Abstract

Frequency-resolved optical gating (FROG) is a technique used to measure ultrafast laser pulses by optically producing a spectrogram, or FROG trace, of the measured pulse. While a great deal of information about the pulse can be gleaned from its FROG trace, quantitative pulse information must be obtained using an iterative two-dimensional phase retrieval algorithm. A general spectrogram/sonogram inversion algorithm called principal components generalized projections (PCGP) that can be applied to pulse measurement schemes, such as FROG, is reviewed. The algorithm is fast, robust, and can invert FROG traces in real time, making commercial pulse measurement systems based on FROG a reality. Measurement rates are no longer algorithm limited; they are data-acquisition limited. Also, because of some of its unique properties, the PCGP algorithm has found applications in measuring attosecond pulses and measuring telecommunications pulses. In addition, the PCGP structures the inversion and measurement process in a way that can allow new insights into convergence properties of spectrogram and sonogram inversion algorithms.

© 2008 Optical Society of America

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2006 (1)

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

2005 (2)

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

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

2004 (1)

C. Dorrer and I. Kang, “Real-time implementation of linear spectrograms for the characterization of high-bit rate optical pulse trains,” IEEE Photon. Technol. Lett. 16, 858-860 (2004).
[CrossRef]

2003 (1)

2001 (1)

2000 (1)

1999 (2)

D. J. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421-431 (1999).
[CrossRef]

D. T. Reid, “Algorithm for complete and rapid retrieval of ultrashort pulse, amplitude and phase from a sonogram,” IEEE J. Quantum Electron. 35, 1584-1589 (1999).
[CrossRef]

1998 (2)

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]

A. Kwok, L. Jusinski, M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271-277 (1998).
[CrossRef]

1997 (4)

1996 (4)

1995 (5)

1994 (5)

1993 (3)

1987 (2)

Altucci, C.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Anton, H.

H. Anton, Elementary Linear Algebra, 2nd ed. (Wiley, 1977).

Avaldi, L.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Backus, S.

Benedetti, E.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Bolton, P. R.

Bullock, A. B.

Calegari, F.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Chu, K.-C. J.

Clement, T.

Clement, T. S.

De Silvestri, S.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Decker, C. D.

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, 3277-3295 (1997).
[CrossRef]

K. W. DeLong, C. L. Ledera, R. Trebino, B. Kohler, and K. R. Wilson, “Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20, 486-488 (1995).
[CrossRef] [PubMed]

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. DeLong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483-485 (1995).
[CrossRef] [PubMed]

K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463-2466 (1995).
[CrossRef]

K. W. DeLong and R. Trebino, “Improved ultrashort phase-retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429-2437 (1994).
[CrossRef]

D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse using frequency-resolved optical gating,” Opt. Lett. 19, 1061-1063 (1994).
[CrossRef] [PubMed]

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

K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206-2215 (1994).
[CrossRef]

Dorrer, C.

C. Dorrer and I. Kang, “Real-time implementation of linear spectrograms for the characterization of high-bit rate optical pulse trains,” IEEE Photon. Technol. Lett. 16, 858-860 (2004).
[CrossRef]

Downer, M. C.

Feit, M. D.

Fienup, J. R.

Fittinghoff, D. N.

A. Kwok, L. Jusinski, M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271-277 (1998).
[CrossRef]

J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Transient-grating frequency-resolved optical gating,” Opt. Lett. 22, 519-521 (1997).
[CrossRef] [PubMed]

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, 3277-3295 (1997).
[CrossRef]

P. R. Bolton, A. B. Bullock, C. D. Decker, M. D. Feit, A. J. P. Megofna, P. E. Young, and D. N. Fittinghoff, “Propagation of intense, ultraviolet laser pulses through metal vapor: refraction-limited behavior for single pulses,” J. Opt. Soc. Am. B 13, 336-346 (1996).
[CrossRef]

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

Flammini, R.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1995).

Fujimoto, J. G.

