Modeling and in-depth analysis of the mid-spatial-frequency error influenced by actual contact pressure distribution in sub-aperture polishing

Lanya Zhang; Lanya Zhang; Songlin Wan; Songlin Wan; Songlin Wan; Songlin Wan; Hanjie Li; Hanjie Li; Hanjie Li; Hao Guo; Hao Guo; Hao Guo; Chaoyang Wei; Chaoyang Wei; Chaoyang Wei; Chaoyang Wei; Dawei Zhang; Dawei Zhang; Jianda Shao; Jianda Shao; Jianda Shao

doi:10.1364/OE.487195

1. Introduction

With the development of modern optics, ultra-precision optical components are widely used in laser fusion, extreme ultraviolet lithography systems, astronomical observation, optoelectronic industries and other fields [1–5]. With the development of the ultra-precision optical manufacturing, computer controlled optical surfacing (CCOS) was first proposed by Itek for optical polishing in the 1970s [6] and also called deterministic polishing [7–10]. To obtain the ultra-precision surface, sub-aperture polishing with small tool is a crucial CCOS technology to achieve deterministic surface shape correction, but the drawback is the generation of the mid-spatial-frequency (MSF) error. The MSF error has not received enough attention in the early development of the optical manufacturing; however, higher demands on surface quality are required in the optical elements used in the extreme ultraviolet lithography systems, laser fusion systems and advanced imaging devices. The MSF errors can cause small angular scattering, affecting image quality and contrast ratio [11]. In high-power systems, MSF errors can even cause focal spot splitting to damage optics [12]. More importantly, once the MSF errors are generated, it is rather difficult to be completely eliminated. Therefore, MSF error has become the primary technical bottleneck that hinders the development of optical polishing.

To address this challenge, study on MSF error generation mechanism and suppressing technology is always the key to solving the problem. In the beginning, the MSF error was simply considered to be introduced due to the periodic path planning [13–16]. Therefore, in 2008, David et al. first proposed pseudo random path which aims to suppress MSF errors caused by periodic paths [17]. Then, some other types of pseudo random paths are further proposed by many researchers to meet different application conditions and improve the effectiveness [18–20]. The theoretical MSF error analysis was promoted since 2010, Tam et al. verifies the increase of path direction can reduce the magnitude of MSF error based on the convolution theory [21]. To obtain a deeper understanding of the MSF error generation mechanism and quantitative prediction under various processing parameters, the quantitative relationship between tool influence function (TIF) size and path interval on MSF error was revealed by B. Zhong et al. [22]. Then, based on the kinematic analysis and polishing material removal theory, the MSF removal characteristic of the dual-axis wheel polishing (DAWP) was analyzed by Ange Lu et al., and the parameters setting criteria is proposed to suppress the MSF error [23]. Furthermore, the MSF error evolution mechanism under the influence of TIF shape, path and tool movement type was further integrated and revealed by Wan et al. [24]. However, in the optics with higher accuracy requirements, there still exists a large proportion of abnormal MSF error which cannot be clearly explained by above theories, such as the phenomena of material removal non-uniformities during tool linear translation (resulting in feed ripples) mentioned by T. Suratwala [25]. Besides, due to the lack of the mechanism understanding and analysis method, it is difficult to conduct in-depth research on these error sources. Therefore, it is crucial to break through the bottlenecks in mechanism and algorithm to reveal the quantitative relationship between the contact pressure distribution and the MSF error, which is of great significance to further suppress the MSF error in ultra-precision optical fabrication.

The organizational structure of this paper is as follows. In section 2, the rotational periodic convolution (RPC) model is introduced to quantitatively predict the MSF error with the influence of the contact pressure. Then, the in-depth analysis is conducted to reveal the influence of contact pressure distribution and tool motion on MSF error in section 3. The experimental results and discussions are presented in section 4, which proves the validity of the theoretical analysis. Finally, the paper is summarized in section 5.

2. Establishment of the model of rotational periodic convolution (RPC)

2.1 Analysis of error sources beyond path convolution effects

In deterministic optical polishing, it is generally accepted that the process follows the Preston equation [26]:

(1)$$dz(x,y) = k \cdot P(x,y) \cdot V(x,y) \cdot \textrm{dt}$$

where dz(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 tool.

Based on the Preston equation, the removal amount can be expressed as the convolution of the TIF per unit time and the dwell time:

(2)$$z(x,y) = R(x,y) \ast T(x,y)$$

where z(x,y) is amount of material removal, R(x,y) is TIF and T(x,y) is dwell time.

