Abstract

We have performed ultrafast pump–probe experiments on a GaAs–AlAs microcavity with a resonance near 1300 nm in the “Original” telecom band. We concentrate on ultimate-fast optical switching of the cavity resonance that is measured as a function of pump-pulse energy. We observe that, at low pump-pulse energies, the switching of the cavity resonance is governed by the instantaneous electronic Kerr effect and is achieved within 300 fs. At high pump-pulse energies, the index change induced by free carriers generated in the GaAs start to compete with the electronic Kerr effect and reduce the resonance frequency shift. We have developed an analytic model that predicts this competition in agreement with the experimental data. To this end, we derive the nondegenerate two- and three-photon absorption coefficients for GaAs. Our model includes a new term in the intensity-dependent refractive index that considers the effect of the probe-pulse intensity, which is resonantly enhanced by the cavity. We calculate the effect of the resonantly enhanced probe light on the refractive index change induced by the electronic Kerr effect for cavities with different quality factors. By exploiting the linear regime where only the electronic Kerr effect is observed, we manage to retrieve the nondegenerate third-order nonlinear susceptibility χ(3) for GaAs from the cavity resonance shift as a function of pump-pulse energy.

© 2012 Optical Society of America

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  1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [CrossRef]
  2. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljaĉić, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express 15, 16161–16176 (2007).
    [CrossRef]
  3. T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Extreme nonlinear optics in a femtosecond enhancement cavity,” Phys. Rev. Lett. 107, 183903 (2011).
    [CrossRef]
  4. A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
    [CrossRef]
  5. J. M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” Top. Appl. Phys. 90, 269–314 (2003).
    [CrossRef]
  6. J. P. Reithmaier, “Strong exciton-photon coupling in semiconductor quantum dot systems,” Semicond. Sci. Technol. 23, 123001 (2008).
    [CrossRef]
  7. P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B 66, 081102(R) (2002).
    [CrossRef]
  8. J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
    [CrossRef]
  9. D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
    [CrossRef]
  10. G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
    [CrossRef]
  11. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
    [CrossRef]
  12. P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J. M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs–AlAs microcavity,” J. Opt. Soc. Am. B 29, A1–A5 (2012).
    [CrossRef]
  13. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
    [CrossRef]
  14. C. Husko, A. D. 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]
  15. E. Yüce, O. Gürlü, and A. Serpengüzel, “Optical modulation with silicon microspheres,” IEEE Photon. Technol. Lett. 21, 1481–1483 (2009).
    [CrossRef]
  16. 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. Photon. 4, 477–483(2010).
    [CrossRef]
  17. J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
    [CrossRef]
  18. T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
    [CrossRef]
  19. O. Boyraz, P. Koonath, V. Raghunathan, and B. Jajali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express 12, 4094–4102 (2004).
    [CrossRef]
  20. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006).
    [CrossRef]
  21. P. J. Harding, T. G. Euser, and W. L. Vos, “Identification of competing ultrafast all-optical switching mechanisms in Si woodpile photonic crystals,” J. Opt. Soc. Am. B 26, 610–619 (2009).
    [CrossRef]
  22. T. G. Euser, P. J. Harding, and W. L. Vos, “Broadband sensitive pump–probe setup for ultrafast optical switching of photonic nanostructures and semiconductors,” Rev. Sci. Instrum. 80, 073104 (2009).
    [CrossRef]
  23. A. Hartsuiker, P. J. Harding, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Kerr and free carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three photon edge of GaAs,” J. Appl. Phys. 104, 083105 (2008).
    [CrossRef]
  24. G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
    [CrossRef]
  25. I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
    [CrossRef]
  26. P. J. Harding, T. G. Euser, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Dynamical ultrafast all-optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. 91, 111103 (2007).
    [CrossRef]
  27. J. P. Mondia, H. W. Tan, S. Linden, and H. M. van Driel, “Ultrafast tuning of two-dimensional planar photonic-crystal waveguides via free-carrier injection and the optical Kerr effect,” J. Opt. Soc. Am. B 22, 2480–2486 (2005).
    [CrossRef]
  28. R. Boyd, Nonlinear Optics (Academic, 1992).
  29. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
    [CrossRef]
  30. D. C. Hutchings and E. W. van Stryland, “Nondegenerate two-photon absorption in zinc blende semiconductors,” J. Opt. Soc. Am. B 9, 2065–2074 (1992).
    [CrossRef]
  31. B. S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1, 67–72 (1984).
    [CrossRef]
  32. C. Aversa, J. E. Sipe, M. Sheik-Bahae, and E. W. van Stryland, “Third-order optical nonlinearities in semiconductors: the two-band model,” Phys. Rev. B 50, 18073–18082 (1994).
    [CrossRef]
  33. C. Aversa and J. E. Sipe, “Nonlinear optical susceptibilities of semiconductors: results with a length-gauge analysis,” Phys. Rev. B 52, 14636–14645 (1995).
    [CrossRef]
  34. P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 306, 566–613 (1900).
    [CrossRef]
  35. P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 308, 369–402 (1900).
    [CrossRef]
  36. A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, 2007).
  37. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
    [CrossRef]
  38. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
    [CrossRef]
  39. E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
    [CrossRef]
  40. B. S. Nair and S. Deepa, Solid State Devices (PHI Learning, 2010).
  41. V. Nathan, A. H. Guenther, and S. S. Mitra, “Review of multiphoton absorption in crystalline solids,” J. Opt. Soc. Am. B 2, 294–316 (1985).
    [CrossRef]
  42. C. Xu and W. W. Webb, “Nonlinear and two-photon-induced fluorescence,” in Topics in Fluoresence Spectroscopy (Plenum, 1997), p. 475.
  43. J. He, Y. Qu, H. Li, J. Mi, and W. Ji, “Three-photon absorption in ZnO and ZnS crystals,” Opt. Express 13, 9235–9247 (2005).
    [CrossRef]