Galatsanos, N. P.

Gregonis, E.

I. Thomann, E. Gregonis, M. Murnane, and H. Kapteyn, “Temporal Characterization of Energy-Tunable EUV Pulses in the Sub-Optical-Cycle Regime Using FROG-CRAB,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CPDB4.
[PubMed]

Gu, X.

Habetler, G. J.

Hunter, J.

Ippen, E. P.

Itatani, J.

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

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing, 1st ed. (Prentice Hall, 1989).

Jusinski, L.

A. Kwok, L. Jusinski, M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271-277 (1998).
[CrossRef]

Kane, D. J.

D. J. Kane, J. Weston, and K.-C. J. Chu, “Real-time inversion of polarization gate frequency resolved optical gating spectrograms,” Appl. Opt. 42, 1140-1144 (2003).
[CrossRef] [PubMed]

D. J. Kane, F. G. Omenetto, and A. J. Taylor, “Convergence test for inversion of frequency-resolved optical gating spectrograms,” Opt. Lett. 25, 1216-1218 (2000).
[CrossRef]

D. J. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421-431 (1999).
[CrossRef]

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]

D. J. Kane, G. Rodriguez, A. J. Taylor, and T. 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]

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, 3277-3295 (1997).
[CrossRef]

T. S. Clement, A. J. Taylor, and D. J. Kane, “Single-shot measurement of the amplitude and phase of ultrashort laser pulses in the violet,” Opt. Lett. 20, 70-72 (1995).
[CrossRef] [PubMed]

D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse using frequency-resolved optical gating,” Opt. Lett. 19, 1061-1063 (1994).
[CrossRef] [PubMed]

R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort laser pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101-1111 (1993).
[CrossRef]

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

D. J. Kane, “New algorithm for the measurement of two ultrashort laser pulses from a single spectrogram,” presented at the Conference on Lasers and Electro-Optics, Baltimore, Md., May 19-23 1997.

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of a femtosecond laser pulse,” presented at Generation and Measurement of Ultrashort Laser Pulses, SPIE/OE Lase, Los Angeles, Calif., January 16-23 1993.

Kang, I.

C. Dorrer and I. Kang, “Real-time implementation of linear spectrograms for the characterization of high-bit rate optical pulse trains,” IEEE Photon. Technol. Lett. 16, 858-860 (2004).
[CrossRef]

Kang, K. S.

C. H. Nam, K. T. Kim, K. S. Kang, D. H. Ko, and J. Y. Park, “Complete temporal reconstruction of attosecond harmonic pulses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CTuW3.
[PubMed]

Kapteyn, H.

I. Thomann, E. Gregonis, M. Murnane, and H. Kapteyn, “Temporal Characterization of Energy-Tunable EUV Pulses in the Sub-Optical-Cycle Regime Using FROG-CRAB,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CPDB4.
[PubMed]

Kapteyn, H. C.

Kim, K. T.

C. H. Nam, K. T. Kim, K. S. Kang, D. H. Ko, and J. Y. Park, “Complete temporal reconstruction of attosecond harmonic pulses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CTuW3.
[PubMed]

Kimmel, M.

Ko, D. H.

C. H. Nam, K. T. Kim, K. S. Kang, D. H. Ko, and J. Y. Park, “Complete temporal reconstruction of attosecond harmonic pulses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CTuW3.
[PubMed]

Kohler, B.

Krumbugel, M. A.

A. Kwok, L. Jusinski, M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271-277 (1998).
[CrossRef]

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, 3277-3295 (1997).
[CrossRef]

Kwok, A.

A. Kwok, L. Jusinski, M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271-277 (1998).
[CrossRef]

Ledera, C. L.

Levi, A.

Mairesse, Y.

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

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

Megofna, A. J. P.

Milane, R. P.

Murnane, M.

I. Thomann, E. Gregonis, M. Murnane, and H. Kapteyn, “Temporal Characterization of Energy-Tunable EUV Pulses in the Sub-Optical-Cycle Regime Using FROG-CRAB,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CPDB4.
[PubMed]

Murnane, M. M.

