Abstract

A noncontact test procedure to obtain the shape of fast concave surfaces is described. A cylindrical null screen with a curved grid drawn on it in such a way that its image, which is formed by reflection on a perfect concave surface, yields a perfect square grid is proposed. The cylindrical null screen design and the surface evaluation algorithm are presented. Experimental results for the testing of an elliptical mirror of 164mm in diameter (f/0.232) are shown.

© 2008 Optical Society of America

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References

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  1. R. Diaz-Uribe and M. Campos-Garcia, “Null screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670-2677(2000).
    [CrossRef]
  2. R. Diaz-Uribe, “Medium precision null screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the case of the large millimeter telescope,” Appl. Opt. 39, 2790-2804 (2000).
    [CrossRef]
  3. M. Campos-Garcia, R. Diaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255-6264 (2004).
    [CrossRef] [PubMed]
  4. M. Avendaño-Alejo and R. Diaz-Uribe, “Testing a fast off-axis parabolic mirror using tilted null-screens,” Appl. Opt. 45, 2607-2614 (2006).
    [CrossRef] [PubMed]
  5. O. G. Rodríguez-Herrera, M. Rosete-Aguilar, and N. C. Bruce, “Scatterometer of visible light for 2D rough surfaces,” Rev. Sci. Instrum. 75, 4820-4823 (2004).
    [CrossRef]
  6. O. G. Rodríguez-Herrera and N. C. Bruce, “Mueller matrix for an ellipsoidal mirror,” Opt. Eng. 45053602-1-053602-7(2006).
    [CrossRef]
  7. R. Diaz-Uribe, R. Bolado-Gomez, M. Campos-García, and M. Avendaño-Alejo, “Medium precision optical testing of a fast concave elliptical mirror by a cylindrical null screen,” Proc. SPIE 6034, 60340Y1-6 (2006).
  8. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, 1990).
  9. M. Campos-García and R. Diaz-Uribe, “Accuracy analysis in laser keratopography,” Appl. Opt. 41, 2065-2073 (2002).
    [CrossRef] [PubMed]
  10. L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53421-430 (2007).
  11. W. Rasban, “Image Processing and Analysis in Java,” National Institutes of Health, USA, ImageJ V. 1.37, http://rsb.info.nih.gov/ij/.
  12. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161-166.
  13. M. Campos-García and R. Díaz-Uribe, “Quantitative shape evaluation of fast aspherics with null screens by fitting two local second order polynomials to the surface normals,” presented at the VI Iberoamerican Conference on Optics (RIAO) / IX Latinamerican meeting on Optics, Lasers and Applications (OPTILAS), Campinas-SP, Brazil, 21-26 October 2007.
  14. D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming (Wiley, 1964), Chap. 6.
  15. V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, Appl. Opt. 47, 644-651 (2008).
    [CrossRef] [PubMed]

2008

2007

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53421-430 (2007).

2006

O. G. Rodríguez-Herrera and N. C. Bruce, “Mueller matrix for an ellipsoidal mirror,” Opt. Eng. 45053602-1-053602-7(2006).
[CrossRef]

R. Diaz-Uribe, R. Bolado-Gomez, M. Campos-García, and M. Avendaño-Alejo, “Medium precision optical testing of a fast concave elliptical mirror by a cylindrical null screen,” Proc. SPIE 6034, 60340Y1-6 (2006).

M. Avendaño-Alejo and R. Diaz-Uribe, “Testing a fast off-axis parabolic mirror using tilted null-screens,” Appl. Opt. 45, 2607-2614 (2006).
[CrossRef] [PubMed]

2004

O. G. Rodríguez-Herrera, M. Rosete-Aguilar, and N. C. Bruce, “Scatterometer of visible light for 2D rough surfaces,” Rev. Sci. Instrum. 75, 4820-4823 (2004).
[CrossRef]

M. Campos-Garcia, R. Diaz-Uribe, and F. Granados-Agustín, “Testing fast aspheric convex surfaces with a linear array of sources,” Appl. Opt. 43, 6255-6264 (2004).
[CrossRef] [PubMed]

2002

2000

Avendaño-Alejo, M.

M. Avendaño-Alejo and R. Diaz-Uribe, “Testing a fast off-axis parabolic mirror using tilted null-screens,” Appl. Opt. 45, 2607-2614 (2006).
[CrossRef] [PubMed]

R. Diaz-Uribe, R. Bolado-Gomez, M. Campos-García, and M. Avendaño-Alejo, “Medium precision optical testing of a fast concave elliptical mirror by a cylindrical null screen,” Proc. SPIE 6034, 60340Y1-6 (2006).

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-Gomez, R.

R. Diaz-Uribe, R. Bolado-Gomez, M. Campos-García, and M. Avendaño-Alejo, “Medium precision optical testing of a fast concave elliptical mirror by a cylindrical null screen,” Proc. SPIE 6034, 60340Y1-6 (2006).

