Abstract

We present a method to compensate for the imaging distortion encountered in interferometric testing of mirrors, which is introduced by interferometer optics as well as from geometric projection errors. Our method involves placing a mask, imprinted with a regular square grid, over the mirror and finding a transformation that relates the grid coordinates to coordinates in the base plane of the parent surface. This method can be used on finished mirrors since no fiducials have to be applied to the surfaces. A critical step in the process requires that the grid coordinates be projected onto the mirror base plane before the regression is performed. We apply the method successfully during a center-of-curvature null test of an F/2 off-axis paraboloid.

© 2009 Optical Society of America

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References

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  1. M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313(2008).
    [CrossRef]
  2. C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).
    [CrossRef]
  3. S. Reichelt, C. Pruss, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158-168 (2002).
    [CrossRef]
  4. F. Schillke, “Critical aspects on testing aspheres in interferometric setups,” Proc. SPIE 3739, 317-324 (1999).
    [CrossRef]
  5. J. E. Hayden and T. S. Lewis, “Distortion correction method for aspheric optical testing,” Proc. SPIE 4101, 57-63 (2000).
  6. H. Anton, Calculus, 2nd ed. (Wiley, 1984), p. 912.
  7. G. A. F. Seber and C. J. Wild, Nonlinear Regression (Wiley, 1988), p. 619.
  8. W. C. Chu, B. Bavarian, C. B. Ahn, Y. S. Kwoh, and E. A. Jonckheere, “Distortion compensation in MR images for robotic stereotactic procedures,” Proc. SPIE 1349, 421-430 (1990).
    [CrossRef]
  9. J. W. Demmel, Applied Numerical Linear Algebra (SIAM, 1997), p. 106.

2008

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313(2008).
[CrossRef]

2006

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).
[CrossRef]

2002

S. Reichelt, C. Pruss, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

2000

J. E. Hayden and T. S. Lewis, “Distortion correction method for aspheric optical testing,” Proc. SPIE 4101, 57-63 (2000).

1999

F. Schillke, “Critical aspects on testing aspheres in interferometric setups,” Proc. SPIE 3739, 317-324 (1999).
[CrossRef]

1990

W. C. Chu, B. Bavarian, C. B. Ahn, Y. S. Kwoh, and E. A. Jonckheere, “Distortion compensation in MR images for robotic stereotactic procedures,” Proc. SPIE 1349, 421-430 (1990).
[CrossRef]

Ahn, C. B.

W. C. Chu, B. Bavarian, C. B. Ahn, Y. S. Kwoh, and E. A. Jonckheere, “Distortion compensation in MR images for robotic stereotactic procedures,” Proc. SPIE 1349, 421-430 (1990).
[CrossRef]

Anton, H.

H. Anton, Calculus, 2nd ed. (Wiley, 1984), p. 912.

Bavarian, B.

W. C. Chu, B. Bavarian, C. B. Ahn, Y. S. Kwoh, and E. A. Jonckheere, “Distortion compensation in MR images for robotic stereotactic procedures,” Proc. SPIE 1349, 421-430 (1990).
[CrossRef]

Bray, M.

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).
[CrossRef]

Burge, J. H.

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313(2008).
[CrossRef]

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).
[CrossRef]

Chu, W. C.

W. C. Chu, B. Bavarian, C. B. Ahn, Y. S. Kwoh, and E. A. Jonckheere, “Distortion compensation in MR images for robotic stereotactic procedures,” Proc. SPIE 1349, 421-430 (1990).
[CrossRef]

Demmel, J. W.

J. W. Demmel, Applied Numerical Linear Algebra (SIAM, 1997), p. 106.

Hayden, J. E.

J. E. Hayden and T. S. Lewis, “Distortion correction method for aspheric optical testing,” Proc. SPIE 4101, 57-63 (2000).

Jonckheere, E. A.

W. C. Chu, B. Bavarian, C. B. Ahn, Y. S. Kwoh, and E. A. Jonckheere, “Distortion compensation in MR images for robotic stereotactic procedures,” Proc. SPIE 1349, 421-430 (1990).
[CrossRef]

Kwoh, Y. S.

W. C. Chu, B. Bavarian, C. B. Ahn, Y. S. Kwoh, and E. A. Jonckheere, “Distortion compensation in MR images for robotic stereotactic procedures,” Proc. SPIE 1349, 421-430 (1990).
[CrossRef]

Lewis, T. S.

J. E. Hayden and T. S. Lewis, “Distortion correction method for aspheric optical testing,” Proc. SPIE 4101, 57-63 (2000).

Novak, M.

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313(2008).
[CrossRef]

Pruss, C.

S. Reichelt, C. Pruss, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

Reichelt, S.

S. Reichelt, C. Pruss, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

Schillke, F.

F. Schillke, “Critical aspects on testing aspheres in interferometric setups,” Proc. SPIE 3739, 317-324 (1999).
[CrossRef]

Seber, G. A. F.

