Abstract

A method for phase measurement in common-path interferometers, believed to be novel, is presented. We use the property of phase reconstruction algorithms, such as the Carré and Hariharan algorithms, that do not require uniform phase across the reference beam. Only the ratio of the phase steps must be the same at each pixel. We show phase measurement and reconstruction in a common-path interferometer by shifting either the tilt or the focus of the reference wave front. We present a theoretical explanation of phase measurement using this property. We also present results from a proof-of-principle experiment using a scatterplate interferometer, in conjunction with the tilt phase-shifting technique, to measure the reflected phase of a test optical element. Furthermore, we present a computer simulation to demonstrate the mathematical validity of this measurement technique using defocus shifting, rather than tilt shifting, in the reference wave front.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara, Optical Shop Testing, 2nd ed. (Academic, Boston, Mass., 1996).
  2. J. Huang, T. Honda, N. Ohyama, J. Tsujiuchi, “Fringe scanning scatterplate interferometer using polarized light,” Opt. Commun. 68, 235–238 (1988).
    [CrossRef]
  3. M. B. North-Morris, J. Van Delden, J. C. Wyant, “Birefringent scatterplate phase shifting interferometer,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 432–433 (1999).
    [CrossRef]
  4. P. Carré, “Installation et utilisation du compateur photoelectique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  5. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  6. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
    [CrossRef] [PubMed]
  7. D. Ghiglia, M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  8. L. Rubin, “Scatterplate interferometry,” Opt. Eng. 19, 815–824 (1980).
    [CrossRef]

1988

J. Huang, T. Honda, N. Ohyama, J. Tsujiuchi, “Fringe scanning scatterplate interferometer using polarized light,” Opt. Commun. 68, 235–238 (1988).
[CrossRef]

1987

1985

1980

L. Rubin, “Scatterplate interferometry,” Opt. Eng. 19, 815–824 (1980).
[CrossRef]

1966

P. Carré, “Installation et utilisation du compateur photoelectique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Carré, P.

P. Carré, “Installation et utilisation du compateur photoelectique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Creath, K.

Eiju, T.

Ghiglia, D.

D. Ghiglia, M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

Hariharan, P.

Honda, T.

J. Huang, T. Honda, N. Ohyama, J. Tsujiuchi, “Fringe scanning scatterplate interferometer using polarized light,” Opt. Commun. 68, 235–238 (1988).
[CrossRef]

Huang, J.

J. Huang, T. Honda, N. Ohyama, J. Tsujiuchi, “Fringe scanning scatterplate interferometer using polarized light,” Opt. Commun. 68, 235–238 (1988).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Academic, Boston, Mass., 1996).

North-Morris, M. B.

M. B. North-Morris, J. Van Delden, J. C. Wyant, “Birefringent scatterplate phase shifting interferometer,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 432–433 (1999).
[CrossRef]

Ohyama, N.

J. Huang, T. Honda, N. Ohyama, J. Tsujiuchi, “Fringe scanning scatterplate interferometer using polarized light,” Opt. Commun. 68, 235–238 (1988).
[CrossRef]

Oreb, B. F.

Pritt, M.

D. Ghiglia, M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

Rubin, L.

L. Rubin, “Scatterplate interferometry,” Opt. Eng. 19, 815–824 (1980).
[CrossRef]

Tsujiuchi, J.

J. Huang, T. Honda, N. Ohyama, J. Tsujiuchi, “Fringe scanning scatterplate interferometer using polarized light,” Opt. Commun. 68, 235–238 (1988).
[CrossRef]

Van Delden, J.

M. B. North-Morris, J. Van Delden, J. C. Wyant, “Birefringent scatterplate phase shifting interferometer,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 432–433 (1999).
[CrossRef]

Wyant, J. C.

M. B. North-Morris, J. Van Delden, J. C. Wyant, “Birefringent scatterplate phase shifting interferometer,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 432–433 (1999).
[CrossRef]

Appl. Opt.

Metrologia

P. Carré, “Installation et utilisation du compateur photoelectique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Commun.

J. Huang, T. Honda, N. Ohyama, J. Tsujiuchi, “Fringe scanning scatterplate interferometer using polarized light,” Opt. Commun. 68, 235–238 (1988).
[CrossRef]

Opt. Eng.

L. Rubin, “Scatterplate interferometry,” Opt. Eng. 19, 815–824 (1980).
[CrossRef]

Other

D. Ghiglia, M. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

M. B. North-Morris, J. Van Delden, J. C. Wyant, “Birefringent scatterplate phase shifting interferometer,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 432–433 (1999).
[CrossRef]

D. Malacara, Optical Shop Testing, 2nd ed. (Academic, Boston, Mass., 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Illustration of phase steps at various points across the five tilt-shifted reference wave fronts.

Fig. 2
Fig. 2

Interferograms of the f/7 mirror taken with a scatterplate interferometer. There is a residual 5 waves of vertical tilt and a residual 0.5 waves of horizontal tilt in the wave front. Tilt is shifted when the scatterplate is translated perpendicular to the optical axis and results in a perceived rotation of the fringes about the hot spot. Note that the contrast is reversed to improve illustration.

Fig. 3
Fig. 3

Reconstructed phase (modulo 2π) reflected from the test optic for the axis of tilt located in the center of the mirror. Note that the curvature error across the center of tilt is corrected.

Fig. 4
Fig. 4

Reconstructed phase (modulo 2π) reflected from the test optic for the axis of tilt located at the edge of the mirror.

Fig. 5
Fig. 5

Contour map of the final reconstructed phase of the test beam, with residual tilt removed. Complete measurement is made when the data from Figs. 3 and 4 are combined.

Fig. 6
Fig. 6

Rms error of six measurements for both the axis of tilt located in the center of the mirror and at the edge of the mirror.

Fig. 7
Fig. 7

Simulated interferograms illustrating a shifted defocus in the reference beam. In each interferogram the focus of the reference wave front was shifted by 0.5 waves. A small amount of random Gaussian noise was added to each of these images in an effort to simulate electronic noise.

Fig. 8
Fig. 8

Reconstructed phase (modulo 2π) by use of simulated defocus interferograms shown in Fig. 7.

Equations (11)

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

Iix, y=Ix, y+Ix, ycosϕx, y+miBx, y.
Bx, y=2πW111xx;
Bx, y=2πW020x2+y2=2πW020ρ2,
tanϕx, y=2 sin Bx, yI-1-I+12I0-I-2+I+2,
sin Bx, y=±1-14I-2-I+2I-1-I+121/2.
miBx3=-2π, -π, 0, π, 2π.
Csx, y=I-1-I+1  sin Bx, ysin ϕx, y,
Ccx, y=I-1+I+1-I-3+I+3 cos Bx, ycos ϕx, y.
Ccx, y=2I0-I-1+I+1 cos ϕx, y1-cos Bx, y.
Di=-5.08 μm, -2.54 μm, 0, +2.54 μm,+5.08 μm,
miBx, y=2π-0.76x, -0.38x, 0, 0.38x, 0.76x.

Metrics