Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials

Guang-ming Dai; Charles E. Campbell; Li Chen; Huawei Zhao; Dimitri Chernyak

doi:10.1364/AO.48.000477

Applied Optics
Vol. 48,
Issue 3,
pp. 477-488
(2009)
•https://doi.org/10.1364/AO.48.000477

Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials

Guang-ming Dai, Charles E. Campbell, Li Chen, Huawei Zhao, and Dimitri Chernyak

Not Accessible

Your library or personal account may give you access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Guang-ming Dai, Charles E. Campbell, Li Chen, Huawei Zhao, and Dimitri Chernyak, "Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials," Appl. Opt. 48, 477-488 (2009)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Related Topics
Table of Contents Category
- Medical Optics and Biotechnology
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: June 10, 2008
- Revised Manuscript: September 15, 2008
- Manuscript Accepted: October 29, 2008
- Published: January 13, 2009
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 4, Iss. 3

Abstract

In wavefront-driven vision correction, ocular aberrations are often measured on the pupil plane and the correction is applied on a different plane. The problem with this practice is that any changes undergone by the wavefront as it propagates between planes are not currently included in devising customized vision correction. With some valid approximations, we have developed an analytical foundation based on geometric optics in which Zernike polynomials are used to characterize the propagation of the wavefront from one plane to another. Both the boundary and the magnitude of the wavefront change after the propagation. Taylor monomials were used to realize the propagation because of their simple form for this purpose. The method we developed to identify changes in low-order aberrations was verified with the classical vertex correction formula. The method we developed to identify changes in high-order aberrations was verified with ZEMAX ray-tracing software. Although the method may not be valid for highly irregular wavefronts and it was only proven for wavefronts with low-order or high-order aberrations, our analysis showed that changes in the propagating wavefront are significant and should, therefore, be included in calculating vision correction. This new approach could be of major significance in calculating wavefront-driven vision correction whether by refractive surgery, contact lenses, intraocular lenses, or spectacles.

Full Article | PDF Article

More Like This

Derivation of the propagation equations for higher order aberrations of local wavefronts

Gregor Esser, Wolfgang Becken, Werner Müller, Peter Baumbach, Josep Arasa, and Dietmar Uttenweiler
J. Opt. Soc. Am. A 28(12) 2442-2458 (2011)

Zernike radial slope polynomials for wavefront reconstruction and refraction

Jayoung Nam, Larry N. Thibos, and D. Robert Iskander
J. Opt. Soc. Am. A 26(4) 1035-1048 (2009)

Wavefront propagation based on the ray transfer matrix and numerical orthogonal Zernike gradient polynomials

Huimin Yin, Zhishan Gao, Qun Yuan, Lu Chen, Jinci Bi, Xin Cao, and Jiangling Huang
J. Opt. Soc. Am. A 36(6) 1072-1078 (2019)

