Multi-Scale Model of Mid-Frequency Errors in Semi-Rigid Tool Polishing of Diamond-Turned Electroless Nickel Mirror

Abstract

Semi-rigid tool polishing is widely used in the high-precision manufacturing of electroless nickel surface due to its stable material removal and high efficiency in correcting mid- and high-frequency profile errors. However, predicting mid-frequency errors remains challenging due to the complexity of their underlying sources. In this study, a theoretical model for semi-rigid tool polishing was developed based on multi-scale contact theory, incorporating a bridging model, rough surface contact, and Hertzian contact mechanics. The model accounts for the effects of tool surface roughness, polishing force, and path spacing. A series of experiments on diamond-turned electroless nickel mirrors was conducted to quantitatively evaluate the model’s feasibility and accuracy. The results demonstrate that the model can effectively predict mid-frequency errors, reveal the material removal mechanisms in semi-rigid polishing, and guide the optimization of process parameters. Ultimately, a surface with mid-frequency errors of 0.59 nm Rms (measured over a 1.26 mm × 0.94 mm window) was achieved, closely matching the predicted value of 0.64 nm.

1. Introduction

Metal-, ceramic-, and composite-based reflectors—typically made of aluminum, copper, or silicon carbide—are desirable for demanding X-ray optical applications due to their long-term stability and high strength-to-weight ratio [1,2,3]. However, these applications require extremely high surface accuracy [4]. In most cases, it is not feasible to directly polish these materials to meet the required specifications [5]. To address this challenge, a layer of electroless nickel (EN) is typically applied to the surface, providing a polishable layer that enables the elimination of surface profile errors through precision finishing [6].

Surface profile errors that degrade optical performance are typically classified into three categories: (1) low-frequency errors (LFEs), which can be described by geometric optics [7]; (2) high-frequency errors (HFEs), which are governed by scattering theory [8]; and (3) mid-frequency errors (MFEs), which lie in the transitional regime between LFEs and HFEs [9]. In X-ray applications, the boundaries of LFEs and HFEs are typically defined as 1 mm⁻¹ and 1 μm⁻¹, respectively [10]. To meet the requirements of X-ray applications [11], the mirrors covered by EN first need to be machined with single-point diamond turning [12] to achieve low-frequency errors at the hundred-nanometer level [13], and mid-/high-frequency errors at the nanometer level [14]. Subsequently, an advanced polishing technology was used to obtain nanometer-level low-frequency errors and sub-nanometer-level mid-/high-frequency errors. Typical polishing technologies include ion-beam figuring [15], magnetorheological finishing [16], and small-tool polishing (STP). However, ion-beam figuring is ineffective at correcting mid-frequency errors (MFEs) with spatial wavelengths in the order of hundreds of micrometers [17], while magnetorheological finishing often introduces high-frequency errors (HFEs) with root-mean-square (Rms) values of several nanometers [18]. Therefore, neither method can independently complete the mirror finishing process and must be combined with STP, which is based on the principle of chemical–mechanical polishing, to effectively control both MFEs and HFEs. In the STP process, a computer-controlled polishing tool—typically spherical, wheel-shaped, or disc-shaped and smaller than the workpiece—is used to polish the entire surface [19]. These tools usually consist of a pitch layer bonded to a metal base and are capable of achieving an Rms surface roughness below 0.3 nm on electroless nickel mirrors [20]. However, the use of these rigid tools can result in surface accuracy degradation and scratches due to localized hard contact between the tool and the workpiece [21].

To address these challenges, researchers have explored the use of elastic polishing tools, which offer greater adaptability to complex surface geometries and improved surface quality compared to rigid tools [22]. Using elastic tools—such as bonnet polishing heads [23] and polishing cloth pads [24]—electroless nickel mirrors with low-frequency errors (LFEs) below 10 nm Rms and high-frequency errors (HFEs) below 0.3 nm Rms can be achieved. However, elastic tools are generally ineffective at eliminating mid-frequency errors (MFEs) with wavelengths greater than 0.1 mm. As a result, this method is typically applied after MFEs have been reduced using fluid jet polishing [25] or rigid tool polishing [26], which inevitably increases process complexity. A polishing tool that combines the MFE-correction capability of rigid tools with the shape-conforming flexibility of elastic tools is therefore essential for improving overall manufacturing efficiency.

A semi-rigid tool—comprising a rigid inner base, an elastic intermediate layer, and an elastic or rigid outer polishing layer—is a key solution for achieving the above objectives [27]. Various studies have been conducted to reduce mid- and high-frequency errors using semi-rigid tools [28]. However, most existing work is experimental in nature and primarily focuses on final polishing outcomes, with the relationship between surface roughness and polishing parameters analyzed only qualitatively. To quantitatively evaluate the influence of polishing parameters on surface quality in semi-rigid tool polishing, it is essential to first model and analyze the material removal process. During semi-rigid tool polishing, the material removal mechanism is similar to that of chemical–mechanical polishing (CMP). LFEs primarily arise from material removal at the macroscopic scale, which can be mathematically described as the convolution of the tool removal function (TRF) and the dwell time [29]. HFEs, on the other hand, result mainly from microscale interactions—specifically, the cutting action of abrasive particles and the chemical etching effect of the polishing fluid [30]. Numerous models have been developed to describe LFEs and HFEs; their predictions generally align well with experimental results. However, MFEs are influenced by both macro- and micro-scale removal processes, leading to varying interpretations of their underlying mechanisms. As a result, the modeling and controlling of MFEs remains a significant challenge [31].

