Abstract

A comparison is made of three methods for modeling the interaction of a laser probe beam with the temperature field of a thermal wave. The three methods include: (1) a new method based on complex ray theory, which allows us to take into account the disturbance of the amplitude and phase of the electric field of the probe beam, (2) the ray deflection averaging theory of Aamodt and Murphy, and (3) the wave theory (WT) of Glazov and Muratikov. To carry out this comparison, it is necessary to reformulate the description of the photodeflection signal in either the WT or the ray deflection averaging theory. It is shown that the differences between calculated signals using the different theories are most pronounced when the radius of the probe beam is comparable with the length of the thermal wave in the region of their interaction. Predictions of the theories are compared with experimental results. A few parameters of the experimental setup are determined through multiparameter fitting of the theoretical curves to the experimental data. A least-squares procedure was chosen as a fitting method. The conclusion is that the calculation of the photodeflection signal in the framework of the complex ray theory is a more accurate approach than the ray deflection averaging theory or the wave one.

© 2007 Optical Society of America

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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. A. Salazar, A. Sanchez-Lavega, and J. Fernandez, "Theory of thermal diffusivity determination by the "mirage" technique in solids," J. Appl. Phys. 65, 4150-4156 (1989).
  3. A. Sanchez-Lavega and A. Salazar, "Thermal diffusivity measurements in opaque solids by the mirage technique in the temperature range from 300 to 1000 K," J. Appl. Phys. 76, 1462-1468 (1994).
    [CrossRef]
  4. P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
    [CrossRef]
  5. M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
    [CrossRef]
  6. J. Bodzenta and M. Pyka, "Photothermal measurement with mirage effect for investigation of LiNbO3 single crystals," J. Phys. IV 137, 259-263 (2006).
    [CrossRef]
  7. M. Commandr and P. Roche, "Characterization of optical coatings by photothermal deflection," Appl. Opt. 35, 5021-5034 (1996).
    [CrossRef]
  8. K. Plamann, D. Fournier, E. Anger, and A. Gicquel, "Photothermal examination of the heat diffusion inhomogeneity in diamond films of sub-micron thickness," Diamond Relat. Mater. 3, 752-756 (1994).
    [CrossRef]
  9. J. Bodzenta, B. Burak, A. Jagoda, and B. Stanczyk, "Thermal conductivity of AIN and AIN-GaN thin films deposited on Si and GaAs substrates," Diamond Relat. Mater. 14, 1169-1174 (2005).
    [CrossRef]
  10. M. A. Schweitzer and J. F. Power, "Optical depth profiling of thin films by impulse mirage effect spectroscopy. Part I: theory," Appl. Spectrosc. 48, 1054-1075 (1994).
    [CrossRef]
  11. K. A. Shailendra, K. L. Narasimhan, S. S. Rajalakshmi, S. S. Chandvankar, and B. M. Arora, "Photothermal deflection spectroscopy of heat-treated GaAs, InP, and InGaAsP alloys," Appl. Phys. Lett. 55, 2512-2513 (1989).
    [CrossRef]
  12. T. Gotoh, S. Nonomura, S. Hirata, and S. Nitta, "Photothermal bending spectroscopy and photothermal deflection spectroscopy of C60 thin films," Appl. Surf. Sci. 113/114, 278-281 (1997).
    [CrossRef]
  13. F. Lepoutre, D. Fournier, and A. C. Boccara, "Nondestructive control of weldings using the mirage detection," J. Appl. Phys. 57, 1009-1015 (1985).
    [CrossRef]
  14. H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov, "New phase interference technique applied for sensitive photothermal microscopy," Appl. Phys. Lett. 57, 1600-1601 (1990).
    [CrossRef]
  15. L. C. Aamodt and J. C. Murphy, "Photothermal measurement using a localized excitation source," J. Appl. Phys. 52, 4903-4914 (1981).
    [CrossRef]
  16. A. L. Glazov and K. L. Muratikov, "Photodeflection and interferometric thermal wave microscopy of solids," Int. J. Optoelectron. 4, 589-597 (1989).
  17. L. C. Aamodt and J. C. Murphy, "Thermal effects in photothermal spectroscopy and photothermal imaging," J. Appl. Phys. 54, 581-591 (1983).
    [CrossRef]
  18. E. L. Lasalle, F. Lepoutre, and J. P. Roger, "Probe beam size effects in photothermal deflection experiments," J. Appl. Phys. 64, 1-5 (1988).
    [CrossRef]
  19. A. L. Glazov and K. L. Muratikov, "Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. "Mirage" effect geometry," Opt. Commun. 84, 283-289 (1991).
  20. A. L. Glazov and K. L. Muratikov, "Calculation of the photodeflection signal in the framework of wave optics," Tech. Phys. 38, 344-352 (1993).
  21. R. J. Bukowski and D. Korte, "Perturbation calculus for eikonal application to analysis of the deflectional signal in photothermal measurements," Opt. Appl. 32, 817-828 (2002).
  22. R. J. Bukowski and D. Korte, "Influence pf probing beam focusing on photothermal signal," J. Phys. IV 109, 19-31 (2003).
  23. R. J. Bukowski and D. Korte, "The deflectional signal analysis in photothermal measurements in the frame of complex geometrical optics," Opt. Appl. 35, 77-92 (2005).
  24. D. K. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, "The complex ray theory of photodeflection signal formation: comparison with the ray theory and the experimental results," J. Appl. Phys. 100, 063501 (2006).
    [CrossRef]
  25. R. J. Bukowski, "Complex geometrical optics application for description of Gaussian beam propagation in optically homogenous media," in Proceedings of Second National Conference "Physical Grounds on Nondestructive Investigation" (Gliwice Division of the Polish Physical Society and Institute of Physics of Silesian University of Technology, 1997), pp. 45-55 (in Polish).
  26. Ju. A. Kravtsov and Ju. I. Orlov, Geometrical Optics of the Nonhomogeneous Media (WNT, 1993) (in Polish).
  27. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford U. Press, 1959).
  28. A. Maitland and M. H. Dunn, Laser Physics (North-Holland, 1969).

