1 Introduction

Advancements in photoelectric technology have led to a rapid growth of extremely accurate optical devices owing to the increasing demands for consumer products in various industries such as aerospace [1], defense, security, medicine [2], and mobile telecommunications. Because such products generally include ultra-high-resolution optical components, the manufacture and development of highly accurate freeform optical components are immediately required to satisfy the fast-growing demand [3, 4]. Conventional processes for polishing optical glass involve semi-automated and, in some instances, manual polishing, both of which are labor- and time-intensive. The requirements of highly accurate surface quality and freeform geometric complexity pose considerable challenges regarding the fabrication of such optical components [5]. However, computer numerical control (CNC) subaperture bonnet polishing [6] is a suitable technology for polishing geometrically complex optical components made of difficult-to-machine materials. Moreover, bonnet polishing improves the peak-to-valley (PV) and root mean square (RMS) values of freeform optical components [7]. A major challenge in bonnet polishing is the prevention of mid-spatial frequency (MSF) errors [8], which often deteriorate the performance of optical systems.

In their study on polishing freeform optical components, Wang et al. [9] present a novel multi-jet polishing process and develop a polishing tool that can potentially perform high-efficiency polishing on medium-to-large surfaces. Additionally, Wang et al. [10] develop a semi-rigid bonnet tool that has the advantages of high efficiency and determinacy for removing material on optical components. Furthermore, Belkhir et al. [11] observe that high-quality optical surfaces can be obtained by controlling the polishing pressure and friction coefficient form against the contact surface. Guo et al. [12] use the R-test to develop a method for measuring and reducing the virtual pivot (VP) error and further investigate the influence of the VP error on surface profiles. Cao et al. [13] present an experimental investigation that attempts to provide a better scientific understanding of material removal and surface generation characteristics in bonnet polishing. Pan et al. [14] modify the tool influence function (TIF) of bonnet polishing based on the interfacial friction coefficient, aiming to improve the modeling accuracy of the TIF such that the modeled results are much closer to the experimental results, thereby demonstrating the effectiveness of the modification. Most researchers focus on high-efficiency material removal and high-quality optical surfaces to achieve better surface generation on optical components fabricated by bonnet polishing, whereas PV and RMS errors directly affect the polishing quality of an optical surface and also accompany low to medium—and even high—spatial frequency errors.

To reduce MSF errors and obtain an extremely high-quality optical surface, the key optimization parameters of the surface texture is theoretically and experimentally investigated via CNC subaperture bonnet polishing on fused silica. In CNC bonnet polishing, the key parameters are feed rate, head speed, track spacing, and tool offset [15]. To further consider the MSF errors accompanying CNC polishing due to the specific tool path, the power spectral density (PSD) index was utilized in the observation and analysis of the magnitude and frequency of MSF errors.

2 Experimental setup

Experiments were designed to improve the quality of optical lens surfaces by optimizing the key polishing parameters and developing a strategy using CNC subaperture bonnet precession polishing. In the experiments, a Zeeko IRP 1000 7-axis CNC polishing machine (Fig. 1) was used to obtain an extremely high-quality optical surface. The polishing head was configured for seven axes, and the interior was inflated with compressed air. The bonnet tool was fitted with a 40-mm-diameter polyurethane polishing pad (LP66) filled with cerium oxide and an inflated flexible bonnet. The LP66 polishing pad [13] is a cellular polyurethane material designed to handle high flatness and surface finishing requirements for optical glass materials. The motion control method was CNC, as it satisfies several polishing requirements for large optical lenses [7], as shown in Fig. 2.

Fig. 1
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ZEEKO IRP 1000 CNC polishing machine

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Fig. 2
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Bonnet polishing method

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Fused silica glass (Nikon, NIFS Series) 40 mm in diameter was used as the workpiece, and the main constituent was SiO2. In addition to high head speed, slurry (comprising an aqueous suspension of cerium oxide abrasives and a specific gravity of 1.03) was used during bonnet polishing for cooling and lubricating the workpiece (Fig. 3) to obtain a high-quality optical lens surface. The LP66 polishing pad was attached to the bonnet (Fig. 4), thereby improving the surface quality of the fused silica glass. Finally, an aspheric stitching interferometer (QED-ASI interferometer) was used to observe and analyze the experimental results.

