## Abstract

Large surface gradient and extensive mid-to-high spatial frequency in continuous phase plates (CPPs) with small structures make it difficult to achieve high-precision fabrication. An ion beam figuring (IBF) technology to fabricate CPPs with such characteristics is proposed in this paper. In order to imprint CPP microstructures with smaller spatial periods even down to 1mm in shorter time, we present a multi-pass IBF approach with different ion beam sizes based on the frequency filtering method. We discuss the selection principle and when to reduce ion beam sizes for different procedures to control dwell time and adequately exert the corrective capability in detail. This filtering method can obtains better surface quality in a faster way compared to the non filtering traditional IBF method. The experimental results verify this optimized method can effectively imprint complex microstructures with spatial period as small as 0.7 mm, surface peak-to-valleys (PV) smaller than 200nm and surface gradient as large as 1.8μm/cm to within 10 nm root-mean-square (RMS) of design specifications, which displays the advantages of our fabrication method.

© 2017 Optical Society of America

## 1. Introduction

Stringent requirements on the imaging quality of modern optical systems are put forward because of performance improvement. However, the wave-front errors both from fabrication and alignment process exist inevitably, thus significantly affect the performance of optical systems. At present, the correction technique by diffractive optical elements is an effective way to compensate the wave-front errors [1,2]. Continuous phase plate (CPP) with continuously varying topographical microstructures is a typical diffractive optical element, and it has widespread applications in modern optics field of beam shaping, compensation and modulation [3,4]. The typical CPPs with great manufacturing difficulties usually have the following characteristics:

- (1) Rich in mid-to-high frequencies within small aperture of tens of millimeters. The spatial periods of microstructures are usually from 5mm down to 0.5mm. This requires the ability to imprint surface topography of different frequencies for the polishing tools, especially spatial periods smaller than 3mm. Since the corrective capability is related to the dimension of polishing tools, this means that the polishing tool can be changed to different sizes conveniently for different frequencies.
- (2) Fast continuous surface changes and large surface gradient within a scale of several millimeters. The PV (peak-to-valleys) of microstructure height is smaller than 500nm while the surface gradient is usually larger than 2μm/cm. This means that the microstructure has highly steep surface morphology. For conventional contacting polishing methods, this requires excellent dynamic performance and high positioning precision for the polishing tool to adapt the drastic changes of surface topography.
- (3) High machining precision. The specific requirements for the dimension accuracy of microstructures are put forward, for example, the RMS (root-mean-square) and PV. In addition, the matching precision between the figured microstructure and the designed ones is usually better than 10nm RMS.

These characteristics especially the reduced dimension of microstructures put forward quite high requirements for the quick-response ability and corrective capability of the polishing tool, which makes the fabrication of this kind of CPPs become a difficult and popular issue.

Magnetorheological Finishing (MRF) shows obvious advantages in the manufacture of large-aperture CPPs, in which the spatial periods of microstructures are usually larger than 4mm, and the PV of structure height is as large as several microns [5,6]. It is the relatively large size of microstructures that reduces the manufacturing difficulty. However, it is still difficult for MRF to manufacture microstructures whose spatial periods smaller than 4mm and surface gradient larger than 2μm/cm due to the sizes limitation of polishing tool. Besides, the stability of MRF removal function is determined by the precisely control of distance between polishing tool and optical surface. This is very unfavorable for the manufacture of highly steep microstructures whose surface PV is less than 500nm. Xu *et al*. develops the ion beam moving etching technology to fabricate diffractive optical elements, but masks in *x-, y-*direction are both needed. Obviously, this is not suitable for imprinting microstructures with arbitrary surface topography changes [7].

Ion beam figuring (IBF) is a high-deterministic technology mainly used for the machining of nano-precision surface accuracy and super-smooth surface quality [8,9]. The removal function with rotationally symmetric Gaussian shape shows good corrective capability for different frequencies. The beam sizes of IBF can be changed easily and even down to 1mm [10]. The diminishing of beam diameter further improves the ability to imprint microstructures with different spatial periods, and the problem that polishing tool cannot adapt to arbitrary surface changes during conventional polishing process can also be solved. In addition, the non-contact processing reduces the requirement for positioning precision in Z-direction. These characteristics of different sizes and good corrective ability make IBF technology a new possible solution for the nano-precision fabrication of CPPs.

