Abstract

We present a new computational algorithm of phase-shifting interferometry that can effectively eliminate the uncertainty errors of reference phases encountered when we obtain multiple interferograms. The algorithm treats the reference phases as additional unknowns and we determine their exact values by analyzing interferograms using the numerical least-squares technique. A series of simulations prove that this algorithm can improve measuring accuracy because it is unaffected by the nonlinear and random errors of phase shifters.

© 1994 Optical Society of America

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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. J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).
  3. J. C. Wyant, K. Creath, “Recent advances in interferometric optical testing,” Laser Focus/Elect. Opt. 21, 118–132 (1985).
  4. C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (Optical Science Center, University of Arizona, Tucson, Ariz., 1981).
  5. Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  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. C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
    [CrossRef] [PubMed]
  8. P. Carré, “Installation et utilisation du comparateur photo-életrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  9. K. Kinnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef] [PubMed]
  10. J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]

1991 (1)

1988 (1)

1987 (1)

1985 (2)

Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

J. C. Wyant, K. Creath, “Recent advances in interferometric optical testing,” Laser Focus/Elect. Opt. 21, 118–132 (1985).

1983 (1)

1982 (1)

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).

1974 (1)

1966 (1)

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

Ai, C.

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Carré, P.

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

Cheng, Y. Y.

Creath, K.

J. C. Wyant, K. Creath, “Recent advances in interferometric optical testing,” Laser Focus/Elect. Opt. 21, 118–132 (1985).

Elssner, K. E.

Frankena, H. J.

Gallagher, J. E.

Grzanna, J.

Herriott, D. R.

Kinnstaetter, K.

Koliopoulos, C. L.

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

Lohmann, A. W.

Merkel, K.

Rosenfeld, D. P.

Schwider, J.

Smorenburg, C.

Spolaczyk, R.

Streibl, N.

van Wingerden, J.

White, A. D.

Wyant, J. C.

C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
[CrossRef] [PubMed]

Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

J. C. Wyant, K. Creath, “Recent advances in interferometric optical testing,” Laser Focus/Elect. Opt. 21, 118–132 (1985).

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).

Appl. Opt. (6)

Laser Focus (1)

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus 18, 65–71 (1982).

Laser Focus/Elect. Opt. (1)

J. C. Wyant, K. Creath, “Recent advances in interferometric optical testing,” Laser Focus/Elect. Opt. 21, 118–132 (1985).

Metrologia (1)

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

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Optical configuration for phase-shifting interferometry. B.S., beam splitter.

Fig. 2
Fig. 2

Ideal two-dimensional surface profile.

Fig. 3
Fig. 3

Convergence paths of reference phases (δ j k → −δ j ).

Fig. 4
Fig. 4

Convergence paths of reference phases (δ j k → − δ j ).

Fig. 5
Fig. 5

Exemplary hysteresis curve of the piezoelectric actuator.

Fig. 6
Fig. 6

Error of the reference phase that is due to piezoelectric nonlinearity.

Fig. 7
Fig. 7

Surface profile errors that are due to reference phase errors.

Tables (1)

Tables Icon

Table 1 Simulation Parameters

Equations (22)

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I ( x , y ) = A 2 ( x , y ) + B 2 ( x , y ) + 2 A ( x , y ) B ( x , y ) cos ϕ ( x , y ) ,
I j ( x , y ) = D ( x , y ) { 1 + γ ( x , y ) sinc ( Δ / 2 ) cos [ ϕ ( x , y ) + δ j ] } = D ( x , y ) { 1 + V ( x , y ) cos [ ϕ ( x , y ) + δ j ] } .
I j = D + D V cos ϕ cos δ j - D V sin ϕ sin δ j .
ϕ = - tan - 1 [ ( j = 1 m I j sin δ j ) / ( j = 1 m I j cos δ j ) ] .
ϕ = tan - 1 2 ( I 4 - I 2 ) I 1 - 2 I 3 + I 5 .
I i j = D i + D i V i cos ( ϕ i + δ j ) ,
I i j = D i + D i V i cos ( ϕ i + δ j ) = D i + D i V i cos ϕ i cos δ j - D i V i sin ϕ i sin δ j = D i + C i cos δ j - S i sin δ j .
n m 3 n + m - 1 or n 1 + 2 m - 3 .
= i = 1 n j = 1 m ( I i j - I ^ i j ) 2 ,
= i = 1 n j = 1 m ( D i + C i cos δ j - S i sin δ j - I ^ i j ) 2 .
= i = 1 n i = j = 1 m j ,
i = j = 1 m ( D i + C i cos δ j - S i sin δ j - I ^ i j ) 2 ,
j = i = 1 m ( D i + C i cos δ j - S i sin δ j - I ^ i j ) 2 .
D i = C i = S i = 0 or i D i = i C i = i S i = 0 ,
δ j = j δ j = 0.
[ D i C i - S i ] = [ m Σ cos δ j Σ sin δ j Σ cos δ j Σ ( cos δ j ) 2 Σ cos δ j sin δ j Σ sin δ j Σ cos δ j sin δ j Σ ( sin δ j ) 2 ] - 1 × [ Σ I ^ i j Σ I ^ i j cos δ j Σ I ^ i j sin δ j ]
j = 1 m .
A 1 cos δ j + A 2 sin δ j + A 3 cos 2 δ j + A 4 sin 2 δ j = 0
A 1 = i = 1 n ( I ^ i j - D i ) S i ,
A 2 = i = 1 n ( I ^ i j - D i ) C i ,
A 3 = i = 1 n - C i S i ,
A 4 = 1 2 i = 1 n ( S i 2 - C i 2 ) .

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