2006 (2)

J. Bodzenta and M. Pyka, "Photothermal measurement with mirage effect for investigation of LiNbO3 single crystals," J. Phys. IV 137, 259-263 (2006).
[CrossRef]

D. K. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, "The complex ray theory of photodeflection signal formation: comparison with the ray theory and the experimental results," J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

2005 (2)

R. J. Bukowski and D. Korte, "The deflectional signal analysis in photothermal measurements in the frame of complex geometrical optics," Opt. Appl. 35, 77-92 (2005).

J. Bodzenta, B. Burak, A. Jagoda, and B. Stanczyk, "Thermal conductivity of AIN and AIN-GaN thin films deposited on Si and GaAs substrates," Diamond Relat. Mater. 14, 1169-1174 (2005).
[CrossRef]

2003 (1)

R. J. Bukowski and D. Korte, "Influence pf probing beam focusing on photothermal signal," J. Phys. IV 109, 19-31 (2003).

2002 (1)

R. J. Bukowski and D. Korte, "Perturbation calculus for eikonal application to analysis of the deflectional signal in photothermal measurements," Opt. Appl. 32, 817-828 (2002).

1997 (3)

R. J. Bukowski, "Complex geometrical optics application for description of Gaussian beam propagation in optically homogenous media," in Proceedings of Second National Conference "Physical Grounds on Nondestructive Investigation" (Gliwice Division of the Polish Physical Society and Institute of Physics of Silesian University of Technology, 1997), pp. 45-55 (in Polish).

T. Gotoh, S. Nonomura, S. Hirata, and S. Nitta, "Photothermal bending spectroscopy and photothermal deflection spectroscopy of C60 thin films," Appl. Surf. Sci. 113/114, 278-281 (1997).
[CrossRef]

M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
[CrossRef]

1996 (1)

1994 (3)

K. Plamann, D. Fournier, E. Anger, and A. Gicquel, "Photothermal examination of the heat diffusion inhomogeneity in diamond films of sub-micron thickness," Diamond Relat. Mater. 3, 752-756 (1994).
[CrossRef]

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

A. Sanchez-Lavega and A. Salazar, "Thermal diffusivity measurements in opaque solids by the mirage technique in the temperature range from 300 to 1000 K," J. Appl. Phys. 76, 1462-1468 (1994).
[CrossRef]

1993 (2)

Ju. A. Kravtsov and Ju. I. Orlov, Geometrical Optics of the Nonhomogeneous Media (WNT, 1993) (in Polish).

A. L. Glazov and K. L. Muratikov, "Calculation of the photodeflection signal in the framework of wave optics," Tech. Phys. 38, 344-352 (1993).

1991 (1)

A. L. Glazov and K. L. Muratikov, "Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. "Mirage" effect geometry," Opt. Commun. 84, 283-289 (1991).

1990 (1)

H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov, "New phase interference technique applied for sensitive photothermal microscopy," Appl. Phys. Lett. 57, 1600-1601 (1990).
[CrossRef]

1989 (3)

A. L. Glazov and K. L. Muratikov, "Photodeflection and interferometric thermal wave microscopy of solids," Int. J. Optoelectron. 4, 589-597 (1989).

A. Salazar, A. Sanchez-Lavega, and J. Fernandez, "Theory of thermal diffusivity determination by the "mirage" technique in solids," J. Appl. Phys. 65, 4150-4156 (1989).

K. A. Shailendra, K. L. Narasimhan, S. S. Rajalakshmi, S. S. Chandvankar, and B. M. Arora, "Photothermal deflection spectroscopy of heat-treated GaAs, InP, and InGaAsP alloys," Appl. Phys. Lett. 55, 2512-2513 (1989).
[CrossRef]

1988 (1)

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

1986 (1)

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

1985 (1)

F. Lepoutre, D. Fournier, and A. C. Boccara, "Nondestructive control of weldings using the mirage detection," J. Appl. Phys. 57, 1009-1015 (1985).
[CrossRef]

1983 (1)

L. C. Aamodt and J. C. Murphy, "Thermal effects in photothermal spectroscopy and photothermal imaging," J. Appl. Phys. 54, 581-591 (1983).
[CrossRef]

1981 (1)

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

1980 (1)

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

1969 (1)

A. Maitland and M. H. Dunn, Laser Physics (North-Holland, 1969).

1959 (1)

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

Aamodt, L. C.

L. C. Aamodt and J. C. Murphy, "Thermal effects in photothermal spectroscopy and photothermal imaging," J. Appl. Phys. 54, 581-591 (1983).
[CrossRef]

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

Anger, E.

K. Plamann, D. Fournier, E. Anger, and A. Gicquel, "Photothermal examination of the heat diffusion inhomogeneity in diamond films of sub-micron thickness," Diamond Relat. Mater. 3, 752-756 (1994).
[CrossRef]

Arora, B. M.

K. A. Shailendra, K. L. Narasimhan, S. S. Rajalakshmi, S. S. Chandvankar, and B. M. Arora, "Photothermal deflection spectroscopy of heat-treated GaAs, InP, and InGaAsP alloys," Appl. Phys. Lett. 55, 2512-2513 (1989).
[CrossRef]

Badoz, J.

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

Bertolotti, M.

M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
[CrossRef]

Boccara, A. C.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

F. Lepoutre, D. Fournier, and A. C. Boccara, "Nondestructive control of weldings using the mirage detection," J. Appl. Phys. 57, 1009-1015 (1985).
[CrossRef]

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

Bodzenta, J.

J. Bodzenta and M. Pyka, "Photothermal measurement with mirage effect for investigation of LiNbO3 single crystals," J. Phys. IV 137, 259-263 (2006).
[CrossRef]

D. K. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, "The complex ray theory of photodeflection signal formation: comparison with the ray theory and the experimental results," J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

J. Bodzenta, B. Burak, A. Jagoda, and B. Stanczyk, "Thermal conductivity of AIN and AIN-GaN thin films deposited on Si and GaAs substrates," Diamond Relat. Mater. 14, 1169-1174 (2005).
[CrossRef]

Bukowski, R. J.

