Isotropic double negative Metamaterial

8/11/2019 Isotropic double negative Metamaterial

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21Isotropic Double-Negative

Materials

Irina VendikSt. Petersburg Electrotechnical

University

Orest G. VendikSt. Petersburg Electrotechnical

University

Mikhail OditSt. Petersburg Electrotechnical

University

. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. wo-Dimensional and Tree-Dimensional Isotropic

Metamaterials Formed by an Array o Cubic Cells withMetallic Planar Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

. L-Based Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. wo-Dimensional Structure o DNG Metamaterial

Based on Resonant Inclusions . . . . . . . . . . . . . . . . . . . . . . . . -. Tree-Dimensional Isotropic DNG Metamaterial

Based on Spherical Resonant Inclusions . . . . . . . . . . . . . . -Symmetry o the Bispherical DNG Structure DNGMedium Composed o Magnetodielectric SphericalInclusions DNG Medium Composed o Dielectric Sphereswith Different Radii (GarnetMaxwell Mixing Rule) DNGMedium Composed o Dielectric Spheres with Different

Radii (Electromagnetic Wave Diffraction Model). Effective Permittivity and Permeability o the

Bispherical L attice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Electric and Magnetic Dipole Moments o SphericalResonators Comparison o the Effective Permittivity andPermeability Obtained with Different Models Results othe Full-Wave Analysis Results o the Experiment

Influence o Distribution o Size and Permittivity oSpherical Particles on DNG Characteristics IsotropicMedium o Coupled Dielectric Spherical Resonators

. Metamaterials or Optical Range . . . . . . . . . . . . . . . . . . . . . -Reerences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

21.1 Introduction

Media with single-negative (SNG) permittivity or SNG permeability and simultaneously negativepermittivity and permeability, which is called double-negative (DNG) media, are under relent-less interest o physicists and microwave engineers []. Tese articial materials are known asmetamaterials. It has been shown that application o SNG materials can sufficiently improve the char-

acteristics o many microwave devices. However, more interesting properties can be realized usingDNG structures. In a limited requency band, such materials exhibit anomalous properties: lens-ing beyond the diffraction limit, wave propagation in subwavelength guiding structures, resonantenhancement o the power radiated by electrically small antennas, etc.

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2009 by Taylor and Francis Group, LLC


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21-2 Theory and Phenomena of Metamaterials

Most theoretical and practical investigations in this area are related to one-dimensional (D)or two-dimensional (D) structures. Te well-known metamaterial structure suggested in [] andexperimentally examined in [] is a combination o two lattices: a lattice o split-ring resonators(SRRs) and a lattice o innitely long parallel wires. Te wires produce the effective negative permit-tivity, and the SRRs are responsible or the effective negative permeability. A combination o SRRs

and metal wires provides an articial material with simultaneously negative effective permeabilityand permittivity [,,]. Tis structure is anisotropic and reveals the negative permittivity and per-meability, i the propagation direction o the electromagnetic wave is orthogonal to the axes o thewires and belongs to the plane o the SRRs. Te unusual electromagnetic properties originate romthese articial structures rather than arising directly rom the materials. It is interesting and useulor practical applications to realize a DNG medium using the intrinsic electromagnetic properties oarticial inclusions orming the material. In many cases, isotropic DNG structures are very attrac-tive or practical uses. For this purpose, structures with embedded three-dimensional (D) resonantinclusions are very promising.

Different ways to create a D and D isotropic DNG medium based on a regular lattice o resonant

inclusions are discussed in this chapter: (a) SNG or DNG metamaterial ormed as a rectangular latticeo cubic unit cells o plane resonant particles on the aces o the cube; (b) a D DNG medium ormedas an array o dielectric cylindrical resonators; and (c) a D DNG medium ormed as a regular latticeo spherical resonant inclusions.

Te characteristic eature o the structures considered is the isotropy o the effective permittiv-ity and permeability. In the case o the cubic symmetry pertaining to the class mm, the secondrank tensors o electromagnetic parameters o the media are diagonal [,]. Tus, the permittivityand permeability tensors o particles arranged in the cubic structure are scalars, eand e. Body-centered and ace-centered structures are characterized by the same orms o the second-order tensoras the simple cubic structure. Hence, the D isotropic metamaterial can be realized as articial struc-

tures designed in orm o a regular array o particles. Te symmetry class o the unit cells arranged inthe periodical structure provides the isotropy o the metamaterial.

In this chapter, we discuss isotropic D and D metamaterials, which differ in the properties o theconstitutive resonant particles. Te ollowing isotropic structures are under consideration:

. SNG and DNG metamaterial ormed by a rectangular/random lattice o isotropic cubicunit cells o particles: SRRs, -particles, and a combination o the SRRs and wire/dipoleparticles

. D DNG metamaterial based on transmission lines (L)

. D DNG medium ormed by an array o dielectric resonators (DRs), providing excitationo the electric and magnetic dipoles

. D DNG medium ormed by a regular lattice o spherical resonant inclusions, providingexcitation o the electric and magnetic dipoles

21.2 Two-Dimensional and Three-Dimensional IsotropicMetamaterials Formed by an Array of Cubic Cells withMetallic Planar Inclusions

Te rst D SNG magnetic structure with high isotropy was described in []. An array o singlecells composed o two intersecting SRRs normal to each other is suggested to demonstrate isotropicmetamaterial. Te single cell is ormed by crossed SRRs (CSRRs). Each SRR is made o two aluminum(Al) strips deposited on the inner and outer aces o the oam ring made rom the dielectric with lowpermittivity. Te dielectric oam has a orm o strip o mm width; the inner radius is mm and


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Isotropic Double-Negative Materials 21-3

the outer radius is mm, and the separation between the inner and outer Al strips ist= mm. It ismachined rom a oam plate to obtain two open rings that are then tted into each other. Te mmwide Al strips are cut rom the mm thick sel-adhesive Al oil. Te gaps at the extremities o thetwo CSRRs are located at the same pole o the spherical structure. Te width o these gaps is mm.Rotating the CSRR in the waveguide around itsz-axis does not affect the transmission coefficientS

measured. Te structure is isotropic in the plane perpendicular to thez-axis. A possibility to obtain amaterial with effective isotropic magnetic properties by building up an array o CSRRs with periodicor random orientations was experimentally conrmed.

Te idea o using cubic cells with planar metallic inclusions on the cube aces is very promising.Each unit cell is ormed by printing perectly conducting plane resonant particles on the aces oa cubic unit cell: SRRs [], -particles [], spider dipoles [], and a combination o SRRs anddipoles [].

Te symmetric SRR was suggested to be used as part o D isotropic structure, which can beconsidered as an isotropic -negative material (Figure .a) []. Te single cell is made o sixplanar SRRs exhibiting rotational symmetry, placed on the aces o the cube so that the whole

particle is invariant with respect to rotations around all Cartesian axes. Te unit cell provides adistinct magnetic SNG response. Te isotropy is experimentally conrmed by the measurement o thetransmission coefficient o the waveguide loaded with one D particle in different orientations. Teexperimental investigation conrmed the isotropy o the -negative material (Figure .a) [,].

