Second-Order Sensitivity Matrix Method for Aligning Off-Axis Telescopes

Fig. 2. Relationship curve between Zernike coefficients and different misalignments. (a) X-axis decenter error; (b) x-axis tilt error; (c) y-axis decenter error; (d) y-axis tilt error

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Fig. 3. Fig. 2 expanded graph of the curve affected greatly. (a) Relationship curve between Z₄, Z₈ and x-axis decenter; (b) relationship curve between Z₄, Z₇ and y-axis decenter; (c) relationship curve between Z₆, Z₇ and x-axis tilt; (d) relationship curve between Z₆, Z₈ and y-axis tilt

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Table 1. Parameters of off-axis optical system
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Table 1. Parameters of off-axis optical system
Item Radius /mm Semi-diameter /mm Thickness /mm Conic
Aperture ∞ 316 1500 0
Primary mirror -2668.8 800 -1166.35 -1
Secondary mirror -336.1 120 1200 -1

Table 2. Goodness of curve fitting
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Table 2. Goodness of curve fitting
Item Fig. 3（a） Fig. 3（b） Fig. 3（c） Fig. 3（d）
Z₄ Z₈ Z₆ Z₇ Z₄ Z₇ Z₆ Z₈
Goodness of fit 0.998 0.997 0.993 0.996 0.997 0.997 0.994 0.995

Table 3. Difference between actual value and calculate value by traditional method

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Table 3. Difference between actual value and calculate value by traditional method

Group No.	XADE /mm		YADE /mm		XATE /mrad		YATE /mrad
Group No.	AV	CV	AV	CV	AV	CV	AV	CV
1	3.0	3.6633	-2.9	-3.4042	3.0	3.6613	2.7	3.0248
2	2.8	3.3278	1.7	1.3406	2.3	2.2692	2.1	1.9371
3	1.6	1.2468	1.3	0.8823	-1.6	-1.2389	1.4	1.0186
4	-0.7	-0.4704	-0.8	-0.5457	1.1	0.7456	0.8	0.5263
5	0.2	0.3017	0.3	0.3214	-0.5	-0.378	0.4	0.3478

Table 4. Difference between actual value and calculate value by new method

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Table 4. Difference between actual value and calculate value by new method

Group No.	XADE /mm		YADE /mm		XATE /mrad		YATE /mrad
Group No.	AV	CV	AV	CV	AV	CV	AV	CV
1	3.0	2.9999	-2.9	-2.8974	3.0	3.0072	2.7	2.7007
2	2.8	2.7998	1.7	1.6962	2.3	2.2974	2.1	2.0968
3	1.6	1.5997	1.3	1.2922	-1.6	-1.6013	1.4	1.3966
4	-0.7	-0.6998	-0.8	-0.7972	1.1	1.0617	0.8	0.7997
5	0.2	0.2003	0.3	0.3011	-0.5	-0.5067	0.4	0.4052

Table 5. Comparison of accuracy between two methods
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Table 5. Comparison of accuracy between two methods
Method X-axis decenter error /mm Y-axis decenter error /mm X-axis tilt error /mrad Y-axis tilt error /mrad
Traditional method 0.4258 0.353 0.3766 0.2665
New method 0.0023 0.0043 0.0177 0.0031

Table 6. Accuracy difference between traditional method and new method

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Table 6. Accuracy difference between traditional method and new method

Type of misalignment	Group No.	Actual value	Traditional method		New method
Type of misalignment	Group No.	Actual value	Calculate value	Error	Calculate value	Error
X-axis decenter error /mm	1	4.0	6.3012	2.3012	4.0038	0.0038
	2	5.0	9.6860	4.686	5.0051	0.0051
	3	6.0	13.8255	7.8255	6.0070	0.007
Y-axis decenter error /mm	1	3.7	5.5684	1.8684	3.6177	-0.0823
	2	-5.2	-10.0917	-4.8917	-5.2823	-0.0823
	3	6.3	15.9132	9.6132	6.2126	-0.0838
X-axis tilt error /mrad	1	-3.8	-5.7484	-1.9484	-3.8186	-0.0186
	2	4.7	8.6559	3.9559	4.7305	0.0305
	3	5.6	12.304	6.704	5.6801	0.0801
Y-axis tilt error /mrad	1	3.0	3.6701	0.6701	3.0029	0.0029
	2	4.4	7.6610	3.261	4.4329	0.0329
	3	-6.1	-14.7609	-8.6609	-6.2079	-0.1079

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Zhu Liu, Qi Peng, Ge Ren, Chunsheng Xiang, Yufeng Tan. Second-Order Sensitivity Matrix Method for Aligning Off-Axis Telescopes[J]. Laser & Optoelectronics Progress, 2022, 59(8): 0811002

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