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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References

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  1. T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, and D. D. Walker, �??The Euro50 Extremely Large Telescope,�?? in Future Giant Telescopes, J. R. P. Angel and R. Gilmozzi, eds., Proc. SPIE 4840, 214-225 (2003).
    [CrossRef]
  2. D. D. Walker, A. P. Doel, R. G. Bingham, D. Brooks, A. M. King, G. Peggs, B. Hughes, S. Oldfield, C. Dorn, H. McAndrews, G. Dando, and D. Riley, �??Design Study Report: The Primary and Secondary Mirrors for the Proposed Euro50 Telescope�?? (2002), <a href="http://www.zeeko.co.uk/papers/dl/New%20Study%20Report%20V%2026.pdf.">http://www.zeeko.co.uk/papers/dl/New%20Study%20Report%20V%2026.pdf</a>
  3. T. Anderson, A. Ardeberg, and M. Owner-Perterson, �??EURO50: Design study of a 50m adaptive optics telescope�?? (2003), <a href="http://www.astro.lu.se/~torben/euro50/publications/white_book80.pdf.">http://www.astro.lu.se/~torben/euro50/publications/white_book80.pdf</a>
  4. P. Dierickx, �??Optical fabrication in the large,�?? in Proceedings of the Backaskog workshop on extremely large telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., no. 57 of ESO conference and workshop proceedings (Lund Observatory and European Southern Observatory, Munchen, 2000), pp. 224-236.
  5. H. M. Martin, D. S. Anderson, J. R. P. Angel, R. H. Nagel, S. C. West, and R. S. Young, "Progress in the stressed-lap polishing of a 1.8-m f/1 mirror," in Advanced Technology Optical Telescopes IV, L. D. Barr, Ed., Proc. SPIE 1236, 682-690 (1990).
    [CrossRef]
  6. A. R. Jones and J. W. Rupp, "Rapid optical fabrication with computer-controlled optical surfacing," Opt. Eng. 30, 1962-1967 (1991).
    [CrossRef]
  7. T. S. Mast and J. E. Nelson, "Fabrication of large optical surfaces using a combination of polishing and mirror bending," in Advanced Technology Optical Telescopes IV, L. D. Barr, Ed., Proc. SPIE 1236, 670-681 (1990).
    [CrossRef]
  8. L. N. Allen, "Progress in ion figuring large optics," in Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, Eds., Proc. SPIE 2428, 237-247 (1995).
    [CrossRef]
  9. F. A. Carbone and D. A. Markle, "Combining ion figuring and SPSI testing to produce a high-quality aspheric 18-inch-diameter f/2 mirror," in Optical Manufacturing and Testing, V. J. Doherty D. V. M. and H. P. Stahl, Eds., Proc. SPIE 2536, 89-98 (1995).
    [CrossRef]
  10. K. Beckstette, M. Küchel, and E. Heynacher, "Large mirror figuring and testing," Astrophys Space Sci. 160, 207-214 (1989).
    [CrossRef]
  11. T. K. Korhonen and T. Lappalainen, "Computer-controlled figuring and testing," in Advanced Technology Optical Telescopes IV, L. D. Barr, Ed., Proc. SPIE 1236, 691-694 (1990).
    [CrossRef]
  12. T. S. Mast, J. E. Nelson, and G. E. Sommargren, �??Primary mirror segment fabrication for CELT,�?? in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE 4003, 43-58 (2000).
    [CrossRef]
  13. D. D. Walker, A. T. Beaucamp, R. G. Bingham, D. Brooks, R. Freeman, S. W. Kim, A. King, G. McCavana, R. Morton, D. Riley, and J. Simms, �??Precessions aspheric polishing : - new results from the development programme,�?? in Optical Manufacturing and Testing V, H. P. Stahl, ed., Proc. SPIE 5180, 15-28 (2004).
    [CrossRef]
  14. 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] [PubMed]
  15. D. D. Walker, A. T. Beaucamp, D. Brooks, V. Doubrovski, M. D. Cassie, C. Dunn, R. R. Freeman, A. King, M. Libert, G. McCavana, R. Morton, D. Riley, and J. Simms, �??New results from the Precessions polishing process scaled to larger sizes,�?? in Optical Fabrication, Metrology, and Material Advancements for Telescopes, E. Atad-Ettedgui and P. Dierickx, eds., Proc. SPIE 5494, 71-80 (2004).
    [CrossRef]
  16. D. D. Walker, A. T. Beaucamp, D. Brooks, V. Doubrovski, M. D. Cassie, C. Dunn, R. R. Freeman, A. King, M. Libert, G. McCavana, R. Morton, D. Riley, and J. Simms, �??Recent developments of Precessions polishing for larger components and free-form surfaces,�?? in Current Developments in Lens Design and Optical Engineering V, P. Z. Mouroulis, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE 5523, 281-289 (2004).
    [CrossRef]
  17. D. W. Kim, Space Optics Laboratory, Yonsei University, 134 Shinchon-dong, Sudaemun-gu, Seoul, Republic of Korea and S. W. Kim are preparing a manuscript to be called �??Novel polishing algorithm for fabrication simulation of 2m class hexagonal mirror segments for extremely large telescopes.�??
  18. H. Lee, Space Optics Laboratory, Yonsei University, 134 Shinchon-dong, Sudaemun-gu, Seoul, Republic of Korea and S. W. Kim are preparing a manuscript to be called �??Non-linear variation of tool influence function in precessing sub-diameter tool polishing.�??
  19. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM Journal of Optimization, 9, 112-147 (1998).
    [CrossRef]

