A theoretical and experimental investigation into tool setting induced form error in diamond turning of micro-lens array

Micro-lens array (MLA) has been widely used for 3D imaging, etc., due to its excellent functional performances. Ultra-precision diamond turning (UPDT) offers a satisfying solution to the high-quality fabrication of MLA with sub-micrometric form accuracy and nanometric surface roughness. However, in UPDT tool setting, errors would deteriorate form accuracy as a crucial factor. This study focuses on discussing the tool setting effect on form error of MLA and proposes a new two-step tool setting method for UPDT. Firstly, a theoretical model was established for form error of MLA under tool setting errors. Moreover, a new two-step tool setting method was developed with high accuracy to control the tool setting errors. Finally, a series of experiments were carried out with different tool setting errors for the MLA fabrication, and its form error would be measured. The theoretical and experimental results are found that the proposed tool setting method is effective with a high-precision accuracy and the tool setting errors would crucially induce periodical form error at the MLA of UPDT. Significantly, the study draws up a comprehensive understanding of the tool setting effect on form accuracy of MLA in UPDT with the further improvement.


Introductions
Due to the excellent functional properties of micro-lens array (MLA), it has been widely used 3D imaging, illumination, sensor devices, etc. [1]. Its form accuracy crucially determines its specific application and functional performance. For the requirement of the MLA fabrication, several non-mechanical methods have been employed such as laser lithography and chemical etching [2]. Nevertheless, these methods are greatly limited by workpiece materials and processing environments [3]. Currently, ultra-precision diamond turning (UPDT) has been developed as a promising means for the high-quality fabrication of MLA with submicrometric form accuracy and nanometric surface roughness [4][5][6]. However, its form accuracy is very sensitive to the machining errors in UPDT [7].
The machining errors, including geometric errors, thermal deformation errors, control errors, and tool setting errors, have a significant influence on the form accuracy of MLA [8,9]. Much research work has been carried out to discuss the machining errors and their effects on form accuracy [10,11]. Liu et al. [12] established a machining error model for a three-axis ultra-precision machine based on the multi-body system theory and revealed the effect of geometric errors on the coordinate distortions and form accuracy. Li et al. [13] presented a systematic methodology of kinematics error for an ultra-precision machine to improve the accuracy of characterized flatness error. Besides, the tool setting errors are crucial issues in UPDT, significantly deteriorating form accuracy [14], named tool setting induced form error. Dai et al. [15] developed a force-based model for the relationship between tool setting errors and cutting force forms to predict the three-dimension form of a convex spherical surface in UPDT. Liu et al. [16] built up a machining error model to investigate the coordinate distortions induced by tool setting errors and form error of MLA. Hereby, different tool setting methods have been developed only technologically to control the form error [17].
Generally, the tool setting methods are classified into automatic types and manual types [17]. The automatic tool setting method is considered suitable for the highprecision manufacturing platforms with high resolution and fast acquisition [18]. Hou et al. [19] proposed a rapid vision method based on the position feedback from the machine to control the tool setting errors up to less than 1 µm with high efficiency. Yu et al. [20] employed an optical probe for the tool setting errors, which could be calculated by the peak-valley value under a certain distance at a tilted flat mirror. However, the automatic instruments are complex and vulnerable to the external disturbances [21]. In practice, the manual tool setting method is widely used with easy integration and technical maturity [17]. Gao et al. [22] developed a trial-cut method for diamond turning of sinusoidal grid surface up to sub-micrometric level tool setting errors. However, the trial-cut method would greatly depend on the operating skills under multiple repeats. The processing is inefficient with the inevitable re-clamping errors. Therefore, the manual tool setting method should be optimized with high accuracy and simple operation.
As aforementioned above, there is still lack of an indepth understanding of the tool setting effect on form error of MLA in UPDT. Therefore, the study focuses on discussing tool setting induced form error under a new two-step tool setting method in UPDT of MLA. Firstly, the tool setting induced form error was established and analyzed mathematically for MLA of UPDT. Secondly, a two-step tool setting method was proposed with high accuracy for UPDT of MLA. The rough tool setting was conducted by an optical instrument, and the fine tool setting was achieved by the designed groove-turning tests. Finally, the UPDT tests were carried out to verify the form error of MLA under different tool setting errors.

