Abstract

We present an experimental study of three calibration methods for spherical reference surfaces in Fizeau interferometry. The ball average, which relies on averaging measurements of a test ball surface over sufficiently many random rotations of the ball, is theoretically an absolute technique but can be very laborious. On the other hand, a recently introduced double-pass technique, comparing the two halves of the reference surface, is able to determine the point-symmetric contribution in as few as three measurements but does not detect the point-antisymmetric portion. Finally, the point-symmetric errors of the reference surface can be captured in a single, so-called “cat’s-eye” measurement. Our study tries to answer the question of which of the techniques is preferable in practice. We find that the answer depends on the required uncertainty, and it appears that the new double-pass technique represents a practical and reasonable trade-off between expediency and accuracy.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2009 (2)

2008 (3)

2006 (1)

N. Gardner and A. Davies, “Self-calibration for microrefractive lens measurements,” Opt. Eng. 45, 033603 (2006).
[CrossRef]

2005 (1)

U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE 5869, 58690S (2005).
[CrossRef]

2002 (1)

M. Sjödahl and B. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
[CrossRef]

1992 (1)

1990 (2)

B. Truax, “Absolute interferometric testing of spherical surfaces,” Proc. SPIE 1400, 61–68 (1990).

K. Creath and J. Wyant, “Absolute measurement of surface roughness,” Appl. Opt. 29, 3823–3827 (1990).
[CrossRef] [PubMed]

1989 (1)

Bramble, A.

Bruning, J.

H. Schreiber and J. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

Burge, J.

P. Zhou and J. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Burke, J.

Burow, R.

Carakos, R.

U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE 5869, 58690S (2005).
[CrossRef]

Creath, K.

Davies, A.

N. Gardner and A. Davies, “Self-calibration for microrefractive lens measurements,” Opt. Eng. 45, 033603 (2006).
[CrossRef]

Elssner, K. E.

Evans, C.

R. Parks, C. Evans, and L. Shao, Calibration of Interferometer Transmission Spheres, Vol.  12 of OSA Technical Digest Series (Optical Society of America, 1998), pp. 80–83.

Gardner, N.

N. Gardner and A. Davies, “Self-calibration for microrefractive lens measurements,” Opt. Eng. 45, 033603 (2006).
[CrossRef]

Green, K.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, “Fabrication and testing of a high-precision concave spherical mirror,” Proc. SPIE 7064, 70640E (2008).
[CrossRef]

Griesmann, U.

U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE 5869, 58690S (2005).
[CrossRef]

Grzanna, J.

Leistner, A.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, “Fabrication and testing of a high-precision concave spherical mirror,” Proc. SPIE 7064, 70640E (2008).
[CrossRef]

Lindlein, N.

Mantel, K.

Oreb, B.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, “Fabrication and testing of a high-precision concave spherical mirror,” Proc. SPIE 7064, 70640E (2008).
[CrossRef]

M. Sjödahl and B. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
[CrossRef]

Parks, R.

R. Parks, C. Evans, and L. Shao, Calibration of Interferometer Transmission Spheres, Vol.  12 of OSA Technical Digest Series (Optical Society of America, 1998), pp. 80–83.

Puhanic, E.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, “Fabrication and testing of a high-precision concave spherical mirror,” Proc. SPIE 7064, 70640E (2008).
[CrossRef]

Schreiber, H.

H. Schreiber and J. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

Schreiner, R.

Schulz, G.

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E.Wolf, ed. (North Holland, 1976), Vol.  13, pp. 128–129.
[CrossRef]

Schwider, J.

R. Schreiner, J. Schwider, N. Lindlein, and K. Mantel, “Absolute testing of the reference surface of a Fizeau interferometer through even/odd decompositions,” Appl. Opt. 47, 6134–6141(2008).
[CrossRef] [PubMed]

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E.Wolf, ed. (North Holland, 1976), Vol.  13, pp. 128–129.
[CrossRef]

Shao, L.

R. Parks, C. Evans, and L. Shao, Calibration of Interferometer Transmission Spheres, Vol.  12 of OSA Technical Digest Series (Optical Society of America, 1998), pp. 80–83.

Sjödahl, M.

M. Sjödahl and B. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
[CrossRef]

Soons, J.

