Abstract

We present a coupling matrix formalism to investigate the effects of periodic and quasi-periodic orders on the photonic bandgap (PBG) structures of coupled-resonator optical waveguides (CROWs) based on microring resonators. For the periodic order case, size-tuned defects are introduced at periodic locations among the regular rings, which are size-untuned, to form a periodic ordered CROW system. The periodic coupled defects result in multiple localization states that lead to the formation of mini-defect bands and mini-PBGs within the PBG of a defect-free CROW. The position and number of such mini-defect bands depend on the size tuning of the defects. For the quasi-periodic order case, the arrangement of the defects and the regular rings in the ring cascade is an intermediate between periodic order and randomness, thus forming a quasi-periodic ordered CROW system. The effects of quasi-periodicity on the PBG structures are illustrated using the Fibonacci sequences, which result in a single high-Q localized state to appear that gradually transits to a mini-band within a wide photonic stop band as the number of lattice cells increases.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  2. D. D.  Smith, H.  Chang, K. A.  Fuller, A. T.  Rosenberger, and R. W.  Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A  69, 063804 (2004).
    [CrossRef]
  3. Y.  Xu, Y.  Li, R. K.  Lee, and A.  Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E  62,7389-7404 (2000).
    [CrossRef]
  4. B. E.  Little, S. T.  Chu, H. A.  Haus, J.  Foresi, and J.-P.  Laine, "Microring resonator channel dropping filters," J. Lightwave Technol.  15, 998-1005 (1997).
    [CrossRef]
  5. Y. M.  Landobasa, S.  Darmawan, and M. K.  Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron.  41, 1410-1418 (2005).
    [CrossRef]
  6. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, "Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects," Opt. Express 15, 11934-11941 (2007).
    [CrossRef] [PubMed]
  7. W.-Y. Chen, V. Van, W. N. Herman, and P.-T. Ho, "Periodic Microring Lattice as a Bandstop Filter," IEEE Photon. Technol. Lett. 18, 2041-2043 (2006).
    [CrossRef]
  8. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B. 21, 1665-1673 (2004).
    [CrossRef]
  9. F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nature Photonics 1, 65 (2007).
    [CrossRef]
  10. Y. Landobasa and M. Chin, "Defect modes in micro-ring resonator arrays," Opt. Express 13, 7800-7815 (2005).
    [CrossRef] [PubMed]
  11. J. E. Heebner, R. W. Boyd, and Q. Park, "SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides," J. Opt. Soc. Am. B 19, 722-731 (2002).
    [CrossRef]
  12. M. M. Sigalas, C. M. Soukoulis, C. T. Chan and D. Turner, "Localization of electromagnetic waves in two-dimensional disordered systems," Phys. Rev. B. 53, 8340 (1996).
    [CrossRef]
  13. J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
    [CrossRef]
  14. J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
    [CrossRef] [PubMed]
  15. H. Hiramoto and M. Kohmoto, "Electronic spectral and wave function properties of one-dimensional quasi-periodic systems: A scaling approach," Int. J. Mod. Phys. B. 6, 281 (1992).
    [CrossRef]
  16. X. Huang, Y. Wang, and C. Gong, "Numerical investigation of light-wave localization in optical Fibonacci superlattices with symmetric internal structure, " J. Phys. Condens. Matter. 11, 7645 (1999).
    [CrossRef]
  17. L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
    [CrossRef]
  18. L. Moretti and V. Mocella, "Two-dimensional photonic aperiodic crystals based on Thue-Morse sequence," Opt. Express 15, 15314-15323 (2007).
    [CrossRef] [PubMed]
  19. K. Mnaymneh and R. C. Gauthier, "Mode localization and band-gap formation in defect-free photonic quasicrystals," Opt. Express 15, 5089-5099 (2007).
    [CrossRef] [PubMed]
  20. S. V. Boriskina, A. Gopinath, and L. Dal Negro, "Optical gap formation and localization properties of optical modes in deterministic aperiodic photonic structures," Opt. Express 16, 18813-18826 (2008).
    [CrossRef]
  21. E. Maciá, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397 (2006).
    [CrossRef]
  22. J.-B. Suck, M. Schreiber, and P. Häussler, Quasicrystals: An Introduction to Structure, Physical Properties and Applications (Springer-Verlag Berlin Heidelberg, 2002), Chap. 2.
  23. H. A. Haus, Waves and fields in optoelectronics (New York: Prentice-Hall, 1984).
  24. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2001), Chap.1.
  25. T.  Kalamakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron.  41, 1419-1425 (2007).
    [CrossRef]
  26. Y.  Chen and S.  Blair, "Nonlinearity enhancement in finite coupled-resonator slow-light waveguides," Opt. Express  12, 3353-3366 (2004).
    [CrossRef] [PubMed]
  27. L. Y. M. Tobing, P. Dumon, R. Baets, and M. Chin, "Boxlike filter response based on complementary photonic bandgaps in two-dimensional microresonator arrays," Opt. Lett. 33, 2512-2514 (2008).
    [CrossRef] [PubMed]
  28. FullWAVE, Rsoft Design Group, Inc., Ossining, NY.
  29. J. Capmany, P. Muñoz, J. D. Domenech, and M. A. Muriel, "Apodized coupled resonator waveguides," Opt. Express 15, 10196-10206 (2007).
    [CrossRef] [PubMed]
  30. C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
    [CrossRef]

