Abstract

A complete and general theoretical description of dual-beam photothermal lensing spectroscopy is given. The results are valid for the most general conditions, that is, for flowing as well as stationary media, and for cw as well as pulsed excitation. For pulsed excitation, the results are valid for arbitrary pulse length. The cw results apply to both modulated as well as unmodulated excitation. Both transverse and collinear geometries are considered.

© 1988 Optical Society of America

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References

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  1. R. Gupta, “Theory of Photothermal Effect in Fluids,” in Photothermal Investigations of Solids and Fluids, J. A. Sell, Ed. (Academic, New York, 1988).
  2. H. L. Fang, R. L. Swofford, “The Thermal Lens in Absorption Spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, Ed. (Academic, New York, 1983).
  3. W. A. Weimer, N. J. Dovichi, “Time-Resolved Crossed-Beam Thermal Lens Measurements as a Noninstrusive Probe of Flow Velocity,” Appl. Opt. 24, 2981 (1985).
    [CrossRef] [PubMed]
  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 (1965).
    [CrossRef]
  5. C. Hu, J. R. Whinnery, “New Thermooptical Measurement Method and A Comparison with Other Methods,” Appl. Opt. 12, 72 (1973).
    [CrossRef] [PubMed]
  6. F. R. Grabiner, D. R. Siebert, G. W. Flynn, “Laser Induced Time-Dependent Thermal Lensing Studies of Vibrational Relaxation: Translational Cooling in CH3F,” Chem. Phys. Lett. 17, 189 (1972).
    [CrossRef]
  7. A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming. I. Theory,” Chem. Phys. 20, 253 (1977).
    [CrossRef]
  8. R. L. Swofford, J. A. Morrell, “Analysis of the Repetitively Pulsed Dual-Beam Thermooptical Absorption Spectrometer,” J. Appl. Phys. 49, 366 (1978).
    [CrossRef]
  9. H. L. Fang, R. L. Swofford, “Analysis of the Thermal Lensing Effect for an Optically Thick Sample—A Revised Model,” J. Appl. Phys. 50, 6609 (1979).
    [CrossRef]
  10. N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory of Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1700 (1984).
    [CrossRef]
  11. W. A. Weimer, N. J. Dovichi, “Time-Resolved Thermal Lens Measurements in Flowing Samples,” Anal. Chem. 57, 2436 (1985).
    [CrossRef]
  12. S. E. Bialkowski, “Photothermal Lens Aberration Effects in Two Laser Thermal Lens Spectrophotometry,” Appl. Opt. 24, 2792 (1985).
    [CrossRef] [PubMed]
  13. J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
    [CrossRef]
  14. S. E. Bialkowski, “Pulsed Laser Photothermal Spectroscopy,” Spectroscopy 1, 26 (1986).
  15. A. Rose, R. Vyas, R. Gupta, “Pulsed Photothermal Deflection Spectroscopy in a Flowing Medium: A Quantitative Investigation,” Appl. Opt. 25, 4626 (1986).
    [CrossRef] [PubMed]
  16. See, for example, A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
    [CrossRef]
  17. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).

1986 (2)

1985 (3)

1984 (1)

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory of Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1700 (1984).
[CrossRef]

1980 (1)

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
[CrossRef]

1979 (1)

H. L. Fang, R. L. Swofford, “Analysis of the Thermal Lensing Effect for an Optically Thick Sample—A Revised Model,” J. Appl. Phys. 50, 6609 (1979).
[CrossRef]

1978 (1)

R. L. Swofford, J. A. Morrell, “Analysis of the Repetitively Pulsed Dual-Beam Thermooptical Absorption Spectrometer,” J. Appl. Phys. 49, 366 (1978).
[CrossRef]

1977 (1)

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

1973 (1)

1972 (1)

F. R. Grabiner, D. R. Siebert, G. W. Flynn, “Laser Induced Time-Dependent Thermal Lensing Studies of Vibrational Relaxation: Translational Cooling in CH3F,” Chem. Phys. Lett. 17, 189 (1972).
[CrossRef]

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 (1965).
[CrossRef]

Bialkowski, S. E.

Dovichi, N. J.

W. A. Weimer, N. J. Dovichi, “Time-Resolved Thermal Lens Measurements in Flowing Samples,” Anal. Chem. 57, 2436 (1985).
[CrossRef]

W. A. Weimer, N. J. Dovichi, “Time-Resolved Crossed-Beam Thermal Lens Measurements as a Noninstrusive Probe of Flow Velocity,” Appl. Opt. 24, 2981 (1985).
[CrossRef] [PubMed]

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory of Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1700 (1984).
[CrossRef]

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
[CrossRef]

Fang, H. L.

