Computing aberration coefficients for plane-symmetric reflective systems: a Lie algebraic approach

A. Barion; M. J. H. Anthonissen; J. H. M. ten Thije Boonkkamp; W. L. IJzerman; W. L. IJzerman

doi:10.1364/JOSAA.487343

Journal of the Optical Society of America A
Vol. 40,
Issue 6,
pp. 1215-1224
(2023)
•https://doi.org/10.1364/JOSAA.487343

Computing aberration coefficients for plane-symmetric reflective systems: a Lie algebraic approach

A. Barion, M. J. H. Anthonissen, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman

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A. Barion, M. J. H. Anthonissen, J. H. M. ten Thije Boonkkamp, and W. L. IJzerman, "Computing aberration coefficients for plane-symmetric reflective systems: a Lie algebraic approach," J. Opt. Soc. Am. A 40, 1215-1224 (2023)

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Abstract

We apply the Lie algebraic method to reflecting optical systems with plane-symmetric freeform mirrors. Using analytical ray-tracing equations, we construct an optical map. The expansion of this map gives us the aberration coefficients in terms of initial ray coordinates. The Lie algebraic method is applied to treat aberrations up to arbitrary order. The presented method provides a systematic and rigorous approach to the derivation, treatment, and composition of aberrations in plane-symmetric systems. We give the results for second- and third-order aberrations and apply them to three single-mirror examples.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Equations (46)

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	Transformation	Analytical RT	Lie
(i)	Propagation in object-space	(3)	(22)
(ii)	Rotation	(8), (12)	(28)
	Reflection	(6), (7)	(28)
	Rotation	(8), (12)	(28)
(iii)	Propagation in image-space	(3)	(22)

Coefficients	Values
$a_{0200}$	$2 c_{02} \sec (θ)$
$a_{2000}$	$2 c_{20} \cos (θ)$
$a_{0201}$	$2 c_{02} \sec (θ) \tan (θ)$
$a_{0300}$	$2 \sec (θ)^{2} (c_{03} - 2 c_{02}^{2} \tan (θ))$
$a_{1110}$	$4 c_{20} \sin (θ)$
$a_{2001}$	$- 2 c_{20} \sin (θ)$
$a_{2100}$	$2 (c_{21} - 4 \cos (θ) c_{20}^{2} \sin (θ) + 2 c_{02} c_{20} \tan (θ))$
$a_{0202}$	$- \frac{1}{2} (- 1 + 3 \cos (2 θ)) c_{02} \sec (θ)^{3}$
$a_{0220}$	$- c_{02} \sec (θ) + 2 c_{20} \sin (θ) \tan (θ)$
$a_{0301}$	$2 \sec (θ)^{4} (- c_{02}^{2} + 3 \cos (2 θ) c_{02}^{2} + c_{03} \sin (2 θ))$
$a_{0400}$	$- \sec (θ)^{5} (c_{02}^{3} + \cos (2 θ) (7 c_{02}^{3} - c_{04}) - c_{04} + 4 c_{02} c_{03} \sin (2 θ))$
$a_{1210}$	$4 c_{02} c_{20} - 2 c_{20}^{2} + 2 \cos (3 θ) c_{20}^{2} \sec (θ) + 4 c_{21} \tan (θ)$
$a_{2002}$	$- \cos (θ) c_{20}$
$a_{2020}$	$- \cos (θ) c_{20}$
$a_{2101}$	$4 c_{02} c_{20}$
$a_{2200}$	$2 (- 4 \cos (θ) c_{02} c_{20}^{2} + 8 c_{02}^{2} c_{20} \sec (θ)^{3} - 4 c_{20} c_{21} \sin (θ) + \sec (θ) (- 12 c_{02}^{2} c_{20} + c_{22} - 6 c_{02}^{2} c_{20} \tan (θ)^{2}))$
$a_{3010}$	$4 \cos (θ)^{2} c_{20}^{2}$
$a_{4000}$	$- 8 \cos (θ)^{3} c_{20}^{3} + 2 \cos (θ) c_{40} - 2 c_{02} c_{20}^{2} \sin (θ) \tan (θ)$

Coefficients	Values
$b_{0003}$	$s_{i m}^{2} (a_{0201} - s_{i m} a_{0300})$
$b_{0021}$	$s_{i m}^{2} (a_{1110} + a_{2001} - s_{i m} a_{2100})$
$b_{0102}$	$s_{i m} (3 s_{i m} a_{0300} - 2 a_{0201})$
$b_{0120}$	$s_{i m} (s_{i m} a_{2100} - a_{1110})$
$b_{0201}$	$a_{0201} - 3 s_{i m} a_{0300}$
$b_{0300}$	$a_{0300}$
$b_{1011}$	$s_{i m} (2 s_{i m} a_{2100} - a_{1110} + 2 a_{2001})$
$b_{1110}$	$a_{1110} - 2 s_{i m} a_{2100}$
$b_{2001}$	$a_{2001} - s_{i m} a_{2100}$
$b_{2100}$	$a_{2100}$

$c_{mn}$	Values
$c_{21}$	0
$c_{03}$	0
$c_{40}$	$- 1 / 8000$
$c_{22}$	$- 1 / 16000$
$c_{04}$	$- 1 / 128000$

$c_{mn}$	Values
$c_{21}$	$2.53388 \times 10^{- 6}$
$c_{03}$	$2.43386 \times 10^{- 6}$
$c_{40}$	$- 6.55111 \times 10^{- 10}$
$c_{22}$	$- 3.77553 \times 10^{- 9}$
$c_{04}$	$- 3.02209 \times 10^{- 9}$

Abstract

Data availability

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Tables (5)

Equations (46)

Journal of the Optical Society of America A