D. K. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, "The complex ray theory of photodeflection signal formation: comparison with the ray theory and the experimental results," J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

R. J. Bukowski and D. Korte, "The deflectional signal analysis in photothermal measurements in the frame of complex geometrical optics," Opt. Appl. 35, 77-92 (2005).

R. J. Bukowski and D. Korte, "Influence pf probing beam focusing on photothermal signal," J. Phys. IV 109, 19-31 (2003).

R. J. Bukowski and D. Korte, "Perturbation calculus for eikonal application to analysis of the deflectional signal in photothermal measurements," Opt. Appl. 32, 817-828 (2002).

R. J. Bukowski, "Complex geometrical optics application for description of Gaussian beam propagation in optically homogenous media," in Proceedings of Second National Conference "Physical Grounds on Nondestructive Investigation" (Gliwice Division of the Polish Physical Society and Institute of Physics of Silesian University of Technology, 1997), pp. 45-55 (in Polish).

Burak, B.

D. K. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, "The complex ray theory of photodeflection signal formation: comparison with the ray theory and the experimental results," J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

J. Bodzenta, B. Burak, A. Jagoda, and B. Stanczyk, "Thermal conductivity of AIN and AIN-GaN thin films deposited on Si and GaAs substrates," Diamond Relat. Mater. 14, 1169-1174 (2005).
[CrossRef]

Carslaw, H. S.

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

Chandvankar, S. S.

K. A. Shailendra, K. L. Narasimhan, S. S. Rajalakshmi, S. S. Chandvankar, and B. M. Arora, "Photothermal deflection spectroscopy of heat-treated GaAs, InP, and InGaAsP alloys," Appl. Phys. Lett. 55, 2512-2513 (1989).
[CrossRef]

Commandr, M.

Dorogan, V.

M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
[CrossRef]

Dunn, M. H.

A. Maitland and M. H. Dunn, Laser Physics (North-Holland, 1969).

Favro, L. D.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Fernandez, J.

A. Salazar, A. Sanchez-Lavega, and J. Fernandez, "Theory of thermal diffusivity determination by the "mirage" technique in solids," J. Appl. Phys. 65, 4150-4156 (1989).

Fournier, D.

K. Plamann, D. Fournier, E. Anger, and A. Gicquel, "Photothermal examination of the heat diffusion inhomogeneity in diamond films of sub-micron thickness," Diamond Relat. Mater. 3, 752-756 (1994).
[CrossRef]

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

F. Lepoutre, D. Fournier, and A. C. Boccara, "Nondestructive control of weldings using the mirage detection," J. Appl. Phys. 57, 1009-1015 (1985).
[CrossRef]

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

Friedrich, K.

H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov, "New phase interference technique applied for sensitive photothermal microscopy," Appl. Phys. Lett. 57, 1600-1601 (1990).
[CrossRef]

Gicquel, A.

K. Plamann, D. Fournier, E. Anger, and A. Gicquel, "Photothermal examination of the heat diffusion inhomogeneity in diamond films of sub-micron thickness," Diamond Relat. Mater. 3, 752-756 (1994).
[CrossRef]

Glazov, A.

H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov, "New phase interference technique applied for sensitive photothermal microscopy," Appl. Phys. Lett. 57, 1600-1601 (1990).
[CrossRef]

Glazov, A. L.

A. L. Glazov and K. L. Muratikov, "Calculation of the photodeflection signal in the framework of wave optics," Tech. Phys. 38, 344-352 (1993).

A. L. Glazov and K. L. Muratikov, "Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. "Mirage" effect geometry," Opt. Commun. 84, 283-289 (1991).

A. L. Glazov and K. L. Muratikov, "Photodeflection and interferometric thermal wave microscopy of solids," Int. J. Optoelectron. 4, 589-597 (1989).

Gotoh, T.

T. Gotoh, S. Nonomura, S. Hirata, and S. Nitta, "Photothermal bending spectroscopy and photothermal deflection spectroscopy of C60 thin films," Appl. Surf. Sci. 113/114, 278-281 (1997).
[CrossRef]

Haupt, K.

H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov, "New phase interference technique applied for sensitive photothermal microscopy," Appl. Phys. Lett. 57, 1600-1601 (1990).
[CrossRef]

Hirata, S.

T. Gotoh, S. Nonomura, S. Hirata, and S. Nitta, "Photothermal bending spectroscopy and photothermal deflection spectroscopy of C60 thin films," Appl. Surf. Sci. 113/114, 278-281 (1997).
[CrossRef]

Inglehart, L. J.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Jaeger, J. C.

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

Jagoda, A.

J. Bodzenta, B. Burak, A. Jagoda, and B. Stanczyk, "Thermal conductivity of AIN and AIN-GaN thin films deposited on Si and GaAs substrates," Diamond Relat. Mater. 14, 1169-1174 (2005).
[CrossRef]

Kim, D. S.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Kobylinska, D. K.

D. K. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, "The complex ray theory of photodeflection signal formation: comparison with the ray theory and the experimental results," J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

Kochowski, S.

D. K. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, "The complex ray theory of photodeflection signal formation: comparison with the ray theory and the experimental results," J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

Korte, D.

R. J. Bukowski and D. Korte, "The deflectional signal analysis in photothermal measurements in the frame of complex geometrical optics," Opt. Appl. 35, 77-92 (2005).

R. J. Bukowski and D. Korte, "Influence pf probing beam focusing on photothermal signal," J. Phys. IV 109, 19-31 (2003).

R. J. Bukowski and D. Korte, "Perturbation calculus for eikonal application to analysis of the deflectional signal in photothermal measurements," Opt. Appl. 32, 817-828 (2002).

