Design method of imaging systems using square-domain orthogonal polynomials freeform surface (<i>invited</i>)

Lijun Zhou; Tong Yang; Dewen Cheng; Yongtian Wang

doi:10.3788/IRLA20230317

Infrared and Laser Engineering, Volume. 52, Issue 7, 20230317(2023)

Design method of imaging systems using square-domain orthogonal polynomials freeform surface (invited)

Lijun Zhou^1,2, Tong Yang^1,2, Dewen Cheng^1,2, and Yongtian Wang^1,2

Author Affiliations

¹School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China

²Beijing Engineering Research Center of Mixed Reality and Advanced Display, Beijing 100081, China

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Figures & Tables(18)

Fig. 1. Sketch of the rectangular surface aperture

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Fig. 2. System layout for design example A. (a) System using XY polynomials surface; (b) System using Chebyshev polynomials surface; (c) System using Legendre polynomials surface and using two types of constraints during optimization

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Fig. 3. MTF (a) and full FOV RMS wavefront error (b) of systems in example A using XY polynomials and Chebyshev polynomials surface types

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Fig. 4. Sag difference between the freeform surface and base surface for the surfaces in example A using XY polynomials and Chebyshev polynomials surface types

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Fig. 5. MTF (a) and full FOV RMS wavefront error (b) of systems in example A using XY polynomials and Legendre polynomials surface types

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Fig. 6. Sag difference between the freeform surface and base surface for the surfaces in example A using XY polynomials and Legendre polynomials surface types

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Fig. 7. System layout for design example B. (a) System using XY polynomials surface type; (b) System using Legendre polynomials surface type and using two types of constraints during optimization

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Fig. 8. MTF and full FOV RMS wavefront error of systems in example B using XY polynomials and Legendre polynomials surface types

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Fig. 9. Sag difference between the freeform surface and base surface for the surfaces in example B using XY polynomials and Legendre polynomials surface types

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Fig. 10. System layout for design example C. (a) System using XY polynomials surface type; (b) System using Legendre polynomials surface type and using two types of constraints during optimization

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Fig. 11. MTF (a) and full FOV RMS wavefront error (b) of systems in example C using XY polynomials and Legendre polynomials surface types

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Fig. 12. Sag difference between the freeform surface and base surface for the surfaces in example C using XY polynomials and Legendre polynomials surface types

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Table 1. 1D Chebyshev polynomial up to the sixth order
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Table 1. 1D Chebyshev polynomial up to the sixth order
n T_n(u)
0 1
1 u
2 2u²−1
3 4u³−3u
4 8u⁴−8u²+1
5 16u⁵−20u³+5u
6 32u⁶−48u⁴+18u²−1

Table 2. 1D Legendre polynomial up to the sixth order

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Table 2. 1D Legendre polynomial up to the sixth order

n	P_n(u)
0	1
1	$\sqrt {\text{3}} u$
2	$(\sqrt {\text{5}} {\text{/2)(3}}{u^2} - 1)$
3	$(\sqrt {\text{7}} {\text{/2)(5}}{u^{\text{3}}} - {\text{3}}u)$
4	$({\text{3/8)(35}}{u^{\text{4}}} - {\text{30}}{u^2} + 3)$
5	$(\sqrt {11} {\text{/8)(63}}{u^{\text{5}}} - {\text{70}}{u^3} + 15 u)$
6	$(\sqrt {1{\text{3}}} {\text{/16)(231}}{u^{\text{6}}} - {\text{315}}{u^{\text{4}}} + 1{\text{0}}5{u^{\text{2}}} - 5)$

Table 3. Surface sag difference and imaging performance for design example A using XY polynomials and Chebyshev polynomials surface types

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Table 3. Surface sag difference and imaging performance for design example A using XY polynomials and Chebyshev polynomials surface types

Surface type	XY polynomials		Chebyshev polynomials
Constraints	None		Sag difference on aperture margins
Constraints	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)
M1	0.667	0.494	0.090	0.090
M2	0.580	2.368	0.254	0.940
Sum	1.247	2.862	0.344	1.030
RMS WFE	0.021λ		0.021λ

