3.1.

Structure of the SR Bonnet

A normal bonnet tool can easily conform to the aspheric or freeform surface, but its material removal rate is not that high due to low contact pressure. Also, the spherical tool contour turns into aspheric under the effect of regular inner pressure as shown in Fig. 1. This makes it difficult to control the contact spot shape and make the tool setting process fussy.

Fig. 1

Schematic diagram of the effect of bonnet tool deformation to the contact spot. The initial bonnet tool has a spherical shape. Under the effect of the inner pressure, the bonnet contour deformed to aspheric shape as shown in the left part of the figure. Also, if the tool contour is spherical, the shape of the contact spot would be a standard circle. But due to the effect of the tool deformation, the contact spot shape becomes irregular.

To enhance the stiffness of the bonnet and maintain a certain flexibility, we embed a thin metal sheet—which has been shaped to fit the inner surface—into the rubber membrane. Figure 2 shows the structure of the SR bonnet. There are three layers stuck together, and from the bonnet’s interior to its exterior, those layers are the following: the metal sheet, the rubber membrane, and the polishing pad. The metal sheet has high stiffness, but due to its thinness, the metal sheet is suitably flexible. Combined with the highly flexible rubber membrane, the metal sheet results in an SR bonnet tool.

Fig. 2

Structure of the semirigid (SR) bonnet.

3.2.

Determination of the Metal Sheet Material Using the FEA Method

The metal sheet can be shaped from common metal materials, such as stainless steel, copper, aluminum, and so on. However, to perfect the SR bonnet, choosing the material that exhibits the best properties is critical. The mold to form the metal sheet is sufficiently expensive that it is unrealistic to make three different molds for the experiment to determine the best material. Therefore, the finite element analysis (FEA) method is adopted. Our purpose is to analyze the distribution and magnitude of the metal sheet pressure and deformation under the same inner pressure. A group of simulation experiments is conducted using three different materials as the metal sheet: stainless steel, copper, and aluminum. Li et al.¹⁹ had built the FEA model for a normal bonnet polishing a Zerodur part, and they used this model to determine the pressure distribution in the contact area. To simplify the problem, the polishing pad is neglected in the model.¹⁹ Hence, we also build a simulation model omitting the effect of the polishing pad. A BK7 part—whose size is $50\times 50{\mathrm{mm}}^2$ and whose thickness is 5 mm—is used in the simulation. Table 1 lists the material properties for this model, and the simulation conditions are shown in Table 2.

Table 1

Material characteristics for model.

Table 2

Simulation conditions.

Figure 3 shows the simulation model for the SR bonnet polishing the BK7 part. Figure 3(a) is the model adopted to simulate the deformation under the effect of inner pressure. The top surface of the bonnet is constrained with zero displacement in all directions. The inner pressure and the symmetric constraint are also applied at the same spot as in the previous discussion. Figure 3(b) is the model used to simulate the pressure distribution over the contact area. The spherical part of the bonnet tool is used to simplify the simulation model. Considering the model’s symmetry, half of the model is adopted to enhance the solution’s efficiency. All DOF for the work-piece’s bottom surface are constrained with zero displacement. The top surface of the SR bonnet is constrained with zero displacement along the $x$-axis and $y$-axis with a $-1\text{-}\mathrm{mm}$ displacement along the $z$-axis. The inner pressure is applied on the bonnet’s inner surface. The symmetric constraint is applied on the symmetrical section.

Fig. 3

(a) Model to simulate bonnet deformation under the effect of inner pressure and (b) simulation model of the SR bonnet polishing the BK7 part.

Figure 4 shows the simulation results for both the pressure distribution and the deformation of different SR bonnets. Figures 4(a) and 4(b) are the simulation results for the SR bonnet with a stainless steel metal sheet. Figures 4(c) and 4(d) show the simulation results for the SR bonnet with a copper metal sheet. The simulation results for the SR bonnet with an aluminum metal sheet are shown in Figs. 4(e) and 4(f). The deformation under the effect of the inner pressure of the SR bonnet with the stainless steel metal sheet is the smallest ($17\mu \mathrm{m}$), and it also can deliver the largest contact pressure (1.327 MPa) among all three materials. Additionally, there is a discontinuous pressure distribution along the contact area’s edge when the metal sheet material is copper or aluminum as shown in Figs. 4(d) and 4(f). This discontinuity mainly results when the $z$-offset reaches 1 mm, and the SR bonnet with a copper or aluminum metal sheet experiences inward-concavity. Larger contact pressure could deliver a higher material removal rate, and the bonnet deformation under the effect of inner pressure is adverse for controlling the size of the contact spot as explained in Fig. 1. Therefore, among all three materials, stainless steel is the superior metal sheet material.

