3D Metamaterial Based on a Regular Array of Resonant Dielectric Inclusions

Transcription

1 RADIOENGINEERING, VOL. 18, NO. 2, JUNE D Metamaterial Baed on a Regular Array o Reonant Dielectric Incluion Irina VENDIK, Mikhail ODIT, Dmitryi KOZLOV St. Peterburg Electrotechnical Univerity, 5 Pro. Popov Str., St. Peterburg, , Ruia IBVendik@eltech.ru, MAOdit@mail.eltech.ru Invited paper Abtract. The 3D regular lattice o bi-pherical dielectric reonant incluion arranged in a cubic lattice a two et o phere made rom the ame dielectric material having dierent radii and embedded in a hot dielectric material with lower dielectric permittivity wa careully invetigated. The magnetic reonance correponding to the irt Mie reonance in the pherical i ollowed by orming a regular array o eective magnetic dipole, and the tructure o the identical pherical dielectric reonator can be deigned a an iotropic μ-negative 3D-metamaterial. For the electric reonance it wa ound experimentally and by the imulation that the reonant repone o the electric dipole wa weakly pronounced and the μ-negative behavior wa remarkably uppreed. To enhance the electric dipole contribution we conidered another kind o the ymmetry o the bi-pherical arrangement o the correponding to the body-centered cubic ymmetry intead o the ymmetry o NaCl analog conidered previouly. Electromagnetic propertie o a volumetric tructure baed on a regular lattice o identical cubic dielectric i alo conidered and analyzed a μ-negative metamaterial. The cubic particle baed 3D-metamaterial i preerable or practical realization a compared with the pherical incluion. Keyword Metamaterial, reonant incluion, Mie reonance, magnetic dipole, electric dipole, iotropic, backward wave, cubic reonator. 1. Introduction Medium with imultaneouly negative permittivity and permeability or o-called double negative medium (DNG) can be ormed by a regular lattice o dielectric reonant incluion, providing excitation o electric and magnetic dipole. Dielectric dik, cylindrical, or pherical reonator are uitable or etablihing the dipole moment. Metamaterial with deired value o permeability ±μ and permittivity ±ε are developed by exciting electric and magnetic reonant mode. In many practical cae, iotropic DNG tructure i very attractive. Dierent way to create the 3D iotropic DNG medium baed on a regular lattice o reonant incluion have been conidered and publihed [1-6]. The 3D regular lattice o bi-pherical dielectric reonant incluion wa uggeted in [2]. In thi tructure, the metamaterial medium i compoed o two et o pherical made o the ame dielectric material embedded in a hot dielectric material. The phere dier by radiu. The dielectric contant o the pherical i much larger than that o the hot material. By combining two et o the phere with uitable radii, dierent mode can be imultaneouly exited in the phere: the magnetic reonance mode giving rie to the magnetic dipole momentum and the electric reonance mode being reponible or the electric dipole momentum. Thee moment create the negative permeability and permittivity in a limited requency range near the reonant requency. In [3-5], it wa uggeted that the dielectric reonator do not interact and or the 3D tructure the repone o the both pherical are uperimpoed. By ull-wave imulation and experimental invetigation, it wa ound that the reonance repone o the magnetic dipole i very eective and the μ -negative iotropic metamaterial can be deigned a a regular array o dielectric pherical with the irt Mie-reonance. At the ame time, the electric dipole correponding to electric reonance mode i weakly pronounced. A a conequence, the ε -negative behavior i blurred [5]. Detailed theoretical decription baed on ull-wave analyi wa perormed in [6] or dierent all-dielectric tructure o metamaterial: ingle-negative (SN) medium baed on the pherical ; bi-pherical DNG medium baed on an array o the dielectric pherical o the ame radiu made o two material diering in dielectric permittivity; a et o identical dic and dic made o two dierent dielectric material etc. The tructure baed on the dic orm 2D SN and DNG metamaterial. Intereting 2D tructure baed on cylindrical reonator array ituated in a parallel-plate waveguide have been dicued and experimentally veriied in [7-8]. In thee tructure, the DNG propertie are provided by magnetic reonance in the cylinder (magnetic dipole, μ -negative behavior) and by ε -negative repone o electromagnetic wave in a parallelplate metallic waveguide with TE n mode below the cut-o requency. Recently the experimental invetigation o iotropic metamaterial baed on reonant dielectric incluion ha