Te same symmetry exhibits the cubic structure based on spider dipoles (Figure .b) providing an-negative response. A periodical spatial arrangement o the cells orms the bulk o the isotropic mag-netic or electric metamaterials. Te negative permeability/permittivity o the single magnetic/electricparticle placed in a rectangular waveguide was extracted rom the measured scattering parameters.Te D regular arrangement o the cubic particles is a sophisticated technological problem. Te con-cept o randomly distributed magnetic particles was thereore checked rst on D and then on D

structures []. Te experiments demonstrated that better isotropy and a wider requency band o themetamaterial can be achieved by a quasiperiodical location and a higher density o the particles [].

Te electrical dipole loaded by a loop inductance (inset in Figure .c) was suggested to provideeffective negative permittivity o the media. Te isotropy o one D cubic unit cell with a single dipoleis documented by its measured transmission coefficient (Figure .c). Experiments with randomdistributions o these particles also exhibited promising results. Measured transmission coefficientso particles inserted in the polystyrene slices consecutively rotated by provided very goodisotropy o the cubic sample, as ollows rom the small dispersion shown in Figure .d.

(a) (b)

y

xz

FIGURE . (a) Volumetric -negative particles composed o C-SRRs and (b) volumetric -negative particles

composed o C-dipoles. (aken rom Baena, J.D., Jelinek, L., Marques, R., and Zehentner, J., Appl. Phys. Lett., ,

, . With permission.)


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(a)

(b) (c) Frequency (GHz)

Mini pass band: Central frequency~6.15 GHz Pass band~0.4 GHz

S21(dB)

60

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05 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8

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r

FIGURE . (a) Geometry o a cubic unit cell o -particles. For graphic clearness, only one pair o -shaped per-ectly conducting particles is shown on two opposite aces o the cubic unit cell. (b) A rectangular lattice o isotropic

cubic unit cells o -particles. (From Simovski C.R. and He, S., Phys. Lett. A, , , . With permission.)

(c) Mini pass band between . and .GHz or our different distances o the source. (aken rom Verney, E., Sauviac,

B., and Simovski, C.R.,Phys. Lett. A, (), , October . With permission.)

(a) (b)

FIGURE . Te design o a ully symmetric unit cell or a one-unit-cell thick slab o an isotropic SRR (a) and a

lef-handed (b) metamaterial. Te interaces are parallel to the lef and right SRRs. (aken rom Koschny, T., Zhang,

L., and Soukoulis, C.M.,Phys. Rev. B, , (R), . With permission.)

Te DNG metamaterial based on the D conguration o SRRs and the continuous wires(Figure .) is another D isotropic metamaterial []. Te isotropic unit cell is based on the our-gap SRRs (Figure .a). Te SRR gaps are lled with a high permittivity dielectric with a relativepermittivity gap= to lower the magnetic resonance requency.

Te design o this type o metamaterial minimizes the mutual interaction o SRRs and wires, a cou-pling o the electric eld to the magnetic resonance, and the cross-polarization scattering amplitudesand effects o the periodicity. Te transmission and reflection coefficients or a slab o the isotropicSRR and the corresponding isotropic lef-handed metamaterial (LHM) o - and -unit-cell thick-ness have been calculated. Te simulated transmission coefficients or unit cell and a slab o cellshave been plotted or different incident angles and polarizations (Figure .). Despite the squareshape o the SRR, the absolute independence o the scattering amplitudes on the orientation o


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SRR

= 5

= 15

0.4 0.45 0.5 0.55 0.6 0.66 0.7

T

T

1

104

108

1012

1016

1

104

108

1012

1016

LHM

TM: = 10

=45

=30

TE: = 1 0

=45=30

0.4

0.45 0.5 0.55 0.6 0.66 0.7

T

T

1

1010

1020

1030

1

104

108

1012

1016

= 45

= 85

0.4 0.45 0.5 0.55 0.6 0.66 0.7

T

T

1

104

108

1012

1016

1

104

108

1012

1016

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1030

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1030

FIGURE . ransmission spectra= t() or a -unit-cell thick slab o the SRR and the LHM metamaterial

or various angles o incidence ()and polarizations (, E and M modes). (aken rom Koschny, T., Zhang, L.,and Soukoulis, C.M.,Phys. Rev. B, , (R), . With permission.)


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the incidence plane or both the transverse electric (E) and transverse magnetic (M) modes, arbi-trary , SRR, and LHM metamaterial slabs o any thickness is conrmed. Analysis o the -unit-cellthick slab o the SRR and the LHM metamaterial revealed the isotropic behavior.

21.3 TL-Based Metamaterials

Among other kinds, the metamaterials based on Ls are o high interest. Te use o loaded L net-works allows the realization o wide-bandand low-loss BWmaterials in the microwave region [,].D isotropic, L-based, BW materials have been proposed in [].

In [], a rotated L method (LM) scheme [] was used to produce a unit cell o isotropicmetamaterial. In the LM representation o discrete electrodynamics, a -port scattering matrix,representing the LM cell, contains all the inormation o the discretized Maxwells equations. Te-port cell can be decomposed into two independent six-port cells by a coordinate transormation.Te two independent hal-cell six ports described by scattering matrices S andS [] are calledA and A cells, respectively. A LM cell that completely samples the electromagnetic eld canbe established by either nesting the six-port structures o the A and A hal-cells or by a cluster oeight hal-cells with alternating A and A cells. Te lumped-element circuit or an A cell containsseries elementsZand shunt elementsY. An elementary metamaterial cell may be conceived on thebasis o rotated LM cells by inserting reactances in series to the six cell ports and our admittancesconnecting the series reactances at a central node, orming a virtual ground. Both hal-cells can beconnected at the virtual ground.

In the general case, the unit cell is composed o composite right-/lef-handed (CRLH) L sections.In a CRLH unit cell, the impedanceZis a series resonator (LR, CL), whereas Yis a parallel resonator(LL , CR). Te right-handed components account or unavoidable parasitics []. Te correspondingunit cell or the rotated LM metamaterial is shown inFigure .a.Te proposed realization o the CRLH rotated LM metamaterial, corresponding to the lumped-element network o Figure .a, is depicted in Figure .b. Shunt inductors are implementedby wires connected to a common center point, and series capacitors are implemented by metalinsulatormetal (MIM) plates located between the adjacent unit cells. Figure .b shows a cluster o nested unit cells. Te plate capacitors are realized in printed circuit board (PCB) with patcheson both sides o the substrate, which ensures accurate CLvalues. Te inductors are realized by rigidwires. Te unit-cell length is cm, the substrate is Rogers B mil, and the lef-handed valuesareLL. nH andCL.pF.

Figure .c shows the unit-cell prototype o the CRLH rotated LM metamaterial. Tis prototype

was measured with a two-port vector network analyzer through baluns (microstrip to parallel-striptransitions) connected at two arbitrary nonaligned ports, whereas the remaining ports are termi-nated with the resistors. Te dispersion diagram depicted in Figure .d shows a good agreementwith circuit simulation results up to . GHz. Te expected two lef-handed and two right-handedrequency bands are clearly visible, thereore veriying the behavior o the rotated LM metamaterial.A simplied planarized implementation, preserving the same network topology, can also be realizedor practical applications.

