Abstract

We describe a method for retrieving spectral phase information from second harmonic interferometric autocorrelation measurements supplemented by the use of the observed spectral intensity. By applying a combination of graduated optimization and genetic algorithms, accurate phase retrieval of laser pulses as short as a few optical cycles was obtained from the measured autocorrelation and spectral intensity. The effectiveness of the combined algorithms is demonstrated on a set of significantly different femtosecond pulse shapes.

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References

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  1. J.-C. Diels, and W. Rodulph, Ultrashort Laser Pulse Phenomena, (Elsevier, 2006), Chap. 9.
  2. T. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses, (Kluwer Academic Publishers, 2000).
  3. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23(10), 792–794 (1998).
    [CrossRef]
  4. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [CrossRef]
  5. K. Naganuma, K. Mogi, and H. Yamada, “General method for ultrashort light pulse chirp Measurement,” IEEE J. Quantum Electron. 25, 6 (1989).
  6. M. Mitchell, An Introduction To Genetic Algorithms, (MIT, 1998).
  7. R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992).
    [CrossRef] [PubMed]
  8. C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
    [CrossRef]
  9. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997).
    [CrossRef]
  10. A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, “Adaptive control of pulse phase in a chirped-pulse amplifier,” Opt. Lett. 23(24), 1915–1917 (1998).
    [CrossRef]
  11. 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(7), 493–495 (1999).
    [CrossRef]
  12. T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).
  13. B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
    [CrossRef]
  14. F. G. Omenetto, A. J. Taylor, M. D. Moores, and D. H. Reitze, “Adaptive control of femtosecond pulse propagation in optical fibers,” Opt. Lett. 26(12), 938–940 (2001).
    [CrossRef]
  15. J. W. Nicholson, F. G. Omenetto, D. J. Funk, and A. J. Taylor, “Evolving FROGS: phase retrieval from frequency-resolved optical gating measurements by use of genetic algorithms,” Opt. Lett. 24(7), 490–492 (1999).
    [CrossRef]
  16. R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74(5), 2670–2674 (2003).
    [CrossRef]
  17. E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene Ring-Opening,” Ultra-fast Phenomena 88, 249–251 (2007).
    [CrossRef]
  18. C.-W. Chen, J. Y. Huang, and C.-L. Pan, “Pulse retrieval from interferometric autocorrelation measurement by use of the population-split genetic algorithm,” Opt. Express 14, 22 (2006).
  19. K.-H. Hong, Y. S. Lee, and C. H. Nam, “Electric-field reconstruction of femtosecond laser pulses from interferometric autocorrelation using an evolutionary algorithm,” Opt. Commun. 271(1), 169–177 (2007).
    [CrossRef]
  20. J.-H. Chung and A. M. Weiner, “Ambiguity of Ultrashort Pulse Shapes Retrieved From the Intensity Autocorrelation and the Power Spectrum,” IEEE J. Sel. Top. Quantum Electron. 7(4), 656–666 (2001).
    [CrossRef]
  21. D. A. Bender, J. W. Nicholson, and M. Sheik-Bahae, “Ultrashort laser pulse characterization using modified spectrum auto-interferometric correlation (MOSAIC),” Opt. Express 16(16), 11782–11794 (2008).
    [CrossRef] [PubMed]
  22. D. A. Bender and M. Sheik-Bahae, “Modified spectrum autointerferometric correlation (MOSAIC) for single-shot pulse characterization,” Opt. Lett. 32(19), 2822–2824 (2007).
    [CrossRef] [PubMed]
  23. A. Rosenfeld, Multiresolution Image Processing and Analysis (Springer, 1984).

2008 (1)

2007 (3)

D. A. Bender and M. Sheik-Bahae, “Modified spectrum autointerferometric correlation (MOSAIC) for single-shot pulse characterization,” Opt. Lett. 32(19), 2822–2824 (2007).
[CrossRef] [PubMed]

E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene Ring-Opening,” Ultra-fast Phenomena 88, 249–251 (2007).
[CrossRef]

K.-H. Hong, Y. S. Lee, and C. H. Nam, “Electric-field reconstruction of femtosecond laser pulses from interferometric autocorrelation using an evolutionary algorithm,” Opt. Commun. 271(1), 169–177 (2007).
[CrossRef]

2006 (1)

C.-W. Chen, J. Y. Huang, and C.-L. Pan, “Pulse retrieval from interferometric autocorrelation measurement by use of the population-split genetic algorithm,” Opt. Express 14, 22 (2006).

