Abstract

The effect of numerical aperture on the fringe spacing in interferometry is analyzed by the use of wave optics. The results are compared with published experimental results, and the influence of apodization of the wave front is discussed. The effects of central obscuration and surface tilt are also considered.

© 1995 Optical Society of America

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References

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  1. E. Inglestam, L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
    [CrossRef]
  2. G. Schulz, K. -E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).
    [CrossRef] [PubMed]
  3. J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507–507 (1956).
    [CrossRef]
  4. C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1957).
    [CrossRef]
  5. G. Schulz, “A theory of double-beam interference,” in Contributions to Interference Microscopy, W. Krug, J. Rienitz, G. Schulz, eds. (Hilger and Watts, London, 1964), pp. 308–325.
  6. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  7. F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 236–238 (1956).
    [CrossRef]
  8. H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
    [CrossRef]
  9. M. B. Dowell, C. A. Hultman, G. M. Rosenblatt, “Determination of slopes of microscopic surface features by Nomarski polarization interferometry,” Rev. Sci. Instrum. 48, 1491–1497 (1977).
    [CrossRef]
  10. J. F. Biegen, “Calibration requirements for Mirau and Linnik microscope interferometers,” Appl. Opt. 28, 1972–1974 (1989).
    [CrossRef] [PubMed]
  11. K. Creath, “Calibration of numerical aperture effects in interferometric microscope objectives,” Appl. Opt. 28, 3333–3338 (1989).
    [CrossRef] [PubMed]
  12. C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
    [CrossRef]
  13. H. Zhou, M. Gu, C. J. R. Sheppard, “Investigation of aberration measurement in confocal microscopy,” to be published in J. Mod. Opt.
  14. H. Kubota, S. Inoue, “Diffraction images in the polarizing microscope,” J. Opt. Soc. Am. 49, 191–198 (1959).
    [CrossRef] [PubMed]
  15. H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration measurement by confocal interferometry,” J. Mod. Opt. 36, 233–250 (1989).
    [CrossRef]
  16. C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
    [CrossRef]
  17. C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
    [CrossRef]
  18. C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A 11, 593–598 (1994).
    [CrossRef]

1994 (1)

1993 (1)

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

1991 (1)

1990 (1)

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

1989 (3)

1987 (1)

1977 (2)

M. B. Dowell, C. A. Hultman, G. M. Rosenblatt, “Determination of slopes of microscopic surface features by Nomarski polarization interferometry,” Rev. Sci. Instrum. 48, 1491–1497 (1977).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

1963 (1)

H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
[CrossRef]

1959 (1)

1958 (1)

E. Inglestam, L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
[CrossRef]

1957 (1)

C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1957).
[CrossRef]

1956 (2)

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507–507 (1956).
[CrossRef]

F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 236–238 (1956).
[CrossRef]

Biegen, J. F.

Bruce, C. F.

C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1957).
[CrossRef]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

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, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

Creath, K.

Dowell, M. B.

M. B. Dowell, C. A. Hultman, G. M. Rosenblatt, “Determination of slopes of microscopic surface features by Nomarski polarization interferometry,” Rev. Sci. Instrum. 48, 1491–1497 (1977).
[CrossRef]

Elssner, K. -E.

Gates, J. W.

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507–507 (1956).
[CrossRef]

Gu, M.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

H. Zhou, M. Gu, C. J. R. Sheppard, “Investigation of aberration measurement in confocal microscopy,” to be published in J. Mod. Opt.

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]

Hultman, C. A.

M. B. Dowell, C. A. Hultman, G. M. Rosenblatt, “Determination of slopes of microscopic surface features by Nomarski polarization interferometry,” Rev. Sci. Instrum. 48, 1491–1497 (1977).
[CrossRef]

Inglestam, E.

E. Inglestam, L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
[CrossRef]

Inoue, S.

Johansson, L. P.

E. Inglestam, L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
[CrossRef]

Kawata, S.

Kawata, Y.

Kubota, H.

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

Mykura, H.

H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
[CrossRef]

Rhead, G. E.

H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
[CrossRef]

Rosenblatt, G. M.

M. B. Dowell, C. A. Hultman, G. M. Rosenblatt, “Determination of slopes of microscopic surface features by Nomarski polarization interferometry,” Rev. Sci. Instrum. 48, 1491–1497 (1977).
[CrossRef]

Schulz, G.

