Abstract

Ion beam figuring (IBF) is established for the final precision figuring of high-performance optical components, where the figuring accuracy is guaranteed by the stability of the removal function and the solution accuracy of the dwell time. In this deterministic method, the figuring process can be represented by a two-dimensional (2D) convolution operation of a constant removal function and the dwell time. However, we have found that the current 2D convolution operation cannot factually describe the IBF process of curved surfaces, which neglects the influences of the projection distortion and the workpiece geometry on the removal function. Consequently, the current 2D convolution algorithm would influence the solution accuracy for the dwell time and reduce the convergence of the figuring process. In this part, based on the material removal characteristics of IBF, a mathematical model of the removal function is developed theoretically and verified experimentally. Research results show that the removal function during IBF of a curved surface is actually a dynamic function in the 2D convolution algorithm. The mathematical modeling of the dynamic removal function provides theoretical foundations for our proposed new algorithm in the next part, and final verification experiments indicate that this algorithm can effectively improve the accuracy of the dwell time solution for the IBF of curved surfaces.

© 2014 Optical Society of America

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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. L. Zhou, Y. Dai, X. Xie, C. Jiao, and S. Li, “Machining reachability in ion beam figuring,” Opt. Precis. Eng. 15, 160–166 (2001).
  3. 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]
  4. T. W. Drueding, T. G. Bifano, and S. C. Fawcett, “Contouring algorithm for ion figuring,” Precis. Eng. 17, 10–21 (1995).
    [CrossRef]
  5. M. Weiser, “Ion beam figuring for lithography optics,” Nucl. Instrum. Methods Phys. Res. B 267, 1390–1393 (2009).
    [CrossRef]
  6. W. Liao, Y. Dai, X. Xie, and L. Zhou, “Morphology evolution of fused silica surface during ion beam figuring of high-slope optical components,” Appl. Opt. 52, 3719–3725 (2013).
    [CrossRef]
  7. L. N. Allen and H. W. Romig, “Demonstration of an ion figuring process,” Proc. SPIE 1333, 22–23 (1990).
    [CrossRef]
  8. T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
    [CrossRef]
  9. 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]
  10. T. Haensel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with a⌝ x; y; z⌊ linear three-axes system,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.
  11. 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]
  12. P. Sigmund, “Theory of sputtering. I. Sputtering yield of amorphous and polycrystalline targets,” Phys. Rev. 184, 383–416, (1969).
    [CrossRef]
  13. R. M. Bradley and J. M. E. Harper, “Theory of ripple topography induced by ion bombardment,” J. Vac. Sci. Technol. A 6, 2390–2395 (1988).
    [CrossRef]
  14. J. Biersack and W. Eckstein, “Sputtering studies with the Monte Carlo program TRIM.SP,” Appl. Phys. A 34, 73–94 (1984).
    [CrossRef]

2013 (1)

2010 (3)

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (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]

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]

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]

M. Weiser, “Ion beam figuring for lithography optics,” Nucl. Instrum. Methods Phys. Res. B 267, 1390–1393 (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 (2)

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

L. N. Allen and H. W. Romig, “Demonstration of an ion figuring process,” Proc. SPIE 1333, 22–23 (1990).
[CrossRef]

1988 (1)

R. M. Bradley and J. M. E. Harper, “Theory of ripple topography induced by ion bombardment,” J. Vac. Sci. Technol. A 6, 2390–2395 (1988).
[CrossRef]

1984 (1)

J. Biersack and W. Eckstein, “Sputtering studies with the Monte Carlo program TRIM.SP,” Appl. Phys. A 34, 73–94 (1984).
[CrossRef]

1969 (1)

P. Sigmund, “Theory of sputtering. I. Sputtering yield of amorphous and polycrystalline targets,” Phys. Rev. 184, 383–416, (1969).
[CrossRef]

Allen, L. N.

L. N. Allen and H. W. Romig, “Demonstration of an ion figuring process,” Proc. SPIE 1333, 22–23 (1990).
[CrossRef]

Arnold, T.

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[CrossRef]

Biersack, J.