Many researchers have suggested that MFEs are primarily influenced by the TRF and the polishing path. For example, Z. He et al. [32] investigated the MFEs generated by different TRF profiles and found that a Gaussian-type TRF, combined with a small path spacing, resulted in lower MFE amplitudes. Some researchers have attempted to reduce MFEs by employing randomized polishing paths [33]. H. Tam conducted both analytical and experimental studies on the influence of various tool path types on MFEs [34]. Their findings indicated that the peak-to-valley (PV) amplitude in the removal map remains essentially unchanged for a given path spacing, regardless of the specific type of tool path used. Other researchers have suggested that, in addition to the polishing path, the surface roughness of the polishing tool also contributes to mid-frequency errors. Accordingly, MFEs can be classified into two categories: PSD-1, which is associated with polishing path spacing, and PSD-2, which is related to the rough contact pressure distribution between the tool and the workpiece [35]. Several models describing tool roughness adopted the Greenwood–Williamson (GW) approach [36] to simulate rough surface contact between the tool and workpiece. In these models, asperity contact is represented based on different height distributions such as exponential [37], Gaussian [38], and periodic distributions [39]. The material removal resulting from rough contact can then be modeled using the Preston equation [40] in combination with Hertzian contact theory [41]. In addition, some researchers have investigated the relationship between pressure distribution and tool surface morphology through experimental methods. For example, Z. Wang et al. [42] modeled both the tool surface and the polished surface profile by statistically analyzing real scanning images. L. Zhang [43] developed an MFE model based on actual contact pressure distributions measured using pressure-sensitive paper. However, these approaches alone are insufficient for accurately predicting mid-frequency errors. To achieve reliable MFE prediction, it is necessary to integrate both types of models.

To achieve this goal, researchers have developed comprehensive polishing models that integrate both Hertzian contact and rough surface contact between the tool and the workpiece [44,45,46,47,48,49,50,51]. In these multi-scale contact models, the pressure exerted by each micro-protrusion on the polishing tool is governed by the overall Hertzian pressure distribution. As early as 2003, J. Seok et al. [44] investigated multi-scale contact in chemical mechanical polishing (CMP) using finite element analysis. In 2016, Z. Cao et al. developed a multi-scale material removal model for computer-controlled bonnet polishing by incorporating both contact pressure distribution and single-particle wear mechanisms [45]. Later, in 2019, A. Lu modeled surface topography and roughness in the dual-axis wheel polishing of optical glass using a combination of single-particle wear theory and rough surface contact analysis [46]. Building on this, W. Yao et al. [47] proposed a multi-scale contact-based material removal model for cylindrical polishing, which extended previous models by including single-particle cutting effects. More recently, F. Meng et al. [48] introduced a multi-scale roughness prediction model for ultrasonic-assisted polishing based on W. Yao’s framework, achieving an error rate of less than 9.43%. In addition to these models, statistical approaches [49], the wear hypothesis [50], and fractal theory [51] have also been applied to study multi-contact behaviors in polishing. These studies collectively highlight the critical importance of multi-scale contact mechanics in accurately modeling mid-frequency errors.

In the above-mentioned multi-scale contact models, it is typically assumed that the polishing process can completely eliminate the mid-frequency errors of the initial surface [52]. As a result, only the MFEs introduced by the polishing tool are considered. However, in practical polishing with semi-rigid and elastic tools, the ability to suppress existing MFEs is often limited [53]. Therefore, the smoothing capability of the polishing tool also significantly affects the final surface MFEs. This smoothing effect arises from variations in contact pressure between surface asperities and valleys during the polishing process. Various studies have been conducted to investigate the smoothing capability of polishing tools through both experimental and theoretical approaches. For example, T. Suratwala et al. [54] experimentally examined the smoothing performance of semi-rigid tools with different polishing and elastic layers. Their results showed that a polishing layer with higher elastic modulus and a thicker elastic layer led to improved MFE suppression. Similarly, P. Lei et al. [55] explored the effect of tool size and found that larger tool sizes enhance the smoothing capability. Compared to experimental studies, theoretical models offer a more quantitative understanding of these effects. Early theoretical work by P.K. Mehta and P.B. Reid [56] introduced a bridging model based on Kirchhoff’s flat plate theory to describe pressure distribution during polishing of sinusoidal surface errors. J. Burge et al. [57] applied finite element modeling to analyze pressure distributions under various tool geometries. S. Liu et al. [58] proposed an MFE prediction model for pitch-based semi-rigid tools and assumed that MFE decreases exponentially with increasing removal depth. Y. Shu et al. [59] investigated the smoothing limitations of different polishing pads using the bridging model and concluded that increased stiffness would lead to a smaller MFE. More recently, X. Li et al. [60] developed a mathematical model for pressure distribution in multi-layer polishing tools, while X. Nie et al. [61] advanced this field by proposing a generalized finite element method model for calculating pressure distribution. These analyses collectively show that pressure differences between surface peaks and valleys reduce significantly as the surface approaches the smoothing limit. Clearly, the smoothing capability of a polishing tool is a key factor that directly influences the final mid-frequency errors. However, existing MFE prediction models based on multiscale contact mechanics have yet to incorporate this factor. Current research on MFE modeling for semi-rigid tool polishing remains insufficient.