Fig. 3
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Slurry used with ZEEKO IRP 1000

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Fig. 4
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Bonnet capped with LP66 polishing pad

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3 Analysis and discussion of experimental results

To examine the relationship of MSF errors with surface ripples, the effects of the tool offset, head speed, track spacing, and surface feed on the ripple error are analyzed. According to Preston [16], the material removal characteristics of bonnet polishing can be described by Eq. (1):

$$ \mathrm{MRR}=\upkappa \times \mathrm{P}\times \mathrm{V} $$
(1)

where MRR denotes the material removal rate in mm3/s, κ is the Preston coefficient (constant), P is the contact pressure between the polishing bonnet and the workpiece in N/mm2, and V is the velocity of the polishing bonnet relative to the workpiece in mm/s. This indicates that the bonnet polishing surface feed rate and track spacing are the key influencing factors of MSF error. If the surface feed rate is high, the contact time on the glass surface is short. Similarly, if the track spacing is large, there are few polishing paths.

3.1 Tool offset

Theoretical and experimental studies on bonnet polishing [14] indicate that the mechanism of material removal is complex, and the tool offset is the dominant factor affecting the volumetric MRR. If the same dwell time is used, the same MRR can be achieved under identical polishing parameters. To further understand how the tool offset parameter affects the surface generation profile and MSF error, the experimental polishing parameters shown in Table 1 are applied.

Table 1 Experimental parameters used for tool offset
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In this experiment, three different tool offset parameters and the raster path are adopted. Other parameters such as the precession angle, head speed, track spacing, and surface feed are also considered. The adopted parameters of a general case are those recommended by Zeeko company, and focus is placed on the effect of the tool offset on the surface quality. To understand how different tool offsets affect the material removal depth and surface profile (including the PSD index), corresponding measurements obtained before and after the experiment are compared. The polishing bonnet pad became considerably worn with increased polishing depth. In the ideal machining range, the fused silica glass demonstrates no special wear patterns other than the normal ones shown in Fig. 5, indicating that the tool offset does not directly affect surface profile generation. The patterns parallel to the y-axis show regular spacing of approximately 3.3 mm. In contrast, the patterns parallel to the x-axis appear tight and unevenly spaced. The surface profiles are examined separately in both the x- and y-axis via the Fourier transform. The x-direction PSD curves in Fig. 6a reveal that the sample polished using the 0.25-mm tool offset shows a higher total PSD than the samples polished using the 0.2-mm and 0.3-mm tool offsets, indicating that the former sample exhibits significant surface rippling and that an increased PV easily dominates the surface profile. Overall, the samples polished using the 0.2-, 0.25-, and 0.3-mm tool offsets exhibit similar PSD curves measured in the x-direction. Moreover, in the PSD curves measured in the y-direction, neither the PV nor the total PSD significantly affects the surface profiles because they are virtually identical. Thus, the tool offset does not significantly affect the surface profiles and is only proportional to the MRR. In this stage of the experiment, a 0.3-mm tool offset is used as the experimental parameter to obtain a high-efficiency MRR.

Fig. 5
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Surface profiles obtained with tool offsets of (a) 0.20, (b) 0.25, and (c) 0.30 mm

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Fig. 6
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PSD analysis diagrams obtained for tool offset experiments. ax-direction and by-direction

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3.2 Head speed

Head speed is a key parameter in bonnet polishing. Lower head speeds cannot provide the ideal material removal profile. Furthermore, extremely high speeds excessively wear the bonnet and causes the polishing machine to vibrate. Thus, the experimental parameters listed in Table 2 are used to examine the relationship between the head speed and the surface profiles.

