Stressed mirror polishing. 2: Fabrication of an off-axis section of a paraboloid

Jerry E. Nelson; George Gabor; Leslie K. Hunt; Jacob Lubliner; Terry S. Mast

doi:10.1364/AO.19.002341

Applied Optics
Vol. 19,
Issue 14,
pp. 2341-2352
(1980)
•https://doi.org/10.1364/AO.19.002341

Stressed mirror polishing. 2: Fabrication of an off-axis section of a paraboloid

Jerry E. Nelson, George Gabor, Leslie K. Hunt, Jacob Lubliner, and Terry S. Mast

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Jerry E. Nelson, George Gabor, Leslie K. Hunt, Jacob Lubliner, and Terry S. Mast, "Stressed mirror polishing. 2: Fabrication of an off-axis section of a paraboloid," Appl. Opt. 19, 2341-2352 (1980)

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Abstract

A mirror has been fabricated with the shape of an off-axis section of a paraboloid by grinding and polishing a sphere into a prestressed blank. The applied stresses were then removed allowing the mirror to spring into the desired paraboloidal shape. The 36-cm diam off-axis section deviated 9.9-μm rms from the polished sphere. The final surface deviated 0.03-μm rms from the desired off-axis section.

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Equations (41)

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					Zernike Coefficients (μ m
n	m	^Znm(ρ,θ)	Cylindrical Monomial Coefficient Relationship	Weight	Desired Deflections	Desired Parabola	First* Polish	Second* Polish	Measured Residuals from Best Fitting Paraboloid

0	0	1	$C_{00} = α_{00} + \frac{α_{20}}{2} + \frac{α_{40}}{3} + . . .$	1
1	−1	ρsinθ	$C_{1 - 1} = α_{1 - 1} + \frac{2}{3} α_{3 - 1} + . . .$	$\frac{1}{4}$		$[\begin{matrix} k = 368.80 cm \\ R = 35.00 cm \\ ϕ = 0.0 \end{matrix}]$	$[\begin{array}{l} k = 368.68 cm \\ R = 35.79 cm \\ ϕ = 0.0043 \end{array}]$	$[\begin{array}{l} k = 368.83 cm \\ R = 35.07 cm \\ ϕ = 0.0060 \end{array}]$
1	1	ρcosθ	$C_{11} = α_{11} + \frac{2}{3} α_{31} + . . .$	$\frac{1}{4}$
2	−2	ρ²sin2θ	$C_{2 - 2} = α_{2 - 2} + \frac{3}{4} α_{4 - 2} + . . .$	$\frac{1}{6}$	0.00	0.00	−0.15	0.24	0.02
2	0	2ρ² − 1	$C_{20} = \frac{α_{20}}{2} + \frac{α_{40}}{2} + . . .$	$\frac{1}{3}$	−8.93	2161.88	2162.59	2161.68	0.00
2	2	ρ²cos2θ	$C_{22} = α_{22} + \frac{3}{4} α_{42} + . . .$	$\frac{1}{6}$	19.34	−19.34	−20.21	−19.42	−0.02
3	−3	ρ³sin3θ	C₃₋₃ = α₃₋₃ + …	$\frac{1}{8}$	0.00	0.00	0.02	0.02	0.01
3	−1	(3ρ³ − 2ρ)sinθ	$C_{3 - 1} = \frac{α_{3 - 1}}{3} + . . .$	$\frac{1}{8}$	0.00	0.00	−0.16	−0.01	−0.05
3	1	(3ρ³ − 2ρ) cosθ	$C_{31} = \frac{α_{31}}{3} + . . .$	$\frac{1}{8}$	6.55	−6.55	−6.71	−6.55	0.01
3	3	ρ³ cos3θ	C₃₃ = α₃₃+…	$\frac{1}{8}$	−0.04	0.04	0.00	0.00	−0.04
4	−4	ρ⁴ sin4θ	C₄₋₄ = α₄₋₄+…	$\frac{1}{10}$	0.00	0.00	0.02	0.02	0.03
4	−2	(4ρ⁴ − 3ρ²) sin2θ	$C_{4 - 2} = \frac{α_{4 - 2}}{4} + . . .$	$\frac{1}{10}$	0.00	0.00	0.02	0.00	0.00
4	0	6ρ⁴ − 6ρ² + 1	$C_{40} = \frac{α_{40}}{6} + . . .$	$\frac{1}{5}$	0.40	0.01	−0.11	0.04	0.03
4	2	(4ρ⁴ − 3ρ²) cos2θ	$C_{42} = \frac{α_{42}}{4} + . . .$	$\frac{1}{10}$	−0.01	0.01	−0.07	0.02	0.00
4	4	ρ⁴ cos4θ	C₄₄ = α₄₄+…	$\frac{1}{10}$	0.00	0.00	−0.01	0.04	0.05
					(Z − Z_desired)_rms		0.55	0.16
					(Z − Z_best) _rms		0.08	0.03

