Abstract

A new technique for measuring the time- and frequency-dependent intensities and phases of correlation functions of broadband, non–transform-limited (noisy) light is presented. It is based on others’ successful previous efforts to measure the intensities and phases of short pulses by frequency-resolved optical gating. It is shown that a simplified algorithm based on the original algorithm for frequency-resolved optical gating is sufficient for the recovery of correlation functions. The first experimental realizations of this technique are presented, and the recovered noisy-light correlation functions are shown. By way of illustration, the dispersion of water is quantified. Applications to interferometrically time-resolved nonlinear optical spectroscopies are discussed.

© 1998 Optical Society of America

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References

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  1. D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
    [CrossRef]
  2. 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]
  3. B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. DeLong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995).
    [CrossRef] [PubMed]
  4. G. Taft, A. Rundquist, M. M. Murnane, H. C. Kapteyn, K. W. DeLong, R. Trebino, and I. P. Christov, “Ultrashort optical waveform measurements using frequency-resolved optical gating,” Opt. Lett. 20, 743–745 (1995).
    [CrossRef] [PubMed]
  5. A. Sullivan, W. E. White, K. C. Chu, J. P. Heritage, K. W. DeLong, and R. Trebino, “Quantitative investigation of optical phase-measuring techniques for ultrashort pulse lasers,” J. Opt. Soc. Am. B 13, 1965–1978 (1996).
    [CrossRef]
  6. K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating using second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
    [CrossRef]
  7. K. W. DeLong, C. L. Ladera, R. Trebino, B. Kohler, and K. Wilson, “Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20, 486–488 (1995).
    [CrossRef] [PubMed]
  8. K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463–2466 (1995).
    [CrossRef]
  9. E. Yudilevich, A. Levi, G. J. Habetler, and H. Stark, “Restoration of signals from their signed Fourier-transform magnitude by the method of generalized projections,” J. Opt. Soc. Am. A 4, 236–246 (1987).
    [CrossRef]
  10. The I stands for interferometric and the superscript (2) indicates the participation of two correlated incoherent fields in the creation of the signal field.
  11. K. W. DeLong, R. Trebino, and D. J. Kane, “A comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B 11, 1595–1608 (1994).
    [CrossRef]
  12. The term monochromatic is used to mean several orders of magnitude narrower than the broadband fields.
  13. R. Loudon, The Quantum Theory of Light, 2nd. ed. (Oxford University, New York, 1990).
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, New York, 1995).
  15. M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency and time resolved coherent Stokes Raman scattering from CS2 using incoherent light,” Chem. Phys. Lett. 263, 185–190 (1996).
    [CrossRef]
  16. M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Time resolved coherent Raman spectroscopy controlled by spectrally tailored noisy light,” J. Raman Spectrosc. 28, 579–587 (1997).
    [CrossRef]
  17. M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency resolved interferometric coherent Raman spectroscopy with incoherent light: Raman frequency shifts, dephasing rate constants, and nonresonant hyperpolarizabilities of mixtures of benzene in hexane,” Chem. Phys. 222, 17–28 (1997).
    [CrossRef]
  18. D. J. Ulness, M. J. Stimson, J. C. Kirkwood, and A. C. Albrecht, “Interferometric downconversion of high frequency molecular vibrations with time-frequency resolved coherent Raman scattering using quasi-cw noisy light: C-H stretching modes of chloroform and benzene,” J. Phys. Chem. 101, 4587–4591 (1997).
    [CrossRef]
  19. D. J. Ulness, J. C. Kirkwood, M. J. Stimson, and A. C. Albrecht, “Theory of coherent Raman scattering with quasi-cw noisy light for a general lineshape function,” J. Chem. Phys. 107, 7127–7137 (1997).
    [CrossRef]
  20. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, New York, 1995).
  21. E. Hanamura, “Coherent and incoherent laser spectroscopy of spatial and temporal fluctuations,” Solid State Commun. 51, 697–700 (1984).
    [CrossRef]
  22. S. Mukamel and E. Hanamura, “Four-wave mixing using partially coherent fields in systems with spatial correlations,” Phys. Rev. A 33, 1099–1108 (1986).
    [CrossRef] [PubMed]
  23. M. A. Dugan and A. C. Albrecht, “Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory,” Phys. Rev. A 43, 3877–3921 (1991).
    [CrossRef] [PubMed]
  24. D. J. Ulness and A. C. Albrecht, “Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density: analytic solution and a diagrammatic approach,” Phys. Rev. A 53, 1081–1095 (1996).
    [CrossRef] [PubMed]
  25. K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating using the method of generalized projections,” Opt. Lett. 19, 2152–2154 (1994).
    [CrossRef] [PubMed]
  26. K. W. DeLong and R. Trebino, “Improved ultrashort-pulse retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429–2437 (1994).
    [CrossRef]
  27. D. N. Fittinghoff, K. W. DeLong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995).
    [CrossRef]
  28. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge University, New York, 1990).
  29. I. Thormählen, J. Straub, and U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
    [CrossRef]