Under the traditional removal calculation method, TIF is usually regarded as the minimum calculation unit, where the TIF can be considered as the average integral of the surface pressure distribution (P (x, y)) multiplied by k, V (x, y) at a fixed point. However, the MSF error caused by the non-uniformity and asymmetry of contact pressure distribution is still be ignored. The error source is shown in Fig. 1.

Fig. 1. (a) actual contact pressure distribution. (b) Actual movement of actual contact pressure distribution. (c) Fixed point motion of actual contact pressure distribution.

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Therefore, to address this problem, a rotational periodic convolution (RPC) model is proposed that using P(x,y) as the minimum calculation unit to obtain more accurate removal amount.

2.2 Rotational periodic convolution (RPC) model

In order to deeply analyze the removal mechanism of taking the contact pressure distribution as the minimum removal unit, Eq. (2) is further modified. For the convolution of the spatial variant TIF, it can be rewritten as the generalized convolution of equation:

(3)$$\begin{array}{l} z(x,y) = \sum\limits_i {{R^i}(x - {x_i},y - {y_i}) \cdot T({x_i},{y_i})} \\ \textrm{with }{R^i}(x,y) = k \cdot P(x,y) \cdot V(x,y) \cdot rot\textrm{ }, rot = \left( {\begin{array}{cc} {\cos \omega t}&{\sin \omega t}\\ { - \sin \omega t}&{\cos \omega t} \end{array}} \right) \end{array}$$

where Rⁱ(x-x_i,y-y_i) represents the TIF changing with position (x_i,y_i). rot represents rotation transformation matrix. t represents the time during the polishing process. ω represents the angular velocity of the tool.

For the sub-aperture polishing with small tool, tool rotation is periodic in polishing, after each rotation period, the contact pressure distribution is returned to the initial state. Therefore, the dwell time can be further rewritten as the generalized convolution between the path fragments in one rotation cycle and positions of the path fragments:

(4)$$\begin{array}{l} T({x_i},{y_i})\textrm{ = }\sum\limits_j {{S^j}({x_i} - {x_j},{y_i} - {y_j})} \cdot Samplin{g_{2D}}\\ \textrm{with S}amplin{g_{2D}} = \sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n ={-} \infty }^\infty {\delta (x - \Delta {X_m},y - \Delta {Y_n})} } \end{array}$$

where S ^j(x_i-x_j,y_i-y_j) represents path fragments. Sampling_2D represents discrete positions of the path fragments. ΔX_m represents the discrete sampling point (i.e., Sampling_2D) interval in the feed direction. ΔY_n represents the discrete sampling point interval in the direction of path interval.

Therefore, the calculation of removal amount can be further modified as follows:

(5)$$\begin{array}{l} z(x,y) = \sum\limits_i {{R^i}(x - {x_i},y - {y_i}) \cdot \sum\limits_j {{S^j}({x_i} - {x_{^j}},{y_i} - {y_j})} \cdot Samplin{g_{2D}}} \\ \textrm{let }{U^j}({x_i},{y_i}) = {S^j}({x_i} - {x_j},{y_i} - {y_j})\\ {\kern 0.8pt} \Rightarrow z(x,y) = \sum\limits_i {\sum\limits_j {{R^i}(x - {x_i},y - {y_i}) \cdot {U^j}({x_i},{y_i}) \cdot Samplin{g_{2D}}} } \\ \textrm{define }eq{u_{TIF}}(x,y) = \sum\limits_i {{R^i}(x - {x_i},y - {y_i}) \cdot {U^j}({x_i},{y_i})} \\ {\kern 1.2cm}eq{u_{TIF}}(x,y) = \sum\limits_i {k \cdot P({x_i},{y_i})} \cdot V({x_i},{y_i}) \cdot rot \cdot {U^j}({x_i},{y_i})\\ {\kern 0.8pt} \Rightarrow z(x,y) = \sum\limits_j {equ_{TIF}^j \cdot Samplin{g_{2D}}} \end{array}$$

where U ^j(x_i,y_i) represents the S ^j(x_i-x_j,y_i-y_j) after spatial translation. equ_TIF is the equivalent rotation convolution kernel, which represents the actual removal amount of feed path for one cycle of tool rotation.

