B-spline representation of optical surfaces and its accuracy in a ray trace algorithm

Philipp Jester; Christoph Menke; Karsten Urban

doi:10.1364/AO.50.000822

B-spline representation of optical surfaces and its accuracy in a ray trace algorithm

Philipp Jester, Christoph Menke, and Karsten Urban

Applied Optics
Vol. 50,
Issue 6,
pp. 822-828
(2011)
•https://doi.org/10.1364/AO.50.000822

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Philipp Jester, Christoph Menke, and Karsten Urban, "B-spline representation of optical surfaces and its accuracy in a ray trace algorithm," Appl. Opt. 50, 822-828 (2011)

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Related Topics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Numerical approximation and analysis (000.4430)
- Geometric optics (080.0080)
- Computation methods (080.1753)
- Aspherics (220.1250)
- Surfaces (240.6700)

About this Article
History
- Original Manuscript: September 16, 2010
- Revised Manuscript: January 6, 2011
- Manuscript Accepted: January 6, 2011
- Published: February 11, 2011

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Abstract

We introduce a representation of aspheric surfaces that is based on a B-spline quasi-interpolation scheme. The scheme is implemented in a ray trace algorithm, and bounds on the approximation error are established. Examples for the reproduction of aspheric surfaces in polynomial description and the ray tracing accuracy are presented. The proposed approach allows the specification of local and global structures and the efficient treatment of measured surface data. The representation gives access to a wavelet analysis, offering extended possibilities for the tolerance analysis of optical systems containing aspheric elements.

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KXY		${KSA}_{1}$		${KSA}_{2}$
b	$5.0 mm$	b	$1.0 mm$	b	$20.0 mm$
$ρ_{x}$	$- 2.15 e^{- 1}$	ρ	1.00	ρ	$- 3.87 e^{- 2}$
$ρ_{y}$	$1.22 e^{- 1}$	—	—	—	—
$κ_{x}$	$- 1.00$	κ	$- 1.00$	κ	0.00
$κ_{y}$	$- 1.00$	—	—	—	—
$c_{10}$	$- 4.05 e^{- 4}$	$c_{1}$	1.50	$c_{1}$	$- 4.17 e^{- 6}$
$c_{12}$	$- 8.13 e^{- 4}$	$c_{2}$	$- 7.00 e^{- 1}$	$c_{2}$	$4.71 e^{- 9}$
$c_{14}$	$5.73 e^{- 4}$	$c_{3}$	$5.00 e^{- 1}$	$c_{3}$	$- 4.94 e^{- 12}$
$c_{21}$	$- 4.59 e^{- 6}$	$c_{4}$	$- 5.00 e^{- 1}$	$c_{4}$	$- 5.42 e^{- 15}$
$c_{23}$	$1.14 e^{- 5}$	—	—	$c_{5}$	$- 4.98 e^{- 18}$
$c_{25}$	$9.64 e^{- 6}$	—	—	$c_{6}$	$- 1.22 e^{- 20}$
$c_{27}$	$4.45 e^{- 7}$	—	—	—	—
$c_{36}$	$- 2.69 e^{- 9}$	—	—	—	—
$c_{38}$	$- 7.96 e^{- 8}$	—	—	—	—
$c_{40}$	$- 8.79 e^{- 8}$	—	—	—	—
$c_{42}$	$- 9.16 e^{- 8}$	—	—	—	—
$c_{44}$	$2.43 e^{- 8}$	—	—	—	—

d	$j = 8$	$j = 9$	$j = 10$
1	$3 s$	$3 s$	$4 s$
2	$3 s$	$4 s$	$6 s$
3	$5 s$	$6 s$	$7 s$
4	$8 s$	$9 s$	$11 s$
5	$12 s$	$14 s$	$15 s$
6	$18 s$	$20 s$	$23 s$

KXY		${KSA}_{1}$		${KSA}_{2}$
b	$5.0 mm$	b	$1.0 mm$	b	$20.0 mm$
$ρ_{x}$	$- 2.15 e^{- 1}$	ρ	1.00	ρ	$- 3.87 e^{- 2}$
$ρ_{y}$	$1.22 e^{- 1}$	—	—	—	—
$κ_{x}$	$- 1.00$	κ	$- 1.00$	κ	0.00
$κ_{y}$	$- 1.00$	—	—	—	—
$c_{10}$	$- 4.05 e^{- 4}$	$c_{1}$	1.50	$c_{1}$	$- 4.17 e^{- 6}$
$c_{12}$	$- 8.13 e^{- 4}$	$c_{2}$	$- 7.00 e^{- 1}$	$c_{2}$	$4.71 e^{- 9}$
$c_{14}$	$5.73 e^{- 4}$	$c_{3}$	$5.00 e^{- 1}$	$c_{3}$	$- 4.94 e^{- 12}$
$c_{21}$	$- 4.59 e^{- 6}$	$c_{4}$	$- 5.00 e^{- 1}$	$c_{4}$	$- 5.42 e^{- 15}$
$c_{23}$	$1.14 e^{- 5}$	—	—	$c_{5}$	$- 4.98 e^{- 18}$
$c_{25}$	$9.64 e^{- 6}$	—	—	$c_{6}$	$- 1.22 e^{- 20}$
$c_{27}$	$4.45 e^{- 7}$	—	—	—	—
$c_{36}$	$- 2.69 e^{- 9}$	—	—	—	—
$c_{38}$	$- 7.96 e^{- 8}$	—	—	—	—
$c_{40}$	$- 8.79 e^{- 8}$	—	—	—	—
$c_{42}$	$- 9.16 e^{- 8}$	—	—	—	—
$c_{44}$	$2.43 e^{- 8}$	—	—	—	—

d	$j = 8$	$j = 9$	$j = 10$
1	$3 s$	$3 s$	$4 s$
2	$3 s$	$4 s$	$6 s$
3	$5 s$	$6 s$	$7 s$
4	$8 s$	$9 s$	$11 s$
5	$12 s$	$14 s$	$15 s$
6	$18 s$	$20 s$	$23 s$

Abstract

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Equations (24)

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