Abstract

Traceable radius of curvature measurements are critical for precision optics manufacturing. An optical bench measurement is repeatable and is the preferred method for low-uncertainty applications. With an optical bench, the displacement of the optic is measured as it is moved between the cat’s eye and the confocal positions, each identified using a figure measuring interferometer. The translated distance is nominally the radius of curvature; however, errors in the motion of the stage add a bias to the measurement, even if the error motions are zero on average. Estimating the bias and resulting measurement uncertainty is challenging. We have developed a new mathematical definition of the radius measurand that intrinsically corrects for error motion biases and also provides a means of representing other terms such as figure error correction, wave-front aberration biases, displacement gauge calibration and their uncertainties. With this formalism, it is no long necessary to design a high-quality radius bench to carry out a precision measurement; rather a lower quality is adequate, provided that error motions are repeatable and characterized and error motion measurement uncertainties are estimated.

© 2005 Optical Society of America

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References

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  1. Sponsored by R. Plympton, Optimax Systems, Inc., Ontario, N.Y.
  2. D. Malacara, Optical Shop Testing (Wiley, 1992).
  3. M. Murty, R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231–235 (1983).
    [CrossRef]
  4. L. Selberg, “Radius measurement by interferometry,” Opt. Eng. 31, 1961–1966 (1992).
    [CrossRef]
  5. T. Schmitz, C. J. Evans, 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.
  6. T. Schmitz, A. Davies, C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 432–447 (2001).
    [CrossRef]
  7. A. Slocum, Precision Machine Design (Prentice-Hall, 1992).
  8. T. Schmitz, C. J. Evans, A. Davies, W. T. Estler, “Displacement uncertainty in interferometric radius measurements,” Ann. CIRP 51/1, 451–454 (2002).
    [CrossRef]
  9. P. R. Bevington, D. K. Robinson, eds., Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (WCB/McGraw-Hill, 1992).
  10. “Guide to the expression of uncertainty in measurement” (International Organization for Standardization, 1995).
  11. B. Taylor, C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” (National Institute of Standards and Technology, 1994).
  12. D. Karodkar, N. Gardner, B. Bergner, A. Davies, “Traceable radius of curvature measurements on a microinterferometer,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed. Proc. SPIE5180, 261–273 (2003).
    [CrossRef]

2002 (1)

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

1992 (1)

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

1983 (1)

M. Murty, R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231–235 (1983).
[CrossRef]

Bergner, B.

D. Karodkar, N. Gardner, B. Bergner, A. Davies, “Traceable radius of curvature measurements on a microinterferometer,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed. Proc. SPIE5180, 261–273 (2003).
[CrossRef]

Davies, A.

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

T. Schmitz, C. J. Evans, 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.

T. Schmitz, A. Davies, C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 432–447 (2001).
[CrossRef]

D. Karodkar, N. Gardner, B. Bergner, A. Davies, “Traceable radius of curvature measurements on a microinterferometer,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed. Proc. SPIE5180, 261–273 (2003).
[CrossRef]

Estler, W. T.

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

Evans, C. J.

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

T. Schmitz, C. J. Evans, 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.

T. Schmitz, A. Davies, C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 432–447 (2001).
[CrossRef]

Gardner, N.

D. Karodkar, N. Gardner, B. Bergner, A. Davies, “Traceable radius of curvature measurements on a microinterferometer,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed. Proc. SPIE5180, 261–273 (2003).
[CrossRef]

Karodkar, D.

D. Karodkar, N. Gardner, B. Bergner, A. Davies, “Traceable radius of curvature measurements on a microinterferometer,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed. Proc. SPIE5180, 261–273 (2003).
[CrossRef]

Kuyatt, C.

B. Taylor, C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” (National Institute of Standards and Technology, 1994).

Malacara, D.

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

Murty, M.

M. Murty, R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231–235 (1983).
[CrossRef]

Schmitz, T.

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

T. Schmitz, C. J. Evans, 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.

T. Schmitz, A. Davies, C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 432–447 (2001).
[CrossRef]

Selberg, L.

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

Shukla, R.

M. Murty, R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231–235 (1983).
[CrossRef]

Slocum, A.

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

Taylor, B.

B. Taylor, C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” (National Institute of Standards and Technology, 1994).

Ann. CIRP (1)

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

Opt. Eng. (2)

M. Murty, R. Shukla, “Measurements of long radius of curvature,” Opt. Eng. 22, 231–235 (1983).
[CrossRef]

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

Other (9)

T. Schmitz, C. J. Evans, 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.

T. Schmitz, A. Davies, C. J. Evans, “Uncertainties in interferometric measurements of radius of curvature,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed., Proc. SPIE4451, 432–447 (2001).
[CrossRef]

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

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

“Guide to the expression of uncertainty in measurement” (International Organization for Standardization, 1995).

B. Taylor, C. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” (National Institute of Standards and Technology, 1994).

D. Karodkar, N. Gardner, B. Bergner, A. Davies, “Traceable radius of curvature measurements on a microinterferometer,” in Optical Manufacturing and Testing IV, H. P. Stahl, ed. Proc. SPIE5180, 261–273 (2003).
[CrossRef]

Sponsored by R. Plympton, Optimax Systems, Inc., Ontario, N.Y.

