Numerical iteration method to reduce the surface shape error of a bendable mirror in synchrotron radiation

Minwei Chen; Lidan Gao; Weifan Sheng; Weifan Sheng; Shaofeng Wang; Fugui Yang

doi:10.1364/AO.451088

Applied Optics
Vol. 61,
Issue 8,
pp. 2096-2102
(2022)
•https://doi.org/10.1364/AO.451088

Numerical iteration method to reduce the surface shape error of a bendable mirror in synchrotron radiation

Minwei Chen, Lidan Gao, Weifan Sheng, Shaofeng Wang, and Fugui Yang

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Minwei Chen, Lidan Gao, Weifan Sheng, Shaofeng Wang, and Fugui Yang, "Numerical iteration method to reduce the surface shape error of a bendable mirror in synchrotron radiation," Appl. Opt. 61, 2096-2102 (2022)

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Abstract

X-ray mirrors with high focusing performance are extensively used in the synchrotron radiation field. Especially for vertical reflecting bendable mirrors, many elements such as gravity, extended parts used for the bending mechanism, etc., usually affect the surface shape precision. There are no effective methods to remove all these errors at this point. However, an iteration method can be adopted to solve this problem. In this paper, a novel, to the best of our knowledge, iteration method on decreasing the error between the practice surface shape and the desired one is proposed. Not only can the precision of the surface shape be realized by this method, but also computational efficiency. Errors induced by gravity can be compensated for by an analytical method, while errors caused by the extended parts should be eliminated by a numerical method. Therefore, two main kinds of errors—gravity and parts of clamping—can be removed by iteration. Some examples are presented to illustrate the advantages of this method by comparison with the regular one.

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Data availability

Some data underlying the results presented in this paper are available in [18]. Other 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.

18. L. Zhang, R. Baker, R. Barrett, P. Cloetens, Y. Dabin, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “Mirror profile optimization for nano-focusing KB mirror,” AIP Conf. Proc. 1234, 801–804 (2010). [CrossRef]

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Variables	Unit	HFM
Object distance $p$	m	36
Image distance $q$	m	0.083
Incidence angle $θ$	mrad	8
Mirror length $L$	m	0.076
Active length $L_{A}$	m	0.056
Width at the center $b_{0}$	m	0.0146
Thickness $t$	m	0.005
Density $π$	$k g / m^{3}$	2329
Material	—	Silicon $⟨ 1, 0, 0 ⟩$

Iteration		1	2	3
Number	$x$ (mm)	$b$ (mm)	$b$ (mm)	$b$ (mm)
Width	−28	13.50	11.28	11.10
	−27	13.25	11.28	11.20
	−26	13.00	11.30	11.32
	−25	12.75	11.27	11.35
	−24	12.50	11.22	11.31
	−20	11.50	10.78	10.79
	−14	10.00	9.79	9.75
	0	7.30	7.61	7.63
	14	5.75	5.73	5.72
	20	5.16	4.97	4.96
	24	4.77	4.49	4.51
	25	4.67	4.37	4.41
	26	4.58	4.24	4.28
	27	4.48	4.08	4.09
	28	4.38	3.91	4.01
Moment (Nm)		0.956 (0.956)	1.046 (0.960)	1.033 (0.960)
Slope error (µrad RMS)		37.85	4.52	0.995
Optimized moment (Nm)		1.046 (0.960)	1.033 (0.960)	1.030 (0.964)
Slope error (µrad RMS)		11.95	0.47	0.37

Variables	Unit	HFM
Object distance $p$	m	40
Image distance $q$	m	0.3
Incidence angle $θ$	mrad	3
Mirror length $L$	m	0.14
Active length $L_{A}$	m	0.1
Width at the center $b_{0}$	m	0.025
Thickness $t$	m	0.005
Density $π$	$k g / m^{3}$	2329
Gravity $g$	$m / s^{2}$	9.80122
Orientation	—	Face-down
Material	—	Silicon $⟨ 1, 0, 0 ⟩$

Iteration		1	2	3	4
Number	$x$ (mm)	$b$ (mm)	$b$ (mm)	$b$ (mm)	$b$ (mm)
Width	−50	16.05	15.42	15.34	15.46
	−48	15.90	15.44	15.48	15.55
	−46	15.75	15.43	15.51	15.61
	−44	15.59	15.36	15.44	15.51
	−40	15.29	15.14	15.17	15.18
	−30	14.53	14.53	14.51	14.51
	−25	14.15	14.15	14.15	14.15
	0	12.50	12.52	12.53	12.53
	25	11.05	11.04	11.05	11.06
	30	10.78	10.78	10.77	10.78
	40	10.25	10.21	10.22	10.21
	44	10.04	10.00	10.04	10.05
	46	9.94	9.87	9.93	9.98
	48	9.83	9.71	9.77	9.83
	50	9.73	9.50	9.51	9.59
Moment (Nm)		0.1743 (0.1743)	0.1795 (0.1772)	0.1792 (0.1773)	0.1791 (0.1775)
Slope error (µrad RMS)		3.4916	0.1152	0.0773	0.0906
Optimized moment (Nm)		0.1795 (0.1772)	0.1792 (0.1773)	0.1791 (0.1775)	0.1791 (0.1777)
Slope error (µrad RMS)		0.1802	0.0507	0.0419	0.0190

Iteration Number	1	2	3	4	5	6
Moment (Nm)	0.1743	0.1743	0.1743	0.1743	0.1743	0.1743
Slope error (µrad RMS)	3.4916	0.2214	0.04984	0.1030	0.0357	0.0198

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Tables (5)

Equations (11)

Applied Optics