Te idea o a superlens based on the L-metamaterial has been discussed in []. Te proposedstructure o a D super-resolution lens consists o two orward-wave (FW) regions and one BWregion. Te D FW networks can be realized with simple Ls and the D BW network with induc-

tively and capacitively loaded Ls. One unit cell o the BW network is shown inFigure .(the unitcell is shown by the dotted line). In the D structure, there are impedances Z and Ls also alongthez-axis (not shown in Figure .). In view o potential generalizations, the loads are representedby series impedancesZ and shunt admittancesY, although or the particular purpose o realizinga BW network, the loads are simple capacitances and inductances. Te unit cell o the FW network


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(a) (b)

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FIGURE . (a) CRLH rotated, LM metamaterial, hal-unit cell []; (b) D CRLH rotated, LM metamaterial

realization, complete structure; (c) D CRLH rotated, LM unit cell with its input and output baluns required

or the differential excitation o the measurement setup; (d) dispersion diagram: measured (solid line) and simulated

(dashed line). (aken rom Zedler, M., Caloz, C., and Russer, P., Circuital and experimental demonstration o a D

isotropic LH metamaterial based on the rotated LM scheme,Microwave Symposium, IEEE/M-S International,

Honolulu, HI, June , , pp. . With permission.)

is the same as inFigure .but without the series impedances Z and shunt admittance Y. In asimplied case,Z=jCandY=jL.

Te derived dispersion equations and analytical expressions or the characteristic impedances orwaves in the FW and BW regions make it possible to nd the condition or a design o such structures.Te ull-wave simulations revealed the subwavelength resolution characteristics o a realizable designwith commercially available lossy materials and components. Tere is a special problem o impedanceand reraction index matching o the FW and BW regions. From the derived dispersion equations,it has been seen that there exists such a requency at which the corresponding isorequency suracesor FW and BW regions coincide. Teoretically, this can provide distortionless ocusing o the prop-agating modes, i the wave impedances o the FW and BW regions are also well matched. Impedancematching becomes even more important when the evanescent modes are taken into account. Teoret-ically, it was shown that the wave impedances can be matched at least within % accuracy or better, ithe characteristic impedances o the Ls are properly tuned. However, rom a practical point o view,accuracy better than % is hardly realizable. It has been shown that decreasing the thickness o the


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TL

TL

TL

TL

TL

TL TL

TL

Y

Z/2 Z/2

Z/2

Z/2

Z/2

zx

y

Z/2

Z/2

Z/2

(1,,)

(,+1,)

(+1,,)

(,,)

(,1,)

FIGURE . Unit cell o a D BW L network (enclosed by the dotted line). Te Ls and impedances along the

z-axis are not shown. Ls have the characteristic impedance Zand the lengthd (dis the period o the structure).

(aken rom Alitalo, P., Maslovski, S., and retyakov, S.,J. Appl. Phys., , , . With permission.)

BW region reduces the negative effect o the impedance mismatch, whereas the amplication o theevanescent modes is preserved.In [], the design and experimental realization o a D superlens based on LC-loaded Ls were

presented. A D prototype was designed (Figure .a). Te structure was excited by a coaxial eed(SMA connectors) connected with the edge o the rst FW region, as shown at the bottom oFigure .b. o change the position o the excitation, our SMA connectors were soldered to thestructure.

Te measured electric eld distributions on the top o the structure are shown inFigure ..Temaximum values o the amplitude occur at the back edge o the BW region (as expected rom the

(a) (b)

0

0

2639526578

z(mm

)

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13 26 39 52 65 78 91 104117x(mm)

BWregion

FIGURE . (a) D prototype o L-based metamaterial; (b) D prototype o L-based metamaterial. (aken rom

Alitalo, P., Maslovski, S., and retyakov, S.,J. Appl. Phys., (), . With permission.)


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BW-region BW-region

(a)

z(m

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78 91 104 117

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FIGURE . Measured amplitude o the vertical component o electric eld on top o the D structure at

f = MHz. Fields are normalized to the maximum value. (a) Symmetrical excitation by two sources atx= mm,

z= . mm, andx = mm, z= . mm. (b) One source atx = mm, z= . mm. (aken rom Alitalo, P.,

Maslovski, S., and retyakov, S.,J. Appl. Phys., (), . With permission.)

theory). In Figure .b, the point o excitation is displaced rom the middle to show that the effectsseen are not caused by reflections rom the side edges. It is clear that both propagating and evanescentmodes are excited in the structure, because the elds do not reveal a signicant decay in the rst FWregion (evanescent modes decay exponentially). Tere is a remarkable growth o the amplitude inthe BW region, since only evanescent modes can be amplied in a passive structure like this. Teexperiment did not show any noticeable reflections at the FW/BW interaces, which implies a goodimpedance matching between the two types o networks.

o realize a D structure, a combination o two and three previously observed D structures wasmanuactured. o connect these layers, vertical sublayers o height . mm were soldered betweenthem. InFigure .b,the geometry o the structure is presented: only bottom horizontal layer and vertical sublayers are shown. Te resulting D structure is isotropic with respect to the wavespropagating inside the Ls. Te distance between adjacent horizontal and vertical nodes remains thesame, and the vertical microstrip lines are also loaded with capacitors in the BW region.

Te experiments with prototypes show BW propagation and amplication o evanescent waves in

the L-based structures.

21.4 Two-Dimensional Structure of DNG Metamaterial Basedon Resonant Inclusions

Attempts to create an isotropic metamaterial resulted in the idea o using resonant inclusions asconstituent particles arranged in a regular structure []. A medium composed o a periodiclattice o resonant particles considered as scatterers generates dielectric polarization and magnetiza-tion according to the distribution o the scatterers and their polarizabilities. A mixture consisting oan array o scatterers embedded in a host media is an effective medium relative to the propagatingwave. When the size o the scatterers is small compared with the wavelength in the host materialand is not small in the material o the scatterers, the effective medium parameters become requencydependent. Within a limited requency range, electric and/or magnetic polarizabilities o inclusionsexhibit a characteristic resonant behavior, and the media yield effective negative permittivity andpermeability.


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As a D isotropic metamaterial, a regular array o dielectric rods is considered. In order to obtaindielectric and magnetic dipoles, one has to excite electric and magnetic resonances into the DRs [].Te D isotropic DNG material consists o an array o dielectric rods with different radii, so that twodifferent types o resonances lead to the DNG response o the medium. Te magnetic resonance isexcited in the roads o smaller diameter, which behave as magnetic dipoles. Te electric resonance is

excited in the roads o larger diameter, which behave as electric dipoles. Te simplied structure o theregular array o the DRs placed between the perect magnetic walls (PMW) and the perect electricwalls (PEW) is shown in Figure .a. Te dispersion diagram has been calculated by analytical ull-wave simulation (Figure .c). Te negative slope o the dispersion characteristic demonstrates theDNG properties o the designed medium. Te transmission coefficient o the structure (Figure .b)reveals a pass band in a limited requency range conditioned by the resonant characteristics o thetwo cylindrical resonators.

