Abstract

In laser spectroscopy, saturation of atomic or molecular transitions cannot be ignored, even at modest laser intensities. The saturation status is customarily diagnosed from measurements of saturation curves describing the dependence of spectroscopic signals on laser intensity. I propose an alternative method that relies on a geometric comparison of the spatial laser profile with images of the spectroscopic quantity under investigation. A single image can be used to determine the saturation status and its associated saturation laser intensity.

© 2005 Optical Society of America

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References

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  1. W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 2003).
    [CrossRef]
  2. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Gordon & Breach, Amsterdam, 1996).
  3. J. W. Daily, “Saturation of fluorescence in flames with a Gaussian laser beam,” Appl. Opt. 17, 225–229 (1978).
    [CrossRef] [PubMed]
  4. J. W. Daily, “Laser-induced fluorescence spectroscopy in flames,” Prog. Energy Combust. Sci. 23, 133–199 (1997).
    [CrossRef]
  5. W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
    [CrossRef] [PubMed]
  6. C. Xu, W. W. Webb, “Measurements of two-photon excitation cross sections of molecular fluorophores with data from 690 to 1050 nm,” J. Opt. Soc. Am. B 13, 481–491 (1996).
    [CrossRef]
  7. M. Marrocco, “Spatial laser-wing suppression in saturated laser-induced fluorescence without spatial discrimination,” Opt. Lett. 28, 2016–2018 (2003).
    [CrossRef] [PubMed]
  8. R. L. Swofford, W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975).
    [CrossRef]
  9. T. Plakhotnik, D. Walser, M. Pirotta, A. Renn, U. P. Wild, “Nonlinear spectroscopy on a single quantum system: two-photon absorption of a single molecule,” Science 271, 1703–1705 (1996).
    [CrossRef]
  10. See the polylogarithmic function at http://mathworld.wolfram.com/Polylogarithm.html .
  11. S. Laporta, E. Remiddi, “The analytical value of the electron (g − 2) at order α3 in QED,” Phys. Lett. B 379, 283–291 (1996).
    [CrossRef]
  12. A. Marcano O., I. Urdaneta, “Fluorescence quantum yield of Rhodamine 101 in the presence of absorption saturation,” Appl. Phys. B 72, 207–213 (2001).
    [CrossRef]
  13. T. F. Johnston, R. H. Brady, W. Proffitt, “Powerful single-frequency ring dye laser spanning the visible spectrum,” Appl. Opt. 21, 2307–2316 (1982).
    [CrossRef] [PubMed]
  14. M. Marrocco, M. D'Apice, S. Giammartini, M. Magaldi, G. P. Romano, “Measurements of molecular carbon radical concentrations by saturated laser-induced fluorescence in hydrocarbon flames at atmospheric pressure,” in Laser Applications in Medicine, Biology, and Environmental Science, G. Mueller, V. V. Tuchin, G. G. Matvienko, C. Werner, V. Y. Panchenko, eds., Proc. SPIE5149, 187–196 (2002).
    [CrossRef]
  15. M. Marrocco, “An alternative approach to temporal laser-wing effects in saturated laser-induced fluorescence,” Appl. Phys. B 77, 65–70 (2003).
    [CrossRef]
  16. B. J. Kirby, R. K. Hanson, “CO2 imaging with saturated planar laser-induced vibrational fluorescence,” Appl. Opt. 40, 6136–6144 (2001).
    [CrossRef]
  17. See polylogarithm integral representations at http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/07/01/01/ .

2003 (2)

M. Marrocco, “Spatial laser-wing suppression in saturated laser-induced fluorescence without spatial discrimination,” Opt. Lett. 28, 2016–2018 (2003).
[CrossRef] [PubMed]

M. Marrocco, “An alternative approach to temporal laser-wing effects in saturated laser-induced fluorescence,” Appl. Phys. B 77, 65–70 (2003).
[CrossRef]

2001 (2)

B. J. Kirby, R. K. Hanson, “CO2 imaging with saturated planar laser-induced vibrational fluorescence,” Appl. Opt. 40, 6136–6144 (2001).
[CrossRef]

A. Marcano O., I. Urdaneta, “Fluorescence quantum yield of Rhodamine 101 in the presence of absorption saturation,” Appl. Phys. B 72, 207–213 (2001).
[CrossRef]

1997 (1)

J. W. Daily, “Laser-induced fluorescence spectroscopy in flames,” Prog. Energy Combust. Sci. 23, 133–199 (1997).
[CrossRef]

1996 (3)

C. Xu, W. W. Webb, “Measurements of two-photon excitation cross sections of molecular fluorophores with data from 690 to 1050 nm,” J. Opt. Soc. Am. B 13, 481–491 (1996).
[CrossRef]

T. Plakhotnik, D. Walser, M. Pirotta, A. Renn, U. P. Wild, “Nonlinear spectroscopy on a single quantum system: two-photon absorption of a single molecule,” Science 271, 1703–1705 (1996).
[CrossRef]

S. Laporta, E. Remiddi, “The analytical value of the electron (g − 2) at order α3 in QED,” Phys. Lett. B 379, 283–291 (1996).
[CrossRef]

1990 (1)

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

1982 (1)

1978 (1)

1975 (1)

R. L. Swofford, W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975).
[CrossRef]

Brady, R. H.