		First Polish		Second Polish
Angle	Lever No.	Force, kg	Couple, kg–cm	Force, kg	Couple, kg–cm
15°	1	−16.78	260.38	−15.75	246.56
30°	2	−17.73	213.92	−16.76	204.70
45°	3	−18.73	144.59	−17.74	141.79
60°	4	−19.07	62.68	−18.13	65.98
75°	5	−18.06	−20.54	−17.28	−13.00
90°	6	−15.29	−95.13	−14.62	−85.56
105°	7	−10.73	−154.32	−9.98	−144.65
120°	8	−4.86	−195.48	−3.88	−187.35
135°	9	1.46	−219.94	2.58	−214.84
150°	10	7.11	−231.82	8.13	−230.66
165°	11	11.03	−236.22	11.73	−238.43
180°	12	12.43	−237.17	12.84	−240.42
195°	13	11.03	−236.22	11.39	−237.21
210°	14	7.11	−2 31.82	7.65	−228.03
225°	15	1.46	−219.94	2.16	−211.07
240°	16	−4.86	−195.48	−4.19	−183.51
255°	17	−10.73	−154.32	−10.26	−142.20
270°	18	−15.29	−95.13	−14.92	−85.27
285°	19	−18.06	−20.54	−17.48	−14.22
300°	20	−19.07	62.68	−17.98	64.86
315°	21	−18.73	144.59	−17.14	142.06
330°	22	−17.73	213.92	−15.93	206.20
345°	23	−16.78	260.38	−15.14	247.96
360°	24	−16.40	276.73	−15.11	261.82

					Zernike Coefficients (μ m
n	m	^Znm(ρ,θ)	Cylindrical Monomial Coefficient Relationship	Weight	Desired Deflections	Desired Parabola	First* Polish	Second* Polish	Measured Residuals from Best Fitting Paraboloid

