Theoretical and experimental comparisons of the smoothing effects for different multi-layer polishing tools during computer-controlled optical surfacing

Xuyu Li; Chaoyang Wei; Simin Zhang; Wendong Xu; Jianda Shao

doi:10.1364/AO.58.004406

Applied Optics
Vol. 58,
Issue 16,
pp. 4406-4413
(2019)
•https://doi.org/10.1364/AO.58.004406

Theoretical and experimental comparisons of the smoothing effects for different multi-layer polishing tools during computer-controlled optical surfacing

Xuyu Li, Chaoyang Wei, Simin Zhang, Wendong Xu, and Jianda Shao

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Xuyu Li, Chaoyang Wei, Simin Zhang, Wendong Xu, and Jianda Shao, "Theoretical and experimental comparisons of the smoothing effects for different multi-layer polishing tools during computer-controlled optical surfacing," Appl. Opt. 58, 4406-4413 (2019)

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Abstract

Optical smoothing is a very effective method to restrict mid-to-high spatial frequency errors generated by computer-controlled sub-aperture polishing technologies. According to Preston’s equation, pressure distribution is the key factor in describing the smoothing effect of a tool. Although the classic bridging model can solve pressure distribution during the smoothing process, it is only suitable for a tool that consists of a rigid base, compliant interlayer, thin metal plate, and pitch polishing layer. In this paper, a mathematical model applicable for any multi-layer polishing tool is proposed, which helps to calculate the pressure distribution dependent on the thickness of the layer, mid-spatial frequency errors at different periods, and the structures of polishing tools. Based on the model, the pressure distributions and smoothing rates of a rigid tool and a semi-flexible tool are deduced and compared in detail. Validity and improvement in accuracy are verified by comparison with the finite element model. In addition, three groups of experiments using a rigid tool and a semi-flexible tool for smoothing 4 mm and 8 mm period errors generated by magnetorheological finishing are carried out to validate the model. The simulation result of the smoothing rates of the errors after every smoothing process matches well with the experiment results.

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Variable Name	Physical or Mathematical Definition	Variable Name	Physical or Mathematical Definition
$E$	Young’s modulus	$σ_{x}$	$x$ -direction normal stress
$E_{k}$	Young’s modulus of layer $k$	$σ_{x k}$	$x$ -direction normal stress of layer $k$
$μ$	Poisson’s ratio	$σ_{z}$	$z$ -direction normal stress
$μ_{k}$	Poisson’s ratio of layer $k$	$σ_{z k}$	$z$ -direction normal stress of layer $k$
$w$	$z$ -direction displacement	$τ_{z x}$	Shear stress between the $x$ direction and $z$ direction
$w_{k}$	$z$ -direction displacement of layer $k$	$τ_{z x k}$	Shear stress between the $x$ direction and $z$ direction of layer $k$
$u$	$x$ -direction displacement	$G$	Shear modulus
$u_{k}$	$x$ -direction displacement of layer $k$	$G_{k}$	Shear modulus of layer $k$
$x_{k}$ , $z_{k}$	Coordinate adopted for layer $k$	$h_{k}$	Thickness of layer $k$

Material Properties	Values
Mirror type	Flat fused silica mirror
Penetrometer	0.12 mm (22°C)
Young’ modulus of the pitch	2.1 Gpa
Poisson’s ratio of the pitch	0.2
Young’s modulus of foam	1e6 Pa
Group number	1	2	3
Structure of the polishing pad	$Steel + pitch$	$Steel + pitch$	$Steel + foam + pitch$
Thicknesses of the pad	$5 mm + 5 mm$	$5 mm + 5 mm$	$5 mm + 5 mm + 5 mm$
Period of the mid-frequency error	4 mm	8 mm	4 mm

Tool Motions	Values
Workpiece diameter	100 mm
Motion mode	Uniform raster scanning (2000 mm/min)
Rotation mode	Single-spin mode (100 r/min)
Pad diameter	40 mm
Nominal applied pressure	0.1 Mpa
Step distance	2 mm

Group number	$k_{1}$	$k_{2}$	$k_{3}$
1	0.12	0.88	0.65
2	0.22	0.78	0.33
3	0.09	0.91	0.61

Variable Name	Physical or Mathematical Definition	Variable Name	Physical or Mathematical Definition
$E$	Young’s modulus	$σ_{x}$	$x$ -direction normal stress
$E_{k}$	Young’s modulus of layer $k$	$σ_{x k}$	$x$ -direction normal stress of layer $k$
$μ$	Poisson’s ratio	$σ_{z}$	$z$ -direction normal stress
$μ_{k}$	Poisson’s ratio of layer $k$	$σ_{z k}$	$z$ -direction normal stress of layer $k$
$w$	$z$ -direction displacement	$τ_{z x}$	Shear stress between the $x$ direction and $z$ direction
$w_{k}$	$z$ -direction displacement of layer $k$	$τ_{z x k}$	Shear stress between the $x$ direction and $z$ direction of layer $k$
$u$	$x$ -direction displacement	$G$	Shear modulus
$u_{k}$	$x$ -direction displacement of layer $k$	$G_{k}$	Shear modulus of layer $k$
$x_{k}$ , $z_{k}$	Coordinate adopted for layer $k$	$h_{k}$	Thickness of layer $k$

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Applied Optics