Abstract

A simple theory is generated for crossed-beam thermal lens and photothermal deflection in flowing sample streams. This theory is used to analyze time-resolved crossed-beam thermal lens data. The result of the regression analysis may be used as a good probe of flow velocity in the 0.2–30-mm/sec range in liquid streams. Faster flow velocity should be measurable in gas streams.

© 1985 Optical Society of America

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References

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  1. W. H. Stevenson, “Principles of Laser Velocimetry,” in Experimental Diagnostics in Gas Phase Combustion Systems,” B. T. Zinn, Ed. Prog. Astronaut. Aeronaut.53, 307 (1977).
  2. H. Mishina, T. Asakura, S. Nagai, “A Laser Doppler Microscope,” Opt. Commun. 11, 99 (1974).
    [CrossRef]
  3. G. C. Herring, W. M. Fairbank, C. Y. She, “Observation and Measurement of Molecular Flow Using Stimulated Raman Gain Spectroscopy,” IEEE J. Quantum. eElectron. QE-17, 1975 (1981).
    [CrossRef]
  4. H. Moosmuller, G. C. Herring, C.-Y. She, “Two-Component Velocity Measurements in a Supersonic Nitrogen Jet with Spatially Resolved Inverse Raman Spectroscopy,” Opt. Lett. 9, 536 (1984).
    [CrossRef] [PubMed]
  5. N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory for Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1770 (1984).
    [CrossRef]
  6. T. G. Nolan, W. A. Weimer, N. J. Dovichi, “Laser-Induced Photothermal Refraction for Small Volume Analysis,” Anal. Chem. 56, 1773 (1984).
    [CrossRef]
  7. T. G. Nolan, N. J. Dovichi, “Crossed-Beam Thermal Lens Measurements for Ultrasensitive Iron Analysis,” IEEE Circuits Devices Mag. submitted.
  8. D. S. Burgi, T. G. Nolan, J. A. Risfelt, N. J. Dovichi, “Photothermal Refraction for Scanning Laser Microscopy,” Opt. Eng. 23, 756 (1984).
    [CrossRef]
  9. W. A. Weimer, N. J. Dovichi, “Time Dependent Crossed-Beam Thermal Lens Measurements,” J. Appl. Phys. submitted.
  10. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spectroscopy and Detection,” Appl. Opt. 20, 1333 (1982).
    [CrossRef]
  11. G. C. Wetsel, S. A. Stotts, “Absolute Measurement of Optical Attenuation,” Appl. Phys. Lett. 42, 931 (1983).
    [CrossRef]
  12. J. A. Sell, “Quantitative Photothermal Deflection Spectroscopy in a Flowing Stream of Gas,” Appl. Opt. 23, 1586 (1984).
    [CrossRef] [PubMed]
  13. A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming,” Chem. Phys. 20, 253 (1977).
    [CrossRef]
  14. N. J. Dovichi, J. M. Harris, “Time-Resolved Thermal Lens Calorimetry,” Anal. Chem. 53, 106 (1981).
    [CrossRef]

1984

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory for Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1770 (1984).
[CrossRef]

T. G. Nolan, W. A. Weimer, N. J. Dovichi, “Laser-Induced Photothermal Refraction for Small Volume Analysis,” Anal. Chem. 56, 1773 (1984).
[CrossRef]

D. S. Burgi, T. G. Nolan, J. A. Risfelt, N. J. Dovichi, “Photothermal Refraction for Scanning Laser Microscopy,” Opt. Eng. 23, 756 (1984).
[CrossRef]

J. A. Sell, “Quantitative Photothermal Deflection Spectroscopy in a Flowing Stream of Gas,” Appl. Opt. 23, 1586 (1984).
[CrossRef] [PubMed]

H. Moosmuller, G. C. Herring, C.-Y. She, “Two-Component Velocity Measurements in a Supersonic Nitrogen Jet with Spatially Resolved Inverse Raman Spectroscopy,” Opt. Lett. 9, 536 (1984).
[CrossRef] [PubMed]

1983

G. C. Wetsel, S. A. Stotts, “Absolute Measurement of Optical Attenuation,” Appl. Phys. Lett. 42, 931 (1983).
[CrossRef]

1982

1981

G. C. Herring, W. M. Fairbank, C. Y. She, “Observation and Measurement of Molecular Flow Using Stimulated Raman Gain Spectroscopy,” IEEE J. Quantum. eElectron. QE-17, 1975 (1981).
[CrossRef]

N. J. Dovichi, J. M. Harris, “Time-Resolved Thermal Lens Calorimetry,” Anal. Chem. 53, 106 (1981).
[CrossRef]

1977

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming,” Chem. Phys. 20, 253 (1977).
[CrossRef]

1974

H. Mishina, T. Asakura, S. Nagai, “A Laser Doppler Microscope,” Opt. Commun. 11, 99 (1974).
[CrossRef]

Amer, N. M.