1997 (4)

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Time resolved coherent Raman spectroscopy controlled by spectrally tailored noisy light,” J. Raman Spectrosc. 28, 579–587 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency resolved interferometric coherent Raman spectroscopy with incoherent light: Raman frequency shifts, dephasing rate constants, and nonresonant hyperpolarizabilities of mixtures of benzene in hexane,” Chem. Phys. 222, 17–28 (1997).
[CrossRef]

D. J. Ulness, M. J. Stimson, J. C. Kirkwood, and A. C. Albrecht, “Interferometric downconversion of high frequency molecular vibrations with time-frequency resolved coherent Raman scattering using quasi-cw noisy light: C-H stretching modes of chloroform and benzene,” J. Phys. Chem. 101, 4587–4591 (1997).
[CrossRef]

D. J. Ulness, J. C. Kirkwood, M. J. Stimson, and A. C. Albrecht, “Theory of coherent Raman scattering with quasi-cw noisy light for a general lineshape function,” J. Chem. Phys. 107, 7127–7137 (1997).
[CrossRef]

1996 (3)

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency and time resolved coherent Stokes Raman scattering from CS2 using incoherent light,” Chem. Phys. Lett. 263, 185–190 (1996).
[CrossRef]

A. Sullivan, W. E. White, K. C. Chu, J. P. Heritage, K. W. DeLong, and R. Trebino, “Quantitative investigation of optical phase-measuring techniques for ultrashort pulse lasers,” J. Opt. Soc. Am. B 13, 1965–1978 (1996).
[CrossRef]

D. J. Ulness and A. C. Albrecht, “Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density: analytic solution and a diagrammatic approach,” Phys. Rev. A 53, 1081–1095 (1996).
[CrossRef] [PubMed]

1995 (5)

1994 (4)

1993 (2)

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

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]

1991 (1)

M. A. Dugan and A. C. Albrecht, “Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory,” Phys. Rev. A 43, 3877–3921 (1991).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

S. Mukamel and E. Hanamura, “Four-wave mixing using partially coherent fields in systems with spatial correlations,” Phys. Rev. A 33, 1099–1108 (1986).
[CrossRef] [PubMed]

1985 (1)

I. Thormählen, J. Straub, and U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

1984 (1)

E. Hanamura, “Coherent and incoherent laser spectroscopy of spatial and temporal fluctuations,” Solid State Commun. 51, 697–700 (1984).
[CrossRef]

Albrecht, A. C.

