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Abstract
Based on deterministic chemical etching, atmospheric pressure plasma processing (APPP) with a high material removal rate and spatial machining resolution, is a promising computer-controlled optical surfacing (CCOS) technique for freeform surface generation. However, the time-variant removal characteristics of APPP induce nonlinearity in the CCOS process, which requires more consideration in the dwell-time calculation. In this paper, the nonlinear dwell-time algorithm based on the concept of controlling volumetric removal is studied. The freeform surface generation by controlling volumetric removal is modeled to provide the theoretical basis for the algorithm. The applicability of the algorithm in freeform generation by APPP with time-varying characteristics is explored through numerical simulations. Finally, a freeform surface is successfully created based on the algorithm and relevant analysis results, which validates the applicability of the algorithm in freeform generation using time-variant tool influence functions.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-end freeform surfaces have been widely applied in many fields, such as advanced opto-instruments, high-power laser systems, and accurate detective devices. However, the ultra-precision fabrication of freeform surfaces creates a major challenge for optical manufacturing techniques owing to their complex structures and special shapes. To meet the stringent requirements of freeform surfaces, various optical machining techniques including, magnetorheological finishing (MRF) [1], bonnet polishing (BP) [2,3], fluid jet polishing (FJP) [4], and ion beam figuring (IBF) [5,6], are investigated to generate freeform surface or correct form error based on the computer-controlled optical surfacing (CCOS) theory. Based on CCOS theory, the freeform surface generation process can be regarded as a linear convolution of the dwell-time and tool influence function (TIF). Hence, conventional dwell-time algorithms, including linear equations and iterative methods, were developed to obtain the dwell-time and control the practical machining process.
However, the conventional dwell-time algorithm is not entirely suitable to reactive plasma-processing techniques [7–9]. The material removal in reactive plasma processing is achieved by chemical etching between reactive radicals and the workpiece material. Because the chemical reaction rate changes with temperature according to the Arrhenius theory, the material removal is significantly affected by the surface temperature. In this way, the dwell time change leads to temperature change, which causes that the TIF to vary with dwell-time at each machining point. Note that, the TIF in reactive plasma-processing has strong time-dependent characteristics, unlike that in BP and MRF which is correlated with the local curvature of the workpiece or long-term removal instability [10–13]. Consequently, the freeform surface generation by reactive plasma processing becomes a nonlinear convolution process, which requires more specific considerations to determine a suitable dwell-time solution. Note that, the distance change between the plasma torch and workpiece or the curvature variation of workpiece may introduce nonlinear influence to freeform surface generation, this paper focuses on the nonlinearity induced by the time-varying TIF. Dai investigated the TIF variation versus dwell-time and proposed a dwell-time algorithm to compensate for the nonlinearity in inductively coupled plasma processing [14]. To generate the typical freeform surface-continuous phase plate, a modified iterative algorithm based on the linear dwell-time method was established [15]. Ji proposed an optimization strategy to eliminate the nonlinear effect in atmospheric-pressure plasma processing (APPP) [16]. Meister established a temperature simulation model to manage the nonlinear error in plasma-jet machining, which is applied for form correction of aspheric surfaces [17]. In contrast, to avoid the effect of the time-variant TIF, Takino proposed a shaping method based on plasma chemical vaporization machining [18]. In this method, the plasma torch scans at a constant feed rate to remove a uniform layer in a certain area, thereby eliminating the nonlinear influence induced by the time-varying TIF.
Recently, we proposed and experimentally demonstrated a concept of controlling volumetric removal for freeform surface generation by reactive plasma-processing [19]. Further, a nonlinear dwell-time algorithm based on the concept was developed for time-variant TIF, which shows considerable simplicity and efficiency in practical optical machining. Ji established a reverse method based on the concept to derive the practical TIF changes considering neighborhood effect in APPP [20], indicating that the concept can be functional in other aspect. However, the concept was only illustrated experimentally, without deep understanding.
