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Fast/slow tool servo (FTS/STS) diamond turning is a very promising technique for the generation of micro-lens array (MLA). However, it is still a challenge to process MLA in large scale due to certain inherent limitations of this technique. In the present study, a novel ultra-precision diamond cutting method, as the end-fly-cutting-servo (EFCS) system, is adopted and investigated for large-scale generation of MLA. After a detailed discussion of the characteristic advantages for processing MLA, the optimal toolpath generation strategy for the EFCS is developed with consideration of the geometry and installation pose of the diamond tool. A typical aspheric MLA over a large area is experimentally fabricated, and the resulting form accuracy, surface micro-topography and machining efficiency are critically investigated. The result indicates that the MLA with homogeneous quality over the whole area is obtained. Besides, high machining efficiency, extremely small volume of control points for the toolpath, and optimal usage of system dynamics of the machine tool during the whole cutting can be simultaneously achieved.
© 2015 Optical Society of America
1. Introduction
Micro-lens array (MLA) based optical system has received intensive attentions for both imaging and non-imaging applications due to its superior performance [1–4]. To satisfy practical applications of the MLA, several fabrication techniques have recently been developed, which can be classified according to material removal nature into physical [5, 6], chemical [7–9], and mechanical [10–12]. Generally, physical and chemical processes are often limited to specific materials and complex processes with high costs, and it is also hard for a majority of these methods to finely obtain lenslets with well-defined intricate shapes. Mechanical machining is more universal and deterministic. In mechanical machining, two typical methods including diamond micro-milling and diamond turning are currently adopted for the generation of MLA on a variety of engineering materials [10, 13]. With diamond micro-milling, each lenslet was treated as an individual and generated by the same process [11, 12, 14, 15]. It is promising to obtain a uniform quality of all the lenslets. However, besides the low machining efficiency, it also sacrifices the flexibility of machining lenslet with intricate shapes [12, 15].
Diamond turning, being dominated by the fast tool servo (FTS) [16, 17] and slow tool servo (STS) [10, 18], is widely regarded as a very promising technique for flexible generation of MLA. Taking advantage of the servo translations along the z-axis of the machine tool, a variety of micro-structured surfaces with complicated shapes can be obtained with high efficiency. For instance, the rotary symmetrical aspheric lenslet array, or even the freeform lenslet array could be generated by adopting FTS/STS with submicron form accuracy [18–20]. More intricately, hierarchical artificial compound eye with three-dimensional (3D) structures, which features a curved primary surface with imposition of MLA, could also be achieved by means of FTS/STS [2, 16, 18]. To enhance the machining capacity of FTS/STS for the MLA, much work has been done towards the determination of optimal cutting conditions. For instance, the tool nose radius and clearance angle compensations were conducted by using computation geometry in [21, 22]. The selection criteria of proper cutting parameters including spindle speed, sampling number, feedrate and tool geometry, were proposed to guarantee an optimal machining process [22, 23]. The optimal toolpath generation strategy for the MLA was also introduced in [24, 25] with consideration of compensations for static error motion of the slide and dynamic errors of the tool servo system. By deliberately selecting the feedrate, the MLA can even be generated on certain brittle materials with crack-free quality and relatively high efficiency [26]. As discussed above, machining capacity of FTS/STS for MLA has been significantly improved with these dedicated efforts.
FTS/STS diamond turning is naturally operated in the Cylindrical coordinate system, namely (ρ, φ, z) [21, 22]. In principle, the cutting operation in this system results in some inherent defects for the generation of the MLA, which can be summarized as follows:
(i) Cutting quality inconsistence. The time-varying radial distance ρ will lead to variable cutting velocity, accordingly inducing the inconsistent cutting conditions at any cutting points. The resulting inconsistent cutting forces will inevitably lead to the inconsistence of structure accuracy, mainly due to the force induced deformation of the cutting loop of the machine tools, especially for the FTS with low loop-stiffness [16]. On the other hand, the intrinsic spindle vibration will generate the periodic and concentric pattern on the machined surface in diamond turning, which is more intensive at the position closer to the center [27, 28]. It was also observed in FTS diamond turning of the MLA [29]. The deteriorated quality at the center of the machined surface cannot be avoided in FTS/STS diamond turning. Although the off-axis turning technique can somewhat alleviate the inconsistence in cutting [30], the improvement is very limited. Besides, it also suffered some inherent defects in turning as discussed below.
(ii) Azimuth sampling conflicts. As pointed in [23, 24, 31], the sampling number is a critical parameter that controls the interpolation error along the forward-cutting direction. In the Cylindrical coordinate system, the commonly adopted constant-angle sampling strategy of the azimuth φ leads to the radial distance dependent interpolation error. To guarantee the form accuracy over the whole surface, numerous sampling points should be adopted, especially for micro-structured surfaces with large apertures [16]. Normally, the large sampling number per revolution is excessively redundant for processing the place close to the rotation center of the workpiece [16]. Although much work has been done on improving the sampling strategy to deal with the radial distance dependent effects, the operation process is complex and the reduction of the volume of data points is limited [31]. Besides, there are still two fatal defects deriving from the sampling requirements: a) Too many controlling points in one revolution will significantly reduce the spindle speed, leading to the extremely low machining efficiency and rapid tool wear [16, 31]; b) The volume of required toolpath data for one cutting will be often too large to be easily operated by the control system [32].
