Abstract

A model is presented for describing ultrafast interferometric second- and third-harmonic generation (SHG and THG) from a gold surface. The model uses an effective four-level density matrix system that accounts for the stepwise resonant excitation channels with finite dephasing and energy relaxation times, and the fast nonresonant channel through the energy continua. By fitting recent experimental results for SHG and THG from a polycrystalline gold surface irradiated with 18fs Ti:sapphire laser pulses, we have extracted a T2=7.9fs dephasing time at 1.56eV above the Fermi energy, compared with a T130fs energy relaxation time at the same energy level. The difference indicates a strong contribution to the dephasing of the polarization from elastic processes. Using these values, we present calculations for other laser pulse durations, from 30fs down to one optical cycle (2.65fs). It is shown that, in particular, interferometric THG from a gold surface can be used to measure and characterize with high accuracy laser pulses as short as one optical cycle.

© 2009 Optical Society of America

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  1. M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
    [CrossRef]
  2. B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and offresonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. 83, 4421-4424 (1999).
    [CrossRef]
  3. Y.-H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, “Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry,” J. Phys. Chem. B 105, 2135-2142 (2001).
    [CrossRef]
  4. C. Voelkmann, M. Reichelt, T. Meier, S. W. Koch, and U. Hofer, “Five-wave mixing spectroscopy of ultrafast electron dynamics at a Si(001) surface,” Phys. Rev. Lett. 92, 127405 (2004).
    [CrossRef] [PubMed]
  5. M. W. Klein, T. Tritschler, and M. Wegener, and S. Linden, “Lineshape of harmonic generation by metallic nanoparticles and metallic photonic crystal slabs,” Phys. Rev. B 72, 115113 (2005).
    [CrossRef]
  6. J. Dai, H. Teng, and C. Guo, “Second- and third-order interferometric autocorrelations based on harmonic generations from metal surfaces,” Opt. Commun. 252, 173-178 (2005).
    [CrossRef]
  7. M. Mauerer, I. L. Shumay, W. Berthold, and U. Hofer, “Ultrafast carrier dynamics in Si(111)7×7 dangling bonds probed by time-resolved second-harmonic generation and two-photon photoemission,” Phys. Rev. B 73, 245305 (2006).
    [CrossRef]
  8. J. A. McGuire, M. B. Raschke, and Y. R. Shen, “Electron dynamics of silicon surface states: second-harmonic hole-burning on Si(111)-(7×7),” Phys. Rev. Lett. 96, 087401 (2006).
    [CrossRef] [PubMed]
  9. U. Bovensiepen, “Coherent and incoherent excitation of the Gd(001) surface on ultrafast timescales,” J. Phys. Condens. Matter 19, 083201 (2007).
    [CrossRef]
  10. M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136-159 (2008).
    [CrossRef]
  11. K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55-58 (2009).
    [CrossRef]
  12. S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339-1342 (1997).
    [CrossRef]
  13. F. Hubenthal, “Ultrafast dephasing time of localized surface plasmon polariton resonance and the involved damping mechanisms in colloidal gold nanoparticles,” Prog. Surf. Sci. 82, 378-387 (2007).
    [CrossRef]
  14. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  15. J. B. Smith and H. Ehrenreich, “Frequency dependence of the optical relaxation time in metals,” Phys. Rev. B 25, 923-930 (1982).
    [CrossRef]
  16. C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
    [CrossRef] [PubMed]
  17. A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10-fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
    [CrossRef]
  18. M. J. Weida, S. Ogawa, H. Nagano, and H. Petek, “Ultrafast interferometric pump-probe correlation measurements in systems with broadened bands or continua,” J. Opt. Soc. Am. B 17, 1443-1451 (2000).
    [CrossRef]
  19. N. A. Papadogiannis, S. D. Moustaizis, P. A. Loukakos, and C. Kalpouzos, “Temporal characterization of ultra short laser pulses based on multiple harmonic generation on a gold surface,” Appl. Phys. B 65, 339-345 (1997).
    [CrossRef]
  20. W. Hubner, K. H. Bennemann, and K. Bohmer, “Theory for the nonlinear optical response of transition metals: polarization dependence as a fingerprint of the electronic structure at surfaces and interfaces,” Phys. Rev. B 50, 17597-17605 (1994).
    [CrossRef]
  21. A. T. Georges and N. E. Karatzas, “Theory of multiple harmonic generation in reflection from a metal surface,” Appl. Phys. B 81, 479-485 (2005).
    [CrossRef]
  22. For typos incurred during the production stage (iNωm printed as iNω or as ImNω), see Appl. Phys. B 81, 725-726 (2005).
    [CrossRef]
  23. See, for example, G. Baym, Lectures on Quantum Mechanics (Benjamin, 1973).
  24. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  25. I. Campillo, J. M. Pitarke, A. Rubio, and P. M. Echenique, “Role of occupied d states in the relaxation of hot electrons in Au,” Phys. Rev. B 62, 1500-1503 (2000).
    [CrossRef]
  26. N. E. Karatzas and A. T. Georges, “Effects of electron relaxation on multiple harmonic generation from metal surfaces with femtosecond laser pulse,” Opt. Commun. 267, 498-504 (2006).
    [CrossRef]
  27. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).
  28. K. L. Moore and T. D. Donnelly, “Probing nonequilibrium electron distribution in gold by use of second-harmonic generation,” Opt. Lett. 24, 990-992 (1999).
    [CrossRef]
  29. S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5, 515-518 (2005).
    [CrossRef] [PubMed]
  30. D. Meshulach, Y. Barad, and Y. Silberberg, “Measurement of ultrashort optical pulses by third-harmonic generation,” J. Opt. Soc. Am. B 14, 2122-2125 (1997).
    [CrossRef]
  31. M. E. Anderson, T. Witting, and I. A. Walmsley, “Gold-SPIDER: spectral phase interferometry for direct electric field reconstruction utilizing sum-frequency generation from a gold surface,” J. Opt. Soc. Am. B 25, A13-A16 (2008).
    [CrossRef]

