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Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes

Dae Wook Kim and Sug-Whan Kim

Open Access Open Access

Abstract

We present a novel simulation technique that offers efficient mass fabrication strategies for 2m class hexagonal mirror segments of extremely large telescopes. As the first of two studies in series, we establish the theoretical basis of the tool influence function (TIF) for precessing tool polishing simulation for non-rotating workpieces. These theoretical TIFs were then used to confirm the reproducibility of the material removal foot-prints (measured TIFs) of the bulged precessing tooling reported elsewhere. This is followed by the reverse-computation technique that traces, employing the simplex search method, the real polishing pressure from the empirical TIF. The technical details, together with the results and implications described here, provide the theoretical tool for material removal essential to the successful polishing simulation which will be reported in the second study.

©2005 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. (a) Gaussian pressure distribution and (b) velocity components overlaid onto concentric speed contours of tool rotation inside the polishing spot (tool-workpiece contact area)

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Fig. 2.
Fig. 2. (a) Three dimensional view of sTIF and (b) cross-sectioned profiles of sTIF in X and Y axis (Δt=6 sec, WT =1000 rpm, PT =0.013 Mpa, α=15 degrees)

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Fig. 3.
Fig. 3. (a) Cross-sectional profiles of sTIFs (100–1000 tool rpm) and (b) depth of measured [14] and theoretical sTIFs

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Fig. 4.
Fig. 4. (a) Cross-sectional profiles of sTIFs (α: 6–20 degrees) and (b) depth of measured [13] and theoretical sTIFs

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Fig. 5.
Fig. 5. (a) Measured sTIFs [14] and (b) theoretical sTIFs (PT : 0.0130–0.0214 Mpa)

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Fig. 6.
Fig. 6. (a) Empirical polishing pressure data (41 data points) inverse-computed from TIFs, (b) theoretical polishing pressure expressed with modified Gaussian function fitted to the data and (c) residual pressure difference between the data and the fitted functions

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Tables (2)

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Table 1. Specifications of three ELTs and KECK primary mirrors

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Table 2. Optimized parameters (PT , σ, and ψ) and standard deviation σ d for the modified Gaussian function fitting

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Equations (8)

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dy = 8 ( f ⁄ D ) 3 k ,
Δ z = κ P V T Δ t ,
P = P T ( exp ( − λ 2 2 σ 2 ) ) ψ ,
V T = [ ( V TRx + V TFx ) 2 + ( V TRy + V TFy ) 2 ] 1 2
Δ z = κ P T ( exp ( − λ 2 2 σ 2 ) ) ψ [ ( V TRx + V TFx ) 2 + ( V TRy + V TFy ) 2 ] 1 2 Δ t
P E ( i ) = Δ z ( i ) κ [ ( V TRx ( i ) + V TFx ) 2 + ( V TRy ( i ) + V TFy ) 2 ] 1 2 Δ t
d ( i ) ( P T , σ , ψ ) = ( P E ( i ) − R ( i ) )
σ d ( P T , σ , ψ ) = [ 1 N ∑ i = 1 N ( d ( i ) − d ̅ ) 2 ] 1 2 = { 1 N ∑ i = 1 N [ d ( i ) − ( 1 N ∑ i = 1 N d ( i ) ) ] 2 } 1 2
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