Abstract

The 3-D theory of impulse response photothermal detection in opaque (i.e., photothermally saturated) solids through the dependence of the surface temperature optical reflectance on the mathematical equivalent of an optical impulse (the Green’s function) is presented. The theory is extended to include the effects of the finite spatial extent of the photothermal laser source. Explicit expressions for the time-dependent temperature field have been obtained in the experimentally important cases of semi-infinite solids and solids of finite thickness in contact with thermally insulating or conducting backings.

© 1988 Optical Society of America

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References

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  1. J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal-Wave Detection and Thin-Film Thickness Measurements with Laser Beam Deflection,” Appl. Opt. 22, 3169 (1983).
    [CrossRef] [PubMed]
  2. A. Mandelis, A. Williams, E. K. M. Siu, “Photothermal Wave Imaging of Metal-Oxide-Semiconductor Field Effect Transistor Structures,” J. Appl. Phys. 63, 92 (1988).
    [CrossRef]
  3. A. Rosencwaig, J. Opsal, W. L. Smith, D. L. Willenborg, “Detection of Thermal Waves Through Optical Reflectance,” Appl. Phys. Lett. 46, 1013 (1985).
    [CrossRef]
  4. G. L. Eesley, “Generation of Nonequilibrium Electron and Lattice Temperatures in Copper by Picosecond Laser Pulses,” Phys. Rev. B 33, 2144 (1986).
    [CrossRef]
  5. C. A. Paddock, G. L. Eesley, “Transient Thermoreflectance from Thin Metal Films,” J. Appl. Phys. 60, 285 (1986).
    [CrossRef]
  6. B. M. Clemens, G. L. Eesley, C. A. Paddock, “Time-Resolved Thermal Transport in Compositionally Modulated Metal Films,” Phys. Rev. B 37, 1085 (1988).
    [CrossRef]
  7. W. P. Leung, A. C. Tam, “Thermal Diffusivity in Thin Films Measured by Noncontact Single-Ended Pulsed-Laser-Induced Thermal Radiometry,” Opt. Lett. 9, 93 (1984).
    [CrossRef] [PubMed]
  8. T. Sawada, M. Kasai, “Non-Destructive Inspection of Stacking Faults and Dislocations of Semiconductor Wafters by Photoacoustic Microscopy (PAM) and Photothermal Beam Deflection (PBD),” in Photoacoustic and Thermal Wave Phenomena in Semiconductors, A. Mandelis, Ed. (North-Holland, New York, 1987), Chap. 1.
  9. S. O. Kanstad, P.-E. Nordal, “Experimental Aspects of Photothermal Radiometry,” Can. J. Phys. 64, 1155 (1986).
    [CrossRef]
  10. A. Mandelis, “Time-Delay-Domain and Pseudorandom-Noise Photoacoustic and Photothermal Wave Processes: A Review of the State of the Art,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 590 (1986).
    [CrossRef]
  11. J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection” (Parts I–III), Rev. Sci. Instrum. 58, 2018, 2024, 2033 (1987).
    [CrossRef]
  12. M. Cardona, Modulation Spectroscopy (Academic, New York, 1969), pp. 117–136.
  13. H. C. Chow, “Theory of Three-Dimensional Photoacoustic Effect with Solids,” J. Appl. Phys. 51, 4053 (1980).
    [CrossRef]
  14. A. Mandelis, B. S. H. Royce, “Time-Domain Photoacoustic Spectroscopy of Solids,” J. Appl. Phys. 50, 4330 (1979).
    [CrossRef]
  15. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 573; A. Sommerfeld, Ann. Phys. 28, 665 (1909).
    [CrossRef]
  16. A. Rosencwaig, J. Opsal, “Thermal Wave Imaging with Thermoacoustic Detection,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 516 (1986).
    [CrossRef]
  17. R. Bellman, R. E. Marshak, G. M. Wing, “Laplace Transform Solution of Two-Medium Neutron Aging Problem,” Philos. Mag. 40, 297 (1949).
  18. H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), Chap. 14.
  19. Ref. 18, Chap. 10.10.V.
  20. Ref. 18, Chap. 10.3.
  21. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1944), p. 393.
  22. P. M. Morse, H. Feshboch, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 812.
  23. W. P. Leung, A. C. Tam, “Techniques of Flash Radiometry,” J. Appl. Phys. 56, 153 (1984).
    [CrossRef]
  24. A. C. Tam, “Pulsed Laser Photoacoustic and Photothermal Detection,” in Photoacoustic and Thermal Wave Phenomena in Semiconductors, A. Mandelis, Ed. (North-Holland, New York, 1987), Chap. 8.
  25. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC, 1964), p. 229.