To develop an accurate MFE prediction model for guiding and optimizing high-efficiency, low-roughness polishing of EN coated X-ray components, this paper presents a semi-empirical model for semi-rigid tool polishing. The model integrates frequency domain analysis and multi-scale contact mechanics, including smoothing capability, rough surface contact, and Hertzian contact mechanics. The influences of path spacing, polishing force, and tool surface roughness on the final MFEs were considered. Model parameters were first calibrated through a series of controlled experiments. Subsequently, polishing tests using a wheel-type semi-rigid tool were performed to validate the model. Both the modeling and experimental results demonstrate that higher tool flexural rigidity, smoother tool profiles, smaller path spacing, and appropriate polishing forces are beneficial for MFE suppression. Based on this model, we successfully reduced the root-mean-square MFE to 0.59 nm on a diamond-turned electroless nickel–phosphorus surface (measured over a 1.26 × 0.95 mm² area), with a predicted value of 0.64 nm. The main objectives of this study are: (1) to reveal the quantitative relationship between MFEs, tool properties, and polishing parameters; and (2) to provide practical guidance for the selection of polishing tools and process parameters.

2. Theory and Modeling of MFEs

This section first analyzed the material removal mechanisms associated with MFEs under three distinct contact behaviors: smoothing effect, rough surface contact, and Hertzian contact. Simplified expressions corresponding to the MFEs induced by each of these contact modes were then derived. The parameters influencing MFEs were categorized into three groups: (1) tool parameters, including tool surface roughness and geometric features; (2) polishing parameters, including path spacing and applied polishing force; and (3) empirical parameters, which are difficult to measure directly and must be obtained through experimental calibration. For modeling purposes, the polishing process was simplified as an input–output system. The input consisted of theoretical parameters and the initial surface topography of the workpiece, while the output includes the predicted final MFE topography and the respective contributions from the three contact mechanisms.

According to the Preston equation [44], the material removal during the polishing process can be expressed as:

$$df(x,y,t)=kp(x,y,t)v(x,y,t)dt$$

(1)

where f(x,y,t) is the surface profile at time t, p(x,y,t) is the real-time pressure distribution, v(x,y,t) is the real-time relative velocity distribution, and k is the Preston coefficient affected by polishing slurry. Given that the polishing velocity remains stable during the process, the velocity distribution was assumed to be constant in this study. As a result, the material removal distribution was determined solely by the pressure distribution.

Figure 1a illustrates the STP process using a semi-rigid, wheel-type polishing tool. The tool consists of a polishing layer, an elastic layer, and a metal base. The elastic layer allows the tool to conform to low-frequency surface errors, while the polishing layer is responsible for smoothing mid- and high-frequency errors, as shown in Figure 1b.

The contact pressure between the tool and the workpiece includes three components:

(1) Hertzian contact pressure—caused by the elastic deformation of the elastic layer;

(2) Bridging contact pressure—caused by the misfit between the polishing layer and the mirror surface;

(3) Rough contact pressure—caused by the surface roughness of the polishing layer.

2.1. Hertzian Contact and Polishing Path-Introduced Marks

Figure 2 illustrates the Hertzian contact and the MFEs introduced by the polishing path. Since the plasticity index of the polishing tool is only about one-tenth of that of the polished mirror, the contact between the tool and the mirror can be considered almost entirely elastic. In the case of a wheel-type tool operating on a planar workpiece, the analytical form of the Hertzian contact [41] pressure distribution is as follows:

$$P(x,y)=P_0{\left(1-({\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}})\right)}^{1/2}$$

(2)

where the maximum pressure P₀ is:

$$P_0={\displaystyle \frac{3F}{2\pi ab}}$$

(3)

where F is the applied force, a is the semi-major axis of the elliptical contact patch, and b is the semi-minor axis of the elliptical contact patch, as shown in Figure 2a.

Consider a simplified case where the curvature of the tool along the minor axis is identical to that of the mirror. In this scenario, b corresponds to the length of the contact area, and a is as follows:

$$a=\sqrt{{\displaystyle \frac{4RF}{\pi b{E}_0}}}$$

(4)

E₀ is the effective elastic modulus of the tool and workpiece, R is the tool radius:

$${\displaystyle \frac{1}{E_0}}={\displaystyle \frac{1}{E_1}}+{\displaystyle \frac{1}{E_2}}$$

(5)

where E₁ and E₂ are the elastic modulus of the tool and workpiece, respectively.

By incorporating the tool path spacing, the MFEs introduced by the tool path can be derived. As shown in Figure 2b, during polishing, the TRF results in noticeable tool marks on the surface. Based on Equations (1)–(5), the side-view profile of the TRF generated by Hertzian contact is as follows:

$$TRF(x)=kv{\displaystyle \frac{2F}{\pi a}}{(1-{\displaystyle \frac{x^2}{a^2}})}^{1/2}$$

(6)

where v is the relative velocity, and k is the Preston coefficient. The tool marks exhibit the same period L as the polishing path, and the height is as follows:

$$H=kv{\displaystyle \frac{2F}{\pi a}}[1-{(1-{\displaystyle \frac{L^2}{4a^2}})}^{1/2}]$$

(7)

Therefore, the polishing path introduced MFE can be expressed as:

$$Z_{path}(t\overrightarrow{r_1})=\begin{array}{cc}{k}_p\sqrt{{\displaystyle \frac{bF}{R\pi }}}[1-{(1-{\displaystyle \frac{{(t-nL)}^2}{a^2}})}^{1/2}]& n-{\displaystyle \frac{1}{2}}\le {\displaystyle \frac{t}{L}}<(n+{\displaystyle \frac{1}{2}}),n\in Z\end{array}$$

(8)

where n is an arbitrary integer, $t\overrightarrow{r_1}$ is the coordinate, t is the distance along the profile as shown in Figure 2b, $\overrightarrow{r_1}$ is the direction vector of polishing path as shown in Figure 2, and k_p is the semi-empirical coefficient related to E₀ and kv.

