## Abstract

Ultra precision optical surfaces can be efficiently manufactured using a computer-controlled optical surfacing (CCOS) process. Based on the chemical reaction, atmospheric pressure plasma processing (APPP) is a promising deterministic CCOS technique and has great application prospect for the figuring processing as well as freeform generation. However, the plasma jet also works as the heat source, leading to the variation of substrate temperature field. This way, the tool influence function (TIF) is continuously changed, which leads to the nonlinear removal characteristic. Especially, it becomes much more complex when considering the neighboring dwell points, because they are thermally interacted. The conventional time-variant TIF model cannot accurately describe the practical TIF changes. In this paper, an innovative reverse analysis method is proposed to derive the practical TIF changes in APPP. First, the special problem of the TIF neighborhood effect is pointed out. The limitation of the conventional TIF model is analyzed with the assisted thermal model. Then, an innovative reverse analysis method is presented to derive the TIF changes from the practical removal, which is demonstrated with the simulation. Further, the proposed method is applied to the analysis of the TIF changes in APPP. To verify its feasibility, the experimental validation is undertaken, which proves its capability of deriving complex TIF changes.

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

## 1. Introduction

It is widely acknowledged that the convolution theorem is of great importance in computer controlled optical surfacing (CCOS), which mathematically accounts for the principle of removal generation [1]. Namely, the removal distribution is essentially equal to the two-dimensional (2D) convolution between the tool influence function (TIF) and dwell time map [2]. In practice, more attention is paid to the deconvolution procedure. For the given TIF and targeting removal distribution, the dwell time map is acquired with all kinds of deconvolution algorithms [2–4]. To verify the theoretical accuracy of the calculated results, the forward convolution is conducted to check the theoretical residual error. However, there are more or less deviations between the theoretical and actual residual error, because the TIF cannot remain absolutely unchanged at every controlled point.

There are many reasons accounting for the TIF changes, and the long-time TIF stability ranks the first. For most cases, it is generally intrinsic and predominant for every small tool in CCOS. To evaluate the TIF stability, the spot method is widely adopted to track the TIF changes with the processing time under the same processing parameters [5]. It is worth noting that the unstable TIF fluctuation is unrepeatable and untraceable, and thus cannot be compensated. Aiming for this, Han et al. [6] investigated the statistical properties of TIF fluctuation by the Monte Carlo simulation, and successfully applied it to the prediction of the processed surface waviness.

Besides, for every small tool, the TIF changes can also be caused by its unique characteristic. In bonnet polishing (BP), the TIFs have been found to be significantly affected by the curvature differences between the pad and the workpiece. Zhong et al. [7] proposed a three-dimensional time-varying TIF model based on the finite element analysis and kinematics analysis methods. In magnetorheological finishing (MRF), the characteristics of TIFs are influenced by the immersion of the optic into the magnetically responsive (MR) fluid ribbon. Menapace et al. [8] pointed that this change can pose issues that limit MRF polishing solution convergence. The increase of imprint depth results in the smaller size and volumetric removal rate (VRR) of TIFs. In atmospheric pressure plasma processing (APPP), the essence of chemical reaction makes the material removal process susceptible to the substrate temperature [9], leading to the strong removal nonlinearity of TIFs [10,11]. To solve this, Su et al. [12] developed an innovative dwell time calculation method to suppress the nonlinearity induced by the time-variant TIF. Compared with the TIF stability, these TIF changes have close connection with a specific influence factor, indicating that they can be predicted and analyzed with the reasonable TIF model. However, considering the complexity of processing condition, the essence of TIF changes cannot be completely disclosed with the simple TIF model, because multiple influence factors may couple together. On account of this, it unavoidably introduces the processing error, and thus impact the figuring efficiency. Moreover, to our best knowledge, there is a lack of the effective method to check the correctness of the established TIF model.

