Abstract

An overview of photothermal deflection principles and applications is given. The modeling of temperature distribution and the calculation of deflection that is due to both the refractive-index gradient and the thermal deformation of the sample are presented. Three configurations usually employed are compared, and their respective advantages are discussed in relation to their application. The calibration for absolute measurement of absorption is detailed, showing that calibration limits the accuracy of measurement. Some examples of specific information obtained by photothermal mapping of absorption are given.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Ristau, X. C. Dang, J. Ebert, “Interface and bulk absorption of oxyde layers and correlation to damage threshold at 1.064 μm,” Nat. Bur. Stand. (U.S.) Spec. Publ. 727, 298–312 (1984).
  2. M. Rahe, E. Oertel, L. Reinhardt, D. Ristau, H. Welling, “Absorption calorimetry and laser induced damage threshold measurements of AR-coated ZnSe and metal mirrors at 10.6 μm,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1441, 113–126 (1990).
  3. R. Chow, S. Falabella, G. E. Loomis, F. Rainer, C. J. Stolz, M. R. Kozlowski, “Absorption and damage thresholds of low-defect-density hafnia deposited with activated oxygen,” in Laser Induced Damage in Optical Materials: 1992, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1848, 349–359 (1992).
  4. M. R. Lange, J. K. MacIver, A. H. Guenther, “Laser damage threshold predictions based on the effects of thermal and optical properties employing a spherical impurity model,” Nat. Bur. Stand. (U.S.) Spec. Publ. 668, 454–465 (1983).
  5. S. M. J. Akhtar, D. Ristau, J. Ebert, “Thermal conductivity of dielectric films and correlation to damage thresholds at 1064 nm,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 345–351 (1986).
  6. A. H. Guenther, J. K. MacIver, “The role of thermal conductivity in the pulsed laser damage sensitivity of optical thin films,” Thin Solid Films 163, 203–214 (1988).
    [CrossRef]
  7. A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
    [CrossRef] [PubMed]
  8. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981).
    [CrossRef] [PubMed]
  9. M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, “Photothermal displacement spectroscopy: an optical probe for solids and surfaces,” Appl. Phys. A 32, 141–154 (1983).
    [CrossRef]
  10. W. C. Mundy, R. S. Hughes, C. K. Carniglia, “Photothermal deflection microscopy of dielectric thin films,” Appl. Phys. Lett. 43, 985–987 (1983).
    [CrossRef]
  11. J. A. Abate, A. W. Schmid, M. J. Guardalben, D. J. Smith, S. D. Jacobs, “Characterization of micron-sized defects by photothermal deflection spectroscopy,” Nat. Bur. Stand. (U.S.) Spec. Publ. 688, 385–392 (1983).
  12. M. Commandré, L. Bertrand, G. Albrand, E. Pelletier, “Measurement of absorption losses of optical thin film components by photothermal deflection spectroscopy,” in Optical Components and Systems, A. Masson, ed., Proc. SPIE805, 128–135 (1987).
  13. M. Commandré, E. Pelletier, “Measurements of absorption losses in TiO2 films by a collinear photothermal deflection technique,” Appl. Opt. 29, 4276–4283 (1990).
    [CrossRef] [PubMed]
  14. M. Commandré, P. Roche, G. Albrand, E. Pelletier, “Photothermal deflection spectroscopy for the study of thin films and optical coatings: measurement of absorption losses and detection of photo-induced changes,” in Optical Thin Films and Applications, R. Herrmann ed., Proc. SPIE1270, 82–93 (1990).
  15. S. E. Watkins, R. Heimlich, R. Reis, “Mapping of absorption in optical coatings,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 246–255 (1991).
  16. Z. L. Wu, M. Reichling, E. Welsch, D. Schäfer, Z. X. Fan, E. Matthias, “Defect characterisation for thin films through thermal wave detection,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 271–281 (1991); M. Reichling, E. Welsch, A. Duparré, E. Matthias, “Photothermal absorption microscopy of defects in ZrO2 and MgF2 single-layer films,” Opt. Eng. 33, 1334–1342 (1994).
    [CrossRef]
  17. Z. L. Wu, M. Reichling, H. Grönbeck, Z. X. Fan, D. Schaefer, E. Matthias, “Photothermal measurement of thermal conductivity of optical coatings,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 331–345 (1991), and references therein; M. Reichling, H. Grönbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914–1922 (1994).
    [CrossRef]
  18. M. Commandré, “Caractérisation de l’absorption dans les composants optiques en couches minces par déflexion photothermique,” Thése de Doctorat d’Etat (Université d’Aix-Marseille, Marseille, 1992).
  19. M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Surface contamination of bare substrates. Mapping of absorption and influence on deposited thin films,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 982–992 (1994).
  20. M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Absorption mapping for characterization of glass surfaces,” Appl. Opt. 34, 2372–2379 (1995).
    [CrossRef] [PubMed]
  21. M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Effects of deposition conditions on thin film bulk and interface absorption,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1253–1262 (1994).
  22. P. Roche, M. Commandré, R. Mollenhauer, F. Flory, “Interpretation of measurements of both losses on guided propagation and absorption from a model of absorbing transition layers,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1286–1296 (1994).
  23. C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).
  24. E. Welsch, “Absorption measurements,” in Thin Films for Optical Coatings, R. E. Hummel, K. H. Guenther, eds. (CRC, Boca Raton, Fla., 1995), Chap. 9, pp. 243–272 and references therein.
  25. M. Commandré, P. Roche, “Characterisation of absorption by photothermal deflection,” in Thin Films for Optical Systems, F. Flory, ed. (Dekker, New York, 1995), Chap. 12, pp. 329–365.
  26. E. Abraham, J. M. Halley, “Some calculations of temperature profiles in thin films with laser heating,” Appl. Phys. A 42, 279–285 (1987).
    [CrossRef]
  27. G. Rousset, F. Charbonnier, F. Lepoutre, “Influence of radiative and convective transfers in a photothermal experiment,” J. Appl. Phys. 56, 2093–2096 (1984).
    [CrossRef]
  28. F. Flory, H. Rigneault, N. N. Maythaveekulchai, F. Zamkotsian, “Characterization by guided wave of instabilities of optical coatings submitted to high-power flux: thermal and third-order nonlinear properties of dielectric thin films,” Appl. Opt. 32, 28, 5628–5639 (1993).
    [CrossRef] [PubMed]
  29. J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal wave detection and thin film thickness measurements with laser beam deflection,” Appl. Opt. 22, 3169–3176 (1983).
    [CrossRef] [PubMed]
  30. D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986).
    [CrossRef] [PubMed]
  31. D. L. Decker, L. G. Koshigoe, E. J. Ashley, “Thermal properties of optical thin film materials,” Nat. Bur. Stand. Spec. Publ. 727, 291–297 (1984).
  32. J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
    [CrossRef]
  33. V. Scheuer, C. Schuchert, T. Tschudi, “The influence of small amounts of impurities in sputtered laser mirrors on their performance,” in Thin Films in Optics, T. T. Tschudi, ed., Proc. SPIE1125, 54–60 (1990).
  34. P. Roche, E. Pelletier, “Characterisation of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
    [CrossRef] [PubMed]
  35. P. A. Temple, “Examination of laser damage sites of transparent surfaces and films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1979, Nat. Bur. Stand. Spec. Publ.568, 333–341 (1979).
  36. F. L. Williams, C. K. Carniglia, B. J. Pond, W. K. Stowell, “Investigation of thin films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 299–308 (1989).
  37. R. C. Estier, N. S. Nogar, R. A. Schmell, “The detection, removal and effect on damage thresholds of cerium impurities on fused silica,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 183–188 (1988).
  38. T. Raj, D. E. McCready, C. K. Carniglia, “Substrate cleaning in vacuum by laser irradiation,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 152–165 (1988).
  39. R. S. Hockett, “Quantitative analysis of surface trace metal contamination on substrates and films by TXRF,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 239–253 (1989).
  40. Very smooth surface finish (French Standard NF S 10-006). The rms roughness is ~0.3 nm.
  41. H. K. Pulker, “Nature of a surface,” in Coating on Glass (Elsevier, New York, 1984), Chap 3, pp. 34–42.
  42. T. S. Izumitani, Optical Glass, American Institute of Physics Translation Series (American Institute of Physics, New York, 1986), Chap. 2, pp. 15–55.
  43. K. Kinosita, “Surface deterioration of optical glasses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 85–143.
    [CrossRef]
  44. P. Roche, M. Commandré, L. Escoubas, J. P. Borgogno, G. Albrand, B. Lazarides, “Substrate effects on absorption of coated surfaces,” Appl. Opt. 35, 5059–5066 (1996).
    [CrossRef] [PubMed]