2012 (3)

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
[CrossRef]

D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
[CrossRef]

P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J. M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs–AlAs microcavity,” J. Opt. Soc. Am. B 29, A1–A5 (2012).
[CrossRef]

2011 (2)

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Extreme nonlinear optics in a femtosecond enhancement cavity,” Phys. Rev. Lett. 107, 183903 (2011).
[CrossRef]

2010 (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. Photon. 4, 477–483(2010).
[CrossRef]

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
[CrossRef]

2009 (5)

P. J. Harding, T. G. Euser, and W. L. Vos, “Identification of competing ultrafast all-optical switching mechanisms in Si woodpile photonic crystals,” J. Opt. Soc. Am. B 26, 610–619 (2009).
[CrossRef]

T. G. Euser, P. J. Harding, and W. L. Vos, “Broadband sensitive pump–probe setup for ultrafast optical switching of photonic nanostructures and semiconductors,” Rev. Sci. Instrum. 80, 073104 (2009).
[CrossRef]

C. Husko, A. D. 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]

E. Yüce, O. Gürlü, and A. Serpengüzel, “Optical modulation with silicon microspheres,” IEEE Photon. Technol. Lett. 21, 1481–1483 (2009).
[CrossRef]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef]

2008 (2)

J. P. Reithmaier, “Strong exciton-photon coupling in semiconductor quantum dot systems,” Semicond. Sci. Technol. 23, 123001 (2008).
[CrossRef]

A. Hartsuiker, P. J. Harding, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Kerr and free carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three photon edge of GaAs,” J. Appl. Phys. 104, 083105 (2008).
[CrossRef]

2007 (4)

I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
[CrossRef]

P. J. Harding, T. G. Euser, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Dynamical ultrafast all-optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. 91, 111103 (2007).
[CrossRef]

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljaĉić, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express 15, 16161–16176 (2007).
[CrossRef]

2006 (1)

2005 (2)

2004 (3)

O. Boyraz, P. Koonath, V. Raghunathan, and B. Jajali, “All optical switching and continuum generation in silicon waveguides,” Opt. Express 12, 4094–4102 (2004).
[CrossRef]

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

2003 (4)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

J. M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” Top. Appl. Phys. 90, 269–314 (2003).
[CrossRef]

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

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

2002 (1)

P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B 66, 081102(R) (2002).
[CrossRef]

1995 (1)

C. Aversa and J. E. Sipe, “Nonlinear optical susceptibilities of semiconductors: results with a length-gauge analysis,” Phys. Rev. B 52, 14636–14645 (1995).
[CrossRef]

1994 (2)

C. Aversa, J. E. Sipe, M. Sheik-Bahae, and E. W. van Stryland, “Third-order optical nonlinearities in semiconductors: the two-band model,” Phys. Rev. B 50, 18073–18082 (1994).
[CrossRef]

T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
[CrossRef]

1992 (1)

1989 (1)

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

1985 (1)

1984 (1)

1972 (1)

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

1900 (2)

P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 306, 566–613 (1900).
[CrossRef]

P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 308, 369–402 (1900).
[CrossRef]

Akahane, Y.

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

Allison, T. K.

T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Extreme nonlinear optics in a femtosecond enhancement cavity,” Phys. Rev. Lett. 107, 183903 (2011).
[CrossRef]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Asano, T.

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

Aversa, C.

C. Aversa and J. E. Sipe, “Nonlinear optical susceptibilities of semiconductors: results with a length-gauge analysis,” Phys. Rev. B 52, 14636–14645 (1995).
[CrossRef]

C. Aversa, J. E. Sipe, M. Sheik-Bahae, and E. W. van Stryland, “Third-order optical nonlinearities in semiconductors: the two-band model,” Phys. Rev. B 50, 18073–18082 (1994).
[CrossRef]

Azoulay, R.

T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
[CrossRef]

Badolato, A.

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
[CrossRef]

Bajcsy, M.

D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
[CrossRef]

Bakker, H. J.

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

Baumberg, J. J.

A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, 2007).

Bazin, M.

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

Bermel, P.

Bloembergen, N.

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

Borselli, M.

Boyd, R.

R. Boyd, Nonlinear Optics (Academic, 1992).

Boyraz, O.

Bravo-Abad, J.

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

Cingöz, A.

T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Extreme nonlinear optics in a femtosecond enhancement cavity,” Phys. Rev. Lett. 107, 183903 (2011).
[CrossRef]

Claudon, J.

P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J. M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs–AlAs microcavity,” J. Opt. Soc. Am. B 29, A1–A5 (2012).
[CrossRef]

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
[CrossRef]

Combrié, S.

C. Husko, A. D. 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]

Ctistis, G.

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
[CrossRef]

Deepa, S.

B. S. Nair and S. Deepa, Solid State Devices (PHI Learning, 2010).

Drude, P.

P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 306, 566–613 (1900).
[CrossRef]

P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 308, 369–402 (1900).
[CrossRef]

English, J. H.

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

Englund, D.

D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
[CrossRef]

I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
[CrossRef]

Euser, T. G.

T. G. Euser, P. J. Harding, and W. L. Vos, “Broadband sensitive pump–probe setup for ultrafast optical switching of photonic nanostructures and semiconductors,” Rev. Sci. Instrum. 80, 073104 (2009).
[CrossRef]

P. J. Harding, T. G. Euser, and W. L. Vos, “Identification of competing ultrafast all-optical switching mechanisms in Si woodpile photonic crystals,” J. Opt. Soc. Am. B 26, 610–619 (2009).
[CrossRef]

P. J. Harding, T. G. Euser, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Dynamical ultrafast all-optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. 91, 111103 (2007).
[CrossRef]

Faraon, A.

D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
[CrossRef]

Flytzanis, C.

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

Forchel, A.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Fushman, I.

