Abstract

A comprehensive investigation of pulsed photothermal deflection spectroscopy in a flowing medium has been carried out. A rigorous solution of the appropriate diffusion equation has been obtained, and experiments have been conducted to verify the theoretical predictions. Absolute measurements of the photothermal deflection were made and no adjustable parameters were used in the theory. Very good agreement between the theory and the experiment was obtained.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. C. Davis, “Trace Detection of Gases Using Phase Fluctuation Optical Heterodyne Spectroscopy,” Appl. Phys. Lett. 36, 515 (1980).
    [CrossRef]
  2. A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130 (1980).
    [CrossRef]
  3. A. C. Boccara, D. Fournier, W. B. Jackson, N. M. Amer, “Sensitive Photothermal Deflection Technique for Measuring Absorption in Optically Thin Media,” Opt. Lett. 5, 377 (1980).
    [CrossRef] [PubMed]
  4. D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ Trace-Gas Detection by Photothermal Deflection Spectroscopy,” Appl. Phys. Lett. 37, 519 (1980).
    [CrossRef]
  5. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal Deflection Spectroscopy and Detection,” Appl. Opt. 20, 1333 (1981).
    [CrossRef] [PubMed]
  6. J. C. Loulergue, A. C. Tam, “Noncontact Photothermal Probe Beam Deflection Measurement of Thermal Diffusivity in a Hot Unconfined Gas,” Appl. Phys. Lett. 46, 457 (1985).
    [CrossRef]
  7. H. Sontag, A. C. Tam, “Time-Resolved Flow-Velocity and Concentration Measurements Using a Traveling Thermal Lens,” Opt. Lett. 10, 436 (1985).
    [CrossRef] [PubMed]
  8. A. C. Tam, H. Sontag, P. Hess, “Photothermal Probe Beam Deflection Monitoring of Photochemical Particulate Production,” Chem. Phys. Lett. 120, 280 (1985).
    [CrossRef]
  9. H. Sontag, A. C. Tam, “Characterization of Vapor and Aerosol Flows by Photothermal Methods,” Can. J. Phys.Sept.1986, to be published.
    [CrossRef]
  10. For a review see A. C. Tam, “Applications of Photoacoustic Sensing Techniques,” Rev. Mod. Phys. 58, 381 (1986).
    [CrossRef]
  11. A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the Photothermal Deflection Technique to Combustion Diagnostics,” Appl. Opt. 21, 2663 (1982).
    [CrossRef] [PubMed]
  12. A. Rose, J. D. Pyrum, G. J. Salamo, R. Gupta, “Photoacoustic Spectroscopy and Photothermal Deflection Spectroscopy: New Tools for Combustion Diagnostics,” in Proceedings, International Conference on Lasers ’82, R. C. Powell, Ed. (STS Press, McLean, VA, 1983).
  13. S. W. Kizirnis, R. J. Brecha, B. N. Ganguly, L. P. Goss, R. Gupta, “Hydroxyl (OH) Distributions and Temperature Profiles in a Premixed Propane Flame Obtained by Laser Deflection Techniques,” Appl. Opt. 23, 3873 (1984).
    [CrossRef] [PubMed]
  14. A. Rose, R. Gupta, “Combustion Diagnostics by Photodeflection Spectroscopy,” in Twentieth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1984).
  15. A. Rose, R. Gupta, “Application of Photothermal Deflection Technique to Flow-Velocity Measurements in a Flame,” Opt. Lett. 10, 532 (1985).
    [CrossRef] [PubMed]
  16. A. Rose, R. Gupta, “Application of Photothermal and Photoacoustic Deflection Techniques to Sooting Flames: Velocity, Temperature, and Concentration Measurements,” Opt. Commun. 56, 303 (1986).
    [CrossRef]
  17. R. Gupta, “A Quantitative Investigation of Pulsed Photothermal and Photoacoustic Deflection Spectroscopy for Combustion Diagnostics,” AIP Conf. Proc. 146, 672 (1986).
    [CrossRef]
  18. J. A. Sell, “Gas Velocity Measurements Using Photothermal Deflection Spectroscopy,” Appl. Opt. 24, 3725 (1985).
    [CrossRef] [PubMed]
  19. V. Zharov, N. M. Amer, “Pulsed Photothermal Deflection Spectroscopy in Flowing Media,” in Technical Digest, Fourth International Topical Meeting on Photoacoustic, Thermal, and Related Science, Ville D’Esterel, Quebec (1985).
  20. J. A. Sell, “Quantitative Photothermal Deflection Spectroscopy in a Flowing Stream of Gas,” Appl. Opt. 23, 1586 (1984).
    [CrossRef] [PubMed]
  21. A. Rose, G. J. Salamo, R. Gupta, “Photoacoustic Deflection Spectroscopy: A New Specie-Specific Method for Combustion Diagnostics,” Appl. Opt. 23, 781 (1984).
    [CrossRef] [PubMed]
  22. A. C. Tam, H. Coufal, “Pulsed Opto-Acoustics: Theory and Applications,” J. Phys. Colloq. C6, 44, 9 (1983).
  23. A. Rose, Y.-X. Nie, R. Gupta, “A Quantitative Investigation of Pulsed Photoacoustic Deflection Spectroscopy in Gaseous Media,” to be submitted to Appl. Opt.
  24. See, for example, M. V. Klein, Optics (Wiley, New York, 1970).
  25. W. A. Weimer, N. J. Dovichi, “Time-Resolved Crossed-Beam Thermal Lens Measurement as a Nonintrusive Probe of Flow Velocity,” Appl. Opt. 24, 2981 (1985).
    [CrossRef] [PubMed]
  26. Y.-X. Nie, K. Hane, R. Gupta, “Measurement of Very Low Gas Flow Velocities by Photothermal Deflection Spectroscopy,” Appl. Opt. 25, 3247 (1986).
    [CrossRef] [PubMed]
  27. W. M. Rutherford, W. J. Roos, K. J. Kaminski, “Experimental Verification of the Thermal Diffusion Column Theory as Applied to the Separation of Isotopically Substituted Nitrogen and Isotopically Substituted Oxygen,” J. Chem. Phys. 50, 5359 (1969).
    [CrossRef]
  28. Reference 7. Our result differs from that of Ref. 7 by a factor of 2. We believe that there is an error in Ref. 7.
  29. A. J. Twarowski, D. S. Klinger, “Multiphoton Absorption Spectra Using Thermal Blooming,” Chem. Phys. 20, 253 (1977).
    [CrossRef]
  30. J. M. Khosrofian, B. A. Garetz, “Measurement of a Gaussian Laser Beam Diameter Through the Direct Inversion of Knife-Edge Data,” Appl. Opt. 22, 3406 (1983).
    [CrossRef] [PubMed]
  31. R. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Boca Raton, FL, 1984).
  32. V. M. Donnelly, F. Kaufman, “Fluorescence Lifetime Studies of NO2. I. Excitation of the Perturbed 2B2 State near 600 nm,” J. Chem. Phys. 66, 4100 (1977).
    [CrossRef]
  33. A. Rose, Y.-X. Nie, R. Gupta, “Laser Beam Profile Measurement by Photothermal Deflection Technique,” Appl. Opt. 25, 1738 (1986).
    [CrossRef] [PubMed]
  34. A. Rose, “Development of Pulsed Photoacoustic and Photothermal Deflection Spectroscopy as Diagnostic Tools for Combustion,” Ph.D. Dissertation, U. Arkansas (1986).

