Abstract

We investigate experimental limitations in the accuracy of Fourier-transform spectral interferometry, a widely used technique for determining the spectral phase difference between two light beams consisting of, for example, femtosecond light pulses. We demonstrate that the spectrometer’s finite spectral resolution, pixel aliasing, and frequency-interpolation error can play an important role, and we provide a new and more accurate recipe for recovering the spectral phase from the experimental data.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” J. Opt. (Paris) 4, 183–196 (1973).
  2. J. Piasecki, B. Colombeau, M. Vampouille, C. Froehly, and J. A. Arnaud, “Nouvelle méthode de mesure de la réponse impulsionnelle des fibres optiques,” Appl. Opt. 19, 3749–3755 (1980).
    [CrossRef] [PubMed]
  3. F. Reynaud, F. Salin, and A. Barthelemy, “Measurement of phase shifts introduced by nonlinear optical phenomena on subpicosecond pulses,” Opt. Lett. 14, 275–277 (1989).
    [CrossRef] [PubMed]
  4. E. Tokunaga, A. Terasaki, and T. Kobayashi, “Induced phase modulation of chirped continuum pulses studied with a femtosecond frequency-domain interferometer,” Opt. Lett. 18, 370–372 (1993).
    [CrossRef] [PubMed]
  5. J.-P. Geindre, P. Audebert, A. Rousse, F. Falliès, J. C. Gauthier, A. Mysyrowicz, A. Dos Santos, G. Hamoniaux, and A. Antonetti, “Frequency-domain interferometer for measuring the phase and amplitude of a femtosecond pulse probing a laser-produced plasma,” Opt. Lett. 19, 1997–1999 (1994).
    [CrossRef] [PubMed]
  6. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995).
    [CrossRef]
  7. D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbugel, K. W. DeLong, R. Trebino, and I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996); “Erratum,” 21, 1313 (1996).
    [CrossRef] [PubMed]
  8. W. J. Walecki, D. N. Fittinghoff, A. L. Smirl, and R. Trebino, “Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry,” Opt. Lett. 22, 81–83 (1997).
    [CrossRef] [PubMed]
  9. S. M. Gallagher, A. W. Albrecht, J. D. Hybl, B. L. Landin, B. Rajaram, and D. M. Jonas, “Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals,” J. Opt. Soc. Am. B 15, 2338–2345 (1998).
    [CrossRef]
  10. C. Dorrer and F. Salin, “Characterization of spectral phase modulation using classical and polarization spectral interferometry,” J. Opt. Soc. Am. B 15, 2331–2337 (1998).
    [CrossRef]
  11. C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999).
    [CrossRef]
  12. D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993); R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
    [CrossRef] [PubMed]
  13. J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993); J. Paye, “How to measure the amplitude and phase of an ultrashort light pulse with an autocorrelator and a spectrometer,” IEEE J. Quantum Electron. 30, 2693–2697 (1994).
    [CrossRef] [PubMed]
  14. K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
    [CrossRef]
  15. C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
    [CrossRef]
  16. L. Lepetit and M. Joffre, “Two-dimensional nonlinear optics using Fourier-transform spectral interferometry,” Opt. Lett. 21, 564–566 (1996).
    [CrossRef] [PubMed]
  17. J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297, 307–313 (1998).
    [CrossRef]
  18. J.-P. Likforman, M. Joffre, and V. Thierry-Mieg, “Measurement of photon echoes by use of Fourier-transform spectral interferometry,” Opt. Lett. 22, 1104–1106 (1997).
    [CrossRef] [PubMed]
  19. M. F. Emde, W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Spectral interferometry as an alternative to time-domain heterodyning,” Opt. Lett. 22, 1338–1340 (1997).
    [CrossRef]
  20. X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B 56, 9738–9743 (1997).
    [CrossRef]
  21. A. L. Smirl, X. Chen, and O. Buccafusca, “Ultrafast time-resolved quantum beats in the polarization state of coherent emission from quantum wells,” Opt. Lett. 23, 1120–1122 (1998).
    [CrossRef]
  22. V. N. Kumar and D. N. Rao, “Using interference in the frequency domain for precise determination of thickness and refractive indices of normal dispersive materials,” J. Opt. Soc. Am. B 12, 1559–1563 (1995).
    [CrossRef]
  23. J. Tignon, M. V. Marquezini, T. Hasche, and D. S. Chemla, “Spectral interferometry of semiconductor nanostructures,” IEEE J. Quantum Electron. 35, 510–522 (1999).
    [CrossRef]
  24. D. Birkedal and J. Shah, “Femtosecond spectral interferometry of resonant secondary emission from quantum wells: resonance Rayleigh scattering in the nonergodic regime,” Phys. Rev. Lett. 81, 2372–2375 (1998).
    [CrossRef]
  25. S. Haacke, S. Schaer, B. Deveaud, and V. Savona, “Interferometric analysis of resonant Rayleigh scattering from two-dimensional excitons,” Phys. Rev. B 61, R5109–R5112 (2000).
    [CrossRef]
  26. C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
    [CrossRef]
  27. C. Dorrer, “Implementation of spectral phase interferometry for direct electric-field reconstruction using a simultaneously recorded reference interferogram,” Opt. Lett. 24, 1532–1534 (1999).
    [CrossRef]
  28. C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J.-P. Rousseau, P. Rousseau, J. P. Chambaret, and F. Salin, “Single-shot real-time characterization of chirped-pulse amplification systems using spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1644–1646 (1999).
    [CrossRef]
  29. L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
    [CrossRef]
  30. In cases such as four-wave mixing in which causality plays a role, the emitted field is asymmetric in time, so that it is usually better to use positive values of τ.
  31. P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry, Vol. 83 of Chemical Analysis (Wiley, New York, 1986).
  32. Note that a similar explanation can be attributed to the strong dependence of the retrieved phase on the exact calibration.11 Even a small α2 in the calibration law will result in a chirp proportional to time delay.
  33. V. Coates and H. Hausdorff, “Interferometric methods of measuring the spectral slit width of spectrometers,” J. Opt. Soc. Am. 45, 425–430 (1955).
    [CrossRef]
  34. V. Kumar and D. Rao, “Interferometric measurement of the modulation transfer function of a spectrometer by using spectral modulations,” Appl. Opt. 38, 660–665 (1999).
    [CrossRef]
  35. For the sake of clarity, we assume that the interferometer is balanced, so that E(t)=E0(t).
  36. A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
    [CrossRef]