I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
[CrossRef]

Gérard, J. M.

P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J. M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs–AlAs microcavity,” J. Opt. Soc. Am. B 29, A1–A5 (2012).
[CrossRef]

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

A. Hartsuiker, P. J. Harding, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Kerr and free carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three photon edge of GaAs,” J. Appl. Phys. 104, 083105 (2008).
[CrossRef]

P. J. Harding, T. G. Euser, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Dynamical ultrafast all-optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. 91, 111103 (2007).
[CrossRef]

J. M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” Top. Appl. Phys. 90, 269–314 (2003).
[CrossRef]

Gérard, J.-M.

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
[CrossRef]

Gossard, A. C.

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

Guenther, A. H.

Gürlü, O.

E. Yüce, O. Gürlü, and A. Serpengüzel, “Optical modulation with silicon microspheres,” IEEE Photon. Technol. Lett. 21, 1481–1483 (2009).
[CrossRef]

Harding, P. J.

P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J. M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs–AlAs microcavity,” J. Opt. Soc. Am. B 29, A1–A5 (2012).
[CrossRef]

T. G. Euser, P. J. Harding, and W. L. Vos, “Broadband sensitive pump–probe setup for ultrafast optical switching of photonic nanostructures and semiconductors,” Rev. Sci. Instrum. 80, 073104 (2009).
[CrossRef]

P. J. Harding, T. G. Euser, and W. L. Vos, “Identification of competing ultrafast all-optical switching mechanisms in Si woodpile photonic crystals,” J. Opt. Soc. Am. B 26, 610–619 (2009).
[CrossRef]

A. Hartsuiker, P. J. Harding, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Kerr and free carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three photon edge of GaAs,” J. Appl. Phys. 104, 083105 (2008).
[CrossRef]

P. J. Harding, T. G. Euser, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Dynamical ultrafast all-optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. 91, 111103 (2007).
[CrossRef]

Hartsuiker, A.

P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J. M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs–AlAs microcavity,” J. Opt. Soc. Am. B 29, A1–A5 (2012).
[CrossRef]

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
[CrossRef]

A. Hartsuiker, P. J. Harding, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Kerr and free carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three photon edge of GaAs,” J. Appl. Phys. 104, 083105 (2008).
[CrossRef]

He, J.

Hennessy, K. J.

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
[CrossRef]

Hofmann, C.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Houh, H. H.

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

Hu, E. L.

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
[CrossRef]

Husko, C.

C. Husko, A. D. 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]

Hutchings, D. C.

Imamoglu, A.

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
[CrossRef]

Izraël, A.

T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
[CrossRef]

Jajali, B.

Jewell, J. L.

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

Ji, W.

Joannopoulos, J. D.

Johnson, P. M.

P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B 66, 081102(R) (2002).
[CrossRef]

Johnson, S. G.

Johnson, T. J.

Kavokin, A. V.

A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, 2007).

Keldysh, L. V.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Koenderink, A. F.

P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B 66, 081102(R) (2002).
[CrossRef]

Koonath, P.

Kuhn, S.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Kulakovskii, V. D.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef]

Kuszelewicz, R.

T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
[CrossRef]

Ladan, F. R.

T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
[CrossRef]

Laussy, F. P.

A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, 2007).

Li, H.

Linden, S.

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

Löffler, A.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Majumdar, A.

D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
[CrossRef]

Malpuech, G.

A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, 2007).

Matsuo, S.

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. Photon. 4, 477–483(2010).
[CrossRef]

McCall, S. L.

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

Mi, J.

Mitra, S. S.

Mondia, J. P.

Mosk, A. P.

Nair, B. S.

B. S. Nair and S. Deepa, Solid State Devices (PHI Learning, 2010).

Nathan, V.

Noda, S.

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

Notomi, M.

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. Photon. 4, 477–483(2010).
[CrossRef]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef]

Nowicki-Bringuier, Y.

A. Hartsuiker, P. J. Harding, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Kerr and free carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three photon edge of GaAs,” J. Appl. Phys. 104, 083105 (2008).
[CrossRef]

P. J. Harding, T. G. Euser, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Dynamical ultrafast all-optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. 91, 111103 (2007).
[CrossRef]

Nozaki, K.

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. Photon. 4, 477–483(2010).
[CrossRef]

Oudar, J. L.

T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
[CrossRef]

Painter, O.

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

Petroff, P.

D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
[CrossRef]

I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
[CrossRef]

Qu, Y.

Raghunathan, V.

Raineri, F.

C. Husko, A. D. 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]

Reinecke, T. L.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Reinhard, A.

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
[CrossRef]

Reithmaier, J. P.

J. P. Reithmaier, “Strong exciton-photon coupling in semiconductor quantum dot systems,” Semicond. Sci. Technol. 23, 123001 (2008).
[CrossRef]

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Reitzenstein, S.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Rivera, T.

T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
[CrossRef]

Rodriguez, A.

Rossi, A. D.

C. Husko, A. D. 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]

Rotenberg, N.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

Sato, T.

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. Photon. 4, 477–483(2010).
[CrossRef]

Scherer, A.

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

Sek, G.

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

Serpengüzel, A.

E. Yüce, O. Gürlü, and A. Serpengüzel, “Optical modulation with silicon microspheres,” IEEE Photon. Technol. Lett. 21, 1481–1483 (2009).
[CrossRef]

Sheik-Bahae, M.

C. Aversa, J. E. Sipe, M. Sheik-Bahae, and E. W. van Stryland, “Third-order optical nonlinearities in semiconductors: the two-band model,” Phys. Rev. B 50, 18073–18082 (1994).
[CrossRef]

Shinya, A.

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. Photon. 4, 477–483(2010).
[CrossRef]

Sipe, J. E.

