## Abstract

Continuous phase plate (CPP), as a key diffractive optical element, is difficult to manufacture owing to its random and small features. In this paper, a novel frequency division combined machining (FDCM) method was proposed to improve polishing efficiency of CPP by optimizing the tool influence functions (TIFs) over targeted frequency bands. In addition, the convergence rate of power spectral density (CR-PSD) was proposed to evaluate the correct ability of TIF in different frequency bands, and to determine the division frequency for the combined processing. Through simulation verification, the combined processing with optimized TIFs by FDCM enabled high precision in less total time than that with single TIF processing. The experimental results verified that the method could imprint a 300 × 300 mm CPP with residual root-mean-square 24.7 nm after approximately 6-h bonnet polishing. Comparing the focal spots of designed and fabricated CPPs, the deviation of their energy concentration within 500 microns is only 0.22%. Hence, bonnet polishing using the FDCM is a new technical option for the production of large-aperture CPPs. Furthermore, the FDCM method shows a significant increase in efficiency, and it could be a generic method for CPP processing through other technologies, including magnetorheological and ion beam finishing.

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

## 1. Introduction

High-power laser facilities utilized for inertial confinement fusion research, such as the National Ignition Facility (NIF, USA) at Lawrence Livermore National Laboratory (LLNL) [1], the Laser Megajoule (LMJ, France) [2], SG-III (China) [3], ISKRA-6 (Russia) [4], and FIREX-II (Japan) [5] require a precise control of the laser beam illumination at the target plane. The large-aperture continuous phase plate (CPP), as a beam shaping optical element, can provide a far-field focal spot distribution with a flat top, steep edge, no sidelobe, which meets the requirements of beam-target coupling. To precisely manipulate the laser focal spot size and the energy intensity uniformity, various CPPs were designed and used in the high-power laser facilities. Neauport et al. [2] described the design and control of a first 383 mm × 398 mm continuous phase-plate prototype. Zhang et al. [6] experimentally studied the multi-frequency modulation (Multi-FM) smoothing by spectral dispersion (SSD), CPP, and polarization smoothing (PS) on the SG-III laser facility, and the far field distribution can be accurately adjusted, controlled, and repeated. Wegner et al. [7] reported that large-aperture (430 mm × 430 mm) CPPs were used to improve the on-target irradiation uniformity which reduced the hydrodynamic instabilities developed during inertial confinement fusion (ICF) implosions. It can be seen that when the CPPs were extensively used in laser fusion facilities for beam smoothing, a good optical performance could be achieved.

The CPP has a complex surface figure with random and small feature structures, whose least spatial period is approximately several millimeters [8,9]. Owing to the complexity of the surface distribution, it is difficult to process effectively using traditional polishing methods. The CPP elements are processed through numerical control (NC) polishing, mainly by using small sized TIFs to quantitatively remove the material. Under the control of process software, the processing figure which is basically consistent with the designed figure is finally completed through deterministic polishing. At present, scholars mainly adopt magnetorheological finishing (MRF) and ion beam figuring (IBF) to obtain high-precision CPP elements. Menapace et al. [8] developed MRF tools and procedures to manufacture large-aperture (430 mm × 430 mm) CPPs whose topography possesses spatial periods of as low as 4 mm and surface peak-to-valleys of as high as 8.6 µm. In addition, they [10] demonstrated that the manufacturing efficiency was directly tied to the physical dimensions of the removal function. The tool influence function (TIF) size should be at least two-thirds smaller than the minimum spatial period to be imprinted onto the surface. Tricard et al. [11] presented MRF imprinting technology for manufacturing large-aperture CPPs, which can generate topographical information at spatial scale-lengths approaching 1 mm and a surface precision of within 30 nm. Xu et al. [12,13] proposed a multi-pass IBF approach with different beam diameters based on the frequency filtering method to improve the machining precision and efficiency of CPPs during IBF.