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Time resolved coherent Raman spectroscopy controlled by spectrally tailored noisy light,” J. Raman Spectrosc. 28, 579–587 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency resolved interferometric coherent Raman spectroscopy with incoherent light: Raman frequency shifts, dephasing rate constants, and nonresonant hyperpolarizabilities of mixtures of benzene in hexane,” Chem. Phys. 222, 17–28 (1997).
[CrossRef]

D. J. Ulness, M. J. Stimson, J. C. Kirkwood, and A. C. Albrecht, “Interferometric downconversion of high frequency molecular vibrations with time-frequency resolved coherent Raman scattering using quasi-cw noisy light: C-H stretching modes of chloroform and benzene,” J. Phys. Chem. 101, 4587–4591 (1997).
[CrossRef]

D. J. Ulness, J. C. Kirkwood, M. J. Stimson, and A. C. Albrecht, “Theory of coherent Raman scattering with quasi-cw noisy light for a general lineshape function,” J. Chem. Phys. 107, 7127–7137 (1997).
[CrossRef]

D. J. Ulness and A. C. Albrecht, “Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density: analytic solution and a diagrammatic approach,” Phys. Rev. A 53, 1081–1095 (1996).
[CrossRef] [PubMed]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency and time resolved coherent Stokes Raman scattering from CS2 using incoherent light,” Chem. Phys. Lett. 263, 185–190 (1996).
[CrossRef]

M. A. Dugan and A. C. Albrecht, “Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory,” Phys. Rev. A 43, 3877–3921 (1991).
[CrossRef] [PubMed]

Christov, I. P.

Chu, K. C.

DeLong, K. W.

A. Sullivan, W. E. White, K. C. Chu, J. P. Heritage, K. W. DeLong, and R. Trebino, “Quantitative investigation of optical phase-measuring techniques for ultrashort pulse lasers,” J. Opt. Soc. Am. B 13, 1965–1978 (1996).
[CrossRef]

K. W. DeLong, C. L. Ladera, R. Trebino, B. Kohler, and K. Wilson, “Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20, 486–488 (1995).
[CrossRef] [PubMed]

K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463–2466 (1995).
[CrossRef]

G. Taft, A. Rundquist, M. M. Murnane, H. C. Kapteyn, K. W. DeLong, R. Trebino, and I. P. Christov, “Ultrashort optical waveform measurements using frequency-resolved optical gating,” Opt. Lett. 20, 743–745 (1995).
[CrossRef] [PubMed]

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. DeLong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995).
[CrossRef] [PubMed]

D. N. Fittinghoff, K. W. DeLong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995).
[CrossRef]

K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating using second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
[CrossRef]

K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating using the method of generalized projections,” Opt. Lett. 19, 2152–2154 (1994).
[CrossRef] [PubMed]

K. W. DeLong and R. Trebino, “Improved ultrashort-pulse retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429–2437 (1994).
[CrossRef]

K. W. DeLong, R. Trebino, and D. J. Kane, “A comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B 11, 1595–1608 (1994).
[CrossRef]

Dugan, M. A.

M. A. Dugan and A. C. Albrecht, “Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory,” Phys. Rev. A 43, 3877–3921 (1991).
[CrossRef] [PubMed]

Fittinghoff, D. N.

Grigull, U.

I. Thormählen, J. Straub, and U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Habetler, G. J.

Hanamura, E.

S. Mukamel and E. Hanamura, “Four-wave mixing using partially coherent fields in systems with spatial correlations,” Phys. Rev. A 33, 1099–1108 (1986).
[CrossRef] [PubMed]

E. Hanamura, “Coherent and incoherent laser spectroscopy of spatial and temporal fluctuations,” Solid State Commun. 51, 697–700 (1984).
[CrossRef]

Heritage, J. P.

Hunter, J.

Kane, D. J.

Kapteyn, H. C.

Kirkwood, J. C.

