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Abstract
Microstructure surfaces are widely and deeply applied in optical fields to decrease the volume of an optical system and realize sophisticated optical requirements. This paper studies the machining of optical micro-structures with micro-scale height on single crystal germanium by elliptical vibration assisted sculpturing (EVAS) method in which two vibration amplitudes keep constant. Due to the superimposition of low-frequency sculpture path and high-frequency elliptical locus, the investigations are focused on cutting plane to describe motion trajectory of the tool tip of cutting tool for desired complex motion path and to calculate the sculpture path of elliptical vibration cutting device to make corresponding path of tool tip be the envelope of target profile. A comparative experiment is firstly conducted to verify that elliptical locus would deteriorate machining accuracy without accurate compensation for the ellipse. Following that, 2D and 3D sinusoidal surfaces and a picture, whose structural heights are all not less than 2.0µm, are experimentally machined on single crystal germanium. Satisfying experimental results demonstrate the feasibility of EVAS method for fabricating optical micro-structures on germanium.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical micro-structure surfaces [1] possess such fascinating functions as increasing field-of-view [2], shaping beam light [3], and shrinking the volume of optical system [4]. To fulfill desired functions and mass production, manufacturing technology for the structured surfaces is required for high machining accuracy and acceptable machining efficiency. Due to high flexibility, broad spectrum of machinable geometries, sub-micron form error and nanometer level surface roughness, ultra-precision diamond cutting is a vital fabrication method for structured surfaces [5]. Diamond cutting is utilized for high value-added parts such as microlens array on crystal materials like germanium, silicon [6], and sophisticated optical mold inserts on high hardness materials like hardened steel [7] and tungsten carbide [8].
For diamond cutting micro-structures on difficult-to-cut materials, machining accuracy limited by single crystal diamond (SCD) tool wear and machining efficiency for removing material in ductile mode are two key factors to be considered. Fortunately, elliptical vibration cutting (EVC) technology developed by Moriwaki and Shamoto [9] possesses desired functions to extend the working life of SCD tool in cutting hardened steel due to suppressing thermo-chemical reaction between tool and steel, and improving machining efficiency in brittle materials as increasing critical depth of cut. In order to cut nano/micro structures on hardened steel, Suzuki et al. [10] innovatively created the control amplitude sculpturing (CAS) method which is a combination of planing cutting method, EVC technology, and fast tool servo (FTS) technology. The EVC device moves along a horizontal straight line. Meanwhile, the vibration amplitude of EVC device in depth-of-cut direction is controlled by upper computer according to the structural height of current position by FTS technology. Zhang et al. [11] gradually improved machining accuracy by investigating cutting trajectory of CAS by compensating for geometry shape of elliptical locus [7] and dynamic contour error of the machine tool [12]. Some nano structured surfaces with a large ratio [11] and micro structured surfaces like low-friction surfaces [13] were precisely machined on hardened steel. Due to closed loop of control, high-precision and high-speed collecting position data system and the calibration of vibration amplitude to control voltage are essential parts of the system which increases the cost and complexity of building the system. As structure height is formed by actively controlling vibration amplitude which means machinable height is limited by maximal mean-to-peak vibration amplitude in depth-of-cut direction, which is 2µm for EL-50∑ resonant type elliptical vibrator. Therefore, it is difficult to utilize CAS method to sculpt micro structures with height over vibration amplitude.
For increasing machinable structural height on difficult-to-cut materials, EVC technology has been integrated into such conventional ultra-precision cutting methods as slow tool servo [14], micro-grooving [15], texturing [16], and planing [17]. During the process, the cutting path for micro structures is accomplished by machine tool, and the advantages of EVC are used to suppress burr formation, extend working life of SCD tool, and improve machining accuracy. In above hybrid cutting methods, vibration amplitudes keep constant. Therefore, this paper categorizes previous hybrid methods into elliptical vibration assisted cutting (EVAC). Compared with above CAS method, EVAC possesses the advantages of cutting structures with lower cost, higher height, simpler machining system, and better extensibility. EVAC has potential for fabricating nano structures on ultra-precision machine tools while the efficiency and accuracy may be lower than those of CAS. Therefore, EVAC is a good choice for cutting micro-structures with micrometric or even larger heights on difficult-to-cut materials.
