Abstract

A technique that compensates for low spatial frequency spurious phase changes during an interference experiment is developed; it permits temporal averaging of multiple phase measurements, made before and after object displacement. The method is tested with phase-stepped real-time holographic interferometry applied to cantilever bending of a piezoelectric bimorph ceramic. Results indicate that temporal averaging of the corrected data significantly reduces the white noise in a phase measurement without incurring systematic errors or sacrificing spatial resolution. White noise is reduced from 3° to less than 1° (λ/360) using these methods.

© 1993 Optical Society of America

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References

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  1. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Markel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
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    [CrossRef] [PubMed]
  8. M. Corke, J. D. C. Jones, A. D. Kersey, D. A. Jackson, “All single-mode fibre optic holographic system with active fringe stabilisation,” J. Phys. E 18, 185–186 (1985).
    [CrossRef]
  9. C. R. Mercer, G. Beheim, “Fiber optic phase stepping system for interferometry,” Appl. Opt. 30, 729–734 (1991).
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  11. B. Ovryn, E. M. Haacke, “Temporal averaging in a turbulent environment: compensation for phase drifts in phase shifting interferometry,” in Laser Interferometry IV Computer-Aided Interferometry, R. J. Prypatniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 221–231 (1991).
  12. B. Ovryn, E. M. Haacke, “Measurement of the piezoelectric effect in bone using quasi-heterodyne holographic interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine, H. Podbielska, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1429, 172–182 (1991).
  13. B. Ovryn, E. M. Haacke, “Phase shifting interferometry,” in Hitemp Review, Rep. CP-10039 (NASA, Washington, D.C., October1989), pp. 38-1–38-9.
  14. B. Ovryn, “Holographic interferometry,” Crit. Rev. Biomed. Eng. 16, 269–322 (1989).
    [PubMed]
  15. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  16. K. Creath, “Phase-measurement interferometry techniques,” Progr. Opt. 26, 350–393 (1989).
  17. P. Carré, “Installation et ultisation due compareteur photoelectronique et interferential de Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  18. W. P. Press, B. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), p. 442.

1991

1989

J. Schwider, “Phase shifting interferometry: reference error reduction,” Appl. Opt. 28, 3889–3892 (1989).
[CrossRef] [PubMed]

B. Ovryn, “Holographic interferometry,” Crit. Rev. Biomed. Eng. 16, 269–322 (1989).
[PubMed]

K. Creath, “Phase-measurement interferometry techniques,” Progr. Opt. 26, 350–393 (1989).

1988

1987

1985

W. T. Estler, “High-accuracy displacement interferometry,” Appl. Opt. 24, 808–815 (1985).
[CrossRef] [PubMed]

M. Corke, J. D. C. Jones, A. D. Kersey, D. A. Jackson, “All single-mode fibre optic holographic system with active fringe stabilisation,” J. Phys. E 18, 185–186 (1985).
[CrossRef]

1983

1980

1974

1966

P. Carré, “Installation et ultisation due compareteur photoelectronique et interferential de Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Beheim, G.

Bobroff, N.

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Carré, P.

P. Carré, “Installation et ultisation due compareteur photoelectronique et interferential de Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Corke, M.

M. Corke, J. D. C. Jones, A. D. Kersey, D. A. Jackson, “All single-mode fibre optic holographic system with active fringe stabilisation,” J. Phys. E 18, 185–186 (1985).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Progr. Opt. 26, 350–393 (1989).

Dandridge, A.

Elssner, K. E.

Estler, W. T.

Flannery, B.

W. P. Press, B. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), p. 442.

Gallagher, J. E.

Grzanna, J.

Haacke, E. M.

B. Ovryn, E. M. Haacke, “Measurement of the piezoelectric effect in bone using quasi-heterodyne holographic interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine, H. Podbielska, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1429, 172–182 (1991).

B. Ovryn, E. M. Haacke, “Phase shifting interferometry,” in Hitemp Review, Rep. CP-10039 (NASA, Washington, D.C., October1989), pp. 38-1–38-9.

B. Ovryn, E. M. Haacke, “Temporal averaging in a turbulent environment: compensation for phase drifts in phase shifting interferometry,” in Laser Interferometry IV Computer-Aided Interferometry, R. J. Prypatniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 221–231 (1991).

