Off-axis three-mirror freeform systems design based on improved W-W differential equations

Sike Chen; Yangjie Wei; YiWei Sun; BaoTing Li; Ji Zhao

doi:10.1364/AO.483753

Applied Optics
Vol. 62,
Issue 15,
pp. 3892-3903
(2023)
•https://doi.org/10.1364/AO.483753

Off-axis three-mirror freeform systems design based on improved W-W differential equations

Sike Chen, Yangjie Wei, YiWei Sun, BaoTing Li, and Ji Zhao

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Sike Chen, Yangjie Wei, YiWei Sun, BaoTing Li, and Ji Zhao, "Off-axis three-mirror freeform systems design based on improved W-W differential equations," Appl. Opt. 62, 3892-3903 (2023)

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Table of Contents Category
- Imaging Systems, Image Processing, and Displays
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About this Article
History
- Original Manuscript: December 16, 2022
- Revised Manuscript: February 26, 2023
- Manuscript Accepted: April 9, 2023
- Published: May 15, 2023

Abstract

Design of an off-axis system using the Wassermann–Wolf (W-W) differential equations can effectively eliminate the spherical aberration and coma problem; however, it is complicated and time consuming to calculate the discrete point coordinates on the freeform mirror surfaces due to multiple numbers of reference system transformation in the design process. This paper presents an improved W-W-differential-equations-based design method for off-axis three-mirror freeform systems. First, to reduce the number of coordinate transformations, a geometric relationship between different optical rays in an off-axis system is established using the distance between the central points of adjacent mirrors. Second, a three-dimensional rotation matrix is used to associate the optical paths passing through adjacent mirrors in different reference coordinate systems, and new simplified W-W differential equations based on the ray vectors are constructed. The experimental results show that our method can easily and effectively design off-axis three-mirror freeform systems with different parameters and structures, and the designed systems have good imaging quality.

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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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Parameter	Specification
Number of mirrors	3
Field angle	$2^{\circ} \times 2^{\circ}$
Entrance pupil diameter	60 mm
Focal length	350 mm
Wave length	486–656 nm

	Secondary Mirror	Tertiary Mirror
$c$	−0.001429	−0.002336
$k$	1.3830	−0.9215
$Z_{1}$ -piston	$- 2.337 \times 10^{- 08}$	$- 8.104 \times 10^{- 08}$
$Z_{3}$ -tilt	$- 4.812 \times 10^{- 07}$	$1.035 \times 10^{- 07}$
$Z_{4}$ -defocus	$- 9.831 \times 10^{- 09}$	$5.153 \times 10^{- 09}$
$Z_{5}$ -astigmatism, primary	$5.103 \times 10^{- 06}$	$- 3.911 \times 10^{- 06}$
$Z_{8}$ -coma, primary	$- 6.983 \times 10^{- 08}$	$4.789 \times 10^{- 08}$
$Z_{9}$ -spherical aberration, primary	$- 5.243 \times 10^{- 11}$	$- 5.808 \times 10^{- 11}$
$Z_{11}$ - trefoil, primary	$- 1.972 \times 10^{- 09}$	$- 5.410 \times 10^{- 10}$
$Z_{12}$ - astigmatism, secondary	$- 3.263 \times 10^{- 12}$	$4.935 \times 10^{- 12}$
$Z_{15}$ - coma, secondary	$- 4.778 \times 10^{- 13}$	$- 4.730 \times 10^{- 13}$
$Z_{16}$ - spherical aberration	$- 2.780 \times 10^{- 14}$	$- 5.596 \times 10^{- 14}$

Parameter	Specification
Number of mirrors	3
Field angle	$2^{\circ} \times 2^{\circ}$
Entrance pupil diameter	55 mm
Focal length	210 mm
Wave length	486–656 nm

