Abstract

Most frequency-domain nonlinear-optical techniques measure only the magnitude of the Fourier transform of a temporal response and, hence, do not uniquely determine the response. We show that, for the commonly used response, h(t) = A exp(−t/τf) + B exp(−t/τs), where Aα/τf and B ≡ (1 − α)/τs, the spectral line shape can always be fitted by two different values of α. A measurement of the optical Kerr transient of carbon disulfide illustrates this ambiguity. We also demonstrate a single-scan method that is free from such ambiguities. It involves adding coherent background with a nonzero quadrature-phase component, obtained simply by proper choice of probe wavelength.

© 1988 Optical Society of America

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  1. R. Trebino, C. E. Barker, A. E. Siegman, IEEE J. Quantum Electron. QE-22, 1413 (1986).
    [CrossRef]
  2. L. J. Rothberg, N. Bloembergen, Phys. Rev. A 30, 2327 (1984).
    [CrossRef]
  3. J. J. Song, J. H. Lee, M. D. Levenson, Phys. Rev. A 17, 1439 (1978).
    [CrossRef]
  4. T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
    [CrossRef]
  5. F. Keilmann, Appl. Phys. 14, 29 (1977).
    [CrossRef]
  6. J. R. Fienup, J. Opt. Soc. Am. 4, 118 (1987).
    [CrossRef]
  7. E. J. Akutowics, Trans. Am. Math. Soc. 83, 179 (1956).
  8. H. Stark, ed., Image Recovery: Theory and Applications (Academic, Orlando, Fla., 1987).
  9. W. Yu, F. Pellegrino, M. Grant, R. R. Alfano, J. Chem. Phys. 64, 2648 (1977).
  10. C. Kalpouzos, W. T. Lotshaw, D. McMorrow, G. A. Kenney-Wallace, J. Phys. Chem. 91, 2028 (1987).
    [CrossRef]
  11. J.-M. Halbout, C. L. Tang, Appl. Phys. Lett. 40, 765 (1982).
    [CrossRef]
  12. S. Ruhman, L. R. Williams, A. G. Joly, B. Kohler, K. A. Nelson, J. Phys. Chem. 91, 2237 (1987).
    [CrossRef]
  13. D. L. Misell, R. E. Burge, A. H. Greenaway, J. Phys. D 7, L27 (1974).
    [CrossRef]
  14. C. L. Mehta, Nuovo Cimento 36, 202 (1965).
    [CrossRef]
  15. D. L. Misell, J. Phys. D 6, 2200 (1963).
    [CrossRef]
  16. C. L. Mehta, J. Opt. Soc. Am. 58, 1233 (1968).
    [CrossRef]
  17. R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, J. Phys. D 7, 61 (1974).
    [CrossRef]
  18. M. D. Levenson, G. L. Eesley, Appl. Phys. 19, 1 (1979).
    [CrossRef]
  19. Ironically, the two frequency-domain methods 3,5 that do involve coherent background contain in-phase, but not quadrature, background.
  20. The existence of this ambiguity and its removal by the addition of quadrature-phase coherent background can also be seen when using Blaschke products.7 Ambiguity removal also follows from the observation that the Fourier-transform magnitude of a real decay is even, and adding iγ breaks this symmetry.
  21. In the course of this work, the fit to the data of this experiment was slightly improved, yielding the parameter values reported herein.

1987 (3)

J. R. Fienup, J. Opt. Soc. Am. 4, 118 (1987).
[CrossRef]

C. Kalpouzos, W. T. Lotshaw, D. McMorrow, G. A. Kenney-Wallace, J. Phys. Chem. 91, 2028 (1987).
[CrossRef]

S. Ruhman, L. R. Williams, A. G. Joly, B. Kohler, K. A. Nelson, J. Phys. Chem. 91, 2237 (1987).
[CrossRef]

1986 (1)

R. Trebino, C. E. Barker, A. E. Siegman, IEEE J. Quantum Electron. QE-22, 1413 (1986).
[CrossRef]

1984 (1)

L. J. Rothberg, N. Bloembergen, Phys. Rev. A 30, 2327 (1984).
[CrossRef]

1982 (1)

J.-M. Halbout, C. L. Tang, Appl. Phys. Lett. 40, 765 (1982).
[CrossRef]

1979 (1)

M. D. Levenson, G. L. Eesley, Appl. Phys. 19, 1 (1979).
[CrossRef]

1978 (2)

J. J. Song, J. H. Lee, M. D. Levenson, Phys. Rev. A 17, 1439 (1978).
[CrossRef]

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

1977 (2)

F. Keilmann, Appl. Phys. 14, 29 (1977).
[CrossRef]

W. Yu, F. Pellegrino, M. Grant, R. R. Alfano, J. Chem. Phys. 64, 2648 (1977).

1974 (2)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, J. Phys. D 7, 61 (1974).
[CrossRef]

D. L. Misell, R. E. Burge, A. H. Greenaway, J. Phys. D 7, L27 (1974).
[CrossRef]

1968 (1)

1965 (1)

C. L. Mehta, Nuovo Cimento 36, 202 (1965).
[CrossRef]

1963 (1)

D. L. Misell, J. Phys. D 6, 2200 (1963).
[CrossRef]

1956 (1)

E. J. Akutowics, Trans. Am. Math. Soc. 83, 179 (1956).

Akutowics, E. J.

