Abstract

Ion beam figuring (IBF) is established for the final precision figuring of optical components. In this deterministic method, the figuring process is represented by a two-dimensional (2D) convolution operation of a constant removal function and the dwell time, where the figuring precision is guaranteed by the stability of the removal function as well as the solution accuracy of the dwell time. However, the current 2D convolution equation cannot factually reflect the IBF process of curved surfaces, which neglects the influence of the projection distortion and the workpiece geometry. Consequently, the current convolution algorithm for the IBF process would influence the solution accuracy for the dwell time and reduce the convergence of the figuring process. In this part, we propose an improved algorithm based on the mathematical modeling of the dynamic removal function in Part A, which provides a more accurate dwell time for IBF of a curved surface. Additionally, simulation analysis and figuring experiments are carried out to verify the feasibility of our proposed algorithm. The final experimental results indicate that the figuring precision and efficiency can be simultaneously improved by this method.

© 2014 Optical Society of America

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References

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  1. R. A. Jones and W. J. Rupp, “Rapid optical fabrication with CCOS,” Proc. SPIE 1333, 34–43 (1990).
    [CrossRef]
  2. Y. Dai, W. Liao, L. Zhou, S. Chen, and X. Xie, “Ion beam figuring of high slope surfaces based on figure error compensation algorithm,” Appl. Opt. 49, 6630–6636 (2010).
    [CrossRef]
  3. L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).
  4. C. Jiao, S. Li, X. Xie, L. Zhou, and W. Duan, “Bayesian principle based dwell time algorithm for ion beam figuring of low gradient mirrors,” J. Mech. Eng. Lab. 45, 253–259 (2009).
    [CrossRef]
  5. T. W. Drueding, T. G. Bifano, and S. C. Fawcett, “Contouring algorithm for ion figuring,” Precis. Eng. 17, 10–21 (1995).
    [CrossRef]
  6. J. F. Wu, Z. W. Lu, H. X. Zhang, and T. S. Wang, “Dwell time algorithm in ion beam figuring,” Appl. Opt. 48, 3930–3937 (2009).
    [CrossRef]
  7. C. Jiao, S. Li, X. Xie, S. Chen, D. Wu, and N. Kang, “Figuring algorithm for high-gradient mirrors with axis-symmetrical removal function,” Appl. Opt. 49, 578–585 (2010).
    [CrossRef]
  8. L. Zhou, “Study on theory and technology in ion beam figuring for optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2008), in Chinese.

2010 (2)

2009 (2)

C. Jiao, S. Li, X. Xie, L. Zhou, and W. Duan, “Bayesian principle based dwell time algorithm for ion beam figuring of low gradient mirrors,” J. Mech. Eng. Lab. 45, 253–259 (2009).
[CrossRef]

J. F. Wu, Z. W. Lu, H. X. Zhang, and T. S. Wang, “Dwell time algorithm in ion beam figuring,” Appl. Opt. 48, 3930–3937 (2009).
[CrossRef]

2001 (1)

L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).

1995 (1)

T. W. Drueding, T. G. Bifano, and S. C. Fawcett, “Contouring algorithm for ion figuring,” Precis. Eng. 17, 10–21 (1995).
[CrossRef]

1990 (1)

R. A. Jones and W. J. Rupp, “Rapid optical fabrication with CCOS,” Proc. SPIE 1333, 34–43 (1990).
[CrossRef]

Bifano, T. G.

T. W. Drueding, T. G. Bifano, and S. C. Fawcett, “Contouring algorithm for ion figuring,” Precis. Eng. 17, 10–21 (1995).
[CrossRef]

Chen, S.

Dai, Y.

Y. Dai, W. Liao, L. Zhou, S. Chen, and X. Xie, “Ion beam figuring of high slope surfaces based on figure error compensation algorithm,” Appl. Opt. 49, 6630–6636 (2010).
[CrossRef]

L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).

Drueding, T. W.