In the above derivations, the material removal can be reconsidered as the convolution of equ_TIF and discrete sampling points. Moreover, in the case of small change in equ_TIF, the material removal can be further simplified to the multiplication calculation in the frequency domain:

(6)$$\begin{aligned} z(x,y) &= equTIF \ast Samplin{g_{2D}}\\ & = {{\cal F}^{ - 1}}[{\cal F}(eq{u_{TIF}}) \times {\cal F}(Samplin{g_{2D}})] \end{aligned}$$

Through the above simplification of the removal amount calculation, the material removal amount can be further accurately calculated.

3. Mechanism analysis of MSF error caused by contact pressure distribution

In this paper, the frequency band of 1mm-10 mm is considered as the interval of MSF error, which is the specification in EUV and X-ray optics system [27], and it is also the most significant part of MSF error deterioration in sub-aperture polishing. In order to conduct in-depth research on the MSF error influenced by the contact pressure distribution, the analysis is mainly conducted in following three parts:

1. the relationship between the contact pressure distribution and MSF error.
2. the relationship between speed ratio and MSF error.
3. the influence of the coupling relationship between the contact pressure distribution and speed ratio (spin velocity/feed speed) on the MSF error.

3.1 Influence of contact pressure distribution on MSF error

3.1.1 Mechanism analysis

The influence of different contact pressure distributions on the MSF error is mathematically analyzed under the condition of uniform removal. In this case, the MSF error is basically equal to the remaining error after removing the uniform removal depth (0 frequency information). Therefore, the MSF error can be expressed as:

(7)$$\begin{array}{l} \textrm{MSF} \approx {\cal F}({z(x,y) - \overline {z(x,y)} } )= {\cal F}(eq{u_{TIF}} - \overline {eq{u_{TIF}}} ) \cdot {\cal F}(Samplin{g_{2D}})\\ \Rightarrow RMS[{{\cal F}({z(x,y) - \overline {z(x,y)} } )} ]= RMS[{{\cal F}(eq{u_{TIF}} - \overline {eq{u_{TIF}}} ) \cdot {\cal F}(Samplin{g_{2D}})} ]\end{array}$$

Because the discrete sampling point (Sampling_2D) contains most of the 0 items and a few of the 1 items, the cross terms of $eq{u_{TIF}} - \overline {eq{u_{TIF}}}$ and Sampling_2D are mostly 0. Therefore, the multiplication of the root mean square (RMS) value of $eq{u_{TIF}} - \overline {eq{u_{TIF}}}$ and the RMS value of Sampling_2D is considered as the RMS value of MSF error. The RMS value of MSF error can be expressed as:

(8)$$\begin{aligned} RMS[{{\cal F}({z(x,y) - \overline {z(x,y)} } )} ]&= RMS[{{\cal F}(eq{u_{TIF}} - \overline {eq{u_{TIF}}} ) \cdot {\cal F}(Samplin{g_{2D}})} ]\\ & \approx RMS[{{\cal F}(eq{u_{TIF}} - \overline {eq{u_{TIF}}} )} ]\cdot RMS[{{\cal F}(Samplin{g_{2D}})} ]\end{aligned}$$

According to the property of Fourier transform, the translation and rotation of the contact pressure distribution will not change the frequency domain amplitude, the relationship between MSF error and contact pressure distribution can be obtained by Eq. (8).

(9)$$\begin{array}{l} \frac{{RMS[{{\cal F}({z(x,y) - \overline {z(x,y)} } )} ]}}{{RMS[{{\cal F}(Samplin{g_{2D}})} ]}} \approx RMS[{{\cal F}(eq{u_{TIF}} - \overline {eq{u_{TIF}}} )} ]\\ \textrm{where}\\ RMS(eq{u_{TIF}} - \overline {eq{u_{TIF}}} ) \propto RMS\left[ {\sum {(P(x,y) - \overline {P(x,y)} ) \cdot V(x,y) \cdot rot} } \right]\\ \textrm{ } \propto RMS[{P(x,y) - \overline {P(x,y)} } ]\\ RMS[{{\cal F}(Samplin{g_{2D}})} ]\propto {H_{z(x,y)}}\\ RMS[{{\cal F}({z(x,y) - \overline {z(x,y)} } )} ]= MS{F_{z(x,y)}}\\ \textrm{define }er{r_{Normal}}\textrm{ = }\frac{{MS{F_{z(x,y)}}}}{{{H_{z(x,y)}}}}\\ \Rightarrow er{r_{Normal}} \propto RMS[{P(x,y) - \overline {P(x,y)} } ]\end{array}$$

where err_Normal represents the normalized MSF error, H_z(x,y) represents the removal depth, and MSF_z(x,y) represents the MSF error of total removal.

