Abstract

A rigorous analysis of ultrashort pulse shaping by the spectral filtering of dispersed frequency components is presented. Particular attention is directed toward the case of two liquid-crystal spatial light modulators used to provide programmable manipulation of both the phase and the amplitude profiles of the shaped waveform in the time domain. Different optical configurations are evaluated and their theoretical and practical effects determined. An important result is that, even with optimal alignment and components, the diffraction arising from spectral filtering necessarily produces a transverse spatial shift that varies linearly along the temporal profile of the shaped waveform. Despite this effect it is shown that the technique can generate arbitrary phase and amplitude temporal profiles (subject to limitations in temporal extent and temporal resolution) for the Gaussian spatial component of the shaped output waveform.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993).
    [CrossRef] [PubMed]
  2. R. Kosloff, A. D. Hammerich, and D. Tannor, Phys. Rev. Lett. 69, 2172 (1992).
    [CrossRef] [PubMed]
  3. J. A. Cina and T. J. Smith, J. Chem. Phys. 98, 9211 (1993).
    [CrossRef]
  4. S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988).
    [CrossRef]
  5. B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
    [CrossRef]
  6. W. S. Warren and A. H. Zewail, J. Chem. Phys. 78, 2279 (1983).
    [CrossRef]
  7. A. M. Weiner, D. E. Leaird, G. P. Wiederrecht, and K. A. Nelson, Science 247, 1317 (1990).
    [CrossRef] [PubMed]
  8. N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
    [CrossRef]
  9. A. M. Weiner, J. P. Heritage, and J. A. Salehi, Opt. Lett. 13, 300 (1988).
    [CrossRef] [PubMed]
  10. A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
    [CrossRef]
  11. A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
    [CrossRef] [PubMed]
  12. M. Haner and W. S. Warren, Opt. Lett. 12, 398 (1987).
    [CrossRef] [PubMed]
  13. M. Haner and W. S. Warren, Appl. Phys. Lett. 52, 1458 (1988).
    [CrossRef]
  14. K. B. Hill and D. J. Brady, Opt. Lett. 18, 1739 (1993).
    [CrossRef] [PubMed]
  15. D. J. Brady, A. G. S. Chen, and G. Rodriguez, Opt. Lett. 14, 868 (1989).
    [CrossRef]
  16. A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, Opt. Lett. 17, 224 (1992).
    [CrossRef] [PubMed]
  17. C. Froehly, B. Colmbeau, and M. Vampouille, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 115–121.
    [CrossRef]
  18. J. P. Heritage, A. M. Weiner, and R. N. Thurston, Opt. Lett. 10, 609 (1985).
    [CrossRef] [PubMed]
  19. A. M. Weiner and D. E. Leaird, Opt. Lett. 15, 51 (1990).
    [CrossRef] [PubMed]
  20. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, J. Opt. Soc. Am. B 5, 1563 (1988).
    [CrossRef]
  21. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, Opt. Lett. 15, 326 (1990).
    [CrossRef] [PubMed]
  22. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, IEEE J. Quantum Electron. 28, 908 (1992).
    [CrossRef]
  23. M. M. Wefers and K. A. Nelson, Opt. Lett. 18, 2032 (1993).
    [CrossRef] [PubMed]
  24. M. M. Wefers and K. A. Nelson, Science 262, 1381 (1993).
    [CrossRef] [PubMed]
  25. M. M. Wefers, H. Kawashima, and K. A. Nelson, in Ultrafast Phenomena IX, G. A. Mourou, A. H. Zewail, P. F. Barbara, and W. H. Knox, eds. (Springer-Verlag, Berlin, 1994); M. M. Wefers and K. A. Nelson, "Generation of high-fidelity programmable ultrafast optical waveforms," Opt. Lett. (to be published).
  26. M. M. Wefers, H. Kawashima, and K. A. Nelton, "Automated multidimensional coherent optical spectroscopy with multiple phase-related femtosecond pulses," J. Chem. Phys. (to be published).
  27. C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, Opt. Lett. 19, 737 (1994).
    [CrossRef] [PubMed]
  28. R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, IEEE J. Quantum Electron. QE-22, 682 (1986).
    [CrossRef]
  29. M. M. Wefers and K. A. Nelson, "Programmable femtosecond multiple pulse generation and spectroscopy," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 39–41.
  30. O. E. Martinez, IEEE J. Quantum Electron. QE-23, 59 (1987).
    [CrossRef]
  31. O. E. Martinez, J. Opt. Soc. Am. B 3, 929 (1986).
    [CrossRef]
  32. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).
  33. M. B. Danailov and I. P. Christov, J. Mod. Opt. 36, 725 (1989).
    [CrossRef]
  34. A. M. Weiner, S. Oudin, D. E. Leaird, and D. H. Reitze, J. Opt. Soc. Am. A 10, 1112 (1993).
    [CrossRef]
  35. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).
  36. D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993).
    [CrossRef] [PubMed]
  37. R. Trebino and D. J. Kane, J. Opt. Soc. Am. A 10, 1101 (1993).
    [CrossRef]
  38. J. L. A. Chilla and O. E. Martinez, Opt. Lett. 16, 39 (1991).
    [CrossRef] [PubMed]
  39. F. Salin, J. Squier, G. Mourou, and G. Vaillancourt, Opt. Lett. 16, 1964 (1991).
    [CrossRef] [PubMed]
  40. D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.
  41. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
    [CrossRef]

Andrejco, M. J.

