Abstract

We describe a noncontact test procedure with which to obtain the shapes of fast convex surfaces. For this, an array of sources is positioned in a straight line and separated in such a way that the image by reflection on the surface consists of a set of equally spaced bright spots. By rotating the surface, we test different meridians such that, after 360°, the entire surface is measured. We present the source array design and the surface evaluation algorithm. We found that, to reduce numerical error in the evaluation of the shape of the surface, a numerical integration must be performed by a method that uses parabolic arcs instead of the traditional method that uses trapezoids. Through some numerical simulations we analyzed the accuracy of the method by introducing random displacements into the measured data. We found that to measure the quality of the surface with accuracy better than 5 μm, we have to measure the coordinates of the centroids on the image plane with an accuracy better than 0.5 pixel, and we to have measure the positions of the linear sources with an accuracy better than 0.5 mm. Experimental results for the testing of a carbon fiber convex sphere of 383.6-mm diameter (f/0.398) are shown.

© 2004 Optical Society of America

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References

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  1. J. H. Burge, “Measurement of large convex aspheres,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 362–373 (1997).
    [CrossRef]
  2. F. Schillke, “Critical aspects on testing aspheres in interferometric setups,” in Optical Fabrication and Testing, R. Geyl, J. Maxwell, eds., Proc. SPIE3739, 317–324 (1999).
    [CrossRef]
  3. Instituto Nacional de Astrofísica, Óptica y Electrónica, “The Large Millimeter Telescope Homepage,” http://www.lmtgtm.org/ .
  4. R. Díaz-Uribe, M. Campos-García, “Null screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677 (2000).
    [CrossRef]
  5. Y. Barbosa, D. Malacara, “Object surface for applying a modified Hartmann test to measure corneal topography,” Appl. Opt. 40, 5778–5786 (2001).
    [CrossRef]
  6. I. E. Funes-Maderey, “Videoqueratometría de campo plano” (“Flat field videokeratometry”), B.A. dissertation (Universidad Nacional Autónoma de México, México, 1998).
  7. M. Campos-García, R. Díaz-Uribe, F. S. Granados-Agustín, D. Sacramento-Solano, “Null test of aspheric convex surface,” in Proceedings of International Symposium on Photonics in Measurement, VDI-Berichte 1694 (VDI Verlag GmbH, Düsseldorf, Germany, 2002), pp. 155–160.
  8. R. Díaz-Uribe, M. Campos-García, F. S. Granados-Agustín, “Testing the optics of the Large Millimeter Telescope (LMT),” in Infrared Spaceborne Remote Sensing X, M. Strojnik, B. F. Andresen, eds., Proc. SPIE4818, 63–70 (2002).
    [CrossRef]
  9. R. Díaz-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]
  10. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, Mass., 1990).
  11. M. Campos-García, R. Díaz-Uribe, “Accuracy analysis in laser keratophography,” Appl. Opt. 41, 2065–2073 (2002).
    [CrossRef] [PubMed]
  12. L. Carmona-Paredes, R. Díaz-Uribe, “Imágenes circulares para pruebas con pantallas nulas,” in Proceedings of Memorias en Extenso; Sesiones de Óptica, C. G. Treviño, ed. (Academia Mexicana de Óptica, León, Gto., Mexico, 2002), pp. 58–60.
  13. L. Carmona-Paredes, R. Díaz-Uribe, “Corrección al cálculo del centroide de las manchas luminosas en la prueba de una superficie esférica,” in Proceedings of Memorias en Extenso; Sesiones de Óptica, C. G. Treviño, ed. (Academia Mexicana de Óptica, León, Gto., Mexico, 2003), pp. 46V03–1–46V03–10.
  14. W. Rasban, National Institutes of Health, USA: Image Processing and Analysis in Java, ImageJ V. 1.312u, http://rsb.info.nih.gov/ij/ .
  15. P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, New York, 1992), pp. 161–166.

2002 (1)

2001 (1)

2000 (2)

Barbosa, Y.

Bevington, P. R.

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

Burge, J. H.

J. H. Burge, “Measurement of large convex aspheres,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 362–373 (1997).
[CrossRef]

Campos-García, M.