In Eq. (9), it can see that a good property of the linear proportional relationship between err_Normal and $RMS[P(x,y) - \overline {P(x,y)} ]$ is proved, hereafter, a series of contact pressure distributions are simulated to verify the correctness of the linear relationship.

3.1.2 Simulation results

To characterize the actual contact surface pressure distribution differences due to the uneven surface of the polishing pad in the simulation, we used small circles with different diameters to fill the polishing pad. Among them, 8 kinds of small circle diameter are randomly selected, which are 1/50, 7/250, 9/250, 11/250, 17/250, 31/250,51/250 and 81/250 of the polishing pad diameter, as shown in Fig. 2.

Fig. 2. $P(x,y) - \overline {P(x,y)} $ of randomly generated contact pressure distribution.

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The simulated contact pressure distribution is used in the RPC model, and the simulation parameters are listed in Table 1. The RPC simulation results are filtered, as shown in Fig. 3.

Fig. 3. Simulate of err_Normal after removal of different contact pressure distribution.

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Table 1. Parameters of simulation

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In Fig. 3 the MSF error is divided into periodic ripples in the feed direction and path interval. The periodic ripples in the feed direction are caused by uneven contact pressure distribution, and the periodically corrugated shape is directly related to the contact pressure distribution shape. The relationship between $RMS[{P(x,y) - \overline {P(x,y)} } ]$ and err_Normal is linear, and the linear fitting results are shown in Fig. 4.

Fig. 4. The Linear relationship between $RMS[{P(x,y) - \overline {P(x,y)} } ]$ and err_Normal.

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In Fig. 4, the goodness of fit parameter for linear fitting is: R-square = 0.7886, which proves to be a good linear relationship, and the correctness of the theoretical analysis is proved by the linear fitting results of $RMS[{P(x,y) - \overline {P(x,y)} } ]$ and err_Normal. It can also be seen that this linear relationship is not strict enough, and the specific reasons for the apparent linear relationship will be analyzed in Section 3.3. According to the linear relationship, to better restrain the MSF error, the pressure distribution on the contact surface of optics and tool should be as uniform as possible.

3.2. Influence of speed ratio on MSF error

In section 3.1, the influence of the contact pressure distribution on equ_TIF has been quantitatively analyzed. In addition to the surface pressure distribution, the factors that affect equ_TIF also include the impact of speed ratio on path fragment. In Section 3.2, the speed ratio (spin velocity/feed speed) is further considered to analyze the effect on equ_TIF.

3.2.1 Mechanism analysis

The interval of discrete positions of the path fragments (Sampling_2D) in the feed direction (X direction) is changed by the length of equ_TIF, which is affected by changes in speed ratio. The center of the tool is usually used as a moving reference in polishing, Then, the relationship between the moving distance of one cycle of tool rotation (ΔX) and the speed ratio is given by the equation:

(10)$$\Delta X = \frac{v}{r}$$

where r represents the spin velocity of the tool and v represents the feed speed of the tool.

In Eq. (10), the moving distance of the tool is inversely proportional to the speed ratio. Therefore, the distribution of two-dimensional random sampling points is changed by different moving distances, the change of uniform motion is shown in Fig. 5.

Fig. 5. Schematic diagram of two-dimensional discrete sampling points in uniform motion.

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The Fourier transform of the discrete sampling point of the two-dimensional period is shown in the equation:

(11)$$\begin{array}{l} Samplin{g_{2D}} = \sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n ={-} \infty }^\infty {\delta (x - m\Delta X,y - n\Delta Y)} } \\ {\cal F}(Samplin{g_{2D}}) = \frac{1}{{\Delta X}} \cdot \frac{1}{{\Delta Y}}\sum\limits_{m ={-} \infty }^\infty {\sum\limits_{n ={-} \infty }^\infty {\delta (u - \frac{m}{{\Delta X}},v - \frac{n}{{\Delta Y}})} } \end{array}$$

where ΔX represents the discrete sampling point interval in the direction of feed, ΔY represents the discrete sample point spacing of the path interval, m represents the number of sample points in the direction of feed, and n represents the number of sample point in the path interval direction.

