Abstract

A theoretical model of a novel time-resolved, photothermal-deflection spectrometry with step optical excitation (a rectangular pulse of a finite duration) is presented. To make the theory practicable, i.e., able to be used to determine the values of optical and thermal properties of the sample by fitting experimental data to the theory, approximations are introduced. The numerical simulations show that when the value of the thermal conductivity of the deflecting medium (e.g., air) is small, the adiabatic approximation for the heat conduction at both front and rear surfaces of a sample is effective for both transparent and opaque samples, while the adiabatic and isothermal approximations for the heat conduction at the front and rear surfaces, respectively, are valid only for an opaque sample. With these approximations, the optical and thermal properties—such as optical absorption coefficient and thermal diffusivity—of solid materials (optically transparent or opaque) can be directly measured by this novel method.

© 2004 Optical Society of America

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References

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  1. A. Mandelis, Principle and Perspectives of Photothermal and Photoacoustic Phenomena (Elsevier, New York, 1992).
  2. D. Almond and P. Patel, Photothermal Science and Techniques (Chapman & Hall, London, 1996).
  3. H. Vargas and L. C. M. Miranda, “Photothermal techniques applied to thermophysical properties measurements,” Rev. Sci. Instrum. 74, 794–799 (2003).
    [Crossref]
  4. A. C. Boccara, D. Fournier, and J. Badoz, “Thermo-optical spectroscopy: Detection by the ‘mirage effect’,” Appl. Phys. Lett. 36, 130–132 (1980).
    [Crossref]
  5. J. Zhao, J. Shen, and C. Hu, “Continuous-wave photothermal deflection spectroscopy with fundamental and harmonic responses,” Opt. Lett. 27, 1755–1757 (2002).
    [Crossref]
  6. M. A. Schweitzer and J. F. Power, “Optical depth profiling of thin films by impulse mirage effect spectroscopy: Part I, theory,” Appl. Spectrosc. 48, 1054–1075 (1994).
    [Crossref]
  7. J. F. Power, S. W. Fu, and M. A. Schweitzer, “Depth profiling of optical absorption in thin films via the mirage effect and a new inverse scattering theory: Part I, principles and methodology,” Appl. Spectrosc. 54, 110–126 (2000).
    [Crossref]
  8. A. Salazar and A. Sánchez-Lavega, “Thermal diffusivity measurements using linear relations from photothermal wave experiments,” Rev. Sci. Instrum. 65, 2896–2900 (1994).
    [Crossref]
  9. J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
    [Crossref]
  10. M. L. Baesso, J. Shen, and R. D. Snook, “Mode mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732–3737 (1994).
    [Crossref]
  11. S. M. Brown, M. L. Baesso, J. Shen, and R. D. Snook, “Thermal diffusivity of skin measured by two photothermal techniques,” Anal. Chim. Acta 282, 711–719 (1993).
    [Crossref]
  12. M. N. Özisik, Boundary Value Problems of Heat Conduction (International Textbook, Scranton, Pa., 1968).
  13. A. Mandelis and B. S. H. Royce, “Time-domain photoacoustic spectroscopy of solids,” J. Appl. Phys. 50, 4330–4338 (1979).
    [Crossref]
  14. A. Mandelis and J. Vanniasinkam, “Theory of nonradiative decay dynamics in intensely pumped solid-state laser media via laser photothermal diagnostics,” J. Appl. Phys. 80, 6107–6119 (1996).
    [Crossref]
  15. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, U.K., 1959).
  16. S. Wu and N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin film,” J. Appl. Phys. 67, 1170–1182 (1990).
    [Crossref]
  17. W. P. Leung and A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
    [Crossref]
  18. J. D. Rudnicki and F. R. McLarnon, “In situ characterization of electrode processes by photothermal-deflection spectroscopy,” in Techniques for Characterization of Electrodes and Electrochemical Processes, R. Varma and J. R. Selman, eds. (Wiley, New York, 1991), pp. 127–166.
  19. R. W. Jones and J. F. McClelland, “Phase references and cell effects in photoacoustic spectroscopy,” Appl. Spectrosc. 55, 1360–1366 (2001).
    [Crossref]
  20. S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis (Wiley, New York, 1996).

