Mathematical modeling and application of removal functions during deterministic ion beam figuring of optical surfaces. Part 1: Mathematical modeling Wenlin Liao,1,2 Yifan Dai,1,2,* Xuhui Xie,1,2 and Lin Zhou1,2 1

College of Mechatronics and Automation, National University of Defense Technology, Changsha 410073, China 2

Hu’nan Key Laboratory of Ultraprecision Machining Technology, Changsha 410073, China *Corresponding author: [email protected] Received 6 March 2014; revised 27 May 2014; accepted 27 May 2014; posted 2 June 2014 (Doc. ID 207505); published 30 June 2014

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 OCIS codes: (220.0220) Optical design and fabrication; (220.4610) Optical fabrication; (220.5450) Polishing. http://dx.doi.org/10.1364/AO.53.004266

1. Introduction

With the development of modern optical systems, the surface accuracy of some key optical components has become a critical factor for the optical performance of a whole system. Nowadays, deterministic figuring technologies are developed to meet the growing demands of such high-precision optical components. As a deterministic figuring method, ion beam figuring (IBF) is highly deterministic, highly stable, and noncontact, which would make it more advantageous 1559-128X/14/194266-09$15.00/0 © 2014 Optical Society of America 4266

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over conventional figuring technologies [1–4]. Problems that exist in conventional figuring processes, such as the edge effect, tool wear, and pressure load, disappear naturally [3,4]. All of these advantages distinguish IBF as the final procedure for highprecision optics manufacture. The typical application of IBF, which can sufficiently embody the machining capability of IBF, is for the improvement of the surface accuracy up to sub-nanometer magnitude in the field of lithography optics fabrication [5,6]. According to the computer-controlled optical surfacing (CCOS) theory, the figuring method always assumes a constant removal function, so the process is represented by a two-dimensional (2D) convolution

equation, and the calculation of the dwell time for IBF process is a deconvolution operation [4]. The CCOS principle described here is applicable to flat optics without additional processing. However, for IBF of curved optics, the surface error Ex; y; z is projected from a three-dimensional (3D) Cartesian frame to a 2D frame, resulting in Ex; y for the deconvolution operation. Furthermore, the removal function is influenced by projection distortion and workpiece profile, exhibiting a dynamic characteristic. Consequently, the current convolution algorithm cannot factually reflect the figuring process, which directly influences the solution accuracy for the dwell time and finally reduces the convergence of the figuring process. All of these problems should be considered seriously during IBF of curved surfaces. During the last decade, many research institutes have developed their own IBF devices, including representative five-axis IBF (FIF) systems developed by Kodak [7], IOM [8], and GFKD [9]. Besides these, IOM [8,10] and GFKD [11] have built low-cost three-axis IBF (TIF) systems and obtained successful processing results by using compensation methods. However, to further improve the surface accuracy, especially for the lithography optics, a more accurate algorithm for the IBF process is required, but no clear insight of this research exists so far. In this part, based on the material removal characteristics of IBF, mathematical models of removal functions for FIF and TIF methods are developed theoretically and verified experimentally, respectively. The research results indicate that the removal function is a dynamic function in the 2D convolution equation, and the current algorithm for the dwell time should be improved to figure higher-precision optical surfaces. With the help of the mathematical modeling of the removal function, we will propose a new algorithm to improve the accuracy of the dwell time solution in the next part. Final experiments validate that the figuring precision and efficiency can be simultaneously improved by our proposed algorithm. 2. Problem Analysis

According to the CCOS principle, the material removal Ex; y is a convolution of the removal function Rx; y and the dwell time Tx; y, given as follows [2–4]: Ex; y  Rx; y ⊗⊗ Tx; y:

(1)

Based on Eq. (1), the accurate solution of the dwell time relies on the assumption that the measured figure error Ex; y contains true information of the expected material removal as well as the related position of every point on the optical surface [11]. Besides this, the figuring method described here assumes a constant removal function during the whole process, and no projection distortion exists [11]. Therefore, a deconvolution operation is involved to calculate the dwell time, which will determine the raster scanning velocities of the ion beam on the

