Abstract

We theoretically investigate coupled Fano structures which combine the characteristics of both directly coupled Fabry-Perot cavities (DCFPC) and a side-coupled integrated spaced sequence of resonators (SCISSOR). Asymmetric and symmetric Fano resonances in a single and doubly-coupled Fano unit are analytically derived based on Fabry-Perot approach. It is found that doubly-coupled Fano units show a special asymmetric EIT-like lineshape. This structure shows an index changing sensitivity of 10−6, about two orders higher than that of the single Fano resonator, which is promising for index sensor application. A unique frequency detuning method of EIT like lineshape is also found in the doubly-coupled Fano units. The multiple coupled Fano unit structure demonstrates potentials for applications of tunable optical filter and slow light, whereas for the latter it shows much higher group delay than that of SCISSORS and DCFPC with the same parameters.

© 2010 OSA

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  1. M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
    [Crossref]
  2. M. Lipson, “Compact electro-optic modulators on a silicon chip,” J. Sel. Top. Quant. Electron. 12(6), 1520–1526 (2006).
    [Crossref]
  3. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
    [Crossref] [PubMed]
  4. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
    [Crossref]
  5. S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).
    [Crossref]
  6. L. Y. Mario, S. Darmawan, and M. K. Chin, “Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching,” Opt. Express 14(26), 12770–12781 (2006).
    [Crossref] [PubMed]
  7. W. Liang, L. Yang, J. K. S. Poon, Y. Huang, K. J. Vahala, and A. Yariv, “Transmission characteristics of a Fabry-Perot etalon-microtoroid resonator coupled system,” Opt. Lett. 31(4), 510–512 (2006).
    [Crossref] [PubMed]
  8. D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structure with intracavity reflections,” J. Sel. Top. Quant. Electron. 15(5), 1420–1426 (2009).
    [Crossref]
  9. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004).
    [Crossref] [PubMed]
  10. Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
    [Crossref] [PubMed]
  11. C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
    [Crossref]
  12. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
    [Crossref]
  13. O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW formation of mid-band zero group velocity,” Opt. Express 17(17), 14817–14824 (2009).
    [Crossref] [PubMed]
  14. J. E. Heebner, R. W. Boyd, and Q.-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19(4), 722–731 (2002).
    [Crossref]
  15. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
    [Crossref] [PubMed]
  16. Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066616 (2003).
    [Crossref]
  17. H. A. Haus, Waves and Fields in Optoelectronics. (Prentice–Hall, New York, 1984).
  18. M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A 71(1), 013803 (2005).
    [Crossref]
  19. S. F. Mingaleev, A. E. Miroshnichenko, and Y. S. Kivshar, “Coupled-resonator-induced reflection in photonic-crystal waveguide structures,” Opt. Express 16(15), 11647–11659 (2008).
    [Crossref] [PubMed]
  20. S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
    [Crossref]
  21. M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82(26), 4648–4650 (2003).
    [Crossref]
  22. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator for linear and nonlinear optics,” J. Opt. Soc. B 21(10), 1818–1832 (2004).
    [Crossref]
  23. T. Kamalakis and T. Sphicopoulos, “Analytical Expressions for the Resonant Frequencies and Modal fields of Finite Coupled Optical Cavity Chains,” IEEE J. Quantum Electron. 41(11), 1419–1425 (2005).
    [Crossref]
  24. Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41(11), 1410–1418 (2005).
    [Crossref]
  25. T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q Nanocavities in Two-dimensional Photonic Crystal Slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
    [Crossref]
  26. Y. Takahashi, H. Hagino, Y. Tanaka, B.-S. Song, T. Asano, and S. Noda, “High-Q nanocavity with a 2-ns photon lifetime,” Opt. Express 15(25), 17206–17213 (2007).
    [Crossref] [PubMed]
  27. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature 1, 65 (2007).
  28. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21(9), 1665–1673 (2004).
    [Crossref]
  29. 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(19), 11934–11941 (2007).
    [Crossref] [PubMed]

2009 (2)

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structure with intracavity reflections,” J. Sel. Top. Quant. Electron. 15(5), 1420–1426 (2009).
[Crossref]

O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW formation of mid-band zero group velocity,” Opt. Express 17(17), 14817–14824 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (5)

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
[Crossref]

Y. Takahashi, H. Hagino, Y. Tanaka, B.-S. Song, T. Asano, and S. Noda, “High-Q nanocavity with a 2-ns photon lifetime,” Opt. Express 15(25), 17206–17213 (2007).
[Crossref] [PubMed]

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature 1, 65 (2007).