2009 (1)

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

2008 (3)

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136-159 (2008).
[CrossRef]

A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10-fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
[CrossRef]

M. E. Anderson, T. Witting, and I. A. Walmsley, “Gold-SPIDER: spectral phase interferometry for direct electric field reconstruction utilizing sum-frequency generation from a gold surface,” J. Opt. Soc. Am. B 25, A13-A16 (2008).
[CrossRef]

2007 (2)

U. Bovensiepen, “Coherent and incoherent excitation of the Gd(001) surface on ultrafast timescales,” J. Phys. Condens. Matter 19, 083201 (2007).
[CrossRef]

F. Hubenthal, “Ultrafast dephasing time of localized surface plasmon polariton resonance and the involved damping mechanisms in colloidal gold nanoparticles,” Prog. Surf. Sci. 82, 378-387 (2007).
[CrossRef]

2006 (3)

M. Mauerer, I. L. Shumay, W. Berthold, and U. Hofer, “Ultrafast carrier dynamics in Si(111)7×7 dangling bonds probed by time-resolved second-harmonic generation and two-photon photoemission,” Phys. Rev. B 73, 245305 (2006).
[CrossRef]

J. A. McGuire, M. B. Raschke, and Y. R. Shen, “Electron dynamics of silicon surface states: second-harmonic hole-burning on Si(111)-(7×7),” Phys. Rev. Lett. 96, 087401 (2006).
[CrossRef] [PubMed]

N. E. Karatzas and A. T. Georges, “Effects of electron relaxation on multiple harmonic generation from metal surfaces with femtosecond laser pulse,” Opt. Commun. 267, 498-504 (2006).
[CrossRef]

2005 (5)

A. T. Georges and N. E. Karatzas, “Theory of multiple harmonic generation in reflection from a metal surface,” Appl. Phys. B 81, 479-485 (2005).
[CrossRef]

For typos incurred during the production stage (iNωm printed as iNω or as ImNω), see Appl. Phys. B 81, 725-726 (2005).
[CrossRef]

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5, 515-518 (2005).
[CrossRef] [PubMed]

M. W. Klein, T. Tritschler, and M. Wegener, and S. Linden, “Lineshape of harmonic generation by metallic nanoparticles and metallic photonic crystal slabs,” Phys. Rev. B 72, 115113 (2005).
[CrossRef]

J. Dai, H. Teng, and C. Guo, “Second- and third-order interferometric autocorrelations based on harmonic generations from metal surfaces,” Opt. Commun. 252, 173-178 (2005).
[CrossRef]

2004 (1)

C. Voelkmann, M. Reichelt, T. Meier, S. W. Koch, and U. Hofer, “Five-wave mixing spectroscopy of ultrafast electron dynamics at a Si(001) surface,” Phys. Rev. Lett. 92, 127405 (2004).
[CrossRef] [PubMed]

2002 (1)

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

2001 (1)

Y.-H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, “Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry,” J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

2000 (2)

I. Campillo, J. M. Pitarke, A. Rubio, and P. M. Echenique, “Role of occupied d states in the relaxation of hot electrons in Au,” Phys. Rev. B 62, 1500-1503 (2000).
[CrossRef]

M. J. Weida, S. Ogawa, H. Nagano, and H. Petek, “Ultrafast interferometric pump-probe correlation measurements in systems with broadened bands or continua,” J. Opt. Soc. Am. B 17, 1443-1451 (2000).
[CrossRef]

1999 (2)

K. L. Moore and T. D. Donnelly, “Probing nonequilibrium electron distribution in gold by use of second-harmonic generation,” Opt. Lett. 24, 990-992 (1999).
[CrossRef]

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and offresonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. 83, 4421-4424 (1999).
[CrossRef]

1998 (1)

M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
[CrossRef]

1997 (3)

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339-1342 (1997).
[CrossRef]

N. A. Papadogiannis, S. D. Moustaizis, P. A. Loukakos, and C. Kalpouzos, “Temporal characterization of ultra short laser pulses based on multiple harmonic generation on a gold surface,” Appl. Phys. B 65, 339-345 (1997).
[CrossRef]

D. Meshulach, Y. Barad, and Y. Silberberg, “Measurement of ultrashort optical pulses by third-harmonic generation,” J. Opt. Soc. Am. B 14, 2122-2125 (1997).
[CrossRef]

1994 (1)

W. Hubner, K. H. Bennemann, and K. Bohmer, “Theory for the nonlinear optical response of transition metals: polarization dependence as a fingerprint of the electronic structure at surfaces and interfaces,” Phys. Rev. B 50, 17597-17605 (1994).
[CrossRef]

1982 (1)

J. B. Smith and H. Ehrenreich, “Frequency dependence of the optical relaxation time in metals,” Phys. Rev. B 25, 923-930 (1982).
[CrossRef]

1972 (1)

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

Aizpurua, J.

