Abstract

A new (N + 1)-bucket algorithm is proposed for phase-stepping systems. It eliminates most of the errors caused by the phase-shifter miscalibration.

© 1993 Optical Society of America

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References

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  1. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
    [CrossRef]
  2. K. Creath, “Phase measurement interferometry techniques,” Prog. Opt. 26, 350–393 (1988).
  3. H. P. Stahl, “Review of phase-measuring interferometry,” in Optical Testing and Metrology HI: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 704–719 (1991).
  4. K. A. Stetson, “Optical heterodyning,” in Handbook on Experimental Mechanics, A. S. Kobayashi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 501–515.
  5. H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253–282 (1989).
    [CrossRef]
  6. 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]
  7. J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1987), p. 414.
  8. K. Creath, “Phase measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 213–220 (1992).
  9. 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]
  10. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. 7, 537–541 (1990).
    [CrossRef]

1990 (1)

1989 (1)

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253–282 (1989).
[CrossRef]

1988 (1)

K. Creath, “Phase measurement interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

1987 (1)

1983 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Brophy, C. P.

Bruning, J. H.

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1987), p. 414.

Burow, R.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

K. Creath, “Phase measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 213–220 (1992).

Eiju, T.

Elssner, K.-E.

Grzanna, J.

Hariharan, P.

Merkel, K.

Oreb, B. F.

Schwider, J.

Spolaczyk, R.

Stahl, H. P.

H. P. Stahl, “Review of phase-measuring interferometry,” in Optical Testing and Metrology HI: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 704–719 (1991).

Stetson, K. A.

K. A. Stetson, “Optical heterodyning,” in Handbook on Experimental Mechanics, A. S. Kobayashi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 501–515.

Tiziani, H. J.

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253–282 (1989).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[CrossRef]

Opt. Quantum Electron. (1)

H. J. Tiziani, “Optical methods for precision measurements,” Opt. Quantum Electron. 21, 253–282 (1989).
[CrossRef]

Prog. Opt. (1)

K. Creath, “Phase measurement interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

Other (4)

H. P. Stahl, “Review of phase-measuring interferometry,” in Optical Testing and Metrology HI: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 704–719 (1991).

K. A. Stetson, “Optical heterodyning,” in Handbook on Experimental Mechanics, A. S. Kobayashi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987), pp. 501–515.

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1987), p. 414.

K. Creath, “Phase measurement interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 213–220 (1992).

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Figures (2)

Fig. 1
Fig. 1

Plot of 1/[2N sin(2π/N)] versus N. It is the amplitude of the phase error proportional to ε for the N-bucket algorithm for ε = 1 in units of 2π rad [Eq. (9)]. As an example, ε = 0.1 and N = 4 yield an amplitude of 0.125 × 0.1 × 2π rad. The asymptotic value is 1/4π ≈ 0.0796.

Fig. 2
Fig. 2

Plot of π/[2N2 sin2(2π/N)] versus N. It is the amplitude of the phase error proportional to ε2 for the (N + 1)-bucket algorithm for ε = 1 in units of 2π rad [approximation (14)]. As an example, ε = 0.1 and N = 4 yield an amplitude of 0.1 × 0.01 × 2π rad. The asymptotic value is 1/8π ≈ 0.0398.

Equations (16)

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α = α ( 1 + ɛ ) ,
tan ( Φ ) = n = 0 N 1 I ( n ) sin ( 2 π n N ) n = 0 N 1 I ( n ) cos ( 2 π n N ) ,
tan ( Φ ) = I ( 1 ) I ( 3 ) I ( 0 ) I ( 2 ) .
tan ( Φ ) = I ( 1 ) I ( 3 ) [ I ( 0 ) + I ( 4 ) ] / 2 I ( 2 ) .
= n = 0 N 1 I ( n ) ζ n
I ( n ) = cos [ Φ + 2 π ( 1 + ɛ ) n N ] ,
= | | exp ( i Φ m ) = 1 2 n = 0 N 1 [ exp ( i Φ ) ζ ( 1 + ɛ ) n + exp ( i Φ ) ζ ( 1 + ɛ ) n ] ζ n = 1 2 exp ( i Φ ) n = 0 N 1 [ ζ ɛ n + exp ( 2 i Φ ) ζ ( 2 + ɛ ) n ] ,
2 | | exp ( i Δ Φ ) 2 | | ( 1 + i Δ Φ ) = n = 0 N 1 [ ζ ɛ n + exp ( 2 i Φ ) ζ ( 2 + ɛ ) n ] .
2 | | exp ( i Δ Φ ) 2 | | ( 1 + i Δ Φ ) N + i π ( N 1 ) ɛ + π ɛ ζ exp ( 2 i Φ ) sin ( 2 π N ) .
Δ Φ = N 1 N π ɛ π ɛ N sin ( 2 π N ) sin ( 2 Φ 2 π N ) .
I ( N ) I ( 0 ) = cos ( Φ + 2 π ɛ ) cos ( Φ ) = ½ { exp ( i Φ ) [ exp ( i 2 π ɛ ) 1 ] + exp ( i Φ ) [ exp ( i 2 π ɛ ) 1 ] } exp ( i Φ ) i π ɛ [ 1 exp ( 2 i Φ ) ] .
π ɛ ζ exp ( 2 i Φ ) sin ( 2 π N ) 2 A i π ɛ exp ( 2 i Φ ) ,
A = i ζ 2 sin ( 2 π N ) = 1 2 [ 1 i ctg ( 2 π N ) ] .
tan ( Φ ) = I ( 0 ) I ( N ) 2 ctg ( 2 π N ) n = 1 N 1 I ( n ) sin ( 2 π n N ) I ( 0 ) + I ( N ) 2 + n = 1 N 1 I ( n ) cos ( 2 π n N ) .
Δ Φ π ɛ + [ π ɛ N sin ( 2 π N ) ] 2 sin ( 2 Φ ) .
[ 4 sin ( 2 π / 4 ) 15 sin ( 2 π / 15 ) ] 2 = 0.43

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