2003 (1)

H. Vargas and L. C. M. Miranda, “Photothermal techniques applied to thermophysical properties measurements,” Rev. Sci. Instrum. 74, 794–799 (2003).
[Crossref]

2002 (1)

2001 (1)

2000 (1)

1996 (1)

A. Mandelis and J. Vanniasinkam, “Theory of nonradiative decay dynamics in intensely pumped solid-state laser media via laser photothermal diagnostics,” J. Appl. Phys. 80, 6107–6119 (1996).
[Crossref]

1994 (4)

A. Salazar and A. Sánchez-Lavega, “Thermal diffusivity measurements using linear relations from photothermal wave experiments,” Rev. Sci. Instrum. 65, 2896–2900 (1994).
[Crossref]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[Crossref]

M. L. Baesso, J. Shen, and R. D. Snook, “Mode mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732–3737 (1994).
[Crossref]

M. A. Schweitzer and J. F. Power, “Optical depth profiling of thin films by impulse mirage effect spectroscopy: Part I, theory,” Appl. Spectrosc. 48, 1054–1075 (1994).
[Crossref]

1993 (1)

S. M. Brown, M. L. Baesso, J. Shen, and R. D. Snook, “Thermal diffusivity of skin measured by two photothermal techniques,” Anal. Chim. Acta 282, 711–719 (1993).
[Crossref]

1990 (1)

S. Wu and N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin film,” J. Appl. Phys. 67, 1170–1182 (1990).
[Crossref]

1984 (1)

W. P. Leung and A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
[Crossref]

1980 (1)

A. C. Boccara, D. Fournier, and J. Badoz, “Thermo-optical spectroscopy: Detection by the ‘mirage effect’,” Appl. Phys. Lett. 36, 130–132 (1980).
[Crossref]

1979 (1)

A. Mandelis and B. S. H. Royce, “Time-domain photoacoustic spectroscopy of solids,” J. Appl. Phys. 50, 4330–4338 (1979).
[Crossref]

Badoz, J.

A. C. Boccara, D. Fournier, and J. Badoz, “Thermo-optical spectroscopy: Detection by the ‘mirage effect’,” Appl. Phys. Lett. 36, 130–132 (1980).
[Crossref]

Baesso, M. L.

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[Crossref]

M. L. Baesso, J. Shen, and R. D. Snook, “Mode mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732–3737 (1994).
[Crossref]

S. M. Brown, M. L. Baesso, J. Shen, and R. D. Snook, “Thermal diffusivity of skin measured by two photothermal techniques,” Anal. Chim. Acta 282, 711–719 (1993).
[Crossref]

Boccara, A. C.

A. C. Boccara, D. Fournier, and J. Badoz, “Thermo-optical spectroscopy: Detection by the ‘mirage effect’,” Appl. Phys. Lett. 36, 130–132 (1980).
[Crossref]

Brown, S. M.

S. M. Brown, M. L. Baesso, J. Shen, and R. D. Snook, “Thermal diffusivity of skin measured by two photothermal techniques,” Anal. Chim. Acta 282, 711–719 (1993).
[Crossref]

Dovichi, N. J.

S. Wu and N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin film,” J. Appl. Phys. 67, 1170–1182 (1990).
[Crossref]

Fournier, D.

A. C. Boccara, D. Fournier, and J. Badoz, “Thermo-optical spectroscopy: Detection by the ‘mirage effect’,” Appl. Phys. Lett. 36, 130–132 (1980).
[Crossref]

Fu, S. W.

Hu, C.

Jones, R. W.

Leung, W. P.

W. P. Leung and A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
[Crossref]

Mandelis, A.

