Abstract

Frequency-resolved optical gating (FROG) and its variations are the only techniques available for measuring complex pulses without a well-characterized reference pulse. We study the performance of the FROG generalized-projections algorithm for retrieving the intensity and phase of very complex ultrashort laser pulses [with time–bandwidth products (TBPs) of up to 100] in the presence of noise. We compare the performance of three versions of FROG: second-harmonic-generation (SHG) FROG, polarization-gate (PG) FROG, and cross-correlation FROG (XFROG), the last of which requires a well-characterized reference pulse. We found that the XFROG algorithm converged in all cases on the first initial guess. The PG FROG algorithm converged for all moderately complex pulses, for 99% of the pulses we tried, and for more than 95% of even the most complex pulses (TBP100). The SHG FROG algorithm converged for 95% of the pulses we tried and for over 80% of even the most complex pulses. We found no additional ambiguities in any of these techniques.

© 2008 Optical Society of America

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2006 (2)

2005 (1)

2004 (1)

2002 (1)

2000 (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929-1960 (2000).
[CrossRef]

1999 (2)

E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. 24, 493-495 (1999).
[CrossRef]

C. J. Bardeen, V. V. Yakovlev, J. A. Squier, K. R. Wilson, S. D. Carpenter, and P. M. Weber, “Effect of pulse shape on the efficiency of multiphoton process: implications for biological microscopy,” J. Biomed. Opt. 4, 362-367 (1999).
[CrossRef]

1998 (3)

1995 (1)

1994 (2)

1993 (1)

1988 (1)

1980 (2)

K. Sala, G. Kenney-Wallace, and G. Hall, “CW autocorrelation measurements of picosecond laser pulses,” IEEE J. Quantum Electron. 16, 990-996 (1980).
[CrossRef]

R. A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 67, 1232-1246 (1980).
[CrossRef]

Akturk, S.

Altes, R. A.

R. A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 67, 1232-1246 (1980).
[CrossRef]

Backus, S.

Baltuska, A.

Bardeen, C. J.

C. J. Bardeen, V. V. Yakovlev, J. A. Squier, K. R. Wilson, S. D. Carpenter, and P. M. Weber, “Effect of pulse shape on the efficiency of multiphoton process: implications for biological microscopy,” J. Biomed. Opt. 4, 362-367 (1999).
[CrossRef]

Beaurepaire, E.

Bowlan, P.

Carpenter, S. D.

C. J. Bardeen, V. V. Yakovlev, J. A. Squier, K. R. Wilson, S. D. Carpenter, and P. M. Weber, “Effect of pulse shape on the efficiency of multiphoton process: implications for biological microscopy,” J. Biomed. Opt. 4, 362-367 (1999).
[CrossRef]

Chowdhury, I. H.

I. H. Chowdhury, X. Xu, and A. M. Weiner, “Laser machining using temporally controlled ultrafast pulses,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2004), paper CThD5.
[PubMed]

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice-Hall, 1995).

Dantus, M.

Débarre, D.

DeLong, K. W.

Fittinghoff, D. N.

Frumker, E.

Gabolde, P.

Goswami, D.

Gu, X.

Hall, G.

K. Sala, G. Kenney-Wallace, and G. Hall, “CW autocorrelation measurements of picosecond laser pulses,” IEEE J. Quantum Electron. 16, 990-996 (1980).
[CrossRef]

Heritage, J. P.

Hillegas, W.

Iaconis, C.

Joffre, M.

Kane, D. J.

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]

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]

Kapteyn, H.

Kenney-Wallace, G.

K. Sala, G. Kenney-Wallace, and G. Hall, “CW autocorrelation measurements of picosecond laser pulses,” IEEE J. Quantum Electron. 16, 990-996 (1980).
[CrossRef]

Kimmel, M.

Kohler, B.

Ladera, C. L.

Lozovoy, V. V.

Maginnis, K.

Majer, D.

Martin, J.-L.

McGresham, K.

Mourou, G.

Murnane, M.

Ogilvie, J. P.

O'Shea, P.

Pastirk, I.

Pshenichnikov, M. S.

Russek, U.

Sala, K.

K. Sala, G. Kenney-Wallace, and G. Hall, “CW autocorrelation measurements of picosecond laser pulses,” IEEE J. Quantum Electron. 16, 990-996 (1980).
[CrossRef]

Salehi, J. A.

Shreenath, A.

Shreenath, A. P.

Silberberg, Y.

Solinas, X.

Squier, J. A.

C. J. Bardeen, V. V. Yakovlev, J. A. Squier, K. R. Wilson, S. D. Carpenter, and P. M. Weber, “Effect of pulse shape on the efficiency of multiphoton process: implications for biological microscopy,” J. Biomed. Opt. 4, 362-367 (1999).
[CrossRef]

Strickland, D.

Tal, E.

Trebino, R.

Tull, J. X.

Vdovin, G.

Walmsley, I. A.

Warren, W. S.

Weber, P. M.

