Sparse bi-step raster path for suppressing the mid-spatial-frequency error by fluid jet polishing

Kuiping Wan; Kuiping Wan; Songlin Wan; Songlin Wan; Songlin Wan; Songlin Wan; Chen Jiang; Chen Jiang; Chaoyang Wei; Chaoyang Wei; Chaoyang Wei; Chaoyang Wei; Jianda Shao; Jianda Shao; Jianda Shao

doi:10.1364/OE.453122

1. Introduction

With the development of modern optical technology, the precision optical components are employed widely in the fields of lithography, astronomical observation, and optoelectronics [1–3]. To meet the urgent processing demand for modern ultra-precision optical components, the most important technology is the computer-controlled optical surfacing (CCOS) first proposed by Itek Inc [4], and then bonnet polishing, magnetorheological finishing and Ion-beam figuring [5–7], etc. were further derived. The common characteristics of these methods are that the size of the polishing tool is much smaller than that of the workpiece, which is prone to generate mid-spatial-frequency (MSF) error [8] (ripple error, spatial-period of error generally between 0.08 mm and 3 mm [9]) due to the periodic path and the convolution effect. In high-precision imaging or lithography systems, MSF error will cause small-angle scattering and reduce imaging contrast, resulting in reduced imaging quality [10]; in high power laser physics area, MSF error can even produce flare spots and damage the optical components [11]. Therefore, it is of great significance to investigate and control the MSF error in optical processing field.

To address this challenge, researchers have conducted in-depth research on the generation mechanism of MSF error to restrain the generation of MSF error during processing [12]. At present, there are two general ways to restrain the MSF error. One is to use the hard tools for the smooth processing [13]; another is processed with the special paths planning [14–18]. The mechanism of the smoothing processing has been revealed by Li et al. [13]; but this method is very time-consuming sacrifices form accuracy. The pseudo-random path first proposed by Dunn and Walker was attempted to reduce MSF error [14], and then improved forms such as the random maze path [15] and 8-directions pseudo-random path [16] were further derived; but these methods also sacrifice the low-spatial-frequency (LSF) error and have strict requirements for machine tools. In addition, reducing the path step is considered another way to restrain the MSF error [17,18]. Hu et al. proposed a step-adaptive Archimedes path that reduces the MSF error by adjusting the path step according to the material removal amount [18]. However, the path step cannot be infinitely reduced owing to the limitation of the machine tool performance, as well as the residual MSF ripple that is sustained even when the path step is reduced to 0.5 mm. Thus, it is urgent to develop a processing method that effectively suppress the MSF error without affecting the surface form of the workpiece.

The fluid jet polishing (FJP) proposed by Faehnle et al. is a promising polishing method, which relies on the pressure mixing of water and abrasive fluid interacting with the workpiece to develop material removal [19]. Compared with other polishing methods, FJP is a non-contact, no tool wear, no workpiece heating processing method and has a high processing accuracy [20,21]. However, the small size of TIF and the low processing efficiency of FJP have made it difficult to be widely used. At present, researchers mainly use it to improve surface roughness. Li et al. removed the grooves generated by single-point diamond processing by FJP [22]; Peng et al. used FJP to uniformly remove the tiny tails after magnetorheological finishing (MRF) processing [23]. These research works have verified the effectiveness of FJP in improving roughness; but the processing time requires 22.3 h processing for a 28 mm diameter workpiece [23], which is still hard to be widely applicated. Wang et al. presented a multi-jet polishing based on FJP, which can implement a higher efficiency [20]; but larger flow rate is needed in multi-jet polishing, the performance requirement of the pump is still facing severe challenges. Hereafter, we investigate the possibility of using FJP to suppress MSF error by combining the spectral characteristics of the TIF and MSF error structure. The proposed method in this paper extends the range of application of FJP and has instructional significance to the suppression of MSF error.

This paper is organized as follows. The processing theory of sparse “Bi-Step Raster Path” (BSRP) are analyzed in Section 2. Then the experiment setup and process are introduced in Section 3. Section 4 illustrates the experiment results to demonstrate the validity of the method. Finally, the paper is summarized in Section 5.

2. Analysis of processing theory

2.1 Theoretical background of suppressing the MSF error by FJP

In deterministic optical polishing, it is generally accepted that the process follows the Preston equation [24]:

(1)$$\textrm{d}z(x,y) = k \cdot P(x,y,t) \cdot V(x,y,t) \cdot \textrm{d}t$$

where z(x, y) is the amount of material removal, and P(x, y, t) is the pressure in the contact region and the V(x, y, t) denotes the relative velocity between the tool head and the workpiece; k is the Preston coefficient, which is determined by the specific processing conditions.

The instantaneous material removal rate of the tool head cannot be determined during the processing; to simplify the process of calculation, the removal amount can be expressed as the convolution of the TIF per unit time and the dwell time:

(2)$$z(x,y) = f(x,y) \otimes g(x,y)$$

where z(x, y) is amount of material removal, f(x, y) is TIF and g(x, y) is dwell time, ${\otimes}$ represent convolution symbol.

It can be derived from Eq. (2), in the actual processing of deterministic processing, the amount of material removal is the convolution of the TIF and the dwell time along the processing path. When processing along the deterministic periodic processing path, small-scale wavy-shaped surface errors will be superimposed due to the convolution effect between adjacent paths, which are called MSF error such as shown in Fig. 1.

