Abstract

The confocal imaging of stratified media (for example, thin-film structures) is investigated. A simple model is introduced for the imaging of a single layer in order to explore the axial resolution attainable. A rigorous model is also described and compared with experimental results from thin surface films. A theoretical treatment of imaging of stratified media with a continuously varying refractive index is presented, and the inverse problem of reconstructing the refractive-index profile from a confocal image is discussed.

© 1994 Optical Society of America

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References

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  1. C. J. R. Sheppard, “Confocal interference microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 389–411.
  2. C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
    [CrossRef]
  3. C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).
  4. C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A11 (to be published).
  5. C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
    [CrossRef] [PubMed]
  6. C. J. R. Sheppard, “General considerations of diffraction theory of 3-D imaging,” Eur. J. Cell Biol. 48, Suppl. 25, 29–32 (1989).
  7. M. Kaveh, M. Soumekh, “Computer-assisted diffraction tomography,” in Image Recovery, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 369–411.
  8. C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1139, 32–39 (1989).
  9. J. T Sheridan, C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 614–632 (1993).
    [CrossRef]
  10. T. Corle, J. Fanton, G. Kino, “Distance measurements by differential confocal optical ranging,” Appl. Opt. 26, 2416–2420 (1987).
    [CrossRef] [PubMed]
  11. C. J. R. Sheppard, M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. 30, 3563–3568 (1991).
    [CrossRef] [PubMed]
  12. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 100–108.
  14. C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
    [CrossRef]
  15. C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).
  16. C. J. R. Sheppard, H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  17. C. J. R. Sheppard, A. R. Carlini, H. J. Matthews, “Three-dimensional imaging of phase steps,” Optik 80, 91–94 (1988).
  18. C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–860 (1981).
    [CrossRef]
  19. C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
    [CrossRef]
  20. H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration measurement by confocal interferometry,” J. Mod. Opt. 36, 233–250 (1989).
    [CrossRef]
  21. H. Kubota, S. Inoué, “Diffraction images in the polarizing microscope,” J. Opt. Soc. Am. 49, 191–198 (1959).
    [CrossRef] [PubMed]
  22. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 5, pp. 247–286.
    [CrossRef]
  23. T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).
  24. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  25. L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by a thin inhomogenous dielectric film,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 17, 41–48 (1968).
  26. L. Sossi, “A method for synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 23, 229–237 (1974).
  27. L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 25, 171–176 (1976).
  28. D. L. Jaggard, Y. Kim, “Accurate one-dimensional inverse scattering using a nonlinear renormalization technique,” J. Opt. Soc. Am. A 2, 1922–1930 (1985).
    [CrossRef]
  29. G. Mazzarella, G. Panariello, “Reconstruction of one-dimensional dielectric profiles,” J. Opt. Soc. Am. A 8, 1622–1628 (1991).
    [CrossRef]

1993

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

J. T Sheridan, C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 614–632 (1993).
[CrossRef]

1992

C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
[CrossRef]

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

1991

C. J. R. Sheppard, M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. 30, 3563–3568 (1991).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).

G. Mazzarella, G. Panariello, “Reconstruction of one-dimensional dielectric profiles,” J. Opt. Soc. Am. A 8, 1622–1628 (1991).
[CrossRef]

1990

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

1989

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration measurement by confocal interferometry,” J. Mod. Opt. 36, 233–250 (1989).
[CrossRef]

C. J. R. Sheppard, “General considerations of diffraction theory of 3-D imaging,” Eur. J. Cell Biol. 48, Suppl. 25, 29–32 (1989).

1988

C. J. R. Sheppard, A. R. Carlini, H. J. Matthews, “Three-dimensional imaging of phase steps,” Optik 80, 91–94 (1988).

1987

1985

1984

C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

1981

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–860 (1981).
[CrossRef]

1978

1976

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 25, 171–176 (1976).

1974

L. Sossi, “A method for synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 23, 229–237 (1974).

1969

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1968

L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by a thin inhomogenous dielectric film,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 17, 41–48 (1968).

1959

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 100–108.

Brakenhoff, G. J.

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

Carlini, A. R.

