Research Article

Vol. 55, No. 4 / February 1 2016 / Applied Optics

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Restraint of path effect on optical surface in magnetorheological jet polishing TAN WANG,1,2 HAOBO CHENG,2,3,* WEIGUO ZHANG,1 HAO YANG,2,3

AND

WENTAO WU2,3

1

Xi’an Institute of Applied Optics, Xi’an 710065, China School of Optoelectronics, Joint Research Center for Optomechatronics Design and Engineering, Beijing Institute of Technology, Beijing 100081, China 3 Shenzhen Research Institute, Beijing Institute of Technology, Shenzhen 518057, China *Corresponding author: [email protected] 2

Received 16 November 2015; revised 28 December 2015; accepted 4 January 2016; posted 5 January 2016 (Doc. ID 253981); published 1 February 2016

Path effect on a polished surface always matches tool motions and manifests as mid-spatial frequency (MSF) errors in magnetorheological jet polishing processing. To control the path effect, extended methods and partconstrained paths are presented in this paper. The extended methods, Zernike extension and Neighbor– Gerchberg extension, are developed for building an extended surface with a weak edge effect in simulation. Under the constraint of the pitch principle, the unicursal part-constrained path is presented to enhance the randomness of the tool path, including path turns and dwell-point positions. Experiments are executed to validate the effectiveness of these measures for diminishing MSF errors. The peak to valley and root mean square of the surface are improved from 0.220λ and 0.047λ (λ
632.8 nm) to 0.064λ and 0.007λ, respectively, while simultaneously restricting the path effect. © 2016 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.55.000935

1. INTRODUCTION In sub-aperture polishing processing [computer-controlled polishing (CCP)] [1,2], high-precision components rely on the iterative process by applying a small footprint along a needed path. By using a small footprint, the low frequency surface component [3–5] can be removed easily, but the ripple errors increase, especially in magnetorheological jet polishing (MJP) [6] processing (with a range of footprints from the millimeter to sub-millimeter scale). If applying a purely repetitive path, the iterative process can tend to produce an obvious path track on the surface, i.e., the distribution of residual errors matches the path track. These ripple errors that belong to mid-spatial frequency (MSF) errors [5,7] are called the path effect here. These errors are serious in some applications such as intense laser systems and high-resolution image formation systems for scattering. According to the theory of CCP [8], the path effect mainly depends on the initial surface, removal function, and adopted paths. To alleviate this problem, two aspects have been proposed [1,4,5,8]. One is reducing the height of errors [8]. For inhibiting the height, some methods are proposed, including employing a steady Gauss-like removal function [9] and designing appropriate pitch for the tool path [8,10,11]. Steady removal is expected to increase the controllability of 1559-128X/16/040935-08$15/0$15.00 © 2016 Optical Society of America

waviness or texture, and the small size of the footprint exerts high removal capability [10]. Moreover, the principle of path pitch is built for a high-precision surface with many processing parameters in a synthetic way. The second proposed aspect is optimizing the period of ripple [1]. To yield a non-uniform distribution of residual error, strengthening the randomness of the path turn [1] and reducing the duplication ratio of path points [10] are the most effective methods. For strengthening the randomness of the path turn, a pseudo-random path is introduced to suppress the path effect. Also, with a modified operation (an appropriate translation or rotation), the small duplication ratio of path points is adopted for reducing the repeatability in different iteration processes. Based on the aforementioned research, some smoothing approaches are presented for optimizing path effect in this paper. To investigate the path effect in a full aperture, the edge effect should be suppressed with an extended surface map during calculation. The Zernike and Neighbor–Gerchberg extensions are then employed for extending the surfaces with different characters in Section 2. To increase the randomness of the tool path, the part-constrained paths, possessing random turns and the positions of dwell-points, are created under the limitation of the pitch principle in Section 3. Our process is completed with the part-constrained path in Section 4.

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Fig. 1. Path effect on the polished surface with different tool paths. (a) Scanning path, (b) Hilbert multi-path.

