Abstract

We address theoretically and numerically pump-probe switching in a nonlinear semiconductor nanocavity where tuning is achieved via a dominant mechanism of free-carrier plasma dispersion. By using coupled-mode approach we give a set of guidelines to optimize the switching performances both in terms of avoiding self-pulsation and keeping switching power to the minimum, ending up by showing that such devices can achieve high-performances with relatively low-power consumption.

© 2013 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  24. T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A86, 063808 (2012).
    [CrossRef]
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    [CrossRef]
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  27. Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
    [CrossRef]

2013 (1)

X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep.3, 01436 (2013).
[CrossRef]

2012 (6)

Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
[CrossRef]

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012).
[CrossRef]

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A86, 063808 (2012).
[CrossRef]

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express20, 7454–7468 (2012).
[CrossRef] [PubMed]

2011 (3)

Y. Dumeige and P. Féron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A84, 043847 (2011).
[CrossRef]

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A83, 051802(R)(2011).
[CrossRef]

V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A83, 043816 (2011).
[CrossRef]

2010 (2)

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010).
[CrossRef]

2009 (5)

A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A79, 043818 (2009).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009).
[CrossRef]

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A80, 033805 (2009).
[CrossRef]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon-waveguide resonators,” Opt. Express17, 22124–22137 (2009).
[CrossRef] [PubMed]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett.35, 55–57 (2009).
[CrossRef]

2008 (1)

2007 (2)

A. Rodriguez, M. Soljaĉić, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express, 15, 7303–7318 (2007).
[CrossRef] [PubMed]

E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett.90, 101118 (2007).
[CrossRef]

2006 (3)

2005 (2)

2002 (1)

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

1982 (1)

K. Ikeda and O. Akimoto, “Instability Leading to Periodic and Chaotic Self-Pulsations in a Bistable Optical Cavity,” Phys. Rev. Lett.48, 617–620 (1982).
[CrossRef]

Absil, P. P.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

Agrawal, G. P.

Akimoto, O.

K. Ikeda and O. Akimoto, “Instability Leading to Periodic and Chaotic Self-Pulsations in a Bistable Optical Cavity,” Phys. Rev. Lett.48, 617–620 (1982).
[CrossRef]

Asano, T.

Bellanca, G.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A83, 051802(R)(2011).
[CrossRef]

Bencheikh, K.

Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
[CrossRef]

Benisty, H.

Biancalana, F.

V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A83, 043816 (2011).
[CrossRef]

Bienstman, P.

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A86, 063808 (2012).
[CrossRef]

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A80, 033805 (2009).
[CrossRef]

Bigot, L.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012).
[CrossRef]

Blasco, J.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Brunstein, M.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012).
[CrossRef]

Cao, T.

Cassette, S.

E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett.90, 101118 (2007).
[CrossRef]

Chen, S.

Combrié, S.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A83, 051802(R)(2011).
[CrossRef]

A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A79, 043818 (2009).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009).
[CrossRef]

S. Combrié, N. V. Q. Tran, A. de Rossi, and H. Benisty, “GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55 μm,” Opt. Lett.33, 1908–1910 (2008).
[CrossRef]

E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett.90, 101118 (2007).
[CrossRef]

Daldosso, N.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Dambre, J.

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A86, 063808 (2012).
[CrossRef]

de Rossi, A.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A83, 051802(R)(2011).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009).
[CrossRef]

A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A79, 043818 (2009).
[CrossRef]

S. Combrié, N. V. Q. Tran, A. de Rossi, and H. Benisty, “GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55 μm,” Opt. Lett.33, 1908–1910 (2008).
[CrossRef]

E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett.90, 101118 (2007).
[CrossRef]

Dumeige, Y.

Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
[CrossRef]

Y. Dumeige and P. Féron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A84, 043847 (2011).
[CrossRef]

Fedeli, J. M.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Fei, Y.

Féron, P.

Y. Dumeige and P. Féron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A84, 043847 (2011).
[CrossRef]

Fiers, M.

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A86, 063808 (2012).
[CrossRef]

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A80, 033805 (2009).
[CrossRef]

Galan, J. V.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Garcia-Ruperez, J.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Garrido, B.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Gautier, P.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Goldhar, J.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

Grigoriev, V.

V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A83, 043816 (2011).
[CrossRef]

Grinberg, P.

Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
[CrossRef]

Grover, R.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

Gu, T.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Guider, R.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Hernandez, S.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Ho, P.-T.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

Hone, J.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Husko, C.

C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009).
[CrossRef]

A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A79, 043818 (2009).
[CrossRef]

Ibrahim, T. A.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

Ikeda, K.