T. W. Drueding, T. G. Bifano, and S. C. Fawcett, “Contouring algorithm for ion figuring,” Precis. Eng. 17, 10–21 (1995).
[CrossRef]

Duan, W.

C. Jiao, S. Li, X. Xie, L. Zhou, and W. Duan, “Bayesian principle based dwell time algorithm for ion beam figuring of low gradient mirrors,” J. Mech. Eng. Lab. 45, 253–259 (2009).
[CrossRef]

Fawcett, S. C.

T. W. Drueding, T. G. Bifano, and S. C. Fawcett, “Contouring algorithm for ion figuring,” Precis. Eng. 17, 10–21 (1995).
[CrossRef]

Jiao, C.

C. Jiao, S. Li, X. Xie, S. Chen, D. Wu, and N. Kang, “Figuring algorithm for high-gradient mirrors with axis-symmetrical removal function,” Appl. Opt. 49, 578–585 (2010).
[CrossRef]

C. Jiao, S. Li, X. Xie, L. Zhou, and W. Duan, “Bayesian principle based dwell time algorithm for ion beam figuring of low gradient mirrors,” J. Mech. Eng. Lab. 45, 253–259 (2009).
[CrossRef]

L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).

Jones, R. A.

R. A. Jones and W. J. Rupp, “Rapid optical fabrication with CCOS,” Proc. SPIE 1333, 34–43 (1990).
[CrossRef]

Kang, N.

Li, S.

C. Jiao, S. Li, X. Xie, S. Chen, D. Wu, and N. Kang, “Figuring algorithm for high-gradient mirrors with axis-symmetrical removal function,” Appl. Opt. 49, 578–585 (2010).
[CrossRef]

C. Jiao, S. Li, X. Xie, L. Zhou, and W. Duan, “Bayesian principle based dwell time algorithm for ion beam figuring of low gradient mirrors,” J. Mech. Eng. Lab. 45, 253–259 (2009).
[CrossRef]

L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).

Liao, W.

Lu, Z. W.

Rupp, W. J.

R. A. Jones and W. J. Rupp, “Rapid optical fabrication with CCOS,” Proc. SPIE 1333, 34–43 (1990).
[CrossRef]

Wang, T. S.

Wu, D.

Wu, J. F.

Xie, X.

Y. Dai, W. Liao, L. Zhou, S. Chen, and X. Xie, “Ion beam figuring of high slope surfaces based on figure error compensation algorithm,” Appl. Opt. 49, 6630–6636 (2010).
[CrossRef]

C. Jiao, S. Li, X. Xie, S. Chen, D. Wu, and N. Kang, “Figuring algorithm for high-gradient mirrors with axis-symmetrical removal function,” Appl. Opt. 49, 578–585 (2010).
[CrossRef]

C. Jiao, S. Li, X. Xie, L. Zhou, and W. Duan, “Bayesian principle based dwell time algorithm for ion beam figuring of low gradient mirrors,” J. Mech. Eng. Lab. 45, 253–259 (2009).
[CrossRef]

L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).

Zhang, H. X.

Zhou, L.

Y. Dai, W. Liao, L. Zhou, S. Chen, and X. Xie, “Ion beam figuring of high slope surfaces based on figure error compensation algorithm,” Appl. Opt. 49, 6630–6636 (2010).
[CrossRef]

C. Jiao, S. Li, X. Xie, L. Zhou, and W. Duan, “Bayesian principle based dwell time algorithm for ion beam figuring of low gradient mirrors,” J. Mech. Eng. Lab. 45, 253–259 (2009).
[CrossRef]

L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).

L. Zhou, “Study on theory and technology in ion beam figuring for optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2008), in Chinese.

Appl. Opt. (3)

J. Mech. Eng. Lab. (1)

C. Jiao, S. Li, X. Xie, L. Zhou, and W. Duan, “Bayesian principle based dwell time algorithm for ion beam figuring of low gradient mirrors,” J. Mech. Eng. Lab. 45, 253–259 (2009).
[CrossRef]

Opt. Precis. Eng. (1)

L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).