Zhou et al. [14] utilized EVAC method for machining three kinds of sinusoidal surfaces on die steel and they focused on tool path in feedrate direction. Kurniawan et al. [16] textured micro-dimple patterns on AISI 1045 steel alloy with EVAC and they mainly researched on surface roughness of the dimple. Yang and Guo [17] studied the optimization algorithm for smoothing abruptly changed velocity of cutting path in cutting micro grating structures. They also machined some other interesting diffractive patterns [18]. Those researchers paid less attention to the calculation of cutting path of EVAC in cutting plane which is one of the key factors to influence machining accuracy. The cutting path is composed of low-frequency conventional cutting trajectory and high-frequency additional elliptical locus. If the cutting path was calculated by simply superimposing two trajectories, machining accuracy would be deteriorated by over-cutting caused by elliptical locus when cutting curved target profile. This problem has been indirectly demonstrated in CAS [7]. To avoid over-cutting in the CAS method, Zhang et al. [7] proposed a compensation algorithm for finely modifying elliptical shape, based on the results without compensation, to make loci tangent to target profile in each vibration period. Unfortunately, the algorithm is invalid in EVAC. The first reason is that elliptical shape keeps constant during EVAC process, and the second one is that the motion trajectory of EVC device is a curve instead of a horizontal straight line in CAS. Therefore, to improve the machining accuracy of EVAC, it is necessary to research the compensation method for elliptical locus to make the motion trajectory of tool tip be the envelope of target profile.
For machining finite/infinite linear micro structures [5] and some complex optical micro structured surfaces like Alvarez microlens array [19] on difficult-to-cut materials, elliptical vibration assisted sculpturing (EVAS) is an ideal fabrication method. To accurately cut micro structures, the motion trajectory of tool tip containing two frequency trajectories in cutting plane is firstly studied to be suitable for a complex curved path. Following that, an algorithm is proposed to calculate sculpture path by compensating for elliptical locus without over-cutting. Some optical micro structures with heights not less than vibration amplitude are experimentally machined on single crystal germanium to verify the feasibility of EVAS method.
2. Principle of elliptical vibration assisted sculpturing
Figure 1 (a) shows a schematic illustration of the whole setup of EVAS. During cutting, the EVC device moves along desired sculpture path accomplished by y-axis and z-axis of ultra-precision machine tool, meanwhile, the tool tip of SCD tool fixed on the device vibrates clockwise along an ellipse generated by two different vibration modes with nearly the same frequency, second-order longitudinal vibration along the z-axis, and fifth-order bending vibration along the y-axis. The center of the ellipse is not only the position of tool tip when it is stable without vibration, but is also regarded as a characteristic point representing current position of EVC device. Therefore, the trajectory of tool tip is composed of low-frequency sculpture path and high-frequency elliptical locus like Fig. 1 (b) where sculpture path determines profile of the structures and the locus influences machining accuracy of the profile.
Fig. 1. Schematic illustration of EVAS method. (a) The layout of the machining system. (b) The cutting process of EVAS method.
The calculation process in EVAS is summarized in Fig. 2. As the EVC device is powered on, the tool tip of SCD tool rotates along an ellipse like Step 1 in Fig. 2. When the device moves along target profile as conventional sculpture shown as Step 1 in Fig. 2. As EVC device moves along the profile, the trajectory of tool tip is the combination of low-frequency profile and high-frequency elliptical locus like dense red curves in Step 2. The combined trajectory of tool tip is defined as motion trajectory. Due to curvature radius of the ellipse, the curvature radius must be compensated to avoid over-cutting between the yellow profile and red motion trajectory. After compensating for the curvature radius, the compensated motion trajectory is defined as cutting path like blue curves in Step 3. And the corresponding centers of modified ellipses of cutting path form sculpture path like black thick dash curve in Step 3.