2. ANALYSIS OF PATH EFFECT Surfacing is modeled by assuming that the footprint of a polishing tool moves with a specific velocity along a certain path. Tool movement velocity is obtained as a function of the dwell time D
x; y. The footprint is described as removal function R
x; y in a sense. This polishing process is presented by a two-dimensional convolution [8,12,13]: ΔE
x; y  R
x; y  D
x; y ZZ R
x − x 0 ; y − y 0  · D
x 0 ; y 0 dx 0 dy 0 ; 

(1)

path

where ΔE
x; y is the redundant error on the surface, and
x − x 0 ; y − y 0  is the transformational relation of coordinate systems between the removal function and workpiece. With Preston’s hypothesis, the removal function R
x; y at point
x; y can be expressed by a linear equation [6,14,15]: R
x; y  kp
x; yv
x; y;

(2)

where k is the Preston coefficient that depends on surface chemistry, workpiece material properties, adopted slurry, etc.; p
x; y is the normal pressure; and v
x; y is the relative velocity between abrasives and the workpiece. In actual fabrication of the MJP process with a high-precision machine, the semi-continuous pattern [8] is always applied for the small and special removal function, e.g., an M-shaped removal function. For this pattern, the continuous motion is adopted in each path interval. With the traditional path (e.g., scanning path, concentric-circles path, and spiral path), the path effect is yielded on the polished surface distinctly with the constant pitch, as shown in Fig. 1(a). Moreover, the polished surface with a small height of residual errors is obtained by the scanning path (2 mm pitch) [8]. For using the scanning path, the periodical banding distribution of errors appears along the y direction. These errors will create two peaks [see Fig. 1(a)] in the power spectral density (PSD) curves. The left peak is located at the 0.5 mm−1 spatial frequency, while the right peak is at about 1 mm−1 . The spatial frequencies of the polished

surface are mainly induced by the path pitch. These indicate that the traditional path with constant distribution will easily generate evident path effect on the workpiece. To suppress the distinct peaks on the PSD curve, the classic fractal path (e.g., Hilbert path, Peano path, and Peano-like path) is adopted [3,10,11]. The result of the run with the Hilbert path is shown in Fig. 1(b) [10]. When using the Hilbert path, the distribution of errors presents a fragment shape with small peak to valley (PV), and the PSD is plotted with no marked peaks. However, the strong vibration and serrated shape are created along the PSD curve for the fragment error. These results show that the classic fractal path can control path effect to some extent and that the path with a disorder character is suited to polishing the high-precision components [1]. 3. STRATEGY TO REDUCE THE PATH EFFECT A. Surface Extension

In actual fabrication, the footprint of the tool always exceeds the edge of the workpieces for full-aperture polishing, which results in tool path extension [16]. However, the error form in the edge area has a discontinuity distribution (i.e., valid data in the inner shape and invalid data outside the workpieces). To implement the effective process, the initial surface error should be extended accordingly, and the extended surface should show a continuous distribution with weak edge effect. At the Lawrence Livermore National Laboratory, the optical surfaces are quantified as a function of spatial frequencies (PSD) [17]: low-spatial frequency errors or figure (v < 0.0303 mm−1 ), MSF errors or ripple (0.0303 mm−1 ≤ v < 8.3333 mm−1 ), and high-spatial frequency errors or roughness (8.3333 mm−1 ≤ v). Based on these errors, two methods of extension are designed for the surface with different characters: Zernike and Neighbor– Gerchberg. As is well known, the low-frequency errors can be described by the Zernike polynomial set [18]. Thus, the Zernike extension method is applied to the surface with abundant lowfrequency errors. As an example, in Fig. 2(a), the initial surface

Research Article

Vol. 55, No. 4 / February 1 2016 / Applied Optics

Fig. 2. Zernike extension. (a) Initial surface, (b) final surface.

(0.374λ PV, 0.077λ RMS) possessing valid data in the initial area should be extended to the final area. Also, the frequency errors of the initial surface are analyzed by ZYGO software (GPI.app), as listed in Table 1. The invalid data is distributed in the gray area. The final area is larger than the initial area, and the white area should be filled with fitted data. First, the final area is zoomed to the area of the unit circle. Second, with the valid data, the coefficients of the Zernike polynomial can be calculated in the final area. Next, the simulation surface in the final area will be built by the coefficients through Zernike polynomial fitting. The final surface, which fills the whole final area, is obtained when the fitted data in the initial area is replaced by the valid data of the initial surface. At last, the final area should be focused on the original size, as shown in