2 112 I. VENDIK, M. ODIT, D. KOZLOV, 3D METAMATERIAL BASED ON A REGULAR ARRAY OF RESONANT DIELECTRIC INCLUSIONS been reported [9-11]. The reonant μ-negative repone wa regitered in the 3D tructure baed on a regular array o dielectric cube in [9] and the 2D array o cube in []. The 3D DNG material wa realized a a et o dielectric pherical regularly ditributed in a metallic wire rame exhibiting cubic ymmetry. The wire rame provide an environment with evanecent wave ollowed by the eective negative permittivity, which in combination with the reonant dielectric giving rie to magnetic dipole and negative permeability lead to a propagating wave with a negative phae velocity i.e. orm the 3D iotropic DNG medium. All thee achievement upport the ruitul idea o a realization o the iotropic metamaterial uing dielectric reonant incluion. In order to improve the perormance o the all dielectric DNG medium with cubic ymmetry baed on pherical dielectric, we ugget the new tructure obtained by changing the ymmetry o the ingle cell o the bi-pherical tructure to enhance the contribution o the electric reonance in the eective dielectric permittivity. In thi paper, we conider the 3D DNG bi-pherical tructure baed on the ingle cell belonging to the bodycentered cubic ymmetry exhibiting higher packing denity or the ame ditance between adjacent reonant a compared with the ace-centered cubic lattice previouly analyzed in [2-5]. The regular 3D-tructure baed on cubic reonant incluion i alo conidered a the candidate or a practical realization o the μ-negative iotropic metamaterial. εe 0 0 μe 0 0 ε = 0 εe 0, μ = 0 μe 0 (1) 0 0 ε 0 0 μ e where the ub-indice e are introduced to tre that the permittivity and permeability are obtained a a reult o averaging electric and magnetic polarization o pherical embedded in the matrix. The reult o averaging the polarization o the pherical embedded in the matrix depend on the volume o the matrix alling on each particle conidered. 2r mall 2r big 2r mall 2r big 2r mall 2r big e с) 2. 3D Iotropic Bi-pherical Metamaterial o Cubic Symmetry with High Packing Denity Single cubic cell o iotropic metamaterial baed on two type o pherical reonator are preented in Fig 1. Dierent way o particle packing are poible. The acecentered NaCl like tructure (Fig. 1, body-centered tructure (Fig. 1 and other type o ace-centered cubic lattice (Fig. 1c) are all member o the cubic ytem o ymmetry pertaining to the cla m3m [3, 12]. In the cae o cubic ymmetry, the econd rank tenor o all phyical parameter o the media are diagonal and have the component o the ame value [12]. Thu the permittivity and permeability tenor are written in the ollowing orm: Fig. 1. Face centered NaCl cell; body-centered ingle cell; c)ace-centered ingle cell. The tructure conidered are characterized by dierent minimal ditance between the o the ame/dierent dimenion. The comparative Tab. 1 how dierence in thee parameter or dierent packing actor or the ame ditance a between adjacent. Here i the crytallographic lattice contant. The volume raction i deined a the ratio o the volume o the phere and the total volume o cube limiting the pace around one phere and v = 4/3 πr 3 / 3 i the volume raction or one phere in the pace limited by the cube with the ide, r i the radiu o the pherical particle. The ditance between the reonant i o crucial importance. Mutual interaction o cloely poitioned lead to hit o the reonance requency and ditortion o the reonant characteritic. The analytical model Structure Face-centered (NaCl-like) Body-centered Face-centered Lattice contant, Minimal ditance between the big Minimal ditance between the mall Minimal ditance between dierent Volume raction or the big 2a 2/2 2/2 /2 4v 2 a / 3 3/2 v 2 a / 2 2/2 2/2 3v Tab. 1. Ditance between or dierent tructure. Volume raction or the mall 4v v v

