Abstract

In surface figure testing, most of them are relative tests, and the reference surface usually limits the accuracy of test results. Absolute calibration is one of the most important and efficient techniques to reach subnanometer accuracy in surface figure testing. An absolute testing method, a shift-rotation method using Zernike polynomials, is presented, which can be used to calibrate both flat and spherical surfaces (concave or convex). Calibration contains at least three position measurements: one basic position, one rotation, and one lateral shift of the test surface. Experiments show that the repeatability of this method is 0.13 nm RMS, and pixel-to-pixel comparison with the two-sphere method is 0.2 nm RMS.

© 2012 Optical Society of America

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References

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  1. G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
    [CrossRef]
  2. G. Schulz and J. Schwider, in Vol. 13 of Progress in Optics, E. Wolf, ed. (Elsevier, 1976), pp. 93–167.
  3. E. E. Bloemhof, Opt. Lett. 35, 2346 (2010).
    [CrossRef]
  4. J. T. Wiersma, “Absolute sphere testing: a tutorial,” OPTI–515R (2011).
  5. M. Schulz, A. Wiegmann, A. Márquez, and C. Elster, Opt. Pura Apl. 41 (4), 325 (2008).
  6. L. A. Selberg, in Optical Fabrication and Testing Workshop, Technical Digest (CD) (Optical Society of America, 1994), pp. 181–184.
  7. R. E. Parks, Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2006).
  8. C. J. Evans, “Method and apparatus for tilt corrected lateral shear in a lateral shear plus rotational shear absolute flat test,” U.S. patent 7,446,883 (November42008).

2010

2008

M. Schulz, A. Wiegmann, A. Márquez, and C. Elster, Opt. Pura Apl. 41 (4), 325 (2008).

2004

G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
[CrossRef]

Bloemhof, E. E.

Dinger, U.

G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
[CrossRef]

Elster, C.

M. Schulz, A. Wiegmann, A. Márquez, and C. Elster, Opt. Pura Apl. 41 (4), 325 (2008).

Evans, C. J.

C. J. Evans, “Method and apparatus for tilt corrected lateral shear in a lateral shear plus rotational shear absolute flat test,” U.S. patent 7,446,883 (November42008).

Fellner, B.

G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
[CrossRef]

Hocky, O.

G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
[CrossRef]

Márquez, A.

M. Schulz, A. Wiegmann, A. Márquez, and C. Elster, Opt. Pura Apl. 41 (4), 325 (2008).

Parks, R. E.

R. E. Parks, Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2006).

Rupp, M.

G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
[CrossRef]

Schulte, S.

G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
[CrossRef]

Schulz, G.

G. Schulz and J. Schwider, in Vol. 13 of Progress in Optics, E. Wolf, ed. (Elsevier, 1976), pp. 93–167.

Schulz, M.

M. Schulz, A. Wiegmann, A. Márquez, and C. Elster, Opt. Pura Apl. 41 (4), 325 (2008).

Schwider, J.

G. Schulz and J. Schwider, in Vol. 13 of Progress in Optics, E. Wolf, ed. (Elsevier, 1976), pp. 93–167.

Seitz1, G.

G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
[CrossRef]

Selberg, L. A.

L. A. Selberg, in Optical Fabrication and Testing Workshop, Technical Digest (CD) (Optical Society of America, 1994), pp. 181–184.

Wiegmann, A.

M. Schulz, A. Wiegmann, A. Márquez, and C. Elster, Opt. Pura Apl. 41 (4), 325 (2008).

Wiersma, J. T.

J. T. Wiersma, “Absolute sphere testing: a tutorial,” OPTI–515R (2011).

Opt. Lett.

Opt. Pura Apl.

M. Schulz, A. Wiegmann, A. Márquez, and C. Elster, Opt. Pura Apl. 41 (4), 325 (2008).

Proc. SPIE

G. Seitz1, S. Schulte, U. Dinger, O. Hocky, B. Fellner, and M. Rupp, Proc. SPIE 5533, 20 (2004).
[CrossRef]

Other

G. Schulz and J. Schwider, in Vol. 13 of Progress in Optics, E. Wolf, ed. (Elsevier, 1976), pp. 93–167.

J. T. Wiersma, “Absolute sphere testing: a tutorial,” OPTI–515R (2011).

L. A. Selberg, in Optical Fabrication and Testing Workshop, Technical Digest (CD) (Optical Society of America, 1994), pp. 181–184.

R. E. Parks, Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2006).

C. J. Evans, “Method and apparatus for tilt corrected lateral shear in a lateral shear plus rotational shear absolute flat test,” U.S. patent 7,446,883 (November42008).

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Figures (4)

Fig. 1.
Fig. 1.

The measurement progress.

Fig. 2.
Fig. 2.

