## Abstract

Single point diamond turning (SPDT) is highly versatile in fabricating axially symmetric form, non-axially-symmetric form and free form surfaces. However, inevitable microstructure known as turning marks left on the surface have limited the mirror’s optical performance. Based on chemical mechanical polishing (CMP) mechanism, smoothing polishing (SP) process is believed to be an effective method to remove turning marks. However, the removal efficiency is relatively low. In this paper, based on Greenwood-Williamson (GW) theory, the factors that limit removal efficiency of SP are discussed in details. Influences of process parameters (work pressure and rotational speed) are firstly discussed. With further analysis, surface spectral characteristics are identified as the inherent factor affecting further efficiency improvement. According to theoretical analysis, the removal efficiency of isotropic surface is nearly 1.8 times higher than anisotropy surface like surface with turning marks. A high efficiency turning marks removal process combining ion beam sputtering (IBS) and SP is proposed in our research. With removal depth exceeding 100 nm, the isotropic aluminum surface can be constructed by IBS so that the efficiency of SP process can be greatly improved. Though deteriorated by IBS, the surface roughness will be rapidly reduced by SP process. Finally, experiments are conducted to verify our analysis. A 3.7 nm roughness surface without turning marks is achieved by new method while direct SP can only reach roughness of 4.3 nm with evident turning marks. Experimental results show that removal efficiency nearly doubled which matches well with the theoretical analysis. Our research not only can be used as a high efficiency turning marks removal and surface quality improvement method but also can be a new method for high precision aluminum optics fabrication.

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

## 1. Introduction

With the advancement in computer control and high precision machine tool manufacturing, Single Point Diamond Turning (SPDT) which possess nanometric sharpness and wear resistance has been one of the most promising techniques for obtaining small and medium-sized aluminum optical components since 1980s [1–3]. Additionally, it offers a potential for mass production because of high fabrication efficiency [4]. Recently, with the rapidly growing demand for precision optical components such as freeform surfaces, SPDT is continuously improved in control and tool design. Currently, SPDT can achieve optical freeform surface with micrometer to submicrometer form accuracy and nanometer range surface roughness [5]. Though with great improvement, SPDT has some inherent drawbacks. The resulting microstructure, which is called turning marks, will lead to light diffraction and stray light that will cause double image and other problems. Great efforts have been made on analyzing the diffraction effects and its drawbacks [6–10]. Therefore, in order to improve optical components performance, further process to eliminate the turning marks without destroying the surface form is required.

Many presently polishing techniques such as magnetorheological finishing (MRF) [11], computer controlled optical surfacing (CCOS) [12], bonnet polishing (BP) [13,14], abrasive jet polishing (AJP) [15] are used to eliminated turning marks. Excellent results are achieved by these techniques. However, the characteristics of soft and high chemical activity make aluminum polishing extremely difficult. Most of the techniques are not quite fit for removing turning marks on aluminum surface. For example, MRF will induce contamination on the surface and will bring new middle and high frequency spatial errors.

The smoothing polishing (SP) process, which has a long-standing history, belongs to traditional polishing in nature, is believed to be an effective method to remove turning marks (high-spatial error). Based on CMP mechanisms, by strictly controlling the machining status, a rigid or semirigid polish pad is used to smoothing the high-spatial error [16–19]. Through former research, the turning marks can be effectively removed by smoothing polishing process with optimized processing parameters [20]. However, there are several issues in SP process. First, SP is a time-consuming process to remove turning marks which will be a great limitation for its application [21]. Secondly, due to the longtime process, the surface form will deteriorate because of the existing edge effects [22,23]. Most of research about SP process concentrated on optimizing pad pressure distribution, working parameters [24,25], etc. There is a great lack of researches about the mechanism of efficiency limitation and needs to be studied urgently.

In the work reported in this paper, a high efficiency process combining ion beam sputtering (IBS) and SP is proposed. This method can remarkably improve the turning marks removal efficiency. In section 2, the key factors limiting SP efficiency are analyzed in detail by theoretical methods. IBS process is introduced to aluminum optics innovatively to create conditions for high efficiency fabricate. In section 3, an experiment is conducted to verify our theoretical results. This method greatly improves the processing efficiency and also extends the application of IBS in aluminum optics manufacture field. This work presents a new idea for high precision and efficiency aluminum optics fabrication.