Fig. 1. Figuring method of IBF: (a) TIF method and (b) FIF method.

target surfaces to correct the surface error. For IBF of a flat surface, since Ex; y is the true surface error at the corresponding point (x; y), and Rx; y keeps stable during the whole process, both of these are able to assure an accurate solution of the dwell time. However, there are many issues that need to be addressed before figuring curved surfaces. Figure 1 shows the frequently used figuring methods of IBF. From the above mentioned figuring methods, we know that the incidence ion beam is kept parallel to the optical axis during the TIF process, while the ion beam perpendicularly bombards the optical surface throughout the FIF process. Obviously, during the TIF process, the incidence angle and beam-current density are influenced by the workpiece profile, and these different process conditions at various dwell points would lead to inconsistency of the removal function. In addition, although a fixed beam is held perpendicularly to the optical surface in the FIF method, maintaining a constant ion beam in 3D space, the projection distortion of the removal function Rx; y in the 2D convolution equation should be taken into account. Figure 2 gives schematic sketches of the removal function distribution at various dwell points. We can easily find that the removal function should be a dynamic function in the 2D convolution equation for these two IBF methods. If the dwell time is still calculated with a constant removal function [Fig. 2(c)], the solution accuracy of the dwell time would be reduced. Therefore, the current algorithm should be improved based on the dynamic removal function. However, it is tedious and even impossible for us to attempt to obtain all the dynamic removal functions through experimental methods. To address this problem, first we should develop a mathematical model of the removal function though theoretical methods. 3. Mathematical Modeling

Figure 3 illustrates the three reference systems: O-XYZ for the workpiece coordinate, P-xyz for the local surface profile, and P-x0 y0 z0 for the ion energy distribution. According to the TIF and FIF methods, 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS

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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.

when the ion beam bombards an optical surface at point P along the −Pz0 direction, the defined planes zPx; POZ, and z0 Px0 are the same plane in the geometrical space. Where P is the dwell point, Pz and Px are the corresponding exterior normal and tangent of the surface profile at point P. In addition, the y and y0 axes in the coordinate systems are determined by the right-hand rule, and the rotation angle θ between the z0 and z axes is the ion incidence angle. A.

Ion Energy Distribution

Based on Sigmund sputtering theory, the material removal from a point Mx; y; z on an arbitrary surface z  hx; y is proportional to the deposited energy [12], where the energy deposition of incident ions follows the Gaussian distribution. In Figs. 3 and 4, when an ion hits the surface at point P along the −z0 axis, the average energy deposited at point Mx; y; z on the surface can be written as [12]

E⊥ x; y; z 

0

0

ε

where ε is the total energy deposited; ρ is the average depth of ion incidence; and α and β are the Gaussian parameters of the distribution parallel and perpendicular to the beam direction, respectively. Generally, ρ, α and β are comparable at the nanometer scale. However, since the local surface profile is defined on another coordinate system, P-xyz, and there is a rotation angle (incidence angle θ) between these two coordinate systems (Fig. 4), coordination transformation is needed. The energy deposited at point M should be derived again, and the spatial distribution of energy deposited under the P-xyz coordinate system can be given as

  x sin θ  z cos θ  ρ2 x cos θ − z sin θ2  y2 × exp − − : 2α2 2β2 2π3∕2 αβ2 ε

Fig. 3. Schematic of ion beam bombarding an optical surface. 4268

  z0  ρ2 x02  y02 Ex ; y ; z   exp − − ; 2α2 2β2 2π3∕2 αβ2 (2) 0

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Fig. 4. Schematic illustration of energy deposition.

