Abstract

We propose a simple scheme to characterize attosecond extreme ultraviolet (XUV) pulses. A broadband ultraviolet (UV) ~ vacuum ultraviolet (VUV) pump pulse creates a coherent superposition of atomic bound states, from which photoionization takes place by the time-delayed attosecond XUV probe pulse. Information on the spectral phase of the XUV pulse can be extracted from the phase offset of the interference beating in the photoelectron spectra using a standard SPIDER (spectral phase interferometry for direct electric-field reconstruction) algorithm. We further discuss the influence of the chirp and polychromaticity of the pump pulse, and show that they do not spoil the reconstruction process. Since our scheme is applicable for various simple atoms such as H, He, and Cs, etc., and capable of characterizing attosecond XUV pulses with a pulse duration of a few hundred attoseconds or even less, it can be an alternative technique to characterize attosecond XUV pulses. Specific numerical examples are presented for the H atom utilizing the 2p and 3p states.

©2010 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  3. R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
    [Crossref]
  4. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794(1998).
    [Crossref]
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    [Crossref]
  6. J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
    [Crossref] [PubMed]
  7. T. Sekikawa, T. Katsura, S. Miura, and S. Watanabe, “Measurement of the Intensity-Dependent Atomic Dipole Phase of a High Harmonic by Frequency-Resolved Optical Gating,” Phys. Rev. Lett. 88, 193902(2002).
    [Crossref] [PubMed]
  8. Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71, 011401(R)(2005).
    [Crossref]
  9. F. Quéré, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt.,  52, 339–360(2005).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  14. Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  20. M. E. Anderson, L. E. E. de Araujo, E.M. Kosik, and I.A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER”, Appl. Phys. B, 70[Suppl.], S85–S93(2000).
  21. N. Rahman “The degenerate two-level system and high-order harmonic generation,” Chem. Phys. Lett. 270, 189–192 (1997).
    [Crossref]
  22. Y. Kobayashi, T. Ohno, T. Sekikawa, Y. Nabekawa, and S. Watanabe, “Pulse width measurement of high-order harmonics by autocorrelation,” Appl. Phys. B 70, 389–394 (2000).
    [Crossref]
  23. S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
    [Crossref] [PubMed]
  24. T. Nakajima, “Effects of the carrier-envelope phase on atomic ionization by the pulse train in the multiphoton ionization regime,” Phys. Rev. A 79, 043414 (2009).
    [Crossref]

2009 (3)

J. Chen and T. Nakajima, “Characterization of Attosecond XUV Pulses from Photoelectron Spectra of Atoms,” Laser Phys. 19, 1586–1590(2009).
[Crossref]

F. Krausz and M. Ivanov,“Attosecond physics,” Rev. Mod. Phys. 81, 163–234(2009).
[Crossref]

T. Nakajima, “Effects of the carrier-envelope phase on atomic ionization by the pulse train in the multiphoton ionization regime,” Phys. Rev. A 79, 043414 (2009).
[Crossref]

2008 (1)

2007 (1)

R. Itakura, “Spectral phase measurement of attosecond pulses using the quantum beat between the P1/2 and P3/2 levels of alkali-metal atoms,” Phys. Rev. A 76, 033810(2007).
[Crossref]

2006 (1)

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

2005 (4)

Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71, 011401(R)(2005).
[Crossref]

F. Quéré, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt.,  52, 339–360(2005).
[Crossref]

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. DiMauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94, 033905(2005).
[Crossref] [PubMed]

E. Cormier, L. Corner, E. M. Kosik, I. A. Walmsley, and A. S. Wyatt, “Spectral Phase Interferometry for Complete Reconstruction of Attosecond Pulses,” Laser Phys.,  15, 1–7(2005).

2004 (1)

P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67, 813–855(2004).
[Crossref]

2002 (4)

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

T. Sekikawa, T. Katsura, S. Miura, and S. Watanabe, “Measurement of the Intensity-Dependent Atomic Dipole Phase of a High Harmonic by Frequency-Resolved Optical Gating,” Phys. Rev. Lett. 88, 193902(2002).
[Crossref] [PubMed]

H. G. Muller, “Reconstruction of attosecond harmonic beating by interference of two-photon transitions,” Appl. Phys. B 74, S17–S21(2002).
[Crossref]

C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B 19, 1019–1029 (2002).
[Crossref]

2001 (2)

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

2000 (3)

M. E. Anderson, L. E. E. de Araujo, E.M. Kosik, and I.A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER”, Appl. Phys. B, 70[Suppl.], S85–S93(2000).

