Strong spatial dispersion in wire media in the very large wavelength limit

Strong spatial dispersion in wire media in the very large wavelength limit P. A. Belov,1,2 R. Marque ´s,3S. I. Maslovski,1I. S. Nefedov,1,4 M. Silveirinha,5C. R. Simovski,1,2 and S. A. Tretyakov1 1Radio Laboratory, Helsinki University of Technology, P.O. Box 3000, FIN-02015 HUT, Finland 2Physics Department, St. Petersburg Institute of Fine Mechanics and Optics, Sablinskaya 14, 197101 St. Petersburg, Russia 3Department of Electronics and Electromagnetism, Facultad de Fı ´sica, University of Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain 4Institute of Radio Engineering and Electronics, Russian Academy of Science, Zelyonaya 38, 410019 Saratov, Russia 5Instituto Superior Te ´cnico, Instituto de Telecomunicac¸o ˜ es, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal 共Received 27 November 2002; published 25 March 2003兲 It is found that there exist composite media that exhibit strong spatial dispersion even in the very large wavelength limit. This follows from the study of lattices of ideally conducting parallel thin wires 共wire media兲. In fact, our analysis reveals that the description of this medium by means of a local dispersive uniaxial dielectric tensor is not complete, leading to unphysical results for the propagation of electromagnetic waves at any frequencies. Since nonlocal constitutive relations have been usually considered in the past as a secondorder approximation, meaningful in the short-wavelength limit, the aforementioned result presents a relevant theoretical interest. In addition, since such wire media have been recently used as a constituent of some discrete artificial media 共or metamaterials兲, the reported results open the question of the relevance of the spatial dispersion in the characterization of these artificial media. DOI: 10.1103/PhysRevB.67.113103 PACS number共s兲: 78.20.Ci, 41.20.Jb, 42.70.Qs, 78.70.Gq Causality imposes that all material media must be dispersive. In most cases this behavior results in local dispersive constitutive relations, i.e., in frequency-dependent constitutive permittivity and permeability tensors. Nonlocal dispersive behavior 共i.e., spatial dispersion兲, which results in constitutive operators depending also on the spatial derivatives of the mean fields 共or, for plane electromagnetic waves, on the wave-vector components兲, is usually considered as a small effect, meaningful in the short-wavelength limit. Specifically, spatial dispersion will always appear when the higher-order terms in the series expansion of the constitutive parameters in power series of the dimensionless parameter a/␭(ais the lattice constant of the crystal and ␭the wavelength inside the medium兲are not neglected.1Thus, it is usually assumed that nonlocal dispersive reations are only meaningful when ␭approaches a. The usually weak natural optical activity of some materials is a well-known example of the application of this principle.1When such principle is translated to the analysis of discrete artificial media, also called metamaterials, it would imply that nonlocal dispersive constitutive relations are only expected to be a small refinement of the local constitutive relations usually considered. However, there is at least a counterexample for this assumption. The parallel wire medium is a medium formed by a regular lattice of ideally conducting wires with small radii compared to the lattice periods and the wavelength, see Fig. 1. It has been known in microwave applications for a long time2–4 as an artificial dielectric, also called rodded medium and quasistatic models of the effective permittivity are available.5 Recently, some attention to