## 2. Theoretical analysis

#### 2.1. Requirements of SP process for aluminum optics

In SP experiment, the mirror is fixed on the mirror holder as shown in Fig. 1. The pad rotates around its center. With a steady pressure imposed on pad base, pad oscillates on the mirror to and fro. The SP equipment is composed of a pressure applying module, a polishing pad (including pad base and lap), slurry, etc. To achieving a perfect SP process, several conditions are necessary.

In processing flows, SP process is used to improve surface quality and remove turning marks without deteriorate surface forms. For aluminum optics with complex surface, there are several restrictions for SP process.

- (1) Pad properties
Intimate contact between the tool and the optical surface is required during SP. Misfit occurs when a rotating rigid tool moves across an asphere. A larger size of pad would make misfit problems more severity. In our earlier research, misfit problems can be ignored by using small size pad. Usually, the size of pad is chosen to be 1/4∼1/16 of optical aperture [26]. Another restriction is pad materials. To avoiding surface shape deterioration, polishing cloth is chosen instead of materials with higher elasticity modulus such as pith or polyurethane (PU).

- (2) Process parameters
Pressure is a critical process parameter which usually affects contacting state of pad and working surface. Aluminum materials has relatively low hardness and elasticity modulus. Moreover, polishing abrasives are easily aggregated. With higher pressure, large particles are easy to be embedded into workpiece which will cause scratches. Based on taguchi experiment conducted by our group, the optimal pressure of 0.02Mpa is chosen [21].

With requirements (small tool size, soft pad material, low work pressure, etc.) mention above, SP process can be effectively used to improve surface quality of aluminum optics and eliminate turning marks. Based on these requirements, there are several conditions such as fluid PH, rotational speed of pad that can be optimized for a better polishing process. According to Preston’s equation, pressure deviation will generate between the peak and valley of turning marks during SP process that will cause material removal deviation. Turning marks will be eliminated based on that principle. However, despite the optimized conditions, actual SP process shows that turning marks on aluminum optical surface can hardly be eliminated during polishing. However, there are great lack of literature about turning marks removal mechanism during SP process and constraints on SP efficiency and need systemic studies.

#### 2.2. SP process efficiency analysis

Preston’s equation which belongs to the macro-scale studies has limitation in revealing turning marks evolution. Firstly, amplitude difference between peak and valley is relatively small (usually several nanometers) which makes pressure deviation hard to obtain; Secondly, Microscopic anisotropy is not taken into account in Preston’s equation while turning marks shows obvious periodicity. Considering these drawbacks, Greenwood-Williamson (GW) theory is chosen to quantify the SP process and predict the turning marks evolution in this paper. Based on Hertz contact theory, GW theory [27] which can reveal microscale material removal phenomenon quite fits our purpose. GW theory divides the contact surface into random rough surface and nominal flat surface. The work surface is assumed to be nominal flat surface which means that the random rough pad surface form is the sum of actual form of pad and work surface. The load is supported by pad asperities (dotted areas). Abrasives attaching on the pad surface remove materials when the relative motion generates between pad and work surface.

The material removal capability can be measured by removal rate of a single abrasive. For simplification, this paper only discusses the removal rate of a single abrasive. Volume removal rate of single abrasive can be expressed as:

where*K*is constant of wear,

*V*is the relative speed of abrasive and workpiece and

*A*is the cross-sectional area of a single abrasive particle in a workpiece that can be expressed as: where

*a*is contact radius and

*r*is abrasive radius.

According to GM theory, the load *L* and the contact area *A* between a smooth surface and rough surface can be express as follow:

*N*is number density of pad asperities,

*A*

_{0}is total contact area,

*d*is the average distance between two surfaces,

*R*is radius of asperity summits as shown in Fig. 2 and

*ϕ*(z) is standard gaussian distribution function of the asperity’s height,

*υ*and

_{p}*υ*are Poisson's ratio of abrasive and polishing pad respectively.

