Abstract

The spectrally resolved low-level absorption of thin films and of solid and liquid surfaces was measured by cavity-enhanced photothermal spectroscopy. The technique is ultrasensitive and can determine surface-specific absorbances α(ω) ~ 10−6 at a power density of 104W/cm2. Both cases of continuous wave and modulated laser light absorption were studied experimentally and are interpreted theoretically. It is shown that it is possible to achieve a spatial resolution of absorbance variations in the few-micrometer range. The thermal diffusivity can also be simultaneously measured by observing the time evolution of the surface temperature during laser irradiation.

© 1992 Optical Society of America

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  1. D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ trace-gas detection by photothermal deflection spectroscopy,” Appl. Phys. Lett. 37, 519–521 (1980).
    [Crossref]
  2. J. A. Sell, “Quantitative photothermal deflection spectroscopy in a flowing stream of gas,” Appl. Opt. 23, 1586–1597 (1984).
    [Crossref] [PubMed]
  3. A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the photothermal deflection technique to combustion diagnostics,” Appl. Opt. 21, 2663–2665 (1982).
    [Crossref] [PubMed]
  4. S. W. Kizirnis, R. J. Brecha, B. N. Ganguly, L. P. Goss, R. Gupta “Hydroxyl (OH) distributions and temperature profiles in a premixed propane flame obtained by laser deflection techniques,” Appl. Opt. 23, 3873–3881 (1984).
    [Crossref] [PubMed]
  5. J. A. Sell, “Photoacoustic and photothermal deflection spectroscopy of propane at CO2 laser wavelengths,” Appl. Opt. 24, 152–153 (1985).
    [Crossref] [PubMed]
  6. A. Rose, Reeta Vyas, R. Gupta, “Pulsed photothermal deflection spectroscopy in a flowing medium: a quantitative investigation,” Appl. Opt. 25, 4626 (1986).
    [Crossref] [PubMed]
  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. A. C. Boccara, D. Fournier, J. Badoz, “Thermo-optical spectroscopy: detection by the mirage effect,” Appl. Phys. Lett. 36, 130–132 (1980).
    [Crossref]
  9. A. Rose, R. Gupta, “Application of photothermal and photoacoustic deflection techniques to sooting flames; velocity, temperature, and concentration measurements,” Opt. Commun. 56, 303–307 (1986).
    [Crossref]
  10. R. W. Pitz, “Low level smoke emission measurements from a flame by photothermal deflection spectroscopy,” Appl. Opt. 29, 2418–2423 (1990).
    [Crossref] [PubMed]
  11. A. J. Campillo, S. J. Petuchowski, C. C. Davis, H. B. Lin, “Fabry–Perot photothermal trace detection,” Appl. Phys. Lett. 41, 327 (1982).
    [Crossref]
  12. J. Stone, “Thermooptical technique for the measurement of absorption loss spectrum in liquids,” Appl. Opt. 12, 1828–1830 (1973).
    [Crossref] [PubMed]
  13. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981).
    [Crossref] [PubMed]
  14. J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580 (1980).
    [Crossref]
  15. A. Mandelis, “Absolute optical absorption coefficient measurements using transverse photothermal deflection spectroscopy,” J. Appl. Phys. 54, 3404 (1983).
    [Crossref]
  16. Q. Yu, S. H. Chen, Z. Rong, Y. Xu, H. A. Schuessler, “Weak absorption measurements of films by means of an injection confocal spherical Fabry–Perot cavity,” Chin. J. Lasers 16, 87–97 (1989).
  17. S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Thermal modeling in cavity-enhanced photothermal spectroscopy,” in Laser Photoionization and Desorption Surface Analysis Techniques, N. S. Nogar, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1208, 149–151 (1990).
  18. S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Spatial resolution of cavity-enhanced photothermal spectroscopy,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 187–190 (1990).
  19. K. H. Fung, H. B. Lin, “Trace gas detection by laser intracavity photothermal spectroscopy,” Appl. Opt. 25, 749 (1986).
    [Crossref] [PubMed]
  20. D. Z. Anderson, J. C. Frisch, C. S. Masser, “Mirror reflectometer based on optical cavity decay time,” Appl. Opt. 23, 1238–1245 (1984).
    [Crossref] [PubMed]
  21. P. Connes, “Augmentation du produit luminosité x Résolution des interféromètres par l’ emploi d’une différence de marche indépendante de l’incidence,” Rev. Opt. 35, 37–43 (1956); M. Hercher, “The spherical mirror Fabry–Perot interferometer,” Appl. Opt. 7, 951–966 (1968).
    [Crossref] [PubMed]
  22. Y. Xu, Z. Rong, Q. Yu, S. H. Chen. “Measurement of intracavity weak loss for an injection Fabry–Perot spherical cavity,” Chin. Phys. 9, 1114–1120 (1989).

