Abstract

With the recent emergence of slow-servo diamond turning, optical designs with surfaces that are not intrinsically rotationally symmetric can be manufactured. In this paper, we demonstrate some important limitations to Zernike polynomial representation of optical surfaces in describing the evolving freeform surface descriptions that are effective for optical design and encountered during optical fabrication. Specifically, we show that the ray grids commonly used in sampling a freeform surface to form a database from which to perform a φ-polynomial fit is limiting the efficacy of computation. We show an edge-clustered fitting grid that effectively suppresses the edge ringing that arises as the polynomial adapts to the fully nonsymmetric features of the surface.

© 2011 OSA

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    [CrossRef] [PubMed]
  4. Y. Tohme and R. Murray, “Principles and Applications of the Slow Slide Servo,” Moore Nanotechnology Systems White Paper (2005).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chap. 22, (Dover, 1972).
  13. G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific Publishing, Singapore, 2007).
  14. B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
    [CrossRef]
  15. R. Platte, Accuracy and Stability of Global Radial Basis Function Methods for the Numerical Solution of Partial Differential Equations, Ph.D. Thesis, (University of Delaware, 2005).

2011 (3)

2010 (2)

J. P. Rolland, C. Dunn, and K. P. Thompson, “An Analytic Expression for the Field Dependence of FRINGE Zernike Polynomial Coefficients in Rotationally Symmetric Optical Systems,” Proc. SPIE 7790, (2010).

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18(13), 13851–13862 (2010).
[CrossRef] [PubMed]

2008 (1)

2007 (1)

1954 (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

1934 (1)

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[CrossRef]

Bhatia, A. B.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

Born, M.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

Cakmakci, O.

Dunn, C.

J. P. Rolland, C. Dunn, and K. P. Thompson, “An Analytic Expression for the Field Dependence of FRINGE Zernike Polynomial Coefficients in Rotationally Symmetric Optical Systems,” Proc. SPIE 7790, (2010).

Flyer, N.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Forbes, G. W.

Fornberg, B.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Foroosh, H.

Fuerschbach, K.

Larsson, E.

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Li, L.

Ma, B.

Moore, B.

Rolland, J. P.

Thompson, K. P.

Wolf, E.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

Zernike, F.

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[CrossRef]

Opt. Express (5)

Physica (1)

F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1(7-12), 689–704 (1934).
[CrossRef]

Proc. Camb. Philos. Soc. (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[CrossRef]

Proc. SPIE (1)

J. P. Rolland, C. Dunn, and K. P. Thompson, “An Analytic Expression for the Field Dependence of FRINGE Zernike Polynomial Coefficients in Rotationally Symmetric Optical Systems,” Proc. SPIE 7790, (2010).

SIAM J. Sci. Comput. (1)

B. Fornberg, E. Larsson, and N. Flyer, “Stable computations with Gaussian radial basis functions,” SIAM J. Sci. Comput. 33(2), 869–892 (2011).
[CrossRef]

Other (6)

R. Platte, Accuracy and Stability of Global Radial Basis Function Methods for the Numerical Solution of Partial Differential Equations, Ph.D. Thesis, (University of Delaware, 2005).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, chap. 22, (Dover, 1972).

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific Publishing, Singapore, 2007).

E. Abbe, Lens system. U.S. Patent No. 697,959, (April, 1902).

M. Born and E. Wolf, Principles of Optics, (Cambridge, 1999).

Y. Tohme and R. Murray, “Principles and Applications of the Slow Slide Servo,” Moore Nanotechnology Systems White Paper (2005).

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