Abstract

We have developed a Fourier-deconvolution-based linewidth-deduction method for nonlinear optical spectroscopy with transform-limited light pulses. The phase-retrieval problem involved in this method was solved with a phase-retrieval procedure based on the maximum-entropy model. Our proposed method can also help one to surpass the resolution limit set by the uncertainty principle when the amplitude line-shape function of the laser source can be fully predetermined.

© 1998 Optical Society of America

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References

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  1. J. K. Kauppinen, D. J. Moffatt, M. R. Hollberg, and H. H. Mantsch, “A new line-narrowing procedure based on Fourier self-deconvolution, maximum entropy, and linear prediction,” Appl. Spectrosc. 45, 411–416 (1991).
    [CrossRef]
  2. J. K. Kauppinen, D. J. Moffatt, H. H. Mantch, and D. G. Gameron, “Fourier self-deconvolution: a method for resolving intrinsically overlapped bands,” Appl. Spectrosc. 35, 271–276 (1981).
    [CrossRef]
  3. See, for example, A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 331–335.
  4. J. Y. Huang and Y. R. Shen, “Sum-frequency generation as a surface probe,” in Laser Spectroscopy and Photochemistry on Metal Surface, H. L. Dai and W. Ho, eds. (World Scientific, Singapore, 1995), Vol. 1, pp. 5–53.
  5. H. Kataoka, S. Maeda, and C. Hirose, “Effects of laser linewidth on the coherent anti-Stokes Raman spectroscopy spectral profile,” Appl. Spectrosc. 36, 565–569 (1982).
    [CrossRef]
  6. F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
    [CrossRef]
  7. H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
    [CrossRef] [PubMed]
  8. E. M. Vartiainen, “Phase retrieval approach for coherent anti-Stokes Raman scattering spectrum analysis,” J. Opt. Soc. Am. B 9, 1209–1214 (1992).
    [CrossRef]
  9. E. M. Vartiainen and K.-E. Peiponen, “Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum,” Phys. Rev. B 50, 1941–1944 (1994).
    [CrossRef]
  10. E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the |χ(3)| spectra of polysilane,” J. Opt. Soc. Am. B 13, 2106–2114 (1996).
    [CrossRef]
  11. P.-K. Yang and J. Y. Huang, “Phase-retrieval problems in infrared-visible sum-frequency generation spectroscopy by the maximum-entropy method,” J. Opt. Soc. Am. B 14, 2443–2448 (1997).
    [CrossRef]
  12. R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
    [CrossRef]
  13. S. R. Greenfield and M. R. Wasielewski, “Near-transform-limited visible and near-IR femtosecond pulses from optical parametric amplification using type II β-barium borate,” Opt. Lett. 20, 1394–1396 (1995).
    [CrossRef] [PubMed]
  14. P. Guyot-Sionnest, P. Dumas, Y. J. Chabal, and G. S. Higashi, “Lifetime of an adsorbate-substrate vibration: H on Si(111),” Phys. Rev. Lett. 64, 2156–2159 (1990).
    [CrossRef] [PubMed]

1997 (2)

P.-K. Yang and J. Y. Huang, “Phase-retrieval problems in infrared-visible sum-frequency generation spectroscopy by the maximum-entropy method,” J. Opt. Soc. Am. B 14, 2443–2448 (1997).
[CrossRef]

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

E. M. Vartiainen and K.-E. Peiponen, “Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum,” Phys. Rev. B 50, 1941–1944 (1994).
[CrossRef]

1993 (1)

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

1990 (1)

P. Guyot-Sionnest, P. Dumas, Y. J. Chabal, and G. S. Higashi, “Lifetime of an adsorbate-substrate vibration: H on Si(111),” Phys. Rev. Lett. 64, 2156–2159 (1990).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

1975 (1)

F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
[CrossRef]

Chabal, Y. J.

P. Guyot-Sionnest, P. Dumas, Y. J. Chabal, and G. S. Higashi, “Lifetime of an adsorbate-substrate vibration: H on Si(111),” Phys. Rev. Lett. 64, 2156–2159 (1990).
[CrossRef] [PubMed]

Delong, K. W.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Dumas, P.

P. Guyot-Sionnest, P. Dumas, Y. J. Chabal, and G. S. Higashi, “Lifetime of an adsorbate-substrate vibration: H on Si(111),” Phys. Rev. Lett. 64, 2156–2159 (1990).
[CrossRef] [PubMed]

Fittinghoff, D. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Gameron, D. G.

Good, R. H.

F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
[CrossRef]

Greenfield, S. R.

Guyot-Sionnest, P.

P. Guyot-Sionnest, P. Dumas, Y. J. Chabal, and G. S. Higashi, “Lifetime of an adsorbate-substrate vibration: H on Si(111),” Phys. Rev. Lett. 64, 2156–2159 (1990).
[CrossRef] [PubMed]

Hasegawa, T.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

Higashi, G. S.