3 RADIOENGINEERING, VOL. 18, NO. 2, JUNE o the diraction o a plane electromagnetic wave on a dielectric pherical particle [5] decribe the eective parameter o the bi-pherical iotropic metamaterial under the aumption that there i no mutual interaction between the. Neglecting mutual coupling i reaonable in cae o appropriate ditance between the. Simulation o the electromagnetic wave outide the pherical demontrate the trong e-time attenuation o the electromagnetic wave outide the phere over the ditance equal to it diameter. At the ame time, to trengthen the DNG eect the hould be placed a cloe a poible to each other. I are poitioned too ar, the DNG eect become negligibly mall. Simulation howed that the optimal ditance between the reonant i achieved or the body-centered tructure (Fig. 1. For thi type o iotropic tructure there i maximum packing denity and the ditance between the adjacent (o dierent kind) i till big enough to neglect the eect o their interaction. Frequency dependent eective permittivity and permeability were calculated or three dierent tructure: μ ( e ) N 3 ( t) ε r ( ) = ε (, ) (, ) 3 p Fep rbig b rbig + εh, (2) a 2 n ε 3 = +. (3) ( e ) p ( t) r ( ) F (, ) (, ) 3 mu rmall a rmall μh a ε h 2 Here a i the ditance between cloet o dierent radii, ε p (μ р ), ε h (μ h ) are the permittivity (permeability) o the particle and the hot material repectively, r mall and r big are the radii o the reonator, i the incident electromagnetic wave requency, a (t) and b (t) are the amplitude o pherical wave unction, F ep and F mu are the reult o integration o the electric and magnetic ield component over the particle volume, N = 0.5, 0.65, 0.35 and n = 0.5, 0.65, 0.65 or the tructure in Fig. 1a, 1b, 1с repectively. In Fig. 2, the reult o calculation o the eective parameter o the bi-pherical metamaterial are preented or three dierent tructure with parameter: ε p = 400, ε h = 1, μ р = μ h = 1, r mall = mm, r big = mm, and = 4 mm, no loe were taken into account. The reult are obtained by a conideration o the diraction o the plane electromagnetic wave on a ingle dielectric pherical particle µ e2,3 µ e Fig. 2. Eective permittivity and permeability or acecentered Na-Cl tructure (ub-index 1) the acecentered tructure (ub-index 2) and body-centered tructure (ub-index 3). In order to demontrate the eectivene o the bodycentered lattice, let u conider the impliied cae o onedimenional tructure (Fig. 3) uing ull-wave analyi. The ingle cell i centered by the pherical particle o bigger value o the radiu r big = 1.05 mm and i urrounded by the o maller value o the radiu r mall =0.748 mm. The ditance between the o the ame ize i =4 mm. To it the condition o the tranlation ymmetry, the one-dimenional tructure i bounded by a perect electric conductor (PEC) and a perect magnetic conductor (PMC). The particle dimenion are choen to provide the magnetic reonance in the maller phere and the electric reonance in the bigger at the ame requency. It reult in creation o magnetic and electric dipole ollowed by the negative eective permeability and eective permittivity in a limited requency range. That lead to appearance o tranmiion o electromagnetic wave in the requency range near the reonant requency (Fig. 4). H k E input port big phere quarter o mall phere PMC PEC output port Fig D tructure o bi-pherical iotropic metamaterial. Radii o mall and big phere are mm and 1.05 mm repectively. Boundary condition are provided by perect magnetic (PMC) and perect electric (PEC) wall. 5 0 ε e2 ε e3 ε e1 S-Parameter, db S 21 backward wave region S Fig. 4. Tranmiion S 21 and relection S 11 coeicient or the tructure depicted in Fig. 3. Gray area i the requency range, where the reonant requencie o two type o reonance coincide and backward wave i oberved.

4 114 I. VENDIK, M. ODIT, D. KOZLOV, 3D METAMATERIAL BASED ON A REGULAR ARRAY OF RESONANT DIELECTRIC INCLUSIONS The ull-wave analyi o the electromagnetic wave propagation conirm the exitence o the backward wave. In Fig. 5, the magnetic ield pattern i hown or our dierent moment o the period o time. It i clearly een that the wave propagate rom the right ide to the let, wherea the incident wave enter the tructure rom the let ide. incident wave phae veloсity t = 0 t = T/8 t = T/4 The 3D volumetric tructure correponding to the body-centered ymmetry i hown in Fig. 6. The mall and big pherical are ituated in dierent plane. The ame i related to the magnetic and electric dipole. The PEC and PMC are ued or providing boundary condition or thi limited in volume tructure. The dimenion o the pherical and pacing between the identical are the ame a in the previou one-dimenional cae. Magnetic ield pattern or the ection plane hown by yellow in Fig. 6 i depicted in Fig. 6 or our dierent moment o the time period. Again the backward wave i oberved. The repone o the tructure to the incident plane electromagnetic wave i preented in Fig. 7. In the limited requency range one can oberve the wave tranmitting