J. Biersack and W. Eckstein, “Sputtering studies with the Monte Carlo program TRIM.SP,” Appl. Phys. A 34, 73–94 (1984).
[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]

Böhm, G.

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[CrossRef]

Bradley, R. M.

R. M. Bradley and J. M. E. Harper, “Theory of ripple topography induced by ion bombardment,” J. Vac. Sci. Technol. A 6, 2390–2395 (1988).
[CrossRef]

Chen, S.

Dai, Y.

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]

Eckstein, W.

J. Biersack and W. Eckstein, “Sputtering studies with the Monte Carlo program TRIM.SP,” Appl. Phys. A 34, 73–94 (1984).
[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]

Fechner, R.

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[CrossRef]

Frost, F.

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[CrossRef]

Haensel, T.

T. Haensel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with a⌝ x; y; z⌊ linear three-axes system,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.

Hänsel, T.

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[CrossRef]

Harper, J. M. E.

R. M. Bradley and J. M. E. Harper, “Theory of ripple topography induced by ion bombardment,” J. Vac. Sci. Technol. A 6, 2390–2395 (1988).
[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.

Meister, J.

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[CrossRef]

Nickel, A.

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[CrossRef]

T. Haensel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with a⌝ x; y; z⌊ linear three-axes system,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.

Romig, H. W.

L. N. Allen and H. W. Romig, “Demonstration of an ion figuring process,” Proc. SPIE 1333, 22–23 (1990).
[CrossRef]

Rupp, W. J.

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

Schindler, A.

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[CrossRef]

T. Haensel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with a⌝ x; y; z⌊ linear three-axes system,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper JWD6.

Sigmund, P.

P. Sigmund, “Theory of sputtering. I. Sputtering yield of amorphous and polycrystalline targets,” Phys. Rev. 184, 383–416, (1969).
[CrossRef]

Weiser, M.

M. Weiser, “Ion beam figuring for lithography optics,” Nucl. Instrum. Methods Phys. Res. B 267, 1390–1393 (2009).
[CrossRef]

Wu, D.

Xie, X.

Zhou, L.

W. Liao, Y. Dai, X. Xie, and L. Zhou, “Morphology evolution of fused silica surface during ion beam figuring of high-slope optical components,” Appl. Opt. 52, 3719–3725 (2013).
[CrossRef]

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).

Appl. Opt. (3)

Appl. Phys. A (1)

J. Biersack and W. Eckstein, “Sputtering studies with the Monte Carlo program TRIM.SP,” Appl. Phys. A 34, 73–94 (1984).
[CrossRef]

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]

J. Vac. Sci. Technol. A (1)

R. M. Bradley and J. M. E. Harper, “Theory of ripple topography induced by ion bombardment,” J. Vac. Sci. Technol. A 6, 2390–2395 (1988).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B (1)

M. Weiser, “Ion beam figuring for lithography optics,” Nucl. Instrum. Methods Phys. Res. B 267, 1390–1393 (2009).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. Sect. A (1)

T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Methods Phys. Res. Sect. A 616, 147–156 (2010).
[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).

Phys. Rev. (1)

P. Sigmund, “Theory of sputtering. I. Sputtering yield of amorphous and polycrystalline targets,” Phys. Rev. 184, 383–416, (1969).
[CrossRef]

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 (2)

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

L. N. Allen and H. W. Romig, “Demonstration of an ion figuring process,” Proc. SPIE 1333, 22–23 (1990).
[CrossRef]

Other (1)

T. Haensel, A. Nickel, and A. Schindler, “Ion beam figuring of strongly curved surfaces with a⌝ x; y; z⌊ linear three-axes system,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/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.

Figuring method of IBF: (a) TIF method and (b) FIF method.

Fig. 2.
Fig. 2.

Schematic sketches of the removal function at different dwell points: (a) Removal function distribution during TIF of a curved surface, (b) Removal function distribution during FIF of a curved surface, and (c) Constant removal function during IBF of a flat surface.

Fig. 3.
Fig. 3.

Schematic of ion beam bombarding an optical surface.

