## Abstract

In the field of ultra-precision manufacturing, the mid-spatial-frequency (MSF) error can severely affect the performance of the optical elements, but it is rather difficult to quantitatively predict the MSF error distribution. In this paper, the piecewise-path convolution (PPC) analysis is established to investigate the characteristic and the mechanism of the MSF error. The path type, tool influence function (TIF), feed rate, movement type, etc. are all considered mathematically in the analysis. This method can quantitatively predict the MSF error distribution. The coupling relationship among the path type, TIF and the MSF error are proved through the filtering theory. Besides, the analysis reveals the mathematical relationship between the tool movement type (orbital motion, radial runout) and the MSF error; the results show that the tool motion can also introduce non-negligible MSF error. Based on the research above, two selection formulae of path type, TIF and polishing parameters are provided for low MSF error polishing, which gives the theoretical guidance for the parameter selection in deterministic polishing. Practical experiments demonstrate the validity of the analysis results and conclusions.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

With the development of the modern optical technology, the ultra-precision optical components are increasingly employed in the fields of astronomical observation, laser fusion and opto-electronic industries [1–5]. Polishing is an essential step to obtain an ultra-precision curved surface. The computer controlled optical surfacing (CCOS) technology first proposed by Itek Inc. is a commonly used technology of optical polishing to improve the surface quality [6]. This technology is the basis to realize deterministic surface form correction and it is widely used in advanced polishing machine, e.g., small-tool polishing, magnetorheological polishing and ion-beam polishing [7–16]. The common feature of these polishing tools is that the tool size is much smaller than that of the workpiece, thus ensuring the convergence efficiency of the form error; but the drawback of using small size tool is the generation of the mid-spatial-frequency (MSF) error. The MSF error did not receive enough attention in the early development of the optical polishing industry; however, with the development of the optical systems, the optical elements used in the extreme ultraviolet lithography systems, laser fusion systems and advanced imaging devices put higher demands on surface quality. MSF error can cause small angle scattering, reduce the image contrast and produce the image defects [17]. In high power laser physics area, MSF error can even produce flare spots and damage the optical components [18]. Therefore, investigating and controlling the MSF error have a great significant in the optical polishing field.

At present, there are two general ways to restrain the MSF error. One is to use the hard tools for the smooth processing; another is to use the pseudo-random path or reduce the path spacing. The mechanism of the smoothing processing has been revealed by Li et al. [19]; but for complex curved surface it is hard to make the hard pad fit the workpiece well, which makes the method invalid [20]. From 2008, the pseudo-random path was first proposed by D. Walker et al. and widely used in bonnet-tool, small-tool and magnetorheological jet polishing, which shown outstanding polishing results [21–23]. Similarly, the Hilbert based path, unicursal random maze path, Lissajous path, etc. are also developed to reduce MSF error [24–28]. Researchers designed pseudo-random path tentatively because they believe that the MSF error is mainly from the regular path planning. Dai et al. [29] found that the MSF error is negatively correlated with the entropy of the path from the experimental results. However, the mechanism of the generation of the MSF error has not been fully revealed, we can only wait for the measuring results rather than quantitatively predict the MSF error before polishing. The research on how the path type, TIF and other polishing parameters affect the MSF error is not deep enough. As a result, it is necessary to further analyze the mechanism of MSF error mathematically, and then quantitatively predict the MSF error and give the general selection formulae of polishing parameters. It has a great significant to restrain the MSF error further and improve the polishing accuracy.

This paper is organized as follows. The piecewise-path convolution (PPC) method is proposed in Section 2. Then the characteristics and mechanism of the MSF error is analyzed in Section 3. Section 4 illustrates the experiment setup and results to demonstrate the validity of the method. Finally, the paper is summarized in Section 5.

## 2. Establishment of the piecewise-path convolution method (PPC)

In the field of deterministic optical polishing, it is widely accepted that the material removal rate follows the Preston equation [30]

where d*z*(

*x*,

*y*) is the material removal depth, and

*k*is the Preston coefficient related to the particular polishing conditions.

*P*(

*x*,

*y*) denotes the pressure in the contact region and

*V*(

*x*,

*y*) is the relative velocity between the workpiece and the pad.