Table 1. Frequency Errors of the Initial Surface in Fig. 2(a) Residual Errors Low-frequency errors (LFE∕λ) Mid- to high-frequency errors (MHFE∕λ) Ratio (LFE/MHFE)

PV

RMS

0.337 0.098 3.439

0.073 0.011 6.636

Table 2. Frequency Errors of the Initial Surface in Fig. 3(a) Residual errors Low-frequency errors (LFE∕λ) Mid- to high-frequency errors (MHFE∕λ) Ratio (LFE/MHFE)

PV

RMS

0.200 0.073 2.740

0.039 0.008 4.875

937

Fig. 2(b), which obviously shows that the final surface possesses continuous distribution. The second method, the Neighbor–Gerchberg extension, can serve the surface with affluent mid- to high-frequency errors (MHFE). The surface with a quadrate outer and circle inner boundary is adopted as an example [see Fig. 3(a)], with the frequency errors listed in Table 2. Comparing the parameters in Tables 1 and 2, the ratios in Table 1 (3.439 for PV and 6.636 for RMS) are larger than those in Table 2 (2.740 for PV and 4.875 for RMS). These ratios indicate that there are more MHFE in the surface that is shown in Fig. 3(a). Using the Zernike extension, the discontinuity distributions are obviously located at the marginal regions, as shown in Fig. 3(b). Thus, the Gerchberg-pupil extension is employed [16,19]. Based on the Fourier transform and Nyquist criterion, the initial surface can be well extended by performing multiple iterations. However, the iterations are time consuming, especially for large matrices. To suppress the number of iterations, the pretreatment, Neighbor-average extrapolation, is utilized on the initial surface before the Gerchberg-pupil extension, which is defined as the Neighbor–Gerchberg extension here. The final surface, shown in Fig. 2(c), is obtained with a smooth shape in six iterations (many dozens of iterations are needed for the Gerchberg-pupil extension). In addition, compared to the Zernike extension, the Neighbor–Gerchberg extension is more time consuming. Therefore, the two methods suit different surfaces in optical surfacing. B. Part-Constrained Path

The path effect caused by the tool path cannot be eliminated, but it could be controlled by increasing the randomness of the path [1]. Two concepts are considered for strengthening path randomness: turning the path segments and lengthening the path segments. In the MJP process, the path should possess other basic characteristics, as follows: (1) The continuous path is always applied. (2) The path can fill in the whole desired area with relatively even distribution, but in local areas with heterogeneous density, i.e., the distribution of the path pitch shows a nonuniform map. (3) The desired area can possess many kinds of shapes in actual manufacture, e.g., one continuous zone and many continuous zones with an arbitrary boundary. (4) To reduce unnecessary damage, a small number of path segments pass the undesired area.

Fig. 3. Surface extension. (a) Initial surface, (b) final surface with Zernike extension, (c) final surface with Neighbor–Gerchberg extension.

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(5) The randomness of dwell-points should be controlled by the principle of path pitch. The distribution of dwell-points displays relative homogeneity to a certain extent. Considering the above requirements in MJP processing, the part-constrained path is designed. 1. Random Turns

The idea behind a tool path is that the dwell-points are arrayed in a certain order, which is also suited to the part-constrained path. According to the above requirements, the diagram for designing random turns is shown as Fig. 4. Some explanations should be given first [see Fig. 4(a)]. The path matrix is composed of valid and invalid dwell-points. The valid dwell-points (e.g., A9 ) are located in the desired area. The invalid dwell-points (e.g., A5 ) are distributed out of the desired area. The current path [(CP) e.g., CPfA9 ; A16 ; A22 ; A27 g] is a part of the final path (FP). The effective current path (EP) is a subset of the CP. The operating point (e.g., A16 ) is currently used to find the embedded point. The embedded point (e.g., A15 ) is a valid dwell-point and will be inserted into the current path. To select the appropriate embedded point, some limiting conditions defined as Terms are utilized [see Fig. 4(b)], as follows: (i) The intersection between valid dwell-points and the current path is empty. The embedded point belongs to valid dwell-points rather than the CP. Conversely, the operating point is an element of the CP. (ii) The embedded point is an arbitrary element in the 8neighborhood of operating point (e.g., embedded point A15 for operating point A16 ). To control the length of the path segment, the embedded point locates in the eight-point