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136-159 (2008).
[CrossRef]

Anderson, M. E.

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

Assion, A.

M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
[CrossRef]

Aussenegg, F. R.

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and offresonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. 83, 4421-4424 (1999).
[CrossRef]

Barad, Y.

Baym, G.

See, for example, G. Baym, Lectures on Quantum Mechanics (Benjamin, 1973).

Bennemann, K. H.

W. Hubner, K. H. Bennemann, and K. Bohmer, “Theory for the nonlinear optical response of transition metals: polarization dependence as a fingerprint of the electronic structure at surfaces and interfaces,” Phys. Rev. B 50, 17597-17605 (1994).
[CrossRef]

Berciaud, S.

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5, 515-518 (2005).
[CrossRef] [PubMed]

Berthold, W.

M. Mauerer, I. L. Shumay, W. Berthold, and U. Hofer, “Ultrafast carrier dynamics in Si(111)7×7 dangling bonds probed by time-resolved second-harmonic generation and two-photon photoemission,” Phys. Rev. B 73, 245305 (2006).
[CrossRef]

Bohmer, K.

W. Hubner, K. H. Bennemann, and K. Bohmer, “Theory for the nonlinear optical response of transition metals: polarization dependence as a fingerprint of the electronic structure at surfaces and interfaces,” Phys. Rev. B 50, 17597-17605 (1994).
[CrossRef]

Bovensiepen, U.

U. Bovensiepen, “Coherent and incoherent excitation of the Gd(001) surface on ultrafast timescales,” J. Phys. Condens. Matter 19, 083201 (2007).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Bryant, G.

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136-159 (2008).
[CrossRef]

Campillo, I.

I. Campillo, J. M. Pitarke, A. Rubio, and P. M. Echenique, “Role of occupied d states in the relaxation of hot electrons in Au,” Phys. Rev. B 62, 1500-1503 (2000).
[CrossRef]

Chang, Q.

Y.-H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, “Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry,” J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Christy, R. W.

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

Cognet, L.

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5, 515-518 (2005).
[CrossRef] [PubMed]

Dai, J.

J. Dai, H. Teng, and C. Guo, “Second- and third-order interferometric autocorrelations based on harmonic generations from metal surfaces,” Opt. Commun. 252, 173-178 (2005).
[CrossRef]

Donnelly, T. D.

Echenique, P. M.

I. Campillo, J. M. Pitarke, A. Rubio, and P. M. Echenique, “Role of occupied d states in the relaxation of hot electrons in Au,” Phys. Rev. B 62, 1500-1503 (2000).
[CrossRef]

Ehrenreich, H.

J. B. Smith and H. Ehrenreich, “Frequency dependence of the optical relaxation time in metals,” Phys. Rev. B 25, 923-930 (1982).
[CrossRef]

Feldmann, J.

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

Franzl, T.

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

Georges, A. T.

A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10-fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
[CrossRef]

N. E. Karatzas and A. T. Georges, “Effects of electron relaxation on multiple harmonic generation from metal surfaces with femtosecond laser pulse,” Opt. Commun. 267, 498-504 (2006).
[CrossRef]

A. T. Georges and N. E. Karatzas, “Theory of multiple harmonic generation in reflection from a metal surface,” Appl. Phys. B 81, 479-485 (2005).
[CrossRef]

Gerber, G.

M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
[CrossRef]

Guo, C.

J. Dai, H. Teng, and C. Guo, “Second- and third-order interferometric autocorrelations based on harmonic generations from metal surfaces,” Opt. Commun. 252, 173-178 (2005).
[CrossRef]

Heberle, A. P.

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339-1342 (1997).
[CrossRef]

Hofer, U.

M. Mauerer, I. L. Shumay, W. Berthold, and U. Hofer, “Ultrafast carrier dynamics in Si(111)7×7 dangling bonds probed by time-resolved second-harmonic generation and two-photon photoemission,” Phys. Rev. B 73, 245305 (2006).
[CrossRef]

C. Voelkmann, M. Reichelt, T. Meier, S. W. Koch, and U. Hofer, “Five-wave mixing spectroscopy of ultrafast electron dynamics at a Si(001) surface,” Phys. Rev. Lett. 92, 127405 (2004).
[CrossRef] [PubMed]

Hubenthal, F.

F. Hubenthal, “Ultrafast dephasing time of localized surface plasmon polariton resonance and the involved damping mechanisms in colloidal gold nanoparticles,” Prog. Surf. Sci. 82, 378-387 (2007).
[CrossRef]

Hubner, W.

