Abstract

A theoretical treatment of photothermal phase shift spectroscopy in a fluid medium for the most general conditions is given. The medium is assumed to be flowing. Results for a stationary medium appear as a special case. Both pulsed and cw excitation are considered. For pulsed excitation, the results are valid for excitation pulses of arbitrary length. For the cw case, modulated excitation is explicitly considered, and the results for unmodulated excitation appear as a special case.

© 1989 Optical Society of America

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References

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  1. J. A. Sell, Photothermal Investigations of Solids and Fluids (Academic, New York, 1988).
  2. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spectroscopy and Detection,” Appl. Opt. 20, 1333–1344 (1981).
    [CrossRef] [PubMed]
  3. For a review see R. Gupta, “Pulsed Photothemal Deflection Spectroscopy in Fluid Media: A Review,” in Proceedings, International Conference on Lasers ’86, R. W. McMillan, Ed. (STS Press, McLean, VA, 1987).
  4. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
    [CrossRef]
  5. For a review see H. L. Fang, R. L. Swofford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983).
  6. R. Vyas, R. Gupta, “Photothermal Lensing Spectroscopy in a Flowing Medium: Theory,” Appl. Opt. 27, 4701–4711 (1988). This article contains references to other recent articles.
    [CrossRef] [PubMed]
  7. J. Stone, “Thermooptical Technique for the Measurement of Absorption Loss Spectrum in Liquids,” Appl. Opt. 12, 1828–1830 (1973).
    [CrossRef] [PubMed]
  8. C. C. Davis, “Trace Detection of Gases Using Phase Fluctuation Optical Heterodyne Spectroscopy,” Appl. Phys. Lett. 36, 515–518 (1980).
    [CrossRef]
  9. C. C. Davis, S. J. Petuchowski, “Phase Fluctuation Optical Heterodyne Spectroscopy of Gases,” Appl. Opt. 20, 2539–2554 (1981).
    [CrossRef] [PubMed]
  10. C. C. Davis, “Radial Laser-induced Soundwave Propagation and Vibrational Relaxation in Carbon Dioxide,” IEEE J. Quantum Electron. QE-18, 999–1003 (1982).
    [CrossRef]
  11. A. J. Campillo, H.-B. Lin, C. J. Dodge, C. C. Davis, “Stark-Effect-Modulated Phase-Fluctuation Optical Heterodyne Interferometer for Trace-Gas Analysis,” Opt. Lett. 5, 424–426 (1980); S. J. Petuchowski, C. C. Davis, “Selective Trace Detection of Asymmetric Rotor Molecules by Stark-Modulated Phase Fluctuation Optical Heterodyne Spectroscopy,” Opt. Commun. 38, 26–30 (1981).
    [CrossRef] [PubMed]
  12. H.-B. Lin, A. J. Campillo, “Photothermal Aerosol Absorption Spectroscopy,” Appl. Opt. 24, 422–433 (1985); D. U. Fluckiger, H.-B. Lin, W. H. Marlow, “Composition Measurement of Aerosols of Submicrometer Particles by Phase Fluctuation Absorption Spectroscopy,” Appl. Opt. 24, 1668–1681 (1985).
    [CrossRef] [PubMed]
  13. W.-K. Lee, A. Gungor, P.-T. Ho, C. C. Davis, “Direct Measurement of Dilute Dye Solution Quantum Yield by Photothermal Laser Heterodyne Interferometry,” Appl. Phys. Lett. 47, 916–918 (1985).
    [CrossRef]
  14. M. L. Swicord, C. C. Davis, “An Optical Method for Investigating the Microwave Absorption Characteristics of DNA and Other Biomolecules in Solutions,” Bioelectromagnetics 4, 21–42 (1983).
    [CrossRef] [PubMed]
  15. A. J. Campillo, S. J. Petuchowski, C. C. Davis, H.-B. Lin, “Fabry-Perot Photothermal Trace Detection,” Appl. Phys. Lett. 41, 327–329 (1982).
    [CrossRef]
  16. R. Gupta, “The Theory of Photothermal Effect in Fluids,” in Photothermal Investigations of Solids and Fluids, J. A. Sell, Ed. (Academic, New York, 1988).
  17. A. Rose, R. Vyas, R. Gupta, “Pulsed Photothermal Deflection Spectroscopy in a Flowing Medium: A Quantitative Investigation,” Appl. Opt. 25, 4626–4643 (1986).
    [CrossRef] [PubMed]