1990 (1)

1989 (2)

Q. Yu, S. H. Chen, Z. Rong, Y. Xu, H. A. Schuessler, “Weak absorption measurements of films by means of an injection confocal spherical Fabry–Perot cavity,” Chin. J. Lasers 16, 87–97 (1989).

Y. Xu, Z. Rong, Q. Yu, S. H. Chen. “Measurement of intracavity weak loss for an injection Fabry–Perot spherical cavity,” Chin. Phys. 9, 1114–1120 (1989).

1986 (3)

1985 (1)

1984 (3)

1983 (1)

A. Mandelis, “Absolute optical absorption coefficient measurements using transverse photothermal deflection spectroscopy,” J. Appl. Phys. 54, 3404 (1983).
[Crossref]

1982 (2)

A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the photothermal deflection technique to combustion diagnostics,” Appl. Opt. 21, 2663–2665 (1982).
[Crossref] [PubMed]

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H. B. Lin, “Fabry–Perot photothermal trace detection,” Appl. Phys. Lett. 41, 327 (1982).
[Crossref]

1981 (1)

1980 (4)

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]

J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580 (1980).
[Crossref]

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ trace-gas detection by photothermal deflection spectroscopy,” Appl. Phys. Lett. 37, 519–521 (1980).
[Crossref]

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

1973 (1)

1956 (1)

P. Connes, “Augmentation du produit luminosité x Résolution des interféromètres par l’ emploi d’une différence de marche indépendante de l’incidence,” Rev. Opt. 35, 37–43 (1956); M. Hercher, “The spherical mirror Fabry–Perot interferometer,” Appl. Opt. 7, 951–966 (1968).
[Crossref] [PubMed]

Aamodt, L. C.

J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580 (1980).
[Crossref]

Amer, N. M.

Anderson, D. Z.

Badoz, J.

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

Boccara, A. C.

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

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

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]

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ trace-gas detection by photothermal deflection spectroscopy,” Appl. Phys. Lett. 37, 519–521 (1980).
[Crossref]

Brecha, R. J.

Campillo, A. J.

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H. B. Lin, “Fabry–Perot photothermal trace detection,” Appl. Phys. Lett. 41, 327 (1982).
[Crossref]

Chen, S. H.

Q. Yu, S. H. Chen, Z. Rong, Y. Xu, H. A. Schuessler, “Weak absorption measurements of films by means of an injection confocal spherical Fabry–Perot cavity,” Chin. J. Lasers 16, 87–97 (1989).

Y. Xu, Z. Rong, Q. Yu, S. H. Chen. “Measurement of intracavity weak loss for an injection Fabry–Perot spherical cavity,” Chin. Phys. 9, 1114–1120 (1989).

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Thermal modeling in cavity-enhanced photothermal spectroscopy,” in Laser Photoionization and Desorption Surface Analysis Techniques, N. S. Nogar, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1208, 149–151 (1990).

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Spatial resolution of cavity-enhanced photothermal spectroscopy,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 187–190 (1990).

Connes, P.

P. Connes, “Augmentation du produit luminosité x Résolution des interféromètres par l’ emploi d’une différence de marche indépendante de l’incidence,” Rev. Opt. 35, 37–43 (1956); M. Hercher, “The spherical mirror Fabry–Perot interferometer,” Appl. Opt. 7, 951–966 (1968).
[Crossref] [PubMed]

Davis, C. C.

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H. B. Lin, “Fabry–Perot photothermal trace detection,” Appl. Phys. Lett. 41, 327 (1982).
[Crossref]

Fournier, D.

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, J. Badoz, “Thermo-optical spectroscopy: detection by the mirage effect,” Appl. Phys. Lett. 36, 130–132 (1980).
[Crossref]

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]

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ trace-gas detection by photothermal deflection spectroscopy,” Appl. Phys. Lett. 37, 519–521 (1980).
[Crossref]

Frisch, J. C.

