Dwell-time algorithm for polishing large optics Chunjin Wang,1,3 Wei Yang,1,* Zhenzhong Wang,1 Xu Yang,1 Chenlin Hu,1 Bo Zhong,2 Yinbiao Guo,1 and Qiao Xu3 1
Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005, China 2
3
Fine Optical Engineering Research Center, Chengdu 610041, China
Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China *Corresponding author: [email protected] Received 16 May 2014; accepted 15 June 2014; posted 19 June 2014 (Doc. ID 211900); published 17 July 2014
The calculation of the dwell time plays a crucial role in polishing precision large optics. Although some studies have taken place, it remains a challenge to develop a calculation algorithm which is absolutely stable, together with a high convergence ratio and fast solution speed even for extremely large mirrors. For this aim, we introduced a self-adaptive iterative algorithm to calculate the dwell time in this paper. Simulations were conducted in bonnet polishing (BP) to test the performance of this method on a real 430 mm × 430 mm fused silica part with the initial surface error PV 1741.29 nm, RMS 433.204 nm. The final surface residual error in the clear aperture after two simulation steps turned out to be PV 11.7 nm, RMS 0.5 nm. The results confirm that this method is stable and has a high convergence ratio and fast solution speed even with an ordinary computer. It is notable that the solution time is usually just a few seconds even on a 1000 mm × 1000 mm part. Hence, we believe that this method is perfectly suitable for polishing large optics. And not only can it be applied to BP, but it can also be applied to other subaperture deterministic polishing processes. © 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.53.004752
1. Introduction
The manufacturing of large optical elements for large telescopes and high-power laser systems presents unique challenges in combining quality of the form and texture achieved, with fast processing speeds. Hence, various computer-controlled optical surfacing (CCOS) processes have been developed in recent decades [1–7], including ion beam finishing [2], magnetorheological finishing [4], fluid jet polishing [5], and bonnet polishing (BP) [7–11]. In the CCOS process, the tool is controlled to dwell at different points in precalculated time to remove various amounts of material. To achieve a high-accuracy surface, calculation of the dwell-time 1559-128X/14/214752-09$15.00/0 © 2014 Optical Society of America 4752
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algorithm plays a crucial role during the process. Hitherto, many algorithms to optimize the dwell time of CCOS have been studied [2,12–25] and are mainly based on two models, which are the discrete convolution model and the linear equation model. Algorithms based on the discrete convolution model mainly include the iterative method, the Fourier transform method, the series expansion method, and the Bayesian method. The iterative method was first introduced in the dwell-time solution by Jones [12] at the Perkin-Elmer Corporation, and then it was used in ion figuring by Allen and Keim [2]. This method has fast speed but with limited accuracy and convergence. However, it laid an important foundation for the development of the dwell-time algorithm. Based on this method, the Fourier transform method was developed by Wilson and McNeil [13,14], which transferred the 2D convolution process in the space
domain into a product problem in the frequency domain. The solution result of the dwell time should be positive, but unfortunately, negative dwell time usually appeared in part of this method’s results. And the result of the computation sometimes fails to converge when the negative part is replaced by zero, which can lead to vibration of the residual error together with acute partial variety. To avoid negative solutions, both Drueding et al. [15] and Shanbhag et al. [16] attempted to solve the dwell time by adopting the series expansion method by adjusting a constant term. Based on the Bayesian principle, Jiao et al. [17] introduced an iterative dwell-time algorithm for planar mirrors deduced from the CCOS principle, but it is time consuming and not suitable for large-scale calculations. Algorithms based on the linear equation method mainly include the least squares QR (LSQR) decomposition regularization method, the truncated singular value decomposition (TSVD) regularization method, and the Tikhonov regularization method. In 1992, Carnal et al. [18] first introduced the linear equation method and solved the dwell time by adopting the LSQR method. It opened a new way to solve the dwell time. But it usually needs to offset the negative dwell-time results. Because of this, Lee and Yang [19] utilized non-negative least squares (NNLS), which is the call function of MATLAB version 5.1, to obtain an optimal real solution. Though this method has many advantages, such as high solution accuracy and no need of coincidence between the removal grid points and the dwell grid points, its computation cost is significant. Zhou et al. [20] used the TSVD method and Deng et al. [21] established a combined optimization of the residual error and the energy of solution with the Tikhonov method to solve the linear equation model. However, both of these methods still need a lot of resources to calculate which makes them not suitable for largescale calculations. Later, Wu et al. [22] and Li et al. [23] revised the linear matrix with a proper optimization operator in order to pursue faster convergence and edge protection, but the element number of the matrix combining the rate of material removal at all dwell points is extremely huge when encountering a large mirror, which means that its requirement for computer RAM would be rather high. Shi [24] chose an iterative method of a large sparse matrix to improve efficiency and accuracy. It is beneficial for realizing the large-scale numerical calculation and enhancing the feasibility of the solution, but it may lack a good balance between time and residual error [25]. Song et al. [25] combined the realization of the dwell time to solve the linear equation, which makes it as accurate as possible when realizing the dwell time. As stated above, two main issues can be concluded for previous dwell-time algorithms. First, algorithms based on the discrete convolution method usually have a fast computation speed, but most of them are not always convergent and stable. Second,
algorithms based on the linear equation method have a high convergence ratio, but their computation speed is too slow when encountering large-scale calculations. Hence, for large optics, there is a need to develop a dwell-time algorithm which possesses fast computation speed, a high convergence ratio, and high stability. In this paper, we introduce a self-adaptive iterative (SAI) dwell-time algorithm and test its feasibility in BP. We first introduce the material removal mechanism in BP. Then, the SAI dwell-time algorithm is demonstrated theoretically which can meet the above requirements and with a computation speed that is quite fast, even when executed on an ordinary computer. After that, simulations are presented on the real surface adopting the actual extracted tool influence function (TIF) to test the SAI algorithm. Finally, the SAI algorithm is discussed, followed by the conclusion. 2. Material Removal Mechanism of BP A. TIF of BP
In BP, its tool is comprised of a spinning inflated bulging rubber membrane with a spherical form, and it is covered with a polishing cloth (often comprised of polyurethane) and operates in the presence of a cerium oxide polishing slurry. The bonnet is brought into contact with the surface of the workpiece and then compresses to create a defined polishing spot. The orientation of the tool-rotation axis can be controlled at a defined inclination angle of typically 10–25 deg (called “precession angle”) with respect to the local normal surface over the entire work-piece. The tool axis then precesses in (typically four) discrete steps about the local normal to the work-piece surface. This process generates an accumulated TIF which is Gaussian-like [26]. Figure 1 shows the polishing model of BP. The size of the contact area is controlled by a z-offset between the bonnet tool and the work-piece. The material removal model of BP is derived from the well-known Preston equation [27], expressed as Δhx; y k · px; y · vx; y;
(1)
Fig. 1. Tool movement model in BP. 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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stimulated further studies on this method. Zhou et al. [33] added a damping factor to control this method avoiding divergence, but this factor is not constant and varies in different cases. This makes it difficult to determine a suitable value for it to confirm the stability of this method and it also makes this method difficult to automate. For that matter, we introduce a modified coefficient to make the damping factor become self-adaptive, as described in the following text. In this dwell-time algorithm, the volumetric removal rate of the removal function denoted as VR is used to determine the initial dwell time and correction of dwell time, defined as ZZ VR Fig. 2. Schematic diagram of the material removal process of BP.