Kravtsov, Ju. A.

Ju. A. Kravtsov and Ju. I. Orlov, Geometrical Optics of the Nonhomogeneous Media (WNT, 1993) (in Polish).

Kuo, P. K.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Lasalle, E. L.

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

Lepoutre, F.

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

F. Lepoutre, D. Fournier, and A. C. Boccara, "Nondestructive control of weldings using the mirage detection," J. Appl. Phys. 57, 1009-1015 (1985).
[CrossRef]

Li Voti, R.

M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
[CrossRef]

Liakhou, G.

M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
[CrossRef]

Lin, M. J.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Maitland, A.

A. Maitland and M. H. Dunn, Laser Physics (North-Holland, 1969).

Muratikov, K.

H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov, "New phase interference technique applied for sensitive photothermal microscopy," Appl. Phys. Lett. 57, 1600-1601 (1990).
[CrossRef]

Muratikov, K. L.

A. L. Glazov and K. L. Muratikov, "Calculation of the photodeflection signal in the framework of wave optics," Tech. Phys. 38, 344-352 (1993).

A. L. Glazov and K. L. Muratikov, "Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. "Mirage" effect geometry," Opt. Commun. 84, 283-289 (1991).

A. L. Glazov and K. L. Muratikov, "Photodeflection and interferometric thermal wave microscopy of solids," Int. J. Optoelectron. 4, 589-597 (1989).

Murphy, J. C.

L. C. Aamodt and J. C. Murphy, "Thermal effects in photothermal spectroscopy and photothermal imaging," J. Appl. Phys. 54, 581-591 (1983).
[CrossRef]

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

Narasimhan, K. L.

K. A. Shailendra, K. L. Narasimhan, S. S. Rajalakshmi, S. S. Chandvankar, and B. M. Arora, "Photothermal deflection spectroscopy of heat-treated GaAs, InP, and InGaAsP alloys," Appl. Phys. Lett. 55, 2512-2513 (1989).
[CrossRef]

Nitta, S.

T. Gotoh, S. Nonomura, S. Hirata, and S. Nitta, "Photothermal bending spectroscopy and photothermal deflection spectroscopy of C60 thin films," Appl. Surf. Sci. 113/114, 278-281 (1997).
[CrossRef]

Nonomura, S.

T. Gotoh, S. Nonomura, S. Hirata, and S. Nitta, "Photothermal bending spectroscopy and photothermal deflection spectroscopy of C60 thin films," Appl. Surf. Sci. 113/114, 278-281 (1997).
[CrossRef]

Orlov, Ju. I.

Ju. A. Kravtsov and Ju. I. Orlov, Geometrical Optics of the Nonhomogeneous Media (WNT, 1993) (in Polish).

Paoloni, S.

M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
[CrossRef]

Plamann, K.

K. Plamann, D. Fournier, E. Anger, and A. Gicquel, "Photothermal examination of the heat diffusion inhomogeneity in diamond films of sub-micron thickness," Diamond Relat. Mater. 3, 752-756 (1994).
[CrossRef]

Power, J. F.

Pyka, M.

J. Bodzenta and M. Pyka, "Photothermal measurement with mirage effect for investigation of LiNbO3 single crystals," J. Phys. IV 137, 259-263 (2006).
[CrossRef]

Rajalakshmi, S. S.

K. A. Shailendra, K. L. Narasimhan, S. S. Rajalakshmi, S. S. Chandvankar, and B. M. Arora, "Photothermal deflection spectroscopy of heat-treated GaAs, InP, and InGaAsP alloys," Appl. Phys. Lett. 55, 2512-2513 (1989).
[CrossRef]

Reyes, C. B.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Roche, P.

Roger, J. P.

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

Salazar, A.

A. Sanchez-Lavega and A. Salazar, "Thermal diffusivity measurements in opaque solids by the mirage technique in the temperature range from 300 to 1000 K," J. Appl. Phys. 76, 1462-1468 (1994).
[CrossRef]

A. Salazar, A. Sanchez-Lavega, and J. Fernandez, "Theory of thermal diffusivity determination by the "mirage" technique in solids," J. Appl. Phys. 65, 4150-4156 (1989).

Sanchez-Lavega, A.

A. Sanchez-Lavega and A. Salazar, "Thermal diffusivity measurements in opaque solids by the mirage technique in the temperature range from 300 to 1000 K," J. Appl. Phys. 76, 1462-1468 (1994).
[CrossRef]

A. Salazar, A. Sanchez-Lavega, and J. Fernandez, "Theory of thermal diffusivity determination by the "mirage" technique in solids," J. Appl. Phys. 65, 4150-4156 (1989).

Schweitzer, M. A.

Shailendra, K. A.

K. A. Shailendra, K. L. Narasimhan, S. S. Rajalakshmi, S. S. Chandvankar, and B. M. Arora, "Photothermal deflection spectroscopy of heat-treated GaAs, InP, and InGaAsP alloys," Appl. Phys. Lett. 55, 2512-2513 (1989).
[CrossRef]

Sibilia, C.

M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
[CrossRef]

Stanczyk, B.

J. Bodzenta, B. Burak, A. Jagoda, and B. Stanczyk, "Thermal conductivity of AIN and AIN-GaN thin films deposited on Si and GaAs substrates," Diamond Relat. Mater. 14, 1169-1174 (2005).
[CrossRef]

Thomas, R. L.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Walther, H. G.

H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov, "New phase interference technique applied for sensitive photothermal microscopy," Appl. Phys. Lett. 57, 1600-1601 (1990).
[CrossRef]

Zhang, S.

