Abstract

A parametric smoothing model is developed to quantitatively describe the smoothing action of polishing tools that use visco-elastic materials. These materials flow to conform to the aspheric shape of the workpieces, yet behave as a rigid solid for short duration caused by tool motion over surface irregularities. The smoothing effect naturally corrects the mid-to-high frequency errors on the workpiece while a large polishing lap still removes large scale errors effectively in a short time. Quantifying the smoothing effect allows improvements in efficiency for finishing large precision optics. We define normalized smoothing factor SF which can be described with two parameters. A series of experiments using a conventional pitch tool and the rigid conformal (RC) lap was performed and compared to verify the parametric smoothing model. The linear trend of the SF function was clearly verified. Also, the limiting minimum ripple magnitude PVmin from the smoothing actions and SF function slope change due to the total compressive stiffness of the whole tool were measured. These data were successfully fit using the parametric smoothing model.

© 2010 OSA

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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. R. E. Wagner and R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt. 13(7), 1683–1689 (1974).
    [CrossRef] [PubMed]
  3. 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(8), 958–964 (2003).
    [CrossRef] [PubMed]
  4. H. M. Pollicove, E. M. Fess, and J. M. Schoen, “Deterministic manufacturing processes for precision optical surfaces,” in Window and Dome Technologies VIII, R. W. Tustison, eds., Proc. SPIE 5078, 90–96 (2003).
  5. J. H. Burge, S. Benjamin, D. Caywood, C. Noble, M. Novak, C. Oh, R. Parks, B. Smith, P. Su, M. Valente, and C. Zhao, “Fabrication and testing of 1.4-m convex off-axis aspheric optical surfaces,” in Optical Manufacturing and Testing VIII, J. H. Burge; O. W. Fähnle and R. Williamson, eds., Proc. SPIE 7426, 74260L1–12 (2009).
  6. 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(24), 21850–21866 (2009).
    [CrossRef] [PubMed]
  7. R. E. Parks, “Alignment of off-axis conic mirrors,” in Optical Fabrication and Testing, OSA Technical Digest Series (Optical Society of America, 1980), paper TuB4.
  8. N. J. Brown, and R. E. Parks, “The polishing-to-figuring transition in turned optics,” SPIE’s 25th Annual International Technical Symposium, (SPIE, 1981)
  9. P. K. Mehta, and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” in Optomechanical Engineering and Vibration Control, E. A. Derby, C. G. Gordon, D. Vukobratovich, P. R. Yoder Jr., and C. H. Zweben, eds., Proc. SPIE 3786, 447 (1999).
  10. M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41(7), 1473–1474 (2002).
    [CrossRef]
  11. M. T. Tuell, “Novel tooling for production of aspheric surfaces,” M.S. Thesis (2002).
  12. M. Johns, “The Giant Magellan Telescope (GMT),” in Extremely Large Telescopes: Which Wavelengths? T. E. Andersen, eds., Proc. SPIE 6986, 698603 1–12 (2008).
  13. J. Nelson, and G. H. Sanders, “The status of the Thirty Meter Telescope project,” in Ground-based and Airborne Telescopes II, L. M. Stepp and R. Gilmozzi, eds., Proc. SPIE 7012, 70121A1–18 (2008).
  14. A. Ardeberg, T. Andersen, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, “From Euro50 towards a European ELT,” in Ground-based and Airborne Telescopes, L. M. Stepp, eds., Proc. SPIE 6267, 626725 1–10 (2006).
  15. A. Heller, “Safe and sustainable energy with LIFE” (2009), https://str.llnl.gov/AprMay09/pdfs/05.09.02.pdf .
  16. R. E. Parks, “Specifications: Figure and Finish are not enough,” in An optical Believe It or Not: Key Lessons Learned, M. A. Kahan, eds., Proc. SPIE 7071, 70710B1–9 (2008).
  17. J. M. Hill, “Optical Design, Error Budget and Specifications for the Columbus Project Telescope,” in Advanced Technology Optical Telescopes IV, L. D. Barr, eds., Proc. SPIE 1236, 86–107 (1990).
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    [CrossRef] [PubMed]
  19. H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, M. T. Tuell, S. C. West, C. Zhao, and T. Zobrist, “Fabrication and testing of the first 8.4 m off-axis segment for the Giant Magellan Telescope,” in Modern Technologies in Space- and Ground-based Telescopes and Instrumentation, E. Atad-Ettedgui and D. Lemke, eds., Proc. SPIE 7739, 77390A (2010).
  20. M. J. Valente, D. W. Kim, M. J. Novak, C. J. Oh, and J. H. Burge, “Fabrication of 4-meter class astronomical optics,” in Modern Technologies in Space- and Ground-based Telescopes and Instrumentation, E. Atad-Ettedgui and D. Lemke, eds., Proc. SPIE 7739, 77392D (2010).
  21. R. P. Chhabra, and J. F. Richardson, Non-Newtonian Flow and Applied Rheology (2nd edition) (Elsevier Ltd, 2008), Chap. 1.
  22. 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(7), 5656–5665 (2009).
    [CrossRef] [PubMed]
  23. M. A. Meyers, and K. K. Chawla, Mechanical Behavior of Materials (2nd edition) (Cambridge University Press, 2009), 124–125.
  24. A. C. Fischer-Cripps, “Multiple-frequency dynamic nanoindentation testing,” J. Mater. Res. 19(10), 2981–2988 (2004).
    [CrossRef]
  25. D. W. Kim and J. H. Burge of the College of Optical Sciences, University of Arizona, 1630 East University Boulevard, Tucson, AZ 85721 are preparing a manuscript to be called “Smoothing performance of various polishing tools.”
  26. B. C. Don Loomis, Crawford, Norm Schenck, and Bill Anderson, Optical Engineering and Fabrication Facility, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, (personal communication, 2009).
  27. N. J. Brown, “Optical polishing pitch,” in Optical Fabrication and Testing, Optical Society of America Workshop (Optical Society of America, 1977).

