Abstract

A method of postprocessing the data from a phase-shifting interferometer is proposed that reduces the influence of vibration or other environmental disturbances occurring during the data-acquisition time. The method is based on spectral analysis of the time-varying intensity data from each of the camera pixels. A correction term is obtained at a frequency three times that of the phase-shifting device. No additional hardware is required, although the phase-shifting algorithm used must have phase steps of π/3. Numerical studies have demonstrated reductions in the rms phase error of more than one and two orders of magnitude for vibration amplitudes of 0.1 and 0.01 rad, respectively, with a commonly used seven-frame algorithm. The influence of factors affecting the technique’s performance—in particular, miscalibration of the phase-shifting device, nonlinearity of the photodetector, and finite integration time—is also investigated.

© 1998 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. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  2. K. Creath, “Phase-measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 213–220 (1991).
    [CrossRef]
  3. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  4. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  5. Y. Surrel, “Phase-stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  6. B. Zhao, Y. Surrel, “Phase-shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
    [CrossRef]
  7. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  8. J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [CrossRef] [PubMed]
  9. P. J. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  10. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  11. P. J. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  12. I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
    [CrossRef]
  13. L. Deck, “Vibration-resistant phase-shifting interferometry,” Appl. Opt. 35, 6655–6662 (1996).
    [CrossRef] [PubMed]
  14. G.-S. Han, S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321–7325 (1994).
    [CrossRef] [PubMed]
  15. J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 1889–3892 (1989).
    [CrossRef]
  16. P. J. de Groot, L. L. Deck, “Numerical simulations of vibration in phase-shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
    [CrossRef] [PubMed]

1996

1995

1994

1993

Y. Surrel, “Phase-stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

1992

1989

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 1889–3892 (1989).
[CrossRef]

1983

Burow, R.

Creath, K.

J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

K. Creath, “Phase-measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 213–220 (1991).
[CrossRef]

de Groot, P. J.

Deck, L.

Deck, L. L.

Elssner, K.-E.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Farrant, D. I.

Grzanna, J.

Han, G.-S.

Hibino, K.

Kato, J.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Kim, S.-W.

Larkin, K. G.

Liu, J. Y.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Merkel, K.

Oreb, B. F.

Schmit, J.

Schreiber, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schwider, J.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 1889–3892 (1989).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

Spolaczyk, R.

Streibl, N.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Surrel, Y.

Yamaguchi, I.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Zhao, B.

B. Zhao, Y. Surrel, “Phase-shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Zöller, A.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

B. Zhao, Y. Surrel, “Phase-shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Other

K. Creath, “Phase-measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 213–220 (1991).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Modulus of the Fourier transform of the intensity signal from a phase-shifting interferometer in the presence of vibration, plotted as a function of nondimensional frequency f=ν/ν0. fN=νN/ν0 is the nondimensional Nyquist frequency for the case of sampled signals. The delta functions labeled S0, S±1, and S±1±1 are defined in Eq. (8).

Fig. 2
Fig. 2

Fourier transform as shown in Fig. 1 for the case fν=2. Superimposition of the S-11 and S1 peaks can be expected to result in large phase errors, owing to the vibration.

Fig. 3
Fig. 3

Rms phase error for the seven-frame algorithm,3 with and without vibration suppression (filled and open circles, respectively). Vibration amplitudes are 0.01, 0.1, and 1.0 radians for (a), (b), and (c), respectively. Solid curves are calculated by the analytical linear approximation from Ref. 11.

Fig. 4
Fig. 4

Rms phase error for the ten-frame algorithm,10 with and without vibration suppression (filled and open circles, respectively). Vibration amplitudes are 0.01, 0.1, and 1.0 radians for (a), (b), and (c), respectively. Solid curves are calculated by the analytical linear approximation from Ref. 11.

Fig. 5
Fig. 5

Wrapped phase map (black represents -π, white represents +π) corresponding to the phase function for tilt fringes (4.7 and 1.3 across the horizontal and the vertical directions, respectively) and spherical aberration (1.5 fringes from the center to the edge of the field along a horizontal or a vertical radius).

Fig. 6
Fig. 6

Phase-error profiles, with and without vibration correction (solid and dashed curves, respectively), along the dashed line of Fig. 5. Seven intensity maps were calculated from Fig. 5 with phase shifts of π/3 and vibration of amplitude 0.1 rad and analyzed by the seven-frame algorithm.3

Fig. 7
Fig. 7

Results as presented in Fig. 3(b) but with the additional effect of a phase-shift error of 5%.

Fig. 8
Fig. 8

Results as presented in Fig. 3(b) but with the additional effect of a third-order detector nonlinearity of 5%.

