## Investigation of Fano resonance in planar metamaterial with perturbed periodicity |

Optics Express, Vol. 21, Issue 1, pp. 992-1001 (2013)

http://dx.doi.org/10.1364/OE.21.000992

Acrobat PDF (3004 KB)

### Abstract

In this paper, we report the formation of sharp Fano resonance in planar metamaterial array with perturbed periodicity. Rigorous sheet impedance theory is given to analyze the electric-magnetic and magnetic-magnetic coupling effects. It is found that periodicity perturbation can provide a general approach for Fano resonance with ultra-strong local field enhancement.

© 2013 OSA

## 1. Introduction

1. 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]

3. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science **308**(5721), 534–537 (2005). [CrossRef] [PubMed]

4. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(4), 046608 (2004). [CrossRef] [PubMed]

5. M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B **75**(7), 075119 (2007). [CrossRef]

6. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science **323**(5912), 366–369 (2009). [CrossRef] [PubMed]

7. Q. Feng, M. Pu, C. Hu, and X. Luo, “Engineering the dispersion of metamaterial surface for broadband infrared absorption,” Opt. Lett. **37**(11), 2133–2135 (2012). [CrossRef] [PubMed]

10. M. Pu, Q. Feng, M. Wang, C. Hu, C. Huang, X. Ma, Z. Zhao, C. Wang, and X. Luo, “Ultrathin broadband nearly perfect absorber with symmetrical coherent illumination,” Opt. Express **20**(3), 2246–2254 (2012). [CrossRef] [PubMed]

*f*/Δ

*f*) was pursued in recent work concerning biological sensing [11

11. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. **10**(7), 2342–2348 (2010). [CrossRef] [PubMed]

12. C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. **106**(10), 107403 (2011). [CrossRef] [PubMed]

13. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. **99**(14), 147401 (2007). [CrossRef] [PubMed]

## 2. Structure and simulation

23. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. **30**(23), 3198–3200 (2005). [CrossRef] [PubMed]

25. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Plasmon hybridization in stacked cut-wire metamaterials,” Adv. Mater. (Deerfield Beach Fla.) **19**(21), 3628–3632 (2007). [CrossRef]

*p*= 5 mm and the thickness of the dielectric spacer is set as

*d*= 0.75 mm. Metal is assumed as perfect conductor while the permittivity of dielectric material is chosen as 12 with loss tangent

*τ*= 0 (the material loss will be considered later).

*x*direction and perfectly matched layer in ±

*z*direction. The structure is assumed to be infinite long in

*y*direction while the electric and magnetic fields are along

*x*and

*y*directions, respectively (i.e. normal incidence). The incident magnetic field is set as unit.

*w*

_{1}=

*w*

_{2}=

*w*

_{3}= 4 mm), the transmission peak at 9.78 GHz is induced by the magnetic resonance. Similar with Fano resonance, the transmission profile is asymmetric and a transmission dip occurs at 11.1 GHz.

*w*

_{1}= 4 mm,

*w*

_{2}= 3.6 mm and

*w*

_{3}= 3.56 mm as an example. Compared with the symmetry case, the line shape around 9.8 GHz becomes more asymmetric. Moreover, ultra sharp transmission dip at the center of a transmission peak is observed at frequencies around 10.8 GHz with Q-factor as large as 100. Interestingly, this is in contrary to the metamaterial analogy of electromagnetic induced transparency (EIT) [26

26. S.-Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B **80**(15), 153103 (2009). [CrossRef]

## 3. Analysis and discussion

### 3.1 Generalized impedance theory

27. P. Tassin, T. Koschny, and C. M. Soukoulis, “Effective material parameter retrieval for thin sheets: Theory and application to graphene, thin silver films, and single-layer metamaterials,” Physica B **407**(20), 4062–4065 (2012). [CrossRef]

*r*and

*t*). In the following, the generalized impedance theory is used to interpret the formation of Fano resonances in the planar metamaterial as shown in Fig. 2.

### 3.2 Fano resonance induced by the interference of magnetic and electric resonances

*w*

_{1}=

*w*

_{2}=

*w*

_{3}= 4 mm), the line shape shown in Fig. 2 is asymmetric. In order to understand the physical meaning behind it, the electric and magnetic sheet impedances are retrieved using Eq. (3). As depicted in Figs. 4(a) and 4(b), the electric admittance is nearly flat in the whole frequency range and magnetic impedance manifests itself as a sharp resonance. We noted that this phenomenon is similar with Fano resonance, which stems from the interference of a discrete state with a continuum.

*Z*

_{e}= 0) artificially and the corresponding transmission coefficient is plotted in Fig. 4(c). Obviously, the asymmetric line shape will transform into symmetric one when the electric resonance is not taken into account. Thus, one can conclude that it is the interference of the broadband electric resonance and narrowband magnetic resonance leads to the asymmetric Fano resonance.

