## Numerical demonstration of a three-dimensional negative-index metamaterial at optical frequencies |

Optics Express, Vol. 19, Issue 1, pp. 289-296 (2011)

http://dx.doi.org/10.1364/OE.19.000289

Acrobat PDF (1271 KB)

### Abstract

As a consequent work of the two-dimensional (2D) dendritic metamaterial which has been systematically studied in our previous work, a model of three-dimensional (3D) sphere-rod shaped structure is presented. Based on Drude model of the dielectric function of silver in the visible region, the parametric curves of electromagnetic response to the incident fields have been retrieved from detailed simulations. It is shown that the simultaneously negative values of permittivity and permeability in the optical range lead to a negative refractive index (NIM) through adjusting structural parameters, only the dimensions of the unit cells satisfy the effective medium theory. We therefore conclude that the proposed model offers a feasible route to fabricating 3D optical NIMs by ‘bottom-up’ approach.

© 2010 OSA

## 1. Introduction

1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

2. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **72**(1), 016623 (2005). [CrossRef] [PubMed]

3. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. **78**(4), 489–491 (2001). [CrossRef]

5. J. Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco Jr, B. I. Wu, J. A. Kong, and M. Chen, “Cerenkov radiation in materials with negative permittivity and permeability,” Opt. Express **11**(7), 723–734 (2003). [CrossRef] [PubMed]

6. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**(5514), 77–79 (2001). [CrossRef] [PubMed]

8. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science **312**(5775), 892–894 (2006). [CrossRef] [PubMed]

9. G. Dolling, M. Wegener, and S. Linden, “Realization of a three-founctional-layer negative-index photonic metamaterial,” Opt. Lett. **32**(5), 551–553 (2007). [CrossRef] [PubMed]

10. U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, “Dual-band negative index metamaterial: double negative at 813 nm and single negative at 772 nm,” Opt. Lett. **32**(12), 1671–1673 (2007). [CrossRef] [PubMed]

7. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. **31**(12), 1800–1802 (2006). [CrossRef] [PubMed]

8. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science **312**(5775), 892–894 (2006). [CrossRef] [PubMed]

11. S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. **95**(13), 137404 (2005). [CrossRef] [PubMed]

12. 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**(20), 203901 (2005). [CrossRef] [PubMed]

13. C. Enkrich, F. Perez-Williard, D. Gerthsen, J. Zhou, T. Koschny, C. M. Soukoulis, M. Wegener, and S. Linden, “Focused-ion-beam nanofabrication of near-infrared magnetic metamaterials,” Adv. Mater. **17**(21), 2547 (2005). [CrossRef]

14. 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**(16), 7188–7197 (2006). [CrossRef] [PubMed]

15. Y. Yao and X. P. Zhao, “Multilevel dendritic structure with simultaneously negative permeability and permittivity,” J. Appl. Phys. **101**(12), 124904 (2007). [CrossRef]

17. X. Zhou and X. P. Zhao, “Resonant condition of unitary dendritic structure with overlapping negative permittivity and permeability,” Appl. Phys. Lett. **91**(18), 181908 (2007). [CrossRef]

18. H. Liu, X. P. Zhao, Y. Yang, Q. W. Li, and J. Lv, “Fabrication of infrared left-handed metamaterials via double template-assisted electrochemical deposition,” Adv. Mater. **20**(11), 2050–2054 (2008). [CrossRef]

19. B. Q. Liu, X. P. Zhao, W. R. Zhu, W. Luo, and X. C. Cheng, “Multiple Pass-Band Optical Left-Handed Metamaterials Based on Random Dendritic Cells,” Adv. Funct. Mater. **18**(21), 3523–3528 (2008). [CrossRef]

20. X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. **95**(7), 071111 (2009). [CrossRef]

21. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature **455**(7211), 376–379 (2008). [CrossRef] [PubMed]