Previous Article Next Article

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Figures (5)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Tables (3)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Equations (63)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Symbol	Expression
$g_{0}$	$4 [c_{1}^{2} + c_{2}^{2} + 3 c_{3}^{2} + 6 c_{4}^{2} + 3 c_{5}^{2} + 6 c_{6}^{2} + 14 c_{7}^{2} + 14 c_{8}^{2} + 6 c_{9}^{2} + 10 c_{10}^{2} + 25 c_{11}^{2} + 30 c_{12}^{2} + 25 c_{13}^{2} + 10 c_{14}^{2} + 2 \sqrt{2} (c_{1} c_{7} + c_{2} c_{8}) + 2 \sqrt{15} (c_{3} c_{11} + 2 c_{4} c_{12} + c_{5} c_{13})]$
$g_{1}$	$4 [\sqrt{6} c_{2} c_{3} + c_{1} (2 \sqrt{5} c_{12} - \sqrt{10} c_{13} + 2 \sqrt{3} c_{4} - \sqrt{6} c_{5}) + 6 \sqrt{5} c_{13} c_{6} - 6 \sqrt{5} c_{14} c_{6} + 4 \sqrt{3} c_{5} c_{6} + 14 \sqrt{10} c_{12} c_{7} - 14 \sqrt{5} c_{13} c_{7} + 8 \sqrt{6} c_{4} c_{7} - 4 \sqrt{3} c_{5} c_{7} + 4 \sqrt{3} c_{3} c_{8} + \sqrt{5} c_{11} (\sqrt{2} c_{2} + 14 c_{8} - 6 c_{9}) + 6 \sqrt{5} c_{10} c_{9} - 4 \sqrt{3} c_{3} c_{9}]$
$g_{2}$	$4 [\sqrt{10} c_{13} c_{2} + c_{1} (\sqrt{10} c_{11} + \sqrt{6} c_{3}) + 2 \sqrt{3} c_{2} c_{4} + \sqrt{6} c_{2} c_{5} + 6 \sqrt{5} c_{10} c_{6} + 6 \sqrt{5} c_{11} c_{6} + 4 \sqrt{3} c_{3} c_{6} + 14 \sqrt{5} c_{11} c_{7} + 4 \sqrt{3} c_{3} c_{7} + 14 \sqrt{5} c_{13} c_{8} + 8 \sqrt{6} c_{4} c_{8} + 4 \sqrt{3} c_{5} c_{8} + 2 \sqrt{5} c_{12} (c_{2} + 7 \sqrt{2} c_{8}) + 6 \sqrt{5} c_{13} c_{9} + 6 \sqrt{5} c_{14} c_{9} + 4 \sqrt{3} c_{5} c_{9}]$
$g_{3}$	$4 [9 \sqrt{6} c_{10} c_{13} + 6 \sqrt{5} c_{12} c_{3} - 3 \sqrt{10} c_{14} c_{3} + 4 \sqrt{3} c_{3} c_{4} + 3 c_{11} (14 \sqrt{3} c_{12} - 3 \sqrt{6} c_{14} + 4 \sqrt{5} c_{4}) + 3 \sqrt{10} c_{10} c_{5} + 2 \sqrt{3} c_{2} c_{6} + 2 \sqrt{3} c_{2} c_{7} + 2 \sqrt{3} c_{1} c_{8} + 5 \sqrt{6} c_{6} c_{8} + 10 \sqrt{6} c_{7} c_{8} - 2 \sqrt{3} c_{1} c_{9} - 5 \sqrt{6} c_{7} c_{9}]$
$g_{4}$	$4 [6 \sqrt{3} c_{10}^{2} + 15 \sqrt{3} c_{11}^{2} + 18 \sqrt{3} c_{12}^{2} + 15 \sqrt{3} c_{13}^{2} + 6 \sqrt{3} c_{14}^{2} + 6 \sqrt{5} c_{11} c_{3} + \sqrt{3} c_{3}^{2} + 12 \sqrt{5} c_{12} c_{4} + 2 \sqrt{3} c_{4}^{2} + 6 \sqrt{5} c_{13} c_{5} + \sqrt{3} c_{5}^{2} + 3 \sqrt{3} c_{6}^{2} + 2 \sqrt{6} c_{1} c_{7} + 7 \sqrt{3} c_{7}^{2} + 2 \sqrt{6} c_{2} c_{8} + 7 \sqrt{3} c_{8}^{2} + 3 \sqrt{3} c_{9}^{2}]$
$g_{5}$	$4 [9 \sqrt{6} c_{10} c_{11} + 42 \sqrt{3} c_{12} c_{13} + 9 \sqrt{6} c_{13} c_{14} + 3 \sqrt{10} c_{10} c_{3} + 12 \sqrt{5} c_{13} c_{4} + 6 \sqrt{5} c_{12} c_{5} + 3 \sqrt{10} c_{14} c_{5} + 4 \sqrt{3} c_{4} c_{5} + 2 \sqrt{3} c_{1} c_{6} - 2 \sqrt{3} c_{1} c_{7} + 5 \sqrt{6} c_{6} c_{7} - 5 \sqrt{6} c_{7}^{2} + 2 \sqrt{3} c_{2} c_{8} + 5 \sqrt{6} c_{8}^{2} + 2 \sqrt{3} c_{2} c_{9} + 5 \sqrt{6} c_{8} c_{9}]$
$g_{6}$	$8 \sqrt{5} c_{1} (c_{13} - c_{14}) + 4 [10 \sqrt{5} c_{11} c_{2} + 54 \sqrt{5} c_{12} c_{6} + 30 \sqrt{3} c_{4} c_{6} + 55 \sqrt{10} c_{13} c_{7} - 28 \sqrt{10} c_{14} c_{7} + 15 \sqrt{6} c_{5} c_{7} + 55 \sqrt{10} c_{11} c_{8} + 15 \sqrt{6} c_{3} c_{8} + 2 \sqrt{5} c_{10} (5 c_{2} + 14 \sqrt{2} c_{8})] / 5$
$g_{7}$	$4 [10 \sqrt{5} c_{1} (\sqrt{2} c_{12} - c_{13}) + 21 \sqrt{10} c_{13} c_{6} - 12 \sqrt{10} c_{14} c_{6} + 5 \sqrt{6} c_{5} c_{6} + 86 \sqrt{5} c_{12} c_{7} - 34 \sqrt{10} c_{13} c_{7} + 30 \sqrt{3} c_{4} c_{7} - 10 \sqrt{6} c_{5} c_{7} + 10 \sqrt{6} c_{3} c_{8} + \sqrt{5} c_{11} (10 c_{2} + \sqrt{2} (34 c_{8} - 21 c_{9})) + 12 \sqrt{10} c_{10} c_{9} - 5 \sqrt{6} c_{3} c_{9}] / 5$
