Modeling and prediction of tool influence function under complex edge in sub-aperture optical polishing

Songlin Wan; Songlin Wan; Songlin Wan; Chaoyang Wei; Chaoyang Wei; Chaoyang Wei; Chaoyang Wei; Chen Hu; Chen Hu; Chen Hu; Haojin Gu; Haojin Gu; Jianda Shao; Jianda Shao; Jianda Shao; Jianda Shao

doi:10.1364/OE.432318

1. Introduction

With the development of modern optical technology, ultra-precision optical components have been widely used in high-power lasers [1–3], astronomical observation [4,5], and ultraviolet lithography systems [6–8]. In these types of technologies, form errors in the edge part of the workpiece are increasingly crucial to the performance of the optical system. The edge surface quality of segmented optics dominates the performance of optical systems, where the total length of their edges is much longer than that of conventional optical systems, such as the European Extremely Large Telescope (E-ELT) [9], the Giant Magellan Telescope (GMT) [10], and the James Webb Space Telescope (JWST) [11]. In addition, various optics (elongated rectangular shape, holed optics, etc.) used in lithography and high-power laser systems also require high edge surface quality to maximize the facility’s useful real-estate [12]. However, the form accuracy in the complex edge region is the most difficult area to control owing to the lack of theoretical modeling guidance, such as the edge regions with an angle of 120° of the hexagonal mirrors and the edge of the circular or elliptical optics with a small radius of curvature [13,14]. On the other hand, various types of 1-layer/multi-layers tools have been developed to satisfy both curved surface fitting and mid-spatial-frequency error smoothing [15]; however, the existing models cannot organically unify the edge TIFs of various tools, which hinders the improvement of polishing accuracy. Therefore, it is important to recognize and characterize the edge removal behavior of various tools in workpieces with complex edge contours.

The main cause of the edge effect is the change in the pressure distribution when the polishing pad overhangs the edge of the workpiece. In previous studies, with the continuous iterative development of many researchers, relatively high-accuracy TIF models suitable for straight-line or approximate straight-line edge TIF prediction have been achieved. Jones established a simplified linear pressure model to describe edge TIFs [16]. However, this linear model cannot quantitatively fit the pressure distribution well, especially in areas near the edge side. Then, a simplified nonlinear model was proposed by Cordero-Davila et al. [17], but it is too rough to describe the pressure distribution accurately. The accuracy of the model is greatly improved by the introduction of finite element analysis (FEA). In 2009, Kim et al. provided a parametric fitting model to describe the edge removal behavior, which matches well with the experimental results [18]. Liu et al. proposed pressure models for lap tools based on the FEA software [19]. In 2016, Wan et al. introduced the basic pressure distribution and correcting factor to describe the edge TIFs analytically, which greatly widens the application range of the model and improves the model accuracy [20]. However, the models mentioned above all attribute the edge pressure distribution to a function related to the overhang ratio in one dimension (1-D) view, which is only suitable for the straight-line edge and cannot be universally applied to various tools. Therefore, the establishment of an accurate edge TIF model suitable for different complex edges and tools can play an important role in edge error convergence and greatly improve the manufacturing capacity of high-end optical equipment.

In this paper, the establishment of the edge TIF model is presented in Section 2, and Section 3 illustrates the experimental setup and results to demonstrate the validity and accuracy of the proposed edge model. Finally, the paper is summarized in Section 4.

2. Modeling of the edge removal behavior

Since the shape of the contact regions becomes much more complex, especially when the tools have an orbital motion, as shown in Fig. 1. It is impossible to obtain a universal TIF model suitable for each shape of the contact region using the traditional function fitting method. Instead, the mechanism of the edge nonlinear pressure distribution must be studied clearly.

Fig. 1. Schematic diagram of the circumstances when the pad overhangs a polygonal or circular workpiece

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2.1 Nonlinear edge kernel function

In essence, this nonlinear pressure phenomenon is a mechanical stretching effect of the unconstrained part to the deformed part, where the stretching effect leads to additional shear stress. Figure 2 depicts a simplified 2-D contact case. According to the material mechanics theory [21], the contact pressure can be expressed as the sum of the normal and shear stresses, as shown in Eq. (1)

(1)$$P(x) = E \cdot \sigma (x) + G \cdot \gamma (x)$$

where E and G are the elastic modulus and shear modulus, respectively. P(x) is the pressure distribution in the 2-D case, σ and γ are the normal strain and shear strain, respectively.