As an obvious truism, the polishing time is mostly dependent on the material removal rate of the polishing tools. However, owing to the limitation of material removal rate of MRF and IBF, the high-precision manufacturing of CPPs in batches remains difficult to achieve. Bonnet polishing (BP) [14] and plasma jet machining (PJM) [15] are two processing methods with high efficiency. Bonnet polishing is an efficient polishing technology, the principle of which is to adjust the pressure, speed, and tool offset of the semi-flexible bonnet to contact the elements. The PJM process is driven by a reactive plasma jet generated at atmospheric pressure, which interacts with the workpiece surface. According to current reports [16,17], the PJM shows the advantage of a high efficiency, but it may introduce a significant amount of heat into the substrate. Meanwhile, the surface texture becomes deteriorated with the surface roughness up to 60 nm in Ra, which cannot satisfy the optical application. Su et al. [17,18] proposed a combined processing chain consisting of PJM and bonnet polishing, in which PJM plays a role in form generation, and bonnet polishing is responsible for rapid surface smoothing, and their combined processing can be used to fabricate the CPP. BP not only has the characteristics of flexibility, but also can obtain stable TIFs with a large adjustable range of size [19,20]. Based on the surface error, the dwell time of a TIF at each position of a surface can be obtained, and the deterministic surface shape correction processing of the optical element can then be realized. Moreover, under the same size of TIF, the removal rate of bonnet polishing is much higher than that of other polishing technologies, such as MRF and IBF [21]. Therefore, BP can improve the processing efficiency of CPP theoretically. In addition, for the surface with a complex periodic structure, the appropriate sized TIF is needed to obtain the high-efficiency and high-deterministic processing. Otherwise, it can cause a waste of processing time [22]. When a large sized removal function is used to correct small form errors, the error cannot be effectively converged. With this in view, a frequency division combined machining (FDCM) method combined with BP was proposed to improve the manufacturing efficiency of CPP elements.

## 2. Tool influence function of bonnet polishing

The theoretical TIFs of different sizes can be obtained according to the TIF model of bonnet polishing presented in our previous studies [19,20]. First, the ANSYS analysis model for the contact between the bonnet and the workpiece is established, and the contact stress distribution between the bonnet and plane workpiece under different tool offsets is calculated. The Hertz formula is used to fit the stress distribution under each set of parameters to determine the fitting coefficients (*A*\*B*\*C*) under different simulation parameters. The fitting coefficients under different tool offset *H* are shown in Table 1.

According to Table 1, the linear expressions of the fitting coefficients (*A\B\C*) on the tool offset *H* can be obtained, and the mathematical expressions of stress *P* can then be obtained:

The bonnet polishing kinematic model is shown in Fig. 1. Here, AO_{1} is the bonnet rotation axis, n is the rotation speed, O_{1} is the intersection of the rotation axis and the element, *ρ* is the angle between the rotation axis and the surface of the element, and O is the center of the contact area, *P* (*P _{x}*,

*P*) is the point in the contact area, and r is the distance between m and O

_{y}_{1}. In addition,

*R*is the bonnet radius,

*H*is the tool offset, and

*V*is the combined velocity point at

_{p}*P*. According to the bonnet polishing kinematics model, the velocity distribution between the bonnet and the workpiece in the contact area can be obtained, as follows:

Where,

The removal distribution *R* can then be obtained based on the pressure distribution *P* and the velocity distribution *V*. Figure 2(a) shows the TIFs of a bonnet contacting with different tool offsets based on a simulation, the conditions are listed as follows: 0.4/0.5/0.6/0.7 mm tool offsets, 1000 rpm rotation speed, 4 s polishing time, and Preston coefficient *K* of 4 × 10^{−14} m^{2} · N^{−1}, reference from a previous study [19]. The actual TIFs of different sizes can be obtained through a spot picking experiment using a self-developed bonnet polishing machine. To obtain small sized TIFs suitable for CPP element processing, an R40 mm bonnet tool was selected during the experiment. Figure 2(b) shows a series of TIFs on different tool offsets based on experiment results. The simulative and experimental conditions and the results are shown in Table 2.

