Mirror deflection on multiple axial supports

Der-Shen Wan; J. R. P. Angel; Robert E. Parks

doi:10.1364/AO.28.000354

Applied Optics
Vol. 28,
Issue 2,
pp. 354-362
(1989)
•https://doi.org/10.1364/AO.28.000354

Mirror deflection on multiple axial supports

Der-Shen Wan, J. R. P. Angel, and Robert E. Parks

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Der-Shen Wan, J. R. P. Angel, and Robert E. Parks, "Mirror deflection on multiple axial supports," Appl. Opt. 28, 354-362 (1989)

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Abstract

A theoretical model based on thin plate theory is developed to calculate mirror deflection on multiple axial supports. Thickness variation due to mirror curvature, sandwich structure, and shear effects is included in this model. Comparison with the finite element analysis shows almost identical results for a three-point support. Surface deflections due to more complicated support patterns (forty-eight- and eighty-four-point support) are also included. A tolerancing model of support forces is developed and applied to the case of eight-four-point support.

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Material	Borosilicate
Density	2.18 g/cm³
Young’s modulus	5.86 × 10¹¹ dyn/cm²
dDiameter	1.8 m
Radius of hole	0.148 m
Thickness at center	24.5 cm
Front plate thickness	2.54 cm
Back plate thickness	2.86 cm
Cell wall thickness	1.9 cm
Cell spacing	19.9 cm
Radius of curvature	360 cm

Material	Fused silica
Young’s modulus	7.3 × 10¹¹ dyn/cm²
Poisson’s ratio	0.16
Density	2.2 g/cm³
Central thickness	32.5 cm
Top plate thickness	2.54 cm
Bottom plate thickness	1.27 cm
Cell wall thickness	0.64 cm
Cell spacing	7.6 cm
Radius of curvature	988 cm
Diameter	186.7 cm

Material	Borosilicate
Density	2.18 g/cm³
Young’s modulus	5.17 × 10¹¹ dyn/cm²
Poisson’s ratio	0.195
Diameter	3.5 m
Radius of hole	0.508 m
Thickness at center	33.2 cm
Front plate thickness	2.5 cm
Back plate thickness	2.5 cm
Cell wall thickness	1.3 cm
Cell spacing	19.2 cm
Radius of curvature	12.27 m

Group	Number of Points	Fractional radius	Offset angle (degree)
1	6	0.3796	8.21
2	6	0.3796	30
3	6	0.3796	51.772
4	6	0.6250	15.298
5	6	0.6250	44.702
6	6	0.8935	0
7	6	0.8935	21.789
8	6	0.8935	38.216

Group	Finite element	Current model
1	0.05049	0.04965
2	0.07015	0.05718
3	0.05049	0.04965
4	0.15452	0.14815
5	0.15452	0.14815
6	0.18908	0.21549
7	0.16537	0.16586
8	0.16537	0.16586

Group	Number of points	Fractional radius	Offset angle (deg)
1	6	0.22592	0
2	6	0.39130	30
3	6	0.45184	0
4	6	0.59772	19.108
5	6	0.59772	40.892
6	6	0.67776	0
7	6	0.78261	30
8	6	0.81457	13.899
9	6	0.81457	46.1
10	6	0.90368	9
11	6	0.94158	23.107
12	6	0.94158	36.892
13	6	0.99431	10.072
14	6	0.99431	49.928

Plate thickness	2.5 cm
Cell wall thickness	1.3 cm
Cell center spacing	28.24 cm
Central thickness	56 cm
Radius of curvature	27 m

Ringa	Force	Ring	Force
1	0.021928	8	0.030582
1	0.021717	8	0.030661
1	0.021717	8	0.030536
2	0.026681	9	0.030536
2	0.026859	9	0.030661
2	0.026681	9	0.030582
3	0.025147	10	0.028266
3	0.025156	10	0.028378
3	0.025155	10	0.028378
4	0.028125	11	0.019114
4	0.028083	11	0.019132
4	0.028212	11	0.019194
5	0.028212	12	0.019193
5	0.028083	12	0.019130
5	0.028126	12	0.019111
6	0.029001	13	0.008443
6	0.028972	13	0.008365
6	0.028972	13	0.008360
7	0.029801	14	0.008360
7	0.029777	14	0.008366
7	0.029801	14	0.008443

Standard deviation (%)	rms (nm)	Peak-to-valley (nm)
0	13.9	98
0.1375	21.7	110.9
0.275	35.6	157.2
0.55	68	299.7

Standard deviation (%)	rms (nm)	Peak-to-valley (nm)
0	13.9	98
0.1375	21.7	110.9
0.275	35.6	157.2
0.55	68	299.7

Abstract

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Figures (17)

Tables (9)

Equations (21)

Applied Optics