G. A. F. Seber and C. J. Wild, Nonlinear Regression (Wiley, 1988), p. 619.

Sprowl, R. A.

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).
[CrossRef]

Tiziani, H. J.

S. Reichelt, C. Pruss, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

Wild, C. J.

G. A. F. Seber and C. J. Wild, Nonlinear Regression (Wiley, 1988), p. 619.

Zhao, C.

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313(2008).
[CrossRef]

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).
[CrossRef]

Proc. SPIE

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313(2008).
[CrossRef]

C. Zhao, R. A. Sprowl, M. Bray, and J. H. Burge, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930K(2006).
[CrossRef]

S. Reichelt, C. Pruss, and H. J. Tiziani, “New design techniques and calibration methods for CGH-null testing of aspheric surfaces,” Proc. SPIE 4778, 158-168 (2002).
[CrossRef]

F. Schillke, “Critical aspects on testing aspheres in interferometric setups,” Proc. SPIE 3739, 317-324 (1999).
[CrossRef]

J. E. Hayden and T. S. Lewis, “Distortion correction method for aspheric optical testing,” Proc. SPIE 4101, 57-63 (2000).

W. C. Chu, B. Bavarian, C. B. Ahn, Y. S. Kwoh, and E. A. Jonckheere, “Distortion compensation in MR images for robotic stereotactic procedures,” Proc. SPIE 1349, 421-430 (1990).
[CrossRef]

Other

J. W. Demmel, Applied Numerical Linear Algebra (SIAM, 1997), p. 106.

H. Anton, Calculus, 2nd ed. (Wiley, 1984), p. 912.

G. A. F. Seber and C. J. Wild, Nonlinear Regression (Wiley, 1988), p. 619.

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Figures (7)

Fig. 1
Fig. 1

Code V model of the pupil distortion and wavefront aberration introduced by the null corrector in the center-of-curvature null test of the off-axis paraboloid.

Fig. 2
Fig. 2

Schematic illustration of the test setup. The x y plane shown is referred to throughout this paper as the base plane of the mirror.

Fig. 3
Fig. 3

Projection errors for rays normally incident on the mirror surface. The OAP segment is indicated on the left with the distortion mask in a plane parallel to the parent ( x , y ) base plane.

Fig. 4
Fig. 4

Geometry of normal projection errors.

Fig. 5
Fig. 5

Distortion mask in its housing.

Fig. 7
Fig. 7

(a) image of the pupil plane mask as recorded by the interferometer before any distortion compensation. (b) Image of the mask remapped into the pupil coordinates. Note that it is square. (c) Mask image after projection back to the mirror surface. The apparent distortion in this image illustrates the lateral displacement of rays due to propagation along surface normals.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

d = [ f ( x e , y e ) f ( x , y ) ] z ^ = d z ^ ,
c = d [ f x ( x , y ) x ^ + f y ( x , y ) y ^ z ^ ] ,
ε = c d = d [ f x ( x , y ) x ^ + f y ( x , y ) y ^ ] ,
x = x + ε x = x d · f x ( x , y ) ,
y = y + ε y = y d · f y ( x , y ) .
[ x y ] = u = u ( u ) .
u = u ( u ) u ( u 0 ) + u u | u = u 0 δ u ,
u u = [ x x x y y x y y ] .
δ u 0 = ( u u | u = u 0 ) 1 [ u m u ( u 0 ) ] .
u 1 = u 0 + δ u 0 ,
f ( i , j ) = a 0 + a 1 i + a 2 j + a 3 i 2 + a 4 j 2 + a 5 i j , g ( i , j ) = b 0 + b 1 i + b 2 j + b 3 i 2 + b 4 j 2 + b 5 i j .
x = [ x 1 x 2 x n ] T , y = [ y 1 y 2 y n ] T , a = [ a 0 a 1 a 2 a 3 a 4 a 5 ] T , b = [ b 0 b 1 b 2 b 3 b 4 b 5 ] T .
Ω = [ 1 i 1 j 1 i 1 2 j 1 2 i 1 j 1 1 i 2 j 2 i 2 2 j 2 2 i 2 j 2 1 i n j n i n 2 j n 2 i n j n ] .
E 2 = ( Ω a x ) T ( Ω a x ) + ( Ω b y ) T ( Ω b y ) .
a = ( Ω T Ω ) 1 Ω T x = Ω + x , b = ( Ω T Ω ) 1 Ω T x = Ω + y ,
x = x + x 89.40 304.8 [ ( x 89.40 ) 2 + y 2 609.6 26.66 ] ,
y = y + y 304.8 [ ( x 89.40 ) 2 + y 2 609.6 26.66 ] .
a = [ 68.30 0.02492 0.4602 9.077 e 5 3.394 e 4 4.243 e 7 ] , b = [ 42.47 0.3686 0.01840 1.719 e 5 5.100 e 6 1.847 e 4 ] .

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