Abstract

In phase-shifting interferometry, many algorithms have been reported that suppress systematic errors caused by, e.g., nonlinear motion of the phase shifter and nonsinusoidal signal waveform. However, when a phase-shifting algorithm is designed to compensate for the systematic phase-shift errors, it becomes more susceptible to random noise and gives larger random errors in the measured phase. The susceptibility of phase-shifting algorithms to random noise is analyzed with respect to their immunity to phase-shift errors and harmonic components of the signal. It is shown that for the most common group of error-compensating algorithms for nonlinear phase shift, both random errors and the effect of high-order harmonic components of the signal cannot be minimized simultaneously. It is also shown that if an algorithm is designed to have extended immunity to nonlinear phase shift, simultaneous minimization becomes possible.

© 1997 Optical Society of America

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Corrections

Kenichi Hibino, "Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry: erratum," Appl. Opt. 36, 5362-5362 (1997)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-36-22-5362

References

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  1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [Crossref] [PubMed]
  2. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.
    [Crossref]
  3. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [Crossref]
  4. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wavefront measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [Crossref]
  5. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [Crossref] [PubMed]
  6. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [Crossref]
  7. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [Crossref]
  8. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [Crossref] [PubMed]
  9. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shiftings for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [Crossref]
  10. Y. Surrel, “Design of algorithms for phase measurements by the use phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [Crossref] [PubMed]
  11. 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]
  12. P. J. de Groot, “Derivation of algorithms for phase-shifting interferometry using concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [Crossref]
  13. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
    [Crossref]
  14. J. Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [Crossref] [PubMed]
  15. N. Ohyama, S. Kinoshita, A. Cornejo-Rodriguez, T. Honda, J. Tsujiuchi, “Accuracy of phase determination with unequal reference phase shift,” J. Opt. Soc. Am. A 5, 2019–2025 (1988).
    [Crossref]
  16. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–540 (1990).
    [Crossref]
  17. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [Crossref]
  18. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
    [Crossref]
  19. K. A. Stetson, W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [Crossref]

1997 (1)

1996 (1)

1995 (4)

1993 (2)

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

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

1992 (1)

1991 (1)

1990 (2)

1988 (1)

1987 (1)

1985 (1)

1983 (1)

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Brangaccio, D. J.

Brohinsky, W. R.

Brophy, C. P.

Bruning, J. H.

Burow, R.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Cornejo-Rodriguez, A.

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 techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.
[Crossref]

de Groot, P. J.

Eiju, T.

Elssner, K. E.

Falkenstorfer, O.

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

Farrant, D. I.

Frankena, H. J.

Freischlad, K.

Gallagher, J. E.

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Hibino, K.

Honda, T.

Kinoshita, S.

Koliopoulos, C. L.

Larkin, K. G.

Merkel, K.

Ohyama, N.

Oreb, B. F.

Rathjen, C.

Rosenfeld, D. P.

Schmit, J.

Schreiber, H.

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

Schwider, J.

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

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

Smorenburg, C.

Spolaczyk, R.

Stetson, K. A.

Streibl, N.

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

Surrel, Y.

Tsujiuchi, J.

White, A. D.

Wingerden, J.

Zoller, A.

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

Appl. Opt. (9)

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

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[Crossref] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[Crossref] [PubMed]

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

Y. Surrel, “Design of algorithms for phase measurements by the use phase stepping,” Appl. Opt. 35, 51–60 (1996).
[Crossref] [PubMed]

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]

P. J. de Groot, “Derivation of algorithms for phase-shifting interferometry using concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[Crossref]

J. Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[Crossref] [PubMed]

K. A. Stetson, W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
[Crossref]

J. Opt. Soc. Am. A (7)

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[Crossref]

Opt. Eng. (1)

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

Other (1)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland Elsevier, Amsterdam, 1988), Vol. 26, pp. 349–393.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Susceptibilities of group II algorithms to random noise as a function of the error-compensating capability to nonlinear phase-shift errors and nonsinusoidal waveforms.

Fig. 2
Fig. 2

Susceptibilities of group II algorithms to random noise when sampled over one period of the phase shift.

Fig. 3
Fig. 3

Susceptibilities of group III algorithms to random noise as a function of the error-compensating capability to nonlinear phase-shift errors and nonsinusoidal waveforms.

Fig. 4
Fig. 4

Susceptibilities of three groups of algorithms to random noise as a function of the number of samples. The Roman figures in the parentheses show the group numbers; P is the order of nonlinearity of the phase shift for which the algorithms compensate.