C. J. R. Sheppard, A. R. Carlini, H. J. Matthews, “Three-dimensional imaging of phase steps,” Optik 80, 91–94 (1988).

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

Connolly, T. J.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Corle, T.

Dobrowolski, J. A.

Fanton, J.

Gu, M.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. 30, 3563–3568 (1991).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A11 (to be published).

Hamilton, D. K.

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration measurement by confocal interferometry,” J. Mod. Opt. 36, 233–250 (1989).
[CrossRef]

Heaton, J. M.

C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

Inoué, S.

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 5, pp. 247–286.
[CrossRef]

Jaggard, D. L.

Kard, P.

L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by a thin inhomogenous dielectric film,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 17, 41–48 (1968).

Kaveh, M.

M. Kaveh, M. Soumekh, “Computer-assisted diffraction tomography,” in Image Recovery, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 369–411.

Kawata, S.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A11 (to be published).

Kawata, Y.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A11 (to be published).

Kim, Y.

Kino, G.

Kubota, H.

Lowe, D.

Mao, X. Q.

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

Matthews, H. J.

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration measurement by confocal interferometry,” J. Mod. Opt. 36, 233–250 (1989).
[CrossRef]

C. J. R. Sheppard, A. R. Carlini, H. J. Matthews, “Three-dimensional imaging of phase steps,” Optik 80, 91–94 (1988).

C. J. R. Sheppard, H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

Mazzarella, G.

Oud, J. L.

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

Panariello, G.

Sheppard, C. J. R.

J. T Sheridan, C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 614–632 (1993).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Aberration compensation in confocal microscopy,” Appl. Opt. 30, 3563–3568 (1991).
[CrossRef] [PubMed]

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration measurement by confocal interferometry,” J. Mod. Opt. 36, 233–250 (1989).
[CrossRef]

C. J. R. Sheppard, “General considerations of diffraction theory of 3-D imaging,” Eur. J. Cell Biol. 48, Suppl. 25, 29–32 (1989).

C. J. R. Sheppard, A. R. Carlini, H. J. Matthews, “Three-dimensional imaging of phase steps,” Optik 80, 91–94 (1988).

C. J. R. Sheppard, H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–860 (1981).
[CrossRef]

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1139, 32–39 (1989).

C. J. R. Sheppard, “Confocal interference microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 389–411.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A11 (to be published).

Sheridan, J. T

Sheridan, J. T.

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1139, 32–39 (1989).

Sossi, L.

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 25, 171–176 (1976).

L. Sossi, “A method for synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 23, 229–237 (1974).

L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by a thin inhomogenous dielectric film,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 17, 41–48 (1968).

Soumekh, M.

M. Kaveh, M. Soumekh, “Computer-assisted diffraction tomography,” in Image Recovery, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 369–411.

Visser, T. D.

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–860 (1981).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 100–108.

Appl. Opt.

Appl. Phys. Lett.

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–860 (1981).
[CrossRef]

Eesti NSV Tead. Akad. Toim. Füüs. Mat.

L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by a thin inhomogenous dielectric film,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 17, 41–48 (1968).

L. Sossi, “A method for synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 23, 229–237 (1974).

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Füüs. Mat. 25, 171–176 (1976).

Eur. J. Cell Biol.

C. J. R. Sheppard, “General considerations of diffraction theory of 3-D imaging,” Eur. J. Cell Biol. 48, Suppl. 25, 29–32 (1989).

J. Microsc.

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 337–390 (1991).

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

J. Mod. Opt.

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration measurement by confocal interferometry,” J. Mod. Opt. 36, 233–250 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

C. J. R. Sheppard, M. Gu, “Axial imaging through an aberrating layer of water in confocal microscopy,” Opt. Commun. 88, 180–190 (1992).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

C. J. R. Sheppard, M. Gu, X. Q. Mao, “Three-dimensional coherent transfer function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81, 281–284 (1991).
[CrossRef]

Optik

C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

C. J. R. Sheppard, A. R. Carlini, H. J. Matthews, “Three-dimensional imaging of phase steps,” Optik 80, 91–94 (1988).

T. D. Visser, J. L. Oud, G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik 90, 17–19 (1992).

Phys. Rev. Lett.