1988 (1)

1986 (1)

1985 (2)

1983 (1)

M. L. Swicord, C. C. Davis, “An Optical Method for Investigating the Microwave Absorption Characteristics of DNA and Other Biomolecules in Solutions,” Bioelectromagnetics 4, 21–42 (1983).
[CrossRef] [PubMed]

1982 (2)

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H.-B. Lin, “Fabry-Perot Photothermal Trace Detection,” Appl. Phys. Lett. 41, 327–329 (1982).
[CrossRef]

C. C. Davis, “Radial Laser-induced Soundwave Propagation and Vibrational Relaxation in Carbon Dioxide,” IEEE J. Quantum Electron. QE-18, 999–1003 (1982).
[CrossRef]

1981 (2)

1980 (2)

1973 (1)

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Amer, N. M.

Boccara, A. C.

Campillo, A. J.

Davis, C. C.

W.-K. Lee, A. Gungor, P.-T. Ho, C. C. Davis, “Direct Measurement of Dilute Dye Solution Quantum Yield by Photothermal Laser Heterodyne Interferometry,” Appl. Phys. Lett. 47, 916–918 (1985).
[CrossRef]

M. L. Swicord, C. C. Davis, “An Optical Method for Investigating the Microwave Absorption Characteristics of DNA and Other Biomolecules in Solutions,” Bioelectromagnetics 4, 21–42 (1983).
[CrossRef] [PubMed]

C. C. Davis, “Radial Laser-induced Soundwave Propagation and Vibrational Relaxation in Carbon Dioxide,” IEEE J. Quantum Electron. QE-18, 999–1003 (1982).
[CrossRef]

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H.-B. Lin, “Fabry-Perot Photothermal Trace Detection,” Appl. Phys. Lett. 41, 327–329 (1982).
[CrossRef]

C. C. Davis, S. J. Petuchowski, “Phase Fluctuation Optical Heterodyne Spectroscopy of Gases,” Appl. Opt. 20, 2539–2554 (1981).
[CrossRef] [PubMed]

C. C. Davis, “Trace Detection of Gases Using Phase Fluctuation Optical Heterodyne Spectroscopy,” Appl. Phys. Lett. 36, 515–518 (1980).
[CrossRef]

A. J. Campillo, H.-B. Lin, C. J. Dodge, C. C. Davis, “Stark-Effect-Modulated Phase-Fluctuation Optical Heterodyne Interferometer for Trace-Gas Analysis,” Opt. Lett. 5, 424–426 (1980); S. J. Petuchowski, C. C. Davis, “Selective Trace Detection of Asymmetric Rotor Molecules by Stark-Modulated Phase Fluctuation Optical Heterodyne Spectroscopy,” Opt. Commun. 38, 26–30 (1981).
[CrossRef] [PubMed]

Dodge, C. J.

Fang, H. L.

For a review see H. L. Fang, R. L. Swofford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983).

Fournier, D.

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Gungor, A.

W.-K. Lee, A. Gungor, P.-T. Ho, C. C. Davis, “Direct Measurement of Dilute Dye Solution Quantum Yield by Photothermal Laser Heterodyne Interferometry,” Appl. Phys. Lett. 47, 916–918 (1985).
[CrossRef]

Gupta, R.

R. Vyas, R. Gupta, “Photothermal Lensing Spectroscopy in a Flowing Medium: Theory,” Appl. Opt. 27, 4701–4711 (1988). This article contains references to other recent articles.
[CrossRef] [PubMed]

A. Rose, R. Vyas, R. Gupta, “Pulsed Photothermal Deflection Spectroscopy in a Flowing Medium: A Quantitative Investigation,” Appl. Opt. 25, 4626–4643 (1986).
[CrossRef] [PubMed]

R. Gupta, “The Theory of Photothermal Effect in Fluids,” in Photothermal Investigations of Solids and Fluids, J. A. Sell, Ed. (Academic, New York, 1988).