H. L. Fang, R. L. Swofford, “Analysis of the Thermal Lensing Effect for an Optically Thick Sample—A Revised Model,” J. Appl. Phys. 50, 6609 (1979).
[CrossRef]

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

Flynn, G. W.

F. R. Grabiner, D. R. Siebert, G. W. Flynn, “Laser Induced Time-Dependent Thermal Lensing Studies of Vibrational Relaxation: Translational Cooling in CH3F,” Chem. Phys. Lett. 17, 189 (1972).
[CrossRef]

Ghatak, A. K.

See, for example, A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

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 (1965).
[CrossRef]

Grabiner, F. R.

F. R. Grabiner, D. R. Siebert, G. W. Flynn, “Laser Induced Time-Dependent Thermal Lensing Studies of Vibrational Relaxation: Translational Cooling in CH3F,” Chem. Phys. Lett. 17, 189 (1972).
[CrossRef]

Gupta, R.

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

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

Harris, J. M.

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
[CrossRef]

Hu, C.

Kliger, D. S.

A. J. Twarowski, D. S. Kliger, “Multiphoton Absorption Spectra Using Thermal Blooming. I. Theory,” Chem. Phys. 20, 253 (1977).
[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 (1965).
[CrossRef]

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 (1965).
[CrossRef]

Morrell, J. A.

R. L. Swofford, J. A. Morrell, “Analysis of the Repetitively Pulsed Dual-Beam Thermooptical Absorption Spectrometer,” J. Appl. Phys. 49, 366 (1978).
[CrossRef]

Nolan, T. G.

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory of Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1700 (1984).
[CrossRef]

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 (1965).
[CrossRef]

Rose, A.

Siebert, D. R.

F. R. Grabiner, D. R. Siebert, G. W. Flynn, “Laser Induced Time-Dependent Thermal Lensing Studies of Vibrational Relaxation: Translational Cooling in CH3F,” Chem. Phys. Lett. 17, 189 (1972).
[CrossRef]

Swofford, R. L.

H. L. Fang, R. L. Swofford, “Analysis of the Thermal Lensing Effect for an Optically Thick Sample—A Revised Model,” J. Appl. Phys. 50, 6609 (1979).
[CrossRef]

R. L. Swofford, J. A. Morrell, “Analysis of the Repetitively Pulsed Dual-Beam Thermooptical Absorption Spectrometer,” J. Appl. Phys. 49, 366 (1978).
[CrossRef]

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

Thyagarajan, K.

See, for example, A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Twarowski, A. J.

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

Vyas, R.

Weimer, W. A.

W. A. Weimer, N. J. Dovichi, “Time-Resolved Thermal Lens Measurements in Flowing Samples,” Anal. Chem. 57, 2436 (1985).
[CrossRef]

W. A. Weimer, N. J. Dovichi, “Time-Resolved Crossed-Beam Thermal Lens Measurements as a Noninstrusive Probe of Flow Velocity,” Appl. Opt. 24, 2981 (1985).
[CrossRef] [PubMed]

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory of Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1700 (1984).
[CrossRef]

Whinnery, J. R.

C. Hu, J. R. Whinnery, “New Thermooptical Measurement Method and A Comparison with Other Methods,” Appl. Opt. 12, 72 (1973).
[CrossRef] [PubMed]

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 (1965).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).

Anal. Chem. (3)

N. J. Dovichi, T. G. Nolan, W. A. Weimer, “Theory of Laser-Induced Photothermal Refraction,” Anal. Chem. 56, 1700 (1984).
[CrossRef]

W. A. Weimer, N. J. Dovichi, “Time-Resolved Thermal Lens Measurements in Flowing Samples,” Anal. Chem. 57, 2436 (1985).
[CrossRef]

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
[CrossRef]

Appl. Opt. (4)

Chem. Phys. (1)

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

Chem. Phys. Lett. (1)

F. R. Grabiner, D. R. Siebert, G. W. Flynn, “Laser Induced Time-Dependent Thermal Lensing Studies of Vibrational Relaxation: Translational Cooling in CH3F,” Chem. Phys. Lett. 17, 189 (1972).
[CrossRef]

J. Appl. Phys. (3)

R. L. Swofford, J. A. Morrell, “Analysis of the Repetitively Pulsed Dual-Beam Thermooptical Absorption Spectrometer,” J. Appl. Phys. 49, 366 (1978).
[CrossRef]

H. L. Fang, R. L. Swofford, “Analysis of the Thermal Lensing Effect for an Optically Thick Sample—A Revised Model,” J. Appl. Phys. 50, 6609 (1979).
[CrossRef]

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 (1965).
[CrossRef]

Spectroscopy (1)

S. E. Bialkowski, “Pulsed Laser Photothermal Spectroscopy,” Spectroscopy 1, 26 (1986).

Other (4)

See, for example, A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).