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

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

K. A. Shailendra, K. L. Narasimhan, S. S. Rajalakshmi, S. S. Chandvankar, and B. M. Arora, "Photothermal deflection spectroscopy of heat-treated GaAs, InP, and InGaAsP alloys," Appl. Phys. Lett. 55, 2512-2513 (1989).
[CrossRef]

H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov, "New phase interference technique applied for sensitive photothermal microscopy," Appl. Phys. Lett. 57, 1600-1601 (1990).
[CrossRef]

Appl. Spectrosc. (1)

Appl. Surf. Sci. (1)

T. Gotoh, S. Nonomura, S. Hirata, and S. Nitta, "Photothermal bending spectroscopy and photothermal deflection spectroscopy of C60 thin films," Appl. Surf. Sci. 113/114, 278-281 (1997).
[CrossRef]

Can. J. Phys. (1)

P. K. Kuo, M. J. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Zhang, L. J. Inglehart, D. Fournier, and A. C. Boccara, "Mirage-effect measurement of thermal diffusivity. Part I: experiment," Can. J. Phys. 64, 1165-1167 (1986).
[CrossRef]

Diamond Relat. Mater. (2)

K. Plamann, D. Fournier, E. Anger, and A. Gicquel, "Photothermal examination of the heat diffusion inhomogeneity in diamond films of sub-micron thickness," Diamond Relat. Mater. 3, 752-756 (1994).
[CrossRef]

J. Bodzenta, B. Burak, A. Jagoda, and B. Stanczyk, "Thermal conductivity of AIN and AIN-GaN thin films deposited on Si and GaAs substrates," Diamond Relat. Mater. 14, 1169-1174 (2005).
[CrossRef]

Int. J. Optoelectron. (1)

A. L. Glazov and K. L. Muratikov, "Photodeflection and interferometric thermal wave microscopy of solids," Int. J. Optoelectron. 4, 589-597 (1989).

J. Appl. Phys. (7)

L. C. Aamodt and J. C. Murphy, "Thermal effects in photothermal spectroscopy and photothermal imaging," J. Appl. Phys. 54, 581-591 (1983).
[CrossRef]

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

F. Lepoutre, D. Fournier, and A. C. Boccara, "Nondestructive control of weldings using the mirage detection," J. Appl. Phys. 57, 1009-1015 (1985).
[CrossRef]

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

A. Salazar, A. Sanchez-Lavega, and J. Fernandez, "Theory of thermal diffusivity determination by the "mirage" technique in solids," J. Appl. Phys. 65, 4150-4156 (1989).

A. Sanchez-Lavega and A. Salazar, "Thermal diffusivity measurements in opaque solids by the mirage technique in the temperature range from 300 to 1000 K," J. Appl. Phys. 76, 1462-1468 (1994).
[CrossRef]

D. K. Kobylinska, R. J. Bukowski, B. Burak, J. Bodzenta, and S. Kochowski, "The complex ray theory of photodeflection signal formation: comparison with the ray theory and the experimental results," J. Appl. Phys. 100, 063501 (2006).
[CrossRef]

J. Phys. IV (2)

R. J. Bukowski and D. Korte, "Influence pf probing beam focusing on photothermal signal," J. Phys. IV 109, 19-31 (2003).

J. Bodzenta and M. Pyka, "Photothermal measurement with mirage effect for investigation of LiNbO3 single crystals," J. Phys. IV 137, 259-263 (2006).
[CrossRef]

Opt. Appl. (2)

R. J. Bukowski and D. Korte, "The deflectional signal analysis in photothermal measurements in the frame of complex geometrical optics," Opt. Appl. 35, 77-92 (2005).

R. J. Bukowski and D. Korte, "Perturbation calculus for eikonal application to analysis of the deflectional signal in photothermal measurements," Opt. Appl. 32, 817-828 (2002).

Opt. Commun. (1)

A. L. Glazov and K. L. Muratikov, "Photodeflection signal formation in thermal wave spectroscopy and microscopy of solids within the framework of wave optics. "Mirage" effect geometry," Opt. Commun. 84, 283-289 (1991).

Rev. Sci. Instrum. (1)

M. Bertolotti, V. Dorogan, G. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, "New photothermal deflection method for thermal diffusivity measurement of semiconductor wafers," Rev. Sci. Instrum. 68, 1521-1526 (1997).
[CrossRef]

Tech. Phys. (1)

A. L. Glazov and K. L. Muratikov, "Calculation of the photodeflection signal in the framework of wave optics," Tech. Phys. 38, 344-352 (1993).

Other (4)

R. J. Bukowski, "Complex geometrical optics application for description of Gaussian beam propagation in optically homogenous media," in Proceedings of Second National Conference "Physical Grounds on Nondestructive Investigation" (Gliwice Division of the Polish Physical Society and Institute of Physics of Silesian University of Technology, 1997), pp. 45-55 (in Polish).

Ju. A. Kravtsov and Ju. I. Orlov, Geometrical Optics of the Nonhomogeneous Media (WNT, 1993) (in Polish).

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

A. Maitland and M. H. Dunn, Laser Physics (North-Holland, 1969).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the experimental configuration.

Fig. 2
Fig. 2

(a) Calculated amplitude and (b) phase of photodeflection signal changes versus the probe beam radius a to the length of the TW in air λ g ratio for different theories: CRT, the complex ray theory; RDAT, the ray deflection averaging theory; WT, the wave theory. The height of the probe beam over the sample h = 800   μm , the angular modulation frequency of the temperature field Ω = 60 rad / s   ( λ g = 5130   μm ) , the detector position z D = 1.5   m , the sample position z l = 0.5   m .

Fig. 3
Fig. 3

(a) Calculated amplitude and (b) phase of photodeflection signal changes versus the probe beam radius a to the length of the TW in air λ g ratio for different theories: CRT, the complex ray theory; RDAT, the ray deflection averaging theory; WT, the wave theory. The height of the probe beam over the sample h = 800   μm , the angular modulation frequency of the temperature field Ω = 60 rad / s ( λ g = 5130   μm ) , the detector position z D = 1.5   m , the sample position was z l = 0.5   m , the probe beam-waist position L = 1.6   m .