Bolado-Gómez, R.

Bruce, N. C.

O. G. Rodríguez-Herrera and N. C. Bruce, “Mueller matrix for an ellipsoidal mirror,” Opt. Eng. 45053602-1-053602-7(2006).
[CrossRef]

O. G. Rodríguez-Herrera, M. Rosete-Aguilar, and N. C. Bruce, “Scatterometer of visible light for 2D rough surfaces,” Rev. Sci. Instrum. 75, 4820-4823 (2004).
[CrossRef]

Campos-Garcia, M.

Campos-García, M.

V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, Appl. Opt. 47, 644-651 (2008).
[CrossRef] [PubMed]

R. Diaz-Uribe, R. Bolado-Gomez, M. Campos-García, and M. Avendaño-Alejo, “Medium precision optical testing of a fast concave elliptical mirror by a cylindrical null screen,” Proc. SPIE 6034, 60340Y1-6 (2006).

M. Campos-García and R. Diaz-Uribe, “Accuracy analysis in laser keratopography,” Appl. Opt. 41, 2065-2073 (2002).
[CrossRef] [PubMed]

M. Campos-García and R. Díaz-Uribe, “Quantitative shape evaluation of fast aspherics with null screens by fitting two local second order polynomials to the surface normals,” presented at the VI Iberoamerican Conference on Optics (RIAO) / IX Latinamerican meeting on Optics, Lasers and Applications (OPTILAS), Campinas-SP, Brazil, 21-26 October 2007.

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. 53421-430 (2007).

Diaz-Uribe, R.

Díaz-Uribe, R.

V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, Appl. Opt. 47, 644-651 (2008).
[CrossRef] [PubMed]

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53421-430 (2007).

M. Campos-García and R. Díaz-Uribe, “Quantitative shape evaluation of fast aspherics with null screens by fitting two local second order polynomials to the surface normals,” presented at the VI Iberoamerican Conference on Optics (RIAO) / IX Latinamerican meeting on Optics, Lasers and Applications (OPTILAS), Campinas-SP, Brazil, 21-26 October 2007.

Dorn, W. S.

D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming (Wiley, 1964), Chap. 6.

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 U. Press, 1990).

Granados-Agustín, F.

McCracken, D. D.

D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming (Wiley, 1964), Chap. 6.

Moreno-Oliva, V. I.

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 U. Press, 1990).

Rasban, W.

W. Rasban, “Image Processing and Analysis in Java,” National Institutes of Health, USA, ImageJ V. 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.

Rodríguez-Herrera, O. G.

O. G. Rodríguez-Herrera and N. C. Bruce, “Mueller matrix for an ellipsoidal mirror,” Opt. Eng. 45053602-1-053602-7(2006).
[CrossRef]

O. G. Rodríguez-Herrera, M. Rosete-Aguilar, and N. C. Bruce, “Scatterometer of visible light for 2D rough surfaces,” Rev. Sci. Instrum. 75, 4820-4823 (2004).
[CrossRef]

Rosete-Aguilar, M.

O. G. Rodríguez-Herrera, M. Rosete-Aguilar, and N. C. Bruce, “Scatterometer of visible light for 2D rough surfaces,” Rev. Sci. Instrum. 75, 4820-4823 (2004).
[CrossRef]

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 U. 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 U. Press, 1990).

Appl. Opt.

Opt. Eng.

O. G. Rodríguez-Herrera and N. C. Bruce, “Mueller matrix for an ellipsoidal mirror,” Opt. Eng. 45053602-1-053602-7(2006).
[CrossRef]

Proc. SPIE

R. Diaz-Uribe, R. Bolado-Gomez, M. Campos-García, and M. Avendaño-Alejo, “Medium precision optical testing of a fast concave elliptical mirror by a cylindrical null screen,” Proc. SPIE 6034, 60340Y1-6 (2006).

Rev. Mex. Fís.

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53421-430 (2007).

Rev. Sci. Instrum.

O. G. Rodríguez-Herrera, M. Rosete-Aguilar, and N. C. Bruce, “Scatterometer of visible light for 2D rough surfaces,” Rev. Sci. Instrum. 75, 4820-4823 (2004).
[CrossRef]

Other

W. Rasban, “Image Processing and Analysis in Java,” National Institutes of Health, USA, ImageJ V. 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.

M. Campos-García and R. Díaz-Uribe, “Quantitative shape evaluation of fast aspherics with null screens by fitting two local second order polynomials to the surface normals,” presented at the VI Iberoamerican Conference on Optics (RIAO) / IX Latinamerican meeting on Optics, Lasers and Applications (OPTILAS), Campinas-SP, Brazil, 21-26 October 2007.

D. D. McCracken and W. S. Dorn, Numerical Methods and Fortran Programming (Wiley, 1964), Chap. 6.

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

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Figures (13)

Fig. 1
Fig. 1

Layout of the testing configuration.