E. J. Akutowics, Trans. Am. Math. Soc. 83, 179 (1956).

Alfano, R. R.

W. Yu, F. Pellegrino, M. Grant, R. R. Alfano, J. Chem. Phys. 64, 2648 (1977).

Barker, C. E.

R. Trebino, C. E. Barker, A. E. Siegman, IEEE J. Quantum Electron. QE-22, 1413 (1986).
[CrossRef]

Bloembergen, N.

L. J. Rothberg, N. Bloembergen, Phys. Rev. A 30, 2327 (1984).
[CrossRef]

Burge, R. E.

D. L. Misell, R. E. Burge, A. H. Greenaway, J. Phys. D 7, L27 (1974).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, J. Phys. D 7, 61 (1974).
[CrossRef]

Eesley, G. L.

M. D. Levenson, G. L. Eesley, Appl. Phys. 19, 1 (1979).
[CrossRef]

Fiddy, M. A.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, J. Phys. D 7, 61 (1974).
[CrossRef]

Fienup, J. R.

J. R. Fienup, J. Opt. Soc. Am. 4, 118 (1987).
[CrossRef]

Grant, M.

W. Yu, F. Pellegrino, M. Grant, R. R. Alfano, J. Chem. Phys. 64, 2648 (1977).

Greenaway, A. H.

D. L. Misell, R. E. Burge, A. H. Greenaway, J. Phys. D 7, L27 (1974).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, J. Phys. D 7, 61 (1974).
[CrossRef]

Halbout, J.-M.

J.-M. Halbout, C. L. Tang, Appl. Phys. Lett. 40, 765 (1982).
[CrossRef]

Ishida, Y.

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

Joly, A. G.

S. Ruhman, L. R. Williams, A. G. Joly, B. Kohler, K. A. Nelson, J. Phys. Chem. 91, 2237 (1987).
[CrossRef]

Kalpouzos, C.

C. Kalpouzos, W. T. Lotshaw, D. McMorrow, G. A. Kenney-Wallace, J. Phys. Chem. 91, 2028 (1987).
[CrossRef]

Keilmann, F.

F. Keilmann, Appl. Phys. 14, 29 (1977).
[CrossRef]

Kenney-Wallace, G. A.

C. Kalpouzos, W. T. Lotshaw, D. McMorrow, G. A. Kenney-Wallace, J. Phys. Chem. 91, 2028 (1987).
[CrossRef]

Kohler, B.

S. Ruhman, L. R. Williams, A. G. Joly, B. Kohler, K. A. Nelson, J. Phys. Chem. 91, 2237 (1987).
[CrossRef]

Lee, J. H.

J. J. Song, J. H. Lee, M. D. Levenson, Phys. Rev. A 17, 1439 (1978).
[CrossRef]

Levenson, M. D.

M. D. Levenson, G. L. Eesley, Appl. Phys. 19, 1 (1979).
[CrossRef]

J. J. Song, J. H. Lee, M. D. Levenson, Phys. Rev. A 17, 1439 (1978).
[CrossRef]

Lotshaw, W. T.

C. Kalpouzos, W. T. Lotshaw, D. McMorrow, G. A. Kenney-Wallace, J. Phys. Chem. 91, 2028 (1987).
[CrossRef]

McMorrow, D.

C. Kalpouzos, W. T. Lotshaw, D. McMorrow, G. A. Kenney-Wallace, J. Phys. Chem. 91, 2028 (1987).
[CrossRef]

Mehta, C. L.

Misell, D. L.

D. L. Misell, R. E. Burge, A. H. Greenaway, J. Phys. D 7, L27 (1974).
[CrossRef]

D. L. Misell, J. Phys. D 6, 2200 (1963).
[CrossRef]

Nelson, K. A.

S. Ruhman, L. R. Williams, A. G. Joly, B. Kohler, K. A. Nelson, J. Phys. Chem. 91, 2237 (1987).
[CrossRef]

Pellegrino, F.

W. Yu, F. Pellegrino, M. Grant, R. R. Alfano, J. Chem. Phys. 64, 2648 (1977).

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, J. Phys. D 7, 61 (1974).
[CrossRef]

Rothberg, L. J.

L. J. Rothberg, N. Bloembergen, Phys. Rev. A 30, 2327 (1984).
[CrossRef]

Ruhman, S.

S. Ruhman, L. R. Williams, A. G. Joly, B. Kohler, K. A. Nelson, J. Phys. Chem. 91, 2237 (1987).
[CrossRef]

Siegman, A. E.

R. Trebino, C. E. Barker, A. E. Siegman, IEEE J. Quantum Electron. QE-22, 1413 (1986).
[CrossRef]

Song, J. J.