1996 (1)

1995 (1)

1993 (1)

1990 (1)

1989 (1)

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

1988 (1)

A. H. Guenther, J. K. MacIver, “The role of thermal conductivity in the pulsed laser damage sensitivity of optical thin films,” Thin Solid Films 163, 203–214 (1988).
[CrossRef]

1987 (1)

E. Abraham, J. M. Halley, “Some calculations of temperature profiles in thin films with laser heating,” Appl. Phys. A 42, 279–285 (1987).
[CrossRef]

1986 (2)

D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986).
[CrossRef] [PubMed]

S. M. J. Akhtar, D. Ristau, J. Ebert, “Thermal conductivity of dielectric films and correlation to damage thresholds at 1064 nm,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 345–351 (1986).

1984 (4)

D. Ristau, X. C. Dang, J. Ebert, “Interface and bulk absorption of oxyde layers and correlation to damage threshold at 1.064 μm,” Nat. Bur. Stand. (U.S.) Spec. Publ. 727, 298–312 (1984).

D. L. Decker, L. G. Koshigoe, E. J. Ashley, “Thermal properties of optical thin film materials,” Nat. Bur. Stand. Spec. Publ. 727, 291–297 (1984).

P. Roche, E. Pelletier, “Characterisation of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
[CrossRef] [PubMed]

G. Rousset, F. Charbonnier, F. Lepoutre, “Influence of radiative and convective transfers in a photothermal experiment,” J. Appl. Phys. 56, 2093–2096 (1984).
[CrossRef]

1983 (5)

J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal wave detection and thin film thickness measurements with laser beam deflection,” Appl. Opt. 22, 3169–3176 (1983).
[CrossRef] [PubMed]

M. R. Lange, J. K. MacIver, A. H. Guenther, “Laser damage threshold predictions based on the effects of thermal and optical properties employing a spherical impurity model,” Nat. Bur. Stand. (U.S.) Spec. Publ. 668, 454–465 (1983).

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, “Photothermal displacement spectroscopy: an optical probe for solids and surfaces,” Appl. Phys. A 32, 141–154 (1983).
[CrossRef]

W. C. Mundy, R. S. Hughes, C. K. Carniglia, “Photothermal deflection microscopy of dielectric thin films,” Appl. Phys. Lett. 43, 985–987 (1983).
[CrossRef]

J. A. Abate, A. W. Schmid, M. J. Guardalben, D. J. Smith, S. D. Jacobs, “Characterization of micron-sized defects by photothermal deflection spectroscopy,” Nat. Bur. Stand. (U.S.) Spec. Publ. 688, 385–392 (1983).

1981 (1)

1980 (1)

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
[CrossRef] [PubMed]

Abate, J. A.

J. A. Abate, A. W. Schmid, M. J. Guardalben, D. J. Smith, S. D. Jacobs, “Characterization of micron-sized defects by photothermal deflection spectroscopy,” Nat. Bur. Stand. (U.S.) Spec. Publ. 688, 385–392 (1983).

Abraham, E.

E. Abraham, J. M. Halley, “Some calculations of temperature profiles in thin films with laser heating,” Appl. Phys. A 42, 279–285 (1987).
[CrossRef]

Akhtar, S. M. J.

S. M. J. Akhtar, D. Ristau, J. Ebert, “Thermal conductivity of dielectric films and correlation to damage thresholds at 1064 nm,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 345–351 (1986).

Albrand, G.

P. Roche, M. Commandré, L. Escoubas, J. P. Borgogno, G. Albrand, B. Lazarides, “Substrate effects on absorption of coated surfaces,” Appl. Opt. 35, 5059–5066 (1996).
[CrossRef] [PubMed]

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Absorption mapping for characterization of glass surfaces,” Appl. Opt. 34, 2372–2379 (1995).
[CrossRef] [PubMed]

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Effects of deposition conditions on thin film bulk and interface absorption,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1253–1262 (1994).

C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).

M. Commandré, L. Bertrand, G. Albrand, E. Pelletier, “Measurement of absorption losses of optical thin film components by photothermal deflection spectroscopy,” in Optical Components and Systems, A. Masson, ed., Proc. SPIE805, 128–135 (1987).

M. Commandré, P. Roche, G. Albrand, E. Pelletier, “Photothermal deflection spectroscopy for the study of thin films and optical coatings: measurement of absorption losses and detection of photo-induced changes,” in Optical Thin Films and Applications, R. Herrmann ed., Proc. SPIE1270, 82–93 (1990).

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Surface contamination of bare substrates. Mapping of absorption and influence on deposited thin films,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 982–992 (1994).

Amer, N. M.

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, “Photothermal displacement spectroscopy: an optical probe for solids and surfaces,” Appl. Phys. A 32, 141–154 (1983).
[CrossRef]

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

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
[CrossRef] [PubMed]

Amra, C.

C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).

Amsden, C. A.

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

Ashley, E. J.