For a review see R. Gupta, “Pulsed Photothemal Deflection Spectroscopy in Fluid Media: A Review,” in Proceedings, International Conference on Lasers ’86, R. W. McMillan, Ed. (STS Press, McLean, VA, 1987).

Ho, P.-T.

W.-K. Lee, A. Gungor, P.-T. Ho, C. C. Davis, “Direct Measurement of Dilute Dye Solution Quantum Yield by Photothermal Laser Heterodyne Interferometry,” Appl. Phys. Lett. 47, 916–918 (1985).
[CrossRef]

Jackson, W. B.

Lee, W.-K.

W.-K. Lee, A. Gungor, P.-T. Ho, C. C. Davis, “Direct Measurement of Dilute Dye Solution Quantum Yield by Photothermal Laser Heterodyne Interferometry,” Appl. Phys. Lett. 47, 916–918 (1985).
[CrossRef]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Lin, H.-B.

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Petuchowski, S. J.

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H.-B. Lin, “Fabry-Perot Photothermal Trace Detection,” Appl. Phys. Lett. 41, 327–329 (1982).
[CrossRef]

C. C. Davis, S. J. Petuchowski, “Phase Fluctuation Optical Heterodyne Spectroscopy of Gases,” Appl. Opt. 20, 2539–2554 (1981).
[CrossRef] [PubMed]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Rose, A.

Sell, J. A.

J. A. Sell, Photothermal Investigations of Solids and Fluids (Academic, New York, 1988).

Stone, J.

Swicord, M. L.

M. L. Swicord, C. C. Davis, “An Optical Method for Investigating the Microwave Absorption Characteristics of DNA and Other Biomolecules in Solutions,” Bioelectromagnetics 4, 21–42 (1983).
[CrossRef] [PubMed]

Swofford, R. L.

For a review see H. L. Fang, R. L. Swofford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983).

Vyas, R.

Whinnery, J. R.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. Lett. (3)

W.-K. Lee, A. Gungor, P.-T. Ho, C. C. Davis, “Direct Measurement of Dilute Dye Solution Quantum Yield by Photothermal Laser Heterodyne Interferometry,” Appl. Phys. Lett. 47, 916–918 (1985).
[CrossRef]

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H.-B. Lin, “Fabry-Perot Photothermal Trace Detection,” Appl. Phys. Lett. 41, 327–329 (1982).
[CrossRef]

C. C. Davis, “Trace Detection of Gases Using Phase Fluctuation Optical Heterodyne Spectroscopy,” Appl. Phys. Lett. 36, 515–518 (1980).
[CrossRef]

Bioelectromagnetics (1)

M. L. Swicord, C. C. Davis, “An Optical Method for Investigating the Microwave Absorption Characteristics of DNA and Other Biomolecules in Solutions,” Bioelectromagnetics 4, 21–42 (1983).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

C. C. Davis, “Radial Laser-induced Soundwave Propagation and Vibrational Relaxation in Carbon Dioxide,” IEEE J. Quantum Electron. QE-18, 999–1003 (1982).
[CrossRef]

J. Appl. Phys. (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Opt. Lett. (1)

Other (4)

For a review see H. L. Fang, R. L. Swofford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983).

For a review see R. Gupta, “Pulsed Photothemal Deflection Spectroscopy in Fluid Media: A Review,” in Proceedings, International Conference on Lasers ’86, R. W. McMillan, Ed. (STS Press, McLean, VA, 1987).

R. Gupta, “The Theory of Photothermal Effect in Fluids,” in Photothermal Investigations of Solids and Fluids, J. A. Sell, Ed. (Academic, New York, 1988).

J. A. Sell, Photothermal Investigations of Solids and Fluids (Academic, New York, 1988).

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Figures (10)

Fig. 1
Fig. 1

(a) Schematic illustration of a typical PTPS experiment. The sample cell is placed in one arm of a Michelson interferometer. The pump beam passes through the cell either collinearly (solid line) or transversely (dotted line); (b) intensity variation observed at the photodetector as a function of the phase difference (ϕAϕB).

Fig. 2
Fig. 2

Pump and probe beam configurations for (a) transverse and (b) collinear geometries.