U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE 5869, 58690S (2005).
[CrossRef]

Spolaczyk, R.

Stuart, W.

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, “Fabrication and testing of a high-precision concave spherical mirror,” Proc. SPIE 7064, 70640E (2008).
[CrossRef]

Truax, B.

B. Truax, “Absolute interferometric testing of spherical surfaces,” Proc. SPIE 1400, 61–68 (1990).

Wang, K.

Wang, Q.

U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE 5869, 58690S (2005).
[CrossRef]

Wyant, J.

Zhou, P.

P. Zhou and J. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Appl. Opt. (4)

Opt. Eng. (2)

N. Gardner and A. Davies, “Self-calibration for microrefractive lens measurements,” Opt. Eng. 45, 033603 (2006).
[CrossRef]

M. Sjödahl and B. Oreb, “Stitching interferometric measurement data for inspection of large optical components,” Opt. Eng. 41, 403–408 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (4)

B. Truax, “Absolute interferometric testing of spherical surfaces,” Proc. SPIE 1400, 61–68 (1990).

U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE 5869, 58690S (2005).
[CrossRef]

J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, “Fabrication and testing of a high-precision concave spherical mirror,” Proc. SPIE 7064, 70640E (2008).
[CrossRef]

P. Zhou and J. Burge, “Limits for interferometer calibration using the random ball test,” Proc. SPIE 7426, 74260U (2009).
[CrossRef]

Other (5)

Optical Perspectives LLC, http://www.optiper.com.

H. Schreiber and J. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 2007), pp. 547–666.
[CrossRef]

A. Jensen, “Absolute calibration method for laser Twyman-Green wave-front testing interferometers,” J. Opt. Soc. Am.63, 1313 (1973) (abstract only).JOSAAH0030-3941
[PubMed]

G. Schulz and J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E.Wolf, ed. (North Holland, 1976), Vol.  13, pp. 128–129.
[CrossRef]

R. Parks, C. Evans, and L. Shao, Calibration of Interferometer Transmission Spheres, Vol.  12 of OSA Technical Digest Series (Optical Society of America, 1998), pp. 80–83.

Supplementary Material (1)

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Figures (14)

Fig. 1
Fig. 1

Jensen test. The Fizeau interferometer with transmission sphere (TS; internal lenses omitted for clarity) is on the left; the reflective surface under test (RS) is on the right. A typical fringe pattern from W 1 ( r , θ ) is also shown; typically, the odd errors of the transmission sphere overwhelm the even errors in the reference surface by at least an order of magnitude.

Fig. 2
Fig. 2

Double-pass test. The Fizeau interferometer with transmission sphere (TS; internal lenses omitted for clarity) is on the left; a flat mirror (M) is used to reflect the beam at the focus, and a beam stop (BS) is inserted behind the transmission sphere assembly.

Fig. 3
Fig. 3

Ball-averaging test. The Fizeau interferometer with transmission sphere (TS; internal lenses omitted for clarity) is on the left; a calibration sphere (CS) is used with averaging over random rotations to approximate a perfect spherical surface.

Fig. 4
Fig. 4

Three surface segments that can be stitched together into a calibration map. Data are from the first transmission sphere (TS1). Scale is 7.5 to + 7.5 nm .

Fig. 5
Fig. 5

Stitching results from different numbers of segments as indicated. Scale is 7.5 to + 7.5 nm .

Fig. 6
Fig. 6

Double-logarithmic plot of rms calibration uncertainty as a function of different N for random BAs. The theoretical slope (plotted without data points) is N 0.5 .

Fig. 7
Fig. 7

Snapshot from an animation showing how a 100-measurement BA is accumulated (Media 1); the frame shows the calibration after 25 of 100 measurements, achieving about half of the total uncertainty reduction.

Fig. 8
Fig. 8

Estimate of W R and its odd and even parts. The scale is 7.5 to + 7.5 nm .

Fig. 9
Fig. 9

Calibration results from second transmission sphere, with N = 100 for the BA, and examples for N = 5 and N = 12 for the DP method. Scale is 7.5 to + 7.5 nm .

Fig. 10
Fig. 10

Calibration results from CE test: top row, Instrument 1; bottom row, Instrument 2; left, TS 1; right, TS2. Scale is 7.5 to + 7.5 nm .