Therefore, theoretical modeling and analysis of both the concept of controlling volumetric removal and the nonlinear dwell-time algorithm need to be conducted to promote their application in freeform surface generation by APPP. In this paper, the nonlinear dwell-time algorithm based on the concept is introduced. Then, the freeform surface generation by controlling volumetric removal is modeled, and the theoretical basis of the nonlinear dwell-time algorithm by the concept is validated. Next, the applicability analysis of the nonlinear dwell-time algorithm is conducted. Finally, a freeform surface using the nonlinear dwell algorithm is generated and its feasibility and applicability are demonstrated.
2. Nonlinear dwell-time algorithm for time-variant TIF
2.1 Time-variant TIF of APPP
As a typical CCOS technique, APPP utilizes chemical etching to achieve deterministic material removal with high material removal rate. Note that, there are several limitations of APPP. it can only be used to fabricate silica-based material. The surface quality after processed is affected by the initial surface and subsurface condition. An ultra-smooth surface is hard to be achieved only using APPP, and a post-finishing method is usually introduced to improve the surface roughness after APPP. The chemically-reactive plasma jet consists of active fluorine, oxygen and other inert particles and is able to etch silica-based material. When impinging on the workpiece, the active particles react with the material, while other particles transfer heat into the workpiece. The workpiece temperature increases with the dwell-time or feed rate of the plasma jet, causing the chemical reaction rates to change. This means that APPP has a time-variant TIF. The time-variant TIF of APPP can be expressed as:
where $a(t)$ and $w(t)$ are the peak removal rate and full width at half maximum (FWHM), respectively, respectively. ${p_i}$ and ${q_i}$ are the coefficients of the time-variant TIF model.Figure 1 shows the typical change law of the time-variant TIF. The change law is obtained based on experimental method, which is illustrated in our previous study [19]. Note that the dwell time is the dwell time of the plasma jet at a machining point, instead of the total machining time. As the dwell-time rises from a small value, both the peak removal rate and FWHM increase rapidly with dwell-time and then reach a constant value. Thus, the change in the time-variant TIF can be divided into three stages: the rapid-change, change-to-stable and stable stages, as shown in Fig. 1. Because of the time dependence of the TIF, the freeform surface generation by APPP becomes a nonlinear convolution process, which is difficult for the conventional dwell-time algorithm for a time-invariant TIF.
2.1 Nonlinear dwell-time algorithm based on controlling volumetric removal
To solve for the dwell-time in APPP, a concept of controlling volumetric removal was proposed. The concept requires two assumptions, (1) the FWHMs of the time-variant TIF at different dwell time is smaller enough to generate the freeform surface with a certain spatial-frequency, which means that all the time-variant TIFs have the capability to generate the freeform surface. (2) The constant spacing is used and smaller enough compared with the spatial-period of the freeform surface and the FWHM of the time-variant TIF, which can meet the requirements to avoid its influence on the freeform generation. These two assumptions are also necessary requirements to achieve an accurate CCOS. It has been proven that a freeform surface can be created deterministically by controlling the volumetric removal. With Fig. 2 as an example, suppose that there are two TIFs with different FWHM and the same volumetric removal rate. Note that, the spacing between machining points is plotted in an enlarged view to make it clear. The actual spacing is smaller enough than the spatial period of the freeform surface and FHWM of the TIFs. When machining with one dwell-time solution, the volumetric removal at each machining point is controlled in the same way, and thus the surfaces generated by the two TIFs are almost identical. Based on this concept, a nonlinear dwell-time algorithm with high efficiency and simplicity was developed in our previous study [19].
