Multiplex path for magnetorheological jet polishing with vertical impinging Tan Wang,1 Haobo Cheng,1,* Yong Chen,1 and Honyuen Tam2 1
School of Optoelectronics, Joint Research Center for Optomechatronics Design and Engineering, Beijing Institute of Technology, Beijing 100081, China
2
Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China *Corresponding author: [email protected] Received 13 December 2013; revised 14 February 2014; accepted 18 February 2014; posted 19 February 2014 (Doc. ID 202730); published 25 March 2014
We report a way to shape surfaces by optimizing the path instead of changing the removal function of a polishing tool for magnetorheological jet polishing (MJP). The M-shaped removal function of MJP generates a track with a W-shaped profile along one path. However, applying two parallel paths with appropriate line spacing can obtain a track with V-shaped profile, which has a removal distribution similar to that by using the Gaussian removal function along one path. Based on this, a multiplex path applying an M-shaped removal function is constructed in an actual process. A transformation model describing the relationship between the M-shaped removal function and the Gaussian removal function is established, which is crucial to determine the velocity function on the multiplex path. By using the M-shaped removal function, we have planned new processing steps by applying the multiplex path and the velocity function for full aperture polishing. Polishing performance is designed and demonstrated on two K9 work-pieces with different multiplex paths. The form error on 23 mm diameter is decreased from 0.256λ PV (λ 632.8 nm) and 0.068λ RMS to 0.038λ PV and 0.005λ RMS with scanning multiplex path. Results indicate that this method of path optimization is suitable for optical manufacturing. © 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.53.002012
1. Introduction
Shaping and polishing complex optical surfaces has always been a challenge in the optical fabrication industry. Specifically, the mechanical interference, and finishing of steep and concave sections of a-spheres, is difficult. A new technique called magnetorheological jet polishing (MJP) has been developed to address this challenge. For example, Kordonski et al. [1] illustrated the basic principles and published further works about MJP, and Tricard et al. [2] indicated this technique was successful in optical polishing,
1559-128X/14/102012-08$15.00/0 © 2014 Optical Society of America 2012
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
and especially went well with components with small volume and thickness, and steep local slopes. The removal model of MJP is constructed based on the principle that “the rate of polishing of the glass is proportional to the rate at which work is done on each unit area of the glass.” This principle indicates that the removal character in a vertical jet presents annular distribution with a W-shaped profile [3,4]. The detailed studies by Booij et al. [5] presented the removal characters, which show that in research of track scanning a nozzle with respect to the surface has a W-shaped profile (two deep trenches with a less deep part in between). According to computercontrolled polishing theory [6], surface error decreases monotonously, and high-frequency error also decreases with a removal function possessing
Gaussian character. Therefore, the Gaussian removal function is applied in the simulation step for obtaining the parameters of precision optics polishing, and the polishing tool possessing the same removal function is always utilized in an actual polishing step with regular tool path. Researchers have proposed many methods to obtain the ideal removal function with Gaussian-like character by the W-shaped removal function, e.g., Fang et al. [6] presented a method that keeps the nozzle fixed while rotating the work-piece, and applying eccentric rotation motion for a normal fixed nozzle, which is discussed in our previous work [7]. However, caution should be considered when these methods are used in designing complex mechanical and liquid systems, which are not easily controlled during polishing. This paper presents a novel approach to achieving the removal distribution of the Gaussian removal function by multiplex path. With this path and the M-shaped removal function, the distribution of material removal applying the Gaussian removal function along a regular path can be obtained easily. The multiplex path is discussed in Section 2. Specifically, the feasibility analysis, the transformation model between the M-shaped removal function for the multiplex path and the corresponding Gaussian removal function for the regular path, and the new processing steps applied on the optimized path are provided. In Section 3, experiments are designed on K9 glasses by using the proposed approach. Finally, a summary of this work is also given.
Ex; y Rx; y Dx; y ZZ Rx − x0 ; y − y0 · Dx0 ; y0 dx0 dy0 :
(2)
path
Here, Ex; y is the amount of removed material on the surface, and x − x0 ; y − y0 is the transformational relation of coordinate systems between the removal function and work-piece. The ideal removal function shows a Gaussian character with peak value in the middle. The Gaussian shape removal function can be given by RG x; y B exp−u · x2 y2 ;
(3)
where B is the peak removal, and u is the parameter of the Gaussian shape. When the tool moves along a part of the target path (also called the regular path, e.g., scanning path, concentric-circles path, spiral path, peano-like path [9], and pseudorandom paths [10]) with constant velocity [see Fig. 1(a)], the distribution of removal material takes a V-shaped profile, which has maximum removal that gradually decreases as the offset distance increases [see Fig. 1(b)]. H Gm and SG are, respectively, the maximum depth and the crosssectional areas of material removal. The removal function of MJP in a normal impact presents an annular distribution with an M-shaped profile, which has a valley in the middle and two
2. Mathematical Model of Optimized Path for MJP A.
Material Removal Model
According to the studies of Kordonski et al. [1] and Shi et al. [8], the removal function Rx; y can be described as R KPV K
F D V K r; μ μS
(1)
where K is a constant that depends on the material properties of the abrasive, work-piece, and other process parameters. P is the shear stress at point x; y, V is the relative velocity between the abrasives and work-piece at point x; y, F is the frictional force between the work-piece and the polishing lap, S is the polishing zone, μ is the coefficient of friction, and Dr is the power of the polishing tool at the surface. Material removal is modeled by assuming that a tool with a removal function Rx; y moves with a specific velocity along a certain path. Tool movement velocity is obtained as a function of the dwell time Dx; y. Therefore, material removal depends on the removal function and the dwell time on a certain part of the surface. This process can be presented by a two-dimensional convolution between the removal and dwell-time functions along a processing path:
Fig. 1. Effect of Gaussian removal function after translation in the x direction. (a) Schematic view of the movement of the footprint. (b) Cross section of the track along the y direction. 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
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Fig. 2. Regular removal function of MJP.