Dielectric cylinders can also be placed in a cutoff parallel plate waveguide []. In this case, thecollective macroscopic behavior o the DR lattice under E resonance gives negative effective per-meability, whereas the parallel plate waveguide below the cutoff requency or the undamental E

modes shows negative effective permittivity, which leads to the lef-handedness. Te D triangu-lar prism o the proposed lef-handed waveguide that is sandwiched by the right-handed parallelplate waveguides provides the numerical and experimental demonstrations o the negative reractionor the propagated waves. Dispersion diagrams obtained show the D isotropic and lef-handedpropagation characteristics o the proposed structure.

Resonance phenomena in metamaterials constructed as an array o dielectric rods are studiedin [] by means o numerical modeling using the nite-difference time-domain (FDD) method.

(a)

a

Port 1 Port 2

x

y

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Frequency

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11

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10.2 10.4 0.0 0.2 0.4 0.6 0.8 1.0

N= 5N= 7

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|S21|

FIGURE . (a) Cross-section o simulation domain. (b) Numerically calculatedS-parameters o a lattice o cylin-

ders[p = ( j.), h = , a = .mm, a = . mm, lattice constants = mm]. (c) Dispersion

curves or a lattice o cylinders calculated or different numbers o cylinders(N).


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f = 16.9 GHzf = 16.0 GHz

Positive beam refraction Negative beam refraction

p = 77, h =7,8

FIGURE . Wave propagation through the prism ormed by a regular array o dielectric cylindrical resonators.

(aken rom Semouchkina, E.A., Semouchkin, G.B., Lanagan, M., and Randall, C.A., IEEE rans. Microwave Teory

echnol., , , April . With permission.)

Te authors suggested that coupling between the resonators could affect the EM response o themetamaterial in a way similar to that observed at the BW propagation in DNG media. Couplingbetween resonators causes resonant mode splitting and promotes the channeling o electromagneticenergy by coupled elds, which contributes to the ormation o the bands with enhanced transmis-

sion. Te all-dielectric metamaterial consists o closely positioned dielectric cylinders embedded in alow-permittivity matrix. Te ability o the all-dielectric metamaterial to provide negative reractionhas been demonstrated by EM simulation o the wave propagation through the prism o metama-terial with a rhombus lattice at requencies close to the rst higher-order resonance o the DRs(Figure .): the negative beam reraction is observed at f=.GHz.

Tere is one more way to provide simultaneous negative effective permittivity and permeabilityo articial media: mutual constitutive particle interaction []. As an example, the clustered dielec-tric particle (CDP) metamaterial, constituted by the periodic repetition o a molecule-like cluster odielectric atom-like particles, is explored. Te structure consists o clusters o coupled DRs (cubes,or example) arranged along a periodic lattice. Te clusters may be seen as molecules, whereas the

DRs may be seen as atoms or particles, by analogy with natural materials. It is thereore expectedthat this CDP structure could exhibit some properties identical to those o natural materials, such aselectromagnetic homogeneity, and in addition metamaterial properties, such as negative reraction,under appropriate design conditions.

21.5 Three-Dimensional Isotropic DNG Metamaterial Basedon Spherical Resonant Inclusions

Isotropy o a DNG structure is provided by the symmetry o the structure and by the symmetry o

the components constituting the structure.

21.5.1 Symmetry of the Bispherical DNG Structure

Let us consider two sets o the spherical particles arranged in the NaCl structure (Figure .). Tisstructure is a member o the cubic system o symmetry and pertains to the class mm. In the case


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2a1 2a2

s

s

s

FIGURE . A cell o the periodic composite medium consisting o two sublattices o dielectric spherical particles

with different radii embedded in a host.

o cubic symmetry, the second-rank tensors o all physical parameters o the media are diagonal andcharacterized by the components o the same values []. Tus, the permittivity and permeabilitytensors have the ollowing orms:

=

e e e

, =

e e e

, (.)

where the subindices eff are introduced to stress that the permittivity and permeability are obtainedas a result o averaging the electric and magnetic polarizations o spherical particles embedded inthe matrix. Body-centered and ace-centered structures are characterized by the same orm o thesecond-order tensor as the simple cubic structure []. For averaging the polarization o sphericalparticles embedded in the matrix, one needs to nd the volume o the matrix alling on each particleconsidered. For a lattice o cubic symmetry, the volume o the unit cell is evaluated as s, wheres isthe distance between the nearest neighbors o the two-component crystal lattice (Figure .).

We should stress that the isotropy o the media considered is valid only or the second-rank tensors.I one considers phenomena like dielectric nonlinearity or electrostriction, which are described by

ourth-rank tensors, the specic anisotropy o the media ormed by the embedded spherical particlesshould be taken into account.

Te idea o using magnetodielectric spherical particles as constitutive particles or articial meta-material belongs to Holloway []. Te modeling o the electromagnetic response o sphericalinclusions embedded in a host material (Figure .a) is based on the generalized Lewins model [].Te spherical particles with radiusaare arranged in a cubic lattice with the lattice constant s.

Te incident electromagnetic plane wave with wavelength propagating in the host materialexcites certain modes in the particles. Tese modes are not strongly eigenmodes o spherical DRs,but they can be specied as HorE modes at the requencies that are close to the spherical cavityeigenrequencies.

In , the isotropic structure suitable or a practical realization was introduced in [](Figure .b). It was suggested that the articial material is composed o two sets o dielectricspheres embedded in a host dielectric material. Te spheres are made rom the same dielectricmaterial and have different radii. Te dielectric constant o the spherical particles is much largerthan that o the host material. Te wavelength inside the sphere is comparable with the diameter o


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15.0

20.0

25.0

0.04 0.06 0.06 0.07

tan=0.00

tan=0.01

tan=0.02

tan=0.04

tan=0.20

k0a k0a

0.08 0.08 0.090.10 0.10

FIGURE . (a) eand e orv = ., h = h = , p = , p = . Te dashed-dotted lines are the

asymptotes. (b) e or v=., h =h =, p =, p =. (aken rom Holloway, C. and Kuester, E.,IEEE rans.

Antennas Propag., , , October . With permission.)

Te unctionF()has a resonance nature. It becomes negative at some requencies and in someranges o . Negative values o unctionF() are necessary (but not sufficient) demand or eandeto be negative (or real values o and o the matrix media and spheres in them) (Figure .a).

Decreasing the inclusion volume ractionvhas the effect o narrowing the band o requencies, orwhich eand ebecome negative. Besidesv, the product o pand pinfluences the bandwidthand location o the resonance. By making this product smaller, the rst resonance o eand e ismoved to larger values o the productk aand the requency bandwidth over which the permittivityand permeability are negative increases.

Losses in material o particles decreases the effect o resonance behavior on the composite. Tiseffect is shown in Figure .b. In this gure, the dependence o the real part o the effective permit-tivity on a normalized requency is shown or several different values o the dielectric loss tangent

o the inclusions dened as tan = pp

. Te dielectric loss tangent o the matrix as well as the mag-

netic loss tangents o both materials are taken to be zero. Notice that, or this example, the real parto the effective permittivity can still be negative or loss tangents o the inclusions, as large as ..However, or larger values, the resonance is damped out and the real part o the effective permittivityremains positive. Tis shows that i the inclusion (i.e., the spherical particle) becomes too lossy, theDNG properties cannot be realized.