1986

For a review see A. C. Tam, “Applications of Photoacoustic Sensing Techniques,” Rev. Mod. Phys. 58, 381 (1986).
[CrossRef]

A. Rose, R. Gupta, “Application of Photothermal and Photoacoustic Deflection Techniques to Sooting Flames: Velocity, Temperature, and Concentration Measurements,” Opt. Commun. 56, 303 (1986).
[CrossRef]

R. Gupta, “A Quantitative Investigation of Pulsed Photothermal and Photoacoustic Deflection Spectroscopy for Combustion Diagnostics,” AIP Conf. Proc. 146, 672 (1986).
[CrossRef]

Y.-X. Nie, K. Hane, R. Gupta, “Measurement of Very Low Gas Flow Velocities by Photothermal Deflection Spectroscopy,” Appl. Opt. 25, 3247 (1986).
[CrossRef] [PubMed]

A. Rose, Y.-X. Nie, R. Gupta, “Laser Beam Profile Measurement by Photothermal Deflection Technique,” Appl. Opt. 25, 1738 (1986).
[CrossRef] [PubMed]

1985

1984

1983

1982

1981

1980

C. C. Davis, “Trace Detection of Gases Using Phase Fluctuation Optical Heterodyne Spectroscopy,” Appl. Phys. Lett. 36, 515 (1980).
[CrossRef]

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

A. C. Boccara, D. Fournier, W. B. Jackson, N. M. Amer, “Sensitive Photothermal Deflection Technique for Measuring Absorption in Optically Thin Media,” Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ Trace-Gas Detection by Photothermal Deflection Spectroscopy,” Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

1977

V. M. Donnelly, F. Kaufman, “Fluorescence Lifetime Studies of NO2. I. Excitation of the Perturbed 2B2 State near 600 nm,” J. Chem. Phys. 66, 4100 (1977).
[CrossRef]

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

1969

W. M. Rutherford, W. J. Roos, K. J. Kaminski, “Experimental Verification of the Thermal Diffusion Column Theory as Applied to the Separation of Isotopically Substituted Nitrogen and Isotopically Substituted Oxygen,” J. Chem. Phys. 50, 5359 (1969).
[CrossRef]

Amer, N. M.

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

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ Trace-Gas Detection by Photothermal Deflection Spectroscopy,” Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

A. C. Boccara, D. Fournier, W. B. Jackson, N. M. Amer, “Sensitive Photothermal Deflection Technique for Measuring Absorption in Optically Thin Media,” Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

V. Zharov, N. M. Amer, “Pulsed Photothermal Deflection Spectroscopy in Flowing Media,” in Technical Digest, Fourth International Topical Meeting on Photoacoustic, Thermal, and Related Science, Ville D’Esterel, Quebec (1985).

Badoz, J.

A. C. Boccara, D. Fournier, J. Badoz, “Thermo-Optical Spectroscopy: Detection by the Mirage Effect,” Appl. Phys. Lett. 36, 130 (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 (1981).
[CrossRef] [PubMed]

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ Trace-Gas Detection by Photothermal Deflection Spectroscopy,” Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

A. C. Boccara, D. Fournier, W. B. Jackson, N. M. Amer, “Sensitive Photothermal Deflection Technique for Measuring Absorption in Optically Thin Media,” Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

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

Brecha, R. J.

Coufal, H.

A. C. Tam, H. Coufal, “Pulsed Opto-Acoustics: Theory and Applications,” J. Phys. Colloq. C6, 44, 9 (1983).

Davis, C. C.

C. C. Davis, “Trace Detection of Gases Using Phase Fluctuation Optical Heterodyne Spectroscopy,” Appl. Phys. Lett. 36, 515 (1980).
[CrossRef]

Donnelly, V. M.

V. M. Donnelly, F. Kaufman, “Fluorescence Lifetime Studies of NO2. I. Excitation of the Perturbed 2B2 State near 600 nm,” J. Chem. Phys. 66, 4100 (1977).
[CrossRef]

Dovichi, N. J.

Fournier, D.

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

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

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ Trace-Gas Detection by Photothermal Deflection Spectroscopy,” Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

A. C. Boccara, D. Fournier, W. B. Jackson, N. M. Amer, “Sensitive Photothermal Deflection Technique for Measuring Absorption in Optically Thin Media,” Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

Ganguly, B. N.

Garetz, B. A.

Gerlach, R.

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ Trace-Gas Detection by Photothermal Deflection Spectroscopy,” Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

Goss, L. P.

Gupta, R.

