Abstract

It is difficult to calculate the wear produced by free-pinned tools because their angular movement is not entirely predictable. We analyze the wear produced with free-pinned ring tools, using both simulations and experiments. We conclude that the wear of an incomplete ring is directly proportional to the ring’s angular size, independently of the mean radius of the ring. We present an algorithm for calculation of the wear produced by free-pinned petal tools, as they can be considered a linear combination of incomplete free-pinned ring tools. Finally, we apply this result to the enhancement of a defective flat surface and to making a concave spheric surface.

© 2005 Optical Society of America

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References

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  1. F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).
  2. C. C. Dévé, Le travail des Verres d’Optique de Precision, Revue d’Optique Theorique et Instrumentale, Paris, 1936; translated by T. L. Tippel as Optical Workshop Principles (Hilger, London, 1945).
  3. M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, L. I. Shevel’kova, “A method of manufacturing aspherical surfaces which deviate only slightly from the sphere,” Sov. J. Opt. Technol. 35, 254–256 (1968).
  4. Yu. K. Lysyannyy, L. S. Tsesnek, “Computation of the contour of a mask tool surface for shaping a concave paraboloid of revolution,” Opt. Technol. 40, 446–448 (1973).
  5. Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, L. N. Khokhlenkov, “The shaping of optical surfaces by the successively corrected mask method,” Sov. J. Opt. Technol. 44, 226–227 (1977).
  6. N. J. Brown, “Axisymmetric aspheric lens figuring for the small shop,” Opt. Eng. 15, 424–427 (1976).
    [CrossRef]
  7. N. J. Brown, “Computationally directed axisymetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
    [CrossRef]
  8. R.-S. Chang, P.-Y. Lee, “Computer simulations of loose abrasive grinding aspherical optical surface by local figuring pitch,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 312–317 (1991).
    [CrossRef]
  9. S. I. Vinokur, “Kinetics of generating plane surfaces,” in Generation of Optical Surfaces,K. J. Kumamin, ed. (Focal, London, 1962), pp. 365–418.
  10. R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, A. Cordero-Dávila, “Análisis cinemático de una máquina pulidora comercial para superficies ópticas,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 2 (2003).
  11. W. D. Dong, E. S. Putilin, Y. V. Rudin, “Modeling the velocity and trajectory of the relative motion of a zone of a workpiece during surface lapping,” J. Opt. Technol. 70, 573–575 (2003).
    [CrossRef]
  12. A. Parra-Flores, A. Cordero-Dávila, J. Cuautle-Cortés, C. Robledo-Sánchez, J. González-García, V. Cabrera-Peláez, “Simulación de desgastes en el pulido de superficies con la ecuación de Preston,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 138 (2003).
  13. A. Cordero-Dávila, J. Gonzalez-Garcia, M. Pedrayes-López, L. A. Aguilar-Chiu, J. Cuautle-Cortés, C. Robledo Sánchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. 43, 1250–1254 (2004).
    [CrossRef] [PubMed]
  14. M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 1–50.
  15. N. R. Draper, H. Smith, Applied Regression Analysis,3rd ed., Wiley Series in Probability and Statistics (Wiley, New York, 1995), pp. 80–83.
  16. A. Bermúdez-López, A. Cordero-Dávila, J. Cuautle-Cortés, “Diseño para la construcción de Herramienta de Pétalo aplicada al pulido de superficies ópticas,” in Program of 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 39 (2003).

2004

2003

1978

N. J. Brown, “Computationally directed axisymetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
[CrossRef]

1977

Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, L. N. Khokhlenkov, “The shaping of optical surfaces by the successively corrected mask method,” Sov. J. Opt. Technol. 44, 226–227 (1977).

1976

N. J. Brown, “Axisymmetric aspheric lens figuring for the small shop,” Opt. Eng. 15, 424–427 (1976).
[CrossRef]

1973

Yu. K. Lysyannyy, L. S. Tsesnek, “Computation of the contour of a mask tool surface for shaping a concave paraboloid of revolution,” Opt. Technol. 40, 446–448 (1973).

