Three-flat test with plates in horizontal posture

Maurizio Vannoni; Giuseppe Molesini

doi:10.1364/AO.47.002133

Applied Optics
Vol. 47,
Issue 12,
pp. 2133-2145
(2008)
•https://doi.org/10.1364/AO.47.002133

Three-flat test with plates in horizontal posture

Maurizio Vannoni and Giuseppe Molesini

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Maurizio Vannoni and Giuseppe Molesini, "Three-flat test with plates in horizontal posture," Appl. Opt. 47, 2133-2145 (2008)

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Abstract

Measuring flats in the horizontal posture with interferometers is analyzed in detail, taking into account the sag produced by gravity. A mathematical expression of the bending is provided for a plate supported at three unevenly spaced locations along the edge. It is shown that the azimuthal terms of the deformation can be recovered from a three-flat measuring procedure, while the pure radial terms can only be estimated. The effectiveness of the iterative algorithm for data processing is also demonstrated. Experimental comparison on a set of three flats in horizontal and upright posture is provided.

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Term	Expression	Meaning
$Z_{0}$	1	piston
$Z_{1}$	$r c os θ$	tilt x
$Z_{2}$	$r \sin θ$	tilt y
$Z_{3}$	$2 r^{2} - 1$	power
$Z_{4}$	$r^{2} \cos 2 θ$	astigmatism x
$Z_{5}$	$r^{2} \sin 2 θ$	astigmatism y
$Z_{6}$	$(3 r^{2} - 2) r \cos θ$	coma x
$Z_{7}$	$(3 r^{2} - 2) r \sin θ$	coma y
$Z_{9}$	$r^{3} \cos 3 θ$	trefoil x
$Z_{10}$	$r^{3} \sin 3 θ$	trefoil y
$Z_{11}$	$(4 r^{2} - 3) r^{2} \cos 2 θ$	secondary astigmatism x
$Z_{12}$	$(4 r^{2} - 3) r^{2} \sin 2 θ$	secondary astigmatism y
$Z_{16}$	$r^{4} \cos 4 θ$	tetrafoil x
$Z_{17}$	$r^{4} \sin 4 θ$	tetrafoil y
$Z_{18}$	$(5 r^{2} - 4) r^{3} \cos 3 θ$	secondary trefoil x
$Z_{19}$	$(5 r^{2} - 4) r^{3} \sin 3 θ$	secondary trefoil y
$Z_{25}$	$r^{5} \cos 5 θ$	pentafoil x
$Z_{26}$	$r^{5} \sin 5 θ$	pentafoil y

Quantity	Value	Units
Young’s modulus	$7.3 \times 10^{10}$	$N m^{- 2}$
Poisson’s ratio	0.17
Density	$2.2 \times 10^{3}$	$kg m^{- 3}$
Acceleration due to gravity	9.8	$N {kg}^{- 1}$
Plate thickness	$1.9 \times 10^{- 2}$	m
Plate radius	( $5.70 \times 10^{- 2}$ )	m
Clear circle radius	( $5.08 \times 10^{- 2}$ )	m

Quantity	Finite Element Analysis	Sag Experiment	3-Flat Algebra	Iterative Algorithm
Peak-to-valley	$16.7 nm$	$15.0 nm$	$15.2 nm$	$16.2 nm$
Power	$5.6 nm$	$4.9 nm$	—	—

Zernike Coefficient	Finite Element Analysis	Sag Experiment	3-Flat Algebra	Iterative Algorithm
$c_{3}$	0.0045	0.0039
$c_{4}$	$- 0.0099$	$- 0.0097$	$- 0.0090$	$- 0.0088$
$c_{10}$	$- 0.0050$	$- 0.0037$	$- 0.0042$	$- 0.0041$
$c_{26}$	0.0010	0.0010	0.0009	0.0008

Term	Expression	Meaning
$Z_{0}$	1	piston
$Z_{1}$	$r c os θ$	tilt x
$Z_{2}$	$r \sin θ$	tilt y
$Z_{3}$	$2 r^{2} - 1$	power
$Z_{4}$	$r^{2} \cos 2 θ$	astigmatism x
$Z_{5}$	$r^{2} \sin 2 θ$	astigmatism y
$Z_{6}$	$(3 r^{2} - 2) r \cos θ$	coma x
$Z_{7}$	$(3 r^{2} - 2) r \sin θ$	coma y
$Z_{9}$	$r^{3} \cos 3 θ$	trefoil x
$Z_{10}$	$r^{3} \sin 3 θ$	trefoil y
$Z_{11}$	$(4 r^{2} - 3) r^{2} \cos 2 θ$	secondary astigmatism x
$Z_{12}$	$(4 r^{2} - 3) r^{2} \sin 2 θ$	secondary astigmatism y
$Z_{16}$	$r^{4} \cos 4 θ$	tetrafoil x
$Z_{17}$	$r^{4} \sin 4 θ$	tetrafoil y
$Z_{18}$	$(5 r^{2} - 4) r^{3} \cos 3 θ$	secondary trefoil x
$Z_{19}$	$(5 r^{2} - 4) r^{3} \sin 3 θ$	secondary trefoil y
$Z_{25}$	$r^{5} \cos 5 θ$	pentafoil x
$Z_{26}$	$r^{5} \sin 5 θ$	pentafoil y

Abstract

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Figures (11)

Tables (4)

Equations (21)

Applied Optics