Fig. 4

(a) Bonnet deformation when the metal sheet material is stainless steel, (b) contact pressure distribution when the metal sheet material is stainless steel, (c) bonnet deformation when the metal sheet material is copper, (d) contact pressure distribution when the metal sheet material is copper, (e) bonnet deformation when the metal sheet material is aluminum, and (f) contact pressure distribution when the metal sheet material is aluminum. * In order to see the deformation clearly, the deformation has been scaled 100 times.

To specifically analyze the contact pressure distribution, the contact pressure on the section line of each simulation result has been extracted and fitted using the following Gaussian-like equation:²⁶

Fig. 5

Pressure distribution on the section line of different SR bonnets with different metal sheet materials.

Note that the maximum contact pressure delivered by the SR bonnet is much larger than the R160 normal bonnet (mentioned in Fig. 7 in Sec. 3.3 of Ref. 19), whose maximum contact pressure is 0.117 Mpa. Hence, the SR bonnet would generate higher material removal efficiency than the normal bonnet. Furthermore, the SR bonnet’s deformation is much smaller than the normal bonnet’s deformation, which is at the same level of polishing as the machine’s accuracy, and thus the deformation error of the tool contour can be ignored during the polishing process, while the shape of the contact spot can be considered to be a standard circle. Therefore, the size of the contact spot can be accurately controlled by controlling the $z$-offset.

3.3.

Determination of the Thickness of Stainless Steel Sheet

The designed thickness of the stainless steel sheet is crucial for the property of the SR bonnet. Therefore, we also conduct five groups of simulation experiments using five different thicknesses for the steel sheet, which are 0.1, 0.15, 0.2, 0.25, and 0.3 mm, respectively. The simulation conditions are the same as previously mentioned. Both the SR bonnet deformation under the effect of inner pressure is simulated using the model as shown in Fig. 3(a) and the bonnet tool compress to the work-piece is simulated adopting the model as shown in Fig. 3(b).

The simulation results have been shown in Fig. 6. Figure 6(a) shows the deformation of the SR bonnet tool under the effect of the inner pressure that varies with the thickness of the stainless steel sheet together with the maximum contact pressure curve. A thicker sheet will lead to a higher stiffness. This results in less deformation of the tool and a larger maximum contact pressure. The maximum von Mises stress of the tool during the tool compress to the work-piece surface is also extracted and compared to the yield limit stress of the stainless steel as shown in Fig. 6(b). It is noted that when the thickness is 0.1 or 0.15 mm, the maximum von Mises stress is larger than the yield limit, which means that the stainless steel sheet of the SR bonnet will fail to return to the original shape. Therefore, these two kinds of thicknesses are not appropriate. But when the thickness of this sheet is too large, the SR bonnet will lose flexibility. In order to maintain a certain flexibility of the SR bonnet, 0.2 mm is finally chosen as the thickness of the stainless steel sheet based on the analysis above.

Fig. 6

(a) The deformation of the SR bonnet tool under the effect of the inner pressure and the maximum contact pressure in each result varying with the thickness of the stainless steel sheet, and (b) the maximum von Mises stress in each result vary with the thickness of the stainless steel sheet.

4.1.

Tool Influence Function (TIF) of the SR Bonnet Tool

Through the aforementioned analysis, we have chosen stainless steel with a thickness of 0.2 mm as the material of the metal sheet for the R80 SR bonnet. It has been manufactured as shown in Fig. 7. Figure 8 displays the experimental prototype designed by our group to test the performance of the SR bonnet. As shown in Fig. 8, the $H$-axis controls the tool rotation of the bonnet tool, the $B$-axis controls the angle between the tool rotation axis and the normal line of the polished surface, and the $A$-axis is designed to make the tool can implement precession motion. In order to observe both its TIF and its TIF stability, a group of experiments to extract four TIFs have been conducted. We use the R80 SR bonnet to polish a BK7 work-piece. The speeds of the $H$-axis and $A$-axis are 500 and 20 rpm, respectively. Cerium oxide slurry (whose weight percentage is 5.54%) is used in the experiment. The dwell time on each spot is 6 s. All other experimental conditions are identical to the conditions listed in Table 2.

Fig. 7

Picture of the R80 SR bonnet.

