Abstract

We describe the application of a vector-based radius approach to optical bench radius measurements in the presence of imperfect stage motions. In this approach, the radius is defined using a vector equation and homogeneous transformation matrix formulism. This is in contrast to the typical technique, where the displacement between the confocal and cat’s eye null positions alone is used to determine the test optic radius. An important aspect of the vector-based radius definition is the intrinsic correction for measurement biases, such as straightness errors in the stage motion and cosine misalignment between the stage and displacement gauge axis, which lead to an artificially small radius value if the traditional approach is employed. Measurement techniques and results are provided for the stage error motions, which are then combined with the setup geometry through the analysis to determine the radius of curvature for a spherical artifact. Comparisons are shown between the new vector-based radius calculation, traditional radius computation, and a low uncertainty mechanical measurement. Additionally, the measurement uncertainty for the vector-based approach is determined using Monte Carlo simulation and compared to experimental results.

© 2008 Optical Society of America

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  1. D. Malacara, Optical Shop Testing (Wiley, 1992).
  2. M. Murty and R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231-235 (1983).
  3. L. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31, 1961-1966 (1992).
    [CrossRef]
  4. T. Schmitz, C. J. Evans, and A. Davies, “An investigation of uncertainties limiting radius measurement performance,” in Proceedings of ASPE Spring Topical Meeting (American Society of Precision Engineering, 2000), p. 27.
  5. T. Schmitz, A. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE 4451, 432-447 (2001).
    [CrossRef]
  6. T. Schmitz, C. Evans, A. Davies, A., and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51 (1), 451-454 (2002).
    [CrossRef]
  7. U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” Ann. CIRP 53, 451-454 (2004).
    [CrossRef]
  8. A. Davies and T. Schmitz, “Defining the measurand in radius of curvature measurements,” in Proceedings of the SPIE 48th Annual International Symposium on Optical Science and Technology, J. Decker and N. Brown, eds. (SPIE, 2003).
  9. T. Schmitz, N. Gardner, M. Vaughn, and A. Davies, “Radius case study: optical bench measurement and uncertainty including stage error motions, in Proceedings of the SPIE 50th Annual International Symposium on Optics and Photonics, J. Decker and N. Brown, eds. (SPIE, 2005).
  10. A. Davies and T. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44, 5884-5893(2005).
    [CrossRef] [PubMed]
  11. J. Denavit and R. Hartenberg, “A kinematic notation for lower-pair mechanisms based on matrices,” J. Appl. Mech. 22, 215-221 (1955).
  12. R. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT Press, 1981).
  13. A. Slocum, Precision Machine Design (Prentice-Hall, 1992).
  14. P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (WCB/McGraw-Hill, 1992).
  15. C. Evans, R. Hocken, and W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Ann. CIRP 45, 617-634 (1996).
    [CrossRef]
  16. B. Taylor and C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Tech. Note 1297 (NIST, 1994).

2005 (1)

2004 (1)

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” Ann. CIRP 53, 451-454 (2004).
[CrossRef]

2002 (1)

T. Schmitz, C. Evans, A. Davies, A., and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51 (1), 451-454 (2002).
[CrossRef]

2001 (1)

T. Schmitz, A. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE 4451, 432-447 (2001).
[CrossRef]

1996 (1)

C. Evans, R. Hocken, and W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Ann. CIRP 45, 617-634 (1996).
[CrossRef]

1992 (1)

L. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31, 1961-1966 (1992).
[CrossRef]

1983 (1)

M. Murty and R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231-235 (1983).

1955 (1)

J. Denavit and R. Hartenberg, “A kinematic notation for lower-pair mechanisms based on matrices,” J. Appl. Mech. 22, 215-221 (1955).

A.,

T. Schmitz, C. Evans, A. Davies, A., and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51 (1), 451-454 (2002).
[CrossRef]

Bevington, P.

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (WCB/McGraw-Hill, 1992).

Davies, A.

A. Davies and T. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44, 5884-5893(2005).
[CrossRef] [PubMed]

T. Schmitz, C. Evans, A. Davies, A., and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51 (1), 451-454 (2002).
[CrossRef]

T. Schmitz, A. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE 4451, 432-447 (2001).
[CrossRef]

T. Schmitz, N. Gardner, M. Vaughn, and A. Davies, “Radius case study: optical bench measurement and uncertainty including stage error motions, in Proceedings of the SPIE 50th Annual International Symposium on Optics and Photonics, J. Decker and N. Brown, eds. (SPIE, 2005).

T. Schmitz, C. J. Evans, and A. Davies, “An investigation of uncertainties limiting radius measurement performance,” in Proceedings of ASPE Spring Topical Meeting (American Society of Precision Engineering, 2000), p. 27.

A. Davies and T. Schmitz, “Defining the measurand in radius of curvature measurements,” in Proceedings of the SPIE 48th Annual International Symposium on Optical Science and Technology, J. Decker and N. Brown, eds. (SPIE, 2003).

Denavit, J.

J. Denavit and R. Hartenberg, “A kinematic notation for lower-pair mechanisms based on matrices,” J. Appl. Mech. 22, 215-221 (1955).

