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.

© 2001 Optical Society of America

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  1. A.M. Weiner, J.P. Heritage, and J.A. Salehi, “Encoding and decoding of femtosecond pulses,” Opt. Lett. 13, 300 (1988).
    [CrossRef] [PubMed]
  2. D. Meshulach and 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, and 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, and K.A. Nelson, “Optical control over femtosecond polarization dynamics,” J. Phys. Chem. Sol. 57, 1425 (1996).
    [CrossRef]
  6. D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive Ultrashort Pulse Compression and Shaping,” Opt. Commun. 138, 345 (1997).
    [CrossRef]
  7. T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and 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, and 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, and 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, and 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, and 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 and 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, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B 72, 627 (2001).
    [CrossRef]
  15. M.M. Wefers and K.A. Nelson, “Programmable Phase and Amplitute Femtosecond Pulseshaping,” Opt. Lett. 18, 2032 (1993).
    [CrossRef] [PubMed]
  16. D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive real-time femtosecond pulse shaping,” J. Opt. Soc. Am. B 15, 1615 (1998).
    [CrossRef]
  17. J. Peatross and A. Rundquist, “Temporal decorrelation of short laser pulses,” J. Opt. Soc. Am. B 15, 216 (1998).
    [CrossRef]
  18. F. Wyrowski and 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 and W.O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1971).
  21. E. Schöneburg, F. Heinzmann, and S. Feddersen, Genetische Algorithmen und Evolutionsstrategien, Addison-Wesley, New York (1994).
  22. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, Second Edition, Cambridge University Press, Cambridge (1986).

2001 (1)

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

2000 (4)

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

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

D. Zeidler, T. Hornung, D. Proch, and 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]

1998 (3)

1997 (3)

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

D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive Ultrashort Pulse Compression and Shaping,” Opt. Commun. 138, 345 (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, 779 (1997).
[CrossRef]

1996 (1)

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

1995 (1)

1993 (3)

1988 (2)

1985 (1)

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

1971 (1)

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

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, 779 (1997).
[CrossRef]

Brenner, K.-H.

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, 779 (1997).
[CrossRef]

Bryngdahl, O.

Dahleh, M.

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

Dugan, M.A.

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

Feddersen, S.

E. Schöneburg, F. Heinzmann, and S. Feddersen, Genetische Algorithmen und Evolutionsstrategien, Addison-Wesley, New York (1994).

Feurer, T.

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

Flannery, B.P.

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

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, 779 (1997).
[CrossRef]

Gerchberg, R.

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

Hacker, M.

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

Heinzmann, F.

E. Schöneburg, F. Heinzmann, and S. Feddersen, Genetische Algorithmen und Evolutionsstrategien, Addison-Wesley, New York (1994).

Heritage, J.P.

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

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

Hornung, T.

D. Zeidler, T. Hornung, D. Proch, and 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]

Kawashima, H.

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

Leaird, D.E.

Meshulach, D.

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

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

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

Motzkus, M.

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

D. Zeidler, T. Hornung, D. Proch, and 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]

Nelson, K.A.

Oudin, S.

Peatross, J.

Press, W.H.

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

Proch, D.

D. Zeidler, T. Hornung, D. Proch, and 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]

Rabitz, H.

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

Reichel, F.

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

Reitze, D.H.

Rice, S.

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

Rundquist, A.

Salehi, J.A.

Saxton, W.O.

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

Schöneburg, E.

E. Schöneburg, F. Heinzmann, and S. Feddersen, Genetische Algorithmen und Evolutionsstrategien, Addison-Wesley, New York (1994).

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, 779 (1997).
[CrossRef]

Silberberg, Y.

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

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

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

Stobrawa, G.

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

Stolen, R.H.

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

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, 779 (1997).
[CrossRef]

Teukolsky, S.A.

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

Thurston, R.N.

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

Tomlinson, W.J.

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

Tull, J.X.

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

Vetterling, W.T.

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

Warren, W.S.

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

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

Wefers, M.M.

Weiner, A.M.

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, and D.H. Reitze, “Shaping of femtosecond pulses using phase-only filters designed by simulated annealing,” J. Opt. Soc. Am. A 10, 1112 (1993).
[CrossRef]

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

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

Wyrowski, F.

Yelin, D.

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

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

Zeidler, D.

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

D. Zeidler, T. Hornung, D. Proch, and 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]

Adv. Magn. Opt. Reson. (1)

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

Appl. Phys. B (3)

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

D. Zeidler, T. Hornung, D. Proch, and 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]

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

Appl. Phys. Lett. (1)

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

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (3)

J. Phys. Chem. Sol. (1)

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

Nature (2)

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

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

Opt. Commun. (1)

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

Opt. Lett. (3)

Optik (1)

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

Rev. Sci. Instrum. (1)

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

Science (1)

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

Other (2)

E. Schöneburg, F. Heinzmann, and S. Feddersen, Genetische Algorithmen und Evolutionsstrategien, Addison-Wesley, New York (1994).

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and 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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