Fig. 3
Fig. 3

Typical pulsed collinear PTPS signal in a stationary medium for several pump–probe separations (expressed in units of a). δVL has been plotted as a function of time, and one division corresponds to δVL = 1. The top curve has been expanded by a factor of 5 for clarity. Parameters used in this computation are t0 = 1 μs, E0 = 6 mJ, α = 0.39 m−1, a = 0.5 mm, l = 1 cm, TA = 300 K, λ = 490 nm, n0 = 1.000294, ρCp = 1218.7 J m−3 k−1, D = 2.035 × 10−5 m2 s−1, υx = 0, and ½V = 1.

Fig. 4
Fig. 4

Comparison of the collinear and transverse PTPS signals in a stationary medium. The transverse signal has been expanded by a factor of 15.3 for ease of comparison. The transverse signal decays more slowly than the collinear signal.

Fig. 5
Fig. 5

Pulsed collinear PTPS signals in a flowing medium (υx = 2 m/s). One division corresponds to δVL = 1. All other parameters used in this computation are given in the caption to Fig. 3.

Fig. 6
Fig. 6

Temporal evolution of the temperature at several positions in the medium. The cw laser was assumed to be modulated at 100 Hz, had power Pav = 1 W, and was turned on at t = 0. x = 0 corresponds to the axis of the pump beam. The top and bottom curves have been expanded by factors of 2 for clarity.

Fig. 7
Fig. 7

Quasi-steady state temperature distributions in the medium (atmospheric pressure of N2 seeded with 1000-ppm NO2) created by an unmodulated cw laser. The laser power was assumed to be 1 W. All other parameters have been given previously in connection with pulsed PTPS. The top two curves have been expanded by the indicated factors for clarity.

Fig. 8
Fig. 8

Continuous wave collinear PTPS signals as a function of the pump–probe distance for four different flow velocities of the medium. Root mean square values of δVL have been plotted for a modulation frequency of 10 Hz and an interaction length of 1 cm. Pav = 1 W, and all other parameters are given in the caption of Fig. 3. One division corresponds to the rms value of δVL = 0.5. The top curve has been expanded by a factor of 5.

Fig. 9
Fig. 9

Variation of the temperature distribution with time in a medium with flow velocity υx = 1 cm/s when the pump beam is modulated at 10 Hz. The four curves represent four different times in the modulation cycle of the pump beam as noted on the diagram. Time t = 1.00 s corresponds to the peak of the laser power, while t = 1.05 s corresponds to the minimum of laser power. All parameters used in this computation are given in the caption to Figs. 3 and 8.

Fig. 10
Fig. 10

Continuous wave transverse PTPS signals computed with the same parameters as in Fig. 8. One division corresponds to the rms value of δVL = 0.05. The top curve has been expanded by a factor of 5 for clarity.

Equations (26)

