Abstract

We present an iterative retrieval algorithm based on data constraint for ultrashort pulse characterization using dispersion scan (d-scan). The proposed algorithm is much faster and leads to a drastic reduction of retrieval times, but, compared to the standard algorithm, it performs less robustly in the retrieval of noisy d-scan traces. The algorithm is tested on several simulated cases and in two different experimental cases in the few-cycle regime.

© 2016 Optical Society of America

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References

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  1. R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
    [Crossref]
  2. S. Akturk, C. D’Amico, and A. Mysyrowicz, “Measuring ultrashort pulses in the single-cycle regime using frequency-resolved optical gating,” J. Opt. Soc. Am. B 25, A63–A69 (2008).
    [Crossref]
  3. 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]
  4. T. Witting, F. Frank, C. A. Arrell, W. A. Okell, J. P. Marangos, and J. W. G. Tisch, “Characterization of high-intensity sub-4-fs laser pulses using spatially encoded spectral shearing interferometry,” Opt. Lett. 36, 1680–1682 (2011).
    [Crossref]
  5. A. Moulet, S. Grabielle, C. Cornaggia, N. Forget, and T. Oksenhendler, “Single-shot, high-dynamic-range measurement of sub-15  fs pulses by self-referenced spectral interferometry,” Opt. Lett. 35, 3856–3858 (2010).
    [Crossref]
  6. 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]
  7. Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71, 011401(R) (2005).
    [Crossref]
  8. A. S. Wyatt, T. Witting, A. Schiavi, D. Fabris, P. Matia-Hernando, I. A. Walmsley, J. P. Marangos, and J. W. G. Tisch, “Attosecond sampling of arbitrary optical waveforms,” Optica 3, 303–310 (2016).
    [Crossref]
  9. V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775–777 (2004).
    [Crossref]
  10. V. Loriot, G. Gitzinger, and N. Forget, “Self-referenced characterization of femtosecond laser pulses by chirp scan,” Opt. Express 21, 24879–24893 (2013).
    [Crossref]
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    [Crossref]
  12. D. Fabris, W. Holgado, F. Silva, T. Witting, J. W. G. Tisch, and H. Crespo, “Single-shot implementation of dispersion-scan for the characterization of ultrashort laser pulses,” Opt. Express 23, 32803–32808 (2015).
    [Crossref]
  13. M. Miranda, C. L. Arnold, T. Fordell, F. Silva, B. Alonso, R. Weigand, A. L’Huillier, and H. Crespo, “Characterization of broadband few-cycle laser pulses with the d-scan technique,” Opt. Express 20, 18732–18743 (2012).
    [Crossref]
  14. K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating based on the method of generalized projections,” Opt. Lett. 19, 2152–2154 (1994).
    [Crossref]
  15. D. J. Kane, G. Rodriguez, A. J. Taylor, and T. S. Clement, “Simultaneous measurement of two ultrashort laser pulses from a single spectrogram in a single shot,” J. Opt. Soc. Am. B 14, 935–943 (1997).
    [Crossref]
  16. D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
    [Crossref]
  17. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  18. D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
    [Crossref]
  19. T. Witting, D. Greening, D. Walke, P. Matia-Hernando, T. Barillot, J. P. Marangos, and J. W. G. Tisch, “Time-domain ptychography of over-octave-spanning laser pulses in the single-cycle regime,” Opt. Lett. 41, 4218–4221 (2016).
    [Crossref]
  20. A. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276–1283 (1983).
    [Crossref]
  21. A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Amplitude and phase characterization of 4.5-fs pulses by frequency-resolved optical gating,” Opt. Lett. 23, 1474–1476 (1998).
    [Crossref]
  22. D. E. Wilcox and J. P. Ogilvie, “Comparison of pulse compression methods using only a pulse shaper,” J. Opt. Soc. Am. B 31, 1544–1554 (2014).
    [Crossref]
  23. P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
    [Crossref]
  24. M. Louisy, C. L. Arnold, M. Miranda, E. W. Larsen, S. N. Bengtsson, D. Kroon, M. Kotur, D. Guénot, L. Rading, P. Rudawski, F. Brizuela, F. Campi, B. Kim, A. Jarnac, A. Houard, J. Mauritsson, P. Johnsson, A. L’Huillier, and C. M. Heyl, “Gating attosecond pulses in a noncollinear geometry,” Optica 2, 563–566 (2015).
    [Crossref]
  25. F. Silva, M. Miranda, B. Alonso, J. Rauschenberger, V. Pervak, and H. Crespo, “Simultaneous compression, characterization and phase stabilization of GW-level 1.4 cycle VIS-NIR femtosecond pulses using a single dispersion-scan setup,” Opt. Express 22, 10181–10191 (2014).
    [Crossref]

2016 (2)

2015 (4)

2014 (2)

2013 (1)

2012 (2)

2011 (1)

2010 (1)

2008 (1)

2005 (1)

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

2004 (1)

2002 (1)

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]

1998 (3)

1997 (1)

1994 (1)

1993 (1)

1983 (1)

A. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276–1283 (1983).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Ahrens, J.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Akturk, S.