Nam, C. H.

C. H. Nam, K. T. Kim, K. S. Kang, D. H. Ko, and J. Y. Park, “Complete temporal reconstruction of attosecond harmonic pulses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CTuW3.
[PubMed]

Nisoli, M.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Omenetto, F. G.

O'Shea, P.

Park, J. Y.

C. H. Nam, K. T. Kim, K. S. Kang, D. H. Ko, and J. Y. Park, “Complete temporal reconstruction of attosecond harmonic pulses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CTuW3.
[PubMed]

Paye, J.

Peatross, J.

Poletto, L.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1995).

Quéré, F.

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

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

Quéré, Fabien

Fabien Quéré (Personal communication).

Ramaswamy, M.

Reid, D. T.

D. T. Reid, “Algorithm for complete and rapid retrieval of ultrashort pulse, amplitude and phase from a sonogram,” IEEE J. Quantum Electron. 35, 1584-1589 (1999).
[CrossRef]

Rodriguez, G.

Rundquist, A.

Sansone, G.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Siders, C. W.

C. W. Siders, A. J. Taylor, and M. C. Downer, “Multi-pulse interferometric frequency-resolved optical gating: real time phase-sensitive imaging of ultrafast dynamics,” Opt. Lett. 22, 624-626 (1997).
[CrossRef] [PubMed]

C. W. Siders, J. L. W. Siders, and A. J. Taylor, “Femtosecond coherent spectroscopy at 800nm: MI-FROG measures high-field ionization rates in gases,” presented at the Ultrafast Phenomena XI, Garmisch-Partenkirchen, Germany, July 12-17 1998.

Siders, J. L. W.

C. W. Siders, J. L. W. Siders, and A. J. Taylor, “Femtosecond coherent spectroscopy at 800nm: MI-FROG measures high-field ionization rates in gases,” presented at the Ultrafast Phenomena XI, Garmisch-Partenkirchen, Germany, July 12-17 1998.

Squier, J.

Stagira, S.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Stark, H.

Sweetser, J. N.

A. Kwok, L. Jusinski, M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271-277 (1998).
[CrossRef]

J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Transient-grating frequency-resolved optical gating,” Opt. Lett. 22, 519-521 (1997).
[CrossRef] [PubMed]

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, 3277-3295 (1997).
[CrossRef]

Taft, G.

Taylor, A. J.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1995).

Thomann, I.

I. Thomann, E. Gregonis, M. Murnane, and H. Kapteyn, “Temporal Characterization of Energy-Tunable EUV Pulses in the Sub-Optical-Cycle Regime Using FROG-CRAB,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CPDB4.
[PubMed]

Trebino, R.

P. O'Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932-934 (2001).
[CrossRef]

A. Kwok, L. Jusinski, M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271-277 (1998).
[CrossRef]

J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Transient-grating frequency-resolved optical gating,” Opt. Lett. 22, 519-521 (1997).
[CrossRef] [PubMed]

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, 3277-3295 (1997).
[CrossRef]

K. W. DeLong, C. L. Ledera, R. Trebino, B. Kohler, and K. R. Wilson, “Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20, 486-488 (1995).
[CrossRef] [PubMed]

K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463-2466 (1995).
[CrossRef]

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. DeLong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483-485 (1995).
[CrossRef] [PubMed]

K. W. DeLong and R. Trebino, “Improved ultrashort phase-retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429-2437 (1994).
[CrossRef]

D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse using frequency-resolved optical gating,” Opt. Lett. 19, 1061-1063 (1994).
[CrossRef] [PubMed]

K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206-2215 (1994).
[CrossRef]

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

R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort laser pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101-1111 (1993).
[CrossRef]

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of a femtosecond laser pulse,” presented at Generation and Measurement of Ultrashort Laser Pulses, SPIE/OE Lase, Los Angeles, Calif., January 16-23 1993.

