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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 16 — Aug. 7, 2006
  • pp: 7188–7197
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Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials

Xin Zhou, Quan H. Fu, Jing Zhao, Yang Yang, and Xiao P. Zhao  »View Author Affiliations


Optics Express, Vol. 14, Issue 16, pp. 7188-7197 (2006)
http://dx.doi.org/10.1364/OE.14.007188


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Abstract

We present the design for a hexagonal cell made of quasi-periodic dendritic arranged collections of plasmonic metallic wires that may exhibit a resonant magnetic collective response. When such quasi-periodic dendritic cells are etched on a host medium, they may provide metamaterials with negative effective permeability. We also show that a clear point image is observed, as expected, with our left-handed metamaterials (LHMs) lens composed of metallic dendritic cells and wire strips. These prominent characteristics of quasi-periodic dendritic cells potentially enable us to prepare infrared or visible domain LHMs by using a general chemical method.

© 2006 Optical Society of America

1. Introduction

Recently, a number of applications in optics, materials science, biophysics, and biology have prompted extensive research in the area of artificial periodic materials, which are also known as left-handed metamaterials (LHMs) [1–2

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

]. LHMs with simultaneously negative dielectric permittivity and magnetic permeability were first reported by Veselago [3

3. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov.Phys. Usp. 10, 509-514 (1968). [CrossRef]

] in 1967 and have been successfully demonstrated in microwave (GHz) frequencies [4

4. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004.) [CrossRef] [PubMed]

]. Through many experiments and theories, researchers who aimed to design “perfect” metamaterials have developed new LHMs that are isotropic, wide band, non-bianisotropic, easy to design, and so forth [5–7

5. M. T. Grzegorczyk and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys.Rev. E 70, 057605 (2004). [CrossRef]

]. It is tempting to extend the approach to visible-light frequencies, where one can expect more applications. New advances in microfabrications make it possible to create composite materials with constituents of different forms and sizes down to nanoscales [5

5. M. T. Grzegorczyk and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys.Rev. E 70, 057605 (2004). [CrossRef]

], which offer a way to ultrahigh frequency LHMs. Despite arguments [9

9. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 264–270 (Oxford, Pergamon,1960).

] concerning whether there is magnetic susceptibility at optical frequencies, Soukoulis et al., using “U-shaped” rings in 200 nm fabricated with the high-resolution electron-beam lithography technique [10

10. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C.M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95,203901 (2005). [CrossRef] [PubMed]

], have approached visible frequencies of negative permeability. Grigorenko, et al. [11

11. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335–338 (2005). [CrossRef] [PubMed]

] have fabricated pairs of gold dots designed at a 10-nm level with negative permeability at visible frequencies. However, designing an isotropic and more easily fabricated negative permeability material at infrared and visible frequencies remains problematic.

The dendritic structure has a fairly high level of symmetry and can be fabricated by using relatively easy chemical methods such as the electrochemical or macromolecular synthesis techniques, and the size can be adjusted from a macroscopic scale to a nanometer scale [12-15

12. Y. Sawada, A. Dougherty, and J. P. Gollub, “Dendritic and fractal patterns in electrolytic metal deposits,” Phys. Rev. Lett. 56, 1260–1263 (1986). [CrossRef] [PubMed]

]. Also notable is the fact that it is easy to design dendritic structures with 3D morphology, and thus achieve 3D isotropic metamaterials. Although the metallic dendritic structure is often employed in antennas or, formerly, FSS [16

16. J. S. Patko and D. H. Werner, “Miniature reconfigurable three-dimensional fractal tree antennas,” IEEETrans. Antennas Propag. 52, 1945–1956 (2004). [CrossRef]

], there has been little research into using dendritic cells as a metamaterial. In this letter, we demonstrate metamaterials with a magnetic response by employing a novel quasi-periodic dendritic structure. We have proved theoretically and experimentally that the dendritic cells can be used to accomplish negative permeability, and we experimentally studied the subwavelength focusing effect by using left-handed metamaterials that are based on dendritic cells and wires.