Generally, to simplify the calculation of the material removal amount, the removal amount is considered as the convolution of discrete sampling points and the TIF. The sampling points are usually distributed equidistantly on the path; and the sampling interval is generally equal to the path interval. The TIF is typically obtained by polishing at a fixed point for a period of time; we call it “static TIF”. The dual-rotation tool is taken as an example, as shown in Fig. 1.

#### 2.1 Analysis of MSF error source

The traditional removal amount calculation method neglects two crucial points in predicting MSF error. First, the discrete sampling points cannot represent the pattern of the path; i.e. for a given sampling points distribution, there are different path patterns, as shown in Fig. 2(a). However, different path patterns can lead to different MSF error even though the sampling point distribution is the same.

Second, the “static TIF” cannot always represent the actual dynamic material removal. For most tools, the “static TIF” is different from the “instantaneous TIFs” (the material removal rate at a moment), the static TIF is the time integral of the instantaneous TIFs. When the tool moves, the change of actual integration path can lead to MSF error, as shown in Fig. 2(b). The dynamic material removal satisfies the traditional model only when the Preston coefficient *k*, pressure distribution *P*(*x*,*y*) and relative velocity *V*(*x*,*y*) are all time-invariant. Non-uniform distribution of contact pressure, multi-axis structure and even the radial runout of the tool will make the traditional model invalid.

Thus, the traditional material removal calculation method is no longer suitable for the study of the MSF error mechanism. We adopted a method to integral the instantaneous TIFs along the actual path instead of the convolution of discrete sampling points and the static TIF.

#### 2.2 Piecewise-path convolution (PPC) method

The integration of the instantaneous TIFs along the actual path has two difficulties. First, the instantaneous TIFs are time-variant in polishing; another is the complexity of the tool motion. We adopted piecewise-path convolution (PPC) method in this section to predict and investigate the MSF error.

To obtain the actual dynamic material removal efficiently, in the short time the material removal amount is regarded as the convolution between the actual integration path and the instantaneous TIF. Then the total material removal can be calculated by summing up the material removal amount in each short time.

*TIF*(

*t*) is the instantaneous TIF at time

*t*;

*L*(

*t*) represents the actual integration path from time (

*i*-1)Δ

*t*to

*i*Δ

*t*;

*T*is the total polishing time. * indicates the convolution operation.

*n*is the number of the piecewise-path; and the calculation results will be more accurate with the increase of

*n*. Generally, the accuracy mainly depends on the error between the piecewise path and the actual path. The convolution operation can be calculated by Fourier transform. For an actual integration path with

*N*pixels, the computational complexity of fast Fourier transform (FFT) is (log

_{2}

*N*+1)/

*N*times that of the spatial convolution, which greatly improves the efficiency.

The instantaneous TIF can be obtained by the Preston equation. The Preston coefficient *k* is regarded as a constant, and the instantaneous pressure distribution *P*(*x*,*y*) and the instantaneous velocity distribution *V*(*x*,*y*) can be expressed as

*ω*

_{1}and

*ω*

_{2}are the angular velocities of the spin and orbital motion, respectively; and the symbol ${\left|\left|{\;}\right|\right|_2}$ means the 2-norm operator.

*P*

_{0}is the original pressure distribution; and the instantaneous pressure distribution changes with the spin motion.

Then, the actual integration path function *L*(*t*) is the superposition of the machine motion *L*_{0}(*t*) and tool motion. For dual-rotation tool, the actual integration path *L*(*t*) can be expressed as

*ρ*is the eccentricity of the polishing tool, and

*θ*

_{0}is the original angle of the tool, and

*ω*

_{2}is the angular velocity of the orbital motion. The introduction of

*ρ*enables Eq. (4) to represent the actual integration paths of the dual-rotation tool and to simulate the radial runout of the tool when

*ρ*is small.

According to Eq. (1) and Eq. (3), the instantaneous TIF of each moment can be gotten. Then the actual integration path can be obtained by Eq. (4). Finally, the total material removal can be calculated by the numerical integration according to Eq. (2). The flow chart is shown in Fig. 3(a) and the motion mode of dual rotation tool is shown in Fig. 3(b).