*ω*is the angular frequency,

*L*and

*C*are the effective inductance and capacitance. The values of

*L*and

*C*can be evaluated by fitting the results retrieved from

*r*and

*t*. For the case of Fig. 4, L and C are 0.708 nH and 0.369 pF, which agrees well with the retrieved results.

### 3.3 Fano resonance induced by coupled magnetic resonances

*L*

_{1},

*L*

_{2},

*L*

_{3},

*C*

_{1},

*C*

_{2}and

*C*

_{3}are the corresponding inductances and capacitances. The resonant frequencies are

*L*

_{1}= 0.295 nH,

*L*

_{2}= 0.273 nH,

*L*

_{3}= 0.27 nH,

*C*

_{1}= 0.896 pF,

*C*

_{2}= 0.8 pF and

*C*

_{3}= 0.794 pF. The corresponding resonant frequencies are 9.79 GHz, 10.77 GHz and 10.87 GHz.

*f*<10.77GHz. Also, For 10.77 GHz <

*f*<10.87 GHz, the third one is out of phase with the others.

*ω*

_{2}and

*ω*

_{3}. Using previous fitted parameters, the sheet current at 10.82 GHz can be calculated as

*j*

_{m}_{1}= 3.3

*iH*,

_{i}*j*

_{m}_{2}= 2000

*i H*, and

_{i}*j*

_{m}_{3}= −2000

*i H*, which agree well with that shown in Fig. 6(a).

_{i}*L = p*is the unit cell length and Δ is the effective thickness of the metamaterial layer. As a result, the magnetic field

*H*can be calculated as:It should be noted that Eq. (13) is only an approximation under the condition Δ<<λ, for which the electric field integration along the thickness direction can be neglected.

_{z}*S*parameters. At

*f*

_{1}= 8 GHz, all the three wire pairs are out of resonance and the maximum magnetic field is only 3.6

*H*

_{i}. At

*f*

_{2}= 9.79 GHz, the first resonator is resonant and the maximum magnetic field increases to 69

*H*

_{i}. At

*f*

_{3}= 10.77 GHz, and

*f*

_{5}= 10.87 GHz, the second and the third wire pairs are resonant, respectively. Since

*f*

_{3}and

*f*

_{5}are spectrally close, a new resonant mode takes place between them. At

*f*

_{4}= 10.8 GHz, the second and the third wire pairs are on resonance with opposite phase. The maximum magnetic field becomes as high as 140

*H*

_{i}.

18. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics **2**(6), 351–354 (2008). [CrossRef]

### 3.4 Influence of loss

8. C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett. **37**(3), 308–310 (2012). [CrossRef] [PubMed]

9. Y. Cui, J. Xu, K. Hung Fung, Y. Jin, A. Kumar, S. He, and N. X. Fang, “A thin film broadband absorber based on multi-sized nanoantennas,” Appl. Phys. Lett. **99**(25), 253101 (2011). [CrossRef]

*p*= 5 mm,

*d*= 0.75 mm,

*w*

_{1}= 4 mm,

*w*

_{2}= 4.12 mm and

*w*

_{3}= 3.9 mm. The permittivity is set as 12 with a higher loss tangent of

*τ*= 0.015. The achieved bandwidth for 90% absorption is about 0.5 GHz, which is larger than that of metamaterial absorber without asymmetry (0.2 GHz) at the same thickness (

*p*= 5 mm,

*w*= 4 mm,

*τ*= 0.025).

## 4. Conclusion

## Acknowledgment

## References and Links

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

2. | R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

3. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science |

4. | R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

5. | M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B |

6. | R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science |

7. | Q. Feng, M. Pu, C. Hu, and X. Luo, “Engineering the dispersion of metamaterial surface for broadband infrared absorption,” Opt. Lett. |

8. | C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett. |

9. | Y. Cui, J. Xu, K. Hung Fung, Y. Jin, A. Kumar, S. He, and N. X. Fang, “A thin film broadband absorber based on multi-sized nanoantennas,” Appl. Phys. Lett. |

10. | M. Pu, Q. Feng, M. Wang, C. Hu, C. Huang, X. Ma, Z. Zhao, C. Wang, and X. Luo, “Ultrathin broadband nearly perfect absorber with symmetrical coherent illumination,” Opt. Express |

11. | N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. |

12. | C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett. |

13. | V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. |

14. | F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant LSPR sensing and a tunable Fano resonance,” Nano Lett. |

15. | A. Christ, O. J. F. Martin, Y. Ekinci, N. A. Gippius, and S. G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett. |

16. | V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. |

17. | N. Papasimakis, V. A. Fedotov, Y. H. Fu, D. P. Tsai, and N. I. Zheludev, “Coherent and incoherent metamaterials and order-disorder transitions,” Phys. Rev. B |

18. | N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics |

19. | E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express |

20. | C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater. |

21. | Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scr. |

22. | B. Gallinet and O. J. F. Martin, “ |

23. | G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. |

24. | V. D. Lam, J. B. Kim, S. J. Lee, Y. P. Lee, and J. Y. Rhee, “Dependence of the magnetic-resonance frequency on the cut-wire width ofcut-wire pair medium,” Opt. Express |