Fung, K. H.

Ganguly, B. N.

Gerlach, R.

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ trace-gas detection by photothermal deflection spectroscopy,” Appl. Phys. Lett. 37, 519–521 (1980).
[Crossref]

Goss, L. P.

Gupta, R.

Jackson, W.

Jackson, W. B.

Kizirnis, S. W.

Lin, H. B.

K. H. Fung, H. B. Lin, “Trace gas detection by laser intracavity photothermal spectroscopy,” Appl. Opt. 25, 749 (1986).
[Crossref] [PubMed]

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H. B. Lin, “Fabry–Perot photothermal trace detection,” Appl. Phys. Lett. 41, 327 (1982).
[Crossref]

Mandelis, A.

A. Mandelis, “Absolute optical absorption coefficient measurements using transverse photothermal deflection spectroscopy,” J. Appl. Phys. 54, 3404 (1983).
[Crossref]

Masser, C. S.

Murphy, J. C.

J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580 (1980).
[Crossref]

Muzny, C.

Petuchowski, S. J.

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H. B. Lin, “Fabry–Perot photothermal trace detection,” Appl. Phys. Lett. 41, 327 (1982).
[Crossref]

Pitz, R. W.

Pyrum, J. D.

Rong, Z.

Y. Xu, Z. Rong, Q. Yu, S. H. Chen. “Measurement of intracavity weak loss for an injection Fabry–Perot spherical cavity,” Chin. Phys. 9, 1114–1120 (1989).

Q. Yu, S. H. Chen, Z. Rong, Y. Xu, H. A. Schuessler, “Weak absorption measurements of films by means of an injection confocal spherical Fabry–Perot cavity,” Chin. J. Lasers 16, 87–97 (1989).

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Spatial resolution of cavity-enhanced photothermal spectroscopy,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 187–190 (1990).

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Thermal modeling in cavity-enhanced photothermal spectroscopy,” in Laser Photoionization and Desorption Surface Analysis Techniques, N. S. Nogar, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1208, 149–151 (1990).

Rose, A.

Salamo, G. J.

Schuessler, H. A.

Q. Yu, S. H. Chen, Z. Rong, Y. Xu, H. A. Schuessler, “Weak absorption measurements of films by means of an injection confocal spherical Fabry–Perot cavity,” Chin. J. Lasers 16, 87–97 (1989).

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Thermal modeling in cavity-enhanced photothermal spectroscopy,” in Laser Photoionization and Desorption Surface Analysis Techniques, N. S. Nogar, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1208, 149–151 (1990).

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Spatial resolution of cavity-enhanced photothermal spectroscopy,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 187–190 (1990).

Sell, J. A.

Stone, J.

Tang, Z. C.

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Spatial resolution of cavity-enhanced photothermal spectroscopy,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 187–190 (1990).

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Thermal modeling in cavity-enhanced photothermal spectroscopy,” in Laser Photoionization and Desorption Surface Analysis Techniques, N. S. Nogar, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1208, 149–151 (1990).

Vyas, Reeta

Xu, Y.

Q. Yu, S. H. Chen, Z. Rong, Y. Xu, H. A. Schuessler, “Weak absorption measurements of films by means of an injection confocal spherical Fabry–Perot cavity,” Chin. J. Lasers 16, 87–97 (1989).

Y. Xu, Z. Rong, Q. Yu, S. H. Chen. “Measurement of intracavity weak loss for an injection Fabry–Perot spherical cavity,” Chin. Phys. 9, 1114–1120 (1989).

Yu, Q.

Y. Xu, Z. Rong, Q. Yu, S. H. Chen. “Measurement of intracavity weak loss for an injection Fabry–Perot spherical cavity,” Chin. Phys. 9, 1114–1120 (1989).

Q. Yu, S. H. Chen, Z. Rong, Y. Xu, H. A. Schuessler, “Weak absorption measurements of films by means of an injection confocal spherical Fabry–Perot cavity,” Chin. J. Lasers 16, 87–97 (1989).

