Abstract

We describe a unique approach for extracting the temporal profile of ultrashort laser pulses from typical autocorrelation measurements. The use of the constraint that intensity is a nonnegative quantity enables an iterative numerical algorithm to reconstruct pulse shapes in a one-dimensional procedure. With the reconstruction of the intensity profile, the Gerchberg–Saxton algorithm can be used to retrieve the phase of the electric field from a spectral measurement. Because these procedures are carried out in one dimension, they are numerically much faster than two-dimensional techniques such as frequency-resolved optical gating. Their high computational efficiency can save substantial time by constructing good trial solutions for the more accurate but slower procedure of frequency-resolved optical gating.

© 1998 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  15. I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Attosecond pulse generation in the single-cycle regime,” Phys. Rev. Lett. 78, 1251–1254 (1997).
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    [Crossref]
  18. W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
    [Crossref] [PubMed]
  19. B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  25. R. W. Gerchberg and W. O. Saxton, Optik 35, 237–246 (1972).
  26. E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
    [Crossref]
  27. Yu. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [Crossref]
  28. T. R. Crimmins and J. R. Fienup, “Ambiguity of phase retrieval for functions with disconnected support,” J. Opt. Soc. Am. 71, 1026–1028 (1981).
    [Crossref]
  29. T. R. Crimmins and J. R. Fienup, “Uniqueness of phase retrieval for functions with sufficiently disconnected support,” J. Opt. Soc. Am. 73, 218–221 (1983).
    [Crossref]
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1997 (1)

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Attosecond pulse generation in the single-cycle regime,” Phys. Rev. Lett. 78, 1251–1254 (1997).
[Crossref]

1996 (3)

V. T. Platonenko, V. V. Strelkov, G. Ferranti, V. Miceli, and E. Fiordilino, “Control of the spectral width and pulse duration of a single high-order harmonic,” Laser Phys. 6, 1164–1167 (1996).

J. K. Rhee, T. S. Sosnowski, A. C. Tien, and T. B. Norris, “Real-time dispersion analyzer of femtosecond laser pulses with use of a spectrally and temporally resolved upconversion technique,” J. Opt. Soc. Am. B 13, 1780–1785 (1996).
[Crossref]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[Crossref] [PubMed]

1995 (2)

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410–3413 (1995).
[Crossref] [PubMed]

1994 (4)

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]

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowicz, “A simple method for the determination of the intensity and phase of ultrashort optical pulses,” Opt. Commun. 113, 79–84 (1994).
[Crossref]

E. Fiordilino and V. Miceli, “Laser pulse shape effects in harmonic generation from a two-level atom,” J. Mod. Opt. 41, 1415–1426 (1994).
[Crossref]

1993 (3)

1991 (2)

1989 (1)

K. Naganuma, K. Mogi, and H. Yamada, “General method of ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[Crossref]

1985 (1)

1983 (2)

T. Andersson and S. T. Eng, “Determination of the pulse response from intensity autocorrelation measurements of ultrashort laser pulses,” Opt. Commun. 47, 228–290 (1983).
[Crossref]

T. R. Crimmins and J. R. Fienup, “Uniqueness of phase retrieval for functions with sufficiently disconnected support,” J. Opt. Soc. Am. 73, 218–221 (1983).
[Crossref]

1982 (1)

1981 (1)

1979 (1)

Yu. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[Crossref]

1978 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237–246 (1972).

1971 (1)

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

1967 (1)

H. P. Weber, “Method for pulsewidth measurement of ultrashort light pulses generated by phase-locked lasers using nonlinear optics,” J. Appl. Phys. 38, 2231–2234 (1967).
[Crossref]

1964 (1)

E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[Crossref]

Andersson, T.

T. Andersson and S. T. Eng, “Determination of the pulse response from intensity autocorrelation measurements of ultrashort laser pulses,” Opt. Commun. 47, 228–290 (1983).
[Crossref]

Banyai, W. C.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

Bardeen, C. J.

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410–3413 (1995).
[Crossref] [PubMed]

Bloom, D. M.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

Bruck, Yu. M.

Yu. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[Crossref]

Che, Jianwei

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Chilla, J. L. A.

Christov, I. P.

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Attosecond pulse generation in the single-cycle regime,” Phys. Rev. Lett. 78, 1251–1254 (1997).
[Crossref]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[Crossref] [PubMed]

Crimmins, T. R.

Dahleh, M.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[Crossref] [PubMed]

DeLong, K. W.

Diels, J.-C.

Eng, S. T.

T. Andersson and S. T. Eng, “Determination of the pulse response from intensity autocorrelation measurements of ultrashort laser pulses,” Opt. Commun. 47, 228–290 (1983).
[Crossref]

Ferranti, G.

V. T. Platonenko, V. V. Strelkov, G. Ferranti, V. Miceli, and E. Fiordilino, “Control of the spectral width and pulse duration of a single high-order harmonic,” Laser Phys. 6, 1164–1167 (1996).