wire media has been paid also in optics 共e.g., Refs. 6,7兲and in the realization of left-handed media8,9 as composite media made from lattices of long conducting wires and split ring resonators10–12 共see discussions corresponding to that in13兲. The electromagnetic reponse of the specific wire medium shown in Fig. 1 is analyzed in Refs. 3,7 following different approaches. Both analysis are carried out for wave propagation perpendicular to the wires and show that, for electric-field polarization along the wires, the medium is characterized 共if a/␭,b/␭Ⰶ1) by a frequencydependent effective dielectric constant given by ␧⫽␧0 冉 1⫺k0 2 k2 冊 ⫽␧0 冉 1⫺ ␻ 0 2 ␻ 2 冊 .共1兲 The constant ␻ 0共the corresponding wave number k0 ⫽ ␻ 0 冑 ␧0 ␮ 0) in Eq. 共1兲plays the role of an equivalent ‘‘plasma frequency.’’ Thus, this medium is often called ‘‘artificial plasma’’ since the ideal 共collisionless兲electron plasma is described by the same equivalent parameter. If the wires FIG. 1. The geometry of wire media: A lattice of parallel ideally conducting thin wires. PHYSICAL REVIEW B 67, 113103 共2003兲 0163-1829/2003/67共11兲/113103共4兲/$20.00 ©2003 The American Physical Society67 113103-1 are assumed to be very thin, so that their polarization in the direction orthogonal to the wires can be neglected, the effective permittivity for electric-field polarization orthogonal to the wires is ␧0. The aforementioned analysis suggests that this wire medium could be modeled as a uniaxial dielectric with the following local permittivity dyadic: ␧ ¯ ¯ ⫽␧z0z0⫹␧0共x0x0⫹y0y0兲,共2兲 whose permittivity in the axial direction, ␧, would be given by Eq. 共1兲. However, it will be shown in the following that this naive hypothesis leads to unphysical results and must be substituted by a nonlocal dispersive relation. In fact, assuming that the medium can be described by the uniaxial dyadic 共2兲, the dispersion equation for extraordinary plane waves (Ez⫽0) with the wave vector (qx,qy,qz)Tin this uniaxial dielectric reads14,15 ␧0共qx 2⫹qy 2兲⫽␧共k2⫺qz 2兲,共3兲 where k⫽ ␻ 冑 ␧0 ␮ 0is the phase constant of the host matrix. On the other hand, these extraordinary waves correspond to the well-known TM 共to z) set of modes, allowed by the invariance of the boundary conditions along z. Thus, for any extraordinary wave traveling with a phase constant qzalong the zaxis, the Ezfield must satisfy the Helmholtz equation 再 ⳵ ⳵ x2⫹ ⳵ ⳵ y2⫹共k2⫺qz 2兲 冎 Ez⫽0, 共4兲 with the boundary condition Ez⫽0 on the wires. It is clear from this equation that any ‘‘plane’’extraordinary wave must satisfy k共qx,qy,qz兲⫽ 冑 k2共qx,qy,0兲⫹qz 2.共5兲 This last result is incompatible with Eq. 共1兲–共3兲, as can be easily seen by substitution of Eq. 共1兲into Eq. 共3兲. However, if we choose ␧共k,qz兲⫽␧0 冉 1⫺k0 2 k2⫺qz 2 冊 共6兲 instead of Eq. 共1兲, then Eqs. 共2兲and 共3兲become compatible with Eq. 共5兲, giving the following dispersion equation for the plane wave: q2⬅qx 2⫹qy 2⫹qz 2⫽k2⫺k0 2,共7兲 where we have assumed that qz⫽k共the case with qz⫽kwill be analyzed at the end of this paper兲. The above rationale suggests that the considered wire media still can be described by the permittivity dyadic 共2兲, but the axial permittivity ␧ must be a nonlocal parameter of the form 共6兲. The conventional expression 共1兲would be only a particular case of Eq. 共6兲, valid for wave propagation in the x-yplane. The main difference between the local uniaxial model, Eq. 共1兲, and the proposed nonlocal model, Eq. 共6兲, for the parallel wire medium is that the nonlocal model predicts a stop band 共at frequencies below ␻ 0⫽k0/ 冑 ␧0 ␮ 0) for extraordinary waves propagating along any direction in the media. On the contrary, Eqs. 