symmetrical peaks at the adjacent margins (see Fig. 2; this regular removal function is obtained by using a cylindrical nozzle with 0.9 mm diameter, and the distance between the two peaks is about 1.4 mm). If this type of tool moves along a path [Fig. 1(a)], then the track has a degenerative Wshaped profile (two deep trenches with a shallow part in between) [5]. One way to obtain the track as shown in Fig. 1 is changing the M-shaped removal function into a Gaussian-like removal function. The common method, which entails taking a nozzle and rotating it around the target point, can yield the ideal removal function as shown in Fig. 3(a), but it is difficult to implement because of the complex mechanical and liquid systems [7]. Therefore, another way to optimize the path is presented and adopted. Instead of moving along the target path with Gaussian-like removal function, the regular footprint of MJP moves along two real paths [see Fig. 3(b)] with appropriate line spacing. With this approach, the distribution applying the Gaussian removal function can be obtained by the M-shaped removal function. This optimized tool path is called the “multiplex path.” Without loss of generality, the linear path is adopted as an example. Real paths 1 and 2 are located on the opposite sides parallel to the target path (distance is
dl). Real paths 1 and 2 form the multiplex path. For the linear path or the path with a large curvature radius, the distance between real path 1 and the target path is dl∕2. The different profiles [see Figs. 4(a) and 4(b)] and depths [see Fig. 4(c)] of the tracks can be obtained by adjusting the dl on the basis of the M-shaped removal function. Figure 4(a) shows that the profile has a U-shaped form for dl 0.6 mm. When dl increases, the profile will change from U-shaped form to V-shaped form. Specifically, the distribution of material removal is V-shaped for dl 0.8 mm [see Fig. 4(b)], which is similar to the distribution in Fig. 1. The profiles gradually change back into a U-shaped form, which is wider and shallower than the form with a smaller dl. In addition, the depth of the track does not depend linearly on dl. More removal materials are present along the target path for dl < 1.6 mm. And the maximum depth gradually decreases as the dl increases. Especially, for dl > 2.2 mm, the maximum depth gradually flattens out with increasing dl. This result is deceptive because the footprints begin to separate. H Mm and SM are, respectively, the maximum depth and the cross-sectional areas of material removal with an M-shaped removal function along the multiplex path [see Fig. 4(b)].
Fig. 3. Methods to create V-shaped profile track by regular removal function of MJP. (a) Optimizing removal function. (b) Using multiplex path. 2014
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Normalized depth
0
where H Gm , SG , H Mm , and SM are given as
-0.2
Z
0.6
H M xA ; y
-0.4
RP1
Rw xA − x01 ; y − y01 Dw1 x01 ; y01 dx01
Z
-0.6 3.6
2.3
1.4
RP2
Rw xA − x02 ; y − y02 Dw2 x02 ; y02 dx02 ; (5)
-0.8 -1
1
2
3
4 5 y position /mm
6
7
H Mm maxH M xA ; y;
8
(6)
Z SM
Normalized depth
0 -0.2
S
Z
M
H G xA ; y
-0.4
HMm
Path
H M xA ; ydy;
(7)
RG xA − x0 ; y − y0 DG x0 ; y0 dx0 ; (8)
-0.6 -0.8 -1
0
1
2
3 4 y position /mm
5
6
7
DG xA ; yA Dw1 xA1 ; yA1 Dw2 xA2 ; yA2 ;
(9)
H Gm maxH G xA ; y;
(10)
Z
-1
SG
H G xA ; ydy:
(11)
Normalized depth
-0.9 -0.8 -0.7 -0.6 -0.5 0.5
1
1.5 2 line spacing dl /mm
2.5
3
Fig. 4. Effect of the M-shaped removal function after translation in the x direction along the multipex path. (a) Cross sections of the tracks with different dl along y. (b) Track profile for dl 0.8 mm. (c) Depth of the tracks with different dl.
Therefore, for dl 0.8 mm, the track of the multiplex path for the M-shaped removal function is equivalent to that of the target path of the Gaussian removal function. In line with this, the Gaussian removal function is the corresponding function of the M-shaped one. B. Transformation Model of Gaussian Removal Function for the M-shaped Removal Function
Here RP1 and RP2 are, respectively, the real paths 1 and 2 of the multiplex path; simultaneously, Dw1 and Dw2 are the dwell-time functions on RP1 and RP2 with the M-shaped removal function Rw . DG is the dwell-time function on the target path with corresponding Gaussian function RG . In particular, for the linear path or the path with a large curvature radius, Dw1 Dw2 DG ∕2. According to Eqs. (4)–(11), the B and u in Eq. (3) can be calculated to express the corresponding Gaussian removal function. If the footprint of MJP across point A [Fig. 3(b)] with constant velocity v along the path has large curvature radius, then H Gm and SG can be described as
nR o H Gm max B · exp−ux2 y2 · Tdx TB uπ 1∕2 : R SG B · exp−ux2 y2 · Tdxdy TB uπ 12
Here, T is the average time, which is equal to 2∕v. B and u can be calculated on the basis of Eqs. (4) and (12): 8 2 < B H Mm
To obtain the corresponding Gaussian shape of the M-shaped removal function, the profile parameters of the two distributions [see Figs. 1(b) and 4(b)] should satisfy the condition
H Gm H Mm ; SG SM
SM T
π : u H 2Mm 2
C.