(3)

2

Fig. 5. Schematic distribution.

illustration

of

beam-current

cosφB  0 sinφB  6 0 1 0 PB   6 4 − sinφB  0 cosφB  0 0 0 2 cosφC  − sinφC  0 6 sinφC  cosφC  0 PC   6 40 0 1 0 0 0 2 3 1 0 0 X0 6 0 1 0 Y0 7 7 PT   6 4 0 0 1 Z0 5; 0 0 0 1

density

3 0 07 7; 05 1 3 0 07 7; 05 1

φB  arcsinuZp ; B.

Beam-Current Density Distribution

During the IBF process, a fixed-current beam is generated by a Kaufman-type ion source to provide a rotationally symmetric Gaussian ion beam, where the beam-current density distribution can be given as Jx0 ; y0 ; z0 

  02 J0 x  y02 ;  exp − 2πσ 2 2σ 2

(4)

where J 0 is the beam current and σ is the beam diameter parameter. Since ions are limited by space charge, the angular divergence effect would occur when the ion beam is extracted from the plasma source to the target. Along the −z0 direction, the current density will decrease and the beam diameter d  6σ will increase. The parameter σ can be given as

JX; Y; Z 

u  uXp ; uYp ; uZp  is a unit vector that represents the ion incidence direction at the dwell point P. Therefore, we can obtain the transformation matrix x0 ; y0 ; z0 ; 10  PT −1 PC −1 PB −1 X; Y; Z; 10 , and the point MX; Y; Z under the P-x0 y0 z0 system can be expressed as follows: 8 0 x  cos φB X − X 0  cos φC  Y − Y 0  sin φC  > > > > − Z − Z0  sin φB < y0  Y − Y 0  cos φC − X − X 0  sin φC : 7 > 0 > z  sin φ X − X  cos φ  Y − Y  sin φ  > B 0 C 0 C > : − Z − Z0  cos φB Substituting Eqs. (7) and (5) into Eq. (4), we can give the current density distribution under the O-XYZ coordinate system:

 J0 fcos φB X − X 0  cos φC  Y − Y 0  sin φC  − Z − Z0  sin φB g2 exp − 2πσ 2 2σ 2  Y − Y 0  cos φC − X − X 0  sin φC 2 − : 2σ 2

σ  σP −

z0 tan γ; 3

(5)

where dP  6σ P is the beam diameter of the removal function at the dwell point PX 0 ; Y 0 ; Z0 , and γ is the half-angle of divergence (Fig. 5). Suppose the coordinates of point M are (X; Y; Z) and (x0 ; y0 ; z0 ) under the workpiece reference system and the ion energy distribution systems, respectively. According to the definition of the coordinate systems, the coordinate relationship can be described with the following expression: X; Y; Z; 10  PB PC PT x0 ; y0 ; z0 ; 10 ; where

φC  arctanuYp ∕uXp ;

(6)

C.

(8)

Mathematical Modeling

In Section 3.A, we have shown that the energy deposition parameters ρ; α, and β are comparable at the nanometer scale, which means that the sputtering effect of a single ion is very limited. Consequently, we can assume that only the ions bombarding the optical surface within the microscale area A (Fig. 3) can contribute to the material removal at point MX; Y; Z, and the current density JX; Y; Z within this microscale area is constant. According to Sigmund’s sputtering theory, the erosion rate at the point M is expressed as [12] Z VX; Y; Z  Λ

A

J ⊥ X; Y; ZE⊥ x; y; zdA;

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(9) 4269

where Λ is a constant of proportionality related to the erosion rate and the deposited energy; J ⊥ and E⊥ , respectively, define the beam-current density and deposited energy when the ion beam perpendicularly bombards the optical surface; J ⊥ dA is the equivalent number of ions bombarding on the area dA; and dA  1  z2x  z2y 1∕2 dxdy. The local current density J ⊥ can be represented by cos θ − zy sin θ J ⊥ X; Y; Z  JX; Y; Z q : 1  Z2X  Z2Y

(10)