2008 (3)

2007 (6)

2006 (2)

W.-Y. Chen, V. Van, W. N. Herman, and P.-T. Ho, "Periodic Microring Lattice as a Bandstop Filter," IEEE Photon. Technol. Lett. 18, 2041-2043 (2006).
[CrossRef]

E. Maciá, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397 (2006).
[CrossRef]

2005 (2)

Y. M.  Landobasa, S.  Darmawan, and M. K.  Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron.  41, 1410-1418 (2005).
[CrossRef]

Y. Landobasa and M. Chin, "Defect modes in micro-ring resonator arrays," Opt. Express 13, 7800-7815 (2005).
[CrossRef] [PubMed]

2004 (5)

J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
[CrossRef] [PubMed]

Y.  Chen and S.  Blair, "Nonlinearity enhancement in finite coupled-resonator slow-light waveguides," Opt. Express  12, 3353-3366 (2004).
[CrossRef] [PubMed]

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B. 21, 1665-1673 (2004).
[CrossRef]

D. D.  Smith, H.  Chang, K. A.  Fuller, A. T.  Rosenberger, and R. W.  Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A  69, 063804 (2004).
[CrossRef]

2002 (1)

2000 (1)

Y.  Xu, Y.  Li, R. K.  Lee, and A.  Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E  62,7389-7404 (2000).
[CrossRef]

1999 (2)

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

X. Huang, Y. Wang, and C. Gong, "Numerical investigation of light-wave localization in optical Fibonacci superlattices with symmetric internal structure, " J. Phys. Condens. Matter. 11, 7645 (1999).
[CrossRef]

1997 (2)

B. E.  Little, S. T.  Chu, H. A.  Haus, J.  Foresi, and J.-P.  Laine, "Microring resonator channel dropping filters," J. Lightwave Technol.  15, 998-1005 (1997).
[CrossRef]

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

1996 (1)

M. M. Sigalas, C. M. Soukoulis, C. T. Chan and D. Turner, "Localization of electromagnetic waves in two-dimensional disordered systems," Phys. Rev. B. 53, 8340 (1996).
[CrossRef]

1992 (1)

H. Hiramoto and M. Kohmoto, "Electronic spectral and wave function properties of one-dimensional quasi-periodic systems: A scaling approach," Int. J. Mod. Phys. B. 6, 281 (1992).
[CrossRef]

Baets, R.

Blair, S.

Boriskina, S. V.

