Abstract

An analytical model that incorporates effects of light scattering was developed for dual-beam photothermal deflection spectroscopy. Thermal gradients induced by a modulated excitation beam deflect an optical probe beam which was treated as being of finite dimensions. Mechanisms by which thermal gradients produce refractive index gradients are discussed, with an explicit expression for dn/dT being derived. Experimental studies with suspensions of small latex particles in Nd3+ solutions demonstrated that the model accurately predicts both the shape of the deflection signal and the attenuation of the signal due to light scattering. The absolute magnitude of the observed signal is approximately predicted by theory.

© 1990 Optical Society of America

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References

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  1. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spectroscopy and Detection,” Appl. Opt. 20, 1333–1344 (1981).
    [CrossRef] [PubMed]
  2. A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130–132 (1980).
    [CrossRef]
  3. A. C. Tam, “Applications of Photoacoustic Sensing Techniques,” Rev. Mod. Phys. 58, 381–431 (1986).
    [CrossRef]
  4. H. L. Fang, R. L. Swafford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983), Chap. 3, p. 175.
  5. A. J. Campillo, H-B. Lin, “Photothermal Spectroscopy of Aerosols,” in Photothermal Investigations of Solids and Fluids, J. A. Sell, Ed. (Academic, New York, 1989), Chap. 10, p. 321.
  6. J. D. Spear, R. E. Russo, R. J. Silva, “Differential Photothermal Deflection Spectroscopy Using a Single Position Sensor,” Appl. Spectrosc. 42, 1103–1105 (1988).
    [CrossRef]
  7. R. Vyas, B. Monson, Y-X. Nie, R. Gupta, “Continuous Wave Photothermal Deflection Spectroscopy in a Flowing Medium,” Appl. Opt. 27, 3914–3920 (1988).
    [CrossRef] [PubMed]
  8. A. Mandelis, B. S. H. Royce, “Fundamental-Mode Laser-Beam Propagation in Optically Inhomogeneous Electrochemical Media with Chemical Species Concentration Gradients,” Appl. Opt. 23, 2892–2901 (1984).
    [CrossRef] [PubMed]
  9. L. C. Aamodt, J. C. Murphy, “Photothermal Measurements Using a Localized Excitation Source,” J. Appl. Phys. 52, 4903–4914 (1981).
    [CrossRef]
  10. E. Legal Lasalle, F. Lepoutre, J. P. Roger, “Probe Beam Size Effects in Photothermal Deflection Experiments,” J. Appl. Phys. 64, 1–5 (1988).
    [CrossRef]
  11. F. A. McDonald, G. C. Wetsel, J. E. Jamieson, “Photothermal Beam-Deflection Imaging of Vertical Interfaces in Solids,” Can. J. Phys. 64, 1265–1268 (1986).
    [CrossRef]
  12. W. Nowacki, Thermoelasticity (Pergamon, New York, 1986).
  13. Previous theoretical and experimental work has been published on dn/∂T in solids. A list of references can be found in R. M. Waxler, D. Horowitz, A. Feldman, “Optical and Physical Parameters of Plexiglas 55 and Lexan,” Appl. Opt. 18, 101–104 (1979).
    [CrossRef] [PubMed]
  14. S. A. Akhmanov, R. V. Kokhlov, A. P. Surhorukov, “Self-Focusing, Self-Defocusing and Self-Modulation of Laser Beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, Eds. (North-Holland, Amsterdam, 1972), p. 1151.
  15. Smithsonian Physical Tables (The Lord Baltimore Press, Baltimore, 1956).
  16. CRC Handbook of Chemistry and Physics (Chemical Rubber Co., Cleveland, 1971).
  17. N. J. Dovichi, T. G. Nolan, W. A. Weiner, “Theory for Lasenduced Photothermal Refraction,” Anal. Chem. 56, 1700–1704 (1984).
    [CrossRef]
  18. D. Solimini, “Loss Measurements of Organic Materials at 6328 Å,” J. Appl. Phys. 37, 3314–3315 (1966).
    [CrossRef]
  19. L. W. Tilton, J. K. Taylor, “Refractive Index and Dispersion of Distilled Water For Visible Radiation at Temperatures 0 to 60°C,” J. Res. Natl. Bur. Stand. Sect. A 20, 419–477 (1938). Values of dn/dT are derived from successive values of n at 1°C intervals, measured at the 656.28-nm spectral emission line of hydrogen.
  20. J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580–4588 (1980).
    [CrossRef]

1988 (3)

1986 (2)

F. A. McDonald, G. C. Wetsel, J. E. Jamieson, “Photothermal Beam-Deflection Imaging of Vertical Interfaces in Solids,” Can. J. Phys. 64, 1265–1268 (1986).
[CrossRef]

