Abstract

We report the testing of a fast off-axis surface based on the null screen principles. Here we design a tilted null screen with drop shaped spots drawn on it in such a way that its image, which is formed by reflection on the test surface, becomes an exact square array of circular spots if the surface is perfect. Any departure from this geometry is indicative of defects on the surface. Here the whole surface is tested at once. The test surface has a radius of curvature of r=20.4mm (F/0.206). The surface departures from the best surface fit are shown; in addition, we show that the errors in the surface shape are below 0.4μm when the errors in the determination of the coordinates of the centroids of the reflected images are less than 1 pixel, and the errors in the coordinates of the spots of the null screen are less than 0.5mm.

© 2009 Optical Society of America

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References

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  1. A. Cornejo-Rodríguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed., 3th. ed. (Wiley, 2007), pp. 317-360.
  2. D. Malacara-Doblado and I. Ghozeil, “Hartmann, Hartmann-Shack, and other screen tests,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 361-397.
    [CrossRef]
  3. A. Cordero-Davila, A. Cornejo-Rodriguez, and O. Cardona-Nunez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt. 31, 2370-2376 (1992).
    [CrossRef] [PubMed]
  4. A. B. Meinel and M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 71-75 (1989).
  5. R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the Large Millimeter Telescope,” Appl. Opt. 39, 2790-2804(2000).
    [CrossRef]
  6. M. Avendano-Alejo and R. Díaz-Uribe, “Testing a fast off-axis parabolic mirror using tilted null-screens,” Appl. Opt. 45, 2607-2614 (2006).
    [CrossRef] [PubMed]
  7. M. Campos-Garcia, R. Díaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255-6264 (2004).
    [CrossRef] [PubMed]
  8. O. Cardona-Nunez, A. Cornejo-Rodr?guez, R. Díaz-Uribe, A. Cordero-Davila, and J. Pedraza-Contreras, “Conic that best fits an off-axis conic section,” Appl. Opt. 25, 3258-3259 (1984).
  9. L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53(5), 421-430 (2007).
  10. M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing fast aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849-859 (2008).
    [CrossRef] [PubMed]
  11. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University Press, 1990).
  12. D. Malacara, “Mathematical representation of an optical surface and its characteristics,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 832-851.
    [CrossRef]
  13. M. Campos-García and R. Díaz-Uribe, “Accuracy analysis in laser keratopography,” Appl. Opt. 41, 2065-2073 (2002).
    [CrossRef] [PubMed]
  14. W. Rasban, ImageJ, Image Processing and Analysis in Java (National Institutes of Health), Vol. 1.37, http://rsb.info.nih.gov/ij/.
  15. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161-166.
  16. C. Menchaca and D. Malacara, “Directional curvature in a conic mirror,” Appl. Opt. 23, 3258-3261 (1984).
    [CrossRef] [PubMed]

2008 (1)

2007 (4)

A. Cornejo-Rodríguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed., 3th. ed. (Wiley, 2007), pp. 317-360.

D. Malacara-Doblado and I. Ghozeil, “Hartmann, Hartmann-Shack, and other screen tests,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 361-397.
[CrossRef]

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53(5), 421-430 (2007).

D. Malacara, “Mathematical representation of an optical surface and its characteristics,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 832-851.
[CrossRef]

2006 (1)

2004 (1)

2002 (1)

2000 (1)

1992 (2)

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161-166.

A. Cordero-Davila, A. Cornejo-Rodriguez, and O. Cardona-Nunez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt. 31, 2370-2376 (1992).
[CrossRef] [PubMed]

1990 (1)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University Press, 1990).

1989 (1)

A. B. Meinel and M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 71-75 (1989).

1984 (2)

O. Cardona-Nunez, A. Cornejo-Rodr?guez, R. Díaz-Uribe, A. Cordero-Davila, and J. Pedraza-Contreras, “Conic that best fits an off-axis conic section,” Appl. Opt. 25, 3258-3259 (1984).

C. Menchaca and D. Malacara, “Directional curvature in a conic mirror,” Appl. Opt. 23, 3258-3261 (1984).
[CrossRef] [PubMed]

Avendano-Alejo, M.

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161-166.

Bolado-Gómez, R.

Campos-Garcia, M.

Campos-García, M.

Cardona-Nunez, O.