A. Mandelis and J. Vanniasinkam, “Theory of nonradiative decay dynamics in intensely pumped solid-state laser media via laser photothermal diagnostics,” J. Appl. Phys. 80, 6107–6119 (1996).
[Crossref]

A. Mandelis and B. S. H. Royce, “Time-domain photoacoustic spectroscopy of solids,” J. Appl. Phys. 50, 4330–4338 (1979).
[Crossref]

McClelland, J. F.

Miranda, L. C. M.

H. Vargas and L. C. M. Miranda, “Photothermal techniques applied to thermophysical properties measurements,” Rev. Sci. Instrum. 74, 794–799 (2003).
[Crossref]

Power, J. F.

Royce, B. S. H.

A. Mandelis and B. S. H. Royce, “Time-domain photoacoustic spectroscopy of solids,” J. Appl. Phys. 50, 4330–4338 (1979).
[Crossref]

Salazar, A.

A. Salazar and A. Sánchez-Lavega, “Thermal diffusivity measurements using linear relations from photothermal wave experiments,” Rev. Sci. Instrum. 65, 2896–2900 (1994).
[Crossref]

Sánchez-Lavega, A.

A. Salazar and A. Sánchez-Lavega, “Thermal diffusivity measurements using linear relations from photothermal wave experiments,” Rev. Sci. Instrum. 65, 2896–2900 (1994).
[Crossref]

Schweitzer, M. A.

Shen, J.

J. Zhao, J. Shen, and C. Hu, “Continuous-wave photothermal deflection spectroscopy with fundamental and harmonic responses,” Opt. Lett. 27, 1755–1757 (2002).
[Crossref]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[Crossref]

M. L. Baesso, J. Shen, and R. D. Snook, “Mode mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732–3737 (1994).
[Crossref]

S. M. Brown, M. L. Baesso, J. Shen, and R. D. Snook, “Thermal diffusivity of skin measured by two photothermal techniques,” Anal. Chim. Acta 282, 711–719 (1993).
[Crossref]

Snook, R. D.

M. L. Baesso, J. Shen, and R. D. Snook, “Mode mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732–3737 (1994).
[Crossref]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[Crossref]

S. M. Brown, M. L. Baesso, J. Shen, and R. D. Snook, “Thermal diffusivity of skin measured by two photothermal techniques,” Anal. Chim. Acta 282, 711–719 (1993).
[Crossref]

Tam, A. C.

W. P. Leung and A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
[Crossref]

Vanniasinkam, J.

A. Mandelis and J. Vanniasinkam, “Theory of nonradiative decay dynamics in intensely pumped solid-state laser media via laser photothermal diagnostics,” J. Appl. Phys. 80, 6107–6119 (1996).
[Crossref]

Vargas, H.

H. Vargas and L. C. M. Miranda, “Photothermal techniques applied to thermophysical properties measurements,” Rev. Sci. Instrum. 74, 794–799 (2003).
[Crossref]

Wu, S.

S. Wu and N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin film,” J. Appl. Phys. 67, 1170–1182 (1990).
[Crossref]

Zhao, J.

Anal. Chim. Acta (1)

S. M. Brown, M. L. Baesso, J. Shen, and R. D. Snook, “Thermal diffusivity of skin measured by two photothermal techniques,” Anal. Chim. Acta 282, 711–719 (1993).
[Crossref]

Appl. Phys. Lett. (1)

A. C. Boccara, D. Fournier, and J. Badoz, “Thermo-optical spectroscopy: Detection by the ‘mirage effect’,” Appl. Phys. Lett. 36, 130–132 (1980).
[Crossref]

Appl. Spectrosc. (3)

J. Appl. Phys. (6)

S. Wu and N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin film,” J. Appl. Phys. 67, 1170–1182 (1990).
[Crossref]

W. P. Leung and A. C. Tam, “Techniques of flash radiometry,” J. Appl. Phys. 56, 153–161 (1984).
[Crossref]

A. Mandelis and B. S. H. Royce, “Time-domain photoacoustic spectroscopy of solids,” J. Appl. Phys. 50, 4330–4338 (1979).
[Crossref]