The purpose of above calculation process is to construct a sculpture path. When EVC device moves along the sculpture path, which is achieved by motional axes of ultra-precision machine tool, the cutting path of tool tip would be the envelope of target profile. While it is hard to directly obtain the sculpture path based on target profile and elliptical parameters. Therefore, the motion trajectory of tool tip is firstly studied, then is the cutting path, and finally is the sculpture path.
Therefore, the motion trajectory of tool tip in EVAS is firstly studied in Section 3. Following that, the reason for over-cutting or under-cutting caused by improper compensation strategy for curvature radius of the ellipse like enlarge view of Step 3 in Fig. 2 is studied, and then a novel algorithm of compensation for the curvature radius is studied in Section 4.
3. Motion trajectory of tool tip in EVAS method
When the target profile is described by function, the motion trajectory can be expressed as Eq. (1)
In order to overcome the disadvantages of Eq. (1) and improve the generality of the calculation process, an algorithm for calculating motion trajectory in EVAS method is proposed based on the linear interpolation fitting method. The profile is firstly divided into a sequence of short line segments with submicron scale length by linear interpolation fitting. Then motion trajectory of each segment is calculated by Eq. (2)
The motion trajectories for different two kinds of target profiles are shown in Fig. 3 where Fig. 3 (a) is a triangle wave described by piecewise function, and its wavelength is 20µm and peak-to-valley height is 1µm, and Fig. 3 (b) is a profile described by random discrete points. The sequence constructed by red squares represents target profile, and blue circles are endpoints of finely divided line segment and the length of each segment is set to a sub-micrometric length such as 0.4µm. The black continuous curve is motion trajectory of tool tip for the profile. Although there are deviations between desired target profile and linear interpolation fitting, fitting accuracy could be controlled by adjusting threshold of the length of segment.
Fig. 3. The motion trajectories of tool tip for two kinds of target profiles. (a) Triangle target profile with 20µm wavelength and 1µm peak-to-valley height. (b) Target profile described by random discrete points.
4. Sculpture path of EVC device in EVAS method
4.1 Investigation of over-cutting in EVAS
In conventional cutting methods like diamond turning and sculpturing, target profile in cutting plane can be regarded as cutting path. In such diamond milling methods as ball-end milling and raster fly-cutting, the cutting path of tool tip is the envelope of target profile, and the trajectory of the center of rotational circle forms an equidistant offset surface in ball-end milling or equidistant offset curve in raster fly-cutting due to constant tool’s swing radius [20].
Above cutting path calculation methods are invalid for calculating sculpture path of EVAS because the curvature radius of elliptical locus is variable along the curve as two vibration amplitudes are different. Therefore, if the compensation value for the locus (Av = 1µm, Bv = 2µm) was selected as mean-to-peak vibration amplitude in depth-of-cut direction, over-cutting or under-cutting phenomenon would happen shown in Fig. 4 where sculpture path is shifted from target profile and cutting path is calculated based on sculpture path. If the sculpture path was shifted from target profile along z-axis as shown in Fig. 4 (a), there would be an over-cutting region except for the peaks and the valleys. If the sculpture path is an equidistant offset curve from target profile, and under-cutting phenomenon would occur like that in Fig. 4 (b). Therefore, the sculpture path is a non-equidistant curve of the target profile.
Fig. 4. Over-cutting and under-cutting phenomena are induced by compensation strategy with a constant offset value. (a) Sculpture path is translation curve from target profile along z-axis direction. (b) The sculpture path is an equidistant curve of the profile.
4.2 Investigation of compensation for elliptical locus
The above investigation has demonstrated that the sculpture path is a non-equidistant curve of the target profile, thereby each sculpture position must be modulated, based on ellipse shape and the slope of the divided line segment, to make cutting path of tool tip tangent to the segment.