Appl. Opt. (10)

J. A. Sell, “Quantitative photothermal deflection spectroscopy in a flowing stream of gas,” Appl. Opt. 23, 1586–1597 (1984).
[Crossref] [PubMed]

A. Rose, J. D. Pyrum, C. Muzny, G. J. Salamo, R. Gupta, “Application of the photothermal deflection technique to combustion diagnostics,” Appl. Opt. 21, 2663–2665 (1982).
[Crossref] [PubMed]

S. W. Kizirnis, R. J. Brecha, B. N. Ganguly, L. P. Goss, R. Gupta “Hydroxyl (OH) distributions and temperature profiles in a premixed propane flame obtained by laser deflection techniques,” Appl. Opt. 23, 3873–3881 (1984).
[Crossref] [PubMed]

J. A. Sell, “Photoacoustic and photothermal deflection spectroscopy of propane at CO2 laser wavelengths,” Appl. Opt. 24, 152–153 (1985).
[Crossref] [PubMed]

A. Rose, Reeta Vyas, R. Gupta, “Pulsed photothermal deflection spectroscopy in a flowing medium: a quantitative investigation,” Appl. Opt. 25, 4626 (1986).
[Crossref] [PubMed]

R. W. Pitz, “Low level smoke emission measurements from a flame by photothermal deflection spectroscopy,” Appl. Opt. 29, 2418–2423 (1990).
[Crossref] [PubMed]

J. Stone, “Thermooptical technique for the measurement of absorption loss spectrum in liquids,” Appl. Opt. 12, 1828–1830 (1973).
[Crossref] [PubMed]

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

K. H. Fung, H. B. Lin, “Trace gas detection by laser intracavity photothermal spectroscopy,” Appl. Opt. 25, 749 (1986).
[Crossref] [PubMed]

D. Z. Anderson, J. C. Frisch, C. S. Masser, “Mirror reflectometer based on optical cavity decay time,” Appl. Opt. 23, 1238–1245 (1984).
[Crossref] [PubMed]

Appl. Phys. Lett. (3)

A. J. Campillo, S. J. Petuchowski, C. C. Davis, H. B. Lin, “Fabry–Perot photothermal trace detection,” Appl. Phys. Lett. 41, 327 (1982).
[Crossref]

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

D. Fournier, A. C. Boccara, N. M. Amer, R. Gerlach, “Sensitive in situ trace-gas detection by photothermal deflection spectroscopy,” Appl. Phys. Lett. 37, 519–521 (1980).
[Crossref]

Chin. J. Lasers (1)

Q. Yu, S. H. Chen, Z. Rong, Y. Xu, H. A. Schuessler, “Weak absorption measurements of films by means of an injection confocal spherical Fabry–Perot cavity,” Chin. J. Lasers 16, 87–97 (1989).

Chin. Phys. (1)

Y. Xu, Z. Rong, Q. Yu, S. H. Chen. “Measurement of intracavity weak loss for an injection Fabry–Perot spherical cavity,” Chin. Phys. 9, 1114–1120 (1989).

J. Appl. Phys. (2)

J. C. Murphy, L. C. Aamodt, “Photothermal spectroscopy using optical beam probing: mirage effect,” J. Appl. Phys. 51, 4580 (1980).
[Crossref]

A. Mandelis, “Absolute optical absorption coefficient measurements using transverse photothermal deflection spectroscopy,” J. Appl. Phys. 54, 3404 (1983).
[Crossref]

Opt. Commun. (1)

A. Rose, R. Gupta, “Application of photothermal and photoacoustic deflection techniques to sooting flames; velocity, temperature, and concentration measurements,” Opt. Commun. 56, 303–307 (1986).
[Crossref]

Opt. Lett. (1)

Rev. Opt. (1)

P. Connes, “Augmentation du produit luminosité x Résolution des interféromètres par l’ emploi d’une différence de marche indépendante de l’incidence,” Rev. Opt. 35, 37–43 (1956); M. Hercher, “The spherical mirror Fabry–Perot interferometer,” Appl. Opt. 7, 951–966 (1968).
[Crossref] [PubMed]

Other (2)

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Thermal modeling in cavity-enhanced photothermal spectroscopy,” in Laser Photoionization and Desorption Surface Analysis Techniques, N. S. Nogar, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1208, 149–151 (1990).

S. H. Chen, Z. C. Tang, Z. Rong, H. A. Schuessler, “Spatial resolution of cavity-enhanced photothermal spectroscopy,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 187–190 (1990).

