Abstract

We demonstrate the adaptation of an iterative Fourier transform algorithm for the calculation of theoretical spectral phase functions required for pulse shaping applications. The algorithm is used to determine the phase functions necessary for the generation of different temporal intensity profiles. The performance of the algorithm is compared to two exemplary standard approaches. i.e. a Genetic Algorithm and a combination of a Simplex Downhill and a Simulated Annealing algorithm. It is shown that the iterative Fourier transform algorithm converges much faster than both alternative methods.

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References

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  1. A.M. Weiner, J.P. Heritage, J.A. Salehi, "Encoding and decoding of femtosecond pulses," Opt. Lett. 13, 300 (1988).
    [CrossRef] [PubMed]
  2. D. Meshulach, Y. Silberberg, "Coherent Quantum Control of Two-Photon Transitions by a Femtosecond Laser Pulse," Nature 396, 239 (1998).
    [CrossRef]
  3. W.S. Warren, H. Rabitz, M. Dahleh, "Coherent Control of Chemical Reactions: The Dream is Alive," Science 259, 1581 (1993).
    [CrossRef] [PubMed]
  4. S. Rice, "Optical control of reactions," Nature 403, 496 (2000).
    [CrossRef] [PubMed]
  5. M.M. Wefers, H. Kawashima, K.A. Nelson, "Optical control over femtosecond polarization dynamics," J. Phys. Chem. Sol. 57, 1425 (1996).
    [CrossRef]
  6. D. Meshulach, D. Yelin, Y. Silberberg, "Adaptive Ultrashort Pulse Compression and Shaping," Opt. Commun. 138, 345 (1997).
    [CrossRef]
  7. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, G. Gerber, "Femtosecond pulse shaping by an evolutionary algorithm with feedback," Appl. Phys. B 65, 779 (1997).
    [CrossRef]
  8. D. Zeidler, T. Hornung, D. Proch, M. Motzkus, "Adaptive compression of tunable pulses from a non-collinear type OPA to below 20fs by feedback-controlled pulse shaping," Appl. Phys. B 70, 125 (2000).
    [CrossRef]
  9. A.M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929 (2000).
    [CrossRef]
  10. A.M. Weiner, S. Oudin, D.E. Leaird, D.H. Reitze, "Shaping of femtosecond pulses using phase-only filters designed by simulated annealing," J. Opt. Soc. Am. A 10, 1112 (1993).
    [CrossRef]
  11. J.X. Tull, M.A. Dugan, W.S. Warren, "High Resolution, Ultrafast Laser Pulse Shaping and its Applications," Adv. Magn. Opt. Reson. 20, 1 (1997).
    [CrossRef]
  12. J.P. Heritage, R.N. Thurston, W.J. Tomlinson, A.M. Weiner, R.H. Stolen, "Spectral Windowing of Frequency-modulated Optical Pulses in a Grating Compressor," Appl. Phys. Lett. 47, 87 (1985).
    [CrossRef]
  13. M.M. Wefers, K.A. Nelson, "Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators," J. Opt. Soc. Am. B 12, 1343 (1995).
    [CrossRef]
  14. G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, F. Reichel, "A new high-resolution femtosecond pulse shaper," Appl. Phys. B 72, 627 (2001).
    [CrossRef]
  15. M.M. Wefers, K.A. Nelson, "Programmable Phase and Amplitute Femtosecond Pulseshaping," Opt. Lett. 18, 2032 (1993).
    [CrossRef] [PubMed]
  16. D. Meshulach, D. Yelin, Y. Silberberg, "Adaptive real-time femtosecond pulse shaping," J. Opt. Soc. Am. B 15, 1615 (1998).
    [CrossRef]
  17. J. Peatross, A. Rundquist, "Temporal decorrelation of short laser pulses," J. Opt. Soc. Am. B 1 , 216 (1998).
    [CrossRef]
  18. F. Wyrowski, O. Bryngdahl, "Iterative Fourier-transform algorithm applied to computer holography," J. Opt. Soc. Am. A 5, 1058 (1988).
    [CrossRef]
  19. K.-H. Brenner, "Method for designing arbitrary two-dimensional continuous phase elements," Opt. Lett. 25, 31 (2000).
    [CrossRef]
  20. R. Gerchberg, W.O. Saxton, "A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures," Optik 35, 237 (1971).
  21. E. Sch�neburg, F. Heinzmann, S. Feddersen, Genetische Algorithmen und Evolutionsstrategien, (Addison-Wesley, New York,1994).
  22. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Second Edition, (Cambridge University Press, Cambridge, 1986).