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

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Figures (7)

Fig. 1
Fig. 1

Schematic of radius of curvature measurement. (a) The part is translated between the confocal and the cat’s eye positions, and the measured displacement is an estimate of the radius of curvature. (b) Translation error in the x direction during the z-direction translation leads to a measured value that is less than the radius.

Fig. 2
Fig. 2

(a) Schematic of reference coordinate system, which is fixed in the laboratory reference frame. The focus of the test beam exiting the interferometer is considered the probe. The vector shown is the location of the probe in the reference coordinate system, rXp. (b) Schematic of the stage coordinate system. The test artifact (a ball in this case) is fixed in this frame and the location is defined by the vector sXA, which locates the center of curvature of the artifact in the stage coordinate system. The stage coordinate system is taken to overlap with the reference coordinate system at the confocal position (the condition shown). Possible biases and uncertainties from this assumption are discussed in the text.

Fig. 3
Fig. 3

(a) Radius measurement with no error motions and with the motion axis parallel to the gauge axis (the reference axis). The gauge axis may be offset from the optical axis to capture Abbe errors. At the confocal position the z-axis coordinate of the probe in the reference frame and the z-axis coordinate of the artifact in the stage frame are both zero. The stage and the origins overlap at this position and the displacement gauge is zeroed. In the absence of error motions, the displacement gauge reading at cat’s eye is equal to the radius of the artifact. (b) Same conditions as in (a), but x-axis straightness is added. Error motions now cause the artifact to translate to the right; therefore the cat’s eye reflection occurs away from the apex of the artifact. The displacement gauge, having been zeroed at confocal, now reads a value at cat’s eye that is less than the true radius.

Fig. 4
Fig. 4

Same conditions as in Fig. 3, but showing the correct definition of the measurand R, which is in terms of the vectors defining the location of the probe and the center of curvature of the artifact in the reference coordinate system. The subaperture of the sphere measured at confocal is highlighted with a heavy gray curve and an X is included to indicate the center of the aperture. The same aperture and center location are indicated at cat’s eye with a heavy black curve and an X.

Fig. 5
Fig. 5

Schematic of a HTM characterization of nominal z motion of a stage relative to a fixed reference coordinate system. (a) The vector sX locates an object fixed in the stage reference frame. (b) The stage is moved to a position d along the reference z axis and the motion can be described by a HTM rTs. Error motion measurements determine the elements of the HTM, and the z axis of the reference coordinate system is the displacement gauge axis so the gauge reads a displacement of d. The location of the object on the stage is located at rXrTssX in the reference coordinate system after the motion.

Fig. 6
Fig. 6

Components of vector R that define the radius of the test artifact. The magnitude of this vector is the radius measurand R. See text for a discussion of the terms.

Fig. 7
Fig. 7

Histogram of the Monte Carlo results for the radius measurement illustration discussed in the text. Random values drawn from Gaussian distributions were used for the variables in Table 2 to calculate the radius measurand as defined by Eq. (8) (100,000 iterations).

Tables (2)

Tables Icon

Table 1 Definition of Symbols and Vector Relationships

Tables Icon

Table 2 Parameter Values Used for the Examples

Equations (8)

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X s A = X r p + ( d x cf , d y cf , d z cf ) .
R = X r p - X r A ce ,
R 2 = R 2 = X r p - X r A ce 2 .
T r s = [ 1 - ɛ z ɛ y δ x ɛ z 1 - ɛ x δ y - ɛ y ɛ x 1 d 0 0 0 1 ] ,
[ x r y r z r 1 ] = [ 1 - ɛ z ɛ y δ x ɛ z 1 - ɛ x δ y - ɛ y ɛ x 1 d 0 0 0 1 ] [ x s y s z s 1 ] .
X r A ce T r s s X A .
R X r p = T r s [ x r p + d x cf y r p + d y cf d z cf 1 ] .
R = [ d ce 2 + δ x 2 + δ y 2 + ( d x cf ) 2 + ( d y cf ) 2 + ( d z cf ) 2 + 2 d x cf δ x + 2 d y cf δ y + 2 d z cf d ce + ɛ z 2 [ y r p 2 + 2 y r p d y cf + ( d y cf ) 2 ] + ɛ z 2 [ x r p 2 + 2 x r p d x cf + ( d x cf ) 2 ] + ɛ y 2 [ x r p 2 + 2 x r p d x cf + ( d x cf ) 2 ] + ɛ x 2 [ y r p 2 + 2 y r p d y cf + ( d x cf ) 2 ] + ɛ x 2 ( d z cf ) 2 + ɛ y 2 ( d z cf ) 2 + 2 ɛ z δ y x r p - 2 ɛ z δ x y r p + 2 ɛ x y r p d ce - 2 ɛ y x r p d ce + 2 d y cf ɛ x d ce - 2 d x cf ɛ y d ce + 2 d y cf ɛ z ( x r p - δ x ) - 2 d x cf ɛ z ( y r p - δ y ) + 2 d z cf ɛ x ( y r p - δ y ) - 2 d z cf ɛ y ( x r p - δ x ) - 2 ɛ x ɛ y ( x r p y r p + x r p d y cf + y r p d x cf + d x cf d y cf ) - 2 ɛ x ɛ z ( x r p d z cf + d x cf d z cf ) - 2 ɛ y ɛ z ( y r p d z cf + d y cf d z cf ) ] 1 / 2 ,

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