Boyd, R. W.

D. D.  Smith, H.  Chang, K. A.  Fuller, A. T.  Rosenberger, and R. W.  Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A  69, 063804 (2004).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q. Park, "SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides," J. Opt. Soc. Am. B 19, 722-731 (2002).
[CrossRef]

Capmany, J.

Chan, C. T.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan and D. Turner, "Localization of electromagnetic waves in two-dimensional disordered systems," Phys. Rev. B. 53, 8340 (1996).
[CrossRef]

Chang, H.

D. D.  Smith, H.  Chang, K. A.  Fuller, A. T.  Rosenberger, and R. W.  Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A  69, 063804 (2004).
[CrossRef]

Chen, W.-Y.

W.-Y. Chen, V. Van, W. N. Herman, and P.-T. Ho, "Periodic Microring Lattice as a Bandstop Filter," IEEE Photon. Technol. Lett. 18, 2041-2043 (2006).
[CrossRef]

Chen, Y.

Chin, M.

Chin, M. K.

Y. M.  Landobasa, S.  Darmawan, and M. K.  Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron.  41, 1410-1418 (2005).
[CrossRef]

Chu, S. T.

B. E.  Little, S. T.  Chu, H. A.  Haus, J.  Foresi, and J.-P.  Laine, "Microring resonator channel dropping filters," J. Lightwave Technol.  15, 998-1005 (1997).
[CrossRef]

Dal Negro, L.

S. V. Boriskina, A. Gopinath, and L. Dal Negro, "Optical gap formation and localization properties of optical modes in deterministic aperiodic photonic structures," Opt. Express 16, 18813-18826 (2008).
[CrossRef]

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Danner, A. J.

C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
[CrossRef]

Darmawan, S.

Y. M.  Landobasa, S.  Darmawan, and M. K.  Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron.  41, 1410-1418 (2005).
[CrossRef]

Domenech, J. D.

Duan, X.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Dumon, P.

Fan, S.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Ferrera, J.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Foresi, J.

B. E.  Little, S. T.  Chu, H. A.  Haus, J.  Foresi, and J.-P.  Laine, "Microring resonator channel dropping filters," J. Lightwave Technol.  15, 998-1005 (1997).
[CrossRef]

Foresi, J. S.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Fuller, K. A.

D. D.  Smith, H.  Chang, K. A.  Fuller, A. T.  Rosenberger, and R. W.  Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A  69, 063804 (2004).
[CrossRef]

Gauthier, R. C.

Gong, C.

X. Huang, Y. Wang, and C. Gong, "Numerical investigation of light-wave localization in optical Fibonacci superlattices with symmetric internal structure, " J. Phys. Condens. Matter. 11, 7645 (1999).
[CrossRef]

Gopinath, A.

Haavisto, J.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Haus, H. A.

B. E.  Little, S. T.  Chu, H. A.  Haus, J.  Foresi, and J.-P.  Laine, "Microring resonator channel dropping filters," J. Lightwave Technol.  15, 998-1005 (1997).
[CrossRef]

Heebner, J. E.

Herman, W. N.

W.-Y. Chen, V. Van, W. N. Herman, and P.-T. Ho, "Periodic Microring Lattice as a Bandstop Filter," IEEE Photon. Technol. Lett. 18, 2041-2043 (2006).
[CrossRef]

Hiramoto, H.

H. Hiramoto and M. Kohmoto, "Electronic spectral and wave function properties of one-dimensional quasi-periodic systems: A scaling approach," Int. J. Mod. Phys. B. 6, 281 (1992).
[CrossRef]

Ho, P.-T.

W.-Y. Chen, V. Van, W. N. Herman, and P.-T. Ho, "Periodic Microring Lattice as a Bandstop Filter," IEEE Photon. Technol. Lett. 18, 2041-2043 (2006).
[CrossRef]

Huang, X.

