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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 16 — Aug. 12, 2013
  • pp: 19061–19081

Fitting freeform shapes with orthogonal bases

G. W. Forbes  »View Author Affiliations

Optics Express, Vol. 21, Issue 16, pp. 19061-19081 (2013)

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Orthogonality is exploited for fitting analytically-specified freeform shapes in terms of orthogonal polynomials. The end result is expressed in terms of FFTs coupled to a simple explicit form of Gaussian quadrature. Its efficiency opens the possibilities for proceeding to arbitrary numbers of polynomial terms. This is shown to create promising options for quantifying and filtering the mid-spatial frequency structure within circular domains from measurements of as-built parts.

© 2013 OSA

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(220.1250) Optical design and fabrication : Aspherics
(220.4610) Optical design and fabrication : Optical fabrication
(220.4830) Optical design and fabrication : Systems design
(220.4840) Optical design and fabrication : Testing

ToC Category:
Optical Design and Fabrication

Original Manuscript: June 11, 2013
Manuscript Accepted: July 19, 2013
Published: August 2, 2013

G. W. Forbes, "Fitting freeform shapes with orthogonal bases," Opt. Express 21, 19061-19081 (2013)

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  1. A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt.51(15), 3054–3058 (2012), doi:. [CrossRef] [PubMed]
  2. I. Kaya and J. P. Rolland, “Hybrid RBF and local ϕ-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2012), doi:. [CrossRef]
  3. P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math.77, 357–363 (2012).
  4. R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:. [CrossRef]
  5. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express20(3), 2483–2499 (2012). [CrossRef] [PubMed]
  6. C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2012), doi:. [CrossRef]
  7. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 15.4.
  8. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010), doi:. [CrossRef] [PubMed]
  9. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978). See 25.4.38.
  10. See Section 12.3 of [7]. Or see Sec. 12.4 in http://apps.nrbook.com/empanel/index.html#
  11. J. H. Hannay and J. F. Nye, “Fibonacci numerical integration on a sphere,” J. Phys. Math. Gen.37(48), 11591–11601 (2004), doi:. [CrossRef]
  12. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010), doi:. [CrossRef] [PubMed]
  13. J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:. [CrossRef]

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