Abstract

Optical bistability with a hybrid silicon-plasmonic configuration consisting of a nonlinear traveling-wave (disk) resonator side-coupled with a bus waveguide is theoretically investigated. The nonlinear response is studied with a theoretical framework combining perturbation theory and temporal coupled-mode theory. For the CW case, a general closed-form expression is derived. The effect of the parameters entering in the expression on the bistability curve is thoroughly investigated, and the physical system is accordingly designed so as to exhibit minimum power threshold for bistability and maximum extinction ratio between bistable states. Finally, the temporal dynamics are assessed. The system can toggle between bistable states in approximately 5 ps and is thus suitable for ultrafast memory and switching applications.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits (Pan Stanford, 2008).
  2. O. Tsilipakos, E. E. Kriezis, and S. I. Bozhevolnyi, “Thermo-optic microring resonator switching elements made of dielectric-loaded plasmonic waveguides,” J. Appl. Phys. 109, 073111 (2011).
    [CrossRef]
  3. J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
    [CrossRef]
  4. K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
    [CrossRef]
  5. C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
    [CrossRef]
  6. A. Kriesch, D. Ploss, J. Wen, P. Banzer, and U. Peschel, “Nonlinear effects in subwavelength plasmonic directional couplers,” in CLEO: Conference on Lasers and Electro-Optics (2012), paper 6327055.
  7. A. Pitilakis and E. E. Kriezis, “Highly nonlinear hybrid silicon-plasmonic waveguides: analysis and optimization,” J. Opt. Soc. Am. B 30, 1954–1965 (2013).
    [CrossRef]
  8. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004).
    [CrossRef]
  9. A. Pannipitiya, I. D. Rukhlenko, and M. Premaratne, “Analytical theory of optical bistability in plasmonic nanoresonators,” J. Opt. Soc. Am. B 28, 2820–2826 (2011).
    [CrossRef]
  10. X. Wang, H. Jiang, J. Chen, P. Wang, Y. Lu, and H. Ming, “Optical bistability effect in plasmonic racetrack resonator with high extinction ratio,” Opt. Express 19, 19415–19421 (2011).
    [CrossRef]
  11. G. Wang, H. Lu, X. Liu, Y. Gong, and L. Wang, “Optical bistability in metal-insulator-metal plasmonic waveguide with nanodisk resonator containing Kerr nonlinear medium,” Appl. Opt. 50, 5287–5290 (2011).
    [CrossRef]
  12. Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).
  13. M. Wu, Z. Han, and V. Van, “Conductor-gap-silicon plasmonic waveguides and passive components at subwavelength scale,” Opt. Express 18, 11728–11736 (2010).
    [CrossRef]
  14. A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, “Nonlinear effects in hybrid plasmonic waveguides,” in ICTON: 14th International Conference on Transparent Optical Networks (IEEE, 2012), paper 6254436.
  15. B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
    [CrossRef]
  16. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
    [CrossRef]
  17. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  18. B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: polarization effects,” J. Opt. Soc. Am. B 27, 956–965 (2010).
    [CrossRef]
  19. O. Tsilipakos and E. E. Kriezis, “Microdisk resonator filters made of dielectric-loaded plasmonic waveguides,” Opt. Commun. 283, 3095–3098 (2010).
    [CrossRef]
  20. D. A. Ketzaki, O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Electromagnetically induced transparency with hybrid silicon-plasmonic traveling-wave resonators,” J. Appl. Phys. 114, 113107 (2013).
    [CrossRef]
  21. J. Bravo-Abad, S. Fan, S. Johnson, J. D. Joannopoulos, and M. Soljačić, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightwave Technol. 25, 2539–2546 (2007).
    [CrossRef]
  22. M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
    [CrossRef]
  23. M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
    [CrossRef]
  24. B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778–1784 (2005).
    [CrossRef]
  25. O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
    [CrossRef]
  26. M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010).
    [CrossRef]
  27. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  28. 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]
  29. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
    [CrossRef]
  30. E. D. Palik, ed., Handbook of Optical Constants of Solids, 1st ed. (Academic, 1985).
  31. B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
    [CrossRef]
  32. V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).
  33. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31, 341–343 (2006).
    [CrossRef]
  34. 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, 2678–2687 (2005).
    [CrossRef]

2013 (3)

A. Pitilakis and E. E. Kriezis, “Highly nonlinear hybrid silicon-plasmonic waveguides: analysis and optimization,” J. Opt. Soc. Am. B 30, 1954–1965 (2013).
[CrossRef]

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).

