Abstract

We examine some computational problems related to the finishing of axisymmetric optical surfaces discussed in a paper by N. J. Brown [Opt. Eng. 17, 602–620 (1978)]. In the spirit of Brown’s paper we develop solution algorithms that are suitable for implementation on today’s personal computers. The use of the algorithms is illustrated by some sample calculations.

© 1996 Optical Society of America

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References

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  1. N. J. Brown, “Computationally directed axisymmetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
  2. F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).
  3. R. E. Wagner, R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt. 12, 1683–1689 (1974).
    [CrossRef]
  4. K. B. Paxton, G. H. Connors, R. J. Cooley, “Uniform polishing of convex aspheres with an elastic lap,” Appl. Opt. 14, 2274–2279 (1975).
    [CrossRef] [PubMed]
  5. D. J. Bajuk, “Computer controlled generation of rotationally symmetric aspheric surfaces,” Opt. Eng. 15, 401–406 (1976).
  6. D. W. Small, S. J. Hoskins, “An automated aspheric polishing machine,” in Optical Manufacturing, Testing and Aspheric Optics, G. M. Sanger, ed., Proc. SPIE645, 66–74 (1986). Reprinted in Selected Papers On Computer-Controlled Optical Surfacing, R. Jones, ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1991).
  7. R. Aspden, R. McDonough, F. Nitchie, “Computer assisted optical surfacing,” Appl. Opt. 11, 2739–2747 (1972).
    [CrossRef] [PubMed]
  8. A. P. Bogdanov, “Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool,” Sov. J. Opt. Technol. 52, 409–413 (1985). Reprinted in R. Jones, ed., Selected Papers On Computer-Controlled Optical Surfacing (SPIE Optical Engineering Press, Bellingham, Wash., 1991).
  9. R. A. Jones, “Optimization of computer controlled polishing,” Appl. Opt. 16, 218–2247 (1977).
    [CrossRef] [PubMed]
  10. T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Model. 18, 25–40 (1993).
    [CrossRef]
  11. C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part i: finite dimensional material removal profile spaces,” IMA J. Numer. Anal. 12, 67–84 (1992).
    [CrossRef]
  12. C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part ii: mollifications,” IMA J. Numer. Anal. 12, 259–269 (1992).
    [CrossRef]
  13. T. A. Porsching, C. A. Hall, “Approximation methods and the computer numerically controlled fabrication of optical surfaces,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 205–215 (1992).
  14. K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), Chap. 6, p. 320.
  15. A. E. Taylor, Introduction to Functional Analysis (Wiley, New York, 1967), Chap. 3, p. 105.

1993 (1)

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Model. 18, 25–40 (1993).
[CrossRef]

1992 (2)

C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part i: finite dimensional material removal profile spaces,” IMA J. Numer. Anal. 12, 67–84 (1992).
[CrossRef]

C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part ii: mollifications,” IMA J. Numer. Anal. 12, 259–269 (1992).
[CrossRef]

1985 (1)

A. P. Bogdanov, “Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool,” Sov. J. Opt. Technol. 52, 409–413 (1985). Reprinted in R. Jones, ed., Selected Papers On Computer-Controlled Optical Surfacing (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

1978 (1)

N. J. Brown, “Computationally directed axisymmetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).

1977 (1)

1976 (1)

D. J. Bajuk, “Computer controlled generation of rotationally symmetric aspheric surfaces,” Opt. Eng. 15, 401–406 (1976).

1975 (1)

1974 (1)

R. E. Wagner, R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt. 12, 1683–1689 (1974).
[CrossRef]

1972 (1)

1927 (1)

F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).

Aspden, R.

Atkinson, K. E.

K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), Chap. 6, p. 320.

Bajuk, D. J.

D. J. Bajuk, “Computer controlled generation of rotationally symmetric aspheric surfaces,” Opt. Eng. 15, 401–406 (1976).

Bennett, T. L.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Model. 18, 25–40 (1993).
[CrossRef]

Bogdanov, A. P.

A. P. Bogdanov, “Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool,” Sov. J. Opt. Technol. 52, 409–413 (1985). Reprinted in R. Jones, ed., Selected Papers On Computer-Controlled Optical Surfacing (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Brown, N. J.

N. J. Brown, “Computationally directed axisymmetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).

Connors, G. H.

Cooley, R. J.

Ernsthausen, J. M.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Model. 18, 25–40 (1993).
[CrossRef]

Hall, C. A.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Model. 18, 25–40 (1993).
[CrossRef]

C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part ii: mollifications,” IMA J. Numer. Anal. 12, 259–269 (1992).
[CrossRef]

C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part i: finite dimensional material removal profile spaces,” IMA J. Numer. Anal. 12, 67–84 (1992).
[CrossRef]

T. A. Porsching, C. A. Hall, “Approximation methods and the computer numerically controlled fabrication of optical surfaces,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 205–215 (1992).

