Test of a slow off-axis parabola at its center of curvature

Robert E. Parks; Chris J. Evans; Lianzhen Shao

doi:10.1364/AO.34.007174

Applied Optics
Vol. 34,
Issue 31,
pp. 7174-7178
(1995)
•https://doi.org/10.1364/AO.34.007174

Test of a slow off-axis parabola at its center of curvature

Robert E. Parks, Chris J. Evans, and Lianzhen Shao

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Robert E. Parks, Chris J. Evans, and Lianzhen Shao, "Test of a slow off-axis parabola at its center of curvature," Appl. Opt. 34, 7174-7178 (1995)

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Abstract

We describe the interferometric testing of a slow (f/16 at the center of curvature) off-axis parabola, intended for use in an x-ray spectrometer, that uses a spherical wave front matched to the mean radius of the asphere. We find the figure error in the off-axis mirror by removing the theoretical difference between the off-axis segment and the spherical reference from the measured wave-front error. This center of curvature test is easy to perform because the spherical reference wave front has no axis and thus alignment is trivial. We confirm that the test results are the same as the double-pass null test for a parabola that uses a plane autocollimating mirror. We also determine that the off-axis section apparently warped as the result of being cut from the symmetric parent part.

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		Measured Surface Departure			Measured Null Test

Zernike Coefficient	Calculated Departurea (nm)	Over Full Apertureb (nm)	Rotated 15° Clockwisec (nm)	Figure Errord	Full Aperture Surfacee (nm)	Rotated 11° Clockwisef (nm)	80% of Full Apertureg (nm)
a ₄ ⁰	−43	−68	−68	−25	−33	−33	−22
a ₃ ³	−1	−137	−184	−183	−145	−173	−87
a ₃−³	0	−123	−10	−10	−94	1	2
a ₃ ¹	−546	−549	−567	−21	−33	−35	3
a ₃−¹	0	−138	9	9	−11	−5	2
a ₂ ²	−1307	−1169	−1342	−35	1	2	74
a ₂−²	0	−660	13	13	3	2	2
A ₂ ⁰	−1433		interferometer alignment sensitive
(Tilt and power removed)
rms		580	580	77	74	74	46
Peak to valley		3602	3602	671	608	608	227

Zernike Term	Zernike Polynomial
U ₀ ⁰	1
U ₁ ¹	r cos θ
U ₁ ⁻¹	r sin θ
U ₂ ²	r ² cos 2θ
U ₂ ⁰	2r ² − 1
U ₂ ⁻²	r ² sin 2θ
U ₃ ³	r ³ cos 3θ
U ₃ ¹	(3r ² − 2)r cos θ
U ₃ ⁻¹	(3r ² − 2)r sin θ
U ₃ ⁻³	r ³ sin 3θ
U ₄ ⁰	6r ⁴ − 6r ² + 1

		Measured Surface Departure			Measured Null Test

Zernike Coefficient	Calculated Departurea (nm)	Over Full Apertureb (nm)	Rotated 15° Clockwisec (nm)	Figure Errord	Full Aperture Surfacee (nm)	Rotated 11° Clockwisef (nm)	80% of Full Apertureg (nm)
a ₄ ⁰	−43	−68	−68	−25	−33	−33	−22
a ₃ ³	−1	−137	−184	−183	−145	−173	−87
a ₃−³	0	−123	−10	−10	−94	1	2
a ₃ ¹	−546	−549	−567	−21	−33	−35	3
a ₃−¹	0	−138	9	9	−11	−5	2
a ₂ ²	−1307	−1169	−1342	−35	1	2	74
a ₂−²	0	−660	13	13	3	2	2
A ₂ ⁰	−1433		interferometer alignment sensitive
(Tilt and power removed)
rms		580	580	77	74	74	46
Peak to valley		3602	3602	671	608	608	227

Zernike Term	Zernike Polynomial
U ₀ ⁰	1
U ₁ ¹	r cos θ
U ₁ ⁻¹	r sin θ
U ₂ ²	r ² cos 2θ
U ₂ ⁰	2r ² − 1
U ₂ ⁻²	r ² sin 2θ
U ₃ ³	r ³ cos 3θ
U ₃ ¹	(3r ² − 2)r cos θ
U ₃ ⁻¹	(3r ² − 2)r sin θ
U ₃ ⁻³	r ³ sin 3θ
U ₄ ⁰	6r ⁴ − 6r ² + 1

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Equations (7)

Applied Optics