21.5.3 DNG Medium Composed of Dielectric Spheres with DifferentRadii (GarnetMaxwell Mixing Rule)

Te particles used in articial isotropic media on electromagnetic resonance spheres [] are imprac-tical because o their simultaneous high values not only o permittivity but o permeability as well.Another model suitable or practical realization has been introduced by Vendik and Gashinova [].In that case, two types o spheres were used with different radii but made rom the same material(Figure .b). In such a structure, one can observe the resonance o the E mode in one type ospherical particles and the resonance o the M mode in other types o spherical particles at the


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same given requency. In this case, it is possible to realize the negative dielectric permittivity as aresult o a contribution o the resonance o the M mode and the negative magnetic permeabilityas a result o a contribution o the resonance o the E mode. Te diameters o the spheres must bemuch smaller than the wavelength in the matrix (host material) but comparable to the wavelength inthe material o the spheres. Ferroelectric single crystal or ceramic samples with dielectric constants

can be used as a material or manuacturing the spherical particles.Te numerical simulation o two-lattice structures by eigenmode analysis and FDD analysishas also been implemented in []. Te resonant behavior o a single sphere embedded in a hostmaterial was investigated. Te cell under simulation with two pairs o PEW and PMW boundinga propagation region is shown in the inset o Figure .. Te eld pattern o the exiting planewave is equivalent to a waveguide excitation o the structure. Tis denition o the elementary cellallows us to obtain a scattering parameter (Figure .a and b) o a two-port device. Extractiono the effective dielectric permittivity and magnetic permeability is based on the transormationoZ-parameters o such an effective L section. Te DNG behavior is observed at f=.GHz(Figure .c and d).

In [,], Lewins equations were applied directly to the system o two sets o spheres, and theelectric polarizability o spheres in the magnetic resonant mode was not taken into account. How-ever, the electrical properties o these spheres can have a signicant effect on the effective permittivityo the composite. Tis affects especially the low-requency limit, which then approaches the classi-cal MaxwellGarnett mixing rule. In [], a new model was presented, which takes this effect intoaccount. Equations are validated both analytically and numerically. Scattering rom a single spherewas calculated both analytically rom the ull Mie theory and numerically.

PMW

1.00

0.75

0.50

0.25

0.0062 63 64 65 66

Frequency (GHz)(a)

|S11

|,|S

21

|

67 68 69 70

PEW

a1 = 100 m

S11S21

60

40

20

0

20

40

62 63 64 65 66

Frequency (GHz)(c)

Re

(ef

f),

Im(

eff)

67 68 69 70

Re (eff)

Im (eff)

1.00

0.75

0.50

0.25

0.0062 63 64 65 66

Frequency (GHz)(b)

|S11

|,|S

21

|

67 68 69 70

a2 = 70.5 m

S11

S12

a1 = 100 m

6

3

0

3

662 63 64 65 66

Frequency (GHz)(d)

Re

(ef

f),

Im(

eff)

67 68 69 70

Re (eff)

Im (eff)

a2= 70.5 m

FIGURE . Scattering parameters and extracted effective permittivity(e)and effective permeability(e )o

a single sphere with = , radiusa = m (a,c), anda = . m (b,d).


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An effective medium model or a composite consisting o two sets o resonating spheres isdened by

e he+ h =

feh

h+ p F()h pF() +

fmh

h+ pF()h pF() , (.)

e e+ = fm

+ F() F() , (.)

fe=

a

s, fm=

a

s, (.)

F() = (sin cos)( ) sin + cos , (.)where

=k app=k appais the radius o spheres with the electric resonanceais the radius o spheres with the magnetic resonancefeand fmare the volume ractions or the corresponding spheres

In Figure ., an example o effective permittivity as a unction o the volume raction o spheresis shown. Te solid line represents egiven by Equation ., and the dashed line,

e, is calculated

using the method in [,], where the electric polarizability o spheres in the magnetic resonance

Re (eff), Equation 21.4

Im (eff), Equation 21.4

Re (eff)

Re (eff)

14.7250

200

150

100

50

0

50

100

150

14.75 14.8

Frequency (GHz)

eff

14.85 14.9

FIGURE . Te effective permittivity eas a unction o the requency calculated by Equation . compared

with the effective permittivity ecalculated without taking into account the electrical polarizability o spheres in

the magnetic resonance. In this case, the second term on the right-hand side o Equation . is zero. p=

( j. ), h = , fe = ., fm = ., a = . mm, a = .

mm. (aken rom Jylh, L.,

Kolmakov, I., Maslovski, S., and retyakov, S., J. Appl. Phys., (), -, . With permission.)


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mode is not taken into account. It can be seen that the resonant requency slightly shifs when theimproved mixing equation is used.

21.5.4 DNG Medium Composed of Dielectric Spheres withDifferent Radii (Electromagnetic Wave Diffraction Model)

A numerical analysis o a bispherical structure [] revealed that the intererence o the adjacentspherical particles is negligibly small. Tis makes it possible to solve the electromagnetic problemor each sphere independent o the influence o all others. We consider the diffraction o a planeelectromagnetic wave on a dielectric sphere using the approach o Stratton []. Some results osolving this problem as applied to the bispherical structure were presented in [,].

Let us consider the diffraction o a plane electromagnetic wave with the amplitude o electric eldElinearly polarized along thex-axis. Te wave propagates along thez-axis (Figure .):

E (z, t) =e xEe i(tk z) , H(z, t) =e y k

oEe i(tk z) . (.)

Te wave numberkis dened later.In order to ulll the boundary conditions on the surace o the spherical particle with respect to

the tangential components o the electric and magnetic elds, expansion o the incident plane wave interms o the spherical unction is used. Te spherical modes inside the sphere and the spherical modespropagating in open space outside the sphere are taken into consideration as well. Te boundary con-ditions give rise to two pairs o nonhomogeneous equations with respect to the complex amplitudeso the spherical unctions inside and outside the spherical particle.

Te elds inside the spherical particle are presented in the ollowing orms:

E(in)=E e i t

n= in n

+

n(n + )a(in)n m onib(in)n n en , (.)H(in)= k

E e

i t

n=in

n + n(n + )b(in)n m en+ia(in)n n on , (.)

wherem on ,m ,nand n oe ,m ,nare spherical wave unctions (odd and even) []. As ar as the incidentwave in open space is linearly polarized, the number m= is taken in Equations . and ..Te wave numbers are dened as

k

=

p , k

=

h , (.)

P

(a) (b)TE111 TM111

z

M

y

x

E

H

k

FIGURE . Spherical particle in the eld o a linearly polarized electromagnetic wave and eld distribution in

the equatorial plane. (a) Dipole momentum o electric polarization o the particle P and dipole momentum o mag-

netization o the particle M. (b) Mode charts o the dominant E and M modes in a spherical resonator with

magnetic walls. Solid and dashed lines show the magnetic and electric eld lines, respectively.


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where and are the dielectric permittivity and permeability o ree space. Te diffracted eldoutside the spheres is given in [].

Te solutions o the system o equations specied by the boundary conditions dene the ampli-tudes o the waves inside the spherical particle in the ollowing orms:

. For the waves o magnetic (transverse electric) type (Er=,Figure .b, lef),a(in)n = jn()[h

()n()] h()n()[jn()]

jn(N)[h()n()]h()n()[Njn(N)] (.)