Alonso, B.

Arnold, C.

Arnold, C. L.

Arrell, C. A.

Baltuška, A.

Barillot, T.

Bengtsson, S. N.

Binhammer, T.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Brizuela, F.

Brügmann, M. H.

Campi, F.

Clement, T. S.

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]

Cornaggia, C.

Crespo, H.

D’Amico, C.

Dantus, M.

DeLong, K. W.

Fabris, D.

Feurer, T.

Fittinghoff, D. N.

Fordell, T.

Forget, N.

Frank, F.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gitzinger, G.

Grabielle, S.

Greening, D.

Guénot, D.

Guo, C.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Harth, A.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Heyl, C.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Heyl, C. M.

Holgado, W.

Houard, A.

Iaconis, C.

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]

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]

Jarnac, A.

Johnsson, P.

Kane, D. J.

Kim, B.

Kohler, B.

Kotur, M.

Krausz, F.

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]

Kroon, D.

L’Huillier, A.

Larsen, E.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Larsen, E. W.

Lorek, E.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Loriot, V.

Louisy, M.

Lozovoy, V. V.

Mairesse, Y.

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

Marangos, J. P.

Matia-Hernando, P.

Mauritsson, J.

M. Louisy, C. L. Arnold, M. Miranda, E. W. Larsen, S. N. Bengtsson, D. Kroon, M. Kotur, D. Guénot, L. Rading, P. Rudawski, F. Brizuela, F. Campi, B. Kim, A. Jarnac, A. Houard, J. Mauritsson, P. Johnsson, A. L’Huillier, and C. M. Heyl, “Gating attosecond pulses in a noncollinear geometry,” Optica 2, 563–566 (2015).
[Crossref]

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Miranda, M.

Morgner, U.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Moulet, A.

Mysyrowicz, A.

Ogilvie, J. P.

Okell, W. A.

Oksenhendler, T.

Pastirk, I.

Pervak, V.

Prochnow, O.

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Pshenichnikov, M. S.

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]

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]

Rading, L.

Rauschenberger, J.

Rodriguez, G.

Rohwer, E.

Rudawski, P.

M. Louisy, C. L. Arnold, M. Miranda, E. W. Larsen, S. N. Bengtsson, D. Kroon, M. Kotur, D. Guénot, L. Rading, P. Rudawski, F. Brizuela, F. Campi, B. Kim, A. Jarnac, A. Houard, J. Mauritsson, P. Johnsson, A. L’Huillier, and C. M. Heyl, “Gating attosecond pulses in a noncollinear geometry,” Optica 2, 563–566 (2015).
[Crossref]

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schiavi, A.

Silva, F.

Spangenberg, D.

Taylor, A. J.

Tisch, J. W. G.

Trebino, R.

Walke, D.

Walmsley, I. A.

Weigand, R.

Weiner, A.

A. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276–1283 (1983).
[Crossref]

Wiersma, D. A.

Wilcox, D. E.

Wilson, K.

Witting, T.

Wyatt, A. S.

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]

Eur. Phys. J. D (1)

P. Rudawski, A. Harth, C. Guo, E. Lorek, M. Miranda, C. Heyl, E. Larsen, J. Ahrens, O. Prochnow, T. Binhammer, U. Morgner, J. Mauritsson, A. L’Huillier, and C. L. Arnold, “Carrier-envelope phase dependent high-order harmonic generation with a high-repetition rate OPCPA-system,” Eur. Phys. J. D 69, 70 (2015).
[Crossref]

IEEE J. Quantum Electron. (1)

A. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. 19, 1276–1283 (1983).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

D. J. Kane, “Real-time measurement of ultrashort laser pulses using principal component generalized projections,” IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998).
[Crossref]

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

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

Opt. Express (5)

Opt. Lett. (8)

K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating based on the method of generalized projections,” Opt. Lett. 19, 2152–2154 (1994).
[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]

T. Witting, F. Frank, C. A. Arrell, W. A. Okell, J. P. Marangos, and J. W. G. Tisch, “Characterization of high-intensity sub-4-fs laser pulses using spatially encoded spectral shearing interferometry,” Opt. Lett. 36, 1680–1682 (2011).
[Crossref]