D. A. Ketzaki, O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Electromagnetically induced transparency with hybrid silicon-plasmonic traveling-wave resonators,” J. Appl. Phys. 114, 113107 (2013).
[CrossRef]

2011 (6)

C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

A. Pannipitiya, I. D. Rukhlenko, and M. Premaratne, “Analytical theory of optical bistability in plasmonic nanoresonators,” J. Opt. Soc. Am. B 28, 2820–2826 (2011).
[CrossRef]

X. Wang, H. Jiang, J. Chen, P. Wang, Y. Lu, and H. Ming, “Optical bistability effect in plasmonic racetrack resonator with high extinction ratio,” Opt. Express 19, 19415–19421 (2011).
[CrossRef]

G. Wang, H. Lu, X. Liu, Y. Gong, and L. Wang, “Optical bistability in metal-insulator-metal plasmonic waveguide with nanodisk resonator containing Kerr nonlinear medium,” Appl. Opt. 50, 5287–5290 (2011).
[CrossRef]

O. Tsilipakos, E. E. Kriezis, and S. I. Bozhevolnyi, “Thermo-optic microring resonator switching elements made of dielectric-loaded plasmonic waveguides,” J. Appl. Phys. 109, 073111 (2011).
[CrossRef]

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

2010 (4)

M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010).
[CrossRef]

M. Wu, Z. Han, and V. Van, “Conductor-gap-silicon plasmonic waveguides and passive components at subwavelength scale,” Opt. Express 18, 11728–11736 (2010).
[CrossRef]

B. A. Daniel and G. P. Agrawal, “Vectorial nonlinear propagation in silicon nanowire waveguides: polarization effects,” J. Opt. Soc. Am. B 27, 956–965 (2010).
[CrossRef]

O. Tsilipakos and E. E. Kriezis, “Microdisk resonator filters made of dielectric-loaded plasmonic waveguides,” Opt. Commun. 283, 3095–3098 (2010).
[CrossRef]

2009 (3)

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
[CrossRef]

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

2008 (1)

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (2)

2004 (1)

2003 (2)

M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

2002 (2)

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

1997 (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]

1994 (1)

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Absil, P. P.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).

Agrawal, G. P.

Almeida, V. R.

Atwater, H. A.

J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
[CrossRef]

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Maes, P. Bienstman, and R. Baets, “Switching in coupled nonlinear photonic-crystal resonators,” J. Opt. Soc. Am. B 22, 1778–1784 (2005).
[CrossRef]

Baker, G.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Banzer, P.

A. Kriesch, D. Ploss, J. Wen, P. Banzer, and U. Peschel, “Nonlinear effects in subwavelength plasmonic directional couplers,” in CLEO: Conference on Lasers and Electro-Optics (2012), paper 6327055.

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Bienstman, P.

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Bozhevolnyi, S. I.

O. Tsilipakos, E. E. Kriezis, and S. I. Bozhevolnyi, “Thermo-optic microring resonator switching elements made of dielectric-loaded plasmonic waveguides,” J. Appl. Phys. 109, 073111 (2011).
[CrossRef]

Bravo-Abad, J.

Cai, W.

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).

Cha, M.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Chen, J.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[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]

Daniel, B. A.

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Diest, K.

J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
[CrossRef]

Dionne, J. A.

J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
[CrossRef]

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Etemad, S.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Fan, S.

Fink, Y.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[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]

Freude, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Gong, Y.

Grover, R.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).

Han, Z.

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]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Ho, P.-T.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).

Ibanescu, M.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Ibrahim, T. A.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).

Jiang, H.

Joannopoulos, J. D.

Johnson, F. G.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Johnson, S.

Johnson, S. G.

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Kang, J. U.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Ketzaki, D. A.

D. A. Ketzaki, O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Electromagnetically induced transparency with hybrid silicon-plasmonic traveling-wave resonators,” J. Appl. Phys. 114, 113107 (2013).
[CrossRef]

Kira, G.