Y. Kobayashi, T. Ohno, T. Sekikawa, Y. Nabekawa, and S. Watanabe, “Pulse width measurement of high-order harmonics by autocorrelation,” Appl. Phys. B 70, 389–394 (2000).
[Crossref]

T. Nakajima, “Possibility of direct determination of the quantum phase of continua utilizing the phase of lasers,” Phys. Rev. A 61, 041403(R)(2000).
[Crossref]

1999 (1)

C. Iaconis and I. A. Walmsley, “Self-Referencing Spectral Interferometry for Measuring Ultrashort Optical Pulses,” IEEE J. Quantum Electron. 35, 501–509(1999).
[Crossref]

1998 (1)

1997 (2)

N. Rahman “The degenerate two-level system and high-order harmonic generation,” Chem. Phys. Lett. 270, 189–192 (1997).
[Crossref]

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Agostini, P.

P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67, 813–855(2004).
[Crossref]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

Anderson, M. E.

M. E. Anderson, L. E. E. de Araujo, E.M. Kosik, and I.A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER”, Appl. Phys. B, 70[Suppl.], S85–S93(2000).

Araujo, L. E. E. de

M. E. Anderson, L. E. E. de Araujo, E.M. Kosik, and I.A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER”, Appl. Phys. B, 70[Suppl.], S85–S93(2000).

Augé, F.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

Balcou, Ph.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

Beauvoir, B. de

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

Blanchet, V.

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

Bouchene, M. A.

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

Breger, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

Buil, S.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Chen, J.

J. Chen and T. Nakajima, “Characterization of Attosecond XUV Pulses from Photoelectron Spectra of Atoms,” Laser Phys. 19, 1586–1590(2009).
[Crossref]

Cormier, E.

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. DiMauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94, 033905(2005).
[Crossref] [PubMed]

E. Cormier, L. Corner, E. M. Kosik, I. A. Walmsley, and A. S. Wyatt, “Spectral Phase Interferometry for Complete Reconstruction of Attosecond Pulses,” Laser Phys.,  15, 1–7(2005).

Corner, L.

E. Cormier, L. Corner, E. M. Kosik, I. A. Walmsley, and A. S. Wyatt, “Spectral Phase Interferometry for Complete Reconstruction of Attosecond Pulses,” Laser Phys.,  15, 1–7(2005).

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. DiMauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94, 033905(2005).
[Crossref] [PubMed]

Degert, J.

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

Delong, K. W.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

DiMauro, L. F.

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. DiMauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94, 033905(2005).
[Crossref] [PubMed]

P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67, 813–855(2004).
[Crossref]

Dorrer, C.

Dühr, O.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Fittinghoff, D. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Furusawa, K.

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

Gaarde, M. B.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Girard, B.

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

Haraguchi, E.

Hasegawa, H.

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

Iaconis, C.

C. Iaconis and I. A. Walmsley, “Self-Referencing Spectral Interferometry for Measuring Ultrashort Optical Pulses,” IEEE J. Quantum Electron. 35, 501–509(1999).
[Crossref]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794(1998).
[Crossref]

Itakura, R.

R. Itakura, “Spectral phase measurement of attosecond pulses using the quantum beat between the P1/2 and P3/2 levels of alkali-metal atoms,” Phys. Rev. A 76, 033810(2007).
[Crossref]

Itatani, J.

F. Quéré, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt.,  52, 339–360(2005).
[Crossref]

Ivanov, M.

F. Krausz and M. Ivanov,“Attosecond physics,” Rev. Mod. Phys. 81, 163–234(2009).
[Crossref]

Johansson, A.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Kane, D. J.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Katsura, T.

