Fractal NegativeEpsilon Metamaterial
M. Palandoken
*(1)
, H. Henke
(1)
(1)Fachgebiet Theoretische Elektrotechnik, Technische Universität Berlin, Germany Einsteinufer1, !"1#$% Berlin, Germanymerih&tetibm1'ee'tu"berlin'e
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In this paper, a novel fractal metamaterial with negative permittivity is presented. It consists of two pairs of fractal Hilbert curves with antisymmetric current distribution at resonance. The resonant current distribution shows that the structure behaves as an electrically small dipole. This approach, which is different from the conventional Complementary Split Ring Resonator (CSRR based approaches, is described and the electric resonance is illustrated. The negative permittivity is derived from the numerically calculated !"parameters for one unit cell"thic# sample under plane wave e$citation. The unit cell si%e is only &o').*+ $ &o').*+ at resonance. The proposed electrically small structure presents therefore a more homogeneous negative"epsilon material for eft"handed etamaterial applications than the standard CSRR based solutions.
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etamaterials are artificially structured, periodic materials that can be engineered to e$hibit e$otic electromagnetic properties, which could not be observed in nature. These materials can be basically classified into two groups, which are negative permeability and negative permittivity materials. The first class was srcinally reali%ed by split"ring resonators (SRR /)0, and the second class by strip wires /0. 1 third class, so"called eft"handed aterials (H, could be reali%ed with the composition of these two fundamental classes.However, rather than strip wires, more compact negative"epsilon materials can be reali%ed with CSRR designs due to 2abinet3s principle /40. The magnetic resonance of SRR is replaced by the electric resonance of CSRR. In the design, the metal strips in SRR are replaced by slots and the slots are replaced by metal strips as the complementary form of SRR. Thus, CSRR based designs have to be etched in the ground plane, which leads the negative permeability material, SRR, to be structured on a different layer to reali%e H. The main contributions of the present paper are, at first, to propose an alternative method to design negative"epsilon materials without complementary structures. This leads more fle$ibility in the feeding and topological design of metamaterial based components. 2ecause of no SRR formed defects in the ground plane , the possible radiation through the slotted ground at resonance could be suppressed with this new design method. Secondly, as a further step in the miniaturi%ation, this novel artificial dielectric is designed with fractal curves, which are mostly utili%ed for the highly homogeni%ed magnetic metamaterial design /50. The use of fractals in the design increases the degree of homogeni%ation further in the view of e$isting alternatives /60. The paper describes the unit cell geometry and design philosophy of proposed negative"epsilon material. 7e illustrate the electric resonance from numerically calculated electric field. The effective anisotropic permittivity is analytically derived from the numerical data. The negative permittivity in the vicinity of resonance fre8uency for the right polari%ation confirms that the proposed structure can be considered as an electrically small artificial dielectric material.
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The geometry of the artificial dielectric material is shown in 9ig. ). The red colour mar#ed section is a second order fractal Hilbert curve with modified side length ratio. It is surrounded closely with another fractal li#e structure in order to decrease the resonance fre8uency by capacitive coupling inbetween. The curve is connected to its mirror image, which forms the second half of the structure. The mirroring process is important in order to alternate the current distribution at resonance. In that way, the structure is effectively e$cited by the incoming electric field, but not the magnetic field. The substrate material is +.6 mm thic# 9R5 with dielectric constant 5.5 and tan(: +.+. The metalli%ation is copper. The copper line width and minimum distance between any two lines are +. mm. The other geometrical parameters are 
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; +.< mm, 
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; mm. The unit cell si%e is a
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; 6 mm, a
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; 6 mm. =nly one side of the substrate is structured with the prescribed fractal geometry.
9781424448852/10/$25.00 ©2010 IEEE
Fig. 1 *eometr, o te negativeepsilon metamaterial NME#&$A/ !&M/A"&'N! AN &!$!!&'N
In order to observe the electric resonance for the negative permittivity, the structure has to be e$cited by a plane wave, which propagates in %"direction and is polari%ed in y"direction. The blue mar#ed overlapping sections in 9ig. ) show the enhanced capacitively coupled sections for the resonance fre8uency reduction. Thus, in the numerical model, >erfect ?lectric Conductor (>?C at two y planes and >erfect agnetic Conductor (>C at two $ planes are imposed as the boundary conditions. Two port simulation is done with 9? based commercial software H9SS. The simulated S" parameters are shown in 9ig. . However, additional S parameter simulations have been carried out for the same proposed structure under remaining $" and %"polari%ed plane waves. These simulations are important to investigate which polari%ation could e$cite the structure as an effective electrically small dipole at the lowest resonance fre8uency for the higher degree of homogeni%ation. 7hile there is no electric or magnetic resonance at a lower fre8uency than .@6 AH% for $" and %"polari%ed plane waves, the above mentioned polari%ation is the right one for more homogeneous negative"epsilon material.