Fig. 4
Fig. 4

(a) Calculated amplitude and (b) phase of photodeflection signal changes versus the detector position z D for different theories: CRT, the complex ray theory; RDAT, the ray deflection averaging theory; WT, the wave theory. The height of the probe beam over the sample h = 1200   μm , the probe beam radius in the waist a = 500   μm , the angular modulation frequency of the temperature field Ω = 600 rad / s ( λ g = 1622   μm ) , the sample position z l = 0.5   m , the probe beam-waist position L = 0.5   m .

Fig. 5
Fig. 5

(a) Calculated amplitude and (b) phase of photodeflection signal changes versus the probe beam-waist position L for different theories: CRT, the complex ray theory; RDAT, the ray deflection averaging theory; WT, the wave theory. The height of the probe beam over the sample h = 800   μm , the probe beam radius in the waist a = 150   μm , the angular modulation frequency of the temperature field Ω = 4500 rad / s ( λ g = 621   μm ) , the sample position z l = 0.5   m , the detector position z D = 1.5   m .

Fig. 6
Fig. 6

Block diagram of the experimental setup.

Fig. 7
Fig. 7

(a) Amplitude and (b) phase of photodeflection signal dependence on the height h of the probe beam over the sample. EXPERIMENT, the experimental results; the solid curve represents the best fitting to the experimental data on the basis of the CRT. The obtained by fitting values of the parameters are shown in the figure. RDAT are the dependences calculated on the grounds of the ray averaging theory for the parameters obtained by fitting. WT are the dependences calculated on the grounds of the WT for the parameters obtained by fitting. The probe beam radius in the place of probe beam interaction with the thermal wave field a ( z l ) = 215   μm , the angular modulation frequency of temperature field Ω = 3391 rad / s ( λ g = 682   μm ) , the sample position z l = 22.5   cm , the detector position z D = 77   cm .

Fig. 8
Fig. 8

(a) Amplitude and the (b) phase of photodeflection signal dependence on modulation frequencies of the temperature field Ω. EXPERIMENT, the experimental results; the solid curve represents the best fitting to the experimental data on the basis of the CRT. The obtained by fitting values of the parameters are shown in the figure. RDAT are the dependences calculated on the grounds of the ray averaging theory for the parameters obtained by fitting. WT are the dependences calculated on the grounds of the WT for the parameters obtained by fitting. The sample position z l = 7.5   cm , the detector position z D = 77   cm .

Fig. 9
Fig. 9

(a) Amplitude and the (b) phase of photodeflection signal dependence on the sample position z l . EXPERIMENT, the experimental results; the solid curve represents the best fitting to the experimental data on the basis of the CRT. The obtained-by fitting values of the parameters are shown in the figure. RDAT are the dependences calculated on the grounds of the ray averaging theory for the parameters obtained by fitting. WT are the dependences calculated on the grounds of the WT for the parameters obtained by fitting. The angular modulation frequency of the temperature field Ω = 1445 rad / s ( λ g = 1095   μm ) , the detector position z D = 77   cm .

Equations (65)

