Abstract

We investigate attosecond streaking measurements, where a spectrogram is described by an ensemble of electron wave packets. Such a description may be required for processes more complex than direct photoemission from an isolated atom; an ensemble of wave packets may also be needed to describe the role of shot-to-shot fluctuations or a non-uniform spatio-temporal profile of attosecond light pulses. Under these conditions, we examine the performance of conventional (FROG) analysis of attosecond streaking measurements.

© 2009 Optical Society of America

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  1. R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
    [CrossRef] [PubMed]
  2. J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
    [CrossRef] [PubMed]
  3. M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
    [CrossRef] [PubMed]
  4. E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
    [CrossRef] [PubMed]
  5. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
    [CrossRef] [PubMed]
  6. V. S. Yakovlev, F. Bammer, and A. Scrinzi, "Attosecond streaking measurements," J. Mod. Opt. 52, 395-410 (2005).
    [CrossRef]
  7. M. Kitzler, N. Milošević, A. Scrinzi, F. Krausz, and T. Brabec, "Quantum Theory Of Attosecond XUV Pulse Measurement By Laser Dressed Photoionization," Phys. Rev. Lett. 88, 173904 (2002).
    [CrossRef] [PubMed]
  8. Y. Mairesse and Quéré, "Frequency-Resolved Optical Gating for Complete Reconstruction of Attosecond Bursts," Phys. Rev. A 71, 011401 (2005).
    [CrossRef]
  9. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
    [CrossRef] [PubMed]
  10. R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "Measuring Ultrashort Laser Pulses In The Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
    [CrossRef]
  11. D. J. Kane, "Recent Progress Toward Real-Time Measurement Of Ultrashort Laser Pulses," IEEE J. Quantum Electron. 35, 421-431 (1999).
    [CrossRef]
  12. A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
    [CrossRef] [PubMed]
  13. M. B. Gaarde and K. J. Schafer, "Generating single attosecond pulses via spatial filtering," Opt. Lett. 31, 3188- 3190 (2006).
    [CrossRef] [PubMed]
  14. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory Of High-Harmonic Generation By Low-Frequency Laser Fields," Phys. Rev. Lett. 49, 2117-2132 (1994).
  15. Z. Zhao and C. D. Lin, "Theory of laser-assisted autoionization by attosecond light pulses," Physical Rev. A, Atomic, molecular, and optical physics 71, 060702 (2005).
    [CrossRef]
  16. J. Gagnon, E. Goulielmakis, and V. S. Yakovlev, "The accurate FROG characterization of attosecond pulses from streaking measurements," Appl. Phys. B, Lasers and optics 92, 25-32 (2008).
    [CrossRef]
  17. M. Chini, H. Wang, S. D. Khan, S. Y. Chen, and Z. H. Chang, "Retrieval of satellite pulses of single isolated attosecond pulses," Appl. Phys. Lett. 94, 161112 (2009).
    [CrossRef]
  18. M. Wickenhauser, J. Burgdörfer, F. Krausz, and M. Drescher, "Time resolved Fano resonances," Phys. Rev. Lett. 94, 023002 (2005).
    [CrossRef] [PubMed]
  19. S. Kazamias and P. Balcou, "Intrinsic chirp of attosecond pulses: Single-atom model versus experiment," Phys. Rev. A 69, 063416 (2004).
    [CrossRef]
  20. P. B. Corkum, "Plasma Perspective On Strong-Field Multiphoton Ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
    [CrossRef] [PubMed]
  21. H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
    [CrossRef] [PubMed]
  22. I. Thomann, A. Bahabad, X. Liu, R. Trebino, M. M. Murnane, and H. C. Kapteyn, "Characterizing isolated attosecond pulses from hollow-core waveguidesusing multi-cycle driving pulses," Opt. Express 17, 4611-4633 (2009).
    [CrossRef] [PubMed]
  23. M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
    [CrossRef]
  24. A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
    [CrossRef] [PubMed]
  25. F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
    [CrossRef] [PubMed]

2009 (2)

M. Chini, H. Wang, S. D. Khan, S. Y. Chen, and Z. H. Chang, "Retrieval of satellite pulses of single isolated attosecond pulses," Appl. Phys. Lett. 94, 161112 (2009).
[CrossRef]

