Correction of remounting errors by masking reference points in small footprint polishing process Tan Wang,1 Haobo Cheng,1,* Yong Chen,1 Yunpeng Feng,1 Zhichao Dong,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 15 August 2013; revised 10 October 2013; accepted 11 October 2013; posted 14 October 2013 (Doc. ID 195816); published 12 November 2013
The remounting accuracy of optical components between measurement and polishing affects the polishing results, especially for small polishing footprint processes such as magnetorheological jet polishing (MJP). In this paper, two important remounting errors (translation and rotation errors) are discussed, and a masking method is proposed to correct these errors. A mathematical model that describes the relationship between remounting errors and reference points is constructed. A mask is created to provide reference points on a sample because such points are important for identifying remounting errors. The remounting errors are then used as bases in correcting the parameters used for actual polishing. Experiments are conducted on a K9 optical sample to validate the proposed approach. After the reference points are obtained by measuring the mask on the sample, the remounting errors are derived. The translation errors are 5.61 mm in the X direction and 6.08 mm in the Y direction; the rotation error is 4.1°. Deviations from the desired positions are eliminated and the desired surface smoothness is obtained after parameter correction. Results indicate that the proposed method is suitable for high-precision polishing. © 2013 Optical Society of America OCIS codes: (220.4610) Optical fabrication; (220.5450) Polishing; (120.6650) Surface measurements, figure; (120.4610) Optical fabrication. http://dx.doi.org/10.1364/AO.52.007851
1. Introduction
State-of-the-art optical finishing techniques for highprecision components rely on the iterative process of subaperture polishing by applying a small removal function to achieve removal deviation at a specified position. One of the important elements of subaperture polishing [1–3] is the accuracy with which polishing footprints are positioned in the desired sample shapes. Generally, the first step for each iterative process in the high-precision manufacture of optical components is the measurement of 3D surfaces. The distribution of deviations on a sample is determined 1559-128X/13/337851-08$15.00/0 © 2013 Optical Society of America
by an interferometer and the parameters of polishing are designed. The sample is then mounted onto a polishing machine for correction. During this operation, remounting errors (primarily translation errors, rotation error, and tip and tilt errors) occur. To implement deterministic polishing, measured sample data should be transferred from a metrology measurement machine into a polishing machine with sufficient accuracy. This step is specifically important for magnetorheological jet polishing (MJP) [4,5], which is a subaperture polishing technique characterized by steady and small footprints (range of lateral size is from millimeter scale to submillimeter scale) [2,6,7]. To minimize remounting errors, researchers have adopted approaches, such as the application of self-centering chucks, calibration of 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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sample edges, and in-process interferometer measurements for full apertures [8]. The first two approaches are adopted mainly for components with inerratic edges, but it is difficult to correct rotation errors of circular samples at a high level. Selfcentering chucks are also difficult to apply in MJP because polishing slurry filters into the inner parts of the chucks. Similarly, interferometers are easily exposed to slurry and difficult to fix onto a polishing machine for optical measurement. This paper presents an approach to optimizing sample remounting, in which a mask is used in MJP. First, a discussion is presented as to the relationship between the remounting accuracy required for sufficient precision polishing and the shape of the removal function. Second, a mathematical model of remounting error is constructed on the basis of reference points. An approach to identifying reference points by using a mask is then presented. Third, some examples for K9 glass are designed to validate the proposed approach, followed by a summary of this work.
In deterministic polishing, material removal from one location on an optical surface pertains to cumulated removal by polishing along adjacent path lines. Therefore, the deviations at points x; y on a surface are completely removed if the convolution of removal function Rx; y and dwell time Dx; y are equal to the surface error. This process can be described as follows [9,10]:
(1)
where Ex; y is the total material removed from the sample. The discrete function can be as [11] Ex; y
XX i
j
i
j
XX
Rx − xi ; y − yj · Dxi ; yj Rij · Dxi ; yj ;
(2)
where Σ is the summation over i or j, i and j are the indices of the sampled points in the X and Y directions, and Rij is the removal rate at point x; y. Surface error and removal function data are obtained by (for example) an interferometer, and polishing path and dwell time are calculated for the actual process. In general, samples are mounted onto different workbenches for measurement and polishing. This process causes remounting errors, such as translation errors, rotation error (Fig. 1), and tip and tilt errors. Here, the tip and tilt errors are ignored because the base surface of the sample agrees well with that of the workbench. xg og yg and xoy are the coordinate systems of polishing and measurement, 7852
respectively. δx and δy are the translation errors along the X and Y directions, respectively, and δθ is the rotation error. The material removal at an arbitrary point px; y on a sample can be illustrated as Eq. (3): Ea x; y
2. Effect of Surface Residual Error Caused by Remounting Accuracy with Different Footprints
Ex; y Rx; y Dx; y ZZ Rx − x0 ; y − y0 · Dx0 ; y0 dx0 dy0 ;
Fig. 1. Remounting errors in optical manufacturing.