Fig. 2. Schematic diagram of the edge stress and deformation for the 2-D case

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The normal strain σ(x) at each contact region is equal to the ratio of the deformation to the thickness of the polishing pad. However, the shear strain γ(x) is difficult to obtain; only the shear strain at the edge (γ₀) can be approximately expressed by the shear angle Δd /Λ. Therefore, the contact pressure can be further rewritten as

(2)$$\begin{aligned} P(x) &= E \cdot \frac{{\Delta d}}{d} + G \cdot {\gamma _0} \cdot \frac{{\gamma (x)}}{{{\gamma _0}}}\\ \textrm{ } &= E \cdot \frac{{\Delta d}}{d} + G \cdot \frac{{\Delta d}}{\Lambda } \cdot \frac{{\gamma (x)}}{{{\gamma _0}}} \end{aligned}$$

where Δd is the deformation of the polishing pad, d is the thickness of the polishing pad, and Λ is the lateral deformation at the edge, as shown in Fig. 2.

Generally, the materials of the polishing pad are isotropic; hereafter, the relationship between the elastic modulus and the shear modulus follows the E = G/2(1+ν) equation, where ν denotes the Poisson’s ratio. The contact pressure can be further rewritten as

(3)$$\begin{aligned} P(x) &= E \cdot \frac{{\Delta d}}{d} + \frac{E}{{2(1 + \nu )}} \cdot \frac{{\Delta d}}{\Lambda } \cdot \frac{{\gamma (x)}}{{{\gamma _0}}}\textrm{ with }G = \frac{E}{{2(1 + \nu )}}\\ \textrm{ } &= E \cdot \frac{{\Delta d}}{d} \cdot \left( {1 + \frac{d}{{2(1 + \nu )\Lambda }} \cdot \frac{{\gamma (x)}}{{{\gamma_0}}}} \right) \end{aligned}$$

In the 2-D contact case, if the polishing pad overhangs both sides of the workpiece, as shown in Fig. 3, the shear strain in each position is the superposition of the shear strains influenced by two sides, which follows the principle of linear superposition of forces.

(4)$$\begin{aligned} P(x) &= E \cdot \frac{{\Delta d}}{d} \cdot \left( {1 + \frac{d}{{2(1 + \nu )\Lambda }} \cdot \frac{{\gamma (x)}}{{{\gamma_0}}} + \frac{d}{{2(1 + \nu )\Lambda }} \cdot \frac{{\gamma (S - x)}}{{{\gamma_0}}}} \right)\\ \textrm{ } &= E \cdot \frac{{\Delta d}}{d} \cdot \left( {1 + ({\delta (x) + \delta (S - x)} )\ast \left( {\frac{d}{{2(1 + \nu )\Lambda }} \cdot \frac{{\gamma (x)}}{{{\gamma_0}}}} \right)} \right)\\ \textrm{ } &= {P_0} \cdot ({1 + ({\delta (x) + \delta (S - x)} )\ast K(x)} )\textrm{ }\\ &\textrm{ with }{P_0} = E \cdot \frac{{\Delta d}}{d},K(x) = \frac{d}{{2(1 + \nu )\Lambda }} \cdot \frac{{\gamma (x)}}{{{\gamma _0}}} \end{aligned}$$

where S is the length of the 2-D workpiece, δ(x) is the impulse function, P₀ is the basic pressure distribution when the pad does not overhang the workpiece, and K(x) is a function defined as the convolution kernel.

Fig. 3. Schematic diagram of the stress and deformation for the 2-D case when the polishing pad overhangs both side of the workpiece

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From the deviation results shown in Eq. (4), the edge pressure distribution can be regarded as the convolution between the edge contour and a nonlinear edge kernel, where the nonlinear edge kernel is only related to the shear properties of the pad materials. We define the edge kernel function K(x), which is related to the distance from the edge point.