where Δhx; y is the removal in a unit time, k is the Preston constant, Px; y is the pressure distribution in the contact area, and vx; y is the relative speed between the tool and the work-piece surface. The TIF of BP is defined as the average material removal in the unit time. The simulation of TIF based on this model has been reported on some literature [28–31]. In the dwell-time calculation process, both the actual TIF and the simulated TIF could be adopted.
Ω
Rx; ydxdy;
(4)
where Ω is the area of each TIF. When the desired material removal matrix H 0 mentioned above is acquired, the initial dwell-time matrix D0 can be expressed as D0 H 0 ∕VR;
(5)
and the surface residual error matrix E0 after the polishing process using D0 could be expressed as E0 H 0 − R D0 :
(6)
BP is a type of deterministic polishing technology. The amount of material removed in BP, Hx; y, is equal to the two-dimensional convolution between the material removal function per unit time Rx; y and the dwell-time function Dx; y, along with the motion track, and can be expressed as
Several computation iterations would be executed using the formerly calculated surface error arrays, until the predicted residual error is sufficiently small. The specific realization steps of this algorithm could be concluded as shown in Fig. 3 and H 0 has been processed to be non-negative adopting the equation
Hx; y Rx; y Dx; y:
H 0 H 0 − minH 0 :
B.
Material Removal Process
(2)
Therefore, the surface residual error Ex; y after the BP process can be expressed as Ex; y H 0 x; y − Hx; y;
(3)
where H 0 x; y is the desired removal of material. This material removal process has been specifically demonstrated in Fig. 2. H 0 x; y is discretized as an n × m matrix, so as to Dx; y, Hx; y, and Ex; y. As shown in Fig. 2, Rx; y is dwelling at one of the dwell points xi ; yj and it will dwell at each point in precalculated time to implement the required material removal. 3. SAI Algorithm
The iterative method has been widely used to calculate the dwell time in subaperture polishing, especially when dealing with large-scale calculations considering its fast computation speed. However, experience indicates that this method is not always stable and sometimes fails to converge [32]. This 4754
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(7)
Furthermore, it could ensure that the acquired dwell time is non-negative. As shown in Fig. 3, Dk is the dwell-time matrix used in the kth iteration, Ek is the surface residual error matrix after the polishing process using Dk, RMS0 is the root mean square value of E0 which also can be expressed as two-norm of E0, RMSk is the root mean square value of Ek, ξ is the damping factor, and ψ is the modified coefficient of ξ. In the previous method [33], ξ ≤ 1, larger ξ would lead to faster convergence speed, whereas, the iteration will be divergent when ξ is too large. Hence, to determine a best-suited value of ξ was not that easy before. In this method, with the help of the modified coefficient ψ, ξ could be defined as 1 and ψ could be any value between 0 and 1. The value of ψ is preferred to be a value near 1 which will make ξ change slowly in each iteration. We therefore foresee that the damping factor would be self-adaptive once it is not suited for the process. In addition, RMSk1 > RMSk is defined as the convergence condition considering that RMS will
concentration, movement error of machine tool, etc., the material removal is not absolutely linear and proportional to the dwell time, and TIF is not constant over the entire surface and with time. Therefore, the simulation experiment is usually adopted to verify the dwell-time calculation algorithm instead of the actual one [22,23,25,32]. In references [22,23,25,32], all the initial surface errors are measured from a real part to make the simulation process closer to reality. In order to verify the effectiveness of the SAI algorithm introduced in this paper, we simulated the algorithm using the actual measurement result as the initial surface error and two TIFs were also the actual measurement results of the polishing spots as shown in Fig. 4, whose volumetric removal rates are 5.269 and 0.207 mm3 ∕ min , respectively. A. Preprocessing of the Measured Surface Error Map
Fig. 3. Flow chart of the self-adaptive iterative algorithm.
become smaller after each iteration in this method. This results in that the final convergent results will deliver the smallest surface residual error. In other dwell-time algorithms, whether the RMS reaches the RMS threshold is taken as the convergence condition [23], but this leads to the fact that the final results may not have the smallest RMS value and may lead the process to be divergent when the preset threshold is smaller than the smallest RMS during the process. It makes it so that we have to change the threshold RMS value every time because a different initial surface error would lead to a different convergence RMS value. Therefore, the method introduced in this paper is self-adaptive and could be easily integrated into the computer program.