From Table 2 and Fig. 3, it can be seen that the size and efficiency of the TIF are directly related to the tool offset and have an approximate linear relationship. For the theoretical TIFs, when the tool offset is 0.4 - 0.7 mm, the TIF has a full length at half maximum (FLHM) of 7.7 - 10 mm, a full width at half maximum (FWHM) of 6.2 - 8 mm and a volume removal rate of 0.165 - 0.498 mm^{3}/min. For the experimental TIFs, when the tool offset is 0.4 - 0.7 mm, the TIF has an FLHM of 7.1 - 10.5 mm, an FWHM of 5.4 - 8.5 mm, and a volume removal rate of 0.165 - 0.502 mm^{3}/min. Comparing between theoretical and experimental TIFs, the maximum deviation in size is 13%, and the maximum deviation in efficiency is 6%. Therefore, the theoretical TIFs are in good agreement with the experimental results. It can be seen that the bonnet polishing has high removal rate with a small TIF size, which lays the foundation for improving the processing efficiency of CPP elements based on bonnet polishing.

## 3. Simulation model of surface topography

The CPP processing workflow includes CPP data processing, simulated processing, and processing and testing, as shown in Fig. 4. CPP data processing is used to compare measured errors with the designed errors to obtain the combined errors. The simulated processing aims to calculate the dwell time, residual error, and CNC code through a convolution iteration of the combined error and TIF. Next, an NC machine with a smaller bonnet is used for processing. The CNC code controls the dwell time of bonnet on the workpiece and achieves an effective convergence of the surface error. After processing, a sub-aperture stitching method is used to measure the CPP, and a high-resolution and high-precision full-aperture surface error of a CPP is then obtained by stitching the algorithm. The combined error is also then obtained by comparing the measured error with the designed error, and the next processing is continued until the combined error is less than the CPP specifications.

The key technology in CPP processing is the construction of a dwell time algorithm. The accurate solution to the dwell time is based on the consistency of the solution model and the removal principle of the bonnet polishing. When the bonnet moves on the workpiece with the planned trajectory, the removal amount of each point on the workpiece is the superposition of the TIF when the TIF moves along the planned trajectory. The removal process is called moving superposition removal. According to the above removal principle, the TIF is integrated along the polishing trajectory in the contact area to calculate the removal distribution of the polishing tools along the polishing trajectory. Figure 5 shows a schematic diagram of a bonnet polishing.

When the tool polishes the surface continuously, the material removal of any point is the integration of the TIF within the dwell time. The material removal distribution can be expressed as the convolution of the dwell time and the TIF.

where the *R*(*x*, *y*) is the material removal amount per unit time, the *T*(*x*, *y*) is the dwell time of the polishing tool in any area of the workpiece, and ** is the symbol of a two-dimensional convolution.

After processing, the residual errors can be expressed as follows:

Where *H _{0}*(

*x*,

*y*) is the initial error distribution, and

*T*(

*x*,

*y*) is the dwell time distribution.

During the actual processing, the actual surface figure is measured using an interferometer. Ideally, *E*(*x*, *y*) is zero, but in practice, the peak-valley (PV) or root mean square (RMS) is allowed to be less than a certain preset value. Without considering the distortion of the TIF, the TIF is known when the process parameters are determined. Therefore, the surface error correction problem becomes the solution of the dwell time under a given TIF, which is the deconvolution of the surface error and TIF:

In actual processing, the commonly used algorithms for solving the dwell time include the least squares algorithm [23], matrix method [24], and convolution iteration method [25]. Additionally, the pulse iteration method [26] does not require complicated mathematical calculations and is commonly used in solving the dwell time. The principle of the pulse iterative method is to idealize the TIF into a pulse function, that is, to concentrate the material removal of the TIF to the center point so as to characterize its removal intensity (*RI*).