A. Rose, R. Gupta, “Application of Photothermal and Photoacoustic Deflection Techniques to Sooting Flames: Velocity, Temperature, and Concentration Measurements,” Opt. Commun. 56, 303 (1986).
[CrossRef]

R. Gupta, “A Quantitative Investigation of Pulsed Photothermal and Photoacoustic Deflection Spectroscopy for Combustion Diagnostics,” AIP Conf. Proc. 146, 672 (1986).
[CrossRef]

Y.-X. Nie, K. Hane, R. Gupta, “Measurement of Very Low Gas Flow Velocities by Photothermal Deflection Spectroscopy,” Appl. Opt. 25, 3247 (1986).
[CrossRef] [PubMed]

A. Rose, Y.-X. Nie, R. Gupta, “Laser Beam Profile Measurement by Photothermal Deflection Technique,” Appl. Opt. 25, 1738 (1986).
[CrossRef] [PubMed]

A. Rose, R. Gupta, “Application of Photothermal Deflection Technique to Flow-Velocity Measurements in a Flame,” Opt. Lett. 10, 532 (1985).
[CrossRef] [PubMed]

S. W. Kizirnis, R. J. Brecha, B. N. Ganguly, L. P. Goss, R. Gupta, “Hydroxyl (OH) Distributions and Temperature Profiles in a Premixed Propane Flame Obtained by Laser Deflection Techniques,” Appl. Opt. 23, 3873 (1984).
[CrossRef] [PubMed]

A. Rose, G. J. Salamo, R. Gupta, “Photoacoustic Deflection Spectroscopy: A New Specie-Specific Method for Combustion Diagnostics,” Appl. Opt. 23, 781 (1984).
[CrossRef] [PubMed]

A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the Photothermal Deflection Technique to Combustion Diagnostics,” Appl. Opt. 21, 2663 (1982).
[CrossRef] [PubMed]

A. Rose, J. D. Pyrum, G. J. Salamo, R. Gupta, “Photoacoustic Spectroscopy and Photothermal Deflection Spectroscopy: New Tools for Combustion Diagnostics,” in Proceedings, International Conference on Lasers ’82, R. C. Powell, Ed. (STS Press, McLean, VA, 1983).

A. Rose, Y.-X. Nie, R. Gupta, “A Quantitative Investigation of Pulsed Photoacoustic Deflection Spectroscopy in Gaseous Media,” to be submitted to Appl. Opt.

A. Rose, R. Gupta, “Combustion Diagnostics by Photodeflection Spectroscopy,” in Twentieth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1984).

Hane, K.

Hess, P.

A. C. Tam, H. Sontag, P. Hess, “Photothermal Probe Beam Deflection Monitoring of Photochemical Particulate Production,” Chem. Phys. Lett. 120, 280 (1985).
[CrossRef]

Jackson, W. B.

Kaminski, K. J.

W. M. Rutherford, W. J. Roos, K. J. Kaminski, “Experimental Verification of the Thermal Diffusion Column Theory as Applied to the Separation of Isotopically Substituted Nitrogen and Isotopically Substituted Oxygen,” J. Chem. Phys. 50, 5359 (1969).
[CrossRef]

Kaufman, F.

V. M. Donnelly, F. Kaufman, “Fluorescence Lifetime Studies of NO2. I. Excitation of the Perturbed 2B2 State near 600 nm,” J. Chem. Phys. 66, 4100 (1977).
[CrossRef]

Khosrofian, J. M.

Kizirnis, S. W.

Klein, M. V.

See, for example, M. V. Klein, Optics (Wiley, New York, 1970).

Klinger, D. S.

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

Loulergue, J. C.

J. C. Loulergue, A. C. Tam, “Noncontact Photothermal Probe Beam Deflection Measurement of Thermal Diffusivity in a Hot Unconfined Gas,” Appl. Phys. Lett. 46, 457 (1985).
[CrossRef]

Muzny, C.

Nie, Y.-X.

Pyrum, J. D.

A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the Photothermal Deflection Technique to Combustion Diagnostics,” Appl. Opt. 21, 2663 (1982).
[CrossRef] [PubMed]

A. Rose, J. D. Pyrum, G. J. Salamo, R. Gupta, “Photoacoustic Spectroscopy and Photothermal Deflection Spectroscopy: New Tools for Combustion Diagnostics,” in Proceedings, International Conference on Lasers ’82, R. C. Powell, Ed. (STS Press, McLean, VA, 1983).

Roos, W. J.

W. M. Rutherford, W. J. Roos, K. J. Kaminski, “Experimental Verification of the Thermal Diffusion Column Theory as Applied to the Separation of Isotopically Substituted Nitrogen and Isotopically Substituted Oxygen,” J. Chem. Phys. 50, 5359 (1969).
[CrossRef]

Rose, A.

A. Rose, R. Gupta, “Application of Photothermal and Photoacoustic Deflection Techniques to Sooting Flames: Velocity, Temperature, and Concentration Measurements,” Opt. Commun. 56, 303 (1986).
[CrossRef]

A. Rose, Y.-X. Nie, R. Gupta, “Laser Beam Profile Measurement by Photothermal Deflection Technique,” Appl. Opt. 25, 1738 (1986).
[CrossRef] [PubMed]

A. Rose, R. Gupta, “Application of Photothermal Deflection Technique to Flow-Velocity Measurements in a Flame,” Opt. Lett. 10, 532 (1985).
[CrossRef] [PubMed]

A. Rose, G. J. Salamo, R. Gupta, “Photoacoustic Deflection Spectroscopy: A New Specie-Specific Method for Combustion Diagnostics,” Appl. Opt. 23, 781 (1984).
[CrossRef] [PubMed]

A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the Photothermal Deflection Technique to Combustion Diagnostics,” Appl. Opt. 21, 2663 (1982).
[CrossRef] [PubMed]

A. Rose, J. D. Pyrum, G. J. Salamo, R. Gupta, “Photoacoustic Spectroscopy and Photothermal Deflection Spectroscopy: New Tools for Combustion Diagnostics,” in Proceedings, International Conference on Lasers ’82, R. C. Powell, Ed. (STS Press, McLean, VA, 1983).