Herein, the process of the nonlinear dwell-time algorithm is explained, and more detailed elaboration can be found in the previous paper [19]. Figure 3 presents a flowchart of the algorithm. First, trench machining experiments are conducted to establish the time-variant model of the TIF. The relationship between the volumetric removal and dwell-time is obtained, as well. Then, a constant TIF is selected according to the time-variant TIF model and used to calculate the initial dwell-time using a linear deconvolution method. Thus, the desired volumetric removal at each point is obtained through forward convolution of the initial dwell-time and the selected TIF. Next, the final dwell-time can be determined by controlling the volumetric removal at each point equal to the desired volumetric removal, according to the obtained relationship between the volumetric removal and dwell-time. In the nonlinear algorithm, the ratio between the spatial period of freeform surface and the FWHM of TIF is usually controlled larger than 1.25 according to previous study [6,21] to guarantee that the TIF can achieve an accurate removal. In addition, the removal capability of a certain TIF is also checked by numerical simulations, even the FWHM satisfies the ratio. Considering the influence of the peak removal rate, the lower limit of the dwell time is used in the nonlinear dwell time algorithm. In this way, the maximum feedrate does not exceed the dynamic motion capability of the machine.
The previous paper demonstrated the feasibility of the nonlinear dwell-time algorithm, but a deep understanding of its theoretical basis and applicability is needed to improve the algorithm and promote its application.
3. Theoretical analysis of the nonlinear dwell-time algorithm
3.1 Surface generation modeling based on controlling volumetric removal
To understand the theoretical basis of the nonlinear dwell-time algorithm, the process of freeform surface generation based on controlling volumetric removal is modeled.
At first, two time-invariant TIFs (TIF1 and TIF2) with the same volumetric removal rate are defined as:
The volumetric removal rate can be calculated by integrating the TIF in the spatial domain, expressed as:
Thus, the volumetric removal rates of the two given TIFs are:
According to the assumption that the given TIFs have identical volumetric removal rates:
To demonstrate the simulation of freeform surface generation, a bisinusoidal surface is used as the desired form. Thus, the desired removal distribution is expressed as:
where $C$ is the additional removal height that ensures that the desired removal is positive. ${A_1}$ and $f$ are the amplitude and spatial frequency of the bisinusoidal structure, respectively. Note that, the bisinusoidal surface described as ${z_1}(x,y)$ has the same spatial frequency f in the $x$ and $y$ directions. In the diagonal direction, the spatial frequency reaches its minimum value of $1/\sqrt 2 \cdot f$. Thus, the above bisinusoidal surface can be used to analyze the generation process of freeform surface generation with a maximum spatial frequency of f.Suppose that ${t_1}^{\prime}(x,y)$ is the dwell-time distribution to realize the desired removal, expressed as Eq. (12). The dwell-time consists of two parts corresponding to the two parts of the desired removal distribution $z^{\prime}(x,y)$. ${t_1}(x,y)$ and $\Delta {t_1}(x,y)$ are related to ${z_1}(x,y)$ and $C$, respectively.
When the uniform layer, corresponding to the additional removal height C, needs to be removed, the machining process can be formulated with respect to the volumetric removal. Given that $Vr{r_1}$=$Vr{r_2}$, Eq. (13) can be obtained. It can be found that the desired additional removal can be perfectly achieved by the two given TIFs with the same dwell-time, which means that the removal of the additional height by the two TIFs has no influence on form error.
