Abstract

The smoothing effect of the rigid lap plays an important role in controlling midspatial frequency errors (MSFRs). At present, the pressure distribution between the polishing pad and processed surface is mainly calculated by Mehta’s bridging model. However, this classic model does not work for the irregular MSFR. In this paper, a generalized numerical model based on the finite element method (FEM) is proposed to solve this problem. First, the smoothing polishing (SP) process is transformed to a 3D elastic structural FEM model, and the governing matrix equation is gained. By virtue of the boundary conditions applied to the governing matrix equation, the nodal displacement vector and nodal force vector of the pad can be attained, from which the pressure distribution can be extracted. In the partial contact condition, the iterative method is needed. The algorithmic routine is shown, and the applicability of the generalized numerical model is discussed. The detailed simulation is given when the lap is in contact with the irregular surface of different morphologies. A well-designed SP experiment is conducted in our lab to verify the model. A small difference between the experimental data and simulated result shows that the model is totally practicable. The generalized numerical model is applied on a Φ500mm parabolic surface. The calculated result and measured data after the SP process have been compared, which indicates that the model established in this paper is an effective method to predict the SP process.

© 2014 Optical Society of America

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References

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  1. T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
    [CrossRef]
  2. J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
    [CrossRef]
  3. D. D. Walker, D. Brooks, A. King, R. Freeman, R. Morton, G. McCavana, and S. W. Kim, “The ‘Precessions’ tooling for polishing and figuring flat, spherical and aspheric surfaces,” Opt. Express 11, 958–964 (2003).
    [CrossRef]
  4. F. Shi, “Study on the key techniques of magneto-rheological finishing for high-precision optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2009) (in Chinese).
  5. L. Zhou, “Study on theory and technology in ion beam figuring for optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2008) (in Chinese).
  6. N. J. Brown and R. E. Parks, “The polishing-to-figuring transition in turned optics,” in SPIE’s 25th Annual International Technical Symposium (SPIE, 1981).
  7. R. A. Jones, “Computer simulation of smoothing during computer-controlled optical polishing,” Appl. Opt. 34, 1162–1169 (1995).
    [CrossRef]
  8. P. K. Mehta and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” Proc. SPIE 3786, 447–459 (1999).
    [CrossRef]
  9. J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
    [CrossRef]
  10. M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41, 1473–1474 (2002).
    [CrossRef]
  11. D. W. Kim, W. H. Park, H. K. An, and J. H. Burge, “Parametric smoothing model for visco-elastic polishing tools,” Opt. Express 18, 22515–22526 (2010).
    [CrossRef]
  12. Y. Shu, D. W. Kim, H. M. Martin, and J. H. Burge, “Correlation-based smoothing model for optical polishing,” Opt. Express 21, 28771–28782 (2013).
    [CrossRef]
  13. J. A. Greenwood and J. B. Williamson, “Contact of nominally flat surfaces,” Proc. Roy. Soc. A 295, 300–319 (1966).
    [CrossRef]
  14. J. J. Vlassak, “A model for chemical-mechanical polishing of a material surface based on contact mechanics,” J. Mech. Phys. Solids 52, 847–873 (2004).
    [CrossRef]
  15. B. Vasilev, S. Bott, R. Rzehak, P. Kücher, and J. W. Bartha, “A feature scale Greenwood–Williamson model predicting pattern-size effects in CMP,” Microelectron. Eng. 91, 159–166 (2012).
    [CrossRef]
  16. D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
    [CrossRef]
  17. M. D. Feit, R. P. DesJardin, W. A. Steele, and T. I. Suratwala, “Optimized pitch button blocking for polishing high-aspect-ratio optics,” Appl. Opt. 51, 8350–8359 (2012).
    [CrossRef]
  18. R. Varshneya, J. E. DeGroote, L. L. Gregg, and S. D. Jacobs, “Characterizing optical polishing pitch,” Proc. SPIE TD02, 87–89 (2003).
  19. S. S. Rao, The Finite Element Method in Engineering (Pergamon, 1989).
  20. J. Alberty, C. Carstensen, S. A. Funken, and R. Klose, “Matlab implementation of the finite element method in elasticity,” Computing 69, 239–263 (2002).
  21. F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).