D. L. Decker, L. G. Koshigoe, E. J. Ashley, “Thermal properties of optical thin film materials,” Nat. Bur. Stand. Spec. Publ. 727, 291–297 (1984).

Bertrand, L.

M. Commandré, L. Bertrand, G. Albrand, E. Pelletier, “Measurement of absorption losses of optical thin film components by photothermal deflection spectroscopy,” in Optical Components and Systems, A. Masson, ed., Proc. SPIE805, 128–135 (1987).

Boccara, A. C.

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, “Photothermal displacement spectroscopy: an optical probe for solids and surfaces,” Appl. Phys. A 32, 141–154 (1983).
[CrossRef]

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

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
[CrossRef] [PubMed]

Borgogno, J. P.

P. Roche, M. Commandré, L. Escoubas, J. P. Borgogno, G. Albrand, B. Lazarides, “Substrate effects on absorption of coated surfaces,” Appl. Opt. 35, 5059–5066 (1996).
[CrossRef] [PubMed]

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Absorption mapping for characterization of glass surfaces,” Appl. Opt. 34, 2372–2379 (1995).
[CrossRef] [PubMed]

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Surface contamination of bare substrates. Mapping of absorption and influence on deposited thin films,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 982–992 (1994).

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Effects of deposition conditions on thin film bulk and interface absorption,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1253–1262 (1994).

Carniglia, C. K.

W. C. Mundy, R. S. Hughes, C. K. Carniglia, “Photothermal deflection microscopy of dielectric thin films,” Appl. Phys. Lett. 43, 985–987 (1983).
[CrossRef]

T. Raj, D. E. McCready, C. K. Carniglia, “Substrate cleaning in vacuum by laser irradiation,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 152–165 (1988).

F. L. Williams, C. K. Carniglia, B. J. Pond, W. K. Stowell, “Investigation of thin films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 299–308 (1989).

Charbonnier, F.

G. Rousset, F. Charbonnier, F. Lepoutre, “Influence of radiative and convective transfers in a photothermal experiment,” J. Appl. Phys. 56, 2093–2096 (1984).
[CrossRef]

Chow, R.

R. Chow, S. Falabella, G. E. Loomis, F. Rainer, C. J. Stolz, M. R. Kozlowski, “Absorption and damage thresholds of low-defect-density hafnia deposited with activated oxygen,” in Laser Induced Damage in Optical Materials: 1992, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1848, 349–359 (1992).

Cleva, F.

C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).

Commandré, M.

P. Roche, M. Commandré, L. Escoubas, J. P. Borgogno, G. Albrand, B. Lazarides, “Substrate effects on absorption of coated surfaces,” Appl. Opt. 35, 5059–5066 (1996).
[CrossRef] [PubMed]

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Absorption mapping for characterization of glass surfaces,” Appl. Opt. 34, 2372–2379 (1995).
[CrossRef] [PubMed]

M. Commandré, E. Pelletier, “Measurements of absorption losses in TiO2 films by a collinear photothermal deflection technique,” Appl. Opt. 29, 4276–4283 (1990).
[CrossRef] [PubMed]

M. Commandré, P. Roche, G. Albrand, E. Pelletier, “Photothermal deflection spectroscopy for the study of thin films and optical coatings: measurement of absorption losses and detection of photo-induced changes,” in Optical Thin Films and Applications, R. Herrmann ed., Proc. SPIE1270, 82–93 (1990).

M. Commandré, L. Bertrand, G. Albrand, E. Pelletier, “Measurement of absorption losses of optical thin film components by photothermal deflection spectroscopy,” in Optical Components and Systems, A. Masson, ed., Proc. SPIE805, 128–135 (1987).

M. Commandré, “Caractérisation de l’absorption dans les composants optiques en couches minces par déflexion photothermique,” Thése de Doctorat d’Etat (Université d’Aix-Marseille, Marseille, 1992).

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Surface contamination of bare substrates. Mapping of absorption and influence on deposited thin films,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 982–992 (1994).

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Effects of deposition conditions on thin film bulk and interface absorption,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1253–1262 (1994).

P. Roche, M. Commandré, R. Mollenhauer, F. Flory, “Interpretation of measurements of both losses on guided propagation and absorption from a model of absorbing transition layers,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1286–1296 (1994).

M. Commandré, P. Roche, “Characterisation of absorption by photothermal deflection,” in Thin Films for Optical Systems, F. Flory, ed. (Dekker, New York, 1995), Chap. 12, pp. 329–365.

Dang, X. C.

D. Ristau, X. C. Dang, J. Ebert, “Interface and bulk absorption of oxyde layers and correlation to damage threshold at 1.064 μm,” Nat. Bur. Stand. (U.S.) Spec. Publ. 727, 298–312 (1984).

Decker, D. L.

D. L. Decker, L. G. Koshigoe, E. J. Ashley, “Thermal properties of optical thin film materials,” Nat. Bur. Stand. Spec. Publ. 727, 291–297 (1984).

Diakomihalis, D.

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

Ebert, J.

D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986).
[CrossRef] [PubMed]

S. M. J. Akhtar, D. Ristau, J. Ebert, “Thermal conductivity of dielectric films and correlation to damage thresholds at 1064 nm,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 345–351 (1986).

D. Ristau, X. C. Dang, J. Ebert, “Interface and bulk absorption of oxyde layers and correlation to damage threshold at 1.064 μm,” Nat. Bur. Stand. (U.S.) Spec. Publ. 727, 298–312 (1984).

Escoubas, L.

Estier, R. C.

R. C. Estier, N. S. Nogar, R. A. Schmell, “The detection, removal and effect on damage thresholds of cerium impurities on fused silica,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 183–188 (1988).

Falabella, S.

R. Chow, S. Falabella, G. E. Loomis, F. Rainer, C. J. Stolz, M. R. Kozlowski, “Absorption and damage thresholds of low-defect-density hafnia deposited with activated oxygen,” in Laser Induced Damage in Optical Materials: 1992, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1848, 349–359 (1992).

Fan, Z. X.

Z. L. Wu, M. Reichling, H. Grönbeck, Z. X. Fan, D. Schaefer, E. Matthias, “Photothermal measurement of thermal conductivity of optical coatings,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 331–345 (1991), and references therein; M. Reichling, H. Grönbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914–1922 (1994).
[CrossRef]

Z. L. Wu, M. Reichling, E. Welsch, D. Schäfer, Z. X. Fan, E. Matthias, “Defect characterisation for thin films through thermal wave detection,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 271–281 (1991); M. Reichling, E. Welsch, A. Duparré, E. Matthias, “Photothermal absorption microscopy of defects in ZrO2 and MgF2 single-layer films,” Opt. Eng. 33, 1334–1342 (1994).
[CrossRef]

Flory, F.

