Abstract

Figuring technology of nonaxisymmetric errors using a spiral path is presented. Based on an approximation of a removal function, a finite-field nonlinear model is deduced from the computer-controlled optics shaping principle. We then present a modified Richardson–Lucy iterative algorithm to deconvolute the dwell time. With a velocity realization method for dwell time on a spiral path, the figuring technology comes into being. Simulations are made to validate these algorithms. Theoretical and simulation studies demonstrate that the figuring technology is a novel method for inexpensive fabrication of precision mirrors.

© 2009 Optical Society of America

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References

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  1. C. J. Jiao, S. Y. Li, X. H. Xie, D. F. Wang, and L. Zhou, “Controllability of removal function in the ion beam figuring process for optics mirrors,” Opt. Technique 34, 651-654 (2008) (in Chinese).
  2. L. N. Allen and R. E. Keim, “An ion figuring system for large optic fabrication,” Proc. SPIE 1168, 33-50(1989).
  3. C. Song, Y. F. Dai, and X. Q. Peng, “Polishing parameters of magnetorheological finishing for high-precision optical surfaces,” Nanotechnol. Precis. Eng. 6, 424-429 (2008) (in Chinese).
  4. L. Yang, “Advanced Technology of Optics Manufacturing" (Science Press China, 2001).
  5. L. Hocheol and Y. Minyang, “Dwell time algorithm for computer-controlled polishing of small axis-symmetrical aspherical lens mold,” Opt. Eng. 40, 1936-1943 (2001).
  6. X. Q. Peng, Y. F. Dai, S. Li, and W. W. You, “Dwell time algorithm for MRF of axis-symmetrical aspherical parts,” J. Natl. Univ. Defense Technol. 26, 89-92 (2004).
  7. C. L. Carnal, C. M. Egert, and K. W. Hylton, “Advanced matrix-based algorithm for ion beam milling of optical components,” Proc. SPIE 1752, 54-62 (1992).
  8. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB (Beijing Publishing House of Electronics Industry, 2005) (in Chinese).
  9. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55-59 (1972).
    [CrossRef]
  10. L. B. Lucy, “An iterative technique for rectification of observed distributions,” Astron. J. 79, 745-754 (1974).
    [CrossRef]
  11. C. J. Jiao, S. Y. Li, and X. H. Xie, “Algorithm for ion beam figuring of low-gradient mirrors,” Appl. Opt. 48, 4090-4096 (2009).
    [CrossRef]
  12. S. Li, C. Jiao, X. Xie, and L. Zhou, “Stitching algorithm for ion beam figuring of optical mirrors,” Science in China Ser. E (to be published).

2009 (1)

2008 (2)

C. J. Jiao, S. Y. Li, X. H. Xie, D. F. Wang, and L. Zhou, “Controllability of removal function in the ion beam figuring process for optics mirrors,” Opt. Technique 34, 651-654 (2008) (in Chinese).

C. Song, Y. F. Dai, and X. Q. Peng, “Polishing parameters of magnetorheological finishing for high-precision optical surfaces,” Nanotechnol. Precis. Eng. 6, 424-429 (2008) (in Chinese).

2004 (1)

X. Q. Peng, Y. F. Dai, S. Li, and W. W. You, “Dwell time algorithm for MRF of axis-symmetrical aspherical parts,” J. Natl. Univ. Defense Technol. 26, 89-92 (2004).

2001 (1)

L. Hocheol and Y. Minyang, “Dwell time algorithm for computer-controlled polishing of small axis-symmetrical aspherical lens mold,” Opt. Eng. 40, 1936-1943 (2001).

1992 (1)

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

1989 (1)

L. N. Allen and R. E. Keim, “An ion figuring system for large optic fabrication,” Proc. SPIE 1168, 33-50(1989).

1974 (1)

L. B. Lucy, “An iterative technique for rectification of observed distributions,” Astron. J. 79, 745-754 (1974).
[CrossRef]

1972 (1)

Allen, L. N.

L. N. Allen and R. E. Keim, “An ion figuring system for large optic fabrication,” Proc. SPIE 1168, 33-50(1989).

Carnal, C. L.

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

Dai, Y. F.

C. Song, Y. F. Dai, and X. Q. Peng, “Polishing parameters of magnetorheological finishing for high-precision optical surfaces,” Nanotechnol. Precis. Eng. 6, 424-429 (2008) (in Chinese).