$g_{8}$	$4 [10 \sqrt{5} c_{1} c_{11} + 10 \sqrt{5} c_{13} c_{2} + 12 \sqrt{10} c_{10} c_{6} + 21 \sqrt{10} c_{11} c_{6} + 5 \sqrt{6} c_{3} c_{6} + 34 \sqrt{10} c_{11} c_{7} + 10 \sqrt{6} c_{3} c_{7} + 34 \sqrt{10} c_{13} c_{8} + 30 \sqrt{3} c_{4} c_{8} + 10 \sqrt{6} c_{5} c_{8} + 2 \sqrt{5} c_{12} (5 \sqrt{2} c_{2} + 43 c_{8}) + 21 \sqrt{10} c_{13} c_{9} + 12 \sqrt{10} c_{14} c_{9} + 5 \sqrt{6} c_{5} c_{9}] / 5$
$g_{9}$	$8 \sqrt{5} c_{1} (c_{10} - c_{11}) + 4 [10 \sqrt{5} c_{14} c_{2} + 28 \sqrt{10} c_{10} c_{7} - 55 \sqrt{10} c_{11} c_{7} - 15 \sqrt{6} c_{3} c_{7} + 28 \sqrt{10} c_{14} c_{8} + 15 \sqrt{6} c_{5} c_{8} + 5 \sqrt{5} c_{13} (2 c_{2} + 11 \sqrt{2} c_{8}) + 54 \sqrt{5} c_{12} c_{9} + 30 \sqrt{3} c_{4} c_{9}) / 5$
$g_{10}$	$8 (10 \sqrt{6} c_{13} c_{3} + 20 c_{10} (2 \sqrt{5} c_{12} + \sqrt{3} c_{4}) + 10 c_{11} (4 \sqrt{10} c_{13} + \sqrt{6} c_{5}) + 9 \sqrt{10} c_{6} c_{8} + 9 \sqrt{10} c_{7} c_{9}] / 5$
$g_{11}$	$4 [30 \sqrt{3} c_{12} c_{3} - 5 \sqrt{6} c_{14} c_{3} + 5 c_{11} (22 \sqrt{5} c_{12} - 5 \sqrt{10} c_{14} + 8 \sqrt{3} c_{4}) + 5 c_{10} (5 \sqrt{10} c_{13} + \sqrt{6} c_{5}) + 9 \sqrt{10} c_{6} c_{8} + 18 \sqrt{10} c_{7} c_{8} - 9 \sqrt{10} c_{7} c_{9}] / 5$
$g_{12}$	$8 \sqrt{5} c_{10}^{2} + 32 \sqrt{5} c_{11}^{2} + 48 \sqrt{5} c_{12}^{2} + 32 \sqrt{5} c_{13}^{2} + 8 \sqrt{5} c_{14}^{2} + 16 \sqrt{3} c_{11} c_{3} + 32 \sqrt{3} c_{12} c_{4} + 16 \sqrt{3} c_{13} c_{5} + (12 c_{6}^{2}) / \sqrt{5} + 12 \sqrt{5} c_{7}^{2} + 12 \sqrt{5} c_{8}^{2} + (12 c_{9}^{2}) / \sqrt{5}$
$g_{13}$	$4 [110 \sqrt{5} c_{12} c_{13} + 25 \sqrt{10} c_{13} c_{14} + 5 c_{10} (5 \sqrt{10} c_{11} + \sqrt{6} c_{3}) + 40 \sqrt{3} c_{13} c_{4} + 30 \sqrt{3} c_{12} c_{5} + 5 \sqrt{6} c_{14} c_{5} + 9 \sqrt{10} c_{6} c_{7} - 9 \sqrt{10} c_{7}^{2} + 9 \sqrt{10} c_{8}^{2} + 9 \sqrt{10} c_{8} c_{9}] / 5$
$g_{14}$	$- 32 \sqrt{10} c_{11}^{2} + 32 \sqrt{10} c_{13}^{2} + 64 \sqrt{5} c_{12} c_{14} - 16 \sqrt{6} c_{11} c_{3} + 32 \sqrt{3} c_{14} c_{4} + 16 \sqrt{6} c_{13} c_{5} - 72 \sqrt{2 / 5} c_{6} c_{7} + 72 \sqrt{2 / 5} c_{8} c_{9}$
$g_{15}$	$16 \sqrt{15} (c_{13} c_{6} + c_{14} c_{7} + c_{10} c_{8} + c_{11} c_{9})$
$g_{16}$	$16 \sqrt{3 / 5} (3 \sqrt{2} c_{12} c_{6} + 5 c_{13} c_{7} - 2 c_{14} c_{7} + 2 c_{10} c_{8} + 5 c_{11} c_{8})$
$g_{17}$	$8 \sqrt{3 / 5} (3 c_{13} c_{6} - c_{14} c_{6} + 9 \sqrt{2} c_{12} c_{7} - 7 c_{13} c_{7} + 7 c_{11} c_{8} + c_{10} c_{9} - 3 c_{11} c_{9})$
$g_{18}$	$8 \sqrt{3 / 5} (c_{10} c_{6} + 3 c_{11} c_{6} + 7 c_{11} c_{7} + 9 \sqrt{2} c_{12} c_{8} + 7 c_{13} c_{8} + 3 c_{13} c_{9} + c_{14} c_{9})$
$g_{19}$	$16 \sqrt{3 / 5} (2 c_{10} c_{7} - 5 c_{11} c_{7} + 5 c_{13} c_{8} + 2 c_{14} c_{8} + 3 \sqrt{2} c_{12} c_{9})$
$g_{20}$	$- 16 \sqrt{15} (c_{11} c_{6} + c_{10} c_{7} - c_{14} c_{8} - c_{13} c_{9})$
$g_{21}$	$160 \sqrt{2 / 7} (c_{10} c_{13} + c_{11} c_{14})$
$g_{22}$	$160 (c_{10} c_{12} + \sqrt{2} c_{11} c_{13}) / \sqrt{7}$
$g_{23}$	$32 (8 c_{11} c_{12} + \sqrt{2} c_{10} c_{13} - \sqrt{2} c_{11} c_{14}) / \sqrt{7}$
$g_{24}$	$8 (c_{10}^{2} + 10 c_{11}^{2} + 18 c_{12}^{2} + 10 c_{13}^{2} + c_{14}^{2}) / \sqrt{7}$
$g_{25}$	$32 (\sqrt{2} c_{10} c_{11} + 8 c_{12} c_{13} + \sqrt{2} c_{13} c_{14}) / \sqrt{7}$
$g_{26}$	$- 80 (\sqrt{2} c_{11}^{2} - \sqrt{2} c_{13}^{2} - 2 c_{12} c_{14}) / \sqrt{7}$
$g_{27}$	$- 160 \sqrt{2 / 7} (c_{10} c_{11} - c_{13} c_{14})$