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ϑ g ( x , t ) = T ( x , t ) T 0 = b g exp [ Ω 2 κ g ( x + h ) ] × cos [ Ω t Ω 2 κ g ( x + h ) + φ g ] .
n ( T ) n 0 + d n d T | T 0 ( T T 0 ) = n 0 + n 0 s T ϑ g ( x , t ) ,
x ( τ ) = ξ ( 1 + i n 0 τ z R C ) ,
y ( τ ) = η ( 1 + i n 0 τ z R C ) ,
z ( τ ) = n 0 τ 1 + ξ 2 + η 2 z R C 2 ,
x 1 ( ξ , τ ) = n 0 2 s T 0 τ ( τ τ ) ϑ g x d τ P ( ξ ) ( n 0 τ z s ) ( z p z l ) ,
P ( ξ ) = n 0 s T b g Ω κ g exp [ ξ Ω 2 κ g ( 1 + i z s z R C ) ] × sin [ Ω t ξ Ω 2 κ g ( 1 + i z s z R C ) + φ g π 4 ] ,
A ( z ) A ( τ = 0 ) [ ( x / ξ ) ( y / η ) ( z / τ ) τ = 0 ( x / ξ ) ( y / η ) ( z / τ ) ] 1 / 2 E 0 z R z R C ( 1 + i z z R C ) 1 [ 1 1 2 P ξ ( τ z s n 0 ) ( z p n 0 z l n 0 ) × ( 1 + i z z R C ) 1 ] ,
ψ = ψ 0 + n 0 2 τ + n 0 2 s T 0 τ ϑ g ( x ( τ ) ) d τ ,
ψ 1 = ψ 1 f + ψ 1 d ,
ψ 1 f = k n 0 s T b g ( z p z l ) exp ( k g x s ) cos ( Ω t k g x s + φ g ) ,
ψ 1 d = z n 0 x z R C 2 P ( ξ ) ( z z s ) ( z p z l ) ( 1 + i z z R C ) 2 .
S n = K d + d y D ( 0 + h 0 ) d x D I ( r D ) = S n d + S n f = A t cos ( Ω t + φ g + φ t ) .
ϕ n = ( 1 n d n d T ) | T 0 z l z p ϑ g ( x , t ) x d z .
S n r = y D = + x D = h + P ( x D , y D ) ϕ n ( x D , y D ) d x D d y D = A r cos ( Ω t + φ g + φ r + π 4 ) ,
Δ u ( x , y , z ) i Δ ϕ u 0 ( x , y , z ) ,
Δ ϕ = 2 π λ d n d T | T 0 z l z p ϑ g ( x , t ) d z .
u ( x D , y D , z D ) = i λ ( z D z s ) + + [ u 0 ( x s , y s , z s ) + Δ u ( x s , y s , z s ) ] exp { i π λ ( z D z s ) × [ ( x D x s ) 2 + ( y D y s ) 2 ] } d x D d y D .
S n w = 2 K d y D = + x D = h + [ I s ( x D , y D , z D ) I 0 ( x D , y D , z D ) ] d x D d y D = A w cos ( Ω t + φ g + φ w ) ,
A t = [ A d cos ( φ g + φ d ) + A f cos ( φ g φ f ) ] 2 + [ A d sin ( φ g + φ d ) + A f sin ( φ g φ f ) ] 2 ,
tan φ t = A d sin ( φ g + φ d ) + A f sin ( φ g φ f ) A d cos ( φ g + φ d ) + A f cos ( φ g φ f ) ,
A d = [ A d A cos φ d A + A d f cos ( φ d f π 4 ) ] 2 + [ A d A sin φ d A + A d f sin ( φ d f π 4 ) ] 2 ,
tan φ d = A d A sin φ d A + A d f sin ( φ d f π / 4 ) A d A cos φ d A + A d f cos ( φ d f π / 4 ) ,
A d A = 2 K d [ Re ( G 1 G 2 ) ] 2 + [ Im ( G 1 + G 2 ) ] 2 ,
  tan φ d A = Im ( G 1 + G 2 ) Re ( G 1 G 2 ) ,
A d f = 2 K d [ Im ( H 1 H 2 ) ] 2 + [ Re ( H 1 + H 2 ) ] 2 ,
tan φ d f = Re ( H 1 + H 2 ) Im ( H 1 H 2 ) ,
A f = 2 K d [ Im ( F 1 + F 2 ) ] 2 + [ Re ( F 2 F 1 ) ] 2 ,
tan φ f = Re ( F 2 F 1 ) Im ( F 1 + F 2 ) ,
G 1 = ν m exp [ ( i 1 ) 2 C γ 2 4 a 2 λ g 2 ] { erf [ ( i 1 ) C γ 2 a λ g ] ± 1 2   erf [ ( i 1 ) C γ 2 a λ g + z R z R 2 + ( L z D ) 2 h a ] + 1 2 } ,
G 2 = ν m exp [ ( i + 1 ) 2 C γ 2 4 a 2 λ g 2 ] { erf [ ( i + 1 ) C γ 2 a λ g ] ± 1 2   erf [ ( i + 1 ) C γ 2 a λ g + z R z R 2 + ( L z D ) 2 h a ] + 1 2 } , 
  H 1 = κ m { π ( i 1 ) C γ 2 a λ g exp [ ( i 1 ) 2 C γ 2 4 a 2 λ g 2 ] [ erf ( ( i 1 ) C γ 2 a λ g + z R z R 2 + ( L z D ) 2 h a ) erf ( ( i 1 ) C γ 2 a λ g ) ] + exp [ 2 π ( i 1 ) ( 1 + i z s z R C ) ( 1 + i z D z R C ) 1 h λ g z R z R 2 + ( L z D ) 2 h 2 a 2 ] ± π ( i + 1 ) C γ 2 a λ g × exp [ ( i + 1 ) 2 C γ 2 4 a 2 λ g 2 ] [ 1 erf ( ( i + 1 ) C γ 2 a λ g ) ] } ,
  H 2 = κ m { π ( i 1 ) C γ 2 a λ g exp [ ( i 1 ) 2 C γ 2 4 a 2 λ g 2 ] [ 1 erf ( ( i 1 ) C γ 2 a λ g ) ] + exp [ 2 π ( i + 1 ) ( 1 + i z s z R C ) × ( 1 + i z D z R C ) 1 h λ g z R z R 2 + ( L z D ) 2 h 2 a 2 ] + π ( i + 1 ) C γ 2 a λ g exp [ ( i + 1 ) 2 C γ 2 4 a 2 λ g 2 ] [ erf ( ( i + 1 ) C γ 2 a λ g ) erf ( ( i + 1 ) C γ 2 a λ g + z R z R 2 + ( L z D ) 2 h a ) ] } ,
  F 1 = γ m exp { [ ( i + 1 ) C γ 2 a λ g ] 2 } { 1 2   erf [ ( i + 1 ) C γ 2 a λ g ] + erf [ ( i + 1 ) C γ 2 a λ g z R z R 2 + ( L z D ) 2 h a ] } ,
  F 2 = γ m exp { [ ( i 1 ) C γ 2 a λ g ] 2 } { 1 2   erf [ ( i 1 ) C γ 2 a λ g ] + erf [ ( i 1 ) C γ 2 a λ g z R z R 2 + ( L z D ) 2 h a ] } ,