K. Ikeda and O. Akimoto, “Instability Leading to Periodic and Chaotic Self-Pulsations in a Bistable Optical Cavity,” Phys. Rev. Lett.48, 617–620 (1982).
[CrossRef]

Joannopoulos, J. D.

Johnson, F. G.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

Johnson, S. G.

Jordana, E.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Kira, G.

Kuramochi, E.

Kwong, D. L.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Lauritano, M.

A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A79, 043818 (2009).
[CrossRef]

Le Cren, E.

Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
[CrossRef]

Lebour, Y.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Levenson, J. A.

Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
[CrossRef]

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012).
[CrossRef]

Lipson, M.

Lo, G. Q.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Maes, B.

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A80, 033805 (2009).
[CrossRef]

Malaguti, S.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A83, 051802(R)(2011).
[CrossRef]

Manolatou, C.

Marti, J.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Martinez, A.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Matsuo, S.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010).
[CrossRef]

McMillan, J. F.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Mitsugi, S.

Noda, S.

Notomi, M.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010).
[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. Express13, 2678–2687 (2005).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett.30, 2575–2577 (2005).
[CrossRef] [PubMed]

Nozaki, K.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010).
[CrossRef]

Pavesi, L.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Petrone, N.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Premaratne, M.

Raineri, F.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009).
[CrossRef]

Ritter, K.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

Rodriguez, A.

Rukhlenko, I. D.

Sagnes, I.

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012).
[CrossRef]

Sanchis, P.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Sato, T.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010).
[CrossRef]

Schuck, C.

X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep.3, 01436 (2013).
[CrossRef]

Segawa, T.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

Shinya, A.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010).
[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. Express13, 2678–2687 (2005).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett.30, 2575–2577 (2005).
[CrossRef] [PubMed]

Soljacic, M.

Song, B.-S.

Spano, R.

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Sun, X.

X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep.3, 01436 (2013).
[CrossRef]

Suzaki, Y.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

Takahashi, R.

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

Tanabe, T.

Tang, H. X.

X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep.3, 01436 (2013).
[CrossRef]

Taniyama, H.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010).
[CrossRef]

Tran, N. V. Q.

S. Combrié, N. V. Q. Tran, A. de Rossi, and H. Benisty, “GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55 μm,” Opt. Lett.33, 1908–1910 (2008).
[CrossRef]

E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett.90, 101118 (2007).
[CrossRef]

Tran, Q. V.

C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009).
[CrossRef]

Tran, Q.V.

A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A79, 043818 (2009).
[CrossRef]

Trillo, S.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A83, 051802(R)(2011).
[CrossRef]

Uesugi, T.

Van, V.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

van der Zande, A.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Van Vaerenbergh, T.

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A86, 063808 (2012).
[CrossRef]

Weidner, E.

E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett.90, 101118 (2007).
[CrossRef]

Wong, C. W.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009).
[CrossRef]

Xu, Q.

Yacomotti, A. M.

Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
[CrossRef]

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012).
[CrossRef]

Yu, M.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Zhang, L.

Zhang, X.

X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep.3, 01436 (2013).
[CrossRef]

Appl. Phys. Lett. (2)

E. Weidner, S. Combrié, A. de Rossi, N. V. Q. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett.90, 101118 (2007).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. V. Tran, F. Raineri, and C. W. Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett.94, 021111 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,”IEEE Photon. Technol. Lett.14, 74–76 (2002).
[CrossRef]

J. Lightwave Technol. (1)

Nano Lett. (1)

A. Martinez, J. Blasco, P. Sanchis, J. V. Galan, J. Garcia-Ruperez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Marti, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett.10, 1506–1511 (2010).
[CrossRef] [PubMed]

Nat. Photonics (2)

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic crystal nanocavity,” Nat. Photonics4, 477–483 (2010).
[CrossRef]

K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6, 248–252 (2012).
[CrossRef]

Nature Photon. (1)

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, “Regenerative oscillation and four-wave mixing in graphene optoelectronics,” Nature Photon.6, 554–559 (2012).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

Phys. Rev. A (8)

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A80, 033805 (2009).
[CrossRef]

V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A83, 043816 (2011).
[CrossRef]

M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity, ”Phys. Rev. A85, 031803(R)(2012).
[CrossRef]

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A86, 063808 (2012).
[CrossRef]

Y. Dumeige and P. Féron, “Stability and time-domain analysis of the dispersive tristability in microresonators under modal coupling,” Phys. Rev. A84, 043847 (2011).
[CrossRef]

A. de Rossi, M. Lauritano, S. Combrié, Q.V. Tran, and C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A79, 043818 (2009).
[CrossRef]

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A83, 051802(R)(2011).
[CrossRef]

Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A85, 063824 (2012).
[CrossRef]

Phys. Rev. Lett. (1)