Therefore, the following analysis of the bisinusoidal surface generation only focuses on ${z_1}(x,y)$ for simplicity. Based on CCOS theory, the bisinusoidal surface generation is the convolution between the TIF and dwell-time, expressed as:
After a two-dimensional Fourier transform of Eq. (14), the surface generation in frequency domain is expressed as:
where ${Z_1}({f_x},{f_y})$, ${R_1}({f_x},{f_y})$, and ${T_1}({f_x},{f_y})$ are Fourier transforms of ${z_1}(x,y)$, ${r_1}(x,y)$, and ${t_1}(x,y)$, respectively; ${f_x}$ and ${f_y}$ denote the frequency-domain coordinates. ${R_1}({f_x},{f_y})$ is calculated as Eq. (16). Thus, the Fourier transforms of ${r_1}(x,y)$ and ${r_2}(x,y)$ can be expressed as:When using the same dwell-time ${t_1}(x,y)$, the surface generated by ${r_2}(x,y)$ can be expressed in the frequency domain as:
By substituting Eq. (17) and Eq. (18) into Eq. (19) and conducting inverse Fourier transforms, the surface generated by ${r_2}(x,y)$ in the spatial domain can be obtained as:
Note that, ${z_1}(x,y)$ and ${z_2}(x,y)$ are named as the desired form and generated form, respectively. The form error induced by controlling volumetric removal is determined by subtracting ${z_2}(x,y)$ from ${z_1}(x,y)$:
For convenience, the spatial period of the bisinusoidal surface is introduced as $T = 1/f$, and the FWHM-change ratio between ${r_1}(x,y)$ and ${r_2}(x,y)$ is defined as $\alpha = {w_2}/{w_1}$. By inserting these two parameters into Eq. (19), the amplitude change ratio between the actual and the desired forms can be obtained as:
Given that the form error is also a bisinusoidal surface, the root mean square (RMS) of the form error can be calculated as,
Generally, the convergence rate is used to evaluate optical fabrication efficiency. Here, the convergence rate between the generated and the desired forms can be expressed as:
Thus, the model of freeform generation by two TIFs with identical volumetric removal is established, which can be used to analyze the freeform generation based on controlling volumetric removal.
3.2 Analysis of surface generation based on controlling volumetric removal
In the above modelling process, the surfaces generated using two different TIFs by controlling volumetric removal are both bisinusoidal surfaces. According to Eq. (19), the generated form is also a bisinusoidal surface with the same spatial period as the desired form, but a different amplitude. The amplitude change ratio is dependent on the spatial period ($T$)and the FWHM-change ratio ($\alpha$), and in dependent of $(x,y)$. It can be said that Eq. (19) provides a theoretical basis for the concept of controlling volumetric removal and also proves the reasonability of the nonlinear dwell-time algorithm. Besides, Eq. (19) also provides the reasonability of the optimization strategy by selecting a suitable TIF to suppress the nonlinearity in APPP [15,16].
The form error between the generated and the desired forms is expressed as Eq. (20). The form error is also a bisinusoidal surface with the same spatial period as the desired form. Compared with the desired form, the form error shows two different distribution patterns. When the ${w_1}$ is less than ${w_2}$, the first coefficient in Eq. (20) is positive, and the form error has the same phase in height with the desired form. On the contrary, the form error has an opposite phase in height with the desired form. Besides, the form error scale depends on the FWHM-change ratio and the ratio between spatial period of the desired form and FWHM (period-FWHM ratio in the following, $T/{w_1}$).
To further analyze the specific change laws of freeform generation, numerical simulations were performed. Based on the typical characteristics of APPP, the simulation parameters are assumed in Table 1. Corresponding to ${w_1}$ and ${w_2}$, the FWHM-change ratio in the range of 0.5–2. Considering the spatial period of freeform surface and FWHM of TIF, the spacing is set as 1 mm. Based on Eq. (21) and Eq. (23), the amplitude change ratio and convergence rate were calculated.
Table 1. Simulation parameters of theoretical analysis of freeform generation
Figure. 4(a) presents the amplitude change ratio between the generated and the desired forms, which shows different variation laws in the coordinate directions of the period-FWHM ratio and FWHM-change ratio. As the period-FWHM ratio changes, the amplitude change ratio varies more significantly than it does for the FWHM-change ratio. And, the amplitude change ratio converges rapidly to 1 when the period-FWHM ratio increases, indicating that the form generated by TIF2 with ${w_2}$ is almost identical to the desired form. The amplitude change ratio is less than 1 when the FWHM-change ratio is less than 1, which means that the form generated by the TIF2 has a smaller amplitude. When the FWHM-change ratio is larger than 1, the result reverses.
