## Abstract

Based on a controllable chemical reaction, atmospheric pressure plasma processing (APPP) can achieve efficient material removal even when the tool influence function (TIF) size is reduced to several millimeters, resulting in its great application potential for generating freeform surfaces. However, the TIF changes with the local dwell time, introducing nonlinearity into processing, because of the influence of the plasma thermal effect on chemical reactions. In this paper, a freeform generation method using a time-variant TIF is presented and validated. First, the time-variant removal characteristics of APPP and its nonlinear influence on freeform surface generation are analyzed. Then, the freeform surface generation concept is proposed based on controlling the local volumetric removal. Consequently, the dwell time calculation method is developed to suppress the nonlinearity induced by the time-variant TIF. Finally, the developed method is evaluated by the simulation and experimental analysis of the complex structure generation process. Results show that the proposed method can reduce the nonlinear influence of the time-variant TIF by reasonably calculating dwell time, promoting the application of APPP in freeform surface generation.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Freeform optics have been widely introduced into many advanced optical systems to provide additional design freedom and ensure the lightweight nature as well as miniaturization of the system. However, the high-quality fabrication of freeform optics presented a challenge to optical manufacturing techniques because of the complex shape or surface structure of freeform optics. Hence, various computer-controlled optical surfacing (CCOS) techniques based on different removal mechanisms were proposed and developed such as fluid jet polishing (FJP) [1], bonnet polishing (BP) [2], ion beam figuring (IBF) [3], and magnetorheological finishing (MRF) [4].

Based on these techniques, many researchers investigated various freeform fabrication methods. FJP uses a pressurized slurry for material removal, with an ability to generate submillimeter freeform structures [5]. Zhu proposed a deterministic freeform generation method using FJP, in which the dwell time is optimized and smoothed via Zernike decomposition to achieve a bi-sinusoidal surface with a 5-mm spatial period and 1.5-µm peak-to-valley (PV) [6]. Cheung studied the material removal model of BP and established a model-based simulation system for structured surface generation [7]. Based on this, Cao developed swing precess bonnet polishing (SWBP) by controlling the tool orientation and tool path, based on which complex three-dimensional structures can be created [8]. Recently, BP was also investigated to polish a stainless-steel freeform surfaces [9] and create a continuous phase plate (CPP) [10]. IBF, which is based on the physical sputtering effect of an energized ion beam, is a highly deterministic fabrication technique. A multipass IBF method was proposed to generate a CPP with a spatial period as low as 1 mm. This method used a frequency filter to imprint structures within certain spatial periods in a single pass, improving the efficiency and accuracy in multiple passes [11]. MRF is used to fabricate large-aperture CPPs. A fuzzy theory was introduced in the process to determine the optimum removal functions [12,13].

In recent years, various reactive plasma processing techniques based on chemical etching in atmospheric pressure were proposed [14–18]. Because of the chemical etching mechanism and various plasma discharge modes, the tool influence function (TIF) can be modified to different shapes, with the size changing from 0.2 mm to 25 mm [19–21]. Hence, reactive plasma processing has strong advantages in freeform generation. Arnold developed plasma jet machining (PJM) to generate a sinusoidal surface with spatial period down to 0.5 mm [20]. Plasma chemical vaporization machining (PCVM) was proposed to create an aspheric mirror with nonsymmetrical structure [22,23]. Atmospheric pressure plasma processing (APPP) was developed and used to fabricate an optical sinusoidal grid with 2 mm spatial period and 100 nm amplitude [24].