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Figures (15)

Fig. 1
Fig. 1

Schematic of the optical measurement system and the detection electronics: BS, beam splitter; M’s, mirrors; PZT, piezoelectric transducer.

Fig. 2
Fig. 2

Arrangement of the mirrors M1 and M2 to form a confocal spherical étalon operated in the traveling wave mode.

Fig. 3
Fig. 3

Relative fringe intensity at location A as a function of the single-pass loss in the intracavity medium with the mirror reflectivity R as the parameter.

Fig. 4
Fig. 4

Relative fringe intensity at location A as a function of the mirror reflectivity, R, with the single-pass loss in the intracavity medium β as the parameter.

Fig. 5
Fig. 5

Optimum values of R for maximum contrast as a function of the single-pass intracavity losses.

Fig. 6
Fig. 6

Achievable sensitivity limit η for small optical path changes versus the noise Δf|IA| of the transmitted output beam. The mirror reflectivity is the parameter.

Fig. 7
Fig. 7

Relative size and sign of the experimentally observed photothermal signal as a function of the locking point.

Fig. 8
Fig. 8

(a) Time-resolved measurement of the photothermal signal for various probe beam heights above the surface with a dielectric coating with α ≈ 10−3 as the sample. (b) Equilibrium values of the photothermal signal versus the probe beam height.

Fig. 9
Fig. 9

Calculated time evolution of the temperature in the medium adjacent to the absorbing surface for both a thin dielectric (α = 10−3) and an aluminum coating (α = 0.1). A pump laser beam of 1 W focused to a spot size of 20 μm and a probe beam height of 1 mm were assumed in the calculation.

Fig. 10
Fig. 10

Lateral extent of the photothermal signal for an aluminum film and a dielectric coating. The signal was observed by moving the pump beam across the sample and waiting at each position until thermal equilibrium was reached.

Fig. 11
Fig. 11

Photothermal signal versus laser power for the dielectric coating. The absorbance was α = 10−3, and the mirror reflectivity was R = 0.98.

Fig. 12
Fig. 12

Absorption versus laser wavelength for (a) the aluminum coating and (b) the dielectric coating. The photothermal signal was calibrated against known standards to obtain this result.

Fig. 13
Fig. 13

Photothermal signal for modulated pump laser light as a function of the modulation frequency. The dashed and the solid curves are the theoretical predictions.

Fig. 14
Fig. 14

Visibility of the photothermal signal versus the normalized pump beam spot size ωo/(ωa + ωb). The parameter in the insert describes different Ronchi gratings where the ratio of the ruling distances ωa/(ωa + ωb) varied as indicated.

Fig. 15
Fig. 15

Compilation of the experimental data for determining the resolution: (a) absorption profile of the Ronchi grating, (b) theoretical prediction of the periodic photothermal signal for different widths of the pump beam and a negligibly narrow probe beam width, (c) observation of the periodic photothermal signal for a pump beam with a waist spot of 40 μm.

Equations (37)