Fig. 1. Schematic diagram of the formation of periodic surface error.

Download Full Size | PDF

As shown in Fig. 2, the MSF error in the region of the fused silica glass surface with a period interval of 1 mm can be found to be sinusoidal-like in its two-dimensional profile. Combining Fig. 1 and Fig. 2, the height of the resulting MSF error is commonly in nanoscale level and the volume removal rate of FJP is usually 0.027 mm³/min, which makes it possible to accurately remove the MSF error peaks without deteriorating the surface form.

Fig. 2. Schematic diagram of the surface structure of MSF error.

Download Full Size | PDF

2.2 Derivation of SSRP and BSRP processing models

Most of MSF errors that occur in actual polishing are periodic structures as shown in Fig. 2; therefore, an intuitive idea is to add a processing path with the same period as the MSF error to ensure that the removal amount is precisely located at the high point of the MSF error to form a sparse raster path with equal steps as shown in Fig. 3; we named it sparse “Single-Step Raster Path” (SSRP) method. Consequently, the challenges of the SSRP method are to accurately determine the path position and the dwell time. In order to determine these key parameters, according to Eq. (2), the material removal in deterministic processing can be described by the convolution of the TIF and the dwell time; but the process involving convolution is hard to obtain high accuracy path parameters; thus, we choose to analyze this process from the frequency spectrum by Fourier transform. According to the time domain convolution theorem, it can be expressed as:

(3)$$\begin{aligned} z(x,y) &= f(x,y) \otimes g(x,y)\\ &= {{{\cal F}}^{ - 1}}[{{\cal F}}(f) \times {{\cal F}}(g)] \end{aligned}$$

where z(x, y), f(x, y), g(x, y), ${{\cal F}}$ and ${\otimes}$ represent the amount of material removal, tool influence function, path function (processing path and dwell time), Fourier transform symbol and convolution symbol, respectively.

Fig. 3. Schematic diagram of the sparse SSRP method to suppress MSF error. (a) Spatial domain representation of the SSRP. (b) Frequency domain representation of the SSRP.

Download Full Size | PDF

In order to determine the exact position of the path, we obtain the position information of the TIF and the initial position of the processing path by introducing a complex argument. In order to both consider the real and imaginary parts of the spectral value, the complex least squares method is used to characterize the magnitude the MSF error. Hereafter, the optimization function of the sparse SSRP method can be expressed as:

(4)$$\begin{aligned} &{G_S}(x,y) = {\bigg \|}{{{\cal F}}[{z_0}]\left|{_{\omega = \frac{{2n\pi }}{{{T_1}}}}} \right.} {\bigg \|}\cdot {e^{i{\varphi _{z\omega }}}} - {\bigg \|}{{{\cal F}}[f]\left|{_{\omega = \frac{{2n\pi }}{{{T_1}}}}} \right.} {\bigg \|}\cdot {e^{i{\varphi _{f\omega }}}} \cdot {\bigg \|}{{{\cal F}}[{g_0}]\left|{_{\omega = \frac{{2n\pi }}{{{T_1}}}}} \right.} {\bigg \|}\cdot {e^{i{\varphi _{g\omega }}}} \cdot {e^{ - i\omega {x_1}}} \cdot {k_0}\\ &\min {S_S}({x_1};{k_0}) = \sum\limits_{n = 1}^\infty {\{{{G_S}(x,y) \cdot \overline {{G_S}(x,y)} } \}} \\ &s.t\;\; {x_1} \in [0,{T_1}];{k_0} \in (0, + \infty ) \end{aligned}$$

where S_S is the sum of squares of SSRP, z₀(x, y) is the data matrix of the MSF error; f(x, y), g₀(x, y) respectively represent the TIF and processing path used to generate the MSF error; φ_zω and φ_fω respectively represent the arguments of the corresponding frequency spectrum values of MSF error and TIF; the || || means modular operation; x₁ and k₀ are the compensated distance and the processing coefficient (consisting of dwell time and other factors) of the SSRP method, respectively.

In order to better guide the processing as well as the subsequent model improvement, it is necessary to make an exploration of the essence of the sparse SSRP method to suppress MSF error. The above analysis shows that the main point of SSRP lies in the variation of its path; thus, we propose to express this process by the change on the spectrum of the processing path. For the MSF error with periodic ripple structure, the sequence of periodic unit impulse (comb function) as shown in Fig. 4(a) is used to represent the processing path, with the expression shown as:

(5)$$\begin{aligned} &{g_0}(x,y) \textrm{ = } {\delta _T}(t) = \sum\limits_{n ={-} \infty }^\infty {\delta (t - n{T_1})} \textrm{ = } \sum\limits_{n ={-} \infty }^\infty {{F_n} \cdot {e^{jn{\omega _1}t}}} \\ &{F_n} = \frac{1}{{{T_1}}}\int\limits_{{T_1}} {{\delta _T}(t) \cdot {e^{ - jn{\omega _1}t}}\textrm{d}t = } \frac{1}{{{T_1}}}{F_0}(\omega )|{_{\omega = n{\omega_1}}} \end{aligned}$$

Fig. 4. Analysis of sparse SSRP and BSRP processing paths under ideal conditions. (a) Time domain representation of the processing path that generates MSF error. (b1) Time domain representation of processing path of the SSRP. (b2) Time domain representation of processing path of the BSRP. (c1) Frequency domain representation of the MSF error. (c2) Frequency domain representation of the processing path that generates MSF error. (d1) Frequency domain representation of processing path of the SSRP method. (d2) Frequency domain representation of processing path of the BSRP method.