Asakura, T.

H. Mishina, T. Asakura, S. Nagai, “A Laser Doppler Microscope,” Opt. Commun. 11, 99 (1974).
[CrossRef]

Boccara, A. C.

Burgi, D. S.

D. S. Burgi, T. G. Nolan, J. A. Risfelt, N. J. Dovichi, “Photothermal Refraction for Scanning Laser Microscopy,” Opt. Eng. 23, 756 (1984).
[CrossRef]

Dovichi, N. J.

T. G. Nolan, W. A. Weimer, N. J. Dovichi, “Laser-Induced Photothermal Refraction for Small Volume Analysis,” Anal. Chem. 56, 1773 (1984).
[CrossRef]

D. S. Burgi, T. G. Nolan, J. A. Risfelt, N. J. Dovichi, “Photothermal Refraction for Scanning Laser Microscopy,” Opt. Eng. 23, 756 (1984).
[CrossRef]

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory for Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1770 (1984).
[CrossRef]

N. J. Dovichi, J. M. Harris, “Time-Resolved Thermal Lens Calorimetry,” Anal. Chem. 53, 106 (1981).
[CrossRef]

T. G. Nolan, N. J. Dovichi, “Crossed-Beam Thermal Lens Measurements for Ultrasensitive Iron Analysis,” IEEE Circuits Devices Mag. submitted.

W. A. Weimer, N. J. Dovichi, “Time Dependent Crossed-Beam Thermal Lens Measurements,” J. Appl. Phys. submitted.

Fairbank, W. M.

G. C. Herring, W. M. Fairbank, C. Y. She, “Observation and Measurement of Molecular Flow Using Stimulated Raman Gain Spectroscopy,” IEEE J. Quantum. eElectron. QE-17, 1975 (1981).
[CrossRef]

Fournier, D.

Harris, J. M.

N. J. Dovichi, J. M. Harris, “Time-Resolved Thermal Lens Calorimetry,” Anal. Chem. 53, 106 (1981).
[CrossRef]

Herring, G. C.

H. Moosmuller, G. C. Herring, C.-Y. She, “Two-Component Velocity Measurements in a Supersonic Nitrogen Jet with Spatially Resolved Inverse Raman Spectroscopy,” Opt. Lett. 9, 536 (1984).
[CrossRef] [PubMed]

G. C. Herring, W. M. Fairbank, C. Y. She, “Observation and Measurement of Molecular Flow Using Stimulated Raman Gain Spectroscopy,” IEEE J. Quantum. eElectron. QE-17, 1975 (1981).
[CrossRef]

Jackson, W. B.

Kliger, D. S.

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming,” Chem. Phys. 20, 253 (1977).
[CrossRef]

Mishina, H.

H. Mishina, T. Asakura, S. Nagai, “A Laser Doppler Microscope,” Opt. Commun. 11, 99 (1974).
[CrossRef]

Moosmuller, H.

Nagai, S.

H. Mishina, T. Asakura, S. Nagai, “A Laser Doppler Microscope,” Opt. Commun. 11, 99 (1974).
[CrossRef]

Nolan, T. G.

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory for Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1770 (1984).
[CrossRef]

D. S. Burgi, T. G. Nolan, J. A. Risfelt, N. J. Dovichi, “Photothermal Refraction for Scanning Laser Microscopy,” Opt. Eng. 23, 756 (1984).
[CrossRef]

T. G. Nolan, W. A. Weimer, N. J. Dovichi, “Laser-Induced Photothermal Refraction for Small Volume Analysis,” Anal. Chem. 56, 1773 (1984).
[CrossRef]

T. G. Nolan, N. J. Dovichi, “Crossed-Beam Thermal Lens Measurements for Ultrasensitive Iron Analysis,” IEEE Circuits Devices Mag. submitted.

Risfelt, J. A.