In this paper, an improved pulse iterative method was proposed to solve the dwell time, as shown in Fig. 6. The characteristics of the improved method were summarized as follows: (a) the convolution of the TIF and the dwell time is related to the polishing path, which can truly reflect the actual moving overlay removal process, and (b) the adjustment factor of the dwell time *β* is added to analyze the surface figure.

Two important parameters were involved in the simulation process, i.e., surface convergence coefficient *α* and dwell time adjustment factor *β*. Here, *α* determines the final precision of the iterative calculation (RMS_{E} < RMS_{0}·*α*), and *β* is used to actively modify the dwell time to obtain a new figure, making it convenient for researchers to analyze the surface figure. A CPP figure is simulated and analyzed by using the above model. In the simulation, the tool offset is 0.4 mm, the turning speed is 1000 rpm, *α* is 0.1, and *β* is set to 0.8 - 1.2. Figure 7 shows the initial surface figure and the simulated surface figures with different coefficients. Table 3 presents the processing precision and processing time with different coefficients. In Fig. 7, when *β *= 1, the RMS of the residual error is the smallest (25.6 nm), which indicates that the dwell time obtained at this time is optimal. When *β* < 1, the simulated time *T* decreases and simulated residual error increases. Comparing Fig. 7(b) and (c) with Fig. 7(a), we can see that the residual errors are similar to the initial error. This indicates that when the simulated time is less than the optimal time, the surface error correction is insufficient. When *β* > 1, the simulated time and residual error increase simultaneously. Comparing Fig. 7(e) and (f) with Fig. 7(a), we can see that the red feature in the residual error corresponds to the blue feature in the initial error. Accordingly, when the simulated time is longer than the optimal time, the surface error correction is excessive. According to the simulation results, the dwell time (i.e. polishing time) directly affects the processing precision. For the given TIF and surface error, the corresponding relationship between the processing time and processing precision can be obtained according to the above simulation model, which lays a theoretical foundation for the optimization analysis of the TIF.

## 4. Frequency division combined machining method

#### 4.1 Convergence rate of power spectral density and division frequency

According to previous study [22], the processing precision is directly related to the TIF size. Therefore, the TIF needs to be optimized according to the surface shape. According to the above simulation process, the TIFs of different sizes in Fig. 2 and the surface figure in Fig. 7(a) were selected to analyze the relationship between processing time and processing precision based on different TIFs. As shown in Fig. 8(a), the polishing precision shows a trend of first decreasing and then increasing with the polishing time. The inflection point is the ultimate precision that can be processed. The ultimate precision corresponding to the 0.4, 0.5, 0.6, 0.7 mm tool offset are 25, 28, 33, and 41 nm, respectively. In the high-power laser facility, a precision specification of CPP is the residual RMS of form error between the machined and designed CPPs, and the RMS is required to be less than 30 nm [11]. To balance the processing precision and efficiency, the TIF of the maximum size that can meet the precision requirements of the CPP elements is first selected. The TIF_{H-0.5} can obtain an RMS of 28 nm within the specifications. According to the simulation processing results of different TIFs, the relationship between the processing time, processing precision, and tool offset can be obtained, as shown in Fig. 8(b).

The processing precision and tool offset show an exponentially decreasing relationship:

The processing time and tool offset show a logarithmic increasing relationship:

With the size of TIF decreasing, the ultimate processing precision of elements can be improved, while the processing time increases significantly. According to Eqs. (8) and (9), the required TIF (or the offset) can be inferred from the requirement of the processing precision.

