Abstract

The optimization of dwell time is an important procedure in deterministic subaperture polishing. We present a modified optimization model of dwell time by iterative and numerical method, assisted by extended surface forms and tool paths for suppressing the edge effect. Compared with discrete convolution and linear equation models, the proposed model has essential compatibility with arbitrary tool paths, multiple tool influence functions (TIFs) in one optimization, and asymmetric TIFs. The emulational fabrication of a Φ200mm workpiece by the proposed model yields a smooth, continuous, and non-negative dwell time map with a root-mean-square (RMS) convergence rate of 99.6%, and the optimization costs much less time. By the proposed model, influences of TIF size and path interval to convergence rate and polishing time are optimized, respectively, for typical low and middle spatial-frequency errors. Results show that (1) the TIF size is nonlinear inversely proportional to convergence rate and polishing time. A TIF size of 1/7 workpiece size is preferred; (2) the polishing time is less sensitive to path interval, but increasing the interval markedly reduces the convergence rate. A path interval of 1/81/10 of the TIF size is deemed to be appropriate. The proposed model is deployed on a JR-1800 and MRF-180 machine. Figuring results of Φ920mm Zerodur paraboloid and Φ100mm Zerodur plane by them yield RMS of 0.016λ and 0.013λ (λ=632.8nm), respectively, and thereby validate the feasibility of proposed dwell time model used for subaperture polishing.

© 2014 Optical Society of America

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References

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  1. R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra, 34–39 (1963).
  2. W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
    [CrossRef]
  3. P. M. Shanbhag, M. R. Feinberg, G. Sandri, M. N. Horenstein, and T. G. Bifano, “Ion-beam machining of millimeter scale optics,” Appl. Opt. 39, 599–611 (2000).
    [CrossRef]
  4. D. D. Walker, D. Brooks, A. King, R. Freeman, R. Morton, G. McCavana, and S. W. Kim, “The ‘Precessions’ tooling for polishing and figuring flat, spherical and aspheric surfaces,” Opt. Express 11, 958–964 (2003).
    [CrossRef]
  5. W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5180, 107–114 (2004).
    [CrossRef]
  6. H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
    [CrossRef]
  7. Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Investigation on removal features of multidistribution fixed abrasive diamond pellets used in the polishing of SiC mirrors,” Appl. Opt. 51, 8373–8382 (2012).
    [CrossRef]
  8. Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Further investigations on fixed abrasive diamond pellets used for diminishing mid-spatial frequency errors of optical mirrors,” Appl. Opt. 53, 327–334 (2014).
    [CrossRef]
  9. R. A. Jones, “Optimization of computer controlled polishing,” Appl. Opt. 16, 218–224 (1977).
    [CrossRef]
  10. S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320–324 (1987).
    [CrossRef]
  11. L. C. Charles, C. M. Egert, and W. H. Kathy, “Advanced matrix based algorithm for ion beam milling of optical components [C],” Proc. SPIE 1752, 54–62 (1992).
    [CrossRef]
  12. W. Deng, L. Zheng, Y. Shi, X. Wang, and X. Zhang, “Dwell-time algorithm based on matrix algebra and regularization method,” Opt. Precis. Eng. 15, 1009–1015 (2007).
  13. L. Zhou, Y. F. Dai, X. H. Xie, C. J. Jiao, and S. Y. Li, “Model and method to determine dwell time in ion beam figuring,” Nanotechnol. Precis. Eng. 5, 107–112 (2007).
  14. J. F. Wu, Z. W. Lu, and H. X. Zhang, “Dwell time algorithm in ion beam figuring,” Appl. Opt. 48, 3930–3937 (2009).
    [CrossRef]
  15. Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Modified subaperture tool influence functions of a flat pitch polisher with reverse-calculated material removal rate,” Appl. Opt. 53, 2455–2464 (2014).
    [CrossRef]
  16. M. Johns, “Giant Magellan Telescope,” Proc. SPIE 5382, 85–92 (2004).
  17. D. C. Zimmerman, “Feasibility studies for the alignment of the Thirty Meter Telescope,” Appl. Opt. 49, 3485–3498 (2010).
    [CrossRef]
  18. A. Cordero-Davila, J. Gonzalez-Garcıa, and M. Pedrayes-Lopez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. 43, 1250–1254 (2004).
    [CrossRef]
  19. D. W. Kim, W. H. Park, S. W. Kim, and J. H. Burge, “Parametric modeling of edge effects for polishing tool influence functions,” Opt. Express 17, 5656–5665 (2009).
    [CrossRef]
  20. D. D. Walker, G. Y. Yu, H. Y. Li, W. Messelink, R. Evans, and A. Beaucamp, “Edges in CNC polishing: from mirror-segments towards semiconductors, paper 1: edges on processing the global surface,” Opt. Express 20, 19787–19798 (2012).
    [CrossRef]
  21. H. Y. Li, D. D. Walker, G. Y. Yu, A. Sayle, W. Messelink, R. Evans, and A. Beaucamp, “Edge control in CNC polishing, paper 2: simulation and validation of tool influence functions on edges,” Opt. Express 21, 370–381 (2013).
    [CrossRef]
  22. P. J. Guo, H. Fang, and J. C. Yu, “Edge effect in fluid jet polishing,” Appl. Opt. 45, 6729–6735 (2006).
    [CrossRef]
  23. T. Wang, H. B. Cheng, Z. C. Dong, and H. Y. Tam, “Removal character of vertical jet polishing with eccentric rotation motion using magnetorheological fluid,” J. Mater. Process. Technol. 213, 1532–1537 (2013).
    [CrossRef]
  24. R. J. Marks, “Gerchberg’s extrapolation algorithm in two dimensions,” Appl. Opt. 20, 1815–1820 (1981).
    [CrossRef]
  25. D. W. Kim, S. W. Kim, and J. H. Burge, “Non-sequential optimization technique for a computer controlled optical surfacing process using multiple tool influence functions,” Opt. Express 17, 21850–21866 (2009).
    [CrossRef]
  26. H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
    [CrossRef]