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(19), 11934–11941 (2007).
[Crossref] [PubMed]

2006 (5)

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q Nanocavities in Two-dimensional Photonic Crystal Slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

M. Lipson, “Compact electro-optic modulators on a silicon chip,” J. Sel. Top. Quant. Electron. 12(6), 1520–1526 (2006).
[Crossref]

L. Y. Mario, S. Darmawan, and M. K. Chin, “Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching,” Opt. Express 14(26), 12770–12781 (2006).
[Crossref] [PubMed]

W. Liang, L. Yang, J. K. S. Poon, Y. Huang, K. J. Vahala, and A. Yariv, “Transmission characteristics of a Fabry-Perot etalon-microtoroid resonator coupled system,” Opt. Lett. 31(4), 510–512 (2006).
[Crossref] [PubMed]

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
[Crossref] [PubMed]

2005 (4)

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
[Crossref] [PubMed]

M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A 71(1), 013803 (2005).
[Crossref]

T. Kamalakis and T. Sphicopoulos, “Analytical Expressions for the Resonant Frequencies and Modal fields of Finite Coupled Optical Cavity Chains,” IEEE J. Quantum Electron. 41(11), 1419–1425 (2005).
[Crossref]

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41(11), 1410–1418 (2005).
[Crossref]

2004 (3)

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator for linear and nonlinear optics,” J. Opt. Soc. B 21(10), 1818–1832 (2004).
[Crossref]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21(9), 1665–1673 (2004).
[Crossref]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004).
[Crossref] [PubMed]

2003 (3)

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[Crossref]

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82(26), 4648–4650 (2003).
[Crossref]

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066616 (2003).
[Crossref]

2002 (2)

2000 (1)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[Crossref] [PubMed]

1999 (2)

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[Crossref]

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Adibi, A.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Akahane, Y.

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q Nanocavities in Two-dimensional Photonic Crystal Slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

Alexandropoulos, D.

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structure with intracavity reflections,” J. Sel. Top. Quant. Electron. 15(5), 1420–1426 (2009).
[Crossref]

Asano, T.

Y. Takahashi, H. Hagino, Y. Tanaka, B.-S. Song, T. Asano, and S. Noda, “High-Q nanocavity with a 2-ns photon lifetime,” Opt. Express 15(25), 17206–17213 (2007).
[Crossref] [PubMed]

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q Nanocavities in Two-dimensional Photonic Crystal Slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

Askari, M.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Boyd, R. W.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator for linear and nonlinear optics,” J. Opt. Soc. B 21(10), 1818–1832 (2004).
[Crossref]

J. E. Heebner, R. W. Boyd, and Q.-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19(4), 722–731 (2002).
[Crossref]

Chak, P.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator for linear and nonlinear optics,” J. Opt. Soc. B 21(10), 1818–1832 (2004).
[Crossref]

Chao, C. Y.

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[Crossref]

Chin, M. K.

L. Y. Mario, S. Darmawan, and M. K. Chin, “Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching,” Opt. Express 14(26), 12770–12781 (2006).
[Crossref] [PubMed]

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41(11), 1410–1418 (2005).
[Crossref]

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Darmawan, S.

L. Y. Mario, S. Darmawan, and M. K. Chin, “Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching,” Opt. Express 14(26), 12770–12781 (2006).
[Crossref] [PubMed]

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41(11), 1410–1418 (2005).
[Crossref]

Fan, S.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
[Crossref] [PubMed]

M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A 71(1), 013803 (2005).
[Crossref]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004).
[Crossref] [PubMed]

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066616 (2003).
[Crossref]

S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).
[Crossref]

Guo, L. J.

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[Crossref]

Hagino, H.

Heebner, J. E.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator for linear and nonlinear optics,” J. Opt. Soc. B 21(10), 1818–1832 (2004).
[Crossref]

J. E. Heebner, R. W. Boyd, and Q.-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19(4), 722–731 (2002).
[Crossref]

Ho, S. T.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Hotchkiss, P. J.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Huang, Y.

Jones, S. C.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Kamalakis, T.