共1兲–共3兲predict propagation of extraordinary waves at any frequency provided qz⬎k⫽ ␻ 冑 ␮ 0␧0. Thus, both models predict qualitatively very different behaviors, even near the cutoff plasma frequency ␻ 0where q2 →0共i.e., a/␭→0). That is, the nonlocality of the proposed constitutive relations affects the electromagnetic response of the medium even in the very large wavelength limit, thus being important for any values of the a/␭ratio inside the medium. Other relevant differences between the predictions of both models will be developed along this paper. The rigorous proof of Eq. 共6兲is based on the local-field approach which is described in detail in Ref. 16. In Ref. 16 the low-frequency stop band of the wire medium has been analyzed, as well as its high-frequency band-gap structure. This analysis reveals that, in the thin wire medium 共Fig. 1兲 and for qz⫽k, two sets of modes can propagate: ordinary 共with Ez⫽0) and extraordinary 共with Ez⫽0) waves. The ordinary waves do not interact with the wires and propagate in the host media. For extraordinary waves, an explicit dispersion equation connecting the wave vector q ⫽(qx,qy,qz)Twith the wave number of the host isotropic matrix k⫽ ␻ 冑 ␧0 ␮ 0has been derived in Ref. 16. It can be written as 1 ␲ ln b 2 ␲ r0 ⫹1 bkx (0) sinkx (0)a coskx (0)a⫺cosqxa ⫹兺 n⫽0 冉 1 bkx (n) sinkx (n)a coskx (n)a⫺cosqxa⫺1 2 ␲ 兩 n 兩 冊 ⫽0. 共8兲 Here kx (n)denotes the xcomponent of nth Floquet mode wave vector: kx (n)⫽⫺j 冑 冉 qy⫹2 ␲ n b 冊 2 ⫹qz 2⫺k2,Re 兵 冑 共兲 其 ⬎0. 共9兲 The other notations are clear from Fig. 1. Numerical solution of this dispersion equation shows that there exists a lowfrequency stop band for all propagation directions 共except for the particular case of qz⫽kthat will be analyzed later兲. Let us simplify the dispersion equation 共8兲for the quasistatic case a,bⰆ ␲ /k. Using the Taylor expansion of sin(x) and cos(x) functions for small arguments we obtain Eq. 共7兲with k0 2⫽2 ␲ /s2 ln s 2 ␲ r0 ⫹F共r兲 ,共10兲 where s⫽ 冑 ab,r⫽a/b, and F共r兲⫽⫺ 1 2lnr⫹兺 n⫽1 ⫹⬁ 冉 coth共 ␲ nr兲⫺1 n 冊 ⫹ ␲ r 6.共11兲 Therefore, we have shown the suitability of the suggested approach for the description of the wire medium, with k0 given by Eq. 共10兲. Parameter k0here plays the role of the BRIEF REPORTS PHYSICAL REVIEW B 67, 113103 共2003兲 113103-2 wave number corresponding to the plasma frequency. More exactly, it indicates the upper edge of the low-frequency stop band. Naturally, k0as a function of two lattice periods aand bis a symmetric function: k0(a,b)⫽k0(b,a). It means that function F(r) has the following property: F(1/r)⫽F(r). For the commonly used case of the square grid (a⫽b), F(1) ⫽0.5275. Expression 共10兲looks similar to the approximate expressions for the plasma frequency developed earlier in Refs. 2–4,6, but for thin wires it is more accurate and takes into account the geometry of the lattice. Notice that the dispersion equation 共7兲for extraordinary waves is indifferent to the direction of the wires’ axis: the wave-vector components qx,qy,qzenter into this equation completely symmetrically. It means that, within the low-frequency stop band, the extraordinary wave decays with the same decay factor along all directions in space. The same can be said for propagation in the first frequency passband. This isotropy of the dispersion equation is rather surprising since the medium is strongly anisotropic. However, it can be shown from the very fundamental facts summarized in Eqs. 共5兲and 共3兲by assuming, as usual, that Eq. 共1兲is valid for extraordinary waves propagating in the x-yplane. It is possible to transit Eq. 共6兲from the spectral domain (q, ␻ ) to the physical domain (r,t). The following nonlocal material equation can be derived from Eq. 