4
:
13
SM
Processing Steps with Multiplex Path for MJP
Figure 5(a) shows a simple diagram of the polishing program that applies the multiplex path for MJP. 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
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Fig. 5. Processing steps with multiplex path. (a) Diagram of the simple simulation steps. (b) Scanning path as an example.
The surface error of the work-piece and the Mshaped removal function of MJP are measured by, e.g., an optical interferometer. The data for the initial surface error and the M-shaped removal function are input to the computer. Then the appropriate target path is calculated [without loss of generality, the scanning path is adopted as an example in Fig. 5(b); d is line spacing of the target path] using the principle in the work of Tam and Cheng [11]. The distribution of the material removal with V-shaped profile is obtained by adjusting the dl through experiment or simulation. Therefore, the corresponding Gaussian removal function of the M-shaped removal function can be calculated through Eq. (4) by using H Mm and SM . The multiplex path is also simultaneously obtained [see Fig. 5(b), which is formed by the real paths]. The dwell-time function on the target path is calculated by using the best-fit equations on the basis of the initial surface error, target path, and Gaussian removal function. The velocity on the target path, which is appropriate for computer numerical control, can be established by this dwell-time function. The corresponding velocity on the multiplex path is computed by using the velocity of the target path via Eq. (9). A polishing program is then implemented on the work-piece with M-shaped removal function, multiplex path, and the corresponding velocity function. It is obvious to see that the Gaussian removal function and the target path are only applied in the simulation step to calculate the parameters for polishing, but the M-shaped removal function and multiplex path are used in the actual polishing step. This approach not only retains the advantages of the Gaussian removal function for precise polishing, but also is easy to implement in actual MJP processing. Changing the M-shaped removal function of MJP into a Gaussian-like removal function is complicated, e.g., the eccentric rotation motion for nozzle and coil, transmitting a steady current and slurry form of the static system to the coil and nozzle, and designing and implementing the gesture and position for fixing the nozzle. Thus, the approach presented in this paper is simpler than the above-mentioned approaches. This is because the regular removal function, the 2016
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
M-shaped removal function, is applied for actual processing. 3. Experimental Validation and Analysis A. Corresponding Gaussian Shape of the Actual M-shaped Removal Function
A polishing liquid containing 2.3 wt. % CeO2 particles (2 μm) was chosen. The diameter of the cylindrical nozzle was 0.95 mm. The pump pressure showed 0.4 Mpa, and the stand-off distance was 66 mm. An annular footprint with a W-shaped profile (see Fig. 6) was obtained for 3 min processing time. The preceding experiment was designed to investigate the dl with material removal, as shown in Fig. 7. The 0.2 mm∕s linear traverse velocity of the nozzle was chosen with dl, ranging from 0.2 to 2.4 mm. It is obvious that the material removal has a V-shaped profile when dl 0.8 mm, and the maximum depth H Mm and width of the profile are about 0.1702λ and 4.3 mm, respectively [see Figs. 7(a) and 7(b), which are similar to the curves at the same condition in Fig. 4]. Therefore, the area SM of the profile is about 0.37λ (the unit of the width is 1 in area computation). On the basis of this finding, determining the corresponding Gaussian removal function [from Eq. (13), B 7.83 × 10−3 λ∕s, u 0.665] is easy: Rx; y 7.83 × 10−3 × exp−0.665 × x2 y2 : (14) With the corresponding Gaussian removal function as expressed in Eq. (14), a simulation is made to obtain the distribution of the track along
Fig. 6. Footprint of a nozzle on K9 optical glass.
Fig. 7. Dependence on the dl of polishing shape. (a) Profile curves of tracks. (b) Depths of tracks with different dl. (c) V-shaped profile curves of tracks for dl 0.8 mm.
the regular path. Comparing to the distribution of the regular footprint moving along the multiplex path [Fig. 7(a), dl 0.8 mm], it is important to notice that profiles possess similar shapes to those shown in Fig. 7(c). Considering the convergence efficiency of polishing and the diameter of the
Table 1.
corresponding Gaussian removal function [Fig. 7(c), approach to 6 mm], the line spacing of the target path (d 2 mm) is adopted for the multiplex path. According to the above analysis, the parameters for polishing processing are exhibited in Table 1.
Parameters of Polishing Processing
Parameter Mass fraction of CeO2 particles in fluid (%) Diameter of CeO2 particle (μm) Diameter of nozzle (mm) Pressure (MPa) Stand-off distance (mm) Peak of corresponding Gaussian removal function (B) (λ∕s) Parameter of corresponding Gaussian removal function (u) Line spacing of the target path (d) (mm) Value of dl in multiplex path (mm)
Value 2.3 2 0.95 0.4 66 7.83 × 10−3 0.665 2 0.8
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B.