Daily, J. W.

J. W. Daily, “Laser-induced fluorescence spectroscopy in flames,” Prog. Energy Combust. Sci. 23, 133–199 (1997).
[CrossRef]

J. W. Daily, “Saturation of fluorescence in flames with a Gaussian laser beam,” Appl. Opt. 17, 225–229 (1978).
[CrossRef] [PubMed]

D'Apice, M.

M. Marrocco, M. D'Apice, S. Giammartini, M. Magaldi, G. P. Romano, “Measurements of molecular carbon radical concentrations by saturated laser-induced fluorescence in hydrocarbon flames at atmospheric pressure,” in Laser Applications in Medicine, Biology, and Environmental Science, G. Mueller, V. V. Tuchin, G. G. Matvienko, C. Werner, V. Y. Panchenko, eds., Proc. SPIE5149, 187–196 (2002).
[CrossRef]

Demtröder, W.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 2003).
[CrossRef]

Denk, W.

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Gordon & Breach, Amsterdam, 1996).

Giammartini, S.

M. Marrocco, M. D'Apice, S. Giammartini, M. Magaldi, G. P. Romano, “Measurements of molecular carbon radical concentrations by saturated laser-induced fluorescence in hydrocarbon flames at atmospheric pressure,” in Laser Applications in Medicine, Biology, and Environmental Science, G. Mueller, V. V. Tuchin, G. G. Matvienko, C. Werner, V. Y. Panchenko, eds., Proc. SPIE5149, 187–196 (2002).
[CrossRef]

Hanson, R. K.

Johnston, T. F.

Kirby, B. J.

Laporta, S.

S. Laporta, E. Remiddi, “The analytical value of the electron (g − 2) at order α3 in QED,” Phys. Lett. B 379, 283–291 (1996).
[CrossRef]

Magaldi, M.

M. Marrocco, M. D'Apice, S. Giammartini, M. Magaldi, G. P. Romano, “Measurements of molecular carbon radical concentrations by saturated laser-induced fluorescence in hydrocarbon flames at atmospheric pressure,” in Laser Applications in Medicine, Biology, and Environmental Science, G. Mueller, V. V. Tuchin, G. G. Matvienko, C. Werner, V. Y. Panchenko, eds., Proc. SPIE5149, 187–196 (2002).
[CrossRef]

Marcano O., A.

A. Marcano O., I. Urdaneta, “Fluorescence quantum yield of Rhodamine 101 in the presence of absorption saturation,” Appl. Phys. B 72, 207–213 (2001).
[CrossRef]

Marrocco, M.

M. Marrocco, “An alternative approach to temporal laser-wing effects in saturated laser-induced fluorescence,” Appl. Phys. B 77, 65–70 (2003).
[CrossRef]

M. Marrocco, “Spatial laser-wing suppression in saturated laser-induced fluorescence without spatial discrimination,” Opt. Lett. 28, 2016–2018 (2003).
[CrossRef] [PubMed]

M. Marrocco, M. D'Apice, S. Giammartini, M. Magaldi, G. P. Romano, “Measurements of molecular carbon radical concentrations by saturated laser-induced fluorescence in hydrocarbon flames at atmospheric pressure,” in Laser Applications in Medicine, Biology, and Environmental Science, G. Mueller, V. V. Tuchin, G. G. Matvienko, C. Werner, V. Y. Panchenko, eds., Proc. SPIE5149, 187–196 (2002).
[CrossRef]

McClain, W. M.

R. L. Swofford, W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975).
[CrossRef]

Pirotta, M.

T. Plakhotnik, D. Walser, M. Pirotta, A. Renn, U. P. Wild, “Nonlinear spectroscopy on a single quantum system: two-photon absorption of a single molecule,” Science 271, 1703–1705 (1996).
[CrossRef]

Plakhotnik, T.