Substituting Eqs. (10) and (3) into Eq. (9), we obtain ZZ ΛεJX; Y; Z VX; Y; Z  cos θ − zy sin θ 2π3∕2 αβ2 A  x sin θ  z cos θ  ρ2 × exp − 2α2  x cos θ − z sin θ2  y2 dxdy: (11) − 2β2 Since the radii of principal curvature Rx and Ry at point P are much larger than a, the surface profile z  hx; y of the area A varies slowly enough that the surface profile can be approximated as [13] hx; y  −

x2 y2 − : Rx Ry

(12)

Combining Eq. (12) with Eq. (11) and working to first order in z, Eq. (11) becomes   ZZ ΛεJX; Y; Z ρ sin θ exp − x VX; Y; Z ≈ α2 2π3∕2 αβ2 A    2    y2 sin θ cos2 θ 2 x  × exp − 2 exp − 2α2 2β2 2β   2  x x y2 sin θ − cos θ  × cos θ − Rx Ry Rx   x sin 2θ ρ cos θ x sin 2θ ×  − dxdy: 2β2 2α2 2α2 (13)

ΛεJX; Y; Z VX; Y; Z ≈ p 2πα2 cos2 θ  β2 sin2 θ   ρ ρ × cos θ − cx − cy Rx Ry   2 ρ cos2 θ × exp − ; (14) 2α2 cos2 θ  β2 sin2 θ where the value of ρ is in nanometers and not comparable in magnitude with Rx and Ry . cx and cy are finite parameters. Hence, cx ρ∕Rx and cy ρ∕Ry are infinitesimals and can be neglected [11]. As a result, Eq. (14) is approximated as ΛεJX; Y; Z cos θ VX; Y; Z ≈ p 2πα2 cos2 θ  β2 sin2 θ   ρ2 cos2 θ × exp − : 2α2 cos2 θ  β2 sin2 θ

Since the ion beam is kept parallel to the optical axis (along the −Z direction) during the whole TIF process, the coordinate transformation angle at point PX; Y; Z can be given by φB  0. Combining these conditions with Eqs. (8) and (15), we can obtain the mathematical model of the removal function during the TIF when the ion beam dwells at point PX 0 ; Y 0 ; Z0    X − X 0 2  Y − Y 0 2 VX; Y; Z ≈ V F exp − . 2σ 2 (16) Where V F is the peak removal rate,  VF 

2 2θ − 2α2 cosρ 2 cos θβ2 sin2 θ

J 0 Λε cos θ × exp p 2πσ 2 2πα2 cos2 θ  β2 sin2 θ

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 ;

(17)

Whereas the ion beam vertically bombards the optical surface throughout the FIF process, we can know φB  arcos1  Z2X  Z2Y −1∕2  at every dwell point. Similarly, the mathematical model of removal function for FIF can be given by

 fcos φB X − X 0  cos φC  Y − Y 0  sin φC  − Z − Z0  sin φB g2 VX; Y; Z ≈ V F exp − 2σ 2  Y − Y 0  cos φC − X − X 0  sin φC 2 : − 2σ 2

The integral in Eq. (13) is quite difficult to evaluate. Following the approach derived by Bradley and Harper (BH model) [13], we straightforwardly give the calculation result:

(15)

(18)

In Eqs. (17) and (18), the incidence angle θ at point MX; Y; Z is θX; Y; Z  arccos juP · uM j∕juP jjuM j; uP and uM denote the normal vectors at the points P and M, respectively.

From the above theoretical analysis, it can be summarized that the removal rate is proportional to the sputtered material of a single incidence ion and the local beam current density. For the former factor, the influence of the workpiece profile can be neglected; it is mainly determined by energy deposition parameters, incidence angle, and target material. However, the latter has an inseparable relationship with the figuring method and workpiece profile, resulting in the change of the removal function at various dwell points. 4. Simulation and Experiment

For the final application of mathematical models, first we should investigate the validity of the above theoretical analyses through simulation and experiment. In this section, all figuring experiments are performed in our self-developed IBF systems under the bombardment of Ar ions. Within the experiments, the processing conditions are fixed at ion energy Eion  1.0 KeV and beam current J ion  30 mA. Additionally, the sample material is fused silica, and the used small removal function is obtained by an ion diaphragm method. A.

