Abstract

We show how the dependence of a nonsaturated excitation process on local, instantaneous intensity can be obtained from temporally and spatially unresolved data by inversion, even for a nonuniform intensity distribution in the probe volume. This treatment is in contrast with the usual approach in which effects of nonuniform excitation are either disregarded or simulated. For exponential intensity profiles, the solution is obtained in the form of Abel inversion. The ill-conditioned nature of the problem is demonstrated, and extension to the study of intensity-dependent line shapes is made.

© 1988 Optical Society of America

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References

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  1. J. H. Eberly, J. Krasinski, in Advances in Multi-Photon Processes and Spectroscopy, S. H. Lin, ed. (Academic, Orlando, Fla., 1984), pp. 1–49.
    [CrossRef]
  2. G. Mainfray, C. Manus, in Multiphoton Ionization of Atoms, S. L. Chin, P. Lambropoulos, eds. (Academic, New York, 1984), pp. 7–34; Y. Gontier, M. Trahin, ibid., pp. 35–64.
  3. W. M. Huo, K. P. Gross, R. L. McKenzie, “Optical Stark effect in the two-photon spectrum of NO,” Phys. Rev. Lett. 54, 1012–1015 (1985).
    [CrossRef] [PubMed]
  4. P. Lambropoulos, X. Tang, “Multiple excitation and ionization of atoms by strong lasers,” J. Opt. Soc. Am. B 4, 821–832 (1987).
    [CrossRef]
  5. Y. Gontier, M. Trahin, “Spatio-temporal effects in resonant multiphoton ionisation of the caesium atom,” J. Phys. B 13, 259–272 (1980).
    [CrossRef]
  6. For partial results, see P. M. Morse, H. Feshbach, Methods of Theoretical Physics I (McGraw-Hill, New York, 1953), p. 475, problem 4.28.
  7. G. R. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), Secs. 16.1 and 16.2.
  8. S. Twomey, Developments in Geomathematics 3—Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).
  9. R. L. Henning, C. D. Capps, G. M. Hess, Inversion Technique Evaluation, (Defense Technical Information Center, Alexandria, Va., 1982).
  10. P. Dierckx, “An algorithm for experimental data deconvolution using spline functions,” J. Comp. Phys. 52, 163–186 (1983).
    [CrossRef]
  11. L. M. Delves, J. Walsh, eds., Numerical Solution of Integral Equations (Clarendon, Oxford, 1974).
  12. C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).
  13. G. Hämmerlin, K.-H. Hoffmann, eds., Constructive Methods for the Practical Treatment of Integral Equations, Vol. 73 of International Series of Numerical Mathematics (Birkhäuser, Basel, 1985).
    [CrossRef]
  14. H. Brunner, P. J. van der Houwen, The Numerical Solution of Volterra Equations (North-Holland, Amsterdam, 1986).

1987 (1)

1985 (1)

W. M. Huo, K. P. Gross, R. L. McKenzie, “Optical Stark effect in the two-photon spectrum of NO,” Phys. Rev. Lett. 54, 1012–1015 (1985).
[CrossRef] [PubMed]

1983 (1)

P. Dierckx, “An algorithm for experimental data deconvolution using spline functions,” J. Comp. Phys. 52, 163–186 (1983).
[CrossRef]

1980 (1)

Y. Gontier, M. Trahin, “Spatio-temporal effects in resonant multiphoton ionisation of the caesium atom,” J. Phys. B 13, 259–272 (1980).
[CrossRef]

Arfken, G. R.

G. R. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), Secs. 16.1 and 16.2.

Baker, C. T. H.

C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).

Brunner, H.

H. Brunner, P. J. van der Houwen, The Numerical Solution of Volterra Equations (North-Holland, Amsterdam, 1986).

Capps, C. D.

R. L. Henning, C. D. Capps, G. M. Hess, Inversion Technique Evaluation, (Defense Technical Information Center, Alexandria, Va., 1982).

Dierckx, P.

P. Dierckx, “An algorithm for experimental data deconvolution using spline functions,” J. Comp. Phys. 52, 163–186 (1983).
[CrossRef]

Eberly, J. H.

J. H. Eberly, J. Krasinski, in Advances in Multi-Photon Processes and Spectroscopy, S. H. Lin, ed. (Academic, Orlando, Fla., 1984), pp. 1–49.
[CrossRef]

Feshbach, H.

For partial results, see P. M. Morse, H. Feshbach, Methods of Theoretical Physics I (McGraw-Hill, New York, 1953), p. 475, problem 4.28.

Gontier, Y.

Y. Gontier, M. Trahin, “Spatio-temporal effects in resonant multiphoton ionisation of the caesium atom,” J. Phys. B 13, 259–272 (1980).
[CrossRef]

Gross, K. P.

W. M. Huo, K. P. Gross, R. L. McKenzie, “Optical Stark effect in the two-photon spectrum of NO,” Phys. Rev. Lett. 54, 1012–1015 (1985).
[CrossRef] [PubMed]

Henning, R. L.