## 3. Analysis of the MSF error characteristics and generation mechanism

The PPC method has the ability to obtain the actual dynamic material removal by considering the integration path and the instantaneous TIFs. Hereafter, the analyses are divided into two parts. First, the influence of the path selection of the machine on MSF error is analyzed by PPC method; the analysis is to reveal the mechanism of reducing MSF error by pseudo-random path. Another is to investigate the influence of the tool movement mode on MSF error; especially the polishing parameters neglected by traditional static model (e.g., feed rate, eccentricity, spin and orbital motion, etc.). In both analyses, the time interval Δ*t* is set to 0.01 s.

#### 3.1 Influence of the path and tool selections of the machine on MSF error

### 3.1.1 Numerical simulation results

Based on the PPC method, a series of simulations are carried out to investigate the MSF error generation mechanism for different paths and TIFs. The MSF errors of regular path and pseudo-random path under different size and shape of the TIFs were simulated and analyzed, respectively. The process parameters and conditions of the simulations are listed in Table 1.

In Table 1, the shapes of the TIFs are chosen two typical shapes of TIFs in optical polishing, i.e. ‘M’ shape and Gaussian shape TIFs. Then the zigzag path is chosen to represent the regular path, while the Archimedes spiral path is used to further demonstrate the typicality of the zigzag path; and the pseudo-random path is used the pattern mentioned in Wang’s work [21], which the scanning intervals of the three paths are all set to be 2 mm. The shapes of the TIFs and the paths are shown in Fig. 4.

Based on the PPC method, the MSF errors of the actual removal amount results under uniform polishing are shown in Fig. 5. Normalized residual error is used to characterize the proportion of MSF error, which is the distribution of the MSF error per unit removal depth.

where*err*is the normalized residual error distribution,

_{Normal}*err*is the distribution of the actual error and

_{actual}*z*(

*x*,

*y*) is the material removal distribution. The purpose of normalization is to eliminate the influence of removal amount on MSF error, where the magnitude of MSF error is linear to the removal amount.

In Fig. 5, it is interesting to find that the spatial-frequency of the residual errors of regular path are always same to the path frequency [Figs. 5(1), 5(2), 5(4), 5(6), 5(8)]; only the amplitude of the error decreases with the increase of the TIF size, and the error amplitude of the ‘M’ shape TIF is larger than that of the Gaussian shape TIF in the same size.

However, the spatial-frequency of residual error of the pseudo-random path changes with the size and shape of the TIF. The residual error is completely different from the pattern of pseudo-random path. For the same shape of the TIF, the spatial-frequency of residual error decreases with the increase of the diameter of the TIF; and for the same size of the TIF, the spatial-frequency of residual error of the ‘M’ shape TIF is larger than that of the Gaussian shape TIF. To make the results clearer, the PSD curve of the simulation results are shown in Fig. 6.

It can be clearly found from the PSD curve that, there are always peaks at the fixed spatial frequencies for regular paths, and the amplitude of the peaks change with the TIFs, as shown in Fig. 6(b). However, when using pseudo-random path, there is no sharp peak in PSD curves; and the MSF error decreases but the LSF error increases. The MSF error seems to be ‘transferred’ into the LSF error, as shown in Figs. 6(d)–6(f). Besides, the spatial frequency of the residual error continuously moves to the lower region as the TIF becomes smoother.

### 3.1.2 Mechanism explanation

The reason why the residual error frequency of the pseudo-random path is independent of the path pattern and changes with the TIF can be explained mathematically. First, based on the convolution theorem, the PPC equation with orbital motion can be simplified by Fourier expression.

*L*is the orbital circle path with the eccentricity

_{orb}*ρ*.

*x*and

*y*are the coordinate value with the center of orbital motion as the origin.

We can think of the polishing work as a filter system; and *TIF* and *L _{orb}* in Eq. (6) are two low-pass filters in the system, where the machine path

*L*

_{0}is the input and the material removal

*z*(

*x*,

*y*) is the output.

The Fourier spectra of different paths *L*_{0} (i.e. input) are shown in Fig. 7; the spectra of the TIFs and the orbital circle paths *L _{orb}* are shown in Fig. 8.

To fabricate a mirror without MSF error, the highest point of spectrum in the MSF area should be lower than the spectral noise floor of the initial surface error of the workpiece. Therefore, we define the cut-off frequency *f _{c}* for the selection of path type and TIF.