In Eq. (11), the frequency domain two-dimensional interval is the reciprocal relationship with the time domain two-dimensional interval. If the sampling interval in the feed direction (ΔX) is less than the path interval direction (ΔY), the feed direction interval in frequency domain sampling will be greater than the path direction. In the unit length, the number of path interval samples is more than the number of feed interval samples. At this time, MSF error is dominated by path interval.

3.2.2 Simulation results

To investigate the influence of speed ratio and path interval in dynamic polishing process, the MSF errors under different processing parameters were simulated and analyzed, respectively. The detailed conditions of the simulations are listed in Table 2.

Table 2. Parameters of simulation

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In Table 2, Fig. 6(a) shows the $P(x,y) - \overline {P(x,y)} $ of contact pressure distribution in simulation. Figures 6 (b)–(d) shows the three different path interval raster paths used in the simulation, as shown in Fig. 6.

Fig. 6. (a) $P(x,y) - \overline {P(x,y)} $ of contact pressure distribution. (b) raster path with 1 mm path interval. (c) raster path with 3 mm path interval. (d) raster path with 5 mm path interval.

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The simulation Fourier filtering results of 1 mm, 3 mm, and 5 mm path intervals are shown in Fig. 7

Fig. 7. (a) err_Norma_l at 1 mm path interval (b) err_Norma_l at 3 mm path interval (c) err_Normal at 5 mm path interval

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In Fig. 7, for the same path interval, the speed ratio is negatively correlated with err_Normal; while for the same speed ratio, the path interval is positively correlated with err_Normal, which is consistent with the previous analysis. According to our model analysis (Eq. (6) and Eq. (11)), when the path interval (ΔY) is much smaller than ΔX, the MSF error is mainly caused by the impact of the speed ratio. The path ripples will be submerged in the MSF error generated by the speed ratio, which makes the 1 mm and 3 mm period ripples not obvious, as shown in Fig. 7(a) and (b). When the path interval is great enough, the impact of path interval on MSF error becomes more obvious due to the increase of ΔY, as shown in Fig. 7(c). The relationship between err_Normal and speed ratio is fitted in different path intervals, as shown in Fig. 8.

Fig. 8. Function fitting of speed ratio and err_Normal.

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In Fig. 8, first, the err_Normal gradually decreases and tends to a stable value when the speed ratio gradually increases, because the influence of equ_TIF on MSF error decreases with the increase of speed ratio. Second, the stable value of err_Normal increases with the increase of path interval, because the MSF error under the influence of path interval gradually increases. It can be found that the MSF error and path interval is monotonic only when the speed ratio and surface pressure distribution are consistent. However, this monotonous relationship can be destroyed when the conditions are not consistent. For example, when the path interval is 1 mm and the speed ratio is 1, the err_Normal can even greater than the circumstance when the path interval is 3 mm and the speed ratio is greater than 3. Therefore, the path interval is not the only cause that affects MSF error, therefore, considering the appropriate speed ratio and path interval is beneficial to restrain the MSF error in polishing.

3.3 Influence of the coupling relationship between the rotational asymmetry degree of contact pressure distribution and speed ratio on MSF error

In section 3.1, the linear relationship between the pressure distribution on the contact surface and the normalized MSF error is proved, but the linear relationship is not rigorous. The coupling relationship between the rotational asymmetry of the contact pressure distribution and the speed ratio may have an impact on the linear relationship. Therefore, this section analyzes whether the coupling relationship affects the MSF error.

3.3.1 Zernike polynomial decomposition contact pressure distribution

We found that the coupling effect between the contact pressure distribution and the rotation speed ratio is mainly affected by the number of the “petals” of the contact pressure distribution, where the ΔX changes with shape of the contact pressure distribution even under the same speed ratio, as shown in Fig. 9.

Fig. 9. Equivalent ΔX of contact pressure distribution with different number of petals.