共6兲using the double Fourier transform: D共x,y,z兲⫽␧0E共x,y,z兲⫹ ␧0k0 2c 2z0 冕 ⫺⬁ t 冕 z⫺c(t⫺t⬘) z⫹c(t⫺t⬘) ⫻Ez共x,y,z⬘,t⬘兲dz⬘dt⬘,共12兲 where c⫽1/ 冑 ␧0 ␮ 0is the speed of light in the host matrix. Here, the area of integration in the z-tplane is the light cone 兩 z⫺z⬘ 兩 ⬍c(t⫺t⬘). In other words, the kernel in the Fourier convolution is u关c(t⫺t⬘)⫺ 兩 z⫺z⬘ 兩 兴, where u(x) is the Heaviside step function. It means that the point (x,y,z) inside the wire medium 共described as a dispersive continuum兲 at moment tis affected by the zcomponents of electric fields coming from the domain „x,y,z⫾c(t⫺t⬘)…surrounding 共along the wire axis兲this point during all the past time (t⬘ ⬍t). This result illustrates the consistency of the reported model from the relativistic standpoint. In the following we will describe some relevant effects in the analyzed parallel wire medium, associated with the spatial dispersion. Refraction and reflection of plane waves at a plane interface show strong differences between the local and nonlocal models. Let us consider an interface between an isotropic dielectric with the permittivity ␧1and a uniaxial dielectric with ␧described by Eq. 共1兲. The interface is in the y-zplane and it is illuminated by a plane wave coming from the isotropic dielectric. The wave vector and electric-field vector lie in x-zplane (qy⫽0,Ey⫽0). The incidence angle of the plane wave is ␪ .If␧1⬎␧0,␧⬍0, and sin2( ␪ ) ⬍␧0/␧1the wave will be completely reflected, but for sin2( ␪ )⬎␧0/␧1some part of the wave will be transmitted through the interface. This transmitted wave will be an extraordinary wave, as it follows from its polarization state, and can be excited at any frequency. This amazing behavior disappears when the nonlocal model summarized in Eq. 共6兲 is used. Indeed, if the nonlocal axial permittivity 共6兲is used for the wire medium, we observe that no transmission inside the wire medium is possible for k⬍k0.Atk⫽k0transmission is possible only in the case of the normal incidence. Only if k⬎k0, a refracted wave appears. Let us next consider the guidance of electromagnetic waves in a parallel-plate waveguide infinite in the xand y directions and bounded by parallel perfectly conducting planes orthogonal to the zaxis. Separation between the conducting walls is d. We assume that this waveguide is filled with a wire medium with the wires along the zdirection. We will consider eigenwave propagation along the xaxis of the TM01 mode (Hy,Ex,Ez⫽0). For waveguides filled by a local uniaxial dielectric with anisotropy axis along the zdirection we have from Eq. 共3兲 ␧0qx 2⫽␧共k2⫺qz 2兲,qx⫽ 冑 ␧ ␧0 冋 k2⫺ 冉 ␲ d 冊 2 册 .共13兲 If ␧⬎0, Eq. 共13兲gives a cutoff for k⬍ ␲ /dand propagation for k⬎ ␲ /d. In contrast, if ␧⬍0, propagation is allowed when k⬍ ␲ /d共and forbidden for k⬎ ␲ /d). Within this passband a backward wave (dq/d ␻ ⬍0) propagates, as one can see in Fig. 2 共thin solid lines兲. This amazing behavior disappears if one fills the waveguide with the analyzed nonlocal wire medium. Using Eq. 共7兲, we have in this case qx 2⫹qz 2⫽k2⫺k0 2,qx⫽ 冑 k2⫺k0 2⫺ 冉 ␲ d 冊 2 ,共14兲 and we obtain the usual frequency behavior: cutoff for k ⬍ 冑 ( ␲ /d)2⫹k0 2and propagation for k⬎ 冑 ( ␲ /d)2⫹k0 2.An increase of the cutoff frequency is observed compared to the FIG. 2. Normalized propagation factors in a parallel-plate waveguide vs normalized frequency for different types of waveguide filling: Thin solid lines, uniaxial dielectric with a negative permittivity; thick solid line, wire medium; dashed line, empty waveguide. The wire medium and the uniaxial dielectric have the same k0 ⫽ ␲ /(2d). BRIEF REPORTS PHYSICAL REVIEW B 67, 113103 共2003兲 113103-3 case when there is no filling medium, as one can see in Fig. 2共thick solid line and dashed line兲. Let us finally analyze the propagation of plane waves along