In the 2-D case, the edge is a point, and the shear strain effect has only one direction from the edge to the inner part of the workpiece. When we extend this conclusion to the 3-D case, the edge becomes a line, and the shear strain effect of each edge point can radiate from the point to all directions inside the workpiece. At this time, the nonlinear factor in the 3-D case is the convolution between the edge contour and the nonlinear edge kernel, as shown in Fig. 4, and the contact pressure can be further rewritten as

(5)$$P(x,y) = {P_0} \cdot ({1 + L(x,y) \ast K(r)} )$$

where K(r) is the edge convolution kernel in the 3-D case, and L(x,y) is the edge contour function.

Fig. 4. The schematic diagram of the edge kernel of 2-D and 3-D cases, respectively

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The derivation above gives an extremely elegant formula that can accurately predict the edge pressure distribution of the polishing pad. The sophisticated parameters related to the overhang ratio in previous edge models can be completely avoided, with the key being to obtain an accurate edge convolution kernel of the polishing tool.

We chose to obtain the nonlinear edge kernel based on FEA software combined with the deconvolution algorithm. To avoid the influence of the linear pressure distribution introduced by the tilt of the pad, the polishing pad in the FEA should be constrained to move only along the z-axis. In the deconvolution process, the nonlinear edge kernel is constrained to be rotationally symmetric, and the algorithm is based on the gradient descent method [22]. The detailed procedure is given in Eq. (6).

(6)$$\begin{array}{l} \min \;{||{{F_{non}}(x,y) - L(x,y) \ast K(r)} ||_2}\\ \;\;\textrm{with}\;\;{F_{non}}(x,y) = P(x,y)/{P_0}(x,y) - 1\\ \textrm{Iterative formula}:\\ \text { (1) }\;e(x,y) = L(x,y) \otimes ({{F_{non}}(x,y) - L(x,y) \ast K(r)} )\\ \text { (2) } \;e(r) = \{{ {e(r)} |e(r) = \bar{e}({x^2} + {y^2} = {r^2})} \}\\ \text { (3) } \;{K_i}(r) = {K_{i - 1}}(r) + {\eta _\textrm{1}} \cdot e(r) \ast Gauss(r;\sigma ) \end{array}$$

where P(x,y) is the pressure distribution where the pad overhangs the workpiece; P₀(x,y) is the basic pressure distribution when the pad does not overhang the workpiece; F_non(x,y) is the nonlinear factor calculated based on the FEA result, L(x,y) is the edge contour, K(r) is the nonlinear edge kernel, which is constrained to be rotational symmetric; ${\ast} $ is the convolution operation; ${\otimes}$ is the correlation operation; $\bar{e}$ is the operation to average e(x, y) of the same radius; Gauss(r;σ) is the Gauss filter, where σ is the standard deviation of the Gaussian function; and η₁ is the stepping ratio coefficient of the iterative formula.

We show several nonlinear edge kernels of two commonly used pads with different overhanging lengths. In the FEA model, a circular tool with different pad layers and a workpiece was created. The polishing pad was set to be flat and the bottom of the workpiece was fixed. A force of 20 N was added at the center of the pad, where the displacement of the tool in the x-y axis was set to 0 to avoid introducing the linear pressure. The boundary condition between the tool and the pads is set as a “bonded” case, and the boundary condition between the pad and the workpiece is set as a “frictional” case with a friction coefficient of 0.2. The elastic moduli and Poisson's ratios of the materials are listed in Table 1, and the results are shown in Fig. 5.

Fig. 5. The nonlinear factor and the nonlinear edge kernels of different pads under different overhanging lengths. (a) FEA meshing result for the 1-layer tool; (b) schematic diagram of the deformation of the 1-layer tool; (c1)-(c3) the nonlinear factors calculated based on the FEA result for the 1-layer tool where the overhanging lengths are 2 mm, 5 mm and 8 mm, respectively; (d1)-(d3) nonlinear edge kernels of the 1-layer tool with different overhanging lengths solved based on Eq. (6); (e) FEA meshing result for the 2-layer tool; (f) schematic diagram of the deformation of the 2-layer tool; (g1)-(g3) the nonlinear factors calculated based on the FEA result for the 2-layer tool where the overhanging lengths are 2 mm, 5 mm and 8 mm, respectively; (h1)-(h3) nonlinear edge kernels of the 2-layer tool with different overhanging lengths solved based on Eq. (6).