A. C. Tam, “Applications of Photoacoustic Sensing Techniques,” Rev. Mod. Phys. 58, 381–431 (1986).
[CrossRef]

1984 (2)

1981 (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]

L. C. Aamodt, J. C. Murphy, “Photothermal Measurements Using a Localized Excitation Source,” J. Appl. Phys. 52, 4903–4914 (1981).
[CrossRef]

1980 (2)

J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580–4588 (1980).
[CrossRef]

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

1979 (1)

1966 (1)

D. Solimini, “Loss Measurements of Organic Materials at 6328 Å,” J. Appl. Phys. 37, 3314–3315 (1966).
[CrossRef]

1938 (1)

L. W. Tilton, J. K. Taylor, “Refractive Index and Dispersion of Distilled Water For Visible Radiation at Temperatures 0 to 60°C,” J. Res. Natl. Bur. Stand. Sect. A 20, 419–477 (1938). Values of dn/dT are derived from successive values of n at 1°C intervals, measured at the 656.28-nm spectral emission line of hydrogen.

Aamodt, L. C.

L. C. Aamodt, J. C. Murphy, “Photothermal Measurements Using a Localized Excitation Source,” J. Appl. Phys. 52, 4903–4914 (1981).
[CrossRef]

J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580–4588 (1980).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, R. V. Kokhlov, A. P. Surhorukov, “Self-Focusing, Self-Defocusing and Self-Modulation of Laser Beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, Eds. (North-Holland, Amsterdam, 1972), p. 1151.

Amer, N. M.

Badoz, J.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Boccara, A. C.

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spectroscopy and Detection,” Appl. Opt. 20, 1333–1344 (1981).
[CrossRef] [PubMed]

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Campillo, A. J.

A. J. Campillo, H-B. Lin, “Photothermal Spectroscopy of Aerosols,” in Photothermal Investigations of Solids and Fluids, J. A. Sell, Ed. (Academic, New York, 1989), Chap. 10, p. 321.

Dovichi, N. J.

N. J. Dovichi, T. G. Nolan, W. A. Weiner, “Theory for Lasenduced Photothermal Refraction,” Anal. Chem. 56, 1700–1704 (1984).
[CrossRef]

Fang, H. L.

H. L. Fang, R. L. Swafford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983), Chap. 3, p. 175.

Feldman, A.

Fournier, D.

W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spectroscopy and Detection,” Appl. Opt. 20, 1333–1344 (1981).
[CrossRef] [PubMed]

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Gupta, R.

Horowitz, D.

Jackson, W. B.

Jamieson, J. E.

F. A. McDonald, G. C. Wetsel, J. E. Jamieson, “Photothermal Beam-Deflection Imaging of Vertical Interfaces in Solids,” Can. J. Phys. 64, 1265–1268 (1986).
[CrossRef]

Kokhlov, R. V.

S. A. Akhmanov, R. V. Kokhlov, A. P. Surhorukov, “Self-Focusing, Self-Defocusing and Self-Modulation of Laser Beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, Eds. (North-Holland, Amsterdam, 1972), p. 1151.

Legal Lasalle, E.

E. Legal Lasalle, F. Lepoutre, J. P. Roger, “Probe Beam Size Effects in Photothermal Deflection Experiments,” J. Appl. Phys. 64, 1–5 (1988).
[CrossRef]

Lepoutre, F.

E. Legal Lasalle, F. Lepoutre, J. P. Roger, “Probe Beam Size Effects in Photothermal Deflection Experiments,” J. Appl. Phys. 64, 1–5 (1988).
[CrossRef]

Lin, H-B.

A. J. Campillo, H-B. Lin, “Photothermal Spectroscopy of Aerosols,” in Photothermal Investigations of Solids and Fluids, J. A. Sell, Ed. (Academic, New York, 1989), Chap. 10, p. 321.

Mandelis, A.

McDonald, F. A.

F. A. McDonald, G. C. Wetsel, J. E. Jamieson, “Photothermal Beam-Deflection Imaging of Vertical Interfaces in Solids,” Can. J. Phys. 64, 1265–1268 (1986).
[CrossRef]

Monson, B.

Murphy, J. C.

L. C. Aamodt, J. C. Murphy, “Photothermal Measurements Using a Localized Excitation Source,” J. Appl. Phys. 52, 4903–4914 (1981).
[CrossRef]

J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580–4588 (1980).
[CrossRef]

Nie, Y-X.

Nolan, T. G.

N. J. Dovichi, T. G. Nolan, W. A. Weiner, “Theory for Lasenduced Photothermal Refraction,” Anal. Chem. 56, 1700–1704 (1984).
[CrossRef]

Nowacki, W.

W. Nowacki, Thermoelasticity (Pergamon, New York, 1986).

Roger, J. P.