A 430 mm × 430 mm aspheric fused silica part whose clear aperture is 400 mm × 400 mm was taken as the work-piece. Figure 5(a) shows the initial surface error map measured using an interferometer. The initial surface error is PV 1741.3 nm, RMS 433.2 nm. There are some data missing at the left side of the error shape and the error map is not at the center of the axis as shown in Fig. 5(a). In the preprocessing step of the surface error map, care was taken to solve these problems including filling the gap data, centering the error map matrix, and eliminating the outliers when necessary. As shown in Fig. 5(b), the existing problems of the measured error map were solved and Eq. (7) was used to make the initial error non-negative. And the initial error map was converted to a matrix and resized to suit the algorithm.
4. Simulations
The dwell-time calculation algorithm is based on the following assumptions [2]: a. Material removal is linear and proportional to dwell time; b. TIF is constant over the entire surface; and c. TIF is constant with time. In the actual polishing process, affected by the initial surface error shape, the instability of the slurry
Fig. 4. Measured TIFs used in the simulation. (a) TIF1, diameter 30 mm and (b) TIF2, diameter 16 mm. 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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PV=1741.3nm,RMS=433.2nm
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(b) Fig. 5. Initial surface error map of the fused silica part. (a) Measurement results from the interferometer and (b) the preprocessed surface error.
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Two simulation polishing processes were conducted in this paper. The first process by adopting a larger size TIF and step size to implement fast surface form correction, and the second one by using a smaller TIF and step size to acquire higher accuracy of the surface form. These two TIFs are shown in Fig. 4, whose diameters are 30 mm and 16 mm, respectively. The raster path was applied and the step size in x and y directions were both 5 mm in the first process. The SAI method as shown in Fig. 3 was adopted in the simulation. The value of the damping factor ξ was defined as 1, and the modified coefficient ψ was 0.95. Both of them had been validated as suited for this method. The first simulation process was executed on an ordinary laptop (CPU: Intel Core i5–3210M 2.50 GHz, RAM: 4 GB). In order to test the solution time, the process was tested five times in MATLAB and the average solution time was 2.4 s. (The simulation process is provided in the accompanying movie 4756
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clip, Media 1.) It indicates that the arithmetic speed of this algorithm is quite fast. The final value of ξ in this process is 1, which means that this value meets the requirement to make the algorithm convergent. The surface residual error after this process is shown in Fig. 6. The whole surface residual error is PV 269.8 nm, RMS 12.7 nm. And the surface residual error in the clear aperture is PV 42.0 nm, RMS 4.0 nm. It is noted that the convergence ratios of both PV and RMS are quite high, which are 6.5 and 34.1, respectively. However, there exists sharp edges due to the edge effect which is a common problem in the dwell-time calculation method [23]. Figure 7(a) shows the calculated dwell-time distribution map. The total dwell time is 37.834 min and its distribution is quite similar to the initial surface error map. It indicates that the dwell time on each point is proportional to its residual error. There are mainly two modes, position dwell mode and velocity dwell mode [32], for realizing the dwell time in the subaperture deterministic polishing process including BP. In velocity dwell mode, the dwell time has to be translated into different velocities at each dwell point. Herein, the minimum velocity, the maximum velocity, and the maximum acceleration or deceleration are defined as 1 mm∕s, 250 mm∕s, and 1250 mm∕s2 , respectively, according to the characteristics of our machine tool. The velocity
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(b) Fig. 6. Surface residual error after the first simulation process using the self-adaptive iterative algorithm. (a) The whole surface and (b) 400 mm × 400 mm of the center part. (The simulation process is provided in the accompanying movie clip, Media 1.)
Unit: s
PV=127.1nm RMS=2.4nm
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(b) Fig. 8. Surface residual error after the second simulation process using the self-adaptive iterative algorithm. (a) The whole surface and (b) 400 mm × 400 mm of the center part.
(b) Fig. 7. Solution results of (a) the dwell-time distribution and (b) the velocity distribution.
distribution result has been carried out as shown in Fig. 7(b) based on the method reported in references [20,25]. In the velocity distribution results, the derived velocity which is larger than the maximum velocity of the machine, will be set as 250 mm∕s, and which is smaller than the minimum velocity, will be set as 1 mm∕s. The velocity distribution map as shown in Fig. 7(b) corresponds well with the dwell-time map in Fig. 7(a), demonstrating that the derived velocity map is correct. To reduce the impact of the edge effect to the area in the clear aperture and improve the surface accuracy, a second simulation process was conducted using TIF2 and the step size decreased to 2 mm both in x and y directions. The surface residual error after the first process as shown in Fig. 6(a) was taken as the initial surface error and it had been preprocessed to be non-negative before processing. The average solution time was 2.7 s, which is a little longer than the first one causing by the increase of surface error matrix’s elements. Figure 8 shows the solution results of the surface residual error. The whole surface error is PV 127.1 nm, RMS 2.4 nm, and the surface error in the clear aperture is PV 11.7 nm, RMS 0.5 nm. The convergence ratios of PV and RMS are 2.1 and 5.3, respectively. If compared to the initial surface error in the first process, they would be 13.7 and 180.5.
5. Discussion
The simulation results prove that the SAI method could be used to calculate the dwell time in polishing large optics, and the convergence ratio is relatively high. In addition, the solution speed is extremely fast even when executed on an ordinary laptop, and the final surface form accuracy is high enough for most industrial applications of optical lenses. For the dwell-time algorithms based on the linear equation method [18–25], although they can acquire surface form accuracy like this or even higher, the solution speed is too slow. For example, in reference [18], the solution of the dwell time is based on the following equation: rnr ×1 Bnr ×nt tnt ×1 ;
(8)
where nr is the point amount of the removal point, nt is the point amount of the dwell point, r is an nr × 1 matrix denoting the material removal amount at all removal point, B is an nr × nt matrix combining the rate of material removal at all dwell points, and t is an nt × 1 matrix denoting the dwell time at all dwell points. For instance, in the situation at the first process of the simulation, both the values of nr and nt are 7396, then the element amount of matrix B would be 5.4700816 × 107 and its value is 2.136750625 × 109. This is unbearable for an ordinary computer or even a computer station, not to speak of solving Eq. (8). Commonly, several minutes 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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or hours are needed to solve a medium-scale case. So the dwell-time algorithms based on the linear equation method are not suited for large-scale calculations. In order to further verify that the SAI method can be used when encountering complicated surface forms or larger optics, another two simulation processes have also been conducted. A.
