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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 11 — May. 26, 2008
  • pp: 7674–7679

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Disorder effects of left-handed metamaterials with unitary dendritic structure cell

X. Zhou, X. P. Zhao, and Y. Liu  »View Author Affiliations


Optics Express, Vol. 16, Issue 11, pp. 7674-7679 (2008)
http://dx.doi.org/10.1364/OE.16.007674


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Abstract

We demonstrate that the left-handed materials (LHMs) with unity dendritic unit cells arrayed in disorder state present still passband and negative refractive. The resonance behavior of LHMs in disturbed periodic lattice, quasi-periodic lattice and random array are experimentally investigated. Employing amended retrieval method, the LHMs with disordered state exhibits a negative index of refraction. Basing on such LHMs lens, the subwavelength imaging experiment give a clearly point image with a full wave at half maximum width of 0.4 λ at 9.3 GHz. Similarly, the power field distribution of “N” shaped antenna is measured beyond the diffraction limit.

© 2008 Optical Society of America

1. Introduction

Left-handed metamaterials (LHMs) are artificially structured composites with simultaneously negative electrical permittivity and magnetic permeability. Generally, the unit cells of LHMs are arrayed in periodic topology, the wire and split-ring resonator (SRR) plays the role of a “meta-atom” within the unit cells [1

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

, 2

J. B. Pendry, A. J. Holden, and D. J. Robbins, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

]. Because of the peculiar physical properties, which no analogise the naturally materials, the LHMs has attracted a tremendous amount of attention. Benefiting from the promotion of investigation for negative refraction [3

R. A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

], “perfect” lens [4

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

], invisible cloak [5

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

], et al. and the improvement in microfabrications, LHMs exhibit an infinite potential in infrared and visible frequencies. The minute structures of unit cell are commonly constructed by “top-down” approach like physics lithography [6

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7, 31–47(2007). [CrossRef] [PubMed]

]. Recently, a “bottom-up” self-assembly technique used for LHMs are repeatedly mentioned in the articles [7-9

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett. 99, 017401 (2007). [CrossRef] [PubMed]

]. In contrast with physics lithography method, the chemosynthesis has many advantages including inexpensive, convenient, mass-fabricable, but it bring on disorder in the cell or unit lattice. In previously, we demonstrated a unitary dendritic structure model, which can be easily fabricated utilized self-assembly technique, has simultaneously negative permittivity and permeability under the resonant condition [8-12

H. Liu and X. P. Zhao, “Magnetic response of dendritic structures at infrared frequencies,” Solid State Commun. 140, 9–13 (2006). [CrossRef]

]. Figure 1(a) show the silver dendritic structures prepared by chemical electro-deposition method at micron-scale [9

H. Liu and X. P. Zhao, “Metamaterials with dendriticlike structure at infrared frequencies,” Appl. Phys. Lett. 90, 191904 (2007). [CrossRef]

], Fig. 1(b) is the similar dendritic structures at millimeter-scale fabricated on PCB sheet with 12-fold quasiperiodic array. It can be seen that the both have analogous topology, but the former consist of disorder array. Some studies have discussed the disorder states of negative permeability medium with size and lattice [13-16

J. Gollub, T. Hand, S. Sajuyigbe, S. Mendonca, S. Cummer, and D. R. Smith, “Characterizing the effects of disorder in metamaterial,” Appl. Phys. Lett. 91, 162907 (2007). [CrossRef]

]. And the effects of misalignment on the left-handed behavior of the metamaterial have been investigated [17

H. Chen, L. Ran, D. Wang, J. Huangfu, Q. Jiang, and J. A. Kong, “Metamaterial with randomized patterns for negative refraction of electromagnetic waves,” Appl. Phys. Lett. 88, 031908 (2006). [CrossRef]

, 18

D. Wang, J. Huangfu, L. Ran, H. Chen, T. M. Grzegorczyk, and J. A. Kong, “Measurement of negative permittivity and permeability from experimental transmission and reflection with effects of cell misalignment,” J. Appl. Phys. 99, 123114 (2006). [CrossRef]

]. However, the disorder effect of LHMs has been a haze. Another, the disorder states of LHMs with unitary structure unit, in which disorder states will simultaneity influence electrical and magnetic coupling between unit cells, have never been reported.

Fig. 1. (a). Scanning electron microscope images of a sliver dendritic structure with 1µm scale. (b) Photograph of dendritic structures etched on PCB slab, the inset is the dimension of the geometry.

