Abstract

The space–time Wigner function is a powerful tool for the analysis of femtosecond optical devices that act on both the spatial and the temporal features of ultrashort pulses. The advantages of this approach and the properties of the space–time Wigner function are reviewed. The matrix formalism that can be used to describe the action of dispersive first-order devices on any femtosecond pulse is presented. The method is used to describe and understand the behavior of an ultrashort light pulse propagating through a pulse shaper. The consequences of the limited resolution in the spectral-to-spatial conversion for the output pulse are fully analyzed.

© 1995 Optical Society of America

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  1. J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds., Ultrafast Phenomena VIII (Springer-Verlag, Berlin, 1993); P. F. Barbara, W. H. Knox, G. A. Mourou, and A. H. Zewail, eds., Ultrafast IX (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  2. P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, "Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion," Opt. Lett. 18, 54–56 (1993).
    [CrossRef] [PubMed]
  3. M. T. Asaki, C. P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M. Murnane, "Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser," Opt. Lett. 18, 977–979 (1993).
    [CrossRef] [PubMed]
  4. S. A. Rice, "New ideas for guiding the evolution of a quantum system," Science 258, 412–413 (1992).
    [CrossRef] [PubMed]
  5. W. S. Warren, H. Rabitz, and M. Dahleh, "Coherent control of quantum dynamics: the dream is alive," Science 259, 1581–1589 (1993).
    [CrossRef] [PubMed]
  6. D. Umstadter, E. Esarey, and J. Kim, "Nonlinear plasma waves resonantly driven by optimized pulse trains," Phys. Rev. Lett. 72, 1224–1227 (1994).
    [CrossRef] [PubMed]
  7. R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Appl. Phys. Lett. 38, 671–673 (1984).
    [CrossRef]
  8. M. Ramaswamy-Paye and J. G. Fujimoto, "Compact dispersion-compensating geometry for Kerr-lens mode-locked femtosecond lasers," Opt. Lett. 19, 1756–1758 (1994).
    [CrossRef] [PubMed]
  9. M. Kempe, U. Stamm, B. Wihelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158–1165 (1992).
    [CrossRef]
  10. Z. L. Horváth and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description," Opt. Commun. 108, 333–342 (1994).
    [CrossRef]
  11. R. Gase, "The Wolf function interacting with multi-layer systems," Opt. Commun. 79, 397–401 (1990).
    [CrossRef]
  12. J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262–2273 (1992).
    [CrossRef]
  13. J. Paye, "Application of the chronocyclic representation of ultrashort light pulses," in Ultrafast Phenomena VIII, J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds. (Springer-Verlag, Berlin, 1993).
    [CrossRef]
  14. D. E. Spence, P. N. Kean, and W. Sibbett, "60-fsec pulse generation from a self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 42–44 (1991).
    [CrossRef] [PubMed]
  15. L. Spinelli, B. Couillaud, N. Goldblatt, and D. K. Negus, "Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self-focusing," in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.
  16. M. Pessot, P. Maine, and G. Mourou "1000 times expansiony/compression of optical pulses for chirped-pulse amplification," Opt. Commun. 62, 419–421 (1987).
    [CrossRef]
  17. We refer the reader to the still unpublished extensive review by A. M. Weiner, "Femtosecond optical pulse shaping and processing," submitted to Progr. Quantum Electron.
  18. C. Froehly, B. Colombeau, and M. Vampouille, "Shaping and analysis of picosecond light pulses," in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1983), Vol. XX.
    [CrossRef]