A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

Brady, D. J.

K. B. Hill and D. J. Brady, Opt. Lett. 18, 1739 (1993).
[CrossRef] [PubMed]

D. J. Brady, A. G. S. Chen, and G. Rodriguez, Opt. Lett. 14, 868 (1989).
[CrossRef]

Bucksbaum, P.

D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.

Carlson, R. J.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

Chen, A. G. S.

D. J. Brady, A. G. S. Chen, and G. Rodriguez, Opt. Lett. 14, 868 (1989).
[CrossRef]

Chilla, J. L. A.

J. L. A. Chilla and O. E. Martinez, Opt. Lett. 16, 39 (1991).
[CrossRef] [PubMed]

Christov, I. P.

M. B. Danailov and I. P. Christov, J. Mod. Opt. 36, 725 (1989).
[CrossRef]

Cina, J. A.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

J. A. Cina and T. J. Smith, J. Chem. Phys. 98, 9211 (1993).
[CrossRef]

Colmbeau, B.

C. Froehly, B. Colmbeau, and M. Vampouille, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 115–121.
[CrossRef]

Dahleh, M.

W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993).
[CrossRef] [PubMed]

Danailov, M. B.

M. B. Danailov and I. P. Christov, J. Mod. Opt. 36, 725 (1989).
[CrossRef]

Du, M.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

Fleming, G. R.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

Fouckhardt, H.

A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
[CrossRef]

Froehly, C.

C. Froehly, B. Colmbeau, and M. Vampouille, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 115–121.
[CrossRef]

Goswami, D.

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, Opt. Lett. 19, 737 (1994).
[CrossRef] [PubMed]

Hammerich, A. D.

R. Kosloff, A. D. Hammerich, and D. Tannor, Phys. Rev. Lett. 69, 2172 (1992).
[CrossRef] [PubMed]

Haner, M.

M. Haner and W. S. Warren, Opt. Lett. 12, 398 (1987).
[CrossRef] [PubMed]

M. Haner and W. S. Warren, Appl. Phys. Lett. 52, 1458 (1988).
[CrossRef]

Hawkins, R. J.

A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
[CrossRef]

Heritage, J. P.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, IEEE J. Quantum Electron. QE-22, 682 (1986).
[CrossRef]

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, J. Opt. Soc. Am. B 5, 1563 (1988).
[CrossRef]

A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, and J. A. Salehi, Opt. Lett. 13, 300 (1988).
[CrossRef] [PubMed]

J. P. Heritage, A. M. Weiner, and R. N. Thurston, Opt. Lett. 10, 609 (1985).
[CrossRef] [PubMed]

Hill, K. B.

K. B. Hill and D. J. Brady, Opt. Lett. 18, 1739 (1993).
[CrossRef] [PubMed]

Hillegas, C. W.

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, Opt. Lett. 19, 737 (1994).
[CrossRef] [PubMed]

Kane, D. J.

D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993).
[CrossRef] [PubMed]

R. Trebino and D. J. Kane, J. Opt. Soc. Am. A 10, 1101 (1993).
[CrossRef]

Kawashima, H.

M. M. Wefers, H. Kawashima, and K. A. Nelton, "Automated multidimensional coherent optical spectroscopy with multiple phase-related femtosecond pulses," J. Chem. Phys. (to be published).

M. M. Wefers, H. Kawashima, and K. A. Nelson, in Ultrafast Phenomena IX, G. A. Mourou, A. H. Zewail, P. F. Barbara, and W. H. Knox, eds. (Springer-Verlag, Berlin, 1994); M. M. Wefers and K. A. Nelson, "Generation of high-fidelity programmable ultrafast optical waveforms," Opt. Lett. (to be published).

Kirschner, E. M.

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, J. Opt. Soc. Am. B 5, 1563 (1988).
[CrossRef]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
[CrossRef]

A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
[CrossRef] [PubMed]

Kohler, B.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.

Kosloff, R.

R. Kosloff, A. D. Hammerich, and D. Tannor, Phys. Rev. Lett. 69, 2172 (1992).
[CrossRef] [PubMed]

Krause, J. L.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

Leaird, D. E.

A. M. Weiner, D. E. Leaird, G. P. Wiederrecht, and K. A. Nelson, Science 247, 1317 (1990).
[CrossRef] [PubMed]

A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
[CrossRef] [PubMed]

A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
[CrossRef]

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, Opt. Lett. 17, 224 (1992).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
[CrossRef]

A. M. Weiner, S. Oudin, D. E. Leaird, and D. H. Reitze, J. Opt. Soc. Am. A 10, 1112 (1993).
[CrossRef]

A. M. Weiner and D. E. Leaird, Opt. Lett. 15, 51 (1990).
[CrossRef] [PubMed]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, Opt. Lett. 15, 326 (1990).
[CrossRef] [PubMed]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

Martinez, O. E.

O. E. Martinez, IEEE J. Quantum Electron. QE-23, 59 (1987).
[CrossRef]

O. E. Martinez, J. Opt. Soc. Am. B 3, 929 (1986).
[CrossRef]

J. L. A. Chilla and O. E. Martinez, Opt. Lett. 16, 39 (1991).
[CrossRef] [PubMed]

Mato, A.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

Mourou, G.

