Abstract

We have reproduced quantitatively the technique commonly used in optical shop to evaluate surface error from comparison between experimental and simulated Ronchigrams. We used this procedure to evaluate, from Ronchigrams of any number of fringes, the curvature radius and/or conic constant of conic surfaces. The error function is calculated without using integration (numerical or polynomial) so the corresponding problems were avoided. Furthermore, when the error function is described with cubic splines, then the local errors are very well reproduced, which is not the case with the polynomial description. We have described the error functions with conical surfaces or with cubic splines, and for the best reproduction of experimental Ronchigram we used genetic algorithms.

© 2011 Optical Society of America

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References

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  1. A. Cornejo-Rodríguez, “Ronchi test,” in Optical Shop Testing, 3rd ed., D.Malacara (ed.) (Wiley and Sons, 2007), pp. 317–360.
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    [CrossRef]
  9. J. González-García, A. Cordero-Dávila, I. Leal-Cabrera, C. I. Robledo-Sánchez, and A. Santiago-Alvarado, “Calculating petal tools using genetic algorithms,” Appl. Opt. 45, 6126–6136 (2006).
    [CrossRef] [PubMed]
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  15. A. U. Rivera-Ortega and A. y Cordero-Dávila, “Evaluación de superficies ópticas utilizando la prueba de Ronchi y la fórmula de aberración transversal de Malacara,” in Program of the LII Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 21, 59 (2009).
  16. C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001).
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  18. J. Rosaura Kantún-Montiel, A. Cordero-Dávila, and J. González-García, “Quantitative surface evaluation by matching experimental and simulated Ronchigram images,” to be presented and published in the 22nd General Congress of the International Commission for Optics (ICO-22).

2010 (1)

A. Cordero-Dávila and J. González-García, “Surface evaluation with Ronchi test by using Malacara formula, genetic algorithms and cubic splines,” Proc. SPIE 7652, 76521F(2010).
[CrossRef]

2009 (1)

A. U. Rivera-Ortega and A. y Cordero-Dávila, “Evaluación de superficies ópticas utilizando la prueba de Ronchi y la fórmula de aberración transversal de Malacara,” in Program of the LII Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 21, 59 (2009).

2008 (1)

2006 (1)

2001 (1)

2000 (3)

1984 (1)

1971 (2)

1964 (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Arasa, J.

Canty, B. R.

Cordero, A.

A. Cornejo and A. Cordero “Wavefront slope measurements in optical testing” in Optical Engineering, D.Malacara and B.J.Thompson (eds.) (Marcel Dekker, 2001), pp. 311–338.

Cordero-Dávila, A.

A. Cordero-Dávila and J. González-García, “Surface evaluation with Ronchi test by using Malacara formula, genetic algorithms and cubic splines,” Proc. SPIE 7652, 76521F(2010).
[CrossRef]

J. González-García, A. Cordero-Dávila, I. Leal-Cabrera, C. I. Robledo-Sánchez, and A. Santiago-Alvarado, “Calculating petal tools using genetic algorithms,” Appl. Opt. 45, 6126–6136 (2006).
[CrossRef] [PubMed]

A. Cordero-Dávila, J. M. Núñez-Alfonso, E. Luna-Aguilar, and C. I. Robledo-Sánchez, “Only one fitting for the bironchigrams,” Appl. Opt. 40, 5600–5609 (2001).
[CrossRef]

J. Rosaura Kantún-Montiel, A. Cordero-Dávila, and J. González-García, “Quantitative surface evaluation by matching experimental and simulated Ronchigram images,” to be presented and published in the 22nd General Congress of the International Commission for Optics (ICO-22).

Cornejo, A.

A. Cornejo and A. Cordero “Wavefront slope measurements in optical testing” in Optical Engineering, D.Malacara and B.J.Thompson (eds.) (Marcel Dekker, 2001), pp. 311–338.

Cornejo-Rodríguez, A.

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

de Boor, C.

C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001).

Donner, K.

Fang, F. Z.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 2007).

González-García, J.

A. Cordero-Dávila and J. González-García, “Surface evaluation with Ronchi test by using Malacara formula, genetic algorithms and cubic splines,” Proc. SPIE 7652, 76521F(2010).
[CrossRef]

J. González-García, A. Cordero-Dávila, I. Leal-Cabrera, C. I. Robledo-Sánchez, and A. Santiago-Alvarado, “Calculating petal tools using genetic algorithms,” Appl. Opt. 45, 6126–6136 (2006).
[CrossRef] [PubMed]

J. Rosaura Kantún-Montiel, A. Cordero-Dávila, and J. González-García, “Quantitative surface evaluation by matching experimental and simulated Ronchigram images,” to be presented and published in the 22nd General Congress of the International Commission for Optics (ICO-22).

Groening, S.

Hu, X. T.

Kantún-Montiel, J. Rosaura

J. Rosaura Kantún-Montiel, A. Cordero-Dávila, and J. González-García, “Quantitative surface evaluation by matching experimental and simulated Ronchigram images,” to be presented and published in the 22nd General Congress of the International Commission for Optics (ICO-22).

Leal-Cabrera, I.

Lindlein, N.

Luna-Aguilar, E.

Malacara-Hernández, D.

D. Malacara-Hernández, “Testing of optical surfaces,” Ph.D. dissertation (The University of Rochester, 1965).

Núñez-Alfonso, J. M.

Pfund, J.

Pizarro, C.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 2007).

Rayces, J. L.

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Rigler, A. K.

Rivera-Ortega, A. U.