Surfacing Polishing with Scanning Multiplex Path
The scanning target path (d 2 mm; diameter of coverage area is 26 mm) was applied for the surfacing polishing of a 23 mm diameter plane K9 optical glass, with 0.256λ PV and 0.068λ RMS, which were measured by a ZYGO interferometer [see Fig. 8(a)]. Based on the target path, the scanning multiplex path with the polishing parameters (in Table 1, d1 d2 dl∕2 0.4 mm, dl 0.8 mm) was selected for polishing [see Fig. 8(b)]. After computing for the dwell time by using the corresponding Gaussian removal function, the figure of the initial surface and the target path, the velocity function on the multiplex path can be obtained using Eq. (9). The polishing process was conducted on the work-piece. In the processing, the polishing tool, possessing an M-shaped removal function, travels along the multiplex path with the velocity function. After three iterations, MJP processing delivers a surface waviness improvement from 0.256λ PV and 0.068λ RMS down to 0.038λ PV and 0.005λ RMS [Fig. 9(a)]. Figures 9(b) and 9(c) show the average power spectral density (PSD) curves along the x (horizontal) and y (vertical) directions, respectively. Contrasting the PSD curves, there is mass of spatial frequency errors before polishing, which indicates that the spatial frequency errors are suppressed well. Moreover, there are two small peaks [in parts 1 and 2; see Fig. 9(c)] in the PSD curves along the y direction of the final surface. For one peak in the part 1, the PSD of the polished surface is approximately equal to that before being polished with about
Fig. 9. Polishing results. (a) Final surface: 0.038λ PV, 0.005λ RMS. (b) Average PSD curves along x direction. (c) Average PSD curves along y direction.
0.51∕mm spatial frequency [Freq; see Fig. 9(c)]. For the other peak in part 2, the spatial frequency is about 1 1∕mm Freq [Fig. 9(c)]. These indicate that the spatial frequencies of the polished surface are mainly induced by the line spacing of the target path (d 2 mm). The case of surface fabrication verifies that the multiplex path is effective and valid. In addition, the figure of the final surface is very homogeneous near the edge of the work-piece. This result is deceptive because there is no edge effect (this was also proposed by Booij et al. [5] and Walker et al. [12]). 4. Conclusions Fig. 8. Experiment on the 23 mm diameter work-piece with grid multiplex path. (a) Form error of original work-piece: 0.256λ PV, 0.068λ RMS. (b) Multiplex path. 2018
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
This paper aims to optimize the MJP process. A multiplex path has been presented instead of optimizing the removal function. The idea behind the multiplex path is that a W-shaped footprint along
two parallel paths, which possess appropriate dl, can obtain a track that has the same character as that when the Gaussian removal function along one path is applied. The distributions of material removal with different dl of the multiplex path are discussed. Results show that the methods using the regular path with a Gaussian removal function and the method using the multiplex path with an M-shaped removal function obtain the same material removal. The constructed transformation model is built, and describes the relationship between the M-shaped removal function and the Gaussian removal function. With the corresponding Gaussian removal function and the regular path, the appropriate dwell time for precise polishing is obtained. The velocity function on the multiplex path for the M-shaped removal function can be obtained by this dwell time. The newly designed polishing program is then implemented on the work-piece with the M-shaped removal function, the multiplex path, and the velocity function in actual processing. On the basis of this approach, we designed and demonstrated deterministic polishing performances to correct plane K9 optical glasses by using the Mshaped removal function. With the scanning multiplex path, the form errors of the K9 optical glass (23 mm diameter) decreased from 0.256λ PV and 0.068λ RMS to 0.038λ PV and 0.005λ RMS. Similarly, the spatial frequency errors are suppressed to a great degree. These results demonstrate that the new polishing method influences the final surface shape as predicted. Therefore, optimizing the path of the polishing process presents an alternative approach to precise optical manufacturing. This work was supported by the National Natural Science Foundation of China (grant no. 61222506) and the Specialized Research Fund
for the Doctoral Program of Higher Education (grant no. 20131101110026). References 1. W. I. Kordonski, A. B. Shorey, and M. Tricard, “Magnetorheological jet (MR JetTM) finishing technology,” J. Fluids Eng. 128, 20–26 (2006). 2. M. Tricard, W. I. Kordonski, and A. B. Shorey, “Magnetorheological jet finishing of conformal, freeform and steep concave optics,” CIRP Annals 55, 309–312 (2006). 3. W. Kordonski and A. Shorey, “Magnetorheological (MR) jet finishing technology,” J. Intell. Mater. Syst. Struct. 18, 1127–1130 (2007). 4. T. Wang, H. B. Cheng, Y. Chen, Y. P. Feng, Z. C. Dong, and H. Y. Tam, “Correction of remounting errors by masking reference points in small footprint polishing process,” Appl. Opt. 52, 7851–7858 (2013). 5. S. M. Booij, H. V. Brug, J. J. M. Braat, and O. W. Fähnle, “Nanometer deep shaping with fluid jet polishing,” Opt. Eng. 41, 1926–1931 (2002). 6. H. Fang, P. J. Guo, and J. C. Yu, “Optimization of the material removal in fluid jet polishing,” Opt. Eng. 45, 053401 (2006). 7. T. Wang, H. B. Cheng, Z. C. Dong, and H. Y. Tam, “Removal character of vertical jet polishing with eccentric rotation motion using magnetorheological fluid,” J. Mater. Process. Technol. 213, 1532–1537 (2013). 8. C. Y. Shi, J. H. Yuan, F. Wu, X. Hou, and Y. J. Wan, “Material removal model of vertical impinging in fluid jet polishing,” Chin. Opt. Lett. 8, 323–325 (2010). 9. H. Y. Tam, H. B. Cheng, and Z. C. Dong, “Peano-like paths for subaperture polishing of optical aspherical surfaces,” Appl. Opt. 52, 3624–3636 (2013). 10. C. R. Dunn and D. D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16, 18942–18949 (2008). 11. H. Y. Tam and H. B. Cheng, “An investigation of the effects of the tool path on the removal of material in polishing,” J. Mater. Process. Technol. 210, 807–818 (2010). 12. D. D. Walker, G. Y. Yu, H. Y. Li, W. Messelink, R. Evans, and A. Beaucamp, “Edges in CNC polishing: from mirrorsegments towards semiconductors, paper 1: edges on processing the global surface,” Opt. Express 20, 19787–19798 (2012).