F. Salin, J. Squier, G. Mourou, and G. Vaillancourt, Opt. Lett. 16, 1964 (1991).
[CrossRef] [PubMed]

Mukamel, S.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

Nelson, K. A.

A. M. Weiner, D. E. Leaird, G. P. Wiederrecht, and K. A. Nelson, Science 247, 1317 (1990).
[CrossRef] [PubMed]

M. M. Wefers and K. A. Nelson, "Programmable femtosecond multiple pulse generation and spectroscopy," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 39–41.

M. M. Wefers and K. A. Nelson, Opt. Lett. 18, 2032 (1993).
[CrossRef] [PubMed]

M. M. Wefers, H. Kawashima, and K. A. Nelson, in Ultrafast Phenomena IX, G. A. Mourou, A. H. Zewail, P. F. Barbara, and W. H. Knox, eds. (Springer-Verlag, Berlin, 1994); M. M. Wefers and K. A. Nelson, "Generation of high-fidelity programmable ultrafast optical waveforms," Opt. Lett. (to be published).

M. M. Wefers and K. A. Nelson, Science 262, 1381 (1993).
[CrossRef] [PubMed]

Nelton, K. A.

M. M. Wefers, H. Kawashima, and K. A. Nelton, "Automated multidimensional coherent optical spectroscopy with multiple phase-related femtosecond pulses," J. Chem. Phys. (to be published).

Oudin, S.

A. M. Weiner, S. Oudin, D. E. Leaird, and D. H. Reitze, J. Opt. Soc. Am. A 10, 1112 (1993).
[CrossRef]

Paek, E. G.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, Opt. Lett. 17, 224 (1992).
[CrossRef] [PubMed]

Patel, J. S.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, Opt. Lett. 15, 326 (1990).
[CrossRef] [PubMed]

Pinkos, D.

D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

Rabitz, H.

S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988).
[CrossRef]

W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993).
[CrossRef] [PubMed]

Raksi, F.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

Reitze, D. H.

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, Opt. Lett. 17, 224 (1992).
[CrossRef] [PubMed]

A. M. Weiner, S. Oudin, D. E. Leaird, and D. H. Reitze, J. Opt. Soc. Am. A 10, 1112 (1993).
[CrossRef]

Rice, S. A.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

Rodriguez, G.

D. J. Brady, A. G. S. Chen, and G. Rodriguez, Opt. Lett. 14, 868 (1989).
[CrossRef]

Romero-Rochin, V.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

Rose-Petruck, C.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

Ruggiero, A. J.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

Saifi, M. A.

A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
[CrossRef]

Salehi, J. A.

A. M. Weiner, J. P. Heritage, and J. A. Salehi, Opt. Lett. 13, 300 (1988).
[CrossRef] [PubMed]

Salin, F.

F. Salin, J. Squier, G. Mourou, and G. Vaillancourt, Opt. Lett. 16, 1964 (1991).
[CrossRef] [PubMed]

Scherer, N. F.

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

Schumacher, D.

D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.

Shi, S.

S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988).
[CrossRef]

Silberberg, Y.

A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
[CrossRef]

Smith, P. W.

A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
[CrossRef]

Smith, T. J.

J. A. Cina and T. J. Smith, J. Chem. Phys. 98, 9211 (1993).
[CrossRef]

Squier, J.

D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.

F. Salin, J. Squier, G. Mourou, and G. Vaillancourt, Opt. Lett. 16, 1964 (1991).
[CrossRef] [PubMed]

Strickland, D.

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, Opt. Lett. 19, 737 (1994).
[CrossRef] [PubMed]

Tannor, D.

R. Kosloff, A. D. Hammerich, and D. Tannor, Phys. Rev. Lett. 69, 2172 (1992).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

Thurston, R. N.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, IEEE J. Quantum Electron. QE-22, 682 (1986).
[CrossRef]

A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
[CrossRef] [PubMed]

J. P. Heritage, A. M. Weiner, and R. N. Thurston, Opt. Lett. 10, 609 (1985).
[CrossRef] [PubMed]

Tominson, W. J.

A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
[CrossRef] [PubMed]

Tomlinson, W. J.

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, IEEE J. Quantum Electron. QE-22, 682 (1986).
[CrossRef]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
[CrossRef]

Trebino, R.

D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993).
[CrossRef] [PubMed]

R. Trebino and D. J. Kane, J. Opt. Soc. Am. A 10, 1101 (1993).
[CrossRef]

Tull, J. X.

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, Opt. Lett. 19, 737 (1994).
[CrossRef] [PubMed]

Vaillancourt, G.

F. Salin, J. Squier, G. Mourou, and G. Vaillancourt, Opt. Lett. 16, 1964 (1991).
[CrossRef] [PubMed]

Vakolev, V. V.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

Vampouille, M.

C. Froehly, B. Colmbeau, and M. Vampouille, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 115–121.
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

Warren, W. S.

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, Opt. Lett. 19, 737 (1994).
[CrossRef] [PubMed]

M. Haner and W. S. Warren, Appl. Phys. Lett. 52, 1458 (1988).
[CrossRef]

M. Haner and W. S. Warren, Opt. Lett. 12, 398 (1987).
[CrossRef] [PubMed]

W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993).
[CrossRef] [PubMed]

W. S. Warren and A. H. Zewail, J. Chem. Phys. 78, 2279 (1983).
[CrossRef]

Wefers, M. M.