F. Flory, H. Rigneault, N. N. Maythaveekulchai, F. Zamkotsian, “Characterization by guided wave of instabilities of optical coatings submitted to high-power flux: thermal and third-order nonlinear properties of dielectric thin films,” Appl. Opt. 32, 28, 5628–5639 (1993).
[CrossRef] [PubMed]

P. Roche, M. Commandré, R. Mollenhauer, F. Flory, “Interpretation of measurements of both losses on guided propagation and absorption from a model of absorbing transition layers,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1286–1296 (1994).

Fournier, D.

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, “Photothermal displacement spectroscopy: an optical probe for solids and surfaces,” Appl. Phys. A 32, 141–154 (1983).
[CrossRef]

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

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
[CrossRef] [PubMed]

Gilman, S. E.

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

Grézes-Besset, C.

C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).

Grönbeck, H.

Z. L. Wu, M. Reichling, H. Grönbeck, Z. X. Fan, D. Schaefer, E. Matthias, “Photothermal measurement of thermal conductivity of optical coatings,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 331–345 (1991), and references therein; M. Reichling, H. Grönbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914–1922 (1994).
[CrossRef]

Guardalben, M. J.

J. A. Abate, A. W. Schmid, M. J. Guardalben, D. J. Smith, S. D. Jacobs, “Characterization of micron-sized defects by photothermal deflection spectroscopy,” Nat. Bur. Stand. (U.S.) Spec. Publ. 688, 385–392 (1983).

Guenther, A. H.

A. H. Guenther, J. K. MacIver, “The role of thermal conductivity in the pulsed laser damage sensitivity of optical thin films,” Thin Solid Films 163, 203–214 (1988).
[CrossRef]

M. R. Lange, J. K. MacIver, A. H. Guenther, “Laser damage threshold predictions based on the effects of thermal and optical properties employing a spherical impurity model,” Nat. Bur. Stand. (U.S.) Spec. Publ. 668, 454–465 (1983).

Halley, J. M.

E. Abraham, J. M. Halley, “Some calculations of temperature profiles in thin films with laser heating,” Appl. Phys. A 42, 279–285 (1987).
[CrossRef]

Heimlich, R.

S. E. Watkins, R. Heimlich, R. Reis, “Mapping of absorption in optical coatings,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 246–255 (1991).

Hockett, R. S.

R. S. Hockett, “Quantitative analysis of surface trace metal contamination on substrates and films by TXRF,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 239–253 (1989).

Hughes, R. S.

W. C. Mundy, R. S. Hughes, C. K. Carniglia, “Photothermal deflection microscopy of dielectric thin films,” Appl. Phys. Lett. 43, 985–987 (1983).
[CrossRef]

Izumitani, T. S.

T. S. Izumitani, Optical Glass, American Institute of Physics Translation Series (American Institute of Physics, New York, 1986), Chap. 2, pp. 15–55.

Jackson, W.

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
[CrossRef] [PubMed]

Jackson, W. B.

Jacobs, S. D.

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

J. A. Abate, A. W. Schmid, M. J. Guardalben, D. J. Smith, S. D. Jacobs, “Characterization of micron-sized defects by photothermal deflection spectroscopy,” Nat. Bur. Stand. (U.S.) Spec. Publ. 688, 385–392 (1983).

Joly, M. R.

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

Kinosita, K.

K. Kinosita, “Surface deterioration of optical glasses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 85–143.
[CrossRef]

Kohn, S.

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, “Photothermal displacement spectroscopy: an optical probe for solids and surfaces,” Appl. Phys. A 32, 141–154 (1983).
[CrossRef]

Koshigoe, L. G.

D. L. Decker, L. G. Koshigoe, E. J. Ashley, “Thermal properties of optical thin film materials,” Nat. Bur. Stand. Spec. Publ. 727, 291–297 (1984).

Kozlowski, M. R.

R. Chow, S. Falabella, G. E. Loomis, F. Rainer, C. J. Stolz, M. R. Kozlowski, “Absorption and damage thresholds of low-defect-density hafnia deposited with activated oxygen,” in Laser Induced Damage in Optical Materials: 1992, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1848, 349–359 (1992).

Lambropoulos, J. C.

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

Lange, M. R.

M. R. Lange, J. K. MacIver, A. H. Guenther, “Laser damage threshold predictions based on the effects of thermal and optical properties employing a spherical impurity model,” Nat. Bur. Stand. (U.S.) Spec. Publ. 668, 454–465 (1983).

Lazarides, B.

Lepoutre, F.

G. Rousset, F. Charbonnier, F. Lepoutre, “Influence of radiative and convective transfers in a photothermal experiment,” J. Appl. Phys. 56, 2093–2096 (1984).
[CrossRef]

Loomis, G. E.

R. Chow, S. Falabella, G. E. Loomis, F. Rainer, C. J. Stolz, M. R. Kozlowski, “Absorption and damage thresholds of low-defect-density hafnia deposited with activated oxygen,” in Laser Induced Damage in Optical Materials: 1992, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1848, 349–359 (1992).

MacIver, J. K.

A. H. Guenther, J. K. MacIver, “The role of thermal conductivity in the pulsed laser damage sensitivity of optical thin films,” Thin Solid Films 163, 203–214 (1988).
[CrossRef]

M. R. Lange, J. K. MacIver, A. H. Guenther, “Laser damage threshold predictions based on the effects of thermal and optical properties employing a spherical impurity model,” Nat. Bur. Stand. (U.S.) Spec. Publ. 668, 454–465 (1983).

Matthias, E.

Z. L. Wu, M. Reichling, E. Welsch, D. Schäfer, Z. X. Fan, E. Matthias, “Defect characterisation for thin films through thermal wave detection,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 271–281 (1991); M. Reichling, E. Welsch, A. Duparré, E. Matthias, “Photothermal absorption microscopy of defects in ZrO2 and MgF2 single-layer films,” Opt. Eng. 33, 1334–1342 (1994).
[CrossRef]

Z. L. Wu, M. Reichling, H. Grönbeck, Z. X. Fan, D. Schaefer, E. Matthias, “Photothermal measurement of thermal conductivity of optical coatings,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 331–345 (1991), and references therein; M. Reichling, H. Grönbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914–1922 (1994).
[CrossRef]

Maure, S.

C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).

Maythaveekulchai, N. N.

McCready, D. E.

T. Raj, D. E. McCready, C. K. Carniglia, “Substrate cleaning in vacuum by laser irradiation,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 152–165 (1988).

Mollenhauer, R.

C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).

P. Roche, M. Commandré, R. Mollenhauer, F. Flory, “Interpretation of measurements of both losses on guided propagation and absorption from a model of absorbing transition layers,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1286–1296 (1994).

Mundy, W. C.

W. C. Mundy, R. S. Hughes, C. K. Carniglia, “Photothermal deflection microscopy of dielectric thin films,” Appl. Phys. Lett. 43, 985–987 (1983).
[CrossRef]

Nogar, N. S.