X. Huang, Y. Wang, and C. Gong, "Numerical investigation of light-wave localization in optical Fibonacci superlattices with symmetric internal structure, " J. Phys. Condens. Matter. 11, 7645 (1999).
[CrossRef]

Huang, Y.

Ippen, E. P.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Joannopoulos, J. D.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Kalamakis, T.

T.  Kalamakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron.  41, 1419-1425 (2007).
[CrossRef]

Kimerling, L. C.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Kohmoto, M.

H. Hiramoto and M. Kohmoto, "Electronic spectral and wave function properties of one-dimensional quasi-periodic systems: A scaling approach," Int. J. Mod. Phys. B. 6, 281 (1992).
[CrossRef]

Laine, J.-P.

B. E.  Little, S. T.  Chu, H. A.  Haus, J.  Foresi, and J.-P.  Laine, "Microring resonator channel dropping filters," J. Lightwave Technol.  15, 998-1005 (1997).
[CrossRef]

Landobasa, Y.

Landobasa, Y. M.

Y. M.  Landobasa, S.  Darmawan, and M. K.  Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron.  41, 1410-1418 (2005).
[CrossRef]

LeBlanc, J.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Lee, R. K.

Y.  Xu, Y.  Li, R. K.  Lee, and A.  Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E  62,7389-7404 (2000).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

Li, E. P.

C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
[CrossRef]

Li, Y.

Y.  Xu, Y.  Li, R. K.  Lee, and A.  Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E  62,7389-7404 (2000).
[CrossRef]

Lim, S. T.

C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
[CrossRef]

Little, B. E.

B. E.  Little, S. T.  Chu, H. A.  Haus, J.  Foresi, and J.-P.  Laine, "Microring resonator channel dropping filters," J. Lightwave Technol.  15, 998-1005 (1997).
[CrossRef]

Maciá, E.

E. Maciá, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397 (2006).
[CrossRef]

Michel, J.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Mnaymneh, K.

Mocella, V.

Mookherjea, S.

Moretti, L.

Muñoz, P.

Muriel, M. A.

Ogawa, K.

C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
[CrossRef]

Paloczi, G.

Park, G. H.

C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
[CrossRef]

Park, Q.

Png, C. E.

C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
[CrossRef]

Poon, J.

Poon, J. K. S.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B. 21, 1665-1673 (2004).
[CrossRef]

Rooks, M.

Rosenberger, A. T.

D. D.  Smith, H.  Chang, K. A.  Fuller, A. T.  Rosenberger, and R. W.  Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A  69, 063804 (2004).
[CrossRef]

Scherer, A.

Scheuer, J.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B. 21, 1665-1673 (2004).
[CrossRef]

J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
[CrossRef] [PubMed]

Sekaric, L.

Sigalas, M. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan and D. Turner, "Localization of electromagnetic waves in two-dimensional disordered systems," Phys. Rev. B. 53, 8340 (1996).
[CrossRef]

Smith, D. D.

D. D.  Smith, H.  Chang, K. A.  Fuller, A. T.  Rosenberger, and R. W.  Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A  69, 063804 (2004).
[CrossRef]

Smith, H. I.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Soukoulis, C. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan and D. Turner, "Localization of electromagnetic waves in two-dimensional disordered systems," Phys. Rev. B. 53, 8340 (1996).
[CrossRef]

Sphicopoulos, T.

T.  Kalamakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron.  41, 1419-1425 (2007).
[CrossRef]

Steinmeyer, G.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Stolfi, M.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Tan, Y. T.

C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
[CrossRef]

Thoen, E. R.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Tobing, L. Y. M.

Turner, D.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan and D. Turner, "Localization of electromagnetic waves in two-dimensional disordered systems," Phys. Rev. B. 53, 8340 (1996).
[CrossRef]

Van, V.

W.-Y. Chen, V. Van, W. N. Herman, and P.-T. Ho, "Periodic Microring Lattice as a Bandstop Filter," IEEE Photon. Technol. Lett. 18, 2041-2043 (2006).
[CrossRef]

Villeneuve, P. R.