1988

A. Mandelis, A. Williams, E. K. M. Siu, “Photothermal Wave Imaging of Metal-Oxide-Semiconductor Field Effect Transistor Structures,” J. Appl. Phys. 63, 92 (1988).
[CrossRef]

B. M. Clemens, G. L. Eesley, C. A. Paddock, “Time-Resolved Thermal Transport in Compositionally Modulated Metal Films,” Phys. Rev. B 37, 1085 (1988).
[CrossRef]

1987

J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection” (Parts I–III), Rev. Sci. Instrum. 58, 2018, 2024, 2033 (1987).
[CrossRef]

1986

A. Rosencwaig, J. Opsal, “Thermal Wave Imaging with Thermoacoustic Detection,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 516 (1986).
[CrossRef]

S. O. Kanstad, P.-E. Nordal, “Experimental Aspects of Photothermal Radiometry,” Can. J. Phys. 64, 1155 (1986).
[CrossRef]

A. Mandelis, “Time-Delay-Domain and Pseudorandom-Noise Photoacoustic and Photothermal Wave Processes: A Review of the State of the Art,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 590 (1986).
[CrossRef]

G. L. Eesley, “Generation of Nonequilibrium Electron and Lattice Temperatures in Copper by Picosecond Laser Pulses,” Phys. Rev. B 33, 2144 (1986).
[CrossRef]

C. A. Paddock, G. L. Eesley, “Transient Thermoreflectance from Thin Metal Films,” J. Appl. Phys. 60, 285 (1986).
[CrossRef]

1985

A. Rosencwaig, J. Opsal, W. L. Smith, D. L. Willenborg, “Detection of Thermal Waves Through Optical Reflectance,” Appl. Phys. Lett. 46, 1013 (1985).
[CrossRef]

1984

1983

1980

H. C. Chow, “Theory of Three-Dimensional Photoacoustic Effect with Solids,” J. Appl. Phys. 51, 4053 (1980).
[CrossRef]

1979

A. Mandelis, B. S. H. Royce, “Time-Domain Photoacoustic Spectroscopy of Solids,” J. Appl. Phys. 50, 4330 (1979).
[CrossRef]

1949

R. Bellman, R. E. Marshak, G. M. Wing, “Laplace Transform Solution of Two-Medium Neutron Aging Problem,” Philos. Mag. 40, 297 (1949).

Bellman, R.

R. Bellman, R. E. Marshak, G. M. Wing, “Laplace Transform Solution of Two-Medium Neutron Aging Problem,” Philos. Mag. 40, 297 (1949).

Cardona, M.

M. Cardona, Modulation Spectroscopy (Academic, New York, 1969), pp. 117–136.

Carslaw, H. S.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), Chap. 14.

Chow, H. C.

H. C. Chow, “Theory of Three-Dimensional Photoacoustic Effect with Solids,” J. Appl. Phys. 51, 4053 (1980).
[CrossRef]

Clemens, B. M.

B. M. Clemens, G. L. Eesley, C. A. Paddock, “Time-Resolved Thermal Transport in Compositionally Modulated Metal Films,” Phys. Rev. B 37, 1085 (1988).
[CrossRef]

Eesley, G. L.

B. M. Clemens, G. L. Eesley, C. A. Paddock, “Time-Resolved Thermal Transport in Compositionally Modulated Metal Films,” Phys. Rev. B 37, 1085 (1988).
[CrossRef]

G. L. Eesley, “Generation of Nonequilibrium Electron and Lattice Temperatures in Copper by Picosecond Laser Pulses,” Phys. Rev. B 33, 2144 (1986).
[CrossRef]

C. A. Paddock, G. L. Eesley, “Transient Thermoreflectance from Thin Metal Films,” J. Appl. Phys. 60, 285 (1986).
[CrossRef]

Feshboch, H.

P. M. Morse, H. Feshboch, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 812.

Jaeger, J. C.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), Chap. 14.

Kanstad, S. O.