Term	Input	Output
Term	Input	Analytical	ZEMAX	Diff
$c_{3}^{- 3}$	$- 0.21030$	$- 0.22703$	$- 0.22690$	0.00013
$c_{3}^{- 1}$	0.08100	0.09221	0.09221	0.00000
$c_{3}^{1}$	0.05900	0.06224	0.06210	$- 0.00014$
$c_{3}^{3}$	0.00470	0.00438	0.00444	0.00006
$c_{4}^{- 4}$	0.04510	0.05715	0.05705	$- 0.00010$
$c_{4}^{- 2}$	$- 0.05380$	$- 0.06240$	$- 0.06237$	0.00003
$c_{4}^{0}$	0.07050	0.06995	0.06977	$- 0.00018$
$c_{4}^{2}$	0.11100	0.11770	0.11760	$- 0.00010$
$c_{4}^{4}$	$- 0.04770$	$- 0.05086$	$- 0.05074$	0.00012
$c_{5}^{- 5}$	0.01690	0.02081	0.02078	$- 0.00003$
$c_{5}^{- 3}$	0.02760	0.02466	0.02463	$- 0.00003$
$c_{5}^{- 1}$	$- 0.02960$	$- 0.02733$	$- 0.02729$	0.00004
$c_{5}^{1}$	0.02940	0.03057	0.03052	$- 0.00005$
$c_{5}^{3}$	$- 0.02170$	$- 0.02240$	$- 0.02237$	0.00003
$c_{5}^{5}$	$- 0.00790$	$- 0.00657$	$- 0.00655$	0.00002
$c_{6}^{- 6}$	0.05750	0.05992	0.05982	$- 0.00010$
$c_{6}^{- 4}$	0.00800	0.01091	0.01089	$- 0.00002$
$c_{6}^{- 2}$	0.01220	0.01091	0.01089	$- 0.00002$
$c_{6}^{0}$	0.02800	0.02816	0.02812	$- 0.00004$
$c_{6}^{2}$	$- 0.00930$	$- 0.00900$	$- 0.00899$	0.00001
$c_{6}^{4}$	$- 0.01790$	$- 0.01848$	$- 0.01845$	0.00003
$c_{6}^{6}$	$- 0.03320$	$- 0.03540$	$- 0.03535$	0.00005
r.m.s.	0.29651	0.31405	0.31432	0.00034