M. M. Wefers and K. A. Nelson, Opt. Lett. 18, 2032 (1993).
[CrossRef] [PubMed]

M. M. Wefers and K. A. Nelson, "Programmable femtosecond multiple pulse generation and spectroscopy," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 39–41.

M. M. Wefers and K. A. Nelson, Science 262, 1381 (1993).
[CrossRef] [PubMed]

M. M. Wefers, H. Kawashima, and K. A. Nelson, in Ultrafast Phenomena IX, G. A. Mourou, A. H. Zewail, P. F. Barbara, and W. H. Knox, eds. (Springer-Verlag, Berlin, 1994); M. M. Wefers and K. A. Nelson, "Generation of high-fidelity programmable ultrafast optical waveforms," Opt. Lett. (to be published).

M. M. Wefers, H. Kawashima, and K. A. Nelton, "Automated multidimensional coherent optical spectroscopy with multiple phase-related femtosecond pulses," J. Chem. Phys. (to be published).

Weiner, A. M.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, J. Opt. Soc. Am. B 5, 1563 (1988).
[CrossRef]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, Opt. Lett. 15, 326 (1990).
[CrossRef] [PubMed]

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, IEEE J. Quantum Electron. QE-22, 682 (1986).
[CrossRef]

A. M. Weiner, S. Oudin, D. E. Leaird, and D. H. Reitze, J. Opt. Soc. Am. A 10, 1112 (1993).
[CrossRef]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
[CrossRef]

A. M. Weiner, D. E. Leaird, G. P. Wiederrecht, and K. A. Nelson, Science 247, 1317 (1990).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, and J. A. Salehi, Opt. Lett. 13, 300 (1988).
[CrossRef] [PubMed]

A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
[CrossRef] [PubMed]

A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
[CrossRef]

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, Opt. Lett. 17, 224 (1992).
[CrossRef] [PubMed]

A. M. Weiner and D. E. Leaird, Opt. Lett. 15, 51 (1990).
[CrossRef] [PubMed]

J. P. Heritage, A. M. Weiner, and R. N. Thurston, Opt. Lett. 10, 609 (1985).
[CrossRef] [PubMed]

Whitnell, A. M.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

Wiederrecht, G. P.

A. M. Weiner, D. E. Leaird, G. P. Wiederrecht, and K. A. Nelson, Science 247, 1317 (1990).
[CrossRef] [PubMed]

Wilson, K. R.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.

Woody, A.

S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988).
[CrossRef]

Wullert, J. R.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, Opt. Lett. 15, 326 (1990).
[CrossRef] [PubMed]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

Yakovlev, V. V.

D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.

Yan, Y.

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

Zewail, A. H.

W. S. Warren and A. H. Zewail, J. Chem. Phys. 78, 2279 (1983).
[CrossRef]

Other

W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993).
[CrossRef] [PubMed]

R. Kosloff, A. D. Hammerich, and D. Tannor, Phys. Rev. Lett. 69, 2172 (1992).
[CrossRef] [PubMed]

J. A. Cina and T. J. Smith, J. Chem. Phys. 98, 9211 (1993).
[CrossRef]

S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988).
[CrossRef]

B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, A. M. Whitnell, K. R. Wilson, V. V. Vakolev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993).
[CrossRef]

W. S. Warren and A. H. Zewail, J. Chem. Phys. 78, 2279 (1983).
[CrossRef]

A. M. Weiner, D. E. Leaird, G. P. Wiederrecht, and K. A. Nelson, Science 247, 1317 (1990).
[CrossRef] [PubMed]

N. F. Scherer, R. J. Carlson, A. Mato, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991).
[CrossRef]

A. M. Weiner, J. P. Heritage, and J. A. Salehi, Opt. Lett. 13, 300 (1988).
[CrossRef] [PubMed]

A. M. Weiner, Y. Silberberg, H. Fouckhardt, D. E. Leaird, M. A. Saifi, M. J. Andrejco, and P. W. Smith, IEEE J. Quantum Electron. 25, 2648 (1989).
[CrossRef]

A. M. Weiner, R. N. Thurston, W. J. Tominson, J. P. Heritage, D. E. Leaird, E. M. Kirschner, and R. J. Hawkins, Opt. Lett. 14, 868 (1989).
[CrossRef] [PubMed]

M. Haner and W. S. Warren, Opt. Lett. 12, 398 (1987).
[CrossRef] [PubMed]

M. Haner and W. S. Warren, Appl. Phys. Lett. 52, 1458 (1988).
[CrossRef]

K. B. Hill and D. J. Brady, Opt. Lett. 18, 1739 (1993).
[CrossRef] [PubMed]

D. J. Brady, A. G. S. Chen, and G. Rodriguez, Opt. Lett. 14, 868 (1989).
[CrossRef]

A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, Opt. Lett. 17, 224 (1992).
[CrossRef] [PubMed]