S. O. Kanstad, P.-E. Nordal, “Experimental Aspects of Photothermal Radiometry,” Can. J. Phys. 64, 1155 (1986).
[CrossRef]

Kasai, M.

T. Sawada, M. Kasai, “Non-Destructive Inspection of Stacking Faults and Dislocations of Semiconductor Wafters by Photoacoustic Microscopy (PAM) and Photothermal Beam Deflection (PBD),” in Photoacoustic and Thermal Wave Phenomena in Semiconductors, A. Mandelis, Ed. (North-Holland, New York, 1987), Chap. 1.

Leung, W. P.

Mandelis, A.

A. Mandelis, A. Williams, E. K. M. Siu, “Photothermal Wave Imaging of Metal-Oxide-Semiconductor Field Effect Transistor Structures,” J. Appl. Phys. 63, 92 (1988).
[CrossRef]

J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection” (Parts I–III), Rev. Sci. Instrum. 58, 2018, 2024, 2033 (1987).
[CrossRef]

A. Mandelis, “Time-Delay-Domain and Pseudorandom-Noise Photoacoustic and Photothermal Wave Processes: A Review of the State of the Art,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 590 (1986).
[CrossRef]

A. Mandelis, B. S. H. Royce, “Time-Domain Photoacoustic Spectroscopy of Solids,” J. Appl. Phys. 50, 4330 (1979).
[CrossRef]

Marshak, R. E.

R. Bellman, R. E. Marshak, G. M. Wing, “Laplace Transform Solution of Two-Medium Neutron Aging Problem,” Philos. Mag. 40, 297 (1949).

Morse, P. M.

P. M. Morse, H. Feshboch, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 812.

Nordal, P.-E.

S. O. Kanstad, P.-E. Nordal, “Experimental Aspects of Photothermal Radiometry,” Can. J. Phys. 64, 1155 (1986).
[CrossRef]

Opsal, J.

A. Rosencwaig, J. Opsal, “Thermal Wave Imaging with Thermoacoustic Detection,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 516 (1986).
[CrossRef]

A. Rosencwaig, J. Opsal, W. L. Smith, D. L. Willenborg, “Detection of Thermal Waves Through Optical Reflectance,” Appl. Phys. Lett. 46, 1013 (1985).
[CrossRef]

J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal-Wave Detection and Thin-Film Thickness Measurements with Laser Beam Deflection,” Appl. Opt. 22, 3169 (1983).
[CrossRef] [PubMed]

Paddock, C. A.

B. M. Clemens, G. L. Eesley, C. A. Paddock, “Time-Resolved Thermal Transport in Compositionally Modulated Metal Films,” Phys. Rev. B 37, 1085 (1988).
[CrossRef]

C. A. Paddock, G. L. Eesley, “Transient Thermoreflectance from Thin Metal Films,” J. Appl. Phys. 60, 285 (1986).
[CrossRef]

Power, J. F.

J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection” (Parts I–III), Rev. Sci. Instrum. 58, 2018, 2024, 2033 (1987).
[CrossRef]

Rosencwaig, A.

A. Rosencwaig, J. Opsal, “Thermal Wave Imaging with Thermoacoustic Detection,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 516 (1986).
[CrossRef]

A. Rosencwaig, J. Opsal, W. L. Smith, D. L. Willenborg, “Detection of Thermal Waves Through Optical Reflectance,” Appl. Phys. Lett. 46, 1013 (1985).
[CrossRef]

J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal-Wave Detection and Thin-Film Thickness Measurements with Laser Beam Deflection,” Appl. Opt. 22, 3169 (1983).
[CrossRef] [PubMed]

Royce, B. S. H.

A. Mandelis, B. S. H. Royce, “Time-Domain Photoacoustic Spectroscopy of Solids,” J. Appl. Phys. 50, 4330 (1979).
[CrossRef]

Sawada, T.

T. Sawada, M. Kasai, “Non-Destructive Inspection of Stacking Faults and Dislocations of Semiconductor Wafters by Photoacoustic Microscopy (PAM) and Photothermal Beam Deflection (PBD),” in Photoacoustic and Thermal Wave Phenomena in Semiconductors, A. Mandelis, Ed. (North-Holland, New York, 1987), Chap. 1.

Siu, E. K. M.