2000 (1)

S. Haacke, S. Schaer, B. Deveaud, and V. Savona, “Interferometric analysis of resonant Rayleigh scattering from two-dimensional excitons,” Phys. Rev. B 61, R5109–R5112 (2000).
[CrossRef]

1999 (8)

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[CrossRef]

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

J. Tignon, M. V. Marquezini, T. Hasche, and D. S. Chemla, “Spectral interferometry of semiconductor nanostructures,” IEEE J. Quantum Electron. 35, 510–522 (1999).
[CrossRef]

C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B 16, 1160–1168 (1999).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
[CrossRef]

C. Dorrer, “Implementation of spectral phase interferometry for direct electric-field reconstruction using a simultaneously recorded reference interferogram,” Opt. Lett. 24, 1532–1534 (1999).
[CrossRef]

C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J.-P. Rousseau, P. Rousseau, J. P. Chambaret, and F. Salin, “Single-shot real-time characterization of chirped-pulse amplification systems using spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1644–1646 (1999).
[CrossRef]

V. Kumar and D. Rao, “Interferometric measurement of the modulation transfer function of a spectrometer by using spectral modulations,” Appl. Opt. 38, 660–665 (1999).
[CrossRef]

1998 (6)

1997 (4)

1996 (1)

1995 (2)

1994 (2)

1993 (1)

1989 (1)

1980 (1)

1973 (1)

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” J. Opt. (Paris) 4, 183–196 (1973).