1968

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, L. I. Shevel’kova, “A method of manufacturing aspherical surfaces which deviate only slightly from the sphere,” Sov. J. Opt. Technol. 35, 254–256 (1968).

1927

F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).

Aguilar-Chiu, L. A.

Bermúdez-López, A.

A. Bermúdez-López, A. Cordero-Dávila, J. Cuautle-Cortés, “Diseño para la construcción de Herramienta de Pétalo aplicada al pulido de superficies ópticas,” in Program of 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 39 (2003).

Brown, N. J.

N. J. Brown, “Computationally directed axisymetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
[CrossRef]

N. J. Brown, “Axisymmetric aspheric lens figuring for the small shop,” Opt. Eng. 15, 424–427 (1976).
[CrossRef]

Cabrera-Peláez, V.

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle-Cortés, C. Robledo-Sánchez, J. González-García, V. Cabrera-Peláez, “Simulación de desgastes en el pulido de superficies con la ecuación de Preston,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 138 (2003).

Chang, R.-S.

R.-S. Chang, P.-Y. Lee, “Computer simulations of loose abrasive grinding aspherical optical surface by local figuring pitch,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 312–317 (1991).
[CrossRef]

Cordero-Dávila, A.

A. Cordero-Dávila, J. Gonzalez-Garcia, M. Pedrayes-López, L. A. Aguilar-Chiu, J. Cuautle-Cortés, C. Robledo Sánchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. 43, 1250–1254 (2004).
[CrossRef] [PubMed]

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle-Cortés, C. Robledo-Sánchez, J. González-García, V. Cabrera-Peláez, “Simulación de desgastes en el pulido de superficies con la ecuación de Preston,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 138 (2003).

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, A. Cordero-Dávila, “Análisis cinemático de una máquina pulidora comercial para superficies ópticas,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 2 (2003).

A. Bermúdez-López, A. Cordero-Dávila, J. Cuautle-Cortés, “Diseño para la construcción de Herramienta de Pétalo aplicada al pulido de superficies ópticas,” in Program of 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 39 (2003).

Cuautle-Cortés, J.

A. Cordero-Dávila, J. Gonzalez-Garcia, M. Pedrayes-López, L. A. Aguilar-Chiu, J. Cuautle-Cortés, C. Robledo Sánchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. 43, 1250–1254 (2004).
[CrossRef] [PubMed]

A. Bermúdez-López, A. Cordero-Dávila, J. Cuautle-Cortés, “Diseño para la construcción de Herramienta de Pétalo aplicada al pulido de superficies ópticas,” in Program of 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 39 (2003).

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle-Cortés, C. Robledo-Sánchez, J. González-García, V. Cabrera-Peláez, “Simulación de desgastes en el pulido de superficies con la ecuación de Preston,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 138 (2003).

Dévé, C. C.

C. C. Dévé, Le travail des Verres d’Optique de Precision, Revue d’Optique Theorique et Instrumentale, Paris, 1936; translated by T. L. Tippel as Optical Workshop Principles (Hilger, London, 1945).

Dong, W. D.

Draper, N. R.

N. R. Draper, H. Smith, Applied Regression Analysis,3rd ed., Wiley Series in Probability and Statistics (Wiley, New York, 1995), pp. 80–83.

Golovanova, M. N.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, L. I. Shevel’kova, “A method of manufacturing aspherical surfaces which deviate only slightly from the sphere,” Sov. J. Opt. Technol. 35, 254–256 (1968).

González-Castillo, R.

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, A. Cordero-Dávila, “Análisis cinemático de una máquina pulidora comercial para superficies ópticas,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 2 (2003).

Gonzalez-Garcia, J.

González-García, J.

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, A. Cordero-Dávila, “Análisis cinemático de una máquina pulidora comercial para superficies ópticas,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 2 (2003).

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle-Cortés, C. Robledo-Sánchez, J. González-García, V. Cabrera-Peláez, “Simulación de desgastes en el pulido de superficies con la ecuación de Preston,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 138 (2003).

Gurevich, L. N.

Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, L. N. Khokhlenkov, “The shaping of optical surfaces by the successively corrected mask method,” Sov. J. Opt. Technol. 44, 226–227 (1977).

Kachkin, S. S.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, L. I. Shevel’kova, “A method of manufacturing aspherical surfaces which deviate only slightly from the sphere,” Sov. J. Opt. Technol. 35, 254–256 (1968).

Khokhlenkov, L. N.

Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, L. N. Khokhlenkov, “The shaping of optical surfaces by the successively corrected mask method,” Sov. J. Opt. Technol. 44, 226–227 (1977).

Krylova, Ye. I.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, L. I. Shevel’kova, “A method of manufacturing aspherical surfaces which deviate only slightly from the sphere,” Sov. J. Opt. Technol. 35, 254–256 (1968).

Lee, P.-Y.

R.-S. Chang, P.-Y. Lee, “Computer simulations of loose abrasive grinding aspherical optical surface by local figuring pitch,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 312–317 (1991).
[CrossRef]

Lysyannyy, Yu. K.

Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, L. N. Khokhlenkov, “The shaping of optical surfaces by the successively corrected mask method,” Sov. J. Opt. Technol. 44, 226–227 (1977).

Yu. K. Lysyannyy, L. S. Tsesnek, “Computation of the contour of a mask tool surface for shaping a concave paraboloid of revolution,” Opt. Technol. 40, 446–448 (1973).

Mantravadi, M. V.

M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 1–50.

Parra-Flores, A.

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, A. Cordero-Dávila, “Análisis cinemático de una máquina pulidora comercial para superficies ópticas,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 2 (2003).

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle-Cortés, C. Robledo-Sánchez, J. González-García, V. Cabrera-Peláez, “Simulación de desgastes en el pulido de superficies con la ecuación de Preston,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 138 (2003).

Pedrayes-López, M.

Preston, F. W.

F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).

Putilin, E. S.

Robledo Sánchez, C.

Robledo-Sánchez, C.

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle-Cortés, C. Robledo-Sánchez, J. González-García, V. Cabrera-Peláez, “Simulación de desgastes en el pulido de superficies con la ecuación de Preston,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 138 (2003).

Rudin, Y. V.

Shevel’kova, L. I.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, L. I. Shevel’kova, “A method of manufacturing aspherical surfaces which deviate only slightly from the sphere,” Sov. J. Opt. Technol. 35, 254–256 (1968).

Smith, H.

N. R. Draper, H. Smith, Applied Regression Analysis,3rd ed., Wiley Series in Probability and Statistics (Wiley, New York, 1995), pp. 80–83.

Tsesnek, L. S.

Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, L. N. Khokhlenkov, “The shaping of optical surfaces by the successively corrected mask method,” Sov. J. Opt. Technol. 44, 226–227 (1977).

Yu. K. Lysyannyy, L. S. Tsesnek, “Computation of the contour of a mask tool surface for shaping a concave paraboloid of revolution,” Opt. Technol. 40, 446–448 (1973).

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, L. I. Shevel’kova, “A method of manufacturing aspherical surfaces which deviate only slightly from the sphere,” Sov. J. Opt. Technol. 35, 254–256 (1968).

Venegas-Pérez, L.

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, A. Cordero-Dávila, “Análisis cinemático de una máquina pulidora comercial para superficies ópticas,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 2 (2003).

Vinokur, S. I.

S. I. Vinokur, “Kinetics of generating plane surfaces,” in Generation of Optical Surfaces,K. J. Kumamin, ed. (Focal, London, 1962), pp. 365–418.

Appl. Opt.

J. Opt. Technol.

J. Soc. Glass Technol.

F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).

Opt. Eng.

N. J. Brown, “Axisymmetric aspheric lens figuring for the small shop,” Opt. Eng. 15, 424–427 (1976).
[CrossRef]

N. J. Brown, “Computationally directed axisymetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
[CrossRef]

Opt. Technol.

Yu. K. Lysyannyy, L. S. Tsesnek, “Computation of the contour of a mask tool surface for shaping a concave paraboloid of revolution,” Opt. Technol. 40, 446–448 (1973).