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Scattering by a one-dimensional rough surface, and surface profile reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef] [PubMed]

Other

M. Kaveh, M. Soumekh, “Computer-assisted diffraction tomography,” in Image Recovery, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 369–411.

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1139, 32–39 (1989).

C. J. R. Sheppard, “Confocal interference microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990), pp. 389–411.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A11 (to be published).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 100–108.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. 5, pp. 247–286.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Axial images of two weakly reflecting planes separated by a distance 2u 0: (a) α = 60°, n 0 < n 1 < n s ; (b) α = 60°, n 0 < n 1 > n s ; (c) α = 90°, n 0 < n 1 < n s ; (d) α = 90°, n 0 < n 1 > n s . For α = 60° and λ = 632.8 nm, z ~ u/10 μm, whereas for α = 90°, z ~ u/20 μm.

Fig. 2
Fig. 2

Intensities midway between the planes (solid curves) and at the planes (dashed curves) for α = 60°: (a) n 0 < n 1 < n s , (b) n 0 < n 1 > n s .

Fig. 3
Fig. 3

Intensities midway between the planes (solid curves) and at the planes (dashed curves) for α = 90°: (a) n 0 < n 1 < n s , (b) n 0 < n 1 > n s .

Fig. 4
Fig. 4

Ratio of the intensity midway between the planes to that at the planes for n 0 < n 1 < n s (solid curves) and n 0 < n 1 > n s (dashed curves): (a) α = 60°, (b) α = 90°. The generalized Rayleigh criterion is shown as a solid line (ratio 0.735). Ratios smaller than 0.735 are resolved.

Fig. 5
Fig. 5

Minimum resolved distance d between planes as a function of NA for n 0 < n 1 < n s (solid curve) and n 0 < n 1 > n s (dashed curve). Regions of spurious resolution have not been included.

Fig. 6
Fig. 6

Images of a thin-film structure of SiO2 on Si calculated by the rigorous theory: (a) sin α = 0.80, d = 2.5 μm; (b) sin α = 0.80, d = 5.5 μm; (c) sin α = 0.95, d = 2.5 μm; (d) sin α = 0.95, d = 5.5 μm.

Fig. 7
Fig. 7

Experimental axial images obtained with a 0.80-NA objective of a SiO2–Si film of thickness (a) 1.0 μm, (b) 2.5 μm, (c) 3.0 μm, (d) 5.5 μm. Horizontal scale: (a), (b), (d) one division represents 0.81 μm, (c) one division represents 0.89 μm.

Fig. 8
Fig. 8

Experimental axial images obtained with a 0.95-NA objective of a SiO2–Si film of thickness (a) 1.0 μm, (b) 2.5 μm, (c) 3.0 μm, (d) 5.5 μm. Horizontal scale: (a), (b), (d) one division represents 0.81 μm, (c) one division represents 0.89 μm.

Fig. 9
Fig. 9

Separation of the peaks versus the film thickness according to the rigorous theory and the approximate theory [Eq. (4.1)] for 0.80 NA. The experimental values are also shown. The resolution limit according to the generalized Rayleigh criterion is marked as t min.

Fig. 10
Fig. 10

Separation of the peaks versus the film thickness according to the rigorous theory and the approximate theory [Eq. (4.1)] for 0.95 NA. The experimental values are also shown. The resolution limit according to the generalized Rayleigh criterion is marked as t min.

Fig. 11
Fig. 11

Amplitude reflection coefficient for an interface between media of refractive indices n 0 and n. Curve (a) is the exact variation for normal incidence derived from the Fresnel theory, whereas curve (b) is given by the approximate formula (1/2)ln(n/n 0).