2010 (1)

2009 (2)

2004 (1)

A. C. Fischer-Cripps, “Multiple-frequency dynamic nanoindentation testing,” J. Mater. Res. 19(10), 2981–2988 (2004).
[CrossRef]

2003 (1)

2002 (1)

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41(7), 1473–1474 (2002).
[CrossRef]

1974 (1)

1963 (1)

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

Anderson, B.

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41(7), 1473–1474 (2002).
[CrossRef]

Brooks, D.

Burge, J. H.

Fischer-Cripps, A. C.

A. C. Fischer-Cripps, “Multiple-frequency dynamic nanoindentation testing,” J. Mater. Res. 19(10), 2981–2988 (2004).
[CrossRef]

Freeman, R.

Jones, R. A.

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

Kim, D. W.

Kim, S. W.

King, A.

McCavana, G.

Morton, R.

Park, W. H.

Shannon, R. R.

Tuell, M. T.

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41(7), 1473–1474 (2002).
[CrossRef]

Wagner, R. E.

Walker, D. D.

Appl. Opt. (1)

J. Mater. Res. (1)

A. C. Fischer-Cripps, “Multiple-frequency dynamic nanoindentation testing,” J. Mater. Res. 19(10), 2981–2988 (2004).
[CrossRef]

Opt. Eng. (1)

M. T. Tuell, J. H. Burge, and B. Anderson, “Aspheric optics: smoothing the ripples with semi-flexible tools,” Opt. Eng. 41(7), 1473–1474 (2002).
[CrossRef]

Opt. Express (4)

Photon. Spectra (1)

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

Other (19)

M. A. Meyers, and K. K. Chawla, Mechanical Behavior of Materials (2nd edition) (Cambridge University Press, 2009), 124–125.

H. M. Martin, R. G. Allen, J. H. Burge, D. W. Kim, J. S. Kingsley, M. T. Tuell, S. C. West, C. Zhao, and T. Zobrist, “Fabrication and testing of the first 8.4 m off-axis segment for the Giant Magellan Telescope,” in Modern Technologies in Space- and Ground-based Telescopes and Instrumentation, E. Atad-Ettedgui and D. Lemke, eds., Proc. SPIE 7739, 77390A (2010).

M. J. Valente, D. W. Kim, M. J. Novak, C. J. Oh, and J. H. Burge, “Fabrication of 4-meter class astronomical optics,” in Modern Technologies in Space- and Ground-based Telescopes and Instrumentation, E. Atad-Ettedgui and D. Lemke, eds., Proc. SPIE 7739, 77392D (2010).