Download Full Size | PDF

The Fourier transform of comb function is shown in Eq. (6).

(6)$$\begin{aligned} &{{\cal F}}[{g_0}(x,y)] = {{\cal F}}[{\delta _T}(t)] = \frac{1}{{{T_1}}}\sum\limits_{n ={-} \infty }^\infty {{F_0}(\omega ) \cdot } {{\cal F}}[{e^{jn{\omega _1}t}}] = \frac{{2\pi }}{{{T_1}}}\sum\limits_{n ={-} \infty }^\infty {\delta (\omega - n{\omega _1})} \\ &\textrm{with}\;\; {F_0}(\omega ) = 1 , {{\cal F}}[{e^{jn{\omega _1}t}}] = 2\pi \delta (\omega - n{\omega _1}),{\omega _1} = \frac{{2\pi }}{{{T_1}}} \end{aligned}$$

where g₀(x, y) represents the processing path used to generate the MSF error, T₁ represents the period of the MSF error, F_n is the Fourier coefficient, and n is an integer.

The idea of the sparse SSRP method is that the material removal falls to the position of the peak of MSF error. From Fig. 4(b1), in an ideal situation, if the comb function in Eq. (5) is shifted to the right by T₁/2, which can precisely reach the peak of the MSF error; that is, if the processing path of FJP is at this position, the MSF error can be partly suppressed, which can be expressed as:

(7)$${g_1}(x,y) = {g_0}(x - {T_1}/2,y) = \sum\limits_{n ={-} \infty }^\infty {\delta (t - n{T_1} - {T_1}/2)} $$

According to the time-shift property of the Fourier transform, the Fourier transform of Eq. (7) is multiplying by e^−jωT1/2 on the spectrum of Eq. (6), and the result is shown in Eq. (8).

(8)$$\begin{aligned} &{{\cal F}}[{g_1}(x,y)] = {{\cal F}}[{g_0}(x,y)] \cdot {e^{\textrm{ - }j\omega {T_1}/2}} = \frac{{2\pi }}{{{T_1}}}\sum\limits_{n ={-} \infty }^\infty {\delta (\omega - n{\omega _1})} \cdot {e^{\textrm{ - }j\omega {T_1}/2}} \\ &{e^{\textrm{ - }j\omega {T_1}/2}} \textrm{ = } \cos (\omega {T_1}/2) - j\sin (\omega {T_1}/2) \textrm{ = } \left\{ \begin{array}{l} 1, \omega = \frac{{2\pi }}{{{T_1}}} \cdot 2n\\ - 1,\omega = \frac{{2\pi }}{{{T_1}}} \cdot (2n + 1) \end{array} \right.; n\textrm{ is a positive integer}\textrm{.}\\ &\textrm{with} \left\{ \begin{array}{l} {e^{jx}} = \cos x + j\sin x\\ {e^{ - jx}} = \cos x - j\sin x \end{array} \right. \end{aligned}$$

After multiplying the spectrum of Fig. 4(c2) by e^−jωT1/2, the spectral values in the odd frequency band become negative; thus, the first positive spectral peak of the MSF error in Fig. 4(c1) can be suppressed by the first negative spectral peak in Fig. 4(d1). It also illustrates the feasibility of analyzing the polishing process in the frequency domain. However, from Fig. 4(c1) and Fig. 4(d1) can be seen that their second spectral peaks are both positive may bring about superimposed effects, which indicates that using the SSRP method can only suppress the first spectral peak on the MSF error frequency domain. This phenomenon may result in the inability to completely remove the MSF error and still have residues.

Therefore, in order to achieve better suppression of MSF error, we proposed the idea of suppressing the MSF error with FJP as shown in Fig. 5. TIF takes two paths at a specific position so that the removal amount accurately falls to the high point of MSF error, and the specific step interval between the adjacent paths is uncertain; thus, we proposed an optimal step, and it is calculated by the constraints to make the best effect of suppressing the MSF error. It also means that the processing path of FJP is the sparse raster path at two different step intervals, which called the sparse “Bi-Step Raster Path” (BSRP) method. For the BSRP method, it can be theoretically applied to more complex scenarios because of the extra path, which may limit its application effect if both paths correspond to the same processing coefficient. Therefore, we further extend the application of the model by envisioning the use of different processing coefficients for each path to achieve better results in MSF error removal. Finally, the process of using the BSRP method to suppress MSF error can be expressed as:

(9)$$\begin{aligned} &{G_B}(x,y) = {\bigg \|}{{{\cal F}}[{z_0}]\left|{_{\omega = \frac{{2n\pi }}{{{T_1}}}}} \right.} {\bigg \|}\cdot {e^{i{\varphi _{z\omega }}}} - {\bigg \|}{{{\cal F}}[f]\left|{_{\omega = \frac{{2n\pi }}{{{T_1}}}}} \right.} {\bigg \|}\cdot {e^{i{\varphi _{f\omega }}}} \cdot {\bigg \|}{{{\cal F}}[{g_0}]\left|{_{\omega = \frac{{2n\pi }}{{{T_1}}}}} \right.} {\bigg \|}\cdot {e^{i{\varphi _{g\omega }}}}\\ &\qquad \qquad \cdot ({e^{ - i\omega ({x_2} - {t_0}/2)}} \cdot {k_1} + {e^{ - i\omega ({x_2} + {t_0}/2)}} \cdot {k_2})\\ &\min {S_B}({x_2};{k_1};{k_2};{t_0}) = \sum\limits_{n = 1}^\infty {\{{{G_B}(x,y) \cdot \overline {{G_B}(x,y)} } \}} \\ &s.t \;\;{x_2} \in [0,{T_1}];{k_1},{k_2} \in (0, + \infty );{t_0} \in (0,{T_1}/2) \end{aligned}$$

where S_B is the sum of squares of BSRP, t₀ is the optimal step distance between the two processing paths to suppress the MSF error; k₁, k₂ are the processing coefficients corresponding to each of the two processing paths.

