Abstract

We show that self-induced oscillations at frequencies above GHz and with a high spectral purity can be obtained in a silicon photonic crystal nanocavity under optical pumping. This self-pulsing results from the interplay between the nonlinear response of the cavity and the photon cavity lifetime. We provide a model to analyze the mechanisms governing the onset of self-pulsing, the amplitudes of both fundamental and harmonic oscillations and their dependences versus input power and oscillation frequency. Theoretically, oscillations at frequencies higher than 50 GHz could be achieved in this system.

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    [CrossRef]
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    [CrossRef] [PubMed]
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  9. J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express18, 15484–15497 (2010).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  24. A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (2011).
    [CrossRef]
  25. T. J. Johnson and O. Painter, “Passive modification of free carrier lifetime in high-Q silicon-on-insulator optics,” 2009 Conference On Lasers and Electro-optics and Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), 1–5, 72–73 (2009).
  26. P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
    [CrossRef] [PubMed]
  27. S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
    [CrossRef]
  28. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express18, 3582–3591 (2010).
    [CrossRef] [PubMed]
  29. If we consider that the transmitted signal is detected by a photodetector, the RF power is proportional to the square of the electric intensity generated by the photodetector, i.e. to the square of the optical intensity.
  30. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett.82, 2954–2956 (2003).
    [CrossRef]
  31. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. and Chem. Ref. Data9, 98 (1980).
  32. T. Tanabe, H. Taniyama, and M. Notomi, “Carrier diffusion and recombination in photonic crystal nanocavity optical switches,” J. Lightwave Tech.26, 1396–1403 (2008).
    [CrossRef]

2012 (4)

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]

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

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A85, 053819 (2012).
[CrossRef]

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

2011 (3)

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (2011).
[CrossRef]

Z. Han, X. Checoury, L.-D. Haret, and P. Boucaud, “High quality factor in a two-dimensional photonic crystal cavity on silicon-on-insulator,” Opt. Lett.36, 1749–1751 (2011).
[CrossRef] [PubMed]

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 (2011).
[CrossRef]

2010 (6)

Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun.283, 4387–4391 (2010).
[CrossRef]

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308(R) (2010).
[CrossRef]

J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express18, 15484–15497 (2010).
[CrossRef] [PubMed]

T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett.96, 101103 (2010).
[CrossRef]

L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express18, 23965–23972 (2010).
[CrossRef] [PubMed]

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express18, 3582–3591 (2010).
[CrossRef] [PubMed]

2009 (2)

2008 (1)

T. Tanabe, H. Taniyama, and M. Notomi, “Carrier diffusion and recombination in photonic crystal nanocavity optical switches,” J. Lightwave Tech.26, 1396–1403 (2008).
[CrossRef]

2007 (2)

2006 (3)

2005 (4)

S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
[CrossRef]

P. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express13, 801–820 (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]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater.4, 207–210 (2005).
[CrossRef]

2004 (1)

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3, 211–219 (2004).
[CrossRef]

2003 (2)

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London)425, 944–947 (2003).
[CrossRef]

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett.82, 2954–2956 (2003).
[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]

1980 (1)

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. and Chem. Ref. Data9, 98 (1980).

Adibi, A.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A85, 053819 (2012).
[CrossRef]

Agrawal, G. P.

Akahane, Y.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater.4, 207–210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London)425, 944–947 (2003).
[CrossRef]

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]

Armaroli, A.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (2011).
[CrossRef]

Asano, T.

Barclay, P.

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 (2011).
[CrossRef]

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (2011).
[CrossRef]

Bencheikh, K.

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

Bigot, L.

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

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

Borselli, M.

Boucaud, P.

Z. Han, X. Checoury, L.-D. Haret, and P. Boucaud, “High quality factor in a two-dimensional photonic crystal cavity on silicon-on-insulator,” Opt. Lett.36, 1749–1751 (2011).
[CrossRef] [PubMed]

Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun.283, 4387–4391 (2010).
[CrossRef]

L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express18, 23965–23972 (2010).
[CrossRef] [PubMed]

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308(R) (2010).
[CrossRef]

Brunstein, M.

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

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

Cao, T.

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics1, 319–330 (2007).
[CrossRef]

Checoury, X.