D. J. Ulness, J. C. Kirkwood, M. J. Stimson, and A. C. Albrecht, “Theory of coherent Raman scattering with quasi-cw noisy light for a general lineshape function,” J. Chem. Phys. 107, 7127–7137 (1997).
[CrossRef]

D. J. Ulness, M. J. Stimson, J. C. Kirkwood, and A. C. Albrecht, “Interferometric downconversion of high frequency molecular vibrations with time-frequency resolved coherent Raman scattering using quasi-cw noisy light: C-H stretching modes of chloroform and benzene,” J. Phys. Chem. 101, 4587–4591 (1997).
[CrossRef]

Kohler, B.

Ladera, C. L.

Levi, A.

Mukamel, S.

S. Mukamel and E. Hanamura, “Four-wave mixing using partially coherent fields in systems with spatial correlations,” Phys. Rev. A 33, 1099–1108 (1986).
[CrossRef] [PubMed]

Murnane, M. M.

Rundquist, A.

Squier, J.

Stark, H.

Stimson, M. J.

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Time resolved coherent Raman spectroscopy controlled by spectrally tailored noisy light,” J. Raman Spectrosc. 28, 579–587 (1997).
[CrossRef]

D. J. Ulness, M. J. Stimson, J. C. Kirkwood, and A. C. Albrecht, “Interferometric downconversion of high frequency molecular vibrations with time-frequency resolved coherent Raman scattering using quasi-cw noisy light: C-H stretching modes of chloroform and benzene,” J. Phys. Chem. 101, 4587–4591 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency resolved interferometric coherent Raman spectroscopy with incoherent light: Raman frequency shifts, dephasing rate constants, and nonresonant hyperpolarizabilities of mixtures of benzene in hexane,” Chem. Phys. 222, 17–28 (1997).
[CrossRef]

D. J. Ulness, J. C. Kirkwood, M. J. Stimson, and A. C. Albrecht, “Theory of coherent Raman scattering with quasi-cw noisy light for a general lineshape function,” J. Chem. Phys. 107, 7127–7137 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency and time resolved coherent Stokes Raman scattering from CS2 using incoherent light,” Chem. Phys. Lett. 263, 185–190 (1996).
[CrossRef]

Straub, J.

I. Thormählen, J. Straub, and U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Sullivan, A.

Taft, G.

Thormählen, I.

I. Thormählen, J. Straub, and U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Trebino, R.

A. Sullivan, W. E. White, K. C. Chu, J. P. Heritage, K. W. DeLong, and R. Trebino, “Quantitative investigation of optical phase-measuring techniques for ultrashort pulse lasers,” J. Opt. Soc. Am. B 13, 1965–1978 (1996).
[CrossRef]

K. W. DeLong, C. L. Ladera, R. Trebino, B. Kohler, and K. Wilson, “Ultrashort-pulse measurement using noninstantaneous nonlinearities: Raman effects in frequency-resolved optical gating,” Opt. Lett. 20, 486–488 (1995).
[CrossRef] [PubMed]

K. W. DeLong, R. Trebino, and W. E. White, “Simultaneous recovery of two ultrashort laser pulses from a single spectrogram,” J. Opt. Soc. Am. B 12, 2463–2466 (1995).
[CrossRef]

G. Taft, A. Rundquist, M. M. Murnane, H. C. Kapteyn, K. W. DeLong, R. Trebino, and I. P. Christov, “Ultrashort optical waveform measurements using frequency-resolved optical gating,” Opt. Lett. 20, 743–745 (1995).
[CrossRef] [PubMed]

B. Kohler, V. V. Yakovlev, K. R. Wilson, J. Squier, K. W. DeLong, and R. Trebino, “Phase and intensity characterization of femtosecond pulses from a chirped-pulse amplifier by frequency-resolved optical gating,” Opt. Lett. 20, 483–485 (1995).
[CrossRef] [PubMed]

D. N. Fittinghoff, K. W. DeLong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995).
[CrossRef]

K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating using second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
[CrossRef]

K. W. DeLong, R. Trebino, and D. J. Kane, “A comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B 11, 1595–1608 (1994).
[CrossRef]

K. W. DeLong and R. Trebino, “Improved ultrashort-pulse retrieval algorithm for frequency-resolved optical gating,” J. Opt. Soc. Am. A 11, 2429–2437 (1994).
[CrossRef]

K. W. DeLong, D. N. Fittinghoff, R. Trebino, B. Kohler, and K. Wilson, “Pulse retrieval in frequency-resolved optical gating using the method of generalized projections,” Opt. Lett. 19, 2152–2154 (1994).
[CrossRef] [PubMed]

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

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]

Ulness, D. J.