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
XX i
j
i
j
XX
Rx − Δxi − xi ; y − Δyj − yj · Dxi ; yj Raij · Dxi ; yj ;
(3)
where Δxi and Δyj indicate the deviations of the actual polishing position relative to the theoretical position on the sample and Raij is the actual removal rate at point px; y. Therefore, the distribution of residual error ΔE on the surface (caused by remounting error) can be described as ΔE E − Ea XX Rij − Raij · Dxi ; yj
i
j
i
j
XX
ΔRij · Dxi ; yj ;
(4)
where ΔRij represents the changes in R within Δx and Δy. These changes can also be called the gradients of the removal function at minimal Δx and Δy. Residual error ΔE is associated primarily with the removal function characters derived from the analysis using Eq. (4). ΔE and ΔRij show a directly proportional relationship. That means two things: One is, for the small areas of removal functions, remounting error should be small under the same residual error ΔE; because the footprint, possessing a small area, generates the high-gradient removal function easily. The other is that, for the high peaks of removal functions that occupy the same area, remounting errors should be small under the same residual error ΔE. As indicated by the analysis above, increasing the remounting accuracy enables the effective acquisition of desired sample shapes; a feature that is particularly important in MJP. Moreover, from the works of Fähnle [2], the remounting accuracy of
the polishing tool onto the sample surface should be smaller than 1∕10th of the full width half-maximum of the removal function for precise manufacture. 3. Strategy for Reducing Remounting Errors A.
Mathematical Model of Remounting Error
B. Mask for Providing Reference Points
Suppose pg is the value of an arbitrary reference point on a sample in the coordinate system of a polishing machine, then p is the value of the same reference point in the coordinate system of a measurement system. The relation between pg and p can be described as the following expression: pg T 1 · T 2 · p;
(5)
where T 1 is the translation matrix along the X and Y axes and T 2 is the rotation matrix. These variables can be expressed as follows: p x; y; 1T pg xg ; yg ; 1T ; 2 3 1 0 δx 6 7 7 T1 6 4 0 1 δy 5 0 0 1 2 3 cos δθ sin δθ 0 6 7 T 2 4 − sin δθ cos δθ 1 5: 0
0
1
Thus, the mathematical model of the remounting error can be described as
xg x cos δθ y sin δθ δx : yg −x sin δθ y cos δθ δy
remounting errors. The mask presents many advantages in that it causes no damage on the initial surface, protects part of the sample surface from slurry, enables easy measurement of the remounting errors, and does not compromise the sample’s mounting position on the machine workbench.
6
According to Eq. (6), δx, δy, and δθ can be calculated using the reference points on the sample. Reference points are crucial for correcting remounting errors. The mask created on the sample is used to effectively provide the reference points for correcting
The mask is used to obtain values for the reference points on the samples in different coordinate systems. The method for correcting remounting errors by using a mask in MJP is shown in Fig. 2. In the measuring process, the sample is mounted onto the measurement workbench to obtain the fullaperture figure. With initial data from the sample, the parts that do not require polishing are identified. The mask is then created on these parts of the sample. The masked sample is then mounted onto the same measurement workbench and measured to obtain the masked data from the sample. Without loss of generality, a quadrilateral mask is used on the sample as an example [see Fig. 3(a) where vertices A1 , A2 , A3 , and A4 are used as reference points]. The origin of the measurement coordinate system can be established at either the central or an arbitrary point on the sample with the masked data. A1 x1 ; y1 , A2 x2 ; y2 , A3 x3 ; y3 , and A4 x4 ; y4 are then obtained in the measurement coordinate system and the initial data of the sample is calibrated by the system. Subsequently, the path for polishing is designed by the initial, calibrated data from the sample. In preparation for polishing, the sample with the mask was placed on the polishing machine and mounted on its workbench, as illustrated in Fig. 3(b). The scanning tool (e.g., laser rangefinder or CCD) was fixed beside the polishing tool to collect the reference points provided by the mask in the coordinate system of the polishing machine (a laser rangefinder is used in this study). A box was designed to protect the scanning tool during polishing. The scanning tool, controlled by the polishing machine, scanned the mask along a certain line. Data distribution on the line can be described as a step function [see Fig. 3(b), where data from the sample do not accord
Fig. 2. Flow chart of the whole process. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 3. Mask is used to provide reference points on the sample. (a) Reference points provided by the mask and (b) the process for measuring the reference points.