At present, the figuring efficiency is generally characterized by the concept of convergence rate [13], which takes the root mean square (RMS) of initial and final form errors into consideration. Despite its simplicity and effectiveness, more detailed information about the TIF changes is missing. For example, what is the changed TIF distribution? And how much does the TIF change?

In order to answer these questions, this paper proposes an innovative reverse analysis method to derive the TIF changes from the practical processing results, which is further applied to APPP to explain its significance. In APPP, the substrate temperature is the key factor to influence the TIFs, and diversified in different processing conditions. Thermally, the removal characteristic of one dwell point is influenced by its neighboring points, which is essentially connected with the neighborhood effect. This way, the removal characteristic of APPP cannot be simply described with the conventional time-variant TIF model. Aiming for this, the TIF changes of APPP is first fully discussed in this paper, and the TIF neighborhood effect is analyzed. Besides, the limitation of the conventional TIF model is also investigated with the assisted thermal model. Then, based on the concept of controlling volumetric removal from our previous research [12], an innovative reverse analysis method is presented, which is demonstrated with the simulation. Further, the TIF changes in APPP is investigated with the proposed reverse analysis method. Finally, the experimental validation is conducted to verify the feasibility of the proposed method.

## 2. TIF neighborhood effect in APPP

#### 2.1 TIF changes caused by the neighborhood effect

In APPP, the plasma jet, which is excited by the radio-frequency (RF) power, not only works as a provider of reactive particles, but also heats the reaction zone. In this procedure, the electrical energy is converted into reactive chemical energy and thermal energy. When the plasma jet dwells at the controlled points, the input heat flux increases as time goes on, and the local substrate surface temperature rises rapidly. Therefore, according to the Arrhenius laws, the TIFs of APPP are susceptible to the surface temperature change due to the essence of chemical reaction. This way, the material removal is accelerated until the thermal balance is achieved, leading to the nonlinear removal characteristic in APPP. Therefore, the TIF changes are mainly attributed to the thermal effect.

Regarding to the application conditions of APPP, the abovementioned thermal effect can be classified into two categories, global and local thermal effect. Figure 1 gives the schematic of thermal effect in APPP. The global thermal effect mainly occurs in the conformal polishing, which aims at rapidly removing the surface/subsurface damage of optics [14]. To conduct this, the dwell time is controlled to be the same at every dwell point. With the constant speed, the plasma jet traverses along the given path to achieve the uniform removal. This way, every dwell point contributes equally to the global thermal state and thermally affects the removal generation. The local thermal effect appears in the figuring process or freeform generation. Compared with the conformal polishing, it focuses on the uneven removal distribution by implementing the different dwell time at every dwell point. With the given removal distribution, the dwell time map needs to be solved with all kinds of deconvolution algorithms. Therefore, the thermal state of every dwell point is different, which indicates that the local TIF constantly changes at every dwell point.

However, for the local thermal effect, the practical thermal state is much more complex. In CCOS, the removal generation highly depends on the interaction between with two neighboring points at the small interval. When the plasma jet dwells at every point, the heat diffuses radially and generates an effective thermal affected area [14]. On this occasion, all the dwell points are thermally coupled together, unavoidably leading to the hidden neighborhood effect, as shown in Fig. 2. When the plasma jet moves along the processing direction, there exists the preheating effect. For example, when the plasma dwells at the point I, the input heat is conducted radially, leading to the preheating effect on the following dwell points. Similarly, the point I has also been preheated by the previous dwell points. Therefore, in practice, the temperature at any dwell point is influenced by the multiple preheating effect, which is defined as the neighborhood effect. Moreover, the preheating effect is different for any dwell point, because the combination of previous dwell time is diversified. This way, it’s nearly impossible to accurately calibrate the exact TIF of every dwell point.