A. Mandelis, A. Williams, E. K. M. Siu, “Photothermal Wave Imaging of Metal-Oxide-Semiconductor Field Effect Transistor Structures,” J. Appl. Phys. 63, 92 (1988).
[CrossRef]

Smith, W. L.

A. Rosencwaig, J. Opsal, W. L. Smith, D. L. Willenborg, “Detection of Thermal Waves Through Optical Reflectance,” Appl. Phys. Lett. 46, 1013 (1985).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 573; A. Sommerfeld, Ann. Phys. 28, 665 (1909).
[CrossRef]

Tam, A. C.

W. P. Leung, A. C. Tam, “Thermal Diffusivity in Thin Films Measured by Noncontact Single-Ended Pulsed-Laser-Induced Thermal Radiometry,” Opt. Lett. 9, 93 (1984).
[CrossRef] [PubMed]

W. P. Leung, A. C. Tam, “Techniques of Flash Radiometry,” J. Appl. Phys. 56, 153 (1984).
[CrossRef]

A. C. Tam, “Pulsed Laser Photoacoustic and Photothermal Detection,” in Photoacoustic and Thermal Wave Phenomena in Semiconductors, A. Mandelis, Ed. (North-Holland, New York, 1987), Chap. 8.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1944), p. 393.

Willenborg, D. L.

A. Rosencwaig, J. Opsal, W. L. Smith, D. L. Willenborg, “Detection of Thermal Waves Through Optical Reflectance,” Appl. Phys. Lett. 46, 1013 (1985).
[CrossRef]

J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal-Wave Detection and Thin-Film Thickness Measurements with Laser Beam Deflection,” Appl. Opt. 22, 3169 (1983).
[CrossRef] [PubMed]

Williams, A.

A. Mandelis, A. Williams, E. K. M. Siu, “Photothermal Wave Imaging of Metal-Oxide-Semiconductor Field Effect Transistor Structures,” J. Appl. Phys. 63, 92 (1988).
[CrossRef]

Wing, G. M.

R. Bellman, R. E. Marshak, G. M. Wing, “Laplace Transform Solution of Two-Medium Neutron Aging Problem,” Philos. Mag. 40, 297 (1949).

Appl. Opt.

Appl. Phys. Lett.

A. Rosencwaig, J. Opsal, W. L. Smith, D. L. Willenborg, “Detection of Thermal Waves Through Optical Reflectance,” Appl. Phys. Lett. 46, 1013 (1985).
[CrossRef]

Can. J. Phys.

S. O. Kanstad, P.-E. Nordal, “Experimental Aspects of Photothermal Radiometry,” Can. J. Phys. 64, 1155 (1986).
[CrossRef]

IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control

A. Mandelis, “Time-Delay-Domain and Pseudorandom-Noise Photoacoustic and Photothermal Wave Processes: A Review of the State of the Art,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 590 (1986).
[CrossRef]

A. Rosencwaig, J. Opsal, “Thermal Wave Imaging with Thermoacoustic Detection,” IEEE Trans. Ultrasonics, Ferroelectrics and Frequency Control UFFC-33, 516 (1986).
[CrossRef]

J. Appl. Phys.

A. Mandelis, A. Williams, E. K. M. Siu, “Photothermal Wave Imaging of Metal-Oxide-Semiconductor Field Effect Transistor Structures,” J. Appl. Phys. 63, 92 (1988).
[CrossRef]

C. A. Paddock, G. L. Eesley, “Transient Thermoreflectance from Thin Metal Films,” J. Appl. Phys. 60, 285 (1986).
[CrossRef]

H. C. Chow, “Theory of Three-Dimensional Photoacoustic Effect with Solids,” J. Appl. Phys. 51, 4053 (1980).
[CrossRef]

A. Mandelis, B. S. H. Royce, “Time-Domain Photoacoustic Spectroscopy of Solids,” J. Appl. Phys. 50, 4330 (1979).
[CrossRef]

W. P. Leung, A. C. Tam, “Techniques of Flash Radiometry,” J. Appl. Phys. 56, 153 (1984).
[CrossRef]

Opt. Lett.

Philos. Mag.

R. Bellman, R. E. Marshak, G. M. Wing, “Laplace Transform Solution of Two-Medium Neutron Aging Problem,” Philos. Mag. 40, 297 (1949).