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

R. Díaz-Uribe, M. Campos-García, “Null screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677 (2000).
[CrossRef]

M. Campos-García, R. Díaz-Uribe, F. S. Granados-Agustín, D. Sacramento-Solano, “Null test of aspheric convex surface,” in Proceedings of International Symposium on Photonics in Measurement, VDI-Berichte 1694 (VDI Verlag GmbH, Düsseldorf, Germany, 2002), pp. 155–160.

R. Díaz-Uribe, M. Campos-García, F. S. Granados-Agustín, “Testing the optics of the Large Millimeter Telescope (LMT),” in Infrared Spaceborne Remote Sensing X, M. Strojnik, B. F. Andresen, eds., Proc. SPIE4818, 63–70 (2002).
[CrossRef]

Carmona-Paredes, L.

L. Carmona-Paredes, R. Díaz-Uribe, “Imágenes circulares para pruebas con pantallas nulas,” in Proceedings of Memorias en Extenso; Sesiones de Óptica, C. G. Treviño, ed. (Academia Mexicana de Óptica, León, Gto., Mexico, 2002), pp. 58–60.

L. Carmona-Paredes, R. Díaz-Uribe, “Corrección al cálculo del centroide de las manchas luminosas en la prueba de una superficie esférica,” in Proceedings of Memorias en Extenso; Sesiones de Óptica, C. G. Treviño, ed. (Academia Mexicana de Óptica, León, Gto., Mexico, 2003), pp. 46V03–1–46V03–10.

Díaz-Uribe, R.

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

R. Díaz-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]

R. Díaz-Uribe, M. Campos-García, “Null screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677 (2000).
[CrossRef]

M. Campos-García, R. Díaz-Uribe, F. S. Granados-Agustín, D. Sacramento-Solano, “Null test of aspheric convex surface,” in Proceedings of International Symposium on Photonics in Measurement, VDI-Berichte 1694 (VDI Verlag GmbH, Düsseldorf, Germany, 2002), pp. 155–160.

R. Díaz-Uribe, M. Campos-García, F. S. Granados-Agustín, “Testing the optics of the Large Millimeter Telescope (LMT),” in Infrared Spaceborne Remote Sensing X, M. Strojnik, B. F. Andresen, eds., Proc. SPIE4818, 63–70 (2002).
[CrossRef]

L. Carmona-Paredes, R. Díaz-Uribe, “Corrección al cálculo del centroide de las manchas luminosas en la prueba de una superficie esférica,” in Proceedings of Memorias en Extenso; Sesiones de Óptica, C. G. Treviño, ed. (Academia Mexicana de Óptica, León, Gto., Mexico, 2003), pp. 46V03–1–46V03–10.

L. Carmona-Paredes, R. Díaz-Uribe, “Imágenes circulares para pruebas con pantallas nulas,” in Proceedings of Memorias en Extenso; Sesiones de Óptica, C. G. Treviño, ed. (Academia Mexicana de Óptica, León, Gto., Mexico, 2002), pp. 58–60.

Flannery, B. P.

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

Funes-Maderey, I. E.

I. E. Funes-Maderey, “Videoqueratometría de campo plano” (“Flat field videokeratometry”), B.A. dissertation (Universidad Nacional Autónoma de México, México, 1998).

Granados-Agustín, F. S.

R. Díaz-Uribe, M. Campos-García, F. S. Granados-Agustín, “Testing the optics of the Large Millimeter Telescope (LMT),” in Infrared Spaceborne Remote Sensing X, M. Strojnik, B. F. Andresen, eds., Proc. SPIE4818, 63–70 (2002).
[CrossRef]

M. Campos-García, R. Díaz-Uribe, F. S. Granados-Agustín, D. Sacramento-Solano, “Null test of aspheric convex surface,” in Proceedings of International Symposium on Photonics in Measurement, VDI-Berichte 1694 (VDI Verlag GmbH, Düsseldorf, Germany, 2002), pp. 155–160.

Malacara, D.

Press, W. H.

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

Robinson, D. K.

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

Sacramento-Solano, D.

M. Campos-García, R. Díaz-Uribe, F. S. Granados-Agustín, D. Sacramento-Solano, “Null test of aspheric convex surface,” in Proceedings of International Symposium on Photonics in Measurement, VDI-Berichte 1694 (VDI Verlag GmbH, Düsseldorf, Germany, 2002), pp. 155–160.