T. Kamalakis and T. Sphicopoulos, “Analytical Expressions for the Resonant Frequencies and Modal fields of Finite Coupled Optical Cavity Chains,” IEEE J. Quantum Electron. 41(11), 1419–1425 (2005).
[Crossref]

Kira, G.

Kivshar, Y. S.

Kuramochi, E.

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
[Crossref] [PubMed]

Landobasa, Y. M.

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41(11), 1410–1418 (2005).
[Crossref]

Lee, R. K.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[Crossref] [PubMed]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[Crossref]

Li, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[Crossref] [PubMed]

Liang, W.

Lipson, M.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
[Crossref] [PubMed]

M. Lipson, “Compact electro-optic modulators on a silicon chip,” J. Sel. Top. Quant. Electron. 12(6), 1520–1526 (2006).
[Crossref]

Loncar, M.

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82(26), 4648–4650 (2003).
[Crossref]

Marder, S. R.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Mario, L. Y.

Mingaleev, S. F.

Miroshnichenko, A. E.

Mitsugi, S.

Momeni, B.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Noda, S.

Y. Takahashi, H. Hagino, Y. Tanaka, B.-S. Song, T. Asano, and S. Noda, “High-Q nanocavity with a 2-ns photon lifetime,” Opt. Express 15(25), 17206–17213 (2007).
[Crossref] [PubMed]

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q Nanocavities in Two-dimensional Photonic Crystal Slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

Norwood, R. A.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Notomi, M.

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
[Crossref] [PubMed]

Park, Q.-H.

Pereira, S.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator for linear and nonlinear optics,” J. Opt. Soc. B 21(10), 1818–1832 (2004).
[Crossref]

Peyghambarian, N.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Pierson, T.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Poon, J. K. S.

Povinelli, M. L.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
[Crossref] [PubMed]

Qiu, Y.

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82(26), 4648–4650 (2003).
[Crossref]

Ren, Z.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Rooks, M.

Sandhu, S.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
[Crossref] [PubMed]

Scherer, A.

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82(26), 4648–4650 (2003).
[Crossref]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[Crossref]

Scheuer, J.

O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW formation of mid-band zero group velocity,” Opt. Express 17(17), 14817–14824 (2009).
[Crossref] [PubMed]

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structure with intracavity reflections,” J. Sel. Top. Quant. Electron. 15(5), 1420–1426 (2009).
[Crossref]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21(9), 1665–1673 (2004).
[Crossref]

Sekaric, L.

Shakya, J.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
[Crossref] [PubMed]

Shinya, A.

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
[Crossref] [PubMed]

Sipe, J. E.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator for linear and nonlinear optics,” J. Opt. Soc. B 21(10), 1818–1832 (2004).
[Crossref]

Song, B.-S.

Y. Takahashi, H. Hagino, Y. Tanaka, B.-S. Song, T. Asano, and S. Noda, “High-Q nanocavity with a 2-ns photon lifetime,” Opt. Express 15(25), 17206–17213 (2007).
[Crossref] [PubMed]

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q Nanocavities in Two-dimensional Photonic Crystal Slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

Sphicopoulos, T.

T. Kamalakis and T. Sphicopoulos, “Analytical Expressions for the Resonant Frequencies and Modal fields of Finite Coupled Optical Cavity Chains,” IEEE J. Quantum Electron. 41(11), 1419–1425 (2005).
[Crossref]

Suh, W.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004).
[Crossref] [PubMed]

Takahashi, Y.

Tanabe, T.

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005).
[Crossref] [PubMed]

Tanaka, Y.

Taniyama, H.

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
[Crossref]

Tay, S.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Thomas, J.

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

Vahala, K. J.

Vainos, N. A.

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structure with intracavity reflections,” J. Sel. Top. Quant. Electron. 15(5), 1420–1426 (2009).
[Crossref]

Vlasov, Y.

Wang, L.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Wang, Z.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004).
[Crossref] [PubMed]

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066616 (2003).
[Crossref]

Weiss, O.

Wu, S. L.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Xia, F.

Xu, Q. F.

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
[Crossref] [PubMed]

Xu, Y.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21(9), 1665–1673 (2004).
[Crossref]

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[Crossref] [PubMed]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[Crossref]

Yang, L.

Yanik, M. F.