Fig. 5. Schematic diagram of the sparse BSRP method to suppress MSF error. (a) Spatial domain representation of the BSRP. (b) Frequency domain representation of the BSRP.

Download Full Size | PDF

In the same way, the change on the spectrum of the processing path mentioned above can be used to explore the essence of the sparse BSRP method to suppress MSF error. Taking the valley position of MSF error as the initial position of the processing path, shift to the processing direction by T₁/2 reach the position of peak of the MSF error, and then shift the distance of t₀/2 to the left and right; so that the position of processing path of the BSRP method is determined, the time domain and frequency domain information can be shown in Eq. (10) and Eq. (11).

(10)$$\begin{aligned} {g_2}(x,y) &= {g_0}(x - ({T_1}/2 - {t_0}/2),y) + {g_0}(x - ({T_1}/2 + {t_0}/2),y)\\ &= \sum\limits_{n ={-} \infty }^\infty {\delta (t - ({T_1}/2 - {t_0}/2))} + \sum\limits_{n ={-} \infty }^\infty {\delta (t - ({T_1}/2 + {t_0}/2))} \end{aligned}$$

(11)$$\begin{aligned} {{\cal F}}[{g_2}(x,y)] &= {{\cal F}}[{g_0}(x,y)] \cdot ({e^{ - j\omega ({T_1}/2 - {t_0}/2)}} + {e^{ - j\omega ({T_1}/2 + {t_0}/2)}})\\ &= \frac{{2\pi }}{{{T_1}}}\sum\limits_{n ={-} \infty }^\infty {\delta (\omega - n{\omega _1})} \cdot {e^{ - j\omega {T_1}/2}} \cdot 2\cos (\omega {t_0}/2) \end{aligned}$$

where g₂(x, y) represents the processing path of the sparse BSRP method, t₀ is the optimal step distance between the two processing paths to suppress MSF error.

Comparing Fig. 4(d1) and Fig. 4(d2), the BSRP method takes one more path than the SSRP method. It appears that the spectrum of the BSRP method is in fact the product of the spectrum of SSRP and the frequency of the cosine function of an angular velocity t₀/2. Therefore, the degree of decay of its spectrum can be controlled and more peaks of the MSF error can be minimized by adjusting t₀. As a result, theoretical derivation and analysis have shown that the use of FJP through the sparse BSRP method probably have a better effect of suppressing the MSF error than sparse SSRP method.

We will follow up with further discussion on the ability of the sparse BSRP method to suppress MSF error, thus it is necessary to elaborate on the determination of several crucial parameters in Eq. (9). When accurately determining the position of the processing path, if only the modulus of the spectrum is considered to calculate the relevant parameters, the position information will be missing; therefore, the spectrum can be represented in the form of modulus and argument in Eq. (4) and Eq. (9). In addition, considering that in the actual processing, because of various errors and the influence of the shape of the TIF (there will be different degrees of center of gravity offset error), the processing path may not be necessary to at the position of the peaks of MSF error to suppress the MSF error to the minimum value. Therefore, we proposed a concept of compensation distance x_i, the position of the path used to generate the MSF error as the initial position (the valley position of MSF error) of the processing path; and then offset a compensation distance (which can be calculated) in the processing direction to reach the exact position of the peak of MSF error as the final position of the processing path. Furthermore, because the TIF of FJP is special, some data processing is required when determining the spectrum information of TIF in Eq. (4) and Eq. (9). In Fig. 6(d), the spectrum information of the original TIF is performed directly cannot express the information of individual frequencies, which is caused by the fact that the size of TIF of FJP is generally small and the number of samples is not sufficient. The TIF essentially is zero in extension area, therefore, the TIF matrix should be expanded with zeros to ensure that the spectrum is not distorted (Fig. 6(b)). The two-dimensional spectrogram obtained after the matrix expansion can express the information of each frequency in detail (Fig. 6(c)). However, the obtained spectrum information by the expanded matrix may be different from the original spectrum of the TIF; it is found that there is indeed a phenomenon of the same trend of change and a multiplicative relationship between the specific values after the uniform sampling rate (Fig. 6(e)). In addition, because of the characteristics of the two-dimensional fast Fourier transform, the obtained frequency spectrum is not the actual frequency spectrum; thus, there is also a transformation relationship to calculate the actual frequency spectrum. Combining the above two aspects, the true spectral amplitude can be found using Eq. (12).

(12)$${Y_R} = \frac{{4Y}}{{M \cdot N \cdot {a^2}}}$$

where YR, Y, M, N and a respectively represent the actual spectrum value, the transformed spectrum value, the number of rows of the expanded matrix, the number of columns of the expanded matrix and the magnification of the original TIF matrix.