X. Q. Peng, Y. F. Dai, S. Li, and W. W. You, “Dwell time algorithm for MRF of axis-symmetrical aspherical parts,” J. Natl. Univ. Defense Technol. 26, 89-92 (2004).

Eddins, S. L.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB (Beijing Publishing House of Electronics Industry, 2005) (in Chinese).

Egert, C. M.

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

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB (Beijing Publishing House of Electronics Industry, 2005) (in Chinese).

Hocheol , L.

L. Hocheol and Y. Minyang, “Dwell time algorithm for computer-controlled polishing of small axis-symmetrical aspherical lens mold,” Opt. Eng. 40, 1936-1943 (2001).

Hylton, K. W.

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

Jiao, C.

S. Li, C. Jiao, X. Xie, and L. Zhou, “Stitching algorithm for ion beam figuring of optical mirrors,” Science in China Ser. E (to be published).

Jiao, C. J.

C. J. Jiao, S. Y. Li, and X. H. Xie, “Algorithm for ion beam figuring of low-gradient mirrors,” Appl. Opt. 48, 4090-4096 (2009).
[CrossRef]

C. J. Jiao, S. Y. Li, X. H. Xie, D. F. Wang, and L. Zhou, “Controllability of removal function in the ion beam figuring process for optics mirrors,” Opt. Technique 34, 651-654 (2008) (in Chinese).

Keim, R. E.

L. N. Allen and R. E. Keim, “An ion figuring system for large optic fabrication,” Proc. SPIE 1168, 33-50(1989).

Li, S.

X. Q. Peng, Y. F. Dai, S. Li, and W. W. You, “Dwell time algorithm for MRF of axis-symmetrical aspherical parts,” J. Natl. Univ. Defense Technol. 26, 89-92 (2004).

S. Li, C. Jiao, X. Xie, and L. Zhou, “Stitching algorithm for ion beam figuring of optical mirrors,” Science in China Ser. E (to be published).

Li, S. Y.

C. J. Jiao, S. Y. Li, and X. H. Xie, “Algorithm for ion beam figuring of low-gradient mirrors,” Appl. Opt. 48, 4090-4096 (2009).
[CrossRef]

C. J. Jiao, S. Y. Li, X. H. Xie, D. F. Wang, and L. Zhou, “Controllability of removal function in the ion beam figuring process for optics mirrors,” Opt. Technique 34, 651-654 (2008) (in Chinese).

Lucy, L. B.

L. B. Lucy, “An iterative technique for rectification of observed distributions,” Astron. J. 79, 745-754 (1974).
[CrossRef]

Minyang, Y.

L. Hocheol and Y. Minyang, “Dwell time algorithm for computer-controlled polishing of small axis-symmetrical aspherical lens mold,” Opt. Eng. 40, 1936-1943 (2001).

Peng, X. Q.

C. Song, Y. F. Dai, and X. Q. Peng, “Polishing parameters of magnetorheological finishing for high-precision optical surfaces,” Nanotechnol. Precis. Eng. 6, 424-429 (2008) (in Chinese).

X. Q. Peng, Y. F. Dai, S. Li, and W. W. You, “Dwell time algorithm for MRF of axis-symmetrical aspherical parts,” J. Natl. Univ. Defense Technol. 26, 89-92 (2004).

Richardson, W. H.

Song, C.

C. Song, Y. F. Dai, and X. Q. Peng, “Polishing parameters of magnetorheological finishing for high-precision optical surfaces,” Nanotechnol. Precis. Eng. 6, 424-429 (2008) (in Chinese).

Wang, D. F.

C. J. Jiao, S. Y. Li, X. H. Xie, D. F. Wang, and L. Zhou, “Controllability of removal function in the ion beam figuring process for optics mirrors,” Opt. Technique 34, 651-654 (2008) (in Chinese).

Woods, R. E.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB (Beijing Publishing House of Electronics Industry, 2005) (in Chinese).

Xie, X.

S. Li, C. Jiao, X. Xie, and L. Zhou, “Stitching algorithm for ion beam figuring of optical mirrors,” Science in China Ser. E (to be published).

Xie, X. H.

C. J. Jiao, S. Y. Li, and X. H. Xie, “Algorithm for ion beam figuring of low-gradient mirrors,” Appl. Opt. 48, 4090-4096 (2009).
[CrossRef]

C. J. Jiao, S. Y. Li, X. H. Xie, D. F. Wang, and L. Zhou, “Controllability of removal function in the ion beam figuring process for optics mirrors,” Opt. Technique 34, 651-654 (2008) (in Chinese).