2013

2012

M. D. Feit, R. P. DesJardin, W. A. Steele, and T. I. Suratwala, “Optimized pitch button blocking for polishing high-aspect-ratio optics,” Appl. Opt. 51, 8350–8359 (2012).
[CrossRef]

B. Vasilev, S. Bott, R. Rzehak, P. Kücher, and J. W. Bartha, “A feature scale Greenwood–Williamson model predicting pattern-size effects in CMP,” Microelectron. Eng. 91, 159–166 (2012).
[CrossRef]

2010

2006

T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

2004

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

J. J. Vlassak, “A model for chemical-mechanical polishing of a material surface based on contact mechanics,” J. Mech. Phys. Solids 52, 847–873 (2004).
[CrossRef]

2003

2002

J. Alberty, C. Carstensen, S. A. Funken, and R. Klose, “Matlab implementation of the finite element method in elasticity,” Computing 69, 239–263 (2002).

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41, 1473–1474 (2002).
[CrossRef]

2001

J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
[CrossRef]

1999

P. K. Mehta and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” Proc. SPIE 3786, 447–459 (1999).
[CrossRef]

1997

D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
[CrossRef]

1995

1966

J. A. Greenwood and J. B. Williamson, “Contact of nominally flat surfaces,” Proc. Roy. Soc. A 295, 300–319 (1966).
[CrossRef]

1927

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

Alberty, J.

J. Alberty, C. Carstensen, S. A. Funken, and R. Klose, “Matlab implementation of the finite element method in elasticity,” Computing 69, 239–263 (2002).

An, H. K.

Anderson, B.

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41, 1473–1474 (2002).
[CrossRef]

J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
[CrossRef]

Asami, T.

T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

Bartha, J. W.

B. Vasilev, S. Bott, R. Rzehak, P. Kücher, and J. W. Bartha, “A feature scale Greenwood–Williamson model predicting pattern-size effects in CMP,” Microelectron. Eng. 91, 159–166 (2012).
[CrossRef]

Beaudoin, S.

D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
[CrossRef]

Benjamin, S.

J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
[CrossRef]

Bibby, T.

D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
[CrossRef]

Borden, M. R.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Bott, S.

B. Vasilev, S. Bott, R. Rzehak, P. Kücher, and J. W. Bartha, “A feature scale Greenwood–Williamson model predicting pattern-size effects in CMP,” Microelectron. Eng. 91, 159–166 (2012).
[CrossRef]

Brooks, D.

Brown, N. J.

N. J. Brown and R. E. Parks, “The polishing-to-figuring transition in turned optics,” in SPIE’s 25th Annual International Technical Symposium (SPIE, 1981).

Burge, J. H.

Y. Shu, D. W. Kim, H. M. Martin, and J. H. Burge, “Correlation-based smoothing model for optical polishing,” Opt. Express 21, 28771–28782 (2013).
[CrossRef]

D. W. Kim, W. H. Park, H. K. An, and J. H. Burge, “Parametric smoothing model for visco-elastic polishing tools,” Opt. Express 18, 22515–22526 (2010).
[CrossRef]

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41, 1473–1474 (2002).
[CrossRef]

J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
[CrossRef]

Cale, T.

D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
[CrossRef]

Campbell, J. H.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Carstensen, C.

J. Alberty, C. Carstensen, S. A. Funken, and R. Klose, “Matlab implementation of the finite element method in elasticity,” Computing 69, 239–263 (2002).

Cho, M.

J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
[CrossRef]

DeGroote, J. E.

R. Varshneya, J. E. DeGroote, L. L. Gregg, and S. D. Jacobs, “Characterizing optical polishing pitch,” Proc. SPIE TD02, 87–89 (2003).

DesJardin, R. P.

Feit, M. D.

M. D. Feit, R. P. DesJardin, W. A. Steele, and T. I. Suratwala, “Optimized pitch button blocking for polishing high-aspect-ratio optics,” Appl. Opt. 51, 8350–8359 (2012).
[CrossRef]

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Freeman, R.

Funken, S. A.

J. Alberty, C. Carstensen, S. A. Funken, and R. Klose, “Matlab implementation of the finite element method in elasticity,” Computing 69, 239–263 (2002).

Greenwood, J. A.

J. A. Greenwood and J. B. Williamson, “Contact of nominally flat surfaces,” Proc. Roy. Soc. A 295, 300–319 (1966).
[CrossRef]

Gregg, L. L.

R. Varshneya, J. E. DeGroote, L. L. Gregg, and S. D. Jacobs, “Characterizing optical polishing pitch,” Proc. SPIE TD02, 87–89 (2003).

Hackel, R. P.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Hawley-Fedder, R. A.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Holland, K.

D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
[CrossRef]

Jacobs, S. D.