In the process of surface shape correction, when the TIF does not match with the surface shape error, the processing time will be wasted, which is related to the correct ability of TIF. Boerret et al. [27] indicated that error wavelengths of two- to three-times smaller than the spot-size can be removed by MRF. Liao et al. [28] shown that the TIF width of IBF at FWHM should be smaller than four-fifths that of the shortest spatial period to be imprinted. This paper proposed a new evaluation factor dealing with the processing precision and processing time, i.e., the convergence rate of the power spectral density (CR-PSD). The CR-PSD can evaluate the correct ability of TIF in different frequency bands, and its curve can used to determine cut-off frequency and cross frequency. The calculation process of CR-PSD is as follows:

where

*E*_{0}is the initial surface shape distribution,*E*_{1}is the residual surface shape distribution of the simulated processing to the ultimate precision, and*E*_{01}is the processing surface shape convergence distribution.Assume that,

$${E_{{01}}}(m) = \Delta x\sum\limits_{n = 0}^{N - 1} {z(n){e^{ - i2\pi mn/N}},} - N/2 \le m \le N/2,$$where*z*(*n*) is the amplitude of each point in the surface distribution,*N*is the total number of sampling points, and Δ*x*is the sampling interval.

From Eq. (13), the CR-PSD and its curve of each TIF can be obtained, as shown in Fig. 9.

This paper defines the cut-off frequency as the frequency at which the CR-PSD is zero. In Fig. 9, the cut-off frequency corresponding to the TIFs with 0.4/0.5/0.6/0.7 mm offsets is 0.111, 0.120, 0.130, and 0.141 mm^{−1}, respectively. The cross frequency is the frequency at the intersection of the CR-PSD curves of different TIFs. The cross frequencies are 0.081, 0.095, and 0.106 mm^{−1}, respectively. According to Fig. 9, compared with the cut-off frequency, the cross frequency defined as the division frequency (DF) is more optimized for the combined processing efficiency. Theoretically, each frequency band then selects a TIF with large CR-PSD for frequency division combined processing, and the overall processing efficiency is also the highest.

#### 4.2 Optimized convergence rate using the FDCM method

Theoretically, the more frequency bands that are divided, and the greater the efficiency is, although this is impossible to achieve practically. Normally, a two-section or three-section frequency is suitable. To ensure the processing precision and efficiency, the minimum TIF of the CPP processing in this paper is TIF_{offset 0.5}. On this basis, the division frequency of the two-section frequency processing is 0.088 mm^{−1}, and the TIFs are TIF_{offset 0.5} and TIF_{offset 0.7}, respectively. The cross-frequency of the three-section frequency processing is 0.081 and 0.095 mm^{−1}, and the TIFs are respectively TIF_{offset 0.5}, TIF_{offset 0.6}, and TIF_{offset 0.7}, as shown in Fig. 10. The specific steps of FDCM are listed as follows:

- (1) Determine the DF
*f*_{1}and TIFs*R*_{1}and*R*_{2}for processing. - (2) The initial surface error
*E*_{0}is low-pass filtered to obtain the low-frequency error*E*_{f1}to be processed. - (3) Use
*R*_{1}to simulate the processing of*E*_{f1}to the ultimate precision, and the shape error after simulation processing is*E*_{R}_{1}. - (4) Calculate the machining error
*E*_{m1}=*E*_{f1}-*E*_{R1}. - (5) Calculate the residual error
*E*_{r1}=*E*_{0}-*E*_{m1}. - (6) If there is only one DF, use
*R*_{2}to simulate the machining*E*_{r1}to the ultimate precision and meet the processing requirements. If there are multiple DFs, refer to steps (2)–(5) for the next frequency simulation processing until the processing is completed within the entire frequency domain.

According to the forementioned FDCM method, the two-section frequency processing and three-section frequency processing were carried out, the results of which are shown in Table 4 and Fig. 11. It is noted that, compared with the single TIF processing, the FDCM reduces the processing time from 429 to 278 h, and the efficiency is increased by 35.2%. This indicates that the FDCM significantly improves the efficiency. In the three-section frequency processing, the processing time of the second and third stages is too short, which is difficult to realize in practice. Therefore, in practice, the number of DFs needs to be determined according to the actual conditions. In this paper, the experimental verification of the CPP was planned to adopt the two-section frequency division combined processing.