Test on Complicated Surface Form
Aiming to test its property of convergence when encountering complicated surface form, we designed a surface form as shown in Fig. 9(a). Its size is also 430 mm × 430 mm. The initial PV and RMS value are 2 μm and 1.2 μm, respectively. TIF1 as shown in Fig. 4(a) was also adopted in this test, and the step size of the raster path both in x and y directions were 0.5 mm. The initial damping factor ξ was still defined as 1, and the modified coefficient ψ was 0.95. The final converged residual surface error is shown in Fig. 9(b). The final damping factor ξ 0.3235(0.3235 0.9522 ), and the RMS of the residual error is 11.3 nm. It is noted that the convergence ratio still reaches 106.2 which is quite high. This proves that the SAI algorithm is effective to this model. It is also interesting to note that the final value of ξ is different from the initial one, which means that this process would be divergent when ξ 1. In order to study the effect of ξ to the simulation process,
Initial error /µ µm
PV=2µm RMS=1.2µm
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a group of ξ (ξ 1; 0.95; 0.952 ; 0.953 ; …; 0.9530 ) have been tested in this model. The final convergent RMS of the residual surface error and the convergence ratio of each ξ have been extracted as shown in Fig. 10. As shown in Fig. 10, the process could be considered divergent when ξ ≥ 0.4877 (0.4877 0.9514 ), because the corresponding convergent RMS is 1100 nm and the convergence ratio is only 1.09. The convergence ratio becomes larger along with the decrease of ξ, until ξ 0.3235 (0.3235 0.9522 ). It means that the value of 0.3235 or close to it would be the best suited for ξ, which can deliver the largest convergence ratio. With the help of the modified coefficient ψ, the changing process of ξ could be done by itself until the process becomes convergent. Without the help of ψ, to determine the suited ξ would be difficult and time consuming. It indicates that the SAI algorithm can make the dwell-time solution process easier to be convergent. B. Test on Large-Size Optics
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(b) Fig. 9. Surface error shape. (a) The initial surface error and (b) the surface residual error of the central 400 mm × 400 mm area after the simulation polishing process when ξ 0.3235 (0.3235 0.9522 ). 4758
Fig. 10. Changing curve of the final surface residual convergent RMS and its convergence ratio vary with the damping factor.
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In order to test the applicability of this algorithm to large-size optics, a simulation process polishing a 1000 mm × 1000 mm surface was also conducted. To demonstrate its high calculation speed, the step size is set as 1 mm both in x and y directions to increase the amount of calculation. TIF1 mentioned above was still selected in this process. Considering that we did not have a mirror with so large a size, a measuring result from a 430 mm × 430 mm mirror was resized to 1000 mm × 1000 mm as shown in Fig. 11(a). The initial error is PV 4.5 μm, RMS 3.6 μm. Figure 11(b) shows the 970 mm × 970 mm of the final surface’s center part whose surface residual error is PV 87.1 nm, RMS 1.3 nm. The convergence ratio is still extremely high. And the average total solution time was only 3.8 s which is just a little longer than that in the simulations aforementioned. (The simulation process is provided in the accompanying movie clip, Media 2.) Viewing the result of this case, we note that it is also applicable to a 10 m × 10 m mirror when the step size is set as 10 mm both in x and y directions through scaling this model. Hence, the SAI algorithm can be successfully used when polishing large-size optics.
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(b) Fig. 11. Surface error shape. (a) The initial surface error and (b) the surface residual error of the central 970 mm × 970 mm area after the simulation polishing process. (The simulation process is provided in the accompanying movie clip, Media 2.)
14. 15. 16.
6. Conclusions
This paper introduces the SAI algorithm based on the iterative method to calculate the dwell time when polishing large optics. It is implemented by adding a modified coefficient to the damping factor to make the damping factor be self-adaptive, and RMSk1 > RMSk is defined as the convergence condition which can bring us the highest precision solution all the time. The simulations in bonnet polishing demonstrate that this algorithm is stable, and has a high convergence ratio and fast solution speed (it usually takes a few seconds) even on an ordinary computer. Moreover, for the property of self-adaptiveness, it is easy to integrate into the computer program. The SAI method is not only applied to BP, but also can be applied to other subaperture deterministic polishing processes for the reason that all of them follow the same basic principle of the material removal. This work was financially supported by major national science and technology projects (Grant No. 2013ZX04006011-206). References 1. R. A. Jones, “Computer control for grinding and polishing,” Photonics Spectra 17, 34–39 (1983). 2. L. N. Allen and R. E. Keim, “An ion figuring system for large optic fabrication,” Proc. SPIE 1168, 33–50 (1989). 3. S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. B. Davison, T. J. Trebisky, S. T. Derigne, and B. B. Hille,
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for extremely large telescopes,” Opt. Express 13, 910–917 (2005). 29. C. Wang, Y. Guo, Z. Wang, R. Pan, and Y. Xie, “Dynamic removal function modeling of bonnet tool polishing on optics elements,” J. Mech. Eng. 49, 19–25 (2013) [in Chinese]. 30. H. Li, D. D. Walker, G. Yu, and W. Zhang, “Modeling and validation of polishing tool influence functions for manufacturing segments for an extremely large telescope,” Appl. Opt. 52, 5781–5787 (2013).