  19. R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682–696 (1986).
    [CrossRef]
  20. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, "Programmable femtosecond pulse shaping by use of a multielement liquid-crysstal phase modulator," Opt. Lett. 15, 326–328 (1990).
    [CrossRef] [PubMed]
  21. M. M. Wefers and K. A. Nelson, "Ultrafast optical waveforms," Science 262, 1381–1382 (1993).
    [CrossRef] [PubMed]
  22. M. M. Wefers and K. A. Nelson, "Programmable phase and amplitude femtosecond pulse shaping," Opt. Lett. 19, 2032–2034 (1994).
  23. C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, "Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses," Opt. Lett. 19, 737–739 (1994).
    [CrossRef] [PubMed]
  24. A. M. Weiner, D. E. Leaird, and D. H. Reitze, "Femtosecond spatial holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
    [CrossRef]
  25. L. Cohen, "Time-frequency distributions—a review," Proc. IEEE 77, 941–981 (1989), and references therein.
    [CrossRef]
  26. C. H. Page, "Instantaneous power spectra," J. Appl. Phys. 23, 103–106 (1952).
    [CrossRef]
  27. R. Gase, "Time-dependent spectrum of linear optical systems," J. Opt. Soc. Am. A 8, 850–859 (1991).
    [CrossRef]
  28. J. H. Eberly and K. Wódkiewicz, "The time-dependent physical spectrum of light," J. Opt. Soc. Am. 67, 1252–1261 (1977).
    [CrossRef]
  29. J. Ville, "Théorie et applications de la notion de signal analytique," Cables Transmiss. 2A, 61–74 (1948).
  30. H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, "The Wigner distribution function and its optical production," Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  31. E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  32. M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  33. R. Gase, H.-E. Ponath, and M. Schubert, "On a temporal-spatial radiation functional and its measurement," Ann. Pys. 43, 487–498 (1986).
    [CrossRef]
  34. M. J. Bastiaans, "The Wigner distribution and its application to optics," in Optics in Four Dimensions—1980, M. A. Machado and L. M. Narducci, eds., Am. Inst. Phys. Proc. 65, 292–312 (1981).
  35. J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136–138 (1978).
    [CrossRef]
  36. M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  37. R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields," Phys. Rev. A 29, 3273–3275 (1984).
    [CrossRef]
  38. A. T. Friberg, "Propagation of a generalized radiance in paraxial optical systems," Appl. Opt. 30, 2443–2446 (1991).
    [CrossRef] [PubMed]
  39. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.
  40. Ref. 39, Chap. 16.
  41. O. E. Martinez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530–2536 (1988).
    [CrossRef]
  42. O. E. Martinez, "Matrix formalism for dispersive laser cavities," IEEE J. Quantum Electron. 25, 296–300 (1989).
    [CrossRef]
  43. A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron. 26, 1148–1157 (1990).
    [CrossRef]
  44. M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725–731 (1989).
    [CrossRef]
  45. J. Paye, "Applications of the chronocyclic representation of ultrashort light pulses," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper MC1.
  46. 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.
  47. Since we submitted this paper, we have been made aware of two preprints by 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, 1343–1362 (1995); "Space-time profiles of shaped ultrafast optical waveforms," submitted to IEEE J. Quantum Electron.
    [CrossRef]
  48. B. H. Kolner and M. Nazarathy, "Temporal imaging with a time lens," Opt. Lett. 14, 630–632 (1989).
    [CrossRef] [PubMed]
  49. O. E. Martinez, "Grating and prism compressors in the case of finite beam size," J. Opt. Soc. Am. B 3, 929–934 (1986).
    [CrossRef]