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Table 1. Material parameters of the polishing tools and workpiece

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It is interesting to note that the nonlinear edge kernels for the two pads are completely different, but for the same tool, the nonlinear edge kernels solved by the algorithm above are extremely similar, as shown in Fig. 5 d1-d3, h1-h3. The results further verify that the nonlinear edge kernel of the same tool can be regarded as a constant, which is inherent to the tool and only related to the tool material/structure.

The shape of the nonlinear edge kernel is also worthy of further research. As for the 1-layer tool, the nonlinear edge kernel is highest in the center, and it decays exponentially to 0. However, for the 2-layer tool, the magnitude of the kernel is much greater than that of the 1-layer tool, and in addition to exponential decay, the kernel fluctuates and a negative value exists. The deformation of the 2-layer pad is different from that of the 1-layer pad, as shown in Fig. 5(b), (f). Because the elastic modulus of pad2 is greater than that of pad1, pad1 cannot provide enough pressure to constrain pad2 to the same deformation; therefore, fluctuations occur close to the edge of pad2, which leads to a different nonlinear edge kernel.

Furthermore, it can be deduced that the fluctuation of the nonlinear edge kernel will gradually weaken with a decrease in the elastic modulus of pad2 or an increase in the elastic modulus of pad1, as shown in Fig. 6. When the elastic modulus of pad2 is less than that of pad1, the fluctuation of the nonlinear edge kernel disappears and the nonlinear effect becomes even smaller than that of the 1-layer tool, which is further confirmed by the FEA analysis results.

Fig. 6. The nonlinear factor of the 2-layer tool with different elastic moduli

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In Fig. 6, the size of the polishing tool is the same as that in the simulation in Fig. 5, and the overhanging length was fixed at 8 mm. The elastic moduli of pad1 in the three results are set as 2.5×10⁷ Pa, 1×10⁸ Pa, and 2.5×10⁹ Pa, respectively, and the elastic moduli of pad2 were always set to 2.5×10⁸ Pa. As the elastic moduli of pad1 increased, the fluctuation of the nonlinear factor shown in Fig. 6 def became weaker and disappeared when the elastic moduli of pad1 were greater than those of pad2, as shown in Fig. 6(g), (h), (i).

Furthermore, we find that the radial profile of the nonlinear edge kernel of different polishing tools can be approximately expressed as the solution of a second-order differential equation. The radial profiles of the nonlinear edge kernel of 1-layer tools are similar to the exponential function (ae^br), and the radial profiles of the nonlinear edge kernel of 2-layer tools are similar to the product of the exponential function and trigonometric function (ae^brsin(cr + d)) or the Bessel function (aJ₀(br + c)). However, this conjecture requires further theoretical research to be demonstrated.

2.2 Linear part of the pressure distribution

Except for the nonlinear part, since the commonly used polishing tools adopt universal joints, the linear part of the pressure distribution cannot be ignored. The polishing tool is tilted when the pad overhangs the edge, and the tilt of the polishing tool maintains the balance of the moment. Since the tool base is generally made of metal, and the elastic modulus is much greater than that of the polishing pad, the pressure distribution influenced by the tool tilt can be considered as linear.

The pressure distribution of universal joint tools obeys the moment equilibrium condition, and the moment balance of the pressure distribution can be expressed as

(7)$$\int\!\!\!\int {x \cdot P(x,y)\textrm{d}x\textrm{d}y = 0} ,\textrm{ }\int\!\!\!\int {y \cdot P(x,y)\textrm{d}x\textrm{d}y = 0}$$

where the x, y in Eq. (7) is the coordinate value of the coordinate system with the center of the polishing pad as the origin.