In this letter, we report the disorder effect of LHMs at microwave frequencies. The disordered states include disturbed periodic lattice, quasi-periodic lattice and random array. The planar subwavelength imaging experiment are operated for proving the left-handed property of disorder samples. Even though the scaling is not applicable in a strict sense, small scale structures may benefit from the conclusions drawn here as the physics remain essentially the same.

2. Amended retrieval method for disordered LHMs

To investigate the effect of disorder in LHMs, we intentionally introduce the disturbed lattice constants into unit cells. The statuses from order to disorder are depicted in four types as shown in Figs. 2(a)–2(c); that are periodic, periodic with random deviation, quasiperiodic and random arrangements. The unitary dendritic structure is selected for the LHMs design owing to its advantages in two-dimensional isotropic and unitary structure with electronic and magnetic resonance [12

X. Zhou and X. P. Zhao, “Resonant condition of unitary dendritic structure with overlapping negative permittivity and permeability,” Appl. Phys. Lett. 91, 181908 (2007). [CrossRef]

]. The periodic sample has the lattice constant l=8.9 mm. Two kinds of the disarranged samples based on the periodic one are defined by the deviation δ, where 0<δ<l/4 and 0<δ<l/2 [(Fig. 2(a)], respectively. The quasiperiodic samples are designed as 8-fold and 12-fold quasiperiodic arrangement [see Fig. 2(b)]. For random arrangements, we define the distance between the cells by a random number Δ, within the ranges of l/2<Δ<3l/2 and 0<Δ<2l, as shown in Fig. 2(c) respectively. The distances between the two cells for each topology must be smaller than the measured wavelength to avoid the diffraction.

The effective electromagnetic parameters of periodic metamaterials can be obtained using standard retrieval method. However, for the disordered case, the retrieval method can be amended with following approximation. We define f(Δ) as the cell fraction function, which can be treated as a probability density distribution function of a unit cell with lattice constant Δ. For random arrangement, e.g. f(Δ) is assumed to be a uniformly distributed function, or max min f(Δ)=1/(Δmax − Δmin). The effective permittivity and permeability can be approximately written as Drude-Lorentz form [1

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

,2

J. B. Pendry, A. J. Holden, and D. J. Robbins, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

],

εeff (ω)≈ ∫[ 1−F (Δ) ωpe2 ω2− ω 0e2+i Γeω]f (Δ)dΔ
(1)
μeff (ω)≈∫ [ 1−F (Δ) ωpm2 ω2− ω 0m2+i Γmω]f (Δ)dΔ
(2)

Where F(Δ) is the fractional volume of the cell occupied by the interior of the dendritic structure. Note that the three resonance parameters enter: plasma frequency ωp , the resonant frequency ω 0, and a damping factor Γ. These parameters are indexed with an “e” for electric or an “m” for magnetic response. In practice, the effective parameters of the metamaterials usually retrieve from S-parameter which obtained from simulation or measurement using network analyzer. Therefore, at first, the effective refractive index neff must be calculated as follow [19

Th. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005). [CrossRef]

]:

neff≈∫ 1 kΔ 𝕊·f (Δ)dΔ
(3)

Here, the expression 𝕊 is the function of S-parameter with transmission and reflection. The wave impedance Z is only the function of S-parameter which has been proposed in reference [19

Th. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005). [CrossRef]

]. Utilized neff and Z, the effective permittivity and permeability also can be deduced.

3. Microwave transmission experiment

We previously demonstrated a dendiritc structure unit for constructing the LHMs [10-12

X. Zhou, Q. H. Fu, J. Zhao, Y. Yang, and X.P. Zhao, “Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials,” Opt. Express 14, 7188–7197 (2006). [CrossRef] [PubMed]

]. The dendritic structures start with eight loop wires in the multilevel branches which form an LC resonator. For an electromagnetic wave incident with its wave vector and electric field parallel to the plane of the dendritic structure and the magnetic field perpendicular to the dendritic structure, the magnetic resonance mode will be as if producing the negative permeability. Similarly, the electric resonance mode, which exhibited the negative permittivity, also resonated due to the two symmetric inductive loops beside the medial axis of the dendritic structure. So that, two effects appeared overlapped frequencies together lead to negative index behavior.

