Abstract

A noniterative phase retrieval method for coherent anti-Stokes Raman scattering spectrum analysis is presented. This method facilitates the computation of the real and imaginary parts of the effective third-order susceptibility χ when only its modulus is known. One obtains this result by approximating the squared modulus |χ|2 by the maximum entropy model and using some a priori information on χ.

© 1992 Optical Society of America

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References

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  1. M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).
  2. S. A. J. Druet, J.-P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron 7, 1–72 (1981).
    [CrossRef]
  3. R. J. Hall, A. C. Eckbreth, “Coherent anti-Stokes Raman spectroscopy (CARS): application to combustion diagnostics,” in Laser Applications, J. E. Ready, R. K. Erf, eds. Academic, Orlando, Fla., 1984), Vol. 5, pp. 213–309.
  4. H. Kataoka, S. Maeda, C. Hirose, “Effects of laser line-width on the CARS profile,” Appl. Spectrosc. 36, 565–569 (1982).
    [CrossRef]
  5. R. E. Teets, “Accurate convolutions of CARS spectra,” Opt. Lett. 9, 226–229 (1984).
    [CrossRef] [PubMed]
  6. L. A. Rahn, R. L. Farrow, R. P. Lucht, “Effects of field statistics on CARS intensities,” Opt. Lett. 9, 223–225 (1984).
    [CrossRef] [PubMed]
  7. R. L. Farrow, L. A. Rahn, “Interpreting coherent anti Stokes Raman spectra measured with multimode Nd:YAG pump lasers,” J. Opt. Soc. Am. B 2, 903–907 (1985).
    [CrossRef]
  8. J. W. Fleming, C. S. Johnson, “A practical analysis for coherent anti-Stokes Raman scattering (CARS) spectra,” J. Raman Spectrosc. 8, 284–290 (1979).
    [CrossRef]
  9. E. M. Vartiainen, K.-E. Peiponen, T. Tsuboi, “Analysis of coherent Raman spectra,” J. Opt. Soc. Am. B 7, 722–725 (1990).
    [CrossRef]
  10. E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurement of the degree of coherence?” Proc. Phys. Soc. London 80, 1269–1272 (1962).
    [CrossRef]
  11. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  12. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  13. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
    [CrossRef]
  14. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  15. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  16. S. Haykin, S. Kesler, “Prediction-error filtering and maximum entropy spectral estimation,” in Nonlinear Methods of Spectral Analysis, 2nd ed., S. Haykin, ed. (Springer-Verlag, Berlin, 1983), pp. 9–72.
  17. J. Laane, W. Kiefer, “Interference effects in the high resolution CARS spectra of gases,” J. Raman Spectrosc. 9, 353–360 (1980).
    [CrossRef]
  18. J. K. Kauppinen, D. J. Moffatt, H. H. Mantsch, D. G. Cameron, “Fourier self-deconvolution: a method for resolving intrinsically overlapped bands,” Appl. Spectrosc. 35, 271–276 (1981).
    [CrossRef]
  19. J. K. Kauppinen, D. J. Moffatt, M. R. Hollberg, H. H. Mantsch, “A new line-narrowing method based on Fourier self-deconvolution, maximum entropy, and linear prediction,” Appl. Spectrosc. 45, 411–416 (1991).
    [CrossRef]
  20. G. L. Eesley, M. D. Levenson, W. M. Tolles, “Optically heterodyned coherent Raman spectroscopy,” IEEE J. Quantum Electron. QE-14, 45–49 (1978).
    [CrossRef]
  21. A. Owyoung, “Coherent Raman gain spectroscopy using cw laser sources,” IEEE J. Quantum Electron. QE-14, 192–203 (1978).
    [CrossRef]
  22. A. Owyoung, “High-resolution cw stimulated Raman spectroscopy in molecular hydrogen,” Opt. Lett. 2, 91–93 (1978).
    [CrossRef] [PubMed]
  23. S. A. Akhmanov, A. F. Bunkin, S. G. Ivanov, N. I. Koroteev, “Coherent ellipsometry of Raman scattering of light,” JETP Lett. 25, 416–421 (1977).
  24. J.-L. Oudar, R. W. Smith, Y. R. Shen, “Polarization-sensitive coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 34, 758–760 (1979).
    [CrossRef]
  25. L. A. Rahn, L. J. Zych, P. L. Mattern, “Background-free CARS studies of carbon monoxide in a flame,” Opt. Commun. 30, 249–252 (1979).
    [CrossRef]

1991 (1)

1990 (1)

1987 (1)

1985 (1)

1984 (2)

1982 (2)

1981 (2)

1980 (1)