2. Dendritic structure

Fig. 1. Morphology of the dendritic geometry, defining the dimension of a dendritic geometry with parameter a, b, c, and θ. Some split irregular polygon rings (SIPRs) are shown in the dendritic geometry.

As shown in Fig. 1, a dendritic structure cell grows with a V-shaped core, which we define as being 1st order. At the end of each arm of the core, there are new V-shaped branches that are 2nd order. Continually, each branch has new higher-order branches. So, it can be 1st order, 2nd order, 3rd order. A split irregular polygonal ring (SIPR) is formed in two branches of the same order. With the increase in the order of branches, the amount of SIPRs will be 1, 2, 4, 8… 2n, and its distribution is quasi-periodic. A circular dendritic cell grown with a star core is more symmetrical and isotropic. It can be regarded as a 2pi, V-shaped, dendritic cell. Electromagnetic resonance in the two-branches, 3rd-order dendritic cell and the 2pi dendritic cell (All dimensions of two series of the dendritic structure and SIPRs are a = 1.4 mm, b = 0.85 mm, c = 1 mm, θ = 45°; the lengths of a unit cell are the same as lattice constant 7.28 mm, as representative) are numerically investigated using CST Microwave studio code, which is based on the finite integration technique with perfect boundary approximation. Our simulations used electric and magnetic boundary conditions on the transverse boundaries and two open ports to simulate the S-parameter response of a single infinite layer medium to a normally incident plane wave. The electromagnetic wave propagates along the x axis with the electric field vector in parallel to the z axis and the magnetic field vector in parallel to the y axis (see Fig. 1). The result of transmission, T spectrum, is shown in Fig. 2. It exhibits a T-dip in the certain frequency range. According to Pendry’s theory, the split ring resonators (SRRs) must have circle current oscillation that can bring magnetic resonance [1

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

], and the permeability can be negative in this magnetic resonance frequency region. Therefore, the response of the dendritic structure must distinguish between electrical resonance and magnetic resonance.

Fig. 2. (a) Simulated and experimental transmission spectrums of the two branches, 3rd-order dendritic cell (black curves), Simulated transmission spectrum of SIPRs (red curve); (b) Simulated and experimental transmission spectrum of the circular dendritic cell (black curve), experimental transmission spectrum of the “closed” circular dendritic cell (blue).

We calculate magnetic field intensity and surface current distribution in the dendritic structure at its resonance frequencies as shown in Fig. 3. The color scales exhibit electric or magnetic field distribution intensity, and the direction of the arrows indicates the surface current vector. It can be clearly seen from Fig. 3(a) that magnetic intensity is strong around the split hexagonal ring within the dendritic structure in top sight, and the surface current mainly distributes over the split hexagonal ring composed of two branches, and thus generates a circle current oscillation, which definitely denotes the magnetic resonance.

Fig. 3. (a) [left panel] Induced magnetic field at a plane perpendicular to the dendritic structure plane, [right panel] Induced surface current at the dendritic structure plane. (b) [left panel] Induced magnetic field at a plane perpendicular to the SIRP plane, [right panel] Induced surface current at the SIRP plane. (c) Induced surface current at the circular dendritic structure plane.

Here the common single-split hexagonal ring, which has the same dimensions contained in the 3rd-order dendritic cell, has a similar magnetic dipole, magnetic field intensity, and surface current distribution, as shown in Fig. 3(b); however, it exhibits a higher resonance frequency than the 3rd-order dendritic cell. Generally, resonance frequency of the dendritic cell depends on the size of its SIPRs; however, additional branches influence the resonance frequency of the unattached SIPRs [see Fig. 2(a)]. This phenomenon can be understood from the equivalent-circuit point of view. The magnetic resonant behavior in the dendritic cell is due to capacitance and inductance in the structure. Like SRRs, the SIPR has its capacitance formed between the split, and its inductance is composed of winding. The additional branches support additional parallel capacitance to SIPRs, as shown in Fig. 2(a). In essence, just a small LC circuit, consisting of an inductance L and a capacitance C, is formed in the branches. Simulations show that the L of the dendritic cell can be increased by increasing the length or the angle of the branches, and the C of the dendritic cell can be increased by increasing the distance of the terminal branch.