A. Rose, R. Gupta, “Combustion Diagnostics by Photodeflection Spectroscopy,” in Twentieth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1984).

A. Rose, Y.-X. Nie, R. Gupta, “A Quantitative Investigation of Pulsed Photoacoustic Deflection Spectroscopy in Gaseous Media,” to be submitted to Appl. Opt.

A. Rose, “Development of Pulsed Photoacoustic and Photothermal Deflection Spectroscopy as Diagnostic Tools for Combustion,” Ph.D. Dissertation, U. Arkansas (1986).

Rutherford, W. M.

W. M. Rutherford, W. J. Roos, K. J. Kaminski, “Experimental Verification of the Thermal Diffusion Column Theory as Applied to the Separation of Isotopically Substituted Nitrogen and Isotopically Substituted Oxygen,” J. Chem. Phys. 50, 5359 (1969).
[CrossRef]

Salamo, G. J.

A. Rose, G. J. Salamo, R. Gupta, “Photoacoustic Deflection Spectroscopy: A New Specie-Specific Method for Combustion Diagnostics,” Appl. Opt. 23, 781 (1984).
[CrossRef] [PubMed]

A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the Photothermal Deflection Technique to Combustion Diagnostics,” Appl. Opt. 21, 2663 (1982).
[CrossRef] [PubMed]

A. Rose, J. D. Pyrum, G. J. Salamo, R. Gupta, “Photoacoustic Spectroscopy and Photothermal Deflection Spectroscopy: New Tools for Combustion Diagnostics,” in Proceedings, International Conference on Lasers ’82, R. C. Powell, Ed. (STS Press, McLean, VA, 1983).

Sell, J. A.

Sontag, H.

A. C. Tam, H. Sontag, P. Hess, “Photothermal Probe Beam Deflection Monitoring of Photochemical Particulate Production,” Chem. Phys. Lett. 120, 280 (1985).
[CrossRef]

H. Sontag, A. C. Tam, “Time-Resolved Flow-Velocity and Concentration Measurements Using a Traveling Thermal Lens,” Opt. Lett. 10, 436 (1985).
[CrossRef] [PubMed]

H. Sontag, A. C. Tam, “Characterization of Vapor and Aerosol Flows by Photothermal Methods,” Can. J. Phys.Sept.1986, to be published.
[CrossRef]

Tam, A. C.

For a review see A. C. Tam, “Applications of Photoacoustic Sensing Techniques,” Rev. Mod. Phys. 58, 381 (1986).
[CrossRef]

H. Sontag, A. C. Tam, “Time-Resolved Flow-Velocity and Concentration Measurements Using a Traveling Thermal Lens,” Opt. Lett. 10, 436 (1985).
[CrossRef] [PubMed]

A. C. Tam, H. Sontag, P. Hess, “Photothermal Probe Beam Deflection Monitoring of Photochemical Particulate Production,” Chem. Phys. Lett. 120, 280 (1985).
[CrossRef]

J. C. Loulergue, A. C. Tam, “Noncontact Photothermal Probe Beam Deflection Measurement of Thermal Diffusivity in a Hot Unconfined Gas,” Appl. Phys. Lett. 46, 457 (1985).
[CrossRef]

A. C. Tam, H. Coufal, “Pulsed Opto-Acoustics: Theory and Applications,” J. Phys. Colloq. C6, 44, 9 (1983).

H. Sontag, A. C. Tam, “Characterization of Vapor and Aerosol Flows by Photothermal Methods,” Can. J. Phys.Sept.1986, to be published.
[CrossRef]

Twarowski, A. J.

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

Weimer, W. A.

Zharov, V.

V. Zharov, N. M. Amer, “Pulsed Photothermal Deflection Spectroscopy in Flowing Media,” in Technical Digest, Fourth International Topical Meeting on Photoacoustic, Thermal, and Related Science, Ville D’Esterel, Quebec (1985).

AIP Conf. Proc.

R. Gupta, “A Quantitative Investigation of Pulsed Photothermal and Photoacoustic Deflection Spectroscopy for Combustion Diagnostics,” AIP Conf. Proc. 146, 672 (1986).
[CrossRef]

Appl. Opt.

J. A. Sell, “Gas Velocity Measurements Using Photothermal Deflection Spectroscopy,” Appl. Opt. 24, 3725 (1985).
[CrossRef] [PubMed]

A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the Photothermal Deflection Technique to Combustion Diagnostics,” Appl. Opt. 21, 2663 (1982).
[CrossRef] [PubMed]

S. W. Kizirnis, R. J. Brecha, B. N. Ganguly, L. P. Goss, R. Gupta, “Hydroxyl (OH) Distributions and Temperature Profiles in a Premixed Propane Flame Obtained by Laser Deflection Techniques,” Appl. Opt. 23, 3873 (1984).
[CrossRef] [PubMed]

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

J. A. Sell, “Quantitative Photothermal Deflection Spectroscopy in a Flowing Stream of Gas,” Appl. Opt. 23, 1586 (1984).
[CrossRef] [PubMed]

A. Rose, G. J. Salamo, R. Gupta, “Photoacoustic Deflection Spectroscopy: A New Specie-Specific Method for Combustion Diagnostics,” Appl. Opt. 23, 781 (1984).
[CrossRef] [PubMed]

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

Y.-X. Nie, K. Hane, R. Gupta, “Measurement of Very Low Gas Flow Velocities by Photothermal Deflection Spectroscopy,” Appl. Opt. 25, 3247 (1986).
[CrossRef] [PubMed]

J. M. Khosrofian, B. A. Garetz, “Measurement of a Gaussian Laser Beam Diameter Through the Direct Inversion of Knife-Edge Data,” Appl. Opt. 22, 3406 (1983).
[CrossRef] [PubMed]

A. Rose, Y.-X. Nie, R. Gupta, “Laser Beam Profile Measurement by Photothermal Deflection Technique,” Appl. Opt. 25, 1738 (1986).
[CrossRef] [PubMed]

Appl. Phys. Lett.