W. Hubner, K. H. Bennemann, and K. Bohmer, “Theory for the nonlinear optical response of transition metals: polarization dependence as a fingerprint of the electronic structure at surfaces and interfaces,” Phys. Rev. B 50, 17597-17605 (1994).
[CrossRef]

Johnson, P. B.

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

Kalpouzos, C.

N. A. Papadogiannis, S. D. Moustaizis, P. A. Loukakos, and C. Kalpouzos, “Temporal characterization of ultra short laser pulses based on multiple harmonic generation on a gold surface,” Appl. Phys. B 65, 339-345 (1997).
[CrossRef]

Karatzas, N. E.

A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10-fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
[CrossRef]

N. E. Karatzas and A. T. Georges, “Effects of electron relaxation on multiple harmonic generation from metal surfaces with femtosecond laser pulse,” Opt. Commun. 267, 498-504 (2006).
[CrossRef]

A. T. Georges and N. E. Karatzas, “Theory of multiple harmonic generation in reflection from a metal surface,” Appl. Phys. B 81, 479-485 (2005).
[CrossRef]

Klein, M. W.

M. W. Klein, T. Tritschler, and M. Wegener, and S. Linden, “Lineshape of harmonic generation by metallic nanoparticles and metallic photonic crystal slabs,” Phys. Rev. B 72, 115113 (2005).
[CrossRef]

Koch, S. W.

C. Voelkmann, M. Reichelt, T. Meier, S. W. Koch, and U. Hofer, “Five-wave mixing spectroscopy of ultrafast electron dynamics at a Si(001) surface,” Phys. Rev. Lett. 92, 127405 (2004).
[CrossRef] [PubMed]

Krenn, J. R.

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and offresonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. 83, 4421-4424 (1999).
[CrossRef]

Lamprecht, B.

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and offresonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. 83, 4421-4424 (1999).
[CrossRef]

Lang, B.

M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
[CrossRef]

Leitner, A.

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and offresonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. 83, 4421-4424 (1999).
[CrossRef]

Liau, Y.-H.

Y.-H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, “Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry,” J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Linden, S.

M. W. Klein, T. Tritschler, and M. Wegener, and S. Linden, “Lineshape of harmonic generation by metallic nanoparticles and metallic photonic crystal slabs,” Phys. Rev. B 72, 115113 (2005).
[CrossRef]

Loukakos, P. A.

N. A. Papadogiannis, S. D. Moustaizis, P. A. Loukakos, and C. Kalpouzos, “Temporal characterization of ultra short laser pulses based on multiple harmonic generation on a gold surface,” Appl. Phys. B 65, 339-345 (1997).
[CrossRef]

Lounis, B.

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5, 515-518 (2005).
[CrossRef] [PubMed]

MacDonald, K. F.

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

Mauerer, M.

M. Mauerer, I. L. Shumay, W. Berthold, and U. Hofer, “Ultrafast carrier dynamics in Si(111)7×7 dangling bonds probed by time-resolved second-harmonic generation and two-photon photoemission,” Phys. Rev. B 73, 245305 (2006).
[CrossRef]

McGuire, J. A.

J. A. McGuire, M. B. Raschke, and Y. R. Shen, “Electron dynamics of silicon surface states: second-harmonic hole-burning on Si(111)-(7×7),” Phys. Rev. Lett. 96, 087401 (2006).
[CrossRef] [PubMed]

Meier, T.

C. Voelkmann, M. Reichelt, T. Meier, S. W. Koch, and U. Hofer, “Five-wave mixing spectroscopy of ultrafast electron dynamics at a Si(001) surface,” Phys. Rev. Lett. 92, 127405 (2004).
[CrossRef] [PubMed]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

Meshulach, D.

Moore, K. L.

Moustaizis, S. D.

N. A. Papadogiannis, S. D. Moustaizis, P. A. Loukakos, and C. Kalpouzos, “Temporal characterization of ultra short laser pulses based on multiple harmonic generation on a gold surface,” Appl. Phys. B 65, 339-345 (1997).
[CrossRef]

Mulvaney, P.

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

Nagano, H.

M. J. Weida, S. Ogawa, H. Nagano, and H. Petek, “Ultrafast interferometric pump-probe correlation measurements in systems with broadened bands or continua,” J. Opt. Soc. Am. B 17, 1443-1451 (2000).
[CrossRef]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339-1342 (1997).
[CrossRef]

Ogawa, S.

M. J. Weida, S. Ogawa, H. Nagano, and H. Petek, “Ultrafast interferometric pump-probe correlation measurements in systems with broadened bands or continua,” J. Opt. Soc. Am. B 17, 1443-1451 (2000).
[CrossRef]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339-1342 (1997).
[CrossRef]

Papadogiannis, N. A.

N. A. Papadogiannis, S. D. Moustaizis, P. A. Loukakos, and C. Kalpouzos, “Temporal characterization of ultra short laser pulses based on multiple harmonic generation on a gold surface,” Appl. Phys. B 65, 339-345 (1997).
[CrossRef]

Pelton, M.

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136-159 (2008).
[CrossRef]

Petek, H.