The full 2pi 3rd-order circular dendritic cell with an 8-arm star core can be regarded as eight SIPRs arranged in a circle. Detailed numerical simulations provide T spectrums and surface current, as shown in Fig. 2(b) (black line) and Fig. 3(c), exhibiting that the 3rd-order circular dendritic cell has a wider resonance frequency and its SIPRs generate magnetic resonance.

The circular dendritic cell possesses a fairly high level of symmetry and approach the isotropy in the plane. Single-split SRR media, which do not possess this symmetry, have cross-polarizing behavior. Although a symmetric structure also dictates its coupling to the magnetic field, the inductive loops aside the cell axis are equivalent and oppositely wound with the capacitor, which acts as a magnetic field gradiometer. A uniform magnetic field cannot drive the fundamental LC resonance [17

17. D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006). [CrossRef]

].

The transmission spectra of the samples are measured in the frequency range 7-12 GHz and, as shown in Fig. 2, each sample resonates a T-dip spectrum, sample 1 and sample 2 exhibit a resonance near 9.5 GHz. To determine whether the resonance peak is induced by magnetic resonance, Soukoulis et al. have found an experimental method to validate the magnetic resonance of SRRs [21

21. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004). [CrossRef] [PubMed]

]. In their experiment, the gap of SRRs is closed, which eliminates the capacitors from the rings of the SRRs, and the induced circulating currents inside the SRRs are still allowed to flow but cannot oscillate independently of the external electromagnetic field anymore. Therefore, the T-dip disappears, and it can be confirmed as magnetic resonance. Similarly, we prepared a dendritic structure in which the split hexagonal ring formed by two branches was closed. The result is that the “closed” dendritic structure no longer has a T-dip at the former resonance frequency [see Fig. 2(b)]. Furthermore, because symmetry properties of the dendritic cell cancel electric moments in each other, we consider that resonance is pure magnetic.

3. Bianisotropic composite modeling for parameter calculating

Generally, procedures for retrieval from S parameters can be used for the determination of the permeability of the metamaterials. Here we calculated the permeability of the dendritic structures by applying bianisotropic composite modeling.

The medium composed of inclusions with dendritic geometries is linear, so it can be described in terms of the bianisotropic constitutive relations [22

22. D. H. Werner and R. Mittra, Frontiers in Electromagnetics, 732–770 (IEEE Press, Piscataway, 2000).

]

{D=εr̿ε0E+jε0μ0χ̿THB=μr̿μ0Hjε0μ0χ̿E,
(1)

where ϵ̿r, μ̿r and χ̿ are effective dielectric permittivity, effective magnetic permeability, and magnetoelectric coupling coefficient, respectively. ϵ̿r, μ̿r and χ̿ can be easily expressed in terms of the polarizabilities aee̿, amm̿, aem̿, and ame̿ of inclusion. The detailed derivation is as follows:

The macroscopic properties of a medium are described by its polarization P and magnetization M

{D=ε0E+P=ε0E+NpB=μ0(H+M)=μ0(H+Nm),
(2)

where p and m represent induced electric and magnetic dipole moments of inclusion, and N is the inclusion number per unit volume.

{p=ε0(aee̿Eloc+aem̿ηHloc)m=amm̿Hloc+ame̿Elocη,
(3)

where η= (μ00)1/2 is the wave impedance of free space.

Considering the Lorentzian field, we can write

{Eloc=E+P(3ε0)Hloc=H+M3.
(4)

Combining expressions (3) and (4), P and M are expressed as functions of E and H. Results are put in expression (2), and the obtained formulas are compared with general constitutive relations (1). Then expressions of ϵ̿r, μ̿r and χ can be derived:

{εr̿=I̿+K̿11[Naee̿+N23aem̿(I̿Namm̿3)1ame̿]μr̿=I̿+K̿21[Namm̿+N23ame̿(I̿Naee̿3)1aem̿]χ̿=jK̿21Name̿̿[I̿+(I̿Naee̿3)1Naee̿3]
(5)

where I̿ is unit dyadic, and

{K1̿=I̿Naee̿3N29aem̿(I̿Namm̿3)1ame̿K2̿=I̿Namm̿3N29ame̿(I̿Naee̿3)1aem̿.
(6)