. For the waves o electric (transverse magnetic) type (Hr=, Figure .b, right),

b(in)n = jn()[h

()n()] h()n()N[jn()]

Njn

(N

)[h

()n

(

)]

h()n

(

)[Njn

(N

)]

(.)

where=k a, ais the radius o the spherical particlejn(z)is the spherical Bessel unctionh()n(z)is the spherical Hankel unction o the rst order

the sign[]means the differentiation with respect to or NN=kk

Figure . represents the distribution o the electromagnetic eld components in line withEquations . through . or the spheres with the dielectric permittivity p= surroundedby the air(h=). Diagrams were plotted or values o polar angles = and =.

Analysis o Equations . and . is ollowed by the two important conclusions:

. At certain requencies, the modulus o the denominators o the ractions (Equations .and .) become minimum and corresponds to the resonance phenomena, but becauseo the complex nature o the Hankel unctions, they do not lead to singularities.

. Te imaginary components o the Hankel unctions determine the quality actor o theresonator, which is nite even in the case o lossless material o the spheres. Physically

this can be explained by the loss caused by the radiation o the diffracted waves outsidethe sphere.

21.6 Effective Permittivity and Permeabilityof the Bispherical Lattice

For the model o a DNG composite arranged rom the magnetodielectric spherical inclusionsintroduced by Holloway, the effective permittivity and permeability are determined on the effectivemedium theory o the electromagnetic response o inclusions embedded in a host material devel-oped by Lewin (see Section ..). In Section .., the effective permittivity and permeabilitywere ound by application o the classical MaxwellGarnet mixing rule. Now, we consider the rame-work o the electromagnetic wave diffraction model. Te effective permittivity and permeability othe bisphere lattice are determined.


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0 0.5 1 1.5 2.0 2.5 0 0.5 1 1.5 2 2.50

100

200

300

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

20

40

60

0

1

2

3

4

5

(a)

(b)

ErEE

HrHH

Radius (mm) Radius (mm)

Radius (mm) Radius (mm)

a1 a1

a2a2

ErEE

HrHH

FIGURE . (a) Electromagnetic eld components distribution or magnetic type waves;Er= inside the sphere.

(b) Electromagnetic eld components distribution or electric type waves;Hr = inside the sphere.

21.6.1 Electric and Magnetic Dipole Moments of Spherical Resonators

Te spherical particle electric dipole momentumD(E)x oriented along thex-axis and the magnetic

dipole momentumD(M)y oriented along they-axis(Figure .a) are calculated as ollows:

D(E)x

=p

Vsph E(in)

(r, ,

) ex

E

dv, (.)

D(M)y =

VsphH(in)(r, , ) ey k

Edv. (.)

While integrating the scalar product o basis vectorser,e , ande ,exandeyshould be taken intoaccount.

Te averaged macroscopic magnetization and averaged macroscopic electric polarization can beound as the corresponding dipole momentum divided by the volume o the cell containing thedipoles []. Tus, one obtains the relative effective permittivity and permeability:

(e)r () = D(E)x ()s E + h , (.)

(e)r () = D(M)y ()s E

k

+ (.)


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eff < 0 Re (eff)

Re (eff)eff < 0

20

10

10

20

eff,e

ff

9.96 10.02 10.08

Frequency (GHz)

0

FIGURE . Te effective permittivity and permeability o bispherical medium versus requency.

Afer calculating the integrals in Equations . and . in accordance with Equations . and.,where the spherical wave unctions should be used [], one obtains

(e)r () = a

sp b(in) (k a) I(k a), (.)

(e)r () = a

s

pa(in) (k a) I(k a). (.)Here,I() is the result o integration over the volume o the particle, aandaare radii o the parti-cles,a > a. Te unctionI() has been approximated in the region < < by the ollowing simpleormula:

I() =.(. ) + .( ) . (.)Te requency dependence o the wave amplitude or the excited modes a

(in) andb

(in) determines

the requency dependence o (e)r ()and (e)r (). Considering the structure composed by two

sublattices o the dielectric spherical particles with different radii, we can adjust these radii to obtainthe same resonant requencies or the H mode in the smaller sphere and the E mode in the

larger sphere. Figure . presents the simulated requency dependence o (e)r

(

)and

(e)r

(

)ora =. mm, a = .mm, s = mm, dielectric permittivity o the particle p = and tan = , and permittivity o the matrix h = .One may see that at the requency slightly above f= GHz both the permittivity (e)r and the

permeability (e)r are negative. Tus, in the rather narrow requency band around f= GHz, the

existence o isotropic DNG has been theoretically substantiated. A negative reraction bandwidthdepends on the permittivity o the spherical particles. Te smaller the value o permittivity o thedielectric spherical particles, the wider is the requency range, where both the effective permittivityand permeability are negative. Te dependence o the negative reraction index bandwidth on thepermittivity o the material constituent particles is presented inFigure ..

21.6.2 Comparison of the Effective Permittivity and PermeabilityObtained with Different Models

Different analytical models or the DNG medium description were introduced to describe thestructures with sets o spherical particles [,,,,,]. Te modeling o the electromagnetic


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0

10

20

30

4050

60

70

80

90

100

200 300 400 500 600 700 800 900 1000

Sphere permittivity

Negativeindex

ban

dwidth(MHz)

10 GHz, tan = 0.001

10 GHz, tan = 0.003

10 GHz, tan = 0.005

15 GHz, tan = 0.001

15 GHz, tan = 0.003

FIGURE . Dependence o negative reractive index bandwidth on spherical particle permittivity or two

resonance requencies, f =GHz f = GHz, and different loss levels.

response o spherical inclusions embedded in a host material [,,] is based on the generalizedLewins model []. Originally, the Lewins model has been specied only or spherical particles withthe same radius a arranged in a cubic lattice with the lattice constant s. Te spheres are assumedto resonate either in the rst or second resonance mode o the Mie theory []. Expansion o themodel or the case o two sublattices o dielectric spherical particles with different radii makes pos-sible the description o the DNG media [,]. Te properties o DNG media required could beobserved in the requency region, where the resonance o theE mode in one set o particles and the

resonance o theHmode in another set o particles are excited simultaneously. Te improved modelo the bispherical structure was presented in []. Te effective permittivity eor a material withtwo types o inclusions having two different electric polarizabilities was calculated rom the general-ized ClaussiusMossotti relation, taking into account the electric polarizabilities o the spheres in themagnetic resonance and in the electric resonance mode. Consideration o the remaining static elec-tric polarizability o spheres in the magnetic resonance modes, which is not equal to zero as in [],is important.

Let us compare the requency dependences o both the effective dielectric permittivity and theeffective magnetic permeability calculated by using different models. Figure .presents an exampleo effective permittivity and permeability as a unction o the requency or three different analytical

models: () Lewins model [], () the improved mixing rule model [], taking into account theelectrical polarizability o spheres in the magnetic resonance, and () the diffraction model [].

