Abstract

To get a high-precision optical surface, the deconvolved process of dwell time was transferred to a matrix equation in which the damped factor and the extra removal amount were introduced to expand the freedom of solution. A path weight factor and a surface error weight factor were used to take the scanning path and the initial surface error into account. Combined with the Gerchberg bandlimited extrapolation algorithm for initial surface error map extension, a high-precision final surface could be obtained within a factual aperture. Two surface error maps were calculated to rms=0.1nm from rms=130.23nm and to rms=0.08nm from rms=282.74nm. The simulations show that a perfect dwell time solution could be obtained by the revised matrix equation and initial surface error map extension with the help of the least squares QR (LSQR) algorithm.

© 2009 Optical Society of America

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References

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  1. N. A. Lynn, E. K. Robert, and S. L. Timothy, “Surface error correction of a Keck 10 m telescope primary mirror segment by ion figuring,” Proc. SPIE 1531, 195-204 (1991).
  2. 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]
  3. L. Aschke, F. Schubert, J. Kegeler, and A. Schindler, “Flatness correction of NZTE mask blank substrates,” Proc. SPIE 4343, 646-653 (2001).
    [CrossRef]
  4. M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).
  5. T. Hänsel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with an (x,y,z) linear three-axis system,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.
  6. S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320-322 (1987).
  7. S. R. Wilson, D. W. Reicher, C. F. Kranenberg, and J. R. McNeil, “Ion beam milling of fused silica for windows fabrication,” Proc. SPIE 1441, 82-85 (1990).
    [CrossRef]
  8. A. Schindler, T. Hänsel, F. Frost, R. Fechner, and G. Seidenkranz, “Ion beam finishing technology for high precision optics production,” in Optical Fabrication and Testing, A.Sawchuk, ed., Vol. 76 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper OTuB5.
  9. H. Fang, P. Guo, and J. Yu, “Dwell function algorithm in fluid jet polishing,” Appl. Opt. 45, 4291-4296 (2006).
    [CrossRef] [PubMed]
  10. C. L. Charles, C. M. Egert, and K. W. Hyltonet, “Advanced matrix-based algorithm for ion beam milling of optical components,” Proc. SPIE 1752, 54-62 (1992).
    [CrossRef]
  11. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Software 8, 43-71 (1982).
    [CrossRef]
  12. J. Robert and Marks, “Gerchberg's extrapolation algorithm in two dimensions,” Appl. Opt. 20, 1815-1820 (1981).
    [CrossRef]

2006 (1)

2004 (1)

M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).

2001 (1)

L. Aschke, F. Schubert, J. Kegeler, and A. Schindler, “Flatness correction of NZTE mask blank substrates,” Proc. SPIE 4343, 646-653 (2001).
[CrossRef]

2000 (1)

1992 (1)

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

1991 (1)

N. A. Lynn, E. K. Robert, and S. L. Timothy, “Surface error correction of a Keck 10 m telescope primary mirror segment by ion figuring,” Proc. SPIE 1531, 195-204 (1991).

1990 (1)

S. R. Wilson, D. W. Reicher, C. F. Kranenberg, and J. R. McNeil, “Ion beam milling of fused silica for windows fabrication,” Proc. SPIE 1441, 82-85 (1990).
[CrossRef]

1987 (1)

S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320-322 (1987).

1982 (1)

C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Software 8, 43-71 (1982).
[CrossRef]

1981 (1)

Antonello, E.

M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).

Aschke, L.

L. Aschke, F. Schubert, J. Kegeler, and A. Schindler, “Flatness correction of NZTE mask blank substrates,” Proc. SPIE 4343, 646-653 (2001).
[CrossRef]

Bifano, T. G.

Charles, C. L.

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

Conconia, P.

M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).

Egert, C. M.

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

Fang, H.

Fechner, R.

A. Schindler, T. Hänsel, F. Frost, R. Fechner, and G. Seidenkranz, “Ion beam finishing technology for high precision optics production,” in Optical Fabrication and Testing, A.Sawchuk, ed., Vol. 76 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper OTuB5.

Feinberg, M. R.

Frost, F.

A. Schindler, T. Hänsel, F. Frost, R. Fechner, and G. Seidenkranz, “Ion beam finishing technology for high precision optics production,” in Optical Fabrication and Testing, A.Sawchuk, ed., Vol. 76 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper OTuB5.

Ghigo, M.

M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).

Guo, P.

Hänsel, T.