To achieve the target, the characteristic point, whose tangent line is parallel to the segment, on motion trajectory is of key importance. The characteristic point is called “tangent point” ${T_i}({y_{T,i}},{z_{T,i}})$. An algorithm is proposed to confirm each ${T_i}$ on elliptical locus for the segment. For the sake of clarity, Fig. 5 shows a schematic for calculating sculpture path of a single segment. The start point and endpoint of the segment are respectively denoted as ${P_i}({y_i},{z_i})$ and ${P_{i + 1}}({y_{i + 1}},{z_{i + 1}})$. Then corresponding endpoints of sculpture path are denoted as ${Q_i}({y_{Q,i}},{z_{Q,i}})$ and ${Q_{i + 1}}({y_{Q,i + 1}},{z_{Q,i + 1}})$.
1st step: Calculate the slope ${k_i}$ of the segment ${P_i}{P_{i + 1}}$ using Eq. (3)
2nd step: Calculate the slope ${k_{vib,i}}$ of the motion trajectory for the segment ${P_i}{P_{i + 1}}$ using Eq.(4).Then Eq. (5) can be simplified as
where ${C_1} = \omega ({A_v}{k_i} - {B_v}\cos \phi )$, ${C_2} = \omega {B_v}\sin \phi$, ${C_3} = {k_i}{v_{yi}} - {v_{zi}}$, $\beta = \arctan ({{{C_2}} / {{C_1}}})$.Then solutions of Eq. (6) are expressed by
As ${T_i}$ must be in the lower half of the ellipse and the time starts from right vertex of the ellipse, thereby select ${t_1}$ or ${t_2}$ whose value belongs to [0,0.5T] as result ${t_0}$. Then the coordinates of ${T_i}({y_{T,i}},{z_{T,i}})$ can be calculated by Eq. (2) as Eq. (8), the blue pentagram in Fig. 5.
5th step: Based on sculpture segment ${Q_i}{Q_{i + 1}}$, employ Eq. (2) to calculate cutting path which is the envelope of the target profile.
Repeat above calculation process for each line segment, and then the whole sculpture path would be obtained and corresponding cutting path of tool tip is the envelope of the target profile. The whole calculation process for sculpture path and cutting path is summarized in Fig. 6.
Employ the proposed method to recalculate the cutting path and sculpture path in Fig. 4, and the results are shown in Fig. 7. By the comparison between the results, the cutting path obtained by proposed method is tangent to the target profile without over-cutting or under-cutting phenomena.
Fig. 7. Different calculation results by different algorithms for the same target profile. (a) The results of proposed algorithm. (b) The enlarged view of (a). (c) The enlarged view of results by shifting along the z-direction. (d) The enlarged view of results by equidistant compensation strategy.
5. Experimental verification
5.1 Experimental setup
Figure 8 shows experimental setup. The EVC device was fixed on the Z-axis of an ultra-precision machine tool having three linear motion axes with nano-positioning accuracy and one rotation motion axis with arc-second accuracy. The material of workpiece was single crystal germanium, and it was glued to the fixture sucked on vacuum chuck. The rake angle of 2# single crystal diamond cutting tool was -25°, its relief angle was 10°, and tool nose radius was 1.023mm. Before each experiment, the surface was machined to be a crack-free plane by conventional turning with 1# negative rake angle diamond tool. Then the spindle was switched to positioning mode to keep the workpiece still, and EVC device was employed to cut desired structured surfaces. The peak-to-valley heights of the following four kinds of structures were all not less than the mean-to-peak vibration amplitude of EVC device in depth-of-cut direction. Experimental parameters were summarized in Table 1.
Table 1. Experimental machining parameters
After the experiment, the machined surface was mainly measured by Zygo NewVIEW 8200 optical profiler, and measured raw data was processed by MetroPro software. Some geometries were captured by a digital camera.
5.2 Experimental results
5.2.1 Verification of cutting path algorithm
The target microstructure is a 2D triangle structure with 130.0µm wavelength and 2.5µm structural height. The triangle structure was selected to test machining accuracy under conventional compensation strategy and proposed compensation strategy for elliptical locus.