_{w}*E*and

_{p}*E*are Modulus of Elasticity of abrasive and polishing pad respectively.

_{w}According to plastic contact theory, the contact force *F _{w}* between abrasive and workpiece and contact force

*F*between abrasive and pad can be expressed as follow respectively:

_{p}*B*is Brinell hardness of workpiece and

_{e}*b*is the radius of the circular indentation into the pad. To be the most efficient material remove state, the pad should be sufficiently hard to make value of

*b*to be close to the abrasive radius

*r*. For simplicity, we will take

*b*=

*r*. On the basis of force balance of abrasives, we have

*F*=

_{w}*F*. we now use this formula and Eqs. (3)(8)(9) to eliminate the

_{p}*a*in Eqs. (1). The removal rate can be rewritten as:

For particular abrasives and pad materials, the parameters *r*, *E _{sp}*,

*B*and

_{e}*K*are constants in Eqs. (10). For particular work surface and pad surface,

*R*and

*σ*are also constant. The remain parameters are highly related to the process parameters such as work pressure and rotate speed. From Eqs. (10), it is obvious that the removal rate increases linearly with the increase of

*V*. Work pressure is strongly associated with parameter

*d*which represents the contact degree of pad and work surface. Statistically speaking,

*d/σ*=3.0 represents a situation where only 0.13% of the pad asperities are in contact with the work surface [28] as shown in Fig. 3(a). With the increase value of

*d*, the contact area between pad and work surface reduces which means the reduction on the work pressure. During SP, pad intimately contact with work surface which means that peak and valley on the surface correspond to different

*d/σ*value. In Fig. 3(b),

*R*has non-linear relationship with

_{v}*d*. The difference of removal rate between peak and valley determines the surface quality improvement and turning marks removal efficiency (

*Re*) which can be written as:

*R*and

_{vp}*R*are the material removal rate of turning marks peak and valley respectively. As the red dotted line shows, with constant peak-to-valley (PV) value,

_{vv}*Re*also decreases non-linearly as

*d*increases (working pressure decreases). Therefore, working pressure should be increase in order to achieving high turning marks removal efficiency which quite fits experimental results. However, according to the requirements in section 2.1, there are many limitations for pressure choosing.

#### 2.3. Frequency spectrum related removal efficiency analysis

In actual SP process, rotational speed and work pressure has been optimized. However, the improvement of turning marks removal efficiency is not obvious. So, there must be other affecting factors. Above analysis is conducted on the assumption that the surface state is isotropic. However, turning marks shows significant directional periodicity. Fortunately, Most of the parameters in Eqs. (10) can be expressed in terms of three quantities, *m _{0}*,

*m*and

_{2}*m*which related to spatial spectral [29].

_{4}*z*(x

*)*represents the profile height evolution. The parameters which will be expressed as: Apparently, SPDT surfaces microtopography embodies anisotropic properties. Figure 4 shows the difference of anisotropic surface (surface with turning marks) and isotropous surface. For isotropous surfaces, the characteristics of spatial frequency remain the same at every direction as shown in Fig. 4(a). Therefore, we can consider the surface error of single spatial frequency of

*f*. As for anisotropic surface (turning marks), considering that periodicity, when abrasives’ moving direction is perpendicular to turning mark ripples, maximum spatial frequency of

_{x}*f*

_{x}is achieved. However, when abrasives motion direction deviates from perpendicular direction (assume the deflection angle is

*θ*), spatial frequency will be

*f*

_{x}cos(

*θ*) as shown in Fig. 4(c). In extreme circumstance which abrasives motion direction is parallel to turning marks, spatial frequency value will be zero. In actual SP process, every direction is traversed by abrasives. Therefore, micromachining state varies greatly and need to be studied.

