Abstract

An interferogram produced by wide-aperture interferometers is studied both theoretically and experimentally. The fringe spacing is shown to increase nonlinearly with the numerical aperture and the fringe envelope to become narrower as the numerical aperture is increased. Phase measurements with wide-aperture interferometers therefore require calibration, and the phase can be measured only over a limited range. A calibration is given for accurate phase measurements, and the range over which the phase can be measured is calculated. Experimental measurements are presented and compared with theory.

© 2000 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. F. R. Tolmon, J. G. Wood, “Fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 236–238 (1956).
    [CrossRef]
  6. C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1956).
    [CrossRef]
  7. J. W. Gates, “Correspondence: fringe spacing in interference microscopes,” J. Sci. Instrum. 33, 507 (1956).
    [CrossRef]
  8. T. Wilson, R. Juskaitis, N. P. Rea, D. K. Hamilton, “Fibre optic interference and confocal microscopy,” Opt. Commun. 110, 1–6 (1994).
    [CrossRef]
  9. D. K. Hamilton, J. R. Sheppard, “Interferometric measurements of the complex amplitude of the defocus signal V(z) in the confocal scanning optical microscope,” J. Appl. Phys. 60, 2708–2712 (1986).
    [CrossRef]
  10. C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
    [CrossRef]
  11. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanetic system,” Proc. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  12. F. C. Chang, G. S. Kino, “325-nm interference microscope,” Appl. Opt. 37, 3471–3479 (1998).
    [CrossRef]

1998 (1)

1994 (1)

T. Wilson, R. Juskaitis, N. P. Rea, D. K. Hamilton, “Fibre optic interference and confocal microscopy,” Opt. Commun. 110, 1–6 (1994).
[CrossRef]

1990 (1)

1989 (1)

1987 (1)

1986 (1)

D. K. Hamilton, J. R. Sheppard, “Interferometric measurements of the complex amplitude of the defocus signal V(z) in the confocal scanning optical microscope,” J. Appl. Phys. 60, 2708–2712 (1986).
[CrossRef]

1981 (1)

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanetic system,” Proc. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1956 (3)

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

C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1956).
[CrossRef]

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

Chang, F. C.

Chim, S. C.

Creath, K.

K. Creath, “Phase-measurements interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
[CrossRef]

Gates, J. W.

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

Hamilton, D. K.

T. Wilson, R. Juskaitis, N. P. Rea, D. K. Hamilton, “Fibre optic interference and confocal microscopy,” Opt. Commun. 110, 1–6 (1994).
[CrossRef]

D. K. Hamilton, J. R. Sheppard, “Interferometric measurements of the complex amplitude of the defocus signal V(z) in the confocal scanning optical microscope,” J. Appl. Phys. 60, 2708–2712 (1986).
[CrossRef]

Juskaitis, R.

T. Wilson, R. Juskaitis, N. P. Rea, D. K. Hamilton, “Fibre optic interference and confocal microscopy,” Opt. Commun. 110, 1–6 (1994).
[CrossRef]

Kino, G. S.

Laeri, F.

Rea, N. P.

T. Wilson, R. Juskaitis, N. P. Rea, D. K. Hamilton, “Fibre optic interference and confocal microscopy,” Opt. Commun. 110, 1–6 (1994).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanetic system,” Proc. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

Sheppard, J. R.

D. K. Hamilton, J. R. Sheppard, “Interferometric measurements of the complex amplitude of the defocus signal V(z) in the confocal scanning optical microscope,” J. Appl. Phys. 60, 2708–2712 (1986).
[CrossRef]

Strand, T. C.

Thornton, B. S.

C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1956).
[CrossRef]

Tolmon, F. R.

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

Wilson, T.

T. Wilson, R. Juskaitis, N. P. Rea, D. K. Hamilton, “Fibre optic interference and confocal microscopy,” Opt. Commun. 110, 1–6 (1994).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanetic system,” Proc. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Wood, J. G.