Modelling of tool setting induced form error of MLA
A typical micro-lens array is employed, whose microaspheric lenslet is expressed with the mathematical description Z(x, y) as [23]:  (1) where S is the shape parameter, C is the curvature parameter, R is the radius of each lenslet, K is the conic constant, and ρ is the radial operator in the local coordinate system. In UPDT of MLA, the tool path was generated in the cylindrical coordinates (x, z, α), where α is the spindle rotation angle and it was transformed into the Cartesian coordinates (xcos(α), xsin(α), z). Under the tool setting errors, consisting of the horizontal tool setting error Δx in the X axis and the vertical tool setting error Δy in the Y axis, there The relationship between L x1 and the deviation of Δx would exist the coordinate distortions (E x , E y , and E z in X, Y, and Z axis, respectively) with their directions and sizes, as shown in Table 1. It indicates that the tool setting errors only affect the coordinate distortions in the X axis and the Y axis. Moreover, the coordinate distortions under different tool setting errors are schematically shown in Fig. 1, where the arrows represent the corresponding directions. In Fig. 1a, when the tool setting error only exists in the X axis, the direction of the coordinate distortions moves along the radial direction and the size of the coordinate distortions is constant as |Δx|. In Fig. 1b, when the tool setting errors only exist in the Y axis, the direction of the coordinate distortions moves along the circumferential direction, and the size of the coordinate distortions is constant as |Δy|. In Fig. 1c, when the tool setting errors exist both in the X axis and the Y axis, the direction of the coordinate distortions move along the radial and circumferential direction, and the size of the coordinate distortions is constant as √ Δx 2 + Δy 2 . Further, according to the mathematical description of micro-aspheric lenslet in Eq. 1, the form error ΔZ(x, y)