Equations (35)

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

U ( z ) = exp [ i u 2 cot 2 ( α 2 ) ] { sin ( u / 2 ) ( u / 2 ) + 2 i u tan 2 ( α 2 ) × [ sin ( u / 2 ) ( u / 2 ) cos ( u / 2 ) ] } ,
u = 4 k z sin 2 ( α / 2 ) ,
I ( z ) = | 0 α r ( θ ) exp ( 2 i k n 0 z cos θ ) sin θ cos θ d θ | 2 .
r ( θ ) = ( r σ r π ) / 2 ,
r σ = r 01 , σ + r 1 s , σ exp ( 2 i β ) 1 + r 01 , σ r 1 s , σ exp ( 2 i β ) ,
r 01 , σ = n 0 cos θ 0 n 1 cos θ 1 n 0 cos θ 0 + n 1 cos θ 1 ,
r 01 , π = n 1 cos θ 0 n 0 cos θ 1 n 1 cos θ 0 + n 0 cos θ 1 ,
r 1 s , σ = n 1 cos θ 1 n s cos θ s n 1 cos θ 1 + n s cos θ s ,
r 1 s , π = n s cos θ 1 n 1 cos θ s n s cos θ 1 + n 1 cos θ s .
f = 0 α ( n 1 2 n 0 2 sin 2 θ ) 1 / 2 sin θ cos θ d θ / 0 α sin θ d θ .
U ( z ) = + 1 2 P 2 ( θ ) [ r σ ( θ ) r π ( θ ) ] × exp ( 2 i k z n 0 cos θ ) sin θ cos θ d θ,
n = n 0 z < 0 = n ( z ) 0 < z < d = n s z > d .
r = 2 i k γ r β 2 β ( 1 r 2 ) ,
β = ( μ 0 0 ) 1 / 2 1 [ n 2 n 0 2 ( 1 c 2 ) ] 1 / 2 σ pol . = ( 0 μ 0 ) 1 / 2 n 2 [ n 2 n 0 2 ( 1 c 2 ) ] 1 / 2 π pol .,
γ = [ n 2 n 0 2 ( 1 c 2 ) ] 1 / 2 .
β 2 β = 1 4 d d z { ln [ n 0 n 2 n 0 2 ( 1 c 2 ) ] } σ pol . = 1 4 d d z [ ln ( n 4 n 0 2 { [ n 2 n 0 2 ( 1 c 2 ) ] } ) ] π pol .
r = 0 d β 2 β exp ( 2 i k 0 z γ d z 1 ) d z .
1 2 [ ( β 2 β ) σ ( β 2 β ) π ] = 1 2 d d z [ ln ( β σ β π ) 1 / 2 ] = 1 2 d d z [ ln ( n n 0 ) ] ,
r = 0 d 1 2 d d z [ ln ( n n 0 ) ] × exp { 2 i k 0 z [ n 2 n 0 2 ( 1 c 2 ) ] 1 / 2 d z 1 } d z .
n = n 0 + Δ n ,
[ n 2 n 0 2 ( 1 c 2 ) ] 1 / 2 = n 0 c + Δ n / c ,
r = 0 d 1 2 d d z [ ln ( n n 0 ) ] exp [ 2 i k ( n 0 c z + 0 z Δ n c d z 1 ) ] .
r = 0 d 1 2 d d z [ ln ( n n 0 ) ] exp ( 2 i k n 0 c z ) d z .
r = + 1 2 d d z [ ln ( n n 0 ) ] exp ( 2 i k n 0 c z ) d z ,
U ( z ) = + P c 2 ( c ) r c ( c ) exp ( 2 i k n 0 c z ) c d c ,
U ( z ) = g ( z ) { 1 2 d d z [ ln ( n n 0 ) ] } ,
g ( z ) = + P c 2 ( c ) exp ( 2 i k n 0 c z ) c d c .
s = 2 c ,
U ( z ) = 1 2 + P c 2 ( s / 2 ) r c ( s / 2 ) exp ( i k n 0 s z ) s d s ,
U ( z ) = + c ( s ) T ( s ) exp ( i k n 0 s z ) d s ,
c ( s ) = P c 2 ( s / 2 )
T ( s ) = 1 2 r c ( s / 2 ) s
= s 4 + d d z [ ln ( n n 0 ) ] exp ( i k n 0 s z ) d z
= i 2 k n 0 + d 2 d z 2 [ ln ( n n 0 ) ] exp ( i k n 0 s z ) d z
2 cos α < s < 2 ,

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