Velotta, R.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1995).

Villoresi, P.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Vozzi, C.

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Walmsley, I. A.

Weston, J.

White, W. E.

Wilson, K. R.

Wong, V.

Yakovlev, V. V.

Yang, Y.

Young, P. E.

Yudilevich, E.

Zeek, Z.

Appl. Opt. (1)

IEEE J. Quantum Electron. (3)

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

D. T. Reid, “Algorithm for complete and rapid retrieval of ultrashort pulse, amplitude and phase from a sonogram,” IEEE J. Quantum Electron. 35, 1584-1589 (1999).
[CrossRef]

D. J. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421-431 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

A. Kwok, L. Jusinski, M. A. Krumbugel, J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Frequency-resolved optical gating using cascaded second-order nonlinearities,” IEEE J. Sel. Top. Quantum Electron. 4, 271-277 (1998).
[CrossRef]

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]

IEEE Photon. Technol. Lett. (1)

C. Dorrer and I. Kang, “Real-time implementation of linear spectrograms for the characterization of high-bit rate optical pulse trains,” IEEE Photon. Technol. Lett. 16, 858-860 (2004).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (6)

J. Opt. Soc. Am. B (5)

Opt. Lett. (12)

J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946-1948 (1993).
[CrossRef] [PubMed]

D. J. Kane, A. J. Taylor, R. Trebino, and K. W. DeLong, “Single-shot measurement of the intensity and phase of a femtosecond UV laser pulse using frequency-resolved optical gating,” Opt. Lett. 19, 1061-1063 (1994).
[CrossRef] [PubMed]

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

J. N. Sweetser, D. N. Fittinghoff, and R. Trebino, “Transient-grating frequency-resolved optical gating,” Opt. Lett. 22, 519-521 (1997).
[CrossRef] [PubMed]

C. W. Siders, A. J. Taylor, and M. C. Downer, “Multi-pulse interferometric frequency-resolved optical gating: real time phase-sensitive imaging of ultrafast dynamics,” Opt. Lett. 22, 624-626 (1997).
[CrossRef] [PubMed]

S. Backus, J. Peatross, Z. Zeek, A. Rundquist, G. Taft, M. M. Murnane, and H. C. Kapteyn, “16-fs, 1-μJ ultraviolet pulses generated by third-harmonic conversion in air,” Opt. Lett. 21, 665-667 (1996).
[CrossRef] [PubMed]

A. J. Taylor, G. Rodriguez, and T. S. Clement, “Determination of n2 by direct measurement of the optical phase,” Opt. Lett. 21, 1812-1814 (1996).
[CrossRef] [PubMed]

D. J. Kane, F. G. Omenetto, and A. J. Taylor, “Convergence test for inversion of frequency-resolved optical gating spectrograms,” Opt. Lett. 25, 1216-1218 (2000).
[CrossRef]

P. O'Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932-934 (2001).
[CrossRef]

T. S. Clement, A. J. Taylor, and D. J. Kane, “Single-shot measurement of the amplitude and phase of ultrashort laser pulses in the violet,” Opt. Lett. 20, 70-72 (1995).
[CrossRef] [PubMed]

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. DeLong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483-485 (1995).
[CrossRef] [PubMed]

K. W. DeLong, C. L. Ledera, R. Trebino, B. Kohler, and K. R. Wilson, “Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20, 486-488 (1995).
[CrossRef] [PubMed]

Phys. Rev. A (1)

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

Rev. Sci. Instrum. (1)

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, 3277-3295 (1997).
[CrossRef]

Science (1)

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443-446 (2006).
[CrossRef] [PubMed]

Other (9)

C. H. Nam, K. T. Kim, K. S. Kang, D. H. Ko, and J. Y. Park, “Complete temporal reconstruction of attosecond harmonic pulses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CTuW3.
[PubMed]

Fabien Quéré (Personal communication).