A. Moulet, S. Grabielle, C. Cornaggia, N. Forget, and T. Oksenhendler, “Single-shot, high-dynamic-range measurement of sub-15  fs pulses by self-referenced spectral interferometry,” Opt. Lett. 35, 3856–3858 (2010).
[Crossref]

V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775–777 (2004).
[Crossref]

D. Spangenberg, E. Rohwer, M. H. Brügmann, and T. Feurer, “Ptychographic ultrafast pulse reconstruction,” Opt. Lett. 40, 1002–1005 (2015).
[Crossref]

T. Witting, D. Greening, D. Walke, P. Matia-Hernando, T. Barillot, J. P. Marangos, and J. W. G. Tisch, “Time-domain ptychography of over-octave-spanning laser pulses in the single-cycle regime,” Opt. Lett. 41, 4218–4221 (2016).
[Crossref]

A. Baltuška, M. S. Pshenichnikov, and D. A. Wiersma, “Amplitude and phase characterization of 4.5-fs pulses by frequency-resolved optical gating,” Opt. Lett. 23, 1474–1476 (1998).
[Crossref]

Optica (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. A (1)

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

Phys. Rev. Lett. (1)

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]

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Figures (8)

Fig. 1.
Fig. 1. Example of (a) simulated and (b) retrieved d-scan traces. (c) Both spectral intensity (blue) and phase (orange) are precisely retrieved, as well as the corresponding pulse in the time domain. (d) Retrieved fields are plotted in lighter colors. Note that in this plot the simulated and retrieved fields overlap almost exactly and cannot be easily distinguished.
Fig. 2.
Fig. 2. Example of (a) simulated and (b) retrieved scans of a double pulse. The retrieved spectral intensity and phase are shown in (c) in light colors, as well as the corresponding spectral intensity for an isolated pulse. The corresponding pulses in the time domain are shown in (d).
Fig. 3.
Fig. 3. Example of (a) simulated trace where a spectral filter was applied, simulating phase-matching and clipping. The algorithm uses the known fundamental spectrum (c) and retrieves a trace (b) very similar to the original trace. The corresponding pulse in the time domain is shown in (d).
Fig. 4.
Fig. 4. Example of (a) simulated trace with added noise and (b) corresponding retrieved trace. The basic algorithm retrieves (c) both spectral amplitude and phase, but especially the spectral amplitude (light-colored line) is not retrieved very accurately. In the time domain (d) the main features are reproduced but much less accurately than in the noiseless cases.
Fig. 5.
Fig. 5. Same as previous case, but with known fundamental spectrum. The algorithm variation that includes projection of the fundamental spectrum was used, yielding a much better reconstruction.
Fig. 6.
Fig. 6. Same as previous case, but using a mixed algorithm. After the retrieval using the basic algorithm, a multidimensional algorithm optimizes the spectral intensity. The retrieved spectral phase is left unchanged, and a generic optimization algorithm minimizes the error between (a) simulated and (b) retrieved scans by applying a filter curve to the previously retrieved spectral intensity.
Fig. 7.
Fig. 7. Characterization of the output from an OPCPA system, with (b) retrieved trace very similar to (a) measured trace. (c) Retrieved spectral intensity and phase (light colored) are compared to the directly measured spectrum (dark color). (d) The shortest pulse (obtained at insertion 0) is 6.3 fs FWHM.
Fig. 8.
Fig. 8. Characterization of the output from a hollow-core fiber compressor. (a) Measured and (b) retrieved trace using the basic algorithm. Light colored lines correspond to the retrieval done with the basic algorithm (i.e., without using the measured spectrum). Dark colored lines correspond to the retrieval using the algorithm variation for noncalibrated traces, which uses the measured spectrum. The retrieved pulse durations in both cases is 3.0 fs FWHM.

Equations (43)