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

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

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Figures (17)

Fig. 1
Fig. 1

Schematic illustration of the photothermal lensing effect.

Fig. 2
Fig. 2

Focal length of a thermal lens.

Fig. 3
Fig. 3

Pump–probe beam configuration for (a) transverse, (b) collinear photothermal lensing spectroscopy.

Fig. 4
Fig. 4

Detection of a photothermal lens.

Fig. 5
Fig. 5

Pulsed collinear PTLS signals in a stationary medium. The signal is plotted as a function of time for various pump–probe separations. The laser energy was assumed to be 6 mJ, and all other parameters used in this calculation are given in the text.

Fig. 6
Fig. 6

Pulsed collinear PTLS signals in a medium flowing with a velocity of 2 m/s. Negative x corresponds to the probe beam being upstream from the pump beam.

Fig. 7
Fig. 7

Pulsed transverse PTLS signals in a stationary medium.

Fig. 8
Fig. 8

Pulsed transverse PTLS signals in a flowing medium.

Fig. 9
Fig. 9

PTLS signal shapes for a long laser pulse (t0 = 0.5 ms) for four values of the flow velocity, as labeled. Time t = 0 corresponds to the start of the laser pulse. The signal has been plotted in arbitrary units.

Fig. 10
Fig. 10

Temporal evolution of the temperature at several positions in the medium. The cw laser was assumed to be modulated at 100 Hz 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. 11
Fig. 11

“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, and its 1/e2 radius was assumed to be 0.5 mm. All other parameters have been given previously in connection with pulsed PTLS. The top two curves have been expanded by the indicated factors for clarity.

Fig. 12
Fig. 12

Transverse cw PTLS signals corresponding to the temperature distribution shown in Fig. 11 for several flow velocities.

Fig. 13
Fig. 13

Root mean square values of transverse PTLS signals for a modulation frequency of 10 Hz.

Fig. 14
Fig. 14

Transverse PTLS signals at four times in the modulation cycle of the laser. t = 1.00 s corresponds to the peak of the laser intensity, and t = 1.05 s corresponds to the off portion of the modulation cycle.

Fig. 15
Fig. 15

Root mean square values of collinear PTLS signals for f = 10 Hz. Interaction length was assumed to be 1 cm.

Fig. 16
Fig. 16

Dependence of the rms transverse PTLS signals on the modulation frequency of the laser in a stationary medium.

Fig. 17
Fig. 17

Curves similar to those shown in Fig. 16 for vx = 10 cm/s.

Equations (52)