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Time resolved coherent Raman spectroscopy controlled by spectrally tailored noisy light,” J. Raman Spectrosc. 28, 579–587 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency resolved interferometric coherent Raman spectroscopy with incoherent light: Raman frequency shifts, dephasing rate constants, and nonresonant hyperpolarizabilities of mixtures of benzene in hexane,” Chem. Phys. 222, 17–28 (1997).
[CrossRef]

D. J. Ulness, M. J. Stimson, J. C. Kirkwood, and A. C. Albrecht, “Interferometric downconversion of high frequency molecular vibrations with time-frequency resolved coherent Raman scattering using quasi-cw noisy light: C-H stretching modes of chloroform and benzene,” J. Phys. Chem. 101, 4587–4591 (1997).
[CrossRef]

D. J. Ulness, J. C. Kirkwood, M. J. Stimson, and A. C. Albrecht, “Theory of coherent Raman scattering with quasi-cw noisy light for a general lineshape function,” J. Chem. Phys. 107, 7127–7137 (1997).
[CrossRef]

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency and time resolved coherent Stokes Raman scattering from CS2 using incoherent light,” Chem. Phys. Lett. 263, 185–190 (1996).
[CrossRef]

D. J. Ulness and A. C. Albrecht, “Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density: analytic solution and a diagrammatic approach,” Phys. Rev. A 53, 1081–1095 (1996).
[CrossRef] [PubMed]

White, W. E.

Wilson, K.

Wilson, K. R.

Yakovlev, V. V.

Yudilevich, E.

Chem. Phys. (1)

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency resolved interferometric coherent Raman spectroscopy with incoherent light: Raman frequency shifts, dephasing rate constants, and nonresonant hyperpolarizabilities of mixtures of benzene in hexane,” Chem. Phys. 222, 17–28 (1997).
[CrossRef]

Chem. Phys. Lett. (1)

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Frequency and time resolved coherent Stokes Raman scattering from CS2 using incoherent light,” Chem. Phys. Lett. 263, 185–190 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. J. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron. 29, 571–579 (1993).
[CrossRef]

J. Chem. Phys. (1)

D. J. Ulness, J. C. Kirkwood, M. J. Stimson, and A. C. Albrecht, “Theory of coherent Raman scattering with quasi-cw noisy light for a general lineshape function,” J. Chem. Phys. 107, 7127–7137 (1997).
[CrossRef]

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

J. Opt. Soc. Am. B (5)

J. Phys. Chem. (1)

D. J. Ulness, M. J. Stimson, J. C. Kirkwood, and A. C. Albrecht, “Interferometric downconversion of high frequency molecular vibrations with time-frequency resolved coherent Raman scattering using quasi-cw noisy light: C-H stretching modes of chloroform and benzene,” J. Phys. Chem. 101, 4587–4591 (1997).
[CrossRef]

J. Phys. Chem. Ref. Data (1)

I. Thormählen, J. Straub, and U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

J. Raman Spectrosc. (1)

M. J. Stimson, D. J. Ulness, and A. C. Albrecht, “Time resolved coherent Raman spectroscopy controlled by spectrally tailored noisy light,” J. Raman Spectrosc. 28, 579–587 (1997).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (3)