1955 (1)

Albrecht, A. W.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297, 307–313 (1998).
[CrossRef]

S. M. Gallagher, A. W. Albrecht, J. D. Hybl, B. L. Landin, B. Rajaram, and D. M. Jonas, “Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals,” J. Opt. Soc. Am. B 15, 2338–2345 (1998).
[CrossRef]

Antonetti, A.

Arnaud, J. A.

Audebert, P.

Barthelemy, A.

Birkedal, D.

D. Birkedal and J. Shah, “Femtosecond spectral interferometry of resonant secondary emission from quantum wells: resonance Rayleigh scattering in the nonergodic regime,” Phys. Rev. Lett. 81, 2372–2375 (1998).
[CrossRef]

Buccafusca, O.

A. L. Smirl, X. Chen, and O. Buccafusca, “Ultrafast time-resolved quantum beats in the polarization state of coherent emission from quantum wells,” Opt. Lett. 23, 1120–1122 (1998).
[CrossRef]

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B 56, 9738–9743 (1997).
[CrossRef]

Chambaret, J. P.

Chemla, D. S.

J. Tignon, M. V. Marquezini, T. Hasche, and D. S. Chemla, “Spectral interferometry of semiconductor nanostructures,” IEEE J. Quantum Electron. 35, 510–522 (1999).
[CrossRef]

Chen, X.

A. L. Smirl, X. Chen, and O. Buccafusca, “Ultrafast time-resolved quantum beats in the polarization state of coherent emission from quantum wells,” Opt. Lett. 23, 1120–1122 (1998).
[CrossRef]

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B 56, 9738–9743 (1997).
[CrossRef]

Chériaux, G.

Coates, V.

Colombeau, B.

de Beauvoir, B.

de Boeij, W. P.

DeLong, K. W.

Deveaud, B.

S. Haacke, S. Schaer, B. Deveaud, and V. Savona, “Interferometric analysis of resonant Rayleigh scattering from two-dimensional excitons,” Phys. Rev. B 61, R5109–R5112 (2000).
[CrossRef]

Dorrer, C.

Dos Santos, A.

Emde, M. F.

Falliès, F.

Fittinghoff, D. N.

W. J. Walecki, D. N. Fittinghoff, A. L. Smirl, and R. Trebino, “Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry,” Opt. Lett. 22, 81–83 (1997).
[CrossRef] [PubMed]

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B 56, 9738–9743 (1997).
[CrossRef]

Froehly, C.

J. Piasecki, B. Colombeau, M. Vampouille, C. Froehly, and J. A. Arnaud, “Nouvelle méthode de mesure de la réponse impulsionnelle des fibres optiques,” Appl. Opt. 19, 3749–3755 (1980).
[CrossRef] [PubMed]

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” J. Opt. (Paris) 4, 183–196 (1973).

Gallagher, S. M.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

S. M. Gallagher, A. W. Albrecht, J. D. Hybl, B. L. Landin, B. Rajaram, and D. M. Jonas, “Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals,” J. Opt. Soc. Am. B 15, 2338–2345 (1998).
[CrossRef]

Gallagher Faeder, S. M.

J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297, 307–313 (1998).
[CrossRef]

Gallmann, L.

Gauthier, J. C.

Geindre, J.-P.

Haacke, S.

S. Haacke, S. Schaer, B. Deveaud, and V. Savona, “Interferometric analysis of resonant Rayleigh scattering from two-dimensional excitons,” Phys. Rev. B 61, R5109–R5112 (2000).
[CrossRef]

Hamoniaux, G.

Hasche, T.

J. Tignon, M. V. Marquezini, T. Hasche, and D. S. Chemla, “Spectral interferometry of semiconductor nanostructures,” IEEE J. Quantum Electron. 35, 510–522 (1999).
[CrossRef]

Hausdorff, H.

Hunter, J.

Hybl, J. D.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297, 307–313 (1998).
[CrossRef]

S. M. Gallagher, A. W. Albrecht, J. D. Hybl, B. L. Landin, B. Rajaram, and D. M. Jonas, “Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals,” J. Opt. Soc. Am. B 15, 2338–2345 (1998).
[CrossRef]

Iaconis, C.

Joffre, M.

Jonas, D. M.