1994 (5)

D. Umstadter, E. Esarey, and J. Kim, "Nonlinear plasma waves resonantly driven by optimized pulse trains," Phys. Rev. Lett. 72, 1224–1227 (1994).
[CrossRef] [PubMed]

M. Ramaswamy-Paye and J. G. Fujimoto, "Compact dispersion-compensating geometry for Kerr-lens mode-locked femtosecond lasers," Opt. Lett. 19, 1756–1758 (1994).
[CrossRef] [PubMed]

Z. L. Horváth and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description," Opt. Commun. 108, 333–342 (1994).
[CrossRef]

M. M. Wefers and K. A. Nelson, "Programmable phase and amplitude femtosecond pulse shaping," Opt. Lett. 19, 2032–2034 (1994).

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, "Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses," Opt. Lett. 19, 737–739 (1994).
[CrossRef] [PubMed]

1993 (4)

1992 (4)

S. A. Rice, "New ideas for guiding the evolution of a quantum system," Science 258, 412–413 (1992).
[CrossRef] [PubMed]

M. Kempe, U. Stamm, B. Wihelmi, and W. Rudolph, "Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems," J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

A. M. Weiner, D. E. Leaird, and D. H. Reitze, "Femtosecond spatial holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[CrossRef]

1991 (3)

1990 (3)

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, "Programmable femtosecond pulse shaping by use of a multielement liquid-crysstal phase modulator," Opt. Lett. 15, 326–328 (1990).
[CrossRef] [PubMed]

A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

R. Gase, "The Wolf function interacting with multi-layer systems," Opt. Commun. 79, 397–401 (1990).
[CrossRef]

1989 (4)

L. Cohen, "Time-frequency distributions—a review," Proc. IEEE 77, 941–981 (1989), and references therein.
[CrossRef]

M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725–731 (1989).
[CrossRef]

B. H. Kolner and M. Nazarathy, "Temporal imaging with a time lens," Opt. Lett. 14, 630–632 (1989).
[CrossRef] [PubMed]

O. E. Martinez, "Matrix formalism for dispersive laser cavities," IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

1988 (1)

O. E. Martinez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

1987 (1)

M. Pessot, P. Maine, and G. Mourou "1000 times expansiony/compression of optical pulses for chirped-pulse amplification," Opt. Commun. 62, 419–421 (1987).
[CrossRef]

1986 (3)

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682–696 (1986).
[CrossRef]

O. E. Martinez, "Grating and prism compressors in the case of finite beam size," J. Opt. Soc. Am. B 3, 929–934 (1986).
[CrossRef]

R. Gase, H.-E. Ponath, and M. Schubert, "On a temporal-spatial radiation functional and its measurement," Ann. Pys. 43, 487–498 (1986).
[CrossRef]

1984 (2)

R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Appl. Phys. Lett. 38, 671–673 (1984).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields," Phys. Rev. A 29, 3273–3275 (1984).
[CrossRef]

1980 (1)

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, "The Wigner distribution function and its optical production," Opt. Commun. 32, 32–38 (1980).
[CrossRef]

1979 (1)

1978 (3)

E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26–30 (1978).
[CrossRef]

J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136–138 (1978).
[CrossRef]

1977 (1)

1952 (1)

C. H. Page, "Instantaneous power spectra," J. Appl. Phys. 23, 103–106 (1952).
[CrossRef]

1948 (1)

J. Ville, "Théorie et applications de la notion de signal analytique," Cables Transmiss. 2A, 61–74 (1948).

Asaki, M. T.

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, "The Wigner distribution function and its optical production," Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26–30 (1978).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution and its application to optics," in Optics in Four Dimensions—1980, M. A. Machado and L. M. Narducci, eds., Am. Inst. Phys. Proc. 65, 292–312 (1981).

Bor, Zs.

Z. L. Horváth and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description," Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Brabec, T.

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, "The Wigner distribution function and its optical production," Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Christov, I. P.

M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725–731 (1989).
[CrossRef]

Cohen, L.

L. Cohen, "Time-frequency distributions—a review," Proc. IEEE 77, 941–981 (1989), and references therein.
[CrossRef]

Colombeau, B.

C. Froehly, B. Colombeau, and M. Vampouille, "Shaping and analysis of picosecond light pulses," in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1983), Vol. XX.
[CrossRef]

Couillaud, B.

L. Spinelli, B. Couillaud, N. Goldblatt, and D. K. Negus, "Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self-focusing," in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Curley, P. F.

Dahleh, M.

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

Danailov, M. B.

M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725–731 (1989).
[CrossRef]

Eberly, J. H.

Esarey, E.

D. Umstadter, E. Esarey, and J. Kim, "Nonlinear plasma waves resonantly driven by optimized pulse trains," Phys. Rev. Lett. 72, 1224–1227 (1994).
[CrossRef] [PubMed]

Fork, R. L.

R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Appl. Phys. Lett. 38, 671–673 (1984).
[CrossRef]

Friberg, A. T.

Froehly, C.

C. Froehly, B. Colombeau, and M. Vampouille, "Shaping and analysis of picosecond light pulses," in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1983), Vol. XX.
[CrossRef]

Fujimoto, J. G.

Garvey, D.

Gase, R.

R. Gase, "Time-dependent spectrum of linear optical systems," J. Opt. Soc. Am. A 8, 850–859 (1991).
[CrossRef]

R. Gase, "The Wolf function interacting with multi-layer systems," Opt. Commun. 79, 397–401 (1990).
[CrossRef]

R. Gase, H.-E. Ponath, and M. Schubert, "On a temporal-spatial radiation functional and its measurement," Ann. Pys. 43, 487–498 (1986).
[CrossRef]

Goldblatt, N.

