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Abstract
A new category of circular pseudo-random paths is proposed in order to suppress repetitive patterns and improve surface waviness on ultra-precision polished surfaces. Random paths in prior research had many corners, therefore deceleration of the polishing tool affected the surface waviness. The new random path can suppress velocity changes of the polishing tool and thus restrict degradation of the surface waviness, making it suitable for applications with stringent mid-spatial-frequency requirements such as photomask blanks for EUV lithography.
© 2017 Optical Society of America
1. Introduction
In CNC polishing, tool paths covering an entire processing area uniformly are needed [1]. Usually, raster tool paths and spiral tool paths are widely used. These paths cover whole areas and are generated with shortest possible total distance in order to shorten processing time. The tool path curves generally get imprinted onto the workpiece surface. Since these paths have repetitive patterns, periodic bumps then appear on the processed surface [2]. However, it may happen that these bumps cause interference of light in optical use. The interference fringes intensify spikes at specific locations, and can thus affect the optical performance of the final system.
Thus, previous research in generating pseudo-random paths was carried out. Dunn et al. generated unicursal pseudo-random paths which spread to 6 directions in a hexagonal area with a masked out hole [3]. Wang et al. also reduced repetitive patterns with rectangular pseudo-random paths, comparing them with raster paths and Hilbert curve paths [4].
However, the tool paths that both Dunn et al. and Wang et al. used have sharp corners, where the feed direction of the tool changes. In previous research, deceleration of polishing tools was not taken into account. Deceleration of tools can be a problem in cutting processes, and many researches have tried to achieve constant feed rate [5]. Their aim in cutting process is mainly to shorten processing time. On the other hand, in polishing process, deceleration can affect the accuracy of polished surfaces directly. When the tool stays at corners for a longer time than over straight sections of the path, they are polished deeper than the other areas because removal depth increases proportionally with dwell time. As a result, the overall surface waviness gets worse.
In this paper, a method for generating circular pseudo-random paths is presented, which avoids the generation of any corners on the surface. Their merits against other types of pseudo-random path is then investigated for various polishing tool feeds.
2. Algorithm for creating circular random paths
When a unicursal path without intersecting points is created, a structure called spanning tree is generally used [6]. A rectangular mesh is generated in the target area, and each element is connected with a spanning tree. A new path is created by tracing around the tree [7]. But in random circular tool paths, this method cannot be used because the structure is a little more complex than rectangular paths. Therefore another method was used, in which the number of path elements increases. This idea was inspired by the cellular development of neurons, which spread structures similar to their original shape toward attractants [8], as shown in Fig. 1.
Figure 2 shows the creation procedure of a circular random path. In this example, the target region consists of 3 × 3 cells to simplify the explanation. Figure 2(a) shows a meshed area and pairs of elements. Pairs are created with each two neighboring cells, except for one of the cells. In this case, cell ● is singled out. Figure 2(b) shows the initial structure of circular random paths. The curved path consists of three elements. The first structure is put on the singled out cell and a neighboring pair. Figure 2(c) shows the path shape after another basic structure is added, with one cell of new structure overlapping the existing path. The random circular path can be created by repeating this operation, as shown in Fig. 2(d).
3. Velocity variation of tool in random paths
In CNC polishing, the relative velocity of the tool against the work table has to decrease at corners of the tool path. In order to estimate the effect of deceleration on the surface waviness, measuring velocity variations in random paths is important. Figure 3 shows the shapes of tool paths used in this paper, and Table 1 shows their specification. Path pitches and circle diameter were dependent on the tool path type, such that the total path length was almost the same for every cases. The 6-direction and 4-direction paths imitate Dunn’s [3] and Wang’s [4] paths. In this section, the three random paths were used for velocity measurements.
Table 1. Parameters of tool paths.
Table 2 shows the specification of the polishing machine. Tool paths were created on XY plane, which for this polishing machine is a ballscrew driven table system. Lead of the ballscrew is 5 mm. Velocity of the table was measured by rotary encoders under a command feed rate of 1000 mm/min. Sampling time was 8 ms, and the tool position and velocity were calculated from the number of encoder rotation steps. Table 3 shows the main servo parameters of the polishing machine.