T. Sekikawa, T. Katsura, S. Miura, and S. Watanabe, “Measurement of the Intensity-Dependent Atomic Dipole Phase of a High Harmonic by Frequency-Resolved Optical Gating,” Phys. Rev. Lett. 88, 193902(2002).
[Crossref] [PubMed]

Keller, U.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Kobayashi, Y.

Y. Kobayashi, T. Ohno, T. Sekikawa, Y. Nabekawa, and S. Watanabe, “Pulse width measurement of high-order harmonics by autocorrelation,” Appl. Phys. B 70, 389–394 (2000).
[Crossref]

Kosik, E. M.

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. DiMauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94, 033905(2005).
[Crossref] [PubMed]

E. Cormier, L. Corner, E. M. Kosik, I. A. Walmsley, and A. S. Wyatt, “Spectral Phase Interferometry for Complete Reconstruction of Attosecond Pulses,” Laser Phys.,  15, 1–7(2005).

Kosik, E.M.

M. E. Anderson, L. E. E. de Araujo, E.M. Kosik, and I.A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER”, Appl. Phys. B, 70[Suppl.], S85–S93(2000).

Krausz, F.

F. Krausz and M. Ivanov,“Attosecond physics,” Rev. Mod. Phys. 81, 163–234(2009).
[Crossref]

Krumbügel, M. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

LHuillier, A.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Mairesse, Y.

F. Quéré, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt.,  52, 339–360(2005).
[Crossref]

Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71, 011401(R)(2005).
[Crossref]

Mauritsson, J.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Midorikawa, K.

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

Miura, S.

T. Sekikawa, T. Katsura, S. Miura, and S. Watanabe, “Measurement of the Intensity-Dependent Atomic Dipole Phase of a High Harmonic by Frequency-Resolved Optical Gating,” Phys. Rev. Lett. 88, 193902(2002).
[Crossref] [PubMed]

Muller, H. G.

H. G. Muller, “Reconstruction of attosecond harmonic beating by interference of two-photon transitions,” Appl. Phys. B 74, S17–S21(2002).
[Crossref]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

Mullot, G.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

Nabekawa, Y.

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

Y. Kobayashi, T. Ohno, T. Sekikawa, Y. Nabekawa, and S. Watanabe, “Pulse width measurement of high-order harmonics by autocorrelation,” Appl. Phys. B 70, 389–394 (2000).
[Crossref]

Nakajima, T.

J. Chen and T. Nakajima, “Characterization of Attosecond XUV Pulses from Photoelectron Spectra of Atoms,” Laser Phys. 19, 1586–1590(2009).
[Crossref]

T. Nakajima, “Effects of the carrier-envelope phase on atomic ionization by the pulse train in the multiphoton ionization regime,” Phys. Rev. A 79, 043414 (2009).
[Crossref]

T. Sekikawa, T. Okamoto, E. Haraguchi, M. Yamashita, and T. Nakajima, “Two-photon resonant excitation of a doubly excited state in He atoms by high-harmonic pulses,” Opt. Express 16, 21922–21929 (2008).
[Crossref] [PubMed]

T. Nakajima, “Possibility of direct determination of the quantum phase of continua utilizing the phase of lasers,” Phys. Rev. A 61, 041403(R)(2000).
[Crossref]

Norin, J.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Ohno, T.

Y. Kobayashi, T. Ohno, T. Sekikawa, Y. Nabekawa, and S. Watanabe, “Pulse width measurement of high-order harmonics by autocorrelation,” Appl. Phys. B 70, 389–394 (2000).
[Crossref]

Okamoto, T.

Okino, T.

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

Paul, P. M.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

Persson, A.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Quéré, F.

Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71, 011401(R)(2005).
[Crossref]

F. Quéré, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt.,  52, 339–360(2005).
[Crossref]

Raarup, M. K.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Rahman, N.

N. Rahman “The degenerate two-level system and high-order harmonic generation,” Chem. Phys. Lett. 270, 189–192 (1997).
[Crossref]

Richman, B. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Schafer, K. J.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Sekikawa, T.