Estler, W. T.

T. Schmitz, C. Evans, A. Davies, A., and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51 (1), 451-454 (2002).
[CrossRef]

C. Evans, R. Hocken, and W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Ann. CIRP 45, 617-634 (1996).
[CrossRef]

Evans, C.

T. Schmitz, C. Evans, A. Davies, A., and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51 (1), 451-454 (2002).
[CrossRef]

C. Evans, R. Hocken, and W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Ann. CIRP 45, 617-634 (1996).
[CrossRef]

Evans, C. J.

T. Schmitz, A. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE 4451, 432-447 (2001).
[CrossRef]

T. Schmitz, C. J. Evans, and A. Davies, “An investigation of uncertainties limiting radius measurement performance,” in Proceedings of ASPE Spring Topical Meeting (American Society of Precision Engineering, 2000), p. 27.

Gardner, N.

T. Schmitz, N. Gardner, M. Vaughn, and A. Davies, “Radius case study: optical bench measurement and uncertainty including stage error motions, in Proceedings of the SPIE 50th Annual International Symposium on Optics and Photonics, J. Decker and N. Brown, eds. (SPIE, 2005).

Griesmann, U.

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” Ann. CIRP 53, 451-454 (2004).
[CrossRef]

Hartenberg, R.

J. Denavit and R. Hartenberg, “A kinematic notation for lower-pair mechanisms based on matrices,” J. Appl. Mech. 22, 215-221 (1955).

Hocken, R.

C. Evans, R. Hocken, and W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Ann. CIRP 45, 617-634 (1996).
[CrossRef]

Kuyatt, C.

B. Taylor and C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Tech. Note 1297 (NIST, 1994).

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1992).

Murty, M.

M. Murty and R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231-235 (1983).

Paul, R.

R. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT Press, 1981).

Robinson, D.

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (WCB/McGraw-Hill, 1992).

Schmitz, T.

A. Davies and T. Schmitz, “Correcting for stage error motions in radius measurements,” Appl. Opt. 44, 5884-5893(2005).
[CrossRef] [PubMed]

T. Schmitz, C. Evans, A. Davies, A., and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51 (1), 451-454 (2002).
[CrossRef]

T. Schmitz, A. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE 4451, 432-447 (2001).
[CrossRef]

T. Schmitz, N. Gardner, M. Vaughn, and A. Davies, “Radius case study: optical bench measurement and uncertainty including stage error motions, in Proceedings of the SPIE 50th Annual International Symposium on Optics and Photonics, J. Decker and N. Brown, eds. (SPIE, 2005).

T. Schmitz, C. J. Evans, and A. Davies, “An investigation of uncertainties limiting radius measurement performance,” in Proceedings of ASPE Spring Topical Meeting (American Society of Precision Engineering, 2000), p. 27.

A. Davies and T. Schmitz, “Defining the measurand in radius of curvature measurements,” in Proceedings of the SPIE 48th Annual International Symposium on Optical Science and Technology, J. Decker and N. Brown, eds. (SPIE, 2003).

Selberg, L.

L. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31, 1961-1966 (1992).
[CrossRef]

Shukla, R.

M. Murty and R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231-235 (1983).

Slocum, A.

A. Slocum, Precision Machine Design (Prentice-Hall, 1992).

Soons, J.

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” Ann. CIRP 53, 451-454 (2004).
[CrossRef]

Taylor, B.

B. Taylor and C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Tech. Note 1297 (NIST, 1994).

Vaughn, M.

T. Schmitz, N. Gardner, M. Vaughn, and A. Davies, “Radius case study: optical bench measurement and uncertainty including stage error motions, in Proceedings of the SPIE 50th Annual International Symposium on Optics and Photonics, J. Decker and N. Brown, eds. (SPIE, 2005).

Wang, Q.

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” Ann. CIRP 53, 451-454 (2004).
[CrossRef]

Ann. CIRP (3)

T. Schmitz, C. Evans, A. Davies, A., and W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51 (1), 451-454 (2002).
[CrossRef]

U. Griesmann, J. Soons, and Q. Wang, “Measuring form and radius of spheres with interferometry,” Ann. CIRP 53, 451-454 (2004).
[CrossRef]

C. Evans, R. Hocken, and W. T. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Ann. CIRP 45, 617-634 (1996).
[CrossRef]

Appl. Opt. (1)

J. Appl. Mech. (1)

J. Denavit and R. Hartenberg, “A kinematic notation for lower-pair mechanisms based on matrices,” J. Appl. Mech. 22, 215-221 (1955).

Opt. Eng. (2)

M. Murty and R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231-235 (1983).

L. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31, 1961-1966 (1992).
[CrossRef]

Proc. SPIE (1)

T. Schmitz, A. Davies, and C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” Proc. SPIE 4451, 432-447 (2001).
[CrossRef]

Other (8)

D. Malacara, Optical Shop Testing (Wiley, 1992).

T. Schmitz, C. J. Evans, and A. Davies, “An investigation of uncertainties limiting radius measurement performance,” in Proceedings of ASPE Spring Topical Meeting (American Society of Precision Engineering, 2000), p. 27.