Te parameters o the constituent materials are p = , h = , tan = , p = h = , a=. mm, a = .mm, and s = mm. Te results are in general similar, but they differ in theresonant requency and the magnitude o effective electromagnetic parameters o the medium. Teresonant requency is slightly shifed in comparison with Lewins model when the improved mixingequation is used and is shifed more remarkably or the diffraction model.

21.6.3 Results of the Full-Wave Analysis

Afer analytical calculations based on the diffraction model, the structure was simulated by ull-waveanalysis []. Te simulated structure consists o quarters o spheres placed in the dielectric material(Figure .). In the case o appropriate boundary conditions, simulation o this model should givethe same results as those or the innite D structure. Four quarters o the spheres o different radiiwere placed in a host medium with the permittivity and the permeability equal to unity bounded with


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Frequency (GHz)

100

50

0

50

100

100

50

0

50

100

9.97 9.99 10.0210.004

Frequency (GHz )

9.99 9.995 10 10.005

Re (eff1)

Im (eff1)

Re (eff3)

Im (eff3)

Re (eff2)

Im (eff2)Re (eff1)Im (eff1)Re (eff2)Im (eff2)

(a) (b)

FIGURE . (a) Te effective permittivity as a unction o the requency or three different analytical models:

() Lewins model [] e ; () the improved mixing rule model (taking into account the electrical polarizability

o spheres in the magnetic resonance), [] e; and () the diffraction model. (b) Te effective permeability as a

unction o the requency or the three different analytical models: () the mixing rule model [,], e ; and ()diffraction model, e. Te parameters o the constituent materials are as ollows: p = , tan =

, h = ;

p =h = , a =. mm, a = . mm, ands = mm.

(a) (b) (c)

=

s

FIGURE . Single cells o single spherical and bispherical structures. Boundary conditions are two PEWs and

two PMWs on opposite sides: (a) large spheres, (b) small spheres, and (c) mixed structure.

two PEWs and two PMWs on the opposite sides, respectively. First, our quarters o the larger sphere(radiusr= . mm, permittivity o particle p=, and loss actor tan = .) and thenour quarters o the smaller sphere(r=.mm, p =, tan =.)were modeled. Ten,the structure consisting o sets o spheres o two radii was modeled.

Te results or scattering matrix elementsS andS are shown inFigure ..Te stop bandis observed near the requency GHz in case o negative permittivity or permeability only. For themedium with the set o both spheres, a narrow pass band near the requency GHz is observed. Terequency range o the electromagnetic wave with an enhanced transmission coefficient correspondsto the DNG characteristics o the structure. Te resonance requency has the same value as the onecalculated analytically.

Field patterns inside the unit cells are presented in Figure .. Te magnetic eld distribution onthe side plane o the structure (Figure .a) and the electric eld distribution on the top plane othe structure (Figure .b) represent the M mode in a larger sphere (electric dipole momentum)and the E mode (magnetic dipole momentum) in a smaller sphere.


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Large spheres

Smallspheres

Set ofspheres

9.92

S12, dB

S11, dB

Large sphere, electric resonance

Mix structureSmall sphere, magnetic resonance

0

10

20

30

40

0

10

20

10.04109.96 10.08

9.92 10.04109.96 10.08

=~

=

+

FIGURE . Simulation results or a mixed structure with the ollowing parameters: p = , h = ;

a =. mm, a =. mm, s= mm, and tan= ..

(a)

2 2

11

(b)

FIGURE . Field patterns inside the unit cell o a structure: (a) magnetic eld distribution on the side plane

(magnetic dipoles) and (b) electric eld distribution on the top plane (electric dipoles).

21.6.4 Results of the Experiment

In order to veriy the resonance behavior o the spheres, an experiment was conducted []. TeNetwork Analyzer Agilent ES was used or the measurement oS-parameters o the spheres.A spherical particle was placed inside the rectangular waveguide. Different samples were used in theexperiment (two large spheres with radius . mm and two small spheres with radius . mm). Teresults o the experiment were compared with the data obtained previously by modeling. Te trans-mission and reflection coefficients or the small sphere (Figure .) reveal the resonance behaviorat the requency . GHz. Here, the gray solid and dashed lines represent the measuredS()andS() parameters, and the black line corresponds to the simulated results. Te transmission andreflection coefficients or the large sphere exhibit two resonances (Figure .): magnetic resonanceat the requency . GHz and electric resonance at . GHz.


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10

20

30

40

50

16.7

S12(21), dB

S11(22), dB

First, magnetic resonance

17.3

0

1

2

3

10 11 12 13 14 15 16 17 18 19 20

10 11 12 13 14 15 16 17 18 19 20

16.7 17.3

FIGURE . S-parameters or the small sphere. Te solid black line represents the simulation results, and the gray

solid and dashed lines correspond to the measured characteristics. p =, r= . mm, tan ..

10

20

30

40

50

60

70

0

1

2

3

4

510 11 12 13 14

First, magnetic resonance

Second, electric resonance

15 16 17 18 19 20

10 11 12 13

S11(22), dB

S12(21), dB

14

11.89 GHz 16.8 GHz

15 16 17 18 19 20

FIGURE . S-parameters or the large sphere. Te solid black line represents the simulation result, and the gray

solid and dashed lines correspond to the measured characteristics. p =, r= . mm, tan ..

In the experiment, the magnetic resonance requency in the small sphere does not coincide withthe electric resonance requency in the large sphere, because the radii were not adjusted accurately,and the possible DNG behavior o the structure consisting o these samples was not observed. Never-theless, the experiment proved the validity o an analytical diffraction model describing the resonancebehavior o the dielectric spheres [].


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21.6.5 Influence of Distribution of Size and Permittivity of SphericalParticles on DNG Characteristics

Let us consider a DNG medium composed o dielectric spheres with different radii described bythe MaxwellGarnet mixing rule (seeSection ..). It has been shown in [,] that the randomdistribution o the spherical particle sizes caused by manuacturing inaccuracy may affect the values

o the effective permittivity and permeability o the DNG medium.In order to take into account the random size distribution o the spheres, one should rewrite

Equation . in the orm []

e he+ h =

h K

k=

fe kG(k)+

Ll=

fmlG( l) , (.)

wherekis the number o spheres in the electric resonance

lis the number o spheres in the magnetic resonance,G

(

) = hp F() h

+p F

(

)Tis is the solution or the effective permittivity o the structure with two sets o spheres, where

one set o spheres is in the magnetic resonance and the other set is in the electric resonance. Teeffective permeability can be calculated in a similar way.

An example o how a normal size distribution, N=

exp (rr)

o spheres with hal-

value widths = eand = mand expectation valuesr = aandr = aaffects the effective materialparameters is presented in Figure .. Te lef-hand side o Figure . describes the spheresthat are normally distributed with the hal-value widths m = e = m. Te expected values o thesphere radii area

=. mm anda

=. mm. Te size distributionNor eis also shown. On the

right-hand side, everything is the same, except the hal-value widths e = m = m.Te hal-value width e = m = m (Figure ., lef) does not increase the loss actor o thestructure signicantly, but in the case o the hal-value width o e = m = m (Figure ., right),the imaginary part becomes remarkably larger and edoes not exhibit negative values.

a1(mm)

N/max

(N)

N/max

(N)

9.5 10

Frequency (GHz)

10.59.510

8

6

4

2

0

1

0.5

03.15 3.2

Re ()Im ()Re ()Im ()

24

6

8

10

10

8

6

4

2

0

1

0.5

03.15 3.2

Re ()Im ()

Re ()

Im ()

24

6

8

10

10

Frequency (GHz)

10.5

a1(mm)

FIGURE . Te effective permittivity as a unction o the requency calculated using Equation . with p =

( j.), h =, and lling ratios fe =%, fm =%. (aken rom Jylh, L., Kolmakov, I., Maslovski, S.,

and retyakov, S.,J. Appl. Phys., (), -, . With permission.)