M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A 71(1), 013803 (2005).
[Crossref]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004).
[Crossref] [PubMed]

Yariv, A.

W. Liang, L. Yang, J. K. S. Poon, Y. Huang, K. J. Vahala, and A. Yariv, “Transmission characteristics of a Fabry-Perot etalon-microtoroid resonator coupled system,” Opt. Lett. 31(4), 510–512 (2006).
[Crossref] [PubMed]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21(9), 1665–1673 (2004).
[Crossref]

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[Crossref] [PubMed]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[Crossref]

Youtsey, C.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Zhao, W.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Zhao, Y. G.

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

Appl. Phys. Lett. (4)

S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).
[Crossref]

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett. 83(8), 1527–1529 (2003).
[Crossref]

S. Tay, J. Thomas, B. Momeni, M. Askari, A. Adibi, P. J. Hotchkiss, S. C. Jones, S. R. Marder, R. A. Norwood, and N. Peyghambarian, “Planar photonic crystals infiltrated with nanopartical/polymer composites,” Appl. Phys. Lett. 91(22), 221109 (2007).
[Crossref]

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical detection,” Appl. Phys. Lett. 82(26), 4648–4650 (2003).
[Crossref]

IEEE J. Quantum Electron. (2)

T. Kamalakis and T. Sphicopoulos, “Analytical Expressions for the Resonant Frequencies and Modal fields of Finite Coupled Optical Cavity Chains,” IEEE J. Quantum Electron. 41(11), 1419–1425 (2005).
[Crossref]

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41(11), 1410–1418 (2005).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q Nanocavities in Two-dimensional Photonic Crystal Slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1123–1134 (2006).
[Crossref]

IEEE. Photon. Tech. Lett. (1)

M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu, L. Wang, Y. G. Zhao, and S. T. Ho, “GaAs microcavity channel-dropping filter based on a race-track resonator,” IEEE. Photon. Tech. Lett. 11(12), 1620–1622 (1999).
[Crossref]

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

J. Opt. Soc. B (2)

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator for linear and nonlinear optics,” J. Opt. Soc. B 21(10), 1818–1832 (2004).
[Crossref]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21(9), 1665–1673 (2004).
[Crossref]

J. Sel. Top. Quant. Electron. (2)

M. Lipson, “Compact electro-optic modulators on a silicon chip,” J. Sel. Top. Quant. Electron. 12(6), 1520–1526 (2006).
[Crossref]

D. Alexandropoulos, J. Scheuer, and N. A. Vainos, “Spectral properties of active racetrack semiconductor structure with intracavity reflections,” J. Sel. Top. Quant. Electron. 15(5), 1420–1426 (2009).
[Crossref]

Nat. Photonics (1)

T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1(1), 49–52 (2007).
[Crossref]

Nature (1)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature 1, 65 (2007).

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. A (1)

M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A 71(1), 013803 (2005).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 066616 (2003).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(55 Pt B), 7389–7404 (2000).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004).
[Crossref] [PubMed]

Q. F. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an On-Chip all-optical analogue to Electromagnetically Induced Transparency,” Phys. Rev. Lett. 96(12), 123901 (2006).
[Crossref] [PubMed]

Other (1)

H. A. Haus, Waves and Fields in Optoelectronics. (Prentice–Hall, New York, 1984).

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Figures (10)

Fig. 1
Fig. 1

The schematic of (a) directly coupled Fabry-Perot cavities (DCFPC), (b) side-coupled integrated spaced sequence of resonators (SCISSOR), and (c) coupled Fano resonance units.

Fig. 2
Fig. 2

(a) The schematic of a single Fano resonance unit (FRU). (b) The reduction of half FRU to effective FP cavity. (c) The equivalent FP configuration for a single FRU.

Fig. 3
Fig. 3

(a) The transmission spectrum in different resonance detuning between side-coupled and FP cavity in lossless case γ0 = 0. (b) The transmission in different loss (γ0) with fixed coupling (γc = 2x10−3) and ω 0 = 1.5ω FP. Both the resonance broadening and the frequency are normalized to (2πc/L). The FP responses are indicated in dashed lines.

Fig. 4
Fig. 4

Two possible coupling configurations: (a) General case and (b) Special case.