1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
2019
School of Optoelectronics, Joint Research Center for Optomechatronics Design and Engineering, Beijing Institute of Technology, Beijing 100081, China
2
Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong, China *Corresponding author: [email protected] Received 13 December 2013; revised 14 February 2014; accepted 18 February 2014; posted 19 February 2014 (Doc. ID 202730); published 25 March 2014
We report a way to shape surfaces by optimizing the path instead of changing the removal function of a polishing tool for magnetorheological jet polishing (MJP). The M-shaped removal function of MJP generates a track with a W-shaped profile along one path. However, applying two parallel paths with appropriate line spacing can obtain a track with V-shaped profile, which has a removal distribution similar to that by using the Gaussian removal function along one path. Based on this, a multiplex path applying an M-shaped removal function is constructed in an actual process. A transformation model describing the relationship between the M-shaped removal function and the Gaussian removal function is established, which is crucial to determine the velocity function on the multiplex path. By using the M-shaped removal function, we have planned new processing steps by applying the multiplex path and the velocity function for full aperture polishing. Polishing performance is designed and demonstrated on two K9 work-pieces with different multiplex paths. The form error on 23 mm diameter is decreased from 0.256λ PV (λ 632.8 nm) and 0.068λ RMS to 0.038λ PV and 0.005λ RMS with scanning multiplex path. Results indicate that this method of path optimization is suitable for optical manufacturing. © 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.53.002012
1. Introduction
Shaping and polishing complex optical surfaces has always been a challenge in the optical fabrication industry. Specifically, the mechanical interference, and finishing of steep and concave sections of a-spheres, is difficult. A new technique called magnetorheological jet polishing (MJP) has been developed to address this challenge. For example, Kordonski et al. [1] illustrated the basic principles and published further works about MJP, and Tricard et al. [2] indicated this technique was successful in optical polishing,
1559-128X/14/102012-08$15.00/0 © 2014 Optical Society of America 2012
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
and especially went well with components with small volume and thickness, and steep local slopes. The removal model of MJP is constructed based on the principle that “the rate of polishing of the glass is proportional to the rate at which work is done on each unit area of the glass.” This principle indicates that the removal character in a vertical jet presents annular distribution with a W-shaped profile [3,4]. The detailed studies by Booij et al. [5] presented the removal characters, which show that in research of track scanning a nozzle with respect to the surface has a W-shaped profile (two deep trenches with a less deep part in between). According to computercontrolled polishing theory [6], surface error decreases monotonously, and high-frequency error also decreases with a removal function possessing
Gaussian character. Therefore, the Gaussian removal function is applied in the simulation step for obtaining the parameters of precision optics polishing, and the polishing tool possessing the same removal function is always utilized in an actual polishing step with regular tool path. Researchers have proposed many methods to obtain the ideal removal function with Gaussian-like character by the W-shaped removal function, e.g., Fang et al. [6] presented a method that keeps the nozzle fixed while rotating the work-piece, and applying eccentric rotation motion for a normal fixed nozzle, which is discussed in our previous work [7]. However, caution should be considered when these methods are used in designing complex mechanical and liquid systems, which are not easily controlled during polishing. This paper presents a novel approach to achieving the removal distribution of the Gaussian removal function by multiplex path. With this path and the M-shaped removal function, the distribution of material removal applying the Gaussian removal function along a regular path can be obtained easily. The multiplex path is discussed in Section 2. Specifically, the feasibility analysis, the transformation model between the M-shaped removal function for the multiplex path and the corresponding Gaussian removal function for the regular path, and the new processing steps applied on the optimized path are provided. In Section 3, experiments are designed on K9 glasses by using the proposed approach. Finally, a summary of this work is also given.
Ex; y Rx; y Dx; y ZZ Rx − x0 ; y − y0 · Dx0 ; y0 dx0 dy0 :
(2)
path
Here, Ex; y is the amount of removed material on the surface, and x − x0 ; y − y0 is the transformational relation of coordinate systems between the removal function and work-piece. The ideal removal function shows a Gaussian character with peak value in the middle. The Gaussian shape removal function can be given by RG x; y B exp−u · x2 y2 ;
(3)
where B is the peak removal, and u is the parameter of the Gaussian shape. When the tool moves along a part of the target path (also called the regular path, e.g., scanning path, concentric-circles path, spiral path, peano-like path [9], and pseudorandom paths [10]) with constant velocity [see Fig. 1(a)], the distribution of removal material takes a V-shaped profile, which has maximum removal that gradually decreases as the offset distance increases [see Fig. 1(b)]. H Gm and SG are, respectively, the maximum depth and the crosssectional areas of material removal. The removal function of MJP in a normal impact presents an annular distribution with an M-shaped profile, which has a valley in the middle and two
2. Mathematical Model of Optimized Path for MJP A.
Material Removal Model
According to the studies of Kordonski et al. [1] and Shi et al. [8], the removal function Rx; y can be described as R KPV K
F D V K r; μ μS
(1)
where K is a constant that depends on the material properties of the abrasive, work-piece, and other process parameters. P is the shear stress at point x; y, V is the relative velocity between the abrasives and work-piece at point x; y, F is the frictional force between the work-piece and the polishing lap, S is the polishing zone, μ is the coefficient of friction, and Dr is the power of the polishing tool at the surface. Material removal is modeled by assuming that a tool with a removal function Rx; y moves with a specific velocity along a certain path. Tool movement velocity is obtained as a function of the dwell time Dx; y. Therefore, material removal depends on the removal function and the dwell time on a certain part of the surface. This process can be presented by a two-dimensional convolution between the removal and dwell-time functions along a processing path:
Fig. 1. Effect of Gaussian removal function after translation in the x direction. (a) Schematic view of the movement of the footprint. (b) Cross section of the track along the y direction. 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
2013
Fig. 2. Regular removal function of MJP.