Astrophys Space Sci. (1)

K. Beckstette, M. Küchel, and E. Heynacher, "Large mirror figuring and testing," Astrophys Space Sci. 160, 207-214 (1989).
[CrossRef]

ESO conference (1)

P. Dierickx, �??Optical fabrication in the large,�?? in Proceedings of the Backaskog workshop on extremely large telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., no. 57 of ESO conference and workshop proceedings (Lund Observatory and European Southern Observatory, Munchen, 2000), pp. 224-236.

Opt. Eng. (1)

A. R. Jones and J. W. Rupp, "Rapid optical fabrication with computer-controlled optical surfacing," Opt. Eng. 30, 1962-1967 (1991).
[CrossRef]

Opt. Express (1)

Proc. SPIE (10)

T. S. Mast and J. E. Nelson, "Fabrication of large optical surfaces using a combination of polishing and mirror bending," in Advanced Technology Optical Telescopes IV, L. D. Barr, Ed., Proc. SPIE 1236, 670-681 (1990).
[CrossRef]

L. N. Allen, "Progress in ion figuring large optics," in Laser-Induced Damage in Optical Materials, H. E. Bennett, A. H. Guenther, M. R. Kozlowski, B. E. Newnam, and M. J. Soileau, Eds., Proc. SPIE 2428, 237-247 (1995).
[CrossRef]

F. A. Carbone and D. A. Markle, "Combining ion figuring and SPSI testing to produce a high-quality aspheric 18-inch-diameter f/2 mirror," in Optical Manufacturing and Testing, V. J. Doherty D. V. M. and H. P. Stahl, Eds., Proc. SPIE 2536, 89-98 (1995).
[CrossRef]

H. M. Martin, D. S. Anderson, J. R. P. Angel, R. H. Nagel, S. C. West, and R. S. Young, "Progress in the stressed-lap polishing of a 1.8-m f/1 mirror," in Advanced Technology Optical Telescopes IV, L. D. Barr, Ed., Proc. SPIE 1236, 682-690 (1990).
[CrossRef]

T. Andersen, A. L. Ardeberg, J. Beckers, A. Goncharov, M. Owner-Petersen, H. Riewaldt, R. Snel, and D. D. Walker, �??The Euro50 Extremely Large Telescope,�?? in Future Giant Telescopes, J. R. P. Angel and R. Gilmozzi, eds., Proc. SPIE 4840, 214-225 (2003).
[CrossRef]