Table 2 Experimental parameters used in head speed experiments
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With increasing head speed, the material removal depth increases [17], and the head speed is proportional to the material removal depth, as shown in Fig. 7. Furthermore, the number of abrasive particles in the slurry also increases for a particular area per unit of time. At the same feed rate and tool offset, the abrasive particles remove an equal amount of material, according to Eq. (1). Therefore, increasing the head speed and the corresponding rise in the number of abrasive particles leads to an increase in the material removal depth. The increased head speed causes significantly higher surface rippling, as shown in Fig. 8. The corresponding PSD index shown in Fig. 9 demonstrates that higher head speeds lead to MSF errors. With increasing head speeds, the MSF increases. In the x-axis PSD curve, the 0.33-mm−1 spatial frequency shows a peak corresponding to a pitch of approximately 3 mm when converted into the transverse axis coordinates of the surface ripple, indicating that a 3-mm pitch regularly appears parallel to the x-axis of the surface ripple, as shown by the blue line. Although higher head speeds generally show higher PSD than the lower head speed curves, at the 0.3-mm−1 spatial frequency, all samples have a higher peak value. The surface ripple also indicates that a 3.3-mm pitch regularly appears parallel to the y-axis. The theoretical analysis and experimental results indicate that a lower head speed cannot guarantee excellent material removal capability. A bonnet polishing system for a large aperture is always designed using large components to ensure system strength and rigidity. This experimental result indicates that head speeds higher than 700 rpm produce significant surface ripples, as shown in Fig. 8.

Fig. 7
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MRR depth plotted as a function of head speed

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Fig. 8
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Surface profiles obtained with the head speeds of (a) 300, (b) 500, (c) 700, (d) 900, and (e) 1000 rpm

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Fig. 9
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PSD analysis diagrams for head speed experiments. ax-direction and by-direction

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However, using a lower head speed produces a shallow ripple when bonnet polishing. As a lower head speed results in a lower MRR, which consequently leads to lower polishing efficiency, the system cannot obtain an ideal result and subsequently worsens the surface quality of optical glass. Based on these results, a 700-rpm head speed is used in the next series of experiments.

3.3 Track spacing

Track spacing forms a superposition on the polishing path. Thus, the most suitable track spacing parameter should be identified to obtain a good optical surface. From Fig. 9, it is clear that the 0.33-mm−1 spatial frequency in the x-direction can be converted into transverse axis coordinates at a pitch of approximately 3 mm, which will regularly generate a surface figure parallel to the x-axis. The surface ripples generated by track spacing form superposition patterns. Because 3 mm is a multiple of 0.75, it is reasonable to deduce that the surface figure generated by track spacing shows superposition.

This phenomenon produces serious surface ripples. Considering that 3 mm is not a multiple of the 0.55-mm track spacing, this spacing is also tested to avoid the superposition effect. The head speeds are set as 300, 500, 700, and 900 rpm. The experimental parameters are listed in Table 3. A higher head speed and longer dwell time on the material increase the MRR (Fig. 10). The combination of different head speeds and track spacing produces different surface profiles. The experimental results reveal that the surface profile is significantly improved in the x-direction when the 0.55-mm track spacing is used. The surface profile is also uniform in the y-direction. However, changing the track spacing parameters does not affect the dwell time or the MRR in the y-direction. The 0.55-mm track spacing and 700-rpm head speed produces a fine surface profile, which has a negligible effect on the surface ripple in the y-direction, as shown in Fig. 11. Consequently, these spacing and head speed settings are used as the polishing parameter values. The spatial frequency in the x-direction PSD curves is observed for different head speeds. For any head speed, a 0.55-mm track spacing induces a significantly improved surface profile compared with a 0.75-mm track spacing, as shown in Fig. 12.

Table 3 Experimental parameters used for track spacing
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Fig. 10
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Removal depth plotted as a function of head speed

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Fig. 11
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Surface profiles obtained for track spacing experiments. Track spacing, 0.75 mm: (a) 300, (b) 500, (c) 700, and (d) 900 rpm. Track spacing, 0.55 mm: (e) 300, (f) 500, (g) 700, and (h) 900 rpm

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Fig. 12
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PSD analysis diagrams for x-direction in track spacing experiments

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It is worth noting that the surface profile parallel to the y-direction is consistent. The 0.75-mm track spacing in the x-direction PSD index has peak values that are suppressed compared with the 0.55-mm track spacing. The 0.75-mm track spacing has a peak value PSD of 10−3 mm × μm2. In contrast, the 0.55-mm track spacing has a peak value PSD index of 10−4 mm × μm2 at a 0.33-mm−1 spatial frequency. Therefore, the 0.55-mm track spacing can suppress the surface ripple at the 0.33-mm−1 spatial frequency.