Fig. 8

Experimental prototype for testing the SR bonnet.

Results are presented in Fig. 9. Figure 9(a) shows one TIF of the SR bonnet tool. Its shape is the same as the normal bonnet’s shape, which is also a Gaussian-like profile.⁵ Figure 9(b) presents the material removal rate of the four TIFs. As shown in Fig. 9(a), the maximum removal depth is $1.95\lambda$ ($\lambda =632.8\mathrm{nm}$) and the dwell time is only 6 s. In Fig. 9(b), the time between each sample is $\sim 15\min$, and the removal rate stability is more than 90%.

Fig. 9

(a) Extracted shape of tool influence function (TIF) and (b) material removal rate of four TIFs.

In addition, a series of polishing spots with different contact sizes have also been generated on a BK7 part. The extracted spot contours have been demonstrated in Fig. 10. Figure 10(a) shows the extracted contour of six different polishing spots. Their diameters are 14, 20, 25, 28, 30, and 36 mm, respectively. Figure 10(b) shows the section contour of these spots. It turns out that a “W” shape of TIF comes out when the diameter of the spot is over 25 mm. This performance is just like the normal bonnet²⁷ and the size of the spot could not exceed the certain limit.

Fig. 10

(a) Contours of the extracted polishing spots with different spot diameters, and (b) section line of each polishing spot. The polishing conditions are as follows: tool radius is 80 mm, precession angle is 23 deg, inner pressure is 0.2 MPa, $H$-axis rotation speed is 500 rpm, $A$-axis rotation speed is 20 rpm, and dwell time is 6 s. ($D$ signifies the diameter of the spot, $\lambda =632.8\mathrm{nm}$).

4.2.

Uniform Removal Test

The previous section has demonstrated the SR bonnet’s TIF stability. However, stability during the polishing process must also be tested before this tool is put into production. Hence, a uniform removal test of the SR bonnet on the BK7 work-piece has also been conducted. This test is carried out on a plane $\mathrm{\Phi }$ 120-mm BK7 glass whose surface has been polished to $0.18\lambda$ peak-to-valley (p-to-v). In order to leave unpolished surfaces from which an absolute removal depth could be established, a $\mathrm{\Phi }$ 80-mm subarea of the part was polished. The $Z$-shaped raster path is used, and the polishing cycle has been executed four times with the precession angle orientated in four polar directions along the local normal (0, 90, 180, and 270 deg).⁵ The raster distance is 1.5 mm. Table 3 shows the other experimental conditions.

Table 3

Uniform removal conditions.

The experimental results are presented in Fig. 11. Figure 11(a) shows the whole surface contour after uniform removal polishing using the R80 SR bonnet, as measured using QED SSI™. The result demonstrates high uniformity, and the p-to-v of the uniform removal area is less than $0.1\lambda$. Figure 11(b) shows the surface roughness result measured using Zygo (Middlefield, Connecticut) NewView 7200. Its Ra is only 2.09 nm after one cycle of the uniform removal process.

Fig. 11

(a) The whole surface contour after uniform removal polishing and (b) the surface roughness measuring result.

4.3.

Prepolishing Test

This process is intended to remove surface and subsurface damage on a part from a precision grinding machine and also to preserve the profile. Typically, material must be removed to a depth of tens of microns. Therefore, a polishing tool with a high removal rate is urgently needed for this process.

The prepolishing test of the SR bonnet has been performed on a BK7 glass. To leave an unpolished region for defining absolute removal, a $60\times 60\text{-}{\mathrm{mm}}^2$ area out of a $100\times 100\text{-}{\mathrm{mm}}^2$ surface is polished. A $Z$-shaped raster path is also adopted, and the precession angle is again orientated in four directions during the entire process. The weight percentage of the cerium oxide in the slurry is 3.04%, and the $H$-axis speed is 750 rpm. Other process conditions are identical to the conditions shown in Table 3. The total prepolishing time is 12 min.

Figure 12(a) shows the initial surface profile after removal from the grinding machine (whose p-to-v is $\sim 3.5\mu \mathrm{m}$), on which obvious tool ripples exist. The profile after the prepolishing process is shown in Fig. 12(b). In 12 min, $\sim 36{\mathrm{mm}}^3$ of material are removed and tool ripples are highly depressed.

Fig. 12

(a) Profilometry of the part directly off the grinder measured using PGI 1250S Form Talysurf™ and (b) form measured after 12-min prepolish using the R80 SR bonnet.