In this paper, we try to verify the manufacturing feasibility of this kind of CPPs by IBF technology, and we mainly focus on three key issues during IBF process. How to select appropriate ion beams to imprint microstructures with smaller spatial periods even down to 1mm? How to improve machining precision of microstructures to make it approach the designed surface topography at the greatest extent? How to balance the machining precision and machining efficiency as the volume removal rate decreases when smaller ion beam adopted? In order to answer these questions, we present a multi-pass IBF approach with different beam diameters based on the low-pass frequency filtering method. We discuss the selection principle and replacement time of different ion beams for different procedures to control machining time and adequately exert the corrective capability in detail. Finally, the fabrication experiment of a small-aperture CPP is carried out and the matching precision better than 10nm RMS is obtained.

## 2. Figuring principle

The ion beam figuring principle of CPPs is similar to that of MRF [5,6]. Firstly, surface error contains material removal information is obtained by adding the height-inverted topographical surface of the designed CPP to the measurement data of the figured CPP surface. Then, the material removal is accomplished by different dwell times of the removal function. This virtual surface error converges towards flatness by the IBF solving algorithm and then surface possessing the designed CPP topography is obtained.

Analysis of our previous exploratory experiments shows that problems exist if we directly adopt the conventional IBF process used for plane or curved surfaces. The reasons for the low-precision and low-efficiency are mainly due to the CPP surface topography and the adaptability of IBF removal function. The slopes and pits in CPP surface lead to the inadaptability within error ranges of higher frequency. This means that even minor machining errors will be accumulated, amplified and directly affect the entire machining process. The machining precision is greatly affected by the size of removal function. Larger errors will be introduced during the figuring of slopes and pits by a large ion beam. Machining efficiency is another important factor to be considered due to the low material removal rate during IBF process. We will give detailed analysis on these issues in the following chapters.

## 3. Analysis of the influence factors

#### 3.1 Corrective capability

During IBF process, surface errors of different frequencies that can be corrected depend strongly on the ion beam diameter, since the corrective capability can be improved by the diminishing of beam diameter to widen the frequency scale of the IBF removal function. The ion beam diameter can be easily changed from tens of millimeters to several millimeters. The shielding diaphragm method is used in our experiments to obtain a small ion beam, as shown in Fig. 1 [10]. Liao presents the cut-off frequency to quantitatively evaluate the corrective capability of the IBF removal function [11].

Therefore, the required ion beam size to imprint microstructures with certain spatial period is:

Where *λ* = 1/*f* is the spatial period of microstructures.

Equation (2) quantitatively analyses the relationship between the beam diameter and the spatial period that can be effectively imprinted. The figuring of CPP microstructures of different frequencies can be realized with the diminishing of beam diameter. In other words, the smaller the beam diameter, the smaller spatial period of microstructures can be imprinted. It is obvious from Eq. (2) that IBF removal function has much better corrective capability than MRF, since the MRF removal function width at FWHH should be smaller than two-thirds that of the shortest spatial period to be imprinted, as shown in Fig. 2 [5].

#### 3.2 Surface gradient and machining precision

Surface gradient of CPPs can modulate the transmission direction of light rays and thus affect the far-field distribution, which is an important factor to be considered in the design of CPPs [12]. It also can affect the adaptability of ion beam to surface topography changes. The irregular 3D-microstructures of CPPs are the superposition of different spatial periods, which has adverse effects on the high-precision fabrication, especially ones with large surface gradient. Therefore, it is necessary to analyze the influence of surface gradient on the machining precision. Any continuous surface can be seen as the superposition of sine waves with different frequencies. The greater the amplitude is, the more obvious the dominant role of corresponding frequency. The corrective capability of IBF removal function is the same in all directions because of the Gaussian shape. For simplicity, we just discuss the correcting effect on one-dimensional case. The surface error to be corrected is *E*(*x*) = *A _{λ}*sin(2π

*x*/

*λ*) and the expected material removal is:

The actual material removal is always more than the expected material removal during figuring process, so the actual residual error is [13]:

Obviously, the residual error is related to the amplitude, the spatial period and the beam diameter. Since the surface gradient is *grad*(*E*) = (2π/*λ*)*A _{λ}cos*(2π

*x/λ*), the maximum gradient is

*G*=

_{max}*2πA*. Therefore, the relationship between residual errors and surface gradient and beam diameter can be expressed as:

_{λ}/λEquation (5) displays how the surface gradient influences residual errors. The greater the surface gradient is, the larger the residual error. A small beam is required to reduce the adverse effects of surface gradient on residual error. Figure 3 is the simulation results of relationship between surface gradient and residual error at different spatial wavelength and ion beam diameter. It shows that with the decrease of spatial wavelength, the larger the surface gradient is, the smaller beam diameter is needed.