When only the linear pressure distribution is considered (i.e., assuming P(x,y)=ax + by + c), different types of edge pressure distributions are obtained by solving Eq. (7), as shown in Fig. 7. In the case of the circular edge, the tilt direction of the linear pressure distribution is always toward the center of the circle; however, for the edge with a corner, the tilt direction of the linear pressure distribution varies with the overhanging length of the polishing pad to each edge.

Fig. 7. The schematic diagram of the linear pressure introduced by the tilt of the tool

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The moment balance equations are suitable for complex edge contours, and the results show that the tilt of the polishing pad plays an important role in the pressure distribution. The nonlinear part only affects the region close to the edge, but the linear part affects the entire contacted region, where the pressure distribution inclines more and significantly with an increase in the overhanging length.

2.3 Generation of the edge TIFs

The nonlinear and linear parts of the pressure distribution for the complex edge contour can all be calculated using the methods proposed above, and the final edge pressure distribution is expressed as the following model: A schematic diagram of the edge pressure model is shown in Fig. 8.

(8)$$\begin{array}{l} P(x,y) = \kappa \cdot \tau ({({L(x,y) \ast K(r)} )\cdot ({ax + by + c} )\cdot {P_0}(x,y)} )\\ \;s.t\;\;\;\;\int\!\!\!\int {x \cdot P(x,y)\textrm{d}x\textrm{d}y} = \textrm{0}\\ \;\;\;\;\;\;\;\;\int\!\!\!\int {y \cdot P(x,y)\textrm{d}x\textrm{d}y} = 0 \;\;\;\;\;\;\;\;\textrm{with}\;\;\tau (t) = \left\{ \begin{array}{l} t\;\;\;\;\;\;\;\;t > 0\\ 0\;\;\;\;\;\;\;t \le 0 \end{array} \right.\\ \;\;\;\;\;\;\;\;\int\!\!\!\int {P(x,y)\textrm{d}x\textrm{d}y} = F \end{array}$$

where κ is a scale factor used to adjust the amplitude of the pressure distribution to maintain the vertical force balance. τ(t) is a function that ensures the nonnegative pressure distribution. L(x,y) is the edge contour, K(r) is the nonlinear edge kernel, ${\ast} $ is the convolution operation, a, b, and c are the parameters representing the linear pressure distribution; P₀(x,y) is the basic pressure distribution when the pad does not overhang the workpiece, and F is the magnitude of the force applied on the polishing pad.

Fig. 8. The schematic diagram of the complex edge pressure model

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In the step of solving edge pressure distribution based on the proposed model, the nonlinear edge kernel K(r) of the tool and the basic pressure distribution P₀(x,y) should be obtained first by FEA method; at the same time, the edge contour L(x,y) should be determined by the actual shape of the workpiece. Hereafter, parameters a, b and c are optimized according to the following optimization function:

(9)$$\begin{array}{l} \;\;\min \;E(a,b,c) = {\left||{\int\!\!\!\int {x \cdot P(x,y)\textrm{d}x\textrm{d}y} } \right||_\textrm{2}} + {\left||{\int\!\!\!\int {y \cdot P(x,y)\textrm{d}x\textrm{d}y} } \right||_\textrm{2}}\\ \textrm{with}\;P(x,y;a,b,c)\textrm{ = }\delta ({({L(x,y) \ast K(r)} )\cdot ({ax + by + c} )\cdot {P_0}(x,y)} )\\ \textrm{Iterative formula}:\\ \text { (1) } \;{a_0} = 0;{b_0} = 0;{c_0} = 1;\\ \text { (2) } \;{a_i}\textrm{ = }{a_{i - 1}} - {\eta _2} \cdot \frac{{\partial E}}{{\partial a}};{b_i}\textrm{ = }{b_{i - 1}} - {\eta _2} \cdot \frac{{\partial E}}{{\partial b}};{c_i}\textrm{ = }{c_{i - 1}} - {\eta _2} \cdot \frac{{\partial E}}{{\partial c}} \end{array}$$

where η₂ is the stepping ratio coefficient of the iterative formula. Finally, the scale factor κ can be determined using the following formula:

(10)$$\kappa = F/\int\!\!\!\int {\delta ({({L(x,y) \ast K(r)} )\cdot ({ax + by + c} )\cdot {P_0}(x,y)} )\textrm{d}x\textrm{d}y}$$

The edge pressure distribution is obtained by the above steps, and the generation of the edge TIFs can be determined by the Preston equation [23]. For the dual-rotation tool, the position of the polishing tool varies continuously; the edge TIFs need to be calculated by integrating the material removal amount along the orbital track, and the flow chart is shown in Fig. 9.