J. C. Loulergue, A. C. Tam, “Noncontact Photothermal Probe Beam Deflection Measurement of Thermal Diffusivity in a Hot Unconfined Gas,” Appl. Phys. Lett. 46, 457 (1985).
[CrossRef]

C. C. Davis, “Trace Detection of Gases Using Phase Fluctuation Optical Heterodyne Spectroscopy,” Appl. Phys. Lett. 36, 515 (1980).
[CrossRef]

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

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ Trace-Gas Detection by Photothermal Deflection Spectroscopy,” Appl. Phys. Lett. 37, 519 (1980).
[CrossRef]

Chem. Phys.

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

Chem. Phys. Lett.

A. C. Tam, H. Sontag, P. Hess, “Photothermal Probe Beam Deflection Monitoring of Photochemical Particulate Production,” Chem. Phys. Lett. 120, 280 (1985).
[CrossRef]

J. Chem. Phys.

V. M. Donnelly, F. Kaufman, “Fluorescence Lifetime Studies of NO2. I. Excitation of the Perturbed 2B2 State near 600 nm,” J. Chem. Phys. 66, 4100 (1977).
[CrossRef]

W. M. Rutherford, W. J. Roos, K. J. Kaminski, “Experimental Verification of the Thermal Diffusion Column Theory as Applied to the Separation of Isotopically Substituted Nitrogen and Isotopically Substituted Oxygen,” J. Chem. Phys. 50, 5359 (1969).
[CrossRef]

J. Phys. Colloq. C6

A. C. Tam, H. Coufal, “Pulsed Opto-Acoustics: Theory and Applications,” J. Phys. Colloq. C6, 44, 9 (1983).

Opt. Commun.

A. Rose, R. Gupta, “Application of Photothermal and Photoacoustic Deflection Techniques to Sooting Flames: Velocity, Temperature, and Concentration Measurements,” Opt. Commun. 56, 303 (1986).
[CrossRef]

Opt. Lett.

Rev. Mod. Phys.

For a review see A. C. Tam, “Applications of Photoacoustic Sensing Techniques,” Rev. Mod. Phys. 58, 381 (1986).
[CrossRef]

Other

H. Sontag, A. C. Tam, “Characterization of Vapor and Aerosol Flows by Photothermal Methods,” Can. J. Phys.Sept.1986, to be published.
[CrossRef]

V. Zharov, N. M. Amer, “Pulsed Photothermal Deflection Spectroscopy in Flowing Media,” in Technical Digest, Fourth International Topical Meeting on Photoacoustic, Thermal, and Related Science, Ville D’Esterel, Quebec (1985).

A. Rose, R. Gupta, “Combustion Diagnostics by Photodeflection Spectroscopy,” in Twentieth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, PA, 1984).

A. Rose, J. D. Pyrum, G. J. Salamo, R. Gupta, “Photoacoustic Spectroscopy and Photothermal Deflection Spectroscopy: New Tools for Combustion Diagnostics,” in Proceedings, International Conference on Lasers ’82, R. C. Powell, Ed. (STS Press, McLean, VA, 1983).

A. Rose, Y.-X. Nie, R. Gupta, “A Quantitative Investigation of Pulsed Photoacoustic Deflection Spectroscopy in Gaseous Media,” to be submitted to Appl. Opt.

See, for example, M. V. Klein, Optics (Wiley, New York, 1970).

Reference 7. Our result differs from that of Ref. 7 by a factor of 2. We believe that there is an error in Ref. 7.

R. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Boca Raton, FL, 1984).

A. Rose, “Development of Pulsed Photoacoustic and Photothermal Deflection Spectroscopy as Diagnostic Tools for Combustion,” Ph.D. Dissertation, U. Arkansas (1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (25)

Fig. 1
Fig. 1

Pump and probe beam configuration for (a) transverse PTDS, (b) collinear PTDS, and for the (c) general case. The gas flow is in the x direction, the pump beam propagates along the z axis, and the probe beam is in the y-z plane.

Fig. 2
Fig. 2

Spatial profile of the heat pulse in a flowing gas at different times after the end of the excitation pulse. The medium was assumed to be N2 at room temperature seeded with 1025 ppm of NO2 (absorption coefficient α = 0.39 m−1) and flowing with velocity v x = 0.5 m/s. The pump laser was assumed to have an energy E0 = 1 mJ in a pulse of t0 = 1-μs duration. (1/e2) radius of the pump beam was 0.35 mm. For the values of all the other parameters, see Appendix A.

Fig. 3
Fig. 3

Diagram showing the relationship between the probe beam path s, perpendicular displacement δ, and the deflection angle ϕ.

Fig. 4
Fig. 4

Photothermal deflection signal shapes in a flowing medium for several different pump-to-probe beam distances. x = 0 corresponds to the case when the centers of the two beams coincide. Negative x corresponds to the probe beam being upstream from the pump beam and positive x corresponds to it being downstream. Flow velocity of the medium (N2 seeded with 1025-ppm NO2) was assumed to be 2.0 m/s. Laser pulse energy was assumed to be 1.65 mJ, beam radius a was 0.33 mm, and all the other parameters used in this computation are given in the caption for Fig. 2.

Fig. 5
Fig. 5

Photothermal deflection signal in a stationary medium (v x = 0) for several probe-to-pump beam distances as indicated above. Laser energy was assumed to be E0 = 1 mJ and all the other parameters used in this calculation are the same as for Fig. 4. Two curves on the bottom have been expanded by the indicated factors for clarity.

Fig. 6
Fig. 6

Dependence of the photothermal deflection signal, in a stationary medium, on the temperature of the medium. In this calculation pump laser radius a = 0.33 mm, pump–probe distance x = a/2, pump energy E0 = 1 mJ, pulse width t0 = 1 μs, absorption coefficient α = 0.39 m−1 (corresponding to 1025-ppm NO2), and the diffusion coefficient D is assumed to vary as T1.7 (see Appendix A for other parameters). Curves at elevated temperatures have been expanded by the indicated factors for clarity.