Other

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

D. Meshulach, Y. Silberberg, "Coherent Quantum Control of Two-Photon Transitions by a Femtosecond Laser Pulse," Nature 396, 239 (1998).
[CrossRef]

W.S. Warren, H. Rabitz, M. Dahleh, "Coherent Control of Chemical Reactions: The Dream is Alive," Science 259, 1581 (1993).
[CrossRef] [PubMed]

S. Rice, "Optical control of reactions," Nature 403, 496 (2000).
[CrossRef] [PubMed]

M.M. Wefers, H. Kawashima, K.A. Nelson, "Optical control over femtosecond polarization dynamics," J. Phys. Chem. Sol. 57, 1425 (1996).
[CrossRef]

D. Meshulach, D. Yelin, Y. Silberberg, "Adaptive Ultrashort Pulse Compression and Shaping," Opt. Commun. 138, 345 (1997).
[CrossRef]

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

D. Zeidler, T. Hornung, D. Proch, M. Motzkus, "Adaptive compression of tunable pulses from a non-collinear type OPA to below 20fs by feedback-controlled pulse shaping," Appl. Phys. B 70, 125 (2000).
[CrossRef]

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

A.M. Weiner, S. Oudin, D.E. Leaird, D.H. Reitze, "Shaping of femtosecond pulses using phase-only filters designed by simulated annealing," J. Opt. Soc. Am. A 10, 1112 (1993).
[CrossRef]

J.X. Tull, M.A. Dugan, W.S. Warren, "High Resolution, Ultrafast Laser Pulse Shaping and its Applications," Adv. Magn. Opt. Reson. 20, 1 (1997).
[CrossRef]

J.P. Heritage, R.N. Thurston, W.J. Tomlinson, A.M. Weiner, R.H. Stolen, "Spectral Windowing of Frequency-modulated Optical Pulses in a Grating Compressor," Appl. Phys. Lett. 47, 87 (1985).
[CrossRef]

M.M. Wefers, K.A. Nelson, "Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators," J. Opt. Soc. Am. B 12, 1343 (1995).
[CrossRef]

G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, F. Reichel, "A new high-resolution femtosecond pulse shaper," Appl. Phys. B 72, 627 (2001).
[CrossRef]

M.M. Wefers, K.A. Nelson, "Programmable Phase and Amplitute Femtosecond Pulseshaping," Opt. Lett. 18, 2032 (1993).
[CrossRef] [PubMed]

D. Meshulach, D. Yelin, Y. Silberberg, "Adaptive real-time femtosecond pulse shaping," J. Opt. Soc. Am. B 15, 1615 (1998).
[CrossRef]

J. Peatross, A. Rundquist, "Temporal decorrelation of short laser pulses," J. Opt. Soc. Am. B 1 , 216 (1998).
[CrossRef]

F. Wyrowski, O. Bryngdahl, "Iterative Fourier-transform algorithm applied to computer holography," J. Opt. Soc. Am. A 5, 1058 (1988).
[CrossRef]

K.-H. Brenner, "Method for designing arbitrary two-dimensional continuous phase elements," Opt. Lett. 25, 31 (2000).
[CrossRef]

R. Gerchberg, W.O. Saxton, "A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures," Optik 35, 237 (1971).

E. Sch�neburg, F. Heinzmann, S. Feddersen, Genetische Algorithmen und Evolutionsstrategien, (Addison-Wesley, New York,1994).

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Second Edition, (Cambridge University Press, Cambridge, 1986).

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Figures (8)

Fig. 1.
Fig. 1.

Scheme of the iterative Fourier transform algorithm (FT - Fourier transformation). The initial phase may be set to any random distribution of numbers.

Fig. 2.
Fig. 2.

A bandwidth limited Gaussian pulse of 47 fs FWHM is phase modulated to produce a) a stretched pulse with 400 fs FWHM, b) a double pulse with a temporal separation of 480 fs and a FWHM of 80 fs each, and c) a triple pulse with ascending amplitude.

Fig. 3.
Fig. 3.

a) The iterative Fourier transform algorithm was used to approximate a rectangular pulse with a FWHM of 300 fs. b) Spectral phase function found by the algorithm.

Fig. 4.
Fig. 4.

Progress of the pulse shape optimization versus the number of iterations for the problem depicted in fig. 3. The different curves correspond to different initial phase patterns.

Fig. 5.
Fig. 5.

a) The GA was used to approximate a rectangular pulse with a FWHM of 300 fs. The algorithm was able to change independently the phase of all pixels. b) Spectral phase function found by the algorithm.

Fig. 6.
Fig. 6.

Progress of the GA versus the number of generations for different runs and different bit depths (30 individuals per generation).

Fig. 7.
Fig. 7.

a) The SASD algorithm was used to approximate a rectangular pulse with a FWHM of 300 fs. The phase function was expressed in terms of a Taylor series with five coefficients. b) Spectral phase function found by the algorithm.

Fig. 8.
Fig. 8.

Progress of the SASD algorithm for different runs.

Equations (3)

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T ( ω ) = E out ( ω ) E in ( ω )
T ( ω ) = 1 .
E out ( ω ) = E in ( ω ) e i Δ ( ω ) .

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