## 2. Numerical simulation

_{p}= 1.37 × 10

^{16}s

^{−1}and collision frequency ω

_{c}= 8.5 × 10

^{13}s

^{−1}[22

22. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

_{11}) and transmission (S

_{21}) coefficients, we first extract refractive index n and impedance z from the two parameters, then the effective permittivity ε and the effective permeability μ can be directly calculated from μ = nz and ε = n/z [23

23. X. D. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco Jr, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(1), 016608 (2004). [CrossRef] [PubMed]

## 3. Results and analysis

### 3.1 Parametric optimization

24. V. D. Lam, J. B. Kim, S. J. Lee, and Y. P. Lee, “Left-handed behavior of combined and fishnet structures,” J. Appl. Phys. **103**(3), 033107 (2008). [CrossRef]

### 3.2 Response of visible region

### 3.2.1 Negative refraction for red-light

*=*65 nm, r

*=*39 nm, t

*=*19 nm, a

*=*140 nm. In Fig. 3 (a), it can be seen that there is a transmission peak of S

_{21}and a reflection trough of S

_{11}at bout 436.3 THz. In the vicinity of this frequency, the real part of the refractive index is negative with simultaneously negative real parts of permeability and permittivity as shown in Figs. 3(b), 3(c), 3(d). The frequency band of the negative refractive index is within the red part of the spectrum (630-780nm). The ratio of the operation wavelength to the lattice constant is equal to 4.9, satisfying the effective medium theory.

### 3.2.2 Negative refraction for blue-light

*=*50 nm, r

*=*39 nm, t

*=*18 nm, a

*=*110 nm. In Fig. 4(a), there is a transmission peak with a high value close to 0.6 and the corresponding reflection trough has a low value of less than 0.05 at about 608.6 THz. As the same of Fig. 3, in the neighboring area of this frequency all the real parts of permeability, permittivity and refractive index simultaneously have a negative value, as illustrated in Figs. 4(b), 4(c), 4(d). The ratio of the operation wavelength over the lattice constant is 4.5, so the system can be regarded as an effective medium. The negative refraction is achieved within the wavelength rang of blue-light (470-500nm).

25. R. A. Depine and A. A. Lakhtakia, “New condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. **41**(4), 315–316 (2004). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

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

2. | A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

3. | R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. |

4. | N. Seddon and T. Bearpark, “Observation of the inverse Doppler effect,” Science |

5. | J. Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco Jr, B. I. Wu, J. A. Kong, and M. Chen, “Cerenkov radiation in materials with negative permittivity and permeability,” Opt. Express |

6. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

7. | G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. |

8. | G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science |

9. | G. Dolling, M. Wegener, and S. Linden, “Realization of a three-founctional-layer negative-index photonic metamaterial,” Opt. Lett. |

10. | U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, “Dual-band negative index metamaterial: double negative at 813 nm and single negative at 772 nm,” Opt. Lett. |

11. | S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. |

12. | 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. |

13. | C. Enkrich, F. Perez-Williard, D. Gerthsen, J. Zhou, T. Koschny, C. M. Soukoulis, M. Wegener, and S. Linden, “Focused-ion-beam nanofabrication of near-infrared magnetic metamaterials,” Adv. Mater. |

14. | 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 |

15. | Y. Yao and X. P. Zhao, “Multilevel dendritic structure with simultaneously negative permeability and permittivity,” J. Appl. Phys. |

16. | Y. Yao, Q. H. Fu, and X. P. Zhao, “Three-level dendritic structure with simultaneously negative permeability and permittivity under normal incidence of electromagnetic wave,” J. Appl. Phys. |

17. | X. Zhou and X. P. Zhao, “Resonant condition of unitary dendritic structure with overlapping negative permittivity and permeability,” Appl. Phys. Lett. |

18. | H. Liu, X. P. Zhao, Y. Yang, Q. W. Li, and J. Lv, “Fabrication of infrared left-handed metamaterials via double template-assisted electrochemical deposition,” Adv. Mater. |

19. | B. Q. Liu, X. P. Zhao, W. R. Zhu, W. Luo, and X. C. Cheng, “Multiple Pass-Band Optical Left-Handed Metamaterials Based on Random Dendritic Cells,” Adv. Funct. Mater. |