J. Laane, W. Kiefer, “Interference effects in the high resolution CARS spectra of gases,” J. Raman Spectrosc. 9, 353–360 (1980).
[CrossRef]

1979 (3)

J.-L. Oudar, R. W. Smith, Y. R. Shen, “Polarization-sensitive coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 34, 758–760 (1979).
[CrossRef]

L. A. Rahn, L. J. Zych, P. L. Mattern, “Background-free CARS studies of carbon monoxide in a flame,” Opt. Commun. 30, 249–252 (1979).
[CrossRef]

J. W. Fleming, C. S. Johnson, “A practical analysis for coherent anti-Stokes Raman scattering (CARS) spectra,” J. Raman Spectrosc. 8, 284–290 (1979).
[CrossRef]

1978 (3)

A. Owyoung, “High-resolution cw stimulated Raman spectroscopy in molecular hydrogen,” Opt. Lett. 2, 91–93 (1978).
[CrossRef] [PubMed]

G. L. Eesley, M. D. Levenson, W. M. Tolles, “Optically heterodyned coherent Raman spectroscopy,” IEEE J. Quantum Electron. QE-14, 45–49 (1978).
[CrossRef]

A. Owyoung, “Coherent Raman gain spectroscopy using cw laser sources,” IEEE J. Quantum Electron. QE-14, 192–203 (1978).
[CrossRef]

1977 (1)

S. A. Akhmanov, A. F. Bunkin, S. G. Ivanov, N. I. Koroteev, “Coherent ellipsometry of Raman scattering of light,” JETP Lett. 25, 416–421 (1977).

1976 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1963 (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

1962 (1)

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurement of the degree of coherence?” Proc. Phys. Soc. London 80, 1269–1272 (1962).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. F. Bunkin, S. G. Ivanov, N. I. Koroteev, “Coherent ellipsometry of Raman scattering of light,” JETP Lett. 25, 416–421 (1977).

Bunkin, A. F.

S. A. Akhmanov, A. F. Bunkin, S. G. Ivanov, N. I. Koroteev, “Coherent ellipsometry of Raman scattering of light,” JETP Lett. 25, 416–421 (1977).

Cameron, D. G.

Druet, S. A. J.

S. A. J. Druet, J.-P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron 7, 1–72 (1981).
[CrossRef]

Eckbreth, A. C.

R. J. Hall, A. C. Eckbreth, “Coherent anti-Stokes Raman spectroscopy (CARS): application to combustion diagnostics,” in Laser Applications, J. E. Ready, R. K. Erf, eds. Academic, Orlando, Fla., 1984), Vol. 5, pp. 213–309.

Eesley, G. L.

G. L. Eesley, M. D. Levenson, W. M. Tolles, “Optically heterodyned coherent Raman spectroscopy,” IEEE J. Quantum Electron. QE-14, 45–49 (1978).
[CrossRef]

Farrow, R. L.

Fienup, J. R.

Fleming, J. W.

J. W. Fleming, C. S. Johnson, “A practical analysis for coherent anti-Stokes Raman scattering (CARS) spectra,” J. Raman Spectrosc. 8, 284–290 (1979).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gonsalves, R. A.

Hall, R. J.

R. J. Hall, A. C. Eckbreth, “Coherent anti-Stokes Raman spectroscopy (CARS): application to combustion diagnostics,” in Laser Applications, J. E. Ready, R. K. Erf, eds. Academic, Orlando, Fla., 1984), Vol. 5, pp. 213–309.

Haykin, S.

S. Haykin, S. Kesler, “Prediction-error filtering and maximum entropy spectral estimation,” in Nonlinear Methods of Spectral Analysis, 2nd ed., S. Haykin, ed. (Springer-Verlag, Berlin, 1983), pp. 9–72.

Hirose, C.

Hollberg, M. R.

Ivanov, S. G.

S. A. Akhmanov, A. F. Bunkin, S. G. Ivanov, N. I. Koroteev, “Coherent ellipsometry of Raman scattering of light,” JETP Lett. 25, 416–421 (1977).

Johnson, C. S.

J. W. Fleming, C. S. Johnson, “A practical analysis for coherent anti-Stokes Raman scattering (CARS) spectra,” J. Raman Spectrosc. 8, 284–290 (1979).
[CrossRef]

Kataoka, H.

Kauppinen, J. K.

Kesler, S.

S. Haykin, S. Kesler, “Prediction-error filtering and maximum entropy spectral estimation,” in Nonlinear Methods of Spectral Analysis, 2nd ed., S. Haykin, ed. (Springer-Verlag, Berlin, 1983), pp. 9–72.

Kiefer, W.