C. Aversa and J. E. Sipe, “Nonlinear optical susceptibilities of semiconductors: results with a length-gauge analysis,” Phys. Rev. B 52, 14636–14645 (1995).
[CrossRef]

C. Aversa, J. E. Sipe, M. Sheik-Bahae, and E. W. van Stryland, “Third-order optical nonlinearities in semiconductors: the two-band model,” Phys. Rev. B 50, 18073–18082 (1994).
[CrossRef]

Soljacic, M.

Song, B.-S.

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

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Stoltz, N.

I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
[CrossRef]

Tan, H. W.

Tanabe, T.

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. Photon. 4, 477–483(2010).
[CrossRef]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[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. Photon. 4, 477–483(2010).
[CrossRef]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef]

Tran, Q. V.

C. Husko, A. D. 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]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

van der Pol, E.

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
[CrossRef]

van Driel, H. M.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

J. P. Mondia, H. W. Tan, S. Linden, and H. M. van Driel, “Ultrafast tuning of two-dimensional planar photonic-crystal waveguides via free-carrier injection and the optical Kerr effect,” J. Opt. Soc. Am. B 22, 2480–2486 (2005).
[CrossRef]

van Stryland, E. W.

C. Aversa, J. E. Sipe, M. Sheik-Bahae, and E. W. van Stryland, “Third-order optical nonlinearities in semiconductors: the two-band model,” Phys. Rev. B 50, 18073–18082 (1994).
[CrossRef]

van Stryland, E. W.

Volz, T.

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
[CrossRef]

Vos, W. L.

P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J. M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs–AlAs microcavity,” J. Opt. Soc. Am. B 29, A1–A5 (2012).
[CrossRef]

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
[CrossRef]

T. G. Euser, P. J. Harding, and W. L. Vos, “Broadband sensitive pump–probe setup for ultrafast optical switching of photonic nanostructures and semiconductors,” Rev. Sci. Instrum. 80, 073104 (2009).
[CrossRef]

P. J. Harding, T. G. Euser, and W. L. Vos, “Identification of competing ultrafast all-optical switching mechanisms in Si woodpile photonic crystals,” J. Opt. Soc. Am. B 26, 610–619 (2009).
[CrossRef]

A. Hartsuiker, P. J. Harding, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Kerr and free carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three photon edge of GaAs,” J. Appl. Phys. 104, 083105 (2008).
[CrossRef]

P. J. Harding, T. G. Euser, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Dynamical ultrafast all-optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. 91, 111103 (2007).
[CrossRef]

P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B 66, 081102(R) (2002).
[CrossRef]

Vuckovic, J.

D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
[CrossRef]

I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
[CrossRef]

Waks, E.

I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
[CrossRef]

Webb, W. W.

C. Xu and W. W. Webb, “Nonlinear and two-photon-induced fluorescence,” in Topics in Fluoresence Spectroscopy (Plenum, 1997), p. 475.

Wherrett, B. S.

Whitaker, N. A.

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

Winger, M.

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (2012).
[CrossRef]

Wong, C. W.

C. Husko, A. D. 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, C.

C. Xu and W. W. Webb, “Nonlinear and two-photon-induced fluorescence,” in Topics in Fluoresence Spectroscopy (Plenum, 1997), p. 475.

Yablonovitch, E.

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

Ye, J.

T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Extreme nonlinear optics in a femtosecond enhancement cavity,” Phys. Rev. Lett. 107, 183903 (2011).
[CrossRef]

Yost, D. C.

T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Extreme nonlinear optics in a femtosecond enhancement cavity,” Phys. Rev. Lett. 107, 183903 (2011).
[CrossRef]

Yüce, E.

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

E. Yüce, O. Gürlü, and A. Serpengüzel, “Optical modulation with silicon microspheres,” IEEE Photon. Technol. Lett. 21, 1481–1483 (2009).
[CrossRef]

Ann. Phys. (2)

P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 306, 566–613 (1900).
[CrossRef]

P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 308, 369–402 (1900).
[CrossRef]

Appl. Phys. Lett. (7)

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

C. Husko, A. D. 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]

I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vuĉković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. 90, 091118 (2007).
[CrossRef]

P. J. Harding, T. G. Euser, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Dynamical ultrafast all-optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. 91, 111103 (2007).
[CrossRef]

G. Ctistis, E. Yüce, A. Hartsuiker, J. Claudon, M. Bazin, J. M. Gérard, and W. L. Vos, “Ultimate fast optical switching of a planar microcavity in the telecom wavelength range,” Appl. Phys. Lett. 98, 161114 (2011).
[CrossRef]

J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–25 (1989).
[CrossRef]

T. Rivera, F. R. Ladan, A. Izraël, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. 64, 869–872 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

E. Yüce, O. Gürlü, and A. Serpengüzel, “Optical modulation with silicon microspheres,” IEEE Photon. Technol. Lett. 21, 1481–1483 (2009).
[CrossRef]

J. Appl. Phys. (1)

A. Hartsuiker, P. J. Harding, Y. Nowicki-Bringuier, J. M. Gérard, and W. L. Vos, “Kerr and free carrier ultrafast all-optical switching of GaAs/AlAs nanostructures near the three photon edge of GaAs,” J. Appl. Phys. 104, 083105 (2008).
[CrossRef]

J. Opt. Soc. Am. B (6)

Nat. Photon. (2)

A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. İmamoğlu, “Strongly correlated photons on a chip,” Nat. Photon. 6, 93–96 (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. Photon. 4, 477–483(2010).
[CrossRef]

Nature (5)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

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

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Opt. Express (4)

Phys. Rev. B (4)

P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B 66, 081102(R) (2002).
[CrossRef]

C. Aversa, J. E. Sipe, M. Sheik-Bahae, and E. W. van Stryland, “Third-order optical nonlinearities in semiconductors: the two-band model,” Phys. Rev. B 50, 18073–18082 (1994).
[CrossRef]