Table 2. Specification of polishing machine.
Table 3. Servo parameters of polishing machine.
Figure 4 shows sections of the measured velocities. From Figs. 4(b) and 4(c), at 60 degree corners of the 6-direction path and conjunct corners in the 4-direction path, the velocity decreased down to 500 mm/min. In the circular random path shown in Fig. 4(a), on the other hand, velocity slowed down to 750 mm/min, which was almost the same as 120 degree corners of the 6-direction path. In conclusion, the velocity measurements showed that circular random paths suffer less from velocity reduction.
4. Polishing simulations with random paths
To estimate the surface waviness after polishing, two types of polishing simulations on plane plates were conducted. One ignores velocity changes to check the effect of the basic path shapes, while the other includes it in order to check this additional influence. The former situation supposes a slow feed while the latter does a fast feed, as shown in Table 4. The tool paths were as shown in Fig. 3, and the center 15 mm × 15 mm area was picked up for analysis. Removal simulations were calculated by convolution of influence function [9]. Figure 5 shows the polishing tool influence function, based on an experimentally measured polishing spot and fitted to a polynomial. Table 5 shows the parameters used to generate the influence function.
Table 4. Command feed rate and number of polishing runs.
Table 5. Parameters of polishing runs.
Figure 6 shows the simulated surface waviness when ignoring velocity change, normalized per units of removal depth. The raster and 4-direction paths feature striped patterns. In the circular random path dotted marks appear, and surface waviness is better than the other random paths. Figure 7 shows 1D fast Fourier transform (FFT) analysis of Fig. 6, in the X and Y-axis directions. In Fig. 7(a), the raster path has a 0.17 nm peak at wavelength 0.98 mm and the 4-direction path has a similar peak at 0.88 mm. The circular random path has a 0.069 nm peak at wavelength 0.98 mm, whose amplitude is less than half compared with those previous peaks. There are smaller peaks around wavelength 0.5 mm, which are harmonics of peaks around 1 mm wavelength. In Fig. 7(b), the circular random path has a 0.066 nm peak at wavelength 0.88 mm, which is 30% smaller than a 0.097 nm peak of the 4-direction path at wavelength 0.91 mm. Figure 8 shows 2D FFT analysis of Fig. 6. From Fig. 8(a), the raster path has peaks only on the Y-axis. From Fig. 8(b), the circular random path has peaks like the vertices of an octagon. From Fig. 8(c), the 6-direction path shows strong amplitudes in all directions. From Fig. 8(d), the 4-direction path has strong amplitudes on the X and Y-axes. In Fig. 8, the circular random path has none of the highest amplitude yellow points seen in other paths. Furthermore, compared with other random paths it has the largest region with amplitude lower than 0.01 nm.
Figure 9 shows the simulated surface waviness when taking into account the effect of velocity change. In all cases the surface waviness gets much worse than Fig. 6, though the circular random path shows the smallest increase. Figure 10 shows 1D FFT analysis of Fig. 9. In Fig. 10, none of the random paths shows significant peaks, and when comparing amplitude, the circular random path has the best overall spectrum.
5. Polishing with random paths at slow feed
Polishing trials were conducted on a photomask blank with the slow feed parameters shown in Table 4. The machine setup is shown in Fig. 11. In these runs, the effect of velocity change can be ignored and the direct relationship between path shape and surface waviness may be identified. Table 6 lists the experimental equipment. The polishing machine was same as section 3, the parameters of polishing were same as Table 5, and the tool paths were the same as Fig. 3. Removal depth and surface waviness were measured with a Fizeau laser interferometer from the center 15 mm × 15 mm area of the polished surfaces.
Table 6. Experimental equipment.
It should be noted that polishing runs in sections 5 and 6 were conducted over a period of several months, during which slurry and polishing pad conditions fluctuated. Through usage, sharpness of the abrasive corners gradually diminishes and the polishing pad also gets worn out. Therefore, material removal rate varied noticeably even in repeat trials with the same tool path and dwell time. For this reason, all waviness results were normalized against the average removal depth of each test.