However, the material removal rate in reactive plasma processing is time-dependent, due to the plasma jet heating the workpiece [25,26]. Note that, the time-dependent removal characteristics in reactive plasma processing is different from that in BP, MRF and soft wheel polishing [27]. The time-variant TIF in BP is caused by different surface curvatures in aspheric surface processing [28], which is also spatial-dependent and has less influence on deconvolution calculation for dwell time. The time-variant TIF in MRF is obtained by varying the plunge depth or wheel velocity in processing. And the time-varying TIF in MRF is used to compensate for velocity truncation problem limited by CNC machine performance [29]. To suppress the center dent caused by velocity truncation, a time-variant tool influence function regime was introduced based on polishing force optimization in soft wheel polishing [27]. In addition, the improved algorithms considering variant TIF were proposed to reduce processing time [30] or improve the adaptation within limit of machine dynamics [31]. As for reactive plasma processing, the TIF changes with the local dwell time. This makes the removal process nonlinear to dwell time, which creating difficulties for dwell time calculation and machining process control. For this reason, an iterative method for dwell time calculation was proposed to generate a CPP [32]. Ji investigated an optimization strategy based on velocity distribution to suppress nonlinearity in reactive plasma processing [33]. Dai proposed a dwell time compensation method for nonlinearity in inductively coupled plasma figuring, in which the removal error is calculated forward and compensated in iterative calculations [34]. In addition, Meister developed a processing simulation method based on a temperature predication model of PJM to control machining error. The method requires computing time on the order of 16–20 hours [35], which may hinder its application in practice.

Therefore, an efficient and succinct approach is required to eliminate nonlinearity induced by the time-variant removal characteristics and promote the application of reactive plasma processing. In this paper, the time-variant removal characteristics of APPP are analyzed. The influence of nonlinearity on freeform surface generation of time-variant TIF is investigated. Based on this, a concept of controlling local volumetric removal for freeform surface generation and a related nonlinear method for dwell time calculation are proposed. Finally, a simulation and experiment are carried out, which effectively prove that the proposed method can eliminate the nonlinear influence of the time-variant TIF on freeform generation.

## 2. Time-variant removal characteristics of APPP

The basic principle of APPP is a controllable chemical etching technique with a reactive fluorine plasma jet, as shown in Fig. 1. The APPP machine tool was built in our laboratory. The reactive fluorine plasma jet is used to etch silica-based material, such as fused silica and silicon carbide [36]. In APPP, a mixture of several neutral gases, including He, O_{2}, and CF_{4}, is introduced into the capacitively coupled plasma torch. The plasma jet is excited by 13.56 MHz radio-frequency (RF) power based on the principle of dielectric barrier discharge. In the plasma excitation process, the RF electric energy transfers into chemical energy of reactive radicals and thermal energy of all particles in the plasma jet. Thus, the plasma jet can be regarded as an energy flow consisting of chemical and thermal energy. As the plasma jet reaches the substrate surface, material removal is realized through chemical etching. Meanwhile, the plasma jet transfers heat to the substrate through forced convection, which makes the substrate temperature more dependent on the local dwell time [34,35]. Because the chemical reaction rate changes with temperature according to the Arrhenius theory, the material removal process is directly influenced by substrate temperature. Thus, the TIF changes with the dwell time of the plasma jet, due to the substrate temperature changing with the dwell time. Therefore, the time-variant TIF model includes the influence of substrate temperature [33], which can describe the removal characteristics of APPP.

Figure 2 illustrates the typical removal profile of APPP, that fits a Gaussian curve perfectly. Note that the Gaussian shaped TIF is superior for correcting form error or generating freeform structures [3]. The peak removal rate and full width at half maximum (FWHM) are used to establish the mathematical model of the Gaussian shaped TIF. Hence, the TIF can be expressed as,

where*a*is the peak removal rate, and

*e*is Euler’s number.

As mentioned above, material removal is influenced by dwell time, which means the peak removal rate and FWHM are not constant. To establish a time-variant model of TIF, a series of trench generation experiments were conducted. Table 1 gives the processing parameters. The substrates are fused silica with a thickness of 3 mm provided by the China Building Materials Academy (Beijing, China). The substrates were pre-polished using a conventional pitch lap with CeO_{2} slurry to about 2 nm Ra. Each trench was generated by moving the plasma jet at a certain feedrate, which corresponds to an equivalent dwell time value.

Because the trench generation process can be regarded as a superposition of TIF at discrete points, the TIF corresponding to an equivalent dwell time was obtained based on a calibration calculation, as shown in Fig. 3. In detail, the FWHM was directly obtained from the section profiles and the peak removal rate was determined by backward calculation, express as [32],

where $depth$ denotes the depth of the generated trench, and*v*denotes the feedrate.

