Design method of freeform off-axis three-mirror reflective imaging systems

Huaixue Zhao; Huaixue Zhao; Limin Gao; Xianglong Mao; Yaxuan Duan; Xun Xue

doi:10.1364/AO.498270

Applied Optics
Vol. 62,
Issue 29,
pp. 7852-7859
(2023)
•https://doi.org/10.1364/AO.498270

Design method of freeform off-axis three-mirror reflective imaging systems

Huaixue Zhao, Limin Gao, Xianglong Mao, Yaxuan Duan, and Xun Xue

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Huaixue Zhao, Limin Gao, Xianglong Mao, Yaxuan Duan, and Xun Xue, "Design method of freeform off-axis three-mirror reflective imaging systems," Appl. Opt. 62, 7852-7859 (2023)

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Table of Contents Category
- Optical Design and Fabrication
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About this Article
History
- Original Manuscript: June 20, 2023
- Revised Manuscript: August 25, 2023
- Manuscript Accepted: September 18, 2023
- Published: October 9, 2023

Abstract

In this paper, a novel, to the best of our knowledge, method is proposed to design a freeform off-axis three-mirror reflective imaging system. A special algorithm is demonstrated to calculate the data points on the unknown freeform surface using the rays from the pupil of the center field. Then the three-dimensional shape of the freeform surface is solved by these characteristic points, which serves as a good starting point for further optimization. The benefit of this design method is demonstrated by designing a freeform off-axis three mirror imaging system with high performance. The final system operates at F/2 with an entrance pupil diameter of 400 mm and a field of view of $2.4^\circ \times 2.4^\circ$. The modulation transfer function (MTF) value of the system reaches 0.6 at 100 lp/mm or higher at all fields of view.

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Parameter	Specifications
Work spectrum	380–760 nm
Field of view	${2.4}^{\circ} \times {2.4}^{\circ}$
Entrance pupil diameter	400 mm
Effective focal length	800 mm
Number of surface	3
Image quality	0.5 at 100 lp/mm

	Radius/mm	Thickness/mm	Conic
Primary mirror	−3634.4727	−599.6880	8.2600
Secondary mirror	−861.2882	599.6880	−0.5861
Tertiary mirror	−1128.7848	−1018.4000	−0.1871

	Radius/mm	Thickness/mm	Conic
Primary mirror	−2997.3572	−600.0000	−3.6809
Secondary mirror	−805.1274	600.0000	0.9189
Tertiary mirror	−1054.8941	−874.4299	0.2278

Term	Coordinate			Normal Direction Vector
1	$x$	$y$	$z$	$α$	$β$	$γ$
2	0.0000	−272.1778	862.1376	0.0000	0.0898	−0.9960
3	12.5888	−272.1855	862.1107	−0.0041	0.0898	−0.9959
4	10.1932	−279.7366	861.4296	−0.0033	0.0922	−0.9957
5	3.8954	−284.4118	861.0077	−0.0013	0.0937	−0.9956
6	−3.8954	−284.4118	861.0077	0.0013	0.0937	−0.9956
7	−10.1932	−279.7366	861.4296	0.0033	0.0922	−0.9957
8	−12.5888	−272.1855	862.1107	0.0041	0.0898	−0.9959
9	−10.1754	−264.6518	862.7900	0.0033	0.0874	−0.9962
10	0.0000	−272.1778	862.1376	0.0000	0.0898	−0.9960
11	…	…	…	…	…	…