#### 2.2 Limitation of the conventional time-variant TIF model for the neighborhood effect

Considering the thermal effect, it has reached a consensus that the spot method by static processing is not suitable to calibrate the TIFs [11,12]. To solve this, the calibration method for the time-variant TIF model is developed to compensate the TIF changes in APPP, where the multiple scanning test with different constant velocities is implemented [10], as shown in Fig. 3. After the interferometric measurement, the depth and the full width at half maximum of every trench section are extracted. According to the step and the different velocities, the equivalent dwell time and peak removal rate are further determined. Then, the time-variant models of peak removal rate and full width at half maximum are obtained by fitting the data exponentially. This way, the temperature is mapped to the dwell time or velocity, because they are closely connected with each other. The established time-variant TIF model can be expressed as [12],

where*a*is the peak removal rate,

*FWHM*is the full width at half maximum. To conduct the figuring process or freeform generation, the modified deconvolution algorithm is further developed to solve the dwell time map, which is expressed as [12],

However, if the neighborhood effect is considered, the conventional time-variant TIF model is limited to accurately analyze the true TIF. In practice, the temperature at any dwell point is influenced by the dwell time combination of previous points along the path, which further determines the TIF. This way, the conventional convolution should be accurately modified as,

*N*is the number of previous dwell points that contribute to the preheating effect on the current dwell point. The larger serial number, the longer distance from the current dwell point. Take the peaks and valleys of the removal distribution in practical processing (figuring process or freeform generation) as an example, and the current dwell time reaches the extremum value in the surrounding area. At the peaks, there exists the relationship, For the scanning test, all the equivalent dwell time of every dwell point is the same, which indicates that the TIFs at the peaks is acquired in the situation of, On this occasion, the preheating effect of the scanning test is much more severe than that of the practical processing. Therefore, compared with the practical processing, the calibrated TIFs at the peaks have larger removal rate. Similarly, the removal rate of the calibrated TIFs at the valleys is smaller than the practical value. From this, the neighborhood effect is not considered in the calibration method, leading to the limitation of the conventional TIF model.

Take the one-dimensional (1D) processing as an example, an assisted thermal model was established to fully investigate the neighborhood effect. Figure 4(a) gives the schematic of APPP, where the substrate is fixed on the worktable. The heat transfer progress in APPP is mainly described by the thermal convection and conduction [14], which can be respectively solved with Newton’s law of cooling and Fourier’s law. The heat input comes from the plasma jet, and depends on the force convection. Surrounded with the ambient air, the substrate is cooled down through natural convection. Besides, there exist two physical contact surfaces, where the heat is transferred by thermal conduction. Based on this, the simplified thermal model was established using COMSOL software as shown in Fig. 4(b). The Gaussian input heat flux was empirically adopted, which can meet the demand for approximately simulating the temperature field [15]. As for the natural convection, the temperature of ambient air was set to be constantly 20 °C. Besides, the thermal contact pair was adopted for the contact surface 1. Considering that the heat is rapidly lost through machine body, the contact surface 2 was assigned with the constant room temperature, which was also 20 °C. Without loss of generality, the substrate material was set to be the fused silica, and its thickness was 3 mm. To acquire the smooth temperature field, the micron-sized mesh grid was adopted in the substrate surface.

In order to couple the temperature field into the removal generation, an explicit relationship between the temperature and the TIFs needs to be defined as shown in Fig. 5. It’s worth noting that this relationship and the Gaussian heat flux are both artificially determined on the basis of previous research [15]. In the low temperature range, the effect of temperature on the removal rate is not obvious. With the temperature increasing, the removal rate is more susceptible to the temperature changes. Besides, the Gaussian heat flux is acquired by the temperature measuring experiment. So far, the transient temperature distribution can be solved for any given dwell time map. Further, the transient practical TIFs at every dwell point can be determined with the simulation. In detail, the explicit relationship between the temperature and the TIFs is given by,

Based on this, the multiple scanning test was first conducted to establish the time-variant TIF model. Figure 6(a) gives the solved relationship between the transient temperature and the velocity. With the increase of scanning velocity, the temperature presents the sharp decrease trend. The conventional time-variant TIF model is given in Fig. 6(b). The peak removal rate and full width at half maximum both rapidly increase as the equivalent dwell time goes up. This variation trend is consistent with the relative experimental result in our previous research [10], which proves that the simulation is reasonable and meaningful.