Experimental results are summarized in Fig. 9. From 2D and 3D topography, two micro structured surfaces are clear and crack-free. The profile in cutting direction with conventional strategy has obvious deviation that measured profile seems to be composed of curves instead of straight lines, and peak and valley are a little round, while the profile with proposed strategy is composed of straight lines, and the peak and valley are sharp. The wavelengths of two structures can be estimated from the profile plot where the actual wavelengths of two structures are the same with 130.6µm with 0.462% relative error. More precise result of the wavelength for periodic structure can be directly obtained from the autocovariance plot where the distance between y-axis and the first peak is the dominant wavelength, thereby the x-coordinate of the white dot in the plot represents actual wavelength with the value of 130.4µm a little less than that in profile plot. The results from the profile plot contain uncertainty caused by manually moving inspectors to find the peak and valley of the profile. As a triangle structure is composed of two oblique straight lines, then the theoretical result in the bearing ratio plot should be an oblique straight line. After linear fitting, the data in bearing ratio plot, the correlation coefficient of linear fitting R2 was chosen to describe the degree of deviation that the much closer to 1, the more accurate the structure is. Therefore, the structure with proposed strategy is much closer to the desired. The experiment shows that the proposed algorithm for sculpture path is feasible.
Fig. 9. Experimental results of triangle structure with 130µm wavelength and 2.5µm structural height. (a) With conventional compensation strategy. (b) With proposed compensation strategy.
5.2.2 Feasibility investigation of machining microstructure described by function
2D sinusoidal surface and 3D sinusoidal surface were selected to test the performances of the algorithm for the structures described by functions. The 2D surface has 2.0µm structural height and 100.0µm wavelength, and cutting parameters for the surface are that pick feed is 5.0µm and the depth-of-cut in each process is 1.0µm. The 3D surface was designed to have 2.0µm peak-to-valley amplitude and 100.0µm wavelength in cutting direction, and 1.0µm peak-to-valley amplitude and 200.0µm wavelength in pick feed direction. The pick feed is 3.0µm and depth-of-cut in each process is 1.0µm.
The experimental results of 2D sinusoidal surface are shown in Fig. 10. The 2D topography and 3D topography show that the structure is clear and removed in ductile mode. From the profile plot in Fig. 10 (c), peak-to-valley amplitude of the structure is 1.9992µm with -0.04% relative error, and wavelength is about 100.34µm with 0.34% relative error. The wavelength is 100.20µm obtained from Fig. 10 (d).
Fig. 10. The measured results of 2D sinusoidal surface. (a) 2D topography. (b) 3D topography. (c) The profile of the slice in (a). (d) Autocovariance plot of the profile in (c).
The measured results of 3D sinusoidal surface are shown in Fig. 11. Four slices are selected to pass through characteristic peaks and valleys in cutting and pick-feed directions, respectively. The 3D topography shows that the structure is crack-free and clear. From Fig. 11 (c), the maximal value and the minimal value of each characteristic slice are in accord with theoretical values. The measured wavelengths in two directions are 100.2µm with 0.2% relative error in cutting direction and 199.7µm with -0.15% relative error in pick feed direction. While there are some deviations near the valleys of slice (3) and (4), authors think the improper compensation strategy, the tool arc was regarded as a standard arc, for the tool nose of negative rake angle tool would be the main reason.
Fig. 11. The measured results of 3D sinusoidal surface. (a) 2D topography. (b) 3D topography. (c) The profile of slices in (a). (d) Autocovariance plot of profiles in (c).
The above experiments verify that proposed sculpture path algorithm possesses the ability to fabricate 2D and 3D microstructure surfaces described by function.
