Calculation of zonal power and astigmatism of a freeform gradient index lens with freeform surfaces

Nicholas S. Kochan; Greg R. Schmidt

doi:10.1364/AO.479142

Applied Optics
Vol. 62,
Issue 12,
pp. 2978-2987
(2023)
•https://doi.org/10.1364/AO.479142

Calculation of zonal power and astigmatism of a freeform gradient index lens with freeform surfaces

Nicholas S. Kochan and Greg R. Schmidt

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Nicholas S. Kochan and Greg R. Schmidt, "Calculation of zonal power and astigmatism of a freeform gradient index lens with freeform surfaces," Appl. Opt. 62, 2978-2987 (2023)

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Abstract

Freeform gradient index (F-GRIN) lenses have been recently shown to enable compact optical design. However, aberration theory is only fully developed for rotationally symmetric distributions with a well-defined optical axis. The F-GRIN has no well-defined optical axis, and rays are continuously perturbed along their trajectory. Optical performance can be understood without abstracting optical function to numerical evaluation. The present work derives freeform power and astigmatism along an axis through a zone of an F-GRIN lens with freeform surfaces. Zonal power and astigmatism can be assessed without tracing any rays, capturing mixed contributions of the F-GRIN and freeform surface. Theory is compared with a commercial design software numerical raytrace evaluation. The comparison shows that the raytrace-free (RTF) calculation represents all raytrace contributions within a margin of error. In one example, it is demonstrated that linear terms of index and surface alone in an F-GRIN corrector can correct the astigmatism of a tilted spherical mirror. Accounting for the induced effects of the spherical mirror, RTF calculation provides the amount of astigmatism correction of the optimized F-GRIN corrector.

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No data were generated or analyzed in the presented research.

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Parameter	Symbol	Value	Unit
Pupil semidiameter	$ρ$	1	mm
Ambient index	$n_{a}$	1.0	–
Base index	$n_{0}$	1.5	–
Zonal thickness	$t$	2.0	mm
Sharma [20] ray step	–	0.01	mm
Linear surface	$A_{S}, B_{S}$	$\pm 0.1$	–
Quadratic surface	$C_{S}, D_{S}, E_{S}$	$\pm 0.002$	${m m}^{- 1}$
Tr. linear GRIN	$A_{G}, B_{G}$	$\pm 0.01$	${m m}^{- 1}$
Tr. quadratic GRIN	$C_{G}, D_{G}, E_{G}$	$\pm 0.0005$	${m m}^{- 2}$
Axial linear GRIN	$L_{G}$	−0.2	${m m}^{- 1}$

	PWR	AST	%dev.	%dev.	nonzero
	(D)	(D)	PWR	AST	vals.	eval. contrib.
T1	1.00	2.00	0.00%	0.00%	$C_{S}$	surface
T2	0.00	2.00	N/A	0.00%	$D_{S}$	surface
T3	1.00	2.00	0.00%	0.00%	$E_{S}$	surface
T4	1.00	2.00	−0.04%	−0.04%	$C_{G}$	GRIN
T5	0.00	2.00	N/A	0.00%	$D_{G}$	GRIN
T6	1.00	2.00	−0.04%	−0.04%	$E_{G}$	GRIN
T7	1.00	2.00	0.15%	0.15%	$A_{S}, A_{G}$	coupling, tr.
T8	0.00	2.00	N/A	1.27%	$A_{S}, B_{G}$	coupling, tr.
T9	1.00	2.00	0.15%	0.15%	$B_{S}, B_{G}$	coupling, tr.
T10	1.00	2.00	0.11%	0.11%	$A_{S}, L_{G}$	coupling, ax.
T11*	2.00	4.00	0.21%	0.21%	$A_{S}, B_{S}, L_{G}$	coupling, ax.
T12	1.00	2.00	0.11%	0.11%	$B_{S}, L_{G}$	coupling, ax.
T13**	1.00	0.00	0.00%	N/A	$C_{S}, E_{S}$	surface
T14**	1.00	0.00	−0.02%	N/A	$C_{G}, E_{G}$	GRIN