with those on the mask] because of the material properties and thickness of the mask. Step points p2i and p1j are the positions where the scanning line cuts the mask edges in the coordinate system of the polishing machine. This way, when the scanning tool moves along a certain path, the step points on every mask edge are derived (Fig. 4, where p11 , p12 , and p13 are on edge l1 ; p21 , p22 , and p23 are on edge l2 ; p31 , p32 , and p33 are on edge l3 ; and p41 , p42 , and p43 are on edge l4 ). Therefore, the edges (li ) of the mask is given by yg axg b
(7)
Nxg ; yg yg − axg b:
(8)
and
The fit coefficients of each mask edge can be determined by solving the following equations: 8∂P 2 P < ∂a N n −2 N n xgn 0 n n P P ; ∂ N 2n −2 N n 0 : ∂b n
9
n
where ∂∕∂a is a partial derivative and n refers to the indices of the points where the scanning path cuts every mask edge. l1 , l2 , l3 , and l4 can be calculated from Eq. (9), and the four vertices (A1 , A2 , A3 , and A4 ) of the mask can be obtained in the coordinate system of the polishing machine. These vertices are expressed as Ag1 xg1 ; yg1 , Ag2 xg2 ; yg2 , Ag3 xg3 ; yg3 , and Ag4 xg4 ; yg4 . The remounting errors (δx, δy, and δθ) can be calculated by A1, A2 , A3 , A4 , Ag1 , Ag2 , Ag3 , and Ag4 by using Eq. (6). Thus, the path for polishing is easily corrected on the basis of the remounting errors. This approach presents potential for application in samples of various shapes. The remounting accuracy of different samples can be optimized using appropriate masks. Small circular masks can be designed for samples with a dispersive distribution of desired errors (Fig. 5). 4. Experimental Validation
Fig. 4. Approach to identifying mask edges in the polishing coordinate system. 7854
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The approach to optimizing remounting accuracy using the mask is demonstrated in MJP on a 50 mm K9 optical glass. In the measuring process, the sample is mounted on the measurement workbench and its full-aperture figure is measured using the ZYGO interferometer, to derive initial data. The PV and RMS are 149.481 and 18.429 nm, respectively. Three parts near the center of the sample and shown in Fig. 6(a), parts 1–3, exhibit many deviations. The following operators aim to remove the redundant material from the three parts. Moreover, the annular
Fig. 5. Small circular masks on the sample for positioning.
footprint with a W-shaped profile [the lateral size is about 4 mm, see Fig. 6(b)] was obtained for a 3 min processing time. Therefore, the error should be less than 0.2 mm [2]. To remove the material from the three desired shapes near the center of the sample, a mask possessing a quadrilateral hole is adopted that can also protect parts of the sample surface from slurry [see Fig. 7(a) where the accuracy of the mask constantly affects the remounting accuracy]. The masked surface is mounted onto the same measurement workbench and measured to obtain masked data [Fig. 7(b)], which is then used to create the
measurement coordinate system. Four vertices are easily computed: A1 −15.03;14.76, A2 14.26;14.87, A3 14.79; −14.89, and A4 −14.31; −14.79 (the distances between each reference points are d12 29.29 mm, d23 29.76 mm, d34 29.09 mm, d41 29.56 mm, and d13 42.05 mm). The initial data from the full-aperture figure are transferred into the measurement coordinate system. Removal function data are obtained while the path [12] and dwell time are calculated [Fig. 7(c)]. In preparation for polishing, the masked sample is mounted onto the workbench of the polishing machine for correction. Remounting accuracy is optimized using the scanning tool (facula diameter is 50 μm, the sampling period is 500 μs), which is carried out by the polishing machine along the raster path (velocity is 1 mm∕s). The step points on each mask edge are obtained and are tabulated in Table 1. The edges of the quadrilateral holes in the mask were calculated from Eqs. (7)–(9): 8 l ∶ − 0.0674x 21.2995 y ; > > < l1 ∶18.3426x g− 364.142 y ; g 2
g
g
l ∶ − 0.0752xg − 8.36 yg ; > > : 3 l4 ∶21.2994xg 200.209 yg ;
:
The four vertices of the hole in the coordinate system of the polishing machine are Ag1 −8.37; 21.86,
Fig. 6. Interferograms of (a) the initial sample shape and (b) the footprint of a nozzle. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 7. Masks for calibration. (a) Mask with a quadrilateral hole. (b) Masked data for obtaining the measurement coordinate system. (c) Calibrated initial data and theoretical path.
Table 1.
