Abstract

A detailed theoretical treatment of a one- (1D) and three-dimensional (3D) photothermal deflection (PTD) technique is presented. Important effects of the probe beam size occur in PTD experiments when the radius of this beam is of the order of magnitude of the thermal diffusion length. The calculation of this effect is checked by experiments in paraffin oil at low modulation frequency as well as for 1D and for 3D. In this last case, we have considered two kinds of deflection: normal and transverse, and we have studied their variation for different values of the pump beam radius. The coincidence between theoretical and experimental curves confirms the validity of our theoretical model.

© 2008 Optical Society of America

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References

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  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]
  2. J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580-4588 (1980).
    [CrossRef]
  3. W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333-1344 (1981).
    [CrossRef] [PubMed]
  4. E. Legal-Lassale, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
    [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. J. H. Rohling, J. Shen, J. Zhou, C. E. Gu, A. N. Medina, and M. L. Baesso, “Application of the diffraction theory for photothermal deflection to the measurement of the temperature coefficient of the refractive index of a binary gas mixture,” J. Appl. Phys. 64, 1-5 (1988).
  7. M. Bertolotti, G. L. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966-982 (1998).
    [CrossRef]
  8. L. C. Aamodt and J. C. Murphy, “Photothermal measurements using a localized excitation source,” J. Appl. Phys. 52, 4903-4914 (1981).
    [CrossRef]
  9. M. Soltanolkotabi and M. H. Naderi, “Three dimensional photothermal deflection and thermal lensing in solids: the effect of modulation frequency,” Jpn. J. Appl. Phys. 43, 611-620 (2004).
    [CrossRef]
  10. N. A. George, “Fiber optic position sensitive detection of photothermal deflection,” Appl. Phys. B 77, 77-80 (2003).
    [CrossRef]
  11. S. I. Yun and H. J. Seo, “Photothermal beam deflection technique for the study of solids,” Chin. J. Phys. 30, 753-765(1992).
  12. T. Ghrib, N. Yacoubi, and F. Saadallah, “Simultaneous determination of thermal conductivity and diffusivity of solid samples using the 'mirage effect' method,” Sens. Actuators A 135, 346-354 (2007).
    [CrossRef]
  13. D. Dangoisse, D. Hennequin, and V. Zehnlé-Dhaoui, Les Laser (DUNOD, 1998).

2007 (1)

T. Ghrib, N. Yacoubi, and F. Saadallah, “Simultaneous determination of thermal conductivity and diffusivity of solid samples using the 'mirage effect' method,” Sens. Actuators A 135, 346-354 (2007).
[CrossRef]

2004 (1)

M. Soltanolkotabi and M. H. Naderi, “Three dimensional photothermal deflection and thermal lensing in solids: the effect of modulation frequency,” Jpn. J. Appl. Phys. 43, 611-620 (2004).
[CrossRef]

2003 (1)

N. A. George, “Fiber optic position sensitive detection of photothermal deflection,” Appl. Phys. B 77, 77-80 (2003).
[CrossRef]

2002 (1)

1998 (1)

M. Bertolotti, G. L. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966-982 (1998).
[CrossRef]

1992 (1)

S. I. Yun and H. J. Seo, “Photothermal beam deflection technique for the study of solids,” Chin. J. Phys. 30, 753-765(1992).

1988 (2)

J. H. Rohling, J. Shen, J. Zhou, C. E. Gu, A. N. Medina, and M. L. Baesso, “Application of the diffraction theory for photothermal deflection to the measurement of the temperature coefficient of the refractive index of a binary gas mixture,” J. Appl. Phys. 64, 1-5 (1988).

E. Legal-Lassale, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
[CrossRef]

1981 (2)

W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333-1344 (1981).
[CrossRef] [PubMed]

L. C. Aamodt and J. C. Murphy, “Photothermal measurements using a localized excitation source,” J. Appl. Phys. 52, 4903-4914 (1981).
[CrossRef]

1980 (2)

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

J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580-4588 (1980).
[CrossRef]

Jpn. J. Appl. Phys. (1)

M. Soltanolkotabi and M. H. Naderi, “Three dimensional photothermal deflection and thermal lensing in solids: the effect of modulation frequency,” Jpn. J. Appl. Phys. 43, 611-620 (2004).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

N. A. George, “Fiber optic position sensitive detection of photothermal deflection,” Appl. Phys. B 77, 77-80 (2003).
[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]

Chin. J. Phys. (1)

S. I. Yun and H. J. Seo, “Photothermal beam deflection technique for the study of solids,” Chin. J. Phys. 30, 753-765(1992).

