 01 010 01 0 0−1 −1 0 00 冥 , 4z⬅ 冤 00−1 −1 0 0 10 01 00 00 10 01 冥 , −1⬅ 冤 01 10 00 001 10 0 00 01 10 冥 .共4兲 They are unitary matrices with the well known property S−1=St. It can be straightforwardly demonstrated that Ffollows the same rule of transformation: F⬘=S·F. Therefore, both Iand Fcan be considered as vectors. In what follows, I and Fwill be called the “current” and the “excitation” vectors, respectively. Therefore, the impedance matrix Zis a second rank tensor, following the transformation rule Z⬘ =S·Z·St. If the CR remains invariant by the transformation S, then Z=S·Z·St.共5兲 This equation gives some relations between the components of Z, which can reduce the number of independent components of Z. Although the current vector Ican be directly solved by multiplying both sides of Eq. 共3兲by Z−1, in order to identify the different resonances of the CR, it is convenient to expand the solution in terms of the eigenvectors of Z. The eigenvalue problem corresponding to Eq. 共3兲is Z·vi=zivi,共6兲 where ziare the eigenvalues, vithe eigenvectors, and the index i=1, ... ,6. The impedance matrix Zcan be expanded in a sum of two terms as Zij = 冉 R+j ␻ L+1 j ␻ C 冊 ␦ ij +Zij共1− ␦ ij兲,共7兲 where ␦ ij is Kronecker’s delta. The first term is the selfimpedance of a single SRR multiplied by the identity, while the second term is the symmetric matrix of mutual impedances. These mutual impedances are purely imaginary numbers since, in the frame of a quasistatic model, they cannot contain a resistive term. Thus, the second term in Eq. 共7兲is a purely imaginary symmetrical matrix. Therefore, its eigenvectors can be chosen in such a way that they form a complete and orthogonal basis that diagonalizes this matrix. Furthermore, since the first summand in Eq. 共7兲is actually a scalar, the eigenvectors of Zij are actually the same as those of Zij共1- ␦ ij兲. Therefore, the eigenvectors of Zcan be chosen in such a way that they form an orthogonal basis for the considered six-dimensional space. Thus, the current and ex- FIG. 4. Definition of the sign of currents in the circuit model for a 3D cubic magnetic resonator. BAENA, JELINEK, AND MARQUÉS PHYSICAL REVIEW B 76, 245115 共2007兲 245115-6 citation vectors can be expanded as a summation of such eigenvectors, I=兺 i 共I·vi兲vi,F=兺 i 共F·vi兲vi.共8兲 By substituting both expressions into Eq. 共3兲and applying Eq. 共6兲,weget I=兺 i F·vi zi vi.共9兲 Therefore, both Fand Ican be expanded in a set of orthogonal modes having mutually proportional excitation and current vectors. From Eq. 共7兲, we can also obtain information about the structure of the eigenvalues zi. These eigenvalues must have the form zi共 ␻ 兲=R+j ␻ L+1 j ␻ C+zc,i共 ␻ 兲=j ␻ L 冋 1− ␻ 0 2 ␻ 2+R j ␻ L +zc,i共 ␻ 兲 j ␻ L 册 ,共10兲 where ␻ 0is the resonance frequency of an isolated SRR 共 ␻ 0 2=1/LC兲, and zc,i共 ␻ 兲the eigenvalues of the second summand in Eq. 共7兲, which are related to the coupling between SRRs. It can be seen in Eq. 共9兲that the ith mode resonates when its eigenvalue approaches zero 共zi⬇0兲. Therefore, the frequency of resonance of the ith mode is given by the relation zi共 ␻ 0,i兲⬇0. If losses and couplings between SRRs are not too strong 共R,zc,iⰆj ␻ L兲, the frequencies of resonance of the CR can be approximated as ␻ 0,i⬇ ␻ 0−zc,i共 ␻ 0兲 2jL .共11兲 In what follows, we will apply this equivalent circuit model to the determination of the resonances and polarizabilities of two CRs made from two well known SRRs: Pendry’s SRR3 and NB-SRR.8 A. Analysis of an anisotropic cube Let us now consider the CR shown in Fig. 1共a兲, made of Pendry’s SRRs. In this section, we are going to get some analytical approximation for its resonances and polarizabilities. Note that the cube possesses inversion symmetry and, thus, magnetoelectric coupling is forbidden, so that ␣ em=0in Eq. 