Fienup, J.

J. Fienup, “Improved synthesis and computational methods for computer-generated holograms,” Ph.D. dissertation (Stanford University, Palo Alto, Calif., 1975).

Fienup, J. R.

Fiordilino, E.

V. T. Platonenko, V. V. Strelkov, G. Ferranti, V. Miceli, and E. Fiordilino, “Control of the spectral width and pulse duration of a single high-order harmonic,” Laser Phys. 6, 1164–1167 (1996).

E. Fiordilino and V. Miceli, “Laser pulse shape effects in harmonic generation from a two-level atom,” J. Mod. Opt. 41, 1415–1426 (1994).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge University, New York, 1992), p. 805.

Fountaine, J. J.

Franco, M. A.

B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowicz, “A simple method for the determination of the intensity and phase of ultrashort optical pulses,” Opt. Commun. 113, 79–84 (1994).
[Crossref]

Fujimoto, J. G.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237–246 (1972).

Godil, A. A.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

Hofstetter, E. M.

E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[Crossref]

Hunter, J.

Ippen, E. P.

Kane, D. J.

Kapteyn, H. C.

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Attosecond pulse generation in the single-cycle regime,” Phys. Rev. Lett. 78, 1251–1254 (1997).
[Crossref]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[Crossref] [PubMed]

Kauffman, M. T.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

Kohler, B.

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Krause, J. L.

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Martinez, O. E.

McMichael, I. C.

Messina, M.

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Miceli, V.

V. T. Platonenko, V. V. Strelkov, G. Ferranti, V. Miceli, and E. Fiordilino, “Control of the spectral width and pulse duration of a single high-order harmonic,” Laser Phys. 6, 1164–1167 (1996).

E. Fiordilino and V. Miceli, “Laser pulse shape effects in harmonic generation from a two-level atom,” J. Mod. Opt. 41, 1415–1426 (1994).
[Crossref]

Mogi, K.

K. Naganuma, K. Mogi, and H. Yamada, “General method of ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[Crossref]

Murnane, M. M.

I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Attosecond pulse generation in the single-cycle regime,” Phys. Rev. Lett. 78, 1251–1254 (1997).
[Crossref]

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[Crossref] [PubMed]

Mysyrowicz, A.

B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowicz, “A simple method for the determination of the intensity and phase of ultrashort optical pulses,” Opt. Commun. 113, 79–84 (1994).
[Crossref]

Naganuma, K.

K. Naganuma, K. Mogi, and H. Yamada, “General method of ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[Crossref]

Nibbering, E. T. J.

B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowicz, “A simple method for the determination of the intensity and phase of ultrashort optical pulses,” Opt. Commun. 113, 79–84 (1994).
[Crossref]

Norris, T. B.

Paye, J.

Peatross, J.

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[Crossref] [PubMed]

Platonenko, V. T.

V. T. Platonenko, V. V. Strelkov, G. Ferranti, V. Miceli, and E. Fiordilino, “Control of the spectral width and pulse duration of a single high-order harmonic,” Laser Phys. 6, 1164–1167 (1996).

Prade, B. S.

B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowicz, “A simple method for the determination of the intensity and phase of ultrashort optical pulses,” Opt. Commun. 113, 79–84 (1994).
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge University, New York, 1992), p. 805.

Rabitz, H.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[Crossref] [PubMed]

Ramaswamy, M.

Rhee, J. K.

Rundquist, A.

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[Crossref] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237–246 (1972).

Schins, J. M.

B. S. Prade, J. M. Schins, E. T. J. Nibbering, M. A. Franco, and A. Mysyrowicz, “A simple method for the determination of the intensity and phase of ultrashort optical pulses,” Opt. Commun. 113, 79–84 (1994).
[Crossref]

Schwentner, N.

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Shank, C. V.

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410–3413 (1995).
[Crossref] [PubMed]

Simoni, F.

Sodin, L. G.

Yu. M. Bruck and L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[Crossref]

Sosnowski, T. S.

Strelkov, V. V.

V. T. Platonenko, V. V. Strelkov, G. Ferranti, V. Miceli, and E. Fiordilino, “Control of the spectral width and pulse duration of a single high-order harmonic,” Laser Phys. 6, 1164–1167 (1996).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge University, New York, 1992), p. 805.

Tien, A. C.

Treacy, E. B.

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

Trebino, R.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge University, New York, 1992), p. 805.

Wang, Q.

C. J. Bardeen, Q. Wang, and C. V. Shank, “Selective excitation of vibrational wave packet motion using chirped pulses,” Phys. Rev. Lett. 75, 3410–3413 (1995).
[Crossref] [PubMed]

Warren, W. S.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[Crossref] [PubMed]

Weber, H. P.