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Vlasov, Y.

Wang, Y.

X. Huang, Y. Wang, and C. Gong, "Numerical investigation of light-wave localization in optical Fibonacci superlattices with symmetric internal structure, " J. Phys. Condens. Matter. 11, 7645 (1999).
[CrossRef]

Xia, F.

Xu, Y.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B. 21, 1665-1673 (2004).
[CrossRef]

Y.  Xu, Y.  Li, R. K.  Lee, and A.  Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E  62,7389-7404 (2000).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

Yariv, A.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B. 21, 1665-1673 (2004).
[CrossRef]

J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
[CrossRef] [PubMed]

Y.  Xu, Y.  Li, R. K.  Lee, and A.  Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E  62,7389-7404 (2000).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

Yi, Y.

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

Appl. Phys. Lett. (2)

L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, "Photon bandgap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004).
[CrossRef]

C. E. Png, G. H. Park, S. T. Lim, E. P. Li, A. J. Danner, K. Ogawa, and Y. T. Tan, "Electrically controlled silicon-based photonic crystal chromatic dispersion compensator with ultralow power consumption," Appl. Phys. Lett. 93, 061111 (2008).
[CrossRef]

IEEE J. Quantum Electron. (2)

T.  Kalamakis and T. Sphicopoulos, "Analytical expressions for the resonant frequencies and modal fields of finite coupled optical cavity chains," IEEE J. Quantum Electron.  41, 1419-1425 (2007).
[CrossRef]

Y. M.  Landobasa, S.  Darmawan, and M. K.  Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron.  41, 1410-1418 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W.-Y. Chen, V. Van, W. N. Herman, and P.-T. Ho, "Periodic Microring Lattice as a Bandstop Filter," IEEE Photon. Technol. Lett. 18, 2041-2043 (2006).
[CrossRef]

Int. J. Mod. Phys. B. (1)

H. Hiramoto and M. Kohmoto, "Electronic spectral and wave function properties of one-dimensional quasi-periodic systems: A scaling approach," Int. J. Mod. Phys. B. 6, 281 (1992).
[CrossRef]

J. Lightwave Technol. (1)

B. E.  Little, S. T.  Chu, H. A.  Haus, J.  Foresi, and J.-P.  Laine, "Microring resonator channel dropping filters," J. Lightwave Technol.  15, 998-1005 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Opt. Soc. Am. B. (1)

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B. 21, 1665-1673 (2004).
[CrossRef]

J. Phys. Condens. Matter. (1)

X. Huang, Y. Wang, and C. Gong, "Numerical investigation of light-wave localization in optical Fibonacci superlattices with symmetric internal structure, " J. Phys. Condens. Matter. 11, 7645 (1999).
[CrossRef]

Nature (1)

J. S.  Foresi, P. R.  Villeneuve, J.  Ferrera, E. R.  Thoen, G.  Steinmeyer, S.  Fan, J. D.  Joannopoulos, L. C.  Kimerling, H. I.  Smith, and E. P.  Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature  390, 143-145 (1997).
[CrossRef]

Nature Photonics (1)

F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nature Photonics 1, 65 (2007).
[CrossRef]

Opt. Express (8)

Opt. Lett. (2)

Phys. Rev. A (1)

D. D.  Smith, H.  Chang, K. A.  Fuller, A. T.  Rosenberger, and R. W.  Boyd, "Coupled-resonator-induced transparency," Phys. Rev. A  69, 063804 (2004).
[CrossRef]

Phys. Rev. B. (1)

M. M. Sigalas, C. M. Soukoulis, C. T. Chan and D. Turner, "Localization of electromagnetic waves in two-dimensional disordered systems," Phys. Rev. B. 53, 8340 (1996).
[CrossRef]

Phys. Rev. E (1)

Y.  Xu, Y.  Li, R. K.  Lee, and A.  Yariv, "Scattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E  62,7389-7404 (2000).
[CrossRef]

Rep. Prog. Phys. (1)

E. Maciá, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397 (2006).
[CrossRef]

Other (4)

J.-B. Suck, M. Schreiber, and P. Häussler, Quasicrystals: An Introduction to Structure, Physical Properties and Applications (Springer-Verlag Berlin Heidelberg, 2002), Chap. 2.