T. K. Korhonen and T. Lappalainen, "Computer-controlled figuring and testing," in Advanced Technology Optical Telescopes IV, L. D. Barr, Ed., Proc. SPIE 1236, 691-694 (1990).
[CrossRef]

T. S. Mast, J. E. Nelson, and G. E. Sommargren, �??Primary mirror segment fabrication for CELT,�?? in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE 4003, 43-58 (2000).
[CrossRef]

D. D. Walker, A. T. Beaucamp, R. G. Bingham, D. Brooks, R. Freeman, S. W. Kim, A. King, G. McCavana, R. Morton, D. Riley, and J. Simms, �??Precessions aspheric polishing : - new results from the development programme,�?? in Optical Manufacturing and Testing V, H. P. Stahl, ed., Proc. SPIE 5180, 15-28 (2004).
[CrossRef]

D. D. Walker, A. T. Beaucamp, D. Brooks, V. Doubrovski, M. D. Cassie, C. Dunn, R. R. Freeman, A. King, M. Libert, G. McCavana, R. Morton, D. Riley, and J. Simms, �??New results from the Precessions polishing process scaled to larger sizes,�?? in Optical Fabrication, Metrology, and Material Advancements for Telescopes, E. Atad-Ettedgui and P. Dierickx, eds., Proc. SPIE 5494, 71-80 (2004).
[CrossRef]

D. D. Walker, A. T. Beaucamp, D. Brooks, V. Doubrovski, M. D. Cassie, C. Dunn, R. R. Freeman, A. King, M. Libert, G. McCavana, R. Morton, D. Riley, and J. Simms, �??Recent developments of Precessions polishing for larger components and free-form surfaces,�?? in Current Developments in Lens Design and Optical Engineering V, P. Z. Mouroulis, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE 5523, 281-289 (2004).
[CrossRef]

SIAM Journal of Optimization (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM Journal of Optimization, 9, 112-147 (1998).
[CrossRef]

Other (4)

D. W. Kim, Space Optics Laboratory, Yonsei University, 134 Shinchon-dong, Sudaemun-gu, Seoul, Republic of Korea and S. W. Kim are preparing a manuscript to be called �??Novel polishing algorithm for fabrication simulation of 2m class hexagonal mirror segments for extremely large telescopes.�??

H. Lee, Space Optics Laboratory, Yonsei University, 134 Shinchon-dong, Sudaemun-gu, Seoul, Republic of Korea and S. W. Kim are preparing a manuscript to be called �??Non-linear variation of tool influence function in precessing sub-diameter tool polishing.�??

D. D. Walker, A. P. Doel, R. G. Bingham, D. Brooks, A. M. King, G. Peggs, B. Hughes, S. Oldfield, C. Dorn, H. McAndrews, G. Dando, and D. Riley, �??Design Study Report: The Primary and Secondary Mirrors for the Proposed Euro50 Telescope�?? (2002), <a href="http://www.zeeko.co.uk/papers/dl/New%20Study%20Report%20V%2026.pdf.">http://www.zeeko.co.uk/papers/dl/New%20Study%20Report%20V%2026.pdf</a>

T. Anderson, A. Ardeberg, and M. Owner-Perterson, �??EURO50: Design study of a 50m adaptive optics telescope�?? (2003), <a href="http://www.astro.lu.se/~torben/euro50/publications/white_book80.pdf.">http://www.astro.lu.se/~torben/euro50/publications/white_book80.pdf</a>

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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)

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)

Fig. 3.
Fig. 3.

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

Fig. 4.
Fig. 4.

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

Fig. 5.
Fig. 5.

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

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

Tables (2)

Tables Icon

Table 1. Specifications of three ELTs and KECK primary mirrors

Tables Icon

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

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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