Abstract

The effect of the quantization of fringe intensity on the phase error in phase-shifting measurements is formulated by a characteristic polynomial method. A numerical simulation is performed, and its result is in good agreement with the analytical one. Several factors influencing the quantization effects are investigated.

© 1997 Optical Society of America

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References

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  1. J. E. Greivenkampm, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1991), Chap. 14, pp. 501–598.
  2. R. S. Sirohi, M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, M. P. Kothiyal, eds. (Marcel Dekker, New York, 1991), pp. 219–246.
  3. D. Shough, “Beyond fringe analysis,” in Interferometry VI, Proc. SPIE2003, 208–223 (1993).
    [CrossRef]
  4. K. Creath, “Phase-measurement interferometry: Beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 213–220 (1992).
    [CrossRef]
  5. J. V. Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shiftinginterferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  6. C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).
  7. C. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  8. P. Carre, “Installation et Utilisation du Comparateur Photoelctrique et Interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  9. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculating algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  10. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  11. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  12. 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]
  13. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift error,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  14. 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]
  15. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  16. 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]
  17. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  18. Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36, 271–276 (1997).
    [CrossRef] [PubMed]

1997

1996

1995

1993

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]

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

1992

1991

1990

1987

1966

P. Carre, “Installation et Utilisation du Comparateur Photoelctrique et Interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Brophy, C.

Bruning, J. H.

J. E. Greivenkampm, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1991), Chap. 14, pp. 501–598.

Carre, P.

P. Carre, “Installation et Utilisation du Comparateur Photoelctrique et Interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

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 (1992).
[CrossRef]

de Groot, P.

Eiju, T.

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.

Greivenkampm, J. E.

J. E. Greivenkampm, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1991), Chap. 14, pp. 501–598.

Hariharan, P.

Hibino, K.

Koliopoulos, C. L.

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

Kothiyal, M. P.

R. S. Sirohi, M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, M. P. Kothiyal, eds. (Marcel Dekker, New York, 1991), pp. 219–246.

Larkin, K. G.

Oreb, B. F.

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]

Shough, D.

D. Shough, “Beyond fringe analysis,” in Interferometry VI, Proc. SPIE2003, 208–223 (1993).
[CrossRef]

Sirohi, R. S.

R. S. Sirohi, M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, M. P. Kothiyal, eds. (Marcel Dekker, New York, 1991), pp. 219–246.

Smorenburg, C.

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.

Wingerden, J. V.

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]

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.

J. Opt. Soc. Am. A

Metrologia

P. Carre, “Installation et Utilisation du Comparateur Photoelctrique et Interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng.

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]

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]

Other

J. E. Greivenkampm, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1991), Chap. 14, pp. 501–598.

R. S. Sirohi, M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, M. P. Kothiyal, eds. (Marcel Dekker, New York, 1991), pp. 219–246.

D. Shough, “Beyond fringe analysis,” in Interferometry VI, Proc. SPIE2003, 208–223 (1993).
[CrossRef]

K. Creath, “Phase-measurement interferometry: Beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 213–220 (1992).
[CrossRef]

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

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Figures (4)

Fig. 1
Fig. 1

Quantization system and the relationship between the quantization signal qn and the phase θ.

Fig. 2
Fig. 2

Presentation of the procedure of the numeric simulation.

Fig. 3
Fig. 3

Curve of the phase error versus the fractional part of the local intensity for a PSM measurement with 5-bit quantization and four fringe patterns.

Fig. 4
Fig. 4

Curve of phase error versus N, where N = 2π/δ, where δ is the phase-shift parameter. R is the quantization levels.

Tables (2)

Tables Icon

Table 1 Quantization Errors in Digital Phase Detection for Some Frequently Used Algorithms given in Refs. 9-18a

Tables Icon

Table 2 Comparison of Our Analytic Result with Brophy’s Resulta

Equations (28)

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IX, Y=IAX, Y+IBX, YcosφX, Y
IkX, Y=IAX, Y+IBX, YcosφX, Y+δk,
δk=kδ, k=0, 1,, M-1,
Ik=IA+IB cosφ+δk.
Qk=INTIk,
ΔIk=Ik-Qk.
Qk=g0+m=1 gm cosmθk+m=1hm sinmθk,
Qk=g0+m=1 gm cosmθk.
gm=1π02π Q cosmθdθ m1, g0=1π0π Qdθ.
αn=0n=0,arccosqn-0.5-IAIB0<αn<π, 1nP-1,πn=P,
P=INTIA+IB-INTIA-IB+1, 0PR-1.
Q=q1=INTIA+IBα0<θ<α1qn=Q1-n+1αn-1<θ<αn.
gm=2πn=0P-1qn+1αnαn+1cosmθdθ=2πmn=1P-1sinmαn m1, g0=qP+1πn=1P-1 αn.
φ*=φ+k=0M-1φ/IkΔIk+OΔI2,
tan φ=k=0M-1bkIk/k=0M-1akIk.
Sφ=k=0M-1ckIk=12IBPζexpiφ,
Px=k=0M-1ckxk.
S*φ*=k=0M-1ckQk=12m=1 gmPζmexpimφ+Pζ-mexp-imφ.
S*φ*=S*expiφ*Sexpiφ1+iΔφ.
Δφ2IBPζm=2k=0M-1gmbk cos φ-ak sin φ×cosmφ+mδk.
σΔφ22IB2Pζ2k=0M-1l=0M-1m=2akal+bkblgm2 costδkl-bkbl-akalcosδk+δl+akbl+bkal×sinδk+δlgmgm+2 cosm+1δkl,
S*φ*=12t=0gtN+1PζtN+1expitN+1φ+gtN-1Pζ-tN+1exp-itN-1φ=12Pζexpiφt=0gtN+1 expitNφ+gtN-1 exp-itNφ.
Δφ1IBt=1gtN+1-gtN-1sintNφ,
σΔφ212IB2t=1gtN+1-gtN-12.
S*φ*=12g1Pζexpiφ+12rPζr×t=1gtN+r expitN+rφ+gtN-r exp-itN-rφ.
σΔφ212IB2rPζrPζt=1gtN+r-gtN-r2,
σΔφ22R-22t=1gtN+1-gtN-12 N=2π/δ3,
gm=-2πmn=1R-1sinm arccosR-2n-1R-21nR-1, m2,

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