Fig. 7
Fig. 7

Dependence of the photothermal deflection signal, in a flowing medium, on the temperature of the medium. Curves at elevated temperatures have been expanded by the factors indicated. The flow velocity of the medium was taken to be 4 m/s, and the probe–pump distance x = 1.5 mm. All the other parameters are the same as in Fig. 6.

Fig. 8
Fig. 8

Dependence of the amplitude of the photothermal deflection signal on temperature of the medium for a few representative values of the flow velocity, as indicated on the curves. In the case of v x ≠ 0, the amplitude of the larger peak (see Fig. 6) has been used. All the parameters used in this calculation were the same as in Figs. 6 and 7.

Fig. 9
Fig. 9

Diagram showing the configuration of the probe and the pump beams for (a) θ > θ0 and (b) θ < θ0.

Fig. 10
Fig. 10

Dependence of the photothermal deflection, signal in a stationary medium, on the angle between the pump and the probe beams. In this calculation the following values were used: a = 0.35 mm, x = a/2, t0 = 1 μs, l = 1 cm, α = 0.39 m−1, and T = 300 K. To display the change in width clearly, the peak values of all the curves have been normalized to the θ = 0 curve. The normalizing factors are 1.2 for θ = 2°, 1.7 for θ = 4°, 2.4 for θ = 6°, and 22.8 for θ = 90°.

Fig. 11
Fig. 11

Dependence of the photothermal deflection, signal in a flowing medium, on the angle between the probe and the pump beams. In this calculation the following values were used: v x = 1 m/s, a = 0.35 mm, x = 1.0 mm, t0 = 1 μs, l = 1 cm, α = 0.39 m−1, and T = 300 K. First peaks of all the curves have been normalized to the θ = 0 value. The normalizing factors are 1.3 for θ = 4°, and 16.3 for θ = 90°.

Fig. 12
Fig. 12

Comparison of the results of the impulse approximation (dotted curve) with the exact results (solid curve) for three values of the laser pulse length t0. When only one curve is shown, the two curves overlap. Parameters used in this calculation were v x = 2 m/s, a = 0.35 mm, x = 2 mm, α = 0.39 m−1, and T = 300 K. Each dotted curve has been shifted to the right by t0/2 to make the centers of the curves match.

Fig. 13
Fig. 13

Percent error in the peak value of the signal introduced by the use of impulse approximation as a function of the observation time of the signal for various values of the laser pulse lengths.

Fig. 14
Fig. 14

Comparison of the results of the impulse approximation (dotted curves) with the exact results (solid curves) for different velocities. The dotted curves have been moved to the right by t0/2, as in Fig. 12. In these curves a = 0.35 mm, t0 = 0.5 ms, θ = 90°, and T = 300 K were used.

Fig. 15
Fig. 15

Comparison of the results of the approximate analytical formula with the exact results for intermediate values of velocity at T = 300 K. In these curves θ = 0 (collinear PTDS), t0 = 1 μs for (a) and (b), and t0 = 1 ms for (c) and (d). Other parameters are given in the diagram. When a single curve is shown, approximate and exact results coincide.

Fig. 16
Fig. 16

Comparison of the results of the approximate analytical formula with the exact results for T = 3000 K. The parameters used in this calculation were v x = 4 m/s, t0 = 1 μs, a = 0.35 mm, and x = 3.0 mm. Curves (a) are for transverse PTDS and curves (b) are for collinear PTDS. Two curves in (b) overlap completely.

Fig. 17
Fig. 17

Schematic illustration of the apparatus. Optical elements (mirrors, lenses, etc.) have been omitted for clarity.

Fig. 18
Fig. 18

Photothermal deflection signals for transverse geometry in a stationary medium (closed cell) for various pump-to-probe distances. Dots represent the experimental signals and they have been plotted on an absolute scale except the bottom curve which has been expanded by a factor of 2 for clarity. Solid lines are the theoretical curves computed from Eq. (21) using the parameters in the text. Theoretical curves have been multiplied by the scaling factors given on each curve to facilitate accurate comparison of the shapes of the theoretical and experimental curves. The scaling factors themselves indicate the degree of agreement between theory and experiment for the amplitude of the signals.

Fig. 19
Fig. 19

Peak photothermal deflection signal for transverse geometry plotted as a function of the probe–pump distance. Circles are the experimental points and the solid line is the theoretical prediction using the parameters given in the text in connection with Fig. 18.

Fig. 20
Fig. 20

Effect of the angle between the pump and the probe beams on the width of the photothermal deflection signal. To permit accurate comparison of the widths of different curves, all the curves have been scaled to the same amplitude. The scaling factors are 1.3 (for θ = 1.4), 2.2 (for θ = 2°), and 8.4 (for θ = 90°). The maximum interaction length occurred for the collinear case and it was ~1 cm (width of the cell).

Fig. 21
Fig. 21

Effect of the angle between the pump and the probe beams on the amplitude of the photothermal deflection signal. Circles are the experimental data points and the solid line is the theoretical curve. The theoretical curve has been normalized to the experimental data at θ = 1° for ease in comparison. These data were taken in a cell different from that corresponding to Fig. 20, and the maximum interaction length (corresponding to θ = 0°) was 19 cm.

Fig. 22
Fig. 22

Photothermal deflection signals for transverse geometry in a flowing medium (open jet of N2 seeded with NO2) for different pump-to-probe separations and a fixed flow velocity (2.1 m/s). Dots are experimental data points and they have been plotted on an absolute scale. The solid lines are the corresponding theoretical curves calculated from Eq. (21) using the parameters given in the text. To permit accurate comparison of signal shapes, the theoretical curves have been multiplied by the scaling factors indicated on each curve.

Fig. 23
Fig. 23

Photothermal deflection signals for coincident pump and probe beams (x = 0) for different flow velocities. Dots represent the experimental signals while the solid lines are the theoretical curves corresponding to the parameters given in the text and scaled by the factors given on each curve.