A. Schindler, T. Hänsel, F. Frost, R. Fechner, and G. Seidenkranz, “Ion beam finishing technology for high precision optics production,” in Optical Fabrication and Testing, A.Sawchuk, ed., Vol. 76 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper OTuB5.

T. Hänsel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with an (x,y,z) linear three-axis system,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.

Horenstein, M. N.

Hyltonet, K. W.

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

Kegeler, J.

L. Aschke, F. Schubert, J. Kegeler, and A. Schindler, “Flatness correction of NZTE mask blank substrates,” Proc. SPIE 4343, 646-653 (2001).
[CrossRef]

Kranenberg, C. F.

S. R. Wilson, D. W. Reicher, C. F. Kranenberg, and J. R. McNeil, “Ion beam milling of fused silica for windows fabrication,” Proc. SPIE 1441, 82-85 (1990).
[CrossRef]

Lynn, N. A.

N. A. Lynn, E. K. Robert, and S. L. Timothy, “Surface error correction of a Keck 10 m telescope primary mirror segment by ion figuring,” Proc. SPIE 1531, 195-204 (1991).

Marks,

McNeil, J. R.

S. R. Wilson, D. W. Reicher, C. F. Kranenberg, and J. R. McNeil, “Ion beam milling of fused silica for windows fabrication,” Proc. SPIE 1441, 82-85 (1990).
[CrossRef]

S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320-322 (1987).

Nickel, A.

T. Hänsel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with an (x,y,z) linear three-axis system,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.

Paige, C.

C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Software 8, 43-71 (1982).
[CrossRef]

Pareschi, G.

M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).

Poletto, L.

M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).

Reicher, D. W.

S. R. Wilson, D. W. Reicher, C. F. Kranenberg, and J. R. McNeil, “Ion beam milling of fused silica for windows fabrication,” Proc. SPIE 1441, 82-85 (1990).
[CrossRef]

Robert, E. K.

N. A. Lynn, E. K. Robert, and S. L. Timothy, “Surface error correction of a Keck 10 m telescope primary mirror segment by ion figuring,” Proc. SPIE 1531, 195-204 (1991).

Robert, J.

Sala, M.

M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).

Sandri, G.

Saunders, M. A.

C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Software 8, 43-71 (1982).
[CrossRef]

Schindler, A.

L. Aschke, F. Schubert, J. Kegeler, and A. Schindler, “Flatness correction of NZTE mask blank substrates,” Proc. SPIE 4343, 646-653 (2001).
[CrossRef]

T. Hänsel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with an (x,y,z) linear three-axis system,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.

A. Schindler, T. Hänsel, F. Frost, R. Fechner, and G. Seidenkranz, “Ion beam finishing technology for high precision optics production,” in Optical Fabrication and Testing, A.Sawchuk, ed., Vol. 76 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper OTuB5.

Schubert, F.

L. Aschke, F. Schubert, J. Kegeler, and A. Schindler, “Flatness correction of NZTE mask blank substrates,” Proc. SPIE 4343, 646-653 (2001).
[CrossRef]

Seidenkranz, G.

A. Schindler, T. Hänsel, F. Frost, R. Fechner, and G. Seidenkranz, “Ion beam finishing technology for high precision optics production,” in Optical Fabrication and Testing, A.Sawchuk, ed., Vol. 76 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper OTuB5.

Shanbhag, P. M.

Timothy, S. L.

N. A. Lynn, E. K. Robert, and S. L. Timothy, “Surface error correction of a Keck 10 m telescope primary mirror segment by ion figuring,” Proc. SPIE 1531, 195-204 (1991).

Wilson, S. R.

S. R. Wilson, D. W. Reicher, C. F. Kranenberg, and J. R. McNeil, “Ion beam milling of fused silica for windows fabrication,” Proc. SPIE 1441, 82-85 (1990).
[CrossRef]

S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320-322 (1987).

Yu, J.

ACM Trans. Math. Software (1)

C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Software 8, 43-71 (1982).
[CrossRef]

Appl. Opt. (3)

Proc. SPIE (6)

L. Aschke, F. Schubert, J. Kegeler, and A. Schindler, “Flatness correction of NZTE mask blank substrates,” Proc. SPIE 4343, 646-653 (2001).
[CrossRef]

M. Ghigo, P. Conconia, M. Sala, E. Antonello, G. Pareschi, and L. Poletto, “Field corrector for the Ultraviolet Italian Sky Surveyor on the International Space Station (UVISS): ion beam figuring and application of the multilayer filters,” Proc. SPIE 5488, 457-480 (2004).