In view of the symmetry of inclusion, we can neglect the magnetoelectric coupling and assume that only the magnetic field component along the y axis induces y-directional magnetization, thus aem̿=ame̿=χ̿=0,amm̿=(0000ammyy0000), and

μr̿=(10001+Nammyy(1Nammyy3)0001).
(7)

Hence, the magnetic response of the medium can be fully described by the following component

μryy=1+Nammyy(1Nammyy3)
(8)

and has nothing to do with the other components of μr̿.

The polarizability ammyy can be extracted as follows:

  1. Obtain surface current density by numerical simulation, and compute the y component my of m;
  2. Read out the y component Hlocy of local magnetic field Hloc from CST software;
  3. Use relation (3) to derive ammyy=my/Hlocy.

Once ammyy is obtained, it is substituted into relation (8) to calculate the effective permeability μryy of the medium. The calculated μryy of sample 2 is illustrated in Fig. 4 and reaches a negative permeability dip near 9.4 GHz.

Fig. 4. Calculated effective permeability for sample 2.

4. Experiment of subwavelength focusing

An experiment was performed to check the possibility of point focusing by our dendritic LHMs lens sample. The dendritic LHMs lens sample is based on sample 2 and wires, the dendritic cells are printed on the front surface of the substrates. There are two layers of dendritic cells along the x axis, 17 layers of dendritic cells along the y axis, and 22 layers of dendritic cells along the z axis. The wires are printed on the back surface of the substrates. The width of the wire strips is 0.5 mm, the length of strips is 130 mm, the periodic arrangement of the wire strips has a lattice constant of 7.28 mm. The wire strips and the dendritic cell are aligned so that the axis of the wire is parallel to the axis of the dendritic cell.

Fig. 5. Measured transmission spectrum of circular dendritic structures combined with wires. Inset: Measured phase spectrum of circular dendritic structures combined with wires and Teflon slab.
Fig. 6. Top view of the experimental setup for flat lens focusing. The left side of the LHMs monopole antenna used as the point source, and the right one used as the receiver.

The transmission measured setup for the LHMs lens also employed two horn antennas, as described in Section 2. The T spectrum in Fig. 5 shows that the dendritic LHMs lens exhibits a clear pass band with the maximum value -1.8 dB between 7.5~10.2 GHz, which matches μ<0 and ϵ<0 simultaneously, and the negative phase advance can be observed in this T peak domain. For example, the presence of the LHMs lens caused a negative phase advance of more than 60° at 8 GHz, contrasted with positive phase increment of near 50° when we inserted the Teflon slab between source and receiving antenna (Fig. 5, inset), similar with Ref. [23

23. A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed material plate,” Phys. Rev. Lett. 92, 077401 (2004). [CrossRef] [PubMed]

]. The focusing experiment facility employed two monopole antennas: the one used as the point source and another used as the receiver. The power distribution at the image plane is measured by scanning and recording the transmission intensity along the lines parallel and perpendicular to the surface of the slab at the focus plane (Fig. 6). Details of the setup can be found in Ref. [23

23. A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed material plate,” Phys. Rev. Lett. 92, 077401 (2004). [CrossRef] [PubMed]

]. The measurement results at 9.4 GHz are shown in Fig. 7. When the point source was set 1 cm away from the LHMs, at the other side a clear point image was focused near the lens. Because our lens was not isotropic in the XY plane, the point image doesn’t move away from the lens, and the image point is slightly stretched [23

23. A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed material plate,” Phys. Rev. Lett. 92, 077401 (2004). [CrossRef] [PubMed]

]. Figure 7(c) exhibits that no image is observed when the LHMs lens is removed.

Fig. 7. (a) Measured result of field amplitude at 9.4 GHz with the receiving antenna moving parallel to the plate, (b) 3D view of Schematic (a), (c) measured intensity distribution along the transverse (Y) direction at the image plane (x = 0 mm) with and without LHMs lens.

