Abstract

We demonstrate a novel algorithmic approach for the second-harmonic-generation (SHG) frequency-resolved optical gating (FROG) ultrashort-pulse-measurement technique that always converges and, for complex pulses, is also much faster. It takes advantage of the Paley-Wiener Theorem to retrieve the precise pulse spectrum—half the desired information—directly from the measured trace. It also uses a multi-grid approach, permitting the algorithm to operate on smaller arrays for early iterations and on the complete array for only the final few iterations. We tested this approach on more than 25,000 randomly generated complex pulses with time-bandwidth products up to 100, yielding SHG FROG traces to which noise was added, and have achieved convergence to the correct pulse in all cases. Moreover, convergence occurs in less than half the time for extremely large traces corresponding to extremely complex pulses.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (3)

2017 (1)

T. Bendory, P. Sidorenko, and Y. C. Eldar, “On the uniqueness of FROG methods,” IEEE Signal Process. Lett. 24(5), 722–726 (2017).
[Crossref]

2016 (4)

M. Rhodes, Z. Guang, and R. Trebino, “Unstable and multiple pulsing can be invisible to ultrashort pulse measurement techniques,” Appl. Sci. (Basel) 7(1), 40 (2016).
[Crossref]

P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18(7), 073009 (2016).
[Crossref]

P.-Y. Wu, H.-H. Lu, C.-Z. Weng, Y.-H. Chen, and S.-D. Yang, “Dispersion-corrected frequency-resolved optical gating,” Opt. Lett. 41(19), 4538–4541 (2016).
[Crossref] [PubMed]

P. Sidorenko, O. Lahav, Z. Avnat, and O. Cohen, “Ptychographic reconstruction algorithm for frequency-resolved optical gating: super-resolution and supreme robustness,” Optica 3(12), 1320–1330 (2016).
[Crossref]

2015 (1)

2014 (1)

2013 (1)

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (1)

V. S. Yakovlev, J. Gagnon, N. Karpowicz, and F. Krausz, “Attosecond streaking enables the measurement of quantum phase,” Phys. Rev. Lett. 105(7), 073001 (2010).
[Crossref] [PubMed]

2008 (1)

2006 (2)

B. von Vacano, W. Wohlleben, and M. Motzkus, “Actively shaped supercontinuum from a photonic crystal fiber for nonlinear coherent microspectroscopy,” Opt. Lett. 31(3), 413–415 (2006).
[Crossref] [PubMed]

S. F. Shu, “Evolving ultrafast laser information by a learning genetic algorithm combined with a knowledge base,” IEEE Photonics Technol. Lett. 18(2), 379–381 (2006).
[Crossref]

2005 (1)

F. Quere, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt. 52(2–3), 339–360 (2005).
[Crossref]

2004 (1)

2003 (1)

2002 (1)

2001 (1)

G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72(5), 627–630 (2001).
[Crossref]

1999 (3)

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, and R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35(4), 441–450 (1999).
[Crossref]

B. Walker, C. Tóth, D. Fittinghoff, T. Guo, D.-E. Kim, C. Rose-Petruck, J. Squier, K. Yamakawa, K. Wilson, and C. Barty, “A 50 EW/cm;2 Ti:sapphire laser system for studying relativistic light-matter interactions,” Opt. Express 5(10), 196–202 (1999).
[Crossref] [PubMed]

1997 (3)

1969 (1)

R. A. Fisher and J. J. A. Fleck, “On the phase characteristics and compression of picosecond pulses,” Appl. Phys. Lett. 15(9), 287–290 (1969).
[Crossref]

Avnat, Z.

Backus, S.

Barry, L. P.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, and R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35(4), 441–450 (1999).
[Crossref]

Bartels, R.

Barty, C.

Bendory, T.

T. Bendory, P. Sidorenko, and Y. C. Eldar, “On the uniqueness of FROG methods,” IEEE Signal Process. Lett. 24(5), 722–726 (2017).
[Crossref]

Bhardwaj, S.

P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18(7), 073009 (2016).
[Crossref]

Birge, J.

Bollond, P. G.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, and R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35(4), 441–450 (1999).
[Crossref]

Bowlan, P.

Chen, Y.-H.

Cohen, O.

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68(9), 3277–3295 (1997).
[Crossref]

Deng, J.

Dikopoltsev, A.

Dudley, J. M.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, and R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35(4), 441–450 (1999).
[Crossref]

Durfee Iii, C. G.