L. Spinelli, B. Couillaud, N. Goldblatt, and D. K. Negus, "Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self-focusing," in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Gordon, J. P.

R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Appl. Phys. Lett. 38, 671–673 (1984).
[CrossRef]

Goswami, D.

Guigay, J.-P.

J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136–138 (1978).
[CrossRef]

Heritage, J. P.

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682–696 (1986).
[CrossRef]

Hillegas, C. W.

Horváth, Z. L.

Z. L. Horváth and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description," Opt. Commun. 108, 333–342 (1994).
[CrossRef]

Huang, C. P.

Kapteyn, H. C.

Kean, P. N.

Kempe, M.

Kim, J.

D. Umstadter, E. Esarey, and J. Kim, "Nonlinear plasma waves resonantly driven by optimized pulse trains," Phys. Rev. Lett. 72, 1224–1227 (1994).
[CrossRef] [PubMed]

Kolner, B. H.

Kostenbauder, A. G.

A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

Krausz, F.

Leaird, D. E.

Lohmann, A. W.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, "The Wigner distribution function and its optical production," Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Maine, P.

M. Pessot, P. Maine, and G. Mourou "1000 times expansiony/compression of optical pulses for chirped-pulse amplification," Opt. Commun. 62, 419–421 (1987).
[CrossRef]

Martinez, O. E.

O. E. Martinez, "Matrix formalism for dispersive laser cavities," IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

O. E. Martinez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

O. E. Martinez, "Grating and prism compressors in the case of finite beam size," J. Opt. Soc. Am. B 3, 929–934 (1986).
[CrossRef]

R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Appl. Phys. Lett. 38, 671–673 (1984).
[CrossRef]

Mourou, G.

M. Pessot, P. Maine, and G. Mourou "1000 times expansiony/compression of optical pulses for chirped-pulse amplification," Opt. Commun. 62, 419–421 (1987).
[CrossRef]

Mukunda, N.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields," Phys. Rev. A 29, 3273–3275 (1984).
[CrossRef]

Murnane, M. M.

Nazarathy, M.

Negus, D. K.

L. Spinelli, B. Couillaud, N. Goldblatt, and D. K. Negus, "Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self-focusing," in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Nelson, K. A.

M. M. Wefers and K. A. Nelson, "Programmable phase and amplitude femtosecond pulse shaping," Opt. Lett. 19, 2032–2034 (1994).

M. M. Wefers and K. A. Nelson, "Ultrafast optical waveforms," Science 262, 1381–1382 (1993).
[CrossRef] [PubMed]

Since we submitted this paper, we have been made aware of two preprints by 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, 1343–1362 (1995); "Space-time profiles of shaped ultrafast optical waveforms," submitted to IEEE J. Quantum Electron.
[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.

Page, C. H.

C. H. Page, "Instantaneous power spectra," J. Appl. Phys. 23, 103–106 (1952).
[CrossRef]

Patel, J. S.

Paye, J.

J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[CrossRef]

J. Paye, "Application of the chronocyclic representation of ultrashort light pulses," in Ultrafast Phenomena VIII, J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds. (Springer-Verlag, Berlin, 1993).
[CrossRef]

J. Paye, "Applications of the chronocyclic representation of ultrashort light pulses," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper MC1.

Pessot, M.

M. Pessot, P. Maine, and G. Mourou "1000 times expansiony/compression of optical pulses for chirped-pulse amplification," Opt. Commun. 62, 419–421 (1987).
[CrossRef]

Ponath, H.-E.

R. Gase, H.-E. Ponath, and M. Schubert, "On a temporal-spatial radiation functional and its measurement," Ann. Pys. 43, 487–498 (1986).
[CrossRef]

Rabitz, H.

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

Ramaswamy-Paye, M.

Reitze, D. H.

A. M. Weiner, D. E. Leaird, and D. H. Reitze, "Femtosecond spatial holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Rice, S. A.

S. A. Rice, "New ideas for guiding the evolution of a quantum system," Science 258, 412–413 (1992).
[CrossRef] [PubMed]

Rudolph, W.

Schmidt, A. J.