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I A max = I 0 [ ( t 1 t 2 η ) / ( 1 - r 1 2 r 2 2 η 4 ) ] 2 , I B max = I 0 [ ( t 1 t 2 r 1 r 2 η 3 ) / ( 1 - r 1 2 r 2 2 η 4 ) ] 2 , I C max = I 0 [ ( t 1 2 r 2 η 2 ) / ( 1 - r 1 2 r 2 2 η 4 ) ] 2 , I D max = I 0 { [ ( t 1 2 r 1 r 2 2 η 4 ) / ( 1 - r 1 2 r 2 2 η 4 ) ] - r 1 } 2 ,
I A max / I 0 = { ( 1 - R ) 2 ( 1 - β ) / [ 1 - ( 1 - β ) 2 R 2 ] 2 } .
d ( d I A max / d β ) / d R = 0 ,
( 1 - β ) 4 ( 6 - 3 R ) R 3 + ( 1 - β 2 ) ( 10 - 12 R ) R - 1 = 0.
I A = I 0 ( 1 - R ) 2 / [ ( 1 - R 2 ) 2 + 4 R 2 sin 2 δ / 2 ] .
d I A / d δ = - 2 I 0 R 2 ( 1 - R ) 2 sin δ / [ ( 1 - R 2 ) 2 + 4 R 2 sin 2 δ / 2 ] 2 .
η = Δ f x = λ Δ f δ / 2 π = λ Δ f I A / 2 π ( d I A / d δ ) z 0 .
z 0 = [ ( 2 A + 3 B ) - { [ ( 2 A + 3 B ) 2 - 8 A B ] / 4 B } 1 / 2 ] ,
η = ( λ / 2 π ) ( Δ f I A / I z 0 ) [ ( 1 - R 2 ) 2 + 4 R 2 z 0 ] / { 4 R 2 [ z 0 ( 1 - z 0 ) ] } 1 / 2 ,
t U 1 - a 1 2 2 U 1 = 0 ,
t U 2 - a 2 2 2 U 2 = f ( r , t ) / ρ 2 C 2 ,
t U 3 - a 3 2 2 U 3 = 0 ,
U 1 z = 0 = U 2 z = 0 = U 3 z = 0 ,
κ 1 U 1 z | z = 0 = κ 2 U 2 z | z = 0 = κ 3 U 3 z | z = 0
U 1 t = 0 = U 2 t = 0 = U 3 t = 0 = 0.
U 2 = 0 [ f ^ ( λ ) / ( λ 2 a 2 2 ) ] [ 1 - exp ( - a 2 2 λ 2 t ) ] J 0 ( λ r ) d λ ,
f ^ ( λ ) = 0 [ f ( r ) / C 2 ρ 2 ] r J 0 ( λ r ) d r ,
t U 1 - a 1 2 2 U 1 = 0 ,
U 1 z = 0 = U 2 z = 0 and U 1 t = 0 = 0 ,
U 1 = U 2 - 0 t 0 erf ( z / 2 a 1 t - τ ) exp [ - a 1 2 λ 2 ( t - τ ) ] × [ ( 1 - a 1 2 / a 2 2 ) exp ( - λ 2 a 2 2 τ ) + a 1 2 / a 2 2 ] f ( λ ) J 0 ( λ r ) d λ d t .
t U 1 - a 1 2 2 U 1 = f ( r , t ) / ρ 2 C 2 , U 1 t = 0 = 0 ,
U 1 = 0 - + [ f ^ ( λ ) / a 1 2 ( μ 2 + λ 2 ) ] × { 1 - exp [ - a 1 2 ( μ 2 + λ 2 ) t ] } exp ( i μ z ) J 0 ( λ r ) d μ d λ .
U 2 = 0 [ f ( λ ) / ( i ω + λ 2 a 2 2 ) ] exp ( i ω t ) J 0 ( λ r ) d λ .
U 1 = 0 [ f ^ ( λ ) / i ω + λ 2 a 2 2 ] exp ( i ω t ) J 0 ( λ r ) exp [ ( i ω / a 1 2 + x 2 ) 1 / 2 z ] d λ .
t U 1 - a 1 2 2 U 1 = f ( r ) exp ( i ω t ) δ ( z ) / ρ 2 C 2
U 1 = 0 { f ^ ( λ ) / a 1 2 [ ( i ω / a 1 2 ) + λ 2 ] 1 / 2 } × exp ( i ω t ) J 0 ( λ r ) exp ( { - i [ ( i ω a 1 2 ) + λ 2 ] 1 / 2 z } ) d λ .
Δ l = ( n / T ) - + U 1 ( t , z , x , y 0 ) d x ,
ω 0 pumb = f λ / π ω ,
ω 0 probe = R 0 λ / 2 π .
I ( x ) = ( 2 I 0 / ω 0 π ) exp ( - 2 x 2 / ω 0 2 ) ,
α ( x ) = n = - + × { a rect [ x + n ( ω a + ω b ) ω a ] + b rect [ x + n ( n + 1 2 ) ( ω a + ω b ) ω b ] } ,
rect ( x - u v ) = { 1 if u - v / 2 x u + v / 2 0 otherwise ,
P ( ξ ) = - + a ( x ) I ( ξ - x ) d x .
P 0 = - + I ( x ) d x .
P ( ξ ) = - + 2 r 0 π exp [ - 2 ( x - ξ ) 2 r 0 2 ] α ( x ) d x .
P max ( ξ = 0 ) = - + α ( x ) I ( x ) d x , P min ( ξ = 1 2 ) = - + α ( x ) I ( x - 1 2 ) d x .
V ( ω 0 ) = ( P max - P min ) / ( P max + P min ) ,

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