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The Zernike polynomial has the ability to decouple the contact pressure distribution. In polar coordinate system, Zernike polynomial coefficient Z_q can be classified according to the number of petals (p). The general terms of petal number classified according to p and the degree of rotational asymmetry of different petal numbers can be expressed as:

(12)$$\begin{array}{l} As{y_p} = \frac{{RMS\left\{ {\left. {\sum\limits_{r = 0}^R {({{Z_{f(p,r)}} + {Z_{f(p,r) + 1}}} )} } \right|f(p,r) + 1 \le Q} \right\}}}{{RMS\left\{ {\sum\limits_{q = 1}^Q {{Z_q}} } \right\}}} \times 100\%\\ \textrm{with }f(p,r) = \frac{{{p^2} + 3p}}{2} + 2{r^2} + (2p + 1)r\\ {\kern 1cm}Q = \frac{{R \times (R + 1)}}{2} \end{array}$$

where Asy_p represents the degree of rotational asymmetry extracted by Zernike polynomials. Z_q represents Zernike coefficient term. R represents the rank of Zernike polynomial. Q represents the total number of Zernike terms under R rank. f (p,r) represents the general term in different petal numbers. Under the same $RMS[{P(x,y) - \overline {P(x,y)} } ]$ condition, the non-uniformity of the contact pressure based on Zernike determines the level of the final MSF error, which decreases with the increase of the number of petals.

3.3.2 Simulation results

The contact pressure distribution of three highly typical petal characteristics was simulated, for example, the p = 1, 2, 3 were extracted with the Zernike polynomial, as shown in Fig. 10.

Fig. 10. Three simulated special contact pressure distributions extracted by Zernike.

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The three contact pressure distributions in Fig. 10 are taken as P(x,y) in the simulation, and the Asy_p (R = 64 or Q = 2080) of each pressure distribution is calculated. The simulation parameters are shown in Table 3.

Table 3. Simulation parameters of three kinds of contact pressure distribution

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In Table 3, a series of simulations of the contact pressure distribution in Fig. 10 was performed using the 1 mm interval raster path in Fig. 6(c). The coupling relationship between special contact pressure distribution and speed ratio is analyzed in three cases, and MSF error is shown in Fig. 11.

Fig. 11. (a) One petal err_Normal of contact pressure distribution under different speed ratio parameters. (b) Two petals err_Normal of contact pressure distribution under different speed ratio parameters. (c) Three petals err_Normal of contact pressure distribution under different speed ratio parameters.

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In Fig. 11, the results of Section 3.3 and Section 3.2 are the same. The err_Normal amplitude is inversely proportional to the speed ratio. The fitting results of err_Normal and speed ratio are shown in Fig. 12.

Fig. 12. err_Normal function fitting of three contact pressure distributions in different speed ratios.

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In Fig. 12, the err_Normal decreases with the decrease of Asy_p and decreases with the increase of speed ratio. For the same surface pressure distribution, the greater the speed ratio, the weaker the influence of the asymmetry of the surface pressure distribution on the MSF error. Besides, for the same speed ratio, the degree of rotational asymmetry of the surface pressure distribution is positively correlated with the MSF error. The result of the three-petal contact pressure distribution tends to be flat, which has little influence on the MSF error. The asymmetry degree of contact pressure distribution of one-petal and two-petal is the main reason for the change of MSF error, the one-petal contact pressure distribution is caused by the inclination of the small tool, and the two-petal contact pressure distribution is caused by the manufacturing process of the tool.

4. Experimental verification

4.1 Experimental setup

For the acquisition of the contact pressure distribution, the contact pressure distribution between the tool and the workpiece is obtained by using FUJIFILM pressure-sensitive paper. The type measurement method is a two-piece type, which is composed of two layers of polyester-based film. One layer is coated with micro-particle color-forming material (A-film), and the other layer is with color-developing material (C-film). The principle is that when the particles are broken under pressure, the color-forming layer reacts with the color-developing layer, and red pinch points appear. The micro-particles are designed to rupture under different pressures, the color density is the reaction pressure. The model used in the experiment is 5lw, and the measuring pressure range is 0.006-0.05 Mpa.

The experimental verification was carried out on the ABB six axis (IRB-6620) manipulator platform, as shown in Fig. 13(a). In order to verify the validity of RPC model, eight K9 glasses with a diameter of 100 mm were polished under different polishing parameters, one of the glasses is shown in Fig. 13(b). Then, two polishing pads with the same diameter but different materials are selected for the experiment. The polishing pads of these two materials are shown in Fig. 13(c) and Fig. 13(d) respectively. In the experiment, different contact pressure distributions of polishing pads are achieved by using different two materials, and the actual contact pressure distributions were obtained by pressure sensitive paper. Then, actual polishing and simulation are conducted under different polishing parameters, and the correctness of the model is verified by comparing the RMS of the introduced MSF error. The influence of different contact pressure distributions and speed ratio on MSF error is studied. The experimental polishing parameters are shown in Table 4.