Koos, C.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Kriesch, A.

A. Kriesch, D. Ploss, J. Wen, P. Banzer, and U. Peschel, “Nonlinear effects in subwavelength plasmonic directional couplers,” in CLEO: Conference on Lasers and Electro-Optics (2012), paper 6327055.

Kriezis, E. E.

A. Pitilakis and E. E. Kriezis, “Highly nonlinear hybrid silicon-plasmonic waveguides: analysis and optimization,” J. Opt. Soc. Am. B 30, 1954–1965 (2013).
[CrossRef]

D. A. Ketzaki, O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Electromagnetically induced transparency with hybrid silicon-plasmonic traveling-wave resonators,” J. Appl. Phys. 114, 113107 (2013).
[CrossRef]

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

O. Tsilipakos, E. E. Kriezis, and S. I. Bozhevolnyi, “Thermo-optic microring resonator switching elements made of dielectric-loaded plasmonic waveguides,” J. Appl. Phys. 109, 073111 (2011).
[CrossRef]

O. Tsilipakos and E. E. Kriezis, “Microdisk resonator filters made of dielectric-loaded plasmonic waveguides,” Opt. Commun. 283, 3095–3098 (2010).
[CrossRef]

A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, “Nonlinear effects in hybrid plasmonic waveguides,” in ICTON: 14th International Conference on Transparent Optical Networks (IEEE, 2012), paper 6254436.

Kuramochi, E.

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]

Lawrence, B. L.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Leuthold, J.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Lipson, M.

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]

Liu, X.

Lu, H.

Lu, Y.

MacDonald, K. F.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Maes, B.

Meth, J.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Michinobu, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Milián, C.

C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

Ming, H.

Mitsugi, S.

Notomi, M.

Pannipitiya, A.

Peschel, U.

A. Kriesch, D. Ploss, J. Wen, P. Banzer, and U. Peschel, “Nonlinear effects in subwavelength plasmonic directional couplers,” in CLEO: Conference on Lasers and Electro-Optics (2012), paper 6327055.

Pitilakis, A.

A. Pitilakis and E. E. Kriezis, “Highly nonlinear hybrid silicon-plasmonic waveguides: analysis and optimization,” J. Opt. Soc. Am. B 30, 1954–1965 (2013).
[CrossRef]

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, “Nonlinear effects in hybrid plasmonic waveguides,” in ICTON: 14th International Conference on Transparent Optical Networks (IEEE, 2012), paper 6254436.

Ploss, D.

A. Kriesch, D. Ploss, J. Wen, P. Banzer, and U. Peschel, “Nonlinear effects in subwavelength plasmonic directional couplers,” in CLEO: Conference on Lasers and Electro-Optics (2012), paper 6327055.

Premaratne, M.

Rukhlenko, I. D.

Samson, Z. L.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Scimeca, M. L.

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Shinya, A.

Skryabin, D. V.

C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

Soljacic, M.

J. Bravo-Abad, S. Fan, S. Johnson, J. D. Joannopoulos, and M. Soljačić, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightwave Technol. 25, 2539–2546 (2007).
[CrossRef]

M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
[CrossRef]

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Stegeman, G.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Stockman, M. I.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Suh, W.

Sweatlock, L. A.

J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
[CrossRef]

Tanabe, T.

Tasolamprou, A. C.

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

Toruellas, W.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Tsilipakos, O.

D. A. Ketzaki, O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Electromagnetically induced transparency with hybrid silicon-plasmonic traveling-wave resonators,” J. Appl. Phys. 114, 113107 (2013).
[CrossRef]

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

O. Tsilipakos, E. E. Kriezis, and S. I. Bozhevolnyi, “Thermo-optic microring resonator switching elements made of dielectric-loaded plasmonic waveguides,” J. Appl. Phys. 109, 073111 (2011).
[CrossRef]

O. Tsilipakos and E. E. Kriezis, “Microdisk resonator filters made of dielectric-loaded plasmonic waveguides,” Opt. Commun. 283, 3095–3098 (2010).
[CrossRef]

A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, “Nonlinear effects in hybrid plasmonic waveguides,” in ICTON: 14th International Conference on Transparent Optical Networks (IEEE, 2012), paper 6254436.