Fig. 6. Fourier transform process of tool influence function. (a) Schematic diagram of the material removal process of FJP. (b) The transform process of matrix expansion of TIF. (c) 2-D spectrogram of TIF. (d) 1-D spectrogram of different sampling rates of TIF. (e) 1-D spectrogram after unified sampling rate.

Download Full Size | PDF

With the spectral information of various parameters, substituting them into Eq. (9), which is to reversely calculate the optimal value of the compensation distance x_i, processing coefficients k_i and optimal step t₀ by gradient descent method. Under the condition that the optimal parameters are obtained, the MSF error can be suppressed to a minimum by the sparse BSRP method.

2.3 Theoretical verification

To verify the ability of the sparse BSRP method to suppress the MSF error, we analyzed its suppression effect on common types of MSF error by means of theoretical modeling. There are several main types of periodic structure of MSF errors that generated in actual processing; and they are divided into the standard sine curve (Type-I), the irregular periodic micro-ripples with narrow peaks and wide valleys (Type-II) and wide peaks and narrow valleys (Type-III). Then, three theoretical examples will be used to analyze the ability of the SSRP and BSRP methods to suppress MSF error. We firstly use the specific function to simulate the MSF error of Type-I with a period of T₁= 1 mm as shown in Fig. 7(a), and then find out its spectral value; because its spectral value decays quickly, thus we need to find out its spectral value of the first two frequency bands (ω = 2π, ω = 4π). The TIF used in the simulation process are all consistent, thus the corresponding spectrum values of the first two frequency bands can be calculated according to Fig. 6 combined with Eq. (12). The position of the first valley of MSF error as the initial position of the processing path and also find the spectral value of its corresponding frequency band. The above results are shown in Table 1.

Fig. 7. Theoretical comparison between sparse SSRP and BSRP methods. (a) Stimulated removal of MSF error of standard sinusoidal (Type-I) by sparse SSRP and BSRP methods. (b) Stimulated removal of MSF error of narrow peaks and wide valleys (Type-II) by sparse SSRP and BSRP methods. (c) Stimulated removal of MSF error of wide peaks and narrow valleys (Type-III) by sparse SSRP and BSRP methods.

Download Full Size | PDF

Table 1. The spectral values used in the theoretical suppression of three types of MSF error

View Table | View all tables in this article

If the sparse SSRP method is used to suppress MSF error, the compensation distance x₁= 0.36 mm and the processing coefficient k₀= 1.8 can be calculated by substituting the corresponding parameter values in Table 1 into Eq. (4) and using the gradient descent method, and the effect that can be achieved is shown in Fig. 7(a). The root mean square (RMS) of MSF error is reduced from the initial 5 nm to 1.4 nm, which is a 72% improvement. When the sparse BSRP method is used to suppress this type of MSF error, the relevant parameters in Table 1 are substituted into Eq. (9) to obtain x₂= 0.27 mm, t₀= 0.18 mm, k₁= 1.25, k₂= 1.1, which can achieve the effect of reducing the RMS of MSF error from 5 nm to 0.8 nm, an improvement of about 84%. It can be seen that the effect achieved by using the BSRP method can be improved by 42.9% on top of the effect achieved by the SSRP method. As shown in Fig. 7(b), we used the same idea to analyze the MSF error of Type II, whose relevant parameters are all shown in Table 1. Suppression of MSF error of this type by the SSRP method can reduce its RMS from 4.1 nm to 1.5 nm, and it can be reduced to 1 nm if the BSRP method is used, which is a 33.3% improvement over the SSRP method. Similarly, for the MSF error of Type III shown in Fig. 7(c), the suppression effect can be improved by 56.9% with the SSRP method and by 77.6% with the BSRP method, which can be seen that the BSRP method improves nearly 48% over the SSRP method. The specific suppression effects are compared in Table 2. By simulating and analyzing the suppression of the above three different types of MSF errors with the SSRP and BSRP methods it can be seen that the BSRP method always has a better suppression effect than the SSRP method, and the difference in the enhancement effect will be greater as the wave peak of MSF error becomes wider; all in all, the sparse BSRP method has more advantages in suppressing MSF error.

Table 2. Comparison of theoretical suppression of three types of MSF error by SSRP and BSRP methods

View Table | View all tables in this article

3. Experimental verification

To further verify the effectiveness of using FJP to suppress MSF error through the sparse BSRP method, the fluid jet polishing equipment as shown in Fig. 8(a) was used as an experimental platform for verification experiments. The polishing fluid used in the experiment is iron oxide abrasive with average particle size of 1.5 µm, deionized water and some dispersants to prepare iron oxide polishing fluid with a concentration of 10 wt.%. The polishing fluid might be deposited in process chamber during the experiment, which would interfere with the experimental results if the workpiece was placed horizontally; thus, the workpiece was fixed to the clamp by tape, and the clamp was then mounted vertically on the process chamber (Fig. 8(c)). Commonly used nozzles are cylindrical and cone-column-shaped, etc. According to Ref. [25], we choose the column-shaped nozzle for better effect in terms of uniform abrasive concentration distribution and jet convergence [25]. A cone-column-shaped nozzle with 1 mm caliber as shown in Fig. 8(b) was designed to suppress the most common MSF error of 1 mm period. The nozzle was mounted on a six-axis robotic arm by means of a flange (Fig. 8(a)), which was capable of ensuring six degrees of freedom of movement. The abrasive was pumped from the recovery dish to the nozzle by a low-pressure diaphragm pump (the pump pressure is generally adjusted to 0.5 MPa), and the nozzle was extruded at a high speed to form a jet beam impacting the surface of the workpiece with the dwell distance between the nozzle and the surface of the workpiece is 10 mm (Fig. 8(d)). The impact angle was set to 45° to ensure the processing efficiency while the surface roughness was not damaged [21], forming a crescent shaped TIF that the size and dimensions are shown in Fig. 9(a), which could be obtained by the fixed-point dwell processing on the fused silica glass using the above experimental conditions. The main parameters of the whole experiment process were kept constant.