S. R. Wilson and J. R. McNeil, “Neutral ion beam figuring of large optical surfaces,” Proc. SPIE 818, 320-322 (1987).

S. R. Wilson, D. W. Reicher, C. F. Kranenberg, and J. R. McNeil, “Ion beam milling of fused silica for windows fabrication,” Proc. SPIE 1441, 82-85 (1990).
[CrossRef]

N. A. Lynn, E. K. Robert, and S. L. Timothy, “Surface error correction of a Keck 10 m telescope primary mirror segment by ion figuring,” Proc. SPIE 1531, 195-204 (1991).

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

Other (2)

A. Schindler, T. Hänsel, F. Frost, R. Fechner, and G. Seidenkranz, “Ion beam finishing technology for high precision optics production,” in Optical Fabrication and Testing, A.Sawchuk, ed., Vol. 76 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), paper OTuB5.

T. Hänsel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with an (x,y,z) linear three-axis system,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.

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Figures (10)

Fig. 1
Fig. 1

Sketch of IBF process.

Fig. 2
Fig. 2

Scanning dwell grids according to uniform scanning path (extending half of the ion beam diameter).

Fig. 3
Fig. 3

Ion beam removal function per unit time.

Fig. 4
Fig. 4

Initial surface error maps on (a) the flat surface and (b)  the parabolic surface.

Fig. 5
Fig. 5

Dwell time without extension on (a) the flat surface and (b) the parabolic surface.

Fig. 6
Fig. 6

Residual surface error maps after IBF on (a) the flat surface and (b) the parabolic surface.

Fig. 7
Fig. 7

Initial surface error maps extending an ion beam diameter on (a) the flat surface and (b) the parabolic surface.

Fig. 8
Fig. 8

Path weight factor map according to a uniform scanning path (extending half of the ion beam diameter).

Fig. 9
Fig. 9

Dwell time extending half of the ion beam diameter on (a) the flat surface and (b) the parabolic surface.

Fig. 10
Fig. 10

Residual surface error maps within a factual aperture after IBF on (a) the flat surface and (b) the parabolic surface.

Equations (13)

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R d ( x , y ) = A ( x , y ) * * t c ( x , y ) ,
R d ( x k , y k ) = Z m ( x k , y k ) Z d ( x k , y k ) ,
R a ( x k , y k ) = i = 1 N t A ( x k ξ i , y k η i ) t c ( ξ i , η i ) ,
R a ( x k , y k ) = r a k , R d ( x k , y k ) = r d k , A ( x k ξ i , y k η i ) = a k i , t c ( ξ i , η i ) = t i ,
( r a 1 r a 2 r a N r ) = ( a 11 a 12 a 1 N t a 21 a 22 a 2 N t a N r 1 a N r 2 a N r N t ) ( t 1 t 2 t N t ) .
( r d 1 r d 2 r d N r 0 0 0 ) = ( a 11 a 12 a 1 N t a 21 a 22 a 2 N t a N r 1 a N r 2 a N r N t W 0 0 0 W 0 0 0 W ) ( t 1 t 2 t N t ) ,
{ k = 1 N r [ i = 1 N t ( a k i t i r d k ) ] 2 / N r + W 2 i = 1 N t t i 2 } 1 / 2 .
( r d 1 + γ 0 r d 2 + γ 0 r d N r + γ 0 0 0 0 ) = ( a 11 a 12 a 1 N t a 21 a 22 a 2 N t a N r 1 a N r 2 a N r N t W P 1 S 1 0 0 0 W P 2 S 2 0 0 0 W P N t S N t ) ( t 1 t 2 t N t ) ,
P i = H · j = 1 N t A ( ξ i ξ j , η i η j ) / i = 1 N t j = 1 N t A ( ξ i ξ j , η i η j ) .
S i = [ ( R d + γ 0 ) max ( R d ( ξ i , η i ) + γ 0 ) ] / ( R d + γ 0 ) max .
{ k = 1 N r [ i = 1 N t ( a k i t i r d k γ 0 ) ] 2 / N r + W 2 i = 1 N t P i 2 S i 2 t i 2 } 1 / 2 .
p s t SVD = i = 1 M σ i μ i T ( r d + γ 0 ) σ i 2 + W 2 ν i ,
u N ( x , y ) = n = 0 N H n u ( x , y ) G T x y 0 ;

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