D. S. Burgi, T. G. Nolan, J. A. Risfelt, N. J. Dovichi, “Photothermal Refraction for Scanning Laser Microscopy,” Opt. Eng. 23, 756 (1984).
[CrossRef]

Sell, J. A.

She, C. Y.

G. C. Herring, W. M. Fairbank, C. Y. She, “Observation and Measurement of Molecular Flow Using Stimulated Raman Gain Spectroscopy,” IEEE J. Quantum. eElectron. QE-17, 1975 (1981).
[CrossRef]

She, C.-Y.

Stevenson, W. H.

W. H. Stevenson, “Principles of Laser Velocimetry,” in Experimental Diagnostics in Gas Phase Combustion Systems,” B. T. Zinn, Ed. Prog. Astronaut. Aeronaut.53, 307 (1977).

Stotts, S. A.

G. C. Wetsel, S. A. Stotts, “Absolute Measurement of Optical Attenuation,” Appl. Phys. Lett. 42, 931 (1983).
[CrossRef]

Twarowski, A. J.

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming,” Chem. Phys. 20, 253 (1977).
[CrossRef]

Weimer, W. A.

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory for Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1770 (1984).
[CrossRef]

T. G. Nolan, W. A. Weimer, N. J. Dovichi, “Laser-Induced Photothermal Refraction for Small Volume Analysis,” Anal. Chem. 56, 1773 (1984).
[CrossRef]

W. A. Weimer, N. J. Dovichi, “Time Dependent Crossed-Beam Thermal Lens Measurements,” J. Appl. Phys. submitted.

Wetsel, G. C.

G. C. Wetsel, S. A. Stotts, “Absolute Measurement of Optical Attenuation,” Appl. Phys. Lett. 42, 931 (1983).
[CrossRef]

Anal. Chem.

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory for Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1770 (1984).
[CrossRef]

T. G. Nolan, W. A. Weimer, N. J. Dovichi, “Laser-Induced Photothermal Refraction for Small Volume Analysis,” Anal. Chem. 56, 1773 (1984).
[CrossRef]

N. J. Dovichi, J. M. Harris, “Time-Resolved Thermal Lens Calorimetry,” Anal. Chem. 53, 106 (1981).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

G. C. Wetsel, S. A. Stotts, “Absolute Measurement of Optical Attenuation,” Appl. Phys. Lett. 42, 931 (1983).
[CrossRef]

Chem. Phys.

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming,” Chem. Phys. 20, 253 (1977).
[CrossRef]

IEEE J. Quantum. eElectron.

G. C. Herring, W. M. Fairbank, C. Y. She, “Observation and Measurement of Molecular Flow Using Stimulated Raman Gain Spectroscopy,” IEEE J. Quantum. eElectron. QE-17, 1975 (1981).
[CrossRef]

Opt. Commun.

H. Mishina, T. Asakura, S. Nagai, “A Laser Doppler Microscope,” Opt. Commun. 11, 99 (1974).
[CrossRef]

Opt. Eng.

D. S. Burgi, T. G. Nolan, J. A. Risfelt, N. J. Dovichi, “Photothermal Refraction for Scanning Laser Microscopy,” Opt. Eng. 23, 756 (1984).
[CrossRef]

Opt. Lett.

Other

W. A. Weimer, N. J. Dovichi, “Time Dependent Crossed-Beam Thermal Lens Measurements,” J. Appl. Phys. submitted.

W. H. Stevenson, “Principles of Laser Velocimetry,” in Experimental Diagnostics in Gas Phase Combustion Systems,” B. T. Zinn, Ed. Prog. Astronaut. Aeronaut.53, 307 (1977).

T. G. Nolan, N. J. Dovichi, “Crossed-Beam Thermal Lens Measurements for Ultrasensitive Iron Analysis,” IEEE Circuits Devices Mag. submitted.

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Figures (8)

Fig. 1
Fig. 1

Photothermal refraction. The pump beam propagates along the y axis, the probe beam propagates along the x axis, and the flow is in the z direction. The signal is detected as a change in the far-field probe beam center intensity.

Fig. 2
Fig. 2

Temperature distribution as a function of distance from the pump beam center in units of pump beam spot size for flowing samples. The family of curves is for time after the pump laser pulse in units of time constant. Impulse excitation is used to generate the temperature distribution.

Fig. 3
Fig. 3

Time-resolved photothermal deflection angle in a flowing tream for impulse excitation. The pump beam is located an offset Z = ½ω upstream from the probe beam. The flow velocity is in units of pump beam spot size divided by time constant, V = w/tc.