Z. Han, X. Checoury, L.-D. Haret, and P. Boucaud, “High quality factor in a two-dimensional photonic crystal cavity on silicon-on-insulator,” Opt. Lett.36, 1749–1751 (2011).
[CrossRef] [PubMed]

Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun.283, 4387–4391 (2010).
[CrossRef]

L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express18, 23965–23972 (2010).
[CrossRef] [PubMed]

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308(R) (2010).
[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 (2011).
[CrossRef]

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (2011).
[CrossRef]

L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express18, 23965–23972 (2010).
[CrossRef] [PubMed]

Corcoran, B.

David, S.

Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun.283, 4387–4391 (2010).
[CrossRef]

de Rossi, A.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (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 (2011).
[CrossRef]

Dinu, M.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett.82, 2954–2956 (2003).
[CrossRef]

Dumeige, Y.

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

Ebnali-Heidari, M.

Eftekhar, A. A.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A85, 053819 (2012).
[CrossRef]

Eggleton, B. J.

El Kurdi, M.

Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun.283, 4387–4391 (2010).
[CrossRef]

Fei, Y.

Foster, M. A.

Gaeta, A. L.

Garcia, H.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett.82, 2954–2956 (2003).
[CrossRef]

Grillet, C.

Grinberg, P.

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

Hagino, H.

Han, Z.

Z. Han, X. Checoury, L.-D. Haret, and P. Boucaud, “High quality factor in a two-dimensional photonic crystal cavity on silicon-on-insulator,” Opt. Lett.36, 1749–1751 (2011).
[CrossRef] [PubMed]

Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun.283, 4387–4391 (2010).
[CrossRef]

L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express18, 23965–23972 (2010).
[CrossRef] [PubMed]

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308(R) (2010).
[CrossRef]

Haret, L.-D.

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.

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3, 211–219 (2004).
[CrossRef]

Johnson, T. J.

T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express14, 817–831 (2006).
[CrossRef] [PubMed]

T. J. Johnson and O. Painter, “Passive modification of free carrier lifetime in high-Q silicon-on-insulator optics,” 2009 Conference On Lasers and Electro-optics and Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), 1–5, 72–73 (2009).

Kaka, S.

S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
[CrossRef]

Katine, J. A.

S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
[CrossRef]

Krauss, T. F.

Kuramochi, E.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

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]

Kwong, D. L.

Levenson, A.

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

Levenson, J. A.

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

Levy, J. S.

Li, H. H.

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. and Chem. Ref. Data9, 98 (1980).

Li, Q.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A85, 053819 (2012).
[CrossRef]

Lin, Q.

Lipson, M.

Malaguti, S.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (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 (2011).
[CrossRef]

McMillan, J. F.

Mitsugi, S.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

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]

Monat, C.

Néel, D.

Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun.283, 4387–4391 (2010).
[CrossRef]

Noda, S.

Notomi, M.

T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett.96, 101103 (2010).
[CrossRef]

T. Tanabe, H. Taniyama, and M. Notomi, “Carrier diffusion and recombination in photonic crystal nanocavity optical switches,” J. Lightwave Tech.26, 1396–1403 (2008).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

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]

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics1, 319–330 (2007).
[CrossRef]

O’Faolain, L.

Painter, O.

Painter, O. J.

Poitras, C. B.

Pufall, M. R.

S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
[CrossRef]

Quochi, F.

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett.82, 2954–2956 (2003).
[CrossRef]

Raineri, F.

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

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

Rippard, W. H.

S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
[CrossRef]

Rossi, A. D.

Russek, S. E.

S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
[CrossRef]

Sagnes, I.

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

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

Salem, R.

Sato, Y.

Shinya, A.

T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett.96, 101103 (2010).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

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]

Silva, T. J.

S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
[CrossRef]

Soljacic, M.

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3, 211–219 (2004).
[CrossRef]

Soltani, M.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A85, 053819 (2012).
[CrossRef]

Song, B. S.

T. Uesugi, B. S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express14, 377–386 (2006).
[CrossRef] [PubMed]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater.4, 207–210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London)425, 944–947 (2003).
[CrossRef]

Srinivasan, K.

Sugiya, T.

Sumikura, H.

T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett.96, 101103 (2010).
[CrossRef]

Takahashi, Y.

Tanabe, T.

T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett.96, 101103 (2010).
[CrossRef]

T. Tanabe, H. Taniyama, and M. Notomi, “Carrier diffusion and recombination in photonic crystal nanocavity optical switches,” J. Lightwave Tech.26, 1396–1403 (2008).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

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]

Tanaka, Y.

Taniyama, H.