Fig. 0 "ransmission (red) and relection (l2e) parameters
The resonance fre8uency is .@6 AH%. In order to understand why the electric resonance is more effectively e$cited, the current distribution has to be investigated along with the electric field distribution at resonance. 1s shown in 9ig. 4.a, the current distributions on the upper and lower half of the structure are antisymmetrical due to the mirror geometry. The induced magnetic fields are oppositely directed and result in a small net out"of"plane magnetic dipole moment. However, as shown in 9ig. 4.b, due to the electrically coupled sections in the middle part, y" directed electric field results the structure to have 8uite similar field distribution as a resonant y"directed electric dipole. The transmission minimum is due to the depolari%ation effect of this e$cited dipole on the incoming electric field.
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(a)(1)Fig. 3 (a) !2race c2rrent distri12tion and (1) electric ield distri12tion on ,4 plane at resonance
Therefore, this artificial dielectric is regarded as a negative permittivity material in a certain fre8uency band. =n the other hand, the e$cited electric dipole is a hori%ontally directed electric dipole rather than vertically directed dipole, which results from the conventional CSRR based reali%ations due to the slots etched in the ground plane. This has the advantage of e$citing the structure effectively with inplane feeding line on the same substrate layer. 1dditionally, to prove analytically the negative permittivity in the vicinity of the resonance fre8uency, the effective parameters are retrieved ne$t. 1s a conventional method, they are retrieved from S parameters of one unit cell thic# sample under plane wave e$citation /<0. However, the calculated effective parameters are only assigned to this one sample. Bue to the inherent periodicity of artificial structures in the propagation direction, we e$ploit an alternative procedure, which has been introduced recently /50. 7e calculate the dispersion diagram of one unit cell with certain phase shifts in the propagation direction with >?C and >C boundary conditions. Then, as a first step, we transform the numerically calculated ! parameters into 12CB parameters asand use the 2loch"9lo8uet theorem to derive 2loch impedance, !
2loch
and comple$ propagation constant, D from 12CB parameters with the period d as in /@0,The effective relative permittivity can then be calculated from 2loch impedance and comple$ propagation constant with free space wave number, #
o
and line impedance, !
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as The effective relative permittivity,
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due to the inherent anisotropy of the structure. 2ecause there is no electric or magnetic resonance in the vicinity of .@6 AH% for $" and %" polari%ed plane waves, the structure is positive"epsilon material for these polari%ations as e$pected from S parameter simulations.
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$'N$/!&'N
1 novel negative"epsilon metamaterial is presented. It is based on fractal Hilbert curves with mirror symmetry. The anisotropic permittivities were retrieved from numerical calculations. The electric resonance and negative permittivity were verified for a certain polari%ation due to inherent anisotropy. The unit cell si%e is appro$imately )'(( of free space wavelength at the resonance fre8uency, (.@6 AH%. The simulation results validate the negative"epsilon property of this electrically small structure. The proposed geometry is an important contribution to design more homogeneous artificial dielectrics than the conventionally used CSRR based structures. Bue to no slotted SRRs in the ground plane as in CSRR based designs, H microwave components with reduced bac# radiation could be designed on the same substrate layer with magnetic metamaterials.
#EFE#EN$E!
/)0 R. Smith, 7. ). >adilla, B. C. Fier, S. C. Gemat"Gasser and S.Schult%, Composite edium with Simultaneously Gegative >ermeability and >ermittivity,
/hysical 0evie1 2etters
, Fol. *5,5)*5, (+++/(0 2 >endry, 1 Holden, B Robbins and 7 Stewart, ow fre8uency plasmons in thin"wire structures,
3ournal' /hys'4 +on(ens' 5atter
18
5@*6"5*+J, )JJ*/40 F. CrnoKevic"2engin , F. Radonic and 2. o#anovic Left"handed microstrip lines with multiple complementary split"ring and spiral resonators,
5icro1' 67t' Technol' 2ett
., vol. 5J, pp. )4J), un. (++@/50. >alando#en, H. Hen#e, L9ractal Spiral Resonator as agnetic etamaterial, in press/609. 2ilotti , 1. Toscano and . Fegni LBesign of spiral and multiple split"ring resonators for the reali%ation of miniaturi%ed metamaterial samples,
8EEE Trans' )ntennas /ro7ag'
, vol. 66, pp. ((6*, 1ug. (++@ /<0 B. R. Smith, B. C. Fier, Th. Moschny, and C. . Sou#oulis, L?lectromagnetic parameter retrieval from inhomogeneous metamaterials,
/hys' 0ev
. ? @), +4<<)@, (++6/@0 Christophe Calo%, Tatsuo Itoh,
Electromagnetic 5etamaterials4 Transmission 2ine Theory an( 5icro1ave )77lications
, 7iley, ohn N Sons, Incorporated, (++6