R. C. Estier, N. S. Nogar, R. A. Schmell, “The detection, removal and effect on damage thresholds of cerium impurities on fused silica,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 183–188 (1988).

Oertel, E.

M. Rahe, E. Oertel, L. Reinhardt, D. Ristau, H. Welling, “Absorption calorimetry and laser induced damage threshold measurements of AR-coated ZnSe and metal mirrors at 10.6 μm,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1441, 113–126 (1990).

Olmstead, M. A.

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, “Photothermal displacement spectroscopy: an optical probe for solids and surfaces,” Appl. Phys. A 32, 141–154 (1983).
[CrossRef]

Opsal, J.

J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal wave detection and thin film thickness measurements with laser beam deflection,” Appl. Opt. 22, 3169–3176 (1983).
[CrossRef] [PubMed]

Pelletier, E.

M. Commandré, E. Pelletier, “Measurements of absorption losses in TiO2 films by a collinear photothermal deflection technique,” Appl. Opt. 29, 4276–4283 (1990).
[CrossRef] [PubMed]

P. Roche, E. Pelletier, “Characterisation of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
[CrossRef] [PubMed]

M. Commandré, L. Bertrand, G. Albrand, E. Pelletier, “Measurement of absorption losses of optical thin film components by photothermal deflection spectroscopy,” in Optical Components and Systems, A. Masson, ed., Proc. SPIE805, 128–135 (1987).

M. Commandré, P. Roche, G. Albrand, E. Pelletier, “Photothermal deflection spectroscopy for the study of thin films and optical coatings: measurement of absorption losses and detection of photo-induced changes,” in Optical Thin Films and Applications, R. Herrmann ed., Proc. SPIE1270, 82–93 (1990).

Pond, B. J.

F. L. Williams, C. K. Carniglia, B. J. Pond, W. K. Stowell, “Investigation of thin films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 299–308 (1989).

Pulker, H. K.

H. K. Pulker, “Nature of a surface,” in Coating on Glass (Elsevier, New York, 1984), Chap 3, pp. 34–42.

Rahe, M.

M. Rahe, E. Oertel, L. Reinhardt, D. Ristau, H. Welling, “Absorption calorimetry and laser induced damage threshold measurements of AR-coated ZnSe and metal mirrors at 10.6 μm,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1441, 113–126 (1990).

Rainer, F.

R. Chow, S. Falabella, G. E. Loomis, F. Rainer, C. J. Stolz, M. R. Kozlowski, “Absorption and damage thresholds of low-defect-density hafnia deposited with activated oxygen,” in Laser Induced Damage in Optical Materials: 1992, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1848, 349–359 (1992).

Raj, T.

T. Raj, D. E. McCready, C. K. Carniglia, “Substrate cleaning in vacuum by laser irradiation,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 152–165 (1988).

Ranier, M.

C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).

Reichling, M.

Z. L. Wu, M. Reichling, H. Grönbeck, Z. X. Fan, D. Schaefer, E. Matthias, “Photothermal measurement of thermal conductivity of optical coatings,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 331–345 (1991), and references therein; M. Reichling, H. Grönbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914–1922 (1994).
[CrossRef]

Z. L. Wu, M. Reichling, E. Welsch, D. Schäfer, Z. X. Fan, E. Matthias, “Defect characterisation for thin films through thermal wave detection,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 271–281 (1991); M. Reichling, E. Welsch, A. Duparré, E. Matthias, “Photothermal absorption microscopy of defects in ZrO2 and MgF2 single-layer films,” Opt. Eng. 33, 1334–1342 (1994).
[CrossRef]

Reinhardt, L.

M. Rahe, E. Oertel, L. Reinhardt, D. Ristau, H. Welling, “Absorption calorimetry and laser induced damage threshold measurements of AR-coated ZnSe and metal mirrors at 10.6 μm,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1441, 113–126 (1990).

Reis, R.

S. E. Watkins, R. Heimlich, R. Reis, “Mapping of absorption in optical coatings,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 246–255 (1991).

Rigneault, H.

Ristau, D.

D. Ristau, J. Ebert, “Development of a thermographic laser calorimeter,” Appl. Opt. 25, 4571–4578 (1986).
[CrossRef] [PubMed]

S. M. J. Akhtar, D. Ristau, J. Ebert, “Thermal conductivity of dielectric films and correlation to damage thresholds at 1064 nm,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 345–351 (1986).

D. Ristau, X. C. Dang, J. Ebert, “Interface and bulk absorption of oxyde layers and correlation to damage threshold at 1.064 μm,” Nat. Bur. Stand. (U.S.) Spec. Publ. 727, 298–312 (1984).

M. Rahe, E. Oertel, L. Reinhardt, D. Ristau, H. Welling, “Absorption calorimetry and laser induced damage threshold measurements of AR-coated ZnSe and metal mirrors at 10.6 μm,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1441, 113–126 (1990).

Roche, P.

P. Roche, M. Commandré, L. Escoubas, J. P. Borgogno, G. Albrand, B. Lazarides, “Substrate effects on absorption of coated surfaces,” Appl. Opt. 35, 5059–5066 (1996).
[CrossRef] [PubMed]

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Absorption mapping for characterization of glass surfaces,” Appl. Opt. 34, 2372–2379 (1995).
[CrossRef] [PubMed]

P. Roche, E. Pelletier, “Characterisation of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
[CrossRef] [PubMed]

M. Commandré, P. Roche, “Characterisation of absorption by photothermal deflection,” in Thin Films for Optical Systems, F. Flory, ed. (Dekker, New York, 1995), Chap. 12, pp. 329–365.

P. Roche, M. Commandré, R. Mollenhauer, F. Flory, “Interpretation of measurements of both losses on guided propagation and absorption from a model of absorbing transition layers,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1286–1296 (1994).

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Effects of deposition conditions on thin film bulk and interface absorption,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1253–1262 (1994).

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Surface contamination of bare substrates. Mapping of absorption and influence on deposited thin films,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 982–992 (1994).

M. Commandré, P. Roche, G. Albrand, E. Pelletier, “Photothermal deflection spectroscopy for the study of thin films and optical coatings: measurement of absorption losses and detection of photo-induced changes,” in Optical Thin Films and Applications, R. Herrmann ed., Proc. SPIE1270, 82–93 (1990).

Rosencwaig, A.

J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal wave detection and thin film thickness measurements with laser beam deflection,” Appl. Opt. 22, 3169–3176 (1983).
[CrossRef] [PubMed]

Rousset, G.

G. Rousset, F. Charbonnier, F. Lepoutre, “Influence of radiative and convective transfers in a photothermal experiment,” J. Appl. Phys. 56, 2093–2096 (1984).
[CrossRef]

Schaefer, D.