C. Froehly, B. Colmbeau, and M. Vampouille, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 115–121.
[CrossRef]

J. P. Heritage, A. M. Weiner, and R. N. Thurston, Opt. Lett. 10, 609 (1985).
[CrossRef] [PubMed]

A. M. Weiner and D. E. Leaird, Opt. Lett. 15, 51 (1990).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, J. Opt. Soc. Am. B 5, 1563 (1988).
[CrossRef]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, Opt. Lett. 15, 326 (1990).
[CrossRef] [PubMed]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

M. M. Wefers and K. A. Nelson, Opt. Lett. 18, 2032 (1993).
[CrossRef] [PubMed]

M. M. Wefers and K. A. Nelson, Science 262, 1381 (1993).
[CrossRef] [PubMed]

M. M. Wefers, H. Kawashima, and K. A. Nelson, in Ultrafast Phenomena IX, G. A. Mourou, A. H. Zewail, P. F. Barbara, and W. H. Knox, eds. (Springer-Verlag, Berlin, 1994); M. M. Wefers and K. A. Nelson, "Generation of high-fidelity programmable ultrafast optical waveforms," Opt. Lett. (to be published).

M. M. Wefers, H. Kawashima, and K. A. Nelton, "Automated multidimensional coherent optical spectroscopy with multiple phase-related femtosecond pulses," J. Chem. Phys. (to be published).

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, Opt. Lett. 19, 737 (1994).
[CrossRef] [PubMed]

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, IEEE J. Quantum Electron. QE-22, 682 (1986).
[CrossRef]

M. M. Wefers and K. A. Nelson, "Programmable femtosecond multiple pulse generation and spectroscopy," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 39–41.

O. E. Martinez, IEEE J. Quantum Electron. QE-23, 59 (1987).
[CrossRef]

O. E. Martinez, J. Opt. Soc. Am. B 3, 929 (1986).
[CrossRef]

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

M. B. Danailov and I. P. Christov, J. Mod. Opt. 36, 725 (1989).
[CrossRef]

A. M. Weiner, S. Oudin, D. E. Leaird, and D. H. Reitze, J. Opt. Soc. Am. A 10, 1112 (1993).
[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

D. J. Kane and R. Trebino, Opt. Lett. 18, 823 (1993).
[CrossRef] [PubMed]

R. Trebino and D. J. Kane, J. Opt. Soc. Am. A 10, 1101 (1993).
[CrossRef]

J. L. A. Chilla and O. E. Martinez, Opt. Lett. 16, 39 (1991).
[CrossRef] [PubMed]

F. Salin, J. Squier, G. Mourou, and G. Vaillancourt, Opt. Lett. 16, 1964 (1991).
[CrossRef] [PubMed]

D. Pinkos, J. Squier, B. Kohler, V. V. Yakovlev, K. R. Wilson, D. Schumacher, and P. Bucksbaum, "Production of programmable amplified, shaped pulses in femtosecond lasers," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 227–229.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1994).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

4-F pulse-shaping apparatus. A grating angularly disperses an ultrashort pulse, and a lens collimates and focuses the laterally dispersed frequency components onto different regions of a spatially varying mask. The mask attenuates or retards selected frequency components. A subsequent lens and grating recombine the spectrally filtered light, producing a shaped waveform in the time domain.

Fig. 2
Fig. 2

Examples of shaped ultrafast waveforms generated from a 70-fs input pulse with our improved pulse-shaping apparatus and measured by cross correlation with an unshaped reference pulse. (a) 800-fs optical square pulse. The structure on the top of the square pulse follows from the limited bandwidth of the input pulse. (b) Phase-related equal-amplitude three-pulse sequence. (c) Phase-related five-pulse sequence. The desired waveform for (c) is given by the dashed curve. In all cases manipulation of both spectral amplitude and spectral phase is required for production of the desired waveforms.

Fig. 3
Fig. 3

8-F pulse-shaping apparatus. In this case two 1:1 telescopes are placed between parallel gratings so that angularly dispersed frequency components are focused and filtered twice. A combination of polarizers (Pol’s) and wave plates is used so the first LC SLM attenuates the frequency profile and the second LC SLM manipulates the relative phase of the frequency profile.

Fig. 4
Fig. 4

Illustration of the effects of pulse shaping by use of a discrete frequency filter with gaps. See text for details. (a) A desired shaped waveform. The dashed curve gives the time window associated with the spatial resolution of the mask. (b) Effects of discrete Fourier sampling and gaps between pixels on the desired waveform if these effects are not compensated for in the design of the mask pattern. Replica waveforms appear, and the waveform is modulated by a sinc envelope function (dashed curve). Additional features associated with the gaps appear in the center of the desired waveform and the replicas. (c) Effects of diffraction included, which yields a Gaussian temporal modulation (dashed curve) of the waveform. (c) Waveform produced by simple use of the discrete Fourier transform of the desired waveform as the frequency filter. (d) Output waveform produced by the mask when the effects described above are compensated for according to the prescription in Section 4.

Fig. 5
Fig. 5

Contour plot of the spatiotemporal amplitude profile of a shaped waveform. The desired waveform consists of three evenly spaced pulses with equal intensities. Because the desired waveform uses more than half of the 10-ps time window given by the spatial resolution of the mask, weak replica pulses outside this time window are apparent. Diffraction from the mask linearly shifts the spatial positions of the different pulses as a function of time. This is a general feature of the shaping of ultrafast waveforms through filtering of spatially dispersed frequency components.