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T ( r , t ) t = D 2 T ( r , t ) - v 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 av 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 + v 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 + v 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 - v 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 - v x ( t - τ ) ] 2 + y 2 } / [ 8 D ( t - τ ) + a 2 ] ) d τ
d d s ( n 0 d δ d s ) = n ( x , y , t ) ,
d d s ( n 0 d δ x d s ) = n x
ϕ x ( x , y ) = 1 n 0 n x d s ,
n ( x , y ) = n ( x , y ) + ( x - x ) ( n x ) x , y + ( y - y ) ( n y ) x , y + ( x - x ) 2 2 ( 2 n x 2 ) x , y + ( y - y ) 2 2 ( 2 n y 2 ) x , y + ( x - x ) ( y - y ) ( 2 n x y ) x , y + .
ϕ x ( x , y ) = 1 n 0 ( n x ) x , y d s + 1 n 0 ( x - x ) ( 2 n x 2 ) x , y d s ,
ϕ x ( x , y ) = ϕ x ( x , y ) + ( x - x ) ϕ x x .
ϕ x ( x , y ) = 1 n 0 ( n x ) x , y d s ,
( x - x ) ϕ x x = 1 n 0 ( x - x ) ( 2 n x 2 ) x , y d s .
d d s ( Δ ϕ x ) = 1 n 0 ( 2 n x 2 ) x , y Δ x ,
1 f x = Δ ϕ x Δ x ,
1 f x = - l R x Δ x = - 1 n 0 ( 2 n x 2 ) x , y l .
1 f x = - ( 2 n x 2 ) x , y l .
n ( x , y , t ) = n 0 + n T | T A T ( x , y , t ) ,
1 f x = - n T ( 2 T x 2 ) x , y l .
1 f x = - n T path ( 2 T x 2 ) x , y d s .
1 f x = - n T l ( 2 T x 2 ) x = x y = 0 ,
1 f y = - n T l ( 2 T y 2 ) x = x y = 0 .
1 f x = - n T ( 2 T x 2 ) x = x d y ,
1 f z = 0.
s ( t ) = P det ( t ) - P det ( 0 ) P det ( 0 ) .
I ( r ) = 2 P π w 2 2 exp ( - 2 r 2 / w 2 2 ) ,
P det = 0 b I ( r ) 2 π r d r 2 P π b 2 π w 2 2 .
s ( t ) = w 2 2 ( 0 ) - w 2 2 ( t ) w 2 2 ( 0 ) ,
q 2 = A q 0 + B C q 0 + D ,
[ A B C D ] = [ 1 z 2 0 1 ] [ 1 0 - 1 / f 1 ] [ 1 z 1 0 1 ] ,
1 q = 1 R - i λ π n w 2 ,
w 2 2 ( t ) = w 0 2 [ ( 1 - z 2 f ) 2 + 1 z 0 2 ( z 1 + z 2 - z 1 z 2 f ) 2 ] ,
s ( t ) = 2 z 2 f ( t ) [ z 0 2 + ( z 1 + z 2 ) 2 ] × [ z 1 2 + z 0 2 + z 1 z 2 - z 2 2 f ( t ) ( z 0 2 + z 1 2 ) ] .
s ( t ) = 2 z 1 f ( t ) .
P det = 2 P π b 2 π w x w y ,
s ( t ) = w x ( 0 ) w y ( 0 ) - w x ( t ) w y ( t ) w x ( 0 ) w y ( 0 ) .
w x ( t ) = w 0 z 2 z 0 [ 1 - z 1 f x ( t ) ]
s ( t ) = z 1 f x ( t ) + z 1 f y ( t ) ,
s L ( x , t ) = 8 α E 0 l z 1 π ρ C p t 0 ( n T ) 0 t 0 1 [ a 2 + 8 D ( t - τ ) ] 2 × { 2 - 4 [ x - v x ( t - τ ) ] 2 [ a 2 + 8 D ( t - τ ) ] } × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ a 2 + 8 D ( t - τ ) ] } d τ ,
lim t 0 0 0 t 0 f ( τ ) d τ = f ( 0 ) t 0
s L ( x , t ) = 8 α E 0 l z 1 π ρ C p ( n T ) 1 ( a 2 + 8 D t ) 2 [ 2 - 4 ( x - v x t ) 2 ( a 2 + 8 D t ) ] × exp [ - 2 ( x - v x t ) 2 / ( a 2 + 8 D t ) ] .
s T ( x , t ) = z 1 f x = - z 1 ( n T ) / ( 2 T x 2 ) d y ,
s T ( x , t ) = 8 α E 0 z 1 2 π ρ C p t 0 ( n T ) 0 t 0 1 [ a 2 + 8 D ( t - τ ) ] 3 / 2 × { 1 - 4 [ x - v x ( t - τ ) ] 2 [ a 2 + 8 D ( t - τ ) ] } × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ a 2 + 8 D ( t - τ ) ] } d τ .
s T ( x , t ) = 8 α E 0 z 1 2 π ρ C p ( n T ) 1 ( a 2 + 8 D t ) 3 / 2 { 1 - 4 ( x - v x t ) 2 ( a 2 + 8 D t ) } × exp [ - 2 ( x - v x t ) 2 / ( a 2 + 8 D t ) ] .
s L ( x , t ) = 8 α P av l z 1 π ρ C p ( n T ) 0 t [ ( 1 + cos ω τ ) ] [ a 2 + 8 D ( t - τ ) ] 2 × { 2 - 4 [ x - v x ( t - τ ) ] 2 [ a 2 + 8 D ( t - τ ) ] } × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ a 2 + 8 D ( t - τ ) ] } d τ .
s T ( x , t ) = 8 α P av z 1 2 π ρ C p ( n T ) 0 t [ ( 1 + cos ω τ ) ] [ a 2 + 8 D ( t - τ ) ] 3 / 2 × { 1 - 4 [ x - v x ( t - τ ) ] 2 [ a 2 + 8 D ( t - τ ) ] } × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ a 2 + 8 D ( t - τ ) } d τ .
T ( x = y = 0 , t ) = α P 0 4 π ρ C p D ln ( 1 + 2 t t c ) ,
s L ( x = y = 0 , t ) = 2 α P l z 1 π ρ C p D a 2 ( n T ) 1 ( 1 + t c / 2 t ) .

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