Transverse ray aberrations determination in&#x00A0;non-axially symmetrical optical imaging systems&#x00A0;using Taylor series expansion

Psang Dain Lin

doi:10.1364/AO.484323

Applied Optics
Vol. 62,
Issue 15,
pp. 4000-4010
(2023)
•https://doi.org/10.1364/AO.484323

Transverse ray aberrations determination in non-axially symmetrical optical imaging systems using Taylor series expansion

Psang Dain Lin

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Psang Dain Lin, "Transverse ray aberrations determination in non-axially symmetrical optical imaging systems using Taylor series expansion," Appl. Opt. 62, 4000-4010 (2023)

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Abstract

Non-axially symmetrical optical systems can provide better solutions to several optical design problems. However, the determination of their aberrations is a challenge due to the great diversity of their configurations. To address this problem, this study extends previous work by the present group to determine the transverse ray aberrations in non-axially symmetrical optical imaging systems through the use of a Taylor series expansion. The expansion converts the coordinates of the incident point, which are highly composite functions, of a general ray on the image plane into polynomial functions. The various order transverse ray aberrations are then extracted directly from the corresponding terms of the polynomials. Notably, the derived expressions are exact since they are determined without any approximations. Furthermore, the proposed method can be equally applied to axially symmetrical systems provided that the object is confined on the meridional plane. Consequently, the proposed method offers a useful and generally applicable approach for the systematic analysis of the transverse ray aberrations in optical systems with the object at either finite or infinite positions.

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The author confirms that the data supporting the findings of this study are either available within the article and its supplementary materials or may be obtained from the authors upon reasonable request.

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Equations (48)

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	Radius $R_{i}$ mm	Separation $V_{i}$ mm	Refractive Index $ξ_{i}$
$i = 1$	33.7	3	1.638543
$i = 2$	0	8.3	1
$i = 3$	−37.8	1	1.603420
$i = 4$	37.8	3	1
$i = 5$	0	5.3	1
$i = 6$	0	3	1.638543
$i = 7$	−30.9	12	1
$i = 8$	0	15	1
$i = 9$	$R_{9}$	$V_{9}$	1
$i = 10$	0		1

Items	Proposed Method	Zemax	% Error
$B_{1} ρ_{max}^{3}$	−0.054537	− 0.054541	0
$B_{2} β_{0} ρ_{max}^{2}$	−0.022490	−0.022719	1.01
$B_{3} β_{0}^{2} ρ_{max}$	0.039953	0.040781	−2.03
$B_{4} β_{0}^{2} ρ_{max}$	−0.051944	−0.053019	2.03
$B_{5} β_{0}^{3}$	0.165451	0.167526 (see Appendix B)	−1.23

${\bar{g}}_{1000} = [\begin{matrix} 0 & 0 .071268 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0010} = [\begin{matrix} 0 & - 73 .173709 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0001} = [\begin{matrix} 0 & 0 & 77 .751197 \end{matrix}]^{T}$
${\bar{g}}_{2000} = [\begin{matrix} 0 & 0 .002635 & 0 \end{matrix}]^{T}$	${\bar{g}}_{1100} = [\begin{matrix} 0 & 0 & - 0 .000802 \end{matrix}]^{T}$	${\bar{g}}_{1010} = [\begin{matrix} 0 & 0 .088119 & 0 \end{matrix}]^{T}$
${\bar{g}}_{1001} = [\begin{matrix} 0 & 0 & 0 .073688 \end{matrix}]^{T}$	${\bar{g}}_{0200} = [\begin{matrix} 0 & 0 .000458 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0110} = [\begin{matrix} 0 & 0 & - 0 .176694 \end{matrix}]^{T}$
${\bar{g}}_{0101} = [\begin{matrix} 0 & - 0 .066317 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0020} = [\begin{matrix} 0 & - 5 .211028 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0011} = [\begin{matrix} 0 & 0 & - 3 .308019 \end{matrix}]^{T}$
${\bar{g}}_{0002} = [\begin{matrix} 0 & - 6 .150234 & 0 \end{matrix}]^{T}$	${\bar{g}}_{3000} = [\begin{matrix} 0 & 0 .000125 & 0 \end{matrix}]^{T}$	${\bar{g}}_{2100} = [\begin{matrix} 0 & 0 & - 0 .000085 \end{matrix}]^{T}$
${\bar{g}}_{2010} = [\begin{matrix} 0 & - 0 .000931 & 0 \end{matrix}]^{T}$	${\bar{g}}_{2001} = [\begin{matrix} 0 & 0 & 0 .000587 \end{matrix}]^{T}$	${\bar{g}}_{1200} = [\begin{matrix} 0 & 0 .000086 & 0 \end{matrix}]^{T}$
${\bar{g}}_{1110} = [\begin{matrix} 0 & 0 & - 0 .002450 \end{matrix}]^{T}$	${\bar{g}}_{1101} = [\begin{matrix} 0 & - 0 .000837 & 0 \end{matrix}]^{T}$	${\bar{g}}_{1020} = [\begin{matrix} 0 & - 0 .470803 & 0 \end{matrix}]^{T}$
${\bar{g}}_{1011} = [\begin{matrix} 0 & 0 & 0 .250245 \end{matrix}]^{T}$	${\bar{g}}_{1002} = [\begin{matrix} 0 & 0 .025209 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0300} = [\begin{matrix} 0 & 0 & - 0 .000071 \end{matrix}]^{T}$
${\bar{g}}_{0210} = [\begin{matrix} 0 & 0 .001344 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0201} = [\begin{matrix} 0 & 0 & - 0 .002955 \end{matrix}]^{T}$	${\bar{g}}_{0120} = [\begin{matrix} 0 & 0 & 0 .062914 \end{matrix}]^{T}$
${\bar{g}}_{0111} = [\begin{matrix} 0 & - 0 .274652 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0102} = [\begin{matrix} 0 & 0 & 0 .217351 \end{matrix}]^{T}$	${\bar{g}}_{0030} = [\begin{matrix} 0 & - 12 .920695 & 0 \end{matrix}]^{T}$
${\bar{g}}_{0021} = [\begin{matrix} 0 & 0 & 31 .716936 \end{matrix}]^{T}$	${\bar{g}}_{0012} = [\begin{matrix} 0 & 9 .554561 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0003} = [\begin{matrix} 0 & 0 & 20 .865541 \end{matrix}]^{T}$

