General equations for the null-screen test for aspherical surfaces with deformation coefficients

Daniel Aguirre-Aguirre; Manuel Campos-García; Rufino Díaz-Uribe; Brenda Villalobos-Mendoza

doi:10.1364/AO.57.010230

General equations for the null-screen test for aspherical surfaces with deformation coefficients

Daniel Aguirre-Aguirre, Manuel Campos-García, Rufino Díaz-Uribe, and Brenda Villalobos-Mendoza

Applied Optics
Vol. 57,
Issue 35,
pp. 10230-10238
(2018)
•https://doi.org/10.1364/AO.57.010230

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Daniel Aguirre-Aguirre, Manuel Campos-García, Rufino Díaz-Uribe, and Brenda Villalobos-Mendoza, "General equations for the null-screen test for aspherical surfaces with deformation coefficients," Appl. Opt. 57, 10230-10238 (2018)

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Abstract

A modified approach to calculate the null screen for testing fast convex/concave aspherical surfaces with deformation coefficients is presented. Introducing the aberration polynomial into the equation of the sagitta, the null screens can be generated in a simple way, in contrast to the conventional design method. This approach is easy to implement for the calculation of the null screen since the equations presented here are easy to program computationally. The validation of this approach was done by analyzing the fast aspherical surface of a convex condenser lens (f/0.18). We obtained a percentage error smaller than 1.3% in the recovery of the coefficients that describe the shape of the surface.

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Parameter	Symbol	Design Values
Diameter	$ϕ$	60.00 mm
1st sur. shape	1st sur.	Aspheric
2nd sur. shape	2nd sur.	Spherical
Radius of curvature	$r_{1}$ (1st sur.)	Inf.
	$r_{2}$ (2nd sur.)	−248.00 mm
Conic constant	k (both surfaces)	0.0

1st Sur. Aspherical Coefficients	Design Values (Zemax Prescription Data)
$D_{0}$ (mm)	$- 2.060476 \times 10^{- 4}$
$D_{1}$ ( ${mm}^{- 1}$ )	$2.339384 \times 10^{- 2}$
$D_{2}$ ( ${mm}^{- 2}$ )	$9.856785 \times 10^{- 5}$
$D_{3}$ ( ${mm}^{- 3}$ )	$- 1.266735 \times 10^{- 5}$
$D_{4}$ ( ${mm}^{- 4}$ )	$6.45506 \times 10^{- 7}$
$D_{5}$ ( ${mm}^{- 5}$ )	$- 7.11 \times 10^{- 9}$

Parameter	Symbol	Value
Cylinder diameter	$R$	66.24 mm
Pinhole-CCD distance	$a$	8.00 mm
Vertex-pinhole distance	$b$	120.59 mm
CCD dimensions	$d_{1} \times d_{2}$	$4.48 mm \times 3.36 mm$

Coefficient	Design Value (Zemax Prescription Data)	Recovered Value	Percentage Error
$D_{0}$ (mm)	$- 2.060476 \times 10^{- 4}$	$- 2.08346 \times 10^{- 4}$	1.12
$D_{1}$ ( ${mm}^{- 1}$ )	$2.339384 \times 10^{- 2}$	$2.31973 \times 10^{- 2}$	0.84
$D_{2}$ ( ${mm}^{- 2}$ )	$9.856785 \times 10^{- 5}$	$9.82867 \times 10^{- 5}$	0.29
$D_{3}$ ( ${mm}^{- 3}$ )	$- 1.266735 \times 10^{- 5}$	$- 1.25929 \times 10^{- 5}$	0.59
$D_{4}$ ( ${mm}^{- 4}$ )	$6.45506 \times 10^{- 7}$	$6.53851 \times 10^{- 7}$	1.29
$D_{5}$ ( ${mm}^{- 5}$ )	$- 7.11 \times 10^{- 9}$	$- 7.15685 \times 10^{- 9}$	0.66

Coefficient	Design Value (mm)	Recovered Value (mm)
$x_{0}$	0.0	−0.2811
$y_{0}$	0.0	0.4199
$z_{0}$	0.0	0.2019
$x_{0}^{'}$	0.0	0.1171
$y_{0}^{'}$	0.0	−0.3002
$z_{0}^{'}$	0.0	−0.2248

	Laser Deflectometry	Our Method
PV (mm)	0.2561	0.0935
rms (mm)	0.0683	0.0179

J	Value	Representation
1	$- 1.23 \times 10^{- 3}$	1
2	$- 4.908 \times 10^{- 4}$	$x$
3	$- 2.021 \times 10^{- 3}$	$y$
4	$- 3.7 \times 10^{- 5}$	$x^{2} - y^{2}$
5	$1.242 \times 10^{- 5}$	$- 1 + 2 y^{2} + 2 x^{2}$
6	$- 8.236 \times 10^{- 6}$	$2 x y$
7	$- 3.147 \times 10^{- 7}$	$x^{3} - 3 x y^{2}$
8	$4.223 \times 10^{- 7}$	$- 2 x + 3 x y^{2} + 3 x^{3}$
9	$9.713 \times 10^{- 7}$	$- 2 y + 3 y^{3} + 3 y x^{2}$
10	$2.98 \times 10^{- 7}$	$3 x^{2} y - y^{3}$
11	$- 2.341 \times 10^{- 9}$	$x^{4} - 6 x^{2} y^{2} + y^{4}$
12	$1.544 \times 10^{- 8}$	$- 3 x^{2} + 3 y^{2} + 4 x^{2} (x^{2} + y^{2}) - 4 y^{2} (x^{2} + y^{2})$
13	$- 6.055 \times 10^{- 9}$	$6 y^{4} + 12 x^{2} y^{2} + 6 x^{4} - 6 y^{2} - 6 x^{2} + 1$
14	$- 9.138 \times 10^{- 10}$	$- 6 x y + 8 x y (x^{2} + y^{2})$
15	$9.38 \times 10^{- 9}$	$4 x^{3} y - 4 x y^{3}$

Coefficient	Design Value (Zemax Prescription Data)	Initial Value	Recovered Value
$D_{0}$ (mm)	$- 2.060476 \times 10^{- 4}$	$- 2.08346 \times 10^{- 4}$	$- 1.925 \times 10^{- 4}$
$D_{1}$ ( ${mm}^{- 1}$ )	$2.339384 \times 10^{- 2}$	$2.31973 \times 10^{- 2}$	$2.552 \times 10^{- 2}$
$D_{2}$ ( ${mm}^{- 2}$ )	$9.856785 \times 10^{- 5}$	$9.82867 \times 10^{- 5}$	$3.87 \times 10^{- 4}$
$D_{3}$ ( ${mm}^{- 3}$ )	$- 1.266735 \times 10^{- 5}$	$- 1.25929 \times 10^{- 5}$	$- 2.933 \times 10^{- 5}$
$D_{4}$ ( ${mm}^{- 4}$ )	$6.45506 \times 10^{- 7}$	$6.53851 \times 10^{- 7}$	$8.845 \times 10^{- 7}$
$D_{5}$ ( ${mm}^{- 5}$ )	$- 7.11 \times 10^{- 9}$	$- 7.15685 \times 10^{- 9}$	$- 1.265 \times 10^{- 8}$

Daniel Aguirre-Aguirre	https://orcid.org/0000-0002-2695-3577
Manuel Campos-García	https://orcid.org/0000-0003-2121-9549
Rufino Díaz-Uribe	https://orcid.org/0000-0002-2111-1303
Brenda Villalobos-Mendoza	https://orcid.org/0000-0002-3310-5197

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