Eldar, Y. C.

T. Bendory, P. Sidorenko, and Y. C. Eldar, “On the uniqueness of FROG methods,” IEEE Signal Process. Lett. 24(5), 722–726 (2017).
[Crossref]

Epstein, J.

Farfan, C. A.

Feurer, T.

G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72(5), 627–630 (2001).
[Crossref]

Fisher, R. A.

R. A. Fisher and J. J. A. Fleck, “On the phase characteristics and compression of picosecond pulses,” Appl. Phys. Lett. 15(9), 287–290 (1969).
[Crossref]

Fittinghoff, D.

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68(9), 3277–3295 (1997).
[Crossref]

Fleck, J. J. A.

R. A. Fisher and J. J. A. Fleck, “On the phase characteristics and compression of picosecond pulses,” Appl. Phys. Lett. 15(9), 287–290 (1969).
[Crossref]

Gagnon, J.

V. S. Yakovlev, J. Gagnon, N. Karpowicz, and F. Krausz, “Attosecond streaking enables the measurement of quantum phase,” Phys. Rev. Lett. 105(7), 073001 (2010).
[Crossref] [PubMed]

Gu, X.

Guang, Z.

M. Rhodes, Z. Guang, and R. Trebino, “Unstable and multiple pulsing can be invisible to ultrashort pulse measurement techniques,” Appl. Sci. (Basel) 7(1), 40 (2016).
[Crossref]

Guo, T.

Hacker, M.

G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72(5), 627–630 (2001).
[Crossref]

Haham, G. I.

Harvey, J. D.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, and R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35(4), 441–450 (1999).
[Crossref]

Honzatko, P.

Itatani, J.

F. Quere, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt. 52(2–3), 339–360 (2005).
[Crossref]

Jacobsen, C.

Jafari, R.

R. Jafari and R. Trebino, “High-speed “multi-grid” pulse-retrieval algorithm for frequency-resolved optical gating,” Opt. Express 26(3), 2643–2649 (2018).
[Crossref] [PubMed]

R. Jafari and R. Trebino, “100% reliable frequency-resolved optical gating pulse-retrieval algorithmic approach,” https://arxiv.org/abs/1811.11100 .

Kane, D. J.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68(9), 3277–3295 (1997).
[Crossref]

Kanka, J.

Kapteyn, H. C.

Karpowicz, N.

V. S. Yakovlev, J. Gagnon, N. Karpowicz, and F. Krausz, “Attosecond streaking enables the measurement of quantum phase,” Phys. Rev. Lett. 105(7), 073001 (2010).
[Crossref] [PubMed]

Kärtner, F. X.

P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18(7), 073009 (2016).
[Crossref]

Keathley, P. D.

P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18(7), 073009 (2016).
[Crossref]

Keusters, D.

Kim, D.-E.

Kimmel, M.

Krausz, F.

V. S. Yakovlev, J. Gagnon, N. Karpowicz, and F. Krausz, “Attosecond streaking enables the measurement of quantum phase,” Phys. Rev. Lett. 105(7), 073001 (2010).
[Crossref] [PubMed]

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68(9), 3277–3295 (1997).
[Crossref]

B. A. Richman, M. A. Krumbügel, and R. Trebino, “Temporal characterization of mid-IR free-electron-laser pulses by frequency-resolved optical gating,” Opt. Lett. 22(10), 721–723 (1997).
[Crossref] [PubMed]

Lahav, O.

Laurent, G.

P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18(7), 073009 (2016).
[Crossref]

Leonhardt, R.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, and R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35(4), 441–450 (1999).
[Crossref]

Lu, H.-H.

Mairesse, Y.

F. Quere, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt. 52(2–3), 339–360 (2005).
[Crossref]

Mannor, S.

Moses, J.

P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18(7), 073009 (2016).
[Crossref]

Moss, D.

Motzkus, M.

B. von Vacano, W. Wohlleben, and M. Motzkus, “Actively shaped supercontinuum from a photonic crystal fiber for nonlinear coherent microspectroscopy,” Opt. Lett. 31(3), 413–415 (2006).
[Crossref] [PubMed]

G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72(5), 627–630 (2001).
[Crossref]

Mourou, G.

Mukhopadhyay, M.

Murnane, M. M.

Nashed, Y. S. G.

O’Shea, P.

Omenetto, F. G.

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

Peterka, T.