Figure 12 shows a comparison of surface waviness on the polished surfaces. A short pass filter with 15 mm cut-off was applied for Fig. 12, because the workpiece substrates were slightly curved. The surface waviness value was then divided by removal depth, in order to obtain the normalized waviness value. In Fig. 12, the raster path, 4-direction path, and 6-direction path show strongly apparent repetitive patterns. In the circular path, even though there is a slight repetitive pattern along Y-axis, dotted marks mainly appeared. When comparing the normalized surface waviness, the circular random path was better than the raster path and 6-direction path, but worse than the 4-direction path.
It should also be noted that for polishing runs with the circular random path, a well-calibrated polishing machine is needed. In a preliminary experiment, the polishing machine had a backlash as small as 7 μm in X-axis. This caused strong repetitive patterns aligned with the Y-axis, as shown in Fig. 13, to appear on the polished surface. This pattern disappeared after controller compensation was entered for the backlash.
Figure 14 shows the 1D FFT analysis of Fig. 12, in the X and Y-axis directions. From Fig. 14(a), the raster path has a 0.78 nm peak near the wavelength 1.0 mm, which is twice larger than the 0.30 nm peak for the circular random path and 0.31 nm peak for the 4-direction path at around wavelength 1.0 mm. In the raster path, there is a peak at wavelength 2 mm, which did not appear in the simulation. The reason seems to be that as the tool went back and forth across the workpiece, it removed more material in one feed direction than the other. Meanwhile, the simulated harmonic peak at wavelength 0.5 mm could not be observed in actual polishing trials. The most likely cause is the amplitude of spectral noise in the actual polishing result, which has a similar amplitude (0.1 nm) to the simulated harmonic peak. From Fig. 14, the circular random path has a 0.42 nm peak and the 6-direction path a 0.44 nm peak at wavelength 0.9 mm, which are 30% smaller than a 0.59 nm peak for the 4-direction path at wavelength 1.4 mm.
Figure 15 shows the 2D FFT analysis. From Fig. 15, the raster path has two peaks on Y-axis, the circular random path has a peak on X-axis at wavelength 1 mm, and 4-direction path has peaks on both the X and Y-axis. In the circular random path, the maximum amplitude was smaller than the raster path and 6-direction path. But the region with amplitude smaller than 0.1 nm is larger than the 4-direction and 6-direction paths. In conclusion, the circular random path suppressed repetition on the polished surface compared with the raster path, but improvement of randomness and surface waviness was only slight when compared with other types of random path.
6. Polishing with random paths at fast feed
Polishing runs at fast feed were conducted to check the influence of velocity change to surface waviness. The tool paths and machine were the same as Chapter 5, and the feed and number of passes matched the conditions shown in Table 4.
Figure 16 shows surface waviness of the polished surfaces. A short pass filter with 15 mm cut-off was applied, and the surface waviness was normalized against removal depth. From Fig. 16, the raster path clearly shows repetitive patterns along the Y-axis, as well as ladder marks along the X-axis. The interval of these marks correspond to offset variations as the tool rotates. The tool used in this experiment had 40 μm run-out, that is 10% of the tool offset value. As offsetting between the tool and workpiece varies over every rotation, larger pressure increases the removal depth, as shown in Fig. 17. For a tool feed of 1000 mm/min and tool rotation of 750 rpm, the spacing of this variation is 1.3 mm, which corresponds to the ladder marks. In Fig. 16, the 4-direction and 6-direction paths also display strongly repetitive patterns, while surface patterns for the circular path are only a little more cluttered than at slow feed.
Figure 18 compares the deviation in surface waviness at slow feed and fast feed, for short-pass filtering under 15 mm and 5 mm. From Fig. 8(a), the raster path dramatically improved at fast feed. That is because the surface was polished many times, little by little, such that removal depth was homogenized over the whole polished area. Surface waviness of the 6-direction and 4-direction path got significantly worse than at slow feed, while the circular random path hardly changed at all. From Fig. 8(b), when comparing waviness of the circular random path under 5 mm cut-off, it got slightly worse at fast feed. It was, however, still better than that of 6-direction and 4-direction paths by several nanometers.