H. A. Haus, Waves and fields in optoelectronics (New York: Prentice-Hall, 1984).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2001), Chap.1.

FullWAVE, Rsoft Design Group, Inc., Ossining, NY.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1.
Fig. 1.

CROW with N mutually coupled unit cells, consisting of defect B at periodic locations.

Fig. 2.
Fig. 2.

The dispersion diagrams of infinite CROW, showing the regions of PBG (shaded) and photonic bands (solid blue line) for different integral value of the size tuning factor γ: (a) γ = 1 (b) γ = 2 (c) γ = 3 (d) γ = 4.

Fig. 3.
Fig. 3.

The dispersion diagram of infinite CROW for γ = 1: (a) Weaker inter-unit cell coupling (tinter 2 < tintra 2): tinter 2 is fixed at 0.3 while increasing tintra 2 to 0.3 (black line), 0.5 (red line), 0.7 (green line) and 0.9 (blue line). (b) Stronger inter-unit cell coupling (tinter 2 > tintra 2): tinter 2 is fixed at 0.9 while reducing tintra 2 to 0.9 (black line), 0.7 (red line), 0.5 (green line) and 0.3 (blue line). Arrows indicate how photonic bands move as ∣F∣ increases, where F = tinter 2 - tintra 2.

Fig. 4.
Fig. 4.

The transformation of the nature of state localization within the PBG from a single high-Q resonant state to a mini-defect band as the number of periodic defects in the CROW is increased in the following sequence: (a) {0 0 0 0 0 0 0 D 0 0 0 0 0 0 0},(b) {0 0 0 0 0 D 0 D 0 D 0 0 0 0 0} (c) {0 0 0 D 0 D 0 D 0 D 0 D 0 0 0} (d) {0 D 0 D 0 D 0 D 0 D 0 D 0 D 0} The ring defect is represented as D with twice the cavity size as the size-untuned or regular ring 0.

Fig. 5.
Fig. 5.

Transmissions and corresponding phase responses for a finite CROW with 7½ unit cells for different defect size tuning factor of γ = 1 (blue), 2 (red) and 3 (green) at r2 = 0.3. At γ = 1, there is no defects and thus only 1 wide PBG forms. Tuning γ above 1 forms mini-PBGs and mini-defect bands in what was formerly a single wide PBG of the original CROW with γ = 1.

Fig. 6.
Fig. 6.

Transmission (top graph) for a CROW with 7½ unit cells at r2 = 0.3 for non-integer size tuning factor γ = 2.3 (red) and 2.7 (blue), which is in good agreement with the dispersion diagram (bottom graph) of an infinite CROW. Arrows indicate directions of movements of defect bands with increasing γ. Similar leftward translations also apply to the primary bands.

Fig. 7.
Fig. 7.

The effects of varying coupling strengths on the primary passbands centered at integer resonance order δ/2π (at resonance) for a CROW with N = 7 for 2 different cases: (a) All rings have similar coupling strength t2 (red: t2 =0.7, blue: t2 = 0.5, green: t2 =0.1) (b) tinter 2 ≠ tintra 2: tinter 2 is fixed at 0.3 while adjusting tintra 2 to 0.3 (blue line), 0.5 (red line) and 0.9 (black line).

Fig. 8.
Fig. 8.

The origin of the mini-PBG at resonance: (a) The effect of increasing tintra 2 while keeping tinter 2 fixed at 0.3 for a single unit cell with 2 identical rings (γ = 1). (b) The effect of increasing the number of unit cells N of a CROW, that has the coupling scheme of tinter 2 = 0.3, tintra 2 = 0.9.