Fig. 6
Fig. 6

Simulations of the time-dependent amplitude profile of the Gaussian spatial mode of a waveform shaped by the use of two LC SLM’s that filter orthogonal polarizations in a 4-F apparatus. The masks are displaced by 1 and 1.5 cm from the focal plane. The desired waveform consists of three equal-amplitude pulse pairs with relative amplitude 2:4:3. (a) The output waveform for a mask pattern derived by discrete sampling of the Fourier transform of the desired waveform, without compensation for impixellation, finite gap effects, and diffraction. (b) The output waveform for a mask pattern determined according to Eq. (66).

Fig. 7
Fig. 7

Simulation of the time-dependent amplitude profile of the Gaussian spatial mode of a waveform shaped by the use of two LC SLM’s that filter nonorthogonal polarizations in a 4-F apparatus and that are positioned symmetrically about the focal plane. The masks are separated, respectively, by 0 mm, 3 mm, 3 cm, and 3 cm in (a), (b), (c), and (d). The desired waveform is the same as in Fig. 6. The mask patterns for (a)–(c) are determined by discrete sampling of the Fourier transform of the desired waveform. In (d) we determined the mask pattern by using the iterative algorithm described by Eq. (67).

Fig. 8
Fig. 8

Simulation of the time-dependent amplitude profile of the Gaussian spatial mode of a waveform shaped by the use of two LC SLM’s with a separation of 3 mm that filter orthogonal polarizations in a 4-F apparatus. The two masks are positioned symmetrically about the focal plane but are laterally translated from each other by 30 μm (i.e., they are no longer in register). The desired waveform is the same in Figs. 6 and 7. The mask pattern for (a) was determined by discrete sampling of the Fourier transform of the desired waveform and assuming that the masks were equivalent to a single device capable of manipulating both spectral phase and amplitude. The mask pattern for (b) was determined according to Eq. (67). This illustrates that iterative improvement of the output waveform can correct for the lack of registration between the two masks.

Equations (74)