T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett.96, 101103 (2010).
[CrossRef]

T. Tanabe, H. Taniyama, and M. Notomi, “Carrier diffusion and recombination in photonic crystal nanocavity optical switches,” J. Lightwave Tech.26, 1396–1403 (2008).
[CrossRef]

Trillo, S.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (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 (2011).
[CrossRef]

Turner-Foster, A. C.

Uesugi, T.

Watanabe, T.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

White, T. P.

Wong, C. W.

Yacomotti, A. M.

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

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

Yegnanarayanan, S.

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A85, 053819 (2012).
[CrossRef]

Yu, M. B.

Zhang, L.

Appl. Phys. Lett. (3)

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett.88, 041112 (2006).
[CrossRef]

T. Tanabe, H. Sumikura, H. Taniyama, A. Shinya, and M. Notomi, “All-silicon sub-Gb/s telecom detector with low dark current and high quantum efficiency on chip,” Appl. Phys. Lett.96, 101103 (2010).
[CrossRef]

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett.82, 2954–2956 (2003).
[CrossRef]

J. Lightwave Tech. (1)

T. Tanabe, H. Taniyama, and M. Notomi, “Carrier diffusion and recombination in photonic crystal nanocavity optical switches,” J. Lightwave Tech.26, 1396–1403 (2008).
[CrossRef]

J. Phys. and Chem. Ref. Data (1)

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. and Chem. Ref. Data9, 98 (1980).

Nat. Mater. (2)

M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3, 211–219 (2004).
[CrossRef]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater.4, 207–210 (2005).
[CrossRef]

Nat. Photonics (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics1, 319–330 (2007).
[CrossRef]

Nature (London) (2)

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London)425, 944–947 (2003).
[CrossRef]

S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, “Mutual phase-locking of microwave spin torque nano-oscillators,” Nature (London)437, 389–392 (2005).
[CrossRef]

Opt. Commun. (1)

Z. Han, X. Checoury, D. Néel, S. David, M. El Kurdi, and P. Boucaud, “Optimized design for 2 × 106 ultra-high Q silicon photonic crystal cavities,” Opt. Commun.283, 4387–4391 (2010).
[CrossRef]

Opt. Express (10)

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]

T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express14, 817–831 (2006).
[CrossRef] [PubMed]

J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express18, 15484–15497 (2010).
[CrossRef] [PubMed]

Y. Takahashi, Y. Tanaka, H. Hagino, T. Sugiya, Y. Sato, T. Asano, and S. Noda, “Design and demonstration of high-Q photonic heterostructure nanocavities suitable for integration,” Opt. Express17, 18093–18102 (2009).
[CrossRef] [PubMed]

P. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express13, 801–820 (2005).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express17, 2944–2953 (2009).
[CrossRef] [PubMed]

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express18, 3582–3591 (2010).
[CrossRef] [PubMed]

T. Uesugi, B. S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express14, 377–386 (2006).
[CrossRef] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express15, 16604–16644 (2007).
[CrossRef] [PubMed]

L.-D. Haret, X. Checoury, Z. Han, P. Boucaud, S. Combrié, and A. D. Rossi, “All-silicon photonic crystal photo-conductor on silicon-on-insulator at telecom wavelength,” Opt. Express18, 23965–23972 (2010).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. A (4)

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 (2011).
[CrossRef]

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

M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A85, 053819 (2012).
[CrossRef]

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A84, 053816 (2011).
[CrossRef]

Phys. Rev. B (1)

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308(R) (2010).
[CrossRef]

Phys. Rev. Lett. (2)

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]

P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett.109, 113903 (2012).
[CrossRef] [PubMed]

Other (2)

T. J. Johnson and O. Painter, “Passive modification of free carrier lifetime in high-Q silicon-on-insulator optics,” 2009 Conference On Lasers and Electro-optics and Quantum Electronics and Laser Science Conference (CLEO/QELS 2009), 1–5, 72–73 (2009).

If we consider that the transmitted signal is detected by a photodetector, the RF power is proportional to the square of the electric intensity generated by the photodetector, i.e. to the square of the optical intensity.

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Figures (4)

Fig. 1
Fig. 1

(a) (top): schematic view of the photonic crystal with the suspended access waveguides and nanotethers. The input light pulse intensity and the oscillating output are also drawn schematically. Bottom: scanning electron microscope view of the entrance of the photonic crystal waveguide (left) and view of an access waveguide (right). (b) Simulated steady-state output power of the cavity as a function of the input power for different detunings of the pump. For the curve at a detuning of −20 pm, the zone of bistability is between P1 = 0.284 mW and P2 = 0.549 mW. Self-pulsation can be observed for input powers larger than P3 = 1.5 mW, above the zone of bistability. The measurement reported in Fig. 3 (b) was performed for an input power P4 = 2 mW.