Fig. 11
Fig. 11

Calibration results from DP test: top row, Instrument 1; bottom row, Instrument 2; left, TS 1; right, TS2. Scale is 7.5 to + 7.5 nm .

Fig. 12
Fig. 12

Calibration results from BA test: top row, Instrument 1; bottom row, Instrument 2; left, TS 1; right, TS2. Scale is 7.5 to + 7.5 nm .

Fig. 13
Fig. 13

Top row: Difference between CE and BA methods for Instrument 2, TS1. Bottom row: Difference between CE and BA methods for Instrument 2, TS2. Left column: comparing only even parts. Right column: using complete BA file (including odd parts). Scale is 10 to + 10 nm .

Fig. 14
Fig. 14

Top row: Difference between DP and BA methods for Instrument 2, TS1. Bottom row: Difference between DP and BA methods for Instrument 2, TS2. Left column: comparing only even parts. Right column: using complete BA file (including odd parts). Scale is 10 to + 10 nm .

Tables (4)

Tables Icon

Table 1 Astigmatism for Even Parts of Double-Pass Calibration Files for all N a

Tables Icon

Table 2 Astigmatism for Even Parts of Ball-Average Calibration Files a

Tables Icon

Table 3 Astigmatism for Even Parts of Calibration Files for the Second Transmission Sphere for Various N a

Tables Icon

Table 4 Differences Between Cat’s-Eye/Double-Pass Calibrations and Ball-Average Calibration a

Equations (12)

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W ( r , θ ) = W ( r , θ + 180 ° ) , W ( r , θ ) + W ( r , θ + 180 ° ) = 2 W e ( r , θ ) ,
W ( r , θ ) = W ( r , θ + 180 ° ) , W ( r , θ ) W ( r , θ + 180 ° ) = 2 W o ( r , θ ) ,
W 1 ( r , θ ) = T R ( r , θ ) + W R ( r , θ ) + W R ( r , θ + 180 ° ) T R ( r , θ + 180 ° ) , W 2 ( r , θ ) = 2 W R ( r , θ ) + 2 W T ( r , θ ) , W 3 ( r , θ ) = 2 W R ( r , θ ) + 2 W T ( r , θ + 180 ° ) ,
W 1 ( r , θ ) = W 1 ( r , θ ) + W 1 ( r , θ + 180 ° ) = T R ( r , θ ) + W R ( r , θ ) + W R ( r , θ + 180 ° ) T R ( r , θ + 180 ° ) + T R ( r , θ + 180 ° ) + W R ( r , θ + 180 ° ) + W R ( r , θ ) T R ( r , θ ) = 4 W R , e ( r , θ ) ,
W 2 , 3 ( r , θ ) = W 2 ( r , θ ) + W 3 ( r , θ + 180 ° ) = 2 W R ( r , θ ) + 2 W T ( r , θ ) + 2 W R ( r , θ + 180 ° ) + 2 W T ( r , θ ) = 4 W R , e ( r , θ ) + 4 W T ( r , θ ) ,
W T ( r , θ ) = 1 4 ( W 2 , 3 ( r , θ ) W 1 ( r , θ ) ) .
W 2 , 1 ( r , θ ) = W 2 ( r , θ ) W 1 ( r , θ ) = 2 W R ( r , θ ) + 2 W T ( r , θ ) T R ( r , θ ) W R ( r , θ ) W R ( r , θ + 180 ° ) + T R ( r , θ + 180 ° ) = 2 W R , o ( r , θ ) + 2 W T ( r , θ ) + 2 T R , o ( r , θ + 180 ° ) ,
W T , e ( r , θ ) = 1 4 ( W 2 , 1 ( r , θ ) + W 2 , 1 ( r , θ + 180 ° ) ) = 1 4 ( 2 W T ( r , θ ) + 2 W T ( r , θ + 180 ° ) ) ,
W DP ( r , θ ) = 2 W R ( r , θ ) + 2 W R ( r , θ + 180 ° ) = 4 W R , e ( r , θ ) ,
W BA ( r , θ ) = 2 W R ( r , θ ) + 2 N n = 1 N W B , n ( r , θ ) ,
σ W = σ R 2 + σ B 2 ,
σ BA = σ R 2 + σ B 2 N .

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