Quere, F.

F. Quere, Y. Mairesse, and J. Itatani, “Temporal characterization of attosecond XUV fields,” J. Mod. Opt. 52(2–3), 339–360 (2005).
[Crossref]

Ratner, J.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

J. Ratner, G. Steinmeyer, T. C. Wong, R. Bartels, and R. Trebino, “Coherent artifact in modern pulse measurements,” Opt. Lett. 37(14), 2874–2876 (2012).
[Crossref] [PubMed]

Reichel, F.

G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72(5), 627–630 (2001).
[Crossref]

Rhodes, M.

M. Rhodes, Z. Guang, and R. Trebino, “Unstable and multiple pulsing can be invisible to ultrashort pulse measurement techniques,” Appl. Sci. (Basel) 7(1), 40 (2016).
[Crossref]

M. Rhodes, M. Mukhopadhyay, J. Birge, and R. Trebino, “Coherent artifact study of two-dimensional spectral shearing interferometry,” J. Opt. Soc. Am. B 32(9), 1881–1888 (2015).
[Crossref]

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

Richman, B. A.

Rose-Petruck, C.

Ross, R.

Segev, M.

Shreenath, A. P.

Shu, S. F.

S. F. Shu, “Evolving ultrafast laser information by a learning genetic algorithm combined with a knowledge base,” IEEE Photonics Technol. Lett. 18(2), 379–381 (2006).
[Crossref]

Siders, C. W.

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

Siders, J. L. W.

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

Sidorenko, P.

Squier, J.

Steinmeyer, G.

M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, “Pulse-shape instabilities and their measurement,” Laser Photonics Rev. 7(4), 557–565 (2013).
[Crossref]

J. Ratner, G. Steinmeyer, T. C. Wong, R. Bartels, and R. Trebino, “Coherent artifact in modern pulse measurements,” Opt. Lett. 37(14), 2874–2876 (2012).
[Crossref] [PubMed]

Stobrawa, G.

G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72(5), 627–630 (2001).
[Crossref]

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68(9), 3277–3295 (1997).
[Crossref]

Tan, H.-S.

Taylor, A. J.

C. W. Siders, J. L. W. Siders, F. G. Omenetto, and A. J. Taylor, “Multipulse interferometric frequency-resolved optical gating,” IEEE J. Quantum Electron. 35(4), 432–440 (1999).
[Crossref]

Thomsen, B. C.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, and R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35(4), 441–450 (1999).
[Crossref]

Thomson, M. D.

J. M. Dudley, L. P. Barry, J. D. Harvey, M. D. Thomson, B. C. Thomsen, P. G. Bollond, and R. Leonhardt, “Complete characterization of ultrashort pulse sources at 1550 nm,” IEEE J. Quantum Electron. 35(4), 441–450 (1999).
[Crossref]

Tóth, C.

Trebino, R.

R. Jafari and R. Trebino, “High-speed “multi-grid” pulse-retrieval algorithm for frequency-resolved optical gating,” Opt. Express 26(3), 2643–2649 (2018).
[Crossref] [PubMed]

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Opt. Lett. (7)

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Phys. Rev. Lett. (1)

V. S. Yakovlev, J. Gagnon, N. Karpowicz, and F. Krausz, “Attosecond streaking enables the measurement of quantum phase,” Phys. Rev. Lett. 105(7), 073001 (2010).
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Rev. Sci. Instrum. (1)

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R. Jafari and R. Trebino, “Pulse-chirp instability and issues for its measurement,” in CLEO: Applications and Technology, (Optical Society of America, 2018), JTh2A. 142.

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Figures (6)