Then, the 1D sinusoidal dwell time map was introduced to simulate the practical figuring process or freeform generation. Its minimum value was 0.1 s, and the spatial wavelength was 40 mm and the amplitude was 0.5 s. The dwell points were assigned from 10 mm to 90 mm at the interval of 1 mm. The processing direction was from left to right. Based on this, the dwell time map was input to the thermal model, generating the transient temperature distribution. To verify the neighborhood effect, the TIF calibration at peaks and valleys (position A and B) in the multiple scanning test was investigated and compared. Figure 7 gives the simulation results of the transient temperature distribution. Detailly, the comparison of the equivalent dwell time map is shown in Fig. 7(b). According to Fig. 7(a) and (c), the difference of the neighborhood effect leads to the temperature deviation for the same point, even if the dwell time is the same. In the conventional time-variant TIF model, the transient temperatures of position A and B are 633.1 K and 457.5 K. However, in the practical processing, they are 629.4 K and 475.4 K, generating the deviations of $- 3.7$ K and 17.9 K respectively. Hence, there exists obvious thermal difference between the practical processing and the calibration procedure of the time-variant TIF model.

Further, according to Eq. (6) and Eq. (7), *a* and *FWHM* were respectively calculated for the 1D sinusoidal dwell time map, which together determined the practical VRR distribution. Simultaneously, the VRR distribution was also predicted with the conventional TIF model. The comparison of two VRR distributions in space domain is given in Fig. 8(b), which proves the reasonability of the above error analysis at the peaks and valleys. The two VRR distributions present the approximately same spatial wavelength but the obvious phase deviation. Compared with the practical value, the predicted VRR distribution with the conventional time-variant TIF model has larger peak-to-valley (PV) value and phase lag along the processing direction. According to the comparison in time domain shown in Fig. 8(c), The practical value is disorderly and fluctuates near the curve of the conventional TIF model, which embodies its limitation. The practical TIFs cannot be accurately predicted and analyzed with the conventional time-variant TIF model.

## 3. Reverse analysis method

#### 3.1 Principle of the reverse analysis

In APPP, the inaccurate prediction of TIF model may result in large processing error, which is also applicable for the other small tools in CCOS. Further, the removal generation cannot be accurately simulated with the experimental calibrated TIF model, leading to the big difference between the theoretical and practical removal distributions. From another aspect, the detailed information about the practical TIF changes is hidden in the residual processing error or practical removal distribution. In this paper, the proposed reverse analysis focuses on how to derive the practical TIF changes at every controlled dwell point from the practical removal, and can be used to check the correctness of the established TIF model.

Take the conformal polishing as an example, as shown in Fig. 9. As a rule, with the even dwell time map, the uniform removal is expected to achieve because the TIFs theoretically keep unchanged. However, affected by some influence factor, the removal rate of the TIF is gradually increased and then decreased, leaving the spherical processing error. With the help of the reverse analysis, the practical TIFs distribution can be derived to compare with the known TIF model. Note that, the long-time TIF stability is not discussed in this paper.

#### 3.2 Characterization of TIF changes

Generally, it is common that the peak removal rate and full width at half maximum of TIFs simultaneously change in APPP. In order to fully describe this, their variation trends should be respectively established. In fact, the VRR is the most essential and comprehensive parameter to characterize the TIF changes, which eclipses the other two parameters. Moreover, this characterization method can also be used to describe the TIF changes of other small tools, including non-Gaussian TIF.