Fig. 4. Analysis of the generated form compared with the desired form. (a) Amplitude change ratio; (b) Convergence rate.
Significantly, the amplitude change ratios near corner A and B with the same period-FWHM ratio show distinct change laws. At corner A, which has the largest FWHM-change ratio, the amplitude change ratio reaches zero, indicating that the form generated by TIF2 approximates a planar surface with negligible amplitude. At corner B, the amplitude change ratio increases to greater than 2, which means that the form error has a larger amplitude than the desired form. The above results demonstrate that the desired form cannot be generated accurately by controlling volumetric removal under conditions corresponding to these two corners.
The convergence rate is calculated to evaluate the form error compared with the desired form, as shown in Fig. 4(b). The convergence rate is distributed in the same pattern as the amplitude change ratio. Because of the larger amplitude of the form error at corner B, the convergence rate is negative, which is meaningless in practice. In consideration of the actual convergence rate in optical machining process, 90% is selected as a reference convergence rate and the corresponding reference line is plotted in Fig. 4(b). Thus, there arise two zones divided by the reference line. For zone I, the convergence rate is less than 90% indicating that a larger form error. Meanwhile, the convergence rate is greater than 90% when the FWHM-change ratio and period-FWHM ratio are located in zone II. In other words, the zone II represents the feasible domain for generating freeform surface by controlling volumetric removal. Furthermore, the freeform surface generated by TIFs that meet the requirements of the feasible domain has acceptable accuracy.
To further elaborate upon freeform surface generation by controlling volumetric removal, the convergence rate with a 30 mm spatial period, corresponding to Table. 1, is shown in Fig. 5. The convergence rate decreases when the FWHM-change ratio deviates from 1. Significantly, when the FWHM-change ratio is greater than 1, the convergence rate decreases more rapidly and reaches a considerably lower level. Specifically, the convergence rate decreases to 65% as the FWHM-change ratio increases to 2. In contrast, when the FWHM-change ratio is less than 1, the convergence rate decreases slowly and finally approaches a stable value. It can be seen that the convergence rate stabilizes at approximately 86% when the FWHM-change ratio is 0.1. To illustrate this phenomenon, the section profiles of the desired form ${z_1}(x,y)$, generated form ${z_2}(x,y)$ and form error $error(x,y)$ corresponding to A–D in Fig. 5 are presented in Fig. 6.
Fig. 6. Section profiles of simulation results with different FWHM-change ratios of (a) 0.1, (b) 0.48, (c) 1.34, (d) 2.
As shown in Fig. 6, when the FWHM-change ratio is less than 1, the generated form at the peaks and valleys is higher and lower than the desired form, respectively. This can be explained by the following. The desired volumetric removal corresponds to TIF1. TIF2, which has the same volumetric removal rate and smaller FWHM, has a narrower removal distribution. This means that TIF2 contributes less removal to neighboring points than TIF1 during freeform surface generation. The dwell-time is larger at the peaks, where the removal by the narrower TIF2 is strengthened, which causes higher removal than desired. In contrast, the dwell-time at the valleys is smaller, and the removal at the valleys relies on contribution from neighboring points. Hence, the generated form at the valleys is lower than the desired form due to the weak influence of TIF2 on neighboring points.
When the FWHM-change ratio is greater than 1, the generated form at the peaks and valleys becomes lower and higher than the desired removal, respectively, because the TIF2 with the larger FWHM contributes more removal to neighboring points. The stronger influence of the wider TIF2 on neighboring points leads to a significant deviation between the generated and the desired forms. For instance, when the FWHM-change ratio reaches 2, the form error approaches approximately to half of the generated form, as shown in Fig. 6(d).
The above analysis does not involve the time-dependent or spatial dependent factors that cause TIF changes. That is, the concept of controlling volumetric removal is applicable to freeform surface generation with time-variant and spatial-variant TIF. In addition, the modeling and analysis thus far is focused on the intrinsic quality of controlling volumetric removal, additional analysis considering the time-varying removal characteristics of APPP is performed in the following section.