To avoid varying material removal depths caused by differences in the surface profile, the collocation of equal material removal depths is investigated. The experimental parameters are provided in Table 3. A 0.75-mm track spacing with 500 rpm and a 0.55-mm track spacing with 300 rpm both result in a 48-nm removal depth. A 0.75-mm track spacing with 700 rpm and a 0.55-mm track spacing with 500 rpm both result in a removal depth of 63 nm. Finally, a 0.75-mm track spacing with 900 rpm and a 0.55-mm track spacing with 700 rpm both result in a removal depth of 77 nm. The PSD curves in the x-direction grouped by equal material removal depth, as shown in Fig. 13, reveal that using the 0.55-mm track spacing can effectively reduce x-direction surface ripple and further provide fine surface quality. In the follow-up experiment, a 0.55-mm track spacing is used.

Fig. 13
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PSD analysis for track spacing experiments at the same MRR

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3.4 Surface feed rate

The surface feed rate directly affects the dwell time and further affects the surface profile of optical glass during bonnet polishing. A higher feed rate can provide higher surface quality but is limited to the 500–4500 mm/min range by the feed rate of the machine. Therefore, the feed rate is set in the range that most operators use for bonnet polishing optical lenses. In this surface feed rate experiment, the previous experimental results after optimization are utilized to obtain the appropriate combination of polishing parameters. Consequently, a 700-rpm head speed, 0.3-mm tool offset, and 0.55-mm track spacing are used as the experimental parameters for the surface feed rate, as shown in Table 4. Although a high-quality surface is achieved, it still exhibits superposition in every pass. This regular and obvious surface ripple directly affects the image quality of optical lenses because it has a closer track spacing that cannot be removed easily by post-processing. The surface ripple appears parallel to the x-direction through superposition from track spacing, as shown in Fig. 14.

Table 4 Experimental parameters for surface feed rate
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Fig. 14
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Surface profiles obtained in surface feed rate experiments. (a) 500, (b) 1000, (c) 2000, (d) 3000, (e) 4000, and (f) 4500 mm/min

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Surface ripples at higher surface feed rates are very shallow because they have a shorter dwell time to maintain the removal depth. For surfaces polished using high surface feed rates, the surface ripple originates from factors such as the bonnet pressure, tool offset, and machine vibration. At lower frequencies, a major amplitude vibration appears parallel to the y-direction of the surface profile. In contrast to the behavior in the x-direction, increasing the surface feed rate improves surface ripples. In addition, the x-direction PSD curve for surface feed rate experiments is observed. When the surface feed rate is 1000 rpm or lower, the PSD curve reaches its highest peak value because superposition induces energy concentration, which increases the PSD, as shown in Fig. 15.

Fig. 15
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x-direction PSD analysis for surface feed rate experiments

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In addition, the y-direction PSD for the surface feed rate is observed; thus, surface feed rates of higher than 4000 mm/min exhibit poor energy concentration performance. At a spatial frequency of 0.3 mm−1, there is a peak value that is mainly induced by the energy concentration at a specific frequency in machine motion. In other words, a 3.3-mm surface ripple from the machine vibration appears parallel to the y-direction, as shown in Fig. 16.

Fig. 16
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y-direction PSD analysis for surface feed rate experiments

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As this result indicates, a lower surface feed rate shows evident surface polishing marks parallel to the x-direction in every pass because of superposition with closer track spacing. Furthermore, a higher surface feed rate suffers from machine vibration caused by the regular vibration frequency, which affects the surface profile. To identify the appropriate surface feed rate, the surface profile is measured with an interferometer for each experiment. The original surface profile is subtracted from the polished surface profile to obtain the actual polishing surface profile, and a Zernike removal filter is used to subtract the first 36 items, including the tilt and original surface profile. Subsequently, the two-direction position and its point data are separately sliced, and the RMS error (RMSE) value is calculated under different surface feed rate parameters. The equation for the RMSE follows:

$$ RMSE=\sqrt{\frac{\sum \limits_{i=1}^n{\left({Z}_{meas,i}-{Z}_{model,i}\right)}^2}{n}}, $$
(2)

where RMSE, n, Zmeas, and Zmodel represent the root mean square error, sample number, measured value, and model value, respectively. Figure 17 shows that lower surface feed rates increase the RMSE in the x-direction because they yield a surface ripple owing to track spacing superposition; higher surface feed rates increase the RMS in the y-direction because they develop a ripple from machine vibration. Therefore, surface feed rates of less than 1000 mm/min or higher than 4000 mm/min are not recommended. Thus, surface feed rates between 1500 and 3500 mm/min are used to avoid worsening the surface profile in any direction.