E. Legal Lasalle, F. Lepoutre, J. P. Roger, “Probe Beam Size Effects in Photothermal Deflection Experiments,” J. Appl. Phys. 64, 1–5 (1988).
[CrossRef]

Royce, B. S. H.

Russo, R. E.

Silva, R. J.

Solimini, D.

D. Solimini, “Loss Measurements of Organic Materials at 6328 Å,” J. Appl. Phys. 37, 3314–3315 (1966).
[CrossRef]

Spear, J. D.

Surhorukov, A. P.

S. A. Akhmanov, R. V. Kokhlov, A. P. Surhorukov, “Self-Focusing, Self-Defocusing and Self-Modulation of Laser Beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, Eds. (North-Holland, Amsterdam, 1972), p. 1151.

Swafford, R. L.

H. L. Fang, R. L. Swafford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983), Chap. 3, p. 175.

Tam, A. C.

A. C. Tam, “Applications of Photoacoustic Sensing Techniques,” Rev. Mod. Phys. 58, 381–431 (1986).
[CrossRef]

Taylor, J. K.

L. W. Tilton, J. K. Taylor, “Refractive Index and Dispersion of Distilled Water For Visible Radiation at Temperatures 0 to 60°C,” J. Res. Natl. Bur. Stand. Sect. A 20, 419–477 (1938). Values of dn/dT are derived from successive values of n at 1°C intervals, measured at the 656.28-nm spectral emission line of hydrogen.

Tilton, L. W.

L. W. Tilton, J. K. Taylor, “Refractive Index and Dispersion of Distilled Water For Visible Radiation at Temperatures 0 to 60°C,” J. Res. Natl. Bur. Stand. Sect. A 20, 419–477 (1938). Values of dn/dT are derived from successive values of n at 1°C intervals, measured at the 656.28-nm spectral emission line of hydrogen.

Vyas, R.

Waxler, R. M.

Weiner, W. A.

N. J. Dovichi, T. G. Nolan, W. A. Weiner, “Theory for Lasenduced Photothermal Refraction,” Anal. Chem. 56, 1700–1704 (1984).
[CrossRef]

Wetsel, G. C.

F. A. McDonald, G. C. Wetsel, J. E. Jamieson, “Photothermal Beam-Deflection Imaging of Vertical Interfaces in Solids,” Can. J. Phys. 64, 1265–1268 (1986).
[CrossRef]

Anal. Chem. (1)

N. J. Dovichi, T. G. Nolan, W. A. Weiner, “Theory for Lasenduced Photothermal Refraction,” Anal. Chem. 56, 1700–1704 (1984).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[CrossRef]

Appl. Spectrosc. (1)

Can. J. Phys. (1)

F. A. McDonald, G. C. Wetsel, J. E. Jamieson, “Photothermal Beam-Deflection Imaging of Vertical Interfaces in Solids,” Can. J. Phys. 64, 1265–1268 (1986).
[CrossRef]

J. Appl. Phys. (4)

D. Solimini, “Loss Measurements of Organic Materials at 6328 Å,” J. Appl. Phys. 37, 3314–3315 (1966).
[CrossRef]

L. C. Aamodt, J. C. Murphy, “Photothermal Measurements Using a Localized Excitation Source,” J. Appl. Phys. 52, 4903–4914 (1981).
[CrossRef]

E. Legal Lasalle, F. Lepoutre, J. P. Roger, “Probe Beam Size Effects in Photothermal Deflection Experiments,” J. Appl. Phys. 64, 1–5 (1988).
[CrossRef]

J. C. Murphy, L. C. Aamodt, “Photothermal Spectroscopy Using Optical Beam Probing: Mirage Effect,” J. Appl. Phys. 51, 4580–4588 (1980).
[CrossRef]

J. Res. Natl. Bur. Stand. Sect. A (1)

L. W. Tilton, J. K. Taylor, “Refractive Index and Dispersion of Distilled Water For Visible Radiation at Temperatures 0 to 60°C,” J. Res. Natl. Bur. Stand. Sect. A 20, 419–477 (1938). Values of dn/dT are derived from successive values of n at 1°C intervals, measured at the 656.28-nm spectral emission line of hydrogen.

Rev. Mod. Phys. (1)

A. C. Tam, “Applications of Photoacoustic Sensing Techniques,” Rev. Mod. Phys. 58, 381–431 (1986).
[CrossRef]

Other (6)

H. L. Fang, R. L. Swafford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983), Chap. 3, p. 175.

A. J. Campillo, H-B. Lin, “Photothermal Spectroscopy of Aerosols,” in Photothermal Investigations of Solids and Fluids, J. A. Sell, Ed. (Academic, New York, 1989), Chap. 10, p. 321.