Schubert, M.

R. Gase, H.-E. Ponath, and M. Schubert, "On a temporal-spatial radiation functional and its measurement," Ann. Pys. 43, 487–498 (1986).
[CrossRef]

Sibbett, W.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.

Simon, R.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields," Phys. Rev. A 29, 3273–3275 (1984).
[CrossRef]

Spence, D. E.

Spielmann, Ch.

Spinelli, L.

L. Spinelli, B. Couillaud, N. Goldblatt, and D. K. Negus, "Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self-focusing," in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

Stamm, U.

Strickland, D.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields," Phys. Rev. A 29, 3273–3275 (1984).
[CrossRef]

Thurston, R. N.

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682–696 (1986).
[CrossRef]

Tomlinson, W. J.

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682–696 (1986).
[CrossRef]

Tull, J. X.

Umstadter, D.

D. Umstadter, E. Esarey, and J. Kim, "Nonlinear plasma waves resonantly driven by optimized pulse trains," Phys. Rev. Lett. 72, 1224–1227 (1994).
[CrossRef] [PubMed]

Vampouille, M.

C. Froehly, B. Colombeau, and M. Vampouille, "Shaping and analysis of picosecond light pulses," in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1983), Vol. XX.
[CrossRef]

Ville, J.

J. Ville, "Théorie et applications de la notion de signal analytique," Cables Transmiss. 2A, 61–74 (1948).

Warren, W. S.

Wefers, M. M.

M. M. Wefers and K. A. Nelson, "Programmable phase and amplitude femtosecond pulse shaping," Opt. Lett. 19, 2032–2034 (1994).

M. M. Wefers and K. A. Nelson, "Ultrafast optical waveforms," Science 262, 1381–1382 (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.

Since we submitted this paper, we have been made aware of two preprints by 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, 1343–1362 (1995); "Space-time profiles of shaped ultrafast optical waveforms," submitted to IEEE J. Quantum Electron.
[CrossRef]

Weiner, A. M.

A. M. Weiner, D. E. Leaird, and D. H. Reitze, "Femtosecond spatial holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, "Programmable femtosecond pulse shaping by use of a multielement liquid-crysstal phase modulator," Opt. Lett. 15, 326–328 (1990).
[CrossRef] [PubMed]

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682–696 (1986).
[CrossRef]

We refer the reader to the still unpublished extensive review by A. M. Weiner, "Femtosecond optical pulse shaping and processing," submitted to Progr. Quantum Electron.

Wihelmi, B.

Wintner, E.

Wódkiewicz, K.

Wolf, E.

Wullert, J. R.

Zhou, J.

Ann. Pys. (1)

R. Gase, H.-E. Ponath, and M. Schubert, "On a temporal-spatial radiation functional and its measurement," Ann. Pys. 43, 487–498 (1986).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. L. Fork, O. E. Martinez, and J. P. Gordon, "Negative dispersion using pairs of prisms," Appl. Phys. Lett. 38, 671–673 (1984).
[CrossRef]

Cables Transmiss. (1)

J. Ville, "Théorie et applications de la notion de signal analytique," Cables Transmiss. 2A, 61–74 (1948).

IEEE J. Quantum Electron. (6)

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682–696 (1986).
[CrossRef]

A. M. Weiner, D. E. Leaird, and D. H. Reitze, "Femtosecond spatial holography," IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

J. Paye, "The chronocyclic representation of ultrashort light pulses," IEEE J. Quantum Electron. 28, 2262–2273 (1992).
[CrossRef]

O. E. Martinez, "Matrix formalism for pulse compressors," IEEE J. Quantum Electron. 24, 2530–2536 (1988).
[CrossRef]

O. E. Martinez, "Matrix formalism for dispersive laser cavities," IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

J. Appl. Phys. (1)

C. H. Page, "Instantaneous power spectra," J. Appl. Phys. 23, 103–106 (1952).
[CrossRef]

J. Mod. Opt. (1)

M. B. Danailov and I. P. Christov, "Time-space shaping of light pulses by Fourier optical processing," J. Mod. Opt. 36, 725–731 (1989).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

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

Opt. Commun. (6)

Z. L. Horváth and Zs. Bor, "Behaviour of femtosecond pulses on the optical axis of a lens. Analytical description," Opt. Commun. 108, 333–342 (1994).
[CrossRef]