Z. L. Wu, M. Reichling, H. Grönbeck, Z. X. Fan, D. Schaefer, E. Matthias, “Photothermal measurement of thermal conductivity of optical coatings,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 331–345 (1991), and references therein; M. Reichling, H. Grönbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914–1922 (1994).
[CrossRef]

Schäfer, D.

Z. L. Wu, M. Reichling, E. Welsch, D. Schäfer, Z. X. Fan, E. Matthias, “Defect characterisation for thin films through thermal wave detection,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 271–281 (1991); M. Reichling, E. Welsch, A. Duparré, E. Matthias, “Photothermal absorption microscopy of defects in ZrO2 and MgF2 single-layer films,” Opt. Eng. 33, 1334–1342 (1994).
[CrossRef]

Scheuer, V.

V. Scheuer, C. Schuchert, T. Tschudi, “The influence of small amounts of impurities in sputtered laser mirrors on their performance,” in Thin Films in Optics, T. T. Tschudi, ed., Proc. SPIE1125, 54–60 (1990).

Schmell, R. A.

R. C. Estier, N. S. Nogar, R. A. Schmell, “The detection, removal and effect on damage thresholds of cerium impurities on fused silica,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 183–188 (1988).

Schmid, A. W.

J. A. Abate, A. W. Schmid, M. J. Guardalben, D. J. Smith, S. D. Jacobs, “Characterization of micron-sized defects by photothermal deflection spectroscopy,” Nat. Bur. Stand. (U.S.) Spec. Publ. 688, 385–392 (1983).

Schuchert, C.

V. Scheuer, C. Schuchert, T. Tschudi, “The influence of small amounts of impurities in sputtered laser mirrors on their performance,” in Thin Films in Optics, T. T. Tschudi, ed., Proc. SPIE1125, 54–60 (1990).

Sinicropi, M. J.

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

Smith, D. J.

J. A. Abate, A. W. Schmid, M. J. Guardalben, D. J. Smith, S. D. Jacobs, “Characterization of micron-sized defects by photothermal deflection spectroscopy,” Nat. Bur. Stand. (U.S.) Spec. Publ. 688, 385–392 (1983).

Stolz, C. J.

R. Chow, S. Falabella, G. E. Loomis, F. Rainer, C. J. Stolz, M. R. Kozlowski, “Absorption and damage thresholds of low-defect-density hafnia deposited with activated oxygen,” in Laser Induced Damage in Optical Materials: 1992, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1848, 349–359 (1992).

Stowell, W. K.

F. L. Williams, C. K. Carniglia, B. J. Pond, W. K. Stowell, “Investigation of thin films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 299–308 (1989).

Temple, P. A.

P. A. Temple, “Examination of laser damage sites of transparent surfaces and films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1979, Nat. Bur. Stand. Spec. Publ.568, 333–341 (1979).

Tschudi, T.

V. Scheuer, C. Schuchert, T. Tschudi, “The influence of small amounts of impurities in sputtered laser mirrors on their performance,” in Thin Films in Optics, T. T. Tschudi, ed., Proc. SPIE1125, 54–60 (1990).

Watkins, S. E.

S. E. Watkins, R. Heimlich, R. Reis, “Mapping of absorption in optical coatings,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 246–255 (1991).

Welling, H.

M. Rahe, E. Oertel, L. Reinhardt, D. Ristau, H. Welling, “Absorption calorimetry and laser induced damage threshold measurements of AR-coated ZnSe and metal mirrors at 10.6 μm,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1441, 113–126 (1990).

Welsch, E.

Z. L. Wu, M. Reichling, E. Welsch, D. Schäfer, Z. X. Fan, E. Matthias, “Defect characterisation for thin films through thermal wave detection,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 271–281 (1991); M. Reichling, E. Welsch, A. Duparré, E. Matthias, “Photothermal absorption microscopy of defects in ZrO2 and MgF2 single-layer films,” Opt. Eng. 33, 1334–1342 (1994).
[CrossRef]

E. Welsch, “Absorption measurements,” in Thin Films for Optical Coatings, R. E. Hummel, K. H. Guenther, eds. (CRC, Boca Raton, Fla., 1995), Chap. 9, pp. 243–272 and references therein.

Willenborg, D. L.

J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal wave detection and thin film thickness measurements with laser beam deflection,” Appl. Opt. 22, 3169–3176 (1983).
[CrossRef] [PubMed]

Williams, F. L.

F. L. Williams, C. K. Carniglia, B. J. Pond, W. K. Stowell, “Investigation of thin films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 299–308 (1989).

Wu, Z. L.

Z. L. Wu, M. Reichling, E. Welsch, D. Schäfer, Z. X. Fan, E. Matthias, “Defect characterisation for thin films through thermal wave detection,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 271–281 (1991); M. Reichling, E. Welsch, A. Duparré, E. Matthias, “Photothermal absorption microscopy of defects in ZrO2 and MgF2 single-layer films,” Opt. Eng. 33, 1334–1342 (1994).
[CrossRef]

Z. L. Wu, M. Reichling, H. Grönbeck, Z. X. Fan, D. Schaefer, E. Matthias, “Photothermal measurement of thermal conductivity of optical coatings,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 331–345 (1991), and references therein; M. Reichling, H. Grönbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914–1922 (1994).
[CrossRef]

Zamkotsian, F.

Appl. Opt. (2)

J. Opsal, A. Rosencwaig, D. L. Willenborg, “Thermal wave detection and thin film thickness measurements with laser beam deflection,” Appl. Opt. 22, 3169–3176 (1983).
[CrossRef] [PubMed]

P. Roche, E. Pelletier, “Characterisation of optical surfaces by measurement of scattering distribution,” Appl. Opt. 23, 3561–3566 (1984).
[CrossRef] [PubMed]

Appl. Opt. (6)

Appl. Phys. A (2)

M. A. Olmstead, N. M. Amer, S. Kohn, D. Fournier, A. C. Boccara, “Photothermal displacement spectroscopy: an optical probe for solids and surfaces,” Appl. Phys. A 32, 141–154 (1983).
[CrossRef]

E. Abraham, J. M. Halley, “Some calculations of temperature profiles in thin films with laser heating,” Appl. Phys. A 42, 279–285 (1987).
[CrossRef]

Appl. Phys. Lett. (1)

W. C. Mundy, R. S. Hughes, C. K. Carniglia, “Photothermal deflection microscopy of dielectric thin films,” Appl. Phys. Lett. 43, 985–987 (1983).
[CrossRef]

J. Appl. Phys. (1)

G. Rousset, F. Charbonnier, F. Lepoutre, “Influence of radiative and convective transfers in a photothermal experiment,” J. Appl. Phys. 56, 2093–2096 (1984).
[CrossRef]

J. Appl. Phys. (1)

J. C. Lambropoulos, M. R. Joly, C. A. Amsden, S. E. Gilman, M. J. Sinicropi, D. Diakomihalis, S. D. Jacobs, “Thermal conductivity of dielectric thin films,” J. Appl. Phys. 66, 4230–4242 (1989).
[CrossRef]

Nat. Bur. Stand. (U.S.) Spec. Publ. (1)

J. A. Abate, A. W. Schmid, M. J. Guardalben, D. J. Smith, S. D. Jacobs, “Characterization of micron-sized defects by photothermal deflection spectroscopy,” Nat. Bur. Stand. (U.S.) Spec. Publ. 688, 385–392 (1983).