Figure 4 shows the the peak removal rate and FWHM after backward calculation. Both the peak removal rate and FWHM increase rapidly with dwell time, and then reach a plateau. When the dwell time is less than 1.5 s, changes in peak removal rate and FWHM are quite obvious, indicating that material removal is a nonlinear process versus dwell time. When dwell time is higher than 2 s, the peak removal rate and FWHM are almost unchanged, thus, material removal becomes a linear process.

By fitting the experimental results with exponential equations, time-variant models of peak removal rate and FWHM were obtained as follows,

Where *t* is the dwell time.

Substituting Eqs. (3)–(4) into Eq. (1), a time-variant model of TIF was obtained as,

## 3. Nonlinearity in freeform surface generation induced by time-variant TIF

Freeform surface generation using APPP is a typical CCOS process, which can be presented mathematically as a convolution of TIF and dwell time. Therefore, the calculation of dwell time is an important step for freeform surface generation. In conventional polishing techniques with time-invariant TIF, the convolution is a linear mathematical process, expressed as Eq. (6). The linear equations method and iterative method are widely used to solve dwell time according to the desired removal. With these linear methods, a constant removal function matrix or convolution kernel based on the time-invariant TIF is created and then used to solve the dwell time. In contrast, freeform surface generation using time-variant TIF in APPP is a nonlinear convolution process, expressed as Eq. (7), due to the time-variant TIF. That is to say, linear methods are not suitable for dwell time calculation in APPP. If a certain constant TIF is given according to the time-variant TIF model, then a dwell time result can be obtained using the above linear methods. But the nonlinear influence on freeform surface generation of the time-variant TIF must be introduced.

where $z(x,y)$ is desired removal and $t(x,y)$ is dwell time.To illustrate the nonlinear influence on freeform surface generation in APPP, a series of simulation experiments were designed and performed. Figure 5 shows the flowchart of the freeform surface generation process simulation.

Step 1: Give a constant TIF corresponding to a certain dwell time, ${r_0}(x,y)$, according to Eq. (5).

Step 2: Calculate the dwell time, ${t_0}(x,y)$, using a linear equation method based on the constant TIF.

Step 3: Determine the actual TIF, ${r_1}(x,y,{t_0}(x,y))$, at each local point based on the dwell time and Eq. (5).

Step 4: Calculate the actual removal, ${z_1}(x,y)$, through forward convolution of the actual TIF and the dwell time.

Step 5: Obtain the nonlinear form error, $\Delta z$, by subtracting the actual removal from the desired.

Table 2 gives the parameters of the simulation experiments. By combination of the FWHM and peak removal rate listed in Table 2, 42 constant TIFs were defined and available for the simulation. The desired freeform surface has 5300 nm PV and 817 nm root mean square (RMS). According to the desired freeform surface, the desired removal is obtained, as shown in Fig. 6(a).

Figure 6 shows the simulation results when the FWHM and peak removal rate of the given constant TIF are 4.1 mm and 12 µm/min. The lateral distribution of the actual removal coincides well with the desired removal. The removal error corresponds strongly to the freeform structure, which has 1382 nm PV and 195 nm RMS. Figure 7(a) illustrates the simulation results in a section, indicating that the removal error is more obvious in height than in the lateral direction. More specially, the removal error increases significantly at peaks and valleys of the freeform structure. It can be concluded that the removal error distribution is correlated with the time-variant characteristics of the TIF. As the dwell time changes in freeform surface generation, the change ratio of the peak removal rate is larger than that of the FWHM, thereby causing more removal error in the height direction.

Figure 7(b) is the dwell time distribution histogram corresponding to Fig. 6(b). The dwell time is basically in accordance with normal distribution, ranging from 0.05 s to 1.5 s. According to Eq. (5), within the dwell time range, the FWHM and peak removal rates change from 3.2 mm to 4.2 mm and 3 µm/min to 14 µm/min, respectively. In addition, the actual TIFs at points with 0.9 s dwell time have approximately 3.9 mm FWHM and 12 µm/min peak removal rate, which is almost the same as the given constant TIF for dwell time calculation. This means that the actual removal at these points with 0.9 s dwell time approximate the desired removal in the dwell time calculation process. Considering the changes to actual TIF, for points with less than 0.9 s dwell time, the actual TIF has weaker removal capability than the given constant TIF. Thus, the actual removal at these points is smaller than the desired removal, which corresponds to the simulation results at valleys in Fig. 7(a). Conversely, the actual removal at points with more than 0.9 s dwell time is larger than the desired removal, corresponding to the simulation results at peaks in Fig. 7(b). Note that such a removal error pattern indicates nonlinear removal error caused by the time-variant TIF in freeform surface generation using APPP.

