Large Angle Positioning Error Correction for Sub-Aperture Stitching Interferometry of Cylindrical X-Ray Mirrors

Die Qin; Yongqian Wu; Yan Xu; Shuai Zhang; Ting Deng

doi:10.3788/LOP202259.1734002

Laser & Optoelectronics Progress, Volume. 59, Issue 17, 1734002(2022)

Large Angle Positioning Error Correction for Sub-Aperture Stitching Interferometry of Cylindrical X-Ray Mirrors

Die Qin^1,2、*, Yongqian Wu^1、**, Yan Xu¹, Shuai Zhang^1,2, and Ting Deng^1,2

Author Affiliations

¹Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, Sichuan , China

²University of Chinese Academy of Sciences, Beijing 100049, China

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Figures & Tables(13)

Fig. 1. Schematic diagram of sub-aperture stitching. (a) Stitching of adjacent sub-apertures; (b) sub-aperture mechanical positioning error

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Fig. 2. Flow chart of the algorithm

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Fig. 3. Simulated cylindrical mirror and sub-apertures. (a) Fitting surface of cylindrical mirror; (b) simulated sub-apertures

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Fig. 4. Sub-aperture compensation effect. (a)-(d) Residuals between sub-aperture surface and original fitting sub-aperture surface when adding -5°, -2°, 2°, 5° angle errors; (e)-(h) residuals between original fitting sub-aperture surface and sub-aperture surface after angle error compensation

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Fig. 5. Simulation of stitching compensation. (a) Original simulation surface; (b) stitching surface after introducing positioning error; (c) stitching surface after positioning error compensation; (d) residuals between original simulation surface and stitching surface after introducing positioning error; (e) residuals between original simulation surface and stitching surface after positioning error compensation

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Fig. 6. Measuring device and tested mirror. (a) 6 inch interferometer; (b) elliptic cylindrical mirror

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Fig. 7. Effect of single sub-aperture compensation. (a) Original sub-aperture $S_{2}$ ; (b) sub-aperture after introducing positioning error $S_{2}^{'}$ ; (c) sub-aperture after positioning error compensation $N_{2}$ ; (d) $S_{2} - S_{2}^{'}$ ; (e) $S_{2} - N_{2}$

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Fig. 8. Effect of full-aperture stitching compensation. (a) Original sub-aperture stitching surface $T_{1}$ ; (b) stitching surface after introducing positioning error $T_{2}$ ; (c) stitching surface after positioning error compensation $T_{3}$ ; (d) $T_{1} - T_{2}$ ; (e) $T_{1} - T_{3}$

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Fig. 9. Stitching surface and long trace profiler measurement results. (a) Stitching surface before positioning error compensation; (b) stitching surface after positioning error compensation; (c) center line measured by long trace profiler; (d) residuals between long trace profiler surface and stitching surface before error compensation; (e) residuals between long trace profiler surface and stitching surface after error compensation

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Table 1. Key parameters of genetic algorithm
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Table 1. Key parameters of genetic algorithm
Parameter Value
Population scale N_p 200
Crossover probability P_c 0.5
Mutation probability P_m 0.1
Iterations G 200
Range of $Δ x$ ， $Δ y$ /pixel ［-10，10］
Range of $Δ θ$ /（°）［-5，5］

Table 2. Simulated cylindrical mirror parameters
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Table 2. Simulated cylindrical mirror parameters
Parameter Value
Equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$
a 300100
b 43.782677
Center coordinate x₀=299900.003191，y₀=1.598171

Table 3. Addition of positioning error and algorithm estimation results

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Table 3. Addition of positioning error and algorithm estimation results

Evaluation index	Addition of positioning error			Algorithm calculation results			Residuals
Evaluation index	θ_add /（°）	x_add/pixel	y_add /pixel	θ_c /（°）	x_c/pixel	y_c /pixel	θ_e /（°）	x_e/pixel	y_e /pixel
	-5	0	0	-4.9969	0.0157	0.0136	-0.0031	-0.0157	-0.0136
	-4	1	1	-4.0009	1.2911	0.9062	0.0009	-0.2911	0.0938
	-3	2	2	-3.0036	2.0384	2.0097	0.0036	-0.0384	-0.0097
	-2	3	3	-2.0023	2.8883	3.1367	0.0023	0.1117	-0.1367
	-1	4	4	-1.0015	3.7140	3.9763	0.0015	0.2860	0.0237
	0	5	5	0.0008	5.1053	4.9998	-0.0008	-0.1053	0.0002
	1	6	6	0.9999	6.1349	5.9788	0.0001	-0.1349	0.0212
	2	7	7	2.0024	6.8761	6.9162	-0.0024	0.1239	0.0838
	3	8	8	3.0012	7.9279	7.9932	-0.0012	0.0721	0.0068
	4	9	9	3.9998	9.1483	8.9674	0.0002	-0.1483	0.0326
	5	10	10	5.0006	10.340	10.1482	-0.0006	-0.3400	-0.1482
PV							0.0067	0.6260	0.2420
RMS							0.0019	0.1830	0.0732

Table 4. Position error addition of sub-apertures
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Table 4. Position error addition of sub-apertures
Sub-aperture θ_add /（°） x_add/pixel y_add /pixel
Sub-aperture 1 0 0 0
Sub-aperture 2 3 2 2
Sub-aperture 3 2 3 2
Sub-aperture 4 5 3 3
Sub-aperture 5 2 1 3

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Die Qin, Yongqian Wu, Yan Xu, Shuai Zhang, Ting Deng. Large Angle Positioning Error Correction for Sub-Aperture Stitching Interferometry of Cylindrical X-Ray Mirrors[J]. Laser & Optoelectronics Progress, 2022, 59(17): 1734002

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