M. J. Weida, S. Ogawa, H. Nagano, and H. Petek, “Ultrafast interferometric pump-probe correlation measurements in systems with broadened bands or continua,” J. Opt. Soc. Am. B 17, 1443-1451 (2000).
[CrossRef]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339-1342 (1997).
[CrossRef]

Pitarke, J. M.

I. Campillo, J. M. Pitarke, A. Rubio, and P. M. Echenique, “Role of occupied d states in the relaxation of hot electrons in Au,” Phys. Rev. B 62, 1500-1503 (2000).
[CrossRef]

Raschke, M. B.

J. A. McGuire, M. B. Raschke, and Y. R. Shen, “Electron dynamics of silicon surface states: second-harmonic hole-burning on Si(111)-(7×7),” Phys. Rev. Lett. 96, 087401 (2006).
[CrossRef] [PubMed]

Reichelt, M.

C. Voelkmann, M. Reichelt, T. Meier, S. W. Koch, and U. Hofer, “Five-wave mixing spectroscopy of ultrafast electron dynamics at a Si(001) surface,” Phys. Rev. Lett. 92, 127405 (2004).
[CrossRef] [PubMed]

Rubio, A.

I. Campillo, J. M. Pitarke, A. Rubio, and P. M. Echenique, “Role of occupied d states in the relaxation of hot electrons in Au,” Phys. Rev. B 62, 1500-1503 (2000).
[CrossRef]

Samson, Z. L.

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

Scherer, N. F.

Y.-H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, “Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry,” J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Shen, Y. R.

J. A. McGuire, M. B. Raschke, and Y. R. Shen, “Electron dynamics of silicon surface states: second-harmonic hole-burning on Si(111)-(7×7),” Phys. Rev. Lett. 96, 087401 (2006).
[CrossRef] [PubMed]

Shumay, I. L.

M. Mauerer, I. L. Shumay, W. Berthold, and U. Hofer, “Ultrafast carrier dynamics in Si(111)7×7 dangling bonds probed by time-resolved second-harmonic generation and two-photon photoemission,” Phys. Rev. B 73, 245305 (2006).
[CrossRef]

Silberberg, Y.

Simon, M.

M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
[CrossRef]

Smith, J. B.

J. B. Smith and H. Ehrenreich, “Frequency dependence of the optical relaxation time in metals,” Phys. Rev. B 25, 923-930 (1982).
[CrossRef]

Sonnichsen, C.

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

Stockman, M. I.

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

Tamarat, P.

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5, 515-518 (2005).
[CrossRef] [PubMed]

Teng, H.

J. Dai, H. Teng, and C. Guo, “Second- and third-order interferometric autocorrelations based on harmonic generations from metal surfaces,” Opt. Commun. 252, 173-178 (2005).
[CrossRef]

Trager, F.

M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
[CrossRef]

Tritschler, T.

M. W. Klein, T. Tritschler, and M. Wegener, and S. Linden, “Lineshape of harmonic generation by metallic nanoparticles and metallic photonic crystal slabs,” Phys. Rev. B 72, 115113 (2005).
[CrossRef]

Unterreiner, A. N.

Y.-H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, “Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry,” J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

Voelkmann, C.

C. Voelkmann, M. Reichelt, T. Meier, S. W. Koch, and U. Hofer, “Five-wave mixing spectroscopy of ultrafast electron dynamics at a Si(001) surface,” Phys. Rev. Lett. 92, 127405 (2004).
[CrossRef] [PubMed]

Voll, S.

M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
[CrossRef]

von Plessen, G.

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

Walmsley, I. A.

Wegener, M.

M. W. Klein, T. Tritschler, and M. Wegener, and S. Linden, “Lineshape of harmonic generation by metallic nanoparticles and metallic photonic crystal slabs,” Phys. Rev. B 72, 115113 (2005).
[CrossRef]

Weida, M. J.

Wilk, T.

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

Wilson, O.

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

Witting, T.

Zheludev, N. I.

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

Appl. Phys. B (3)

N. A. Papadogiannis, S. D. Moustaizis, P. A. Loukakos, and C. Kalpouzos, “Temporal characterization of ultra short laser pulses based on multiple harmonic generation on a gold surface,” Appl. Phys. B 65, 339-345 (1997).
[CrossRef]

A. T. Georges and N. E. Karatzas, “Theory of multiple harmonic generation in reflection from a metal surface,” Appl. Phys. B 81, 479-485 (2005).
[CrossRef]

For typos incurred during the production stage (iNωm printed as iNω or as ImNω), see Appl. Phys. B 81, 725-726 (2005).
[CrossRef]

Chem. Phys. Lett. (1)

M. Simon, F. Trager, A. Assion, B. Lang, S. Voll, and G. Gerber, “Femtosecond time-resolved second-harmonic generation at the surface of alkali metal clusters,” Chem. Phys. Lett. 296, 579-584 (1998).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. Chem. B (1)

Y.-H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, “Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry,” J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

J. Phys. Condens. Matter (1)

U. Bovensiepen, “Coherent and incoherent excitation of the Gd(001) surface on ultrafast timescales,” J. Phys. Condens. Matter 19, 083201 (2007).
[CrossRef]