through the DNG tructure. Hence, the increaing packing actor lead to enhanced electric dipole contribution a compared with our previouly obtained reult or NaCl tructure [4], [5]. t = 3T/8 Fig. 5. Magnetic ield pattern or our dierent moment o the period o time. The incident wave enter the tructure rom the let ide, wherea the wave inide tructure propagate rom the right ide to the let. Sectional plane S-Parameter, db S 11 S 21 Input port backward wave region E Incident wave k H Fig. 7. Tranmiion S 21 and relection S 11 coeicient or the modeled tructure depicted in Fig. 5. Gray area i the requency range where two type o reonance coincide and backward wave i oberved. Incident wave 1 2 t = 0 t = T/8 3 Phae velocity 4 t = T/4 t = 3T/8 Fig. 6. Bi-pherical tructure o metamaterial with bodycentered cubic ymmetry. Radii o mall and big phere are mm and 1.05 mm repectively. Ditance between the ame i 4 mm. Magnetic ield pattern or the ection plane hown by yellow in ( or 4 dierent moment o the period o time. 3. 3D Metamaterial Baed on Cubic Dielectric Reonant Incluion A it wa hown, the 3D regular lattice baed on reonant pherical incluion exhibit propertie o the DNG material. Unortunately the practical realization o uch a tructure i very problematic. A we know, there i the only publication demontrating the experimental reult o meauring tranmiion characteritic o the DNG medium baed on the pherical incluion inerted in the wire rame [11]. The experimental invetigation o the 3D tructure [9] and 2D tructure [] o metamaterial baed on cubic reonant dielectric incluion conirmed the μ- negative propertie o thee artiicial media. In Fig. 8, the regular lattice o the dielectric cube i hown. The cubic are ditantly poitioned and the mutual coupling i negligibly mall. In thi cae the requency repone o the tructure i the ame a or the material baed on the identical pherical incluion: the top-band occur near the reonant requency, which correpond to the μ-negative behavior o the material. The magnetic ield ditribution in the reonant or the irt magnetic reonance in the pherical and cubic i hown in Fig. 9. Evidently the magnetic dipole in the both cae i nearly the ame. The reult o ull-wave imulation o the tranmi-

5 RADIOENGINEERING, VOL. 18, NO. 2, JUNE ion through thi tructure are preented in Fig. exhibiting the top-band near the reonant requency. a d Fig. 8. The lattice o dielectric high permittivity cube in the dielectric matrix o lower permittivity. The ollowing parameter o cube are ued: ize a = 3 mm, εr = 00, dielectric cube tan δ = 0.001, the ditance between the cube d = 1 mm. layer ued, the higher i the reonant requency and the wider i the reonant repone. That mean that the coupling between the cubic inluence the characteritic o the material. The regular lattice o the cubic can be realized a a multilayer tructure (or example multilayer tructure baed on low-temperature coired ceramic, LTCC). In general cae, the 3D lattice can be deigned being regular in plane with deired pacing between the layer. Thi pacing i determined by the ceramic layer thickne, wherea the lattice period in plane i determined by the mak dimenion. The reult o modeling o the reonant characteritic o the tructure baed on the cubic dielectric reonator with the cube edge 1 mm and the ditance between the cube 3 mm in the plane are hown or dierent number o the layer. The pacing between the layer i 1.2 mm. The reonant characteritic are plit due to trong coupling between the layer containing regular 2D lattice o the cube (Fig. 11). The more number o the layer containing the cubic reonant incluion i ued, the more eective i the relection rom thi artiicial material. That can be ued a an eective impedance urace without uing any conducting or magnetic material. We can expect that the cubic incluion baed metamaterial will be realized in multilayer technology in the nearet uture. Fig. 9. Magnetic reonance in the dielectric pherical particle the dielectric cubic particle. 4 layer 3 layer 2 layer 1 layer h d direction o wave incidence S12, db S11, db S12, db n=2 n=3 n=4 Fig.. 2 layer dielectric cube tructure. Tranmiion and relection coeicient or thi tructure d = 5 mm, h = 5 mm. It wa hown by the imulation that or the tructure with a inite number o the the reonant repone depend on the number o layer: the bigger number o the Fig. 11. Dielectric cube tructure with everal number o cube layer. Tranmiion coeicient or dierent number n o layer, d = 3 mm, h = 1.2 mm.