C. Aversa and J. E. Sipe, “Nonlinear optical susceptibilities of semiconductors: results with a length-gauge analysis,” Phys. Rev. B 52, 14636–14645 (1995).
[CrossRef]

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J.-M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82, 195330 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

D. Englund, A. Majumdar, M. Bajcsy, A. Faraon, P. Petroff, and J. Vuĉković, “Ultrafast photon–photon interaction in a strongly coupled quantum dot-cavity system,” Phys. Rev. Lett. 108, 093604 (2012).
[CrossRef]

T. K. Allison, A. Cingöz, D. C. Yost, and J. Ye, “Extreme nonlinear optics in a femtosecond enhancement cavity,” Phys. Rev. Lett. 107, 183903 (2011).
[CrossRef]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef]

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

Rev. Sci. Instrum. (1)

T. G. Euser, P. J. Harding, and W. L. Vos, “Broadband sensitive pump–probe setup for ultrafast optical switching of photonic nanostructures and semiconductors,” Rev. Sci. Instrum. 80, 073104 (2009).
[CrossRef]

Semicond. Sci. Technol. (1)

J. P. Reithmaier, “Strong exciton-photon coupling in semiconductor quantum dot systems,” Semicond. Sci. Technol. 23, 123001 (2008).
[CrossRef]

Top. Appl. Phys. (1)

J. M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” Top. Appl. Phys. 90, 269–314 (2003).
[CrossRef]

Other (4)

R. Boyd, Nonlinear Optics (Academic, 1992).

A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy, Microcavities (Oxford University, 2007).

B. S. Nair and S. Deepa, Solid State Devices (PHI Learning, 2010).

C. Xu and W. W. Webb, “Nonlinear and two-photon-induced fluorescence,” in Topics in Fluoresence Spectroscopy (Plenum, 1997), p. 475.

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Figures (8)

Fig. 1.
Fig. 1.

(a) Scanning electron micrograph of our microcavity. The GaAs λ layer is indicated with white arrows and is sandwiched between two GaAs–AlAs Bragg stacks. The GaAs substrate is visible at the bottom. The GaAs layers appear dark gray, while the AlAs layers appear light gray. (b) Measured (black symbols) and calculated (red line) reflectivity spectra of the microcavity. The stop band of the Bragg stacks extends from 7072 to 8498 cm 1 . Fabry–Perot fringes are visible on both sides of stop band. Within the stop band, a narrow trough at 7794.2 cm 1 (1282 nm) indicates the cavity resonance. From the linewidth ( Δ ω = 20 ± 3 cm 1 , full width at half-maximum) of the cavity resonance, we derive a quality factor Q = 390 ± 60 corresponding to a cavity storage time of τ cav = 0.3 ± 0.045 ps . The calculations are performed with a transfer matrix model.

Fig. 2.
Fig. 2.

(a) Schematic of the setup. The probe beam path is shown in blue; the pump beam path in red. The time delay between the pump and the probe pulses is adjusted through a delay stage. The reflected signal from the cavity is spectrally resolved and detected with a spectrometer. The frequency of the probe beam is resonant with the cavity, and the bandwidth of the probe beam is broader than the cavity linewidth. (b) Schematic energy diagrams for the two-photon carrier excitation processes possible in our experiment [see Eq. (4)]. Two-photon absorption is largely suppressed by the judicious tuning of the pump and the probe frequencies relative to the semiconductor bandgap energy. (c) Schematic energy diagrams for all possible three-photon processes in the experiment that may result in free-carrier generation [see Eq. (5)].

Fig. 3.
Fig. 3.

Transient reflectivity spectra for three different pump–probe delays. The spectra are obtained at 84 pJ / μm 2 pump-pulse energy. The gray curves show the fit to the cavity resonance from which the cavity resonance frequency ( ω res ) is determined as the minimum.

Fig. 4.
Fig. 4.

Resonance frequency versus time delay ( Δ t ) between pump and probe at different pump-pulse energies. The resonance frequency redshifts due to the instantaneous electronic Kerr effect only at temporal overlap ( Δ t = 0 ± 15 fs ) of pump–probe (shaded with light color). The blueshift of the cavity resonance due to free-carriers is observed when the pump-pulse energy is increased (shaded with dark color). The dotted horizontal line shows the unswitched resonance frequency at ω 0 = 7805.6 cm 1 .

Fig. 5.
Fig. 5.

Transient reflectivity versus wavenumber for three different pump-pulse energies. The spectra are obtained at pump–probe coincidence ( Δ t = 0 ± 15 fs ). The gray curves show the fit to the cavity resonance from which the cavity resonance frequency ( ω res ) is determined as the minimum, indicated as ticks.

Fig. 6.
Fig. 6.

Instantaneous negative shift of the resonance frequency versus pump-pulse energy. Black circles show the measured results with a 10% error bar. At low pump-pulse energies (light shaded region), we only observe electronic Kerr effect since the resonance frequency decreases linearly with the pump-pulse energy (magenta dashed line). The competing blueshift of free carriers is observed beyond 70 pJ / μm 2 , and the region with both Kerr and free-carrier excitation is darker shaded. The right ordinate shows the calculated refractive index change. The red solid curve indicates the modeled index change as a function of the pump-pulse energy for general nondegenerate pump–probe light beams.

Fig. 7.
Fig. 7.

Refractive index change versus pump-pulse energy calculated for three different quality factors. The positive index change due to the electronic Kerr effect is more pronounced with low quality factor cavities with fast dynamics. We assumed cavities with resonance ω res = 7812 cm 1 pumped at ω pu = 4165 cm 1 as in our experiment. The pump-pulse duration is taken as τ P = 140 ± 10 fs , whereas the probe-pulse duration is set by τ cav .

Fig. 8.
Fig. 8.