To further elaborate the relationship between removal error and the given constant TIF, simulation results with different TIFs are summarized and illustrated in Fig. 8. The peak removal rate and the FWHM of the given constant TIF in Table 2 are taken as the horizontal and vertical axis, respectively. As presented in Fig. 8(a), the removal error changes with the peak removal rate and FWHM of the given constant TIF, and has a comparatively small value near the diagonal. This indicates that the removal error caused by the time-variant TIF can be reduced to a certain content, if an appropriate TIF is given for dwell time calculation. Under this condition, the minimum RMS of the removal error was less than 100 nm. Note that, the above analysis can prove the fitness of the optimization strategy based on constant TIF to suppress nonlinearity in APPP [33]. However, selecting a suitable constant TIF is only an approximate solution for dwell time calculation, which does not address the nonlinear problem in freeform surface generation using APPP.

In comparing the removal error in Fig. 8(a) and the volumetric removal rate in Fig. 8(b), it can be seen that removal error decreases when the volumetric removal rate approaches about 0.18 mm^{3}/min. This suggests that the volumetric removal rate might be a major factor affecting the nonlinear removal error when using constant TIF to calculate dwell time.

## 4. Dwell time calculation for time-variant TIF

Now, a dwell time calculation method that considers the nonlinear influence of time-variant TIF is proposed. In the existing dwell time calculation method, a deconvolution between the desired removal in height and TIF is performed to obtain dwell time, which can be seen as controlling the removal depths at discrete points. In contrast, the proposed method calculates dwell time by controlling local volumetric removal, while considering the nonlinearity of time-variant TIF.

#### 4.1 Concept of controlling local volumetric removal

To create a freeform structure, material in a certain distribution area needs to be removed. After machining, the heights at all points are used to evaluate the removal result, which corresponds to the desired removal in height for calculating dwell time. From a different perspective, the local volumetric removal at each point can represent the machining process. If local volumetric removal occurs in a suitable distribution, the desired freeform structure might also be generated. This is termed the concept of controlling local volumetric removal.

For instance, when a uniform layer needs to be removed with a time-invariant TIF, the dwell time is calculated by deconvolution of the removal in height and the TIF, shown in Fig. 9. In contrast, dwell time can also be obtained based on the concept of controlling local volumetric removal. First, the total processing time is obtained by dividing the total volumetric removal ($H \cdot L$) by the volumetric removal rate of the TIF. Then the dwell time at each point can be determined by distributing the total processing time among all the machining points. It should be noted that the second method does not conduct a deconvolution, and is relatively simple.

To further study the feasibility of controlling local volumetric removal in freeform surface generation, a one-dimensional curved surface was simulated. The simulation controls volumetric removal at local points to create a curved surface. First, three different TIFs with identical volumetric removal rate were defined, as shown in Fig. 10. The TIFs are Gaussian, sinusoidal, and two-term Gaussian, respectively, with different FWHMs and peak removal rates, expressed as Eqs. (8)–(10). Note that, the normal direction is set as unit length for calculation simplicity. Second, one dwell time for sinusoidal distribution was provided, shown in Fig. 11(a). The dwell time distributes in the range of 10 mm to 90 mm, with discrete spacing of 1 mm. Then, three removal profiles were calculated by forward convolution of the three TIFs and the dwell time. The calculated removal area ranges from 0 mm to 100 mm. Because the TIFs have identical volumetric removal rate, the volumetric removal at local points is controlled to be the same. Consequently, highly consistent surfaces were created with the three TIFs, as shown in Fig. 11(b). Taking the surface profile created by TIF1 as a reference, the removal error of the three TIFs was obtained, shown in Fig. 11(c) and 11(d). Slight deviations exist among the three generated surfaces in the machining region between 10 mm and 90 mm, of approximately 2 nm PV. Close to the edge of the machining region, the removal deviation becomes significantly higher, which can be attributed to the edge effect in CCOS. The simulation results prove the concept of controlling volumetric removal is reasonable in freeform surface generation. That is, a freeform surface can be generated with certainty when the local volumetric removal is determined.