Phys. Rev. B

B. M. Clemens, G. L. Eesley, C. A. Paddock, “Time-Resolved Thermal Transport in Compositionally Modulated Metal Films,” Phys. Rev. B 37, 1085 (1988).
[CrossRef]

G. L. Eesley, “Generation of Nonequilibrium Electron and Lattice Temperatures in Copper by Picosecond Laser Pulses,” Phys. Rev. B 33, 2144 (1986).
[CrossRef]

Rev. Sci. Instrum.

J. F. Power, A. Mandelis, “Photopyroelectric Thin Film Instrumentation and Impulse Response Detection” (Parts I–III), Rev. Sci. Instrum. 58, 2018, 2024, 2033 (1987).
[CrossRef]

Other

M. Cardona, Modulation Spectroscopy (Academic, New York, 1969), pp. 117–136.

H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids (Oxford U.P., London, 1959), Chap. 14.

Ref. 18, Chap. 10.10.V.

Ref. 18, Chap. 10.3.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1944), p. 393.

P. M. Morse, H. Feshboch, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 812.

T. Sawada, M. Kasai, “Non-Destructive Inspection of Stacking Faults and Dislocations of Semiconductor Wafters by Photoacoustic Microscopy (PAM) and Photothermal Beam Deflection (PBD),” in Photoacoustic and Thermal Wave Phenomena in Semiconductors, A. Mandelis, Ed. (North-Holland, New York, 1987), Chap. 1.

A. C. Tam, “Pulsed Laser Photoacoustic and Photothermal Detection,” in Photoacoustic and Thermal Wave Phenomena in Semiconductors, A. Mandelis, Ed. (North-Holland, New York, 1987), Chap. 8.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC, 1964), p. 229.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 573; A. Sommerfeld, Ann. Phys. 28, 665 (1909).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Three-dimensional geometry for impulse response (Green’s function) heat conduction generation due to an (effective) laser pulse; R2 = r2 + (zz0)2, where the exciting source is assumed to be at the sample surface (z0 = 0).

Fig. 2
Fig. 2

Impulse response (Green’s function) from a 3-D semi-infinite solid sample with thermal diffusivity α2 = 6 × 10−8 m2/s (PVDF polymer) using Eq. (25) at pump–probe beam offset distances r = 0.1 μm (1), 4 μm (2), and 6 μm (3). All curves have been normalized to unity peak value.

Fig. 3
Fig. 3

Impulse response (Green’s function) from a 3-D semi-infinite solid sample with α2 = 5 × 10−7 m2/s (a glass), including pump beam size effects, Eq. (47), at pump–probe beam offset distance r = 0.1 μm. Pump beam waist size w0 = 1 μm (1), 10 μm (2), and 1 mm (3).

Fig. 4
Fig. 4

Graphic depiction of Green’s function solution Eq. (50) by the method of images.

Fig. 5
Fig. 5

Three-dimensional (a) and 1-D (b) photothermal impulse response of a sample with adiabatic boundaries at z = 0 and z = −l.

Fig. 6
Fig. 6

Theoretical profiles of the transition from 3-D to 1-D photothermal impulse response of a 9-μm thick PVDF film (α2 = 6 × 10−8 m2/s) at r = 1 × 10−7μm as a function of beamwaist size: w0 = 0.1 μm (1), 10 μm (2), 25 μm (3), and 100 μm (4).

Fig. 7
Fig. 7

Theoretical predictions of the effect of the value of sample thermal diffusivity on the decay profile of the 1-D temperature field: α2 = 8 × 10−8 m2/s (1); 4 × 10−7 m2/s (2); and 8 × 10−7 m2/s (3). Other parameters are r = 1 × 10−7 m, l = 20 μm, w0 = 1 mm; n = 100.

Fig. 8
Fig. 8

One-dimensional theoretical predictions of the effect of the presence of a heat sink in contact with a solid of variable thickness: l = 100 μm (1); 5 μm (2); 3 μm (3); and 1 μm (4). Other parameters are r = 1 × 10−7 m, w0 = 1 mm, α2 = 6 × 10−8 m2/s; n = 100.

Equations (69)

Equations on this page are rendered with MathJax. Learn more.