symmetrical peaks at the adjacent margins (see Fig. 2; this regular removal function is obtained by using a cylindrical nozzle with 0.9 mm diameter, and the distance between the two peaks is about 1.4 mm). If this type of tool moves along a path [Fig. 1(a)], then the track has a degenerative Wshaped profile (two deep trenches with a shallow part in between) [5]. One way to obtain the track as shown in Fig. 1 is changing the M-shaped removal function into a Gaussian-like removal function. The common method, which entails taking a nozzle and rotating it around the target point, can yield the ideal removal function as shown in Fig. 3(a), but it is difficult to implement because of the complex mechanical and liquid systems [7]. Therefore, another way to optimize the path is presented and adopted. Instead of moving along the target path with Gaussian-like removal function, the regular footprint of MJP moves along two real paths [see Fig. 3(b)] with appropriate line spacing. With this approach, the distribution applying the Gaussian removal function can be obtained by the M-shaped removal function. This optimized tool path is called the “multiplex path.” Without loss of generality, the linear path is adopted as an example. Real paths 1 and 2 are located on the opposite sides parallel to the target path (distance is
dl). Real paths 1 and 2 form the multiplex path. For the linear path or the path with a large curvature radius, the distance between real path 1 and the target path is dl∕2. The different profiles [see Figs. 4(a) and 4(b)] and depths [see Fig. 4(c)] of the tracks can be obtained by adjusting the dl on the basis of the M-shaped removal function. Figure 4(a) shows that the profile has a U-shaped form for dl 0.6 mm. When dl increases, the profile will change from U-shaped form to V-shaped form. Specifically, the distribution of material removal is V-shaped for dl 0.8 mm [see Fig. 4(b)], which is similar to the distribution in Fig. 1. The profiles gradually change back into a U-shaped form, which is wider and shallower than the form with a smaller dl. In addition, the depth of the track does not depend linearly on dl. More removal materials are present along the target path for dl < 1.6 mm. And the maximum depth gradually decreases as the dl increases. Especially, for dl > 2.2 mm, the maximum depth gradually flattens out with increasing dl. This result is deceptive because the footprints begin to separate. H Mm and SM are, respectively, the maximum depth and the cross-sectional areas of material removal with an M-shaped removal function along the multiplex path [see Fig. 4(b)].
Fig. 3. Methods to create V-shaped profile track by regular removal function of MJP. (a) Optimizing removal function. (b) Using multiplex path. 2014
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Normalized depth
0
where H Gm , SG , H Mm , and SM are given as
-0.2
Z
0.6
H M xA ; y
-0.4
RP1
Rw xA − x01 ; y − y01 Dw1 x01 ; y01 dx01
Z
-0.6 3.6
2.3
1.4
RP2
Rw xA − x02 ; y − y02 Dw2 x02 ; y02 dx02 ; (5)
-0.8 -1
1
2
3
4 5 y position /mm
6
7
H Mm maxH M xA ; y;
8
(6)
Z SM
Normalized depth
0 -0.2
S
Z
M
H G xA ; y
-0.4
HMm
Path
H M xA ; ydy;
(7)
RG xA − x0 ; y − y0 DG x0 ; y0 dx0 ; (8)
-0.6 -0.8 -1
0
1
2
3 4 y position /mm
5
6
7
DG xA ; yA Dw1 xA1 ; yA1 Dw2 xA2 ; yA2 ;
(9)
H Gm maxH G xA ; y;
(10)
Z
-1
SG
H G xA ; ydy:
(11)
Normalized depth
-0.9 -0.8 -0.7 -0.6 -0.5 0.5
1
1.5 2 line spacing dl /mm
2.5
3
Fig. 4. Effect of the M-shaped removal function after translation in the x direction along the multipex path. (a) Cross sections of the tracks with different dl along y. (b) Track profile for dl 0.8 mm. (c) Depth of the tracks with different dl.
Therefore, for dl 0.8 mm, the track of the multiplex path for the M-shaped removal function is equivalent to that of the target path of the Gaussian removal function. In line with this, the Gaussian removal function is the corresponding function of the M-shaped one. B. Transformation Model of Gaussian Removal Function for the M-shaped Removal Function
Here RP1 and RP2 are, respectively, the real paths 1 and 2 of the multiplex path; simultaneously, Dw1 and Dw2 are the dwell-time functions on RP1 and RP2 with the M-shaped removal function Rw . DG is the dwell-time function on the target path with corresponding Gaussian function RG . In particular, for the linear path or the path with a large curvature radius, Dw1 Dw2 DG ∕2. According to Eqs. (4)–(11), the B and u in Eq. (3) can be calculated to express the corresponding Gaussian removal function. If the footprint of MJP across point A [Fig. 3(b)] with constant velocity v along the path has large curvature radius, then H Gm and SG can be described as
nR o H Gm max B · exp−ux2 y2 · Tdx TB uπ 1∕2 : R SG B · exp−ux2 y2 · Tdxdy TB uπ 12
Here, T is the average time, which is equal to 2∕v. B and u can be calculated on the basis of Eqs. (4) and (12): 8 2 < B H Mm
To obtain the corresponding Gaussian shape of the M-shaped removal function, the profile parameters of the two distributions [see Figs. 1(b) and 4(b)] should satisfy the condition
H Gm H Mm ; SG SM
SM T
π : u H 2Mm 2
C.