Fig. 4.
Fig. 4.

Schematic illustration of energy deposition.

Fig. 5.
Fig. 5.

Schematic illustration of beam-current density distribution.

Fig. 6.
Fig. 6.

Experimental results of the removal function: (a) beam diameter and (b) peak removal rate variation with target distance.

Fig. 7.
Fig. 7.

Experimental results of removal function variation with incidence angle.

Fig. 8.
Fig. 8.

Experimental results of normalized peak removal rate variation with incidence angle.

Fig. 9.
Fig. 9.

Removal function profile at different incidence angle from the images in Fig. 7 and simulation results: (a) longitudinal profile and (b) transverse profile.

Fig. 10.
Fig. 10.

Removal function distribution at various dwell points. First row: experimental results of (a) TIF method and (b) FIF method. Second row: simulation results of (c) TIF method and (d) FIF method.

Equations (24)

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

E(x,y)=R(x,y)T(x,y).
E(x,y,z)=ε(2π)3/2αβ2exp((z+ρ)22α2x2+y22β2),
E(x,y,z)=ε(2π)3/2αβ2×exp((xsinθ+zcosθ+ρ)22α2(xcosθzsinθ)2+y22β2).
J(x,y,z)=J02πσ2exp(x2+y22σ2),
σ=σPz3tanγ,
[X,Y,Z,1]=[PB][PC][PT][x,y,z,1],
[PB]=[cos(φB)0sin(φB)00100sin(φB)0cos(φB)00001],
[PC]=[cos(φC)sin(φC)00sin(φC)cos(φC)0000100001],
[PT]=[100X0010Y0001Z00001],
φB=arcsin(uZp),
φC=arctan(uYp/uXp),
{x=cosφB[(XX0)cosφC+(YY0)sinφC](ZZ0)sinφBy=(YY0)cosφC(XX0)sinφCz=sinφB[(XX0)cosφC+(YY0)sinφC](ZZ0)cosφB.
J(X,Y,Z)=J02πσ2exp({cosφB[(XX0)cosφC+(YY0)sinφC](ZZ0)sinφB}22σ2[(YY0)cosφC(XX0)sinφC]22σ2).
V(X,Y,Z)=ΛAJ(X,Y,Z)E(x,y,z)dA,
J(X,Y,Z)=J(X,Y,Z)cosθzysinθ1+ZX2+ZY2.
V(X,Y,Z)=ΛεJ(X,Y,Z)(2π)3/2αβ2A(cosθzysinθ)×exp((xsinθ+zcosθ+ρ)22α2(xcosθzsinθ)2+y22β2)dxdy.
h(x,y)=x2Rxy2Ry.
V(X,Y,Z)ΛεJ(X,Y,Z)(2π)3/2αβ2Aexp(ρsinθα2x)×exp(y22β2)exp((sin2θ2α2+cos2θ2β2)x2)×(cosθxRxsinθcosθ(x2Rx+y2Ry)×(xsin2θ2β2+ρcosθ2α2xsin2θ2α2))dxdy.
V(X,Y,Z)ΛεJ(X,Y,Z)2π(α2cos2θ+β2sin2θ)×(cosθcxρRxcyρRy)×exp(ρ2cos2θ2(α2cos2θ+β2sin2θ)),
V(X,Y,Z)ΛεJ(X,Y,Z)cosθ2π(α2cos2θ+β2sin2θ)×exp(ρ2cos2θ2(α2cos2θ+β2sin2θ)).
V(X,Y,Z)VFexp((XX0)2+(YY0)22σ2).
VF=J0Λεcosθ×exp(ρ2cos2θ2(α2cos2θ+β2sin2θ))2πσ22π(α2cos2θ+β2sin2θ),
V(X,Y,Z)VFexp({cosφB[(XX0)cosφC+(YY0)sinφC](ZZ0)sinφB}22σ2[(YY0)cosφC(XX0)sinφC]22σ2).
RV=αcosθα2cos2θ+β2sin2θ×exp(ρ2cos2θ2(α2cos2θ+β2sin2θ)+ρ22α2),

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