C. J. Bardeen, V. V. Yakovlev, J. A. Squier, K. R. Wilson, S. D. Carpenter, and P. M. Weber, “Effect of pulse shape on the efficiency of multiphoton process: implications for biological microscopy,” J. Biomed. Opt. 4, 362-367 (1999).
[CrossRef]

Weiner, A. M.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929-1960 (2000).
[CrossRef]

A. M. Weiner, J. P. Heritage, and J. A. Salehi, “Encoding and decoding of femtosecond pulses,” Opt. Lett. 13, 300-302 (1988).
[CrossRef] [PubMed]

I. H. Chowdhury, X. Xu, and A. M. Weiner, “Laser machining using temporally controlled ultrafast pulses,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2004), paper CThD5.
[PubMed]

Wiersma, D.

Wilson, K.

Wilson, K. R.

C. J. Bardeen, V. V. Yakovlev, J. A. Squier, K. R. Wilson, S. D. Carpenter, and P. M. Weber, “Effect of pulse shape on the efficiency of multiphoton process: implications for biological microscopy,” J. Biomed. Opt. 4, 362-367 (1999).
[CrossRef]

Windeler, R. S.

Xu, L.

Xu, X.

I. H. Chowdhury, X. Xu, and A. M. Weiner, “Laser machining using temporally controlled ultrafast pulses,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2004), paper CThD5.
[PubMed]

Yakovlev, V. V.

C. J. Bardeen, V. V. Yakovlev, J. A. Squier, K. R. Wilson, S. D. Carpenter, and P. M. Weber, “Effect of pulse shape on the efficiency of multiphoton process: implications for biological microscopy,” J. Biomed. Opt. 4, 362-367 (1999).
[CrossRef]

Zeek, E.

IEEE J. Quantum Electron. (1)

K. Sala, G. Kenney-Wallace, and G. Hall, “CW autocorrelation measurements of picosecond laser pulses,” IEEE J. Quantum Electron. 16, 990-996 (1980).
[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. Acoust. Soc. Am. (1)

R. A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 67, 1232-1246 (1980).
[CrossRef]

J. Biomed. Opt. (1)

C. J. Bardeen, V. V. Yakovlev, J. A. Squier, K. R. Wilson, S. D. Carpenter, and P. M. Weber, “Effect of pulse shape on the efficiency of multiphoton process: implications for biological microscopy,” J. Biomed. Opt. 4, 362-367 (1999).
[CrossRef]

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

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

Opt. Express (2)

Opt. Lett. (9)

A. M. Weiner, J. P. Heritage, and J. A. Salehi, “Encoding and decoding of femtosecond pulses,” Opt. Lett. 13, 300-302 (1988).
[CrossRef] [PubMed]

E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. 24, 493-495 (1999).
[CrossRef]

W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, “Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses,” Opt. Lett. 19, 737-739 (1994).
[CrossRef] [PubMed]

E. Frumker, E. Tal, Y. Silberberg, and D. Majer, “Femtosecond pulse-shape modulation at nanosecond rates,” Opt. Lett. 30, 2796-2798 (2005).
[CrossRef] [PubMed]

X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O'Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174-1176 (2002).
[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]

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

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 projection,” Opt. Lett. 19, 2152-2154 (1994).
[CrossRef] [PubMed]

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

Rev. Sci. Instrum. (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71, 1929-1960 (2000).
[CrossRef]

Other (3)

I. H. Chowdhury, X. Xu, and A. M. Weiner, “Laser machining using temporally controlled ultrafast pulses,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2004), paper CThD5.
[PubMed]

L. Cohen, Time-Frequency Analysis (Prentice-Hall, 1995).

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, 2002).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Schematic of a SHG FROG apparatus. A pulse is split into two, one pulse gates the other in the SHG crystal, and the relative delay is varied. The nonlinear-optical signal pulse spectrum is then measured versus delay. In PG FROG, the nonlinearity is polarization gating in glass, and crossed polarizers are used. In XFROG, an independent (previously measured) reference pulse is used instead of one of the unknown pulses. Other FROG geometries exist and use other nonlinear-optical processes.

Fig. 2
Fig. 2

Generalization-projections algorithm for FROG.

Fig. 3
Fig. 3

Moderately complex pulse with TBP = 4.7 . (a) Original SHG FROG trace with noise, (b) retrieved trace, (c) generated and retrieved spectral intensity and phase, (d) generated and retrieved temporal intensity and phase. In (c) and (d) (and in all subsequent analogous figures), the generated pulse is indicated by curves and the retrieved pulse by dots. Good convergence has occurred here. The FROG trace is well contained in the image window. For better display, the shown FROG traces were cut from the generated one, which had a wavelength range from 327.675 to 514.385 nm and a delay range from 768 to 762 fs . The maximum value of three rows and columns along the perimeter of the image window is 0.88% of the peak value after background subtraction, so the trace was only slightly cropped.