R. P. Chhabra, and J. F. Richardson, Non-Newtonian Flow and Applied Rheology (2nd edition) (Elsevier Ltd, 2008), Chap. 1.

D. W. Kim and J. H. Burge of the College of Optical Sciences, University of Arizona, 1630 East University Boulevard, Tucson, AZ 85721 are preparing a manuscript to be called “Smoothing performance of various polishing tools.”

B. C. Don Loomis, Crawford, Norm Schenck, and Bill Anderson, Optical Engineering and Fabrication Facility, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, (personal communication, 2009).

N. J. Brown, “Optical polishing pitch,” in Optical Fabrication and Testing, Optical Society of America Workshop (Optical Society of America, 1977).

R. E. Parks, “Alignment of off-axis conic mirrors,” in Optical Fabrication and Testing, OSA Technical Digest Series (Optical Society of America, 1980), paper TuB4.

N. J. Brown, and R. E. Parks, “The polishing-to-figuring transition in turned optics,” SPIE’s 25th Annual International Technical Symposium, (SPIE, 1981)

P. K. Mehta, and P. B. Reid, “A mathematical model for optical smoothing prediction of high-spatial frequency surface errors,” in Optomechanical Engineering and Vibration Control, E. A. Derby, C. G. Gordon, D. Vukobratovich, P. R. Yoder Jr., and C. H. Zweben, eds., Proc. SPIE 3786, 447 (1999).

H. M. Pollicove, E. M. Fess, and J. M. Schoen, “Deterministic manufacturing processes for precision optical surfaces,” in Window and Dome Technologies VIII, R. W. Tustison, eds., Proc. SPIE 5078, 90–96 (2003).

J. H. Burge, S. Benjamin, D. Caywood, C. Noble, M. Novak, C. Oh, R. Parks, B. Smith, P. Su, M. Valente, and C. Zhao, “Fabrication and testing of 1.4-m convex off-axis aspheric optical surfaces,” in Optical Manufacturing and Testing VIII, J. H. Burge; O. W. Fähnle and R. Williamson, eds., Proc. SPIE 7426, 74260L1–12 (2009).

M. T. Tuell, “Novel tooling for production of aspheric surfaces,” M.S. Thesis (2002).

M. Johns, “The Giant Magellan Telescope (GMT),” in Extremely Large Telescopes: Which Wavelengths? T. E. Andersen, eds., Proc. SPIE 6986, 698603 1–12 (2008).

J. Nelson, and G. H. Sanders, “The status of the Thirty Meter Telescope project,” in Ground-based and Airborne Telescopes II, L. M. Stepp and R. Gilmozzi, eds., Proc. SPIE 7012, 70121A1–18 (2008).

A. Ardeberg, T. Andersen, J. Beckers, M. Browne, A. Enmark, P. Knutsson, and M. Owner-Petersen, “From Euro50 towards a European ELT,” in Ground-based and Airborne Telescopes, L. M. Stepp, eds., Proc. SPIE 6267, 626725 1–10 (2006).

A. Heller, “Safe and sustainable energy with LIFE” (2009), https://str.llnl.gov/AprMay09/pdfs/05.09.02.pdf .

R. E. Parks, “Specifications: Figure and Finish are not enough,” in An optical Believe It or Not: Key Lessons Learned, M. A. Kahan, eds., Proc. SPIE 7071, 70710B1–9 (2008).

J. M. Hill, “Optical Design, Error Budget and Specifications for the Columbus Project Telescope,” in Advanced Technology Optical Telescopes IV, L. D. Barr, eds., Proc. SPIE 1236, 86–107 (1990).

Supplementary Material (1)

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Figures (8)

Fig. 1
Fig. 1

Schematic tool structures of three different tool types [18].

Fig. 2
Fig. 2

3D RC lap structure (exploded and cut in half) [18].

Fig. 3
Fig. 3

Smoothing effect simulation using an infinitely rigid tool (Media 1).

Fig. 4
Fig. 4

Storage modulus and loss factor tanδ for fused silica (left) and Silly-PuttyTM (right) as a function of applied stress frequency from the literature [24].

Fig. 5
Fig. 5

The sinusoidal ripple profiles (before and after smoothing), which shows the values to determine the smoothing factor SF in Eq. (22).

Fig. 6
Fig. 6

The ripple generating pitch tool with a grid of circles (left) and a grey scale surface map of the Pyrex substrate with sinusoidal ripples and reference area to measure the nominal removal depth in a rectangular box (right).