4. Applicability of the nonlinear dwell-time algorithm in APPP
Section 3 provides the theoretical basis for controlling volumetric removal, based on the analysis of freeform surface generation by two TIFs. In freeform surface generation by APPP, the time-variant TIF changes with the dwell-time at each point. Thus, the freeform surface generation involves more than two TIFs with various FWHMs and peak removal rates, as illustrated in Fig. 7. Hence, the reasonability of the nonlinear dwell-time algorithm in APPP should be clarified. Besides, among the TIFs involved in freeform surface generation shown in Fig. 7, the TIF used to calculate the desired volumetric removal in the nonlinear dwell-time algorithm may influence the performance of the algorithm. Considering the change law of the time-variant TIF shown in Fig. 1, the TIFs involved in freeform surface generation appear in different stages corresponding to the dwell-time ranges, indicating that the dwell-time range may affect the accuracy of the algorithm. These three issues regarding the applicability of the nonlinear dwell-time algorithm in APPP are investigated via simulation.
4.1 Simulation design
The bisinusoidal form, expressed as Eq. (11), is used as the desired form to simulate freeform surface generation by APPP using the nonlinear dwell-time algorithm. Figure 8 presents a flowchart of the simulation approach. The detailed process is as follows:
Step 1: Give the desired form. The parameters including spatial period ($T$) and amplitude (${A_1}$) need to be provided.
Step 2: Establish the time-variant TIF model, expressed as Eq. (1) through experiments. The model coefficients are listed in Table 2, under the experimental conditions listed in Table 4.
Table 2. Simulation parameters of freeform generation by APPP
Table 3. Simulation parameters of the bisinusoidal form generation by APPP
Table 4. Experimental parameters
Step 3: Calculate the dwell-time according to the nonlinear dwell-time algorithm. One important point is selecting a TIF according to the time-variant TIF model, as shown in Fig. 8.
Step 4: Determine the actual TIF at each machining point during freeform surface generation based on the dwell-time and time-variant TIF model.
Step 5: Calculate the generated form through the superposition of local removal, which is determined by the actual TIFs and the dwell-time solution.
Step 6: Subtract the generated form from the desired form to obtain the form error.
Table 3 lists the parameters of the simulation experiments. The three simulation groups correspond to the three issues mentioned above. Group 1 is used to verify the reasonability of the nonlinear dwell-time algorithm via simulation with different spatial periods of the desired form, which correspond to different period-FWHM ratios. Group 2 is designed to select the optimal TIF to calculate the desired volumetric removal. In Group 3, the desired form has different amplitudes, which results in varying dwell-times; thus, the corresponding change in the time-variant TIF differs. Note that the FWHM of the selected TIF is determined based on the results of Group 2. In addition, the peak removal rate of the selected TIF is set as 15 µm/min in all simulations, as it has no influence on calculating the desired volumetric removal.
4.2 Simulation results and analysis
Figure 9 presents the simulation result of Group 1, when the spatial period of the desired form is 30 mm. The removal error is mainly located at the peaks and valleys, which is more obvious according to the cross-sectional profiles shown in Fig. 9(b). Compared with that in Fig. 6, the form error profile is not a perfect sinusoidal curve with a slight raised area at the valley, which is caused by the time-variant TIF of APPP. This suggests that the freeform generation using time-variant TIFs in APPP is slightly different from that using two constant TIFs.
Fig. 9. Simulation result with 30 mm spatial period corresponding to Group 1. (a) Form error; (b) Cross-sectional profiles; (c) Dwell-time; (d) Histogram of dwell-time.