J. Appl. Phys. (5)

J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580-4588 (1980).
[CrossRef]

E. Legal-Lassale, F. Lepoutre, and J. P. Roger, “Probe beam size effects in photothermal deflection experiments,” J. Appl. Phys. 64, 1-5 (1988).
[CrossRef]

J. H. Rohling, J. Shen, J. Zhou, C. E. Gu, A. N. Medina, and M. L. Baesso, “Application of the diffraction theory for photothermal deflection to the measurement of the temperature coefficient of the refractive index of a binary gas mixture,” J. Appl. Phys. 64, 1-5 (1988).

M. Bertolotti, G. L. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, “Analysis of the photothermal deflection technique in the surface reflection scheme: theory and experiment,” J. Appl. Phys. 83, 966-982 (1998).
[CrossRef]

L. C. Aamodt and J. C. Murphy, “Photothermal measurements using a localized excitation source,” J. Appl. Phys. 52, 4903-4914 (1981).
[CrossRef]

Opt. Lett. (1)

Sens. Actuators A (1)

T. Ghrib, N. Yacoubi, and F. Saadallah, “Simultaneous determination of thermal conductivity and diffusivity of solid samples using the 'mirage effect' method,” Sens. Actuators A 135, 346-354 (2007).
[CrossRef]

Other (1)

D. Dangoisse, D. Hennequin, and V. Zehnlé-Dhaoui, Les Laser (DUNOD, 1998).

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Figures (17)

Fig. 1
Fig. 1

Schematic representation of the probe beam deflection.

Fig. 2
Fig. 2

Schematic representation of the different media browsed by the heat.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4(a)
Fig. 4(a)

Theoretical and experimental curves giving the variations of the logarithm of the (a) amplitude.

Fig. 4(b)
Fig. 4(b)

Theoretical and experimental curves giving the variations (b) phase according to the square root modulation frequency for different values of Z 0 in the uniform electrical heating case.

Fig. 5(a)
Fig. 5(a)

Theoretical and experimental curves giving the variations of the logarithm of the (a) amplitude.

Fig. 5(b)
Fig. 5(b)

Theoretical and experimental curves giving the variations of the logarithm of the (b) phase according to the square root modulation frequency for different values of R S in the uniform electrical heating case.

Fig. 6(a)
Fig. 6(a)

Theoretical and experimental curves giving the variations of the logarithm of the (a) amplitude.

Fig. 6(b)
Fig. 6(b)

Theoretical and experimental curves giving the variations of the logarithm of the (b) phase according to the square root modulation frequency for different values of Z 0 in the uniform optical heating case by a halogen lamp.

Fig. 7(a)
Fig. 7(a)

Theoretical and experimental curves giving the variations of the logarithm of the (a) amplitude.

Fig. 7(b)
Fig. 7(b)

Theoretical and experimental curves giving the variations of the logarithm of the (b) phase according to the square root modulation frequency for different values of R S in the uniform optical heating case by a halogen lamp.

Fig. 8(a)
Fig. 8(a)

Experimental curves and corresponding theoretical curves giving the variations of the logarithm of the (a) amplitude.

Fig. 8(b)
Fig. 8(b)

Experimental curves and corresponding theoretical curves giving the variations of the logarithm of the (b) phase according to the square root modulation frequency for two values of Z 0 in the uniform optical heating case by a halogen lamp.

Fig. 9(a)
Fig. 9(a)

Experimental curves and corresponding theoretical curves giving the variations of the logarithm of the (a) amplitude.

Fig. 9(b)
Fig. 9(b)

Experimental curves and corresponding theoretical curves giving the variations of the logarithm of the (b) phase for the normal deflection according to the square root modulation frequency for different values of R P .

Fig. 10(a)
Fig. 10(a)

Experimental curves and corresponding theoretical curves giving the variations of the logarithm of the (a) amplitude.

Fig. 10(b)
Fig. 10(b)

Experimental curves and corresponding theoretical curves giving the variations of the logarithm of the (b) phase for the transverse deflection according to the square root modulation frequency for different values of R P .