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31. C. Wang, Z. Wang, X. Yang, Z. Sun, Y. Peng, Y. Guo, and Q. Xu, “Modeling of the static tool influence function of bonnet polishing based on FEA,” Int. J. Adv. Manuf. Technol. (to be published). 32. H. Fang, P. Guo, and J. Yu, “Dwell function algorithm in fluid jet polishing,” Appl. Opt. 45, 4291–4296 (2006). 33. 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].
Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005, China 2
3
Fine Optical Engineering Research Center, Chengdu 610041, China
Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China *Corresponding author: [email protected] Received 16 May 2014; accepted 15 June 2014; posted 19 June 2014 (Doc. ID 211900); published 17 July 2014
The calculation of the dwell time plays a crucial role in polishing precision large optics. Although some studies have taken place, it remains a challenge to develop a calculation algorithm which is absolutely stable, together with a high convergence ratio and fast solution speed even for extremely large mirrors. For this aim, we introduced a self-adaptive iterative algorithm to calculate the dwell time in this paper. Simulations were conducted in bonnet polishing (BP) to test the performance of this method on a real 430 mm × 430 mm fused silica part with the initial surface error PV 1741.29 nm, RMS 433.204 nm. The final surface residual error in the clear aperture after two simulation steps turned out to be PV 11.7 nm, RMS 0.5 nm. The results confirm that this method is stable and has a high convergence ratio and fast solution speed even with an ordinary computer. It is notable that the solution time is usually just a few seconds even on a 1000 mm × 1000 mm part. Hence, we believe that this method is perfectly suitable for polishing large optics. And not only can it be applied to BP, but it can also be applied to other subaperture deterministic polishing processes. © 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.53.004752
1. Introduction
The manufacturing of large optical elements for large telescopes and high-power laser systems presents unique challenges in combining quality of the form and texture achieved, with fast processing speeds. Hence, various computer-controlled optical surfacing (CCOS) processes have been developed in recent decades [1–7], including ion beam finishing [2], magnetorheological finishing [4], fluid jet polishing [5], and bonnet polishing (BP) [7–11]. In the CCOS process, the tool is controlled to dwell at different points in precalculated time to remove various amounts of material. To achieve a high-accuracy surface, calculation of the dwell-time 1559-128X/14/214752-09$15.00/0 © 2014 Optical Society of America 4752
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algorithm plays a crucial role during the process. Hitherto, many algorithms to optimize the dwell time of CCOS have been studied [2,12–25] and are mainly based on two models, which are the discrete convolution model and the linear equation model. Algorithms based on the discrete convolution model mainly include the iterative method, the Fourier transform method, the series expansion method, and the Bayesian method. The iterative method was first introduced in the dwell-time solution by Jones [12] at the Perkin-Elmer Corporation, and then it was used in ion figuring by Allen and Keim [2]. This method has fast speed but with limited accuracy and convergence. However, it laid an important foundation for the development of the dwell-time algorithm. Based on this method, the Fourier transform method was developed by Wilson and McNeil [13,14], which transferred the 2D convolution process in the space
domain into a product problem in the frequency domain. The solution result of the dwell time should be positive, but unfortunately, negative dwell time usually appeared in part of this method’s results. And the result of the computation sometimes fails to converge when the negative part is replaced by zero, which can lead to vibration of the residual error together with acute partial variety. To avoid negative solutions, both Drueding et al. [15] and Shanbhag et al. [16] attempted to solve the dwell time by adopting the series expansion method by adjusting a constant term. Based on the Bayesian principle, Jiao et al. [17] introduced an iterative dwell-time algorithm for planar mirrors deduced from the CCOS principle, but it is time consuming and not suitable for large-scale calculations. Algorithms based on the linear equation method mainly include the least squares QR (LSQR) decomposition regularization method, the truncated singular value decomposition (TSVD) regularization method, and the Tikhonov regularization method. In 1992, Carnal et al. [18] first introduced the linear equation method and solved the dwell time by adopting the LSQR method. It opened a new way to solve the dwell time. But it usually needs to offset the negative dwell-time results. Because of this, Lee and Yang [19] utilized non-negative least squares (NNLS), which is the call function of MATLAB version 5.1, to obtain an optimal real solution. Though this method has many advantages, such as high solution accuracy and no need of coincidence between the removal grid points and the dwell grid points, its computation cost is significant. Zhou et al. [20] used the TSVD method and Deng et al. [21] established a combined optimization of the residual error and the energy of solution with the Tikhonov method to solve the linear equation model. However, both of these methods still need a lot of resources to calculate which makes them not suitable for largescale calculations. Later, Wu et al. [22] and Li et al. [23] revised the linear matrix with a proper optimization operator in order to pursue faster convergence and edge protection, but the element number of the matrix combining the rate of material removal at all dwell points is extremely huge when encountering a large mirror, which means that its requirement for computer RAM would be rather high. Shi [24] chose an iterative method of a large sparse matrix to improve efficiency and accuracy. It is beneficial for realizing the large-scale numerical calculation and enhancing the feasibility of the solution, but it may lack a good balance between time and residual error [25]. Song et al. [25] combined the realization of the dwell time to solve the linear equation, which makes it as accurate as possible when realizing the dwell time. As stated above, two main issues can be concluded for previous dwell-time algorithms. First, algorithms based on the discrete convolution method usually have a fast computation speed, but most of them are not always convergent and stable. Second,
algorithms based on the linear equation method have a high convergence ratio, but their computation speed is too slow when encountering large-scale calculations. Hence, for large optics, there is a need to develop a dwell-time algorithm which possesses fast computation speed, a high convergence ratio, and high stability. In this paper, we introduce a self-adaptive iterative (SAI) dwell-time algorithm and test its feasibility in BP. We first introduce the material removal mechanism in BP. Then, the SAI dwell-time algorithm is demonstrated theoretically which can meet the above requirements and with a computation speed that is quite fast, even when executed on an ordinary computer. After that, simulations are presented on the real surface adopting the actual extracted tool influence function (TIF) to test the SAI algorithm. Finally, the SAI algorithm is discussed, followed by the conclusion. 2. Material Removal Mechanism of BP A. TIF of BP
In BP, its tool is comprised of a spinning inflated bulging rubber membrane with a spherical form, and it is covered with a polishing cloth (often comprised of polyurethane) and operates in the presence of a cerium oxide polishing slurry. The bonnet is brought into contact with the surface of the workpiece and then compresses to create a defined polishing spot. The orientation of the tool-rotation axis can be controlled at a defined inclination angle of typically 10–25 deg (called “precession angle”) with respect to the local normal surface over the entire work-piece. The tool axis then precesses in (typically four) discrete steps about the local normal to the work-piece surface. This process generates an accumulated TIF which is Gaussian-like [26]. Figure 1 shows the polishing model of BP. The size of the contact area is controlled by a z-offset between the bonnet tool and the work-piece. The material removal model of BP is derived from the well-known Preston equation [27], expressed as Δhx; y k · px; y · vx; y;
(1)
Fig. 1. Tool movement model in BP. 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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stimulated further studies on this method. Zhou et al. [33] added a damping factor to control this method avoiding divergence, but this factor is not constant and varies in different cases. This makes it difficult to determine a suitable value for it to confirm the stability of this method and it also makes this method difficult to automate. For that matter, we introduce a modified coefficient to make the damping factor become self-adaptive, as described in the following text. In this dwell-time algorithm, the volumetric removal rate of the removal function denoted as VR is used to determine the initial dwell time and correction of dwell time, defined as ZZ VR Fig. 2. Schematic diagram of the material removal process of BP.