Nat. Bur. Stand. (U.S.) Spec. Publ. (2)

D. Ristau, X. C. Dang, J. Ebert, “Interface and bulk absorption of oxyde layers and correlation to damage threshold at 1.064 μm,” Nat. Bur. Stand. (U.S.) Spec. Publ. 727, 298–312 (1984).

M. R. Lange, J. K. MacIver, A. H. Guenther, “Laser damage threshold predictions based on the effects of thermal and optical properties employing a spherical impurity model,” Nat. Bur. Stand. (U.S.) Spec. Publ. 668, 454–465 (1983).

Nat. Bur. Stand. Spec. Publ. (1)

D. L. Decker, L. G. Koshigoe, E. J. Ashley, “Thermal properties of optical thin film materials,” Nat. Bur. Stand. Spec. Publ. 727, 291–297 (1984).

Natl. Inst. Stand. Technol. Spec. Publ. (1)

S. M. J. Akhtar, D. Ristau, J. Ebert, “Thermal conductivity of dielectric films and correlation to damage thresholds at 1064 nm,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 345–351 (1986).

Opt. Lett. (1)

A. C. Boccara, D. Fournier, W. Jackson, N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377–379 (1980).
[CrossRef] [PubMed]

Thin Solid Films (1)

A. H. Guenther, J. K. MacIver, “The role of thermal conductivity in the pulsed laser damage sensitivity of optical thin films,” Thin Solid Films 163, 203–214 (1988).
[CrossRef]

Other (24)

M. Rahe, E. Oertel, L. Reinhardt, D. Ristau, H. Welling, “Absorption calorimetry and laser induced damage threshold measurements of AR-coated ZnSe and metal mirrors at 10.6 μm,” in Laser-Induced Damage in Optical Materials: 1990, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1441, 113–126 (1990).

R. Chow, S. Falabella, G. E. Loomis, F. Rainer, C. J. Stolz, M. R. Kozlowski, “Absorption and damage thresholds of low-defect-density hafnia deposited with activated oxygen,” in Laser Induced Damage in Optical Materials: 1992, H. E. Bennett, L. L. Chase, A. H. Guenther, B. Newman, M. J. Soileau, eds., Proc. SPIE1848, 349–359 (1992).

M. Commandré, L. Bertrand, G. Albrand, E. Pelletier, “Measurement of absorption losses of optical thin film components by photothermal deflection spectroscopy,” in Optical Components and Systems, A. Masson, ed., Proc. SPIE805, 128–135 (1987).

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Effects of deposition conditions on thin film bulk and interface absorption,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1253–1262 (1994).

P. Roche, M. Commandré, R. Mollenhauer, F. Flory, “Interpretation of measurements of both losses on guided propagation and absorption from a model of absorbing transition layers,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1286–1296 (1994).

C. Amra, M. Ranier, C. Grézes-Besset, S. Maure, F. Cleva, R. Mollenhauer, G. Albrand, “Loss anomalies in multilayer planar waveguides,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 1005–1020 (1994).

E. Welsch, “Absorption measurements,” in Thin Films for Optical Coatings, R. E. Hummel, K. H. Guenther, eds. (CRC, Boca Raton, Fla., 1995), Chap. 9, pp. 243–272 and references therein.

M. Commandré, P. Roche, “Characterisation of absorption by photothermal deflection,” in Thin Films for Optical Systems, F. Flory, ed. (Dekker, New York, 1995), Chap. 12, pp. 329–365.

M. Commandré, P. Roche, G. Albrand, E. Pelletier, “Photothermal deflection spectroscopy for the study of thin films and optical coatings: measurement of absorption losses and detection of photo-induced changes,” in Optical Thin Films and Applications, R. Herrmann ed., Proc. SPIE1270, 82–93 (1990).

S. E. Watkins, R. Heimlich, R. Reis, “Mapping of absorption in optical coatings,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 246–255 (1991).

Z. L. Wu, M. Reichling, E. Welsch, D. Schäfer, Z. X. Fan, E. Matthias, “Defect characterisation for thin films through thermal wave detection,” in Laser-Induced Damage in Optical Materials: 1991, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 271–281 (1991); M. Reichling, E. Welsch, A. Duparré, E. Matthias, “Photothermal absorption microscopy of defects in ZrO2 and MgF2 single-layer films,” Opt. Eng. 33, 1334–1342 (1994).
[CrossRef]

Z. L. Wu, M. Reichling, H. Grönbeck, Z. X. Fan, D. Schaefer, E. Matthias, “Photothermal measurement of thermal conductivity of optical coatings,” in Laser-Induced Damage in Optical Materials, H. E. Bennett, L. L. Chase, A. H. Guenther, B. E. Newnam, M. J. Soileau, eds., Proc. SPIE1624, 331–345 (1991), and references therein; M. Reichling, H. Grönbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914–1922 (1994).
[CrossRef]

M. Commandré, “Caractérisation de l’absorption dans les composants optiques en couches minces par déflexion photothermique,” Thése de Doctorat d’Etat (Université d’Aix-Marseille, Marseille, 1992).

M. Commandré, P. Roche, J. P. Borgogno, G. Albrand, “Surface contamination of bare substrates. Mapping of absorption and influence on deposited thin films,” in Optical Interference Coatings, F. Abelés, ed., Proc. SPIE2253, 982–992 (1994).

V. Scheuer, C. Schuchert, T. Tschudi, “The influence of small amounts of impurities in sputtered laser mirrors on their performance,” in Thin Films in Optics, T. T. Tschudi, ed., Proc. SPIE1125, 54–60 (1990).

P. A. Temple, “Examination of laser damage sites of transparent surfaces and films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1979, Nat. Bur. Stand. Spec. Publ.568, 333–341 (1979).

F. L. Williams, C. K. Carniglia, B. J. Pond, W. K. Stowell, “Investigation of thin films using total internal reflection microscopy,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 299–308 (1989).

R. C. Estier, N. S. Nogar, R. A. Schmell, “The detection, removal and effect on damage thresholds of cerium impurities on fused silica,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 183–188 (1988).

T. Raj, D. E. McCready, C. K. Carniglia, “Substrate cleaning in vacuum by laser irradiation,” in Laser Induced Damage in Optical Materials: 1988, Nat. Inst. Stand. Technol. Spec. Publ.775, 152–165 (1988).