Table 4. Surface sag difference and imaging performance for design example A using XY polynomials and Legendre polynomials surface types

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Table 4. Surface sag difference and imaging performance for design example A using XY polynomials and Legendre polynomials surface types

Surface type	XY polynomials		Legendre polynomials
Constraints	None		Sag difference on aperture margins		Square sum of surface coefficients		Both types of constraints
Constraints	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)
M1	0.667	0.494	0.097	0.100	0.100	0.112	0.097	0.097
M2	0.580	2.368	0.156	0.710	0.156	0.601	0.156	0.747
Sum	1.247	2.862	0.253	0.810	0.256	0.713	0.253	0.844
RMS WFE	0.021λ		0.020λ		0.020λ		0.020λ

Table 5. Surface sag difference and imaging performance for design example B using XY polynomials and Legendre polynomials surface types

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Table 5. Surface sag difference and imaging performance for design example B using XY polynomials and Legendre polynomials surface types

Surface type	XY polynomials		Legendre polynomials
Constraints	None		Sag difference on aperture margins		Square sum of surface coefficients		Both types of constraints
Constraints	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)
M1	1.759	5.259	0.116	0.381	0.170	0.681	0.108	0.360
M3	0.860	1.339	0.219	0.447	0.209	0.428	0.216	0.436
Sum	2.619	6.598	0.335	0.828	0.379	1.109	0.324	0.796
RMS WFE	0.008λ		0.008λ		0.008λ		0.008λ

Table 6. Surface sag difference and imaging performance for design example C using XY polynomials and Legendre polynomials surface types

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Table 6. Surface sag difference and imaging performance for design example C using XY polynomials and Legendre polynomials surface types

Surface type	XY polynomials		Legendre polynomials
Constraints	None		Sag difference on aperture margins		Square sum of surface coefficients		Both types of constraints
Constraints	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)	SDPV/mm	MND/(°)
M1	1.941	2.013	1.979	1.874	1.851	2.061	1.734	1.373
M2	0.900	7.600	0.033	0.585	0.050	1.078	0.033	0.591
M3	6.525	5.070	0.645	0.536	0.500	0.448	0.540	0.457
Sum	9.366	14.683	2.657	2.995	2.401	3.587	2.307	2.421
RMS WFE	0.047λ		0.050λ		0.045λ		0.051λ

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Lijun Zhou, Tong Yang, Dewen Cheng, Yongtian Wang. Design method of imaging systems using square-domain orthogonal polynomials freeform surface (invited)[J]. Infrared and Laser Engineering, 2023, 52(7): 20230317

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Paper Information

Category:

Received: May. 29, 2023

Accepted: --

Published Online: Aug. 16, 2023

The Author Email:

DOI:10.3788/IRLA20230317

Topics

laser devices and laser physics

Lasers and Laser Optics

Laser physics

laser manufacturing

Instrumentation, Measurement and Metrology

Table 1. 1D Chebyshev polynomial up to the sixth order

Table 1. 1D Chebyshev polynomial up to the sixth order

Table 2. 1D Legendre polynomial up to the sixth order

Table 2. 1D Legendre polynomial up to the sixth order

Table 3. Surface sag difference and imaging performance for design example A using XY polynomials and Chebyshev polynomials surface types

Table 3. Surface sag difference and imaging performance for design example A using XY polynomials and Chebyshev polynomials surface types

Table 4. Surface sag difference and imaging performance for design example A using XY polynomials and Legendre polynomials surface types

Table 4. Surface sag difference and imaging performance for design example A using XY polynomials and Legendre polynomials surface types

Table 5. Surface sag difference and imaging performance for design example B using XY polynomials and Legendre polynomials surface types

Table 5. Surface sag difference and imaging performance for design example B using XY polynomials and Legendre polynomials surface types

Table 6. Surface sag difference and imaging performance for design example C using XY polynomials and Legendre polynomials surface types

Table 6. Surface sag difference and imaging performance for design example C using XY polynomials and Legendre polynomials surface types