where Δhx; y is the removal in a unit time, k is the Preston constant, Px; y is the pressure distribution in the contact area, and vx; y is the relative speed between the tool and the work-piece surface. The TIF of BP is defined as the average material removal in the unit time. The simulation of TIF based on this model has been reported on some literature [28–31]. In the dwell-time calculation process, both the actual TIF and the simulated TIF could be adopted.
Ω
Rx; ydxdy;
(4)
where Ω is the area of each TIF. When the desired material removal matrix H 0 mentioned above is acquired, the initial dwell-time matrix D0 can be expressed as D0 H 0 ∕VR;
(5)
and the surface residual error matrix E0 after the polishing process using D0 could be expressed as E0 H 0 − R D0 :
(6)
BP is a type of deterministic polishing technology. The amount of material removed in BP, Hx; y, is equal to the two-dimensional convolution between the material removal function per unit time Rx; y and the dwell-time function Dx; y, along with the motion track, and can be expressed as
Several computation iterations would be executed using the formerly calculated surface error arrays, until the predicted residual error is sufficiently small. The specific realization steps of this algorithm could be concluded as shown in Fig. 3 and H 0 has been processed to be non-negative adopting the equation
Hx; y Rx; y Dx; y:
H 0 H 0 − minH 0 :
B.
Material Removal Process
(2)
Therefore, the surface residual error Ex; y after the BP process can be expressed as Ex; y H 0 x; y − Hx; y;
(3)
where H 0 x; y is the desired removal of material. This material removal process has been specifically demonstrated in Fig. 2. H 0 x; y is discretized as an n × m matrix, so as to Dx; y, Hx; y, and Ex; y. As shown in Fig. 2, Rx; y is dwelling at one of the dwell points xi ; yj and it will dwell at each point in precalculated time to implement the required material removal. 3. SAI Algorithm
The iterative method has been widely used to calculate the dwell time in subaperture polishing, especially when dealing with large-scale calculations considering its fast computation speed. However, experience indicates that this method is not always stable and sometimes fails to converge [32]. This 4754
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(7)
Furthermore, it could ensure that the acquired dwell time is non-negative. As shown in Fig. 3, Dk is the dwell-time matrix used in the kth iteration, Ek is the surface residual error matrix after the polishing process using Dk, RMS0 is the root mean square value of E0 which also can be expressed as two-norm of E0, RMSk is the root mean square value of Ek, ξ is the damping factor, and ψ is the modified coefficient of ξ. In the previous method [33], ξ ≤ 1, larger ξ would lead to faster convergence speed, whereas, the iteration will be divergent when ξ is too large. Hence, to determine a best-suited value of ξ was not that easy before. In this method, with the help of the modified coefficient ψ, ξ could be defined as 1 and ψ could be any value between 0 and 1. The value of ψ is preferred to be a value near 1 which will make ξ change slowly in each iteration. We therefore foresee that the damping factor would be self-adaptive once it is not suited for the process. In addition, RMSk1 > RMSk is defined as the convergence condition considering that RMS will
concentration, movement error of machine tool, etc., the material removal is not absolutely linear and proportional to the dwell time, and TIF is not constant over the entire surface and with time. Therefore, the simulation experiment is usually adopted to verify the dwell-time calculation algorithm instead of the actual one [22,23,25,32]. In references [22,23,25,32], all the initial surface errors are measured from a real part to make the simulation process closer to reality. In order to verify the effectiveness of the SAI algorithm introduced in this paper, we simulated the algorithm using the actual measurement result as the initial surface error and two TIFs were also the actual measurement results of the polishing spots as shown in Fig. 4, whose volumetric removal rates are 5.269 and 0.207 mm3 ∕ min , respectively. A. Preprocessing of the Measured Surface Error Map
Fig. 3. Flow chart of the self-adaptive iterative algorithm.
become smaller after each iteration in this method. This results in that the final convergent results will deliver the smallest surface residual error. In other dwell-time algorithms, whether the RMS reaches the RMS threshold is taken as the convergence condition [23], but this leads to the fact that the final results may not have the smallest RMS value and may lead the process to be divergent when the preset threshold is smaller than the smallest RMS during the process. It makes it so that we have to change the threshold RMS value every time because a different initial surface error would lead to a different convergence RMS value. Therefore, the method introduced in this paper is self-adaptive and could be easily integrated into the computer program.