I. Thomann, E. Gregonis, M. Murnane, and H. Kapteyn, “Temporal Characterization of Energy-Tunable EUV Pulses in the Sub-Optical-Cycle Regime Using FROG-CRAB,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CPDB4.
[PubMed]

A. K. Jain, Fundamentals of Digital Image Processing, 1st ed. (Prentice Hall, 1989).

H. Anton, Elementary Linear Algebra, 2nd ed. (Wiley, 1977).

C. W. Siders, J. L. W. Siders, and A. J. Taylor, “Femtosecond coherent spectroscopy at 800nm: MI-FROG measures high-field ionization rates in gases,” presented at the Ultrafast Phenomena XI, Garmisch-Partenkirchen, Germany, July 12-17 1998.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1995).

D. J. Kane, “New algorithm for the measurement of two ultrashort laser pulses from a single spectrogram,” presented at the Conference on Lasers and Electro-Optics, Baltimore, Md., May 19-23 1997.

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of a femtosecond laser pulse,” presented at Generation and Measurement of Ultrashort Laser Pulses, SPIE/OE Lase, Los Angeles, Calif., January 16-23 1993.

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Figures (11)

Fig. 1
Fig. 1

Schematic of an SHG FROG device. A beam splitter splits the input E ( t ) into probe and gate beams. The two beams are focused into an SHG crystal. The spectrum of the second harmonic is collected as a function of delay.

Fig. 2
Fig. 2

All FROG trace inversion algorithms work by iterating between two constraints related by a transformation with an inverse. The algorithm’s performance is determined by how it obtains the estimate of E ( t ) for the next iteration.

Fig. 3
Fig. 3

Ideal operation of a FROG inversion algorithm.

Fig. 4
Fig. 4

Steps utilized in the PCGPA for constructing a FROG trace from the outer product of two vectors. Both SHG and PG are shown. The top image plots show the outer product. The next image results from the row rotation [Eq. (11)]. A correctly oriented time-domain FROG trace can be obtained by rearranging the columns. Fourier transforming the columns produces the FROG traces shown in the bottom image plots. Adapted from [30].

Fig. 5
Fig. 5

Schematic of the PCGPA. The transformation from the outer product to the time-domain FROG trace (and vice versa) is one-to-one and invertable.

Fig. 6
Fig. 6

Data taken using the original “femtosecond oscilloscope” as discussed in the text. The FROG trace is in the upper left corner. This pulse was retrieved using the PCGPA as the inversion engine implemented on the DSP card. In only 1 s (20 iterations) the algorithm converged to a FROG trace error of less than 0.5% for a 64 × 64 FROG trace. Also important to the operation of the femtosecond oscilloscope is the stability of the algorithm; results did not significantly change even after thousands of iterations. Adapted from [20].

Fig. 7
Fig. 7

Schematic of a PG FROG device. The inset shows the gating of the pulse. The pulse is split into two replicas. The probe is time delayed relative to the gate. Both the probe and the gate are focused into a nonlinear medium, such as quartz. The gate instantaneously induces birefringence in the quartz, which causes the probe’s polarization to slightly rotate. The rotated polarized light “leaks” from the polarizer and is spectrally resolved as a function of delay to form the FROG trace.

Fig. 8
Fig. 8

Series of synthetic SHG FROG traces (square root of intensity versus frequency and time delay to show more detail) with cubic spectral phase and various amounts of noise and distortions. The first FROG trace is an ideal FROG trace with no distortions. The second FROG trace has shot noise added. (It has a per pixel rms deviation of 12 V 1 2 , where V is the pixel value varying from 0 to 62,500.) The third trace has additive noise added (per pixel rms deviation of 2% of the maximum pixel value). The fourth trace has both additive and shot noise. The fifth is a modestly filtered version of the fourth trace. The sixth is also a filtered version of the fourth trace but is visibly distorted due to filtering. After pulse retrieval, the FROG trace error, and the retrieved pulse error are (1) 0%, 0%; (2) 0.8%, 0.14%; (3) 2.0%, 3.5 %; (4) 2.2%, 2.7%; (5) 0.44%, 3.9%; (6) 0.3%, 8.5%. Adapted from [43].