#### 3.3 Dwell time and machining efficiency

Although the diminishing of beam diameter enhances the corrective capability to higher spatial frequency and larger surface gradient, the volume removal rate of ion beam decreases rapidly (as shown in Table 1), and thus the dwell time increases accordingly. As the dwell time is linear to the volume removal rate, it is obvious from Table 1 that the dwell time required for beam *d _{6σ}* = 4.1mm and

*d*= 3.1mm are 7.2 and 8.3 times that of beam

_{6σ}*d*= 8.2mm under the same removal depth. This indicates that the ion beam size has great influence on the machining efficiency and we should make a rational choice of the beam size to balance the machining precision as well as the efficiency. Besides, too long figuring time will affect the stability of ion beam and thus has adversely affects on the machining precision.

_{6σ}## 4. Process optimization and figuring experiments

#### 4.1 Multi-pass IBF process with different beam diameters

Surface errors tend to be high-spatial frequency when figured surface closes to the designed CPP surface gradually. However, the removal function dimensions essentially limit the sampling bandwidth of the surface topography due to Nyquist critical sampling issues [5]. Trying to imprint shorter spatial periods (higher frequency) than the smallest practical removal function dimensions (cut-off frequency) results in longer polishing time and greater material removal since less and less of the removal function is effectively used for imprinting the surface topography. Therefore, the balance of machining precision and efficiency lie in the reasonable selection of beam diameters. In order to imprint microstructures of more spatial periods even down to 1mm, it is necessary to determine which sizes of ion beam are selected in IBF process. The selection principle of ion beam is based on the spatial frequency distribution of the designed CPP surface. Take the designed CPP surface in Fig. 5(a) as example, the spatial frequency distribution is firstly analyzed through the power spectral density (PSD) analysis, as shown in Fig. 4. We consider the spatial frequency at the turning point as the dominant frequency (*f* = 0.2mm^{−1}). According to the cut-off frequency analysis, ion beam of *d _{6σ}*<10mm should be selected. Considering the corrective capability and volume removal rate, ion beam of

*d*= 8.2mm is selected for the first figuring stage to imprint microstructure of spatial periods larger than 5mm. After that, a Gaussian high-pass filter with 5mm bandwidth is adopted to filter the designed CPP surface and then the second dominant frequency of CPP surface is analyzed by the PSD analysis. Figure 4 shows that the turning point is at

_{6σ}*f*= 0.4mm

^{−1}. As results, ion beam of

*d*= 4.1mm is selected for the second figuring stage to imprint spatial periods larger than 2.5mm. Similarly, the third dominant frequency is

_{6σ}*f*= 0.65mm

^{−1}and ion beam of

*d*= 3.1mm is selected for the third figuring stage to imprint spatial periods larger than 1.5mm.

_{6σ}For a given removal function, the machining efficiency is different for different spatial frequencies. The figuring convergence rate is rapid for low-frequency error while it can be neglected near the cut-off frequency. In these instances, less and less of the removal function is effectively used for imprinting the topography. This leads to longer polishing times and greater material removal due to “collateral” polishing. This polishing spoils the imprinting efficiency and results in an increase in uniform material removal to attain the desired topography. Therefore, it becomes a key issue when to change the ion beam. Liao proposed a method for determining the replacement time of different ion beams, which can be a good reference here [11]. Defining the surface error ratio to evaluate the degree of figuring difficulty, it can be given as:

Where *RMS _{E}* is the RMS of the entire spatial frequency error and

*RMS*is the RMS of the spatial errors whose frequency higher than the cut-off frequency.

_{HE}According to analysis of the simulation and experimental results, we give the evaluation criterion of when to reduce the beam diameter as follows:

A larger *k* value means easier to reach a higher surface accuracy, but it takes more dwell time as cost. The selected value should be relatively large in the rough machining stage while small in the finish machining stage, so that it can improve the total machining convergence ratio in a short time. When the value of *k* is down to 1.5 or the change ratio of *k* is smaller than 5% after several reiterations, the ion beam should be changed to a smaller one. The replacement time of ion beams according to the *k* value, the arrangements of dwell time and iterative numbers in CPP figuring process are shown in Tables 2–4. Combining the corrective capability of IBF removal function and analysis of surface error, we can reasonably select the suitable ion beam and plan the figuring process, which control dwell time and imprint smaller spatial periods at the same time.