*err*represents the initial surface error of the workpiece. The cut-off frequency

_{initial}*f*is related to the TIF, orbital motion and path type. When the cut-off frequency

_{c}*f*is lower than the lower limit of MSF range, then the MSF error can be fully eliminated. The cut-off frequency is influenced by the filter (TIF,

_{c}*L*) and the path type, as shown in Fig. 8. To make the

_{orb}*f*lower, we can make the TIF smoother or larger based on the Fourier transform property [Figs. 8(d), 8(e), 8(f)]. Smoother TIF can be achieved by making the pressure distribution smoother, such as adjust the curvature of the pad [31]. The

_{c}*L*can also be regarded as a filter, but the spectrum value of

_{orb}*L*in each frequency is much higher than that of the TIFs, as shown in Fig. 8(g). This means that adding orbital motion to polishing tools is basically helpless to reduce MSF error.

_{orb}The regular paths spectra all have significant sharp peaks, as shown in Figs. 7(a), 7(b); this is because the periodic signal can be represented by the sum of the discrete harmonics (Fourier series). These peak amplitudes are very high, which makes it difficult to be filtered to below the spectral noise floor. If the MSF error wants to be eliminated by regular paths, the scanning interval should be short enough and the TIF should be smooth or large enough. But for some tool (i.e. magnetorheological tool, jet tool) whose TIF is small, the suitable scanning interval could be very short, and the accuracy and rigidity of the polisher may not be enough.

As for pseudo-random path, the peaks in the spectrum are much weaker than that of the regular path, as shown in Fig. 7(c). This is mainly due to the non-periodicity of pseudo-random paths, which cannot be expressed by Fourier series. When the peaks in the path spectrum are re-distributed to the nearby frequency area, the residual error spectrum after filtered will also be evenly distributed. In this case, the amplitude of spectrum is much lower than the sharp peaks, the MSF error is easier to be filtered by TIF. However, the spectrum amplitude of LSF part of the pseudo-random path is higher than that of the regular path; thus, the MSF error seems to be transferred to the low frequency error. Besides, there are eight ‘bands’ in the high frequency region of the spectrum for this kind of pseudo-random path; it represents eight directions of the path we choose. When the cut-off frequency is high enough into this region, the residual error with the path pattern can still appear.

Above all, in order to reduce the MSF error, it is necessary to optimize the spectra of the tool path and TIF based on the cut-off frequency analysis. The spectrum of tool path should be distributed as evenly as possible, and the TIF (basically depending on the pressure distribution) should be as smooth as possible.

#### 3.2 Influence of the tool movement mode on MSF error

### 3.2.1 Numerical simulation results

The MSF error generated by the path type is investigated deeply in Section 3.1, but the influence of the tool movement mode has not been considered. Thus in this section, the influence of the tool movement mode on MSF error is studied based on the PPC method.

Traditional static model neglects the polishing parameters such as feed rate, eccentricity, spin and orbital motion. To investigate the influence of these parameters in dynamic polishing process, the MSF errors under different processing parameters were simulated and analyzed, respectively. The detail conditions of the simulations are listed in Table 2.

In Table 2, the regular path is used in the No. 1-8 and the pseudo-random path shown in Fig. 4(e) is used in the No. 9-11. No. 1-3 are used to analyze the influence of the feed rate on the MSF error. No. 4 and No. 5 are used to analyze the influence of the eccentricity and the angular velocity on MSF error, respectively. Besides, No. 6-8 are used to simulate the influence of the radial runout on the MSF error, where their eccentricities are set to be 0.5 mm.

The total simulation is calculated based on the PPC method, and the instantaneous TIFs are obtained based on Eq. (3), where the diameter of the pad is set to be 40 mm and the pressure distribution is regarded as rotational symmetry and convex.

The simulation results of the test No. 1-5 results are shown in Fig. 9.

When taking the orbital motion into account, the actual integration path is completely different from the machine path. The MSF error is still similar to the regular path when the feed rate is 1 mm/s, but the amplitude of the MSF error increases 10 times and its pattern starts to be different from the machine path with the increase of the feed rate, as shown in Figs. 9(b2), 9(c2), 9(d2). The actual integration path is the shape of helical curve, and the pitch of the helical curve is proportional to the feed rate, inversely proportional to the angular velocity and independent of the eccentricity. Therefore the MSF errors are almost the same between Nos. 3 and 5 and they are also the same between the Nos. 2 and 4. It can be seen that the orbital motion can worsen the MSF error with the increase of feed rate.