  ν m = 2 π i s T b g P l ( z D z s ) ( z p z l ) λ g 2 ( 1 + i z D / z R C ) 2 ,
  κ m = s T b g k g 2 π a z D P l ( z D z s ) ( z p z l ) z R 2 + ( L z D ) 2 [ 2 i z R C 2 ( 1 + i z D z R C ) 2 ] 1 ,
  γ m = 1 4 n 0 s T b g k P l ( z p z l ) ,
  C γ = 2 π z R ( 1 + i z s z R C ) ( 1 + i z D z R C ) 1 z R 2 + ( L z D ) 2 ,
A r = [ Re ( Q 1 Q 3 Q 2 + Q 4 ) ] 2 + Im [ ( Q 1 Q 3 + Q 2 Q 4 ) ] 2 ,
tan φ r = Im ( Q 1 Q 3 + Q 2 Q 4 ) Re ( Q 1 Q 3 Q 2 + Q 4 ) ,
Q 1 = ζ exp { [ π ( 1 + i ) ] 2 a 2 λ g 2 } { 1 erf [ π ( 1 + i ) a λ g ] } ,
Q 2 = ζ exp { [ π ( 1 i ) ] 2 a 2 λ g 2 } { 1 + erf [ π ( 1 i ) a λ g ] } ,
Q 3 = ζ exp { [ π ( 1 + i ) ] 2 a 2 λ g 2 } { erf [ π ( 1 + i ) a λ g + h a ] + erf [ π ( 1 + i ) a λ g ] } ,
Q 4 = ζ exp { [ π ( 1 i ) ] 2 a 2 λ g 2 } { erf [ π ( 1 i ) a λ g + h a ] erf [ π ( 1 i ) a λ g ] } ,
ζ = 2 π 4 i a 2 s T b g k g ( z p z l ) .
A w = 2 K d [ Re ( P 1 + P 2 + P 3 + P 4 ) ] 2 + [ Im ( P 3 + P 4 P 1 P 2 ) ] 2 ,
tan φ w = Im ( P 1 + P 2 P 3 P 4 ) Re ( P 1 + P 2 + P 3 + P 4 ) ,
P 1 = π σ m R α 2 R 1 R 2 { ( δ 0 + i δ 1 ) cos [ k ( z l L ) ] ( δ 1 i δ 0 ) sin [ k ( z l L ) ] } exp [ 4 π 2 a 2 λ g 2 ( α 1 i α 2 ) i k ( z l L ) π 2 k 2 a 2 ( a λ g ) 2 ( α 1 i α 2 ) 2 × ( z D z s ) 2 R 1 1 ] { 1 erf [ i R 1 h + i π k a a λ g ( α 1 i α 2 ) ( z D z s ) 1 R 1 1 ] } ,
P 2 = π σ m R β 2 R 4 R 2 { ( δ 0 i δ 1 ) cos [ k ( z l L ) ] ( δ 1 + i δ 0 ) × sin [ k ( z l L ) ] } exp [ 4 π 2 a 2 λ g 2 ( α 3 i α 4 ) i k ( z l L ) π 2 k 2 a 2 ( a λ g ) 2 ( α 3 i α 4 ) 2 × ( z D z s ) 2 R 4 1 ] { 1 erf [ i R 4 h i π a k a λ g × ( α 3 i α 4 ) ( z D z s ) 1 R 4 1 ] } ,
P 3 = π σ m R α 2 R 3 R 2 { ( δ 0 + i δ 1 ) cos [ k ( z l L ) ] ( δ 1 i δ 0 ) sin [ k ( z l L ) ] } exp [ 4 π 2 a 2 λ g 2 ( α 3 + i α 4 ) i k ( z l L ) π 2 k 2 a 2 ( a λ g ) 2 ( α 3 + i α 4 ) 2 × ( z D z s ) 2 R 3 1 ] { 1 erf [ i R 3 h i π a k a λ g × ( α 3 + i α 4 ) ( z D z s ) 1 R 3 1 ] } ,
P 4 = π σ m R β 2 R 5 R 2 { ( δ 0 i δ 1 ) cos [ k ( z l L ) ] ( δ 1 + i δ 0 ) × sin [ k ( z l L ) ] } exp [ 4 π 2 a 2 λ g 2 ( α 1 + i α 2 ) i k ( z l L ) π 2 k 2 a 2 ( a λ g ) 2 ( α 1 + i α 2 ) 2 × ( z D z s ) 2 R 5 1 ] { 1 erf [ i R 5 h + i π a k a λ g × ( α 1 + i α 2 ) ( z D z s ) 1 R 5 1 ] } ,
σ m = k z R 2 π a P l π z l L i z R z D z s [ z R 2 + L 2 3 ( z R 2 + L 2 ) 1 z l 2 L 2 + ( z R 2 + L 2 ) 3 z l 4 L 4 ] 1 ,
χ m = s T k b g 2 π n 0 2 a P l π z R 3 ( z p z l ) ( z D z s ) 2 ( z l 2 + z R 2 ) × [ 1 + ( z l L z R ) 2 ] 2 { 1 [ k 2 ( z D z s ) 1 × ( 1 + ( z l L z R ) 2 ) 2 z l L k n 0 ] 2 } 1 ,
δ 0 = χ m ( z R 2 + L 2 ) 2 ( z R 2 + L 2 z l L ) 2 + z R 2 z l 2 { 2 z R ( z D z s ) k × [ 1 + ( z l L ) 2 z R 2 ] ( 1 2 z l L z R 2 + L 2 ) + k n 0 2 z R ( z l L ) × [ 1 + ( z l L ) 2 z R 2 ] 1 } ,
δ 1 = χ m k 2 [ 1 + ( z l L z R ) 2 ] 1 [ ( z D z s ) 1 n 0 z R 2 ( z l L ) 2 + n 0 ( z R 2 + L 2 ) 2 ( z R 2 + L 2 z l L ) 2 + z R 2 z l 2 ] ,
R 1 = k ( z D z s ) { ( z D z s ) 1 [ z R 2 n 0 + k a 2 ( i α 2 α 1 ) ] + i 2 [ 1 + ( z l L ) z R ] } ,
R 2 = k 2 ( z D z s ) { z R n 0 ( z D z s ) i [ 1 + ( z l L ) z R ] } ,
R 3 = k ( z D z s ) { ( z D z s ) 1 [ z R 2 n 0 + k a 2 ( α 4 + i α 3 ) ] i 2 [ 1 + ( z l L ) z R ] } ,
R 4 = k ( z D z s ) { ( z D z s ) 1 [ z R 2 n 0 + k a 2 ( α 4 i α 3 ) ] i 2 [ 1 + ( z l L ) z R ] } ,
R 5 = k ( z D z s ) { ( z D z s ) 1 [ z R 2 n 0 + k a 2 ( α 1 + i α 2 ) ] i 2 [ 1 + ( z l L ) z R ] } ,
α 1 = 1 2 [ 1 + ( z l L z R ) 2 ] { 1 [ z R n 0 ( z D z s ) [ 1 + ( z l L z R ) 2 ] ( z l L ) z R ] 2 } 1 ,
α 2 = α 1 { z R n 0 ( z D z s ) [ 1 + ( z l L z R ) 2 ] 1 } ,
α 3 = 1 2 [ 1 ( z l L z R ) 2 ] { 1 [ z R n 0 ( z D z s ) [ 1 ( z l L z R ) 2 ] z l L z R ] 2 } 1 ,
α 4 = α 3 [ 1 ( z l L z R ) 2 ] { z R n 0 ( z D z s ) [ 1 ( z l L z R ) 2 ] 1 } .

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