2014

2013

H. Y. Li, D. D. Walker, G. Y. Yu, A. Sayle, W. Messelink, R. Evans, and A. Beaucamp, “Edge control in CNC polishing, paper 2: simulation and validation of tool influence functions on edges,” Opt. Express 21, 370–381 (2013).
[CrossRef]

T. Wang, H. B. Cheng, Z. C. Dong, and H. Y. Tam, “Removal character of vertical jet polishing with eccentric rotation motion using magnetorheological fluid,” J. Mater. Process. Technol. 213, 1532–1537 (2013).
[CrossRef]

2012

2010

2009

2007

W. Deng, L. Zheng, Y. Shi, X. Wang, and X. Zhang, “Dwell-time algorithm based on matrix algebra and regularization method,” Opt. Precis. Eng. 15, 1009–1015 (2007).

L. Zhou, Y. F. Dai, X. H. Xie, C. J. Jiao, and S. Y. Li, “Model and method to determine dwell time in ion beam figuring,” Nanotechnol. Precis. Eng. 5, 107–112 (2007).

2006

2005

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

2004

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5180, 107–114 (2004).
[CrossRef]

A. Cordero-Davila, J. Gonzalez-Garcıa, and M. Pedrayes-Lopez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. 43, 1250–1254 (2004).
[CrossRef]

M. Johns, “Giant Magellan Telescope,” Proc. SPIE 5382, 85–92 (2004).

2003

2000

1999

W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
[CrossRef]

1992

L. C. Charles, C. M. Egert, and W. H. Kathy, “Advanced matrix based algorithm for ion beam milling of optical components [C],” Proc. SPIE 1752, 54–62 (1992).
[CrossRef]

1987

S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320–324 (1987).
[CrossRef]

1981

1977

Beaucamp, A.

Bifano, T. G.

Brooks, D.

Burge, J. H.

Charles, L. C.

L. C. Charles, C. M. Egert, and W. H. Kathy, “Advanced matrix based algorithm for ion beam milling of optical components [C],” Proc. SPIE 1752, 54–62 (1992).
[CrossRef]

Cheng, H. B.

Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Further investigations on fixed abrasive diamond pellets used for diminishing mid-spatial frequency errors of optical mirrors,” Appl. Opt. 53, 327–334 (2014).
[CrossRef]

Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Modified subaperture tool influence functions of a flat pitch polisher with reverse-calculated material removal rate,” Appl. Opt. 53, 2455–2464 (2014).
[CrossRef]

T. Wang, H. B. Cheng, Z. C. Dong, and H. Y. Tam, “Removal character of vertical jet polishing with eccentric rotation motion using magnetorheological fluid,” J. Mater. Process. Technol. 213, 1532–1537 (2013).
[CrossRef]

Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Investigation on removal features of multidistribution fixed abrasive diamond pellets used in the polishing of SiC mirrors,” Appl. Opt. 51, 8373–8382 (2012).
[CrossRef]

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Cheng, K.