H. P. Weber, “Method for pulsewidth measurement of ultrashort light pulses generated by phase-locked lasers using nonlinear optics,” J. Appl. Phys. 38, 2231–2234 (1967).
[Crossref]

White, W. E.

Whitnell, R. M.

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Wilson, K. R.

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Yakovlev, V. V.

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Yamada, H.

K. Naganuma, K. Mogi, and H. Yamada, “General method of ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[Crossref]

Yan, C.

Yan, Yijing

B. Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum control of wave packet evolution with tailored femtosecond pulses,” Phys. Rev. Lett. 74, 3360–3363 (1995).
[Crossref] [PubMed]

Zhou, J.

I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Naganuma, K. Mogi, and H. Yamada, “General method of ultrashort light pulse chirp measurement,” IEEE J. Quantum Electron. 25, 1225–1233 (1989).
[Crossref]

IEEE Trans. Inf. Theory (1)

E. M. Hofstetter, “Construction of time-limited functions with specified autocorrelation functions,” IEEE Trans. Inf. Theory IT-10, 119–126 (1964).
[Crossref]

J. Appl. Phys. (2)

H. P. Weber, “Method for pulsewidth measurement of ultrashort light pulses generated by phase-locked lasers using nonlinear optics,” J. Appl. Phys. 38, 2231–2234 (1967).
[Crossref]

E. B. Treacy, “Measurement and interpretation of dynamic spectrograms of picosecond light pulses,” J. Appl. Phys. 42, 3848–3858 (1971).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1

Schematic of an autocorrelation measurement using second-harmonic generation.

Fig. 2
Fig. 2

Schematic of the TIVI method, which is able to recover the intensity temporal profile.

Fig. 3
Fig. 3

(a) Autocorrelation traces arising from Gaussian, squared hyperbolic secant, and squared sinc pulses. (b) Original pulses (thin solid curves) and reconstructed pulses obtained by the TIVI method (dashed curves).

Fig. 4
Fig. 4

(a)–(c) Autocorrelation traces arising from three arbitrarily chosen pulse shapes. (d)–(f) Original pulses (thin solid curves) and reconstructed pulses obtained by the TIVI method (dashed curves). Note that in (d) the reconstruction method failed to recover the small bump on the left.

Fig. 5
Fig. 5

(a) Autocorrelation trace of an arbitrarily chosen pulse shape. (b) The same autocorrelation trace with a random additive noise of 25% (fluctuation range). (c) Original pulse (thin solid curve) and reconstructed pulse obtained by the TIVI method (dashed curve) without noise. (d) Original pulse (thin solid curve) and reconstructed pulse obtained by the TIVI method (dashed curve) in the presence of noise.

Fig. 6
Fig. 6

(a), (b) Autocorrelation traces arising from pulses for which reconstruction fails. (c), (d) Original pulses (thin solid curves) and reconstructed pulses obtained by the TIVI method (dashed curves).

Fig. 7
Fig. 7

(a) Errors calculated for the curves seen in Fig. 4(d) (solid curve), Fig. 4(e) (long-dashed curve), and Fig. 4(f) (short-dashed curve). (b) Errors calculated for the curves seen in Fig. 6(c) (solid curve) and Fig. 6(d) (dashed curve).

Fig. 8
Fig. 8

(a) Cross-correlation trace arising from the two pulses seen in Fig. 6. (b), (c) Original pulses (thin solid curves) and reconstructed pulses (dashed curves) obtained by the TIVI method while utilizing the information in the cross-correlation trace.

Fig. 9
Fig. 9

Schematic of the Gerchberg–Saxton algorithm applied to the problem of recovering the phase of the electric field.

Fig. 10
Fig. 10

(a) Original amplitude (thin solid curve) and reconstructed amplitude (dashed curve) obtained by the TIVI method. (b) Original phase (thin solid curve) and reconstructed phase (dashed curve) obtained by the Gerchberg–Saxton algorithm. (c) Original amplitude (thin solid curve) and reconstructed amplitude (dashed curve) obtained by the FROG algorithm. (d) Original phase (thin solid curve) and reconstructed phase (dashed curve) obtained by the FROG algorithm.

Fig. 11
Fig. 11

(a) Synthesized FROG trace. (b) FROG trace generated after applying TIVI–GS to the autocorrelation trace and power spectrum with 10% noise, respectively. (c) Trace generated by the FROG algorithm using the result of TIVI–GS as a trial solution.

Equations (6)

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Sig(τ)=-I(t)I(t-τ)dt.
Sig(ω)F{Sig(τ)}=|I(ω)|2.
I(t)=F-1{Sig(ω) exp[iϕ(ω)]}.
Sig12(τ)=-I1(t)I2(t-τ)dt,
Sig12(ω)F{Sig12(τ)}=I1(ω)I2*(ω).
Sig11(ω)Sig22(ω)=|Sig12(ω)|2.

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