4
:
13
SM
Processing Steps with Multiplex Path for MJP
Figure 5(a) shows a simple diagram of the polishing program that applies the multiplex path for MJP. 1 April 2014 / Vol. 53, No. 10 / APPLIED OPTICS
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Fig. 5. Processing steps with multiplex path. (a) Diagram of the simple simulation steps. (b) Scanning path as an example.
The surface error of the work-piece and the Mshaped removal function of MJP are measured by, e.g., an optical interferometer. The data for the initial surface error and the M-shaped removal function are input to the computer. Then the appropriate target path is calculated [without loss of generality, the scanning path is adopted as an example in Fig. 5(b); d is line spacing of the target path] using the principle in the work of Tam and Cheng [11]. The distribution of the material removal with V-shaped profile is obtained by adjusting the dl through experiment or simulation. Therefore, the corresponding Gaussian removal function of the M-shaped removal function can be calculated through Eq. (4) by using H Mm and SM . The multiplex path is also simultaneously obtained [see Fig. 5(b), which is formed by the real paths]. The dwell-time function on the target path is calculated by using the best-fit equations on the basis of the initial surface error, target path, and Gaussian removal function. The velocity on the target path, which is appropriate for computer numerical control, can be established by this dwell-time function. The corresponding velocity on the multiplex path is computed by using the velocity of the target path via Eq. (9). A polishing program is then implemented on the work-piece with M-shaped removal function, multiplex path, and the corresponding velocity function. It is obvious to see that the Gaussian removal function and the target path are only applied in the simulation step to calculate the parameters for polishing, but the M-shaped removal function and multiplex path are used in the actual polishing step. This approach not only retains the advantages of the Gaussian removal function for precise polishing, but also is easy to implement in actual MJP processing. Changing the M-shaped removal function of MJP into a Gaussian-like removal function is complicated, e.g., the eccentric rotation motion for nozzle and coil, transmitting a steady current and slurry form of the static system to the coil and nozzle, and designing and implementing the gesture and position for fixing the nozzle. Thus, the approach presented in this paper is simpler than the above-mentioned approaches. This is because the regular removal function, the 2016
APPLIED OPTICS / Vol. 53, No. 10 / 1 April 2014
M-shaped removal function, is applied for actual processing. 3. Experimental Validation and Analysis A. Corresponding Gaussian Shape of the Actual M-shaped Removal Function
A polishing liquid containing 2.3 wt. % CeO2 particles (2 μm) was chosen. The diameter of the cylindrical nozzle was 0.95 mm. The pump pressure showed 0.4 Mpa, and the stand-off distance was 66 mm. An annular footprint with a W-shaped profile (see Fig. 6) was obtained for 3 min processing time. The preceding experiment was designed to investigate the dl with material removal, as shown in Fig. 7. The 0.2 mm∕s linear traverse velocity of the nozzle was chosen with dl, ranging from 0.2 to 2.4 mm. It is obvious that the material removal has a V-shaped profile when dl 0.8 mm, and the maximum depth H Mm and width of the profile are about 0.1702λ and 4.3 mm, respectively [see Figs. 7(a) and 7(b), which are similar to the curves at the same condition in Fig. 4]. Therefore, the area SM of the profile is about 0.37λ (the unit of the width is 1 in area computation). On the basis of this finding, determining the corresponding Gaussian removal function [from Eq. (13), B 7.83 × 10−3 λ∕s, u 0.665] is easy: Rx; y 7.83 × 10−3 × exp−0.665 × x2 y2 : (14) With the corresponding Gaussian removal function as expressed in Eq. (14), a simulation is made to obtain the distribution of the track along
Fig. 6. Footprint of a nozzle on K9 optical glass.
Fig. 7. Dependence on the dl of polishing shape. (a) Profile curves of tracks. (b) Depths of tracks with different dl. (c) V-shaped profile curves of tracks for dl 0.8 mm.
the regular path. Comparing to the distribution of the regular footprint moving along the multiplex path [Fig. 7(a), dl 0.8 mm], it is important to notice that profiles possess similar shapes to those shown in Fig. 7(c). Considering the convergence efficiency of polishing and the diameter of the
Table 1.
corresponding Gaussian removal function [Fig. 7(c), approach to 6 mm], the line spacing of the target path (d 2 mm) is adopted for the multiplex path. According to the above analysis, the parameters for polishing processing are exhibited in Table 1.
Parameters of Polishing Processing
Parameter Mass fraction of CeO2 particles in fluid (%) Diameter of CeO2 particle (μm) Diameter of nozzle (mm) Pressure (MPa) Stand-off distance (mm) Peak of corresponding Gaussian removal function (B) (λ∕s) Parameter of corresponding Gaussian removal function (u) Line spacing of the target path (d) (mm) Value of dl in multiplex path (mm)
Value 2.3 2 0.95 0.4 66 7.83 × 10−3 0.665 2 0.8
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B.