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297, 307–313 (1998).
[CrossRef]

S. M. Gallagher, A. W. Albrecht, J. D. Hybl, B. L. Landin, B. Rajaram, and D. M. Jonas, “Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals,” J. Opt. Soc. Am. B 15, 2338–2345 (1998).
[CrossRef]

Keller, U.

Kobayashi, T.

Kumar, V.

Kumar, V. N.

Lacourt, A.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” J. Opt. (Paris) 4, 183–196 (1973).

Landin, B. L.

Le Blanc, C.

Lepetit, L.

Likforman, J.-P.

Marquezini, M. V.

J. Tignon, M. V. Marquezini, T. Hasche, and D. S. Chemla, “Spectral interferometry of semiconductor nanostructures,” IEEE J. Quantum Electron. 35, 510–522 (1999).
[CrossRef]

Matuschek, N.

Mysyrowicz, A.

Piasecki, J.

Pshenichnikov, M. S.

Rajaram, B.

Ranc, S.

Rao, D.

Rao, D. N.

Reynaud, F.

Rousse, A.

Rousseau, J.-P.

Rousseau, P.

Salin, F.

Savona, V.

S. Haacke, S. Schaer, B. Deveaud, and V. Savona, “Interferometric analysis of resonant Rayleigh scattering from two-dimensional excitons,” Phys. Rev. B 61, R5109–R5112 (2000).
[CrossRef]

Schaer, S.

S. Haacke, S. Schaer, B. Deveaud, and V. Savona, “Interferometric analysis of resonant Rayleigh scattering from two-dimensional excitons,” Phys. Rev. B 61, R5109–R5112 (2000).
[CrossRef]

Shah, J.

D. Birkedal and J. Shah, “Femtosecond spectral interferometry of resonant secondary emission from quantum wells: resonance Rayleigh scattering in the nonergodic regime,” Phys. Rev. Lett. 81, 2372–2375 (1998).
[CrossRef]

Smirl, A. L.

Steinmeyer, G.

Sutter, D. H.

Terasaki, A.

Thierry-Mieg, V.

Tignon, J.

J. Tignon, M. V. Marquezini, T. Hasche, and D. S. Chemla, “Spectral interferometry of semiconductor nanostructures,” IEEE J. Quantum Electron. 35, 510–522 (1999).
[CrossRef]

Tokunaga, E.

Trebino, R.

Vampouille, M.

Vienot, J. C.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” J. Opt. (Paris) 4, 183–196 (1973).

Walecki, W. J.

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B 56, 9738–9743 (1997).
[CrossRef]

W. J. Walecki, D. N. Fittinghoff, A. L. Smirl, and R. Trebino, “Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry,” Opt. Lett. 22, 81–83 (1997).
[CrossRef] [PubMed]

Walmsley, I. A.

White, W. E.

Wiersma, D. A.

Appl. Opt. (2)

Chem. Phys. Lett. (1)

J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297, 307–313 (1998).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. Tignon, M. V. Marquezini, T. Hasche, and D. S. Chemla, “Spectral interferometry of semiconductor nanostructures,” IEEE J. Quantum Electron. 35, 510–522 (1999).
[CrossRef]

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron. 35, 501–509 (1999).
[CrossRef]

J. Chem. Phys. (1)

A. W. Albrecht, J. D. Hybl, S. M. Gallagher, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[CrossRef]

J. Opt. (Paris) (1)

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de réponse impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimentales et applications,” J. Opt. (Paris) 4, 183–196 (1973).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (6)

Opt. Lett. (12)

J.-P. Likforman, M. Joffre, and V. Thierry-Mieg, “Measurement of photon echoes by use of Fourier-transform spectral interferometry,” Opt. Lett. 22, 1104–1106 (1997).
[CrossRef] [PubMed]

M. F. Emde, W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “Spectral interferometry as an alternative to time-domain heterodyning,” Opt. Lett. 22, 1338–1340 (1997).
[CrossRef]

W. J. Walecki, D. N. Fittinghoff, A. L. Smirl, and R. Trebino, “Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry,” Opt. Lett. 22, 81–83 (1997).
[CrossRef] [PubMed]