T. Plakhotnik, D. Walser, M. Pirotta, A. Renn, U. P. Wild, “Nonlinear spectroscopy on a single quantum system: two-photon absorption of a single molecule,” Science 271, 1703–1705 (1996).
[CrossRef]

Proffitt, W.

Remiddi, E.

S. Laporta, E. Remiddi, “The analytical value of the electron (g − 2) at order α3 in QED,” Phys. Lett. B 379, 283–291 (1996).
[CrossRef]

Renn, A.

T. Plakhotnik, D. Walser, M. Pirotta, A. Renn, U. P. Wild, “Nonlinear spectroscopy on a single quantum system: two-photon absorption of a single molecule,” Science 271, 1703–1705 (1996).
[CrossRef]

Romano, G. P.

M. Marrocco, M. D'Apice, S. Giammartini, M. Magaldi, G. P. Romano, “Measurements of molecular carbon radical concentrations by saturated laser-induced fluorescence in hydrocarbon flames at atmospheric pressure,” in Laser Applications in Medicine, Biology, and Environmental Science, G. Mueller, V. V. Tuchin, G. G. Matvienko, C. Werner, V. Y. Panchenko, eds., Proc. SPIE5149, 187–196 (2002).
[CrossRef]

Strickler, J. H.

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Swofford, R. L.

R. L. Swofford, W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975).
[CrossRef]

Urdaneta, I.

A. Marcano O., I. Urdaneta, “Fluorescence quantum yield of Rhodamine 101 in the presence of absorption saturation,” Appl. Phys. B 72, 207–213 (2001).
[CrossRef]

Walser, D.

T. Plakhotnik, D. Walser, M. Pirotta, A. Renn, U. P. Wild, “Nonlinear spectroscopy on a single quantum system: two-photon absorption of a single molecule,” Science 271, 1703–1705 (1996).
[CrossRef]

Webb, W. W.

Wild, U. P.

T. Plakhotnik, D. Walser, M. Pirotta, A. Renn, U. P. Wild, “Nonlinear spectroscopy on a single quantum system: two-photon absorption of a single molecule,” Science 271, 1703–1705 (1996).
[CrossRef]

Xu, C.

Appl. Opt. (3)

Appl. Phys. B (2)

M. Marrocco, “An alternative approach to temporal laser-wing effects in saturated laser-induced fluorescence,” Appl. Phys. B 77, 65–70 (2003).
[CrossRef]

A. Marcano O., I. Urdaneta, “Fluorescence quantum yield of Rhodamine 101 in the presence of absorption saturation,” Appl. Phys. B 72, 207–213 (2001).
[CrossRef]

Chem. Phys. Lett. (1)

R. L. Swofford, W. M. McClain, “The effect of spatial and temporal laser beam characteristics on two-photon absorption,” Chem. Phys. Lett. 34, 455–460 (1975).
[CrossRef]

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

Opt. Lett. (1)

Phys. Lett. B (1)

S. Laporta, E. Remiddi, “The analytical value of the electron (g − 2) at order α3 in QED,” Phys. Lett. B 379, 283–291 (1996).
[CrossRef]

Prog. Energy Combust. Sci. (1)

J. W. Daily, “Laser-induced fluorescence spectroscopy in flames,” Prog. Energy Combust. Sci. 23, 133–199 (1997).
[CrossRef]

Science (2)

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

T. Plakhotnik, D. Walser, M. Pirotta, A. Renn, U. P. Wild, “Nonlinear spectroscopy on a single quantum system: two-photon absorption of a single molecule,” Science 271, 1703–1705 (1996).
[CrossRef]

Other (5)

See the polylogarithmic function at http://mathworld.wolfram.com/Polylogarithm.html .

W. Demtröder, Laser Spectroscopy (Springer-Verlag, Berlin, 2003).
[CrossRef]

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species (Gordon & Breach, Amsterdam, 1996).

M. Marrocco, M. D'Apice, S. Giammartini, M. Magaldi, G. P. Romano, “Measurements of molecular carbon radical concentrations by saturated laser-induced fluorescence in hydrocarbon flames at atmospheric pressure,” in Laser Applications in Medicine, Biology, and Environmental Science, G. Mueller, V. V. Tuchin, G. G. Matvienko, C. Werner, V. Y. Panchenko, eds., Proc. SPIE5149, 187–196 (2002).
[CrossRef]

See polylogarithm integral representations at http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/07/01/01/ .

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Figures (10)

Fig. 1
Fig. 1

Total signal Σ (normalized to its maximum) as a function of the coordinate y for z = 0 and n = 1. Five values of μ = I0/Isat = 0.1, 1, 10, 100, 1000 are considered. The laser profile is derived from Eq. (3) and coincides with Eq. (4).