Hoskins, S. J.

D. W. Small, S. J. Hoskins, “An automated aspheric polishing machine,” in Optical Manufacturing, Testing and Aspheric Optics, G. M. Sanger, ed., Proc. SPIE645, 66–74 (1986). Reprinted in Selected Papers On Computer-Controlled Optical Surfacing, R. Jones, ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Jones, R. A.

McDonough, R.

Nitchie, F.

Paxton, K. B.

Porsching, T. A.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Model. 18, 25–40 (1993).
[CrossRef]

C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part i: finite dimensional material removal profile spaces,” IMA J. Numer. Anal. 12, 67–84 (1992).
[CrossRef]

C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part ii: mollifications,” IMA J. Numer. Anal. 12, 259–269 (1992).
[CrossRef]

T. A. Porsching, C. A. Hall, “Approximation methods and the computer numerically controlled fabrication of optical surfaces,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 205–215 (1992).

Preston, F. W.

F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).

Shannon, R. R.

R. E. Wagner, R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt. 12, 1683–1689 (1974).
[CrossRef]

Small, D. W.

D. W. Small, S. J. Hoskins, “An automated aspheric polishing machine,” in Optical Manufacturing, Testing and Aspheric Optics, G. M. Sanger, ed., Proc. SPIE645, 66–74 (1986). Reprinted in Selected Papers On Computer-Controlled Optical Surfacing, R. Jones, ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Taylor, A. E.

A. E. Taylor, Introduction to Functional Analysis (Wiley, New York, 1967), Chap. 3, p. 105.

Wagner, R. E.

R. E. Wagner, R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt. 12, 1683–1689 (1974).
[CrossRef]

Appl. Opt. (4)

IMA J. Numer. Anal. (2)

C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part i: finite dimensional material removal profile spaces,” IMA J. Numer. Anal. 12, 67–84 (1992).
[CrossRef]

C. A. Hall, T. A. Porsching, “Approximation methods in the computer numerically controlled fabrication of optical surfaces. Part ii: mollifications,” IMA J. Numer. Anal. 12, 259–269 (1992).
[CrossRef]

J. Soc. Glass Technol. (1)

F. W. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).

Math. Comput. Model. (1)

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Model. 18, 25–40 (1993).
[CrossRef]

Opt. Eng. (2)

N. J. Brown, “Computationally directed axisymmetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).

D. J. Bajuk, “Computer controlled generation of rotationally symmetric aspheric surfaces,” Opt. Eng. 15, 401–406 (1976).

Sov. J. Opt. Technol. (1)

A. P. Bogdanov, “Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool,” Sov. J. Opt. Technol. 52, 409–413 (1985). Reprinted in R. Jones, ed., Selected Papers On Computer-Controlled Optical Surfacing (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Other (4)

D. W. Small, S. J. Hoskins, “An automated aspheric polishing machine,” in Optical Manufacturing, Testing and Aspheric Optics, G. M. Sanger, ed., Proc. SPIE645, 66–74 (1986). Reprinted in Selected Papers On Computer-Controlled Optical Surfacing, R. Jones, ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

T. A. Porsching, C. A. Hall, “Approximation methods and the computer numerically controlled fabrication of optical surfaces,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 205–215 (1992).

K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978), Chap. 6, p. 320.

A. E. Taylor, Introduction to Functional Analysis (Wiley, New York, 1967), Chap. 3, p. 105.

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Figures (9)

Fig. 1
Fig. 1

Brown’s single-spindle machine.

Fig. 2
Fig. 2

Fixed rectangular coordinate system F and polar coordinates.

Fig. 3
Fig. 3

Defining parameters for single-spindle machine.

Fig. 4
Fig. 4

Average removal rate, γ ≡ 0.

Fig. 5
Fig. 5

Removal rates for Table 1.

Fig. 6
Fig. 6

Errors ē 0 and ē 1.

Fig. 7
Fig. 7

Errors ē 1 and ē 2.

Fig. 8
Fig. 8

Errors ē 2 and ē 3.

Fig. 9
Fig. 9

Shapes F, f 0, and f 3.