Fig. 2
Fig. 2

Variables involved in the design of the cylindrical null screen.

Fig. 3
Fig. 3

Normal evaluation (rays r r and r i do not necessarily lie in the X - Z plane).

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

(a) Flat-printed null screen with 100 × 100 lines on the grid for qualitatively testing the elliptical mirror, and (b) the screen after it is inserted into an acrylic cylinder.

Fig. 7
Fig. 7

Qualitative analysis. (a) The resultant image of the screen of Fig. 6 after reflection on the test surface, where the image resembles a wrinkled sheet of paper because of the surface deformations, (b) detail of the image near to the center of the surface showing an almost perfect square grid, and (c) detail of the image corresponding to the rim of the surface showing that the lines of the square array are not straight, we observe doubling imaging effects at the rim.

Fig. 8
Fig. 8

Images of the tested mirror showing surface deformations. The images were obtained with a null screen square array of: (a)  10 × 10 lines, and (b)  100 × 100 lines (defocused image of the screen of Fig. 7).

Fig. 9
Fig. 9

(a) Flat-printed null screen with drop shaped for quantitatively testing the elliptical mirror, (b) the same screen wrapped around an acrylic cylinder, and (c) the resultant image of the screen after reflection on the test surface.

Fig. 10
Fig. 10

Integration paths on the x - y plane. All paths start at the same point z o .

Fig. 11
Fig. 11

Evaluated surface.

Fig. 12
Fig. 12

Differences in sagitta between the measured surface and the best fitting conic. Trapezoidal method.

Fig. 13
Fig. 13

Differences in sagitta between the measured surface and the best fitting conic. Polynomial method.

Tables (2)

Tables Icon

Table 1 Design Parameters for the Test of the Ellipsoidal Mirror

Tables Icon

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

Equations (24)

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ρ 2 = a ( r Q b ) [ a 2 r 2 + ρ 1 2 b ( Q b 2 r ) ] 1 / 2 a 2 Q + ρ 1 2 ρ 1 , z 2 = ρ 2 ρ 1 a + b ,
ρ 3 = R , z 3 = a ρ 2 2 + a ( Q z 2 r ) 2 + 2 ρ 1 ρ 2 ( Q z 2 r ) ρ 1 ρ 2 2 ρ 1 ( Q z 2 r ) 2 + 2 ρ 2 a ( Q z 2 r ) ( R ρ 2 ) + z 2 ,
ρ 1 = ( n 2 + m 2 ) 1 / 2 l , tan ϕ = m n
b = a D d + β ,
β = { D 2 8 r for     Q = 0 r Q [ 1 ( 1 Q D 2 4 r 2 ) 1 / 2 ] for     Q 0 .
z i = f z 3 max f z 3 max .
a = s + f l ( D / d + 1 ) + [ s 2 + f l 2 ( D / d + 1 ) 2 + 2 f l s ( D / d 1 ) ] 1 / 2 2 D / d ,
a f l ,
X = R ϕ , Y = z 3 .
z z o = p o p ( n x n z d x + n y n z d y ) ,
N = r r r i | r r r i | ,
r r = ( X , Y , a ) ( X 2 + Y 2 + a 2 ) 1 / 2 .
x s = a ( Q b r ) [ a 2 r 2 b ( Q b 2 r ) ( X 2 + Y 2 ) ] 1 / 2 Q a 2 + X 2 + Y 2 X , y s = a ( Q b r ) [ a 2 r 2 b ( Q b 2 r ) ( X 2 + Y 2 ) ] 1 / 2 Q a 2 + X 2 + Y 2 Y , z s = a ( Q b r ) [ a 2 r 2 b ( Q b 2 r ) ( X 2 + Y 2 ) ] 1 / 2 Q a 2 + X 2 + Y 2 a + b .
r i = ( x s R , y s y 3 , z s z 3 ) [ ( x s R ) 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 ( X 2 + Y 2 + a 2 ) 1 / 2 + R x s [ ( x s R ) 2 + ( y s y 3 ) 2 + ( z s z 3 ) 2 ] 1 / 2 , δ y = Y ( X 2 + Y 2 + a 2 ) 1 / 2 + y 3 y s [ ( x s R ) 2 + ( y s y 3 ) 2 + ( z s z 3 ) 2 ] 1 / 2 , δ z = a ( X 2 + Y 2 + a 2 ) 1 / 2 + z 3 z s [ ( x s R ) 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 = r { r 2 Q [ ( x x o ) 2 + ( y y o ) 2 ] } 1 / 2 Q + A x + B y + z o ,
N a = f ( x , y , z ) | f ( x , y , z ) | | P 2 ,
f ( x , y , z ) = Q z 2 2 r z + x 2 + y 2 ,
( X , Y ) ( X + δ X , Y + 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 { ( n x a n z a n x n z ) d x + ( n y a n z a n y n z ) d 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 ) .

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