To clarify the influence on the form error of the difference, Fig. 10 summarizes the form error RMS corresponding to the simulations in Group 1. The simulated form error decreases with the spatial period of the desired form. Overall, the form error is low, indicating that the nonlinear dwell-time algorithm based on controlling volumetric removal is still suitable to freeform surface generation in APPP. In the simulation result shown in Fig. 9(c) and (d), the dwell-time ranges from 0.3 s to 1.2 s, thus the FWHM of the related TIF changes from 3.6 mm to 4 mm. If the desired form is assumed to be created by only one of the TIFs, the form error can be calculated using the theoretical model (Eq. (23); the theoretical calculation results are also plotted in Fig. 10. Because TIFs with different FWHMs are involved in the freeform generation by APPP, the simulated form error is located in the range of the theoretical results. This also means that the freeform surface generation by APPP with time-variant TIF can be regarded as a combined result of all involved TIFs.
Figure 11(a) shows the relationship between form error and the FWHM of the selected TIF in Step 3, which correspond to simulation results of Group 2. It can be seen that the curve of the change in form error has a V shape. As the FWHM increases from 3 mm to 4.1 mm, the form error decreases from 35 nm to 5 nm; the form error then rises to approximately 35 nm when FWHM increases to 5 mm. Based on this, the TIF with 4.1 mm FWHM can be selected as the optimal TIF to determine the desired volumetric removal in the nonlinear dwell-time algorithm. In addition, the dwell-time distribution of simulation with 4.1 mm FWHM are presented in Fig. 11(b). The dwell-time is distributed in the range of 0.25–1.28 s and the corresponding FWHM of TIFs involved in surface generation is within the range of 3.6–4.1 mm, indicating that the form error can be minimized by selecting the TIF with the maximum FWHM among the involved TIFs. Considering that the convergence rate decreases more slowly when the FWHM-change ratio is smaller than 1 shown in Fig. 5, the TIF with 4.1 mm FWHM can be regarded as ${r_1}(x,y)$ from Section 3.1, and the FWHM within 3.6–4.1 mm corresponds to a FWHM-change ratio of less than 1. Consequently, the optimal FWHM of the selected TIF can be determined as the maximum FWHM within the dwell-time range, thus a high calculation accuracy can be expected. Moreover, by taking the convergence rate as reference, the TIFs with FWHM ranging from 3.4 mm to 4.8 mm can be selected to achieve a high convergence rate above 97%, demonstrating the superior applicability of the nonlinear dwell-time algorithm.
Fig. 11. Simulation results of Group 2. (a) Relationship between form error RMS and spatial period; (b) Dwell-time histogram.
In the simulations for Group 3, the optimal TIF with the maximum FWHM within the dwell-time range is selected to calculate the desired volumetric removal. Figure 12 presents the form error RMS and the dwell-time range with respect to the desired form amplitude. A roughly linearly increasing tendency is observed in form error with the desired form amplitude. According to Eq. (21), the amplitude of the form error generated by controlling volumetric removal is proportional to that of the desired form; thus, a linearly increase tendency is expected. In addition, the dwell-time ranges cover multiple stages of changes in the time-variant TIF, as shown in Fig. 1. For example, the dwell-time range corresponds to the rapid-change stage of the time-variant TIF when the desired form amplitude is 3 µm. In the rapid-change stage, the FWHM and peak removal rate of TIF increases rapidly, which can introduce strong nonlinearity into the freeform surface generation. Hence, it can be concluded that the linear increase is strongly correlated with the inherent quality when using the nonlinear dwell-time algorithm, instead of the time-varying TIF in APPP. Nonetheless, the form error remains at a low level with different dwell-time ranges corresponding to different stages of the time-variant TIF, indicating that the dwell-time range has negligible impact on the nonlinear dwell-time algorithm.