Fig. 17
figure 17

RMSE analysis of surface feed rate

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3.5 Experiment for equal amounts of material removal

To equally distribute the superposition paths, the polishing passes are divided from 180° into three part, such that the rotation table is placed once every 60°. This process enables repeated multiple distributions of different feed rates to obtain extremely high-quality optical glass surfaces [18]. The experimental parameters for equal amounts of material removal are listed in Table 5 and Fig. 18. In the experiments with equal amounts of material removal and a surface analysis, the material removal depth is found to range from 850 to 920 nm with an error of approximately 6.6%, as shown in Fig. 19, suggesting that the optimization parameters have good reliability. QED-ASI interferometer is used to analyze the surface profiles of the bonnet-polished optical glasses.

Table 5 Experimental parameters for equal amounts of material removal
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Fig. 18
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Path design for equal amounts of material removal

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Fig. 19
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Comparison for equal amounts of material removal

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This measurement provides the surface profile data after subtracting the original surface profile from the polished surface profile. The same dwell time is applied to a unit region to ensure removal of an equal amount of material [19]. The surface profiles with an equal amount of material removed are shown in Fig. 20.

Fig. 20
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Surface profile of the samples for equal amounts of material removal. (a) 1500 mm/min × 3, (b) 2000 mm/min × 4, (c) 2500 mm/min × 5, (d) 3000 mm/min × 6, and (e) 3500 mm/min × 7

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The surface profile data and the PSD index of the equal material removal amounts shown in Figs. 20 and 21 indicate that the PV, and RMS values are greatly improved when the feed rate is optimized. As shown in Fig. 20a, a 1500-mm/min feed rate is used to achieve excellent surface quality. The greater uniformity of the colors implies that excellent PV and RMS values of 29.9 nm and 1.6 nm, respectively, are obtained. The PSD analysis additionally indicates that the 1500-mm/min feed rate results in lower PSD values; along with the three times single pass machining, this implies that a weak energy concentration and a smooth surface are obtained. Thus, the PSD index demonstrates that the 1500-mm/min feed rate provides excellent performance compared with the other feed rates, as shown in Fig. 21.

Fig. 21
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PSD for equal amounts of material removal

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Using the Zernike removal filter and extracting five different positions of the section point data of the optical surface, the RMS deviation is analyzed using Eq. (2). The RMS analysis results are shown in Fig. 22. To evaluate the surface quality of the optical surface performance, the RMS analysis plot is used. The combination of parameters with the 1500-mm/min feed rate for the polishing head provides a lower RMSE when obtaining smooth surface quality and excellent surface profile accuracy. Based on the experimental analysis and results for equal material removal, the combination of parameters with the 1500-mm/min feed rate as the final optimized combination is recommended in the bonnet polishing process. This method can be applied to remove the sub-surface failure layer and obtain a smooth surface.

Fig. 22
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RMSE analysis for equal amounts of material removed

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4 Experimental verification

To examine the repeatability of optimized parameters, three runs of repeatability experiments were performed on the optical glass with diameter 100 mm. As shown in Fig. 23, the results of PV and RMS values in repeatability experiments are similar.

Fig. 23
figure 23

Surface texture of repeatability test for (a) first, (b) second, and (c) third run

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The repeatability statistical results, as listed in Table 6, agree with the outcome mentioned above; it also reveals that a fairly smooth surface texture could be obtained. These results lead to the summary that the optimized parameters can be proved to be reliable in repeated experiments.

Table 6 Repeatability statistical polishing results with optimized parameters
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In the next stage, to verify the reliability of the optimized parameters, the previous and current experiments both with and without optimized parameters are compared. Based on the previous experimental analysis and discussion, the optimization parameters are listed in Table 7.