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T ( r , t ) t = D 2 T ( r , t ) υ x T ( r , t ) x + 1 ρ C p Q ( r , t ) ,
Q ( r , t ) = α I ( r , t ) = [ 2 α E 0 π a 2 t 0 exp ( 2 r 2 / a 2 ) for 0 t t 0 , 0 for t > t 0
Q ( r , t ) = 2 α P a v π a 2 [ exp ( 2 r 2 / a 2 ) ] ( 1 + cos ω t ) ,
T ( x , y , t ) | t ¯ 0 = 0 ; T ( x , y , t ) | t ¯ 0 = 0 , T ( x , y , t ) | x ¯ ± = 0 ; T ( x , y , t ) | y ¯ ± = 0 ,
T ( x , y , t ) = + + 0 Q ( ξ , η , τ ) G ( x / ξ , y / η , t / τ ) d ξ d η d τ ,
D x y 2 G + υ x G x + G t = 1 ρ C p δ ( x ξ ) δ ( y η ) δ ( t τ ) ,
G = H τ ( t ) 4 π ρ C p D ( t τ ) exp ( { x [ ξ + υ x ( t τ ) ] } 2 / [ 4 D ( t τ ) ] ) × exp { ( y η ) 2 / [ 4 D ( t τ ) ] } ,
T ( x , y , t ) = 2 α E 0 π t 0 ρ C p 0 t 0 1 [ 8 D ( t τ ) + a 2 ] × exp { 2 { [ x υ x ( t τ ) ] 2 + y 2 } / [ 8 D ( t τ ) + a 2 ] } d τ for t > t 0
T ( x , y , t ) = 2 α P av π ρ C p 0 t ( 1 + cos ω τ ) [ 8 D ( t τ ) + a 2 ] × exp ( 2 { [ x υ x ( t τ ) ] 2 + y 2 } / [ 8 D ( t τ ) + a 2 ] ) d τ
a = A sin ( k x ω t + ϕ A ) b = B sin [ k x ω t + ϕ B + γ ( t ) ] .
I ( t ) = A 2 + B 2 + 2 A B cos [ ( ϕ A ϕ B ) γ ( t ) ] .
γ ( x , y , t ) = 4 π λ path Δ n ( x , y , t ) d s ,
Δ n ( x , y , t ) = ( n 0 1 ) T ( x , y , t ) T A ,
γ ( x , y , t ) = 4 π λ ( n 0 1 ) T A path T ( x , y , t ) d s .
δ V ( x , y , t ) 2 A B sin [ 4 π λ ( n 0 1 ) T A path T ( x , y , t ) d s ] .
V = V max V min 4 A B .
δ V ( x , y , t ) = 1 2 V sin [ 4 π λ ( n 0 1 ) T A path T ( x , y , t ) d s ] .
δ V L ( x , y , t ) = 1 2 V sin [ 4 π λ ( n 0 1 ) T A path T ( x , y , t ) d z ] = 1 2 V sin [ 4 π λ ( n 0 1 ) T A T ( x , y , t ) ] ,
δ V T ( x , t ) = 1 2 V sin [ 4 π λ ( n 0 1 ) T A T ( x , y , t ) d y ] .
δ V L ( x , y , t ) = 1 2 V sin [ 4 π λ ( n 0 1 ) l T A 2 α E 0 π t 0 ρ C p × 0 t 0 exp ( 2 { [ x υ x ( t τ ) ] 2 + y 2 } / [ a 2 + 8 D ( t τ ) ] ) [ a 2 + 8 D ( t τ ) ] d τ ] .
δ V L ( x , y , t ) = 1 2 V sin ( 4 π λ ( n 0 1 ) l T A 2 α E 0 π ρ C p × exp { 2 [ ( x υ x t ) 2 + y 2 ] / ( a 2 + 8 D t ) } [ a 2 + 8 D t ] ) .
δ V T ( x , t ) = 1 2 V sin [ 4 π λ ( n 0 1 ) l T A 2 α E 0 2 π t 0 ρ C p × 0 t 0 exp ( 2 { [ x υ x ( t τ ) ] 2 + y 2 } / [ a 2 + 8 D ( t τ ) ] ) [ a 2 + 8 D ( t τ ) ] 1 / 2 d τ ] .
δ V T ( x , t ) = 1 2 V sin [ 4 π λ ( n 0 1 ) T A 2 α E 0 2 π ρ C p × exp [ 2 ( x υ x t ) 2 / ( a 2 + 8 D t ) ] ( a 2 + 8 D t ) 1 / 2 ] .
δ V L ( x , y , t ) = 1 2 V sin [ 4 π λ ( n 0 1 ) l T A × 2 α P av π ρ C p 0 t ( 1 + cos ω τ ) [ a 2 + 8 D ( t τ ) ] × exp ( 2 { [ x υ x ( t τ ) ] 2 + y 2 } / [ a 2 + 8 D ( t τ ) ] ) d τ ] .
δ V T ( x , t ) = 1 2 V sin ( 4 π λ ( n 0 1 ) T A 2 α P av 2 π ρ C p 0 t ( 1 + cos ω τ ) [ a 2 + 8 D ( t τ ) ] 1 / 2 × exp { 2 [ x υ x ( t τ ) ] 2 / [ a 2 + 8 D ( t τ ) ] } d τ ) .
T ( x = 0 , y = 0 , t ) = α P 0 4 π ρ C ρ D ln ( 1 + 2 t t c ) ,

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