2.2. Bridging Contact and Smooth Limitation

Figure 3 illustrates the smoothing limitation during STP. According to bridging theory, the polishing pad can be modeled as a thin plate that follows Kirchhoff’s thin plate equation. When pressure q is applied to the polishing layer, its deformation satisfies the following equation [56]:

$$\left\{\begin{array}{l}{\nabla }^{4}w={\displaystyle \frac{q}{D}}-{\displaystyle \frac{{\nabla }^{2}q}{D_s}}\\ D_s={\displaystyle \frac{Eh}{2(1+\nu )}}\\ D={\displaystyle \frac{E{h}^3}{12(1-{\nu }^2)}}\end{array}\right.$$

(9)

where E is the Young’s modulus of the polishing layer, ν is the Poisson’s ratio of the polishing layer, q is the pressure distribution acting on the polishing layer, $\nabla$ is the Laplacian operator, w is the profile of the deformed polishing layer, D is the flexural rigidity, and Ds is the transverse shear stiffness. For diamond-turned surfaces, the mirror surface exhibits a one-dimensional periodic profile, with the relative velocity oriented perpendicular to the direction of periodicity, as shown in Figure 3a. Under this assumption, the deformation of the polished surface will retain the same period length l as the original surface profile.

During polishing, the high points of the surface profile, as shown in Figure 3b, bear most of the pressure due to direct contact with the polishing layer, while the low points experience zero pressure because no contact occurs. This pressure difference leads to preferential material removal at the high points, resulting in smoothing of the surface profile. As polishing continues, the low points of the mirror eventually come into contact with the corresponding low points of the deformed polishing layer. At this stage, the normal pressures at both the high and low points equalize. With further polishing, the surface profile tends to stabilize, as illustrated in Figure 3c. Therefore, the final surface profile z_s after polishing corresponds to the maximum deformation w_x of the polishing layer:

$$z_s=w_x$$

(10)

Considering the deformation within a single cycle, the high points of the surface profile can be treated as supporting beams for the thin plate. Due to the obvious one-dimensional periodic structure of the surface profile after turning, the bending of the polishing layer can be approximated as the bending of a beam in one-dimensional conditions. Accordingly, the deformation w_x can be obtained as [56]:

$$w_x={\displaystyle \frac{q{\left(\raisebox{1ex}{$x^2$}\!\left/ \!\raisebox{-1ex}{$l^2$}\right.-1\right)}^2{l}^4}{24D}}$$

(11)

From Equation (11), it can be observed that the maximum deformation, w_x, is inversely proportional to the flexural rigidity D and the fourth power of the spatial frequency 1/l, and directly proportional to the applied pressure q. Therefore, the following conclusions can be drawn:

• When the spatial frequency of the error is sufficiently high, the smoothing effect can reduce the profile error to nearly zero;
• The smoothing limit is directly proportional to the applied pressure q, and inversely proportional to the flexural rigidity D;
• This implies that the smoothing effect described by the bridging model can be considered analogous to a low-pass filter.

In our model, the pressure q is the pressure of Hertzian contact. Considering that the polishing tool would pass through the whole surface, the q can be regarded as the maximum pressure P₀ in Section 2.1. Accordingly, the smoothing limit is:

$$Z_{smootheffect}=\sqrt{{\displaystyle \frac{F}{bR\pi }}}{\displaystyle \sum _f{k}_f{a}_f{Z}_f}$$

(12)

where a_fZ_f is the Fourier expansion of initial surface profile Z₁:

$$Z_1={\displaystyle \sum _f{a}_f{Z}_f}$$

(13)

where k_f is the smooth coefficient:

$$k_f=\left\{\begin{array}{l}\begin{array}{cc}0& f\ge f_h\end{array}\\ \begin{array}{cc}{\displaystyle \frac{p}{DRms(Z_1)}}& f<f_h\end{array}\end{array}\right.$$

(14)

where f_h is the cutoff frequency of smooth effect, p is the scale factor, and Rms(Z₁) is the root-mean-square (Rms) value of initial surface profile. The f_h and p can be obtained by experiments, the details of which will be discussed in Section 3 and Section 4.

2.3. Rough Contact Introduced Mid-Frequency Errors

The pressure distribution arising from rough surface contact is the primary cause of profile errors with spatial frequencies higher than f_h. According to the GW model, the rough surface of the polishing tool can be considered as a surface composed of multi-scale asperities, as illustrated in Figure 4a. During the polishing process, the material removal can be modeled as the action of an equivalent polishing tool with a one-dimensional fractal profile, as shown in Figure 4b. The profile of the mirror is set as Z(x,y), and the profile of the equivalent polishing tool is set as Z_p(x,y). Under normal conditions, Z(x,y) << Z_p(x,y). According to Hertzian contact theory, the contact pressure between the mirror and the polishing tool is as follows [44]:

$$P={\displaystyle \frac{4}{3}}{P}_H{E}_0\sqrt{R_{pc}}{(Z(x,y)-Z_p(x,y))}^{{\displaystyle \frac{3}{2}}}\approx {\displaystyle \frac{4}{3}}{P}_H{E}_0\sqrt{R_{pc}}{(-Z_p(x,y))}^{{\displaystyle \frac{3}{2}}}$$