From Fig. 4(d), the surface topography can be expressed as follow:

Spatial spectral parameters can be given as:*L*is sampling length, which is on millimeter scale. Considering that

*A*is below ten nanometers, the later quadratic term can be ignored at a wide value range of

_{e}*k*

_{1}. The removal rate in Eqs. (10) can be expressed in spatial domain:

*k*

_{1}. Thus Eqs. (12) can be written as:

*C*and

_{p}*C*are corresponding

_{v}*C*value of peak and valley of turning marks respectively. With the stable machining parameters, the only variable in Eqs. (25) is spatial frequency

*k*

_{1}. Combining Eqs. (20) (25),

*Re*can be expressed as a function of deflection angle

*θ*as shown in Eqs. (26). Figure 5 is the normalized curve of

*Re*as function of

*θ*. The dotted line, which we refer as anisotropic removal efficiency, shows the removal efficiency variation with deflection angle. The solid line in Fig. 5 which we refer as isotropous removal efficiency stands for removal efficiency of isotropic surfaces. For isotropic surfaces, spatial frequency maintains a constant

*f*

_{x}at every deflection angel. Thus, the removal efficiency maintains maximum value.

Furthermore, in one processing cycle, abrasives traverse with same velocity at every direction on the surface during SP. Therefore, the removal efficiency in one cycle which we refer as unit removal efficiency (URE) should be the area bounded by the curve and the axes. By calculating curve integral, the URE of isotropous surface is 1.8 times higher than anisotropic surfaces. That explains the phenomenon that even though the machining parameters are optimized, the removal efficiency of turning marks is still relatively low. Based on theoretical analysis, we also find that constructing isotropic surfaces can greatly improve the processing efficiency. Many figuring processes which are usually used to improve surface form after SPDT will turn anisotropic surface into isotropous surface in a certain extent. However, they will bring contamination and other problems that bring the difficulties no less than turning marks removal. In order to achieve a high efficiency process with little contamination and form misconvergence, we propose IBS as an isotropous surface constructing method.

#### 2.4. Introduction of IBS

Admittedly, IBS is the highest precision process because of its stable, non-contact and contamination-free characteristics. When the atoms of work surface receive enough energy, it will be sputtered from the surface. Figure 6 illustrate the machining process of IBS.

During IBS, the surface error of different frequencies that can be eliminated strongly depend on the beam diameter. Considering the removal function of IBS is a rotationally symmetric Gaussian shape, for a certain beam diameter *d _{6σ}*, corresponding cut-off frequency of surface error correction can be given as follows [30]:

^{-1}∼200mm

^{-1}. In order to eliminated such surface error, ion beam with diameter of 0.01mm∼0.04 mm should be choose. With diameter of 0.04 mm, the requirements for position error and axis dynamic properties will be restrict which makes removal of turning marks hard for IBS.

However, unlike commonly used materials in IBS such as fused silicon, aluminum alloy is polycrystalline alloy with multiple ingredients. Based on several works conducted by our group, morphology evolution regularity of SPDT aluminum surface during IBS can be regulated [31]. Figure 7 shows the surface roughness evolution with the increase of IBS removal depth. At the first section (maximum removal depth of 100nm-110 nm), the roughness value increases rapidly and amplitude of turning marks is enhanced because more energy is deposited on the trough area leading to a higher etching rate. Typical morphology of first section is attached to the dotted box. It is obvious that turning marks are still evident on the surface. At the second section (removal depth of 100nm-300 nm), the roughness value increasing rate slows down and turning marks are fading. In this section, etching depth reaches grain surface, protruding grains cut off turning marks which leads to a distinct typical morphology evolution. As shown in solid line box, turning marks morphology evolves into relief morphology (emerging grains). As disappearance of periodic turning marks, the surface shows isotropic state on this section. With increasing removal depth, the relief morphology and isotropic state remains. Based on such unique characteristics, when IBS removal depth exceeds 100 nm, an isotropic surface can be achieved.

Normally speaking, IBS has limited ability on correction of high frequency error such as turning marks because that process often results surface morphology retention or even deepening as referred at the first section. However, because of the polycrystalline properties of aluminum alloy, despite of high frequency error deterioration, surface morphology has changed radically. This characteristic makes it possible that obtain isotropic surfaces with high precision form and low surface contamination. Also, it can promote the application of IBS in the field of aluminum optics processing which will be a great boost for improving the precision of aluminum optics.