Subsequently, the microwave experiments are employed to investigate the response behaviors of the dendritic metamaterials. The dendritic LHM sample used in the experiments presented here consists of a two-dimensional array of dendritic structure unit, fabricated by a shadow mask/etching technique on 1 mm thick FR4 circuit board material (ε=4.6) with the 0.035 mm thick covered copper film. The dimensions of the dendritic have depicted in Fig. 1(b, inset). The area of each disorder LHM sample is 80×80 mm2. Transmission properties of a mono-layers dendritic structure were measured over the frequency range of 7–11 GHz using a network analyzer AV3618 and a pair of standard gain horn antennas in microwave absorbing surrounding serving as a source and receiver. In the transmission measurements, the microwaves were incident along the sample surface, and the electric field of the incident wave was polarized parallel to the dendritic structure plane. Transmission measurements were calibrated to the transmission between the horns with the sample removed.

Fig. 2. Schematic illustrations of (a) periodic array with deviation 0<δ<l/2, (b) 12-fold quasiperiodic array and (c) random array with 0<Δ<2l; (d), (e) and (f) Transmission spectrums of each type of disorder samples, respectively.

In Figs. 2(d)-2(f), the results of the transmission experiment are plotted versus each type of order and disorder LHM samples. The order sample (δ=0) has a pass peak of -4.1 dB at 9.2 GHz. As the deviation δ in the samples is increased, the passband is widened and the response decreased. The quasiperiodic samples slightly reduce the resonant frequencies compared with order one, and the magnitude of the response also decreased. For the random array samples, the passbands are still presence, although the passbands are more flat.

It is reported that the coupling between two or more SRRs to be a complex mechanism and dependent on their particular arrangement topology [16

X. P. Zhao, Q. Zhao, L. Kang, J. Song, and Q. H. Fu, “Defect effect of split ring resonators in left-handed metamaterials,” Phys. Lett. A 346, 87–91 (2005). [CrossRef]

, 20

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002). [CrossRef]

]. Analogously, while introducing random arrangement, the coupling between LHMs cells could be complicated. Nevertheless, our results demonstrate that the random array samples still have the passband with Left-handed character.

Fig. 3. Retrieved effective refractive index with ordered and disordered sample.

Basing on commercial finite-integration time-domain algorithm, the calculated reflection and transmission of metamaterial structure unit with periodic conditions is good agreement with the measurement [6

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7, 31–47(2007). [CrossRef] [PubMed]

,12

X. Zhou and X. P. Zhao, “Resonant condition of unitary dendritic structure with overlapping negative permittivity and permeability,” Appl. Phys. Lett. 91, 181908 (2007). [CrossRef]

]. So, the effective index for each disorder sample can be retrieved using Eq. (3) from simulated data, as plotted in Fig. 3. All the samples have negative refractive region, e.g. the random array sample (l/2<Δ<3l/2) has the neff of −0.4 at 8.9 GHz. It’s clear that the absolute value of neff decreased when the sample turn to disorder.

4. Subwavelength imaging experiment

Pendry introduced the term perfect lens, a lens that is capable of overcoming the diffraction limit provided that ε=μ=−1. Although fabricating a lossless and perfectly matched negative refractive index material is difficult, it is still possibility to focus electromagnetic waves smaller than a half-wavelength which is so-called overcoming the diffraction limit. We prepare a LHMs lens utilized random array samples (l/2<Δ<3l/2). The volume of the lens along X, Y and Z axis is 150 mm×100 mm×30 mm and the distance between each sheet is 5 mm. The imaging experiment facility employed two monopole antennas: the one used as the point source and another used as the receiver. The power distribution is measured by scanning and recording the transmission intensity along the lines parallel to the surface of the lens (along X axis) at the focus plane. We place the source and receive antenna at the distances of 5 mm away from lens sides, move the receiver in 2.5 mm step [10

X. Zhou, Q. H. Fu, J. Zhao, Y. Yang, and X.P. Zhao, “Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials,” Opt. Express 14, 7188–7197 (2006). [CrossRef] [PubMed]

]. The measured frequencies are swept from 6.5 to 10.5 GHz stepped in 200 MHz. As shown in Fig. 4(a), the white curve is the measured transmission spectrum. Because of the multi-layer coupling and little varied dielectric of substrates with various batches, the passband of the lens move to near 9.3 GHz. The color scales depict the field power distribution along X axis at each point frequency. It clearly shows that the full width at half maximum of the beam shrink to 13 mm at 9.3 GHz, namely the resolution of the random LHMs lens is about 0.4 λ.

