Abstract

Negative exponentially distributed intensities of speckle fields seem unfavorable in terms of precision metrology, if interferometric setups are involved with a saturable photodetector and an analog-to-digital converter that imposes a finite resolution. By spatial integration, extended detector apertures modify the intensity distribution toward a less awkward function. However, because the detector aperture also integrates over points of rapidly changing speckle phases, this is done at the expense of a lower modulation of measured intensity during phase shift. An optimum set of parameters is calculated here, consisting of values for the lens aperture, the mean speckle intensity, and the beam ratio. The remaining phase-measurement error assumes its minimum of 10.6 mrad when the space–bandwidth product of the lens–detector system (thus concerning the lens aperture) is 0.31, the mean speckle intensity is 1/11 of the saturation intensity, and the reference intensity is four times higher than the mean speckle intensity. The 90° phase-shift algorithms with either three, four, or five frames turned out to be quite powerful, even with interference signals of rather poor modulation. Not needing a very small lens aperture is interesting, because stopping down the lens is a trade-off with the limited power of the laser.

© 1997 Optical Society of America

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References

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  1. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
    [CrossRef] [PubMed]
  2. K. Creath, “Phase-shifting speckle interferometry,” in Speckles, H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).
  3. T. Maack, R. Kowarschik, “Camera influence on the phase-measurement accuracy of a phase-shifting speckle interferometer,” Appl. Opt. 35, 3514–3524 (1996).
    [CrossRef] [PubMed]
  4. M. Lehmann, “Optimization of wavefield intensities in phase-shifting speckle interferometry,” Opt. Commun. 118, 199–206 (1995).
    [CrossRef]
  5. G. Å. Slettemoen, J. C. Wyant, “Maximal fraction of acceptable measurements in phase-shifting speckle interferometry: a theoretical study,” J. Opt. Soc. Am. A 3, 210–214 (1986).
    [CrossRef]
  6. G. Å. Slettemoen, “Electronic speckle pattern interferometric system based on a speckle reference beam,” Appl. Opt. 19, 616–623 (1980).
    [CrossRef] [PubMed]
  7. G. Å. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
    [CrossRef]
  8. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.
  9. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.
    [CrossRef]
  10. K. Creath, “Temporal phase measurement techniques,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Publishing, Bristol, UK, 1993), pp. 94–140.
  11. J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–76.
  12. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
    [CrossRef]
  13. A. A. Scribot, “First-order probability density function of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
    [CrossRef]
  14. A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 203–253.
  15. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).
  16. T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–276 (1974).
  17. C. P. Brophy, “Effect of intensity cross correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  18. U. Vry, A. F. Fercher, “Higher-order statistical properties of speckle fields and their application to rough-surface interferometry,” J. Opt. Soc. Am. A 3, 988–1000 (1986).
    [CrossRef]

1996

1995

M. Lehmann, “Optimization of wavefield intensities in phase-shifting speckle interferometry,” Opt. Commun. 118, 199–206 (1995).
[CrossRef]

1990

1986

1985

1980

1979

G. Å. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
[CrossRef]

1974

A. A. Scribot, “First-order probability density function of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–276 (1974).

1970

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

Brophy, C. P.

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Creath, K.

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
[CrossRef] [PubMed]

K. Creath, “Phase-shifting speckle interferometry,” in Speckles, H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).

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

K. Creath, “Temporal phase measurement techniques,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Publishing, Bristol, UK, 1993), pp. 94–140.

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

Ennos, A. E.

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 203–253.

Fercher, A. F.

Goodman, J. W.

J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–76.

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Kowarschik, R.

Lehmann, M.

M. Lehmann, “Optimization of wavefield intensities in phase-shifting speckle interferometry,” Opt. Commun. 118, 199–206 (1995).
[CrossRef]

Maack, T.

McKechnie, T. S.

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–276 (1974).

Scribot, A. A.

A. A. Scribot, “First-order probability density function of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

Slettemoen, G. Å.

Vry, U.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

Wyant, J. C.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Acta

G. Å. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
[CrossRef]

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

Opt. Commun.

A. A. Scribot, “First-order probability density function of speckle measured with a finite aperture,” Opt. Commun. 11, 238–241 (1974).
[CrossRef]

M. Lehmann, “Optimization of wavefield intensities in phase-shifting speckle interferometry,” Opt. Commun. 118, 199–206 (1995).
[CrossRef]

Optik

T. S. McKechnie, “Measurement of some second order statistical properties of speckle,” Optik 39, 258–276 (1974).

Other

K. Creath, “Phase-shifting speckle interferometry,” in Speckles, H. Arsenault, ed., Proc. SPIE556, 337–346 (1985).

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 203–253.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

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

K. Creath, “Temporal phase measurement techniques,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Publishing, Bristol, UK, 1993), pp. 94–140.

J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–76.

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Figures (6)

Fig. 1
Fig. 1

Schematic of the setup. The lens aperture creates a subjective speckle pattern of the rough object surface in the detector plane. The point of divergence of the reference beam (not depicted) is assumed to coincide with the center of the lens aperture.

Fig. 2
Fig. 2

Autocorrelation function Sa π2 Ω2/8 of a circular lens aperture as a function of the spatial frequency, which is given in units of Ω (Ω is the frequency corresponding to the radius of the aperture and is identical with the cutoff frequency of the lens). The complete representation is obtained by rotation of the curve about the ordinate axis.

Fig. 3
Fig. 3

Contrast of intensity modulation of an integrated speckle interferogram versus the space–bandwidth products. The bisecting line of the angle corresponds to detector apertures with aspect ratios equal to one.

Fig. 4
Fig. 4

(a) Contrast of intensity modulation of an integrated speckle interferogram along the line of constant active detector area 4bx by = (0.2λR/a)2. The curve starts at an aspect ratio of one. (b) Enlarged detail of (a).

Fig. 5
Fig. 5

Plots of the PDF’s of Eq. (4) for two different speckle patterns, where the exponential one represents the situation of a fully resolved pattern and the other one depicts the case of an integrated pattern with (ba)/(2λR) = 0.31.

Fig. 6
Fig. 6

Contour plot of standard deviation σΦ of the difference Φ of speckle phases φ1 and φ2 with (ba)/(2λR) = 0.31. The x axis is the beam ratio r and the y axis is the number of times the photodetector’s saturation intensity exceeds the mean speckle intensity. The latter is identical with t, which is the inverse of the modulation of the camera. The contours are labeled with the corresponding standard deviations given in milliradians.

Tables (1)

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Table 1 Comparison of Theoretical and Computer-Simulated Resultsa

Equations (24)

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φmeas=arctanI4-I2I1-I3,
φmeas=arctan2I2-I42I3-I5-I1.
Is=sIs, Ir=rIs, It=tIs,
pss=mmsm-1 exp-msΓmfor s00for s<0,
1m=-+Sbx, yμax, y2 dxdy
Sbx, y=14bxby1-x2bx1-y2by for x2bx and y2by,
b=1in the aperture0outside the aperture
-+ fx, ydxdy=1
a=1in the apertureOoutside the aperture,
μa2=4J122πλadR2πλadR2,
d2=x2+y2.
I=Ir+Is+2ν IrIs cosφ,
ν=-+ Saξ, ημbξ, ηdξdη,
Saξ, η=8π2Ω2arccosωΩ-ωΩ1-ωΩ2
ω2=ξ2+η2,
Ω=aλR.
μb=sin2πbxξ2πbxξsin2πbyη2πbyη.
σΦ2=2σφ2.
σΦ=10.6 mrad
ba2λR=0.31,
t=11.0
r=4.0
m=2.6
ν=0.56.

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