Fig. 24
Fig. 24

Photothermal deflection signals similar to those shown in Fig. 23 but for x = 0.21 mm. Dots represent experimental points and the solid lines are the theoretical curves scaled by the indicated factors.

Fig. 25
Fig. 25

Diagram showing the effect of the relative sizes of the pump and the probe beams on peak photothermal deflection signal. Crosses are experimental points and the solid lines are the theoretical curves. Theoretical curves have been scaled so that their peaks match the experimental values. Curve (a) is for the case when the pump beam diameter is very large compared with the probe beam diameter. In this case the probe beam is able to resolve the spatial mode structure of the pump beam which is non-Gaussian. In (b) the pump beam has been focused down to ~0.4 mm and the probe beam is no longer able to resolve the structure. When the pump beam size becomes about equal to (or smaller than) the probe beam, the experimental curve becomes broader than the theoretical curve as shown in (c).

Equations (50)

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

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 .
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 ( ± / ξ ; y / η ; t / τ ) = 0 , G ( x / ξ ; ± / η ; t / τ ) = 0 , G ( x / ξ ; y / η ; 0 / τ ) = 0.
( ω x 2 + ω y 2 ) D G F - i ω x v x G F + G F t = 1 2 π ρ C p × exp [ i ( ω x ξ + ω y η ) ] δ ( t - τ ) .
( ω x 2 + ω y 2 ) D G F L - i ω x v x G F L + s G F L = 1 2 π ρ C p exp [ i ( ω x ξ + ω y η ) ] × exp ( - s τ ) ,
G F L = exp ( i ω x ξ ) exp ( i ω y η ) exp ( - s τ ) 2 π ρ C p [ D ( ω x 2 + ω y 2 ) - i ω x v x + s ] .
G F = exp ( i ω x ξ ) exp ( i ω y η ) H τ ( t ) 2 π ρ C p exp { [ i ω x v x - ( ω x 2 + ω y 2 ) D ] ( t - τ ) } .
H τ ( t ) = { 0     for 0 t < τ , 1     for 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 exp ( - 2 { [ x - v x ( t - τ ) ] 2 + y 2 } / [ 8 D ( t - τ ) + a 2 ] ) [ 8 D ( t - τ ) + a 2 ] d τ for t > t 0 .
T ( x , y , t ) x = - 8 α E 0 π t 0 ρ C p 0 t 0 x - v x ( t - τ ) [ 8 D ( t - τ ) + a 2 ] 2 × exp ( - 2 { [ x - v x ( t - τ ) ] 2 + y 2 } / { 8 D ( t - τ ) + a 2 } ) d τ for t > t 0 .
d d s ( n 0 d δ d s ) = n ( r , t ) ,
n ( r , t ) = n 0 + n T | T A T ( r , t ) ,
d δ d s = 1 n 0 n T path T ( r , t ) d s ,
ϕ ( x , y , t ) = 1 n 0 n T path T ( x , y , t ) x d s .
ϕ T ( x , t ) = 1 n 0 n T T ( x , y , t ) x d y .
ϕ T ( x , t ) = - 1 n 0 n T 8 α E 0 π t 0 ρ C p 0 t 0 d τ × ( - exp { - 2 y 2 / [ 8 D ( t - τ ) + a 2 ] } d y ) × x - v x ( t - τ ) [ 8 D ( t - τ ) + a 2 ] 2 × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ 8 D ( t - τ ) + a 2 ] } .
ϕ T ( x , t ) = - 1 n 0 n T 8 α E 0 2 π t 0 ρ C p 0 t 0 [ x - v x ( t - τ ) ] [ 8 D ( t - τ ) + a 2 ] 3 / 2 × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ 8 D ( t - τ ) + a 2 ] } d τ for t > t 0 .
ϕ L ( x , y , t ) = 1 n 0 n T T ( x , y , t ) x d z .
ϕ L ( x , y , t ) = - l n 0 n T 8 α E 0 π t 0 ρ C p 0 t 0 [ x - v x ( t - τ ) ] [ 8 D ( t - τ ) + a 2 ] 2 × exp ( - 2 { [ x - v x ( t - τ ) ] 2 + y 2 } / [ 8 D ( t - τ ) + a 2 ] ) d τ ,
ϕ ( x , y , t ) = 1 n 0 n T T ( x , y , t ) x ( d y 2 + d z 2 ) 1 / 2 ,
ϕ ( x , θ , t ) = 1 n 0 n T 1 sin θ - l 2 tan θ + l 2 tan θ T ( x , y , t ) x d y ,
ϕ ( x , θ , t ) = 1 n 0 n T 1 cos θ - l 2 l 2 T ( x , z tan θ , t ) x d z ,
l tan θ 0 = 2 ( a 2 + 8 D t ) 1 / 2 ,
ϕ ( x , θ , t ) = - 1 n 0 n T 8 α E 0 2 π t 0 ρ C p 1 sin θ 0 t 0 [ x - v x ( t - τ ) ] [ 8 D ( t - τ ) + a 2 ] 3 / 2 × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ 8 D ( t - τ ) + a 2 ] } d τ for 2 θ 0 θ π / 2 and t > t 0 ,