l1 (mm)
Step Points on each Edge of the Mask
l2 (mm)
l3 (mm)
l4 (mm)
xp1
yp1
xp2
yp2
xp3
yp3
xp4
yp4
−10.63 −7.63 −4.63 −1.63 1.37 4.37 7.37 10.37 13.37
15.67 15.544 15.251 15.035 14.86 14.656 14.463 14.274 14.077
14.679 14.839 15.021 15.171 15.401 15.519 15.656 15.803
−9 −6 −3 0 3 6 9 12
−10.63 −7.63 −4.63 −1.63 1.37 4.37 7.37 10.37 13.37
−13.945 −14.151 −14.397 −14.603 −14.840 −15.065 −15.292 −15.516 −15.743
−14.524 −14.398 −14.273 −13.121 −13.98 −13.848 −13.701 −13.536
−9 −6 −3 0 3 6 9 12
Ag2 20.94; 19.89, Ag3 19.32; −9.81, and Ag4 −9.76; −7.63 (the distances between each reference point are dg12 29.38 mm, dg23 29.75 mm, dg34 29.16 mm, dg41 29.52 mm, and dg13 42.07 mm). Comparing the corresponding distances of the four vertices in different coordinate systems (the measurement and polishing coordinate systems), the measurement with the use of the scanning tool provides an effective mean. According to Eq. (6), with the values of the four vertices in the measurement and polishing coordinate systems, the remounting errors are computed by the least squares method: δx 5.61 mm; δy 6.08 mm
parameters listed in Table 2. In three iterations, the material in the three parts [Fig. 6(a)] is removed [Fig. 9(b)]. The PV and RMS decrease to 113.746 and 16.789 nm, respectively, indicating that the proposed method can obtain desired sample shapes. Comparing the simulation result [Fig. 9(a)] with the theoretical values of the paths (Fig. 8), it is obvious that using a masking method to modify remounting accuracy can be applicable in precise optical manufacturing.
20
and δθ 4.1°:
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APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
y position / mm
15
With these remounting errors, the path distribution in the measurement coordinate system is corrected into the polishing coordinate system by Eq. (6), as shown in Fig. 8. Lines 1 and 2 indicate the theoretical values of the reference points and paths. The simulation result of polishing with these values is shown in Fig. 9(a). Lines 3 and 4 indicate the actual values of the reference points and paths for polishing in the polishing coordinate system. A CNC machine controls the nozzle that removes redundant material at desired places along the corrected path and according to the processing
10 5 0 -5
1 2 3 4
-10 -15 -10
0
10
20
x position /mm
Fig. 8. Theoretical and corrected parameters for polishing.
Fig. 9. Final sample shape. (a) Simulation result without using the masking method. (b) Actual polishing result using the masking method.
Table 2.
Parameters of the Polishing Process
Parameter
Value
Mass fraction of CeO2 particles in fluid (%) Diameter of CeO2 particle (μm) Diameter of nozzle (mm) Pressure (MPa) stand-off distance (mm)
3 2 0.95 0.6 50
5. Conclusion
Successful CCP requires high remounting accuracy, which is especially important when using polishing techniques in which the removal function with small areas or high peaks are used (e.g., MJP). A mathematical model that describes the relationship between remounting errors and reference points on samples is constructed. To identify reference points, the appropriate mask is used. This is an approach that presents a number of advantages: self-centering chucks and in-process interferometer measurements for remounting are unnecessary, no damage on the initial surface, parts of sample surfaces are protected from slurry, the reference points for correcting remounting errors are easily provided and implemented, and the position of the mounted sample
remains undisturbed. Using a mathematical model, remounting errors are obtained by these reference points on various samples. An experiment on 50 mm K9 optical glass was conducted to verify the effectiveness of the proposed approach. In accordance with the initial figure of the sample, a mask with a quadrilateral hole is used to determine the reference point values in the measurement and polishing coordinate systems. The remounting errors are then obtained (δx 5.61 mm, δy 6.08 mm, and δθ 4.1°). The polishing path is corrected for polishing and the redundant material at desired locations is accurately removed, thereby creating the desired sample shapes. The PV and RMS decrease from 149.481 to 113.746 nm and 18.429 to 16.789 nm, respectively, because of the removal of the desired parts from the sample surface. The experimental results indicate that optimizing remounting accuracy using a mask is suitable for shaping and polishing precision optics. This work was supported by the National Natural Science Foundation of China (Grant Nos. 61128012, 61061160503, and 61222506), the Key Laboratory of Photoelectronic Imaging Technology and System, BIT, Ministry of Education of China (Grant No. 2013OEIOF06). 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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References 1. 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). 2. O. W. Fähnle, “Abrasive jet polishing approaches to the manufacture of micro-optics with complex shapes,” in Imaging and Applied Optics Technical Papers, OSA Technical Digest (online) (Optical Society of America, 2012), paper OM3D.4. 3. C. Y. Shi, J. H. Yuan, F. Wu, and Y. J. Wan, “Ultra-precision figuring using submerged jet polishing,” Chin. Opt. Lett. 9, 092201 (2011). 4. W. I. Kordonski and A. B. Shorey, “Magnetorheological (MR) jet finishing technology,” J. Intell. Mater. Syst. Struct. 18, 1127–1130 (2007). 5. M. Tricard, W. I. Kordonski, and A. B. Shorey, “Magnetorheological jet finishing of conformal, freeform and steep concave optics,” Ann. CIRP 55, 309–312 (2006). 6. T. Wang, H. B. Cheng, Z. C. Dong, and H. Y. Tam, “Removal character of vertical jet polishing with eccentric rotation
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7. 8. 9. 10. 11.
