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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 8 — Apr. 16, 2007
  • pp: 5218–5226

Shape specification for axially symmetric optical surfaces

G. W. Forbes  »View Author Affiliations

Optics Express, Vol. 15, Issue 8, pp. 5218-5226 (2007)

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Advances in fabrication and testing are allowing aspheric optics to have greater impact through their increased prevalence and complexity. The most widely used characterization of surface shape is numerically deficient, however. Furthermore, with regard to tolerancing and to constraints for manufacturability, this representation is poorly suited for design purposes. Effective alternatives are therefore presented for working with rotationally symmetric surfaces that are either (i) strongly aspheric or (ii) constrained in terms of the slope in the departure from a best-fit sphere.

© 2007 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

Original Manuscript: February 20, 2007
Revised Manuscript: April 12, 2007
Manuscript Accepted: April 12, 2007
Published: April 13, 2007

G. W. Forbes, "Shape specification for axially symmetric optical surfaces," Opt. Express 15, 5218-5226 (2007)

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  1. See, for example, the discussion and references in H. Chase, "Optical design with rotationally symmetric NURBS", SPIE Proceedings 4832, 10-24 (2002) and A. W. Greynolds, "Superconic and subconic surface descriptions in optical design," Proc. SPIE 4832, 1-9 (2002). Such matters are also treated within the manuals for commercial optical design software. [CrossRef]
  2. G. H. Spencer and M. V. R.K. Murty, "General ray-tracing procedure," J. Opt. Soc. Am. 52, 672-678 (1962), see Eq. (16). [CrossRef]
  3. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), see 22.2.1.
  4. E. H. Doha, "On the coefficients of differentiated expansions and derivatives of Jacobi polynomials," J. Phys. A: Math. Gen. 35, 3467-3478 (2002). [CrossRef]
  5. E. W. Weisstein, "Jacobi Polynomial" from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiPolynomial.html, see esp. Eqs. (10-14).
  6. B. Y. Ting and Y. L. Luke, "Conversion of polynomials between different polynomial bases," IMA J. Numer. Anal.  1, 229-234 (1981). [CrossRef]
  7. A. J. E. M. Janssen and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," J. Microlithogr. Microfabr. Microsyst. 5, 1-3 (2006). (Note that the Zernikes are also Jacobi polynomials.) [CrossRef]

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