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

E ˜ ( ω ) = | E ˜ ( ω ) | exp [ i ψ ( ω ) ]
Φ ( ω , z ) = exp [ i z k ( ω ) ] .
S ( ω , z ) = F { F 1 { E ˜ ( ω ) Φ ( ω , z ) } 2 } ,
I ( ω , z ) = | S ( ω , z ) | 2 .
M ( ω ) = L + L I ( ω , z ) d z .
E ˜ l = 0 ( ω ) .
Φ ( ω , z ) = exp [ i z k ( ω ) ] ,
U ˜ l ( ω , z ) = E ˜ l ( ω ) Φ ( ω , z ) ,
U l ( t , z ) = F 1 { U ˜ l ( ω , z ) } ,
U l SHG ( t , z ) = U l 2 ( t , z ) ,
S l ( ω , z ) = F { U l SHG ( t , z ) } .
S l ( ω , z ) = S meas ( ω , z ) exp { i arg [ S l ( ω , z ) ] } ,
U l SHG ( t , z ) = F 1 { S l ( ω , z ) } .
P l ( t , z ) = U l SHG ( t , z ) U l * ( t , z ) ,
U l ( t , z ) = | P l ( t , z ) | 1 / 3 exp { i arg [ P l ( t , z ) ] } .
U ˜ l ( ω , z ) = F { U l ( t , z ) } ,
U ˜ l ( ω , z ) = U ˜ l ( ω , z ) Φ * ( ω , z ) .
E ˜ l + 1 ( ω ) = 1 Δ z z U ˜ l ( ω , z ) d z ,
G 2 = 1 N j N k j , k ( I meas ( ω j , z k ) μ I retr ( ω j , z k ) ) 2 ,
μ = j , k I meas ( ω j , z k ) I retr ( ω j , z k ) j , k I retr ( ω j , z k ) .
E ˜ l ( ω ) = z U ˜ l ( ω , z ) w ( z ) d z z w ( z ) d z ,
w ( z ) = ω S meas ( ω , z ) d ω .
P l ( t , z ) = U l THG ( t , z ) [ U l * ( t , z ) ] 2
U l ( t , z ) = | P l ( t , z ) | 1 / 5 exp { i arg [ P l ( t , z ) ] } .
I meas ( ω , z ) = I retr ( ω , z ) R ( ω ) ,
G 2 = 1 N j N k j , k ( I meas ( ω j , z k ) μ ( ω j ) I retr ( ω j , z k ) ) 2 .
μ ( ω j ) = k I meas ( ω j , z k ) I retr ( ω j , z k ) k I retr ( ω j , z k ) .
E ˜ l ( ω ) = | E ˜ meas ( ω ) | exp { i arg [ E ˜ l ( ω ) ] } .
I meas ( ω , z ) = I meas ( ω , z ) / μ ( ω ) ,
S meas ( ω , z ) = | I meas ( ω , z ) | ,
S l ( ω , z ) = S meas ( ω , z ) exp { i arg [ S l ( ω , z ) ] } sgn [ I meas ( ω , z ) ] .
L L d z | S ( ω , z ) | 2 = A ( ω ) log 2 L ( ω ) + B ( ω ) + O ( L 1 ) ,
( ω ) = τ 2 | k ( ω 2 ) | , A ( ω ) = 1 2 π | k ( ω 2 ) | | E ˜ ( ω 2 ) | 4 ,
B ( ω ) = d ω 1 π [ | E ˜ ( ω 2 + ω 1 ) | 2 | E ˜ ( ω 2 ω 1 ) | 2 | k ( ω 2 + ω 1 ) k ( ω 2 ω 1 ) | | E ˜ ( ω 2 ) | 4 e 2 τ 2 ω 1 2 | 2 ω 1 k ( ω 2 ) | ] .
d ω 1 2 π E ˜ ( ω 2 + ω 1 ) E ˜ ( ω 2 ω 1 ) e i z [ k ( ω 2 + ω 1 ) + k ( ω 2 ω 1 ) ] .
L L d z | S ( ω , z ) | 2 = d ω 1 d ω 2 ( 2 π ) 2 L L d z G ( ω , ω 1 , ω 2 , z ) ,
G ( ω , ω 1 , ω 2 , z ) = e i z [ k ( ω 2 + ω 1 ) + k ( ω 2 ω 1 ) k ( ω 2 + ω 2 ) k ( ω 2 ω 2 ) ] E ˜ ( ω 2 + ω 1 ) E ˜ ( ω 2 ω 1 ) E ˜ * ( ω 2 + ω 2 ) E ˜ * ( ω 2 ω 2 ) .
δ ( k ( ω 2 + ω 1 ) + k ( ω 2 ω 1 ) k ( ω 2 + ω 2 ) k ( ω 2 ω 2 ) ) .
δ ( ω 2 ω 1 ) + δ ( ω 2 + ω 1 ) | k ( ω 2 + ω 1 ) k ( ω 2 ω 1 ) | .
d ω 1 π | E ˜ ( ω 2 + ω 1 ) | 2 | E ˜ ( ω 2 ω 1 ) | 2 | k ( ω 2 + ω 1 ) k ( ω 2 ω 1 ) | ,
G ( ω , ω 1 , ω 2 , z ) | E ˜ ( ω 2 ) | 4 e i z ( ω 1 2 ω 2 2 ) k ( ω 2 ) τ 2 ( ω 1 2 + ω 2 2 )
+ | E ˜ ( ω 2 ) | 4 e i z ( ω 1 2 ω 2 2 ) k ( ω 2 ) τ 2 ( ω 1 2 + ω 2 2 ) .
1 2 A ( ω ) L L d z 1 z 2 + 2 ( ω ) = A ( ω ) log L + L 2 + 2 ( ω ) ( ω ) ,

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