#### 4.2 Frequency filtering method

The frequency filtering method is the further optimization of the multi-pass IBF process, and mainly aims to obtain higher machining precision and efficiency. The surface morphology changes and surface gradient of CPP microstructures with different spatial periods are quite different. And this difference leads to completely different error patterns when a same removal function is used. The problems of the machining precision and efficiency can’t be thoroughly solved just by the multi-pass IBF process.

To supply a better solution, the low-pass frequency filtering method is presented. It is possible to obtain higher precision by iterative machining of the filtered surface, since the actual machining errors is smaller due to the lack of mid-to-high spatial frequency errors. Suppose the matching error between the filtered surface and the desired surface is *E*_{1}, and the machining error of the filtered surface is *E*_{2}, thus the total error of the filtering method is *E*_{1} + *E*_{2}. Suppose the machining error from the direct process is *E*_{0}. In the process of CPPs with sharp morphology changes, machining error *E*_{2} will be much less than *E*_{0} since surface gradient decreased by the filtering method. And the matching error *E*_{1} also can be controlled small. Consequently, it is likely to achieve the effect of *E*_{1} + *E*_{2}< *E*_{0} in the actual figuring process.

Based on this idea, a Gaussian low-pass filter is selected to filter the mid-to-high spatial frequencies of the desired CPP surface, as shown in Fig. 5(a). And Fig. 5(b) is the low frequencies surface to be figured after a filter with 5mm bandwidth. Compare Fig. 5(a) and Fig. 5(b), we can see that the mid-to-high spatial frequency have been filtered out, and a relatively smooth surface only with low-frequency is obtained. The low frequencies surface is figured virtually in the IBF simulation software. The high frequencies residual are shown in Fig. 5(c), and the error value is 124.9nm PV, 11.3nm RMS. The residual error output of the IBF simulation process of Fig. 5(b) is 24.9nm PV, 2.9nm RMS, when a beam of *d*_{6}* _{σ}* = 8.2mm is used, as shown in Fig. 5(d). The total expected residual error is 143.2nm PV, 13.8nm RMS, which is the superposition of Figs. 5(c) and 5(d), as shown in Fig. 5(e). However, the total residual error after the direct process by the same ion beam is 155.0nm PV, 16.6nm RMS, as shown in Fig. 5(f). The simulation results display that, the filtering method can filter out the mid-to-high spatial frequency errors, decrease the surface gradient and reduce the machining difficulty, thus smaller PV and RMS value can be obtained.

Another advantage of the frequency filtering method is to improve the machining efficiency, especially surfaces rich in high-spatial frequency errors. This is because the dwell time algorithm is based on the principle of obtaining optimal surface, without taking into account the impact of machining efficiency. Actually, surface errors of all frequency ranges will affect the calculation results of figuring time. In the case of frequency smaller than the cut-off frequency, much more extral material will be removed and much more figuring time will be spend without obvious precision improvement. Taking surface in Fig. 6(a) as an example, the initial surface is filtered by a Gaussian low-pass filter with 2.5mm bandwidth, and then machined by a beam of *d*_{6}* _{σ}* = 4.1mm. The dwell time density functions calculated from surface errors without and with a low-pass filter are shown in Figs. 6(b) and 6(c), respectively. Although residual errors of the two methods are almost the same (about 100nm PV, 5nm RMS), the total figuring time is decreased from 640 min to 450 min, 30% saved, and the peak dwell time is reduced from 0.35s to 0.21s, 40% reduced with the low-pass filter.

The multi-pass IBF process combined with the frequency filtering method incorporates different removal functions that maximize the material removal over the topographical frequencies being imprinted as well as the corrective capability of ion beams. The detailed IBF machining process is shown in Fig. 7. Larger removal functions are used early to figure the surface profile with low frequency. Smaller removal functions are used to perform final correction of topography with higher frequency and larger surface gradient. Higher precision surface can be obtained as long as the filtering frequency is suitable selected. This method makes full use of the high removal efficiency of the large removal function and the high corrective capability of the small removal function. Consequently, this approach can improve the machining efficiency and achieve the fast convergence of machining precision.