P. Guyot-Sionnest, P. Dumas, Y. J. Chabal, and G. S. Higashi, “Lifetime of an adsorbate-substrate vibration: H on Si(111),” Phys. Rev. Lett. 64, 2156–2159 (1990).
[CrossRef] [PubMed]

Hirose, C.

Hollberg, M. R.

Huang, J. Y.

Iwasa, Y.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

Kataoka, H.

Kauppinen, J. K.

Kishida, H.

E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the |χ(3)| spectra of polysilane,” J. Opt. Soc. Am. B 13, 2106–2114 (1996).
[CrossRef]

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

Koda, T.

E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the |χ(3)| spectra of polysilane,” J. Opt. Soc. Am. B 13, 2106–2114 (1996).
[CrossRef]

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

Krumbügel, M. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Maeda, S.

Mantch, H. H.

Mantsch, H. H.

Moffatt, D. J.

Peiponen, K.-E.

Richman, B. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Ridener, F. L.

F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
[CrossRef]

Sweetser, J. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Tokura, Y.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

Trebino, R.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Vartiainen, E. M.

Wasielewski, M. R.

Yang, P.-K.

Appl. Spectrosc. (3)

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

Opt. Lett. (1)

Phys. Rev. B (2)

E. M. Vartiainen and K.-E. Peiponen, “Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum,” Phys. Rev. B 50, 1941–1944 (1994).
[CrossRef]

F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
[CrossRef]

Phys. Rev. Lett. (2)

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

P. Guyot-Sionnest, P. Dumas, Y. J. Chabal, and G. S. Higashi, “Lifetime of an adsorbate-substrate vibration: H on Si(111),” Phys. Rev. Lett. 64, 2156–2159 (1990).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and B. A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Other (2)

See, for example, A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 331–335.

J. Y. Huang and Y. R. Shen, “Sum-frequency generation as a surface probe,” in Laser Spectroscopy and Photochemistry on Metal Surface, H. L. Dai and W. Ho, eds. (World Scientific, Singapore, 1995), Vol. 1, pp. 5–53.

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Figures (3)

Fig. 1
Fig. 1

Effects of a finite laser bandwidth on an error-phase function with a varying broadening ratio of (a) σ/γ=0.1, (b) σ/γ=1.0, and (c) σ/γ=2.0. The dot points in the left column denote the simulated spectra, and their predicted results from the maximum-entropy model are represented by the solid curves. The dotted curves in the right column show the error-phase functions for the corresponding simulated spectra on the left side. The parameter M used in the calculations with the maximum-entropy model was chosen to be 50.

Fig. 2
Fig. 2

(a) Broadened spectrum with σ/γ=1.0 (dot points) and the spectra after deconvolution (solid curve). (b) Real parts and (c) imaginary parts of the nonlinear susceptibility before (dotted curve) and after (solid curve) deconvolution. The corresponding inverse Fourier transforms are shown in (d) and (e) with the same types of curves. The dashed curves in (c) and (d) represent the inverse Fourier transform of the amplitude line-shape function of the light source. The parameter M used in the calculations with the maximum-entropy model has a value of 50.

Fig. 3
Fig. 3

Comparison of the intrinsic (open circles) and the deconvoluted (solid curves) spectra with different degrees of broadening (a) σ/γ=0.5; (b) σ/γ=1.0; (c) σ/γ=1.5; (d) σ/γ=2.0.

Equations (13)

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E(ω)=G(ω)*E(ω)=-G(ω)E(ω-ω)dω,
F-1{E(ω)}=F-1{G(ω)}F-1{E(ω)}.
E(ω)=FF-1{E(ω)}F-1{G(ω)},
Ps(ωs=ων+ωIR)=χ(2)(-ωs; ων, ωIR)E(ων)E(ωIR),
Is(ωs=ων+ωIR0)|G(ωIR)×χ(2)(-ωs; ων, ωIR0-ωIR)|2dωIRI(ων)I(ωIR0).
Is(ωs=ων+ωIR0)G(ωIR)×χ(2)(-ωs; ων, ωIR0-ωIR)×dωIR2I(ων)I(ωIR0).
hf1f2 log S(f )df.
Sˆ(ν)=|β|21+k=1Mak exp(i2πkν)2.
R(0)R(-1)R(-M)R(1)R(0)R(1-M)R(M)R(M-1)R(0) 1a1aM=|β|200,
R(m)=01S(ν)exp(-i2πmν)dν.
χ^(n)(ν)=|β|exp[iϕ(ν)]1+k=1Mak exp(i2πkν).
G(ν)=12πσ exp-12 ν2σ2,
χ(2)(ν)=A(ν0-ν)-iγ.

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