A. Cordero-Davila, A. Cornejo-Rodriguez, and O. Cardona-Nunez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt. 31, 2370-2376 (1992).
[CrossRef] [PubMed]

O. Cardona-Nunez, A. Cornejo-Rodr?guez, R. Díaz-Uribe, A. Cordero-Davila, and J. Pedraza-Contreras, “Conic that best fits an off-axis conic section,” Appl. Opt. 25, 3258-3259 (1984).

Carmona-Paredes, L.

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53(5), 421-430 (2007).

Cordero-Davila, A.

A. Cordero-Davila, A. Cornejo-Rodriguez, and O. Cardona-Nunez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt. 31, 2370-2376 (1992).
[CrossRef] [PubMed]

O. Cardona-Nunez, A. Cornejo-Rodr?guez, R. Díaz-Uribe, A. Cordero-Davila, and J. Pedraza-Contreras, “Conic that best fits an off-axis conic section,” Appl. Opt. 25, 3258-3259 (1984).

Cornejo-Rodriguez, A.

A. Cordero-Davila, A. Cornejo-Rodriguez, and O. Cardona-Nunez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt. 31, 2370-2376 (1992).
[CrossRef] [PubMed]

O. Cardona-Nunez, A. Cornejo-Rodr?guez, R. Díaz-Uribe, A. Cordero-Davila, and J. Pedraza-Contreras, “Conic that best fits an off-axis conic section,” Appl. Opt. 25, 3258-3259 (1984).

Cornejo-Rodríguez, A.

A. Cornejo-Rodríguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed., 3th. ed. (Wiley, 2007), pp. 317-360.

Díaz-Uribe, R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University Press, 1990).

Ghozeil, I.

D. Malacara-Doblado and I. Ghozeil, “Hartmann, Hartmann-Shack, and other screen tests,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 361-397.
[CrossRef]

Granados-Agustín, F.

Malacara, D.

D. Malacara, “Mathematical representation of an optical surface and its characteristics,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 832-851.
[CrossRef]

C. Menchaca and D. Malacara, “Directional curvature in a conic mirror,” Appl. Opt. 23, 3258-3261 (1984).
[CrossRef] [PubMed]

Malacara-Doblado, D.

D. Malacara-Doblado and I. Ghozeil, “Hartmann, Hartmann-Shack, and other screen tests,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 361-397.
[CrossRef]

Meinel, A. B.

A. B. Meinel and M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 71-75 (1989).

Meinel, M. P.

A. B. Meinel and M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 71-75 (1989).

Menchaca, C.

Pedraza-Contreras, J.

O. Cardona-Nunez, A. Cornejo-Rodr?guez, R. Díaz-Uribe, A. Cordero-Davila, and J. Pedraza-Contreras, “Conic that best fits an off-axis conic section,” Appl. Opt. 25, 3258-3259 (1984).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University Press, 1990).

Rasban, W.

W. Rasban, ImageJ, Image Processing and Analysis in Java (National Institutes of Health), Vol. 1.37, http://rsb.info.nih.gov/ij/.

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161-166.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University Press, 1990).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University Press, 1990).

Appl. Opt. (8)

Opt. Eng. (1)

A. B. Meinel and M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 71-75 (1989).

Rev. Mex. Fís. (1)

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53(5), 421-430 (2007).

Other (6)

A. Cornejo-Rodríguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed., 3th. ed. (Wiley, 2007), pp. 317-360.

D. Malacara-Doblado and I. Ghozeil, “Hartmann, Hartmann-Shack, and other screen tests,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 361-397.
[CrossRef]

W. Rasban, ImageJ, Image Processing and Analysis in Java (National Institutes of Health), Vol. 1.37, http://rsb.info.nih.gov/ij/.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161-166.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge University Press, 1990).

D. Malacara, “Mathematical representation of an optical surface and its characteristics,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), pp. 832-851.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Layout of the testing configuration.

Fig. 2
Fig. 2

Normal evaluation (rays R and I do not necessarily lie in the x–z plane).

Fig. 3
Fig. 3

Rms differences in sagitta obtained in a simulation when a random displacement is added to the coordinates of the centroids of the spots at the CCD.

Fig. 4
Fig. 4

Rms differences in sagitta obtained in a simulation when a random displacement is added to the coordinates of the positions of the spots of the null screen.

