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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 19700–19712

Robust, efficient computational methods for axially symmetric optical aspheres

G.W. Forbes  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 19700-19712 (2010)
http://dx.doi.org/10.1364/OE.18.019700


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Abstract

Whether in design or the various stages of fabrication and testing, an effective representation of an asphere’s shape is critical. Some algorithms are given for implementing tailored polynomials that are ideally suited to these needs. With minimal coding, these results allow a recently introduced orthogonal polynomial basis to be employed to arbitrary orders. Interestingly, these robust and efficient methods are enabled by the introduction of an auxiliary polynomial basis.

© 2010 Optical Society of America

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

History
Original Manuscript: June 1, 2010
Revised Manuscript: July 12, 2010
Manuscript Accepted: July 12, 2010
Published: September 1, 2010

Citation
G. W. Forbes, "Robust, efficient computational methods for axially symmetric optical aspheres," Opt. Express 18, 19700-19712 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19700


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References

  1. G. W. Forbes, "Shape specification for axially symmetric optical surfaces," Opt. Express 15, 5218-5226 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218. [CrossRef] [PubMed]
  2. G. W. Forbes, and C. P. Brophy, "Designing cost-effective systems that incorporate high-precision aspheric optics," SPIE Optifab (2009) TD06-25 (1). Available at http://www.qedmrf.com.
  3. C. du Jeu, "Criterion to appreciate difficulties of aspherical polishing," Proc. SPIE 5494, 113-121 (2004), doi:10.1117/12.551420. [CrossRef]
  4. G. W. Forbes, "Robust and fast computation for the polynomials of optics," Opt. Express 18, 13851-13862 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13851. [CrossRef] [PubMed]
  5. E. W. Weisstein, "Jacobi Polynomial" from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiPolynomial.html, see esp. Eqs. (10)-(14).
  6. M. Abramowitz, and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.
  7. C. W. Clenshaw, "A Note on the Summation of Chebyshev Series," Math. Tables Other Aids Comput. 9, 118-120 (1955), http://www.jstor.org/stable/2002068.
  8. F. J. Smith, "An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation," Math. Comput. 19, 33-36 (1965). [CrossRef]
  9. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 2.9.
  10. A useful overview is given at http://en.wikipedia.org/wiki/Discrete_cosine_transform.
  11. G. W. Forbes, "Can you make/measure this asphere for me?", Frontiers in Optics, OSA Technical Digest (2009), http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2009-FThH1.

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