Two-step tool setting method
The tool setting errors would produce a significant impact on the machined surfaces [24,25]. The surface residuals under different tool setting errors at the turning center are depicted in Fig. 2. The horizontal tool setting error Δx includes the tool-not-to-center error (Δx > 0) and the tool-past-center error (Δx < 0). The former could be further divided into r < Δx or 0 < Δx < r (the diamond tool nose radius r), and the corresponding surface residuals are shown in Fig. 2 a and b. The latter would not cause a surface residual upon a machined surface. The vertical tool setting error Δy contains the tool-below-center error (Δy < 0) and the tool-abovecenter error (Δy > 0). The former would result in a central cylindrical residual at a machined surface in Fig. 2c, and the latter would make a central cone residual at a machined surface in Fig. 2d. In a practical tool setting process, surface residuals would be more complicated under different tool setting errors.
To control the tool setting errors, a two-step method was developed, which contained a rough tool setting process and a fine tool setting process. The rough tool setting process was achieved by an optical instrument on the ultra-precision machine. By fitting three points from the optical tool edge image, the position of the diamond tool tip was obtained and then transmitted to the machine-tool coordinate system. The process has the advantage of easy operation and fast realization. However, limited by the spatial resolution of instrument and external disturbances, its accuracy was at ± 5 μm for the horizontal and vertical tool setting errors.
Further to improve tool setting accuracy, a new fine tool setting method was proposed by the groove-turning tests. The process began from the initial position with a horizontal tool setting error Δx and a vertical tool setting error Δy after the rough tool setting process, respectively. For the control of the horizontal tool setting error, it is realized by the first groove-turning test at a moving distance L x1 in the X axis, as schematically shown in Fig. 3. Consequently, the horizontal tool setting error Δx is expressed as where R 1 is the median radius of the first groove. The width w 1 of the first groove is expressed as where r is the tool nose radius and d is the depth of cut in Z axis. Under the condition of L x1 much more than Δy, Eq. 4 could be further simplified and represented as Under the rough tool setting condition of 5 μm for the vertical tool setting error Δy, the deviation of Δx affected    by different L x1 is obtained, as shown in Fig. 4. It shows that the deviation gradually decreases with the increase of L x1 . Significantly, the deviation is less than 0.2 μm when L x1 is more than 100 μm. Surface topography measurement was conducted on a white light interferometer (Bruker Contour GT-X), whose measurement error is ± 65 nm with the Gaussian distribution along the horizontal and vertical directions. The measurement error would produce a fitting-induced error for the R 1 , which would influence the calculation of Δx. Hereby, a statistical analysis was made to identify the relationship between L x1 and Δx, where the leastsquare method was applied for the simulated fitting process with an arc of 200 μm. The relationship between L x1 from 100 μm to 1000 μm and the deviation of Δx was calculated under 10,000 datasets, and the results are shown in Fig. 5. In Fig. 5a, the mean deviation of Δx increases with the increase of L x1 within 0.2 μm when L x1 is less than 500 μm. Figure 5b-d shows the deviation of Δx when L x1 is from 300 to 500 μm. It should be noted that the deviation of Δx is more concentrated at ± 0.2 μm when L x1 is 300 μm.
The horizontal tool setting error Δx could be identified by the first groove-turning test. To control the vertical tool setting error, it is achieved by the second grooveturning test at the moving distance L x2 in the X axis and the    Fig. 6. The vertical tool setting error Δy is expressed as where R 2 is the median radius of the second groove. Based on the width w 1 of the first groove (Eq. 4), the width w 2 of the second groove is expressed as Under the assumption of Δx of 5 μm and L x2 of 100 μm, the deviation of Δy under different L y (form 50 μm to 200 μm) was established, and the results are shown in Fig. 7. It shows that the deviation Δy is well kept at 0.2 μm when L y is 100 μm comparing with other conditions.
Overall, for the horizontal tool setting error, the moving distance L x1 should be limited at one from 100 to 300 μm to control the deviation of Δx within 0.2 μm. For the vertical tool setting error, the moving distances L x2 and L y should be also equal at one from 100 to 300 μm. Especially, the moving distance should also be selected based on the width of two grooves to avoid interference.

Tool setting experiments
The rough tool setting process was achieved by an optical instrument equipped on the employed ultra-precision machine (Precitech Freeform TL), as shown in Fig. 8a. By fitting three points from the detected optical tool edge image, the process could be automatically finished. The schematic diagram of the tool setting error distribution is shown in Fig. 8b. The fine tool setting process was achieved by the groove-turning tests. A new single crystal diamond tool was used during the cutting processes, as shown in Fig. 9, and its geometric parameters are listed at Table 2. The fine tool setting process are as follows: For the first groove-turning test, the moving distance L x1 was selected at 235 μm, and the depth of cut was 3 μm; for the second groove-turning test, the moving distances L x2 and L y were set at 100 μm, and the depth of cut was 3 μm.

MLA cutting experiments
The parameters of micro-aspheric lenslet are listed at Table 3. There are 4 × 4 lenslets with the rectangular distribution for the designed MLA, and the simulated surface topography is shown in Fig. 10a. Figure 10b shows the tool path for MLA in UPDT generated by the equal angle discretization method [26]. Its cutting experiments were carried out on the five-axis ultra-precision machine (Precitech Freeform TL). The workpiece material is the isotropic oxygen free copper at the sample aperture of 10 mm.
The spindle speed was set as 25 rpm and the feed rate was set as 0.25 mm/min. After finishing the two-step tool setting process, the tool setting errors were set manually in Table 4 for the experiments. Different tool setting errors distribute in the error zone, as shown in Fig. 8b. The tool setting error is set large enough at 10 μm for the purpose of ignoring other machining effects such as spindle vibration as far as possible. To obtain a large-scale surface topography, the measurements were performed on the white light interferometer (Bruker Contour GT-X) by stitching.