F. Reynaud, F. Salin, and A. Barthelemy, “Measurement of phase shifts introduced by nonlinear optical phenomena on subpicosecond pulses,” Opt. Lett. 14, 275–277 (1989).
[CrossRef] [PubMed]

E. Tokunaga, A. Terasaki, and T. Kobayashi, “Induced phase modulation of chirped continuum pulses studied with a femtosecond frequency-domain interferometer,” Opt. Lett. 18, 370–372 (1993).
[CrossRef] [PubMed]

J.-P. Geindre, P. Audebert, A. Rousse, F. Falliès, J. C. Gauthier, A. Mysyrowicz, A. Dos Santos, G. Hamoniaux, and A. Antonetti, “Frequency-domain interferometer for measuring the phase and amplitude of a femtosecond pulse probing a laser-produced plasma,” Opt. Lett. 19, 1997–1999 (1994).
[CrossRef] [PubMed]

A. L. Smirl, X. Chen, and O. Buccafusca, “Ultrafast time-resolved quantum beats in the polarization state of coherent emission from quantum wells,” Opt. Lett. 23, 1120–1122 (1998).
[CrossRef]

C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[CrossRef]

L. Lepetit and M. Joffre, “Two-dimensional nonlinear optics using Fourier-transform spectral interferometry,” Opt. Lett. 21, 564–566 (1996).
[CrossRef] [PubMed]

C. Dorrer, “Implementation of spectral phase interferometry for direct electric-field reconstruction using a simultaneously recorded reference interferogram,” Opt. Lett. 24, 1532–1534 (1999).
[CrossRef]

C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J.-P. Rousseau, P. Rousseau, J. P. Chambaret, and F. Salin, “Single-shot real-time characterization of chirped-pulse amplification systems using spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1644–1646 (1999).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
[CrossRef]

Phys. Rev. B (2)

S. Haacke, S. Schaer, B. Deveaud, and V. Savona, “Interferometric analysis of resonant Rayleigh scattering from two-dimensional excitons,” Phys. Rev. B 61, R5109–R5112 (2000).
[CrossRef]

X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B 56, 9738–9743 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

D. Birkedal and J. Shah, “Femtosecond spectral interferometry of resonant secondary emission from quantum wells: resonance Rayleigh scattering in the nonergodic regime,” Phys. Rev. Lett. 81, 2372–2375 (1998).
[CrossRef]

Other (7)

For the sake of clarity, we assume that the interferometer is balanced, so that E(t)=E0(t).

In cases such as four-wave mixing in which causality plays a role, the emitted field is asymmetric in time, so that it is usually better to use positive values of τ.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry, Vol. 83 of Chemical Analysis (Wiley, New York, 1986).

Note that a similar explanation can be attributed to the strong dependence of the retrieved phase on the exact calibration.11 Even a small α2 in the calibration law will result in a chirp proportional to time delay.

D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993); R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
[CrossRef] [PubMed]

J. Paye, M. Ramaswamy, J. G. Fujimoto, and E. P. Ippen, “Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation,” Opt. Lett. 18, 1946–1948 (1993); J. Paye, “How to measure the amplitude and phase of an ultrashort light pulse with an autocorrelator and a spectrometer,” IEEE J. Quantum Electron. 30, 2693–2697 (1994).
[CrossRef] [PubMed]

D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbugel, K. W. DeLong, R. Trebino, and I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996); “Erratum,” 21, 1313 (1996).
[CrossRef] [PubMed]

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 (7)

Fig. 1
Fig. 1

Magnitude of the fast Fourier transform of the experimental interference spectrum, |I(k)|, plotted as a function of ξ=k/α1, for three different values of the time delay.

Fig. 2
Fig. 2

Fourier transform of the same experimental data as those used in Fig. 1, except that a linear interpolation of the frequency axis has first been performed to provide the FFT procedure with an array of evenly spaced data points in frequency domain.