Vallaitis, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Van, V.

M. Wu, Z. Han, and V. Van, “Conductor-gap-silicon plasmonic waveguides and passive components at subwavelength scale,” Opt. Express 18, 11728–11736 (2010).
[CrossRef]

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).

Vorreau, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Wang, G.

Wang, L.

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).

G. Wang, H. Lu, X. Liu, Y. Gong, and L. Wang, “Optical bistability in metal-insulator-metal plasmonic waveguide with nanodisk resonator containing Kerr nonlinear medium,” Appl. Opt. 50, 5287–5290 (2011).
[CrossRef]

Wang, P.

Wang, X.

Wen, J.

A. Kriesch, D. Ploss, J. Wen, P. Banzer, and U. Peschel, “Nonlinear effects in subwavelength plasmonic directional couplers,” in CLEO: Conference on Lasers and Electro-Optics (2012), paper 6327055.

Wu, M.

Xiang, Y.

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).

Xu, J.

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).

Xu, Q.

Yanik, M. F.

M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
[CrossRef]

Ying, C.

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).

Yioultsis, T. V.

D. A. Ketzaki, O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Electromagnetically induced transparency with hybrid silicon-plasmonic traveling-wave resonators,” J. Appl. Phys. 114, 113107 (2013).
[CrossRef]

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

Zhang, X.

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).

Zheludev, N. I.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Adv. Mater. (1)

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

AIP Advances (1)

Y. Xiang, X. Zhang, W. Cai, L. Wang, C. Ying, and J. Xu, “Optical bistability based on Bragg grating resonators in metal-insulator-metal plasmonic waveguides,” AIP Advances 3, 012106 (2013).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

C. Milián and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739–2741 (2003).
[CrossRef]

Electron. Lett. (1)

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal P-toluene sulphonate (PTS) at 1600  nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

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

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Topics Quantum Electron. 8, 705–713 (2002).

J. Appl. Phys. (2)

D. A. Ketzaki, O. Tsilipakos, T. V. Yioultsis, and E. E. Kriezis, “Electromagnetically induced transparency with hybrid silicon-plasmonic traveling-wave resonators,” J. Appl. Phys. 114, 113107 (2013).
[CrossRef]

O. Tsilipakos, E. E. Kriezis, and S. I. Bozhevolnyi, “Thermo-optic microring resonator switching elements made of dielectric-loaded plasmonic waveguides,” J. Appl. Phys. 109, 073111 (2011).
[CrossRef]

J. Lightwave Technol. (2)

J. Bravo-Abad, S. Fan, S. Johnson, J. D. Joannopoulos, and M. Soljačić, “Modeling nonlinear optical phenomena in nanophotonics,” J. Lightwave Technol. 25, 2539–2546 (2007).
[CrossRef]

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. A (1)

J. Opt. Soc. Am. B (4)

Nano Lett. (1)

J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “PlasMOStor: a metal-oxide-Si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
[CrossRef]

Nat. Photonics (2)

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Opt. Commun. (1)

O. Tsilipakos and E. E. Kriezis, “Microdisk resonator filters made of dielectric-loaded plasmonic waveguides,” Opt. Commun. 283, 3095–3098 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. E (1)

M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Rep. Prog. Phys. (1)

M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010).
[CrossRef]

Other (5)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

E. D. Palik, ed., Handbook of Optical Constants of Solids, 1st ed. (Academic, 1985).

A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, “Nonlinear effects in hybrid plasmonic waveguides,” in ICTON: 14th International Conference on Transparent Optical Networks (IEEE, 2012), paper 6254436.

S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits (Pan Stanford, 2008).

A. Kriesch, D. Ploss, J. Wen, P. Banzer, and U. Peschel, “Nonlinear effects in subwavelength plasmonic directional couplers,” in CLEO: Conference on Lasers and Electro-Optics (2012), paper 6327055.

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 (9)

Fig. 1.
Fig. 1.

(a) Cross section of nonlinear CGS waveguide. The gap between silver and silicon layers is occupied by the nonlinear polymer DDMEBT. The heights of the three layers comprising the guiding ridge (silver, DDMEBT, Si) are hAg=100nm, hD=50nm, and hSi=340nm, respectively. The waveguide width w is 200 nm. (b) Distribution of electric field norm for the fundamental mode (TM00) at 1.55 μm.