Precis. Eng. (1)

T. W. Drueding, T. G. Bifano, and S. C. Fawcett, “Contouring algorithm for ion figuring,” Precis. Eng. 17, 10–21 (1995).
[CrossRef]

Proc. SPIE (1)

R. A. Jones and W. J. Rupp, “Rapid optical fabrication with CCOS,” Proc. SPIE 1333, 34–43 (1990).
[CrossRef]

Other (1)

L. Zhou, “Study on theory and technology in ion beam figuring for optical surfaces,” Ph.D. dissertation (National University of Defense Technology, 2008), in Chinese.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of figuring process: (a) for TIF method and (b) for FIF method.

Fig. 2.
Fig. 2.

Removal function distribution during figuring process: (a) constant removal function during IBF of flat surface, (b) dynamic removal function during TIF, and (c) dynamic removal function during FIF.

Fig. 3.
Fig. 3.

Schematic of moving system position during IBF of curved surface.

Fig. 4.
Fig. 4.

Simulation results by using the current algorithm for dwell time: (a) original surface error, (b) removal function, (c) surface error after edge extension, and (d) calculated dwell time within the actual surface region.

Fig. 5.
Fig. 5.

Simulation results by using improved algorithm for dwell time: (a) calculated dwell time for TIF method, (b) difference between the dwell time of Fig. 4(d) and that of panel (a) of this figure, (c) calculated dwell time for FIF method, and (d) difference between the dwell time of Fig. 4(d) and that of panel (c) of this figure.

Fig. 6.
Fig. 6.

Experimental results of TIF of spherical convex surfaces. (a)–(c) indicate the experimental figuring process by using the current algorithm for dwell time: (a) original surface error of sample A (26.8 nm RMS), (b) experimental result after first iteration (11.0 nm RMS), and (c) final result after sixth iteration (9.0 nm RMS). (d) and (e) indicate the experimental figuring process by using the improved algorithm for dwell time: (d) original surface error of sample B (24.9 nm RMS) and (e) final result after three iterations (1.9 nm RMS). (f) shows the comparison of these two figuring processes.

Fig. 7.
Fig. 7.

Experimental results of FIF of spherical convex surfaces. First row shows the figuring process by using the current algorithm for dwell time: (a) original surface error of sample B1 (5.8 nm RMS), (b) simulation result for this iteration (1.42 nm RMS), and (c) actual result after a run (2.5 nm RMS). Second row shows the figuring process by using improved algorithm for dwell time: (d) original surface error of sample B2 (5.5 nm RMS), (e) simulation result for this iteration (1.43 nm RMS), and (f) actual result after a run (1.7 nm RMS).

Equations (8)

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

E(xm,yn)=i=1Mj=1NR(xmxi,ynyj)T(xi,yj),
emn=E(xm,yn),rmnij=R(xmxi,ynyj),tij=T(xi,yj),
[e11e1Ne21e2NemneMN]=[r1111r111Nr1121r112Nr11ijr11MNr1N11r1N1Nr1N21r1N2Nr1Nijr1NMNr2111r211Nr2121r212Nr21ijr21MNr2N11r2N1Nr2N21r2N2Nr2Nijr2NMNrmn11rmn1Nrmn21rmn2NrmnijrmnMNrMN11eMN1NeMN21eMN2NeMNijeMNMN][t11t1Nt21t2NtijtMN].
ηmn=emnemn,
minη2=minE(x,y)E(x,y)2=min(m=1Mn=1Nemnemn2).
E(xm,yn)=i=1Mj=1NRd(xmxi,ynyj)T(xi,yj),
PQ=(Δx,Δy,Δz,ΔA,ΔB),
v=1t(Δx2+Δy2+Δz2+ΔA2+ΔB2).

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