(15)

where R_pc is the radius of protrusions and P_H is the pressure of Hertzian contact. Considering that the polishing tool would pass through the whole surface, the P_H can be regarded as the maximum pressure P₀, as shown in Section 2.1. During the polishing process, the polishing tool cyclically removes material from the mirror surface along a predefined path. When the tool passes over the surface for the i-th time, the surface profile modification introduced by rough contact can be expressed as:

$$Z_i(x,y)=k_s(Z_{i-1}(x,y)-{\displaystyle \frac{4}{3}}k{P}_0{E}_0\sqrt{R_{pc}}{(-Z_p(x,y))}^{{\displaystyle \frac{3}{2}}}v)$$

(16)

where k_s is the smooth coefficient caused by smooth effect and chemical etching. It is evident that:

$$\begin{array}{l}{Z}_{roughcontact}=Z_{i-1}(x,y)=-{\displaystyle \frac{k_s}{1-k_s}}k{\displaystyle \frac{4}{3}}{P}_0{E}_0\sqrt{R_{pc}}{(-Z_p(x,y))}^{{\displaystyle \frac{3}{2}}}v\\ =k_E\sqrt{{\displaystyle \frac{F}{bR\pi }}}{Z}_p{(x,y)}^{{\displaystyle \frac{3}{2}}}v\end{array}$$

(17)

where k_E is the constant related to the elastic coefficient and k_s, the Z_i(x,y) is equal to Z_i-1(x,y), indicating that the surface roughness has stabilized.

2.4. MFEs Predication Model

Figure 5 illustrates the flowchart of the MFE prediction model. Based on the previous analysis, the parameters influencing the MFEs can be categorized into three types:

• Tool parameters, including tool roughness, tool radius, and tool width;
• Polishing parameters, including path spacing and polishing force;
• Empirical parameters obtained by experiments, including f_h, k_p, k_f and k_E.

First, it is necessary to obtain empirical parameters f_h, k_p, k_f and k_E through experiments. This will be discussed in Section 3. Then, the parameters and the initial surface profile are inputted into the model to predicate the final surface profile. After inputting the initial surface profile of the workpiece along with the above parameters, the first step is to compute the contact area and maximum contact pressure using Hertzian contact theory. Then, the mid-frequency error introduced by the tool path, Z_path, is calculated using Equation (8). Next, a low-pass filter is applied to the initial surface profile to simulate the smoothing effect. The filtered profile is then used to calculate the smoothing-induced error, Z_smootheffect, according to Equation (13).

The equivalent profile of the polishing tool cannot be obtained directly. Therefore, it is assumed that a linear proportional relationship exists between the Rms value of the equivalent tool profile Z_p and that of the actual tool surface profile Z_t. This proportionality factor is influenced by both the rotational speed and the translational speed of the polishing tool. Based on this assumption, the surface profile introduced by rough contact, Z_roughcontact, can be reconstructed using fractal theory and the corresponding Rms value. The Rms value of Z_roughcontact can be calculated as:

$$Rms(Z_{roughcontact})=Rms(k_E{P}_0{Z}_t{(x,y)}^{{\displaystyle \frac{3}{2}}}v)$$

(18)

Typically, the Z_roughcontact can be expressed using the Weierstrass–Mandelbrot (WM) function, which is widely used to describe fractal profiles:

$$Z_{roughcontact}(x,y)=G^{D-1}{\displaystyle \sum _{n=n1}^{\infty }\frac{\cos 2\pi {\gamma }^n(x)}{{\gamma }^{(2-D)n}}}$$

(19)

where D is the fractal dimension, G is the scale coefficient, γ is a constant that is always taken as 1.5, n1 is a coefficient related to the lowest truncation frequency of the profile, and x is the coordinates.

Finally, the overall MFEs can be obtained by summing the contributions from all individual sources. The Rms value of the final MFEs is then calculated based on this cumulative result. By inputting different sets of processing parameters into the model, the relationship between these parameters and the resulting MFEs can be systematically analyzed. This enables guidance for the selection and optimization of polishing parameters.

2.5. Error Analysis

The mode’s errors mainly arise from two factors: model simplification and empirical parameters. In this model, only the multi-contact behavior is discussed. For a real polishing process, the chemical etching and fluid effect would also influence the final topography. Chemical etching and fluid effects mainly affect the HFEs. However, there is still a non-ignorable influence on the surface morphology with a period close to micrometers. In the frequency domain analyzed in this article, although the effects of these two are less significant compared to the roughness generated by contact, they are still difficult to ignore and introduce predication errors when the roughness is small enough, especially for Z_roughcontact.

Empirical parameters are obtained from experiments. Measurement errors could lead to errors in these parameters and cause predication error. The measurement error will have a greater impact when the roughness is improved. Therefore, this model will exhibit significant errors when the roughness is small enough.

3. Experiment Design

Two types of experiments were conducted in this study: (1) empirical parameter acquisition experiments; and (2) model validation experiments. Diamond-turned electroless nickel plates with dimensions of 40 × 40 mm² were used as the workpieces for polishing. The polishing experiments were carried out using a 6-axis industrial robot (IRB 1600, ABB, Zurich, Switzerland), as shown in Figure 6a. A wheel-type polishing tool was employed, as illustrated in Figure 6b. The polishing tool consists of a 50 mm diameter aluminum alloy wheel, a 5 mm thick foam elastic layer, and a 1.2 mm thick polyurethane polishing layer, as detailed in Figure 6c. The foam layer has a hardness of 40 HA, and the width of the polishing layer is 10 mm. The surface profile of diamond-turned EN plates exhibits distinct directionality [62]. Figure 6d illustrates the polishing scenario, where the orientation of the mirror’s periodic surface profile is parallel to the rotation direction of the polishing tool. The blue dashed line is the polishing path, which is also expressed by the yellow solid line in Figure 2a.