#### 4.3 Figuring experiments

Figure 8 shows the CPP surface morphology to be processed, whose PV is about 200nm and the maximum surface gradient is 3.1μm/cm appearing only on the surface edge. The CPP surface can be divided into two parts according to its location on the substrate (76mm × 62mm). One part is the mountainous pattern located within the approximate ellipse area (50mm × 35mm). The other part is a plane surface outside the ellipse area. All figuring experiments are performed in our self-developed IBF systems (2.5 × 10^{−3} Pa pressure) under the bombardment of Ar + ions at normal incidence. Within the experiments, the figuring conditions are fixed at an ion source energy *E _{ion}* = 800eV and beam current

*J*= 25mA.

_{ion}Tables 2–4 describe the three figuring stages, in which three different diameter beams of *d*_{6}* _{σ} =* 8.2mm, 4.1mm and 3.1mm are used, respectively. The first stage is the rough figuring, with a large beam is used to realize the rapid correcting of low frequency errors. After two iterations, the surface error ratio reaches

*k*<1.5, which indicates that the removal function almost has no effective corrective capability. The second and third stages are finishing figuring. Matching errors between the desired surface and the figured surface is gradually evolved into mid-to-high spatial frequency, and thus smaller ion beam is required to improve machining precision. The reason for the slow convergence of the surface error ratio in the third stage mainly includes two aspects. One is the material removal is large while the volume removal rate of a small beam decreases sharply, result in a substantial increase of figuring time. In order to reduce the adverse influence of the removal function stability on machining precision, figuring time of each iterative is controlled within about 200min. Another reason is that the corrective capability of the ion beam reduces accordingly when the error frequencies approach to be higher, and thus leads to a slow convergence of the machining precision. That’s also the reason why we don’t continue to reduce the beam diameter in the third figuring stage even though the ∆

*k*/

*k*<5% (Table 4).

As shown in Fig. 8, the surface accuracy of the ellipse area is 23.8nm RMS, 166.1nm PV, and the final matching error between the actual figured surface and the desired CPP surface is 8.1nm RMS, 112.4nm PV. The PV of matching errors is not a good metric to assess the quality of the imprinting. Since PV is computed from only two data points out of possible thousands available, the optic appears worse than it actually is. The RMS calculated from all the data on the optical surface can give a better indication of the overall optic performance. Besides, the imprinting error mainly stems from minor mismatches near the peaks and valleys where collateral polishing causes feature washout because of large slope gradient. Comparison of Figs. 8(d) and 8(e) show that surface gradient can be imprinted reaches about 1.8μm/cm, which has the same distribution with the desired CPP surface apart from the surface edge. The plane surface outside the ellipse area obtains nano-precision (1.4nm RMS) especially the average height error of the boundary area around the ellipse area is within the range of ± 4nm, as shown in Fig. 8(g). Figure 9 analyses the spatial frequency evolution of the CPP structures in different figuring stages through the PSD analysis. Figure 9(a) is the PSD analysis of the matching errors and it shows that surface error with spatial frequency less than 0.6mm^{−1} can be effectively corrected, which is the cut-off frequency of beam *d _{6σ}* = 3.1mm (0.66mm

^{−1}). The coincidence of the PSD curves of different iterations within the frequency range of

*f*<1.6mm

^{−1}indicates that there are no other new machining errors introduced during IBF processes. However, PSD curves begin to appear separation when the frequency larger than 1.6 mm

^{−1}, which indicates new higher frequency errors are introduced, and the reasons will be analyzed in the next section. Figure 9(b) is the PSD analysis of the figured CPP surface. It shows that spatial period larger than 5mm (

*f*= 0.2mm

^{−1}) can be completely imprinted and spatial period as small as 0.7mm (

*f*= 1.5mm

^{−1}) can be partial imprinted on the substrate. There is no obvious increase in PSD amplitude within range of

*f*>1.5mm

^{−1}indicates that the spatial period smaller than 0.7mm cannot be effectively figured by ion beam larger than 3.1mm.

## 5. Analysis of the machining errors

Machining errors of each step will be accumulated, amplified and directly influence the next figuring stage due to the topographical complexity of CPPs. Therefore, a reasonable IBF process must be based on the elimination or strictly control of machining errors. There are mainly three kinds of errors during IBF process, the alignment accuracy of the ion beam with respect to the optic in the IBF plant, the measurement accuracy in the interferometer and the fluctuant error of ion beam stability.

The first kind of error will lead to a translation of the material removal, and thus destroy the existing surface topography. Obviously, the figuring process of CPPs with sharp slope changes is more sensitive to this error. The linear positioning accuracy and repeatability accuracy of the IBF machine system is 5.9μm and 1.6μm, respectively [8]. Besides, our research team presented an identification-compensation method to further improve the alignment accuracy of the ion beam to the optic [11]. Usually the alignment accuracy during figuring process can reach a high level about 10µm after redoing this procedure several times.