Fig. 1
Fig. 1 SHG FROG marginals.
Fig. 2
Fig. 2 (a) Frequency marginal of a SHG FROG trace, MSHG(ω) = S(ω)*S(ω). Plots of (b) real, and (c) imaginary parts of s±(t) = ± √ F −1 [MSHG(ω)] for a randomly generated complex pulse. The root with positive (negative) real component is shown in purple (blue), respectively. Note that the imaginary components of the roots can have either positive or negative values. As a result, the real part is always continuous, but the imaginary part may not be. But both components of the actual complex function s(t) must be continuous. Therefore, the discontinuity of the imaginary part almost always determines sign of the correct roots when the real values are close to zero. Conversely, the continuity of real values can be applied when the imaginary points are close to zero. The gray dotted lines in the two plots pass in between temporal points t = −25fs and t = −22.5fs. If one begins at t = 0 and moves in the decreasing time direction on purple curves, when transitioning from t = −22.5fs to t = −25fs, the correct sign of the root must change to satisfy the C condition. Analogous arguments apply to all of the derivatives of s(t), and such higher order discontinuities are also apparent in the plots.
Fig. 3
Fig. 3 Schematic of steps for retrieving the spectrum of the pulse directly from the frequency marginal of its SHG FROG trace.
Fig. 4
Fig. 4 Examples of spectra retrieved directly from noisy SHG FROG trace frequency marginals in our study. The simulated spectrum and the retrieved spectrum from the frequency marginal are shown in solid green and dashed black lines, respectively. First column: TBP = 5, and second column: TBP = 20. (a,b) The result corresponding to value ΔS in first quartile, (c,d) second quartile, and (e,f) third quartile, respectively. (g) ΔS = 0.17, and (h) ΔS = 0.16. Note, however, that these “incorrect spectra” still retain the rough structure of the spectrum and still provided excellent initial guesses for the iteration and yielded convergence to the correct pulse once the full array was used.
Fig. 5
Fig. 5 Typical SHG FROG pulse-retrieval result with a G error = 0.0018 and a trace size of 512 × 512 for a pulse with TBP = 20 and 0.5% multiplicative noise. (a) The simulated trace. (b) The retrieved trace. (c,d) The simulated temporal/spectral intensity and phase are shown in orange/light green and cyan/ magenta, respectively. The retrieved temporal/spectral intensity and phase are shown in dashed red/dark green and dashed dark blue/dark purple, respectively.
Fig. 6
Fig. 6 (a) Experimental SHG FROG trace of a pulse with TBP = 4.1 used to study the performance of RANA approach. (b) The retrieved trace. (c) The spectrum measured by the spectrometer. (d) The spectrum retrieved directly from the SHG trace. Note that this spectrum actually differs slightly from the correct spectrum, but this does not affect the results. A 512 × 512 trace is used to obtain better temporal and spectral resolutions for this complex pulse. Both approaches converge to a G error = 0.014 and G’ error = 0.28 (G’ is normalized by the trace area, rather than the number of points and so can be compared to the trace noise). Both G values indicate excellent agreement, as does visual inspection of the traces. (e,f) retrieved temporal and spectral intensity and phase, respectively.

Tables (4)

Tables Icon

Table 1 Parameters of the RANA approach used here. The value of the G error (rms difference between the measured and retrieved traces) for determining convergence, number of initial guesses (IGs), and number of iterations that are used on each grid in RANA approach for pulses with TBPs of 2.5 to 100.

Tables Icon

Table 2 Performance of spectrum retrieval from the frequency marginal of the FROG traces with 0.5% multiplicative noise.

Tables Icon

Table 3 Performance of GP algorithm and RANA approach on SHG traces with different complexity and size.

Tables Icon

Table 4 Performance of standard GP algorithm with an initial guess of the spectral intensity obtained from the frequency marginal of SHG FROG trace.

Equations (12)

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I FROG SHG (ω,τ)= | + E(t)E(tτ)exp(iωt)dt | 2 .
M(τ) I FROG (ω,τ) dω
M(ω) I FROG (ω,τ) dτ
F 1 { M SHG (ω) }= ( F 1 { S(ω) } ) 2 s (t) 2 .
s ± =± F 1 { M SHG (ω) }
Δ 0± = s ± ( t i+1 )s( t i ),
Δ 1± =[ s ± ( t i+1 )s( t i ) ][ s( t i )s( t i1 ) ],
Δ 2± ={ [ s ± ( t i+1 )s( t i ) ][ s( t i )s( t i1 ) ] }{ [ s( t i )s( t i1 ) ][ s( t i1 )s( t i2 ) ] }.
ε ± α | Δ 0± | 2 +β | Δ 1± | 2 +γ | Δ 2± | 2 ,
ΔS= 1 N i=1 N ( S i simulated - S i retrieved ) 2 .
TBP= + t 2 I(t)dt ( + tI(t)dt ) 2 + I(t)dt × + (ω ω 0 ) 2 S(ω ω 0 )dω ( + (ω ω 0 )S(ω ω 0 )dω ) 2 + S(ω ω 0 )dω
I FROG noisy ( ω i , τ j )= I FROG ( ω i , τ j )( 1+ m ij α )

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