	Secondary Mirror	Tertiary Mirror
$c$	−0.005286	−0.004108
$k$	1.3810	−0.2823
$Z_{1}$ -piston	$- 5.939 \times 10^{- 06}$	$3.062 \times 10^{- 06}$
$Z_{3}$ -tilt	$1.730 \times 10^{- 06}$	$- 2.292 \times 10^{- 07}$
$Z_{4}$ -defocus	$- 5.702 \times 10^{- 06}$	$2.549 \times 10^{- 06}$
$Z_{5}$ -astigmatism, primary	$2.300 \times 10^{- 04}$	$- 3.976 \times 10^{- 05}$
$Z_{8}$ -coma, primary	$7.641 \times 10^{- 07}$	$- 1.119 \times 10^{- 07}$
$Z_{9}$ -spherical aberration, primary	$- 1.385 \times 10^{- 09}$	$9.825 \times 10^{- 11}$
$Z_{11}$ - trefoil, primary	$4.252 \times 10^{- 07}$	$1.474 \times 10^{- 07}$
$Z_{12}$ - astigmatism, secondary	$8.312 \times 10^{- 09}$	$5.469 \times 10^{- 10}$
$Z_{15}$ - coma, secondary	$- 3.625 \times 10^{- 11}$	$- 3.723 \times 10^{- 12}$
$Z_{16}$ - spherical aberration	$9.618 \times 10^{- 15}$	$- 3.388 \times 10^{- 15}$

Parameter	Specification
Number of mirrors	3
Field angle	$4^{\circ} \times 4^{\circ}$
Entrance pupil diameter	60 mm
Focal length	400 mm
Wave length	486–656 nm

	Secondary Mirror	Tertiary Mirror
$c$	−0.001416	−0.002068
$k$	0.0218	−0.0659
$Z_{1}$ -piston	$2.219 \times 10^{- 06}$	$1.590 \times 10^{- 08}$
$Z_{3}$ -tilt	$- 2.407 \times 10^{- 07}$	$1.145 \times 10^{- 07}$
$Z_{4}$ -defocus	$2.335 \times 10^{- 06}$	$- 1.644 \times 10^{- 07}$
$Z_{5}$ -astigmatism, primary	$7.614 \times 10^{- 06}$	$- 6.784 \times 10^{- 06}$
$Z_{8}$ -coma, primary	$- 6.978 \times 10^{- 08}$	$5.843 \times 10^{- 08}$
$Z_{9}$ -spherical aberration, primary	$- 1.285 \times 10^{- 10}$	$5.431 \times 10^{- 11}$
$Z_{11}$ - trefoil, primary	$- 3.442 \times 10^{- 09}$	$- 1.852 \times 10^{- 09}$
$Z_{12}$ - astigmatism, secondary	$- 1.764 \times 10^{- 11}$	$- 1.230 \times 10^{- 11}$
$Z_{15}$ - coma, secondary	$- 5.112 \times 10^{- 13}$	$- 2.586 \times 10^{- 13}$
$Z_{16}$ - spherical aberration	$- 7.631 \times 10^{- 14}$	$- 5.571 \times 10^{- 14}$

	Secondary Mirror	Tertiary Mirror
$c$	−0.004165	−0.003449
$k$	1.6795	−1.580
$Z_{1}$ -piston	$5.553 \times 10^{- 05}$	$- 1.450 \times 10^{- 05}$
$Z_{3}$ -tilt	$9.888 \times 10^{- 08}$	$6.268 \times 10^{- 07}$
$Z_{4}$ -defocus	$4, 709 \times 10^{- 05}$	$- 1.905 \times 10^{- 05}$
$Z_{5}$ -astigmatism, primary	$1.184 \times 10^{- 04}$	$- 3.589 \times 10^{- 05}$
$Z_{8}$ -coma, primary	$- 9.303 \times 10^{- 08}$	$4.175 \times 10^{- 08}$
$Z_{9}$ -spherical aberration, primary	$- 9.440 \times 10^{- 10}$	$1.269 \times 10^{- 09}$
$Z_{11}$ - trefoil, primary	$2.402 \times 10^{- 07}$	$1.840 \times 10^{- 07}$
$Z_{12}$ - astigmatism, secondary	$3.487 \times 10^{- 09}$	$1.125 \times 10^{- 09}$
$Z_{15}$ - coma, secondary	$- 8.663 \times 10^{- 13}$	$- 3.227 \times 10^{- 13}$
$Z_{16}$ - spherical aberration	$- 5.440 \times 10^{- 14}$	$- 2.196 \times 10^{- 14}$

Abstract

Data availability

Cited By

Figures (15)

Tables (8)

Equations (23)

Applied Optics