Fig. 4. (a). (Left scale) Intensity profiles of field power near the LHM lens as a function of frequency. (Right scale, white line) Transmission spectrum of the LHMs lens. (b). Illustrations of the setup of “N” focusing experiment. (c). Field pattern observed at 9.3GHz in an X-Y plane 1mm above the LHMs lens surface.

To test the two-dimensional imaging performance of the material, we constructed an antenna from a pair of anti-parallel wires, bent into the shape of the letter “N” [(Fig. 4(b)]. The lens is place at 1 mm over the “N” antenna, and the transmitted field was measured by scanning a 2.5 mm diameter loop probe in X-Y plane, about 1mm above the surface of the LHMs material, in 2.5 mm step. Figure 4(c) clearly shows that the lens does indeed act as an image transfer device for the “N” shaped distributed power field.

5. Conclusions

We systematically studied the disorder effect of LHMs on the microwave resonance behaviors. The disordered states include disturbed periodic lattice, quasi-periodic lattice and random array, as representative. All the samples presented the expected passbands at certain frequency region. The effective refractive index curves of samples were calculated by amended retrieval method. The results shown that, while introducing disorder, the passband of the LHMs is widened but the response is decreased, simultaneously, the absolute value of neff also decreased. However, even the units are arrayed in random state (l/2<Δ<3l/2), the sample still has negative neff of −0.4. In standard subwavelength focusing experiment, we measured a point image with a full wave at half maximum width of 0.4 λ at 9.3 GHz basing on such random arrayed LHMs lens. Similarly, the power field distribution of “N” shaped antenna could be measured beyond the diffraction limit. Benefited with the study in disorder phenomena, our results could open other routeway for the development of bottom-up assembling LHMs.

Acknowledgments

We acknowledge support from the National Natural Science Foundation of China under Grant No. 50632030, National Basic Research Program of China under Sub project No. 2004CB719805, and NPU Foundation for Fundamental Research No. WO18101.

References and links

1.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

2.

J. B. Pendry, A. J. Holden, and D. J. Robbins, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

3.

R. A. Shelby, D.R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

4.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

5.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

6.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7, 31–47(2007). [CrossRef] [PubMed]

7.

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, “Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum,” Phys. Rev. Lett. 99, 017401 (2007). [CrossRef] [PubMed]

8.

H. Liu and X. P. Zhao, “Magnetic response of dendritic structures at infrared frequencies,” Solid State Commun. 140, 9–13 (2006). [CrossRef]

9.

H. Liu and X. P. Zhao, “Metamaterials with dendriticlike structure at infrared frequencies,” Appl. Phys. Lett. 90, 191904 (2007). [CrossRef]

10.

X. Zhou, Q. H. Fu, J. Zhao, Y. Yang, and X.P. Zhao, “Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials,” Opt. Express 14, 7188–7197 (2006). [CrossRef] [PubMed]

11.

Y. Yao and X. P. Zhao, “Multilevel dendritic structure with simultaneously negative permeability and permittivity,” J. Appl. Phys. 101, 124904 (2007). [CrossRef]

12.

X. Zhou and X. P. Zhao, “Resonant condition of unitary dendritic structure with overlapping negative permittivity and permeability,” Appl. Phys. Lett. 91, 181908 (2007). [CrossRef]

13.

J. Gollub, T. Hand, S. Sajuyigbe, S. Mendonca, S. Cummer, and D. R. Smith, “Characterizing the effects of disorder in metamaterial,” Appl. Phys. Lett. 91, 162907 (2007). [CrossRef]

14.

K Aydin, K Guven, N. Katsarakis, C. M. Soukoulis, and E. Ozbay, “Effect of disorder on magnetic resonance band gap of split-ring resonator structures,” Opt. Express 12, 5896–5901 (2004). [CrossRef] [PubMed]

15.

C. Mitsumata and S. Tomita, “Negative permeability of magnetic nanocomposite films for designing left-handed metamaterials,” Appl. Phys. Lett. 91, 223104 (2007). [CrossRef]

16.

X. P. Zhao, Q. Zhao, L. Kang, J. Song, and Q. H. Fu, “Defect effect of split ring resonators in left-handed metamaterials,” Phys. Lett. A 346, 87–91 (2005). [CrossRef]

17.

H. Chen, L. Ran, D. Wang, J. Huangfu, Q. Jiang, and J. A. Kong, “Metamaterial with randomized patterns for negative refraction of electromagnetic waves,” Appl. Phys. Lett. 88, 031908 (2006). [CrossRef]

18.