Half-Angle of Divergence

To determine the half-angle of divergence in Eq. (5), the target-distance-dependent beam diameter and removal rate are investigated through experiments in which the ion beam perpendicularly bombards a flat surface, with the target distance increasing from 6 to 18 mm. Experimental results in Fig. 6(a) verify the linear relationship between the beam diameter d  6σ and target distance l. From the linear fitting, the proportional coefficient K  0.037 is obtained. Substituting this coefficient into Eq. (5), we can derive the half-angle of divergence γ  6.3°. In addition, Eq. (17) indicates that the peak removal rate is inversely proportional to the square of the beam diameter (V F ∝ 1∕d2 ), resulting from the change of target distance. Here we optionally choose the experimental removal function (d  3.8 mm and V F  79.1 nm∕ min ) at a 10 mm target distance as

the conditions for simulation. Using these values and Eq. (17) for the calculation of V F yields an agreement with experimental data [Fig. 6(b)], which indicates the validity of the calculated half-angle of divergence under this processing condition. B. Peak Removal Rate

As shown in Eq. (17), when the ion energy and beam current are held constant, the peak removal rate is mainly influenced by the incidence angle for figuring a curved surface. Besides, since the ion beam possesses a dimension, the target distance of various bombarding points would be different, which would lead to a change of the removal function. To investigate the peak removal rate and removal function shape with respect to incidence angle, experiments are performed on a flat fused silica surface at an off-normal angle. Since the coefficients J 0, Λ, and ε in Eq. (17) are hard to determine, it is difficult for us to directly obtain detailed simulation results of the removal function. Fortunately, the relationships between the removal rate and these coefficients are linear, and we can define the normalized removal rate RV  V F θ∕V F 0 to simplify this problem. According to Eq. (17), the normalized removal rate is given as α cos θ RV  p 2 α cos2 θ  β2 sin2 θ   ρ2 cos2 θ ρ2  ; × exp − 2α2 cos2 θ  β2 sin2 θ 2α2

(19)

where the energy deposition parameters α, β, and ρ are constant coefficients under fixed ion energy. From Eq. (19), we can know that RV is a function of a single variable θ. The removal function under normal bombardment is easily obtained through experimental methods; hence, we can derive the peak removal rates at different incidence angles as long as these energy deposition parameters are given. TRIM.SP software [14] is utilized to simulate the energy deposition of Ar ions perpendicularly

Fig. 6. Experimental results of the removal function: (a) beam diameter and (b) peak removal rate variation with target distance. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS

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Fig. 7. Experimental results of removal function variation with incidence angle.

bombarding the SiO2 target with an energy of 1.0 KeV. The result shows that the average incidence depth of ions is ρ  4.2 nm, and the Gaussian distribution parameters are α  1.9 nm and β  1.2 nm. Usually, these parameters need to be revised to better fit the experimental data. In our work, a simple and effective approach is applied in which the parameters are approximated as β  α. Figure 7 shows the removal functions at different incidence angles with the increase of the ion incidence angle from 0° to 70°, from which we can observe that the removal rate and removal function shape strongly depend on the ion incidence angle. At near normal incidence angles, the removal function maintains a rotationally symmetric Gaussian shape, whereas with the increase of the incidence angle it becomes approximately elliptical in shape with the dimension elongated along the ion incidence direction. Moreover, the removal rate for angles within the [30° 70°] range is obviously larger than that near normal incidence angles. Figure 8 displays a comparison of the angledependent normalized peak removal rate of the theoretical model, experimental data, and the simulation result of TRIM.SP. The simulation values described here are calculated by V  Eθ cosθ∕E0, where Eθ is the sputtering yield obtained from TRIM. The comparative results show that the peak removal rate increases with the increase of the incidence angle, achieves a maximum at 60°, and then decreases rapidly. In comparison with experimental data and the simulation results, the calculated values of the theoretical model are overall in good agreement. Figure 9 indicates the vertical and horizontal profile sections of the removal function at different incidence angles. An obvious change occurs with the increase of the vertical dimension at larger incidence angles [Fig. 9(a)], but the horizontal dimension of the removal function keeps constant for all incidence angles [Fig. 9(b)]. It is worthwhile to mention that a comet-tail-like shape can be observed when the incidence angle θ ≥ 30° [Figs. 7 and 9(a)], which 4272