R. L. Henning, C. D. Capps, G. M. Hess, Inversion Technique Evaluation, (Defense Technical Information Center, Alexandria, Va., 1982).

Hess, G. M.

R. L. Henning, C. D. Capps, G. M. Hess, Inversion Technique Evaluation, (Defense Technical Information Center, Alexandria, Va., 1982).

Huo, W. M.

W. M. Huo, K. P. Gross, R. L. McKenzie, “Optical Stark effect in the two-photon spectrum of NO,” Phys. Rev. Lett. 54, 1012–1015 (1985).
[CrossRef] [PubMed]

Krasinski, J.

J. H. Eberly, J. Krasinski, in Advances in Multi-Photon Processes and Spectroscopy, S. H. Lin, ed. (Academic, Orlando, Fla., 1984), pp. 1–49.
[CrossRef]

Lambropoulos, P.

Mainfray, G.

G. Mainfray, C. Manus, in Multiphoton Ionization of Atoms, S. L. Chin, P. Lambropoulos, eds. (Academic, New York, 1984), pp. 7–34; Y. Gontier, M. Trahin, ibid., pp. 35–64.

Manus, C.

G. Mainfray, C. Manus, in Multiphoton Ionization of Atoms, S. L. Chin, P. Lambropoulos, eds. (Academic, New York, 1984), pp. 7–34; Y. Gontier, M. Trahin, ibid., pp. 35–64.

McKenzie, R. L.

W. M. Huo, K. P. Gross, R. L. McKenzie, “Optical Stark effect in the two-photon spectrum of NO,” Phys. Rev. Lett. 54, 1012–1015 (1985).
[CrossRef] [PubMed]

Morse, P. M.

For partial results, see P. M. Morse, H. Feshbach, Methods of Theoretical Physics I (McGraw-Hill, New York, 1953), p. 475, problem 4.28.

Tang, X.

Trahin, M.

Y. Gontier, M. Trahin, “Spatio-temporal effects in resonant multiphoton ionisation of the caesium atom,” J. Phys. B 13, 259–272 (1980).
[CrossRef]

Twomey, S.

S. Twomey, Developments in Geomathematics 3—Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).

van der Houwen, P. J.

H. Brunner, P. J. van der Houwen, The Numerical Solution of Volterra Equations (North-Holland, Amsterdam, 1986).

J. Comp. Phys. (1)

P. Dierckx, “An algorithm for experimental data deconvolution using spline functions,” J. Comp. Phys. 52, 163–186 (1983).
[CrossRef]

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

J. Phys. B (1)

Y. Gontier, M. Trahin, “Spatio-temporal effects in resonant multiphoton ionisation of the caesium atom,” J. Phys. B 13, 259–272 (1980).
[CrossRef]

Phys. Rev. Lett. (1)

W. M. Huo, K. P. Gross, R. L. McKenzie, “Optical Stark effect in the two-photon spectrum of NO,” Phys. Rev. Lett. 54, 1012–1015 (1985).
[CrossRef] [PubMed]

Other (10)

For partial results, see P. M. Morse, H. Feshbach, Methods of Theoretical Physics I (McGraw-Hill, New York, 1953), p. 475, problem 4.28.

G. R. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), Secs. 16.1 and 16.2.

S. Twomey, Developments in Geomathematics 3—Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).

R. L. Henning, C. D. Capps, G. M. Hess, Inversion Technique Evaluation, (Defense Technical Information Center, Alexandria, Va., 1982).

L. M. Delves, J. Walsh, eds., Numerical Solution of Integral Equations (Clarendon, Oxford, 1974).

C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).

G. Hämmerlin, K.-H. Hoffmann, eds., Constructive Methods for the Practical Treatment of Integral Equations, Vol. 73 of International Series of Numerical Mathematics (Birkhäuser, Basel, 1985).
[CrossRef]

H. Brunner, P. J. van der Houwen, The Numerical Solution of Volterra Equations (North-Holland, Amsterdam, 1986).

J. H. Eberly, J. Krasinski, in Advances in Multi-Photon Processes and Spectroscopy, S. H. Lin, ed. (Academic, Orlando, Fla., 1984), pp. 1–49.
[CrossRef]

G. Mainfray, C. Manus, in Multiphoton Ionization of Atoms, S. L. Chin, P. Lambropoulos, eds. (Academic, New York, 1984), pp. 7–34; Y. Gontier, M. Trahin, ibid., pp. 35–64.

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Figures (6)

Fig. 1
Fig. 1

(a) One- and (b) two-photon kernels versus relative intensity for exponential intensity profiles with shape parameter β.

Fig. 2
Fig. 2

Effect of EDF on the intensity dependence of the cross section. (a) Actual cross section. (b) EDF’s for five peak-intensities I0i, i = 1, …, 5 [Eq. (3.3) with n = 2 and β = 1.5]. (c) Experimental cross section, along with the actual cross section for comparison.

Fig. 3
Fig. 3

Minimum and maximum eigenvalues of direct-inversion matrix as a function of composite shape parameter β; number of data points is N: (a) one-photon and (b) two-photon excitation.