To better understand its superiority, a series of simulations were primarily designed and performed, as shown in Fig. 10. It was assumed that a sinusoidal dwell time map $t({x,y} )$ was acquired with the deconvolution algorithm, and expressed as,

*A*is the amplitude, and

*T*is the spatial wavelength. This way, the nominal removal distribution $E({x,y} )$ was determined with a nominal Gaussian TIF${R_{nominal}}({x,y} )$, which was hypothetically obtained with the calibration experiment. Then, the changed TIF ${R_{changed}}({x,y} )$ was generated by changing the

*a*and

*FWHM*, and the errors were both controlled from$- 10\%$ to $10\%$. The processing error $\Delta E({x,y} )$ was calculated by,

To evaluate this, two key parameters are first defined. The value of *p*_{1} is the ratio of the RMS of $\Delta E({x,y} )$ to the RMS of $E({x,y} )$,which represents the processing error. The value of *p*_{2} is the ratio of the changed VRR to the nominal VRR. Figure 11 gives the simulation results versus the TIF changes. It is obvious that the independent changes of *a* and *FWHM* have the strong impact on the values of *p*_{1} and *p*_{2}. Moreover, there exists a potential connection between the two distributions. The relationship between *p*_{1} and *p*_{2} is given in Fig. 11(c). When the value of *p*_{2} is equal to 1, the VRR keeps unchanged, and the value of *p*_{1} reaches the minimum. Whether the value of *p*_{2} is larger or smaller than 1, the value of *p*_{1} increases rapidly with the change of the *p*_{2}, which suggests that the VRR changes are the major influence factor affecting the processing error. From this, a more interesting conclusion can be drawn. In practice, to achieve the high precision of removal generation, necessary attention should be paid to the VRR changes instead of how the TIF changed. Thus, in this paper, the proposed reverse analysis method is aimed at solving the practical VRR distribution of the TIFs.

#### 3.3 Reverse analysis method based on controlling volumetric removal

In previous research, Su et al. [12] pointed out that the specific removal distribution can be generated by controlling the local volumetric removal (LVR). On the basis of the convolution theorem, the removal generation can be reexplained from another perspective. As long as the LVR is determined, the final removal distributions with different TIFs are all approximately equal. Note that, in this procedure, the TIFs can change randomly in any form, if the change occurs within a reasonable range. In Fig. 11(c), it can be seen that there are multiple groups of point sets along the horizontal axis. Each group of points present the approximately equal VRR and processing error. Regarding to the same dwell time map, this can be essentially explained by the approximately unchanged LVR distribution.

Therefore, the concept of controlling volumetric removal is further extended and applied to the reverse analysis method as shown in Fig. 12. The practical removal generation can be divided into the unchanged procedure with the nominal TIF and the changed procedure with the practical changed TIF. Based on controlling volumetric removal, the same practical LVR distribution can link the two procedures together.

The more detailed process is given as follows, which starts with the given targeting removal distribution$E({x,y} )$. Then, a nominal TIF ${R_{nominal}}({x,y} )$ is acquired with the calibration experiment, and the corresponding nominal dwell time map ${t_{nominal}}({x,y} )$ can be determined using any conventional deconvolution method.

When the nominal dwell time map ${t_{nominal}}({x,y} )$ is executed, the TIFs constantly changes because of some specific influence factor, leading to the processing error. The experimental substrate is measured before and after the figuring processing. Based on this, the practical removal distribution ${E_{practical}}({x,y} )$ can be acquired by subtracting the two surfaces. Note that, to achieve better performance, the two surfaces should be primarily matched to eliminate the misalignment with the help of some algorithms or fiducial information [16]. After this, dwell time map is calculated again with the nominal TIF. From another perspective, ${t_{practical}}({x,y} )$ is essentially the equivalent dwell time map when the TIF changes are transformed into the dwell time map. This way, the practical LVR distribution can be determined as, where ${S_{nominal}}$ is the nominal VRR, and ${V_{practical}}({x,y} )$ is the practical LVR distribution. To control the same LVR distribution, the practical VRR distribution ${S_{practical}}({x,y} )$ can be acquired by,#### 3.4 Simulation demonstration