J. Laane, W. Kiefer, “Interference effects in the high resolution CARS spectra of gases,” J. Raman Spectrosc. 9, 353–360 (1980).
[CrossRef]

Koroteev, N. I.

S. A. Akhmanov, A. F. Bunkin, S. G. Ivanov, N. I. Koroteev, “Coherent ellipsometry of Raman scattering of light,” JETP Lett. 25, 416–421 (1977).

Laane, J.

J. Laane, W. Kiefer, “Interference effects in the high resolution CARS spectra of gases,” J. Raman Spectrosc. 9, 353–360 (1980).
[CrossRef]

Levenson, M. D.

G. L. Eesley, M. D. Levenson, W. M. Tolles, “Optically heterodyned coherent Raman spectroscopy,” IEEE J. Quantum Electron. QE-14, 45–49 (1978).
[CrossRef]

M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).

Lucht, R. P.

Maeda, S.

Mantsch, H. H.

Mattern, P. L.

L. A. Rahn, L. J. Zych, P. L. Mattern, “Background-free CARS studies of carbon monoxide in a flame,” Opt. Commun. 30, 249–252 (1979).
[CrossRef]

Moffatt, D. J.

Oudar, J.-L.

J.-L. Oudar, R. W. Smith, Y. R. Shen, “Polarization-sensitive coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 34, 758–760 (1979).
[CrossRef]

Owyoung, A.

A. Owyoung, “Coherent Raman gain spectroscopy using cw laser sources,” IEEE J. Quantum Electron. QE-14, 192–203 (1978).
[CrossRef]

A. Owyoung, “High-resolution cw stimulated Raman spectroscopy in molecular hydrogen,” Opt. Lett. 2, 91–93 (1978).
[CrossRef] [PubMed]

Peiponen, K.-E.

Rahn, L. A.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Shen, Y. R.

J.-L. Oudar, R. W. Smith, Y. R. Shen, “Polarization-sensitive coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 34, 758–760 (1979).
[CrossRef]

Smith, R. W.

J.-L. Oudar, R. W. Smith, Y. R. Shen, “Polarization-sensitive coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 34, 758–760 (1979).
[CrossRef]

Taran, J.-P. E.

S. A. J. Druet, J.-P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron 7, 1–72 (1981).
[CrossRef]

Teets, R. E.

Tolles, W. M.

G. L. Eesley, M. D. Levenson, W. M. Tolles, “Optically heterodyned coherent Raman spectroscopy,” IEEE J. Quantum Electron. QE-14, 45–49 (1978).
[CrossRef]

Tsuboi, T.

Vartiainen, E. M.

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Wolf, E.

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurement of the degree of coherence?” Proc. Phys. Soc. London 80, 1269–1272 (1962).
[CrossRef]

Zych, L. J.

L. A. Rahn, L. J. Zych, P. L. Mattern, “Background-free CARS studies of carbon monoxide in a flame,” Opt. Commun. 30, 249–252 (1979).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J.-L. Oudar, R. W. Smith, Y. R. Shen, “Polarization-sensitive coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 34, 758–760 (1979).
[CrossRef]

Appl. Spectrosc. (3)

IEEE J. Quantum Electron. (2)

G. L. Eesley, M. D. Levenson, W. M. Tolles, “Optically heterodyned coherent Raman spectroscopy,” IEEE J. Quantum Electron. QE-14, 45–49 (1978).
[CrossRef]

A. Owyoung, “Coherent Raman gain spectroscopy using cw laser sources,” IEEE J. Quantum Electron. QE-14, 192–203 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

J. Raman Spectrosc. (2)

J. W. Fleming, C. S. Johnson, “A practical analysis for coherent anti-Stokes Raman scattering (CARS) spectra,” J. Raman Spectrosc. 8, 284–290 (1979).
[CrossRef]

J. Laane, W. Kiefer, “Interference effects in the high resolution CARS spectra of gases,” J. Raman Spectrosc. 9, 353–360 (1980).
[CrossRef]

JETP Lett. (1)

S. A. Akhmanov, A. F. Bunkin, S. G. Ivanov, N. I. Koroteev, “Coherent ellipsometry of Raman scattering of light,” JETP Lett. 25, 416–421 (1977).

Opt. Acta (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Opt. Commun. (1)

L. A. Rahn, L. J. Zych, P. L. Mattern, “Background-free CARS studies of carbon monoxide in a flame,” Opt. Commun. 30, 249–252 (1979).
[CrossRef]

Opt. Lett. (3)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Proc. Phys. Soc. London (1)

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurement of the degree of coherence?” Proc. Phys. Soc. London 80, 1269–1272 (1962).
[CrossRef]

Prog. Quantum Electron (1)

S. A. J. Druet, J.-P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron 7, 1–72 (1981).
[CrossRef]

Other (3)

R. J. Hall, A. C. Eckbreth, “Coherent anti-Stokes Raman spectroscopy (CARS): application to combustion diagnostics,” in Laser Applications, J. E. Ready, R. K. Erf, eds. Academic, Orlando, Fla., 1984), Vol. 5, pp. 213–309.