Case	d	$\bar{x}$	$σ_{x}$	$\bar{y}$	$σ_{y}$
Normal	3.5	$- 0.056$	$- 0.039$	0.585	0.519
Normal	12.5	$- 0.198$	$- 0.141$	2.256	1.678
Irregular	3.5	$- 0.289$	$- 0.319$	2.902	2.575
Irregular	12.5	$- 1.302$	$- 1.138$	10.363	9.196

Symbol	Expression
$g_{0}$	$4 [c_{1}^{2} + c_{2}^{2} + 3 c_{3}^{2} + 6 c_{4}^{2} + 3 c_{5}^{2} + 6 c_{6}^{2} + 14 c_{7}^{2} + 14 c_{8}^{2} + 6 c_{9}^{2} + 10 c_{10}^{2} + 25 c_{11}^{2} + 30 c_{12}^{2} + 25 c_{13}^{2} + 10 c_{14}^{2} + 2 \sqrt{2} (c_{1} c_{7} + c_{2} c_{8}) + 2 \sqrt{15} (c_{3} c_{11} + 2 c_{4} c_{12} + c_{5} c_{13})]$
$g_{1}$	$4 [\sqrt{6} c_{2} c_{3} + c_{1} (2 \sqrt{5} c_{12} - \sqrt{10} c_{13} + 2 \sqrt{3} c_{4} - \sqrt{6} c_{5}) + 6 \sqrt{5} c_{13} c_{6} - 6 \sqrt{5} c_{14} c_{6} + 4 \sqrt{3} c_{5} c_{6} + 14 \sqrt{10} c_{12} c_{7} - 14 \sqrt{5} c_{13} c_{7} + 8 \sqrt{6} c_{4} c_{7} - 4 \sqrt{3} c_{5} c_{7} + 4 \sqrt{3} c_{3} c_{8} + \sqrt{5} c_{11} (\sqrt{2} c_{2} + 14 c_{8} - 6 c_{9}) + 6 \sqrt{5} c_{10} c_{9} - 4 \sqrt{3} c_{3} c_{9}]$
$g_{2}$	$4 [\sqrt{10} c_{13} c_{2} + c_{1} (\sqrt{10} c_{11} + \sqrt{6} c_{3}) + 2 \sqrt{3} c_{2} c_{4} + \sqrt{6} c_{2} c_{5} + 6 \sqrt{5} c_{10} c_{6} + 6 \sqrt{5} c_{11} c_{6} + 4 \sqrt{3} c_{3} c_{6} + 14 \sqrt{5} c_{11} c_{7} + 4 \sqrt{3} c_{3} c_{7} + 14 \sqrt{5} c_{13} c_{8} + 8 \sqrt{6} c_{4} c_{8} + 4 \sqrt{3} c_{5} c_{8} + 2 \sqrt{5} c_{12} (c_{2} + 7 \sqrt{2} c_{8}) + 6 \sqrt{5} c_{13} c_{9} + 6 \sqrt{5} c_{14} c_{9} + 4 \sqrt{3} c_{5} c_{9}]$
$g_{3}$	$4 [9 \sqrt{6} c_{10} c_{13} + 6 \sqrt{5} c_{12} c_{3} - 3 \sqrt{10} c_{14} c_{3} + 4 \sqrt{3} c_{3} c_{4} + 3 c_{11} (14 \sqrt{3} c_{12} - 3 \sqrt{6} c_{14} + 4 \sqrt{5} c_{4}) + 3 \sqrt{10} c_{10} c_{5} + 2 \sqrt{3} c_{2} c_{6} + 2 \sqrt{3} c_{2} c_{7} + 2 \sqrt{3} c_{1} c_{8} + 5 \sqrt{6} c_{6} c_{8} + 10 \sqrt{6} c_{7} c_{8} - 2 \sqrt{3} c_{1} c_{9} - 5 \sqrt{6} c_{7} c_{9}]$
$g_{4}$	$4 [6 \sqrt{3} c_{10}^{2} + 15 \sqrt{3} c_{11}^{2} + 18 \sqrt{3} c_{12}^{2} + 15 \sqrt{3} c_{13}^{2} + 6 \sqrt{3} c_{14}^{2} + 6 \sqrt{5} c_{11} c_{3} + \sqrt{3} c_{3}^{2} + 12 \sqrt{5} c_{12} c_{4} + 2 \sqrt{3} c_{4}^{2} + 6 \sqrt{5} c_{13} c_{5} + \sqrt{3} c_{5}^{2} + 3 \sqrt{3} c_{6}^{2} + 2 \sqrt{6} c_{1} c_{7} + 7 \sqrt{3} c_{7}^{2} + 2 \sqrt{6} c_{2} c_{8} + 7 \sqrt{3} c_{8}^{2} + 3 \sqrt{3} c_{9}^{2}]$