Fig. 4
Fig. 4

Impulse-response time-resolved photothermal refraction focal length in a flowing stream. The pump and probe beam axes intersect. The flow velocity is in units of pump beam spot size divided by time constant V = w/tc.

Fig. 5
Fig. 5

Trapezoidal excitation function. t1 is the time necessary to unblock the pump beam, and t2 is one-half of the chopping period.

Fig. 6
Fig. 6

Time-resolved photothermal refraction data in a flowing stream. A periodically chopped pump beam is used as the excitation source. The data are shown at the 68% confidence interval. The least-squares fit of Eq. (12) to the data is shown as the smooth curve: (a) static sample; (b) flow velocity = 1.9 mm/sec; (c) flow velocity = 3.8 mm/sec; (d) flow velocity = 7.6 mm/sec; (e) flow velocity = 14 mm/sec. The same axes scales are used in all plots.

Fig. 7
Fig. 7

Plot of measured flow rate vs predicted flow rate from the regression analysis. Flow rate was the only parameter used in the analysis.

Fig. 8
Fig. 8

Plot of flow rate determined from the time-resolved photothermal refraction regression analysis vs distance across a flowing stream in a square duct. The smooth curve is the least-squares fit of a parabola to the data.

Equations (13)

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Δ T = 2 . 303 E C 2 π k t c ( 1 + 2 t / t c ) exp [ 2 ( X 2 + Z 2 ) / ω 2 1 + 2 t / t c ] ,
t c = ω 2 C p ρ 4 k ,
Δ T = 2 . 303 E C 2 π k t c ( 1 + 2 t / t c ) exp { 2 [ X 2 + ( Z + V t ) 2 ] / ω 2 1 + 2 t / t c } ,
Φ = d n d T path Δ T Z d X ,
Φ = 4 . 606 E C d n / d T ( Z + V t ) ( 2 π ) 1 / 2 k t c ω ( 1 + 2 t / t c ) 3 / 2 × exp [ 2 ( Z + V t ) 2 / ω 2 1 + 2 t / t c ] .
offset = Φ X 2 ,
Δ I = exp ( 2 offset 2 / ω 2 ) .
1 / f i = d n / d T path 2 Δ T Z 2 d X .
1 / f i = 4 . 606 E C d n / d T ( 2 π ) 1 / 2 k t c ω ( 1 + 2 t / t c ) 3 / 2 [ 1 4 ( Z + V t ) / ω 2 1 + 2 t / t c ] × exp [ 2 ( Z + V t ) 2 / ω 2 1 + 2 t / t c ] .
Δ I = Z 1 / f i ,
Δ I i ( t ) = 4 . 606 E C d n / d T ( 2 π ) 1 / 2 k t c ω ( 1 + 2 t / t c ) 3 / 2 [ 1 4 ( Z + V t ) / ω 2 1 + 2 t / t c ] × exp [ 2 ( Z + V t ) 2 / ω 2 1 + 2 t / t c ] .
= 0 t Δ I i ( τ ) ( t τ t 1 ) d τ + ξ 0 < t < t 1 , = 0 t Δ I i ( τ ) d τ + t t 1 t Δ I i ( τ ) ( t τ t 1 ) d τ + ξ t 1 < t < t 2 , Δ I trap ( t ) = 0 t t 2 Δ I i ( τ ) ( t 2 + t 1 + τ t t 1 ) d τ + t t 2 t t 1 Δ I i ( τ ) d τ + t t 1 t Δ I i ( τ ) ( t τ t 1 ) d τ + ξ t 2 < t < t 1 + t 2 , = t t 1 t 2 t t 2 Δ I i ( τ ) ( t 2 + t 1 + τ t t 1 ) d τ + t t 2 t t 1 Δ I i ( τ ) d τ + t t 1 t Δ I i ( τ ) ( t τ t 1 ) d τ + ξ t 1 + t 2 < t ,
ξ = n = 1 [ t + ( 2 n 1 ) t 2 t 1 t + ( 2 n 1 ) t 2 Δ I i ( τ ) ( t 2 + t 1 + τ t t 1 ) d τ + t + ( 2 n 1 ) t 2 t + 2 n t 2 t 1 Δ I i ( τ ) d τ + t + 2 n t 2 t 1 t + 2 n t 2 Δ I i ( τ ) ( t τ t 1 ) d τ ] .

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