S. Haykin, S. Kesler, “Prediction-error filtering and maximum entropy spectral estimation,” in Nonlinear Methods of Spectral Analysis, 2nd ed., S. Haykin, ed. (Springer-Verlag, Berlin, 1983), pp. 9–72.

M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).

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Figures (3)

Fig. 1
Fig. 1

(a) Simulated squared modulus |χ|2 (solid curve) and its MEM approximation (dotted curve) showing three resonances at normalized frequencies 0.4, 0.6, and 0.8. The corresponding parameters used in Eq. (21) are Ar = 0.01, 0.02, and 0.01, Γr = 0.02, and χN = 0.5. In (b) and (c) are shown the curves of Re χ and Im χ, respectively, obtained from Eq. (21) (solid curves) and from the present calculations (dotted curves) using Eq. (16) with νR = 0.7996.

Fig. 2
Fig. 2

(a) Squeezed spectrum of the curve in Fig. 1(a). Actual (solid) and approximated (dotted) curves of (b) Re χ and (c) Im χ obtained with the squeezing procedure and using Eq. (12) with ν0 = 0.

Fig. 3
Fig. 3

(a) Measured CARS spectrum of the nitrogen Q branch (solid curve) and its MEM approximation (dotted curve). In (b) and (c) are shown the approximated Re χ and Im χ, respectively. In (c) the arrow points out the frequency ν0 used in Eq. (12).

Equations (28)

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I ( ω 1 ω 2 ) | χ ( ω 1 ω 2 ) | 2
ν = ( δ δ min ) / ( δ max δ min ) ,
| χ ( ν ) | 2 = b 0 | 1 + n = 1 M a n exp ( i 2 π n ν ) | 2 = | β A M ( ν ) | 2 ,
[ R ( 0 ) R ( 1 ) R ( M ) R ( 1 ) R ( 0 ) R ( 1 M ) R ( M ) R ( M 1 ) R ( 0 ) ] [ 1 a 1 a M ] = [ b 0 0 0 ] ,
R ( m ) = 0 1 | χ ( ν ) | 2 exp ( i 2 π m ν ) d ν , | m | M .
χ ( ν ) = β / A M ( ν ) ( Type I ) ,
χ ( ν ) = β * / A M * ( ν ) ( Type II ) ,
χ ( ν ) = β * / A M ( ν ) ( Type III ) ,
χ ( ν ) = β / A M * ( ν ) ( Type IV ) .
β ( 1 ) = β ( 2 ) = β ( 3 ) * = β ( 4 ) * ,
χ = β / A M ( ν ) or β * A M * ( ν ) .
χ ( ν ) = β * A M * ( ν ) ,
Re [ χ ( ν ) ] = β A M ( ν ) + β A M ( ν ) | A M ( ν ) | 2 ,
Im [ χ ( ν ) ] = β A M ( ν ) β A M ( ν ) | A M ( ν ) | 2 ,
A M ( ν ) = Re [ A M ( ν ) ] = 1 + n = 1 M [ a n cos ( 2 π n ν ) + a n sin ( 2 π n ν ) ] , A M ( ν ) = Im [ A M ( ν ) ] = n = 1 M [ a n sin ( 2 π n ν ) + a n cos ( 2 π n ν ) ] ,
Im [ χ ( ν 0 ) ] = 0 ,
β = | β | cos ϕ 0 ,
β = | β | sin ϕ 0 ,
tan ϕ 0 = β / β = A M ( ν 0 ) / A M ( ν 0 ) .
ν { Im [ χ ( ν ) ] } ν = ν R = 0
ν R = ( ω R δ min ) / ( δ max δ min ) ,
β = | β | cos ϕ R ,
β = | β | sin ϕ R ,
tan ϕ R = ν [ A M ( ν ) | A M ( ν ) | 2 ] ν = ν R ν [ A M ( ν ) | A M ( ν ) | 2 ] ν = ν R .
| χ ( ω 1 ω 2 ) | 2 = | χ N + r A r ω R ( ω 1 ω 2 ) i Γ r | 2 ,
R ( m ) = 1 N n = 0 N 1 | χ n | 2 exp ( i 2 π m n N ) ,
| R ( m ) | / | R ( 0 ) < 10 3 for every m M .
0 1 Re [ χ ( ν ) ] d ν | |

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