T. Sekikawa, T. Okamoto, E. Haraguchi, M. Yamashita, and T. Nakajima, “Two-photon resonant excitation of a doubly excited state in He atoms by high-harmonic pulses,” Opt. Express 16, 21922–21929 (2008).
[Crossref] [PubMed]

T. Sekikawa, T. Katsura, S. Miura, and S. Watanabe, “Measurement of the Intensity-Dependent Atomic Dipole Phase of a High Harmonic by Frequency-Resolved Optical Gating,” Phys. Rev. Lett. 88, 193902(2002).
[Crossref] [PubMed]

Y. Kobayashi, T. Ohno, T. Sekikawa, Y. Nabekawa, and S. Watanabe, “Pulse width measurement of high-order harmonics by autocorrelation,” Appl. Phys. B 70, 389–394 (2000).
[Crossref]

Shimizu, T.

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

Stock, S.

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

Sweetser, J. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Toma, E. S.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

Trebino, R.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Wahlström, C.-G.

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

Walmsley, I. A.

E. Cormier, L. Corner, E. M. Kosik, I. A. Walmsley, and A. S. Wyatt, “Spectral Phase Interferometry for Complete Reconstruction of Attosecond Pulses,” Laser Phys.,  15, 1–7(2005).

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. DiMauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94, 033905(2005).
[Crossref] [PubMed]

C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction,” J. Opt. Soc. Am. B 19, 1019–1029 (2002).
[Crossref]

C. Iaconis and I. A. Walmsley, “Self-Referencing Spectral Interferometry for Measuring Ultrashort Optical Pulses,” IEEE J. Quantum Electron. 35, 501–509(1999).
[Crossref]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794(1998).
[Crossref]

Walmsley, I.A.

M. E. Anderson, L. E. E. de Araujo, E.M. Kosik, and I.A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER”, Appl. Phys. B, 70[Suppl.], S85–S93(2000).

Watanabe, S.

T. Sekikawa, T. Katsura, S. Miura, and S. Watanabe, “Measurement of the Intensity-Dependent Atomic Dipole Phase of a High Harmonic by Frequency-Resolved Optical Gating,” Phys. Rev. Lett. 88, 193902(2002).
[Crossref] [PubMed]

Y. Kobayashi, T. Ohno, T. Sekikawa, Y. Nabekawa, and S. Watanabe, “Pulse width measurement of high-order harmonics by autocorrelation,” Appl. Phys. B 70, 389–394 (2000).
[Crossref]

Wyatt, A. S.

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. DiMauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94, 033905(2005).
[Crossref] [PubMed]

E. Cormier, L. Corner, E. M. Kosik, I. A. Walmsley, and A. S. Wyatt, “Spectral Phase Interferometry for Complete Reconstruction of Attosecond Pulses,” Laser Phys.,  15, 1–7(2005).

Yamanouchi, K.

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

Yamashita, M.

Zamith, S.

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

Appl. Phys. B (2)

H. G. Muller, “Reconstruction of attosecond harmonic beating by interference of two-photon transitions,” Appl. Phys. B 74, S17–S21(2002).
[Crossref]

Y. Kobayashi, T. Ohno, T. Sekikawa, Y. Nabekawa, and S. Watanabe, “Pulse width measurement of high-order harmonics by autocorrelation,” Appl. Phys. B 70, 389–394 (2000).
[Crossref]

Appl. Phys. B, (1)

M. E. Anderson, L. E. E. de Araujo, E.M. Kosik, and I.A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using SPIDER”, Appl. Phys. B, 70[Suppl.], S85–S93(2000).

Chem. Phys. Lett. (1)

N. Rahman “The degenerate two-level system and high-order harmonic generation,” Chem. Phys. Lett. 270, 189–192 (1997).
[Crossref]

IEEE J. Quantum Electron. (1)

C. Iaconis and I. A. Walmsley, “Self-Referencing Spectral Interferometry for Measuring Ultrashort Optical Pulses,” IEEE J. Quantum Electron. 35, 501–509(1999).
[Crossref]

J. Mod. Opt. (1)

F. Quéré, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt.,  52, 339–360(2005).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Phys. (2)

E. Cormier, L. Corner, E. M. Kosik, I. A. Walmsley, and A. S. Wyatt, “Spectral Phase Interferometry for Complete Reconstruction of Attosecond Pulses,” Laser Phys.,  15, 1–7(2005).