Schillke, F.

F. Schillke, “Critical aspects on testing aspheres in interferometric setups,” in Optical Fabrication and Testing, R. Geyl, J. Maxwell, eds., Proc. SPIE3739, 317–324 (1999).
[CrossRef]

Teukolsky, S. A.

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

Vetterling, W. T.

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

Appl. Opt. (4)

Other (11)

L. Carmona-Paredes, R. Díaz-Uribe, “Imágenes circulares para pruebas con pantallas nulas,” in Proceedings of Memorias en Extenso; Sesiones de Óptica, C. G. Treviño, ed. (Academia Mexicana de Óptica, León, Gto., Mexico, 2002), pp. 58–60.

L. Carmona-Paredes, R. Díaz-Uribe, “Corrección al cálculo del centroide de las manchas luminosas en la prueba de una superficie esférica,” in Proceedings of Memorias en Extenso; Sesiones de Óptica, C. G. Treviño, ed. (Academia Mexicana de Óptica, León, Gto., Mexico, 2003), pp. 46V03–1–46V03–10.

W. Rasban, National Institutes of Health, USA: Image Processing and Analysis in Java, ImageJ V. 1.312u, http://rsb.info.nih.gov/ij/ .

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

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

J. H. Burge, “Measurement of large convex aspheres,” in Optical Telescopes of Today and Tomorrow, A. L. Ardeberg, ed., Proc. SPIE2871, 362–373 (1997).
[CrossRef]

F. Schillke, “Critical aspects on testing aspheres in interferometric setups,” in Optical Fabrication and Testing, R. Geyl, J. Maxwell, eds., Proc. SPIE3739, 317–324 (1999).
[CrossRef]

Instituto Nacional de Astrofísica, Óptica y Electrónica, “The Large Millimeter Telescope Homepage,” http://www.lmtgtm.org/ .

I. E. Funes-Maderey, “Videoqueratometría de campo plano” (“Flat field videokeratometry”), B.A. dissertation (Universidad Nacional Autónoma de México, México, 1998).

M. Campos-García, R. Díaz-Uribe, F. S. Granados-Agustín, D. Sacramento-Solano, “Null test of aspheric convex surface,” in Proceedings of International Symposium on Photonics in Measurement, VDI-Berichte 1694 (VDI Verlag GmbH, Düsseldorf, Germany, 2002), pp. 155–160.

R. Díaz-Uribe, M. Campos-García, F. S. Granados-Agustín, “Testing the optics of the Large Millimeter Telescope (LMT),” in Infrared Spaceborne Remote Sensing X, M. Strojnik, B. F. Andresen, eds., Proc. SPIE4818, 63–70 (2002).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Experimental setup for testing fast large convex aspheres.

Fig. 2
Fig. 2

Exact ray-tracing procedure for designing the mask.

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

Percentage error of the components of the approximated normals to the test surface against the sagittal deviations between a spherical surface (k = 0) and a prolate ellipsoid (k = -0.2) used as a reference surface.

Fig. 5
Fig. 5

Numerical integration error for (a) trapezoids and (b) parabolic arcs.

Fig. 6
Fig. 6

Differences in sagittae obtained when the surface is evaluated with trapezoids (open symbols) and parabolic arcs (filled symbols). The circular symbols correspond to differences in sagittae calculated for 15 sources; the square symbols, for 20 sources.

Fig. 7
Fig. 7

Differences in sagittae obtained when a random displacement is added to the coordinates of the centroids of the spots at the CCD. Filled circles, differences in sagittae for η = 0.5 pixels; open circles, for η = 1.0 pixels.

Fig. 8
Fig. 8

Differences in sagittas obtained when a random displacement is added to the coordinates of the positions of the holes of the mask. Filled circles, values of displacement parameter η′ = 0.5 mm; open circles, to η′ = 1 mm.

Fig. 9
Fig. 9

Carbon fiber spherical shell tested in the research reported here, with its mold behind it.

Fig. 10
Fig. 10

(a) Images of the luminous spots of the sources reflected by the test surface. (b) The sizes of the windows used to evaluate the centroids are not necessarily the same because the holes at the mask are positioned at different distances and the roughness of the test surface produces irregular image spots.