12.
motion using magnetorheological fluid,” J. Mater. Process. Technol. 213, 1532–1537 (2013). W. I. Kordonski, A. B. Shorey, and M. Tricard, “Magnetorheological jet (MR Jet) finishing technology,” J. Fluids Eng. 128, 20–26 (2006). H. H. van Brug, M. Groeneveld, S. M. Booij, and J. J. M. Braat, “In-process measurements of material removal in fluid jet polishing,” Proc. SPIE 4778, 243–250 (2002). S. M. Booij, O. W. Fähnle, and J. J. M. Braat, “Shaping with fluid jet polishing by footprint optimization,” Appl. Opt. 43, 67–69 (2004). R. A. Jones, “Computer simulation of smoothing during computer-controlled optical polishing,” Appl. Opt. 34, 1162– 1169 (1995). W. A. C. M. Messelink, R. Waeger, T. Wons, M. Meeder, K. C. Heiniger, and O. W. Fähnle, “Prepolishing and finishing of optical surfaces using fluid jet polishing,” Proc. SPIE 5869, 586908 (2005). 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).
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 15 August 2013; revised 10 October 2013; accepted 11 October 2013; posted 14 October 2013 (Doc. ID 195816); published 12 November 2013
The remounting accuracy of optical components between measurement and polishing affects the polishing results, especially for small polishing footprint processes such as magnetorheological jet polishing (MJP). In this paper, two important remounting errors (translation and rotation errors) are discussed, and a masking method is proposed to correct these errors. A mathematical model that describes the relationship between remounting errors and reference points is constructed. A mask is created to provide reference points on a sample because such points are important for identifying remounting errors. The remounting errors are then used as bases in correcting the parameters used for actual polishing. Experiments are conducted on a K9 optical sample to validate the proposed approach. After the reference points are obtained by measuring the mask on the sample, the remounting errors are derived. The translation errors are 5.61 mm in the X direction and 6.08 mm in the Y direction; the rotation error is 4.1°. Deviations from the desired positions are eliminated and the desired surface smoothness is obtained after parameter correction. Results indicate that the proposed method is suitable for high-precision polishing. © 2013 Optical Society of America OCIS codes: (220.4610) Optical fabrication; (220.5450) Polishing; (120.6650) Surface measurements, figure; (120.4610) Optical fabrication. http://dx.doi.org/10.1364/AO.52.007851
1. Introduction
State-of-the-art optical finishing techniques for highprecision components rely on the iterative process of subaperture polishing by applying a small removal function to achieve removal deviation at a specified position. One of the important elements of subaperture polishing [1–3] is the accuracy with which polishing footprints are positioned in the desired sample shapes. Generally, the first step for each iterative process in the high-precision manufacture of optical components is the measurement of 3D surfaces. The distribution of deviations on a sample is determined 1559-128X/13/337851-08$15.00/0 © 2013 Optical Society of America
by an interferometer and the parameters of polishing are designed. The sample is then mounted onto a polishing machine for correction. During this operation, remounting errors (primarily translation errors, rotation error, and tip and tilt errors) occur. To implement deterministic polishing, measured sample data should be transferred from a metrology measurement machine into a polishing machine with sufficient accuracy. This step is specifically important for magnetorheological jet polishing (MJP) [4,5], which is a subaperture polishing technique characterized by steady and small footprints (range of lateral size is from millimeter scale to submillimeter scale) [2,6,7]. To minimize remounting errors, researchers have adopted approaches, such as the application of self-centering chucks, calibration of 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
7851
sample edges, and in-process interferometer measurements for full apertures [8]. The first two approaches are adopted mainly for components with inerratic edges, but it is difficult to correct rotation errors of circular samples at a high level. Selfcentering chucks are also difficult to apply in MJP because polishing slurry filters into the inner parts of the chucks. Similarly, interferometers are easily exposed to slurry and difficult to fix onto a polishing machine for optical measurement. This paper presents an approach to optimizing sample remounting, in which a mask is used in MJP. First, a discussion is presented as to the relationship between the remounting accuracy required for sufficient precision polishing and the shape of the removal function. Second, a mathematical model of remounting error is constructed on the basis of reference points. An approach to identifying reference points by using a mask is then presented. Third, some examples for K9 glass are designed to validate the proposed approach, followed by a summary of this work.