Tables (1)

Tables Icon

Table 1 Machine Parameters

Equations (45)

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

Φ ( r , t ) = K p ( r ) | q | ,
Φ ( r , t ) = 0 .
ϕ ( r ) = 1 T 1 T 0 T 0 T 1 Φ ( r , t ) d t ,
R = ( cos θ ) i + ( sin θ ) j ,
T = ( sin θ ) i + ( cos θ ) j .
p = x i + y j = r R .
p ˙ = r ˙ R + r R ˙ = r ˙ R + r θ ˙ T .
r = r cos ( Ω t t 0 ) i + r sin ( Ω t t 0 ) j ,
r ˙ = r Ω T ,
q = p ˙ r ˙ = r ˙ R + r ( θ ˙ Ω ) T .
q = ( r ˙ cos θ r θ ˙ sin θ + r Ω sin θ ) i + ( r ˙ sin θ + r θ ˙ cos θ r Ω cos θ ) j = ( x ˙ + Ω y ) i + ( y ˙ Ω x ) j .
| q | = [ ( x ˙ + Ω y ) 2 + ( y ˙ Ω x ) 2 ] 1 / 2 .
x = x cos α y sin α y = x sin α + y cos α,
x = x cos ( Ω t ) y sin ( Ω t ) ,
y = x sin ( Ω t ) + y cos ( Ω t ) .
( x x c ) 2 + ( y y c ) 2 s 2 .
| q | = [ ( x ˙ c + Ω y c ) 2 + ( y ˙ c Ω x c ) 2 ] 1 / 2 ,
ϕ ( r , T ) = 1 T 0 T Φ ( r , t ) d t ,
d d T [ T ϕ ( r , T ) ] = Φ ( r , T ) .
ϕ ( r , 0 ) = Φ ( r , 0 ) .
ϕ n + 1 ( r ) = t n ϕ n ( r ) + h Φ n + 1 / 2 ( r ) t n + 1 , n = 0 , 1 , , N 1 ,
ϕ n ( r ) = ϕ ( r , t n ) + O ( h 2 ) ,
Do i = 0, , I 1 ϕ ( x i ) = 0 End Do Do n = 0 , , N 1 t n = n h τ = t n + ( h / 2 ) Compute x c ( τ ) , y c ( τ ) , x ˙ c ( τ ) , y ˙ c ( τ ) Do i = 0 , , I 1 x = x i cos ( Ω τ ) y = x i sin ( Ω τ ) If ( x , y ) G then q = { [ x ˙ c ( τ ) + Ω y c ( τ ) ] 2 + [ y ˙ c ( τ ) Ω x c ( τ ) ] 2 } 1 / 2 Φ = K p q Else Φ = 0 End If ϕ ( x i ) = ( t n ϕ ( x i ) + h Φ ) / ( t n + h ) End Do End Do
γ = a sin ( λ t ) ,
x c = A cos γ B , y c = A sin γ .
x ˙ c = A γ ˙ sin γ, y ˙ c = A γ ˙ cos γ,
γ ˙ = a λ cos ( λ t ) .
ϕ ( x ) = { π cos 1 ( ( x ) 2 3 2 x ) , 0 0 x 1 1 < x 3 . 3 < x
S J ( x ) = j = 1 J τ j ϕ ( j ) ( x ) .
e 0 ( x ) = f 0 ( x ) F ( x ) ,
e J ( x ) = e 0 ( x ) S J ( x ) .
e J ( x i ) = 0
e J ( x i ) = τ i 1 ϕ ( i 1 ) ( x i ) + τ i ϕ ( i ) ( x i ) + τ i + 1 ϕ ( i + 1 ) ( x i ) .
e j ( x ) = e j 1 ( x ) τ j ϕ ( j ) ( x ) .
[ f ] = def min g ,
[ e j 1 ( x ) τ j ϕ ( j ) ( x ) ] = min t 0 [ e j 1 ( x ) t ϕ ( j ) ( x ) ] .
f = [ 0 R f 2 ( x ) d x ] 1 / 2 ,
[ f ] = [ 0 R f ¯ 2 ( x ) d x ] 1 / 2 ,
f ¯ ( x ) = def f ( x ) 1 R 0 R f ( x ) d x .
[ e j 1 ( x ) t ϕ ( j ) ( x ) ] 2 = 0 R e ¯ j 1 2 ( x ) d x 2 τ 0 R e ¯ j 1 ( x ) ϕ ¯ ( j ) × ( x ) d x + τ 2 0 R ϕ ¯ ( j ) 2 ( x ) d x ,
τ j = max [ 0 , ( e ¯ j 1 , ϕ ¯ ( j ) ) / ( ϕ ¯ ( j ) , ϕ ¯ ( j ) ) ] ,
( f , g ) = def 0 R f ( x ) g ( x ) d x .
F ( x ) = 0 . 5 x 2 , 0 x R = 1 ,
f 0 ( x ) = 1 ( 1 x 2 ) 1 / 2 .
e 0 ( x ) = 1 ( 1 x 2 ) 1 / 2 0 . 5 x 2 , 0 x 1 .

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