Fig. 8. Experimental platform for fluid jet polishing. (a) Experimental platform consisting of six-axis robot and fluid jet polishing equipment. (b) Structure and parameters of cone-column-shaped nozzle. (c) Mounting and fixing of workpiece. (d) The dwelling distance and angle of incidence of the jet beam impacting the workpiece.

Download Full Size | PDF

Fig. 9. The main parameters of TIF and processing path of experiment. (a) The shape and size of the tool influence function of FJP. (b) Processing path for suppressing the MSF error of the upper part of the quartz glass by FJP with sparse BSRP method.

Download Full Size | PDF

The experiment was carried out on a piece of fused silica glass with a diameter of 30 mm. Firstly, a jet beam was used to complete the pretreatment experiment of processing periodic micro-corrugations on the entire surface of the glass to reproduce the MSF error with period interval of 1mm. Hereafter, we used the sparse BSRP method to suppress the MSF error of the upper half of the reproduced surface of the workpiece by FJP, the processing path as shown as Fig. 9(b). Taking the position of a valley in the middle of the MSF error as the initial position of the processing path, the compensation distance x₂= 0.339 mm and the optimal step distance between two adjacent paths is t₀= 0.149 mm can be calculated by Eq. (9). In the experiment, the upper part of the workpiece was processed by the BSRP with scan step at 0.149 mm and 0.851 mm. The feed rates of the two processing paths are 0.592 mm/s and 0.609 mm/s respectively, and the total processing time to complete the processing path is 36 min.

4. Results and discussion

To explore the ability of FJP to suppress MSF error with the sparse BSRP method, the surface form errors of the workpiece were measured by laser interferometer Zygo DynaFiz during the experiments. Figure 10 shows the surface form data of the workpiece before and after the experiment. In the FJP, the high-speed abrasive particle flow produces a shear slip effect on the surface of the workpiece to remove the material. Compared with Fig. 10(a1), there are obvious periodic stripes on the surface of the workpiece in Fig. 10(b1). After applying the BSRP method to polish the upper part of the pre-processed workpiece, it can be clearly seen from Fig. 10(c1) that the MSF error can be well suppressed. The measurement results showed that the root mean square (RMS) value of the workpiece changed from 0.007λ (4.502 nm) to 0.01λ (6.507 nm), and the peak-to-valley (PV) value changed from 0.076λ (47.777 nm) to 0.083λ (52.411 nm). This phenomenon mainly because the upper part of the workpiece has no obvious peaks and valleys, while the lower half part still has obvious periodic ripples. The interferometer is full-aperture overall measurement; therefore, the overall data will deteriorate. In addition, the workpiece is fixed on the clamp by tape (Fig. 8(c)), it needs to be removed for measurement after the pretreatment experiment is completed and then fixed again for subsequent experiments. The slight difference in the fixed position of tape will also affect the measurement results. We extract the processing area separately for analysis, which forms a clear contrast effect as shown in Fig. 10(b2) and (c2). There are obvious periodic ripples before processing; however, after suppressing MSF error by the BSRP method, the periodic structure is no longer visible. At the same time, we found that the RMS values before and after processing are the same (0.007λ), which also shows that the use of BSRP to suppress MSF error by FJP will not destroy the surface form. In summary, the change of the surface form errors data is within an acceptable range, which is basically the consistent. In addition, the measurement results in Fig. 10(a2) and Fig. 10(c3) show that the surface roughness of the workpiece before and after processing changes from Ra 0.35 nm to Ra 0.39 nm (basically no change), indicating that the MSF error of the workpiece is suppressed while the surface quality is still ensured by FJP with the sparse BSRP method.

Fig. 10. The experimental results of the BSRP. (a1) Initial form map of the workpiece. (a2) Surface roughness of workpiece before processing. (b1) Form error map of the workpiece replicated by FJP. (b2) Form error map of the upper part of the workpiece to be processed on the workpiece surface. (c1) Form error map of the workpiece after suppressing the MSF error by the BSRP. (c2) Form error map of the upper part of the workpiece after suppressing the MSF error by the BSRP. (c3) Surface roughness of workpiece after processing. (d) PSD curves of the workpiece surface after pretreatment and after applying the BSRP to suppress the MSF error (the range of f_L axis is 0mm⁻¹ to 5mm⁻¹).

Download Full Size | PDF

To further analyze the effectiveness of the sparse BSRP method to suppress the MSF error, power spectrum density (PSD) analysis was done before and after processing on the workpiece surface as shown in Fig. 10(d). The result showed that the spectral peak of the periodic structure was obviously eliminated, the PSD result further confirmed the effectiveness of the BSRP method to suppress the MSF error. The above measurement and analysis results clearly show that the application of FJP through the BSRP method can effectively suppress the MSF error, and the surface form is not deteriorated. Because the material removal of the sparse BSRP method is precisely covered around the peak position of the periodic ripples, there is a little removal of MSF error, which greatly reduces the processing time. At present, the experimental results achieve the suppression of MSF error with period around 1 mm by FJP, as for other MSF error periods, the caliber of the nozzle should be changed to meet the requirement, and more in-depth researches can be conducted in the future.