5. Conclusions

In conclusion, we have demonstrated the negative permeability in a novel quasi-periodic dendritic structure. We consider that the quasi-periodic SIPRs contained within the dendritic structure induced magnetic resonance, and we apply bianisotropic composite modeling to calculate its negative permeability. We measured the left-handed T peak by using the dendritic cells combined with wires. In the planar focusing experiment, we got a definite subwavelength point image near the dendritic LHMs lens. In further study, in optical and infrared frequencies, nano-dendritic material can be synthesized using a facilitative chemical method, and many negative refraction phenomena observed in the microwave regime can also be found in the optical frequency. We anticipate that the quasi-periodic dendritic structure will promote the future study of potential applications for metamaterials.

Acknowledgments

We acknowledge support from the National Nature Science Foundation of China for Distinguished Young Scholars under Grant No. 50025207 and the National Basic Research Program of China under sub-project 2004CB719805.

References and links

1.

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

2.

J. B. Pendry, “Perfect cylindrical lenses,” Opt. Express 11, 755–760 (2003). [CrossRef] [PubMed]

3.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov.Phys. Usp. 10, 509-514 (1968). [CrossRef]

4.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004.) [CrossRef] [PubMed]

5.

M. T. Grzegorczyk and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys.Rev. E 70, 057605 (2004). [CrossRef]

6.

M. T. Grzegorczyk and J. A. Kong, “Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns,” Appl. Phys. Lett. 84, 1537–1539 (2004). [CrossRef]

7.

I. Bulu, H. Caglayan, and E. Ozbay, “Experimental demonstration of labyrinth-based left-handed metamaterials,” Opt. Express 13, 10238–10247 (2005). [CrossRef] [PubMed]

8.

J. Rothman, M. Klaui, L. Lopez-Diaz, C. A. F. Vaz, A. Bleloch, J. A. C. Bland, Z. Cui, and R. Speaks, “Observation of a bi-domain state and nucleation free switching in Mesoscopic Ring Magnets,” Phys. Rev. Lett. 86, 1098–1101 (2001). [CrossRef] [PubMed]

9.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 264–270 (Oxford, Pergamon,1960).

10.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C.M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95,203901 (2005). [CrossRef] [PubMed]

11.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335–338 (2005). [CrossRef] [PubMed]

12.

Y. Sawada, A. Dougherty, and J. P. Gollub, “Dendritic and fractal patterns in electrolytic metal deposits,” Phys. Rev. Lett. 56, 1260–1263 (1986). [CrossRef] [PubMed]

13.

Y. J. Song, Y. Yang, C. J. Medforll, E. Pereira, A. K. Singh, Y. B. Jiang, C. J. Brinker, F. V. Swol, and J. A. Shelnutt, “Controlled synthesis of 2-D and 3-D dendritic platinum nanostructures,” J. Am. Chem. Soc. 126, 635–645 (2004). [CrossRef] [PubMed]

14.

X. Q. Wang, H. Itoh, K. Naka, and Y. Chujo, “Tetrathiafulvalene-assisted formation of silver dendritic nanostructures in acetonitrile,” Langmuir 19, 6242–6246 (2003). [CrossRef]

15.

J. J. Flory, “Molecular size distribution in three-dimensional polymersVI. Branched polymers containing A -R-Bf-1-type units,” J. Am. Chem. Soc. 74, 2718–2718 (1952). [CrossRef]

16.

J. S. Patko and D. H. Werner, “Miniature reconfigurable three-dimensional fractal tree antennas,” IEEETrans. Antennas Propag. 52, 1945–1956 (2004). [CrossRef]

17.

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109 (2006). [CrossRef]

18.

F. L. Zhang, Q Zhao, Y. H. Liu, C. R. Luo, and X. P. Zhao, “Behaviour of hexagon split ring resonators andleft-handed metamaterials,” Chin. Phys. Lett. 21, 1330–1332 (2004). [CrossRef]

19.

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.

X. P. Zhao, Q. Zhao, F. L. Zhang, W. Zhao, and Y. H. Liu, “Stopband phenomena in the passband of left-handed metamaterials,” Chin. Phys. Lett. 23, 99–102 (2006). [CrossRef]

21.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004). [CrossRef] [PubMed]

22.