Figure 19 shows X-axis profiles along the edge of the polished areas. The plotted depth shows the average of several profiles, symmetric to the center of the polished area, and normalized to make the depth at center 1 μm. The circular random path shows the smoothest edge transition. The raster path has a depression, mainly due to deceleration of the tool feed at corners. The 6-direction path has a strongly non-uniform removal depth, and the 4-direction is bumpier than the circular random path.
Figure 20 shows 1D FFT analysis of Fig. 16, in the X and Y-axis directions. From Fig. 20(a), the raster path has a 0.54 nm peak at wavelength 0.98 mm, and the 4-direction path has a 1.5 nm peak at wavelength 1.8 mm. The circular random path and 6-direction path have no significant peaks. From Fig. 20(b), the raster path has a peak at wavelength 1.3 mm due to the ladder marks. In the circular random path, a 0.93 nm peak exists at wavelength 1.8 mm. The 4-direction path features this peak, as well as several other more. Figure 21 shows 2D FFT analysis of Fig. 16. From Fig. 21(a), the raster path has two peaks on Y-axis. From Fig. 21(b), the circular random path has a peak at wavelength about 2 mm at 40 degree from X-axis. From Fig. 21(c), the 6-direction path has a peak at 140 degree from X-axis. From Fig. 21(d), the 4-direction path has some peaks at wavelength 1 mm. In Fig. 21, the maximum amplitude of the circular random path is the smallest, and the region with amplitude less than 0.1 nm is again larger than other random paths.
Figure 22 shows a comparison of peak amplitudes along the X and Y-axis. The maximum peak amplitude was picked up near wavelengths of 1 mm and 2 mm. From Fig. 22(a), at wavelength about 1 mm the circular random path had smallest peak amplitude at both slow and fast feed. From Fig. 22(b), at wavelength about 2 mm the circular random path also had smallest peak amplitude at slow feed. At fast feed, even though it was larger than that of the 6-direction path, it was slightly smaller than that of the raster path.
Table 7 summarizes Figs. 18 and 22. In conclusion, when only comparing the rms of surface waviness, at fast feed circular random paths perform worse than raster paths. However, when considering repetitive patterns, circular random paths are able to suppress repetitive patterns while still containing surface waviness degradation in comparison with other pseudo-random paths.
Table 7. Comparison of surface roughness and peak amplitude.
7. Conclusion
In this paper, a new pseudo-random path based on circular elements was proposed, in order to suppress repetitive patterns on polished surfaces. In polishing simulations, when velocity change was ignored the peak amplitude in FFT analysis of surface waviness was 30% smaller compared with a raster path and other pseudo-random paths. When the effect of velocity change was considered, the simulated surface waviness of a circular random path was significantly better than the other two random paths. In polishing runs at slow feed, however, peak amplitude was almost the same for all random paths. But at fast feed, surface waviness with the random circular path was 40% better and peak amplitude was slightly smaller than other random paths. Circular random paths are thus a promising tool pathing method for final corrective finishing of next generation photomask blanks for EUV lithography, after chemical/mechanical polishing [10], as repetitive pattern amplitude needs to be strictly controlled in order to achieve the required feature resolution on wafers.
8. Future work
Dunn’s [3], Wang’s [4] and also this paper work were all performed in the XY plane. On slightly curved surfaces, projection of these tool paths provides a good enough distribution of polishing points [11]. But when surface curvature becomes significant, in previous research tool paths called ‘peano-like paths’ were created by scanning of surfaces with u and w coordinates [12]. More work is needed in future in order to apply pseudo-random paths on non-planar surfaces such as spheres, aspheres and freeforms.
Funding
Japan Society for the Promotion of Science (17K14571).
Acknowledgments
The authors also acknowledge support from Zeeko Ltd. for loaning the IRP200 polishing machine on which experiments were conducted.
References and links
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