$g_{5}$	$4 [9 \sqrt{6} c_{10} c_{11} + 42 \sqrt{3} c_{12} c_{13} + 9 \sqrt{6} c_{13} c_{14} + 3 \sqrt{10} c_{10} c_{3} + 12 \sqrt{5} c_{13} c_{4} + 6 \sqrt{5} c_{12} c_{5} + 3 \sqrt{10} c_{14} c_{5} + 4 \sqrt{3} c_{4} c_{5} + 2 \sqrt{3} c_{1} c_{6} - 2 \sqrt{3} c_{1} c_{7} + 5 \sqrt{6} c_{6} c_{7} - 5 \sqrt{6} c_{7}^{2} + 2 \sqrt{3} c_{2} c_{8} + 5 \sqrt{6} c_{8}^{2} + 2 \sqrt{3} c_{2} c_{9} + 5 \sqrt{6} c_{8} c_{9}]$
$g_{6}$	$8 \sqrt{5} c_{1} (c_{13} - c_{14}) + 4 [10 \sqrt{5} c_{11} c_{2} + 54 \sqrt{5} c_{12} c_{6} + 30 \sqrt{3} c_{4} c_{6} + 55 \sqrt{10} c_{13} c_{7} - 28 \sqrt{10} c_{14} c_{7} + 15 \sqrt{6} c_{5} c_{7} + 55 \sqrt{10} c_{11} c_{8} + 15 \sqrt{6} c_{3} c_{8} + 2 \sqrt{5} c_{10} (5 c_{2} + 14 \sqrt{2} c_{8})] / 5$
$g_{7}$	$4 [10 \sqrt{5} c_{1} (\sqrt{2} c_{12} - c_{13}) + 21 \sqrt{10} c_{13} c_{6} - 12 \sqrt{10} c_{14} c_{6} + 5 \sqrt{6} c_{5} c_{6} + 86 \sqrt{5} c_{12} c_{7} - 34 \sqrt{10} c_{13} c_{7} + 30 \sqrt{3} c_{4} c_{7} - 10 \sqrt{6} c_{5} c_{7} + 10 \sqrt{6} c_{3} c_{8} + \sqrt{5} c_{11} (10 c_{2} + \sqrt{2} (34 c_{8} - 21 c_{9})) + 12 \sqrt{10} c_{10} c_{9} - 5 \sqrt{6} c_{3} c_{9}] / 5$
$g_{8}$	$4 [10 \sqrt{5} c_{1} c_{11} + 10 \sqrt{5} c_{13} c_{2} + 12 \sqrt{10} c_{10} c_{6} + 21 \sqrt{10} c_{11} c_{6} + 5 \sqrt{6} c_{3} c_{6} + 34 \sqrt{10} c_{11} c_{7} + 10 \sqrt{6} c_{3} c_{7} + 34 \sqrt{10} c_{13} c_{8} + 30 \sqrt{3} c_{4} c_{8} + 10 \sqrt{6} c_{5} c_{8} + 2 \sqrt{5} c_{12} (5 \sqrt{2} c_{2} + 43 c_{8}) + 21 \sqrt{10} c_{13} c_{9} + 12 \sqrt{10} c_{14} c_{9} + 5 \sqrt{6} c_{5} c_{9}] / 5$
$g_{9}$	$8 \sqrt{5} c_{1} (c_{10} - c_{11}) + 4 [10 \sqrt{5} c_{14} c_{2} + 28 \sqrt{10} c_{10} c_{7} - 55 \sqrt{10} c_{11} c_{7} - 15 \sqrt{6} c_{3} c_{7} + 28 \sqrt{10} c_{14} c_{8} + 15 \sqrt{6} c_{5} c_{8} + 5 \sqrt{5} c_{13} (2 c_{2} + 11 \sqrt{2} c_{8}) + 54 \sqrt{5} c_{12} c_{9} + 30 \sqrt{3} c_{4} c_{9}) / 5$
$g_{10}$	$8 (10 \sqrt{6} c_{13} c_{3} + 20 c_{10} (2 \sqrt{5} c_{12} + \sqrt{3} c_{4}) + 10 c_{11} (4 \sqrt{10} c_{13} + \sqrt{6} c_{5}) + 9 \sqrt{10} c_{6} c_{8} + 9 \sqrt{10} c_{7} c_{9}] / 5$
$g_{11}$	$4 [30 \sqrt{3} c_{12} c_{3} - 5 \sqrt{6} c_{14} c_{3} + 5 c_{11} (22 \sqrt{5} c_{12} - 5 \sqrt{10} c_{14} + 8 \sqrt{3} c_{4}) + 5 c_{10} (5 \sqrt{10} c_{13} + \sqrt{6} c_{5}) + 9 \sqrt{10} c_{6} c_{8} + 18 \sqrt{10} c_{7} c_{8} - 9 \sqrt{10} c_{7} c_{9}] / 5$