Equations (24)

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d d s ( n 0 d r 0 d s ) = n ,
Ψ = 1 n 0 path d n d T T d s ,
Ψ = 1 n 0 d n d T - + T d x ,
Ψ n = 1 n 0 d n d T - + d T d z d x , Ψ t = 1 n 0 d n d T - + d T d y d x .
Ψ = 1 n 0 d n d T f - L / 2 L / 2 d T f d z d x ,
Ψ = - L n 0 d n d T f σ f T 0 exp ( - σ f z ) = | Ψ ( z ) | exp ( j Φ ) ,
Ψ ( z , t ) = - L n 0 d n d T f 2 μ f | T 0 | exp ( - z 0 / μ f ) exp [ j ( θ + π 4 - z 0 μ f ) ] exp ( j ω t ) .
Ψ beam = I Ψ ray d Σ , d Σ = d y d z ,
I ( y , z ) = I 0 exp { - 2 R s 2 [ ( y - y 0 ) 2 + ( z - z 0 ) 2 ] } ,
Ψ beam ( y 0 , z 0 ) = - L n 0 d n d T f σ f T 0 - + exp [ - 2 R s 2 ( y - y 0 ) 2 ] d y 0 + exp [ - 2 R s 2 ( z - z 0 ) 2 ] exp ( - σ f z ) d z .
2 T f z 2 = 1 D f T f t i f     0 z l f , 2 T s z 2 = 1 D s T s t - A [ 1 + exp ( j ω t ) ] exp ( α z ) i f     - l s z 0 , 2 T b z 2 = 1 D b T b t i f     - l s - l b z - l s ,
T f ( z ) = T 0 exp ( - σ f z ) i f     0 z l f , T s ( z ) = U exp ( σ s z ) + V exp ( - σ s z ) - E exp ( α z ) i f     - l s z 0 , T b ( z ) = W exp [ σ b ( z + l s ) ] i f     - l b - l s z - l s ;
T f ( z = 0 ) = T s ( z = 0 ) , T s ( z = l s ) = T b ( z = l s ) ;
Φ f ( z = 0 ) = Φ s ( z = 0 ) , Φ s ( z = l s ) = Φ b ( z = l s ) .
T 0 = - E [ ( 1 - r ) ( 1 + b ) exp ( σ s l s ) - ( 1 + r ) ( 1 - b ) exp ( - σ s l s ) + 2 ( r - b ) exp ( - α l s ) ] / [ ( 1 + g ) ( 1 + b ) exp ( σ s l s ) - ( 1 - g ) ( 1 - b ) exp ( - σ s l s ) ] ,
2 T s z 2 = 1 D s T t - I 0 K s [ 1 + exp ( j ω t ) ] ,
T 0 = E [ ( 1 + b ) exp ( σ s l s ) - ( 1 - b ) exp ( - σ s l s ) - 2 b ] / [ ( 1 + g ) ( 1 + b ) exp ( σ s l s ) - ( 1 - g ) ( 1 - b ) exp ( - σ s l s ) ] ,
2 T f ( r , z ) - 2 π j f D f T f ( r , z ) = 0 , 2 T s ( r , z ) - 2 π j f D s T s ( r , z ) = - Q ( r , z ) K S , 2 T b ( r , z ) - 2 π j f D b T b ( r , z ) = 0 ,
d 2 t f d z 2 - ( 2 π j f D f + δ 2 ) t f = 0 , d 2 t s d z 2 - ( 2 π j f D s + δ 2 ) t s = - α p K s exp [ - ( δ R p ) 2 / 8 + α z ] , d 2 t b d z 2 - ( 2 π j f D b + δ 2 ) t b = 0 ,
t f = A exp ( - β f z ) , t s = B exp ( β s z ) + C exp ( - β s z ) + I exp ( α z ) , t b = E exp [ β b ( z + l s ) ] ,
t i = t j , K i t i z = K j t j z ( on the i j interface ) ,
A = I [ ( 1 - r ) ( 1 + b ) exp ( β s l s ) - ( 1 + r ) ( 1 - b ) exp ( - β s l s ) + 2 ( r - b ) exp ( - α l s ) ] / [ ( 1 + g ) ( 1 + b ) exp ( β s l s ) - ( 1 - g ) ( 1 - b ) exp ( - β s l s ) ] ,
ψ n = - 1 π n 0 d n d T f 0 + cos ( δ y ) β f A exp ( - β f z ) d δ , ψ t = - 1 π n 0 d n d T f 0 + sin ( δ y ) A exp ( - β f z ) δ d δ .
Ψ n ( y 0 , z 0 ) = - 1 π n 0 d n d T f 0 + β f A { - + cos ( δ y ) exp [ - 2 R s 2 ( y - y 0 ) 2 ] d y × 0 + exp [ - 2 R s 2 ( z - z 0 ) 2 ] exp ( - β f z ) d z } d δ Ψ t ( y 0 , z 0 ) = - 1 π n 0 d n d T f 0 + δ A { - + sin ( δ y ) exp [ - 2 R s 2 ( y - y 0 ) 2 ] d y × 0 + exp [ - 2 R s 2 ( z - z 0 ) 2 ] exp ( - β f z ) d z } d δ .

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