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Cordero-Davila, A.

Dai, Y. F.

L. Zhou, Y. F. Dai, X. H. Xie, C. J. Jiao, and S. Y. Li, “Model and method to determine dwell time in ion beam figuring,” Nanotechnol. Precis. Eng. 5, 107–112 (2007).

Deng, W.

W. Deng, L. Zheng, Y. Shi, X. Wang, and X. Zhang, “Dwell-time algorithm based on matrix algebra and regularization method,” Opt. Precis. Eng. 15, 1009–1015 (2007).

Dong, Z. C.

Egert, C. M.

L. C. Charles, C. M. Egert, and W. H. Kathy, “Advanced matrix based algorithm for ion beam milling of optical components [C],” Proc. SPIE 1752, 54–62 (1992).
[CrossRef]

Evans, R.

Fang, H.

Feinberg, M. R.

Feng, Z. J.

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Freeman, R.

Golini, D.

W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
[CrossRef]

Gonzalez-Garcia, J.

Guo, P. J.

Horenstein, M. N.

Jiao, C. J.

L. Zhou, Y. F. Dai, X. H. Xie, C. J. Jiao, and S. Y. Li, “Model and method to determine dwell time in ion beam figuring,” Nanotechnol. Precis. Eng. 5, 107–112 (2007).

Johns, M.

M. Johns, “Giant Magellan Telescope,” Proc. SPIE 5382, 85–92 (2004).

Jones, R. A.

R. A. Jones, “Optimization of computer controlled polishing,” Appl. Opt. 16, 218–224 (1977).
[CrossRef]

R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra, 34–39 (1963).

Kathy, W. H.

L. C. Charles, C. M. Egert, and W. H. Kathy, “Advanced matrix based algorithm for ion beam milling of optical components [C],” Proc. SPIE 1752, 54–62 (1992).
[CrossRef]

Kim, D. W.

Kim, S. W.

King, A.

Kordonski, W.

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5180, 107–114 (2004).
[CrossRef]

W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
[CrossRef]

Li, H. Y.

Li, S. Y.

L. Zhou, Y. F. Dai, X. H. Xie, C. J. Jiao, and S. Y. Li, “Model and method to determine dwell time in ion beam figuring,” Nanotechnol. Precis. Eng. 5, 107–112 (2007).

Lu, Z. W.

Marks, R. J.

McCavana, G.

McNeil, J. R.

S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320–324 (1987).
[CrossRef]

Messelink, W.

Morton, R.

Park, W. H.

Pedrayes-Lopez, M.

Sandri, G.

Sayle, A.

Sekeres, A.

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5180, 107–114 (2004).
[CrossRef]

Shanbhag, P. M.

Shi, Y.

W. Deng, L. Zheng, Y. Shi, X. Wang, and X. Zhang, “Dwell-time algorithm based on matrix algebra and regularization method,” Opt. Precis. Eng. 15, 1009–1015 (2007).

Shorey, A.

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5180, 107–114 (2004).
[CrossRef]

Tam, H. Y.

Walker, D. D.

Wang, T.

T. Wang, H. B. Cheng, Z. C. Dong, and H. Y. Tam, “Removal character of vertical jet polishing with eccentric rotation motion using magnetorheological fluid,” J. Mater. Process. Technol. 213, 1532–1537 (2013).
[CrossRef]

Wang, X.

W. Deng, L. Zheng, Y. Shi, X. Wang, and X. Zhang, “Dwell-time algorithm based on matrix algebra and regularization method,” Opt. Precis. Eng. 15, 1009–1015 (2007).

Wang, Y. T.

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

Wang, Y. W.

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Wilson, S. R.

S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320–324 (1987).
[CrossRef]

Wu, J. F.

Xie, X. H.

L. Zhou, Y. F. Dai, X. H. Xie, C. J. Jiao, and S. Y. Li, “Model and method to determine dwell time in ion beam figuring,” Nanotechnol. Precis. Eng. 5, 107–112 (2007).

Yam, Y.

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

Yu, G. Y.

Yu, J. C.

Zhang, H. X.

Zhang, X.

W. Deng, L. Zheng, Y. Shi, X. Wang, and X. Zhang, “Dwell-time algorithm based on matrix algebra and regularization method,” Opt. Precis. Eng. 15, 1009–1015 (2007).