Fig. 2
Fig. 2

(a) Period of the oscillations as a function of the cavity effective volume Veff and the free-carrier lifetime, for a cavity with a quality factor Q=130000, a detuning of −20 pm and an input power P4 = 2 mW. The oscillations are non-attenuated in the zone of positive gain (inside the region defined by the black thick line). (b) Period of the oscillations as a function of the quality factor and input power for a free-carrier lifetime of 0.2 ns and an effective volume Veff = 5.25 μm3.

Fig. 3
Fig. 3

Experimental (a and b) and simulated (c and d) output power as a function of time for different detunings between the laser wavelength and the resonance wavelength of the cavity. The input optical pulses have a length of 10 ns. Two types of cavities have been investigated. The first cavity has a quality factor of 9 × 104 and a resonance wavelength of 1590 nm. After the cavity had been immersed in a 3:1 HNO3:H2O2 solution, the quality factor of the cavity increased, rising from 9 × 104 to 1.3 × 105, and the resonance wavelength shifted to 1585.638 nm. Measurements (a) before and (b) after the nitric acid treatment. Corresponding modelings, (c) before nitric acid treatment (Q =9×104 and τfc = 0.3 ns) and (d) after the nitric acid treatment (Q = 1.3 × 105 and τfc = 0.2 ns). In Fig. 3(a), a parasitic extrinsic slow modulation of the signal is observed (period around 2 ns). This parasitic effect due to the measurement set-up was significantly suppressed for the subsequent measurements of the cavity [Fig. 3(b)].

Fig. 4. :
Fig. 4. :

(a) Measured and simulated spectra of the oscillations with thermal effects for a detuning of −20 pm. (b) Comparison between the simulated and the analytically calculated spectra of the cavity with a continuous input power and without the thermal effects. (c) Ratio of the fundamental harmonic amplitude E1 to the average energy in the cavity E0 as a function of the input power and the detuning (analytical expression). (d) Ratio of the second harmonic amplitude E2 to the fundamental amplitude E1(analytical expression).

Equations (37)