R. Gase, "The Wolf function interacting with multi-layer systems," Opt. Commun. 79, 397–401 (1990).
[CrossRef]

M. Pessot, P. Maine, and G. Mourou "1000 times expansiony/compression of optical pulses for chirped-pulse amplification," Opt. Commun. 62, 419–421 (1987).
[CrossRef]

J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136–138 (1978).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26–30 (1978).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, "The Wigner distribution function and its optical production," Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Opt. Lett. (8)

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Generalized rays in first order optics: transformation properties of Gaussian Schell-model fields," Phys. Rev. A 29, 3273–3275 (1984).
[CrossRef]

Phys. Rev. Lett. (1)

D. Umstadter, E. Esarey, and J. Kim, "Nonlinear plasma waves resonantly driven by optimized pulse trains," Phys. Rev. Lett. 72, 1224–1227 (1994).
[CrossRef] [PubMed]

Proc. IEEE (1)

L. Cohen, "Time-frequency distributions—a review," Proc. IEEE 77, 941–981 (1989), and references therein.
[CrossRef]

Science (3)

M. M. Wefers and K. A. Nelson, "Ultrafast optical waveforms," Science 262, 1381–1382 (1993).
[CrossRef] [PubMed]

S. A. Rice, "New ideas for guiding the evolution of a quantum system," Science 258, 412–413 (1992).
[CrossRef] [PubMed]

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

Other (11)

J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds., Ultrafast Phenomena VIII (Springer-Verlag, Berlin, 1993); P. F. Barbara, W. H. Knox, G. A. Mourou, and A. H. Zewail, eds., Ultrafast IX (Springer-Verlag, Berlin, 1994).
[CrossRef]

L. Spinelli, B. Couillaud, N. Goldblatt, and D. K. Negus, "Starting and generation of sub-100 fs pulses in Ti:Al2O3 by self-focusing," in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

We refer the reader to the still unpublished extensive review by A. M. Weiner, "Femtosecond optical pulse shaping and processing," submitted to Progr. Quantum Electron.

C. Froehly, B. Colombeau, and M. Vampouille, "Shaping and analysis of picosecond light pulses," in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1983), Vol. XX.
[CrossRef]

J. Paye, "Application of the chronocyclic representation of ultrashort light pulses," in Ultrafast Phenomena VIII, J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail, eds. (Springer-Verlag, Berlin, 1993).
[CrossRef]

M. J. Bastiaans, "The Wigner distribution and its application to optics," in Optics in Four Dimensions—1980, M. A. Machado and L. M. Narducci, eds., Am. Inst. Phys. Proc. 65, 292–312 (1981).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.

Ref. 39, Chap. 16.

J. Paye, "Applications of the chronocyclic representation of ultrashort light pulses," in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper MC1.

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.

Since we submitted this paper, we have been made aware of two preprints by 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, 1343–1362 (1995); "Space-time profiles of shaped ultrafast optical waveforms," submitted to IEEE J. Quantum Electron.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Representation of a simulated input pulse with a Gaussian spatial mode (FWHM, 1 mm) and a secant hyperbolic temporal mode (FWHM, 100 fs). The central wavelength of the pulse is 800 nm. The six two-dimensional integrals of the space–time Wigner function are represented [see Eqs. (13)(18)] as functions of their respective variables.

Fig. 2
Fig. 2

Representation of the pulse of Fig. 1 after the first grating of the pulse shaper. The grating has 600 grooves/mm and is used in Littrow configuration.

Fig. 3
Fig. 3

Schematic of a pulse shaper setup.

Fig. 4
Fig. 4

Representation of the pulse of Fig. 1 before the mask of the pulse shaper. The focal length of the lens is 427 mm.

Fig. 5
Fig. 5

Representation of the pulse of Fig. 1 after the mask of the pulse shaper.

Fig. 6
Fig. 6

Representation of the pulse of Fig. 1 at the output of the pulse shaper.