Zheng, L.

W. Deng, L. Zheng, Y. Shi, X. Wang, and X. Zhang, “Dwell-time algorithm based on matrix algebra and regularization method,” Opt. Precis. Eng. 15, 1009–1015 (2007).

Zhou, L.

L. Zhou, Y. F. Dai, X. H. Xie, C. J. Jiao, and S. Y. Li, “Model and method to determine dwell time in ion beam figuring,” Nanotechnol. Precis. Eng. 5, 107–112 (2007).

Zimmerman, D. C.

Appl. Opt.

P. M. Shanbhag, M. R. Feinberg, G. Sandri, M. N. Horenstein, and T. G. Bifano, “Ion-beam machining of millimeter scale optics,” Appl. Opt. 39, 599–611 (2000).
[CrossRef]

Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Investigation on removal features of multidistribution fixed abrasive diamond pellets used in the polishing of SiC mirrors,” Appl. Opt. 51, 8373–8382 (2012).
[CrossRef]

Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Further investigations on fixed abrasive diamond pellets used for diminishing mid-spatial frequency errors of optical mirrors,” Appl. Opt. 53, 327–334 (2014).
[CrossRef]

R. A. Jones, “Optimization of computer controlled polishing,” Appl. Opt. 16, 218–224 (1977).
[CrossRef]

J. F. Wu, Z. W. Lu, and H. X. Zhang, “Dwell time algorithm in ion beam figuring,” Appl. Opt. 48, 3930–3937 (2009).
[CrossRef]

Z. C. Dong, H. B. Cheng, and H. Y. Tam, “Modified subaperture tool influence functions of a flat pitch polisher with reverse-calculated material removal rate,” Appl. Opt. 53, 2455–2464 (2014).
[CrossRef]

D. C. Zimmerman, “Feasibility studies for the alignment of the Thirty Meter Telescope,” Appl. Opt. 49, 3485–3498 (2010).
[CrossRef]

A. Cordero-Davila, J. Gonzalez-Garcıa, and M. Pedrayes-Lopez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt. 43, 1250–1254 (2004).
[CrossRef]

P. J. Guo, H. Fang, and J. C. Yu, “Edge effect in fluid jet polishing,” Appl. Opt. 45, 6729–6735 (2006).
[CrossRef]

R. J. Marks, “Gerchberg’s extrapolation algorithm in two dimensions,” Appl. Opt. 20, 1815–1820 (1981).
[CrossRef]

Int. J. Mach. Tools Manuf.

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Int. J. Mod. Phys. B

W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
[CrossRef]

J. Mater. Process. Technol.

T. Wang, H. B. Cheng, Z. C. Dong, and H. Y. Tam, “Removal character of vertical jet polishing with eccentric rotation motion using magnetorheological fluid,” J. Mater. Process. Technol. 213, 1532–1537 (2013).
[CrossRef]

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

Nanotechnol. Precis. Eng.

L. Zhou, Y. F. Dai, X. H. Xie, C. J. Jiao, and S. Y. Li, “Model and method to determine dwell time in ion beam figuring,” Nanotechnol. Precis. Eng. 5, 107–112 (2007).

Opt. Express

Opt. Precis. Eng.

W. Deng, L. Zheng, Y. Shi, X. Wang, and X. Zhang, “Dwell-time algorithm based on matrix algebra and regularization method,” Opt. Precis. Eng. 15, 1009–1015 (2007).

Proc. SPIE

M. Johns, “Giant Magellan Telescope,” Proc. SPIE 5382, 85–92 (2004).

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5180, 107–114 (2004).
[CrossRef]

S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320–324 (1987).
[CrossRef]

L. C. Charles, C. M. Egert, and W. H. Kathy, “Advanced matrix based algorithm for ion beam milling of optical components [C],” Proc. SPIE 1752, 54–62 (1992).
[CrossRef]

Other

R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra, 34–39 (1963).

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Figures (17)

Fig. 1.
Fig. 1.

Extended tool paths. (a) Concentric circle path. (b) Raster path. (c) Hilbert path. (d) Moore path.

Fig. 2.
Fig. 2.

Data extension examples. (a) Origin data, Φ100mm. (b) Extended data by Gaussian method, Φ120mm. (b) Extended data by neighbor-average, Φ120mm. (c) Extended data by Gerchberg band-limited extrapolation, Φ120mm.