W. Nowacki, Thermoelasticity (Pergamon, New York, 1986).

S. A. Akhmanov, R. V. Kokhlov, A. P. Surhorukov, “Self-Focusing, Self-Defocusing and Self-Modulation of Laser Beams,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-Dubois, Eds. (North-Holland, Amsterdam, 1972), p. 1151.

Smithsonian Physical Tables (The Lord Baltimore Press, Baltimore, 1956).

CRC Handbook of Chemistry and Physics (Chemical Rubber Co., Cleveland, 1971).

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Figures (10)

Fig. 1
Fig. 1

Simplified 2-D geometry for the collinear PDS experiment in the presence of light scattering.

Fig. 2
Fig. 2

More detailed 3-D geometry in a useful collinear PDS experiment. The model must account for changing beam diameters and finite overlap angle ϕ.

Fig. 3
Fig. 3

Simulated deflection signal for beam offset x0 = −28 μm. Other parameters are described in the text.

Fig. 4
Fig. 4

Simulated peak-to-peak amplitude of deflection signal vs beam offset x0. The beam waist radius w0p was set at 50 μm.

Fig. 5
Fig. 5

Simulated peak-to-peak amplitude of the deflection signal vs probe beam waist radius w0p. For each value of w0p, the beam offset x0 was optimized for the maximum signal amplitude.

Fig. 6
Fig. 6

Computed values for the clarity factor vs scattering coefficient αs for three different focusing states of the optical beams: z0 = z0p = 0.2 cm (solid line); z0 = z0p = 0.5 cm (broken line); z0 = z0p = 0.8 cm (dashed line).

Fig. 7
Fig. 7

Schematic diagram of the apparatus used for collinear PDS experiments.

Fig. 8
Fig. 8

Oscilloscope trace of observed deflection signal. The vertical scale units are 4.7 × 10−5 rad/div. Horizontal units are 10 ms/div.

Fig. 9
Fig. 9

Photothermal deflection spectra for three samples with different scattering coefficients: αs = 0 (I); αs = 1.6 cm−1 (II); αs = 2.9 cm−1 (III). Beams are focused in the middle of the sample cell (z0 = z0p = 0.5 cm). The peak-to-peak amplitude (μrad) is normalized with respect to P0, the excitation power (mW) entering the sample.

Fig. 10
Fig. 10

Clarity factor vs scattering coefficient αs taken from normalized spectra for z0 = z0p = 0.2 cm (square data points) and z0 = z0p = 0.5 cm (circular data points). Theoretical curves from Fig. 6 are repeated for comparison.

Tables (1)

Tables Icon

Table I Comparison of Experimental Values for dn/dT with Those Given by Eq. (10); Sources for Data are In Parentheses

Equations (15)

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C . F . = 0 z P 0 exp ( - α s z ) d z 0 z P 0 d z = [ 1 - exp ( - α s Z ) ] α s Z .
2 T - 1 α th d T d t = - Q ( x , y , z , t ) κ ,
Q = 2 α a P 0 exp ( - α s Z ) π w 2 [ exp ( - 2 r 2 w 2 ) ]             for j / f t ( j + 1 2 ) / f , Q = 0             for t < 0 , ( j + 1 2 ) / f < t < ( j + 1 ) / f ,
w 2 = w 0 2 { 1 + [ λ ( z - z 0 ) π w 0 2 n ] 2 } ,
d T d r = α a P 0 exp ( - α s Z ) 2 π r κ [ exp ( - 2 r 2 w 2 ) - exp ( - 2 r 2 w 2 + 8 α th t ) ]             0 t 1 / 2 f , d T d r = α a P 0 exp ( - α s Z ) 2 π r κ [ exp ( - 2 r 2 w 2 + 8 α th ( t - 1 / 2 f ) ) - exp ( - 2 r 2 w 2 + 8 α th t ) ]             t 1 / 2 f .
d T d r = j = 0 N α a P 0 exp ( - α s Z ) 2 π r κ ( - 1 ) j { exp ( - 2 r 2 w 2 ) - exp [ - 2 r 2 w 2 + 8 α th ( t - j / 2 f ) ] }             t 0 ,
n = T d n d T ,
1 ρ ( n 2 - 1 n 2 + 2 ) = constant ,
- 1 ρ d ρ d T ( n 2 - 1 ) + 6 n ( n 2 + 2 ) d n d T = 0.
d n d T = - β 6 n ( n 2 - 1 ) ( n 2 + 2 ) .
d n d T - ( n - 1 ) T .
θ = 0 Z n n d z .
d θ d z = - d x - I p n n d y - d x - I p d y ,
w p 2 = w 0 p 2 { 1 + [ λ p ( z - z 0 p ) π w 0 p 2 n ] 2 } .
θ = 0 Z d z - d y - 2 exp [ - 2 ( x 2 + y 2 ) / w p 2 ] n w p 2 π n d x .

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