I. Thomann, A. Bahabad, X. Liu, R. Trebino, M. M. Murnane, and H. C. Kapteyn, "Characterizing isolated attosecond pulses from hollow-core waveguidesusing multi-cycle driving pulses," Opt. Express 17, 4611-4633 (2009).
[CrossRef] [PubMed]

2008 (2)

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

2007 (2)

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (4)

M. Wickenhauser, J. Burgdörfer, F. Krausz, and M. Drescher, "Time resolved Fano resonances," Phys. Rev. Lett. 94, 023002 (2005).
[CrossRef] [PubMed]

V. S. Yakovlev, F. Bammer, and A. Scrinzi, "Attosecond streaking measurements," J. Mod. Opt. 52, 395-410 (2005).
[CrossRef]

Y. Mairesse and Quéré, "Frequency-Resolved Optical Gating for Complete Reconstruction of Attosecond Bursts," Phys. Rev. A 71, 011401 (2005).
[CrossRef]

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

2004 (3)

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

S. Kazamias and P. Balcou, "Intrinsic chirp of attosecond pulses: Single-atom model versus experiment," Phys. Rev. A 69, 063416 (2004).
[CrossRef]

2003 (1)

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

2002 (3)

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

M. Kitzler, N. Milošević, A. Scrinzi, F. Krausz, and T. Brabec, "Quantum Theory Of Attosecond XUV Pulse Measurement By Laser Dressed Photoionization," Phys. Rev. Lett. 88, 173904 (2002).
[CrossRef] [PubMed]

2001 (1)

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

1999 (1)

D. J. Kane, "Recent Progress Toward Real-Time Measurement Of Ultrashort Laser Pulses," IEEE J. Quantum Electron. 35, 421-431 (1999).
[CrossRef]

1997 (1)

R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "Measuring Ultrashort Laser Pulses In The Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
[CrossRef]

1994 (1)

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory Of High-Harmonic Generation By Low-Frequency Laser Fields," Phys. Rev. Lett. 49, 2117-2132 (1994).

1993 (1)

P. B. Corkum, "Plasma Perspective On Strong-Field Multiphoton Ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

Aquila, L. A.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

Attwood, D. T.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

Auge, F.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

Bahabad, A.

Balcou, P.

S. Kazamias and P. Balcou, "Intrinsic chirp of attosecond pulses: Single-atom model versus experiment," Phys. Rev. A 69, 063416 (2004).
[CrossRef]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory Of High-Harmonic Generation By Low-Frequency Laser Fields," Phys. Rev. Lett. 49, 2117-2132 (1994).

Baltuška, A.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

Bammer, F.

V. S. Yakovlev, F. Bammer, and A. Scrinzi, "Attosecond streaking measurements," J. Mod. Opt. 52, 395-410 (2005).
[CrossRef]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

Bauer, D.

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

Becker, W.

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

Blumel, L.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

Brabec, T.

M. Kitzler, N. Milošević, A. Scrinzi, F. Krausz, and T. Brabec, "Quantum Theory Of Attosecond XUV Pulse Measurement By Laser Dressed Photoionization," Phys. Rev. Lett. 88, 173904 (2002).
[CrossRef] [PubMed]

Breger, P.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

Burgdörfer, J.

M. Wickenhauser, J. Burgdörfer, F. Krausz, and M. Drescher, "Time resolved Fano resonances," Phys. Rev. Lett. 94, 023002 (2005).
[CrossRef] [PubMed]

Cavalieri, A. L.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

Chang, Z. H.

M. Chini, H. Wang, S. D. Khan, S. Y. Chen, and Z. H. Chang, "Retrieval of satellite pulses of single isolated attosecond pulses," Appl. Phys. Lett. 94, 161112 (2009).
[CrossRef]

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

Chen, S. Y.

M. Chini, H. Wang, S. D. Khan, S. Y. Chen, and Z. H. Chang, "Retrieval of satellite pulses of single isolated attosecond pulses," Appl. Phys. Lett. 94, 161112 (2009).
[CrossRef]

Chini, M.

M. Chini, H. Wang, S. D. Khan, S. Y. Chen, and Z. H. Chang, "Retrieval of satellite pulses of single isolated attosecond pulses," Appl. Phys. Lett. 94, 161112 (2009).
[CrossRef]

Corkum, P. B.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
[CrossRef] [PubMed]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory Of High-Harmonic Generation By Low-Frequency Laser Fields," Phys. Rev. Lett. 49, 2117-2132 (1994).