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

F ( k , ω ) = ( 2 π ) - 1 f ( x , t ) exp [ i ( k x - ω t ) ] d x d t ,
f ( x , t ) = ( 2 π ) - 1 F ( k , ω ) exp [ - i ( k x - ω t ) ] d k d ω ,
A n = 1 N j = 0 N - 1 B j exp ( i 2 π j n / N ) B j = 1 N j = 0 N - 1 A n exp ( - i 2 π j n / N ) ,
h ( x ) = f ( x ) g ( x ) = - d x f ( x - x ) g ( x ) .
e in ( x , t ) = g ( x ) e in ( t ) exp ( i ω ¯ t ) ,
Δ ϕ = ω Δ n ( V ) l / c ,
B n = exp [ i Δ ϕ ( V n ) ] .
B n = exp [ i Δ ϕ ( V n ) / 2 ] { x ^ cos [ Δ ϕ ( V n ) / 2 ] + i y ^ sin [ Δ ϕ ( V n ) / 2 ] } .
B n = exp [ i Δ ϕ ( V n ) ] exp [ i Δ ϕ ( V n ) / 2 ] sin [ Δ ϕ ( V n ) / 2 ] .
B n = exp [ i Δ ϕ ( 1 ) ] exp [ i Δ ϕ ( 2 ) / 2 ] sin [ Δ ϕ ( 2 ) / 2 ] ,
B n = exp { i [ Δ ϕ ( 1 ) + Δ ϕ ( 2 ) ] / 2 } cos { [ Δ ϕ ( 1 ) - Δ ϕ ( 2 ) ] / 2 } ,
λ = d ( sin θ i + sin θ d ) ,
x = f tan [ θ d ( λ ) - θ d ( λ 0 ) ] .
x = f [ θ d ω | ω = ω ¯ ( ω - ω ¯ ) + 1 2 2 θ d ω 2 | ω = ω ¯ ( ω - ω ¯ ) 2 + ] ,
θ d ω | ω = ω ¯ = - 2 π c ω ¯ 2 d cos θ d ( ω ¯ ) ,
2 nd - order term 1 st - order term = - ( ω - ω ¯ ) ω ¯ ( 1 + λ 2 d tan θ d cos θ d ) .
x ( ω ) = - 2 π c f ω ¯ 2 d cos θ d ( ω ¯ ) ( ω - ω ¯ ) α ( ω - ω ¯ ) .
P ( ω ) = P ( ω ¯ ) [ 1 - a ( ω - ω ¯ ) 2 ] ,
a = 1 2 ( θ d ω | ω = ω ¯ ) 2 = 2 π 2 c 2 ω ¯ 4 d 2 cos 2 θ d ( ω ¯ ) .
E out ( x , ω ) = g ( x ) E in ( Ω ) m ( α Ω ) ,
m ( x ) = ( x - x 0 ) n = - N / 2 N / 2 - 1 ( [ B n ( x - n w ) ] squ ( x / r w ) + { B g [ x - ( n + 1 / 2 ) w ] } squ [ x / ( 1 - r ) w ] ) ,
squ ( x ) = { 1 x 1 / 2 0 x > 1 / 2 .
M ( k ) = exp ( i k x 0 ) ( sin ( r k w ) π k × { n = - A rem ( n , N ) ( k - n 2 π N w ) + B g sin [ ( 1 - r ) k w ] π k } × [ n = - ( - 1 ) n ( k - n 2 π w ) ] ) ,
A ( ω , x , y ) = E in ( Ω ) m ( x ) u 00 ( x - α Ω , y ) ,
u 00 ( x , y ) = ( 2 / π w 0 2 ) 1 / 2 exp [ - ( x 2 + y 2 ) / w 0 2 ] .
A ( ω , x , y ) = m , n A m n ( ω ) u m n ( x - α Ω , y ) .
A 00 ( ω ) = ( 2 / π w 0 2 ) 1 / 2 E ( Ω ) - m ( x ) × exp [ - 2 ( x - α Ω ) 2 / w 0 2 ] d x .
e out ( x , y , t ) = u 00 ( x , y ) exp ( i ω ¯ t ) × { e in ( t ) [ M ( - t / α ) g ( t ) ] } ,
g ( t ) = exp ( - w 0 2 t 2 / 8 α 2 ) .
e out 00 ( t ) = - C n e in ( t + τ n ) ,
C n = N π n exp ( i 2 π x 0 n N w ) exp [ - ( π w 0 n N w ) 2 / 2 ] × { A rem ( n , N ) sin ( r π n N ) + 0 , rem ( n , N ) B g ( - 1 ) n / N × sin [ ( 1 - r ) π n N ] } ,
C n = A rem ( n , N ) sin ( π n / N ) ( π n / N ) exp [ - ( π w 0 n w N ) 2 / 2 ] .
- N τ / 2 = - α π / w t < N τ / 2 = α π / w .
C 0 = r A 0 + ( 1 - r ) B g .
e ( x , z = - F ) = g ( x ) e ( x , z = F ) = g ¯ ( 2 π x / λ F ) ,
E ¯ ( k , z = z 0 ) = E ¯ ( k , z = 0 ) exp ( - i z 0 λ 4 π k 2 ) .
E ( x , ω ) grating = b 1 E in ( δ x , ω ) exp ( i γ Ω x ) ,
e in ( x , t ) = e in ( t ) exp ( i ω ¯ t ) exp ( - x 2 / a 2 ) .
E 1 ( x , ω ) = b 1 E in ( Ω ) exp ( - δ 2 a 2 x 2 + i γ Ω x ) ,
E ¯ 2 ( k , ω ) = b 2 E in ( Ω ) exp [ - ( δ λ F 2 π a ) 2 k 2 - i λ γ F 2 π Ω k ] .
E ¯ 3 ( k , ω ) = b 3 [ E ¯ 2 ( k , ω ) exp ( - i z 0 λ 4 π k 2 ) M ( k ) ] × exp ( + i z 0 λ 4 π k 2 ) .
M ( k ) = n = - c n ( k - n 2 π N w ) ,
c n = N π n exp ( i 2 π x 0 n L ) { A rem ( n , N ) sin ( r π n N ) + 0 , rem ( n , N ) B g ( - 1 ) n / N sin [ ( 1 - r ) π n N ] } .
E ¯ 3 ( k , ω ) = b 3 E in ( Ω ) n = - ( c n exp { - [ ( δ λ F 2 π a ) 2 + i z 0 λ 4 π ] × ( k - n 2 π N w ) 2 } exp [ - i Ω γ λ F 2 π ( k - n 2 π N w ) ] × exp ( i z 0 λ 4 π k 2 ) ) .
E 4 ( x , ω ) = b 4 E in ( Ω ) n = - { c n exp [ - ( δ 2 a 2 + i π z 0 λ F 2 ) × ( x - n λ F N w ) 2 ] exp [ - i Ω γ ( x - n λ F N w ) ] × exp ( i π z 0 λ F 2 k 2 ) } .