5.2.3 Feasibility investigation of machining microstructure described by discrete points
Bing Dwen Dwen, the mascot of the Beijing 2022 Winter Olympic Games, was selected as machined target to test the performance of the algorithm for the structures described by discrete points. The colorful image was firstly converted into a grayscale picture. The strongest gray intensity was set as maximal depth, which was 3µm, and the weakest gray intensity was set as zero. The distance between each pixel of the picture was set as 5µm. Therefore, the grayscale picture was converted into a 3D matrix containing the information of position and structural depth. The cutting process was along the raster path like those in previous experiments, while each cutting path must be calculated individually according to different slices. Each selected slice contains X-, Y- and Z-coordinates, and X-coordinate was a constant, and Y- and Z-coordinates were variable. The cutting path for the slice was calculated by proposed algorithm and the tool nose radius was also compensated like previous experiment.
Machined results are shown in Fig. 12(a) captured by digital camera, and Bing Dwen Dwen can be easily identified from the contour. There are a lot of fine white hairs in black region covering eyes so that the height of this region changes rapidly resulting in inevitable over-cutting in pick-feed direction. Therefore, there are lots of irregular cutting marks in this region forming diffuse reflection which makes it look bright. The 3D topography of the whole body, measured in stitch mode, is shown in Fig. 12 (b) and the details of the topography are finer than those in Fig. 12(a). The blue color represents black color in colorful image. Such details as the small red heart in Bing Dwen Dwen’s left hand in Fig. 12 (c) and the logo of the Olympic Games in Fig. 12 (d) are much clearer than those in Fig. 12 (b). However, the letters “Beijing 2022” are blurred, and so are the Olympic rings. Because the details of those symbols are too close to avoid over-cutting in pick-feed direction.
Fig. 12. Machined results of the 3D structure described by discrete points. (a) Machined surface (b) Measured 3D topography. (c) Enlarge view of Bing Dwen Dwen’s left hand. (d) Enlarge view of Bing Dwen Dwen’s belly.
The experimental results show that the algorithm is also valid for the structure described by discrete points. However, the problem of over-cutting caused by sharply changing details should be carefully treated in this surface type.
6. Conclusions
This paper has investigated the algorithms for calculating three kinds of basic paths of elliptical vibration assisted sculpturing (EVAS) method. The first path is the motion trajectory of tool tip, which is a curve formed by discrete target profile and continuous elliptical locus. Then, the cutting path of the tool tip is investigated by compensating for the curvature radius of the ellipse according to the shape of the ellipse and the current slope of the profile. The cutting path is the envelope of the profile. Finally, the centers of each ellipse of the cutting path forms sculpture path, which is a non-equidistant curve of the target profile. Experimental results show that
- (1) The proposed algorithms can be employed to calculate the sculpture paths for different kinds of micro-structured surfaces such as 2D triangle structure, 3D sinusoidal surface and 3D discrete surface.
- (2) The machining accuracy in cutting direction can be improved by proposed algorithms for the sculpture path, while in pick-feed direction the tool nose compensation for negative rake angle tool should be paid attention [21].
- (3) The heights of machined structures are all not less than the mean-to-peak vibration amplitude in depth-of-cut direction, which means that the maximal machinable structure height unfreezes the limitation from the vibration amplitude of EVC device in EVAS. Therefore, EVAS method can be employed to fabricate micro structures with tens or even hundreds of micrometers height which can not be achieved by CAS method due to the restriction of vibration amplitude. While for the nano structures, the machining efficiency of EVAS is much lower than that of CAS, because fast tool servo technology is integrated into CAS.
In future research, EVC technology can be transferred into slow slide servo to improve the machining efficiency of cutting freeform surfaces on such difficult-to-cut materials as Stavax. The proposed algorithm still works but the compensation strategy for elliptical locus should be converted from normal direction compensation to Z-direction compensation due to the Y-axis of machine tool keeping still in slow slide servo. Moreover, the compensation strategy for non-zero rake angle tool in feed-pick direction would be integrated into the EVAS method.
Funding
National Key Research and Development Program of China (2018YFA0703400).
Disclosures
The authors declare no conflicts of interest.
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.
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