(a) Function f ( t ) versus x in Eq. (B11), which can be approximated as a Dirac delta function; (b) function g ( t ) in Eq. (B12), which can be approximated as two-dimensional Dirac delta function. The absorption probability is proportional to the functions f ( t ) and g ( t ) .

Tables (2)

Tables Icon

Table 1. Parameters Used in Our Model

Tables Icon

Table 2. Coefficients Calculated and Determined from Measurementsa

Equations (49)

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Δ n = 6 π χ ( 3 ) n 0 [ | E cav | 2 + 2 | E pu | 2 ]  Kerr q 2 2 n 0 ϵ 0 m opt * ω pr 2 [ N eh ( 2 ) + N eh ( 3 ) ] free carriers .
P Tot ( ω ) = ϵ 0 χ ( 1 ) E ˜ ( ω ) + ϵ 0 χ ( 2 ) E ˜ 2 ( ω ) + ϵ 0 χ ( 3 ) E ˜ 3 ( ω ) .
P Tot ( ω pr ) = ϵ 0 χ ( 1 ) E cav e i ω pr t + 3 ϵ 0 χ ( 3 ) E cav 3 e i ( ω pr + ω pr ω pr ) t + 6 ϵ 0 χ ( 3 ) E cav E pu 2 e i ( ω pu ω pu + ω pr ) t ,
N eh ( 2 ) = R n g ( 2 ) N atm τ int = N atm τ int 8 π 3 | μ m g μ n m | 2 2 n 0 2 c 2 [ I cav 2 ρ f ( ω n g = 2 ω pr ) 3 in Fig. 2(b) + I cav I pu ω pr ω pu ρ f ( ω n g = ω pr + ω pu ) ( 1 ω pr 2 + 2 ω pr ω pu + 1 ω pu 2 ) 2 in Fig. 2(b) ] ,
N eh ( 3 ) = R l g ( 3 ) N atm τ int = N atm τ int 16 π 4 | μ m g μ n m μ l n | 2 3 n 0 3 c 3 [ I pu 3 ρ f ( ω l g = 3 ω pu ) 4 ω pu 1 in Fig. 2(c) + I cav 3 ρ f ( ω l g = 3 ω pr ) 4 ω pr 4 in Fig. 2(c) + I cav I pu 2 ω pr ω pu 2 ρ f ( ω l g = ω pr + 2 ω pu ) ( 1 4 ω pu 4 + 1 ω pu 2 ( ω pr + ω pu ) 2 + 1 ω pr 2 ( ω pr + ω pu ) 2 2 in Fig. 2(c) + 1 ω pr ω pu 2 ( ω pr + ω pu ) + 1 ω pu 3 ( ω pr + ω pu ) + 2 ω pr ω pu ( ω pr + ω pu ) 2 ) 2 in Fig. 2(c) + I cav 2 I pu ω pr 2 ω pu ρ f ( ω l g = 2 ω pr + ω pu ) ( 1 4 ω pr 4 + 1 ω pu 2 ( ω pr + ω pu ) 2 + 1 ω pr 2 ( ω pr + ω pu ) 2 3 in Fig. 2(c) + 1 ω pr 2 ω pu ( ω pr + ω pu ) + 1 ω pr 3 ( ω pr + ω pu ) + 2 ω pr ω pu ( ω pr + ω pu ) 2 ) 3 in Fig. 2(c) ] .
n ( ω pr ) = n 0 ( ω pr ) + n 2 ( ω pr : ω pr , ω pu ) E ˜ 2 ,
E ˜ ( t ) = E pr ( ω pr ) e i ω pr t + E pu ( ω pu ) e i ω pu t + c.c. ,
E ˜ 2 ( ω pr , ω pu ) = 2 E pr ( ω pr ) E pr * ( ω pr ) + 2 E pu ( ω pu ) E pu * ( ω pu ) = 2 ( | E pr | 2 + | E pu | 2 ) .
n ( ω pr ) = n 0 ( ω pr ) + 2 n 2 ( ω pr : ω pr , ω pu ) ( | E pr | 2 + | E pu | 2 ) .
P Tot ( ω ) = ϵ 0 χ ( 1 ) E ˜ ( ω ) + ϵ 0 χ ( 2 ) E ˜ 2 ( ω ) + ϵ 0 χ ( 3 ) E ˜ 3 ( ω ) .
P Tot ( ω pr ) = ϵ 0 χ ( 1 ) E pr e i ω pr t + 3 ϵ 0 χ ( 3 ) E pr 3 e i ( ω pr + ω pr ω pr ) t + 6 ϵ 0 χ ( 3 ) E pr E pu 2 e i ( ω pu ω pu + ω pr ) t = ϵ 0 E pr e i ω pr t ( χ ( 1 ) + 3 χ ( 3 ) E pr 2 + 6 χ ( 3 ) E pu 2 ) χ eff ,
χ eff = χ ( 1 ) + 3 χ ( 3 ) | E pr | 2 + 6 χ ( 3 ) | E pu | 2 .
n 2 ( ω pr ) = 1 + 4 π χ eff ,
n 0 ( ω pr ) 2 + 4 n 2 ( ω pr : ω pr , ω pu ) n 0 ( ω pr ) ( | E pr | 2 + | E pu | 2 ) + 4 n 2 ( ω pr : ω pr , ω pu ) 2 ( | E pr | 2 + | E pu | 2 ) 2 = 1 + 4 π χ ( 1 ) + 12 π χ ( 3 ) | E pr | 2 + 24 π χ ( 3 ) | E pu | 2 .
n 0 ( ω pr ) 2 = 1 + 4 π χ ( 1 ) ,
n 2 ( ω pr : ω pr , ω pu ) = 3 π χ ( 3 ) n 0 ( ω pr ) ( | E pr | 2 + 2 | E pu | 2 ) ( | E pr | 2 + | E pu | 2 ) .
n ( ω pr ) = n 0 ( ω pr ) + 6 π χ ( 3 ) n 0 ( ω pr ) ( | E pr | 2 + 2 | E pu | 2 ) .