Fig. 13. (a) ABB Six axis robot. (b) k9 material glass. (c) Damping cloth tool. (d) polyurethane tool.

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Table 4. Experimental conditions of RPC method validation.

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4.2 Experimental results

A pressure-sensitive film was used to measure the pressure distribution on the contact surface between Figs. 13(c) and 13(d) material tools and the workpiece. The measurement results were restored by using MATLAB software, as shown in Figs. 14(a) and 14(b).

Fig. 14. (a) On the left is the pressure distribution of the contact surface between the damping cloth and the glass, and on the right is the MATLAB software recovery. (b) On the left is the pressure distribution of the contact surface between the polyurethane and the glass, and on the right is the MATLAB software recovery.

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The damping cloth and polyurethane material tools were used to polish the optics respectively, and the restored contact pressure distribution was taken as the P(x,y) in the simulation. The optics are measured by using a laser interferometer (VeriFire XPZ, Zygo) with 4-inch standard planar mirror, and the MSF error of the predicted and the measured results is obtained through FFT bandpass filtering in the 1-10 mm frequency band. In order to study the MSF error introduced in polishing, the initial MSF error needs to be removed when calculating the normalized MSF error. The distribution of the initial MSF error and the introduced MSF error are commonly independent and can be regarded as incoherent superposition. Therefore, the RMS of the polished MSF error can be thought of as the sum of the RMS of the initial MSF error and the introduced MSF error. Hereafter, the MSF error of the introduced part can be calculated by the following equation:

(13)$$RM{S_{introduce}}_d = \sqrt {RM{S_{total}}^2 - RM{S_{initial}}^2}$$

where RMS_initial is the RMS value of the initial MSF error, and RMS_total is the RMS value of the surface MSF error after polishing.

The experimental and simulation err_Normal comparison results are shown in Fig. 15.

Fig. 15. (a) The err_Normal of experimental results for the damping cloth material. (b) the err_Normal of simulation results for the damping cloth material. (c) the err_Normal of experimental results for the polyurethane material. (d) the err_Normal of simulation results for the polyurethane material.

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In Fig. 15, the MSF error distribution after experiment and simulation is similar. In addition, experimental and simulation results show that the err_Normal decreases as the speed ratio increases. Therefore, the RPC model is proved to be effective in predicting MSF error distribution. Further analysis verifies the effectiveness of the RPC model in quantitatively predicting MSF error.

The comparison of simulation and experimental results of err_Normal between different speed ratios are shown in the Fig. 16.

Fig. 16. Comparison of experimental and simulation err_Normal data

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In Fig. 16, it can be seen that the err_Normal of polyurethane is larger than that of damping cloth, and the err_Normal decreases in trend as the speed ratio increases. To better verify the effectiveness of the proposed RPC model, the contact pressure distribution of polyurethane and damping cloth is extracted from 1-63 by Asy_p.

In Fig. 17, 1–63 petals of Zernike are extracted from the contact pressure distribution. Obviously, the overall rotational asymmetry of polyurethane material is greater than that of damping cloth material. Therefore, the MSF error result of polyurethane material after polishing is greater than that of damping cloth material.

Fig. 17. Asy_p corresponding to damping cloth and polyurethane 1–63 petals.

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The error rate (rate_error) is defined to prove the effectiveness of the RPC model, and can be expressed by the equation:

(14)$$rat{e_{error}} = \frac{{|RMS(Exp\textrm{ - }er{r_{Normal}}) - RMS(Sim\textrm{ - }er{r_{Normal}})|}}{{RMS(Sim\textrm{ - }er{r_{Normal}})}} \times 100\%$$

where RMS(Exp-err_Normal) represents the error size of MSF measured in the experiment. RMS(Sim-err_Normal) represents the error size of MSF calculated in the simulation.

In this paper, the rate_error between the experiment and simulation of polyurethane and damping cloth is shown in Table 5.

Table 5. The error rate between the experiment and simulation

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In Table 5, the error rates are less than 15%, which shows the correctness of the model both in predicting the morphology of MSF and quantitatively predicting the size of MSF error. Through the establishment of the RPC model, it provides guiding suggestions for actual polishing.