Fig. 5
Fig. 5

The lossless transmission of single Fano unit (dash line) and doubly-coupled Fano units (solid line) as a function of mirror reflectivity in (a) special case and (b) general case (d = L). The resonances frequency are ω 0 = 0.15(2πc/L) and ω FP = 0.25(2πc/L).

Fig. 6
Fig. 6

The transmission T 2u as a function of ω 0 in: (a) special case (d = 0) and (b) general case (d = L). The following parameters are kept constant: r p = 0.4 and ω FP = 0.25(2πc/L). The ω 0n is the normalized resonance frequency, ω 0 = ω 0n(2πc/L).

Fig. 7
Fig. 7

The comparison between DCFPC, SCISSOR, and Fano units (d = 0) in (a) ω 0ω FP, ω 0 = 0.22(2πc/L) and (b) ω 0 = ω FP. The dashed lines indicate the spectrum of a single unit of each type (DCFPC, SCISSOR, and Fano unit). Unless specified, the parameters by-default are N = 20, ω FP = 0.25(2πc/L), r P = 0.4, γC = 2x10−3(2πc/L).

Fig. 8
Fig. 8

Calculated photonic bands of SCISSOR (SB), DCFPC (CB), and coupled Fano units (FB). The transmission spectrum of coupled Fano units is shown in the right panel. When |(ω-ω FP)/ ω FP|>0.6, the CB and FB follows the same line.

Fig. 9
Fig. 9

The bandstructure of coupled Fano units in different normalized ω 0.

Fig. 10
Fig. 10

The transmission and group delay of DCFPC, SCISSORS and multiple coupled Fano units in general configuration. The parameters used are: ω 0n = 0.13, ω FPn = 0.25, N = 10, γcn = 2x10−3, r P = 0.4. The ω 0, ω FP, γc are all normalized to 2πc/L.

Equations (11)

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

t s a 2 ' a 1 ' b 1 ' b 2 ' = ( ω ω 0 ) + i γ 0 ( ω ω 0 ) + i γ , r s b 1 ' a 1 ' a 2 ' b 2 ' = i γ c ( ω ω 0 ) + i γ
t u = t p 2 t s exp ( i 2 δ ) [ 1 r s r p exp ( i 2 δ ) ] 2 r p 2 t s 2 exp ( i 4 δ ) , r u = r p + r p t p 2 ( 1 2 r s ) exp ( i 4 δ ) [ 1 r s r p exp ( i 2 δ ) ] 2 r p 2 t s 2 exp ( i 4 δ ) .
t u t F P = t p 2 exp ( i 1 2 ϕ F P ) 1 r p 2 exp ( i ϕ F P ) , r u r F P = r p + r p t p 2 exp ( i ϕ F P ) 1 r p 2 exp ( ϕ F P ) ,
t u [ t p exp ( i δ ) 1 r p exp ( i 2 δ ) ] 2 t s , r u r p + r p [ t p exp ( i 2 δ ) 1 r p exp ( i 2 δ ) ] 2 ( 1 2 r s ) ,
t 2 u = exp ( i θ ) t u 1 t u 2 1 r u 1 r u 2 exp ( i 2 θ ) , r 2 u = r u 1 + t u 1 2 r u 2 exp ( i 2 θ ) 1 r u 1 r u 2 exp ( i 2 θ ) ,
M q = 1 t q ( t q 2 r q 2 r q r q 1 ) ,
( a 2 b 2 ) M u ( a 1 b 1 ) = M p M L M s M L M P ( a 1 b 1 ) ,
M s 1 t s ( t s 2 r s 2 r s r s 1 ) = ( 1 i γ c / ( ω ω 0 + i γ 0 ) i γ c / ( ω ω 0 + i γ 0 ) i γ c / ( ω ω 0 + i γ 0 ) 1 + i γ c / ( ω ω 0 + i γ 0 ) ) , M p 1 t p ( t p 2 r p 2 r p r p 1 ) = 1 i 1 r p 2 ( 1 r p r p 1 ) , M L = ( exp ( i δ ) 0 0 exp ( i δ ) ) .
T u | t u | 2 = | 1 / M 22 | 2 , R u | r u | 2 = | M 11 / M 22 | 2 .
( a 2 ' b 2 ' ) = M u M P 1 ( a 1 b 1 ) = exp ( i κ Λ ) ( a 1 b 1 ) ,
κ Λ = cos 1 [ 1 2 Tr( M u M P 1 ) ] .

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