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

Appl. Opt. (4)

Appl. Phys. Lett. (1)

C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
[CrossRef]

J. Appl. Phys. (1)

D. K. Hamilton, J. R. Sheppard, “Interferometric measurements of the complex amplitude of the defocus signal V(z) in the confocal scanning optical microscope,” J. Appl. Phys. 60, 2708–2712 (1986).
[CrossRef]

J. Sci. Instrum. (3)

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

C. F. Bruce, B. S. Thornton, “Obliquity effects in interference microscopes,” J. Sci. Instrum. 34, 203–204 (1956).
[CrossRef]

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

Opt. Commun. (1)

T. Wilson, R. Juskaitis, N. P. Rea, D. K. Hamilton, “Fibre optic interference and confocal microscopy,” Opt. Commun. 110, 1–6 (1994).
[CrossRef]

Proc. Soc. London Ser. A (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems: II. Structure of the image field in an aplanetic system,” Proc. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other (1)

K. Creath, “Phase-measurements interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometries of interference microscopes where object and reference surfaces are illuminated with converging beams: Michelson, Mirau, and Linnik. The drawback in the Michelson and the Mirau configurations is that the beam splitter can introduce strong aberrations in wide-aperture systems, especially the spherical aberration. Because reference and object beams share a common path over most of their length, these layouts are less sensitive to vibrations than is the Linnik configuration. In addition, only one lens is required, although its working distance must be long enough to accommodate the beam splitter. The Linnik configuration requires two identical objectives, and, because of the long beam paths involved, it must be built massively to avoid vibration problems. Nevertheless, the great advantage of this configuration is that objectives with high numerical apertures can be used. We chose this configuration in our experiments described in Section 3.

Fig. 2
Fig. 2

Plots of the function F NA(z) for different numerical-aperture values [from Eq. (5)]. The displacement of the reference mirror (along z) is normalized by the optical wavelength.

Fig. 3
Fig. 3

(a) Theoretical fringe spacing variation with numerical aperture from exact and approximate theories (normalized by the optical wavelength). (b) Theoretical FWHM of the fringe envelope from exact and approximate theories (normalized by the optical wavelength).

Fig. 4
Fig. 4

Experimental arrangement: a homemade Linnik-type interference microscope illuminated with a He–Ne laser whose beam has been spatially filtered and expanded to fill the whole back aperture of the objective lenses.

Fig. 5
Fig. 5

Experimental and theoretical interferograms obtained with our Linnik-type interference microscope with 0.5-NA objective lenses. In the theoretical curve no parameter was adjusted to fit the experimental data.

Fig. 6
Fig. 6

Theoretical and experimental fringe spacing versus numerical aperture, normalized by the optical wavelength. No parameter was adjusted to fit the experimental data.

Fig. 7
Fig. 7

Theoretical and experimental FWHM of the fringe envelope versus numerical aperture, normalized by the optical wavelength. No parameter was adjusted to fit the experimental data.

Tables (1)

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Table 1 Corrective Factor βa and Measurable Range of Heightb

Equations (16)

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Iz=AO2+AR2+2AOAR cos2kz cos θ+ϕ,
Λ=λ/2 cos θ.
Iz=AO2+AR2+2AOARFNAz,
FNAz=2sin2 θmax0θmaxcos2kz cos θ+ϕ×cos θ sin θdθ.
FNAz=2sin2 θmax×cos2kz+ϕ-cos2kz cos θmax+ϕ4k2z2+sin2kz+ϕ-cos θmax sin2kz cos θmax+ϕ2kz.
FNAz=cos2kz+ϕ,
FNAz2sin2 θmax0θmaxcos2kz cos θ+ϕsin θdθ,
FNAz21+cos θmaxsinkz1-cos θmaxkz1-cos θmax×coskz1+cos θmax+ϕ.
FNAzγNAzcos2αNAkz+ϕ,
γNAz  sinkz1-cos θmaxkz1-cos θmax,
αNA = 1+cos θmax/2.
Λλ2αNA=λ221+cos θmax=λ221+cosarcsin NA,
FWHMγNA0.6λ1-cos θmax=0.6λ1-cosarcsin NA.
β1αNA=21+cosarcsin NA.
1αNA=1+14NA2+18NA4+564NA6+ONA7.
β=1+0.25NA2+0.1NA4+0.086NA6.

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