D. Wang, J. Huangfu, L. Ran, H. Chen, T. M. Grzegorczyk, and J. A. Kong, “Measurement of negative permittivity and permeability from experimental transmission and reflection with effects of cell misalignment,” J. Appl. Phys. 99, 123114 (2006). [CrossRef]

19.

Th. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B 71, 245105 (2005). [CrossRef]

20.

P. Gay-Balmaz and O. J. F. Martin, “Electromagnetic resonances in individual and coupled split-ring resonators,” J. Appl. Phys. 92, 2929–2936 (2002). [CrossRef]

OCIS Codes
(160.4760) Materials : Optical properties
(260.2110) Physical optics : Electromagnetic optics
(260.5740) Physical optics : Resonance

ToC Category:
Metamaterials

History
Original Manuscript: March 20, 2008
Revised Manuscript: May 4, 2008
Manuscript Accepted: May 4, 2008
Published: May 12, 2008

Citation
X. Zhou, X. P. Zhao, and Y. Liu, "Disorder effects of left-handed metamaterials with unitary dendritic structure cell," Opt. Express 16, 7674-7679 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7674


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References

  1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
  2. J. B. Pendry, A. J. Holden, and D. J. Robbins, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
  3. R. A. Shelby, D.R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
  4. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  5. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
  6. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-47 (2007). [CrossRef] [PubMed]
  7. C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, "Design of an artificial three-dimensional composite metamaterial with magnetic resonances in the visible range of the electromagnetic spectrum," Phys. Rev. Lett. 99, 017401 (2007). [CrossRef] [PubMed]
  8. H. Liu and X. P. Zhao, "Magnetic response of dendritic structures at infrared frequencies," Solid State Commun. 140, 9-13 (2006). [CrossRef]
  9. H. Liu and X. P. Zhao, "Metamaterials with dendriticlike structure at infrared frequencies," Appl. Phys. Lett. 90, 191904 (2007). [CrossRef]
  10. X. Zhou, Q. H. Fu, J. Zhao, Y. Yang, and X. P. Zhao, "Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials," Opt. Express 14, 7188-7197 (2006). [CrossRef] [PubMed]
  11. Y. Yao and X. P. Zhao, "Multilevel dendritic structure with simultaneously negative permeability and permittivity," J. Appl. Phys. 101, 124904 (2007). [CrossRef]
  12. X. Zhou and X. P. Zhao, "Resonant condition of unitary dendritic structure with overlapping negative permittivity and permeability," Appl. Phys. Lett. 91, 181908 (2007). [CrossRef]
  13. J. Gollub, T. Hand, S. Sajuyigbe, S. Mendonca, S. Cummer, and D. R. Smith, "Characterizing the effects of disorder in metamaterial," Appl. Phys. Lett. 91, 162907 (2007). [CrossRef]
  14. K. Aydin, K. Guven, N. Katsarakis, C. M. Soukoulis, and E. Ozbay, "Effect of disorder on magnetic resonance band gap of split-ring resonator structures," Opt. Express 12, 5896-5901 (2004). [CrossRef] [PubMed]
  15. C. Mitsumata and S. Tomita, "Negative permeability of magnetic nanocomposite films for designing left-handed metamaterials," Appl. Phys. Lett. 91, 223104 (2007). [CrossRef]
  16. X. P. Zhao, Q. Zhao, L. Kang, J. Song, and Q. H. Fu, "Defect effect of split ring resonators in left-handed metamaterials," Phys. Lett. A 346, 87-91 (2005). [CrossRef]
  17. H. Chen, L. Ran, D. Wang, J. Huangfu, Q. Jiang, and J. A. Kong, "Metamaterial with randomized patterns for negative refraction of electromagnetic waves," Appl. Phys. Lett. 88, 031908 (2006). [CrossRef]
  18. D. Wang, J. Huangfu, L. Ran, H. Chen, T. M. Grzegorczyk, and J. A. Kong, "Measurement of negative permittivity and permeability from experimental transmission and reflection with effects of cell misalignment," J. Appl. Phys. 99, 123114 (2006). [CrossRef]
  19. Th. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71, 245105 (2005). [CrossRef]
  20. P. Gay-Balmaz and O. J. F. Martin, "Electromagnetic resonances in individual and coupled split-ring resonators," J. Appl. Phys. 92, 2929-2936 (2002). [CrossRef]

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