Δ R ( r , z = 0 , t ) = R 0 + ( R T ) T = T 0 T 2 ( r , z = 0 , t ) ,
2 T i ( r , z , t ) - 1 α i T i t ( r , z , t ) = 0 ,
T i ( r , z , 0 ) = 0
2 T ¯ i ( r , z , s ) - q i 2 T ¯ i ( r , z , s ) = 0 ,
q i ( s ) ( s / α i ) 1 / 2 .
T ¯ 1 ( r , 0 , s ) = T ¯ 2 ( r , 0 , s )
T ¯ 2 ( r , - l , s ) = T ¯ 3 ( r , - l , s ) ,
k 1 T ¯ 1 z ( r , 0 , s ) = k 2 T ¯ 2 z ( r , 0 , s ) ,
k 2 T ¯ 2 z ( r , - l , s ) = k 3 T ¯ 3 z ( r , - l , s ) ,
T ¯ 1 ( r , z , s ) = 0 J 0 ( k r ) A ( k ) exp ( - σ 1 z ) d k ; z 0 ,
T ¯ 2 ( r , z , s ) = 0 J 0 ( k r ) [ C ( k ) exp ( σ 2 z ) + D ( k ) exp ( - σ 2 z ) + k exp ( - σ 2 z ) 4 π α 2 σ 2 ] d k ; 0 z - l ,
T ¯ 3 ( r , z , s ) = 0 B ( k ) J 0 ( k r ) exp [ σ 3 ( z + l ) ] d k ; z - l ,
σ i ( s ) [ k 2 + q i 2 ( s ) ] 1 / 2 ,
A - C - D = E ,
B - C exp ( - σ 2 l ) - D exp ( σ 2 l ) = E exp ( - σ 2 l ) ,
- k 1 σ 1 A - k 2 σ 2 C + k 2 σ 2 D = - k 2 σ 2 E ,
k 3 σ 3 B - k 2 σ 2 C exp ( - σ 2 l ) + k 2 σ 2 D exp ( σ 2 l ) = - k 2 σ 2 E exp ( - σ 2 l ) ,
E k 4 π α 2 σ 2 .
A ( k ) = k ( 1 + b 32 ) [ exp ( σ 2 l ) + exp ( - σ 2 l ) ] 2 π α 2 σ 2 [ ( 1 + b 32 ) ( 1 + b 12 ) exp ( σ 2 l ) - ( 1 - b 32 ) ( 1 - b 12 ) exp ( - σ 2 l ) ] ,
b i j ( s ) k i σ i ( s ) / k j σ j ( s ) .
( 1 - x ) - 1 = n = 0 x n ;             x < 1
A ( k ) = k [ 1 + exp ( - 2 σ 2 l ) ] 2 π α 2 σ 2 ( 1 + b 12 ) n = 0 ζ n exp ( - 2 n σ 2 l ) ,
ζ ( 1 - b 32 ) ( 1 - b 12 ) ( 1 + b 32 ) ( 1 + b 12 ) .
k 2 k 1
ζ 1 - b 32 1 + b 32
T ¯ 1 ( r , z , s ) = 1 2 π α 2 0 [ 1 + exp ( - 2 σ 2 l ) σ 2 ] n = 0 ζ n × exp ( - 2 n σ 2 l - σ 1 z ) J 0 ( k r ) k d k             z 0
T ¯ 1 ( r , z , s ) = 1 2 π α 2 { 0 exp ( - σ 1 z ) σ 2 J 0 ( k r ) k d k + 0 exp ( - 2 σ 2 l - σ 1 z ) σ 2 J 0 ( k r ) k d k + n = 1 0 ( ζ n σ 2 ) exp ( - 2 n σ 2 l - σ 1 z ) J 0 ( k r ) k d k + n = 1 0 ( ζ n σ 2 ) × exp [ - 2 ( n + 1 ) σ 2 l - σ 1 z ] J 0 ( k r ) k d k } .
T ¯ 1 ( r , z , s ) = 1 2 π α 2 0 F ( z , s ; k ) J 0 ( k r ) k d k ,
T ¯ 1 ( r , z , s ) = 1 2 π α 2 0 exp ( - z k 2 + q 1 2 ) k 2 + q 2 2 J 0 ( k r ) k d k .
T ¯ 1 ( r , 0 , s ) = 1 2 π α 2 0 J 0 ( k r ) k d k k 2 + q 2 2 .