A. Mandelis and J. Vanniasinkam, “Theory of nonradiative decay dynamics in intensely pumped solid-state laser media via laser photothermal diagnostics,” J. Appl. Phys. 80, 6107–6119 (1996).
[Crossref]

J. Shen, M. L. Baesso, and R. D. Snook, “Three-dimensional model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry and time-resolved measurements of thin film samples,” J. Appl. Phys. 75, 3738–3748 (1994).
[Crossref]

M. L. Baesso, J. Shen, and R. D. Snook, “Mode mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths,” J. Appl. Phys. 75, 3732–3737 (1994).
[Crossref]

Opt. Lett. (1)

Rev. Sci. Instrum. (2)

A. Salazar and A. Sánchez-Lavega, “Thermal diffusivity measurements using linear relations from photothermal wave experiments,” Rev. Sci. Instrum. 65, 2896–2900 (1994).
[Crossref]

H. Vargas and L. C. M. Miranda, “Photothermal techniques applied to thermophysical properties measurements,” Rev. Sci. Instrum. 74, 794–799 (2003).
[Crossref]

Other (6)

A. Mandelis, Principle and Perspectives of Photothermal and Photoacoustic Phenomena (Elsevier, New York, 1992).

D. Almond and P. Patel, Photothermal Science and Techniques (Chapman & Hall, London, 1996).

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, U.K., 1959).

M. N. Özisik, Boundary Value Problems of Heat Conduction (International Textbook, Scranton, Pa., 1968).

J. D. Rudnicki and F. R. McLarnon, “In situ characterization of electrode processes by photothermal-deflection spectroscopy,” in Techniques for Characterization of Electrodes and Electrochemical Processes, R. Varma and J. R. Selman, eds. (Wiley, New York, 1991), pp. 127–166.

S. E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis (Wiley, New York, 1996).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the photothermal-deflection spectrometry of a thin-slab solid sample in a one-dimensional treatment.

Fig. 2
Fig. 2

Temperature rise in the samples during the active heating interval with the heating beam pulse duration τp=1τs for different time t. (a) Glassy carbon in air, (b) glassy carbon in water, (c) lime glass in air, (d) lime glass in water. Solid curves, case (1); diamonds, case (2); squares, case (3).

Fig. 3
Fig. 3

Temperature rise in the deflecting media with the heating beam pulse duration τp=0.5τs for different time t; lg4αgτs. (a) Glassy carbon in air, (b) glassy carbon in water, (c) lime glass in air, and (d) lime glass in water. Solid curves, case (1); diamonds, case (2); squares, case (3).

Fig. 4
Fig. 4

Time-resolved, photothermal-deflection signals with the heating beam pulse duration τp=0.5τs for different positions in the deflecting media; lg4αgτs. All results are normalized to -d/nn/T. (a) Glassy carbon in air, (b) glassy carbon in water, (c) lime glass in air, (d) lime glass in water. Solid curves, case (1); diamonds, case (2); squares, case (3).

Tables (2)

Tables Icon

Table 1 Optical and Thermal Properties of Glassy Carbon and Lime Glass

Tables Icon

Table 2 Parameters Used in the Simulations

Equations (50)