Surfacing Polishing with Scanning Multiplex Path
The scanning target path (d 2 mm; diameter of coverage area is 26 mm) was applied for the surfacing polishing of a 23 mm diameter plane K9 optical glass, with 0.256λ PV and 0.068λ RMS, which were measured by a ZYGO interferometer [see Fig. 8(a)]. Based on the target path, the scanning multiplex path with the polishing parameters (in Table 1, d1 d2 dl∕2 0.4 mm, dl 0.8 mm) was selected for polishing [see Fig. 8(b)]. After computing for the dwell time by using the corresponding Gaussian removal function, the figure of the initial surface and the target path, the velocity function on the multiplex path can be obtained using Eq. (9). The polishing process was conducted on the work-piece. In the processing, the polishing tool, possessing an M-shaped removal function, travels along the multiplex path with the velocity function. After three iterations, MJP processing delivers a surface waviness improvement from 0.256λ PV and 0.068λ RMS down to 0.038λ PV and 0.005λ RMS [Fig. 9(a)]. Figures 9(b) and 9(c) show the average power spectral density (PSD) curves along the x (horizontal) and y (vertical) directions, respectively. Contrasting the PSD curves, there is mass of spatial frequency errors before polishing, which indicates that the spatial frequency errors are suppressed well. Moreover, there are two small peaks [in parts 1 and 2; see Fig. 9(c)] in the PSD curves along the y direction of the final surface. For one peak in the part 1, the PSD of the polished surface is approximately equal to that before being polished with about
Fig. 9. Polishing results. (a) Final surface: 0.038λ PV, 0.005λ RMS. (b) Average PSD curves along x direction. (c) Average PSD curves along y direction.
0.51∕mm spatial frequency [Freq; see Fig. 9(c)]. For the other peak in part 2, the spatial frequency is about 1 1∕mm Freq [Fig. 9(c)]. These indicate that the spatial frequencies of the polished surface are mainly induced by the line spacing of the target path (d 2 mm). The case of surface fabrication verifies that the multiplex path is effective and valid. In addition, the figure of the final surface is very homogeneous near the edge of the work-piece. This result is deceptive because there is no edge effect (this was also proposed by Booij et al. [5] and Walker et al. [12]). 4. Conclusions Fig. 8. Experiment on the 23 mm diameter work-piece with grid multiplex path. (a) Form error of original work-piece: 0.256λ PV, 0.068λ RMS. (b) Multiplex path. 2018
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This paper aims to optimize the MJP process. A multiplex path has been presented instead of optimizing the removal function. The idea behind the multiplex path is that a W-shaped footprint along
two parallel paths, which possess appropriate dl, can obtain a track that has the same character as that when the Gaussian removal function along one path is applied. The distributions of material removal with different dl of the multiplex path are discussed. Results show that the methods using the regular path with a Gaussian removal function and the method using the multiplex path with an M-shaped removal function obtain the same material removal. The constructed transformation model is built, and describes the relationship between the M-shaped removal function and the Gaussian removal function. With the corresponding Gaussian removal function and the regular path, the appropriate dwell time for precise polishing is obtained. The velocity function on the multiplex path for the M-shaped removal function can be obtained by this dwell time. The newly designed polishing program is then implemented on the work-piece with the M-shaped removal function, the multiplex path, and the velocity function in actual processing. On the basis of this approach, we designed and demonstrated deterministic polishing performances to correct plane K9 optical glasses by using the Mshaped removal function. With the scanning multiplex path, the form errors of the K9 optical glass (23 mm diameter) decreased from 0.256λ PV and 0.068λ RMS to 0.038λ PV and 0.005λ RMS. Similarly, the spatial frequency errors are suppressed to a great degree. These results demonstrate that the new polishing method influences the final surface shape as predicted. Therefore, optimizing the path of the polishing process presents an alternative approach to precise optical manufacturing. This work was supported by the National Natural Science Foundation of China (grant no. 61222506) and the Specialized Research Fund
for the Doctoral Program of Higher Education (grant no. 20131101110026). References 1. W. I. Kordonski, A. B. Shorey, and M. Tricard, “Magnetorheological jet (MR JetTM) finishing technology,” J. Fluids Eng. 128, 20–26 (2006). 2. M. Tricard, W. I. Kordonski, and A. B. Shorey, “Magnetorheological jet finishing of conformal, freeform and steep concave optics,” CIRP Annals 55, 309–312 (2006). 3. W. Kordonski and A. Shorey, “Magnetorheological (MR) jet finishing technology,” J. Intell. Mater. Syst. Struct. 18, 1127–1130 (2007). 4. T. Wang, H. B. Cheng, Y. Chen, Y. P. Feng, Z. C. Dong, and H. Y. Tam, “Correction of remounting errors by masking reference points in small footprint polishing process,” Appl. Opt. 52, 7851–7858 (2013). 5. S. M. Booij, H. V. Brug, J. J. M. Braat, and O. W. Fähnle, “Nanometer deep shaping with fluid jet polishing,” Opt. Eng. 41, 1926–1931 (2002). 6. H. Fang, P. J. Guo, and J. C. Yu, “Optimization of the material removal in fluid jet polishing,” Opt. Eng. 45, 053401 (2006). 7. T. Wang, H. B. Cheng, Z. C. Dong, and H. Y. Tam, “Removal character of vertical jet polishing with eccentric rotation motion using magnetorheological fluid,” J. Mater. Process. Technol. 213, 1532–1537 (2013). 8. C. Y. Shi, J. H. Yuan, F. Wu, X. Hou, and Y. J. Wan, “Material removal model of vertical impinging in fluid jet polishing,” Chin. Opt. Lett. 8, 323–325 (2010). 9. H. Y. Tam, H. B. Cheng, and Z. C. Dong, “Peano-like paths for subaperture polishing of optical aspherical surfaces,” Appl. Opt. 52, 3624–3636 (2013). 10. C. R. Dunn and D. D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16, 18942–18949 (2008). 11. H. Y. Tam and H. B. Cheng, “An investigation of the effects of the tool path on the removal of material in polishing,” J. Mater. Process. Technol. 210, 807–818 (2010). 12. D. D. Walker, G. Y. Yu, H. Y. Li, W. Messelink, R. Evans, and A. Beaucamp, “Edges in CNC polishing: from mirrorsegments towards semiconductors, paper 1: edges on processing the global surface,” Opt. Express 20, 19787–19798 (2012).
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