B. Ovryn, E. M. Haacke, “Compensation for spurious phase changes in phase shifting interferometry using Fourier extrapolation,” in Hitemp Review, Rep. CP-10051 (NASA, Washington, D.C., October1990), pp. 68-1–68-8.

Herriott, D. R.

Jackson, D. A.

M. Corke, J. D. C. Jones, A. D. Kersey, D. A. Jackson, “All single-mode fibre optic holographic system with active fringe stabilisation,” J. Phys. E 18, 185–186 (1985).
[CrossRef]

D. A. Jackson, R. Priest, A. Dandridge, A. B. Tveten, “Elimination of drift in a single-mode optical fiber interferometer using a piezoelectrically stretched coiled fiber,” Appl. Opt. 19, 2926–2929 (1980).
[CrossRef] [PubMed]

Jones, J. D. C.

M. Corke, J. D. C. Jones, A. D. Kersey, D. A. Jackson, “All single-mode fibre optic holographic system with active fringe stabilisation,” J. Phys. E 18, 185–186 (1985).
[CrossRef]

Kersey, A. D.

M. Corke, J. D. C. Jones, A. D. Kersey, D. A. Jackson, “All single-mode fibre optic holographic system with active fringe stabilisation,” J. Phys. E 18, 185–186 (1985).
[CrossRef]

Kinnstaetter, K.

Lohmann, A.

Markel, K.

Mercer, C. R.

Ovryn, B.

B. Ovryn, “Holographic interferometry,” Crit. Rev. Biomed. Eng. 16, 269–322 (1989).
[PubMed]

B. Ovryn, E. M. Haacke, “Compensation for spurious phase changes in phase shifting interferometry using Fourier extrapolation,” in Hitemp Review, Rep. CP-10051 (NASA, Washington, D.C., October1990), pp. 68-1–68-8.

B. Ovryn, E. M. Haacke, “Temporal averaging in a turbulent environment: compensation for phase drifts in phase shifting interferometry,” in Laser Interferometry IV Computer-Aided Interferometry, R. J. Prypatniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 221–231 (1991).

B. Ovryn, E. M. Haacke, “Phase shifting interferometry,” in Hitemp Review, Rep. CP-10039 (NASA, Washington, D.C., October1989), pp. 38-1–38-9.

B. Ovryn, E. M. Haacke, “Measurement of the piezoelectric effect in bone using quasi-heterodyne holographic interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine, H. Podbielska, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1429, 172–182 (1991).

Press, W. P.

W. P. Press, B. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), p. 442.

Priest, R.

Rosenfeld, D. P.

Schwider, J.

Spolaczyk, R.

Streibl, N.

Teukolsky, S. A.

W. P. Press, B. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), p. 442.

Tveten, A. B.

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Vetterling, W. T.

W. P. Press, B. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), p. 442.

White, A. D.

Appl. Opt.

Crit. Rev. Biomed. Eng.

B. Ovryn, “Holographic interferometry,” Crit. Rev. Biomed. Eng. 16, 269–322 (1989).
[PubMed]

J. Phys. E

M. Corke, J. D. C. Jones, A. D. Kersey, D. A. Jackson, “All single-mode fibre optic holographic system with active fringe stabilisation,” J. Phys. E 18, 185–186 (1985).
[CrossRef]

Metrologia

P. Carré, “Installation et ultisation due compareteur photoelectronique et interferential de Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Progr. Opt.

K. Creath, “Phase-measurement interferometry techniques,” Progr. Opt. 26, 350–393 (1989).

Other

W. P. Press, B. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), p. 442.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

B. Ovryn, E. M. Haacke, “Compensation for spurious phase changes in phase shifting interferometry using Fourier extrapolation,” in Hitemp Review, Rep. CP-10051 (NASA, Washington, D.C., October1990), pp. 68-1–68-8.

B. Ovryn, E. M. Haacke, “Temporal averaging in a turbulent environment: compensation for phase drifts in phase shifting interferometry,” in Laser Interferometry IV Computer-Aided Interferometry, R. J. Prypatniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 221–231 (1991).

B. Ovryn, E. M. Haacke, “Measurement of the piezoelectric effect in bone using quasi-heterodyne holographic interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine, H. Podbielska, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1429, 172–182 (1991).

B. Ovryn, E. M. Haacke, “Phase shifting interferometry,” in Hitemp Review, Rep. CP-10039 (NASA, Washington, D.C., October1989), pp. 38-1–38-9.