A 430 mm × 430 mm aspheric fused silica part whose clear aperture is 400 mm × 400 mm was taken as the work-piece. Figure 5(a) shows the initial surface error map measured using an interferometer. The initial surface error is PV 1741.3 nm, RMS 433.2 nm. There are some data missing at the left side of the error shape and the error map is not at the center of the axis as shown in Fig. 5(a). In the preprocessing step of the surface error map, care was taken to solve these problems including filling the gap data, centering the error map matrix, and eliminating the outliers when necessary. As shown in Fig. 5(b), the existing problems of the measured error map were solved and Eq. (7) was used to make the initial error non-negative. And the initial error map was converted to a matrix and resized to suit the algorithm.
4. Simulations
The dwell-time calculation algorithm is based on the following assumptions [2]: a. Material removal is linear and proportional to dwell time; b. TIF is constant over the entire surface; and c. TIF is constant with time. In the actual polishing process, affected by the initial surface error shape, the instability of the slurry
Fig. 4. Measured TIFs used in the simulation. (a) TIF1, diameter 30 mm and (b) TIF2, diameter 16 mm. 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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PV=1741.3nm,RMS=433.2nm
(a) µm 200 1.5 100
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(b) Fig. 5. Initial surface error map of the fused silica part. (a) Measurement results from the interferometer and (b) the preprocessed surface error.
500 0 -500 -200
Two simulation polishing processes were conducted in this paper. The first process by adopting a larger size TIF and step size to implement fast surface form correction, and the second one by using a smaller TIF and step size to acquire higher accuracy of the surface form. These two TIFs are shown in Fig. 4, whose diameters are 30 mm and 16 mm, respectively. The raster path was applied and the step size in x and y directions were both 5 mm in the first process. The SAI method as shown in Fig. 3 was adopted in the simulation. The value of the damping factor ξ was defined as 1, and the modified coefficient ψ was 0.95. Both of them had been validated as suited for this method. The first simulation process was executed on an ordinary laptop (CPU: Intel Core i5–3210M 2.50 GHz, RAM: 4 GB). In order to test the solution time, the process was tested five times in MATLAB and the average solution time was 2.4 s. (The simulation process is provided in the accompanying movie 4756
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Solution of the Dwell Time Using the SAI Algorithm
0 200 -200
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(a) PV=42.0nm RMS=4.0nm Residual error /nm
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clip, Media 1.) It indicates that the arithmetic speed of this algorithm is quite fast. The final value of ξ in this process is 1, which means that this value meets the requirement to make the algorithm convergent. The surface residual error after this process is shown in Fig. 6. The whole surface residual error is PV 269.8 nm, RMS 12.7 nm. And the surface residual error in the clear aperture is PV 42.0 nm, RMS 4.0 nm. It is noted that the convergence ratios of both PV and RMS are quite high, which are 6.5 and 34.1, respectively. However, there exists sharp edges due to the edge effect which is a common problem in the dwell-time calculation method [23]. Figure 7(a) shows the calculated dwell-time distribution map. The total dwell time is 37.834 min and its distribution is quite similar to the initial surface error map. It indicates that the dwell time on each point is proportional to its residual error. There are mainly two modes, position dwell mode and velocity dwell mode [32], for realizing the dwell time in the subaperture deterministic polishing process including BP. In velocity dwell mode, the dwell time has to be translated into different velocities at each dwell point. Herein, the minimum velocity, the maximum velocity, and the maximum acceleration or deceleration are defined as 1 mm∕s, 250 mm∕s, and 1250 mm∕s2 , respectively, according to the characteristics of our machine tool. The velocity
50 0 -50 -200
200 0
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y /m
m
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(b) Fig. 6. Surface residual error after the first simulation process using the self-adaptive iterative algorithm. (a) The whole surface and (b) 400 mm × 400 mm of the center part. (The simulation process is provided in the accompanying movie clip, Media 1.)
Unit: s
PV=127.1nm RMS=2.4nm
y/mm
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(b) Fig. 8. Surface residual error after the second simulation process using the self-adaptive iterative algorithm. (a) The whole surface and (b) 400 mm × 400 mm of the center part.
(b) Fig. 7. Solution results of (a) the dwell-time distribution and (b) the velocity distribution.
distribution result has been carried out as shown in Fig. 7(b) based on the method reported in references [20,25]. In the velocity distribution results, the derived velocity which is larger than the maximum velocity of the machine, will be set as 250 mm∕s, and which is smaller than the minimum velocity, will be set as 1 mm∕s. The velocity distribution map as shown in Fig. 7(b) corresponds well with the dwell-time map in Fig. 7(a), demonstrating that the derived velocity map is correct. To reduce the impact of the edge effect to the area in the clear aperture and improve the surface accuracy, a second simulation process was conducted using TIF2 and the step size decreased to 2 mm both in x and y directions. The surface residual error after the first process as shown in Fig. 6(a) was taken as the initial surface error and it had been preprocessed to be non-negative before processing. The average solution time was 2.7 s, which is a little longer than the first one causing by the increase of surface error matrix’s elements. Figure 8 shows the solution results of the surface residual error. The whole surface error is PV 127.1 nm, RMS 2.4 nm, and the surface error in the clear aperture is PV 11.7 nm, RMS 0.5 nm. The convergence ratios of PV and RMS are 2.1 and 5.3, respectively. If compared to the initial surface error in the first process, they would be 13.7 and 180.5.