K. Ikeda and O. Akimoto, “Instability Leading to Periodic and Chaotic Self-Pulsations in a Bistable Optical Cavity,” Phys. Rev. Lett.48, 617–620 (1982).
[CrossRef]

Sci. Rep. (1)

X. Sun, X. Zhang, C. Schuck, and H. X. Tang, “Nonlinear optical effects of ultrahigh-Q silicon photonic nanocavities immersed in superfluid helium,” Sci. Rep.3, 01436 (2013).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Typical stationary responses [Eqs. (6)] for pump (Ep vs. Pp, black curve) and probe (ηs vs. Pp, red curve). We highlight the bistable jumps (arrows) occurring at branch point energies E b ±, and the critical pump point [ P p c, E p c, Eq. (7)] for highest probe efficiency ηs = 1. Dot-dashed portions stand for unstable branches. Here δp = −3, δs = −4, and τ = 1.

Fig. 2
Fig. 2

(a) Level plot of power P p c (dB units) from Eq. (8) in the plane of detunings (δp, δs). In the bistable region (left of the vertical dashed line δ p = δ p c), the curves labelled δ p + (green) and δ p (red) delimit the domains corresponding to the three branches of the stationary response shown in Fig. 1 (LB/UB stand for the lower/upper branch, NSB stands for the negative-slope branch). The optimum (MPE ηs = 1 with minimal Pp) is achieved for pump detunings δ p u (yellow dot-dashed curve). (b)–(e) Corresponding steady-state pump (black) and probe efficiency (red) responses for fixed probe detuning δs = −3 (CR = 10) and increasing values of |δp|. The dots indicate the optimum operation points (the red and black dots give MPE ηs = 1 and corresponding critical values P p = P p c, E p = E p c, respectively): (b) Pump detuning values δp = −1.2, −1.6, −2.1 (Pc = 7.3, 5.1, 3.1) in the UB region; (c) Optimum operation at δ p = δ p u = 2.2 (minimal P p c = 2.8); (d)–(e) MPE at δ p + = 3.1 and δ p = 15, respectively. Here τ = 1.

Fig. 3
Fig. 3

Temporal dynamics of pump (solid black) and probe (solid red) energies for δs = −3 and pump detuning: (a)–(c) δp = −2.2 (peak power Pp = 3); (b)–(d) δp = −3.1 (peak power Pp = 6). The left column cases (a)–(b) and right column cases (c)–(d) are relative to τ = 1 and τ = 4, respectively. The (blue) dashed line is the driving pump Pp(t) (Ps is a cw signal).

Fig. 4
Fig. 4

As in Fig. 3 for δs = −3, δp = −15, and τ = 1, contrasting two dynamical behaviors: (a) pump self-switching (dark curve) towards a SP state [the inset shows the corresponding limit cycle in the phase plane Re(ap) − Im(ap)], obtained with maximum driving power Pp = 250 (slightly larger than knee value P p +); (b) stable behavior for Pp = 248.5 (slightly lower than P p +), with MPE reached with pump on the lower branch (note the different vertical scale in the two plots).

Fig. 5
Fig. 5

Effect of TPA: level plots of (a) MPE; (b) CR, in the plane (α, δs) for fixed optimum value of pump detuning ( δ p u = 2.2) and τ = 1 (labels UB and NSB stand for uppe and negative slope branch, respectively). (c)–(d) Temporal pump-probe dynamics for α = 0.1 and τ = 1, contrasting: (c) stable probe switching for δp = −2.2 (MPE of 0.55 and a CR of 5.5) and (d) SP-dominated dynamics for δp = −3.1.

Equations (10)

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

a p t = i ( δ p + n ) a p a p α | a p | 2 a p γ n a p + P p ,
a s t = i ( δ s + n ) a s a s 2 α | a p | 2 a s γ n a s + P s ,
n t = | a p | 4 n τ ,
a t = i n a a α | a | 2 a γ n a + P ; n t = | a | 4 n τ .
a j t = i ( δ j + n ) a j a j + P j , j = p , s ; n t = | a p | 4 n τ .
P p = E p [ 1 + ( δ p + τ E p 2 ) 2 ] ; η s = 1 1 + ( δ s + τ E p 2 ) 2 ,
E b ± = 3 δ p ± 4 δ p 2 5 5 τ ,
( E p c ) 2 = δ s τ ; P p c = | δ s | τ [ 1 + ( δ p δ s ) 2 ] .
δ p ± = 3 δ s ± 4 δ s 2 1 .
P p = E p [ ( 1 + α E p ) 2 + ( δ p + τ E p 2 ) 2 ] , η s = 1 ( 1 + 2 α E p ) 2 + ( δ s + τ E p 2 ) 2 .

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