J. Chen and T. Nakajima, “Characterization of Attosecond XUV Pulses from Photoelectron Spectra of Atoms,” Laser Phys. 19, 1586–1590(2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (4)

T. Nakajima, “Possibility of direct determination of the quantum phase of continua utilizing the phase of lasers,” Phys. Rev. A 61, 041403(R)(2000).
[Crossref]

Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71, 011401(R)(2005).
[Crossref]

R. Itakura, “Spectral phase measurement of attosecond pulses using the quantum beat between the P1/2 and P3/2 levels of alkali-metal atoms,” Phys. Rev. A 76, 033810(2007).
[Crossref]

T. Nakajima, “Effects of the carrier-envelope phase on atomic ionization by the pulse train in the multiphoton ionization regime,” Phys. Rev. A 79, 043414 (2009).
[Crossref]

Phys. Rev. Lett. (5)

S. Zamith, J. Degert, S. Stock, B. de Beauvoir, V. Blanchet, M. A. Bouchene, and B. Girard,“Observation of Coherent Transients in Ultrashort Chirped Excitation of an Undamped Two-Level System,” Phys. Rev. Lett. 87, 033001(2001).
[Crossref] [PubMed]

Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Yamanouchi, and K. Midorikawa, “Inter-ferometric autocorrelation of an attosecond pulse train in the single-cycle regime,” Phys. Rev. Lett. 97, 153904 (2006).
[Crossref] [PubMed]

E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. DiMauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94, 033905(2005).
[Crossref] [PubMed]

J. Norin, J. Mauritsson, A. Johansson, M. K. Raarup, S. Buil, A. Persson, O. Dühr, M. B. Gaarde, K. J. Schafer, U. Keller, C.-G. Wahlström, and A. LHuillier, “Time-Frequency Characterization of Femtosecond Extreme Ultraviolet Pulses,” Phys. Rev. Lett. 88, 193901(2002).
[Crossref] [PubMed]

T. Sekikawa, T. Katsura, S. Miura, and S. Watanabe, “Measurement of the Intensity-Dependent Atomic Dipole Phase of a High Harmonic by Frequency-Resolved Optical Gating,” Phys. Rev. Lett. 88, 193902(2002).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67, 813–855(2004).
[Crossref]

Rev. Mod. Phys. (1)

F. Krausz and M. Ivanov,“Attosecond physics,” Rev. Mod. Phys. 81, 163–234(2009).
[Crossref]

Rev. Sci. Instrum. (1)

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Science (1)

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292, 1689–1692(2001).
[Crossref] [PubMed]

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Figures (6)