5. Discussion
The simulation results prove that the SAI method could be used to calculate the dwell time in polishing large optics, and the convergence ratio is relatively high. In addition, the solution speed is extremely fast even when executed on an ordinary laptop, and the final surface form accuracy is high enough for most industrial applications of optical lenses. For the dwell-time algorithms based on the linear equation method [18–25], although they can acquire surface form accuracy like this or even higher, the solution speed is too slow. For example, in reference [18], the solution of the dwell time is based on the following equation: rnr ×1 Bnr ×nt tnt ×1 ;
(8)
where nr is the point amount of the removal point, nt is the point amount of the dwell point, r is an nr × 1 matrix denoting the material removal amount at all removal point, B is an nr × nt matrix combining the rate of material removal at all dwell points, and t is an nt × 1 matrix denoting the dwell time at all dwell points. For instance, in the situation at the first process of the simulation, both the values of nr and nt are 7396, then the element amount of matrix B would be 5.4700816 × 107 and its value is 2.136750625 × 109. This is unbearable for an ordinary computer or even a computer station, not to speak of solving Eq. (8). Commonly, several minutes 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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or hours are needed to solve a medium-scale case. So the dwell-time algorithms based on the linear equation method are not suited for large-scale calculations. In order to further verify that the SAI method can be used when encountering complicated surface forms or larger optics, another two simulation processes have also been conducted. A.
Test on Complicated Surface Form
Aiming to test its property of convergence when encountering complicated surface form, we designed a surface form as shown in Fig. 9(a). Its size is also 430 mm × 430 mm. The initial PV and RMS value are 2 μm and 1.2 μm, respectively. TIF1 as shown in Fig. 4(a) was also adopted in this test, and the step size of the raster path both in x and y directions were 0.5 mm. The initial damping factor ξ was still defined as 1, and the modified coefficient ψ was 0.95. The final converged residual surface error is shown in Fig. 9(b). The final damping factor ξ 0.3235(0.3235 0.9522 ), and the RMS of the residual error is 11.3 nm. It is noted that the convergence ratio still reaches 106.2 which is quite high. This proves that the SAI algorithm is effective to this model. It is also interesting to note that the final value of ξ is different from the initial one, which means that this process would be divergent when ξ 1. In order to study the effect of ξ to the simulation process,
Initial error /µ µm
PV=2µm RMS=1.2µm
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(a) PV=345.9nm RMS=11.3nm Residual error /nm
a group of ξ (ξ 1; 0.95; 0.952 ; 0.953 ; …; 0.9530 ) have been tested in this model. The final convergent RMS of the residual surface error and the convergence ratio of each ξ have been extracted as shown in Fig. 10. As shown in Fig. 10, the process could be considered divergent when ξ ≥ 0.4877 (0.4877 0.9514 ), because the corresponding convergent RMS is 1100 nm and the convergence ratio is only 1.09. The convergence ratio becomes larger along with the decrease of ξ, until ξ 0.3235 (0.3235 0.9522 ). It means that the value of 0.3235 or close to it would be the best suited for ξ, which can deliver the largest convergence ratio. With the help of the modified coefficient ψ, the changing process of ξ could be done by itself until the process becomes convergent. Without the help of ψ, to determine the suited ξ would be difficult and time consuming. It indicates that the SAI algorithm can make the dwell-time solution process easier to be convergent. B. Test on Large-Size Optics
x/ 0 mm
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(b) Fig. 9. Surface error shape. (a) The initial surface error and (b) the surface residual error of the central 400 mm × 400 mm area after the simulation polishing process when ξ 0.3235 (0.3235 0.9522 ). 4758
Fig. 10. Changing curve of the final surface residual convergent RMS and its convergence ratio vary with the damping factor.
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In order to test the applicability of this algorithm to large-size optics, a simulation process polishing a 1000 mm × 1000 mm surface was also conducted. To demonstrate its high calculation speed, the step size is set as 1 mm both in x and y directions to increase the amount of calculation. TIF1 mentioned above was still selected in this process. Considering that we did not have a mirror with so large a size, a measuring result from a 430 mm × 430 mm mirror was resized to 1000 mm × 1000 mm as shown in Fig. 11(a). The initial error is PV 4.5 μm, RMS 3.6 μm. Figure 11(b) shows the 970 mm × 970 mm of the final surface’s center part whose surface residual error is PV 87.1 nm, RMS 1.3 nm. The convergence ratio is still extremely high. And the average total solution time was only 3.8 s which is just a little longer than that in the simulations aforementioned. (The simulation process is provided in the accompanying movie clip, Media 2.) Viewing the result of this case, we note that it is also applicable to a 10 m × 10 m mirror when the step size is set as 10 mm both in x and y directions through scaling this model. Hence, the SAI algorithm can be successfully used when polishing large-size optics.
Initial error /µ µm
PV=4.5µm RMS=3.6µm
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11. 12. 13.
(b) Fig. 11. Surface error shape. (a) The initial surface error and (b) the surface residual error of the central 970 mm × 970 mm area after the simulation polishing process. (The simulation process is provided in the accompanying movie clip, Media 2.)
14. 15. 16.
6. Conclusions
This paper introduces the SAI algorithm based on the iterative method to calculate the dwell time when polishing large optics. It is implemented by adding a modified coefficient to the damping factor to make the damping factor be self-adaptive, and RMSk1 > RMSk is defined as the convergence condition which can bring us the highest precision solution all the time. The simulations in bonnet polishing demonstrate that this algorithm is stable, and has a high convergence ratio and fast solution speed (it usually takes a few seconds) even on an ordinary computer. Moreover, for the property of self-adaptiveness, it is easy to integrate into the computer program. The SAI method is not only applied to BP, but also can be applied to other subaperture deterministic polishing processes for the reason that all of them follow the same basic principle of the material removal. This work was financially supported by major national science and technology projects (Grant No. 2013ZX04006011-206). References 1. R. A. Jones, “Computer control for grinding and polishing,” Photonics Spectra 17, 34–39 (1983). 2. L. N. Allen and R. E. Keim, “An ion figuring system for large optic fabrication,” Proc. SPIE 1168, 33–50 (1989). 3. S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. B. Davison, T. J. Trebisky, S. T. Derigne, and B. B. Hille,
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