A. U. Rivera-Ortega and A. y Cordero-Dávila, “Evaluación de superficies ópticas utilizando la prueba de Ronchi y la fórmula de aberración transversal de Malacara,” in Program of the LII Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 21, 59 (2009).

Robledo-Sánchez, C. I.

Royo, S.

Santiago-Alvarado, A.

Schwider, J.

Sick, B.

Stacy, J. E.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 2007).

Tomàs, N.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 2007).

Vogl, T. P.

y Cordero-Dávila, A.

A. U. Rivera-Ortega and A. y Cordero-Dávila, “Evaluación de superficies ópticas utilizando la prueba de Ronchi y la fórmula de aberración transversal de Malacara,” in Program of the LII Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 21, 59 (2009).

Zhang, X. D.

Appl. Opt. (8)

Opt. Acta (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Opt. Express (1)

Proc. SPIE (1)

A. Cordero-Dávila and J. González-García, “Surface evaluation with Ronchi test by using Malacara formula, genetic algorithms and cubic splines,” Proc. SPIE 7652, 76521F(2010).
[CrossRef]

Program of the LII Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. (1)

A. U. Rivera-Ortega and A. y Cordero-Dávila, “Evaluación de superficies ópticas utilizando la prueba de Ronchi y la fórmula de aberración transversal de Malacara,” in Program of the LII Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 21, 59 (2009).

Other (6)

C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge University, 2007).

J. Rosaura Kantún-Montiel, A. Cordero-Dávila, and J. González-García, “Quantitative surface evaluation by matching experimental and simulated Ronchigram images,” to be presented and published in the 22nd General Congress of the International Commission for Optics (ICO-22).

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

A. Cornejo and A. Cordero “Wavefront slope measurements in optical testing” in Optical Engineering, D.Malacara and B.J.Thompson (eds.) (Marcel Dekker, 2001), pp. 311–338.

D. Malacara-Hernández, “Testing of optical surfaces,” Ph.D. dissertation (The University of Rochester, 1965).

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Figures (12)

Fig. 1
Fig. 1

Intersecting coordinates of the fringes of a Ronchigram with the X axis.

Fig. 2
Fig. 2

A schematic diagram showing the configuration used in the Ronchi test.

Fig. 3
Fig. 3

Ronchigram recorded of a 20 cm diameter spherical surface of f / 1.13 and l f = l r = 44.6 cm .

Fig. 4
Fig. 4

Irradiance graph on the X axis of the Ronchigram of the f / 1.13 of the mirror.

Fig. 5
Fig. 5

Gaussian fit made of the zero-order fringe of the Ronchigram recorded of the f / 1.13 of the mirror.

Fig. 6
Fig. 6

Ronchigrams recorded of a 14 cm diameter mirror with a 2.14 f-number, with (a) nine fringes, (b) eleven fringes, and (c) thirteen fringes.

Fig. 7
Fig. 7

Ronchigram used to obtain the best fit conic surface for the surface being tested. The data of the surface are 14 cm diameter and at a distance of 59.38 cm from the source to the vertex.

Fig. 8
Fig. 8

Comparison of simulated and experimental Ronchigrams during the polishing process for the manufacture of optical surfaces.

Fig. 9
Fig. 9

Graphic irradiance on the X axis of the Ronchigram of the 14 cm diameter elliptical surface. Graphic used to calculate the best conic fit for the surface.

Fig. 10
Fig. 10

Surface used to calculate error. Sphere in the process of parabolization.

Fig. 11
Fig. 11

Graph of irradiance on the X axis of the Ronchigram of the sphere in the process of parabolization.

Fig. 12
Fig. 12

Error graph of the sphere in process of parabolization. (a) Error calculated with cubic splines and genetic algorithm; (b) error calculated with polynomial fit and genetic algorithm.

Tables (3)

Tables Icon

Table 1 Results Obtained from the Coordinates of the Experimental and Simulated Fringes of the f / 1.13 Mirror

Tables Icon

Table 2 Results Obtained from the Genetic Algorithm of the Three Ronchigrams Recorded of the f / 2.14 Mirror

Tables Icon

Table 3 Results Obtained from the Coordinates of the Experimental Fringes of the 14 cm Diameter Elliptical Surface

Equations (10)

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

S 2 = i = 1 N f ( X Ei X Si ) 2 ,
Z S ( ρ ) = Z I ( ρ ) + Δ Z ( ρ ) ,
Z I ( ρ ) = 0 ,
Δ Z ( ρ ) = c X 2 1 + 1 ( k + 1 ) c 2 X 2 .
T ( ρ ) = ( l f + l r 2 Δ Z ) [ 1 ( d Δ Z d ρ ) 2 ] + 2 d Δ Z d ρ [ ρ ( l r Δ Z ) ( l f Δ Z ) ρ ] l f Δ Z ρ [ 1 ( d Δ Z d ρ ) 2 ] + 2 d Δ Z d ρ .
Z S ( ρ ) = Z I ( ρ ) + Δ Z ( ρ , α 1 , α 2 , α M ) ,
Δ Z ( ρ ) = A 2 ρ 2 + A 4 ρ 4 + A 6 ρ 6 + A 8 ρ 8 ,
Z I ( X ) = c X 2 1 + 1 ( k + 1 ) c 2 X 2 .
Δ Z j ( X ) = A 0 j + A 1 j X + A 2 j X 2 + A 3 j X 3 ,
Δ ( x ) = Δ ( x , Δ 1 , Δ 2 , , Δ N f ) .

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