Fig. 9. The flow chart of the TIF calculation for sub-aperture tool under the complex edge

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Based on the TIF calculation method shown in the flow chart, we show several modeling edge pressure distributions and TIFs calculated by this model, the FEA pressure distributions are listed for comparison, as shown in Fig. 10. The diameter of the tool was 20 mm, the angular workpiece and circular workpiece were adopted, and the single-layer and multi-layer pads were analyzed.

Fig. 10. The modeling results of edge pressure distributions and TIFs under different tools and the workpieces, respectively. (a1-a2) the edge pressure distribution calculated by FEA and proposed model, where the edge angle is 120° and the pad overhangs 5 mm and 8 mm on two sides, respectively; (a3) schematic diagram of the 1-layer tool circumstance; (a4) the difference between the modeling result and FEA result; (b1-b2) the TIF of the tool with a 5 mm radius orbital motion in the 120° angle edge and the motion diagram; (c1-c2) the edge pressure distribution calculated by FEA and proposed model, where the edge is a circle with a 25 mm radius and the pad overhangs 5 mm; (c3) schematic diagram of the 2-layer tool circumstance; (c4) the difference between the modeling result and FEA result; (d1-d2) the TIF of the tool with a 5 mm radius orbital motion in the circular edge and the motion diagram.

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The modeling results are basically consistent with the FEA results, and only large differences appear in the edge part caused by the noise of the FEA meshing, which verifies the prediction accuracy of the model at the complex edge contour.

For the 1-layer pad, the magnitude of the nonlinear kernel is relatively small, which makes the linear part of the edge pressure distribution more inclined to maintain moment balance; however, for the 2-layer pad, the magnitude of the nonlinear kernel becomes much greater, and the nonlinear part maintains most of the moment balance, which leads to a relatively flat linear part, as shown in Fig. 10(a1), (a2), (c1), (c2). The shape of the modeling TIF also shows the phenomenon; the inner part of the TIF of the 1-layer pad has an obvious inclination, but the TIF of the 2-layer pad has a significant nonlinear removal at the edge, and the inner part does not change significantly.

3. Experimental demonstration

3.1 Experimental setup

An experimental demonstration was conducted on a dual-rotation polishing machine, as shown in Fig. 11(a). To further verify the correctness and accuracy of the proposed model, two workpieces with different shapes were designed to realize several different edge contours, as shown in Fig. 11(b) and (c). The first workpiece has a trapezoidal shape, which is part of the 90 mm×90 mm square, and there are both 90° and 120° angles in the workpiece. The second workpiece was circular with a radius of 50 mm. Both workpieces are pre-polished by ring polishing machine, and the initial form errors of both workpieces are shown in Fig. 11(d) and (e), respectively.

Fig. 11. The photos of the dual-rotation polishing machine and the workpieces

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Two 25 mm diameter two-layer pads with different materials were used in the experiments, and various overhanging lengths were adopted in the two workpieces to demonstrate the accuracy of the proposed model; the constant 20 N force is added on to the tool by air cylinder. Detailed working conditions are listed in Table 2.

Table 2. Edge TIF experiment conditions

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3.2 Experimental results

After obtaining the basic pressure distribution P₀ and the nonlinear edge kernel K(r) by FEA, the modeling edge TIFs can be calculated based on the flow chart shown in Fig. 9 and are shown in Fig. 12 and Fig. 13. The experimental TIFs were calculated by subtracting the two measurement results, where the form error of each mirror before and after polishing was measured by a laser interferometer (Zygo DynaFiz). A commonly used error metric was used to assess the overall performance of the TIF model proposed in this study.