Tables (1)

Tables Icon

Table 1 Coordinate Transformation Matrices for Different Devices and Their Corresponding Scaling Factors

Equations (53)

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E ( x , z , t ) = E ( x , z , t ) exp i ( k 0 z - ω 0 t ) ,
E ˜ ( x , ω ) = E ( x , t ) exp ( i ω t ) d t .
E ^ ( ξ , t ) = 1 2 π E ( x , t ) exp ( - i ξ x ) d x .
E ˜ ^ ( ξ , ω ) = 1 2 π E ( x , t ) exp [ - i ( ξ x - ω t ) ] d x d t .
W ST ( x , ξ , t , ω ) = 1 2 π E ( x + x 2 , t + t 2 ) × E * ( x - x 2 , t - t 2 ) × exp [ i ( - ξ x + ω t ) ] d x d t .
W ST ( x , ξ , t , ω ) = 1 ( 2 π ) 2 E ˜ ( x + x 2 , ω + ω 2 ) × E ˜ * ( x - x 2 , ω - ω 2 ) × exp [ i ( - ξ x - ω t ) ] d x d ω ,
W ST ( x , ξ , t , ω ) = E ^ ( ξ + ξ 2 , t + t 2 ) × E ˜ * ( ξ - ξ 2 , t - t 2 ) × exp [ i ( ξ x + ω t ) ] d ξ d t ,
W ST ( x , ξ , t , ω ) = 1 2 π E ˜ ^ ( ξ + ξ 2 , ω + ω 2 ) × E ˜ ^ * ( ξ - ξ 2 , ω - ω 2 ) × exp [ i ( ξ x - ω t ) ] d ξ d ω .
1 2 π W ST ( x , ξ , t , ω ) d ω = 1 2 π E ( x + x 2 , t ) × E * ( x - x 2 , t ) × exp ( - i ξ x ) d x = W S ( x , ξ , t ) ,
W ST ( x , ξ , t , ω ) d t = 1 2 π E ˜ ( x + x 2 , ω ) × E ˜ * ( x - x 2 , ω ) × exp ( - i ξ x ) d x = W S ( x , ξ , ω ) .
W ST ( x , ξ , t , ω ) d ξ = E ( x , t + t 2 ) E * ( x , t - t 2 ) × exp ( i ω t ) d t = W T ( x , t , ω ) ,
1 2 π W ST ( x , ξ , t , ω ) d ω = E ^ ( ξ , t + t 2 ) E ^ * ( ξ , t - t 2 ) × exp ( i ω t ) d t = W T ( ξ , t , ω ) ,
1 2 π W ST ( x , ξ , t , ω ) d ξ d ω = E ( x , t ) 2 ,
W ST ( x , ξ , t , ω ) d ξ d t = E ˜ ( x , ω ) 2 ,
1 ( 2 π ) 2 W ST ( x , ξ , t , ω ) d x d ω = E ^ ( ξ , t ) 2 ,
1 2 π W ST ( x , ξ , t , ω ) d x d t = E ˜ ^ ( ξ , ω ) 2 ,
1 2 π W S T ( x , ξ , t , ω ) d t d ω = 1 2 π W S ( x , ξ , ω ) d ω = W S ( x , ξ , t ) d t = W ¯ S ( x , ξ ) ,
1 2 π W ST ( x , ξ , t , ω ) d x d t = 1 2 π W T ( x , t , ω ) d x = W T ( ξ , t , ω ) d ξ = W ¯ T ( t , ω ) .
1 ( 2 π ) 2 W ST ( x , ξ , t , ω ) d x d ξ d t ω = .
E out ( x , t ) = 1 2 π χ ST ( x , t ) E in ( x - x , t - t ) d x d t .
E ˜ ^ out ( ξ , ω ) = χ ˜ ^ ST ( ξ , ω ) E ˜ ^ in ( ξ , ω ) .
W out ST ( x , ξ , t , ω ) = 1 2 π W χ ST ( x , ξ , t , ω ) × W in ST ( x - x , ξ , t - t , ω ) d x d t ,
W χ ST ( x , ξ , t , ω ) = 1 2 π χ ST ( x + x 2 , t + t 2 ) × χ ST * ( x - x 2 , t - t 2 ) × exp [ i ( - ξ x + ω t ) ] d x d t ,
E out ( x , t ) = E in ( x , t ) exp ( i k 0 n d ) exp ( - i k 0 x 2 2 f ) ,