共2兲. It can also be seen by inspection that the considered CR is invariant under the rotation 4y·4x. By applying this spatial symmetry, the impedance matrix is reduced to the form Z= 冤 Z11 Z12 Z13 Z14 −Z14 −Z13 Z12 Z11 Z14 Z13 −Z13 −Z14 Z13 Z14 Z11 Z12 −Z14 −Z13 Z14 Z13 Z12 Z11 −Z13 −Z14 −Z14 −Z13 −Z14 −Z13 Z11 Z12 −Z13 −Z14 −Z13 −Z14 Z12 Z11 冥 ,共12兲 where there are only four independent components. The corresponding eigenvalues and its orthonormal eigenvectors are shown in Table II. It is worth noting that the eigenvectors can be classified in even and odd modes: for even 共odd兲modes, the currents I2n-1 and I2nare parallel 共antiparallel兲. Once the eigenvalue problem is solved, the next step is to write an explicit expression for the excitation vector Fand introduce this expression in Eq. 共9兲, in order to get the currents over the SRRs. To begin with, we will assume that the CR is excited by a homogeneous external magnetic field Bext=共Bx ext,By ext,Bz ext兲, and there is no external electric field. Then, the excitation vector is written as Fm=−j ␻ A共Bx ext,Bx ext,By ext,By ext,Bz ext,Bz ext兲,共13兲 where Ais the effective area of the SRR. By introducing Eq. 共13兲into Eq. 共9兲, the current vector Iis calculated. Finally, the magnetic dipole components of the CR are obtained from mx=共I1+I2兲A,my=共I3+I4兲A, and mz=共I5+I6兲A. The resulting expression for the magnetic polarizability tensor is TABLE II. Eigenvalues and a complete set of orthonormal eigenvectors of the impedance matrix 关Eq. 共12兲兴 corresponding to anisotropic cubic resonators with symmetries −1and 4y·4x, as, for instance, the structures shown in Figs. 1共a兲and 1共b兲. Eigenvalues ziEigenvectors vi 1 2共−1,−1,1,1,0,0兲 Even modes 共magnetic兲 Z11+Z12−Z13−Z14 1 2冑3共1,1,1,1,2,2兲 Z11+Z12+2Z13+2Z14 1 冑6共−1,−1,−1,−1,1,1兲 1 2冑3共1,−1,−2,2,1,−1兲 Odd modes 共electric兲 Z11−Z12−Z13+Z14 1 2共1,−1,0,0,−1,1兲 Z11−Z12+2Z13−2Z14 1 冑6共−1,1,−1,1,−1,1兲 TOWARDS A SYSTEMATIC DESIGN OF ISOTROPIC BULK…PHYSICAL REVIEW B 76, 245115 共2007兲 245115-7 ␣ mm =−j ␻ A22 3 冤 1 Z11 +Z12 −Z13 −Z14 冢 2−11 −121 112 冣 +1 Z11 +Z12 +2Z13 +2Z14 冢 11−1 11−1 −1 −1 1 冣 冥 .共14兲 This magnetic polarizability tensor is anisotropic and exhibits two resonances, at those frequencies where Z11+Z12 −Z13−Z14⬇0orZ11+Z12+2Z13+2Z14⬇0. Only the even resonances of Table II appear in Eq. 共14兲because the excitation vector and the odd eigenvectors are orthogonal, i.e., Fm·vi odd=0 in Eq. 共9兲. Just in the limit of no coupling between SRRs 共Zij=0 for i⫽j兲, both resonances converge to the single SRR resonance and ␣ mm becomes a scalar, as mentioned in Refs. 14–16. However, it will be shown in the Sec. IV that this coupling cannot be neglected in most practical configurations. Apart from the magnetic excitation studied above, it is well known that Pendry’s SRR can be excited by an external electric field.5,6,40 Therefore, an electric response is also expected for this particular CR. The electric excitation of the rings on the cube is sketched in Fig. 5. The external electric field Eext can excite an SRR only if it has a nonvanishing component contained in the plane of the particle and orthogonal to the imaginary line passing through the slits of the rings.5,6,40 Therefore, only two SRRs are excited by each Cartesian component of Eext. Thus, for an external and homogeneous electric field of arbitrary direction, the excitation vector has the form Fe⬀共Ez ext,−Ez ext,−Ex ext,Ex ext,−Ey ext,Ey ext兲.共15兲 Taking into account the sign of the charges induced over the rings, it is clear that the electric dipole has to be proportional to p⬀ 冢 I3−I4 I5−I6 I2−I1 冣 .共16兲 The proportionality constants of Eq. 共15兲and 共16兲are given in the Appendix, Eqs. 共A2兲and 共A9兲. Finally, from Eq. 