#### 4.2 Nonlinear method for dwell time based on controlling local volumetric removal

Based on the concept of controlling local volumetric removal, a dwell time calculation method was proposed, as shown in Fig. 12. Since a freeform surface can be generated with certainty when the local volumetric removal is determined, the dwell time can be calculated by making the volumetric removal of time-variant TIF equal to the desired volumetric removal of a constant TIF. In this way, the nonlinear convolution (described by Eq. (7)) is decoupled into a linear process and a nonlinear process. The linear process can be regarded as the freeform surface generation process using a constant TIF, in which the desired volumetric removal is determined by linear deconvolution calculation. The nonlinear process represents the freeform surface generation process using a time-variant TIF, in which the relationship between volumetric removal and dwell time is obtained by trench calibration experiment. Then, the two processes are coupled again by the same desired volumetric removal required for freeform surface generation, to determine the final dwell time. In this way, the calculation method takes into account the nonlinear effect of time-variant TIF, and also avoids complex mathematical calculation. The detailed steps are as follows.

According to the calibration experiments by trench generation described in Section 2, a time-variant TIF model is established, expressed as Eq. (5). By integrating the TIF model in the space-time domain, the volumetric removal model versus dwell time can be determined as Eq. (11). Although this step involves the nonlinear influence of time-variant TIF, a nonlinear process, it is still a simple forward calculation. Note that Eq. (11) describes the relationship between the local volumetric removal of the time-variant TIF and the dwell time.

Based on the time-variant TIF model (Eq. (5)), the TIF remains unchanged when dwell time is greater than 2 s, which is herein named the unchanged TIF. The constant TIF is set as the unchanged TIF, expressed as,

The dwell time corresponding to the constant TIF is calculated, by solving Eq. (13) using conventional methods.

Conducting a forward convolution between the constant TIF and the solved dwell time, the desired volumetric removal can be determined as,

To obtain the desired removal $z(x,y)$, the local volumetric removal is required to be equal to ${V_{desired}}(x,y)$. Therefore, the final dwell time can be obtained by solving Eq. (15). When $V(t)$ is a fundamental function, the inverse of $V(t)$ can be derived to calculate the dwell time at all points. More generally, $V(t)$ can be discretized to obtain a series of data points including the volumetric removal and the corresponding dwell time. Thus, the dwell time at all points can be quickly determined by interpolating with ${V_{desired}}(x,y)$.

Compared with the dwell time calculation method in previous study on reactive plasma processing [33–35], the proposed method transfers the desired removal depth into the desired volumetric removal to determine the dwell time without conducting nonlinear deconvolution calculation or iterative calculation, based on the concept of controlling the local volumetric removal. Hence, the proposed method is simpler and more efficient than that in previous study.

## 5. Validation

In this section, a simulation and experiment were conducted to verify the proposed nonlinear method for dwell time calculation.

#### 5.1 Simulation demonstration

For comparison with the conventional dwell time calculation method, the removal in Fig. 6(a) is used as the desired removal. Using the flowchart in Fig. 5, but calculating the dwell time using the nonlinear method, the freeform surface generation was simulated and analyzed. Figure 13(a) presents the desired volumetric removal calculated by Eq. (14), which shows an approximate pattern with the desired removal in height. The dwell time is also similar to that calculated using the linear method, shown in Fig. 13(b). In contrast with Fig. 6(d), the removal error using the nonlinear method is significantly reduced to 6 nm PV and 0.4 nm RMS, as shown in Fig. 13(c). In addition, there is no obvious pattern corresponding to peaks and valleys in the removal error, indicating that the nonlinear influence on freeform surface generation has been compensated.