Yang, L.

L. Yang, “Advanced Technology of Optics Manufacturing" (Science Press China, 2001).

You, W. W.

X. Q. Peng, Y. F. Dai, S. Li, and W. W. You, “Dwell time algorithm for MRF of axis-symmetrical aspherical parts,” J. Natl. Univ. Defense Technol. 26, 89-92 (2004).

Zhou, L.

C. J. Jiao, S. Y. Li, X. H. Xie, D. F. Wang, and L. Zhou, “Controllability of removal function in the ion beam figuring process for optics mirrors,” Opt. Technique 34, 651-654 (2008) (in Chinese).

S. Li, C. Jiao, X. Xie, and L. Zhou, “Stitching algorithm for ion beam figuring of optical mirrors,” Science in China Ser. E (to be published).

Appl. Opt. (1)

Astron. J. (1)

L. B. Lucy, “An iterative technique for rectification of observed distributions,” Astron. J. 79, 745-754 (1974).
[CrossRef]

J. Natl. Univ. Defense Technol. (1)

X. Q. Peng, Y. F. Dai, S. Li, and W. W. You, “Dwell time algorithm for MRF of axis-symmetrical aspherical parts,” J. Natl. Univ. Defense Technol. 26, 89-92 (2004).

J. Opt. Soc. Am. (1)

Nanotechnol. Precis. Eng. (1)

C. Song, Y. F. Dai, and X. Q. Peng, “Polishing parameters of magnetorheological finishing for high-precision optical surfaces,” Nanotechnol. Precis. Eng. 6, 424-429 (2008) (in Chinese).

Opt. Eng. (1)

L. Hocheol and Y. Minyang, “Dwell time algorithm for computer-controlled polishing of small axis-symmetrical aspherical lens mold,” Opt. Eng. 40, 1936-1943 (2001).

Opt. Technique (1)

C. J. Jiao, S. Y. Li, X. H. Xie, D. F. Wang, and L. Zhou, “Controllability of removal function in the ion beam figuring process for optics mirrors,” Opt. Technique 34, 651-654 (2008) (in Chinese).

Proc. SPIE (2)

L. N. Allen and R. E. Keim, “An ion figuring system for large optic fabrication,” Proc. SPIE 1168, 33-50(1989).

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

Science in China Ser. E (1)

S. Li, C. Jiao, X. Xie, and L. Zhou, “Stitching algorithm for ion beam figuring of optical mirrors,” Science in China Ser. E (to be published).

Other (2)

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB (Beijing Publishing House of Electronics Industry, 2005) (in Chinese).

L. Yang, “Advanced Technology of Optics Manufacturing" (Science Press China, 2001).

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Figures (8)

Fig. 1
Fig. 1

Conventional scanning modes in the CCOS-based figuring method: (a) linear and (b) spiral.

Fig. 2
Fig. 2

Schematic illustration of the spiral figuring process with the nonaxisymmetric removal function.

Fig. 3
Fig. 3

Theoretical diagram for a nonlinear model of the spiral figuring process: (a) sector division and (b) transformation diagram of the removal function.

Fig. 4
Fig. 4

Schematic illustration of dwell time realization.

Fig. 5
Fig. 5

Determination of sector number N: (a) initial surface error, (b) removal function map, (c) relative error curve deduced from the angle disturbance.

Fig. 6
Fig. 6

Simulation result from a finite-field nonlinear model: (a) rms convergence curve, (b) deconvoluted dwell time with N = 36, (c) predicted residual error with N = 36.

Fig. 7
Fig. 7

Predicted residual error in the linear scanning process.

Fig. 8
Fig. 8

Predicted residual error with the CEH model on a spiral path.

Equations (11)

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R ( θ ) = rotz ( R , θ ) ,
R i = rotz ( R , β i ) .
E i = M i E , T i = M i T ,
E = i = 1 N R i T i ,
T k + 1 = T k ( R ( x , y ) R d x d y E R T k ) .
T k + 1 = T k ( i = 1 N R i ( x , y ) R d x d y E i i = 1 N R i T i k ) .
ρ = k ( θ ) .
t ( ρ , θ ) = T ( ρ , θ ) ρ p ( θ ) d θ ,
v ( ρ , θ ) = ( d θ ) 2 + ( k ( θ ) d θ ) 2 t ( ρ , θ ) .
v ( ρ , θ ) 1 T ( ρ , θ ) ρ p ( θ ) .
E m = R t T s .

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