25. | N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Plasmon hybridization in stacked cut-wire metamaterials,” Adv. Mater. (Deerfield Beach Fla.) |

26. | S.-Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B |

27. | P. Tassin, T. Koschny, and C. M. Soukoulis, “Effective material parameter retrieval for thin sheets: Theory and application to graphene, thin silver films, and single-layer metamaterials,” Physica B |

**OCIS Codes**

(160.3918) Materials : Metamaterials

(230.4555) Optical devices : Coupled resonators

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: November 13, 2012

Revised Manuscript: December 22, 2012

Manuscript Accepted: December 28, 2012

Published: January 9, 2013

**Citation**

Mingbo Pu, Chenggang Hu, Cheng Huang, Changtao Wang, Zeyu Zhao, Yanqin Wang, and Xiangang Luo, "Investigation of Fano resonance in planar metamaterial with perturbed periodicity," Opt. Express **21**, 992-1001 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-992

Sort: Year | Journal | Reset

### References

- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001). [CrossRef] [PubMed]
- R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.64(5), 056625 (2001). [CrossRef] [PubMed]
- N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science308(5721), 534–537 (2005). [CrossRef] [PubMed]
- R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(4), 046608 (2004). [CrossRef] [PubMed]
- M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B75(7), 075119 (2007). [CrossRef]
- R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science323(5912), 366–369 (2009). [CrossRef] [PubMed]
- Q. Feng, M. Pu, C. Hu, and X. Luo, “Engineering the dispersion of metamaterial surface for broadband infrared absorption,” Opt. Lett.37(11), 2133–2135 (2012). [CrossRef] [PubMed]
- C. Wu and G. Shvets, “Design of metamaterial surfaces with broadband absorbance,” Opt. Lett.37(3), 308–310 (2012). [CrossRef] [PubMed]
- Y. Cui, J. Xu, K. Hung Fung, Y. Jin, A. Kumar, S. He, and N. X. Fang, “A thin film broadband absorber based on multi-sized nanoantennas,” Appl. Phys. Lett.99(25), 253101 (2011). [CrossRef]
- M. Pu, Q. Feng, M. Wang, C. Hu, C. Huang, X. Ma, Z. Zhao, C. Wang, and X. Luo, “Ultrathin broadband nearly perfect absorber with symmetrical coherent illumination,” Opt. Express20(3), 2246–2254 (2012). [CrossRef] [PubMed]
- N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett.10(7), 2342–2348 (2010). [CrossRef] [PubMed]
- C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum fano resonance,” Phys. Rev. Lett.106(10), 107403 (2011). [CrossRef] [PubMed]
- V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett.99(14), 147401 (2007). [CrossRef] [PubMed]
- F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant LSPR sensing and a tunable Fano resonance,” Nano Lett.8(11), 3983–3988 (2008). [CrossRef] [PubMed]
- A. Christ, O. J. F. Martin, Y. Ekinci, N. A. Gippius, and S. G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett.8(8), 2171–2175 (2008). [CrossRef] [PubMed]
- V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett.104(22), 223901 (2010). [CrossRef] [PubMed]
- N. Papasimakis, V. A. Fedotov, Y. H. Fu, D. P. Tsai, and N. I. Zheludev, “Coherent and incoherent metamaterials and order-disorder transitions,” Phys. Rev. B80(4), 041102 (2009). [CrossRef]
- N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics2(6), 351–354 (2008). [CrossRef]
- E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express17(10), 8548–8551 (2009). [CrossRef] [PubMed]
- C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. A. Yanik, H. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nat. Mater.11(1), 69–75 (2011). [CrossRef] [PubMed]
- Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of Fano resonances,” Phys. Scr.74(2), 259–266 (2006). [CrossRef]
- B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B83(23), 235427 (2011). [CrossRef]
- G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett.30(23), 3198–3200 (2005). [CrossRef] [PubMed]
- V. D. Lam, J. B. Kim, S. J. Lee, Y. P. Lee, and J. Y. Rhee, “Dependence of the magnetic-resonance frequency on the cut-wire width ofcut-wire pair medium,” Opt. Express15(25), 16651–16656 (2007). [CrossRef] [PubMed]
- N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Plasmon hybridization in stacked cut-wire metamaterials,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3628–3632 (2007). [CrossRef]
- S.-Y. Chiam, R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, and A. A. Bettiol, “Analogue of electromagnetically induced transparency in a terahertz metamaterial,” Phys. Rev. B80(15), 153103 (2009). [CrossRef]
- P. Tassin, T. Koschny, and C. M. Soukoulis, “Effective material parameter retrieval for thin sheets: Theory and application to graphene, thin silver films, and single-layer metamaterials,” Physica B407(20), 4062–4065 (2012). [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.