$g_{12}$	$8 \sqrt{5} c_{10}^{2} + 32 \sqrt{5} c_{11}^{2} + 48 \sqrt{5} c_{12}^{2} + 32 \sqrt{5} c_{13}^{2} + 8 \sqrt{5} c_{14}^{2} + 16 \sqrt{3} c_{11} c_{3} + 32 \sqrt{3} c_{12} c_{4} + 16 \sqrt{3} c_{13} c_{5} + (12 c_{6}^{2}) / \sqrt{5} + 12 \sqrt{5} c_{7}^{2} + 12 \sqrt{5} c_{8}^{2} + (12 c_{9}^{2}) / \sqrt{5}$
$g_{13}$	$4 [110 \sqrt{5} c_{12} c_{13} + 25 \sqrt{10} c_{13} c_{14} + 5 c_{10} (5 \sqrt{10} c_{11} + \sqrt{6} c_{3}) + 40 \sqrt{3} c_{13} c_{4} + 30 \sqrt{3} c_{12} c_{5} + 5 \sqrt{6} c_{14} c_{5} + 9 \sqrt{10} c_{6} c_{7} - 9 \sqrt{10} c_{7}^{2} + 9 \sqrt{10} c_{8}^{2} + 9 \sqrt{10} c_{8} c_{9}] / 5$
$g_{14}$	$- 32 \sqrt{10} c_{11}^{2} + 32 \sqrt{10} c_{13}^{2} + 64 \sqrt{5} c_{12} c_{14} - 16 \sqrt{6} c_{11} c_{3} + 32 \sqrt{3} c_{14} c_{4} + 16 \sqrt{6} c_{13} c_{5} - 72 \sqrt{2 / 5} c_{6} c_{7} + 72 \sqrt{2 / 5} c_{8} c_{9}$
$g_{15}$	$16 \sqrt{15} (c_{13} c_{6} + c_{14} c_{7} + c_{10} c_{8} + c_{11} c_{9})$
$g_{16}$	$16 \sqrt{3 / 5} (3 \sqrt{2} c_{12} c_{6} + 5 c_{13} c_{7} - 2 c_{14} c_{7} + 2 c_{10} c_{8} + 5 c_{11} c_{8})$
$g_{17}$	$8 \sqrt{3 / 5} (3 c_{13} c_{6} - c_{14} c_{6} + 9 \sqrt{2} c_{12} c_{7} - 7 c_{13} c_{7} + 7 c_{11} c_{8} + c_{10} c_{9} - 3 c_{11} c_{9})$
$g_{18}$	$8 \sqrt{3 / 5} (c_{10} c_{6} + 3 c_{11} c_{6} + 7 c_{11} c_{7} + 9 \sqrt{2} c_{12} c_{8} + 7 c_{13} c_{8} + 3 c_{13} c_{9} + c_{14} c_{9})$
$g_{19}$	$16 \sqrt{3 / 5} (2 c_{10} c_{7} - 5 c_{11} c_{7} + 5 c_{13} c_{8} + 2 c_{14} c_{8} + 3 \sqrt{2} c_{12} c_{9})$
$g_{20}$	$- 16 \sqrt{15} (c_{11} c_{6} + c_{10} c_{7} - c_{14} c_{8} - c_{13} c_{9})$
$g_{21}$	$160 \sqrt{2 / 7} (c_{10} c_{13} + c_{11} c_{14})$
$g_{22}$	$160 (c_{10} c_{12} + \sqrt{2} c_{11} c_{13}) / \sqrt{7}$
$g_{23}$	$32 (8 c_{11} c_{12} + \sqrt{2} c_{10} c_{13} - \sqrt{2} c_{11} c_{14}) / \sqrt{7}$
$g_{24}$	$8 (c_{10}^{2} + 10 c_{11}^{2} + 18 c_{12}^{2} + 10 c_{13}^{2} + c_{14}^{2}) / \sqrt{7}$
$g_{25}$	$32 (\sqrt{2} c_{10} c_{11} + 8 c_{12} c_{13} + \sqrt{2} c_{13} c_{14}) / \sqrt{7}$
$g_{26}$	$- 80 (\sqrt{2} c_{11}^{2} - \sqrt{2} c_{13}^{2} - 2 c_{12} c_{14}) / \sqrt{7}$
$g_{27}$	$- 160 \sqrt{2 / 7} (c_{10} c_{11} - c_{13} c_{14})$