2003 (1)

R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74(5), 2670–2674 (2003).
[CrossRef]

2001 (3)

F. G. Omenetto, A. J. Taylor, M. D. Moores, and D. H. Reitze, “Adaptive control of femtosecond pulse propagation in optical fibers,” Opt. Lett. 26(12), 938–940 (2001).
[CrossRef]

J.-H. Chung and A. M. Weiner, “Ambiguity of Ultrashort Pulse Shapes Retrieved From the Intensity Autocorrelation and the Power Spectrum,” IEEE J. Sel. Top. Quantum Electron. 7(4), 656–666 (2001).
[CrossRef]

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

2000 (1)

T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).

1999 (2)

1998 (2)

1997 (2)

C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
[CrossRef]

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997).
[CrossRef]

1992 (1)

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992).
[CrossRef] [PubMed]

1989 (1)

K. Naganuma, K. Mogi, and H. Yamada, “General method for ultrashort light pulse chirp Measurement,” IEEE J. Quantum Electron. 25, 6 (1989).

1982 (1)

Backus, S.

Bardeen, C. J.

C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
[CrossRef]

Baumert, T.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997).
[CrossRef]

Beach, N. M.

Bender, D. A.

Brixner, T.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997).
[CrossRef]

Bucksbaum, P. H.

E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene Ring-Opening,” Ultra-fast Phenomena 88, 249–251 (2007).
[CrossRef]

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

Carpenter, S. D.

C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
[CrossRef]

Carroll, E.

E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene Ring-Opening,” Ultra-fast Phenomena 88, 249–251 (2007).
[CrossRef]

Chen, C.-W.

C.-W. Chen, J. Y. Huang, and C.-L. Pan, “Pulse retrieval from interferometric autocorrelation measurement by use of the population-split genetic algorithm,” Opt. Express 14, 22 (2006).

Chung, J.-H.

J.-H. Chung and A. M. Weiner, “Ambiguity of Ultrashort Pulse Shapes Retrieved From the Intensity Autocorrelation and the Power Spectrum,” IEEE J. Sel. Top. Quantum Electron. 7(4), 656–666 (2001).
[CrossRef]

Efimov, A.

Florean, A.

E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene Ring-Opening,” Ultra-fast Phenomena 88, 249–251 (2007).
[CrossRef]

Funk, D. J.

Gerber, G.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997).
[CrossRef]

Hong, K.-H.

K.-H. Hong, Y. S. Lee, and C. H. Nam, “Electric-field reconstruction of femtosecond laser pulses from interferometric autocorrelation using an evolutionary algorithm,” Opt. Commun. 271(1), 169–177 (2007).
[CrossRef]

Hornung, T.

T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).

Huang, J. Y.

C.-W. Chen, J. Y. Huang, and C.-L. Pan, “Pulse retrieval from interferometric autocorrelation measurement by use of the population-split genetic algorithm,” Opt. Express 14, 22 (2006).

Iaconis, C.

Ina, H.

Judson, R. S.

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992).
[CrossRef] [PubMed]

Kano, S. S.

R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74(5), 2670–2674 (2003).
[CrossRef]

Kapteyn, H.

Kobayashi, S.

Kompa, K.-L.

T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).

Krause, J. L.

Lee, Y. S.

K.-H. Hong, Y. S. Lee, and C. H. Nam, “Electric-field reconstruction of femtosecond laser pulses from interferometric autocorrelation using an evolutionary algorithm,” Opt. Commun. 271(1), 169–177 (2007).
[CrossRef]

Maginnis, K.

Meier, R.

T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).

Mizoguchi, R.

R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74(5), 2670–2674 (2003).
[CrossRef]

Mogi, K.

K. Naganuma, K. Mogi, and H. Yamada, “General method for ultrashort light pulse chirp Measurement,” IEEE J. Quantum Electron. 25, 6 (1989).

Moores, M. D.

Motzkus, M.

T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).

Mourou, G.

Murnane, M.

Naganuma, K.

K. Naganuma, K. Mogi, and H. Yamada, “General method for ultrashort light pulse chirp Measurement,” IEEE J. Quantum Electron. 25, 6 (1989).

Nam, C. H.