G. Schulz, K. -E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).
[CrossRef] [PubMed]

G. Schulz, “A theory of double-beam interference,” in Contributions to Interference Microscopy, W. Krug, J. Rienitz, G. Schulz, eds. (Hilger and Watts, London, 1964), pp. 308–325.

Sheppard, C. J. R.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high aperture systems,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

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

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

H. Zhou, M. Gu, C. J. R. Sheppard, “Investigation of aberration measurement in confocal microscopy,” to be published in J. Mod. Opt.

Thornton, B. S.

C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1957).
[CrossRef]

Tolmon, F. R.

F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 236–238 (1956).
[CrossRef]

Wood, J. G.

F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 236–238 (1956).
[CrossRef]

Zhou, H.

H. Zhou, M. Gu, C. J. R. Sheppard, “Investigation of aberration measurement in confocal microscopy,” to be published in J. Mod. Opt.

Appl. Opt. (3)

J. Microsc. (1)

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

J. Mod. Opt. (2)

C. J. R. Sheppard, T. J. Connolly, M. Gu, “Imaging and reconstruction for rough surface scattering in the Kirchhoff approximation by confocal microscopy,” J. Mod. Opt. 40, 2407–2421 (1993).
[CrossRef]

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. (1)

J. Opt. Soc. Am. A (2)

J. Sci. Instrum. (5)

F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 236–238 (1956).
[CrossRef]

H. Mykura, G. E. Rhead, “Errors in surface topography measurements with high aperture interference microscopies,” J. Sci. Instrum. 40, 313–315 (1963).
[CrossRef]

J. W. Gates, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507–507 (1956).
[CrossRef]

C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1957).
[CrossRef]

E. Inglestam, L. P. Johansson, “Corrections due to aperture in transmission interference microscopes,” J. Sci. Instrum. 35, 15–17 (1958).
[CrossRef]

Opt. Acta (1)

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Rev. Sci. Instrum. (1)

M. B. Dowell, C. A. Hultman, G. M. Rosenblatt, “Determination of slopes of microscopic surface features by Nomarski polarization interferometry,” Rev. Sci. Instrum. 48, 1491–1497 (1977).
[CrossRef]

Other (2)

G. Schulz, “A theory of double-beam interference,” in Contributions to Interference Microscopy, W. Krug, J. Rienitz, G. Schulz, eds. (Hilger and Watts, London, 1964), pp. 308–325.

H. Zhou, M. Gu, C. J. R. Sheppard, “Investigation of aberration measurement in confocal microscopy,” to be published in J. Mod. Opt.

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Figures (7)

Fig. 1
Fig. 1

Variation in the amplitude of the apodization for different values of parameter n. The value n = ½ applies for a perfect system satisfying the sine condition.

Fig. 2
Fig. 2

Variation in the NA factor (the height change per half-fringe) with numerical aperture for different values of parameter n.

Fig. 3
Fig. 3

Variation in NA factor f with parameter n for different numerical apertures.

Fig. 4
Fig. 4

Variation in NA factor f with distance from the focal plane: (a) n = ½, (b) n = 1. For n = 0, factor f is independent of focus position.

Fig. 5
Fig. 5

Effect of a central obscuration on the NA factor for a perfect system satisfying sine condition n = ½.

Fig. 6
Fig. 6

Cross sections through the coherent transfer function for a system satisfying sine condition NA = 0.95; K is a normalized radial spatial frequency.

Fig. 7
Fig. 7

Effect of surface tilt on the NA factor for a system satisfying sine condition NA = 0.95.

Tables (1)

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Table 1 Values for Particular Numerical Apertures

Equations (6)

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U ( z ) = 0 α P 1 ( θ ) P 2 ( θ ) R ( θ ) exp ( - 2 i k z cos θ ) sin θ d θ ,
P ( θ ) = cos n θ ,
U ( z ) = 0 α exp ( - 2 i k z cos θ ) sin θ cos θ d θ .
f = ( 2 n + 2 2 n + 1 ) ( 1 - cos 2 n + 1 α 1 - cos 2 n + 2 α ) .
ɛ = sin α 0 sin α ,
f = ( 2 n + 2 2 n + 1 ) ( cos 2 n + 1 α 0 - cos 2 n + 1 α cos 2 n + 2 α 0 - cos 2 n + 2 α ) .

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