Term	Input	Output
Term	Input	Analytical	ZEMAX	Diff
$c_{3}^{- 3}$	$- 0.21030$	$- 0.22703$	$- 0.22690$	0.00013
$c_{3}^{- 1}$	0.08100	0.09221	0.09221	0.00000
$c_{3}^{1}$	0.05900	0.06224	0.06210	$- 0.00014$
$c_{3}^{3}$	0.00470	0.00438	0.00444	0.00006
$c_{4}^{- 4}$	0.04510	0.05715	0.05705	$- 0.00010$
$c_{4}^{- 2}$	$- 0.05380$	$- 0.06240$	$- 0.06237$	0.00003
$c_{4}^{0}$	0.07050	0.06995	0.06977	$- 0.00018$
$c_{4}^{2}$	0.11100	0.11770	0.11760	$- 0.00010$
$c_{4}^{4}$	$- 0.04770$	$- 0.05086$	$- 0.05074$	0.00012
$c_{5}^{- 5}$	0.01690	0.02081	0.02078	$- 0.00003$
$c_{5}^{- 3}$	0.02760	0.02466	0.02463	$- 0.00003$
$c_{5}^{- 1}$	$- 0.02960$	$- 0.02733$	$- 0.02729$	0.00004
$c_{5}^{1}$	0.02940	0.03057	0.03052	$- 0.00005$
$c_{5}^{3}$	$- 0.02170$	$- 0.02240$	$- 0.02237$	0.00003
$c_{5}^{5}$	$- 0.00790$	$- 0.00657$	$- 0.00655$	0.00002
$c_{6}^{- 6}$	0.05750	0.05992	0.05982	$- 0.00010$
$c_{6}^{- 4}$	0.00800	0.01091	0.01089	$- 0.00002$
$c_{6}^{- 2}$	0.01220	0.01091	0.01089	$- 0.00002$
$c_{6}^{0}$	0.02800	0.02816	0.02812	$- 0.00004$
$c_{6}^{2}$	$- 0.00930$	$- 0.00900$	$- 0.00899$	0.00001
$c_{6}^{4}$	$- 0.01790$	$- 0.01848$	$- 0.01845$	0.00003
$c_{6}^{6}$	$- 0.03320$	$- 0.03540$	$- 0.03535$	0.00005
r.m.s.	0.29651	0.31405	0.31432	0.00034

Abstract

Cited By

Figures (5)

Tables (3)

Equations (63)

Applied Optics