Fig. 2
Fig. 2

Saturation parameter μ as a function of the spatial width δ normalized to the beam waist w0 of the Gaussian laser beam. The values of δ are determined for X(y, 0) = 2(0, 0)exp(−1). Note that the ratio δ/w0 = 2−1/2 ≅ 0.707 for μ → 0 because of the different definition of w0.

Fig. 3
Fig. 3

Measured laser profile (symbols) and its best Gaussian fit (solid curve). The relative deviation η between the experimental points and best fit is discussed in the text. The laser wings where the deviation is the largest are indicated in the upper plot.

Fig. 4
Fig. 4

Measured laser profile (symbols) and new best Gaussian fit (solid curve) based on the decomposition discussed in the text. The dashed curve corresponds to the background Λb. The relative deviation η between the experimental points and best fit is discussed in the text and does not show the pronounced and broad peaks of Fig. 3.

Fig. 5
Fig. 5

LIF images of Rh101 for (a) Ep = 0.3 µJ, (b) Ep = 0.12 µJ, and (c) Ep = 0.54 µJ. The two vertical dotted lines mark the limits of the inner length of the quartz cuvette. Outside, the observable residual fluorescence light is given by reflection of the cuvette walls. An arrow denotes the z position used to extract the y profiles.

Fig. 6
Fig. 6

Experimental profiles taken from Fig. 5. The laser profile is shown for comparison. The dashed curve refers to the background function Λb found for Ep = 0.12 µJ. The vertical dotted lines mark the y limits outside which the background was calculated.

Fig. 7
Fig. 7

Corrected experimental profiles of Fig. 5. The curves are drawn according to Eq. (5).

Fig. 8
Fig. 8

Comparison between Fig. 2 and experimental results.

Fig. 9
Fig. 9

Rayleigh and LIF images: (a) measured Rayleigh image of the laser beam, (b) measured LIF image, (c) calculated Rayleigh image of the laser beam, (d) calculated LIF image. The laser beam propagates from left to right and is focused to ≈300. The arrow indicates the horizontal position of the z coordinate used in Fig. 10.

Fig. 10
Fig. 10

Comparison between the measured Rayleigh (open circles) and LIF (solid circles) profile for the z coordinate chosen from Fig. 9. The calculated profiles are from Eq. (3) (dashed curve) and Eq. (5) (solid curve) as discussed in the text. The horizontal axis refers to the pixel number of Fig. 9. The two horizontal lines show the heights used for the measurements of spatial parameters w0 and δ.

Equations (11)

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( y , z ) = k t 0 t 0 + T line of sight S [ ξ , t , I ( r , t ) , I sat ] d x d t .
I ( r , t ) = 2 P ( t ) π w 2 ( z ) exp [ 2 ( x 2 + y 2 ) / w 2 ( z ) ] ,
Λ ( y , z ) = k 0 t 0 t 0 + T line of sight I ( r , t ) d x d t = k 0 g 0 I 0 w 0 ( π 2 ) 1 / 2 exp [ 2 y 2 / w 2 ( z ) ] [ 1 + ( z / z 0 ) 2 ] 1 / 2 ,
I 0 I sat ( y , z ) = 0 exp [ 2 n y 2 / w 2 ( z ) ] [ 1 + ( z / z 0 ) 2 ] n 1 / 2 ,
( y , z ) = 0 T [ 1 + ( z + z 0 ) 2 ] 1 / 2 µ n g 0 , n × L i 1 / 2 { µ n exp [ 2 n y 2 / w 2 ( z ) ] [ 1 + ( z / z 0 ) 2 ] n } ,
δ w 0 0.76 + 0.32 log ( μ + 0.61 ) .
( y , z ) = k α I sat n T × line of sight 1 1 + [ I sat / I 0 ( z ) ] n exp [ 2 n ( x 2 + y 2 ) / w 2 ( z ) ] d x ,
( y , z ) = k α I sat n T w ( z ) ( 2 n ) 1 / 2 h ( y , z ) 0 q 1 / 2 exp ( q ) + h ( y , z ) d q ,
L i 1 / 2 { ψ } = ψ π 1 / 2 0 q 1 / 2 exp ( q ) ψ d q ,
( y , z ) = π 1 / 2 k α I sat n T w ( z ) ( 2 n ) 1 / 2 L i 1 / 2 [ h ( y , z ) ] ;
( y , z ) = 0 T [ 1 + ( z / z 0 ) 2 ] 1 / 2 µ n g 0 , n × L i 1 / 2 { µ n exp [ 2 n y 2 / w 2 ( z ) ] [ 1 + ( z / z 0 ) 2 ] n } ,

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