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d A d t = A 2 τ + i Δ ω A γ F C A N V eff A γ T P A | A | 2 A 2 h ¯ ω + P in τ in
d N d t = N τ f c + 1 2 γ T P A | A | 4 ( h ¯ ω ) 2
Ω ~ τ f c γ T P A γ i F C A 2 ( h ¯ ω ) 2 V eff × T max ( P in 2 τ ) 2 5 Δ ω 5
C 0 ~ 1 Ω τ f c × ( ( 1 + Δ ω Ω ) 3 γ T P A τ f c E 0 4 h ¯ ω )
γ F C A = γ r F C A + i γ i F C A = c 2 n R eff ( σ r i 2 ω c σ i ) × ( ω r ω ) 2
d | A | d t = | A | 2 τ γ r F C A | A | N V eff γ T P A | A | 3 2 h ¯ ω + cos ( φ ) P in τ in
d φ d t = Δ ω γ i F C A N V eff sin ( φ ) | A | P in τ in
d N d t = N τ f c + 1 2 γ T P A | A | 4 ( h ¯ ω ) 2
N 0 = τ f c γ T P A | A 0 | 4 2 ( h ¯ ω ) 2 ,
tan ( φ 0 ) = Δ ω τ f c γ T P A γ i F C A | A 0 | 4 2 ( h ¯ ω ) 2 V eff 1 2 τ + γ T P A | A 0 | 2 2 h ¯ ω + τ f c γ T P A γ r F C A | A 0 | 4 2 ( h ¯ ω ) 2 V eff ,
( | A 0 | 2 τ + γ T P A | A 0 | 3 2 h ¯ ω + τ f c γ T P A γ r F C A | A 0 | 5 2 ( h ¯ ω ) 2 V eff ) 2 + ( Δ ω | A 0 | τ f c γ T P A γ i F C A | A 0 | 5 2 ( h ¯ ω ) 2 V eff ) 2 P in τ in = 0
d d t ( δ | A | δ φ δ N ) = M ( δ | A | δ φ δ N ) ,
M = ( 1 2 τ γ r F C A N 0 V eff 3 γ T P A | A 0 | 2 2 h ¯ ω sin ( φ 0 ) P in τ in γ r F C A | A 0 | V eff sin ( φ 0 ) | A 0 | 2 P in τ in cos ( φ 0 ) | A 0 | P in τ in γ i F C A V eff 2 γ T P A | A 0 | 3 ( h ¯ ω ) 2 0 1 τ f c )
| A 0 | ~ 2 ( h ¯ ω ) 2 V eff τ fc γ T P A γ i F C A ( P in τ in ) 1 / 2 5 × ( 1 Δ ω 5 × 2 ( h ¯ ω ) 2 V eff τ f c γ T P A γ i F C A ( τ in P in ) 2 5 ) 1
Ω ~ τ f c γ T P A γ i F C A 2 ( h ¯ ω ) 2 V eff × T max ( P in 2 τ ) 2 5 Δ ω 5 .
d A d t = A 2 τ + i Δ ω A γ F C A A N V eff i γ T A Δ T γ T P A | A | 2 A 2 h ¯ ω + P in τ in
d N d t = N τ f c + 1 2 γ T P A | A | 4 ( h ¯ ω ) 2
( ρ S i C p S i V eff T ) d Δ T d t = Δ T R T + h ¯ ω ( γ T P A | A | 4 ( h ¯ ω ) 2 + 2 γ r F C A | A | 2 h ¯ ω N V eff )
| A ( t ) | 2 A 0 ( 2 ) + 2 A 1 ( 2 ) cos ( Ω t + φ A 1 ( 2 ) )
d N d t N τ f c + 1 2 γ T P A ( h ¯ ω ) 2 [ ( A 0 ( 2 ) ) 2 + 2 ( A 1 ( 2 ) ) 2 + 4 A 0 ( 2 ) A 1 ( 2 ) cos ( Ω t + φ A 1 ( 2 ) ) ]
d A d t ( R + F ( t ) ) A + P in τ in
R = 1 2 τ i Δ ω + γ F C A N 0 V eff + γ T P A A 0 ( 2 ) 2 h ¯ ω
F ( t ) = 2 γ F C A N 1 V eff cos ( Ω t + φ N 1 ) + 2 γ T P A A 1 ( 2 ) 2 h ¯ ω cos ( Ω t + φ A 1 ( 2 ) )
A ( t ) = K e R t 0 t F ( t ) d t 0 t e R t + 0 t F ( t ) d t d t
e 0 t F ( t ) d t e α i F C A sin ( θ ) = i 3 n I n ( α i F C A ) e i n θ
0 t e R t + 0 t F ( t ) d t d t = e R t [ I 0 ( α i F C A ) R i I 1 ( α i F C A ) ( e i θ R + i Ω e i θ R i Ω ) ]
A = K ( I 0 ( α i F C A ) R i I 1 ( α i F C A ) ( e i θ R + i Ω e i θ R i Ω ) ) e α i F C A sin ( θ )
E = | A | 2 = K 2 | I 0 ( α i F C A ) R i I 1 ( α i F C A ) ( e i θ R + i Ω e i θ R i Ω ) | 2
E = ( E 0 + 2 E 1 cos ( Ω t + φ 1 ) + 2 E 2 cos ( 2 Ω t + φ 2 ) )
E 0 = K 2 ( I 0 ( α i F C A ) 2 | R | 2 + 2 | I 1 ( α i F C A ) | 2 ( Ω 2 + | R | 2 ) ( | R | 2 Ω 2 ) 2 + ( 2 Ω Re ( R ) ) 2 )
E 1 = K 2 2 Ω Im ( R ) I 0 ( α i F C A ) | I 1 ( α i F C A ) | ( | R | 2 Ω 2 ) 2 + ( 2 Ω Re ( R ) ) 2
E 2 = K 2 | I 1 ( α i F C A ) | 2 ( | R | 2 Ω 2 ) 2 + ( 2 Ω Re ( R ) ) 2
E 2 E 1 | I 1 ( α i F C A ) | 2 2 Ω Im ( R ) I 0 ( α i F C A ) | I 1 ( α i F C A ) |
E 2 E 1 C 0 × E 1 E 0
C 0 = γ i F C A N 0 V eff | R | 2 Ω 2 Im ( R ) 1 + Ω 2 τ f c 2
C 0 ~ γ i F C A N 0 τ f c Ω 2 V eff ~ 1 Ω τ f c × ( 1 + Δ ω Ω )
C 0 ~ 1 Ω τ f c × ( ( 1 + Δ ω Ω ) 3 γ T P A τ f c E 0 4 h ¯ ω )

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