T 1 ( r , 0 , t ) = 1 4 ( π α 2 t ) 3 / 2 exp ( - r 2 / 4 α 2 t ) .
μ s ( t ) 2 α 2 t
T ¯ 1 ( r , 0 , s ) = 1 2 π α 2 { 0 1 σ 2 J 0 ( k r ) k d k + 0 exp ( - 2 σ 2 l ) σ 2 J 0 ( k r ) k d k + n = 1 0 exp ( - 2 n σ 2 l ) σ 2 J 0 ( k r ) k d k + n = 1 0 exp [ - 2 ( n + 1 ) σ 2 l ] σ 1 J 0 ( k r ) k d k } .
T 1 ( r , 0 , t ) = exp ( - r 2 / 4 α 2 t ) 4 ( π α 2 t ) 3 / 2 × n = 0 ( exp [ - ( 2 n l ) 2 / 4 α 2 t ] + exp { - [ 2 ( n + 1 ) l ] 2 / 4 α 2 t } ) .
T ¯ 2 H ( r , z , s ) = 0 J 0 ( k r ) [ C ( k ) exp ( σ 2 z ) + D ( k ) exp ( - σ 2 z ) ] d k
T 2 z ( r , 0 , t ) = 1 4 π r δ ( r ) δ ( t ) .
T ˜ 2 H z ( k , 0 , t ) = 2 π δ ( t ) 0 J 0 ( k r ) δ ( r ) 4 π r r d r = ½ δ ( t ) ,
T ¯ ˜ 2 H z ( k , 0 , s ) = ½ ,
T ¯ 2 H z ( r , 0 , s ) = 1 2 0 J 0 ( k r ) k d k .
T ¯ 2 H z ( r , - l , s ) = 0.
σ 2 [ C ( k ) - D ( k ) ] = k 2 ,
C ( k ) exp ( - σ 2 l ) = D ( k ) exp ( σ 2 l ) ,
C ( k ) k 2 σ 2 [ 1 1 - exp ( - 2 σ 2 l ) ] ,
D ( k ) = k 2 σ 2 [ exp ( - 2 σ 2 l ) 1 - exp ( - 2 σ 2 l ) ] .
T 2 ( r , z , t ) = exp ( - r 2 / 4 α 2 t ) 4 ( π α 2 t ) 3 / 2 n = 0 { exp [ - ( 2 n l - z ) 2 / 4 α 2 t ] + exp { - [ 2 ( n + 1 ) l + z ] 2 / 4 α 2 t } } ,             0 z - l .
T ¯ 1 ( r , 0 , s ) = 1 2 π α 2 { I ¯ 1 ( r , 0 , s ) + I ¯ 2 l ( r , 0 , s ) + n = 1 ( - 1 ) n I ¯ 2 n l ( r , 0 , s ) + n = 1 ( - 1 ) n I ¯ 2 ( n + 1 ) l ( r , 0 , s ) } ,
T 1 ( r , 0 , t ) = exp ( - r 2 / 4 α 2 t ) 4 ( π α 2 t ) 3 / 2 n = 0 ( - 1 ) n ( exp [ - ( 2 n l ) 2 / 4 α 2 t ] + exp { - [ 2 ( n + 1 ) l ] 2 / 4 α 2 t } ) .
Q ( r , z , t ) β P 0 exp ( - r 2 / w 0 2 ) exp ( - β z ) δ ( t ) ,
T 1 d ( r , 0 , t ) = 0 - l 0 0 2 π 0 T 1 ( r , 0 , t ; r 0 , 0 , t 0 ) Q ( r 0 , z 0 , t 0 ) r 0 d r 0 d z 0 d θ 0 d t 0 = β P 0 4 ( π α 2 ) 3 / 2 0 0 0 2 π exp [ - ( r - r 0 ) 2 / 4 α 2 ( t - t 0 ) ] δ ( t 0 ) ( t - t 0 ) 3 / 2 exp ( - r 0 2 / w 0 2 ) r 0 d r 0 d θ 0 d t 0 - l 0 exp ( β z 0 ) d z 0 = P 0 4 ( π α 2 t ) 3 / 2 0 2 π 0 exp [ - ( r - r 0 ) 2 / 4 α 2 t ] exp ( - r 0 2 / w 0 2 ) r 0 d r 0 d θ 0 ,
( r - r 0 ) 2 = r 2 + r 0 2 - 2 r r 0 cos ( θ 0 - θ ) ,