5. Conclusion

In this paper, the idea is proposed to suppress the MSF error by FJP with the sparse BSRP method. The feasibility of suppressing the MSF error with FJP is theoretically analyzed starting from the sources of MSF error generation. Based on the classical theory of deterministic processing, the Fourier transform is used to analyze the ability of the sparse SSRP and BSRP method to suppress the MSF error; the mathematical models are proposed to derive the compensation distance and optimal step between adjacent paths based on the spectrum information of MSF error, TIF and processing paths. The results of the two methods to suppress different types of periodic structure MSF errors through theoretical simulation show that the BSRP method is more effective than the SSRP method to suppress MSF error.

The experiments prove that the application of the FJP through the sparse BSRP method can well suppress the MSF error while the surface form and surface roughness are not deteriorated, which is more efficient than the traditional way of suppressing the periodic ripples error. The idea proposed in this study extends the range of application of FJP and provides an effective method to suppress the MSF error, which are beneficial to the development of optical processing.

Funding

Outstanding Member of Youth Innovation Promotion Association of the Chinese Academy of Sciences; Development Project of Scientific Instruments and Equipment of the Chinese Academy of Sciences; Key projects of the Joint Fund for Astronomy of National Natural Science Funding of China (U1831211); Natural Science Foundation of Shanghai (21ZR1472000); Shanghai Sailing Program (20YF1454800).

Disclosures

K. Wan, C. Wei and S. Wan are inventors on a patent application relating to the bi-step raster path described in this work.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. D. Walker, A. Beaucamp, V. Doubrovski, C. Dunn, R. Freeman, G. McCavana, R. Morton, D. Riley, J. Simms, and X. Wei, “New results extending the precessions process to smoothing ground aspheres and producing freeform parts,” Proc. SPIE 5869, 58690E (2005). [CrossRef]

2. Y. Xu, J. Lu, X. Xu, C. C. A. Chen, and Y. Lin, “Study on high-efficient sapphire wafer processing by coupling SG-mechanical polishing and GLA-CMP,” Int. J. Mach. Tool Manufact. 130-131, 12–19 (2018). [CrossRef]

3. M. Tricard, P. Dumas, and G. Forbes, “Subaperture approaches for asphere polishing and metrology,” Proc. SPIE 5638, 284–299 (2005). [CrossRef]

4. R. A. Jones, “Optimization of computer-controlled polishing,” Appl. Opt. 16(1), 218–224 (1977). [CrossRef]

5. C. F. Cheung, L. B. Kong, L. T. Ho, and S. To, “Modelling and simulation of structure surface generation using computer controlled ultra-precision polishing,” Precis. Eng. 35(4), 574–590 (2011). [CrossRef]

6. G. Junkang, A. Beaucamp, and S. Ibaraki, “Virtual pivot alignment method and its influence to profile error in bonnet polishing,” Int. J. Mach. Tool Manufact. 122, 18–31 (2017). [CrossRef]

7. M. Das, V. Jain, and P. Ghoshdastidar, “Fluid flow analysis of magnetorheological abrasive flow finishing (MRAFF) process,” Int. J. Mach. Tool Manufact. 48(3-4), 415–426 (2008). [CrossRef]

8. W. Zhu and A. Beaucamp, “Compliant grinding and polishing: a review,” Int. J. Mach. Tool Manufact. 158, 103634 (2020). [CrossRef]

9. P. A. Baisden, L. J. Atherton, R. A. Hawley, T. A. L. J. A. Menapace, and P. E. Miller, “Large optics for the national ignition facility,” Fusion Sci. Technol. 69(1), 295–351 (2016). [CrossRef]

10. J. M. Tamkin, W. J. Dallas, and T. D. Milster, “Theory of point-spread function artifacts due to structured mid-spatial frequency surface errors,” Appl. Opt. 49(25), 4814–4824 (2010). [CrossRef]

11. D. M. Aikens, A. Roussel, and M. Bray, “Derivation of preliminary specifications for transmitted wavefront and surface roughness for large optics used in inertial confinement fusion,” Proc. SPIE 2633, 350–360 (1995). [CrossRef]

12. B. Zhong, H. Huang, X. Chen, W. Deng, and J. Wang, “Modelling and simulation of mid-spatial-frequency error generation in CCOS,” J. Eur. Opt. Soc-Rapid. 14(1), 4 (2018). [CrossRef]

13. X. Li, C. Wei, S. Zhang, W. Xu, and J. Shao, “Theoretical and experimental comparisons of the smoothing effects for different multi-layer polishing tools during computer-controlled optical surfacing,” Appl. Opt. 58(16), 4406–4413 (2019). [CrossRef]

14. C. Dunn and D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16(23), 18942–18949 (2008). [CrossRef]

15. C. Wang, Z. Wang, and Q. Xu, “Unicursal random maze tool path for computer-controlled optical surfacing,” Appl. Opt. 54(34), 10128 (2015). [CrossRef]

16. T. Wang, H. Cheng, W. Zhang, H. Yang, and W. Wu, “Restraint of path effect on optical surface in magnetorheological jet polishing,” Appl. Opt. 55(4), 935–942 (2016). [CrossRef]