Furthermore, the dwell time profiles at the same location in Fig. 6(b) and Fig. 13(b) were extracted to investigate the compensation process, shown in Fig. 13(d). The two profiles show apparent deviations at peaks and valleys. In specific, the dwell time at peaks calculated using the nonlinear method is smaller than that calculated using the linear method. While the deviations are the opposite at valleys. Such deviations successfully compensate the excessive removal at peaks and the lack of removal at valleys, which is consistent with the analysis based on Fig. 7 in Section 2.

The above simulation results prove that the proposed nonlinear method based on controlling local volumetric removal can determine the dwell time reasonably for time-variant TIF. It should be pointed out that the method is potentially applicable for spatial-variant TIF as well. Reconsidering the flowchart in Fig. 12, the desired volumetric removal is obtained similarly to the linear process. Calibration experiments need to be performed to establish the relationship between local volumetric removal and the spatial position. Hence, dwell time at a certain spatial position can be determined by controlling the actual local volumetric removal equal to the desired volumetric removal.

#### 5.2 Experimental validation

An experimental validation was carried out on the machine tool shown in Fig. 1. The processing parameters are the same as those in Table 1. The substrate is the same as that in Section 2. The substrate form was measured by an interferometer before and after the freeform surface generation experiment. Note that, the freeform surface was generated in a single cycle of APPP. The plasma jet was placed at the start point on the substrate with naked eyes, and then the processing began and completed.

The desired freeform surface has 761 nm RMS and 4350 nm PV, given in Fig. 14(a), which corresponds to the middle region of Fig. 6(a). To generate the desired form, the dwell time was solved by the nonlinear method and transferred into feedrate. Thus, the APPP torch scanned on the substrate in raster trajectory, for 120 min. Figure 14(b) shows the generated freeform surface, which has 741 nm RMS and 4200 PV. By registration with the desired form, the form error was obtained, as shown in Fig. 14(c). There exists low- and medium-frequency error in comparison with the desired form. Because the volumetric removal rate has fluctuations of 10–14% based on experimental experience, deviation between the fitted TIF model and the actual TIF is inevitable. Thus, the desired removal cannot be controlled perfectly in the calculation, which causes the medium-frequency error. The low-frequency error is caused by long-term removal instability. For these reasons, the RMS of the form error is about 171 nm. The convergence ratio (defined as the ratio of the RMS of the desired form to that of the form error [3]) is about 4.5, which is higher than 2–3 obtained after a single figuring process for freeform surface generation [3,11,37]. In addition, a power spectrum density (PSD) analysis of the desired form and the actual form is performed in the direction of length and width respectively, shown in Fig. 14(d). The solid and dotted lines represent the desired form and the actual form, respectively. The blue and red lines represent the length direction and the width direction, respectively. The PSD curves of the actual form are very close to those of the desired form, indicating that all spatial components of the desired form have been generated. These experimental results demonstrate that the nonlinear method can solve dwell time reasonably, to suppress the nonlinear influence of the time-variant TIF on freeform surface generation.

## 6. Conclusions

In this study, a freeform surface generation approach using time-variant TIF in APPP is proposed and investigated. The time-dependent removal characteristics of APPP and its nonlinear influence on freeform surface generation were analyzed. The conventional linear method for dwell time calculation was found to cause more and less material removal at peaks or valleys of freeform surfaces and is no longer applicable for APPP. Hence, a concept of controlling local volumetric removal was proposed to describe the freeform surface generation process. Based on this concept, a novel nonlinear method for dwell time calculation was developed, by decoupling the freeform surface generation into a linear process and a nonlinear process. The linear process transfers the desired removal in height into the desired volumetric removal based on the conventional linear method. The nonlinear process uses trench generation experiments to establish the relationship between local volumetric removal and dwell time. Dwell time is determined by coupling the two processes through controlling the local volumetric removal equal to the desired volumetric removal. Simulation and experiment of a CPP structure generation were carried out, which demonstrated the reasonability of dwell time calculation using the nonlinear method.

## Funding

National Natural Science Foundation of China (51905130); Natural Science Foundation of Heilongjiang Province (LH2020E039); China Postdoctoral Science Foundation (2019TQ0078); National Science and Technology Planning Project (2019ZX04021001).

## Disclosures

The authors declare no conflicts of interest.

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