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Now, we consider the influence o the size distribution or the DNG medium model using themodel o the electromagnetic wave diffraction on dielectric spheres with different radii. Accordingto Equations . and ., the spherical particle radius influences the values o the effective permit-tivity and permeability. Dielectric sphere radius variation affects the value o the resonance requency,which corresponds to the low-requency threshold o the negative index range. Let us estimate how

the resonant requency depends on the radius o the constituent spherical particles. Te electricalradius o the sphere was dened previously as

N=k a, (.)wherek=pis the propagation constant. Let us rewrite Equation . in this way:

f= N a p , (.)

where fis the requency o the electromagnetic wave.

Values o the electrical radius o the resonant spheres can be calculated rom Equations .and . or the minimum modulus o the denominator in Equations . and .. For a given a,the values o the electrical radius providing magnetic or electric resonance the resonance requencycan be calculated using Equation ..

Dependence o the resonant requency on the sphere radius is shown in Figure .. Tis graphrepresents the dependence o the resonant requency on the spherical particle radius or two valueso particle permittivity, and . According to Figure ., the negative index bandwidth ora DNG medium with spherical inclusions permittivity equal to should be MHz or a GHzresonant requency. Tis implies that the spherical particle radius accuracy should be.m inthis case.

In line with Equation ., the resonance requency is also influenced by the permittivity o thedielectric material o the particles.Figure .represents the dependence o the resonant requencyon the spherical particle permittivity or two different values o radii, and . mm. o avoid re-quency spreading beyond the negative index bandwidth o MHz, the tolerance o the permittivityo material should be.%.

With regard to the possibility o the practical realization o such an articial metamaterial, weshould mention that recent technologies allow the production o dielectric spheres with accuracy

0 0.5 1 1.5 2.00

5

10

15

20

E-mode, p=400

H-mode, p=400

E-mode, p=1000

H-mode, p=1000

r

f

Sphere radius (mm)

Resonance

frequency

(GHz)

FIGURE . Dependence o resonance requency on sphere radius or particles with p= and

p= , f= MHz, r= m.


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E-mode, r = 0.5 mmH-mode, r = 0.5 mm

Permittivity

Resonance

frequency

(GHz)

E-mode, r= 1 mmH-mode, r= 1 mm

200 400 600 800 1000 12000

5

10

15

20

p

f

FIGURE . Dependence o resonance requency on spherical particle permittivity or particles withr= .mm

andr= mm; f =MHz, p = .

r= m. At the same time, the achievable accuracy o permittivity o the dielectric material withr> is about %%. Despite this, it is really possible to select samples with desired values opermittivity among a large number o manuactured samples.

21.6.6 Isotropic Medium of Coupled Dielectric Spherical ResonatorsWhen dielectric spherical resonators are placed close to each other, they begin to interact. Tecoupling between resonators leads to the ormation o new electromagnetic eld distribution in themedia outside the spheres. It becomes possible to get electric and magnetic dipole responses usingonly one type o sphere. Te magnetic dipole comes rom the rst Mie resonance in a dielectricsphere. Te electric dipole is ormed by the sphere interaction. Electric and magnetic dipole existenceprovides a DNG response o the media [].

A D plane structure consisting o closely positioned dielectric spheres has been modeled.I the distance between the spheres is large, there is no wave propagation on a resonant requency

(Figure .a). By decreasing the spacing between the spheres, splitting o the resonance curveoccurs (Figure .b). Te pass band appears near the resonant requency.Figure .represents the phase diagram o the structure considered. Te transverse magnetic

eld component in the ree space is shown on the lef side o the picture. Te right side represents themagnetic eld pattern or the structure containing the regular array o dielectric spheres. It is clearly

010203040

5060

6(a) (b)

6.5 7Frequency (GHz)

7.5 8

0

5

10

15

2025

6 6.5

S21

7Frequency (GHz)

7.5 8

FIGURE . ransmission coefficient or (a) ar and (b) closely positioned spheres.


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Vphase

Vphase

Incidentwave

180 210 240 270 300 330 360

360 270 180 90 0 90 180 270 360 450 360 270 180 90 0 90 180270360450deg.

360

342320298276253231209187165143

12199.277.155.133.111.0

0

deg360

342320298276253231209187165143121

99.277.155.133.111.0

0

FIGURE . Phase distribution in a D plane o dielectric spheres.

seen rom the magnied part o the picture that the phase response o the propagated electromagneticinside the array o the spheres is positive, whereas the phase response outside the structure is negative.Tis means that there is BW propagation in the structure considered.

21.7 Metamaterials for Optical Range

Denitely, one o the most challenging problems is producing a metamaterial or the optical re-quency range. Invisibility, superlenses, and light rays manipulationthese properties in the opticalrange are extremely promising. A model o metamaterial that can exhibit a negative reractive indexband in excess o % in a broad requency range rom the deep inrared to the terahertz region, hasbeen investigated in []. Te avored realization o the structure considered is a periodic crystalwherein polaritonic spheres and Drude-like or plasma spheres are arranged on two interpenetratingsimple cubic lattices. When the differences between the spheres are ignored, the resulting structure

is a ace-centered cubic structure. Te sublattice o polaritonic spheres possesses negative magneticpermeability in certain requency regions, whereas the sublattice o the Drude-like spheres possessesnegative electric permittivity. Both phenomena are the results o the strong single-sphere Mie res-onances. By a suitable choice o materials and parameters, a common region can be ound, withina broad requency range rom the deep inrared to the terahertz region, where both unctions arenegative and the structure exhibits a negative reractive index band in excess o the % bandwidth.

A new concept o metafluidsliquid metamaterials based on clusters o metallic nanoparticlesor articial plasmonic molecules (APMs)has been introduced in []. APMs comprising ournanoparticles in a tetrahedral arrangement have isotropic electric and magnetic responses and areanalyzed using the plasmon hybridization method, an electrostatic eigenvalue equation, and vecto-rial nite-element, requency-domain, electromagnetic simulations. It has been demonstrated thata colloidal solution o plasmonic tetrahedral nanoclusters can act as an optical medium with verylarge, small, or even negative effective permittivity, e, and substantial effective magnetic suscepti-bility,e= e , in the visible or near-inrared bands. Te electric and magnetic responses o thetetramer allow one to construct an effective mediumwith a completely isotropic electric and magnetic


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response. Electromagnetic simulations indicate that achieving e< and e< in colloidal solu-tions o articial molecules should be possible using either sufficiently high concentrations o goldclusters or materials with low-loss negative permittivity.

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Isotropic double negative Metamaterial

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