17. C. Wang, W. Yang, S. Ye, Z. Wang, P. Yang, Y. Peng, Y. Guo, and Q. Xu, “Restraint of tool path ripple based on the optimization of tool step size for sub-aperture deterministic polishing,” Int. J. Adv. Manuf. Technol. 75(9-12), 1431–1438 (2014). [CrossRef]

18. H. Hu, Y. Dai, and X. Peng, “Restraint of tool path ripple based on surface error distribution and process parameters in deterministic finishing,” Opt. Express 18(22), 22973–22981 (2010). [CrossRef]

19. O. W. Fähnle, H. V. Brug, and H. J. Frankena, “Fluid jet polishing of optical surfaces,” Appl. Opt. 37(28), 6771 (1998). [CrossRef]

20. C. Wang, C. Cheung, L. Ho, M. Liu, and W. Lee, “A novel multi-jet polishing process and tool for high-efficiency polishing,” Int. J. Mach. Tool Manufact. 115, 60–73 (2017). [CrossRef]

21. H. Fang, P. Guo, and J. Yu, “Surface roughness and material removal in fluid jet polishing,” Appl. Opt. 45(17), 4012 (2006). [CrossRef]

22. Z. Li, J. Wang, and X. Peng, “Removal of single point diamond-turning marks by abrasive jet polishing,” Appl. Opt. 50(16), 2458 (2011). [CrossRef]

23. W. Peng, S. Li, C. Guan, X. Shen, Y. Dai, and Z. Wang, “Improvement of magnetorheological finishing surface quality by nanoparticle jet polishing,” Opt. Eng. 52(4), 043401 (2013). [CrossRef]

24. F. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 11, 214–256 (1927).

25. C. Shi, J. Yuan, F. Wu, and Y. Wan, “Nozzle design of fluid jet polishing,” Opto-Elec Eng. 35(12), 1003–1501 (2008).

Types of MSF	ω = 2π			ω = 4π
Types of MSF	$F [z]$ /mm·mm	$F [f]$ /nm·mm/min	$F [g_{0}]$ /mm·mm	$F [z]$ /mm·mm	$F [f]$ /nm·mm/min	$F [g_{0}]$ /mm·mm
Type-I	2.0027×10³·e^−i1.5394	0.1032·e^−i1.5394	1.0008×10⁴·eⁱ^1.6665	2.6747·eⁱ^1.6336	0.0291·e^−i2.7909	1.0003×10⁴·e⁻ⁱ^2.9501
Type-II	2.0027×10³·eⁱ^1.6028		1.0008×10⁴·e⁻ⁱ^1.4782	500.9289·e⁻ⁱ^3.0735		1.0003×10⁴·e⁻ⁱ^2.9564
Type-III	2.0027×10³·e⁻ⁱ^1.6665		1.0008×10⁴·eⁱ^1.6665	500.9289·eⁱ^0.0681		1.0003×10⁴·e⁻ⁱ^2.9501

RMS/nm	Type I	Type II	Type III
Initial RMS (R₀)	5	4.1	5.8
SSRP processing (R₁)	1.4	1.5	2.5
BSRP processing (R₂)	0.8	1.0	1.3
(R₁-R₀)/R₀×100%	−72%	−63.4%	−56.9%
(R₂-R₀)/R₀×100%	−84%	−75.6%	−77.6%
(R₂-R₁)/R₁×100%	−42.9%	−33.3%	−48%

Types of MSF	ω = 2π			ω = 4π
Types of MSF	$F [z]$ /mm·mm	$F [f]$ /nm·mm/min	$F [g_{0}]$ /mm·mm	$F [z]$ /mm·mm	$F [f]$ /nm·mm/min	$F [g_{0}]$ /mm·mm
Type-I	2.0027×10³·e^−i1.5394	0.1032·e^−i1.5394	1.0008×10⁴·eⁱ^1.6665	2.6747·eⁱ^1.6336	0.0291·e^−i2.7909	1.0003×10⁴·e⁻ⁱ^2.9501
Type-II	2.0027×10³·eⁱ^1.6028		1.0008×10⁴·e⁻ⁱ^1.4782	500.9289·e⁻ⁱ^3.0735		1.0003×10⁴·e⁻ⁱ^2.9564
Type-III	2.0027×10³·e⁻ⁱ^1.6665		1.0008×10⁴·eⁱ^1.6665	500.9289·eⁱ^0.0681		1.0003×10⁴·e⁻ⁱ^2.9501

RMS/nm	Type I	Type II	Type III
Initial RMS (R₀)	5	4.1	5.8
SSRP processing (R₁)	1.4	1.5	2.5
BSRP processing (R₂)	0.8	1.0	1.3
(R₁-R₀)/R₀×100%	−72%	−63.4%	−56.9%
(R₂-R₀)/R₀×100%	−84%	−75.6%	−77.6%
(R₂-R₁)/R₁×100%	−42.9%	−33.3%	−48%

Sparse bi-step raster path for suppressing the mid-spatial-frequency error by fluid jet polishing

Abstract

1. Introduction

2. Analysis of processing theory

2.1 Theoretical background of suppressing the MSF error by FJP

2.2 Derivation of SSRP and BSRP processing models

2.3 Theoretical verification

3. Experimental verification

4. Results and discussion

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Tables (2)

Equations (12)

Optics Express