R. Varshneya, J. E. DeGroote, L. L. Gregg, and S. D. Jacobs, “Characterizing optical polishing pitch,” Proc. SPIE TD02, 87–89 (2003).

Jones, R. A.

Kim, D. W.

Kim, S. W.

King, A.

Klose, R.

J. Alberty, C. Carstensen, S. A. Funken, and R. Klose, “Matlab implementation of the finite element method in elasticity,” Computing 69, 239–263 (2002).

Kohama, Y.

T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

Kücher, P.

B. Vasilev, S. Bott, R. Rzehak, P. Kücher, and J. W. Bartha, “A feature scale Greenwood–Williamson model predicting pattern-size effects in CMP,” Microelectron. Eng. 91, 159–166 (2012).
[CrossRef]

Lee, J.

D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
[CrossRef]

Martin, H. M.

McCavana, G.

Mehta, P. K.

P. K. Mehta and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” Proc. SPIE 3786, 447–459 (1999).
[CrossRef]

Menapace, J. A.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Miura, T.

T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

Morton, R.

Murakami, K.

T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

Ohkubo, Y.

T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

Park, W. H.

Parks, R. E.

N. J. Brown and R. E. Parks, “The polishing-to-figuring transition in turned optics,” in SPIE’s 25th Annual International Technical Symposium (SPIE, 1981).

Preston, F. W.

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

Rao, S. S.

S. S. Rao, The Finite Element Method in Engineering (Pergamon, 1989).

Reid, P. B.

P. K. Mehta and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” Proc. SPIE 3786, 447–459 (1999).
[CrossRef]

Riley, M. O.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Runkel, M.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Rzehak, R.

B. Vasilev, S. Bott, R. Rzehak, P. Kücher, and J. W. Bartha, “A feature scale Greenwood–Williamson model predicting pattern-size effects in CMP,” Microelectron. Eng. 91, 159–166 (2012).
[CrossRef]

Shi, F.

F. Shi, “Study on the key techniques of magneto-rheological finishing for high-precision optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2009) (in Chinese).

Shu, Y.

Smith, K.

J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
[CrossRef]

Steele, W. A.

Stolz, C. J.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Suratwala, T. I.

Suzuki, K.

T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

Tuell, M. T.

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41, 1473–1474 (2002).
[CrossRef]

Valente, M.

J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
[CrossRef]

Varshneya, R.

R. Varshneya, J. E. DeGroote, L. L. Gregg, and S. D. Jacobs, “Characterizing optical polishing pitch,” Proc. SPIE TD02, 87–89 (2003).

Vasilev, B.

B. Vasilev, S. Bott, R. Rzehak, P. Kücher, and J. W. Bartha, “A feature scale Greenwood–Williamson model predicting pattern-size effects in CMP,” Microelectron. Eng. 91, 159–166 (2012).
[CrossRef]

Vlassak, J. J.

J. J. Vlassak, “A model for chemical-mechanical polishing of a material surface based on contact mechanics,” J. Mech. Phys. Solids 52, 847–873 (2004).
[CrossRef]

Walker, D. D.

Wang, D.

D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
[CrossRef]

Whitman, P. K.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Williamson, J. B.

J. A. Greenwood and J. B. Williamson, “Contact of nominally flat surfaces,” Proc. Roy. Soc. A 295, 300–319 (1966).
[CrossRef]

Yu, J.

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Zhou, L.

L. Zhou, “Study on theory and technology in ion beam figuring for optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2008) (in Chinese).

Appl. Opt.

Computing

J. Alberty, C. Carstensen, S. A. Funken, and R. Klose, “Matlab implementation of the finite element method in elasticity,” Computing 69, 239–263 (2002).

J. Electrochem. Soc.

D. Wang, J. Lee, K. Holland, T. Bibby, S. Beaudoin, and T. Cale, “Von Mises stress in chemical-mechanical polishing processes,” J. Electrochem. Soc. 144, 1121–1127 (1997).
[CrossRef]

J. Mech. Phys. Solids

J. J. Vlassak, “A model for chemical-mechanical polishing of a material surface based on contact mechanics,” J. Mech. Phys. Solids 52, 847–873 (2004).
[CrossRef]

J. Soc. Glass Technol.

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

Microelectron. Eng.

B. Vasilev, S. Bott, R. Rzehak, P. Kücher, and J. W. Bartha, “A feature scale Greenwood–Williamson model predicting pattern-size effects in CMP,” Microelectron. Eng. 91, 159–166 (2012).
[CrossRef]

Opt. Eng.