Fig. 9.
Fig. 9.

The transmission of a CROW with around 7½ unit cells of configuration ((ABB)7A)) with tuning factor: (a) γ = 2 (b) γ = 3. For each respective γ, the dispersion diagram of infinite CROW is shown below the corresponding transmission graph. The PBGs are marked by the shaded regions while the photonic bands are represented by red lines.

Fig. 10.
Fig. 10.

The PBG structures before (shown in blue graph) and after (shown in red graph) a new defect G is added at the center of the periodic CROW. Addition of a new defect G changes the CROW sequence from {0 D 0 D 0 D 0 D 0 D 0 D 0} to {0 D 0 D 0 D G D 0 D 0 D 0}and generates a new localized state in each of the mini-PBGs as shown in the red graph.

Fig. 11.
Fig. 11.

A quasi-periodic CROW of 5 rings, following the Fibonacci sequence C4 = {B, A, A, B, A}.

Fig. 12.
Fig. 12.

Transmission spectra of quasi-periodic CROW following the Fibonacci sequence of C3 (blue), C4 (red), C5 (green) and C6 (pink), for the case of γ = 2.

Fig. 13.
Fig. 13.

Transmissions of quasi-periodic CROW following the Fibonacci sequence of C4 at r2 = 0.9 for three different size tuning ratios: γ = 2 (blue), 3 (red) and 4 (green). There are (γ-1) localized states within each wide PBG centered at half-integer resonance order.

Fig. 14.
Fig. 14.

The effects of losses on transmissivity: (a) Periodic CROW structure with 3½ unit cells of sequence {A, B, A, B, A, B, A}, with γ = 2 at r2 = 0.3. (b) Quasi-periodic CROW based on the Fibonacci sequence C4 = {B, A, A, B, A} with γ = 2 and r2 = 0.9.

Fig. 15.
Fig. 15.

FDTD simulated normalized transmission at the output port of a quasi-periodic CROW based on the Fibonacci sequence C4 = { B, A, A, B, A }, with γ = 2.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

[ b 0,2 c 1,1 ] T = [ m ij ] [ a 0,2 d 1,1 ] T
a 1,1 = c 1,1 ( τ 1 ) 1 / 2 exp ( i δ 1 / 2 ) , d 1,1 = b 1,1 ( τ 1 ) 1 / 2 exp ( i δ 1 / 2 )
[ a n , 2 b n , 2 ] T = [ S 2 ij ] [ S 1 ij ] [ a n 1 , 2 b n 1 , 2 ] T = [ U ij ] [ a n 1 , 2 b n 1 , 2 ] T
[ a N , 2 b N , 2 ] = [ U 11 C N 1 ( x ) C N 2 ( x ) U 12 C N 1 ( x ) U 21 C N 1 ( x ) U 22 C N 1 ( x ) C N 2 ( x ) ] [ a 0,1 b 0,1 ] = [ M ij ] [ a 0,2 b 0,2 ] T
[ a n , 2 b n , 2 ] T = exp ( i K Λ ) [ a n 1 , 2 b n 1 , 2 ] T
exp ( i K Λ ) = 0.5 ( U 11 + U 12 ) ± i 1 [ 0.5 ( U 11 + U 12 ) ] 2 = cos θ ± i sin θ
cos ( ) = 0.5 ( U 11 + U 12 ) = { r u cos [ 0.5 ( δ 1 δ 2 ) ] cos [ 0.5 ( δ 1 + δ 2 ) ] } t u 1
T = c N + 1,1 / a 0,2 exp ( i ϕ T ) = i t int er / ( r int er M 12 + M 22 )
T = it inter t u C sin ( θ ) / [ ( r int er 2 D r u [ E + F ] + G ) sin ( N θ ) + t u C sin ( [ N 1 ] θ ) ]

Metrics