Fig. 4
Fig. 4

Very complex pulse with TBP = 40.6 . (a) Original SHG FROG trace with noise, (b) retrieved trace, (c) generated and retrieved spectral intensity and phase, (d) generated and retrieved temporal intensity and phase. Good convergence has occurred here. The generated trace (here and in later figures also) appears somewhat darker due to the additive noise applied to it (and whose mean has been subtracted prior to running the algorithm); this subtraction and the algorithm combine to remove most of the added noise. Indeed, note the identical structure in both pulses and traces. For better display, the shown FROG trace is cut from the generated one with a wavelength range from 326.503 to 512.02 nm and a delay range from 1536 to 1530 fs . The FROG trace is well contained in the image window. The maximal value of the three rows and columns along the perimeter of the image window is 1.25% of the peak value after background subtraction.

Fig. 5
Fig. 5

Extremely complex pulse with TBP = 94.3 . (a) Original SHG FROG trace, (b) retrieved FROG trace (with noise), (c) generated and retrieved spectral intensity and phase, (d) generated and retrieved temporal intensity and phase. Good convergence has occurred here. The FROG trace is well contained in the image window. For better display, the shown FROG trace is cut from the generated one with a wavelength range from 327.348 to 514.385 nm and a delay range from 3072 to 3066 fs . The maximal value of three rows and columns along the perimeter of the image window is 1.11% of the peak value after background subtraction.

Fig. 6
Fig. 6

A pulse for which convergence has not been achieved. (a) Generated SHG FROG trace with noise, (b) retrieved trace, (c) generated and retrieved spectral intensity and phase, (d) generated and retrieved temporal intensity and phase. Note the discrepancies between the generated and retrieved pulses. For better display, the shown FROG trace is cut from the generated one with a wavelength range from 327.464 to 514.385 nm and a delay range from 1530 to 1536 fs . The FROG trace is not as well contained in the image window as in the previous examples, perhaps the reason for the poor convergence. The maximal value of the three rows and columns along the perimeter of the image window is 1.51% of the peak value after background subtraction.

Fig. 7
Fig. 7

Histogram of FROG errors for 30 pulses with a TBP value from 30 to 40, showing a clear delineation between converging (FROG error < 1 % ) and nonconverging (FROG error > 1 % ) cases.

Fig. 8
Fig. 8

Number of initial guesses required for correct pulse retrieval in SHG FROG versus TBP for the pulses in our analysis. Note that most pulses can be retrieved in SHG FROG using only a few initial guesses, but some (shown as requiring ten pulses) cannot.

Fig. 9
Fig. 9

Statistical analysis of the performance of the GP algorithm in SHG FROG. In most cases, when convergence is not achieved after one initial guess, convergence is achieved after a few more tries, but not always.

Fig. 10
Fig. 10

Example of PG FROG for measuring a complex pulse (here a pulse with TBP = 15.5 ). (a) Generated FROG trace with noise, (b) retrieved trace, (c) generated and retrieved spectral intensity and phase, (d) generated and retrieved temporal intensity and phase. For better display, the shown FROG trace is cut from the generated one with a wavelength range from 327.464 to 514.385 nm and a delay range from 1536 to 1530 fs . The PG FROG trace is well contained in the image window. The maximal value of three columns along the perimeter of the image window is 0.89% of the peak value after background subtraction.

Fig. 11
Fig. 11

Number of initial guesses required for correct pulse retrieval in PG FROG versus TBP. Note that most pulses can be retrieved using only one initial guess, and nearly all can be retrieved after two or three.

Fig. 12
Fig. 12

Statistical analysis of the performance of the GP algorithm in PG FROG.

Fig. 13
Fig. 13

Example of XFROG for measuring complex pulses (here a pulse with TBP = 66 ). (a) Generated FROG trace with noise, (b) retrieved trace, (c) generated and retrieved spectral intensity and phase, (d) generated and retrieved temporal intensity and phase. For better display, the shown FROG trace is cut from the generated one with a wavelength range from 343.02 to 480.065 nm and a delay range from 2048 to 2040 fs . The XFROG trace is well contained in the image window. The maximal value of three columns along the perimeter of the image window is 1.013% of the peak value.

Fig. 14
Fig. 14

Statistical analysis of the performance of the XFROG GP algorithm. Convergence is always achieved after only one initial guess, even for extremely complicated pulses.

Fig. 15
Fig. 15

Logarithmic plots of the generated and retrieved pulses from Fig. 4c (the spectrum of the pulse) and Fig. 4d (the temporal intensity of the pulse).

Equations (6)

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I FROG ( ω , τ ) = E ( t ) E g ( t τ ) exp ( i ω t ) d t 2 ,
TBP rms = t rms ω rms ,
where t rms 2 = t t 2 = t 2 t 2 ,
t 2 = t 2 I ( t ) d t ,
ω rms 2 = A ( t ) 2 d t + A ( t ) 2 ϕ ( t ) 2 .
I FROG ( η ¯ ) ( ω i , τ j ) = I FROG ( ω i , τ j ) + η i j α η ¯ ,

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