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41, 1473–1474 (2002).
[CrossRef]

Opt. Express

Proc. Roy. Soc. A

J. A. Greenwood and J. B. Williamson, “Contact of nominally flat surfaces,” Proc. Roy. Soc. A 295, 300–319 (1966).
[CrossRef]

Proc. SPIE

P. K. Mehta and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” Proc. SPIE 3786, 447–459 (1999).
[CrossRef]

J. H. Burge, B. Anderson, S. Benjamin, M. Cho, K. Smith, and M. Valente, “Development of optimal grinding and polishing tools for aspheric surfaces,” Proc. SPIE 4451, 153–164 (2001).
[CrossRef]

R. Varshneya, J. E. DeGroote, L. L. Gregg, and S. D. Jacobs, “Characterizing optical polishing pitch,” Proc. SPIE TD02, 87–89 (2003).

T. Miura, K. Murakami, K. Suzuki, Y. Kohama, Y. Ohkubo, and T. Asami, “Nikon EUVL development progress summary,” Proc. SPIE 6151, 615105 (2006).
[CrossRef]

J. H. Campbell, R. A. Hawley-Fedder, C. J. Stolz, J. A. Menapace, M. R. Borden, P. K. Whitman, J. Yu, M. Runkel, M. O. Riley, M. D. Feit, and R. P. Hackel, “NIF optical material and fabrication technologies: an overview,” Proc. SPIE 5341, 84–101 (2004).
[CrossRef]

Other

S. S. Rao, The Finite Element Method in Engineering (Pergamon, 1989).

F. Shi, “Study on the key techniques of magneto-rheological finishing for high-precision optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2009) (in Chinese).

L. Zhou, “Study on theory and technology in ion beam figuring for optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2008) (in Chinese).

N. J. Brown and R. E. Parks, “The polishing-to-figuring transition in turned optics,” in SPIE’s 25th Annual International Technical Symposium (SPIE, 1981).

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Figures (12)

Fig. 1.
Fig. 1.

Sketch map of polishing process: (a) CCOS process and (b) contact status of the polishing pad and processed surface.

Fig. 2.
Fig. 2.

Meshed FEM model of the polishing pad (not to scale): (a) front view and (b) top view.

Fig. 3.
Fig. 3.

Iteration flow chart of the generalized numerical model.

Fig. 4.
Fig. 4.

Deformation of the pad’s lower surface when the lap is in full contact with a single-frequency sine-wave error: (a) data in full view and (b) data from y=10mm.

Fig. 5.
Fig. 5.

Pressure distribution of the pad’s lower surface when the lap is in full contact with a single-frequency sine-wave error: (a) data in full view and (b) data from y=10mm.

Fig. 6.
Fig. 6.

Calculated results of the pad’s lower surface when the lap is in partial contact with a single-frequency sine-wave error: (a) deformation and (b) pressure distribution.

Fig. 7.
Fig. 7.

Typical MSFR features after rough finishing process: (a) surface map (b) trianglelike error, (c) Gauss-like error, and (d) rectanglelike error.

Fig. 8.
Fig. 8.

Pressure distribution result of three kinds of MSFR: (a) triangular error, (b) sine error, and (c) Gaussian error.

Fig. 9.
Fig. 9.

Verification experiment: (a) sketch map of SP process and (b) surface error after 54-min SP process.

Fig. 10.
Fig. 10.

Experiment data (solid line) and simulated result (solid square marker) of the MSFR evolution in every SP stage.

Fig. 11.
Fig. 11.

SP process of a Φ500mm parabolic surface.

Fig. 12.
Fig. 12.

Comparison of the calculated SP result and measured data: (a) measured data after 3 h SP process, (b) calculated pressure distribution in the SP process, (c) calculated result after SP process, and (d) error morphology evolution along Profile 3. Note that the blank holes in (b) and (c) are due to the ghost image during the measurement.

Tables (2)

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Table 1. Material Properties and Geometriesa

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Table 2. Process Parameters of the SP Experiment

Equations (9)

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[F]=[K][Φ],[K]=v[B]T[D][B]dv,
[Fall]=[Kall][Φall].
{[Φm]=errorz[Φb]=0ux=0=0,vy=0=0[Fb]=[Fm],[Fb]=f·sbsall,
[Fall]=[Kall][Φall].
[Φall]=[Kall]1·[Fall].
[Fall]=[Kall]·[Kall]1·[Fall].
z[Fb]=0.
errors=Assin(2πTs)x,
errorg(x)={e(x5)/0.72,x[0,10)e(x15)/0.72,x[10,20)e(x25)/0.72,x[20,30].

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