In all experiments, the rotational and translational speeds of the polishing tool were maintained constant to ensure process consistency. The polishing wheel rotated at 100 revolutions per minute (rev/min), while its translational speed was set to 5 mm/s. The polishing slurry consisted of SiO₂ abrasives with an average particle size of 100 nm, suspended in deionized water. The material removal depth was controlled at 1 μm in each experiment to ensure that the surface roughness reached a stable, convergent value.

During the polishing process, the directions of tool translation and rotation are arranged to be perpendicular, as shown in Figure 6b,d, in order to minimize the influence of translation speed on the TRF. Consequently, the orientations of Z_path and Z_roughcontact are also perpendicular to each other, and the smoothing effect Z_roughcontact becomes oriented perpendicular to Z_path.

3.1. Parameters Acquisition Experiments

The purpose of this experimental section is to determine the empirical parameters k_p, k_f and k_E. Ideally, these three effects would be studied independently through separate experiments. However, in practical polishing processes, the smoothing effect, tool path effect, and rough contact effect are inevitably coupled. Therefore, decoupling these three contributors to MFEs after polishing became a critical challenge in this study. According to the work of Xia et al. [63], the anisotropic component can be extracted based on two-dimensional power spectral density analysis for polished surfaces. As discussed in Section 2, Z_smootheffect, Z_path and Z_roughcontact each exhibit distinct anisotropic characteristics, making them separable using this method. By applying this approach to the polished surface profiles, the individual contributions of each effect can be isolated and the corresponding empirical parameters k_p, k_f and k_E can be accurately extracted. The empirical parameter f_h can be obtained by analyzing the power spectral density after polishing. The highest frequency corresponding to the periodic structure that remains on the surface after polishing can be considered as f_h.

Two polishing tools of identical dimensions but with different polishing layers were used in this study. The detailed tool parameters are listed in

Table 1. Tool parameters used for the parameter acquisition experiments.

	Tool A	Tool B
Aluminum wheel radius	25 mm	25 mm
Foam layer thickness	5 mm	5 mm
Polishing layer thickness	1.20 mm	1.20 mm
Polishing layer material	LP87	SL4387
Rms tool roughness	1.81 μm	1.70 μm

. Tool A was equipped with an LP87 polishing material (hardness: 96 HA; density: 49 lb/ft², Universal Photonics, New York, NY, USA), while Tool B used SL4387 (hardness: 69 HD; density: 68 lb/ft², Universal Photonics, New York, NY, USA). A polishing force of 8 N was applied in both cases. Prior to polishing, both tools were dressed using identical single-point diamond-turning parameters to ensure consistent initial surface profiles. The surface topographies of the tools were then measured using a white-light interferometer (ContourGT, Bruker, Karlsruhe, Germany). The measured profiles are presented in Figure 7.

After polishing, the surface profiles were measured using a white-light interferometer with a measurement window of 1.26 mm × 0.94 mm. To minimize the influence of positioning errors on the measurement results, points were uniformly selected within the polished area. The overlap between adjacent measurements was 25% of the measurement window, resulting in a total scanned area of 5.00 mm × 1.60 mm. Measurements were performed at 10 locations, as shown in Figure 8. The model accuracy was then evaluated by averaging the results from these 10 measurements.

A high-pass filter with a cutoff frequency of 5 mm⁻¹ was first applied to isolate the component Z_roughcontact from the measured results. Subsequently, two-dimensional power spectral density (2D PSD) analysis was employed to separate Z_smootheffect and Z_path, as shown in Figure 9. For a surface composed of anisotropic features in different orientations (e.g., directions R₁, and R₂ in Figure 9a), the 2D PSD was first computed, as shown in Figure 9b. Then, orientation-specific, one-dimensional PSDs were extracted by averaging along directions R₁ and R₂ from the 2D PSD (Figure 9c). Finally, the root-mean-square values of each component were calculated based on their respective 1D PSDs.

3.2. Model Validation Experiments

The objective of this set of experiments was to validate the proposed MFE prediction model. In this phase, simulations and experimental results were compared under varying input parameters, including polishing force, path spacing, and tool surface roughness.

Tool A was employed to investigate the influence of polishing force F. According to the theoretical analysis, Z_smootheffect and Z_roughcontact are proportional to F^0.5, while Z_path is inversely proportional to F^0.5. During the experiments, the path spacing was fixed at 0.5 mm; the used polishing forces were 5 N, 8 N, 10 N, and 12 N.

To examine the influence of tool roughness, polishing experiments were conducted using tool B and two additional tools (tools C and D), which shared the same size and polishing layer material as tool B. The path spacing was fixed at 0.5 mm, and the polishing force was maintained at 10 N. The surface profiles of tools C and D are shown in Figure 10a and 10b, respectively. The roughness values of tools B, C, and D are summarized in

Table 2. Tool parameters of tool B, C and D.

	Tool B	Tool C	Tool D
Aluminum wheel radius	25 mm	25 mm	25 mm
Foam layer thickness	5 mm	5 mm	5 mm
Polishing layer thickness	1.20 mm	1.20 mm	1.20 mm
Polishing layer material	SL4387	SL4387	SL4387
Rms tool roughness	1.70 μm	0.87 μm	0.96 μm

.