6 116 I. VENDIK, M. ODIT, D. KOZLOV, 3D METAMATERIAL BASED ON A REGULAR ARRAY OF RESONANT DIELECTRIC INCLUSIONS 4. Concluion A double negative artiicial material baed on bipherical dielectric pherical reonator ha been invetigated. A new tructure with body-centered unit cell o higher packaging actor ha been introduced in order to trengthen the DNG eect. The iotropy o the metamaterial main challenging advantage or the material manuacturing ha been kept. Backward wave exitence in the new tructure ha been conirmed by the numerical analyi. A concept o metamaterial baed on cubic dielectric incluion i more preerable or a practical realization. Full wave imulation reult revealed the μ-negative behavior o the tructure baed on the cubic dielectric reonator. The multilayer ceramic technology can be ued or manuacturing o thi metamaterial. Reerence [1] HOLLOWAY, C., KUESTER, E. A double negative compoite medium compoed o magnetodielectric pherical embedded in a matrix. IEEE Tran. on Antenna and Propagation, 2003, vol. 51, no., p [2] VENDIK, O. G., GASHINOVA, M. S. Artiicial double negative (DNG) media compoed by two dierent dielectric phere lattice embedded in a dielectric matrix. In Proceeding o European Microwave Conerence EuMC34. Pari (France), 2004, p [3] VENDIK, I., VENDIK, O., GASHINOVA, M. Artiicial dielectric medium poeing imultaneouly negative permittivity and magnetic permeability. Tech. Phy. Lett., 2006, vol. 32, p [4] VENDIK, I., VENDIK, O., KOLMAKOV, I., ODIT, M. Modeling iotropic DNG media or microwave application. Opto- Electronic Review, 2006, vol. 14, p [5] ODIT, M., VENDIK, I., VENDIK, O. 3D iotropic metamaterial baed on dielectric reonant phere. In Proceeding o Metamaterial Rome (Italy), 2007, p [6] AHMADI, A., MOSALLAEI, H. Phyical coniguration and perormance modeling o all-dielectric metamaterial. Phy. Rev. B, 2008, vol. 77, 0454 (1-11). [7] SEMOUCHKINA, E. A., SEMOUCHKIN, G. B., LANAGAN, M., RANDALL, C. A. FDTD tudy o reonance procee in metamaterial. IEEE Tran. Microwave Theory Tech., 2005, vol. 53, p [8] UEDA, T., LAI1, A., ITOH, T. Negative reraction in a cut-o parallel-plate waveguide loaded with two-dimenional lattice o dielectric reonator. In Proceeding o European Microwave Conerenc EuMC , p [9] QIAN ZHAO, LEI KANG, DU, B., ZHAO, H., XIE, Q., HUANG, X., LI, B., ZHOU, J., LI, L. Experimental demontration o iotropic negative permeability in a three-dimenional dielectric compoite. Phy. Rev. Lett., 2008, vol. 1, p (1-4). [] SHIBUYA, K., TAKANO, K., MATSUMOTO, N., IZUMI, K., MIYAZAKI, H., JIMBA, Y., HANGYO, M. Terahertz metamaterial compoed o TiO 2 cube array. In Proc. Metamaterial Pamplona (Spain), 2008, p [11] XIAOBING CAI, RUI ZHU, GENGKAI HU Experimental tudy or metamaterial baed on dielectric reonator and wire rame. Metamaterial, 2008, vol. 2, no. 4, December 2008, p [12] NYE, J. F. Phyical Propertie o Crytal. Oxord at the Clarendon Pre, About Author... Irina B. VENDIK received the electronic engineer diploma and the candidate o Sc. (Ph.D.) degree rom the Leningrad Electrical Engineering Intitute (now St. Peterburg Electrotechnical Univerity), St.-Peterburg, Ruia, in 1959 and 1964 repectively, and the D. Sc. (Phy.) degree rom A.F. Ioe Phyicotechnical Intitute, St.-Peterburg, Ruia, in She i currently a Proeor in the Dept. o Microelectronic and Radio Engineering, and the Head o the Microwave Microelectronic Laboratory, St.- Peterburg Electrotechnical Univerity, St.-Peterburg, Ruia. Her general reearch interet have been in olid tate phyic and microwave electronic. The lat year he ha been a leader o the group o the ETU involved in FP6 and FP7 project o the EC. I. Vendik i a member o IEEE (USA) rom 1996 and EuMA (2006). I. Vendik i honored educator o the Ruian Federation (1999). Mikhail ODIT wa born in 1982 in St. Peterburg. He wa graduated in Radio Engineering rom St. Peterburg Electrotechnical Univerity "LETI" in Hi reearch interet are concentrated in the ield o artiicial material and microwave application o metamaterial. The lat year he wa involved to the project o Network o Excellence Metamorphoe FP6 EC. Dmitryi KOZLOV wa born in Boriov in Hi reearch interet include metamaterial or microwave application.