K.-H. Hong, Y. S. Lee, and C. H. Nam, “Electric-field reconstruction of femtosecond laser pulses from interferometric autocorrelation using an evolutionary algorithm,” Opt. Commun. 271(1), 169–177 (2007).
[CrossRef]

Nicholson, J. W.

Omenetto, F. G.

Onda, K.

R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74(5), 2670–2674 (2003).
[CrossRef]

Pan, C.-L.

C.-W. Chen, J. Y. Huang, and C.-L. Pan, “Pulse retrieval from interferometric autocorrelation measurement by use of the population-split genetic algorithm,” Opt. Express 14, 22 (2006).

Pearson, B. J.

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

Proch, D.

T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).

Rabitz, H.

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992).
[CrossRef] [PubMed]

Reitze, D. H.

Russek, U.

Sension, R. J.

E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene Ring-Opening,” Ultra-fast Phenomena 88, 249–251 (2007).
[CrossRef]

Seyfried, V.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997).
[CrossRef]

Sheik-Bahae, M.

Strehle, M.

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997).
[CrossRef]

Takeda, M.

Taylor, A. J.

Vdovin, G.

Wada, A.

R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74(5), 2670–2674 (2003).
[CrossRef]

Walmsley, I. A.

Warren, W. S.

C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
[CrossRef]

Weber, P. M.

C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
[CrossRef]

Weinacht, T. C.

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

Weiner, A. M.

J.-H. Chung and A. M. Weiner, “Ambiguity of Ultrashort Pulse Shapes Retrieved From the Intensity Autocorrelation and the Power Spectrum,” IEEE J. Sel. Top. Quantum Electron. 7(4), 656–666 (2001).
[CrossRef]

White, J. L.

E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene Ring-Opening,” Ultra-fast Phenomena 88, 249–251 (2007).
[CrossRef]

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

Wilson, K. R.

C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
[CrossRef]

Yakovlev, V. V.

C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
[CrossRef]

Yamada, H.

K. Naganuma, K. Mogi, and H. Yamada, “General method for ultrashort light pulse chirp Measurement,” IEEE J. Quantum Electron. 25, 6 (1989).

Zeek, E.

Zeidler, D.

T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).

Appl. Phys. B (2)

T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, “Femtosecond pulse shaping by an evolutionary algorithm with feedback,” Appl. Phys. B 65(6), 779–782 (1997).
[CrossRef]

T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, and M. Motzkus, “Optimal control of one- and two-photon transitions with shaped femtosecond pulses and feedback,” Appl. Phys. B 71, 277–284 (2000).

Chem. Phys. Lett. (1)

C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S. Warren, “Feedback quantum control of molecular electronic population transfer,” Chem. Phys. Lett. 280(1-2), 151–158 (1997).
[CrossRef]

IEEE J. Quanum. Electron. (1)

K. Naganuma, K. Mogi, and H. Yamada, “General method for ultrashort light pulse chirp Measurement,” IEEE J. Quantum Electron. 25, 6 (1989).

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

J.-H. Chung and A. M. Weiner, “Ambiguity of Ultrashort Pulse Shapes Retrieved From the Intensity Autocorrelation and the Power Spectrum,” IEEE J. Sel. Top. Quantum Electron. 7(4), 656–666 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

K.-H. Hong, Y. S. Lee, and C. H. Nam, “Electric-field reconstruction of femtosecond laser pulses from interferometric autocorrelation using an evolutionary algorithm,” Opt. Commun. 271(1), 169–177 (2007).
[CrossRef]

Opt. Express (2)

C.-W. Chen, J. Y. Huang, and C.-L. Pan, “Pulse retrieval from interferometric autocorrelation measurement by use of the population-split genetic algorithm,” Opt. Express 14, 22 (2006).

D. A. Bender, J. W. Nicholson, and M. Sheik-Bahae, “Ultrashort laser pulse characterization using modified spectrum auto-interferometric correlation (MOSAIC),” Opt. Express 16(16), 11782–11794 (2008).
[CrossRef] [PubMed]

Opt. Lett. (6)

Phys. Rev. A (1)

B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, “Coherent control using adaptive learning algorithms,” Phys. Rev. A 63(6), 063412 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

R. S. Judson and H. Rabitz, “Teaching lasers to control molecules,” Phys. Rev. Lett. 68(10), 1500–1503 (1992).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

R. Mizoguchi, K. Onda, S. S. Kano, and A. Wada, “Thinning-out in optimized pulse shaping method using genetic algorithm,” Rev. Sci. Instrum. 74(5), 2670–2674 (2003).
[CrossRef]

Ultra-fast Phenomena (1)

E. Carroll, A. Florean, J. L. White, P. H. Bucksbaum, and R. J. Sension, “Control of 1,3-Cyclohexadiene Ring-Opening,” Ultra-fast Phenomena 88, 249–251 (2007).
[CrossRef]

Other (4)

M. Mitchell, An Introduction To Genetic Algorithms, (MIT, 1998).

J.-C. Diels, and W. Rodulph, Ultrashort Laser Pulse Phenomena, (Elsevier, 2006), Chap. 9.

T. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses, (Kluwer Academic Publishers, 2000).

A. Rosenfeld, Multiresolution Image Processing and Analysis (Springer, 1984).

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Figures (6)

Fig. 1
Fig. 1

The scheme of the GA. Each parent will reproduce itself as one of the individuals in the next generation without any change to genes, which is called inheritance. Each of them also produces a certain number of individuals of the next generation by mutation and crossover.

Fig. 2
Fig. 2

First example of phase retrieval by the combined GOA and GA. (a) The red solid curve is the calculated SHIAC with optimized phase from GOA. (b) The red solid curve is the calculated SHIAC with retrieved phase from GA with GOA’s result as starting point. In (a) and (b), the black dotted curve is the experimental SHIAC after adjustment. The retrieved autocorrelation is symmetric, therefore only half of the curve was plotted. (c) The measured spectral intensity (black), the retrieved phase from GOA (blue) and the retrieved spectral phase from GA (red). (d) The evolution of minimum error for each generation. The unit of the error is the same as the unit in (a) and (b).

Fig. 3
Fig. 3

Second example of phase retrieval by the combined GOA and GA. (a) The red solid curve is the calculated SHIAC with optimized phase from GOA. (b) The red solid curve is the calculated SHIAC with retrieved phase from GA with GOA’s results as starting point. In both (a) and (b), the black dotted curve is the experimental SHIAC after adjustment. The retrieved autocorrelation is symmetric, therefore only half of the curve was plotted. (c) The measured spectral intensity (black), the retrieved phase from GOA (blue) and the retrieved spectral phase from GA (red). (d) The evolution of minimum error for each generation. The unit of the error is the same as the unit in (a) and (b).

Fig. 4
Fig. 4

(a) Retrieved electric field from combined GOA + GA in example 1. (b) Retrieved electric field from combined GOA - GA in example 2.

Fig. 5
Fig. 5

(a) Comparison of GA with and without GOA with the data in example 1. (b) Comparison of GA with and without GOA with the data in example 2. The unit of the errors is the same as the unit of (a) and (b) in Fig. 2 and Fig. 3. The scales of (a) and (b) are made different due to the different ranges of the data.

Fig. 6
Fig. 6

GA with and without crossover. The red dashed lines are the minimum errors of each generation in the six runs of GA without using crossover and the black solid line are that of GA with crossover. There is no obvious difference between the results of GA with or without crossover. There is an unlucky case in the GA with crossover whose minimum errors are higher than those of any other runs.

Tables (2)

Tables Icon

Table 1 Results of GOA and GOA + GA for example 1 and 2

Tables Icon

Table 2 GOA + GA’s results for noise-free simulated pulses

Equations (7)

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

φ p ( ω ) = a 2 ( ω ω 0 ) 2 + a 3 ( ω ω 0 ) 3 + a 4 ( ω ω 0 ) 4 ,
I a c _ i n u s e ( τ ) = I a c ( b τ ) ,
Δ = i = 1 N τ [ I a c _ c a l ( τ i ) I a c _ i n u s e ( τ i ) ] 2 / N τ ,
φ ( g , i ) ( ω ) = φ p , ( g , i ) ( ω , a 2 , ( g , i ) , a 3 , ( g , i ) , a 4 , ( g , i ) ) + φ r , ( g , i ) ( ω ) ,
E ( g , i ) ( t ) = | E ˜ ( r ( g , i ) + ω ) | e i φ ( g , i ) ( ω ) e i ω t d ω .
I a c _ c a l ( g , i ) ( τ ) = | [ E ( g , i ) ( t τ ) + E ( g , i ) ( t ) ] 2 | 2 d t .
Δ ( g , i ) = i = 1 N τ [ I a c _ c a l ( g , i ) ( τ i ) I a c _ i n u s e ( τ i ) ] 2 / N τ ,

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