To verify the influence of path spacing, polishing experiments were conducted with path spacings of 0.25 mm, 0.40 mm, 0.50 mm, and 0.60 mm, while maintaining a polishing force of 10 N. According to the analysis, a smoother polishing tool yields better polishing results; therefore, tool C was selected for these experiments.

4. Results and Discussion

4.1. Parameters Acquisition

In this part, the value of f_h, k_p, k_f and k_E were determined from experimental results for the MFE model. Two EN mirrors were polished using Tool A and Tool B, respectively. The first step was to obtain the f_h through one-dimensional PSD analysis in Z_smootheffect direction. Figure 11 shows the one-dimensional PSDs of the surfaces polished with tool A and tool B. From the PSD curve, it can be seen that there are structures with periods greater than 200 μm on both surfaces. The f_h can be obtained as 5.7 mm⁻¹.

Then, the evolution of surface roughness, Z_path, Z_roughcontact and Z_smootheffect was recorded, as shown in Figure 12. When the removal depth reached 1 μm, the surface MFEs converged to a stable value. For the surface polished with Tool A (Figure 12a), the Z_smootheffect and Z_path results were comparable, whereas for the surface polished with Tool B, Z_path dominated the surface profile. This difference arose from the significantly higher hardness of the polishing layer of Tool B compared with that of Tool A.

By substituting the measured results listed in

Table 3. The Z_path, Z_roughcontact and Z_smootheffect with Tool A and Tool B.

	Tool A	Tool B
Rms(Zpath)	0.95 nm	2.41 nm
Rms(Zroughcontact)	0.33 nm	0.51 nm
Rms(Zsmootheffect)	0.69 nm	0.35 nm

into Equations (8) and (18), the k_p and k_E of tool A can be calculated as 55.9 and 1.49, respectively. The k_p and k_E of tool B can be calculated as 151.3 and 2.28, respectively.

By substituting the measured results listed in Table 3 into Equation (14), the k_f of tool A can be calculated as follows:

$$k_f=\left\{\begin{array}{l}\begin{array}{cc}0& f\ge f_h\end{array}\\ \begin{array}{cc}{\displaystyle \frac{6.53}{Rms(Z_1)}}& f<f_h\end{array}\end{array}\right.$$

(20)

The k_f of tool B can be calculated as follows:

$$k_f=\left\{\begin{array}{l}\begin{array}{cc}0& f\ge f_h\end{array}\\ \begin{array}{cc}{\displaystyle \frac{3.47}{Rms(Z_1)}}& f<f_h\end{array}\end{array}\right.$$

(21)

Using the calculated parameters, the modeled profiles for Tool A and Tool B are presented in Figure 13a and 13b, respectively. Figure 13c,d shows the measured surface profiles of the polished samples. The predicted profiles have a similar structure to the measured data.

According to the analysis in Section 2, when the initial Z_smootheffect is sufficiently small, its change during polishing can be neglected. To verify this conclusion, another EN mirror with a smaller initial roughness was polished using Tool A; the evolution of the MFEs is shown in Figure 14. The initial Z_smootheffect was 0.57 nm Rms and remained stable throughout polishing. This result is consistent with the analysis.

4.2. Model Validation

In this section, simulations were conducted using the empirical parameters k_p, k_f and k_E obtained in Section 4.1 to verify the accuracy of the proposed model. The Rms value of the MFEs is used as the evaluation metric. The validation study focused on three aspects: the influence of polishing force, tool roughness, and path spacing.

4.2.1. The Influence of Polishing Force

To verify the accuracy of our model, the predicted results were compared with experimental data and with the predictions of an MFEs model that considers only the path influence. The results are shown in Figure 15. The prediction errors of both models are close to zero when the polishing force is 8 N. For other polishing forces, the errors of our model are much smaller. This is because, in the parameter acquisition experiment in Section 4.1, a polishing force of 8 N was used. The prediction errors of our model are less than 0.1 nm, as listed in

Table 4. Comparison between measured results and the model-predicted results.

Force (N)	5	8	10	12
Measure MFEs (nm)	1.33 ± 0.04	1.25 ± 0.03	1.23 ± 0.02	1.24 ± 0.03
Predicted MFEs (nm)	1.37	1.26	1.24	1.25
Error of predication (nm)	−0.04 ± 0.04	0.00 ± 0.03	−0.01 ± 0.02	−0.01 ± 0.02

. Both the measured data and our model predictions consistently show that increasing the polishing force initially reduces the MFE; however, further increases lead to a rise in MFE. The MFEs model that considers only the path influence predicts that the MFE monotonically decreases with the increase in B. This indicates that our model is closer to the real situation.

According to the analysis in Section 2, the polishing force influences Z_path, Z_roughcontact and Z_smootheffect. Therefore, the values predicted by the model were compared with experimental results, as shown in Figure 16. Both the Z_roughcontact and Z_smootheffect increase with the increase in polishing force, while the Z_path decreases. The figure demonstrates good agreement between predicted and measured data, confirming the accuracy of our model.

4.2.2. The Influence of Path Spacing

The experiments in this section were performed using Tool B. Figure 17 presents the measured results alongside the model predictions. The model indicates that the final MFEs decrease as the path spacing is reduced. When the polishing spacing is sufficiently large, the MFE caused by the path dominates the overall MFEs (as shown in Figure 13b,d). In this case, our model’s predictions closely match those of the path-only model, with errors less than 0.2 nm compared to the measurements (Figure 17b).

When the polishing spacing is below 0.3 mm, Z_roughcontact and Z_smootheffect become significant relative to Z_path. Under these conditions, the path-only model underestimates the MFEs compared to experimental results, whereas the error between our model and the measurements still remains below 0.1 nm.