P. B. Corkum, "Plasma Perspective On Strong-Field Multiphoton Ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

Delong, K.W.

R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "Measuring Ultrashort Laser Pulses In The Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
[CrossRef]

Drescher, M.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

M. Wickenhauser, J. Burgdörfer, F. Krausz, and M. Drescher, "Time resolved Fano resonances," Phys. Rev. Lett. 94, 023002 (2005).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Echenique, P. M.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

Fittinghoff, D. N.

R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "Measuring Ultrashort Laser Pulses In The Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
[CrossRef]

Gaarde, M. B.

Gagnon, J.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

J. Gagnon, E. Goulielmakis, and V. S. Yakovlev, "The accurate FROG characterization of attosecond pulses from streaking measurements," Appl. Phys. B, Lasers and optics 92, 25-32 (2008).
[CrossRef]

Gilbertson, S.

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

Gohle, C.

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

Goulielmakis, E.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

J. Gagnon, E. Goulielmakis, and V. S. Yakovlev, "The accurate FROG characterization of attosecond pulses from streaking measurements," Appl. Phys. B, Lasers and optics 92, 25-32 (2008).
[CrossRef]

Gullikson, E. M.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

Hänsch, T. W.

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

Heinzmann, U.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Hendel, S.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

Hentschel, M.

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Hofstetter, M.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

Holzwarth, R.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

Horvath, B.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

Itatani, J.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
[CrossRef] [PubMed]

Ivanov, M. Y.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
[CrossRef] [PubMed]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory Of High-Harmonic Generation By Low-Frequency Laser Fields," Phys. Rev. Lett. 49, 2117-2132 (1994).

Kane, D. J.

D. J. Kane, "Recent Progress Toward Real-Time Measurement Of Ultrashort Laser Pulses," IEEE J. Quantum Electron. 35, 421-431 (1999).
[CrossRef]

Kapteyn, H. C.

Kazamias, S.

S. Kazamias and P. Balcou, "Intrinsic chirp of attosecond pulses: Single-atom model versus experiment," Phys. Rev. A 69, 063416 (2004).
[CrossRef]

Khan, S. D.

M. Chini, H. Wang, S. D. Khan, S. Y. Chen, and Z. H. Chang, "Retrieval of satellite pulses of single isolated attosecond pulses," Appl. Phys. Lett. 94, 161112 (2009).
[CrossRef]

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

Kienberger, R.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Kim, D.

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

Kim, J.

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

Kitzler, M.

M. Kitzler, N. Milošević, A. Scrinzi, F. Krausz, and T. Brabec, "Quantum Theory Of Attosecond XUV Pulse Measurement By Laser Dressed Photoionization," Phys. Rev. Lett. 88, 173904 (2002).
[CrossRef] [PubMed]

Kleineberg, U.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Krausz, F.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

M. Wickenhauser, J. Burgdörfer, F. Krausz, and M. Drescher, "Time resolved Fano resonances," Phys. Rev. Lett. 94, 023002 (2005).
[CrossRef] [PubMed]

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
[CrossRef] [PubMed]

M. Kitzler, N. Milošević, A. Scrinzi, F. Krausz, and T. Brabec, "Quantum Theory Of Attosecond XUV Pulse Measurement By Laser Dressed Photoionization," Phys. Rev. Lett. 88, 173904 (2002).
[CrossRef] [PubMed]

Krumbügel, M. A.

R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "Measuring Ultrashort Laser Pulses In The Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
[CrossRef]

L’Huillier, A.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory Of High-Harmonic Generation By Low-Frequency Laser Fields," Phys. Rev. Lett. 49, 2117-2132 (1994).

Lewenstein, M.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory Of High-Harmonic Generation By Low-Frequency Laser Fields," Phys. Rev. Lett. 49, 2117-2132 (1994).

Li, C. Q.

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

Lin, C. D.

Z. Zhao and C. D. Lin, "Theory of laser-assisted autoionization by attosecond light pulses," Physical Rev. A, Atomic, molecular, and optical physics 71, 060702 (2005).
[CrossRef]

Lindner, F.