5. Conclusion

A rotational periodic convolution (RPC) model is introduced to predict and investigate the MSF error; the MSF error distribution can be accurately predicted through this method. In addition, the influence mechanism of contact pressure distribution and rotational speed ratio on MSF error is analyzed mathematically. The relationship between contact pressure distribution and MSF error (i.e., the linear relationship between $RMS[{P(x,y) - \overline {P(x,y)} } ]$ and err_Normal) is clarified, and the influence of speed ratio on MSF error is clearly analyzed in mechanism. Besides, Zernike polynomial fitting is proposed to deeply analyze the influence of rotational asymmetry of contact pressure distribution on MSF error. Through theoretical analysis, it provides guidance for practical processing (i.e., make a tool with good uniformity of contact pressure distribution with optics, and the speed ratio of the tool is as large as possible.)

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 RPC model has the ability to quantitatively predict the MSF error and guide the polishing works.

Funding

National Key Research and Development Program of China (2022YFB3403403); Member of Youth Innovation Promotion Association of the Chinese Academy of Sciences (2022246); Natural Science Foundation of Shanghai (21ZR1472000); National Natural Science Youth Foundation of China (62205352); Shanghai Sailing Program (20YF1454800, 22YF1454800).

Acknowledgments

The authors would like to thank the referees for their valuable suggestions and comments that have helped improve the paper.

Disclosures

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

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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No.	1	2	3	4	5	6	7	8
$R M S (p (x, y) - \bar{p (x, y)})$	0.4369	0.4169	0.3933	0.3856	0.4008	0.3723	0.3843	0.4085
Path type	raster path
Diameter	25 mm
r (rpm)	180
Feed speed(mm/s)	15
err_Normal	0.0147	0.0091	0.0081	0.0074	0.0068	0.0064	0.0051	0.0103

No.	1	2	3	4	5	6	7	8
Speed ratio	1	2	3	4	5	9	11	16
Feed speed	15 mm/s
$R M S (p (x, y) - \bar{p (x, y)})$	0.4628
Path interval	1 mm
err_Normal	0.0312	0.0206	0.0194	0.0165	0.0160	0.0164	0.0163	0.0160
Path interval	3 mm
err_Normal	0.0587	0.0360	0.0316	0.0276	0.0286	0.0278	0.0291	0.0281
Path interval	5 mm
err_Normal	0.0889	0.0687	0.0644	0.0613	0.0629	0.0618	0.0628	0.0619

No.	1	2	3	4	5	6	7	8
Speed ratio	1	2	3	4	8	11	13	16
Feed velocity (mm/s)	15
Path type	raster path
Path interval (mm)	1
Asy_p (p = 1)	64.3%
err_Normal	0.1301	0.0961	0.0477	0.0707	0.0298	0.0181	0.0193	0.0115
Asy_p (p = 2)	55.6%
err_Normal	0.0701	0.0827	0.0374	0.0358	0.0214	0.0141	0.0085	0.0078
Asy_p (p = 3)	40.6%
err_Normal	0.0281	0.0349	0.0200	0.0248	0.0122	0.0105	0.0104	0.0073

Experiment	No.1	No.2	No.3	No.4
Tool material	Polyurethane	Polyurethane	Polyurethane
Tool material	Damping cloth	Damping cloth	Damping cloth	Damping cloth
Diameter	25 mm
r (rpm)	30	60	105	200
Feed velocity	15 mm/s
Workpiece	100 mm diameter K9 material glass
Polishing slurry	Cerium oxide

Speed ratio	2	4	7	13.3
Damping cloth	9.8%	13.2%	14.2%	14.5%
Polyurethane	14.8%	8.4%	12.1%	9.8%

Modeling and in-depth analysis of the mid-spatial-frequency error influenced by actual contact pressure distribution in sub-aperture polishing

Abstract

1. Introduction

2. Establishment of the model of rotational periodic convolution (RPC)

2.1 Analysis of error sources beyond path convolution effects

2.2 Rotational periodic convolution (RPC) model

3. Mechanism analysis of MSF error caused by contact pressure distribution

3.1 Influence of contact pressure distribution on MSF error

3.1.1 Mechanism analysis

3.1.2 Simulation results

3.2. Influence of speed ratio on MSF error

3.2.1 Mechanism analysis

3.2.2 Simulation results

3.3 Influence of the coupling relationship between the rotational asymmetry degree of contact pressure distribution and speed ratio on MSF error

3.3.1 Zernike polynomial decomposition contact pressure distribution

3.3.2 Simulation results

4. Experimental verification

4.1 Experimental setup

4.2 Experimental results

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

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Figures (17)

Tables (5)

Equations (14)

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