Fig. 7
Fig. 7

Measured ripple profiles as tool smoothes out the ripples: pitch tool (left) and RC lap (right) [18]. (Note: The initial ripple magnitude PV was about 0.4μm for both cases.)

Fig. 8
Fig. 8

Measured smoothing factor SF vs. initial ripple magnitude Pini for pitch tool and RC lap. (Note: The solid line represents the linear fit using the parametric smoothing model. Two parameters C1 and C2 were used to fit the measured data as shown in Table 3.)

Tables (3)

Tables Icon

Table 1 Experimental set-up for the smoothing experiment

Tables Icon

Table 2 Operating condition for the pitch tool and RC lap

Tables Icon

Table 3 Compressive stiffness κelastic and two parameter values for parametric smoothing model

Equations (25)

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e r r o r ( x ) = P V ( 1 sin ( 2 π ξ x ) ) ,
P ( x ) = P n o m i n a l + e r r o r ( x ) 1 D p l a t e ( 2 π ξ ) 4 + 1 D s _ p l a t e ( 2 π ξ ) 2 + 1 κ t o t a l ,
D p l a t e = E p l a t e t p l a t e 3 / 12 ( 1 ν p l a t e 2 ) ,
D s _ p l a t e = E p l a t e t p l a t e / 2 ( 1 ν p l a t e ) ,
S t o r a g e     m o d u l u s : E ' = σ 0 ε 0 cos δ ,
L o s s m o d u l u s : E " = σ 0 ε 0 sin δ ,
ε = ε 0 sin ( t ω ) ,
σ = σ 0 sin ( t ω + δ )       .
tan δ = E " E '       .
P ( x ) = P n o m i n a l + κ t o t a l e r r o r ( x ) .
1 κ t o t a l = 1 κ e l a s t i c + 1 κ o t h e r s ,
P ( x ) = P n o m i n a l + κ t o t a l e r r o r ( x ) = P n o m i n a l + 1 1 κ e l a s t i c + 1 κ o t h e r s e r r o r ( x )         .
κ e l a s t i c = σ 0 Δ L = ε 0 E ' / cos δ Δ L = { Δ L / L } E ' / cos δ Δ L = E ' L cos δ ,
ω = 2 π T = 2 π ( 1 / ξ V t o o l _ m o t i o n ) = 2 π ξ V t o o l _ m o t i o n ,
V t o o l _ m o t i o n = 2 π 30 20 60 = 62.8     [ m m / sec ]         .
ω = 2 π ξ V t o o l _ m o t i o n = 2 π 0.083 62.8 = 32.77       [ r a d i a n s / s e c ]         .
P ( x ) = P n o m i n a l + E ' L cos δ e r r o r ( x ) 2500 + 0.003 × 10 9 8 × 10 3 1 1 × 10 6 ( 1 sin ( 2 π 0.085 x ) ) ​ ​                                                                                                           = 2500 + 375 ( 1 sin ( 2 π 0.085 x ) )       [ P a s c a l ] ,
Δ z ( x ) = R P r e s t o n P ( x ) V t o o l _ w o r k p i e c e ( x ) Δ t ( x ) ,
P a d d = P P n o m i n a l = 1 1 κ e l a s t i c + 1 κ o t h e r s P V i n i ,
Δ P V = P V i n i P V a f t e r = R P r e s t o n P a d d V t o o l _ w o r k p i e c e Δ t         .
n o m i n a l _ r e m o v a l _ d e p t h = R P r e s t o n P n o m i n a l V t o o l _ w o r k p i e c e Δ t       .
S F Δ P V n o m i n a l _ r e m o v a l _ d e p t h = 1 P n o m i n a l ( 1 κ e l a s t i c + 1 κ o t h e r s ) P V i n i         .
S F 1 P n o m i n a l ( 1 κ e l a s t i c ( ω ) + 1 C 1 ) ( P V i n i C 2 ) ,
P i t c h t o o l :       κ e l a s t i c = E ' L cos δ = 2.5 × 10 9 8 × 10 3 1 = 312500     [ P a / μ m ] ,
R C l a p :       κ e l a s t i c = E ' L cos δ = 0.003 × 10 9 8 × 10 3 1 = 375     [ P a / μ m ] ,

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