R. S. Hockett, “Quantitative analysis of surface trace metal contamination on substrates and films by TXRF,” in Laser Induced Damage in Optical Materials: 1989, Nat. Inst. Stand. Technol. Spec. Publ.801, 239–253 (1989).

Very smooth surface finish (French Standard NF S 10-006). The rms roughness is ~0.3 nm.

H. K. Pulker, “Nature of a surface,” in Coating on Glass (Elsevier, New York, 1984), Chap 3, pp. 34–42.

T. S. Izumitani, Optical Glass, American Institute of Physics Translation Series (American Institute of Physics, New York, 1986), Chap. 2, pp. 15–55.

K. Kinosita, “Surface deterioration of optical glasses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 4, pp. 85–143.
[CrossRef]

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 (13)

Fig. 1
Fig. 1

Different experimental configurations for PD measurement: (a) classical mirage effect configuration in which the probe beam is parallel to the sample surface; (b) transmission configuration in which the deflection of the transmitted probe beam is measured; (c) reflection configuration in which the deflection of the reflected beam is measured.

Fig. 2
Fig. 2

Definition of notation used in the text: n i and k i are the real and the imaginary parts of the complex index of medium i, respectively (i = 0 to 2), K i is the thermal conductivity of medium i, ρ i is the mass density of medium i, C i is the heat capacity per unit mass of medium i, ν is the modulation frequency, ω = 2πν is the modulation pulsation of the pump beam, μ i is the thermal diffusion length of medium i defined by μ i = (K i i C i πν)1/2, R FA and R FS are the thermal resistances at the air–film and film–substrate interfaces, respectively, e is the film thickness, and 2a is the diameter at 1/e 2 of the Gaussian pump beam.

Fig. 3
Fig. 3

Modulated temperature distribution in the three media: isotherms calculated with classical values of thermal parameters for the film (e = 0.5 μm, k 1 = 10−4, K 1 = 0.1 W/mK, fused-silica substrate, P 0 = 1 W, a = 25 μm): (a) ν = 50 Hz, (b) ν = 5000 Hz.

Fig. 4
Fig. 4

Components of PD that is due to the refractive-index gradient (dashed curve) and to thermoelastic deformation (solid curve), calculated in the reflection configuration for classical values of parameters (e = 0.5 μm, BK7 substrate), versus modulation frequency.

Fig. 5
Fig. 5

PD experimental setup in the transmission configuration.

Fig. 6
Fig. 6

Influence of thermal conductivity of thin film on total PD (which is due to index gradients) in the transmission configuration for modulation frequencies of 6, 500, and 1000 Hz: (a) for a film of thickness e = 0.05 μm (≈λ/4 for a high-index film), (b) for a film of thickness e = 0.5 μm (≈10 λ/4).

Fig. 7
Fig. 7

Influence of film thickness on total PD in the transmission configuration for modulation frequencies of 6, 500, and 1000 Hz (K 1 = 0.1 W/m K).

Fig. 8
Fig. 8

Influence of the temperature coefficient of the refractive index of the film on total PD in the transmission configuration for modulation frequencies of 10, 500, and 1000 Hz (e = 0.5 μm, K 1 = 0.1 W/m K).

Fig. 9
Fig. 9

Simultaneous mappings of (a) absorption, (b) total integrated scattering (TIS) on the same area (500 μm × 700 μm) on a BK7 bare substrate: a defect can scatter but not absorb and vice versa.

Fig. 10
Fig. 10

Absorption mapping of the same area on a BK7 bare substrate (a) before, (b) after cleaning by the use of a conventional procedure through an automatic cleaning apparatus. The spatial distribution of absorption is completely modified, whereas the average surface absorptance decreases from 54 down to 40 ppm.

Fig. 11
Fig. 11

Influence of polishing quality on minimum, mean, and maximum values of absorption. Example of results for two fused-silica substrates polished in different conditions and successively cleaned with increasingly effective cleaning procedures are given: soft manual cleaning (thin solid lines) then ultrasonic automatic cleaning (heavy solid lines). For each sample, measurements are performed on the same area.

Fig. 12
Fig. 12

Influence of polishing quality on minimum, mean, and maximum values of surface absorptance for two BK7 substrates polished under different conditions and successively cleaned with increasingly effective cleaning procedures: soft (thin solid lines) then ultrasonic (heavy solid lines) automatic cleaning. For each sample, measurements are performed on the same area.

Fig. 13
Fig. 13

Histogram of mean values of surface absorptance for 52 fused-silica substrates and 31 BK7 substrates, all T3 polished40 and cleaned under the same conditions (ultrasonic automatic cleaning). Generally the defects of absorptance higher than some 100 ppm can be seen through a Nomarski microscope.

Tables (3)

Tables Icon

Table 1 Comparison among Absorptances Measured by PD and by33 LM for Four SiO2 Films of Increasing Thicknesses

Tables Icon

Table 2 Comparison among Absorption Measured by PD and by33 LM for Four Ta2O5 Films of Increasing Thicknesses

Tables Icon

Table 3 Mean Surface Absorptance Values (ppm) before and after Cleaning for T3 Polished40 Substrates

Equations (13)

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

2 T 0 ( r , z , t ) - ρ 0 C 0 K 0 T 0 t = 0 ,
2 T 1 ( r , z , t ) - ρ 1 C 1 K 1 T 1 t = - Q 1 ( r , z , t ) K 1 ,
2 T 2 ( r , z , t ) - ρ 2 C 2 K 2 T 2 t = 0 ,
T 0 ( z = 0 ) = T 1 ( z = 0 ) - R F A Φ F A ,
T 1 ( z = e ) = T 2 ( z = e ) + R F S Φ F S ,
Φ F S = - K 1 ( T 1 z ) z = e = - K 2 ( T 2 z ) z = e ,
Φ F A = K 0 ( T 0 z ) z = 0 = K 1 ( T 1 z ) z = 0 .
Q 1 ( r , z , t ) = 4 π k 1 λ n 1 n 0 P 0 π a 2 exp ( - 2 r 2 / a 2 ) × exp ( j ω t ) | E 1 ( z ) E 0 | 2 ,
T i ( r , z ) = 2 π 0 σ J 0 ( 2 π σ r ) T ˜ i ( σ , z ) d σ ,             ( i = 0 to 2 ) ,
θ Y , i = 1 n i n i T path i d Z [ T i Y ( X , Y , Z , t ) ] X = X 0 , Y = Y 0 ,
( 1 - 2 ν i ) 2 u + ( · u ) = 2 ( 1 + ν i ) α i T             ( i = 1 , 2 ) ,
θ Y 2 u z Y ( X 0 , Y 0 , Z = 0 ) + 2 u z Y ( X 0 , Y 0 , Z = e )
θ Y ( n 1 - n 0 ) u z Y ( X 0 , Y 0 , Z = 0 ) + ( n 2 - n 1 ) u z Y ( X 0 , Y 0 , Z = e )

Metrics