In deterministic polishing, material removal from one location on an optical surface pertains to cumulated removal by polishing along adjacent path lines. Therefore, the deviations at points x; y on a surface are completely removed if the convolution of removal function Rx; y and dwell time Dx; y are equal to the surface error. This process can be described as follows [9,10]:
(1)
where Ex; y is the total material removed from the sample. The discrete function can be as [11] Ex; y
XX i
j
i
j
XX
Rx − xi ; y − yj · Dxi ; yj Rij · Dxi ; yj ;
(2)
where Σ is the summation over i or j, i and j are the indices of the sampled points in the X and Y directions, and Rij is the removal rate at point x; y. Surface error and removal function data are obtained by (for example) an interferometer, and polishing path and dwell time are calculated for the actual process. In general, samples are mounted onto different workbenches for measurement and polishing. This process causes remounting errors, such as translation errors, rotation error (Fig. 1), and tip and tilt errors. Here, the tip and tilt errors are ignored because the base surface of the sample agrees well with that of the workbench. xg og yg and xoy are the coordinate systems of polishing and measurement, 7852
respectively. δx and δy are the translation errors along the X and Y directions, respectively, and δθ is the rotation error. The material removal at an arbitrary point px; y on a sample can be illustrated as Eq. (3): Ea x; y
2. Effect of Surface Residual Error Caused by Remounting Accuracy with Different Footprints
Ex; y Rx; y Dx; y ZZ Rx − x0 ; y − y0 · Dx0 ; y0 dx0 dy0 ;
Fig. 1. Remounting errors in optical manufacturing.
APPLIED OPTICS / Vol. 52, No. 33 / 20 November 2013
XX i
j
i
j
XX
Rx − Δxi − xi ; y − Δyj − yj · Dxi ; yj Raij · Dxi ; yj ;
(3)
where Δxi and Δyj indicate the deviations of the actual polishing position relative to the theoretical position on the sample and Raij is the actual removal rate at point px; y. Therefore, the distribution of residual error ΔE on the surface (caused by remounting error) can be described as ΔE E − Ea XX Rij − Raij · Dxi ; yj
i
j
i
j
XX
ΔRij · Dxi ; yj ;
(4)
where ΔRij represents the changes in R within Δx and Δy. These changes can also be called the gradients of the removal function at minimal Δx and Δy. Residual error ΔE is associated primarily with the removal function characters derived from the analysis using Eq. (4). ΔE and ΔRij show a directly proportional relationship. That means two things: One is, for the small areas of removal functions, remounting error should be small under the same residual error ΔE; because the footprint, possessing a small area, generates the high-gradient removal function easily. The other is that, for the high peaks of removal functions that occupy the same area, remounting errors should be small under the same residual error ΔE. As indicated by the analysis above, increasing the remounting accuracy enables the effective acquisition of desired sample shapes; a feature that is particularly important in MJP. Moreover, from the works of Fähnle [2], the remounting accuracy of
the polishing tool onto the sample surface should be smaller than 1∕10th of the full width half-maximum of the removal function for precise manufacture. 3. Strategy for Reducing Remounting Errors A.
Mathematical Model of Remounting Error
B. Mask for Providing Reference Points
Suppose pg is the value of an arbitrary reference point on a sample in the coordinate system of a polishing machine, then p is the value of the same reference point in the coordinate system of a measurement system. The relation between pg and p can be described as the following expression: pg T 1 · T 2 · p;
(5)
where T 1 is the translation matrix along the X and Y axes and T 2 is the rotation matrix. These variables can be expressed as follows: p x; y; 1T pg xg ; yg ; 1T ; 2 3 1 0 δx 6 7 7 T1 6 4 0 1 δy 5 0 0 1 2 3 cos δθ sin δθ 0 6 7 T 2 4 − sin δθ cos δθ 1 5: 0
0
1
Thus, the mathematical model of the remounting error can be described as
xg x cos δθ y sin δθ δx : yg −x sin δθ y cos δθ δy
remounting errors. The mask presents many advantages in that it causes no damage on the initial surface, protects part of the sample surface from slurry, enables easy measurement of the remounting errors, and does not compromise the sample’s mounting position on the machine workbench.
6
According to Eq. (6), δx, δy, and δθ can be calculated using the reference points on the sample. Reference points are crucial for correcting remounting errors. The mask created on the sample is used to effectively provide the reference points for correcting
The mask is used to obtain values for the reference points on the samples in different coordinate systems. The method for correcting remounting errors by using a mask in MJP is shown in Fig. 2. In the measuring process, the sample is mounted onto the measurement workbench to obtain the fullaperture figure. With initial data from the sample, the parts that do not require polishing are identified. The mask is then created on these parts of the sample. The masked sample is then mounted onto the same measurement workbench and measured to obtain the masked data from the sample. Without loss of generality, a quadrilateral mask is used on the sample as an example [see Fig. 3(a) where vertices A1 , A2 , A3 , and A4 are used as reference points]. The origin of the measurement coordinate system can be established at either the central or an arbitrary point on the sample with the masked data. A1 x1 ; y1 , A2 x2 ; y2 , A3 x3 ; y3 , and A4 x4 ; y4 are then obtained in the measurement coordinate system and the initial data of the sample is calibrated by the system. Subsequently, the path for polishing is designed by the initial, calibrated data from the sample. In preparation for polishing, the sample with the mask was placed on the polishing machine and mounted on its workbench, as illustrated in Fig. 3(b). The scanning tool (e.g., laser rangefinder or CCD) was fixed beside the polishing tool to collect the reference points provided by the mask in the coordinate system of the polishing machine (a laser rangefinder is used in this study). A box was designed to protect the scanning tool during polishing. The scanning tool, controlled by the polishing machine, scanned the mask along a certain line. Data distribution on the line can be described as a step function [see Fig. 3(b), where data from the sample do not accord
Fig. 2. Flow chart of the whole process. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 3. Mask is used to provide reference points on the sample. (a) Reference points provided by the mask and (b) the process for measuring the reference points.
