Alternative computation of the Seidel aberration coefficients using the Lie algebraic method

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

doi:10.1364/JOSAA.465900

Journal of the Optical Society of America A
Vol. 39,
Issue 9,
pp. 1603-1615
(2022)
•https://doi.org/10.1364/JOSAA.465900

Alternative computation of the Seidel aberration coefficients using the Lie algebraic method

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, "Alternative computation of the Seidel aberration coefficients using the Lie algebraic method," J. Opt. Soc. Am. A 39, 1603-1615 (2022)

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Abstract

We give a brief introduction to Hamiltonian optics and Lie algebraic methods. We use these methods to describe the operators governing light propagation, refraction, and reflection in phase space. The method offers a systematic way to find aberration coefficients of any order for arbitrary rotationally symmetric optical systems. The coefficients from the Lie method are linked to the Seidel aberration coefficients. Furthermore, the property of summing individual surface contributions is preserved by the Lie algebraic theory. Two examples are given to validate the proposed methodology with good results.

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Data availability

Data underlying the results presented in this paper are available in [24].

24. M. J. Kidger, Fundamental Optical Design (SPIE, 2001).

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	Matrix (First Order)	Lie (First Order)	Lie (Third Order)
Ray propagation	$(\begin{matrix} I_{2 \times 2} & \frac{d}{n} I_{2 \times 2} \\ O_{2 \times 2} & I_{2 \times 2} \end{matrix})$	$\exp (- d [h_{2}, \cdot])$ (49)	$\exp (- d [h_{4}, \cdot])$ (49)
Refraction	$(\begin{matrix} I_{2 \times 2} & O_{2 \times 2} \\ 2 β (n - n^{'}) I_{2 \times 2} & I_{2 \times 2} \end{matrix})$	$\exp ([g_{2}, \cdot])$ (55)	$\exp ([g_{4}, \cdot])$ (57), (60)

		$S_{I}$	$S_{I I}$	$S_{I I I}$	$S_{I V}$	$S_{V}$
Surface 1	Kidger [24]	0.191312	0.007686	0.000309	0.000491	0.000032
Surface 1	Lie	0.191312	0.007687	0.000309	0.000491	0.000032
Surface 2	Kidger [24]	0.005631	−0.001252	0.000278	−0.000130	−0.000033
Surface 2	Lie	0.005631	−0.001252	0.000278	−0.000130	−0.000033
Surface 3	Kidger [24]	−0.009703	−0.000634	−0.000041	0.000701	0.000043
Surface 3	Lie	−0.009703	−0.000634	−0.000041	0.000701	0.000043
Surface 4	Kidger [24]	−0.183099	−0.007941	−0.000344	−0.000902	−0.000054
Surface 4	Lie	−0.183100	−0.007942	−0.000344	−0.000902	−0.000054
Total	Kidger [24]	0.004140	−0.002141	0.000201	0.000160	−0.000012
Total	Lie	0.004140	−0.002141	0.000201	0.000160	−0.000012

		$S_{I}$	$S_{I I}$	$S_{I I I}$	$S_{I V}$	$S_{V}$
Surface 1	Kidger [24]	−0.258533	−0.220745	0.017983	−0.017983	0.000000
Surface 1	Lie	−0.258534	−0.220610	0.017961	−0.017961	0.000000
Surface 2	Kidger [24]	0.258574	0.220580	0.001553	0.041525	−0.002062
Surface 2	Lie	0.258574	0.220446	0.001552	0.041475	−0.002058
Total	Kidger [24]	0.000040	−0.000164	0.019537	0.023542	−0.002062
Total	Lie	0.000040	−0.000164	0.019513	0.023513	−0.002058

	Matrix (First Order)	Lie (First Order)	Lie (Third Order)
Ray propagation	$(\begin{matrix} I_{2 \times 2} & \frac{d}{n} I_{2 \times 2} \\ O_{2 \times 2} & I_{2 \times 2} \end{matrix})$	$\exp (- d [h_{2}, \cdot])$ (49)	$\exp (- d [h_{4}, \cdot])$ (49)
Refraction	$(\begin{matrix} I_{2 \times 2} & O_{2 \times 2} \\ 2 β (n - n^{'}) I_{2 \times 2} & I_{2 \times 2} \end{matrix})$	$\exp ([g_{2}, \cdot])$ (55)	$\exp ([g_{4}, \cdot])$ (57), (60)

		$S_{I}$	$S_{I I}$	$S_{I I I}$	$S_{I V}$	$S_{V}$
Surface 1	Kidger [24]	0.191312	0.007686	0.000309	0.000491	0.000032
Surface 1	Lie	0.191312	0.007687	0.000309	0.000491	0.000032
Surface 2	Kidger [24]	0.005631	−0.001252	0.000278	−0.000130	−0.000033
Surface 2	Lie	0.005631	−0.001252	0.000278	−0.000130	−0.000033
Surface 3	Kidger [24]	−0.009703	−0.000634	−0.000041	0.000701	0.000043
Surface 3	Lie	−0.009703	−0.000634	−0.000041	0.000701	0.000043
Surface 4	Kidger [24]	−0.183099	−0.007941	−0.000344	−0.000902	−0.000054
Surface 4	Lie	−0.183100	−0.007942	−0.000344	−0.000902	−0.000054
Total	Kidger [24]	0.004140	−0.002141	0.000201	0.000160	−0.000012
Total	Lie	0.004140	−0.002141	0.000201	0.000160	−0.000012

Abstract

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Journal of the Optical Society of America A