	Standard Zernike Polynomials
Term	$n$	$m$	$z_{j} (ρ, θ)$ in Polar Coordinate System	$z_{j} (x, y)$ in Cartesian Coordinate System	Aberration Type
1	0	0	1	1	Piston
2	1	1	$2 ρ \cdot \cos (θ)$	$2 x$	Distortion-tilt ( $x$ axis)
3	1	1	$2 ρ \cdot \sin (θ)$	$2 y$	Distortion-tilt ( $y$ axis)
4	2	0	$\sqrt{3} (2 ρ^{2} - 1)$	$\sqrt{3} ({2 x}^{2} {+ 2 y}^{2} - 1)$	Defocus
5	2	2	$\sqrt{6} ρ^{2} \cdot \sin (2 \cdot θ)$	$2 \sqrt{6} x y$	Astigmatism, primary (axis at 45°)
6	2	2	$\sqrt{6} ρ^{2} \cdot \cos (2 \cdot θ)$	$\sqrt{6} (x^{2} - y^{2})$	Astigmatism, primary (axis at 0°)
7	3	1	$\sqrt{8} (3 ρ^{3} - 2 ρ) \cdot \sin (θ)$	$\sqrt{8} y ({3 x}^{2} {+ 3 y}^{2} - 2)$	Coma, primary ( $y$ axis)
8	3	1	$\sqrt{8} \cdot (3 \cdot ρ^{3} - 2 \cdot ρ) \cdot \cos (θ)$	$\sqrt{8} x ({3 x}^{2} {+ 3 y}^{2} - 2)$	Coma, primary ( $x$ axis)
…	…	…	…	…	…

$a_{j}$	Value	$a_{j}$	Value	$a_{j}$	Value	$a_{j}$	Value
1	8.5549E $+$ 02	11	3.7630E-02	21	8.0236E-04	31	−4.6863E-05
2	−9.0949E-13	12	2.7219E-02	22	2.2620E-03	32	4.6851E-14
3	−2.4387E-01	13	−7.3896E-13	23	−1.4921E-13	33	9.0050E-07
4	−1.0941E $+$ 01	14	3.4818E-04	24	6.6250E-04	34	−2.1760E-14
5	−2.7285E-12	15	1.6485E-12	25	1.7697E-13	35	9.2767E-07
6	2.0320E-01	16	−1.2079E-13	26	3.1281E-04	36	−3.3751E-14
7	−9.0207E-02	17	1.0443E-03	27	−1.5632E-13	37	8.9098E-05
8	−1.0232E-12	18	4.9027E-13	28	1.0470E-05
9	5.7768E-02	19	2.0147E-03	29	2.5214E-04
10	2.9559E-12	20	−2.5580E-13	30	2.6201E-14

	Surface Type	Radius/mm	Thickness/mm	Conic
Primary mirror	Zernike standard sag	Infinity	−1609.6	0
Secondary mirror	Even asphere	−1960.1283	1609.6	3.9632
Tertiary mirror	Even asphere	−2119.0234	−1751.1	1.4514

$a_{j}$	Value	$a_{j}$	Value	$a_{j}$	Value	$a_{j}$	Value
1	0.00E $+$ 00	11	−4.25E-03	21	−2.12E-04	31	−1.95E-05
2	2.12E-02	12	−2.84E-03	22	−3.35E-04	32	−3.61E-06
3	−3.02E $+$ 00	13	4.46E-03	23	7.21E-05	33	−8.12E-06
4	−4.41E-01	14	4.48E-03	24	−2.42E-05	34	3.03E-06
5	3.69E-02	15	−2.70E-03	25	−8.72E-05	35	5.68E-06
6	−3.71E-02	16	4.21E-04	26	1.44E-04	36	1.10E-05
7	3.61E-04	17	−1.33E-03	27	−1.07E-04	37	−1.69E-06
8	9.52E-03	18	−9.18E-04	28	1.75E-04
9	−1.95E-02	19	−1.40E-03	29	−3.55E-05
10	−1.56E-02	20	−1.17E-04	30	1.81E-06

$a_{j}$	Secondary Mirror	Tertiary Mirror
1	0	0
2	−2.80E-11	−9.58E-14
3	−1.01E-16	−2.18E-19
4	3.00E-21	2.10E-25
5	−4.98E-26	−2.27E-31
6	3.04E-31	5.52E-38

Abstract

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Tables (9)

Equations (14)

Applied Optics