with those on the mask] because of the material properties and thickness of the mask. Step points p2i and p1j are the positions where the scanning line cuts the mask edges in the coordinate system of the polishing machine. This way, when the scanning tool moves along a certain path, the step points on every mask edge are derived (Fig. 4, where p11 , p12 , and p13 are on edge l1 ; p21 , p22 , and p23 are on edge l2 ; p31 , p32 , and p33 are on edge l3 ; and p41 , p42 , and p43 are on edge l4 ). Therefore, the edges (li ) of the mask is given by yg axg b
(7)
Nxg ; yg yg − axg b:
(8)
and
The fit coefficients of each mask edge can be determined by solving the following equations: 8∂P 2 P < ∂a N n −2 N n xgn 0 n n P P ; ∂ N 2n −2 N n 0 : ∂b n
9
n
where ∂∕∂a is a partial derivative and n refers to the indices of the points where the scanning path cuts every mask edge. l1 , l2 , l3 , and l4 can be calculated from Eq. (9), and the four vertices (A1 , A2 , A3 , and A4 ) of the mask can be obtained in the coordinate system of the polishing machine. These vertices are expressed as Ag1 xg1 ; yg1 , Ag2 xg2 ; yg2 , Ag3 xg3 ; yg3 , and Ag4 xg4 ; yg4 . The remounting errors (δx, δy, and δθ) can be calculated by A1, A2 , A3 , A4 , Ag1 , Ag2 , Ag3 , and Ag4 by using Eq. (6). Thus, the path for polishing is easily corrected on the basis of the remounting errors. This approach presents potential for application in samples of various shapes. The remounting accuracy of different samples can be optimized using appropriate masks. Small circular masks can be designed for samples with a dispersive distribution of desired errors (Fig. 5). 4. Experimental Validation
Fig. 4. Approach to identifying mask edges in the polishing coordinate system. 7854
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The approach to optimizing remounting accuracy using the mask is demonstrated in MJP on a 50 mm K9 optical glass. In the measuring process, the sample is mounted on the measurement workbench and its full-aperture figure is measured using the ZYGO interferometer, to derive initial data. The PV and RMS are 149.481 and 18.429 nm, respectively. Three parts near the center of the sample and shown in Fig. 6(a), parts 1–3, exhibit many deviations. The following operators aim to remove the redundant material from the three parts. Moreover, the annular
Fig. 5. Small circular masks on the sample for positioning.
footprint with a W-shaped profile [the lateral size is about 4 mm, see Fig. 6(b)] was obtained for a 3 min processing time. Therefore, the error should be less than 0.2 mm [2]. To remove the material from the three desired shapes near the center of the sample, a mask possessing a quadrilateral hole is adopted that can also protect parts of the sample surface from slurry [see Fig. 7(a) where the accuracy of the mask constantly affects the remounting accuracy]. The masked surface is mounted onto the same measurement workbench and measured to obtain masked data [Fig. 7(b)], which is then used to create the
measurement coordinate system. Four vertices are easily computed: A1 −15.03;14.76, A2 14.26;14.87, A3 14.79; −14.89, and A4 −14.31; −14.79 (the distances between each reference points are d12 29.29 mm, d23 29.76 mm, d34 29.09 mm, d41 29.56 mm, and d13 42.05 mm). The initial data from the full-aperture figure are transferred into the measurement coordinate system. Removal function data are obtained while the path [12] and dwell time are calculated [Fig. 7(c)]. In preparation for polishing, the masked sample is mounted onto the workbench of the polishing machine for correction. Remounting accuracy is optimized using the scanning tool (facula diameter is 50 μm, the sampling period is 500 μs), which is carried out by the polishing machine along the raster path (velocity is 1 mm∕s). The step points on each mask edge are obtained and are tabulated in Table 1. The edges of the quadrilateral holes in the mask were calculated from Eqs. (7)–(9): 8 l ∶ − 0.0674x 21.2995 y ; > > < l1 ∶18.3426x g− 364.142 y ; g 2
g
g
l ∶ − 0.0752xg − 8.36 yg ; > > : 3 l4 ∶21.2994xg 200.209 yg ;
:
The four vertices of the hole in the coordinate system of the polishing machine are Ag1 −8.37; 21.86,
Fig. 6. Interferograms of (a) the initial sample shape and (b) the footprint of a nozzle. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 7. Masks for calibration. (a) Mask with a quadrilateral hole. (b) Masked data for obtaining the measurement coordinate system. (c) Calibrated initial data and theoretical path.