Fig. 4
Fig. 4

Eigenvalues of direct-inversion matrix versus central intensity of the corresponding basis vector for different numbers of data points (β = 1.5, n = 2).

Fig. 5
Fig. 5

Example of direct inversion. Experimental data are from Fig. 2, corrupted with 0.5% rms noise in 20 consecutive runs.

Fig. 6
Fig. 6

Effect of EDF on intensity-dependent line shape. (a) Actual line shape. (b) Experimental line shape. The projections onto the Iν planes correspond to data at zero-intensity (ν0) line center. Projections onto the Iν planes indicate the positions of shifted (open circles) and zero-intensity (filled circles) line center.

Equations (40)

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R ( I ) = σ n ( I ) I n .
I = I ( x ) I 0 F ( I 0 ; x ) .
S ( I 0 ) = S 0 P ( x ) R [ I ( x ) ] d 4 x .
V 4 ( I 0 ; I ) I 0 F ( l 0 ; x ) > I P ( x ) d 4 x ,
K n ( I 0 ; I ) - ( I / I 0 ) n I V 4 ( I 0 ; I ) / κ ( I 0 ) ,
0 I 0 K n ( I 0 ; I ) d I = 1.
R ^ ( I 0 ) I 0 n σ ^ n ( I 0 ) S ( I 0 ) / S 0 κ ( I 0 ) .
lim I 0 { R ^ ( I ) / R ( I ) } = lim I 0 { σ ^ n ( I ) / σ n ( I ) } = 1.
σ ^ n ( I 0 ) = 0 I 0 K n ( I 0 ; I ) σ n ( I ) d I .
V ˜ 4 ( f ) F ( x ) > f P ( x ) d 4 x = V 4 ( I 0 ; f I 0 ) ,
K ˜ n ( f ) - f n d d f V ˜ 4 ( f ) / κ = I 0 K n ( I 0 ; f I 0 ) ,
0 1 K ˜ n ( f ) d f = 1.
K n ¯ ( I 0 ¯ ; I ) = 0 P ( I 0 ¯ ; I 0 ) K n ( I 0 ; I ) d I 0 .
F ( x ) = i = 1 4 exp { - | x i a i ± | μ i } ,
P ( x ) = p ,
K ˜ n ( f ) = n β Γ ( β ) f n - 1 { ln 1 f } β - 1 ,
β i = 1 4 1 μ i .
κ = p n - β i = 1 4 { ( a i + + a i - ) Γ ( 1 + 1 / μ i ) } .
M i 0 1 K ˜ n ( f ) f i d f = ( 1 + i / n ) - β .
R ^ ( ω 0 ) = n β Γ ( β ) - ω 0 ( ω 0 - ω ) β - 1 R ( ω ) d ω ,
R ( ω ) = d d ω - ω ω ( ω - ω 0 ) - β ( d d ω 0 ) k R ^ ( ω 0 ) d ω 0 / n β Γ ( 1 - β ) .
R ( I ) = ( I d d I ) k 0 I d I 0 ( ln I I 0 ) - β d R ^ ( I 0 ) d I 0 / n β Γ ( 1 - β ) .
R ( I ) = ( I n d d I ) β R ^ ( I ) ,             β = integer .
n ^ ( I 0 ) d [ ln S ( I 0 ) ] d ( ln I 0 ) = d [ ln R ^ ( I 0 ) ] d ( ln I 0 )
n ( I ) d [ ln R ( I ) ] d ( ln I ) ,
n ( I ) = n ^ ( I ) + d [ ln A ( β ) ( I ) ] d ( ln I ) ,
A ( β ) ( I ) { n ^ ( I ) + I d d I } β 1.
σ ^ n ( I 0 ) = σ 0 ( 1 + γ ^ 1 I 0 + γ ^ 2 I 0 2 + ) .
σ n ( I 0 ) = σ 0 ( 1 + γ 1 I + γ 2 I 2 + ) ,
γ i = γ ^ i / M i ,             i = 1 , 2 , ,
M i 0 I 0 K n ( I 0 ; I ) ( I / I 0 ) i d I .
γ i = γ ^ i ( 1 + i / n ) β .
A i j 0 I 0 i K n ( I 0 i ; I ) ϕ j ( I ) d I = P ( x ) F h ( I 0 i ; x ) ϕ j [ I 0 i F ( I 0 i ; x ) ] d 4 x / κ ( I 0 i ) .
σ n ( I ) p = Λ T ( I ) Λ - 1 / 2 U T s ,
C k ( I ) k T ( I ) d I
ϕ j ( I ) = [ Λ - 1 / 2 U T k ( I ) ] j .
I j I ϕ j 2 ( I ) d I .
ν ^ c ( I 0 ) = ν 0 ( 1 + δ ^ 1 I 0 + δ ^ 2 I 0 2 + ) .
ν c ( I ) = ν 0 ( 1 + δ 1 I + δ 2 I 2 + ) .
δ i = δ ^ i / M i .

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