Two-step tool setting process
After the two-step tool setting process, the workpiece was detached from the vacuum sucker, and the surface topographies of the turned grooves were measured by the white light interferometer, as shown in Fig. 11a. The objective lens was selected at 50 × with the measurement area of 0.32 mm × 0.24 mm. Based on the measured surface topography datasets, the median radii (R 1 and R 2 ) of two turned grooves were fitted by the least-square method, as shown in Fig. 11b. Consequently, the horizontal tool setting error Δx and the vertical tool setting error Δy are calculated by Eq. 4 and Eq. 6. Table 5 shows the calculated tool setting errors by three repeat measurements, and the results are stable under the low standard deviation, which demonstrates the effectiveness of the proposed tool setting method. By adjusting the positions of the diamond tool tip to the spindle in the ultraprecision machine coordinates, the tool setting errors could be controlled finally.

Tool setting induced form error
The MLA cutting experiments were performed on the ultraprecision machine under the selected tool setting errors of Table 5. Surface topographies were measured by the white light interferometer. By removing the ideal form, the experimental and simulated tool setting errors induced form error of MLA were obtained in Fig. 12. The coordinates for surface height together with color-bar were set uniformly at ± 0.6 μm. Figure 12a1-d1 shows the experimental results,  Fig. 12a2-d2 presents the simulated results under the corresponding tool setting errors. In Fig. 12a1, under the horizontal tool setting error at Δx = 10 μm, the generated pattern of MLA form error moves along the radial direction. In Fig. 12b), under the vertical tool setting error at Δy = 10 μm, the generated pattern of MLA form error moves along the circumferential direction. In Fig. 12c1, under the horizontal and vertical tool setting errors at Δx = 10 μm and Δy = 10 μm, the generated pattern of MLA form error is towards the radial and circumferential direction. It should be noticed that the directions agree with the directions of the coordinate distortions in the aforementioned-above results of Fig. 1. In Fig. 12d1, under no tool setting errors, the significant pattern cannot be found. It also means that the tool setting method is high accuracy. Besides, at the boundary of micro-aspheric lenslets, there would exist minor tool servo induced form error, especially for the lenslets far away from the turning center [27]. The deviation between the simulated and experimental results was partly contributed to the material effects such as material recovery and swelling during the machining process. Significantly, there is a good consistency of the tool setting errors induced form error between the theoretical and experimental results, which well demonstrates the effectiveness of the proposed model.

Conclusions
Ultra-precision diamond turning (UPDT) provides a highefficient and low-cost solution for fabricating micro-lens array (MLA) with sub-micrometric form accuracy and nanometric surface roughness. However, the tool setting error is a crucial issue to deteriorate surface quality. In this study, a theoretical model was established to investigate tool setting induced form error in UPDT of MLA. Moreover, a new two-step tool setting method was proposed for tool setting process. Finally, a series of experiments were conducted to verify the proposed method considering the tool setting effect. Main conclusions are drawn as follows: 1) In UPDT of MLA, the directions of coordinate distortions move along the radial and circumferential directions under the horizontal and vertical tool setting error, respectively. Also, the directions of the coordinate distortions are well corresponding to the directions of the generated patterns of MLA form error.
2) The proposed two-step tool setting method includes the rough tool setting process and the fine tool setting process. The rough tool setting process was realized by an optical instrument, and the fine tool setting process was achieved by the groove-turning tests based on the statistical analysis. Significantly, the method is efficient with high accuracy and simple operation.
Future work will be focused on developing the tool setting method for diamond milling and fly cutting process and investigating the tool setting induced form error for different types of micro-structured functional surfaces in ultraprecision machining.