To verify the proposed method, the whole reverse analysis procedure is demonstrated with the simulation. Without loss of generality, the nominal TIF was defined as the Gaussian shape. The practical removal distribution with experiment was simulated by changing the Gaussian parameters (*a* and *FWHM*) of TIFs. It was assumed that these two parameters both change simultaneously with the same error. Considering the diversity, the practical changes of two parameters was generated with the sinusoidal function. If the change percentage was *p*, the two parameters were determined to be,

*A*is the amplitude and${T_f}$ is the spatial wavelength of removal distribution. $\lambda$ is the minimum material removal. In this part, the iteration method was adopted for the calculation of two dwell time maps. Compared with matrix method, it can achieve better smoothness and lower computation cost. The critical parameters are listed in Table 1.

Figure 13 gives the critical simulation results. Because of the TIF changes within error ${\pm} 5\%$, as shown in Fig. 13(d), there exists an obvious processing error between the targeting and practical removal distributions, as shown in Fig. 13(a), (b) and (c). And it is closely connected with the changed VRR distribution. With the practical removal distribution, the VRR distribution is solved with the proposed reverse analysis method, as shown in Fig. 13(e). Compared with the simulated practical VRR distribution in Fig. 13(d), the solved VRR distribution shows high fidelity. The analysis error is the difference of the two VRR distributions, and shows the strong connection with the targeting removal distribution. And this is an intrinsic problem, and will be discussed in the following application part. To characterize the analysis error, the ratio *q* of the RMS of Fig. 13(f) to the RMS of Fig. 13(d) is adopted. In this simulation, it’s only 4.8% and can meet the accuracy demand for practice.

Note that, in this simulation, there is no connection between the removal distribution and TIF changes. In another word, the proposed reverse analysis method has great potentials for deriving TIF changes, including the space-variant or time-variant TIF changes for any other CCOS technologies.

## 4. Analysis of the TIF changes in APPP based on the reverse analysis method

For the 1D sinusoidal dwell time map in Section 2.2, the forward convolution was further conducted with the calculated *a* and *FWHM*, as shown in Fig. 14. The obtained simulated trench was hypothetically considered as the practical experimental results and the removal distribution was extracted to conduct the reverse analysis. In this analysis, the nominal TIF is determined when the thermal balance is achieved.

In order to avoid the edge effect of deconvolution algorithm, the calculation result focuses on the range from 15 mm to 75 mm, as shown in Fig. 15. Compared with Fig. 8, the calculation results with the proposed reverse analysis method are more accurate than the conventional TIF model. Further, the ratio *q* in Section 3.4 is still adopted to characterize the analysis error. For the conventional TIF model, the predicted results of every point are not accurate, and more or less deviate from the practical value because of the phase difference. By contrast, the analysis error only occurs nearby the valleys for the proposed reverse analysis method. This is because of the selected nominal TIF has larger full width at half maximum. For the peaks of the dwell time map, there exists smaller difference between the practical and nominal TIF. Correspondingly, the same LVR is controlled more effectively. Therefore, the analysis error is closely connected with the selection of the nominal TIF. According to the ratio *q*, the analysis error is reduced from 23.7% to 4.6%, which indicates the better analysis performance of the proposed reverse analysis method. This is also connected with the dwell time map or removal. The essence of the reverse analysis method only focuses on controlling the same LVR, instead of removal accuracy in lateral. For the large spatial wavelength, the TIF is relatively small. This way, it’s easier to control the same LVR accurately.