Fig. 11
Fig. 11

Assembly in a radial pattern of the coordinates of the positions of the centroid spots.

Fig. 12
Fig. 12

Differences in sagittas between the measured surface and the best-fitting sphere. Here the radii of curvature for the best sphere are (a) r = 309.97 mm and (b) r = 309.66 mm.

Fig. 13
Fig. 13

Spots diagrams at the best image plane for several object positions. The images are located at (a) the innermost position; (b), (c) the middle; and (d) the outermost image.

Tables (5)

Tables Icon

Table 1 Design Parameters for the Test of the Carbon Fiber Spherical Shell

Tables Icon

Table 2 Differences in Sagittas between the Actual Surface and the Evaluated Surface Using Trapezoids and Parabolic Arcs

Tables Icon

Table 3 Differences in Sagittas Obtained with Randomly Displaced Coordinates of the Centroids

Tables Icon

Table 4 Differences in Sagittas Obtained with Randomly Displaced Coordinates of the Holes in the Mask

Tables Icon

Table 5 Least-Squares Fits

Equations (26)

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

x2=aQb+r-a2r2+x12bQb+2r1/2a2Q+x12 x1,z2=x2x1 a-b;
x3=R,z3=ax22+aQz2-r2-2Qz2-rx1x2+aQz2-ar-x1x22-x1Qz2-r2+2x2x1x2+aQz2-ar×-R+x2+z2,
x1=nl,
b=aDd-β,
z-z0=- nxnzdx+nynzdy,
N=rr-ri|rr-ri|,
rr=X, Y, -aX2+Y2+a21/2.
xs= -aQb+r±a2r2-bQb+2rX2+Y21/2Qa2+X2+Y2 X,ys= -aQb+r±a2r2-bQb+2rX2+Y21/2Qa2+X2+Y2 Y,zs= aQb+ra2r2-bQb+2rX2+Y21/2Qa2+X2+Y2 ×a-b.
ri=xs-R, ys-y3, zs-z3xs-R2+ys-y32+zs-z321/2.
N=δx, δy, δzδx2+δy2+δz21/2,
δx=XX2+Y2+a21/2+R-xsxs-R2+ys-y32+zs-z321/2,δy=YX2+Y2+a21/2+y3-ysxs-R2+ys-y32+zs-z321/2,δz=-aX2+Y2+a21/2+z3-zsxs-R2+ys-y32+zs-z321/2.
Na=fx, y, z|fx, y, z|P2,
fx, y, z=Qz2-2rz+x2+y2
zi+1=-u=x,yi=1nai,u3Δui3+bi,u2Δui2+ci,uΔui+zi,
ai,u=ui+2ni+1,u-ni,u+ui+1ni,u-ni+2,u+uini+2,u-ni+1,u/Λi,u,bi,u=-ui+22ni+1,u-ni,u+ui+12ni,u-ni+2,u+ui2ni+2,u-ni+1,u/Λi,u,ci,u=uiui+2ui+2-uini+1,u+ui+12ui+2ni,u-uini+2,u+ui+1ui2ni+2,u-ui+22ni,u/Λi,u,
Λi,u=ui-ui+1ui-ui+2ui+1-ui+2;
nk,u=nu/nz|k
x1, y1x1+δx1, y1+δy1.
δx=η2-2 ln r11/2 cos2πr2,δy=η2-2 ln r11/2 sin2πr2,
Δz- nxanza-nxnzdx+nyanza-nynzdy,
z=r-r2-x-x02+y-y021/2+z0,
fx=ax2+bx+c.
axi2+bxi+c=yi,axi+12+bxi+1+c=yi+1,axi+22+bxi+2+c=yi+2.
a=xi+2yi+1-yi+xi+1yi-yi+2+xiyi+2-yi+1/Λ,b=-xi+22yi+1-yi+xi+12yi-yi+2+xi2yi+2-yi+1/Λ,c=xixi+2xi+2-xiyi+1+xi+12xi+2yi-xiyi+2+xi+1xi2yi+2-xi+22yi/Λ,
Λ=xi-xi+1xi-xi+2xi+1-xi+2.
I=xixi+1ax2+bx+cdx=a3xi+1-xi3+b2xi+1-xi2+cxi+1-xi.

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