Fig. 1.
Fig. 1. Level scheme of hydrogen atom. The broadband pump (VUV) pulse with an appropriate central photon energy creates a coherent superposition in states |1〉 and |2〉, from which photoionization takes place by the time-delayed attosecond probe (XUV) pulse. If the pump pulse is very broad, it may also excite the next neighboring state, |3〉. The possible ionization continua, |f〉, are εs and εd where ε represents the energy of photoelectrons.
Fig. 2.
Fig. 2. Representative results of the spectral phase error [-β̃(δ, ω)] contained in the reconstructed spectral phase distribution, [-ϕ detector xuv(ω)], as a function of probe photon energy. The solid blue, dashed green, and dot-dashed red curves represent results for the three different angle widths of δ =5°, 10°, and 20°, respectively.
Fig. 3.
Fig. 3. Probe pulse shapes employed for the error estimation of the reconstructed pulses. The temporal intensity (blue solid line) and temporal phase (green dashed line) are shown for the (a) transform-limited, (b) linearly chirped, and (c) quadratically chirped pulses. In each graph the inset shows the corresponding pulse spectrum (black solid line) and spectral phase (red dashed line). All pulses are assumed to have the identical Gaussian spectrum with a FWHM of 20 eV and the central photon energy of 30 eV.
Fig. 4.
Fig. 4. Comparison of reconstructed probe pulses by noisy beat signals with two different energy shears, (a)–(f) ΔEe = 0.069 eV and (g)–(l) ΔEe = 1.9 eV. The solid and dashed lines in each graph correspond to the input pulse and reconstructed pulse under noise with α = 0.2. The definitions and the physical meanings of εI and εϕ are given in the text. (a) and (g) Temporal intensity for the transform-limited (TL) pulse input. (b) and (h) Temporal phase for the TL pulse input. (c) and (i) Temporal intensity for the linearly chirped pulse input. (d) and (j) Temporal phase for the linearly chirped pulse input. (e) and (k) Temporal intensity for the quadratically chirped pulse input. (f) and (l) Temporal phase for the quadratically chirped pulse input. The central photon energy and the spectral bandwidth of the probe pulse is h¯ω 0 xuv = 30 eV and 20 eV, respectively.
Fig. 5.
Fig. 5. Population of each excited state as a function of time under the weak pumping condition. The pump pulse is a (a) transform-limited pulse (ξ = 0), (b) positively chirped pulse (ξ = 2), and (c) negatively chirped pulse (ξ = -2). The solid lines (blue for |C 1|2 and green for |C 2|2) are the numerical results obtained by numerically solving the time-dependent Schrödinger equation for H atoms, and the dashed lines (red for |C 1|2 and cyan for |C 2|2) are the analytical results obtained by Eq. (24). The pump pulse parameters employed for all the calculations are, I 0 = 1011W/cm2 for the peak intensity, τ TL pump = 2 τ p 1 n 2 = 2 fs for the transform-limited pulse duration (FWHM), and h¯ω 0 pump = 11.4 eV for the central photon energy, respectively.
Fig. 6.
Fig. 6. Reconstruction of the electric field for the case in which the pump pulse coherently excites three states instead of two. (a) Numerical simulation of the photoelectron spectra as functions of delay time and electron energy. Note that the signal contains the contribution of three beats, i.e., 2p-3p, 3p-4p, and 2p-4p beats. (b) Beat signals at the photoelectron energy of ε= 28 eV (blue solid line) and 26 eV (green dashed line). (c) The Fourier transform of the beat signal shown in graph (b) at ε = 28 eV. (d) Applying the frequency filter, only the selected spectral peak for 2p-3p is seen. (e) The Inverse-Fourier transform of the selected 2p-3p peak in the time domain. (f) The reconstructed temporal envelope function (blue solid line) and the temporal phase (green dashed line), with the corresponding spectral intensity (black solid line) and spectral phase (red dashed line) shown in inset.

Equations (40)