With beam voltage of 700 eV, beam current of 15 mA and beam diameter of 16.4 mm, the material removal rate of IBS is 40×10^{−3} mm^{3}/min. With raster scan as tool path, processing time of around 20 min is needed to acquire material removal depth of 100 nm for a plane workpiece with diameter of 100 mm. The processing time of IBS is far less than the processing time of subsequent SP process. Moreover, with higher beam energy and current, the process time of IBS will further deceases.

Based on the above studies, we proposed a high efficiency turning marks removal method combining IBS and SP that we refer as IBS-SP process. IBS is used to construct an isotropic surface so the efficiency of SP process can be greatly improved.

## 3. Experiments

#### 3.1. Single point diamond turning

In our experiment, planar aluminum mirrors made from Al6061 were turned by single point diamond turning lathe (Precitech Nanoform 350). The diameter of aluminum sample is 100 mm. Turning parameters are shown in Table 1. Surface roughness is measured by White Light Interferometry (WLI) at 20× lens with a scan size of 0.47mm×0.35 mm. The original surface roughness is around 6.23 nm.

#### 3.2. Ion beam sputtering

The IBS experiments are performed in our self-developed IBS systems (KDIBF650L-VT) under the bombardment of Ar^{+} ions at normal incidence with the work pressure of 2.5×10^{−3}Pa. IBS parameters are shown in Table 2.

On the aluminum surface, an inclined plane with max etching depth of 1000 nm is fabricated. The size of inclined plane is 50mm×50 mm. In our former research, surface roughness of aluminum optics during IBS is highly dependent on the etching depth. In our experiment, the max surface roughness is around 45nm∼47 nm.

#### 3.3. Smoothing polishing

Polishing experiments were carried on a four-axis CCOS CNC polishing machine. The parameters we choose are shown in Table 3.

From our former research, this set of parameters can achieve lowest surface roughness. A raster scan is conducted as pad path. The whole surface is processed under the same condition. The single processing time is approximately 1 hour.

## 4. Results and discussion

#### 4.1. IBS-SP process

Figure 8 shows the roughness evolution of sample at different steps of IBS-SP process. Before IBS, turning marks is evident on the surface in Fig. 8(a). The initial surface roughness is 6.298 nm Ra. After IBS, surface roughness deteriorates to 46 nm. Almost all the SPDT marks are cleared away as shown in Fig. 8(b) which confirm the effectiveness of constructing isotropic surfaces. Due to preferential sputtering of second phase, we can find plenty of pits on the surface after IBS. After first SP, surface roughness value decreases rapidly. The number of pits decreases noteworthily. From Fig. 8(d), after second SP, the surface roughness value decreases to 11 nm. Transverse dimension of pits expands greatly and adjacent pits extent and merge. Also, the depth of pits is significantly reduced. We can deduce that the surface can be improved by further by optimizing the experimental parameters and control the material removal amount.

#### 4.2. Comparison of two methods

In order to prove the efficiency improvement, we compared the IBS-SP process with direct SP. For consistency in the initial states, we use IBS to process the surface roughness to 9 nm. In that state, relief morphology starts to emerge. Turning marks are faint and can hardly been seen on the surface as shown in Fig. 11(a). Figure 9 shows roughness evolution of direct SPDT surface SP process. The initial surface roughness is 6.2 nm. After fourth iteration, the surface roughness decreases to 4.2 nm. However, the last two iterations indicate that roughness convergence rate has slowed considerably. Although the roughness reduces to 4.2 nm, turning marks is still evident on the surface.

To validate the turning marks removal ability, power spectrum density (PSD) analysis is introduced to visualize the spatial frequency as shown in Fig. 10. The contour line perpendicular to the turning marks is extracted and processed by a Fourier transform. Consequently, PSD can be expressed as follow:

where*ω*is spatial frequency, Δω is frequency interval,

_{i}*E*(

*ω*) is the Fourier transform of contour line.

