Abstract

A method to reduce the sensitivity of phase-shifting interferometry to external vibrations is described. The returning interferogram is amplitude split to form two series of interferograms, taken simultaneously and with complementary properties, one with high temporal and low spatial resolution and the other with low temporal and high spatial resolution. The high-temporal-resolution data set is used to calculate the true phase increment between interferograms in the high-spatial-resolution data set, and a generalized phase-extraction algorithm then includes these phase increments when the topographical phases in the high-spatial-resolution data set are calculated. The measured topography thereby benefits from the best qualities of both data sets, providing increased vibration immunity without sacrificing high spatial resolution.

© 1996 Optical Society of America

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References

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  1. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing,” D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.
  2. R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).
  3. P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29, 3271–3279 (1990).
    [CrossRef] [PubMed]
  4. C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Tech. 5, 648–652 (1994).
    [CrossRef]
  5. I. Kong, S. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
    [CrossRef]
  6. G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991).
    [CrossRef]
  7. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  8. J. Schwider, R. Burow, K.-E. Elssner, J. Gizanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  9. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  10. 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]
  11. K. Kinnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef] [PubMed]
  12. C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
    [CrossRef] [PubMed]
  13. P. de Groot, L. Deck, “Numerical simulations of vibration in phase shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
    [CrossRef] [PubMed]

1996 (1)

1995 (3)

1994 (2)

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Tech. 5, 648–652 (1994).
[CrossRef]

C. Joenathan, “Phase-measuring interferometry: new methods and error analysis,” Appl. Opt. 33, 4147–4155 (1994).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

1988 (1)

1984 (2)

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1983 (1)

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing,” D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Burow, R.

de Groot, P.

Deck, L.

Elssner, K.-E.

Farrell, C. T.

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Tech. 5, 648–652 (1994).
[CrossRef]

Gizanna, J.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing,” D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Joenathan, C.

Kim, S.

I. Kong, S. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
[CrossRef]

Kinnstaetter, K.

Kong, I.

I. Kong, S. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
[CrossRef]

Lai, G.

Lohmann, A. W.

Merkel, K.

Moore, R.

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

Player, M. A.

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Tech. 5, 648–652 (1994).
[CrossRef]

Schwider, J.

Smythe, R.

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

Spolaczyk, R.

Streibl, N.

Wizinowich, P. L.

Yatagai, T.

Appl. Opt. (6)

J. Opt. Soc. Am. A (2)

Meas. Sci. Tech. (1)

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Tech. 5, 648–652 (1994).
[CrossRef]

Opt. Eng. (3)

I. Kong, S. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).
[CrossRef]

R. Smythe, R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984).

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Other (1)

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing,” D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

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Figures (14)

Fig. 1
Fig. 1

Diagram of the apparatus necessary to implement the two-camera PSI method for large-aperture, Fizeau-type interferometers.

Fig. 2
Fig. 2

Sampling sequence in which the slow data-set size is equal to five frames, and the fast:slow camera ratio is equal to five. The clear boxes correspond to the frames acquired by the fast camera; the shaded boxes correspond to the frames acquired by the slow camera. The box widths correspond to the frame integration time. Twenty-five fast camera frames are required for this sequence.

Fig. 3
Fig. 3

Calculated P–V phase error for two different phase-extraction algorithms as a function of the phase interval. Identical curves are obtained with 2CPSI, depending on which algorithm is used for fast phase extraction. A phase offset of 0.01 rad has been added to both curves for clarity.

Fig. 4
Fig. 4

Vibration-sensitivity spectra at small vibration amplitudes for PSI with data acquired at the slow camera rate (Std. PSI), PSI with data acquired at the fast camera rate (Fast PSI), and two-camera PSI for a slow camera data-set size of five frames and a 5:1 fast:slow camera rate ratio (2CPSI 5,5). The phase error is normalized to the vibration amplitude, and the frequency is normalized to the slow camera rate. Also indicated are the boundaries of the low-, medium-, and high-frequency regimes discussed in the text.

Fig. 5
Fig. 5

Close-up of the vibration sensitivity for 2CPSI 5,5, standard PSI, and Fast PSI in the low-frequency regime.