0 2 π exp [ - ( 2 r r 0 / 4 α 2 t ) cos ( θ 0 - θ ) ] d θ 0 = I 0 ( r r 0 2 α 2 t ) ,
T 1 d ( r , 0 , t ) = P 0 exp ( - r 2 / 4 α 2 t ) 4 ( π α 2 t ) 3 / 2 0 exp ( - r 0 2 / Ω 0 2 ) I 0 ( r r 0 4 α 2 t ) r 0 d r 0 ,
1 Ω 0 2 1 w 0 2 + 1 4 α 2 t .
T 1 d ( r , 0 , t ) = P 0 w 0 2 2 π ( π α 2 t ) 1 / 2 ( 4 α 2 t + w 0 2 ) exp [ - r 2 / ( 4 α 2 t + w 0 2 ) ] .
A t 3 / 2 exp ( - r 2 / 4 α 2 t ) ,
T 1 d ( r , 0 , t ) w 0 2 4 α 2 t P 0 w 0 2 8 ( π α 2 t ) 3 / 2 exp ( - r 2 / 4 α 2 t ) .
T 1 d ( r , 0 , t ) w 0 2 4 α 2 t P 0 2 π ( π α 2 t ) 1 / 2 exp ( - r 2 / w 0 2 ) .
T 2 ( r , z , t ) = exp ( - r 2 / 4 α 2 t ) 8 ( π a 2 t ) 3 / 2 { exp ( - z 2 / 4 α 2 t ) + n = 0 ( exp [ - ( 2 n l - z ) 2 / 4 α 2 t ] + exp { - [ 2 ( n + 1 ) l - z ] 2 / 4 α 2 t } + 2 exp { - [ 2 ( n + 1 ) l + z ] 2 / 4 α 2 t } ) } .
T 2 ( r , z , t ) = exp ( - r 2 / 4 α 2 t ) 8 ( π α 2 t ) 3 / 2 [ exp ( - z 2 / 4 α 2 t ) + n = 0 ( - 1 ) n ( exp [ - ( 2 n l - z ) 2 / 4 α 2 t ] - exp { - [ 2 ( n + 1 ) l - z ] 2 / 4 α 2 t } - 2 exp { - [ 2 ( n + 1 ) l + z ] 2 / 4 α 2 t } ) ] .
lim z 0 T 2 ( r , z , t ) = T 2 ( r , 0 , t ) = T 1 ( r , 0 , t ) ; [ given by Eq . ( 40 ) ] .
0 J 0 ( k r ) [ 0 exp ( - w k 2 + q 2 2 ) d w ] k d k ,
L - 1 [ I ¯ 1 ( r , 0 , s ) ] I 1 ( r , 0 , t ) = 0 J 0 ( k r ) { 0 L - 1 [ exp ( - w k 2 + q 2 2 ) ] d w } k d k .
L - 1 [ exp ( - w k 2 + q 2 2 ) ] = w ( 4 π α 2 t ) 1 / 2 exp [ - ( w 2 4 α 2 t + α 2 k 2 t ) ]
I 1 ( r , 0 , t ) = 1 ( 4 π α 2 t 3 ) 1 / 2 0 k d k J 0 ( k r ) × exp ( - α 2 k 2 t ) [ 0 w exp ( - w 2 / 4 α 2 t ) d w ] = 2 α 2 t ( 4 π α 2 t 3 ) 1 / 2 0 J 0 ( k r ) exp ( - α 2 k 2 t ) k d k .
0 J 0 ( k r ) exp ( - p k 2 ) k d k = exp ( - r 2 / 4 p ) 2 p ,
I 1 ( r , 0 , t ) = 1 ( 4 π α 2 t 3 ) 1 / 2 exp ( - r 2 / 4 α 2 t ) .
I ¯ G = 0 J 0 ( k r ) [ 0 exp [ - ( w + G ) k 2 + q 2 2 ] d w ] k d k .
I G ( r , 0 , t ) = 1 ( 4 π α 2 t 3 ) 1 / 2 0 J 0 ( k r ) exp ( - α 2 k 2 t ) k d k × { 0 ( w + G ) exp [ - ( w + G ) 2 / 4 α 2 t ] d w }
I G ( r , 0 , t ) = 1 ( 4 π α 2 t 3 ) 1 / 2 exp ( - r 2 / 4 α 2 t ) exp ( - G 2 / 4 α 2 t ) .

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