neighborhood range of adjacent dwell-points in the current path (e.g., A15 for A9 and A16 ). (iii) The new current path (e.g., CPfA9 ; A15 ; A16 ; A22 ; A27 g) never crosses itself when the embedded point is inserted into the current path (e.g., CPfA9 ; A16 ; A22 ; A27 g). This embedded point should be deleted from the set of valid dwell-points. For the specific flow of the algorithm, the initial surface error of the workpiece and the M-shaped removal function of MJP are measured e.g., by an optical interferometer. With initial surface and removal function data, the path matrix can be built by the principle of path pitch [8]. The initial vector (IP, N elements) is composed of valid dwell-points in an arbitrary order. The iteration starts with the empty current path (CPfg) and empty effective current path (EPfg), as illustrated in Fig. 4(c). (1) Finding the start path point. The first point A is a random element of IP. The A is filled in CP and EP; the new CPfCPfg; Ag and EPfEPfg; Ag are obtained. Simultaneously, the new IP is yielded by deleting the element A; IP  DelfIP  Ag. (2) Selecting the operating point and embedded point in a random way. The operating point (P) is selected randomly in vector EP, and the position of P
P p  in vector CP is searched, P p  indexfCP  Pg. Under the restrictions of Term, the embedded point is determined in the set of IP. If there is no appropriate embedded point, this operating point will be removed from the vector EP, and the next step is (4). (3) Searching the embedded position in the current path and creating the new current path. The embedded position can be found by the P p in step (2). For the P p  1, the operating point locates at the forefront of CP, and the new

Fig. 4. Designing random turns. (a) Parameter specification, (b) term specification, (c) algorithm flow.

Research Article CP  fA; CPg. The operating point is the last number of CP, and the new CPfCP; Ag is obtained. For another condition, the embedded position is determined by the comparison between the distances [APb and APl , e.g., A9 A15 and A15 A22 in Fig. 4(b)] under the limitation of Term. If the embedded position does not exist, the operating point will be removed from the vector EP, and the new CP; IP
IP  DelfIP  Ag) and EP
EP  fEP; Ag can be obtained. (4) Judging the iteration. If the element number of CP is N , the iteration terminates. Otherwise, step (1) will be run when vector EP is empty or step (2) is implemented. As an example, a path with random turns is designed for the surface (size is 36 mm × 40 mm, peak is 0.237λ, valley is −0.075λ) shown in Fig. 3(c). Figure 5(a) shows the desired area with interior perforation that is planed with the surface greater than −0.025λ. Furthermore, the path possessing 0.5 mm pitch can fill the full desired area with a random form. If the errors that are larger than 0λ need to be removed, the boundary of the desired area shows a complex shape, as illustrated in Fig. 5(b). With the 0.5 mm pitch, the path is created through the above algorithm. It is obvious to see that the path not only meets the basic requirements for optical manufacture but also strengthens the randomness of turns. 2. Random Positions of Dwell-Points

In our previous work, an effective approach to controlling the path effect is reducing the duplication ratio of paths in different processes [10]. To enhance the capability of smoothing, integral translation is implemented on the path points, but the relative positions between each point do not change. Further, a new approach that can improve the randomness of dwell-point position is designed here. Simultaneously, the randomness is constrained by the principle of path pitch [8].

Fig. 5. Simulation for random turns. (a) Desired area with interior perforation, (b) two separated desired areas.

Vol. 55, No. 4 / February 1 2016 / Applied Optics

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It is well known that the appropriate pitch for precision polishing relies on the initial surface and removal function. The quantitative analysis is built on our previous work and is called the principle of path pitch [8]. For simplicity, the processing condition in all the lines of the scanning path [removal profile of one track is f
x, pitch is d , and total depth is D; see Fig. 6] is evenly spaced straight lines on a planar surface with a semicontinuous pattern. Therefore, the formulation of residual error E rf on the polished surface can be expressed as below [8]: E rf
x 

m X

f
x − id  · T i ;

(3)

in

where T i is the dwell time. n and m are integers, which should satisfy the following conditions: x − wpt ∕2 x − wpt ∕2 ≤n<  1; d d x  wpt ∕2 x  wpt ∕2 −1