_{i}Because of existing periodic turning marks, there is a protruding peak on around 80 mm^{-1} which tally well with the SPDT parameters. The PSD curves reveal that after fourth SP process, the surface quality has been improved. Even though that the amplitude of peak decreases with iterations, the peak of turning marks is still evident after fourth SP process which means that turning marks removal efficiency of direct SP is relatively low which is quite match with our theoretical analysis. It is worth mentioning that there is not much difference in PSD curves of the last three iterations which is matches well with the decrease of roughness convergence rate.

As for IBS-SP process, after fourth iteration, surface roughness decreases to 3.7 nm which is less than direct polishing value. The faint turning marks disappears rapidly at first two iterations. It is worth mentioning that on the second iteration, the roughness deteriorates to 6.3 nm which is close to the initial roughness of direct SPDT surface SP. In another word, two iterations of proposed method can achieve a better surface quality than four iterations of direct SP. The efficiency is highly improved. Also, it can prove the necessity of constructing isotropic surface.

Figure 12 shows the corresponding PSD curves. There is no evident peak on curve of IBS surface which verifies the isotropic surface constructing ability of IBS. The turning marks are removed during IBS process. Moreover, peak of turning marks is no longer appeared on the surface in the following iterations. The decrease of PSD curves with iteration is obvious. After fourth SP process, the PSD curve of proposed process is lower than direct SP which means a better surface quality is achieved by proposed process.

The results of roughness convergence rate comparison are shown in Fig. 13. IBS-SP process has remarkable higher convergence rate and can achieve lower surface roughness rapidly. Roughness value is highly related to the surface errors which can also be used as an evaluation criteria for removal efficiency. It is obvious that higher roughness convergence rate is achieved by IBS-SP process. This also verifies the superiority of our proposed method indirectly. With the increasing iterations, amplitude of turning marks are reducing as observed in Fig. 10. According to Eqs. (25), the removal efficiency is dynamically reduced during iterations which leads to the reduction of convergence rate of direct SP as shown in Fig. 13.

#### 4.3. Discussion

Direct SP of SPDT surfaces can achieve a good surface qualities and removal of turning marks. But the process will be time-consuming. In our former experiment, 13 to 15 hours need to be spent to roughly remove the turning marks for a planar mirror of *Φ*100. In this paper, after four iterations (each iteration for 1 hour), the turning marks is still evident on the surface. By introducing IBS (constructing isotropic surface), the machining process is highly improved. Experimental results show that roughness convergence rate of IBS-SP process is nearly two times of direct SP’s. It is worth mentioning that it quite matches with our theoretical analysis in section 2.3 which is that the URE of isotropous surface is 1.8 times higher than anisotropic surface’s. Though IBS will deteriorate the surface quality, synthesizing the removal of turning marks and roughness decreasing rate, proposed process is still superior than direct SP polishing.

Also, IBS is a widely used highest precision machining method. Comparing with other figuring methods such as MRF, it has the advantages of non-contact, no edge effects and no contamination. The isotropous surface can be constructed with removal depth exceeding 100 nm. Therefore, an isotropous surface is constructed naturally during the IBS figuring process. Therefore, our proposed method not only can be used to remove turning marks and improve surface quality efficiently but also can be a new method for high precision aluminum optics fabrication.

## 5. Conclusions

In this paper, the efficiency improvement limitation of SP is discussed in details. The influence of process parameters (work pressure and speed) is first discussed. The work pressure and speed have positive correlation and positive linear function relation with removal efficiency respectively. However, despite the optimized parameters, the efficiency is still relatively low. With further analysis, we found that surface spectral characteristics is inherent factor affecting efficiency. Based on our analysis, it is revealed that:

- (1) Removal efficiency decreases exponentially with spatial frequency
*k*_{1}and amplitude*A*._{e} - (2) With the same amplitude and maximum spatial frequency, the unit removal efficiency of isotropic surface is nearly 1.8 times higher than anisotropy surface.

## Funding

National Natural Science Foundation of China (51991371); Postgraduate Scientific Innovation Fund of Hunan Province.

## Disclosures

The authors declare no conflicts of interest.

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