(11)$$\Delta = \frac{{\sum {|{TI{F_{measure}} - TI{F_{model}}} |} }}{{\sum {|{TI{F_{measure}}} |} }}$$

Fig. 12. The comparisons between the measurement TIFs and the modeling TIFs under the corner edges

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Fig. 13. The comparisons between the measurement TIFs and the modeling TIFs under the circular edges

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It represents the relative statistical deviation between the experimental and modeling results. The calculation results are presented in Table 3

Table 3. Edge TIF experiment conditions

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The relative errors shown in Table.3 are below 21%, which means that the model has the ability to quantitatively predict the TIFs. However, the relative deviation of the statistics shown in Eq. (11) cannot represent the overall error distribution of the 2-D TIF; therefore, in order to quantitatively assess the performance of the proposed edge TIF model, we must additionally first show the modeling TIFs and experimental TIFs and the corresponding error distributions in a 2-D perspective. The error distribution is defined as

(12)$$err(x,y) = \frac{{|{TI{F_{measure}} - TI{F_{model}}} |}}{{\textrm{6} \cdot rms(TI{F_{measure}})}}$$

It represents the distribution of the deviation rate between the modeling results and the practical experimental results at each position of the TIF. The denominator of Eq. (12) is calculated according to the 6-sigma rule to ensure the stability of the numerical value.

The results shown in Fig. 12 and Fig. 13 show the accurate prediction ability of the proposed model. The residual error distributions are basically all below 20% in both types of workpieces, and the edge profiles are also in good agreement, which further verifies that the model can quantitatively provide guidance to the actual polishing process. In practical polishing processing, the edge material removal could be accurately obtained by the convolution between the path planning and the modeling edge TIFs.

It should be noted that the fluctuation of the pressure distribution at the edge caused by the multi-layer pad is reflected clearly in the measurement TIFs (i.e., abnormal areas near the edge with smaller material removal amount). The fluctuations are more obvious in the TIFs of tool A because the elastic modulus of its middle layer is smaller. This phenomenon is precisely predicted by the proposed model for different overhanging lengths, which further proves that the nonlinear edge kernel is inherent to the tool and only related to the tool material/structure. In addition, the tilt of the TIF with the increase in the overhanging ratio is mainly caused by the linear part of the model, which was also verified in the experiment. Moreover, the model also predicts the phenomenon that the TIFs of the two pads are different in shape (i.e., the TIFs of tool A is ‘M’ shaped and the TIFs of tool B is Gaussian shaped). This is mainly due to the basic pressure distribution of two pads are influenced by tool shape (i.e., the tool A is concave and tool B is convex), which further verifies that the error caused by different tool shapes can be well compensated by the basic pressure distribution term.

For the remaining residual TIF errors, we think they mainly arise from four aspects. First, owing to the numerical error in FEA and inaccurate tool parameter information (pad surface form, pad material parameters, etc.), the prediction accuracy of the basic pressure distribution and the nonlinear edge kernel is not sufficiently high; second, owing to the size tolerance of the universal joint, the application point of the force deviates from the center of the tool during the movement, which causes errors in the calculation of the moment balance equation; third, the loss of the polishing slurry can reduce the actual edge removal amount at the edge, and the tool speed discontinuity may also influence the removal distribution; the fourth reason is introduced by the measurement error of the interferometer, where the fringes in the roll-down edge parts are extremely dense and result in a phase unwrapping error. However, the errors mentioned above require further in-depth modeling studies and analyses.

4. Conclusion

In this paper, a novel edge TIF model is proposed to accurately describe the edge removal behavior of multi-layer tools in a workpiece with a complex edge contour. The model was established based on physical analysis and without empirical parameters. The pressure distribution is divided into three parts in our model: the nonlinear part is first represented as the convolution between the edge contour function and the proposed nonlinear edge kernel, in which the edge kernel is only related to the structure/material of the polishing tool and can be obtained in advance. The core of the linear part is that the universal joint pad follows the moment balance condition, and the basic pressure distribution is used for the pressure distortion compensation caused by the uneven surface form of the polishing tool.