Fig. 9
Fig. 9

Results as presented in Fig. 3(b) but with the additional effect of a camera exposure time equal to the camera interframe time (τ=1).

Fig. 10
Fig. 10

Phase error versus vibration frequency under the same conditions as for Fig. 9 but with compensation for the attenuation of the f=3 peak, owing to the finite integration time.

Fig. 11
Fig. 11

Low-pass filter function B(f ) that is due to the finite integration time of the camera for the case τ=1 (exposure time equal to the interframe time) and fs1. Vibration at frequencies fν=fs±2, 2 fs±2,  causes spectral lines A, A, A, B, B, and B. BB are aliased onto the signal frequency, f=1, and A A are aliased onto the correction frequency, f=3.

Fig. 12
Fig. 12

Phase-error spectral response for a 33-frame phase-shifting algorithm with camera exposure time equal to the camera interframe time (τ=1) and vibration amplitude of 0.1 rad. Results with and without vibration suppression are shown with filled and open circles, respectively, with solid curves calculated by the analytical linear approximation from Ref. 11.

Tables (2)

Tables Icon

Table 1 Sampling Coefficients for the Seven-Frame Larkin–Oreb Algorithma

Tables Icon

Table 2 Sampling Coefficients for the Ten-Frame Surrel Algorithma

Equations (45)

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s(t)=I0{1+V cos[Φ+ϕ(t)+r cos(2πννt+α)]},
ϕ(t)2πν0t,
s(t)=s0+s1(t)+s-1(t)+s11(t)+s1-1(t)+s-11(t)+s-1-1(t),
s0=I0,
s±1(t)=I0V2exp(±iΦ)exp(±i2πν0t),
s1±1(t)=ir2s1(t)exp(±iα)exp(±i2πννt),
s-1±1(t)=-ir2s-1(t)exp(±iα)exp(±i2πννt).
S(ν)=-s(t)exp(-i2πνt)dt.
S(ν)=S0(ν)+S1(ν)+S-1(ν)+S11(ν, νν)+S1-1(ν, νν)+S-11(ν, νν)+S-1-1(ν, νν),
S0(ν)=2AVδ(ν),
S±1(ν)=A exp(±iΦ)δ(νν0),
S1±1(ν, νν)=irA2exp[i(Φ±α)]δ(ν-ν0νν),
S-1±1(ν, νν)=-irA2exp[i(-Φ±α)]δ(ν+ν0νν).
Φˆ=tan-1Im[S(ν0)]Re[S(ν0)].
e=-ir2AW(0)2exp[i(-Φ+α2)],
S(3ν0)=S11(3ν0, 2ν0)+S-11(3ν0, 4ν0)
=ir2AW(0)2exp[i(Φ+α2)]-ir4AW(0)2exp[i(-Φ+α4)].
x,yRA(x, y)exp[-2iΦ(x, y)]=0.
X1=x,yR[S*(x, y, ν0)]2/|S(x, y, ν0)|,
X2=x,yR|S(x, y, ν0)|,
X3=x,yRS(x, y, 3ν0)S*(x, y, ν0)/|S(x, y, ν0)|,
arg(X1)-arg{[S*(x, y, ν0)]2/|S(x, y, ν0)|}<,
|X1|X2<γ,
eˆ(x, y)=-X3X2S*(x, y, ν0),
Φˆ=tan-1Im[S(ν0)-eˆ]Re[S(ν0)-eˆ].
S(kΔν)=j=0N-1s(jΔt)wj exp(-2πijk/N),
j, k=0, 1, 2  N-1,
ΔνΔt=1/N.
S(ν0)=j=0N-1s(jΔt)wj exp(-iϕj),
S(3ν0)=j=0N-1s(jΔt)wj exp(-3iϕj).
Φˆ=tan-1j=0N-1s(jΔt)bjj=0N-1s(jΔt)aj,
aj+ibj=wj exp(-iϕj),
S(ν0)=j=0N-1s(jΔt)aj+ij=0N-1s(jΔt)bj.
bj(3)=bj cos(2ϕj)-aj sin(2ϕj),
aj(3)=aj cos(2ϕj)+bj sin(2ϕj),
S(3ν0)=j=0N-1s(jΔt)aj(3)+ij=0N-1s(jΔt)bj(3).
j=0N-1aj cos(pϕj)=-j=0N-1bj sin(pϕj),
j=0N-1aj sin(pϕj)=j=0N-1bj cos(pϕj)
fN=fs/2,
S(3ν0)=ir2AW(0)2exp[i(Φ+α2)]-ir2AW(0)2exp[i(-Φ-α2)].
s=s+κs3,
B(f)=sin(πfτ/fs)f sin(πτ/fs),
wj=1,j=1, 2, 3  32,
=0.5+2.514i,j=0,
=0.5-2.514i,j=33.

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