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

Liu, X.

Mairesse, Y.

Y. Mairesse and Quéré, "Frequency-Resolved Optical Gating for Complete Reconstruction of Attosecond Bursts," Phys. Rev. A 71, 011401 (2005).
[CrossRef]

Mashiko, H.

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

Miloševic, D. B.

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

Miloševic, N.

M. Kitzler, N. Milošević, A. Scrinzi, F. Krausz, and T. Brabec, "Quantum Theory Of Attosecond XUV Pulse Measurement By Laser Dressed Photoionization," Phys. Rev. Lett. 88, 173904 (2002).
[CrossRef] [PubMed]

Moon, E.

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

Muller, H. G.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

Muller, N.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

Mullot, G.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

Murnane, M. M.

Paul, P. M.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

Quéré, F.

Y. Mairesse and Quéré, "Frequency-Resolved Optical Gating for Complete Reconstruction of Attosecond Bursts," Phys. Rev. A 71, 011401 (2005).
[CrossRef]

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
[CrossRef] [PubMed]

Richman, B. A.

R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "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.

Schatzel, M. G.

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

Schmidt, B.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

Schultze, M.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

Scrinizi, A.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Scrinzi, A.

V. S. Yakovlev, F. Bammer, and A. Scrinzi, "Attosecond streaking measurements," J. Mod. Opt. 52, 395-410 (2005).
[CrossRef]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

M. Kitzler, N. Milošević, A. Scrinzi, F. Krausz, and T. Brabec, "Quantum Theory Of Attosecond XUV Pulse Measurement By Laser Dressed Photoionization," Phys. Rev. Lett. 88, 173904 (2002).
[CrossRef] [PubMed]

Shakya, M. M.

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

Sweetster, J. N.

R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "Measuring Ultrashort Laser Pulses In The Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
[CrossRef]

Thomann, I.

Toma, E. S.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

Trebino, R.

I. Thomann, A. Bahabad, X. Liu, R. Trebino, M. M. Murnane, and H. C. Kapteyn, "Characterizing isolated attosecond pulses from hollow-core waveguidesusing multi-cycle driving pulses," Opt. Express 17, 4611-4633 (2009).
[CrossRef] [PubMed]

R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "Measuring Ultrashort Laser Pulses In The Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
[CrossRef]

Udem, T.

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

Uiberacker, M.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Uphues, T.

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

Walther, H.

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

Wang, H.

M. Chini, H. Wang, S. D. Khan, S. Y. Chen, and Z. H. Chang, "Retrieval of satellite pulses of single isolated attosecond pulses," Appl. Phys. Lett. 94, 161112 (2009).
[CrossRef]

Westerwalbesloh, T.

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Wickenhauser, M.

M. Wickenhauser, J. Burgdörfer, F. Krausz, and M. Drescher, "Time resolved Fano resonances," Phys. Rev. Lett. 94, 023002 (2005).
[CrossRef] [PubMed]

Yakovlev, V.

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

Yakovlev, V. S.

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

V. S. Yakovlev, F. Bammer, and A. Scrinzi, "Attosecond streaking measurements," J. Mod. Opt. 52, 395-410 (2005).
[CrossRef]

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

J. Gagnon, E. Goulielmakis, and V. S. Yakovlev, "The accurate FROG characterization of attosecond pulses from streaking measurements," Appl. Phys. B, Lasers and optics 92, 25-32 (2008).
[CrossRef]

Yudin, G. L.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
[CrossRef] [PubMed]

Zhao, Z.

Z. Zhao and C. D. Lin, "Theory of laser-assisted autoionization by attosecond light pulses," Physical Rev. A, Atomic, molecular, and optical physics 71, 060702 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

M. Chini, H. Wang, S. D. Khan, S. Y. Chen, and Z. H. Chang, "Retrieval of satellite pulses of single isolated attosecond pulses," Appl. Phys. Lett. 94, 161112 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. J. Kane, "Recent Progress Toward Real-Time Measurement Of Ultrashort Laser Pulses," IEEE J. Quantum Electron. 35, 421-431 (1999).
[CrossRef]

J. Mod. Opt. (1)

V. S. Yakovlev, F. Bammer, and A. Scrinzi, "Attosecond streaking measurements," J. Mod. Opt. 52, 395-410 (2005).
[CrossRef]