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

ψ coh ( t ) = k C k exp [ i ω k t ] k ( k = 1,2 ) ,
f ε l Ω = m l i l exp [ i σ l ( ε ) ] Y l m l * ( Ω ) R ε l l m l ,
M f ε l Ω τ = f ε l Ω E xuv ( ω ) r ψ coh ( τ ) ,
E xuv ( ω ) = I xuv ( ω ) exp [ i ϕ xuv ( ω ) ] ,
S ε Θ τ = dP ε Ω τ d Ω
= l M f ε l Ω τ 2
= l , k , k C k * C k M fk * ε l Θ M fk ε l Θ exp [ i ( ω k ω k ) τ ]
S 11 ε Θ τ + S 22 ε Θ τ + S 21 ε Θ τ ,
S 21 ε Θ τ 𝓡 s 1 𝓡 s 2 C 1 * C 2 I xuv ( ω ) I xuv ( ω ω 21 ) cos [ ω 21 τ + ϕ c Δ ϕ 21 xuv ( ω ) ] ,
ω = ( ε + E IP ) h ¯ ω 1 ,
Δ ϕ 21 xuv ( ω ) = ϕ xuv ( ω ω 21 ) ϕ xuv ( ω ) .
ϕ xuv ( ω ) 1 ω 21 ω min ω ω 21 2 Δ ϕ 21 xuv ( ω i ) Δ ω i .
S 21 ε Θ τ S 21 ε Θ 0 τ + B ( ε , τ ) ( 3 cos 2 Θ 1 ) 2 + A ( ε , τ ) ( 3 cos 2 Θ 1 ) ,
B ( ε , τ ) = 5 4 𝓡 d 1 𝓡 d 2 C 1 * C 2 I xuv ( ω ) I xuv ( ω ω 21 ) cos [ ω 21 τ + ϕ c Δ ϕ 21 xuv ( ω ) ] ,
A ( ε , τ ) = 5 2 C 1 * C 2 I xuv ( ω ) I xuv ( ω ω 21 )
× { ( 𝓡 s 1 𝓡 d 2 + 𝓡 d 1 𝓡 s 2 ) cos [ σ d ( ε ) σ s ( ε ) ] cos [ ω 21 τ + ϕ c Δ ϕ 21 xuv ( ω ) ]
( 𝓡 s 1 𝓡 d 2 𝓡 d 1 𝓡 s 2 ) sin [ σ d ( ε ) σ s ( ε ) ] sin [ ω 21 τ + ϕ c Δ ϕ 21 xuv ( ω ) ] } .
P 21 ( ε , τ ) = Ω S 21 ε Θ τ d Ω
= 2 δ Θ 0 δ Θ 0 + δ S 21 ε Θ τ sin Θ d Θ
𝓡 s 1 𝓡 s 2 C 1 * C 2 I xuv ( ω ) I xuv ( ω ω 21 )
× cos [ ω 21 τ + ϕ c ( Δ ϕ 21 xuv ( ω ) + Δ β ˜ ( δ , ω ) ) ] ,
ϕ dector xuv ( δ , ω ) = ϕ xuv ( ω ) + β ˜ ( δ , ω ) ,
β ˜ δ ω 1 ω 21 ω min ω ω 21 2 Δ β ˜ ( δ , ω i ) Δ ω i .
ϕ xuv ( ω ) = a ( ω ω 0 xuv ) 2 + b ( ω ω 0 xuv ) 3 ,
S noisy xuv ( I xuv ( ω ) I xuv ( ω ω 21 ) + I n ) cos [ ω 21 τ + ϕ c Δ ϕ noisy xuv ( ω ) ] + S n
Δ ϕ noisy xuv ( ω ) = Δ ϕ 21 xuv ( ω ) + α n η ,
ε I = 1 N j = 1 N [ I input ( t j ) I noisy ( t j ) ] 2 ,
ε ϕ = 1 N j = 1 N I input 2 ( t j ) [ ϕ input ( t j ) ϕ noisy ( t j ) ] 2 1 N j = 1 N I input 2 ( t j ) ,
E pump ( t ) = E 0 exp [ i ω 0 pump t t 2 2 τ p 2 ( 1 i ξ ) ] ,
C ˙ 0 ( t ) = i μ 01 2 h ¯ E pump ( t ) C 1 ( t ) exp [ i ω 1 t ] + i μ 02 2 h ¯ E pump ( t ) C 2 ( t ) exp [ i ω 2 t ]
C ˙ 1 ( t ) = i μ 10 2 h ¯ E pump ( t ) C 0 ( t ) exp [ i ω 1 t ]
C ˙ 2 ( t ) = i μ 20 2 h ¯ E pump ( t ) C 0 ( t ) exp [ i ω 2 t ] ,
C k ( t ) = i μ k 0 2 h ¯ t E pump ( t ) exp [ i ω k t ] dt ( k = 1,2 ) .
C k i μ k 0 2 h ¯ E ˜ pump ( ω k ) ( k = 1,2 ) ,
E ˜ pump ( ω k ) = E 0 τ p 1 2 π exp [ τ p 2 ( 1 ) 2 ( ω k ω 0 pump ) 2 ] .
S 21 ε Θ 0 τ 𝓡 s 1 𝓡 s 2 𝓡 10 𝓡 20 I pump ( ω 1 ) I pump ( ω 2 )
× I xuv ( ω ) I xuv ( ω ω 21 ) cos [ ω 21 τ + ϕ c Δ ϕ 21 xuv ( ω ) ] ,
I pump ( ω k ) = 2 π E 0 2 τ p 2 1 + ξ 2 exp [ τ p 2 ( ω k ω 0 pump ) 2 ] .
ϕ c = τ p 2 ξ ω 21 ( ω av ω 0 pump ) ,
r 12 = s 1 10 s 2 20 exp [ τ p 2 ω 21 ( ω av ω 0 pump ) ] I xuv ( ω ) I xuv ( ω ω 21 ) .

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