Simulation of the impact of errors on shift-rotation method using Zernike polynomials. In the simulation, the RMS of the test surface is 2.7 nm, with 0.1 pixel of shift error, 0.1° of rotation error, and 0.1 nm RMS of noise.

Fig. 3.
Fig. 3.

The measurement uncertainty matrix of shift-rotation.

Fig. 4.
Fig. 4.

Comparison of shift-rotation method using Zernike polynomials and two-sphere method (81 Zernike term fitting).

Equations (15)

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W t i = 1 K a i Z i ,
W 1 = W r + i = 1 K a i Z i + t x 1 Z 1 + t y 1 Z 3 + p 1 Z 4 ,
W 2 = W r + i = 1 K a i Z i ( x + Δ x , y ) + t x 2 Z 2 + t y 2 Z 3 + p 2 Z 4 .
W 3 = W r + i = 1 K a i Z i ( x , y + Δ y ) + t x 3 Z 2 + t y 3 Z 3 + p 3 Z 4 .
W 4 = W r + i = 1 K a i Z i ( r , θ + Δ θ ) + t x 4 Z 2 + t y 4 Z 3 + p 4 Z 4 .
D x W = i = 1 K a i Δ Z i x + ( t x 2 t x 1 ) Z 2 + ( t y 2 t y 1 ) Z 3 + ( p 2 p 1 ) Z 4 .
D x W = i = 1 K a i Δ Z i x + t t x 1 Z 2 + t t y 1 Z 3 + p p 1 Z 4 .
D y W = i = 1 K a i Δ Z i y + t t x 2 Z 2 + t t y 2 Z 3 + p p 2 Z 4 ,
D r W = i = 1 K a i Δ Z i r + t t x 3 Z 2 + t t y 3 Z 3 + p p 3 Z 4 .
G X = R .
G = [ Δ Z 1 x ( 1 , 1 ) Δ Z 2 x ( 1 , 1 ) Δ Z K x ( 1 , 1 ) Δ Z 1 x ( 1 , 2 ) Δ Z 2 x ( 1 , 2 ) Δ Z K x ( 1 , 2 ) Δ Z 1 x ( M , N 1 ) Δ Z 2 x ( M , N 1 ) Δ Z K x ( M , N 1 ) Z 2 ( 1 , 1 ) Z 3 ( 1 , 1 ) Z 4 ( 1 , 1 ) Z 2 ( 1 , 2 ) Z 3 ( 1 , 2 ) Z 4 ( 1 , 2 ) Z 2 ( M , N 1 ) Z 3 ( M , N 1 ) Z 4 ( M , N 1 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Δ Z 1 y ( 1 , 1 ) Δ Z 2 y ( 1 , 1 ) Δ Z K y ( 1 , 1 ) Δ Z 1 y ( 1 , 2 ) Δ Z 2 y ( 1 , 2 ) Δ Z K y ( 1 , 2 ) Δ Z 1 y ( M 1 , N ) Δ Z 2 y ( M 1 , N ) Δ Z K y ( M 1 , N ) 0 0 0 0 0 0 0 0 0 0 0 0 Z 2 ( 1 , 1 ) Z 3 ( 1 , 1 ) Z 4 ( 1 , 1 ) Z 2 ( 1 , 2 ) Z 3 ( 1 , 2 ) Z 4 ( 1 , 2 ) Z 2 ( M 1 , N ) Z 3 ( M 1 , N ) Z 4 ( M 1 , N ) 0 0 0 0 0 0 0 0 0 0 0 0 Δ Z 1 r ( 1 , 1 ) Δ Z 2 r ( 1 , 1 ) Δ Z K r ( 1 , 1 ) Δ Z 1 r ( 1 , 2 ) Δ Z 2 r ( 1 , 2 ) Δ Z K r ( 1 , 2 ) Δ Z 1 r ( M , N ) Δ Z 2 r ( M , N ) Δ Z K r ( M , N ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Z 2 ( 1 , 1 ) Z 3 ( 1 , 1 ) Z 4 ( 1 , 1 ) Z 2 ( 1 , 2 ) Z 3 ( 1 , 2 ) Z 4 ( 1 , 2 ) Z 2 ( M , N ) Z 3 ( M , N ) Z 4 ( M , N ) ] ,
R = [ D x W ( 1 , 1 ) D x W ( 1 , 2 ) D x W ( M , N 1 ) D y W ( 1 , 1 ) D y W ( 1 , 2 ) D y W ( M 1 , N ) D r W ( 1 , 1 ) D r W ( 1 , 2 ) D r W ( M , N ) ] T ,
x = [ a 1 a 2 a K t t x 1 t t y 1 p 1 t t x 2 t t y 2 p 2 t t x 3 t t y 3 p 3 ] T .
G Δ X = Δ R .
Δ S test = k shift · S test · E shift / Q shift ,

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