ϕ ( x , θ , t ) = - 1 n 0 n T 8 α E 0 π t 0 ρ C p 1 cos θ 0 t 0 [ x - v x ( t - τ ) ] [ 8 D ( t - τ ) + a 2 ] 2 × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ 8 D ( t - τ ) + a 2 ] } × ( - l / 2 + l / 2 exp { - 2 z 2 tan 2 θ / [ 8 D ( t - τ ) + a 2 ] } d z ) d τ for 0 θ 2 θ 0 and t > t 0 .
ϕ ( x , θ , t ) = { ϕ T / sin θ for 2 θ 0 θ π / 2 , ϕ L / cos θ for 0 θ < θ 0 .
T ( x , y , t ) x = α E 0 x 2 π t 0 ρ C p D ( x 2 + y 2 ) × ( exp { - 2 ( x 2 + y 2 ) / [ a 2 + 8 D ( t - t 0 ) ] } - exp { - 2 ( x 2 + y 2 ) / [ a 2 + 8 D t ] } ) .
ϕ T ( x , t ) = - 1 n 0 ( n T ) α E 0 2 t 0 ρ C p D ( erf { [ 2 x 2 a 2 + 8 D ( t - t 0 ) ] 1 / 2 } - erf [ ( 2 x 2 a 2 + 8 D t ) 1 / 2 ] ) ,
ϕ L ( x , y , t ) = l n 0 ( n T ) α E 0 x 2 π t 0 ρ C p D ( x 2 + y 2 ) × ( exp { - 2 ( x 2 + y 2 ) / [ a 2 + 8 D ( t - t 0 ) ] } - exp [ - 2 ( x 2 + y 2 ) / ( a 2 + 8 D t ) ] ) .
T ( x , y , t ) x = 2 α E 0 exp ( - 2 y 2 / a 2 ) π t 0 ρ C p a 2 v x ( exp { - 2 [ x - v x ( t - t 0 ) ] 2 / a 2 } - exp [ - 2 ( x - v x t ) 2 / a 2 ] ) .
ϕ T ( x , t ) = 1 n 0 ( n T ) 2 α E 0 2 π t 0 ρ C p a v x ( exp { - 2 [ x - v x ( t - t 0 ) ] 2 / a 2 } - exp [ - 2 ( x - v x t ) 2 / a 2 ] ) ,
ϕ L ( x , y , t ) = l n 0 n T 2 α E 0 exp ( - 2 y 2 / a 2 ) π t 0 ρ C p a 2 v x × ( exp { - 2 [ x - v x ( t - t 0 ) ] 2 / a 2 } - exp [ - 2 ( x - v x t ) 2 / a 2 ] ) .
lim t 0 0 0 t 0 f ( τ ) d τ = f ( 0 ) t 0 .
T ( x , y , t ) = 2 α E 0 π ρ C p ( 8 D t + a 2 ) × exp { - 2 [ ( x - v x t ) 2 + y 2 ] / [ a 2 + 8 D t ] } ,
T ( x , y , t ) x = - 8 α E 0 π ρ C p ( x - v x t ) ( a 2 + 8 D t ) 2 × exp { - 2 [ ( x - v x t ) 2 + y 2 ] / ( a 2 + 8 D t ) } .
ϕ T ( x , t ) = - 1 n 0 n T 8 α E 0 2 π ρ C p ( x - v x t ) ( 8 D t + a 2 ) 3 / 2 × exp [ - 2 ( x - v x t ) 2 / ( a 2 + 8 D t ) ] ,
ϕ L ( x , y , t ) = - l n 0 n T 8 α E 0 π ρ C p ( x - v x t ) ( 8 D t + a 2 ) 2 × exp { - 2 [ ( x - v x t ) 2 + y 2 ] / ( a 2 + 8 D t ) } .
T ( x , y , t ) x = 2 α E 0 π t 0 ρ C p [ 4 D x + v x a 2 + 4 D v x ( t - t 0 ) ] × [ exp ( - 2 { [ x - v x ( t - t 0 ) ] 2 + y 2 } / [ a 2 + 8 D ( t - t 0 ) ] ) - exp { - 2 [ ( x - v x t ) 2 + y 2 ] / ( a 2 + 8 D t ) } ] .
ϕ T ( x , t ) = 1 n 0 n T 2 α E 0 2 π t 0 ρ C p [ 4 D x + v x a 2 + 4 D v x ( t - t 0 ) ] × ( [ a 2 + 8 D ( t - t 0 ) ] 1 / 2 × exp ( - 2 { [ x - v x ( t - t 0 ) ] 2 } / [ a 2 + 8 D ( t - t 0 ) ] ) - ( a 2 + 8 D t ) 1 / 2 exp { - 2 [ ( x - v x t ) 2 ] / ( a 2 + 8 D t ) } ) ,
ϕ L ( x , y , t ) = l n 0 n T 2 α E 0 π t 0 ρ C p [ 4 D x + v x a 2 + 4 D v x ( t - t 0 ) ] × ( exp ( - 2 { [ x - v x ( t - t 0 ) ] 2 + y 2 } / [ a 2 + 8 D ( t - t 0 ) ] ) - exp { - 2 [ ( x - v x t ) 2 + y 2 ] / ( a 2 + 8 D t ) } ) .
n T = ( n 0 - 1 ) T 0 T 2 ,
D = k ρ C p .
Q ( x , y , t ) = { 2 α E 0 π a b t 0 exp [ - 2 ( x 2 / a 2 + y 2 / b 2 ) ] for 0 t t 0 , 0 for t > t 0 ,
T ( x , y , t ) = 2 α E 0 π t 0 ρ C p 0 t 0 exp { - 2 [ x - v x ( t - τ ) ] 2 / [ a 2 + 8 D ( t - τ ) ] } [ a 2 + 8 D ( t - τ ) ] 1 / 2 × exp { - 2 y 2 / [ b 2 + 8 D ( t - τ ) ] } [ b 2 + 8 D ( t - τ ) ] 1 / 2 d τ             for t > t 0 .
ϕ T ( x , t ) = - 1 n 0 n T 8 α E 0 2 π t 0 ρ C p 0 t 0 [ x - v x ( t - τ ) ] [ a 2 + 8 D ( t - τ ) ] 3 / 2 × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ a 2 + 8 D ( t - τ ) ] } d τ ,
ϕ L ( x , y , t ) = - l n 0 n T 8 α E 0 π t 0 ρ C p 0 t 0 [ x - v x ( t - τ ) ] [ a 2 + 8 D ( t - τ ) ] 3 / 2 × exp { - 2 [ x - v x ( t - τ ) ] 2 / [ a 2 + 8 D ( t - τ ) ] } × exp { - 2 y 2 / [ b 2 + 8 D ( t - τ ) ] } [ b 2 + 8 D ( t - τ ) ] 1 / 2 d τ .

Metrics