Fig. 2.
Fig. 2.

NLCGS-based disk resonator coupled to a CGS bus waveguide through coupling gap g.

Fig. 3.
Fig. 3.

Effect of normalized detuning δ¯ on the optical response. Power transmission is plotted as a function of pin for δ¯={1.2|δ¯th|, 1.7|δ¯th|, 2.2|δ¯th|} while the quality factor ratio is fixed at rQ=1 (critical coupling). As |δ|¯ increases, bistability manifests for higher input powers.

Fig. 4.
Fig. 4.

Effect of quality factor ratio rQ on the optical response. (a) rQ1 regime with δ¯=5.2. (b) rQ1 regime with δ¯=10.4. Deviating from critical coupling results in the elevation of the hysteresis loop from the zero-transmission level, leading to poor ER between bistable states.

Fig. 5.
Fig. 5.

Uncoupled disk resonator based on the nonlinear CGS waveguide. The system is treated as an eigenvalue problem in the linear regime with the 3D-VFEM. (a) Nonlinear feedback parameter κ and effective mode volume Veff as functions of R. (b) Intrinsic quality factor Qi and characteristic power P0 as functions of R. The optimum radius is 1 μm corresponding to a characteristic power of 215 mW.

Fig. 6.
Fig. 6.

Resonant mode of azimuthal order m=9 for a disk resonator with R=1μm. The resonant wavelength is 1553 nm. The real part of the dominant E-field component is plotted (a) at the xy plane halfway inside the polymer layer and (b) at the xz plane containing the resonator axis.

Fig. 7.
Fig. 7.

NLCGS-based disk resonator coupled to CGS bus waveguide. The system is treated as an eigenvalue problem in the linear regime with the 3D-VFEM. (a) Loaded quality factor Ql as a function of coupling gap. As the coupling gap increases, Ql approaches the unloaded (intrinsic) value marked with a dashed line. (b) External quality factor Qe and quality factor ratio rQ=Qi/Qe as functions of coupling gap. The critical coupling condition is satisfied for a coupling gap of 225 nm.

Fig. 8.
Fig. 8.

Bistability curve for a 1-μm radius NLCGS disk resonator coupled to a straight CGS waveguide through a 225-nm gap. The operating wavelength is 1555.3 nm, 2.3 nm higher than the resonant wavelength of the unperturbed resonator. For Pin=PA=1.12W the system exhibits bistable states A and A with theoretically infinite ER. Points B and C are used in Fig. 9 for toggling between them.

Fig. 9.
Fig. 9.

Temporal response of the optimum nonlinear disk resonator system (R=1μm, g=225nm, λ=1555.3nm). The input power is initially set to PA with the system resting at the high-output state (A). The first pulse toggles the system to the low-output state following the route ABA on the bistability curve. The second pulse toggles the system back to the high-output state through the A CA route. In both cases, the system settles at the new state in less than 5 ps, corresponding to approximately 3τl. Points A, B, A, and C are clearly marked in Fig. 8.

Equations (12)

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

Δωω0=c0(ω0c0)3κn2maxW,
κ(c0ω0)3×V13n2(r)n2(r)[(E0·E0)(E0*·E0*)+2|E0|4]dV[Vn2(r)E0·E0*dV]2n2max,
Veff(Vε(r)|E(r)|2dV)2Vε(r)2|E(r)|4dV.
dadt=j(ω0+Δω)a1τia1τea+j2τesi,
st=si+j2τea.
t=stsi=j(ωω0Δω)+(1τi1τe)j(ωω0Δω)+(1τi+1τe).
T=|t|2=PoutPin=(1rQ)2+(δ¯τiΔω)2(1+rQ)2+(δ¯τiΔω)2,
W=QiPinPoutω0.
τiΔω=PinPoutP0,
P0=12(ω0c0)2κQi2n2max
poutpin=(1rQ)2+(δ¯+pinpout)2(1+rQ)2+(δ¯+pinpout)2.
da˜dt=j(δ+2τi2P0|a˜|2)a˜1τla˜+j2τes˜i.

Metrics