S. Mukamel and E. Hanamura, “Four-wave mixing using partially coherent fields in systems with spatial correlations,” Phys. Rev. A 33, 1099–1108 (1986).
[CrossRef] [PubMed]

M. A. Dugan and A. C. Albrecht, “Radiation-matter oscillations and spectral line narrowing in field-correlated four-wave mixing. I. Theory,” Phys. Rev. A 43, 3877–3921 (1991).
[CrossRef] [PubMed]

D. J. Ulness and A. C. Albrecht, “Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density: analytic solution and a diagrammatic approach,” Phys. Rev. A 53, 1081–1095 (1996).
[CrossRef] [PubMed]

Solid State Commun. (1)

E. Hanamura, “Coherent and incoherent laser spectroscopy of spatial and temporal fluctuations,” Solid State Commun. 51, 697–700 (1984).
[CrossRef]

Other (6)

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, New York, 1995).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge University, New York, 1990).

The I stands for interferometric and the superscript (2) indicates the participation of two correlated incoherent fields in the creation of the signal field.

The term monochromatic is used to mean several orders of magnitude narrower than the broadband fields.

R. Loudon, The Quantum Theory of Light, 2nd. ed. (Oxford University, New York, 1990).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, New York, 1995).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup: The broadband source is a commercially available dye laser tuned to zeroth-order dispersion or a homebuilt cavity-free laser. The emission from the broadband source is split in an interferometer where a computer-controllable delay is introduced into one of the beams (B) relative to the other (B). The beams leave the interferometer and join a third, the monochromatic beam (M). The three beams are focused into the sample, and the signal (S) with wave vector kS=kB+kB-kM is spatially filtered and directed into a spectrometer. The spectrum of the signal is detected by a CCD chip and recorded for many values of the interferometric delay (τ).

Fig. 2
Fig. 2

(a) Raw U-I(2)FROG data averaged over five scans. The wave-mixing medium is CCl4. The constant background (clearly exposed in the spectra taken at delay times ranging from -500 fs to -200 fs and from 200 fs to 500 fs) reduces the contrast ratio of (a) but has been subtracted from the spectra at all of the delay times in (b) to enhance the contrast. In (c) is shown the U-I(2)FROG trace that is generated by the correlation function recovered by application of the U-I(2)FROG algorithm to the data in (a). In (d) the time-dependent intensity (filled circles) and phase (open circles) of the correlation function are shown.

Fig. 3
Fig. 3

Intensities (filled circles) and phases (open circles) of the correlation functions recovered from the U-I(2)FROG experiments with (a) water and (b) glass as the wave-mixing media.

Fig. 4
Fig. 4

(a) U-I(2)FROG trace generated from the recovered correlation function of the emission from the homebuilt broadband source. Notice that the horizontal axis spans twice the range as that in Fig. 2 because the spectrum of the homebuilt source is much broader. (b) Time dependent intensity (filled circles) and phase (open circles) of the correlation function. The correlation function of this light has a much shorter temporal duration than that of the commercial broadband source [cf. Fig. 2(d)].

Fig. 5
Fig. 5

Results of two experiments designed to demonstrate the effects of unbalanced dispersion in the interferometer on noisy light correlation functions. (a) Background-subtracted data obtained by performing the U-I(2)FROG experiment in CCl4 with a 5-cm cuvette of water placed in the B arm of the interferometer. (b) U-I(2)FROG trace generated from the correlation function and (c) recovered by application of the algorithm to the data in (a). (d) Background-subtracted data obtained by performing the U-I(2)FROG experiment in CCl4 with a 5-cm cuvette of water placed in the B arm of the interferometer. (e) U-I(2)FROG trace generated from the correlation function and (f) recovered by application of the algorithm to the data in (d). Notice the opposite slopes of the U-I(2)FROG traces and the accompanying opposite signs of the phases between the two experiments.