Table 1.
l1 (mm)
Step Points on each Edge of the Mask
l2 (mm)
l3 (mm)
l4 (mm)
xp1
yp1
xp2
yp2
xp3
yp3
xp4
yp4
−10.63 −7.63 −4.63 −1.63 1.37 4.37 7.37 10.37 13.37
15.67 15.544 15.251 15.035 14.86 14.656 14.463 14.274 14.077
14.679 14.839 15.021 15.171 15.401 15.519 15.656 15.803
−9 −6 −3 0 3 6 9 12
−10.63 −7.63 −4.63 −1.63 1.37 4.37 7.37 10.37 13.37
−13.945 −14.151 −14.397 −14.603 −14.840 −15.065 −15.292 −15.516 −15.743
−14.524 −14.398 −14.273 −13.121 −13.98 −13.848 −13.701 −13.536
−9 −6 −3 0 3 6 9 12
Ag2 20.94; 19.89, Ag3 19.32; −9.81, and Ag4 −9.76; −7.63 (the distances between each reference point are dg12 29.38 mm, dg23 29.75 mm, dg34 29.16 mm, dg41 29.52 mm, and dg13 42.07 mm). Comparing the corresponding distances of the four vertices in different coordinate systems (the measurement and polishing coordinate systems), the measurement with the use of the scanning tool provides an effective mean. According to Eq. (6), with the values of the four vertices in the measurement and polishing coordinate systems, the remounting errors are computed by the least squares method: δx 5.61 mm; δy 6.08 mm
parameters listed in Table 2. In three iterations, the material in the three parts [Fig. 6(a)] is removed [Fig. 9(b)]. The PV and RMS decrease to 113.746 and 16.789 nm, respectively, indicating that the proposed method can obtain desired sample shapes. Comparing the simulation result [Fig. 9(a)] with the theoretical values of the paths (Fig. 8), it is obvious that using a masking method to modify remounting accuracy can be applicable in precise optical manufacturing.
20
and δθ 4.1°:
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y position / mm
15
With these remounting errors, the path distribution in the measurement coordinate system is corrected into the polishing coordinate system by Eq. (6), as shown in Fig. 8. Lines 1 and 2 indicate the theoretical values of the reference points and paths. The simulation result of polishing with these values is shown in Fig. 9(a). Lines 3 and 4 indicate the actual values of the reference points and paths for polishing in the polishing coordinate system. A CNC machine controls the nozzle that removes redundant material at desired places along the corrected path and according to the processing
10 5 0 -5
1 2 3 4
-10 -15 -10
0
10
20
x position /mm
Fig. 8. Theoretical and corrected parameters for polishing.
Fig. 9. Final sample shape. (a) Simulation result without using the masking method. (b) Actual polishing result using the masking method.
Table 2.
Parameters of the Polishing Process
Parameter
Value
Mass fraction of CeO2 particles in fluid (%) Diameter of CeO2 particle (μm) Diameter of nozzle (mm) Pressure (MPa) stand-off distance (mm)
3 2 0.95 0.6 50
5. Conclusion
Successful CCP requires high remounting accuracy, which is especially important when using polishing techniques in which the removal function with small areas or high peaks are used (e.g., MJP). A mathematical model that describes the relationship between remounting errors and reference points on samples is constructed. To identify reference points, the appropriate mask is used. This is an approach that presents a number of advantages: self-centering chucks and in-process interferometer measurements for remounting are unnecessary, no damage on the initial surface, parts of sample surfaces are protected from slurry, the reference points for correcting remounting errors are easily provided and implemented, and the position of the mounted sample
remains undisturbed. Using a mathematical model, remounting errors are obtained by these reference points on various samples. An experiment on 50 mm K9 optical glass was conducted to verify the effectiveness of the proposed approach. In accordance with the initial figure of the sample, a mask with a quadrilateral hole is used to determine the reference point values in the measurement and polishing coordinate systems. The remounting errors are then obtained (δx 5.61 mm, δy 6.08 mm, and δθ 4.1°). The polishing path is corrected for polishing and the redundant material at desired locations is accurately removed, thereby creating the desired sample shapes. The PV and RMS decrease from 149.481 to 113.746 nm and 18.429 to 16.789 nm, respectively, because of the removal of the desired parts from the sample surface. The experimental results indicate that optimizing remounting accuracy using a mask is suitable for shaping and polishing precision optics. This work was supported by the National Natural Science Foundation of China (Grant Nos. 61128012, 61061160503, and 61222506), the Key Laboratory of Photoelectronic Imaging Technology and System, BIT, Ministry of Education of China (Grant No. 2013OEIOF06). 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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