20. | X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. |

21. | J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature |

22. | P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B |

23. | X. D. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco Jr, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

24. | V. D. Lam, J. B. Kim, S. J. Lee, and Y. P. Lee, “Left-handed behavior of combined and fishnet structures,” J. Appl. Phys. |

25. | R. A. Depine and A. A. Lakhtakia, “New condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. |

**OCIS Codes**

(260.5740) Physical optics : Resonance

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: November 9, 2010

Revised Manuscript: December 9, 2010

Manuscript Accepted: December 10, 2010

Published: December 22, 2010

**Citation**

Boyi Gong and Xiaopeng Zhao, "Numerical demonstration of a three-dimensional negative-index metamaterial at optical frequencies," Opt. Express **19**, 289-296 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-289

Sort: Year | Journal | Reset

### References

- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001). [CrossRef]
- N. Seddon and T. Bearpark, “Observation of the inverse Doppler effect,” Science 302(5650), 1537–1540 (2003). [CrossRef] [PubMed]
- J. Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco, B. I. Wu, J. A. Kong, and M. Chen, “Cerenkov radiation in materials with negative permittivity and permeability,” Opt. Express 11(7), 723–734 (2003). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
- G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31(12), 1800–1802 (2006). [CrossRef] [PubMed]
- G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]
- G. Dolling, M. Wegener, and S. Linden, “Realization of a three-founctional-layer negative-index photonic metamaterial,” Opt. Lett. 32(5), 551–553 (2007). [CrossRef] [PubMed]
- U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, “Dual-band negative index metamaterial: double negative at 813 nm and single negative at 772 nm,” Opt. Lett. 32(12), 1671–1673 (2007). [CrossRef] [PubMed]
- S. Zhang, W. J. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef] [PubMed]
- 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(20), 203901 (2005). [CrossRef] [PubMed]
- C. Enkrich, F. Perez-Williard, D. Gerthsen, J. Zhou, T. Koschny, C. M. Soukoulis, M. Wegener, and S. Linden, “Focused-ion-beam nanofabrication of near-infrared magnetic metamaterials,” Adv. Mater. 17(21), 2547 (2005). [CrossRef]
- 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(16), 7188–7197 (2006). [CrossRef] [PubMed]
- Y. Yao and X. P. Zhao, “Multilevel dendritic structure with simultaneously negative permeability and permittivity,” J. Appl. Phys. 101(12), 124904 (2007). [CrossRef]
- Y. Yao, Q. H. Fu, and X. P. Zhao, “Three-level dendritic structure with simultaneously negative permeability and permittivity under normal incidence of electromagnetic wave,” J. Appl. Phys. 105(2), 024911 (2009). [CrossRef]
- X. Zhou and X. P. Zhao, “Resonant condition of unitary dendritic structure with overlapping negative permittivity and permeability,” Appl. Phys. Lett. 91(18), 181908 (2007). [CrossRef]
- H. Liu, X. P. Zhao, Y. Yang, Q. W. Li, and J. Lv, “Fabrication of infrared left-handed metamaterials via double template-assisted electrochemical deposition,” Adv. Mater. 20(11), 2050–2054 (2008). [CrossRef]
- B. Q. Liu, X. P. Zhao, W. R. Zhu, W. Luo, and X. C. Cheng, “Multiple Pass-Band Optical Left-Handed Metamaterials Based on Random Dendritic Cells,” Adv. Funct. Mater. 18(21), 3523–3528 (2008). [CrossRef]
- X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009). [CrossRef]
- J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]
- P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
- X. D. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(1), 016608 (2004). [CrossRef] [PubMed]
- V. D. Lam, J. B. Kim, S. J. Lee, and Y. P. Lee, “Left-handed behavior of combined and fishnet structures,” J. Appl. Phys. 103(3), 033107 (2008). [CrossRef]
- R. A. Depine and A. A. Lakhtakia, “New condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41(4), 315–316 (2004). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.