Laser Photonics Rev. (1)

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photonics Rev. 2, 136-159 (2008).
[CrossRef]

Nano Lett. (1)

S. Berciaud, L. Cognet, P. Tamarat, and B. Lounis, “Observation of intrinsic size effects in the optical response of individual gold nanoparticles,” Nano Lett. 5, 515-518 (2005).
[CrossRef] [PubMed]

Nat. Photonics (1)

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

Opt. Commun. (2)

J. Dai, H. Teng, and C. Guo, “Second- and third-order interferometric autocorrelations based on harmonic generations from metal surfaces,” Opt. Commun. 252, 173-178 (2005).
[CrossRef]

N. E. Karatzas and A. T. Georges, “Effects of electron relaxation on multiple harmonic generation from metal surfaces with femtosecond laser pulse,” Opt. Commun. 267, 498-504 (2006).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (7)

I. Campillo, J. M. Pitarke, A. Rubio, and P. M. Echenique, “Role of occupied d states in the relaxation of hot electrons in Au,” Phys. Rev. B 62, 1500-1503 (2000).
[CrossRef]

M. Mauerer, I. L. Shumay, W. Berthold, and U. Hofer, “Ultrafast carrier dynamics in Si(111)7×7 dangling bonds probed by time-resolved second-harmonic generation and two-photon photoemission,” Phys. Rev. B 73, 245305 (2006).
[CrossRef]

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

J. B. Smith and H. Ehrenreich, “Frequency dependence of the optical relaxation time in metals,” Phys. Rev. B 25, 923-930 (1982).
[CrossRef]

W. Hubner, K. H. Bennemann, and K. Bohmer, “Theory for the nonlinear optical response of transition metals: polarization dependence as a fingerprint of the electronic structure at surfaces and interfaces,” Phys. Rev. B 50, 17597-17605 (1994).
[CrossRef]

M. W. Klein, T. Tritschler, and M. Wegener, and S. Linden, “Lineshape of harmonic generation by metallic nanoparticles and metallic photonic crystal slabs,” Phys. Rev. B 72, 115113 (2005).
[CrossRef]

A. T. Georges and N. E. Karatzas, “Modeling of ultrafast interferometric three-photon photoemission from a metal surface irradiated with sub-10-fs laser pulses,” Phys. Rev. B 77, 085436 (2008).
[CrossRef]

Phys. Rev. Lett. (5)

C. Sonnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

J. A. McGuire, M. B. Raschke, and Y. R. Shen, “Electron dynamics of silicon surface states: second-harmonic hole-burning on Si(111)-(7×7),” Phys. Rev. Lett. 96, 087401 (2006).
[CrossRef] [PubMed]

S. Ogawa, H. Nagano, H. Petek, and A. P. Heberle, “Optical dephasing in Cu(111) measured by interferometric two-photon time-resolved photoemission,” Phys. Rev. Lett. 78, 1339-1342 (1997).
[CrossRef]

C. Voelkmann, M. Reichelt, T. Meier, S. W. Koch, and U. Hofer, “Five-wave mixing spectroscopy of ultrafast electron dynamics at a Si(001) surface,” Phys. Rev. Lett. 92, 127405 (2004).
[CrossRef] [PubMed]

B. Lamprecht, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Resonant and offresonant light-driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation,” Phys. Rev. Lett. 83, 4421-4424 (1999).
[CrossRef]

Prog. Surf. Sci. (1)

F. Hubenthal, “Ultrafast dephasing time of localized surface plasmon polariton resonance and the involved damping mechanisms in colloidal gold nanoparticles,” Prog. Surf. Sci. 82, 378-387 (2007).
[CrossRef]

Other (3)

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

See, for example, G. Baym, Lectures on Quantum Mechanics (Benjamin, 1973).

R. W. Boyd, Nonlinear Optics (Academic, 2003).

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Figures (5)

Fig. 1
Fig. 1

Plot of the model surface potential V ( z ) (black curve) normalized to its bulk value. The relevant energy levels for SHG and THG are shown in black. The four channels correspond to (a) stepwise resonant, (b) nonresonant, (c) one-photon resonant, and (d) two-photon resonant harmonic generation. The arrows show stepwise resonant (solid curve) and nonresonant (dashed curve) laser photon excitation. The dashed–dotted arrows show SHG, while the downward dashed arrows show THG.

Fig. 2
Fig. 2

Plot of the normalized second- (a) and third-harmonic (b) interferometric energy fluence as a function of the time delay between two 18 fs laser subpulses of 10 10 W cm 2 combined peak intensity. The light gray curve is for the normalized total harmonic fluence, while the dark gray is for the normalized fluence from the nonresonant channel (b), which gives the ideal autocorrelations of the laser pulse. Panels (c) and (d) show the total intensities (light gray curve) of the harmonics versus time, in the case of zero time delay between the two laser subpulses. The curves labeled σ 00 (dark gray), σ 01 (long-dashed curve), σ 02 (dashed–dotted curve), and σ 03 (dashed–double-dotted curve) give the partial intensities, as per Eqs. (9, 10), of the nonresonant, one-, two-, and three-photon resonant channels, respectively.