## 5. Experimental verification

To verify the abovementioned analysis, the experiment was conducted in our self-developed APPP system. In practice, we have no idea about the true practical VRR distribution in APPP. However, the conventional TIF model can be established as a reference, which is compared with the calculation results with the proposed reverse analysis method. Therefore, with the same processing parameters, as shown in Table 2, the multiple scanning test was first conducted with different velocities. And then, to represent the practical processing, a velocity-variant trench was generated with the a complex 1D uneven dwell time map. The illustration of APPP system and experimental results is given in Fig. 16.

With the multiple scanning test, the conventional time-variant TIF model was established by extracting the depth and full width at half maximum of every trench. Further, the discrete points of calculated VRR were fitted with the exponential function to establish the conventional time-variant TIF model, as shown in Fig. 17. As for the velocity-variant trench, the practical removal distribution was primarily acquired by subtracting the measured surface from the initial surface, as shown in Fig. 18. Then, the reverse analysis was conducted to derive the practical VRR distribution.

In order to demonstrate the neighborhood effect, the removal distribution was first predicted with the experimentally established time-variant TIF model by the multiple scanning test, and compared with the experimental practical removal distribution. The removal comparison results are shown in Fig. 19. The obvious phase deviation occurs along the processing direction, which embodies the neighborhood effect. According to the analysis in Section 2.2, the removal deviation is essentially accounted by the VRR prediction error of the conventional time-variant TIF model.

Further, the two VRR distributions acquired with the two methods are compared together, which is given in Fig. 20. For better comparison, Fig. 20(a) gives the simulation results from Fig. 8 and Fig. 15, which hides the unknown practical value. For the peaks of the dwell time map, high precision can be achieved to obtain the true VRR, which is also embodied by the simulation results. Regarding to the large analysis error nearby the valleys, the solved VRR with the reverse analysis method is little smaller than the practical value (shown in Fig. 15), and even smaller than the VRR acquired with the conventional TIF model.

With the experimental results, the two VRR distributions with the reverse analysis method and the conventional TIF model are given in Fig. 20(b) and (c), it’s obviously that there exists the phase deviation between the two, which is in accord with the simulation results as shown in Fig. 20(a). It is worth noting that the absolute values are meaningless for comparison, because the this is a reference-based and self-check procedure. The experimental results show the reliability of the proposed simulation and reverse analysis method. Besides, it is believed that this method also has a wide application prospect of deriving the TIF changes in other CCOS technologies.

## 6. Conclusions

In this paper, an innovative reverse analysis method was proposed and applied to the investigation of the TIF neighborhood effect in APPP. With the simulation and experiment results, the following main conclusions can be drawn:

- (1) Regarding to the neighborhood effect, the neighboring dwell points are thermally coupled together, making the practical TIF changes much more complex. The conventional time-variant TIF model is limited in the application of predicting the VRR distribution in practical processing. Compared with the true practical value, the predicted VRR distribution has larger PV value and phase lag along the processing direction.
- (2) To accurately analyze the practical TIF changes of APPP, an innovative reverse analysis method was proposed to derive the VRR changes from the practical removal, which is implemented by controlling the same LVR distribution. Its capability of deriving TIF changes was demonstrated by the simulation, and the analysis error is achieved to be 4.8%.
- (3) Take the 1D sinusoidal dwell time map as an example, the practical VRR distribution in APPP can be effectively derived by conducting the reverse analysis. Compared with the conventional TIF model, the analysis error of the practical VRR changes can be dramatically reduced from 23.7% to 4.6%.
- (4) With the velocity-variant trench acquired by the experiment, the proposed reverse analysis method was conducted to derive the practical VRR changes, and compared with the conventional time-variant TIF model. The comparison results between the simulation and experiment show the reliability of the proposed reverse analysis method. For general purpose, the proposed method can be adopted for any CCOS technologies to derive the TIF changes.

## Funding

National Natural Science Foundation of China (51905130, 52105488); Natural Science Foundation of Heilongjiang Province (LH2020E039).

## Acknowledgments

The authors would like to sincerely thank the reviewers for their valuable comments on this work.

## 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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