the wire medium for the particular case of qz⫽k.In this case, the nonlocal permittivity along the zaxis 共6兲becomes infinite. To avoid the singularity problem we use material equation of the form E⫽␧ ¯ ¯ ⫺1D. In this case, the Maxwell equations have plane-wave solutions for all frequencies. For these waves the transverse wave vector q⬜⫽(qx,qy)Tis arbitrary. The waves are transverse with respect to the wire axis: Hz⫽0 and Ez⫽0. The electric field is parallel to the transverse wave vector, q⬜⫻E⫽0. Such waves can be interpreted as transmission-line modes propagating along the parallel wires. In fact, a set of Ninfinite parallel wires can be viewed as a system of coupled transmission lines. This system can support Ndegenerate transmission-line waves with Hz⫽0, Ez⫽0, and phase constant qz⫽k. The electric field of these waves can be obtained from E⫽⫺共ux ⳵ x⫹uy ⳵ y兲 ␾ 共x,y兲exp共⫺jkz兲,共15兲 where ␾ (x,y) is a quasielectrostatic potential taking constant but arbitrary values ␾ n(n⫽1,2,...,N) at each wire. In fact, the plane wave with transverse wave number q⬜and qz⫽k corresponds to the transmission-line wave with ␾ n ⫽ ␾ 0exp(⫺jq⬜•rn), where rn⫽(xn,yn)Tis the location of the nth wire in the transverse x-yplane. In summary, it has been shown that parallel wire media possess very strong spatial dispersion effects at any frequencies, including the very large wavelength limit. An analytical model for the nonlocal permittivity dyadic of these media has been presented and discussed. Inconsistency of the local model for parallel wire media with nonvanishing wavevector component along the wires has been shown. Dramatic differences in the predicted behavior of that media, arising from the use of the conventional local and/or the nonlocal model for the permittivity are shown. Finally, the proposed nonlocal model for the permittivity has been found to be also suitable for the description of the transmission-line modes of the structure. We feel that the reported results open the question of the role of spatial dispersion in the adequate characterization of discrete metamaterials as effective media, at least if arbitrary directions of propagation and/or polarization of the electromagnetic field should be considered in the analysis. In addition, an example has been presented of an effective medium in which spatial dispersion is important at any frequency, in contrast with some commonly assumed ideas about the physical relevance of this effect. 1L. Landau, E. Lifschitz, and L. Pitaevski, Electrodynamics of Continuous Media 共Pergamon Press, Oxford, 1984兲. 2J. Brown, Prog. Dielectr. 2, 195 共1960兲. 3W. Rotman, IRE Trans. Antennas Propag. 10,82共1962兲. 4R. King, D. Thiel, and K. Park, IEEE Trans. Antennas Propag. 31, 471 共1983兲. 5S. Maslovski, S. Tretyakov, and P. Belov, Microwave Opt. Technol. Lett. 35,47共2002兲. 6J. Pendry, A. Holden, W. Steward, and I. Youngs, Phys. Rev. Lett. 76, 4773 共1996兲. 7J. Pitarke, F. Garcia-Vidal, and J. Pendry, Phys. Rev. B 57,15261 共1998兲. 8V. Veselago, Sov. Phys. Usp. 10, 509 共1968兲,关Usp. Fiz. Nauk 92, 517 共1967兲兴. 9J. Pendry, Phys. Rev. Lett. 85, 3966 共2000兲. 10D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 共2000兲. 11R. Shelby, D. Smith, and S. Schultz, Science 292,77共2001兲. 12D.R. Smith and N. Knoll, Phys. Rev. Lett. 85, 2933 共2000兲. 13A. Pokrovsky and A. Efros, Phys. Rev. Lett. 89, 093901 共2002兲. 14V. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas 共Pergamon, Oxford, 1964兲. 15I.V. Lindell, S. Tretyakov, K. Nikoskinen, and S. Ivonen, Microwave Opt. Technol. Lett. 31, 129 共2001兲. 16P. Belov, S. Tretyakov, and A. Viitanen, J. Electromagn. Waves Appl. 16, 1153 共2002兲. BRIEF REPORTS PHYSICAL REVIEW B 67, 113103 共2003兲 113103-4