D. H. Werner and R. Mittra, Frontiers in Electromagnetics, 732–770 (IEEE Press, Piscataway, 2000).

23.

A. N. Lagarkov and V. N. Kissel, “Near-perfect imaging in a focusing system based on a left-handed material plate,” Phys. Rev. Lett. 92, 077401 (2004). [CrossRef] [PubMed]

OCIS Codes
(160.3900) Materials : Metals
(160.4670) Materials : Optical materials
(220.3630) Optical design and fabrication : Lenses
(260.0260) Physical optics : Physical optics
(350.4010) Other areas of optics : Microwaves

ToC Category:
Metamaterials

History
Original Manuscript: June 2, 2006
Revised Manuscript: July 13, 2006
Manuscript Accepted: July 13, 2006
Published: August 7, 2006

Citation
Xin Zhou, Quan H. Fu, Jing Zhao, Yang Yang, and Xiao P. Zhao, "Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials," Opt. Express 14, 7188-7197 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7188


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References

  1. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  2. J. B. Pendry, "Perfect cylindrical lenses," Opt. Express 11, 755-760 (2003). [CrossRef] [PubMed]
  3. V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
  4. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
  5. M. T. Grzegorczyk and J. A. Kong, "Left-handed materials composed of only S-shaped resonators," Phys. Rev. E 70, 057605 (2004). [CrossRef]
  6. M. T. Grzegorczyk and J. A. Kong, "Experimental confirmation of negative refractive index of a metamaterial composed of Ω-like metallic patterns," Appl. Phys. Lett. 84, 1537-1539 (2004). [CrossRef]
  7. I. Bulu, H. Caglayan, and E. Ozbay, "Experimental demonstration of labyrinth-based left-handed metamaterials," Opt. Express 13, 10238-10247 (2005). [CrossRef] [PubMed]
  8. J. Rothman, M. Klaui, L. Lopez-Diaz, C. A. F. Vaz, A. Bleloch, J. A. C. Bland, Z. Cui, and R. Speaks, "Observation of a bi-domain state and nucleation free switching in Mesoscopic Ring Magnets," Phys. Rev. Lett. 86, 1098-1101 (2001). [CrossRef] [PubMed]
  9. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 264-270 (Oxford, Pergamon, 1960).
  10. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95,203901 (2005). [CrossRef] [PubMed]
  11. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005). [CrossRef] [PubMed]
  12. Y. Sawada, A. Dougherty, and J. P. Gollub, "Dendritic and fractal patterns in electrolytic metal deposits," Phys. Rev. Lett. 56, 1260-1263 (1986). [CrossRef] [PubMed]
  13. Y. J. Song, Y. Yang, C. J. Medforll, E. Pereira, A. K. Singh, Y. B. Jiang, C. J. Brinker, F. V. Swol, and J. A. Shelnutt, "Controlled synthesis of 2-D and 3-D dendritic platinum nanostructures," J. Am. Chem. Soc. 126, 635-645 (2004). [CrossRef] [PubMed]
  14. X. Q. Wang, H. Itoh, K. Naka, and Y. Chujo, "Tetrathiafulvalene-assisted formation of silver dendritic nanostructures in acetonitrile," Langmuir 19, 6242-6246 (2003). [CrossRef]
  15. J. J. Flory, "Molecular size distribution in three-dimensional polymersVI. Branched polymers containing A -R-Bf-1-type units," J. Am. Chem. Soc. 74, 2718-2718 (1952). [CrossRef]
  16. J. S. Patko and D. H. Werner, "Miniature reconfigurable three-dimensional fractal tree antennas," IEEE Trans. Antennas Propag. 52, 1945-1956 (2004). [CrossRef]
  17. D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88, 041109 (2006). [CrossRef]
  18. F. L. Zhang, Q Zhao, Y. H. Liu, C. R. Luo, and X. P. Zhao, "Behaviour of hexagon split ring resonators and left-handed metamaterials," Chin. Phys. Lett. 21, 1330-1332 (2004). [CrossRef]
  19. 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]
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