Equations (48)

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Ix, y, α=s0x, y+k=1skx, ycoskα-φkx, y,
φ=arctanr=1mbrIrr=1marIr,
αr=α0r1+1+2α0rπ+3α0rπ2+···+pα0rπp-1 for r=1, 2, , m,
Δφ=φ-φ1=osk+oq+oqsk+o2+osk2,
r=1mar sin kα0r=0 for k=1, , j,
r=1mar cos kα0r=δk, 1 for k=0, 1, , j,
r=1mbr sin kα0r=δk, 1 for k=1, , j,
r=1mbr cos kα0r=0 for k=0, 1, , j,
r=1mα0rqar sin α0r+br cos α0r=0 for q=1, 2, , p,
r=1mα0rqar cos α0r-br sin α0r=0 for q=1, 2, , p.
r=1marα0rq sin kα0r=0,
r=1marα0rq cos kα0r=0,
r=1mbrα0rq sin kα0r=0,
r=1mbrα0rq cos kα0r=0,
δφi=12r=1mar2+br2δI/s1,
δφp=δα2r=1mar2+br2k=1 ksks1,
ar=2j+2cos2πj+2r-j+32, br=2j+2sin2πj+2r-j+32×for r=1, 2, , j+2,
r=1j+2ar2+br2=2j+2.
r=1mar2+br2min=2m,
ar=2mcos α0r, br=2msin α0r.
φ=arctan1/2I1-Ij+3cot2π/j+2+r=1j+1 Ir+1 sin2πr/j+2-I1+Ij+3/2-r=1j+1 Ir+1 cos2πr/j+2,
b1=-bj+3=1j+2cot2πj+2, br+1=2j+2sin2πrj+2 for r=1, 2, , j+1, a1=aj+3=-1j+2, ar+1=-2j+2cos2πrj+2 for r=1, 2, , j+1.
r=1j+3ar2+br2=2j+2j+32+12cot22πj+2.
φ=arctan14I1+I2-Ij+3-Ij+4sin3π/j+2sin22π/j+2+r=2j+3 Ir sin2πj+2r-j+5214I1-I2-Ij+3+Ij+4cos3π/j+2sin22π/j+2+r=2j+3 Ir cos2πj+2r-j+52,
r=1j+4ar2+br2=2j+2×1+14j+2sin22π/j+21sin22π/j+2+4 cos4πj+2.
ar=2rj+22cos2πrj+2for 1rj+2,=22j+4-rj+22cos2πrj+2for j+3r2j+3,br=2rj+22sin2πrj+2for 1rj+2,=-22j+4-rj+22sin2πrj+2for j+3r2j+3,
a2+b2=2j+223+13j+22.
L=r=1mar2+br2+k=1jν1kar sin kα0r+k=0jν2kar cos kα0r-δk, 1+k=1jν3kbr sin kα0r-δk, 1+k=0jν4kbr cos kα0r+q=1pμ1qα0rqar cos α0r-br sin α0r+q=1pμ2qα0rqar sin α0r+br cos α0r,
Δφ=δarctanr=1mbrIrr=1marIr=r=1mδIr1+j=1m bjIj/j=1m ajIj2Irr=1m brIrr=1m arIr,-r=1mδIrs1br cos φ1-ar sin φ1=-r=1mδIrs1ar2+br2 cosφ1+βr,
δφi2=Δφ2=1s12r=1mδIr2ar2+br2cos2φ1+βr,
δφi=12δIs1r=1mar2+br2,
δIr=-k=1 kskδαr sinkαr-φk.
δφp2=r=1mδαr2ar2+br2×k=1ksks1sinkαr-φk2 cos2φ1+βr.
δφp=δα2r=1mar2+br2k=1 ksks1.
L=r=1mar2+br2+k=0jμ1kar sin kα0r+μ2kar cos kα0r-δk, 1++μ3kbr sin kα0r-δk, 1+μ4kbr cos kα0r,
2ar+k=0jμ1k sin kα0r+μ2k cos kα0r=0 for r=1, , m,
2br+k=0jμ3k sin kα0r+μ4k cos kα0r=0 for r=1, , m.
μ1k=0,
r=1msin kα0r cos kα0r=0 for any integer k.
μ2k=-2δk, 1/r=1mcos2 kα0r,
μ4k=0,
μ3k=-2δk, 1/r=1msin2 kα0r,
ar=cos α0r/j=1mcos2 α0j for r=1, , m,
br=sin α0r/j=1msin2 α0j for r=1, , m.
r=1mar2+br2=1r=1mcos2 αr0r=1msin2 αr02m,
r=1mcos2 αr0r=1msin2 αr012r=1mcos2 αr0+sin2 αr0.
ar=2mcos α0r for r=1, , m,
br=2msin α0r forr=1, , m.

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