## Resonant modes of 12-fold symmetric defect free photonic quasicrystal |

Optics Express, Vol. 22, Issue 2, pp. 2007-2012 (2014)

http://dx.doi.org/10.1364/OE.22.002007

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### Abstract

This work investigates the resonant modes of a 12-fold symmetric defect free photonic quasicrystal (PQC) nanorod array using finite difference time domain (FDTD) simulation. Localized modes can exist in PQC without introducing defects due to the lack of translational symmetry. The resonant modes of the unit cell PQC and the one time expanded PQC from unit cell are systematically examined. The resonant spectrum is that of a single rod modified by the interaction among PQC nanorods. The mode confinement is contributed by guided resonance and destructive interference scattering. The self-scaling similarity of resonant spectrum and mode profile are also investigated.

© 2014 Optical Society of America

## 1. Introduction

1. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: quasiperiodic media,” Phys. Rev. Lett. **58**(23), 2436–2438 (1987). [CrossRef] [PubMed]

9. K. Mnaymneh and R. C. Gauthier, “Mode localization and band-gap formation in defect-free photonic quasicrystals,” Opt. Express **15**(8), 5089–5099 (2007). [CrossRef] [PubMed]

10. G. Gumbs and M. K. Ali, “Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices,” Phys. Rev. Lett. **60**(11), 1081–1084 (1988). [CrossRef] [PubMed]

11. M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing Action due to the Two-Dimensional Quasiperiodicity of Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev. Lett. **92**(12), 123906 (2004). [CrossRef] [PubMed]

17. S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, Y.-H. Lee, and S.-B. Kim, “Photonic quasicrystal single-cell cavity mode,” Appl. Phys. Lett. **86**(3), 031101 (2005). [CrossRef]

11. M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing Action due to the Two-Dimensional Quasiperiodicity of Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev. Lett. **92**(12), 123906 (2004). [CrossRef] [PubMed]

18. S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, and W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express **20**(11), 12457–12462 (2012). [CrossRef] [PubMed]

## 2. Structure model

19. M. Oxborrow and C. L. Henley, “Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals,” Phys. Rev. B Condens. Matter **48**(10), 6966–6998 (1993). [CrossRef] [PubMed]

18. S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, and W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express **20**(11), 12457–12462 (2012). [CrossRef] [PubMed]

## 3. Simulation results and discussion

18. S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, and W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express **20**(11), 12457–12462 (2012). [CrossRef] [PubMed]

20. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B **60**(8), 5751–5758 (1999). [CrossRef]

## 4. Summary

## Acknowledgments

## References and links

1. | M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: quasiperiodic media,” Phys. Rev. Lett. |

2. | W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. |

3. | T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B Condens. Matter |

4. | L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. |

5. | W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature |

6. | A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, and G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Mater. |

7. | T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature |

8. | S. Walter and S.-W. Daniel, “Photonic and phononic quasicrystals,” J. Phys. D Appl. Phys. |

9. | K. Mnaymneh and R. C. Gauthier, “Mode localization and band-gap formation in defect-free photonic quasicrystals,” Opt. Express |

10. | G. Gumbs and M. K. Ali, “Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices,” Phys. Rev. Lett. |

11. | M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing Action due to the Two-Dimensional Quasiperiodicity of Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev. Lett. |

12. | M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature |

13. | K. Nozaki and T. Baba, “Lasing Characteristics of 12-Fold Symmetric Quasi-periodic Photonic Crystal Slab Nanolasers,” Jpn. J. Appl. Phys. |

14. | K. Nozaki and T. Baba, “Quasiperiodic photonic crystal microcavity lasers,” Appl. Phys. Lett. |

15. | J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, and H. M. Gibbs, “Excitonic polaritons in Fibonacci quasicrystals,” Opt. Express |

16. | L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, and D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics |

17. | S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, Y.-H. Lee, and S.-B. Kim, “Photonic quasicrystal single-cell cavity mode,” Appl. Phys. Lett. |

18. | S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, and W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express |

19. | M. Oxborrow and C. L. Henley, “Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals,” Phys. Rev. B Condens. Matter |

20. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: September 17, 2013

Revised Manuscript: December 13, 2013

Manuscript Accepted: January 7, 2014

Published: January 23, 2014

**Citation**

Minfeng Chen, Yun-Jing Li, Yuh-Jen Cheng, Yai-Chung Chang, and Chun-Yen Chang, "Resonant modes of 12-fold symmetric defect free photonic quasicrystal," Opt. Express **22**, 2007-2012 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-2007

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### References

- M. Kohmoto, B. Sutherland, K. Iguchi, “Localization of optics: quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987). [CrossRef] [PubMed]
- W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994). [CrossRef] [PubMed]
- T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B Condens. Matter 50(6), 4220–4223 (1994). [CrossRef] [PubMed]
- L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90(5), 055501 (2003). [CrossRef] [PubMed]
- W. Man, M. Megens, P. J. Steinhardt, P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005). [CrossRef] [PubMed]
- A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener, G. von Freymann, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Mater. 5(12), 942–945 (2006). [CrossRef] [PubMed]
- T. Matsui, A. Agrawal, A. Nahata, Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007). [CrossRef] [PubMed]
- S. Walter, S.-W. Daniel, “Photonic and phononic quasicrystals,” J. Phys. D Appl. Phys. 40(13), R229–R247 (2007). [CrossRef]
- K. Mnaymneh, R. C. Gauthier, “Mode localization and band-gap formation in defect-free photonic quasicrystals,” Opt. Express 15(8), 5089–5099 (2007). [CrossRef] [PubMed]
- G. Gumbs, M. K. Ali, “Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices,” Phys. Rev. Lett. 60(11), 1081–1084 (1988). [CrossRef] [PubMed]
- M. Notomi, H. Suzuki, T. Tamamura, K. Edagawa, “Lasing Action due to the Two-Dimensional Quasiperiodicity of Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev. Lett. 92(12), 123906 (2004). [CrossRef] [PubMed]
- M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000). [CrossRef] [PubMed]
- K. Nozaki, T. Baba, “Lasing Characteristics of 12-Fold Symmetric Quasi-periodic Photonic Crystal Slab Nanolasers,” Jpn. J. Appl. Phys. 45(8A), 6087–6090 (2006). [CrossRef]
- K. Nozaki, T. Baba, “Quasiperiodic photonic crystal microcavity lasers,” Appl. Phys. Lett. 84(24), 4875–4877 (2004). [CrossRef]
- J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, H. M. Gibbs, “Excitonic polaritons in Fibonacci quasicrystals,” Opt. Express 16(20), 15382–15387 (2008). [CrossRef] [PubMed]
- L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, H. E. Beere, D. A. Ritchie, D. S. Wiersma, “Quasi-periodic distributed feedback laser,” Nat. Photonics 4(3), 165–169 (2010). [CrossRef]
- S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, Y.-H. Lee, S.-B. Kim, “Photonic quasicrystal single-cell cavity mode,” Appl. Phys. Lett. 86(3), 031101 (2005). [CrossRef]
- S. P. Chang, K. P. Sou, C. H. Chen, Y. J. Cheng, J. K. Huang, C. H. Lin, H. C. Kuo, C. Y. Chang, W. F. Hsieh, “Lasing action in gallium nitride quasicrystal nanorod arrays,” Opt. Express 20(11), 12457–12462 (2012). [CrossRef] [PubMed]
- M. Oxborrow, C. L. Henley, “Random square-triangle tilings: A model for twelvefold-symmetric quasicrystals,” Phys. Rev. B Condens. Matter 48(10), 6966–6998 (1993). [CrossRef] [PubMed]
- S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, L. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]

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