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Figures (11)

Fig. 1
Fig. 1

Photograph of the ceramic bimorph and steel control. The ceramic is clamped in a vice and the control is held to the vice by a magnet.

Fig. 2
Fig. 2

Schematic of the object surrounded by a stationary control. The numbers represent pixels, and the dotted region, 240 × 34 pixels, corresponds to the area for phase reconstruction. The circular, black region represents a cross section of the connecting wire.

Fig. 3
Fig. 3

Displacement curve through the center of the bimorph (from bottom to top) for an applied potential of 1.5 V. The curve represents one column of data without spatial or temporal averaging. Each pixel represents 0.14 mm.

Fig. 4
Fig. 4

Phase for the 36 data sets before displacement. The left and right regions represent the phase on the control, while the center region is the ceramic. Each pixel represents 0.17 mm.

Fig. 5
Fig. 5

Phase at three points: one point is from the left control (dotted–dashed curve), one is from the right control (dashed curve), and one is from the object (solid curve), for each of the 72 data sets. The 72 data sets require 14 min. to collect and store on disk. So that the curves are plotted with a common origin, a phase offset is subtracted from each of the three curves; the offsets for the left and right controls and for the object points are 770°, 1249°, and 925°, respectively.

Fig. 6
Fig. 6

Thirty-sixth data set before displacement (dotted–dashed curve) and the first data set after displacement. The solid curve represents the displacement after a potential of 1.0 V is applied to the object.

Fig. 7
Fig. 7

Phase difference between the last data set before displacement and the first data set after displacement (solid curve). The rms phase error in the control region is 2.90°. Averaging of the data without phase correcting (dotted–dashed curve) lowers the random noise to an rms error of 0.8° but introduces a systematic error.

Fig. 8
Fig. 8

(a) Phase difference between the first data set and subsequent data sets evaluated at a point on the left side of the control for the data sets after object displacement. (b) Same subtraction as in 8(a), after phase correcting all the data sets to the first of the 36 data sets. The phase drift has a mean of −0.06° and a standard deviation of 0.94°.

Fig. 9
Fig. 9

(a) Phase difference between temporally adjacent data sets after displacement (evaluated at the same point as that used in Fig. 8). The mean and the standard deviation of the data points are −1.87° and 26.36°, respectively. (b) Phase difference between temporally adjacent data sets after phase correcting to the first data set [same point as in 9(a)]. The mean and standard deviation of the data points are −0.01° and 1.41°, respectively.

Fig. 10
Fig. 10

Phase difference after phase correcting and averaging the data (solid curve). The dotted–dashed curve shows the phase difference after phase correcting, averaging, and constraining the control. The rms phase error is 0.8°. The rounded edges near the periphery are an artifact of the low-pass filtering.

Fig. 11
Fig. 11

(a) Two rows of data from the object, each represented by a curve. The data were taken 9 mm above the base of the bimorph. The two curves are separated by 0.14 mm. (b) Same data as in (a), after phase correcting and averaging but before the control correction. (c) Same data as in (a) and (b), after phase correction and control constraint.

Equations (12)

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ϕ ( x , y , t ) = Ψ ( x , y ) + k ( x , y , t ) ,
γ ( x , y , t ) = k ( x , y , t + Δ t ) - k ( x , y , t ) .
ϕ ( x , y , t ) = ϕ ( x , y , t + Δ t ) - γ ( x , y , t ) .
f = F - 1 H F ,
β ( x , y , t i ) = f B ( x , y , t i ) - f B ( x , y , t r B ) ,
α ( x , y , t i ) = f A ( x , y , t i ) - f A ( x , y , t r A ) ,
B ( x , y , t i ) = B ( x , y , t i ) - β ( x , y , t i ) ,
A ( x , y , t i ) = A ( x , y , t i ) - α ( x , y , t i ) .
Δ ϕ ( x , y ) = A ( x , y , t i ) - B ( x , y , t i ) .
Δ ϕ ( x , y ) = R ( x , y , t i ) - f E gap [ R ( x , y , t i ) - A ( x , y , t i ) + B ( x , y , t n ) ] ,
Δ ϕ c ( x , y ) = Δ ϕ ( x , y ) - f E obj Δ ϕ ( x , y ) .
Δ ϕ ( x , y ) = R ( x , y , t i ) - f R ( x , y , t i ) + f A ( x , y , t 1 ) - f B ( x , y , t n ) .

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