A. Davies and T. Schmitz, “Defining the measurand in radius of curvature measurements,” in Proceedings of the SPIE 48th Annual International Symposium on Optical Science and Technology, J. Decker and N. Brown, eds. (SPIE, 2003).

T. Schmitz, N. Gardner, M. Vaughn, and A. Davies, “Radius case study: optical bench measurement and uncertainty including stage error motions, in Proceedings of the SPIE 50th Annual International Symposium on Optics and Photonics, J. Decker and N. Brown, eds. (SPIE, 2005).

R. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT Press, 1981).

A. Slocum, Precision Machine Design (Prentice-Hall, 1992).

P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (WCB/McGraw-Hill, 1992).

B. Taylor and C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Tech. Note 1297 (NIST, 1994).

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Figures (18)

Fig. 1
Fig. 1

Confocal and cat’s eye interferometric null positions for optical bench radius measurements. In the absence of all errors, the displacement between these two positions gives the test optic radius of curvature over the clear aperture of the phase measuring interferometer.

Fig. 2
Fig. 2

In the presence of stage error motions, depicted simply as a y straightness error in the z motion here, a biased radius value is obtained from the linear transducer displacement.

Fig. 3
Fig. 3

Coordinate frames for vector radius definition. The reference frame is fixed and has the origin o r . The stage frame with origin o s is attached to the moving stage, which carries the test optic.

Fig. 4
Fig. 4

Illustration of radius measurement on an optical bench. The reference and stage coordinate frames are identified for the (a) confocal and (b) cat’s eye positions.

Fig. 5
Fig. 5

Vector-based radius definition. The radius is not defined by the linear transducer value only, but by the vector equation X t c e r + R = X p r .

Fig. 6
Fig. 6

Photograph of the optical bench. The linear slide and stage are used to position the test optic relative to the probe (the phase measuring interferometer converging wavefront focus—not visible). The displacement is recorded using the digital encoder attached to the slide.

Fig. 7
Fig. 7

Setup for rotational error measurements. The tilt coefficients from the interferometric phase maps were used to calculate ε x ( z ) and ε y ( z ) .

Fig. 8
Fig. 8

Rotational errors for stage z motion. The errors were measured using a transmission/return flat setup on the phase measuring interferometer. The mean values and standard deviations from ten repeated tests are shown.

Fig. 9
Fig. 9

Setup for the ε z ( z ) measurements.

Fig. 10
Fig. 10

Measurements results for ε z ( z ) . The data were collected using an electronic level system.

Fig. 11
Fig. 11

First setup for y straightness of z motion, δ y ( z ) , measurement.

Fig. 12
Fig. 12

Straightness errors in the x direction for z motion, δ x ( z ) .

Fig. 13
Fig. 13

Straightness errors in the y direction for z motion, δ y ( z ) .

Fig. 14
Fig. 14

Illustration of probe location in reference coordinate frame.

Fig. 15
Fig. 15

Measurement sequence for determining confocal and cat’s eye locations by linear regression. A sequence of phase maps is recorded at z positions on each side of confocal (1 to 2) and cat’s eye (4 to 5). The best fit line is used to determine the null location. The 3 to 4 step is translation between confocal and cat’s eye. Note that the z direction is positive to the left.

Fig. 16
Fig. 16

Monte Carlo simulations results for unbiased radius estimate using the vector-based definition.

Fig. 17
Fig. 17

Summary of Zerodur sphere radius measurements results.

Fig. 18
Fig. 18

Sensitivity of mean radius value to uncertainty in the alignment angle between the stage and interferometer axes.

Equations (9)

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T s r = [ 1 ε z ( z ) ε y ( z ) δ x ( z ) ε z ( z ) 1 ε x ( z ) δ y ( z ) ε y ( z ) ε x ( z ) 1 d 0 0 0 1 ] ,
R 2 = | R | 2 = | X p X t c e | 2 = r | r X p T s r r X t | 2 s .
X t s = X p r + [ d x c f d y c f d z c f 1 ] = [ x p r + d x c f y p r + d y c f d z c f 1 ] ,
R 2 = | R | 2 = | [ x p r y p r 0 1 ] [ 1 ε z ( z ) ε y ( z ) δ x ( z ) ε z ( z ) 1 ε x ( z ) δ y ( z ) ε y ( z ) ε x ( z ) 1 d 0 0 0 1 ] [ x p r + d x c f y p r + d y c f d z c f 1 ] | 2 ,
R 2 = [ d x c f + ε z ( z ) ( y p r + d y c f ) ε y ( z ) d z c f δ x ( z ) ] 2 + [ d y c f ε z ( z ) ( x p r + d x c f ) + ε x ( z ) d z c f δ y ( z ) ] 2 + [ d z c f + ε y ( z ) ( x p r + d x c f ) ε x ( z ) ( y p r + d y c f ) d ] 2 .
I 1 ( z ) = M ( z ) + S ( z ) ,
I 2 ( z ) = M ( z ) + S ( z ) ,
M ( z ) = I 1 ( z ) I 2 ( z ) 2 .
u = π 180 3 d .

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