0	0	1	$C_{00} = α_{00} + \frac{α_{20}}{2} + \frac{α_{40}}{3} + . . .$	1
1	−1	ρsinθ	$C_{1 - 1} = α_{1 - 1} + \frac{2}{3} α_{3 - 1} + . . .$	$\frac{1}{4}$		$[\begin{matrix} k = 368.80 cm \\ R = 35.00 cm \\ ϕ = 0.0 \end{matrix}]$	$[\begin{array}{l} k = 368.68 cm \\ R = 35.79 cm \\ ϕ = 0.0043 \end{array}]$	$[\begin{array}{l} k = 368.83 cm \\ R = 35.07 cm \\ ϕ = 0.0060 \end{array}]$
1	1	ρcosθ	$C_{11} = α_{11} + \frac{2}{3} α_{31} + . . .$	$\frac{1}{4}$
2	−2	ρ²sin2θ	$C_{2 - 2} = α_{2 - 2} + \frac{3}{4} α_{4 - 2} + . . .$	$\frac{1}{6}$	0.00	0.00	−0.15	0.24	0.02
2	0	2ρ² − 1	$C_{20} = \frac{α_{20}}{2} + \frac{α_{40}}{2} + . . .$	$\frac{1}{3}$	−8.93	2161.88	2162.59	2161.68	0.00
2	2	ρ²cos2θ	$C_{22} = α_{22} + \frac{3}{4} α_{42} + . . .$	$\frac{1}{6}$	19.34	−19.34	−20.21	−19.42	−0.02
3	−3	ρ³sin3θ	C₃₋₃ = α₃₋₃ + …	$\frac{1}{8}$	0.00	0.00	0.02	0.02	0.01
3	−1	(3ρ³ − 2ρ)sinθ	$C_{3 - 1} = \frac{α_{3 - 1}}{3} + . . .$	$\frac{1}{8}$	0.00	0.00	−0.16	−0.01	−0.05
3	1	(3ρ³ − 2ρ) cosθ	$C_{31} = \frac{α_{31}}{3} + . . .$	$\frac{1}{8}$	6.55	−6.55	−6.71	−6.55	0.01
3	3	ρ³ cos3θ	C₃₃ = α₃₃+…	$\frac{1}{8}$	−0.04	0.04	0.00	0.00	−0.04
4	−4	ρ⁴ sin4θ	C₄₋₄ = α₄₋₄+…	$\frac{1}{10}$	0.00	0.00	0.02	0.02	0.03
4	−2	(4ρ⁴ − 3ρ²) sin2θ	$C_{4 - 2} = \frac{α_{4 - 2}}{4} + . . .$	$\frac{1}{10}$	0.00	0.00	0.02	0.00	0.00
4	0	6ρ⁴ − 6ρ² + 1	$C_{40} = \frac{α_{40}}{6} + . . .$	$\frac{1}{5}$	0.40	0.01	−0.11	0.04	0.03
4	2	(4ρ⁴ − 3ρ²) cos2θ	$C_{42} = \frac{α_{42}}{4} + . . .$	$\frac{1}{10}$	−0.01	0.01	−0.07	0.02	0.00
4	4	ρ⁴ cos4θ	C₄₄ = α₄₄+…	$\frac{1}{10}$	0.00	0.00	−0.01	0.04	0.05
					(Z − Z_desired)_rms		0.55	0.16
					(Z − Z_best) _rms		0.08	0.03

		First Polish		Second Polish
Angle	Lever No.	Force, kg	Couple, kg–cm	Force, kg	Couple, kg–cm
15°	1	−16.78	260.38	−15.75	246.56
30°	2	−17.73	213.92	−16.76	204.70
45°	3	−18.73	144.59	−17.74	141.79
60°	4	−19.07	62.68	−18.13	65.98
75°	5	−18.06	−20.54	−17.28	−13.00
90°	6	−15.29	−95.13	−14.62	−85.56
105°	7	−10.73	−154.32	−9.98	−144.65
120°	8	−4.86	−195.48	−3.88	−187.35
135°	9	1.46	−219.94	2.58	−214.84
150°	10	7.11	−231.82	8.13	−230.66
165°	11	11.03	−236.22	11.73	−238.43
180°	12	12.43	−237.17	12.84	−240.42
195°	13	11.03	−236.22	11.39	−237.21
210°	14	7.11	−2 31.82	7.65	−228.03
225°	15	1.46	−219.94	2.16	−211.07
240°	16	−4.86	−195.48	−4.19	−183.51
255°	17	−10.73	−154.32	−10.26	−142.20
270°	18	−15.29	−95.13	−14.92	−85.27
285°	19	−18.06	−20.54	−17.48	−14.22
300°	20	−19.07	62.68	−17.98	64.86
315°	21	−18.73	144.59	−17.14	142.06
330°	22	−17.73	213.92	−15.93	206.20
345°	23	−16.78	260.38	−15.14	247.96
360°	24	−16.40	276.73	−15.11	261.82

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