Fig. 3
Fig. 3

Same as Fig. 2, but for a 10 fs laser pulse width. In (b) the additional black dotted curve corresponds to the normalized photocharge density from 3PPE. As in Fig. 2, panels (c) and (d) show the total intensities (light gray curve) of the harmonics versus time, in the case of zero time delay between the two laser subpulses. The curves labeled σ 00 (dark gray), σ 01 (long-dashed curve), σ 02 (dashed–dotted curve), and σ 03 (dashed–double-dotted curve) give the partial intensities, as per Eqs. (9, 10), of the nonresonant, one-, two-, and three-photon resonant channels, respectively.

Fig. 4
Fig. 4

Same as Fig. 3, but for a 2.65 fs laser pulse width.

Fig. 5
Fig. 5

Same as Fig. 3, but for a 30 fs laser pulse width.

Equations (30)

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J ̂ P = t P ̂ = e 2 m ( p ̂ ρ ̂ + ρ ̂ p ̂ ) e 2 m A ρ ̂ ,
P 2 ω ( t ) = σ 02 ( 2 ) μ 20 e 2 f ( z ) i 2 ω c m A ( t ) σ 01 ( 1 ) + l 2 σ 0 l ( 2 ) μ l 0 e 2 f ( z ) i 2 ω c m A ( t ) k 1 σ 0 k ( 1 ) ,
P 3 ω ( t ) = σ 03 ( 3 ) μ 30 e 2 f ( z ) i 3 ω c m A ( t ) σ 02 ( 2 ) + m 3 σ 0 m ( 3 ) μ m 0 e 2 f ( z ) i 3 ω c m A ( t ) l 2 σ 0 l ( 2 ) ,
σ 0 k ( 1 ) = 1 2 σ 00 Ω 0 k ( 1 ) ( ω c ω k 0 ) ,
σ 0 l ( 2 ) = 1 2 σ 00 Ω 0 l ( 2 ) + σ 01 ( 1 ) Ω 1 l ( 1 ) ( 2 ω c ω l 0 ) ,
σ 0 m ( 3 ) = 1 2 σ 00 Ω 0 m ( 3 ) + σ 01 ( 1 ) Ω 1 m ( 2 ) + σ 02 ( 2 ) Ω 2 m ( 1 ) ( 3 ω c ω m 0 ) .
Ω 0 l ( 2 ) = 1 2 k 1 Ω 0 k ( 1 ) Ω k l ( 1 ) ( ω c ω k 0 ) + V 0 l ( 2 ) ,
Ω 0 m ( 3 ) = 1 4 k 1 l 2 Ω 0 k ( 1 ) Ω k l ( 1 ) Ω l m ( 1 ) ( 2 ω c ω l 0 ) ( ω c ω k 0 ) 1 2 k 1 Ω 0 k ( 1 ) V k m ( 2 ) ( ω c ω k 0 ) 1 2 l 2 V 0 l ( 2 ) Ω l m ( 1 ) ( 2 ω c ω l 0 ) .
P 2 ω ( t ) = σ 02 ( 2 ) ( t ) μ 20 + σ 00 ( t ) α nr ( 2 ) E 2 ( t ) + σ 01 ( 1 ) ( t ) ξ 10 ( 1 ) E ( t )
P 3 ω ( t ) = σ 03 ( 3 ) ( t ) μ 30 + σ 00 ( t ) α nr ( 3 ) E 3 ( t ) + σ 01 ( 1 ) ( t ) ξ 10 ( 2 ) E 2 ( t ) + σ 02 ( 2 ) ( t ) ζ 20 ( 1 ) E ( t ) ,
α nr ( 2 ) E 2 ( t ) = 1 2 l 2 Ω 0 l ( 2 ) μ l 0 ( 2 ω c ω l 0 ) + e 2 f ( z ) i 4 ω c m A ( t ) k 1 Ω 0 k ( 1 ) ( ω c ω k 0 ) ,
α nr ( 3 ) E 3 ( t ) = 1 2 m 3 Ω 0 m ( 3 ) μ m 0 ( 3 ω c ω m 0 ) + e 2 f ( z ) i 6 ω c m A ( t ) l 2 Ω 0 l ( 2 ) ( 2 ω c ω l 0 ) ,
ξ 10 ( 1 ) E ( t ) = 1 2 l 2 Ω 1 l ( 1 ) μ l 0 ( 2 ω c ω l 0 ) e 2 f ( z ) i 2 ω c m A ( t ) ,