Fig. 6
Fig. 6

Vibration-sensitivity spectra for different fast:slow camera speed ratios in 2CPSI for small amplitude vibrations. A PSI analysis of data acquired at a speed 5 times faster than standard (Fast PSI) is shown for comparison.

Fig. 7
Fig. 7

Vibration-sensitivity spectra for different slow data-set sizes in 2CPSI.

Fig. 8
Fig. 8

Vibration-sensitivity spectra for chirped 2CPSI 5,5 compared with 2CPSI 5,5 and standard PSI for small-amplitude vibrations.

Fig. 9
Fig. 9

Diagram of the experimental apparatus used to test the 2CPSI method.

Fig. 10
Fig. 10

Vibration-sensitivity spectra comparisons between simulation (curves) and experiment (points) for 2CPSI, Std. PSI, and Fast PSI.

Fig. 11
Fig. 11

Vibration-sensitivity spectrum comparison between simulation and experiment for 2CPSI with an SDS size of 11 frames and a 5:1 fast:slow frame rate ratio.

Fig. 12
Fig. 12

Vibration-sensitivity spectrum comparison between simulation and experiment for 2CPSI with an SDS size of five frames and a 5:1 fast:slow frame rate ratio but with the SDS frame separation set to nine frames.

Fig. 13
Fig. 13

Measurement of an 86-nm step-height standard in the presence of vibration with a 70-nm amplitude and a normalized frequency of 0.4 when 2CPSI 5,5, Std. PSI, and Fast PSI are used. The performance of 2CPSI 5,5 was equivalent to a fast PSI. For clarity, +20 nm was added to the Fast PSI profile, and 20 nm was subtracted from the 2CPSI 5,5 profile.

Fig. 14
Fig. 14

Measurement of an 86-nm step-height standard in the presence of vibration with a 70-nm amplitude and normalized frequency of 0.4 when 2CPSI 5,5, Std. PSI, and Fast PSI are used. The performance of 2CPSI 5,5 was equivalent to Fast PSI. For clarity, +20 nm was added to the Fast PSI profile, and 20 nm was subtracted from the 2CPSI profile.

Equations (13)

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I [ x , y , δ ( t ) ] = I ( x , y ) + V ( x , y ) cos [ ϕ ( x , y ) δ ( t ) ] ,
I I ( x , y , δ ) = I ( x , y ) + V ( x , y ) cos [ ϕ ( x , y ) δ i ] .
I j ( x , y , Δ j ) = I S ( j ) ( x , y , δ ) ,
S ( j ) = 2 + 5 j ,
Φ i ( x , y ) = PSI [ I ( x , y , δ ) , i ] ,
PSI [ I ( x , y , δ ) , i ] = tan 1 { 2 [ I i 1 ( x , y , δ ) I i + 1 ( x , y , δ ) ] 2 I i ( x , y , δ ) I i 2 ( x , y , δ ) I i + 2 ( x , y , δ ) }
Φ i = Φ i + 2 π round [ Φ i ( x , y ) Φ i 1 ( x , y ) 2 π ] for i = 1 , , N F ,
Δ j = Φ S ( j ) Φ S ( 0 ) for j = 0 , , N S 1 .
| N S cos ( Δ j ) sin ( Δ j ) cos ( Δ j ) cos 2 ( Δ j ) cos ( Δ j ) sin ( Δ j ) sin ( Δ j ) cos ( Δ j ) sin ( Δ j ) sin 2 ( Δ j ) | | a 0 ( x , y ) a 1 ( x , y ) a 2 ( x , y ) | = | I j ( x , y ) I j ( x , y ) cos ( Δ j ) I j ( x , y ) sin ( Δ j ) | ,
ϕ ( x , y ) = tan 1 [ a 2 ( x , y ) a 1 ( x , y ) ] .
N F = A + j = 1 N S 1 δ j δ j 1 δ = A + j = 0 N S 1 M j ,
I m j = t j t j + P I 0 { 1 + C cos [ 4 ν t π λ + θ m + F sin ( 2 πω t + φ ) ] } d t ,
E rms = [ 1 M 1 N m = 1 M n = 1 N ( θ m 2 C θ m T ) 2 ] 1 / 2 ,

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