The experiments were conducted in corner edge workpieces with different angles and circular edge workpieces using different polishing tools. The 2-D modeling error distribution maps are first illustrated to show the modeling accuracy at each position of the TIF, and the results show that the residual error distributions are basically all below 20%, which reaches a high prediction accuracy. The construction mechanism of the model determines that it has the ability to quantitatively predict the TIFs under other complex edges (straight line, curve, polyline, etc.) and various tools (different structure/material/shape), which is universal for sub-aperture polishing and has great significance for the prediction and compensation of the edge effect in complex-shaped optics.

Funding

Natural Science Foundation of Shanghai (21ZR1472000); Shanghai Sailing Program (20YF1454800); Key projects of the Joint Fund for Astronomy of National Natural Science Funding of China (U1831211); Development Project of Scientific Instruments and Equipment of the Chinese Academy of Sciences; Outstanding Member of Youth Innovation Promotion Association of the Chinese Academy of Sciences.

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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1-layer pad	Material	Dimension (mm)	Elastic modulus(Pa)	Poisson's ratio
polishing tool	304 stainless steel	Φ20×4	1.2×10¹¹	0.35
polishing pad	polyurethane	Φ20×2	7.1×10⁷	0.13
workpiece	Fused silica	70×70	8.2×10¹⁰	0.21

2-layer pad	Material	Dimension (mm)	Elastic modulus(Pa)	Poisson's ratio
polishing tool	304 stainless steel	Φ20×4	1.2×10¹¹	0.35
polishing pad1	foamed silica gel	Φ20×2	3.3×10⁶	0.03
polishing pad2	polishing cloth	Φ20×0.5	2.5×10⁸	0.19
workpiece	Fused silica	70×70	8.2×10¹⁰	0.21

Experiments	No.1	No.2	No.3	No.5	No.6
Tool material	A: foamed silica gel (E=3.3 MPa) + polishing cloth			B: synthetic rubber (E=86 MPa) + polishing cloth
Motion mode	Eccentricity: 0.4; spin velocity: 150 rpm; orbital velocity: −100 rpm
Force	20 N
Edge type	None	120°	90°	None	50 mm radius circle
Polishing slurry	10% w.t. Fe₂O₃

Experiments	No.1	No.2	No.3	No.4	No.5	No.6	No.7	No.8
Preston coefficient k	0.0105				0.0146
Relative deviation	15.3%	19.9%	21.0%	21.5%	11.9%	16.1%	18.6%	19.6%

1-layer pad	Material	Dimension (mm)	Elastic modulus(Pa)	Poisson's ratio
polishing tool	304 stainless steel	Φ20×4	1.2×10¹¹	0.35
polishing pad	polyurethane	Φ20×2	7.1×10⁷	0.13
workpiece	Fused silica	70×70	8.2×10¹⁰	0.21

2-layer pad	Material	Dimension (mm)	Elastic modulus(Pa)	Poisson's ratio
polishing tool	304 stainless steel	Φ20×4	1.2×10¹¹	0.35
polishing pad1	foamed silica gel	Φ20×2	3.3×10⁶	0.03
polishing pad2	polishing cloth	Φ20×0.5	2.5×10⁸	0.19
workpiece	Fused silica	70×70	8.2×10¹⁰	0.21

Experiments	No.1	No.2	No.3	No.5	No.6
Tool material	A: foamed silica gel (E=3.3 MPa) + polishing cloth			B: synthetic rubber (E=86 MPa) + polishing cloth
Motion mode	Eccentricity: 0.4; spin velocity: 150 rpm; orbital velocity: −100 rpm
Force	20 N
Edge type	None	120°	90°	None	50 mm radius circle
Polishing slurry	10% w.t. Fe₂O₃

Modeling and prediction of tool influence function under complex edge in sub-aperture optical polishing

Abstract

1. Introduction

2. Modeling of the edge removal behavior

2.1 Nonlinear edge kernel function

2.2 Linear part of the pressure distribution

2.3 Generation of the edge TIFs

3. Experimental demonstration

3.1 Experimental setup

3.2 Experimental results

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Tables (3)

Equations (12)

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