Fig. 3.
Fig. 3.

Sketch map of proposed dwell time model.

Fig. 4.
Fig. 4.

Distribution of convergence speed factor for a concentric circle tool path.

Fig. 5.
Fig. 5.

Iterative flow of the proposed dwell time model.

Fig. 6.
Fig. 6.

Schematic illustration of multiple TIFs in one optimization process.

Fig. 7.
Fig. 7.

Schematic illustration of asymmetric TIF in polishing process with spiral tool path.

Fig. 8.
Fig. 8.

Results of simulation: (a) original error map, PV=4.6535λ, RMS=0.9744λ, aperture Φ200mm. (b) Used TIF with aperture 10×10mm. (c) Residual error map PV=0.1216λ, RMS=0.0041λ. (d) Dwell time map.

Fig. 9.
Fig. 9.

Surface gradient maps in simulation: (a) before polishing, gradient PV=117.99μrad, gradient RMS=33.95μrad and (b) after polishing, gradient PV=14.678μrad, gradient RMS=2.270μrad.

Fig. 10.
Fig. 10.

Power spectrum density curves before and after polishing: (a) x direction; (b) y direction.

Fig. 11.
Fig. 11.

Two typical surface forms used in simulations: (a) Surface 1, with PV=8.05λ, RMS=2.31λ; (b) Surface 2, with PV=13.602λ, RMS=0.829λ.

Fig. 12.
Fig. 12.

Relationship curves of TIF size with residual RMS and polishing time under identical ratio of TIF size and path interval for (a) Surface 1; (b) Surface 2.

Fig. 13.
Fig. 13.

Relationship curves of path interval with residual RMS and polishing time under identical TIF size for (a) Surface 1; (b) Surface 2.

Fig. 14.
Fig. 14.

(a) Real picture of JR-1800. (b) Polishing process of Φ920mm parabolic mirror.

Fig. 15.
Fig. 15.

Data for the Φ920mm parabolic mirror. (a) Surface form before fifth polishing, indicating PV=0.362λ, RMS=0.028λ. (b) Used TIF with peak removal rate of 0.22λ/min. (c) Dwell time map optimized by proposed dwell time model. (d) Residual error predicted by proposed dwell time model, PV=0.0258λ, RMS=0.0048λ. (e) Practical residual error after fifth polishing, PV=0.208λ, RMS=0.016λ. (f) Interferometric fringe after fifth polishing.

Fig. 16.
Fig. 16.

(a) Diagram of MRF-180 machine. (b) Two-axis motion model of the MRF tool. (c) MR fluid after magnetorheological effect. (d) Top view of Gaussian-like TIF generated by the tool. (e) Section profile of the TIF.

Fig. 17.
Fig. 17.

Data for the Φ100mm Zerodur plane. (a) Surface form polishing, PV=0.476λ, RMS=0.062λ. (b) TIF used with peak removal rate of 0.86λ/min. (c) Dwell time map optimized by proposed dwell time model. (d) Residual error predicted by proposed dwell time model, PV=0.031λ, RMS=0.0052λ. (e) Practical residual error after polishing, PV=0.085λ, RMS=0.013λ. (f) interferometric fringe after polishing.

Tables (3)

Tables Icon

Table 1. Calculated Time Analysis by Proposed Model

Tables Icon

Table 2. Simulation Parameters and Results for TIF Size

Tables Icon

Table 3. Simulation Parameters and Results for Path Interval

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

dz(x,y)=K·P(x,y)·V(x,y),
TIF(x,y)=1111K·P(x,y)·V(x,y)dxdy.
R(x,y)=TIF(x,y)T(x,y).
OMF=C1·RMS2+C2·GRMS2.
m2m1=spot_rglass_r,
Ls=2·spot_r.
ri=j=1niTIF(xixj,yiyj)·t(xj,yj).
PWFi=1j=1niTIF(xixj,yiyj),
NPWFi=PWFimax(PWF).
μi=ek(xi,yi)e(k1)(xi,yi)=NPWFimax(ni).
tk(xi,yi)=ek(xi,yi)/max(TIF)=μie(k1)(xi,yi)/max(TIF),
ek(x,y)=e(k1)(x,y)j=1niTIF(xxj,yyj)·tk(xj,yj).
Z3=1,Z4=1,Z7=1,Z8=1,Z9=1.
CRRMS=0.97440.00410.9744=99.6%,
CRGRMS=33.962.2733.96=93.3%.

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