The measurement error directly influences the accurate solution of material removal between the desired CPP surface and the figured surface. The surface data at the surface edge are usually absent or indistinct in measurement and the lateral resolution of the interferometer Zygo GPI XP is 0.09mm/pixel, implying that only 1 pixel deviation would arouse considerable errors. Though a mark point on the optic can be reference to ensure the same state during each measurement, the measurement error is inevitable. So we adopt the surface matching method based on regional features to obtain accurate material removal information, in which the measured surface is numerically fitted to the desired surface, using the RMS difference between the actual surface topography and the parameterized prescription as a minimization metric [14]. This method can reduce the matching error to a maximum of about ten microns. The mathematical model is as follows:

The constraint conditions are the variation range defined by *A* in six degrees of freedom.

Where *A* is surface data obtained through the coordinate transformation of measurement data in a range of six degrees of freedom, and *B* is the theoretical surface data.

A higher lateral resolution of the interferometer can obviously improve the measurement accuracy. However, it should be pointed out that we pay more attention to the relative error between two measurements rather than the absolute error of one measurement. Besides, the smallest size of the ion beam is about 3mm, which is 30 times larger than the lateral resolution of the interferometer. Therefore, there is no need to use an interferometer with higher lateral resolution.

The ion beam stability directly influences the convergence rate of the figuring process. So it is critical to guarantee the long-time stability during hours. Actually, with the significant increase in dwell time, higher requirements for the long-time stability with respect to the beam intensity and shape are proposed, especially when a small ion beam is adopted. In order to obtain ion beam with different sizes, high intensity and good Gaussian shape, we have optimized the ion source based on the commercial ones and achieved good results [15]. We usually utilize the spot method to evaluate the long-time stability of ion beam. As shown in Fig. 10, we use ion beam bombarding the appointed spots every half hour and each spot for 3 min. The fluctuant error of beam intensity can be controlled within ± 2% in 4 hours and the shape is nearly standard Gaussian shape. Besides, the figuring time of each iterative is controlled within about 200min to reduce the adverse influence of the beam instability on machining precision.

Anyway, taking all these factors into account comprehensively, the alignment accuracy of the ion beam with respect to the measured data matrix of the optic can be controlled better than 50µm. However, it is still quite difficult to guarantee the alignment accuracy to near 10µm. As shown in Fig. 11, the machining error caused by the alignment accuracy is just 1.3nm RMS, 13.9nm PV, when the alignment accuracy is (*δ _{x}*,

*δ*) = (50µm, 50µm). This indicates that the adverse influence caused by the alignment accuracy can be ignored during the figuring process. However, the errors with frequency larger than 1.6 mm

_{y}^{−1}in Fig. 9(a) are introduced mainly by the ion beam size. And a removal function with diameter of

*d*= 0.3mm is required to correct frequency larger than 1.6 mm

_{6σ}^{−1}. It is not practical considering the very low machining efficiency of this small ion beam. The manufacturing error as an unavoidable and fixed system error will be corrected and eliminated in the alignment process of the optical system.

## 6. Summary

The spatial period and surface gradient of microstructure district the high-precision fabrication of CPPs. In this paper, we solve the machinability problem of one kind of CPPs by IBF technology. In order to improve the machining precision and efficiency, we present a multi-pass IBF approach with different beam sizes based on the frequency filtering method. We discuss the selection principle and replacement time of ion beams for different figuring stages to imprint smaller spatial periods and control dwell time in detail. The theoretical analysis and experimental results demonstrate that the machining precision and efficiency is directly tied to the physical dimension of the ion beam. And this filtering method with different ion beam sizes can obtain better surface quality in a faster way compared to the non filtering traditional IBF method. The research uncovers the advantages of reasonable selection and arrangement of different removal functions to imprint microstructures of small spatial periods onto optics surface. Overall, IBF’s sub-aperture polishing characteristics make it possible to imprint complex microstructures with spatial period as small as 0.7 mm, surface PV smaller than 200nm and surface gradient as high as 1.8μm/cm to within 10nm RMS. The experimental results verify the reliability and advantages of our optimized fabrication method.

## Funding

This work was supported by the National Natural Science Foundation of China (No.91323302, No.91523101, No. 61505259).

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