Fig. 5
Fig. 5

Picture of the actual experimental setup.

Fig. 6
Fig. 6

Tilted null screen.

Fig. 7
Fig. 7

Image of the tilted null screen after reflection on the off-axis mirror.

Fig. 8
Fig. 8

Calculated centroids of the image.

Fig. 9
Fig. 9

Selected integration paths for the integration.

Fig. 10
Fig. 10

Reconstruction of the test surface.

Fig. 11
Fig. 11

Differences in sagitta between the measured surface and the best fitting off-axis conic.

Tables (2)

Tables Icon

Table 1 Design Parameters for Testing the Off-Axis Mirror

Tables Icon

Table 2 Parameters Resulting from the Least Square Fitting of the Sagitta Data

Equations (24)

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z ( x , y ) = γ β + ( β 2 α γ ) 1 / 2 ,
α = c ( 1 + k cos 2 θ ) , β = ( 1 + k sin 2 θ ) 1 / 2 c k x     sin θ cos θ , γ = c ( 1 + k sin 2 θ ) x 2 + c y 2 ,
tan θ = c x c [ 1 ( k + 1 ) c 2 x c 2 ] 1 / 2 ,
x x 1 x 1 = y y 1 y 1 = z a b a .
R = I 2 ( I · N ) N = ( R x , R y , R z ) ,
N = ( z x , z y , 1 ) z x 2 + z y 2 + 1 = ( z / x , z / y , 1 ) ( z / x ) 2 + ( z / y ) 2 + 1 ,
A p x 3 + B p y 3 + C p z 3 = D p ,
A p x 3 + C p z 3 = D p .
ϕ = arctan ( C p A p ) .
x 3 = x + R x M a , y 3 = y + R y M a , z 3 = z + R z M a ,
M a = [ a ( x + z tan ϕ ) R x + R z tan ϕ ] ,
z z o = p o p ( n x n z d x + n y n z d y ) ,
N = R I | R I | ,
R = ( x 1 , y 1 , a ) ( x 2 1 + y 2 1 + a 2 ) 1 / 2 .
I = ( x s x 3 , y s y 3 , z s z 3 ) [ ( x s x 3 ) 2 + ( y s y 3 ) 2 + ( z s z 3 ) 2 ] 1 / 2 .
N = ( δ x , δ y , δ z ) ( δ x 2 + δ y 2 + δ z 2 ) 1 / 2 ,
δ x = x 1 ( x 2 1 + y 2 1 + a 2 ) 1 / 2 + x 3 x s [ ( x s x 3 ) 2 + ( y s y 3 ) 2 + ( z s z 3 ) 2 ] 1 / 2 , δ y = y 1 ( x 2 1 + y 2 1 + a 2 ) 1 / 2 + y 3 y s [ ( x s x 3 ) 2 + ( y s y 3 ) 2 + ( z s z 3 ) 2 ] 1 / 2 , δ z = a ( x 2 1 + y 2 1 + a 2 ) 1 / 2 + z 3 z s [ ( x s x 3 ) 2 + ( y s y 3 ) 2 + ( z s z 3 ) 2 ] 1 / 2 .
z m = i = 1 m 1 { ( n x i n z i + n x i + 1 n z i + 1 ) ( x i + 1 x i ) 2 + ( n y i n z i + n y i + 1 n z i + 1 ) ( y i + 1 y i ) 2 } + z o ,
z = c x ( x x o ) 2 2 + c y ( y y o ) 2 2 + a 3 ( x x o ) 2 ( y y o ) + a 4 ( y y o ) 3 + a 5 ( x x o ) 4 + a 6 ( x x o ) 2 ( y y o ) 2 + a 7 ( y y o ) 4 + A x + B y + z o ,
N a = f ( x , y , z ) | f ( x , y , z ) | | P 2 ,
( x 1 , y 1 ) ( x 1 + δ x , y 1 + δ y ) .
δ x = η 2 ( 2 ln r 1 ) 1 / 2 cos ( 2 π r 2 ) , δ y = η 2 ( 2 ln r 1 ) 1 / 2     sin ( 2 π r 2 ) , δ z = η 2 ( 2 ln r 3 ) 1 / 2     cos ( 2 π r 2 ) ,
δ z { ( n x a n z a n x n z ) d x + ( n y a n z a n y n z ) d y } ,
c y c 2 = c x 3 .

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