Nature (4)

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuška, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Atomic transient recorder," Nature 427, 817-821 (2004).
[CrossRef] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinizi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, "Time-resolved atomic inner-shell spectroscopy," Nature 419, 803-807 (2002).
[CrossRef] [PubMed]

A. L. Cavalieri, N. Muller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, "Attosecond spectroscopy in condensed matter," Nature 449, 1029-1032 (2007).
[CrossRef] [PubMed]

A. Baltuška, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and K. F, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003).
[CrossRef] [PubMed]

New J. Phys. (1)

M. Schultze, E. Goulielmakis, M. Uiberacker, M. Hofstetter, J. Kim, D. Kim, F. Krausz, and U. Kleineberg, "Powerful 170-Attosecond XUV Pulses Generated With Few-Cycle Laser Pulses And Broadband Multilayer Optics," New J. Phys. 9, 243 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

S. Kazamias and P. Balcou, "Intrinsic chirp of attosecond pulses: Single-atom model versus experiment," Phys. Rev. A 69, 063416 (2004).
[CrossRef]

Y. Mairesse and Quéré, "Frequency-Resolved Optical Gating for Complete Reconstruction of Attosecond Bursts," Phys. Rev. A 71, 011401 (2005).
[CrossRef]

Phys. Rev. Lett. (7)

M. Kitzler, N. Milošević, A. Scrinzi, F. Krausz, and T. Brabec, "Quantum Theory Of Attosecond XUV Pulse Measurement By Laser Dressed Photoionization," Phys. Rev. Lett. 88, 173904 (2002).
[CrossRef] [PubMed]

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, "Attosecond Streak Camera," Phys. Rev. Lett. 88, 173903 (2002).
[CrossRef] [PubMed]

P. B. Corkum, "Plasma Perspective On Strong-Field Multiphoton Ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

H. Mashiko, S. Gilbertson, C. Q. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. H. Chang, "Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers," Phys. Rev. Lett. 100, 103906 (2008).
[CrossRef] [PubMed]

M. Wickenhauser, J. Burgdörfer, F. Krausz, and M. Drescher, "Time resolved Fano resonances," Phys. Rev. Lett. 94, 023002 (2005).
[CrossRef] [PubMed]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory Of High-Harmonic Generation By Low-Frequency Laser Fields," Phys. Rev. Lett. 49, 2117-2132 (1994).

F. Lindner, M. G. Schatzel, H. Walther, Baltuška, E. Goulielmakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and P. G. G, "Attosecond Double-Slit Experiment," Phys. Rev. Lett. 95, 040401 (2005).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

R. Trebino, K.W. Delong, D. N. Fittinghoff, J. N. Sweetster, M. A. Krumbügel, and B. A. Richman, "Measuring Ultrashort Laser Pulses In The Time-Frequency Domain Using Frequency-Resolved Optical Gating," Rev. Sci. Instrum. 68, 3277-3295 (1997).
[CrossRef]

Science (3)

E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuška, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct Measurement Of Light Waves," Science 305, 1267-1269 (2004).
[CrossRef] [PubMed]

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, L. A. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, "Single-Cycle Nonlinear Optics," Science 320, 1614-1617 (2008).
[CrossRef] [PubMed]

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller, and A. P, "Observation of a train of attosecond pulses from high harmonic generation," Science 292, 1689-1692 (2001).
[CrossRef] [PubMed]

Other (2)

Z. Zhao and C. D. Lin, "Theory of laser-assisted autoionization by attosecond light pulses," Physical Rev. A, Atomic, molecular, and optical physics 71, 060702 (2005).
[CrossRef]

J. Gagnon, E. Goulielmakis, and V. S. Yakovlev, "The accurate FROG characterization of attosecond pulses from streaking measurements," Appl. Phys. B, Lasers and optics 92, 25-32 (2008).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Simulated spectrograms for uncertainties in various parameters for a single pulse: (a) 17.47eV bandwidth uncertainty; (b) 15eV central energy uncertainty; (c) 100000as2 GDD uncertainty; (d) 900as delay uncertainty; (e) no uncertainty; (f) pulse duration retrieved by the attosecond FROG algorithm as a function of the spectrogram bandwidth (6). The values of these uncertainties do not represent realistic experimental conditions; rather, their values were chosen so that they would have a considerable effect on the spectrogram and on the retrieval.