共A9兲 of the Appendix, we get the following analytical formula for the electric polarizability tensor: ␣ ee =32deff 2 3j ␻␲ 2 冤 1 Z11 −Z12 −Z13 +Z14 冢 −21−1 1−2−1 −1 −1 −2 冣 +1 Z11 −Z12 +2Z13 −2Z14 冢 −1 −1 1 −1 −1 1 11−1 冣 冥 ,共17兲 where deff is an effective distance between the metal strips on each SRR.5,6Clearly, this electric polarizability tensor is anisotropic and exhibits two resonances, at those frequencies where Z11−Z12−Z13+Z14⬇0orZ11−Z12+2Z13−2Z14⬇0. Only the odd resonances of Table II appear in Eq. 共17兲be- cause the excitation vector and the even eigenvectors are orthogonal, i.e., Fe·vi even=0 in Eq. 共9兲. In summary, it has been shown that the coupling between the faces of the CR made of Pendry’s SRRs shown in Fig. 1共a兲splits the original resonance of a single SRR in four new resonances. Besides, both magnetic and electric polarizability tensors are anisotropic, as can be seen from Eqs. 共14兲and 共17兲. Finally, it is worth mentioning that the even modes can also be called magnetic modes because they have a resonant magnetic moment and can be only excited by an external magnetic field but not by an external electric field. Similarly, the odd modes are electric modes because they present a resonant electric dipole, which can be only excited by an external electric field. The reported conclusions are quite relevant for our analysis because they show that a cubic arrangement of Pendry’s SRRs will not be only anisotropic but it will also show several different resonances around the isolated SRR resonance, thus destroying any possibility of a single-resonance Lorentzian behavior of the metamaterial. B. Analysis of an isotropic cube It was already shown in Sec. II B that, in order to ensure an isotropic behavior, the CR has to be invariant at least under the tetrahedron symmetry group T=兵具1,4x·4y,4y·4x典其. The T-CR shown in Fig. 1共c兲, made of six NB-SRRs,8,13 is a good example of particle obeying this symmetry. By using the symmetry transformations and the rule 关Eq. 共5兲兴, its impedance matrix can be significantly reduced to Z= 冤 Z11 Z12 Z13 −Z13 Z13 −Z13 Z12 Z11 −Z13 Z13 −Z13 Z13 Z13 −Z13 Z11 Z12 Z13 −Z13 −Z13 Z13 Z12 Z11 −Z13 Z13 Z13 −Z13 Z13 −Z13 Z11 Z12 −Z13 Z13 −Z13 Z13 Z12 Z11 冥 ,共18兲 where there are just three independent components. All eigenvalues and a complete set of orthonormal eigenvectors of FIG. 5. Electric excitation of the cubic resonator made of Pendry’s SRRs. BAENA, JELINEK, AND MARQUÉS PHYSICAL REVIEW B 76, 245115 共2007兲 245115-8 this matrix are shown in Table III. If we compare Tables III and II, we immediately find some similarities. In both cases, the eigenvectors can be classified into even and odd types, and the eigenvalues in Table III can be obtained from those of Table II by making Z14=−Z13. Furthermore, the odd and even subspaces are kept, and only the two subspaces of the even eigenvectors are unified into a single subspace due to the eigenvalue degeneration induced by the additional rotation symmetry 4x·4y. Let us now analyze the resonances and polarizabilities of the T-CR by following the procedure developed in the previous subsection. By considering a homogeneous external magnetic excitation, the corresponding excitation vector 关Eq. 共13兲兴 can only excite the even modes, thus leading to the isotropic magnetic polarizability tensor ␣ mm =−2j ␻ A2 Z11 +Z12 冤 100 010 001 冥 ,共19兲 which, in fact, corresponds to the substitution Z14=−Z13 in Eq. 共14兲. The self-impedance in Eq. 共19兲is given by Z11 =R+j ␻ L+共j ␻ C兲−1, with R,L, and Cbeing the resistance, self-inductance, and self-capacitance of a single NB-SRR.8 The mutual impedance in Eq. 