Fig. 9
Fig. 9

Time and frequency marginals of FROG traces 2–4 depicted in Fig. 8. Agreement between all frequency and time marginals is excellent. Adapted from [43].

Fig. 10
Fig. 10

SVD analysis of the FROG traces depicted in Fig. 8. Any deviation from a straight line indicates either distortion or a lack of convergence of the algorithm. All of the plots are normalized to have a maximum value of 1. Adapted from [43].

Fig. 11
Fig. 11

Experimental SHG FROG trace (a) square root of the intensity, (b) retrieved pulse, and (c) SVD analysis. The SHG PCGPA was used to invert the FROG trace. Very little differences occur between the pulse and gate indicating that the FROG trace is nearly symmetric. The slight curvature in the normalized weight plot indicates only a small amount of distortion. If we assume the distortion is only due to bandwidth limitation of the doubling crystal and compensate for it, we can achieve nearly noise-limited convergence [dotted curve in (c)]. Adapted from [43].

Equations (23)

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E ( t ) = Re [ I ( t ) exp ( i ω 0 t i φ ( t ) ) ] ,
E sig ( t , τ ) = E ( t ) Γ [ E ( t τ ) ] ,
I FROG ( ω , τ ) = E ( t ) Γ [ E ( t τ ) ] exp ( i ω t ) d t 2 .
I FROG ( ω , τ ) ϕ ( ω , τ ) = E ( t ) Γ [ E ( t τ ) ] exp ( i ω t ) d t
ε FROG = [ 1 N 2 i = 1 N j = 1 N [ I CALC ( ω i , τ j ) I FROG ( ω i , τ j ) ] 2 ] 1 2 ,
Z = t , τ = 1 N E sig ( t , τ ) E ( t ) Γ [ E ( t τ ) ] 2 ,
I FROG ( ω , τ ) = E ( t ) G ( t τ ) exp ( i ω t ) d t 2 ,
E probe = [ E 1 , E 2 , E 3 , , E N ] ,
G gate = [ G 1 , G 2 , G 3 , , G N ] .
[ E 1 G 1 E 1 G 2 E 1 G 3 E 1 G 4 E 1 G N E 2 G 1 E 2 G 2 E 2 G 3 E 2 G 4 E 2 G N E 3 G 1 E 3 G 2 E 3 G 3 E 3 G 4 E 3 G N E 4 G 1 E 4 G 2 E 4 G 3 E 4 G 4 E 4 G N E N G 1 E N G 2 E N G 3 E N G 4 E N G N ] .
[ E 1 G 1 E 1 G 2 E 1 G 3 E 1 G 4 E 1 G N E 2 G 2 E 2 G 3 E 2 G 4 E 2 G 5 E 2 G 1 E 3 G 3 E 3 G 4 E 3 G 5 E 3 G 6 E 3 G 2 E 4 G 4 E 4 G 5 E 4 G 6 E 4 G 7 E 4 G 3 E N G N E N G 1 E N G 2 E N G 3 E N G N 2 ] ,
τ = 0 , τ = Δ t , τ = 2 Δ t , τ = 3 Δ t τ = Δ t .
O = U × W × V T ,
ε 2 = i , j = 1 N E outer i , j E probe i E gate j 2 ,
O O T P i = λ i P i ,
O T O G i = λ i G i ,
O = i = 1 N λ i P i G i T ,
O O T x 0 = i = 1 N κ i λ i P i ,
( O O T ) p x 0 = i = 1 N κ i λ i p P i .
O k i j = probe k i gate k j + gate k i probe k j
O k i j = probe k i gate k j + Γ 1 ( gate k ) i Γ ( probe k ) j ,
Δ t = N t N f Δ f ,
Δ f = 1 N resampled Δ t ,

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