## Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials

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

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

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

*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]

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]

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

## 2. Dendritic structure

^{n}, 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]

*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.

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]

*x*direction and eight cells in the

*y*direction. Both dendritic cells with a v-shaped core (sample 1) and circular dendritic cells with an 8-arm star core (sample 2) are fabricated for transmission measurement.

*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]

## 3. Bianisotropic composite modeling for parameter calculating

_{r}, μ̿

_{r}and χ̿ are effective dielectric permittivity, effective magnetic permeability, and magnetoelectric coupling coefficient, respectively. ϵ̿

_{r}, μ̿

_{r}and χ̿ can be easily expressed in terms of the polarizabilities

*a*̿,

_{ee}*a*̿,

_{mm}*a*̿, and

_{em}*a*̿ of inclusion. The detailed derivation is as follows:

_{me}*and magnetization*

**P**

**M****and**

*p***represent induced electric and magnetic dipole moments of inclusion, and**

*m**N*is the inclusion number per unit volume.

_{0}/ϵ

_{0})

^{1/2}is the wave impedance of free space.

**and**

*P***are expressed as functions of**

*M**and*

**E****. Results are put in expression (2), and the obtained formulas are compared with general constitutive relations (1). Then expressions of ϵ̿**

*H*_{r}, μ̿

_{r}and χ can be derived:

*I*̿ is unit dyadic, and

*y*axis induces

*y*-directional magnetization, thus

_{r}̿.

*a*can be extracted as follows:

_{mmyy}- Obtain surface current density by numerical simulation, and compute the
*y*component*m*of_{y};**m** - Read out the
*y*component*H*of local magnetic field_{locy}from CST software;**H**_{loc} - Use relation (3) to derive
*a*=_{mmyy}*m*/_{y}*H*._{locy}

*a*is obtained, it is substituted into relation (8) to calculate the effective permeability

_{mmyy}*μ*of the medium. The calculated

_{ryy}*μ*of sample 2 is illustrated in Fig. 4 and reaches a negative permeability dip near 9.4 GHz.

_{ryy}## 4. Experiment of subwavelength focusing

*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.

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]

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]

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]

## 5. Conclusions

## Acknowledgments

## References and links

1. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

2. | J. B. Pendry, “Perfect cylindrical lenses,” Opt. Express |

3. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov.Phys. Usp. |

4. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

5. | M. T. Grzegorczyk and J. A. Kong, “Left-handed materials composed of only S-shaped resonators,” Phys.Rev. E |

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. |

7. | I. Bulu, H. Caglayan, and E. Ozbay, “Experimental demonstration of labyrinth-based left-handed metamaterials,” Opt. Express |

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. |

9. | L. D. Landau and E. M. Lifshitz, |

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. |

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 |

12. | Y. Sawada, A. Dougherty, and J. P. Gollub, “Dendritic and fractal patterns in electrolytic metal deposits,” Phys. Rev. Lett. |

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. |

14. | X. Q. Wang, H. Itoh, K. Naka, and Y. Chujo, “Tetrathiafulvalene-assisted formation of silver dendritic nanostructures in acetonitrile,” Langmuir |

15. | J. J. Flory, “Molecular size distribution in three-dimensional polymersVI. Branched polymers containing A -R-Bf-1-type units,” J. Am. Chem. Soc. |

16. | J. S. Patko and D. H. Werner, “Miniature reconfigurable three-dimensional fractal tree antennas,” IEEETrans. Antennas Propag. |

17. | D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. |

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. |

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 |

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. |

21. | T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. |

22. | D. H. Werner and R. Mittra, |

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. |

**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

- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- J. B. Pendry, "Perfect cylindrical lenses," Opt. Express 11, 755-760 (2003). [CrossRef] [PubMed]
- V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
- M. T. Grzegorczyk and J. A. Kong, "Left-handed materials composed of only S-shaped resonators," Phys. Rev. E 70, 057605 (2004). [CrossRef]
- 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]
- I. Bulu, H. Caglayan, and E. Ozbay, "Experimental demonstration of labyrinth-based left-handed metamaterials," Opt. Express 13, 10238-10247 (2005). [CrossRef] [PubMed]
- 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]
- L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 264-270 (Oxford, Pergamon, 1960).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. S. Patko and D. H. Werner, "Miniature reconfigurable three-dimensional fractal tree antennas," IEEE Trans. Antennas Propag. 52, 1945-1956 (2004). [CrossRef]
- D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88, 041109 (2006). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- D. H. Werner and R. Mittra, Frontiers in Electromagnetics, 732-770 (IEEE Press, Piscataway, 2000).
- 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]

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