共19兲can be approached as Z12⬇j ␻ M12, where M12 is the mutual inductance between two NB-SRRs placed on opposite sides of the CR. This approximation is reasonable since the electric field is concentrated inside the gap of the NB-SRR, while the magnetic field created by a NB-SRR spreads out in space. Using these relations, the frequency of resonance of the CR can be calculated as ␻ res⬇关C共M12+L兲兴−1/2, while the magnetic polarizability tensor takes the form ␣ mm ⬇2 ␻ 2CA2 1− ␻ 2共L+M12兲C+j ␻ RC 冤 100 010 001 冥 .共20兲 This formula shows a Lorentzian-like magnetic response, similar to that of the single planar NB-SRR but isotropic in three dimensions. With regard to the behavior of the considered CR under an external electric excitation, since the NB-SRRs cannot be excited by an external electric field,8the present theory predicts the absence of any resonant response to this excitation 共of course, the CR will exhibit a nonresonant electric polarizability due to the static electric moments induced on each ring, which is of no interest in the frame of the present discussion兲. However, experiments reported in Ref. 13 have shown that the considered CR exhibits a weak magnetoelectric coupling at the same resonant frequency, ␻ res=关C共M12 +L兲兴−1/2,asinEq.共20兲. Therefore, this phenomenon does not affect neither the isotropy nor the single-resonance behavior of the considered CR. Since the tetrahedron symmetry group, T, does not include the inversion transformation, this result is not forbidden for T-CRs. Although the origin of this effect will be qualitatively explained below in Sec. IV, it can be advanced that it is basically due to a second order electric interaction between SRRs, which is ignored in the equivalent circuit approximation developed in this section. Actually, the presence of this second order effect near the CR resonance shows how important the analysis of the spatial symmetries is in order to predict the behavior of metamaterial resonators: it seems that any effect not forbidden by symmetry will actually appear in practice, regardless of the equivalent circuit models. It is worth recalling here that this magnetoelectric coupling disappears if the T-CRs are arranged in an fcc lattice with b=2a, as explained at the end of Sec. II B. IV. EXPERIMENTS For the experimental verification of the theory developed in the previous sections, some anisotropic and isotropic CRs were manufactured. Each CR was inserted into a standard WR430 waveguide, as shown in Fig. 6. The testing procedure starts from the fact that the particle is isotropic if their polarizability tensors are invariant by any rotation. Therefore, all CRs were subjected to several rotations and the transmission coefficient through the waveguide was measured using a network analyzer HP-8510. If the measured CR were isotropic, then the measured transmission coefficient would remain invariant after rotations. Namely, two anisotropic cubes made of Pendry’s SRRs and omega particles and two isotropic cubes made of C2-SRRs 共actually NB-SRRs8兲and C4-SRRs 共see insets in Fig. 7兲were implemented. All SRRs were etched on Arlon 250-LX substrate with dielectric constant ␧r=2.43, loss tangent tan ␦ ⬍0.002, and thickness t=0.49 mm. In order to check the similarity between the SRRs belonging to the same CR, their resonance frequencies were measured by placing TABLE III. Eigenvalues and a complete set of orthonormal eigenvectors of the impedance matrix 关Eq. 共18兲兴 corresponding to isotropic cubic resonators symmetric under the tetrahedron group T=兵具1,4x·4y,4y·4x典其, as, for instance, the structure shown in Fig. 1共c兲. Eigenvalues ziEigenvectors vi 1 冑2共0,0,1,1,0,0兲 Even modes Z11+Z12 1 冑2共0,0,0,0,1,1兲 1 冑2共1,1,0,0,0,0兲 1 2冑3共1,−1,−2,2,1,−1兲 Z11−Z12−2Z13 Odd modes 1 2共1,−1,0,0,−1,1兲 Z11−Z12+4Z13 1 冑6共−1,1,−1,1,−1,1兲 TOWARDS A SYSTEMATIC DESIGN OF ISOTROPIC BULK…PHYSICAL REVIEW B 76, 245115 共2007兲 245115-9