J. J. Song, J. H. Lee, M. D. Levenson, Phys. Rev. A 17, 1439 (1978).
[CrossRef]

Souma, H.

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

Tang, C. L.

J.-M. Halbout, C. L. Tang, Appl. Phys. Lett. 40, 765 (1982).
[CrossRef]

Trebino, R.

R. Trebino, C. E. Barker, A. E. Siegman, IEEE J. Quantum Electron. QE-22, 1413 (1986).
[CrossRef]

Williams, L. R.

S. Ruhman, L. R. Williams, A. G. Joly, B. Kohler, K. A. Nelson, J. Phys. Chem. 91, 2237 (1987).
[CrossRef]

Yajima, T.

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

Yu, W.

W. Yu, F. Pellegrino, M. Grant, R. R. Alfano, J. Chem. Phys. 64, 2648 (1977).

Appl. Phys. (2)

F. Keilmann, Appl. Phys. 14, 29 (1977).
[CrossRef]

M. D. Levenson, G. L. Eesley, Appl. Phys. 19, 1 (1979).
[CrossRef]

Appl. Phys. Lett. (1)

J.-M. Halbout, C. L. Tang, Appl. Phys. Lett. 40, 765 (1982).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. Trebino, C. E. Barker, A. E. Siegman, IEEE J. Quantum Electron. QE-22, 1413 (1986).
[CrossRef]

J. Chem. Phys. (1)

W. Yu, F. Pellegrino, M. Grant, R. R. Alfano, J. Chem. Phys. 64, 2648 (1977).

J. Opt. Soc. Am. (2)

C. L. Mehta, J. Opt. Soc. Am. 58, 1233 (1968).
[CrossRef]

J. R. Fienup, J. Opt. Soc. Am. 4, 118 (1987).
[CrossRef]

J. Phys. Chem. (2)

C. Kalpouzos, W. T. Lotshaw, D. McMorrow, G. A. Kenney-Wallace, J. Phys. Chem. 91, 2028 (1987).
[CrossRef]

S. Ruhman, L. R. Williams, A. G. Joly, B. Kohler, K. A. Nelson, J. Phys. Chem. 91, 2237 (1987).
[CrossRef]

J. Phys. D (3)

D. L. Misell, R. E. Burge, A. H. Greenaway, J. Phys. D 7, L27 (1974).
[CrossRef]

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, J. Phys. D 7, 61 (1974).
[CrossRef]

D. L. Misell, J. Phys. D 6, 2200 (1963).
[CrossRef]

Nuovo Cimento (1)

C. L. Mehta, Nuovo Cimento 36, 202 (1965).
[CrossRef]

Phys. Rev. A (3)

L. J. Rothberg, N. Bloembergen, Phys. Rev. A 30, 2327 (1984).
[CrossRef]

J. J. Song, J. H. Lee, M. D. Levenson, Phys. Rev. A 17, 1439 (1978).
[CrossRef]

T. Yajima, H. Souma, Y. Ishida, Phys. Rev. A 17, 324 (1978).
[CrossRef]

Trans. Am. Math. Soc. (1)

E. J. Akutowics, Trans. Am. Math. Soc. 83, 179 (1956).

Other (4)

H. Stark, ed., Image Recovery: Theory and Applications (Academic, Orlando, Fla., 1987).

Ironically, the two frequency-domain methods 3,5 that do involve coherent background contain in-phase, but not quadrature, background.

The existence of this ambiguity and its removal by the addition of quadrature-phase coherent background can also be seen when using Blaschke products.7 Ambiguity removal also follows from the observation that the Fourier-transform magnitude of a real decay is even, and adding iγ breaks this symmetry.

In the course of this work, the fit to the data of this experiment was slightly improved, yielding the parameter values reported herein.

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Figures (2)

Fig. 1
Fig. 1

Experimental results with negligible quadrature-phase coherent background. (a) Diffraction efficiency versus frequency difference, ω = ω1ω2. The solid line indicates the best fit, obtained for two different values of α: − 0.91 and +0.31. (b) Plot of χ2(α) versus α for the data in (a). Observe the presence of two minima with about the same minimum value, indicating ambiguity in the fit for the parameter α.

Fig. 2
Fig. 2

Experimental results with substantial quadrature-phase coherent background. (a) Diffraction efficiency versus frequency difference, ω = ω1ω2. The solid line indicates the best fit, obtained for a single value of α of +0.26. The dashed line indicates a fit using α = −0.91. (A negative baseline has been added to both curves to force their asymptotes to zero.) (b) Plot of χ2(α) versus a for the data in (a). Observe the presence of only one minimum, indicating that ambiguity in the fit for the parameter α has been removed.

Equations (3)

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h ( t ) = θ ( t ) [ A exp ( - t / τ f ) + B exp ( - t / τ s ) ] ,
Γ α β γ ( ω ) 2 = | 1 + i ω τ α ( 1 + i ω τ f ) ( 1 + i ω τ s ) + β + i γ | 2 ,
α = - 2 τ f τ s - τ f - 2 ( β - i γ ) τ s + τ f τ s - τ f - α ,

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