Fig. 3
Fig. 3

Spectral phase obtained from the interference spectrum between two pulses separated by 5 ps. The phase-retrieval techniques used are (a) plain FFT, (b) linear interpolation, (c) the technique described in Subsection 3.D, and (d) zero-filling interpolation. The curves have been vertically shifted for clarity.

Fig. 4
Fig. 4

Time-domain determination of the correlation function, f(t-τ), by use of (a) a plain FFT of the data, (b) a linear interpolation before the FFT, (c) the technique described in Subsection 3.D, and (d) the zero-filling method. (c) and (d) cannot be distinguished because the difference between the two curves is within the line thickness.

Fig. 5
Fig. 5

(a) Blow-up of a particular spectral region of the interference spectra obtained for τ=3 ps (lower curve) and τ=9 ps (upper curve). (b) Amplitude of the FFT of the above data, plotted as a function of ξ. The curve corresponding to τ=9 ps has been multiplied by a factor of 10.

Fig. 6
Fig. 6

(a) FFT computed from a series of experimental interference spectra obtained for different values of τ. (b) FFT computed from a series of numerically computed interference spectra obtained for different values of τ and by use of the experimental laser spectrum.

Fig. 7
Fig. 7

ξ-domain apparatus function of the spectrometer obtained with a narrow spectral line (thin solid curve) or with spectral interferences (dashed curve). The thick solid curve shows the aliased apparatus function deduced after periodization (i.e., after adding the dotted curve), thus simulating the convolution product with P(ξ). The apparatus function that should be used for correction of interference spectra is the dashed curve.

Equations (17)

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

I(ω)=|E0(ω)+E(ω)exp(iωτ)|2=|E0(ω)|2+|E(ω)|2+E0*(ω)E(ω)×exp(iωτ)+c.c.,
I(ω)=|E0(ω)|2+|E(ω)|2+f(ω)exp(iωτ)+c.c.
F.T.-1I(ω)=E0*(-t)E0(t)+E*(-t)E(t)+f(t-τ)+f(-t-τ)*.
ω(x)=ω0+α1x+12α2x2+16α3x3+ .
I(k)=F.T.-1I(x)=I(x)exp(-ikx)dx=N.I.T.+f[ω(x)]exp[iω(x)τ]×exp(-ikx)dx+c.c.,
I(k)=N.I.T.+f[ω(x)]exp[iω(x)τ]×exp(-ikx)dx+c.c.=N.I.T.+exp(iω0τ)(F.T.-1{f[ω(x)]×exp[iΦτ(x)]})(ξ-τ)+[c.c.](-ξ-τ)=N.I.T.+exp(iω0τ)fτ(ξ-τ)+exp(-iω0τ)fτ(-ξ-τ)*,
exp(iω0τ)f[ω(x)]exp[iΦ(x)]exp(iα1xτ)
=f[ω(x)]exp[iω(x)τ].
Si=xi-a/2xi+a/2R(x)I[ω(x)]dx=-+P(x-xi){R(x)I[ω(x)]}dx={P(x)R(x)I[ω(x)]}(xi),
S(x)=(x){P(x)R(x)I[ω(x)]},
S(ξ)=(ξ)(P(ξ)R(ξ)T.F.-1{I[ω(x)]}),
I(ω)+N(ω)=|E0(ω)|2+|E(ω)|2+f(ω)×exp(iωτ)+c.c.+N(ω).
ginter(ω)=g(ωa)+[g(ωb)-g(ωa)]×(ω-ωa)/(ωb-ωa).
ginter(ω)-g(ω)-12(ω-ωa)(ω-ωb) 2gω2=k(ω) 2gω2,
g(ω)=|E0(ω)||E(ω)|cos[ωτ+Δφ(ω)].
2gω2|E0(ω)||E(ω)| 2 cos[ωτ+Δφ(ω)]ω2=-|E0(ω)||E(ω)|φω+τ2 cos[ωτ+Δφ(ω)]+2φω2 sin[ωτ+Δφ(ω)].
N(ω)=-|E0(ω)||E(ω)|cos[ωτ+Δφ(ω)]×φω+τ2k(ω).

Metrics