	$A_{S}$	$B_{S}$	$C_{S}$	$D_{S}$	$E_{S}$	$A_{G}$	$B_{G}$	$C_{G}$	$D_{G}$	$E_{G}$	$L_{G}$
T1	0	0	−0.002	0	0	0	0	0	0	0	0
T2	0	0	0	−0.002	0	0	0	0	0	0	0
T3	0	0	0	0	−0.002	0	0	0	0	0	0
T4	0	0	0	0	0	0	0	−0.0005	0	0	0
T5	0	0	0	0	0	0	0	0	−0.0005	0	0
T6	0	0	0	0	0	0	0	0	0	−0.0005	0
T7	−0.1	0	0	0	0	0.01	0	0	0	0	0
T8	−0.1	0	0	0	0	0	−0.01	0	0	0	0
T9	0	−0.1	0	0	0	0	0.01	0	0	0	0
T10	−0.1	0	0	0	0	0	0	0	0	0	−0.2
T11	−0.1	−0.1	0	0	0	0	0	0	0	0	−0.2
T12	0	−0.1	0	0	0	0	0	0	0	0	−0.2
T13	0	0	−0.001	0	−0.001	0	0	0	0	0	0
T14	0	0	0	0	0	0	0	−0.00025	0	−0.00025	0

Parameter	Symbol	Value	Unit
Pupil semidiameter	$ρ$	1	mm
Ambient index	$n_{a}$	1.0	–
Base index	$n_{0}$	1.5	–
Zonal thickness	$t$	2.0	mm
Sharma [20] ray step	–	0.01	mm
Linear surface	$A_{S}, B_{S}$	$\pm 0.1$	–
Quadratic surface	$C_{S}, D_{S}, E_{S}$	$\pm 0.002$	${m m}^{- 1}$
Tr. linear GRIN	$A_{G}, B_{G}$	$\pm 0.01$	${m m}^{- 1}$
Tr. quadratic GRIN	$C_{G}, D_{G}, E_{G}$	$\pm 0.0005$	${m m}^{- 2}$
Axial linear GRIN	$L_{G}$	−0.2	${m m}^{- 1}$

	PWR	AST	%dev.	%dev.	nonzero
	(D)	(D)	PWR	AST	vals.	eval. contrib.
T1	1.00	2.00	0.00%	0.00%	$C_{S}$	surface
T2	0.00	2.00	N/A	0.00%	$D_{S}$	surface
T3	1.00	2.00	0.00%	0.00%	$E_{S}$	surface
T4	1.00	2.00	−0.04%	−0.04%	$C_{G}$	GRIN
T5	0.00	2.00	N/A	0.00%	$D_{G}$	GRIN
T6	1.00	2.00	−0.04%	−0.04%	$E_{G}$	GRIN
T7	1.00	2.00	0.15%	0.15%	$A_{S}, A_{G}$	coupling, tr.
T8	0.00	2.00	N/A	1.27%	$A_{S}, B_{G}$	coupling, tr.
T9	1.00	2.00	0.15%	0.15%	$B_{S}, B_{G}$	coupling, tr.
T10	1.00	2.00	0.11%	0.11%	$A_{S}, L_{G}$	coupling, ax.
T11*	2.00	4.00	0.21%	0.21%	$A_{S}, B_{S}, L_{G}$	coupling, ax.
T12	1.00	2.00	0.11%	0.11%	$B_{S}, L_{G}$	coupling, ax.
T13**	1.00	0.00	0.00%	N/A	$C_{S}, E_{S}$	surface
T14**	1.00	0.00	−0.02%	N/A	$C_{G}, E_{G}$	GRIN

Abstract

Data availability

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Figures (7)

Tables (3)

Equations (20)

Applied Optics