Fig. 2.
Fig. 2.

Simulated spectrograms for uncertainties in various parameters for a double pulse: (a) 600as uncertainty in the relative timing; (b) uncertainty of 3p/2 in the relative phase; (c) 50% uncertainty in the relative intensity; (d) 900as delay uncertainty; (e) no uncertainty; (f) shows a plot of the retrieved relative intensity between the two pulses versus the fringe contrast in the spectrogram due to different amounts of uncertainty in the relative phase, timing and intensity.

Fig. 3.
Fig. 3.

(a) Streaked spectra taken at the peak of the vector potential (delay Δt s=0) for different central energies, with the inset showing their spectral shift as a function of central energy according to (7); (b) Electron wave packet retrieved by the attosecond FROG algorithm, with an inset showing the retrieved spectrogram, for a central energy uncertainty of 15eV.

Fig. 4.
Fig. 4.

(a) Streaked spectra taken at the zero of the vector potential (delay Δt s=-600as) for different bandwidths of the electron wave packet, with the inset showing the streaked bandwidth as a function of original bandwidth according to (12), with γ=-6244as2; (b) Retrieved electron wave packet, with an inset showing the retrieved spectrogram, for a bandwidth uncertainty of 17.47eV.

Fig. 5.
Fig. 5.

(a) Streaked spectra taken at the zero of the vector potential (delay Δt s=-600as) for different values of GDD, with the inset showing the streaked bandwidth as a function of original GDD according to (12) with δ=10.5eV; (b) Retrieved electron wave packet, with an inset showing the retrieved spectrogram, for a GDD uncertainty of 100 000 as2.

Fig. 6.
Fig. 6.

(a) centre of the streaked spectra, pcA Lt s), as a function of delay between the electron wave packet and laser field (solid line), spectral fringe positions as a function of delay for three values of the relative timing Δt X (dashed lines), given by pc (Bt s+ Δt X)-Bt s))/Δt X; (b), (c) and (d) show spectrograms evaluated for relative timings of Δt X=0.9fs, 1.2fs and 1.5fs respectively.

Fig. 7.
Fig. 7.

(a) Unstreaked spectra for different values of the relative timing; (b) retrieved double-pulse electron wave packet, with an inset showing the retrieved spectrogram, for a 600 as uncertainty in the relative timing.

Fig. 8.
Fig. 8.

(a) Unstreaked spectra for different values of the relative phase; (b) retrieved double-pulse electron wave packet, with an inset showing the retrieved spectrogram, for an uncertainty of 3π/2 in the relative phase.

Fig. 9.
Fig. 9.

(a) Unstreaked spectra for different values of the relative intensity; (b) retrieved double-pulse electron wave packet, with an inset showing the retrieved spectrogram, for a 50% uncertainty in the relative intensity.

Fig. 10.
Fig. 10.

Retrieved isolated electron wave packet (a), and double-pulse wave packet (b), with an inset showing the corresponding retrieved spectrogram, assuming a delay uncertainty of 900as.

Equations (15)

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

S(p,Δts)|d(p)|2=|χ(t)G(p,t+Δts)exp(ip22t)dt|2,
G(p,t)=d(p+AL(t))d(p)exp(it(pAL(t')+12AL2(t'))dt'),
χ˜(p22+WΩX)=2 i ψ˜(p,t)exp(ip22t)d(p)
χ1(t)=exp (ln(4)t2τX2) exp (i(εct+β2t2))
χ2(t)=χ1(t)+i2χ1(t+TL2)
δˉ=ε2ε2 .
Δ εpcAM12AM2
χ(t)=exp (ln(4)t2τ2) exp (i(εct+β2t2)) ,
AR(t)(ER)(ttR),
δs1τ(ln2(16)+(βpcER)2τ4)12.
βs=pcER
δsδη(ln2(16)+δ4(γη2δ4βs)2)12
ς(Δts,ΔtX)=Δts(p1AL(t)+12AL2(t))dt
Δts+ΔtX(p2AL(t)+12AL2(t))dt
ς(Δts,ΔtX)p2B(Δts+ΔtX)p1B(Δts),withdB(t)dt=AL(t).

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