E out ( x , ω ) = b 5 E 4 ( x / δ , ω ) exp ( i γ Ω x / δ ) .
e out ( x , t ) = b 6 exp ( i ω ¯ t ) n = - { c n exp [ - i π z 0 λ w 2 ( n N ) 2 ] × e in ( t + n τ ) exp [ - ( x - n χ ) 2 a 2 ] × exp [ i 2 π z 0 δ F w ( n N ) x ] } ,
e out 00 ( t ) = n = - c n e in ( t + n τ ) exp { - n 2 2 ( χ a ) 2 × [ 1 + ( π λ a 2 z 0 χ 2 w 2 N 2 ) 2 ] } .
e out ( x , t ) = exp ( i ω ¯ t ) n = - c n e in ( t + n τ ) × exp [ - ( x - n χ ) 2 a 2 ] .
x / t = - χ / τ = - c d cos ( θ i ) / λ ,
χ ¯ = N τ x / t = δ λ F / w .
tan ( ρ ) = Δ k ¯ x k ¯ z = z 0 λ δ f w .
e out ( x , t ) = exp ( i ω ¯ t ) n = - [ e in ( t + τ n ) exp [ - ( x - n χ ) 2 a 2 ] × exp [ - i π z 0 ( 1 ) λ w 2 ( n N ) 2 ] × exp [ - i 2 π z 0 ( 1 ) δ F w ( n N ) x ] × n = - ( c n + n ( 1 ) c n ( 2 ) exp { i π [ z 0 ( 1 ) - z 0 ( 2 ) ] w 2 ( n N ) 2 } × exp { - i 2 π [ z 0 ( 1 ) - z 0 ( 2 ) ] δ F w ( n N ) x } ) ] ,
e out ( x , t ) = exp ( i ω ¯ t ) n = - { e in ( t + τ n ) exp [ - ( x - n χ ) 2 a 2 ] × exp [ - i 2 π z 0 ( 1 ) δ F w ( n N ) x ] × exp [ - i π z 0 ( 1 ) λ w 2 ( n N ) 2 ] n = - c n + n ( 1 ) c n ( 2 ) } .
e out 00 ( t ) = n = - n = - [ c n ( 1 ) c n ( 2 ) e in [ t + ( n - n ) τ ] × exp [ i π λ ( z 0 ( 1 ) - z 0 ( 2 ) ) w 2 n n N 2 ] × exp ( - χ 2 2 a 2 { ( n - n ) 2 + ( π λ a 2 χ 2 w 2 ) 2 × [ z 0 ( 1 ) n - z 0 ( 2 ) n N ] 2 } ) ] .
e out ( x , t ) = exp ( i ω ¯ t ) n = - ( e in ( t + τ n ) exp [ - ( x - n χ ) 2 a 2 ] × exp [ - i π z 0 λ w 2 ( n N ) 2 ] exp ( i 2 π z 0 δ F w n N x ) × n = - { c n - n ( 1 ) c n ( 2 ) exp [ - i π λ z ¯ w 2 ( n N ) 2 ] × exp ( i π 2 z ¯ δ F w n N x ) } ) ,
e out 00 ( t ) = n = - n = - ( c n ( 1 ) c n ( 2 ) e in [ t + ( n + n ) τ ] × exp ( i π λ z ¯ w 2 n n N 2 ) × exp { - χ 2 2 a 2 [ ( n + n ) 2 + ( π λ a 2 χ 2 w 2 ) 2 × ( z 0 ( n + n ) + z ¯ n N ) 2 ] } ) .
e out ( x , t ) = exp ( i ω ¯ t ) n = - ( e in ( t + τ n ) exp [ - ( x - n χ ) 2 a 2 ] × n = - { c n - n ( 1 ) c n ( 2 ) exp [ - i π λ z ¯ w 2 ( n N ) 2 ] × exp ( i 2 π z ¯ α F w n N x ) } ) .
e out ( x , t ) = exp ( i ω ¯ t ) n = - ( e in ( t + τ n ) exp [ - ( x - n χ ) 2 a 2 ] × n = - { c n - n ( 1 ) c n ( 2 ) exp [ - i π λ z ¯ w 2 ( n N ) 2 ] × exp ( i 2 π λ z ¯ w 2 n N 2 ) } ) .
e out ( x , t ) = exp ( i ω ¯ t ) n = - ( e in ( t + n τ ) exp [ - ( x - n χ ) 2 a 2 ] × { c n ( 1 ) exp [ - i π z 0 ( 1 ) λ w 2 ( n N ) 2 ] × exp [ i 2 π z 0 ( 1 ) δ F w ( n N ) x ] + c n ( 2 ) exp [ - i π z 0 ( 2 ) λ w 2 ( n N ) 2 ] × exp [ i 2 π z 0 ( 2 ) δ F w ( n N ) x ] } ) .
e out 00 ( t ) = n = - e in ( t + n τ ) exp ( - n 2 χ 2 2 a 2 ) × ( c n ( 1 ) exp { - n 2 2 ( χ a ) 2 [ π λ a 2 z 0 ( 1 ) χ 2 w 2 N 2 ] 2 } + c n ( 2 ) exp { - n 2 2 ( χ a ) 2 [ π λ a 2 z 0 ( 2 ) χ 2 w 2 N 2 ] 2 } ) .
e out 00 ( t ) = n = - [ c n ( 1 ) + c n ( 2 ) 2 ] e in ( t + n τ ) × exp { - n 2 χ 2 2 a 2 [ 1 + ( π λ a 2 z ¯ 2 χ 2 w 2 N 2 ) 2 ] } .
m ( x ) = H ( - x / α ) / E in ( x / α - ω ¯ ) ,
e ( x , t ) = h ( t ) F ( x - t δ / γ ) ,
e out 00 ( t ) = - C n e in ( t + τ n ) ,
A n = η C n - 0 , n B g ( 1 - r ) sin ( r π n ) r π exp [ - ( π w 0 n w N ) 2 / 2 ]
A n ( j + 1 ) = A n ( j ) + C n - C n ( j ) p .
B n = exp ( i φ n ) ,             0 φ n 2 π .
I n desired A n 2 = | 1 p k = - p / 2 p / 2 - 1 exp ( i φ k ) exp ( i 2 π n k / p ) | 2 .
I n = A n 2 = inverse discrete Fourier transform of B n 2 = const .
F = p w 2 π α ,
e desired ( t ) = e in ( t ) n = 0 N - 1 H n ( t - T n ) ,
E in ( ω ) n = 0 N - 1 H n exp ( i ω T n ) = E in ( ω ) n = - N / 2 N / 2 - 1 C n exp ( i ω n τ ) .
C n = τ 2 π - π / τ π / τ [ n = 0 N H n exp ( i ω T n ) ] exp ( - i ω τ n ) d ω .

Metrics