${\bar{g}}_{1000} = [\begin{matrix} 0 & 0 .236530 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0100} = [\begin{matrix} 0 & 0 & - 0 .316486 \end{matrix}]^{T}$	${\bar{g}}_{0010} = [\begin{matrix} 0 & 0 .190844 & 0 \end{matrix}]^{T}$
${\bar{g}}_{2000} = [\begin{matrix} 0 & 0 .000602 & 0 \end{matrix}]^{T}$	${\bar{g}}_{1100} = [\begin{matrix} 0 & 0 & - 0 .000384 \end{matrix}]^{T}$	${\bar{g}}_{1010} = [\begin{matrix} 0 & - 0 .004628 & 0 \end{matrix}]^{T}$
${\bar{g}}_{1001} = [\begin{matrix} 0 & 0 & 0 .001793 \end{matrix}]^{T}$	${\bar{g}}_{0200} = [\begin{matrix} 0 & - 0 .000129 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0110} = [\begin{matrix} 0 & 0 & 0 .000039 \end{matrix}]^{T}$
${\bar{g}}_{0101} = [\begin{matrix} 0 & - 0 .000260 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0020} = [\begin{matrix} 0 & 0 .007616 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0011} = [\begin{matrix} 0 & 0 & - 0 .002186 \end{matrix}]^{T}$
${\bar{g}}_{0002} = [\begin{matrix} 0 & 0. 001358 & 0 \end{matrix}]^{T}$	${\bar{g}}_{3000} = [\begin{matrix} 0 & - 0 .000006 & 0 \end{matrix}]^{T}$	${\bar{g}}_{2100} = [\begin{matrix} 0 & 0 & 0 .000005 \end{matrix}]^{T}$
${\bar{g}}_{2010} = [\begin{matrix} 0 & 0 .000081 & 0 \end{matrix}]^{T}$	${\bar{g}}_{2001} = [\begin{matrix} 0 & 0 & - 0 .000027 \end{matrix}]^{T}$	${\bar{g}}_{1200} = [\begin{matrix} 0 & - 0 .000005 & 0 \end{matrix}]^{T}$
${\bar{g}}_{1110} = [\begin{matrix} 0 & 0 & - 0 .000043 \end{matrix}]^{T}$	${\bar{g}}_{1101} = [\begin{matrix} 0 & 0 .000042 & 0 \end{matrix}]^{T}$	${\bar{g}}_{1020} = [\begin{matrix} 0 & - 0 .000357 & 0 \end{matrix}]^{T}$
${\bar{g}}_{1011} = [\begin{matrix} 0 & 0 & 0 .000201 \end{matrix}]^{T}$	${\bar{g}}_{1002} = [\begin{matrix} 0 & - 0 .000099 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0300} = [\begin{matrix} 0 & 0 & 0 .000006 \end{matrix}]^{T}$
${\bar{g}}_{0210} = [\begin{matrix} 0 & 0 .000020 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0201} = [\begin{matrix} 0 & 0 & - 0 .000069 \end{matrix}]^{T}$	${\bar{g}}_{0120} = [\begin{matrix} 0 & 0 & 0 .000086 \end{matrix}]^{T}$
${\bar{g}}_{0111} = [\begin{matrix} 0 & - 0 .000170 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0102} = [\begin{matrix} 0 & 0 & 0 .000265 \end{matrix}]^{T}$	${\bar{g}}_{0030} = [\begin{matrix} 0 & 0 .000480 & 0 \end{matrix}]^{T}$
${\bar{g}}_{0021} = [\begin{matrix} 0 & 0 & - 0 .000364 \end{matrix}]^{T}$	${\bar{g}}_{0012} = [\begin{matrix} 0 & 0 .000357 & 0 \end{matrix}]^{T}$	${\bar{g}}_{0003} = [\begin{matrix} 0 & 0 & - 0 .000323 \end{matrix}]^{T}$

	Radius $R_{i}$ mm	Separation $V_{i}$ mm	Refractive Index $ξ_{i}$
$i = 1$	33.7	3	1.638543
$i = 2$	0	8.3	1
$i = 3$	−37.8	1	1.603420
$i = 4$	37.8	3	1
$i = 5$	0	5.3	1
$i = 6$	0	3	1.638543
$i = 7$	−30.9	12	1
$i = 8$	0	15	1
$i = 9$	$R_{9}$	$V_{9}$	1
$i = 10$	0		1

Abstract

Data availability

Cited By

Figures (5)

Tables (4)

Equations (48)

Applied Optics