5. Experiments of freeform surface generation by APPP
To check the applicability of the nonlinear dwell-time algorithm based on controlling volumetric removal, a microstructured freeform surface was generated on a fused silica substrate, and the analysis results in Section 3 and 4 were applied. Figure 13 presents the experimental setup: a lab-built APPP machining system with a plasma torch. The distance between the plasma torch and the workpiece surface is 2 mm to avoid its influence on TIF. The chamber temperature is about 24 °C and keeps unchanged in experiments. The working gases, including He, O2 and CF4, are excited by 13.56-MHz radio-frequency power to generate fluorine plasma. The processing parameters are listed in Table 4.
According to the flowchart in Fig. 3, the time-variant model of TIF was established by trench machining experiment. The coefficients in Eqs. (1–3) were obtained, as listed in Table 2. The relationship between the volumetric removal and dwell-time was also determined. Figures 14(a) and (b) show the desired form and its power spectrum density (PSD) analysis results. From the PSD curve, it can be found the main spatial frequency component of the desired form is less than 0.08 mm−1, corresponding to spatial period of 14 mm approximately. According to the convergence rate map in Fig. 4(b), if the 4.2 mm FWHM is taken as ${w_1}$ and $\alpha$ is assumed to range from 0.7 to 1.2, the convergence rate can be approximately expected greater than 94%. In other words, the fabrication condition of the desired form is located within the feasible domain of the concept of controlling volumetric removal.
Fig. 14. Experimental results of micro-structured freeform surface generation. (a) Desired form; (b) PSD of desired form and generated form; (c) Dwell-time distribution; (d) Measurement result of generated form; (e) Form error.
Then, the dwell-time was calculated using the nonlinear dwell-time algorithm. Considering that the optimal TIF in the nonlinear algorithm has the maximum FWHM in the corresponding dwell-time range, the TIF with 4.2 mm FWHM was used initially to calculate the dwell-time. Then, the maximum FWHM of the optimal TIF was determined to be 4 mm and the dwell-time was finally calculated. Figure 14(c) shows the final distribution of the dwell-time. It can be found that the TIF corresponding to the maximum dwell-time is 4 mm, which matches the analysis result shown in Fig. 11. Thus, the microstructured surface was fabricated on the APPP machine. The initial surface is a plane surface, and the form error was obtained by subtracting initial form of the plane surface from the measured form. Thus, the starting form error is the opposite of the desired form shown in Fig. 14(a). The RMS of the starting form error is about 198 nm. Figure 14(d) presents the interferometer measurement result, which indicates that the microstructures of the desired form were successfully generated, and the form error is shown in Fig. 14(e). In addition to the inherent quality of the nonlinear dwell-time algorithm, the form error is also attributed to other inevitable factors, such as machining instability and calibration error of the TIF model. Nonetheless, the RMS of the form error is less than 50 nm, which proves the superior feasibility of the nonlinear dwell-time algorithm in freeform generation by APPP using a time-variant TIF.
6. Conclusions
In this paper, the nonlinear dwell-time algorithm based on controlling volumetric removal for freeform surface generation by APPP using a time-variant TIF was investigated. The freeform surface generation by controlling volumetric removal was modeled and analyzed. The form error was found to have a similar pattern to the desired form. When the period-FWHM ratio between the desired form and TIF increases, the form error amplitude decreases rapidly and ultimately reaches a low level, which provides a theoretical basis for the nonlinear dwell-time algorithm. Through simulations based on the theoretical model, the feasible domain of the concept of controlling volumetric removal was obtained. Furthermore, considering the influence of time-variant TIF, the applicability of the nonlinear dwell-time algorithm in freeform surface generation by APPP was proven via simulations of bisinusoidal surface generation. A fabrication experiment of a microstructured freeform surface was performed, and the form error was controlled to less than 50 nm RMS. This validates the superior feasibility of the nonlinear dwell-time algorithm by controlling volumetric removal for freeform surface generation by APPP using a time-variant TIF.
Funding
National Natural Science Foundation of China (51905130, 52105488); Natural Science Foundation of Heilongjiang Province (LH2020E039).
Disclosures
The authors declare no conflicts of interest.
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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