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can be attributed to the difference of target distance within the ion bombarding area, leading to the unsymmetrical profile along the vertical direction. Additionally, in Fig. 9(a) a removal function example at 70° indicates that the mathematical model is consistent with the experimental result. C.

Removal Function on a Curved Surface

To investigate the influence of the surface profile on the removal function, an ion beam with a dimension of d6σ  4.2 mm, where the removal function dimension is determined though experiment on a planar surface and the target distance is set at 12 mm, is utilized to bombard two spherical convex surfaces. The aperture of the target surfaces is 21.3 mm, and the radius of curvature is 16 mm, which means that the maximal incidence angle is 41.7°. Figure 10(a) shows the removal functions at different dwell points during the TIF process. The first group removal function T0 is made on the center position, where the ion beam perpendicularly bombards the surface at the dwell point; the second group (T11 − T14 ) and third group (T21 –T28 ) are made where

Fig. 8. Experimental results of normalized peak removal rate variation with incidence angle.

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.

the incidence angle θ  15° and θ  30°, respectively. It can be clearly observed that, with the dwell point approaching the surface edge, the peak removal rate increases rapidly and the rotationally symmetric Gaussian removal function transforms to a unsymmetrical shape. Figure 10(b) shows that the 2D removal function is also dynamic during the FIF process, where the removal function F0 on the surface center is obviously different from the removal

functions (F11 –F18 ) made at the same dwell points of the last group of TIF. As we know, since ion beam perpendicularity bombards the surface at every dwell point, the actual removal function in 3D space is constant during FIF. However, resulting from the projection distortion, the dimension of the removal functions (F11 –F18 ) is compressed along the radius radiation direction, exhibiting a dynamic character at various dwell points. Figures 10(c) and 10(d) show

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. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS

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the mathematical model of removal function at the corresponding dwell points during the TIF process and the FIF process, respectively, and indicate that the mathematical model is in agreement with experimental results. Summarized from the above two figuring experiments, we can conclude that the removal function is dynamic in the 2D convolution equation, which must be considered seriously for the accurate calculation of dwell time. 5. Conclusion

In a deterministic method, the figuring process is represented by a two-dimensional (2D) convolution operation of the constant removal function and the dwell time. However, when processing curved surfaces, we find that the removal function exhibits dynamic characteristic in the 2D convolution equation, which results from the problem of projection distortion and the influence of workpiece geometry. Consequently, the current equation cannot factually reflect the figuring process, directly influencing the solution accuracy for the dwell time and then reducing the convergence of the figuring process. In this part, we develop a mathematical model of the removal function for the TIF and the FIF figuring methods through theoretical modeling and experimental validation. In next part, we will propose an improved convolution equation based on the dynamic removal function from the mathematical model, and figuring experiments will be carried out to verify the feasibility of our proposed method. This work was supported by the National Natural Science Foundation of China (Grant Nos. 91023042, 91323302, and 51175504), the Ministry of Science and Technology “973” Plan (Grant No. 2011CB013200), and the Postgraduate Scientific Innovation Fund of Hunan Province.

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Mathematical modeling and application of removal functions during deterministic ion beam figuring of optical surfaces. Part 1: Mathematical modeling.

Ion beam figuring (IBF) is established for the final precision figuring of high-performance optical components, where the figuring accuracy is guarant...
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