Figure 18a shows a representative measured surface profile at a path spacing of 0.25 mm, with the corresponding model prediction shown in Figure 18b. The periodic features caused by the path in the measured profile are less continuous and pronounced than in the prediction. We speculate that this discrepancy arises from random polishing errors, resulting in measured values that are smaller than predicted.

4.2.3. The Influence of Tool Roughness

Experiments were conducted using three different polishing tools: Tool B, Tool C, and Tool D. Considering that tool roughness affects only Z_roughcontact, only the Z_roughcontact predicted by our model was compared with experimental results, as shown in Figure 18. Both the simulation and experimental results indicate that Z_roughcontact will monotonically increase with the increase in tool roughness, as shown in Figure 19a.

Although the error of the model is still less than 0.1 nm, as shown in Figure 19b, the relative error has reached nearly 30%. Figure 20 shows the one-dimension PSDs in the Z_roughcontact direction of the surface polished with tool C and tool D. Evidently, the high-frequency part shows an upward tilt, which means that the measurement noise cannot be ignored. Therefore, the relative error of the model prediction may be caused by various factors, including measurement noise, model simplification, human errors, etc. Considering that the final error value is not very large, we still believe that the model’s prediction results are reliable.

4.3. Further Discussion: Engineering Application of Model

In the ultra-precision manufacturing of EN mirrors for X-ray applications, LFEs, MFEs, and HFEs must all be controlled to the nanometer or sub-nanometer level. Traditional computer-controlled optical surfacing methods can achieve nanometer-level LFEs and sub-nanometer-level HFEs, but face challenges in effectively suppressing MFEs. Semi-rigid tool polishing has been widely adopted for mitigating MFEs. When combined with CCOS, semi-rigid tool polishing enables full-spectrum surface error correction, eliminating the need for additional polishing processes.

The MFE prediction model established in this study incorporates both macro- and micro-scale contact behaviors. It not only enables accurate prediction of final MFEs based on selected process parameters but also provides theoretical guidance and supports process optimization for semi-rigid tool polishing, facilitating a balance between machining accuracy and efficiency. For instance, Figure 21 illustrates the predicted MFEs under various processing conditions of Tool A. If the target MFE is set to be less than 1.30 nm Rms, a corresponding threshold plane can be drawn in Figure 20. The parameter sets yielding predicted MFEs below this threshold can then be selected for the polishing process. By integrating the TRF model, the optimal polishing parameters that meet the MFE requirement with the highest remove rate can be determined.

Figure 21 marks the parameters for the optimal material removal rate (polishing force of 12 N, path gap of 0.5 mm) and optimal roughness (polishing force of 5 N, path gap of 0.25 mm) within the limitations of our processing instrument. Model simulations predicted that the “highest remove rate” parameters yield an MFE of 1.22 nm (Rms), while the “best roughness” parameters yield an MFE of 0.64 nm (Rms). Experiments were conducted using these parameters; the results are shown in Figure 22. The measured results show that the “highest remove rate” parameters yield an MFE of 1.24 nm (Rms), while the “best roughness” parameters yield an MFE of 0.59 nm (Rms). The experimental results are in close agreement with the predictions.

5. Conclusions

Semi-rigid tool polishing is widely used to remove the mid-frequency profile errors on the electroless nickel surface that are introduced by the diamond-turning process. In this paper, an MFE prediction model for the semi-rigid tool polishing of a diamond-turned electroless nickel surface is developed based on the multi-scale contact theory to guide the polishing parameters and tool selection. The main conclusions are as follows:

• The MFE prediction model was developed by incorporating a Hertzian contact, rough surface contact, and bridging model. The key parameters were categorized into three groups: tool parameters, polishing parameters, and empirical parameters. The empirical parameters are associated with the structural properties of the polishing tool and the characteristics of the polishing slurry. Once these empirical parameters are determined through polishing experiments, the model can be used to guide the optimized selection of polishing and tool parameters for improved process performance;
• The effects of tool roughness, polishing force, and path spacing were systematically investigated in this study. The model indicates that reduced tool roughness and smaller path spacing contribute to MFEs. In contrast, the relationship between the polishing force and MFEs is non-linear: The MFEs initially decrease with increasing force but begin to rise once the force is larger than a certain value;
• Polishing experiments were carried out on diamond-turned electroless nickel plates to validate the proposed model. The predicted results show good agreement with the experimental data. Guided by this model, the “best roughness” polishing parameter within the limitations of our processing instrument was selected for the polishing experiments. The surface roughness achieved was 0.59 nm (Rms), measured over a 1.26 mm × 0.94 mm area, which closely matched the predicted value of 0.64 nm (Rms).

This study provides preliminary verification that the smoothing effect of the polishing tool, the rough surface contact between the tool and the mirror, and the Hertzian contact between the tool and the mirror all contribute to the mid-frequency errors of the final polished surface. Compared with models that consider only Hertzian contact between the mold and the mirror, our model demonstrates a significant improvement in predictive accuracy, particularly for tools with lower hardness. Although all experiments in this work were conducted using wheel-shaped tools, the model can be readily extended to polishing tools of other geometries. Furthermore, while this study explained the influence of smoothing effects and tool–mirror rough surface contact on MFEs, thereby deepening the understanding of their origins, the evaluation of the spatial-frequency ranges associated with each effect remains largely empirical. In addition, the velocity was regarded as stable in this study and the influence of velocity was not discussed. Future work will focus on quantitatively determining the spatial-frequency ranges and the influence of velocity.