ξ 10 ( 2 ) E 2 ( t ) = 1 2 m 3 Ω 1 m ( 2 ) μ m 0 ( 3 ω c ω m 0 ) + e 2 f ( z ) i 6 ω c m A ( t ) l 2 Ω 1 l ( 1 ) ( 2 ω c ω l 0 ) ,
ζ 20 ( 1 ) E ( t ) = 1 2 m 3 Ω 2 m ( 1 ) μ m 0 ( 3 ω c ω m 0 ) e 2 f ( z ) i 3 ω c m A ( t ) ,
α nr ( 2 ) E 2 ( t ) = 1 4 k 1 l 2 [ Ω 0 k ( 1 ) Ω k l ( 1 ) μ l 0 ( 2 ω c ω l 0 ) ( ω c ω k 0 ) + Ω 0 k ( 1 ) μ k l Ω l 0 ( 1 ) ( ω c ω l 0 ) ( ω c ω k 0 ) + μ 0 k Ω k l ( 1 ) Ω l 0 ( 1 ) ( ω c + ω l 0 ) ( 2 ω c + ω k 0 ) ] 1 2 l 2 [ V 0 l ( 2 ) μ l 0 ( 2 ω c ω l 0 ) + μ 0 l V l 0 ( 2 ) ( 2 ω c ω l 0 ) ] + e 2 f ( z ) i 4 ω c m A ( t ) k 1 [ Ω 0 k ( 1 ) ( ω c ω k 0 ) + Ω k 0 ( 1 ) ( ω c ω k 0 ) ] .
d d t σ 00 = Im [ Ω 01 ( 1 ) * σ 01 ( 1 ) ] + Im [ Ω 02 ( 2 ) * σ 02 ( 2 ) ] + Im [ Ω 03 ( 3 ) * σ 03 ( 3 ) ] + γ 1 nr σ 11 + γ 2 nr σ 22 + γ 3 nr σ 33 γ 0 ee σ 00 Re [ γ 01 ee σ 01 ( 1 ) ] Re [ γ 02 ee σ 02 ( 2 ) ] ,
[ d d t + 1 2 Γ 01 ] σ 01 ( 1 ) = i 2 ( σ 00 σ 11 ) Ω 01 ( 1 ) i 2 σ 02 ( 2 ) Ω 12 ( 1 ) * i 2 σ 03 ( 3 ) Ω 13 ( 2 ) * ,
[ d d t + γ 1 ] σ 11 = Im [ Ω 01 ( 1 ) * σ 01 ( 1 ) ] + Im [ Ω 12 ( 1 ) * σ 12 ( 1 ) ] + Im [ Ω 13 ( 2 ) * σ 13 ( 2 ) ] ,
[ d d t + 1 2 Γ 02 ] σ 02 ( 2 ) = i 2 ( σ 00 σ 22 ) Ω 02 ( 2 ) i 2 σ 01 ( 1 ) Ω 12 ( 1 ) + i 2 Ω 01 ( 1 ) σ 12 ( 1 ) i 2 σ 03 ( 3 ) Ω 23 ( 1 ) * ,
[ d d t + 1 2 Γ 12 ] σ 12 ( 1 ) = i 2 ( σ 11 σ 22 ) Ω 12 ( 1 ) i 2 σ 13 ( 2 ) Ω 23 ( 1 ) * + i 2 Ω 01 ( 1 ) * σ 02 ( 2 ) ,
[ d d t + γ 2 ] σ 22 = Im [ Ω 12 ( 1 ) * σ 12 ( 1 ) ] + Im [ Ω 23 ( 1 ) * σ 23 ( 1 ) ] Im [ Ω 02 ( 2 ) * σ 02 ( 2 ) ] ,
[ d d t + 1 2 Γ 03 ] σ 03 ( 3 ) = i 2 ( σ 00 σ 33 ) Ω 03 ( 3 ) i 2 σ 02 ( 2 ) Ω 23 ( 1 ) + i 2 Ω 01 ( 1 ) σ 13 ( 2 ) i 2 σ 01 ( 1 ) Ω 13 ( 2 ) + i 2 Ω 02 ( 2 ) σ 23 ( 1 ) ,
[ d d t + 1 2 Γ 13 ] σ 13 ( 2 ) = i 2 ( σ 11 σ 33 ) Ω 13 ( 2 ) i 2 σ 12 ( 1 ) Ω 23 ( 1 ) + i 2 Ω 12 ( 1 ) σ 23 ( 1 ) + i 2 Ω 01 ( 1 ) * σ 03 ( 3 ) ,
[ d d t + 1 2 Γ 23 ] σ 23 ( 1 ) = i 2 ( σ 22 σ 33 ) Ω 23 ( 1 ) + i 2 Ω 12 ( 1 ) * σ 13 ( 2 ) + i 2 Ω 02 ( 2 ) * σ 03 ( 3 ) ,
[ d d t + γ 3 ] σ 33 = Im [ Ω 23 ( 1 ) * σ 23 ( 1 ) ] Im [ Ω 13 ( 2 ) * σ 13 ( 2 ) ] Im [ Ω 03 ( 3 ) * σ 03 ( 3 ) ] ,
P N ω ( t ) = δ s 1 4 π 3 k i 2 d k i d Ω k i P FD ( k i ) [ 1 P FD ( k i ) ] [ P N ω ( t ) ] k i , Ω i ,
I N ω ( t ) = 2 c ϵ 0 ( N ω c ) 2 | F | 2 ( c ϵ 0 ) 2 | P N ω ( t ) | 2 ,
E N ω ( τ ) = I N ω ( t , τ ) d t .
g ( N ) ( τ ) = E N ω ( τ ) E N ω ( ) .

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