Table 7 Optimization parameters used in verification experiments
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The optimization parameters include the tool offset, head speed, track spacing, and surface feed rate. Other polishing parameters, such as the polishing pad, precession angle, and polish path, are fixed. Therefore, the LP66 polishing pad, a 20° precession angle, and the raster path are adopted as fixed parameters. When referring to the analysis and discussion of the experimental results (Section 3.1), the tool offset is set to 0.3 mm as an optimization parameter, as it can obtain a higher-efficiency MRR compared with the tool offsets of 0.2 mm and 0.3 mm. For the optimization parameter of head speed (Section 3.2), 700 rpm is selected because it provides an excellent PV and PSD curve. Using a lower head speed results in a lower MRR; on the contrary, a head speed above 700 rpm would produce serious surface ripple and degrade the surface quality. The optimization parameter of track spacing was adapted as 0.55 mm (Section 3.3) due to the track spacing of 0.75 mm possesses an obvious noise at the frequency of 0.33/mm and a higher total PSD index and that means worse surface quality. As a result, we used 0.55-mm track spacing as an optimization parameter. The optimization parameter of the surface feed rate (Section 3.4) is set at 1500 mm/min because a lower surface feed rate would lead to superposition-induced energy concentration. Figure 24 shows the measurement results of the surface texture tests via QED-ASI interferometer. The results show that various types of surface texture can be generated using different polishing parameters. The optimized parameters directly affect the surface texture quality. The surface texture obtained using the optimized parameters in LP66 bonnet polishing is shown in Fig. 24a. This implies that there is less residual surface texture and it approaches the same lane. In contrast, in the previous experiment as shown in Fig. 24b, obvious middle spatial frequency texture is left. In the numerical analysis, the RMS value of surface texture obtained by using optimized parameters is 1.6 nm, whereas RMS value of surface obtained without using the optimized parameters is 18.2 nm. In this verification stage, we considerably reduced the residual error on optical glass and obtained a high-quality surface texture. The evidence can also express that using a strategy of multi-polishing runs with different direction paths can further improve RMS value, for example, from mean value 3.2 nm obtained from repeat test to 1.6 nm obtained by previous study. This is because that when one polishing process is divided into two polishing processes with different direction path, the features left by first polishing run can be crushed into smaller parts by second polishing run. It will make the RMS value become smaller, and that means better surface quality.

Fig. 24
figure 24

Comparison of surface texture in LP66 bonnet polishing (a) with optimized parameters and (b) without optimized parameters

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5 Conclusions

In this study, theoretical and experimental investigations are conducted to determine optimized parameter combinations for improving the surface texture quality. Bonnet polishing is used to determine the effects of various polishing parameters on the surface texture quality of fused silica obtained using the LP66 bonnet. The main findings are summarized as follows:

  1. 1.

    Although the tool offset affects the MRR, it does not affect the surface texture during bonnet polishing. The large tool offset producing the highest MRR was selected to reduce the polishing time and provide excellent polishing efficiency.

  2. 2.

    When the polishing machine is used to meet the requirements of the large workpiece, the machine vibrations affect the surface texture. Therefore, 700 rpm was selected as the optimized parameter because it demonstrates the best combination of material removal efficiency and excellent surface texture.

  3. 3.

    To avoid the machine vibration caused by the fixed frequency texture of the surface, the track spacing was optimized to 0.55 mm.

  4. 4.

    When the feed rate is less than 1000 mm/min, the reciprocal overlapping of the polishing paths results in significant material removal. Therefore, the feed rate should be in the 1500–3500 mm/min range.

These experimental results demonstrate that the LP66 polishing pad provides high efficiency, high quality, and excellent performance with the optimal parameters; a smooth surface of optical glass with a 1.6-nm RMS value can be achieved after polishing. Through a series of optimization experiments, irregular surface profiles could be effectively removed, and MSF errors could also be reduced. The repeatability statistical results indicate that the experiment results for PV and RMS are similar and a smooth surface texture can be obtained. These results demonstrate the applicability of LP66 in CNC bonnet polishing for extremely accurate freeform optical components. This process can be used in high-powered laser and UV-lithographic optics applications.