## 5. Experiment on CPP manufacturing

A 330 mm × 330 mm × 3 mm CPP substrate was selected for the experiment, as shown in Fig. 12(a). The experiment was carried out using two TIFs, i.e., TIF_{offset 0.7} and TIF_{offset 0.5}. Table 5 presents the experimental conditions of the CPP processing. After the experiment, a 4D FizCam 2000 EP interferometer combined with a mobile platform was used to measure the CPP, as shown in Fig. 12(b). The elements were measured in nine areas (3 × 3), and the nine sub-aperture surface figures were then spliced to obtain a full-scale surface figure. The dimensions of the evaluated aperture of the CPP are 300 mm × 300 mm.

Figure 13 shows the designed and fabricated figures of CPP. In Fig. 13, the RMS of the designed figure is 237 nm, whereas the RMS of the fabricated figure is 230.6 nm, the deviation of which is only 2.7%. Figure 14 shows the simulated and actual residual errors of the CPP. In Fig. 14, the RMS of the simulated residual error is 25.6 nm, while the RMS of the actual residual error is 24.7 nm, with a deviation of only 3.9%. The experimental results show that the CPP manufactured using bonnet polishing not only meets the design specifications (RMS ≤ 30 nm), but also takes only 6 h for a 300 mm × 300 mm CPP element, which greatly improves the processing efficiency.

According to the surface shape error distributions of the designed and fabricated CPPs, the designed and fabricated characteristics of the focal spots are obtained through a far-field focal spot analysis of the CPP element. The focal spot analysis parameters are an input laser wavelength of 351 nm, 20-order super Gaussian, beam diameter of 300 mm, and focal length of 4 m. Figure 15 shows the focal spots of the designed and fabricated CPPs. In addition, Fig. 16 demonstrates the contours of the designed and fabricated focal spots in two directions (0° and 90°). It can be seen from Figs. 15 and 16 that the shapes of the focal spots are both elliptical, with a long-to-short axis ratio of 1.74. In addition, the size and energy concentration of the designed and fabricated focal spots are both extremely consistent. The deviation of their energy concentration within a 500-micron ellipse on the horizontal axis is only 0.22% (i.e., 99.28% designed, 99.50% fabricated). It can be seen that, based on the performance of the focal spot, the large-aperture CPP fabricated by the methods proposed in this paper has a high processing accuracy, and its focal spot is in good agreement with the design, which meets the application requirements of a high-power laser facility.

## 6. Conclusions

To improve the processing efficiency of CPP, a novel CPP manufacturing method, FDCM, using bonnet polishing was proposed. The detailed conclusions are as follows.

- 1. An improved impulse iteration algorithm was introduced to solve the dwell time for the surface topography simulation of CPP. Based on the algorithm, the relationship between the processing precision and time under different TIFs was simulated and analyzed.
- 2. A new evaluation factor of the correct ability of TIF in different frequency bands was proposed, i.e., CR-PSD, to determine the DF for combined processing. Theoretically, selecting the TIF with a large CR-PSD in each frequency band achieves the highest processing efficiency.
- 3. This paper proposed an FDCM method to improve polishing efficiency by optimizing the TIF over targeted frequency bands. Compared with the single TIF processing, the FDCM improves the efficiency by approximately 35%.
- 4. The experimental results verified that this optimized method could efficiently imprint a 300 mm × 300 mm CPP with an RMS of 24.7 nm after 6-h bonnet polishing. Comparing the far-field focal spots of the designed and fabricated CPPs, the deviation of their energy concentration within a 500-micron ellipse on the horizontal axis was only 0.22%.
- 5. The high-efficiency and high-precision processing of CPP by BP with FDCM method was realized, providing a new technical option for the production of large-aperture CPPs.

## Funding

National Natural Science Foundation of China (51905509); Youth Talent Fund of Laser Fusion Research Center, China Academy of Engineering Physics (RCSPD4-2020-3).

## Disclosures

The authors declare no conflicts of interest.

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