As for the influence of the radial runout on the MSF error, the simulation results of test No. 6-8 are shown in Fig. 10.

The radial runout of the tool is equivalent to a tool with a small eccentricity. When the eccentricity is only 0.5 mm, the actual integration paths of Nos. 7 and 8 are similar to cycloid curve instead of the helical curve. At the same time, the actual velocity of the tool becomes non-uniform; the actual feed rate is slower at the cusps of the cycloid, as shown in Figs. 10(b1) and 10(c1). The phenomena make that the residual MSF error still increases 10 times as the feed rate increases from 1 mm/s to 10 mm/s; and the increasing speed is faster than that of the orbital motion. This means that the radial runout of the tool is also an important factor in amplifying the MSF error.

As for the pseudo-random path, the corresponding MSF errors of test No. 9-11 are shown in Fig. 11.

In Fig. 11, it can be found that the residual MSF error remains at a low level when the feed rate is 1 mm/s. But when the feed rate is increased into 10 mm/s, the residual MSF errors both increase 5∼10 times whether the eccentricity is 5 mm or 0.5 mm, as shown in Figs. 11(b2) and 11(c2). This means that the orbital motion or the radial runout can also worsen the MSF error for the pseudo-random path at a high feed rate.

### 3.2.2 Mechanism explanation

The main reason for the MSF error caused by the orbital motion or the radial runout is that the actual integration path becomes helical or even cycloid curve. The schematic diagram is shown in Fig. 12.

The MSF errors caused by the orbital motion and the radial runout are mainly due to the difference between the actual integration path and theoretical path. The actual path is a helical or even cycloid curve with the increase of the feed rate; but for the convolution of the static TIFs, the theoretical path is the superposition of independent orbital circle path (*L _{orb}*). When the pitch of the helical curve (Δ

*d*) is small, the actual integration path is approximately equal to the theoretical path; but this approximation is no longer effective when Δ

*d*is large enough, which causes the MSF errors. For this reason, in order to make the approximation effective, we set constraints on the polishing parameters.

*v*is the feed rate and

_{m}*S*is the scanning interval of the polishing path. When the eccentricity is greater than 1/10 of the scanning interval, the pitch of the actual integration path is specified to be less than the scanning interval and eccentricity. Based on the PPC method, the amplification ratio of MSF error is no more than 2 times under this condition, which ensures the effective control of MSF error.

## 4. Experimental demonstration

#### 4.1 Experimental setup

The experimental demonstration is conducted on a dual-rotation polishing machine, as shown in Fig. 13(a). To verify the effectiveness of the PPC method in predicting the MSF error, the deterministic polishing experiments are conducted on four Fused silica mirrors with 100 mm diameter under different polishing parameters, respectively. One of the mirrors is shown in Fig. 13(b). The initial surfaces of the mirrors are polished with full-aperture polishing tool in order to eliminate the MSF error. Then we choose two tools with different materials and diameters polished with pseudo-random path and zigzag path, and investigate the influence of path selection on MSF error, respectively. The two tools are shown in Figs. 13(c) and 13(d), respectively. Influences of the orbital motion and the radial runout of the tool are also demonstrated in the experiments, where the radial runout of the tool is measured by the roundness tester (0.3 mm). The working conditions of the experiments are shown in Table 3; and the corresponding static TIFs are shown in Figs. 14(a)–14(c), where the pressure distribution of each tool is calculated based on the Preston equation and velocity distribution. The pressure distributions are filtered by median filter in order to reduce the numerical calculation error. The pressure distribution shape formula can be expressed as

The calculation results are shown in Figs. 14(d) and 14(e), which the non-symmetry shape is mainly due to the special form of the polishing pad. The time interval Δ*t* used in the PPC analysis is set to 0.01 s.

#### 4.2 Verification of the PPC method

The form error of each mirror before and after polishing is measured by a laser interferometer Zygo GPI XP/D. The MSF error of each workpiece can be obtained by the band-pass filter. The modeling results are also obtained by PPC method. The comparisons between the experimental results and the modeling results are shown in Fig. 15. The MSF errors of the form error after polishing with 70% effective aperture are chosen to eliminate the influence of the edge effect.