Fig. 6
Fig. 6

Intensities (filled circles) and phases (open circles) of the correlation functions described in Fig. 5 in their frequency-domain representations. (a) Recovered correlation function from an U-I(2)FROG experiment in which a 5-cm cuvette of water is introduced to the B arm of the interferometer. (b) Recovered correlation function from an U-I(2)FROG experiment in which the 5-cm cuvette of water is introduced to the B arm of the interferometer.

Equations (24)

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EB(t)=pB(t)exp(-iω¯Bt),
ΓB,B(τ)=pB(t)pB*(t-τ)limT 1T -T/2T/2dtpB(t)pB*(t-τ).
ΓB,B(τ)=-dωB exp[-i(ωB-ω¯B)τ]J(ωB),
E(3)(t, τ)EB(t)EB(t-τ)EM*(t).
E(3)*(t, τ)EB*(t)EB*(t-τ)EM(t).
I(t, ωS, τ)-dt exp[iωS(t-t)]×E(3)(t, τ)E(3)*(t, τ)+c.c.
limT 1/T -T/2T/2dt-dωSδ(ωD-ωS)
IU-I(2)FROG(ωD, τ)lim T 1T -T/2T/2dt-dt× exp[iωD(t-t)]E(3)(t, τ)×E(3)*(t, τ)+c.c.RelimT 1T -T/2T/2dt-dt× exp[iωD(t-t)]×E(3)(t, τ)E(3)*(t, τ).
IU-I(2)FROG(ωD, τ)Re-dt exp(iωDt)E(3)(t, τ)×E(3)*(t+t, τ).
IU-I(2)FROG(ωD, τ)Re-dt exp(iωDt)EB(t)×EB(t-τ)EM*(t)EB*(t+t)×EB*(t+t-τ)EM(t+t).
EB(t)EB(t-τ)EM*(t)EB*(t+t)EB*(t+t-τ)
×EM(t+t)=EM*(t)EM(t+t)EB(t)EB(t-τ)×EB*(t+t)EB*(t+t-τ)exp(-iωMt)EB(t)EB(t-τ)EB*(t+t)×EB*(t+t-τ).
EB(t)EB(t-τ)EB*(t+t)EB*(t+t-τ)=EB(t)EB*(t+t-τ)EB*(t+t)EB(t-τ)+EB(t)EB*(t+t)EB(t-τ)×EB*(t+t-τ)=EB(t)EB*(t+t-τ)EB*(t)EB(t-t-τ)+EB(t)EB*(t+t)EB*(t)EB(t-t)=exp[i(2ω¯B)t][ΓB,B(-t+τ)ΓB,B*(t+τ)+ΓB,B(-t)ΓB,B*(t)],
IU-I(2)FROG(ωD, τ)
Re-dt exp[i(-ωD-ωM+2ω¯B)t]×[ΓB,B(-t+τ)ΓB,B*(t+τ)+ΓB,B(-t)ΓB,B*(t)].
IU-I(2)FROG(ωD, τ)
Re-dt exp[i(-ωD-ωM+2ω¯B)t]×ΓB,B(-t+τ)ΓB,B*(t+τ)+IU-I(2)FROG(ωD, ).
G=1N2 ωD,τNII(2)FROG(ω, τ)-II(2)FROG(ω, )-Re-dω exp(iωt)ΓB,B(-t+τ)×ΓB,B*(t+τ)21/2
dZdΓB,B(t0)=-4ΓB,B(-t0+2τ)[F*(-t0+τ,τ)-ΓB,B(t0)ΓB,B*(-t0+2τ)].
Γ˜B,B(ω)=J(ω)cos[ϕ(ω)],
ϕ(ω)=ϕ0+ϕ1ω+ϕ2ω2+ .
T(ω0)=dϕ(ω)dωω0=2ϕ2ω0+ .
ϕ(ν˜)=ϕ0+ϕ1ν˜+ϕ2ν˜2+ .
dndν607nm=ϕ2(10.4cm)(2πc)=0.0390fs.

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