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I(t)=I0,(tτp,activeheatinginterval)0,(t>τp,relaxationinterval).
Q(x, t)=βI(t)exp(-βx).
2Ts(x, t)x2-1αs Ts(x, t)t=-Q(x, t)ks,
Tg2(x, t)x2-1αg Tg(x, t)t=0.
Ti(x, 0)=0.
Ts(0, t)=Tg(0, t),
Ts(ls, t)=Tg(ls, t),
ks Ts(x, t)xx=0=kg Tg(x, t)xx=0,
ks Ts(x, t)xx=ls=kg Tg(x, t)xx=ls.
kg Tg(x, t)xx=0kg Tg(0, t)-Tg(-lg, t)lg=kglgTg(0, t).
ks Ts(x, t)xx=0-h1Ts(0, t)=0.
ks Ts(x, t)xx=ls+h2Ts(ls, t)=0.
G(x, x, t, τ)=m=1AmZ(x, x)exp[-αsδm2(t-τ)],
Am=2(ks2δm2+h22)(ks2δm2+h12)[ls(ks2δm2+h22)+ksh2]+ksh1(ks2δm2+h22),
Z(x, x)=(ksδm cos δmx+h1 sin δmx)(ksδm cos δmx+h1 sin δmx).
tan δmls=δmks(h1+h2)ks2δm2-h1h2.
Ts,a(x, t)=0t0ls Q(x, τ)ρscsG(x, x, t, τ)dxdτ,
tτp,
Ts,r(x, t)=0τp0ls Q(x, τ)ρscsG(x, x, t, τ)dxdτ,
t>τp.
Ts,a(x, t)=m=1Bmksumls cos umxls+h1 sin umxls×1-exp-umπ2 tτs,tτp,
Ts,r(x, t)=m=1Bmksumls cos umxls+h1 sin umxls×expumπ2 τpτs-1exp-umπ2 tτs,
t>τp,
Bm=βI0Amls2ksum2(um2+β2ls2){ksum×[exp(-βls)(um sin um-βls cos um)+βls]-h1ls[exp(-βls)(um cos um+βls sin um)-um]}.
Ts,a(0, t)=m=1 Bmksumls 1-exp-umπ2 tτs,
tτp,
Ts,r(0, t)=m=1 Bmksumls expumπ2 τpτs
-1exp-umπ2 tτs,t>τp.
Tg(x, t)=2π x/2αgtϕt-x24αgξ2exp(-ξ2)dξ
Tg,a(x, t)=m=1 Bmksum2ls 2 erfc-x2αgt-exp-umπ2 tτs×expjαsαgumxlserfc-x2αgt-j umπ tτs+exp-jαsαgumxlserfc-x2αgt+j umπ tτs,tτp,
Tg,r(x, t)=Tg,a(x, t)-Tg,a(x, t-τp),
t>τp.
φ(x, t)=-1n nT T(x, t)xd,
φa(x, t)=1n nTdm=1jksBmum22ls2 αsαg exp-umπ2 tτs×expjαsαgumxls×erfc-x2αgt-j umπ tτs-exp-jαsαgumxls×erfc-x2αgt+j umπ tτs,
tτp,
φr(x, t)=φa(x, t)-φa(x, t-τp),t>τp.
δm=(2m-1)π/2ls,m=1, 2, 3,.
δm=mπ/ls,m=1, 2, 3,.
δm=mπ/ls,m=1, 2, 3,.
G(x, x, t, τ)
=1ls 1+2m=1 exp-m2 tτscos mπxls cos mπxls.
Ta(x, t)=qt+m=1Em[1-exp(-m2t/τs)]cos mπxls,
Tr(x, t)=qτp+m=1Em[exp(m2τp/τs)-1]×exp(-m2t/τs)cos mπxls.
Em=2β2I0[1-(-1)m exp(-βls)]ksls(mπ/ls)2[(mπ/ls)2+β2],
q=I0ρscsls[1-exp(-βls)].
Tg,a(x, t)=qt1+x22αgterfc-x2αgt+xπαgt exp-x24αgt+m=1 Em2 2 erfc-x2αgt-exp-m2 tτs×expjmπαsαgxlserfc-x2αgt-jmtτs+exp-jmπαsαgxlserfc-x2αgt+jmtτs,tτp
Tg,r(x, t)=Tg,a(x, t)-Tg,a(x, t-τp),t>τp.
φa(x, t)=1n nTdqt-xαgt erfc-x2αgt-2παgt exp-x24αgt+m=1Dm exp-m2 tτs×expjmπαsαgxlserfc-x2αgt-jmtτs-exp-jmπαsαgxlserfc-x2αgt+jmtτs,tτp,
φr(x, t)=φa(x, t)-φa(x, t-τp),t>τp.
Dm=j2 dn nT mπlsEm,

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