d a m ( N ) d t = ( i ) 1 l a l N 1 V ˜ l m e i ω l m t ,
V ˜ m g = μ m g [ E pr ( ω pr ) e i ω pr t + E pu ( ω pu ) e i ω pu t + E pr * ( ω pr ) e i ω pr t + E pu * ( ω pu ) e i ω pu t ] ,
d a m ( 1 ) d t = ( i ) 1 μ m g [ E pr ( ω pr ) e i ( ω m g ω pr ) t + E pu ( ω pu ) e i ( ω m g ω pu ) t + E pr * ( ω pr ) e i ( ω pr + ω m g ) t + E pu * ( ω pu ) e i ( ω pu + ω m g ) t ] .
a m ( 1 ) ( t ) = ( i ) 1 μ m g 0 t d t [ E pr ( ω pr ) e i ( ω m g ω pr ) t + E pu ( ω pu ) e i ( ω m g ω pu ) t ] = μ m g E pr ( ω m g ω pr ) [ e i ( ω m g ω pr ) t 1 ] + μ m g E pu ( ω m g ω pu ) [ e i ( ω m g ω pu ) t 1 ] .
V ˜ n m = μ n m [ E pr ( ω pr ) e i ω pr t + E pu ( ω pu ) e i ω pu t + E pr * ( ω pr ) e i ω pr t + E pu * ( ω pu ) e i ω pu t ] .
d a n ( 2 ) d t = ( i ) 1 m a m ( 1 ) × V ˜ n m e i ω m n t .
d a n ( 2 ) d t = μ m g μ n m i 2 [ E pr 2 ( ω m g ω pr ) [ e i ( ω m g + ω n m 2 ω pr ) t e i ( ω n m ω pr ) t ] + E pr E pu ( ω m g ω pu ) [ e i ( ω m g + ω n m ω pr ω pu ) t e i ( ω n m ω pr ) t ] + E pr E pu ( ω m g ω pr ) [ e i ( ω m g + ω n m ω pr ω pu ) t e i ( ω n m ω pu ) t ] + E pu 2 ( ω m g ω pu ) [ e i ( ω m g + ω n m 2 ω pu ) t e i ( ω n m ω pu ) t ] ] .
a n ( 2 ) ( t ) = t μ m g μ n m 2 [ E pr 2 [ e i ( ω n g 2 ω pr ) t 1 ] ω pr ( ω n g 2 ω pr ) t + E pu 2 [ e i ( ω n g 2 ω pu ) t 1 ] ω pu ( ω n g 2 ω pu ) t + ( E pr E pu ω pu + E pr E pu ω pr ) e i ( ω n g ω pr ω pu ) t 1 ( ω n g ω pr ω pu ) t ] .
a n ( 2 ) ( t ) = t μ m g μ n m 2 [ A ( e i x 1 ) x + B ( e i y 1 ) y + C ( e i z 1 ) z ] .
p n ( 2 ) ( t ) = | a n ( 2 ) ( t ) | 2 = t 2 | μ m g μ n m | 2 4 × [ A 2 ( 1 e i x e i x + 1 ) x 2 + B 2 ( 1 e i y e i y + 1 ) y 2 + C 2 ( 1 e i z e i z + 1 ) z 2 + A B ( e i ( x y ) e i x e i y + 1 ) x y + A B ( e i ( x y ) e i x e i y + 1 ) x y + A C ( e i ( x z ) e i x e i z + 1 ) x z + A C ( e i ( x z ) e i x e i z + 1 ) x z + B C ( e i ( y z ) e i y e i z + 1 ) y z + B C ( e i ( y z ) e i y e i z + 1 ) y z ] .
f ( t ) = t 2 ( 2 e i x e i x ) x 2 = t 2 2 ( 1 cos x ) x 2 ,
g ( t ) = t 2 [ e i ( x y ) e i x e i y + 1 x y + e i ( x y ) e i x e i y + 1 x y ] = t 2 [ 2 + 2 cos ( x y ) 2 cos x 2 cos y x y ] .
lim t f ( t ) = 2 π t δ ( ω n g 2 ω pr ) .
lim t g ( t ) = 4 π t δ ( ω n g 2 ω pr ) δ ( ω n g ω pr ω pu ) .
0 ρ f ( ω n g ) d ω n g = 1 ,
0 ρ f ( ω n g ) 2 π δ ( ω n g 2 ω pr ) d ω n g = ρ f ( ω n g = 2 ω pr ) .
p n ( 2 ) ( t ) = 2 π t | μ m g μ n m | 2 4 × [ A 2 ρ f ( ω n g = 2 ω pr ) + B 2 ρ f ( ω n g = ω pr + ω pu ) + C 2 ρ f ( ω n g = 2 ω pu ) ] .
R n g ( 2 ) = p n ( 2 ) ( t ) t .
σ n g ( 2 ) = R n g ( 2 ) I 2 ,
N eh ( 2 ) = R n g ( 2 ) N atm τ int ,
N eh ( 2 ) = N atm τ int 8 π 3 | μ m g μ n m | 2 2 n 0 2 c 2 [ I pu 2 ρ f ( ω n g = 2 ω pu ) 1 in Fig. 2(b) + I pr 2 ρ f ( ω n g = 2 ω pr ) 3 in Fig. 2(b) + I pr I pu ω pr ω pu ρ f ( ω n g = ω pr + ω pu ) ( 1 ω pr 2 + 2 ω pr ω pu + 1 ω pu 2 ) 2 in Fig. 2(b) ] .
V ˜ l n = μ l n [ E pr ( ω pr ) e i ω pr t + E pu ( ω pu ) e i ω pu t + E pr * ( ω pr ) e i ω pr t + E pu * ( ω pu ) e i ω pu t ] .