Sov. J. Opt. Technol.

Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, L. N. Khokhlenkov, “The shaping of optical surfaces by the successively corrected mask method,” Sov. J. Opt. Technol. 44, 226–227 (1977).

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, L. I. Shevel’kova, “A method of manufacturing aspherical surfaces which deviate only slightly from the sphere,” Sov. J. Opt. Technol. 35, 254–256 (1968).

Other

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle-Cortés, C. Robledo-Sánchez, J. González-García, V. Cabrera-Peláez, “Simulación de desgastes en el pulido de superficies con la ecuación de Preston,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 138 (2003).

M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 1–50.

N. R. Draper, H. Smith, Applied Regression Analysis,3rd ed., Wiley Series in Probability and Statistics (Wiley, New York, 1995), pp. 80–83.

A. Bermúdez-López, A. Cordero-Dávila, J. Cuautle-Cortés, “Diseño para la construcción de Herramienta de Pétalo aplicada al pulido de superficies ópticas,” in Program of 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 39 (2003).

C. C. Dévé, Le travail des Verres d’Optique de Precision, Revue d’Optique Theorique et Instrumentale, Paris, 1936; translated by T. L. Tippel as Optical Workshop Principles (Hilger, London, 1945).

R.-S. Chang, P.-Y. Lee, “Computer simulations of loose abrasive grinding aspherical optical surface by local figuring pitch,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 312–317 (1991).
[CrossRef]

S. I. Vinokur, “Kinetics of generating plane surfaces,” in Generation of Optical Surfaces,K. J. Kumamin, ed. (Focal, London, 1962), pp. 365–418.

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, A. Cordero-Dávila, “Análisis cinemático de una máquina pulidora comercial para superficies ópticas,” in Program of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl.49, 2 (2003).

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Figures (18)

Fig. 1
Fig. 1

Points of contact between the tool and the glass that were used in our calculation of wear.

Fig. 2
Fig. 2

Schematic of the tool form commonly used in the polishing of optical surfaces: (a) solid, (b) complete ring, (c) segmented ring, and (d) petal.

Fig. 3
Fig. 3

Normalized graph of the simulated wear profile produced by a tool of two rings over the diameter of the glass.

Fig. 4
Fig. 4

Normalized graph of the simulated wear profile produced by a tool of three incomplete rings of angular sizes 360°, 240°, and 120° over the diameter of the glass.

Fig. 5
Fig. 5

Interferograms obtained in the first experiment with the double-ring tool oscillating about the rotating center of the glass: (a) initial, (b) 20 min.

Fig. 6
Fig. 6

Loci of 95% confidence bands with the experimental two-ring tool.

Fig. 7
Fig. 7

Loci of 95% confidence bands with the experimental three-ring tool.

Fig. 8
Fig. 8

Speed of wear versus angular size of a segmented three-ring tool.

Fig. 9
Fig. 9

Interferogram of the defective flat surface to be corrected.

Fig. 10
Fig. 10

Profile of the skeletonized central fringe, showing the desired wear.

Fig. 11
Fig. 11

Comparison of desired and simulated wear profiles.

Fig. 12
Fig. 12

Schematic of the tool calculated to achieve the desired wear.

Fig. 13
Fig. 13

Interferogram of the surface after polishing with the calculated tool.

Fig. 14
Fig. 14

Schematic of a calculated petal tool.

Fig. 15
Fig. 15

Comparison of the desired wear profile and that obtained with the petal tool calculated over the diameter of the surface.

Fig. 16
Fig. 16

Photograph of the tool used to hyperbolize a spherical glass blank of 14-cm diameter.

Fig. 17
Fig. 17

Ronchigram and simulation of the hyperbole produced with the petal tool.

Fig. 18
Fig. 18

Depths obtained with the Ronchi test on the surface.

Equations (4)

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h = 0 τ A p v d t ,
h j = A p Δ t i = 1 N t v i j ,
Δ t = τ / N t
α j + 1 = α j + Δ α k ,

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