W out ST ( x , ξ , t , ω ) = W in ST ( x , ξ + k 0 f x , t , ω ) .
E out ( x , t ) = E in ( x , t ) exp [ - i Φ m cos ( ω m t ) ] ,
E out ( x , t ) = E in ( x , t ) exp [ - Φ m ( 1 - ω m 2 t 2 ) ] ,
W out ST ( x , ξ , t , ω ) = W in ST ( x , ξ , t , ω + Φ m ω m 2 t ) .
E ˜ ^ ( ξ , ω , z ) = E ˜ ^ ( ξ , ω , 0 ) exp ( i k z ) exp ( - i ξ 2 2 k z ) ,
k ( ω ) = k 0 + k 0 ω + ½ k 0 ω 2 ,
E ˜ ^ ( ξ , ω , z ) = E ˜ ^ ( ξ , ω , 0 ) exp [ i ( k 0 + k 0 ω + ½ k 0 ω 2 ) z ] × exp ( - i ξ 2 2 k 0 z ) .
W z ST ( x , ξ , t , ω ) = W 0 ST ( x - z k 0 ξ , ξ , t - k 0 z - k 0 z ω , ω ) .
E out ( x , t ) = E in ( α x , t - β x ) ,
α = cos θ i cos θ d ,             β = 2 π p d cos θ d ω 0 ,
W out ST ( x , ξ , t , ω ) = 1 α W in ST ( α x , ξ α - β α ω , t - β x , ω ) .
W out ST ( X ) = m W in ST ( M · X ) ,
M · X = [ m x x m x ξ m x t m x ω m ξ x m ξ ξ m ξ t m ξ ω m t x m t ξ m t t m t ω m ω x m ω ξ m ω t m ω ω ] [ x ξ t ω ] .
W 1 ST ( X ) = m 1 W 0 ST ( M 1 X ) .
W 2 ST ( X ) = m 2 W 1 ST ( M 2 X ) = m 1 m 2 W 0 ST ( M 1 M 2 X ) .
W out ST ( X ) = m 1 m 2 m n W in ST ( M 1 M 2 M n X ) .
M prop ( f ) M lens ( f ) M prop ( f ) = [ 0 - f / k 0 0 0 k 0 / f 0 0 0 0 0 1 0 0 0 0 1 ] .
W Before mask ST ( x , ξ , t , ω ) = 1 α W in ST ( - α f k 0 ξ , k 0 α f x - β α ω , t + β f k 0 ξ , ω ) .
E in ( x , t ) = F ( x ) E out ( x , t ) .
W Mask S ( x , ξ ) = 1 2 π F ( x + x 2 ) F * ( x - x 2 ) × exp ( - i ξ x ) d x .
W After mask ST ( x , ξ , t , ω ) = 1 α W Mask S ( x , ξ ) × W in ST [ - α f k 0 ( ξ - ξ ) , k 0 α f x - β α ω , t + β f k 0 ( ξ - ξ ) , ω ] d ξ .
W grating 2 ST ( x , ξ , t , ω ) = k 0 α f W Mask S ( - f ξ k 0 , k 0 x f ) × W in ST [ - α ( x - x ) , - ξ α - β ω α , t + β ( x - x ) , ω ] d x ,
W Output ST ( x , ξ , t , ω ) = k 0 f W Mask S [ - f k 0 ( α ξ - β ω ) , k 0 x f ] × W in ST ( - x + α x , - ξ , t - β x α , ω ) d x .
W Output ST ( x , ξ , t , ω ) = k 0 β f W Mask S [ β f k 0 ( ω - α β ξ ) , k 0 β f t ] × W in ST [ - ( x - α β t ) , - ξ , t - t , ω ] d t .
W Output ST ( x , ξ , t , ω ) = W Mask T ( t , ω - α β ξ ) × W in ST [ - ( x - α β t ) , - ξ , t - t , ω ] d t ,
W Mask T ( t , ω ) = k 0 β f W Mask S ( β f k 0 ω , k 0 β f t ) .
α β = d cos θ i ω 0 2 π p .
Δ t Mask β α Δ x Pulse ,             Δ ω Mask α β Δ ξ Pulse ,
Δ ξ Mask α k 0 f Δ x Pulse ,             Δ x Mask α f k 0 Δ ξ Pulse ,

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