To ensure the comparability of MSF errors, the MSF error of each experiment is also represented as the proportion of MSF error to the removal amount. Because the feed rate changes in the deterministic polishing, the feed rate distributions in simulations are set to be the same as the experimental distributions.

The results shown in Fig. 15 show the accurate prediction ability of PPC method. As the tool size increases, the period of the residual error of No. 1-2 experiments also increases, as shown in Figs. 15(a3) and 15(b3). Besides, the period of each modeling results matches well with that of the experimental results, which proves that the pseudo-random path can change the period of the residual error through the TIF. In No. 2 experiments, the period of the residual error reaches more than 10 mm, the error is no longer MSF error according to general standard (0.08 - 3 mm). It can be believed that the period of residual error could continue to increase as TIF size be larger or the shape of the TIF becomes smoother.

As for the influence of the orbital motion and radial runout on the MSF error, it is interesting to find that the pattern of the MSF error in No. 3 experiment changes with the position. This is because the feed rate of the outer ring is slower than that of the inner ring; when the feed rate is slow (the minimum feed rate in No. 3 experiment is about 3 mm/s), the residual MSF error of the radial runout tool is similar to the zigzag path; however as the increasing of the feed rate (the maximum feed rate in No. 3 experiment is about 7 mm/s), the MSF error is no longer similar to the path. The phenomenon perfectly demonstrates the accuracy of the PPC method. But this phenomenon does not appear in No. 4 experiment; this is mainly due to the fact that when the eccentricity is large, the MSF error pattern changes slowly with the feed rate. In a word, the results of No. 3-4 experiments show that the orbital motion and the radial runout of the tool can truly introduce MSF error, thus the eccentricity and the radial runout should meet the constraints of Eq. (8) to minimize their impact on MSF error.

Besides, the amplitudes of experimental MSF errors are basically consistent with the theoretical results. However, the results of Nos. 2 and 4 experiments are slightly different from the theoretical results; these are mainly due to the burr points of the pressure distribution and the influence of some uncontrollable environment factors in polishing. The burr points are caused by the sharp uneven bumps on the polishing pad, which can produce circular rings error and scratches on the TIF and workpiece, respectively. Besides, the grooves on the pitch tool and the complicated slurry distribution caused by orbital motion can further introduce removal errors for Nos. 2 and 4 experiments. However, such errors are hard to be predicted at present and further researches and analyses are needed in the future.

## 5. Conclusion

A piecewise-path convolution (PPC) method is introduced to predict and investigate the MSF error; the MSF error distribution can be accurately predicted through this method. Besides, the mechanism of reducing MSF error by path type and TIF is successfully explained mathematically by using filtering theory. The cut-off frequency *f _{c}* formula is defined for the selection of the path type and TIF, and the MSF error can be eliminated by ensuring the

*f*lower than the range of MSF. We can achieve the requirement through non-periodic paths, regular paths with smaller scanning interval and larger tool / smoother TIF. For contact tool (i.e. small-tool, bonnet tool, etc.), the TIF smoothness can be improved by smoothing the contact pressure distribution, for example, ensuring that the tool curvature is greater than the workpiece curvature.

_{c}Besides, the influence of the tool movement on the MSF error is also investigated by the PPC method. The simulation results show that the orbital motion or the radial runout can also introduce non-negligible MSF error, and the MSF error increases with the increase of the feed rate or the decrease of the orbital motion. We proposed constraints on the polishing parameters (feed rate, angular velocity, etc.) to reduce the MSF error caused by the orbital motion or the radial runout; this condition is of essence for controlling MSF error.

The experiments further prove the validity of the analysis results. The MSF errors of after polishing are in good agreement with the simulation results in both the pattern and amplitude. This means that the PPC method has the ability to quantitatively predict the MSF error and guide the polishing works. Furthermore, this method has the potential to predict and analyze the MSF error of other polishing tools. Besides, the selection formulae are instructive for the setting of processing parameters for low MSF error polishing

## Funding

Youth Innovation Promotion Association; Joint Fund of Astronomy (U1831211).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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