d a l ( 3 ) d t = ( i ) 1 m n a n ( 2 ) × V ˜ l n e i ω l n t .
a l ( 3 ) ( t ) = t μ l n μ n m μ m g 3 [ E pr 3 2 ω pr 2 ( e i ( ω l g 3 ω pr ) t 1 ) ( ω l g 3 ω pr ) t + E pr 2 E pu 2 ω pr 2 ( e i ( ω l g 2 ω pr ω pu ) t 1 ) ( ω l g 2 ω pr ω pu ) t + E pr 2 E pu ω pu ( ω pr + ω pu ) ( e i ( ω l g 2 ω pr ω pu ) t 1 ) ( ω l g 2 ω pr ω pu ) t + E pr E pu 2 ω pu ( ω pr + ω pu ) ( e i ( ω l g ω pr 2 ω pu ) t 1 ) ( ω l g ω pr 2 ω pu ) t + E pr 2 E pu ω pr ( ω pr + ω pu ) ( e i ( ω l g 2 ω pr ω pu ) t 1 ) ( ω l g 2 ω pr ω pu ) t + E pr E pu 2 ω pr ( ω pr + ω pu ) ( e i ( ω l g ω pr 2 ω pu ) t 1 ) ( ω l g ω pr 2 ω pu ) t + E pu 2 E pr 2 ω pu 2 ( e i ( ω l g 2 ω pu ω pr ) t 1 ) ( ω l g 2 ω pu ω pr ) t + E pu 3 2 ω pu 2 ( e i ( ω l g 3 ω pu ) t 1 ) ( ω l g 3 ω pu ) t ] .
x = ( ω l g 3 ω pr ) t , y = ( ω l g 2 ω pr ω pu ) t , z = ( ω l g 2 ω pu ω pr ) t , w = ( ω l g 3 ω pu ) t , A = E pr 3 2 ω pr 2 , B = E pr 2 E pu 2 ω pr 2 + E pr 2 E pu ω pu ( ω pr + ω pu ) + E pr 2 E pu ω pr ( ω pr + ω pu ) , C = E pu 2 E pr 2 ω pu 2 + E pu 2 E pr ω pu ( ω pr + ω pu ) + E pu 2 E pr ω pr ( ω pr + ω pu ) , D = E pu 3 2 ω pu 2 ,
a l ( 3 ) ( t ) = t μ m g μ n m μ l n 3 [ A ( e i x 1 ) x + B ( e i y 1 ) y + C ( e i z 1 ) z + D ( e i w 1 ) w ] .
p l ( 3 ) ( t ) = | a l ( 3 ) ( t ) | 2 = t 2 | μ m g μ n m μ l n | 2 6 × [ A 2 ( 1 e i x e i x + 1 ) x 2 + B 2 ( 1 e i y e i y + 1 ) y 2 + C 2 ( 1 e i z e i z + 1 ) z 2 + D 2 ( 1 e i w e i w + 1 ) w 2 + A B ( e i ( x y ) e i x e i y + 1 ) x y + A B ( e i ( x y ) e i x e i y + 1 ) x y + A C ( e i ( x z ) e i x e i z + 1 ) x z + A C ( e i ( x z ) e i x e i z + 1 ) x z + A D ( e i ( x w ) e i x e i w + 1 ) x w + A D ( e i ( x w ) e i x e i w + 1 ) x w + B C ( e i ( y z ) e i y e i z + 1 ) y z + B C ( e i ( y z ) e i y e i z + 1 ) y z + B D ( e i ( y w ) e i y e i w + 1 ) y w + B D ( e i ( y w ) e i y e i w + 1 ) y w + C D ( e i ( z w ) e i z e i w + 1 ) z w + C D ( e i ( z w ) e i z e i w + 1 ) z w ] .
p l ( 3 ) ( t ) = 2 π t | μ m g μ n m μ l n | 2 6 [ A 2 ρ f ( ω l g = 3 ω pr ) + B 2 ρ f ( ω l g = 2 ω pr + ω pu ) + C 2 ρ f ( ω l g = ω pr + 2 ω pu ) + D 2 ρ f ( ω l g = 3 ω pu ) ] .
R l g ( 3 ) = p l ( 3 ) ( t ) t .
σ l g ( 3 ) = R l g ( 3 ) I 3 ,
N eh ( 3 ) = R l g ( 3 ) N atm τ int ,
N eh ( 3 ) = R l g ( 3 ) N atm τ int = N atm τ int 16 π 4 | μ m g μ n m μ l n | 2 3 n 0 3 c 3 [ I pu 3 ρ f ( ω l g = 3 ω pu ) 4 ω pu 1 in Fig. 2(c) + I pr 3 ρ f ( ω l g = 3 ω pr ) 4 ω pr 4 in Fig. 2(c) + I pr I pu 2 ω pr ω pu 2 ρ f ( ω l g = ω pr + 2 ω pu ) ( 1 4 ω pu 4 + 1 ω pu 2 ( ω pr + ω pu ) 2 + 1 ω pr 2 ( ω pr + ω pu ) 2 2 in Fig. 2(c) + 1 ω pr ω pu 2 ( ω pr + ω pu ) + 1 ω pu 3 ( ω pr + ω pu ) + 2 ω pr ω pu ( ω pr + ω pu ) 2 ) 2 in Fig. 2(c) + I pr 2 I pu ω pr 2 ω pu ρ f ( ω l g = 2 ω pr + ω pu ) ( 1 4 ω pr 4 + 1 ω pu 2 ( ω pr + ω pu ) 2 + 1 ω pr 2 ( ω pr + ω pu ) 2 3 in Fig. 2(c) + 1 ω pr 2 ω pu ( ω pr + ω pu ) + 1 ω pr 3 ( ω pr + ω pu ) + 2 ω pr ω pu ( ω pr + ω pu ) 2 ) 3 in Fig. 2(c) ] .

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