(iii) Tracking bandwidth limitation. The MLA is often uniformly distributed in the Cartesian coordinate system of the workpiece. When transferred to the turning system, there are more lenslets in one revolution at the position with larger radial distance ρ, requiring much higher tracking bandwidth of the servo system. The radial distance dependent requirement of the tracking bandwidth will sometimes restrict the selection of the spindle speed considering the aperture of the MLA. It leads to the insufficient usage of FTS/STS system dynamics when cutting the inner space, suggesting that the cutting operation cannot always keep working in its optimal status during one cutting. Moreover, the phase-lag effects of the servo system, which is highly dependent on the working frequency, will not only result in deterioration of the form accuracy of each lenslet, but also induce distortions of whole array structure [25, 33].
Obviously, the aforementioned irreconcilable defects significantly deteriorate the quality of FTS/STS diamond turned MLA, and block further developments of mechanical machining for MLA. Motivated by this, a novel single point diamond cutting process, as the end-fly-cutting-servo (EFCS) system, is adopted and demonstrated for large-scale fabrication of the MLA. The EFCS system mainly combining the concepts of FTS/STS and end-face fly-cutting was firstly proposed in [34] and further investigated in [35] for the generation of hierarchical micro-nanostructures. Shifting the focus on the generation of the secondary nano-pyramids to the fabrication of MLA, characteristic advantages of the EFCS system for solving the aforementioned inherent problems in FTS/STS turning of MLA are especially identified and demonstrated in the present study.
2. Principle for large-scale generation of the MLA
In the EFCS system, the concepts of FTS/STS and fly-cutting are synthesized to complement each other to enhance the machining capacity. Figure 1(a) illustrates the hardware configuration of the proposed EFCS system. It totally consists of four-axis motions, namely x-, y-, z- and c-axis. Generally, the diamond tool is installed on the fixture and then attached on the spindle (c-axis), whereas the workpiece is clamped on the tool holder and linearly moves with the slide [34, 35]. By exchanging positions of the diamond tool and the workpiece in FTS/STS, the cutting operation is transferred from the Cylindrical coordinate system to the Cartesian coordinate system. In view of the tool motions, it is more like the fly-cutting. However, by adopting end-face cutting, the point removal mechanism of fly-cutting is replaced by line removal mechanism, which significantly enhances the machining efficiency of the cutting process. By inheriting translational servo motion along the z-axis for the workpiece, intricately shaped micro-structures can be flexibly generated.
Fig. 1 Configuration of the EFCS System, (a) hardware configuration, (b) the horizontal cutting and (c) the vertical cutting.
With the planar MLA, the structure possesses rigorous periodicity. Thus, the whole MLA can be subdivided into a set of basic array cells as shown in Figs. 1(b) and 1(c), and the fabrication for the cell is treated as a basic cutting process. By repeating the basic process through the raster motions, the MLA can be well generated covering the whole surface with arbitrary apertures. Normally, there are two cutting modes in the EFCS system, namely the horizontal cutting and vertical cutting as shown in Figs. 1(b) and 1(c), respectively. With the horizontal cutting, the main cutting direction is along the x-axis, and the y-axis motion is just adopted for side feeding. On the other hand, with the vertical cutting, the main cutting direction and the side-feeding direction are along the y- and x-axis directions, respectively. Generally, there is no difference between the two cutting modes for the generation of MLA.
Due to the unique configuration adopted in the EFCS system, the following superior advantages for processing MLA can be accordingly achieved: i) the distance between the cutting point and the rotation axis always keeps as a constant, resulting in the naturally consistent cutting velocity and working frequency at any revolution; ii) the constant-angle sampling of the azimuth will simultaneously result in the constant arc-length for the construction of the machined surface, leading to a relatively homogeneous distribution of interpolation errors at any cutting point; iii) the homogeneous cutting condition within one basic cutting process and its repetition provide a solid background for achieving uniform quality of the whole MLA; iv) taking advantage of the repetition of the BCPs, only the control points of the toolpath for one basic array cell are required to be stored and processed, leading to a significant reduction of the volume of control points for the toolpath.
3. Optimal toolpath determination for the EFCS systems
Since the whole MLA will be generated by repeating the basic cutting process for a basic array cell, only the determination of the toolpath for the array cell is required. By adopting geometry computation, the optimal toolpath generation strategy is detailed with consideration of the geometry and installation pose of the diamond tool.
3.1 Description of the diamond tool
In the EFCS system, the projected toolpath on the workpiece is a set of concentric circles with constant rotation radius Rd [35]. The relative positions between the diamond tool and the workpiece are illustrated in Fig. 2(a), where ow-xwywzw and ot-xtytzt denotes the fixed Cartesian coordinate system of the basic array cell and the cutter location point (CLP) of the diamond tool, respectively. The rake face is initially set as horizontal by using the linear variable differential transformer. Practically, it is hard to adjust the rake face and the spindle axis to be in the same plane, and there will always be an angle discrepancy α as shown in Fig. 2(a).
Fig. 2 Schematic of the cutting system, (a) the relative positions between the tool and the basic cell, and (b) the local spherical coordinate system of the tool. Note: the angle α here is negative.
Assume that a round-edged cutting tool with nose radius RT and rake angle γ0 is employed. The tool edge can be better described in its local Spherical coordinate system as shown in Fig. 2(b), where it can be expressed by [35]
where θmin and θmax denote the lower and upper angle boundary of the cutting edge, respectively.To conduct numerical calculation, the cutting edge is uniformly discretized into (N0 + 1) points. The i-th point at the edge as shown in Fig. 2(b) is represented by , and the corresponding tangent vector can be obtained by [21]
3.2 Determination of the CLP
Similar to the operation in FTS/STS, the rotational angle of the spindle is also uniformly discretized into (Ns + 1) points. With the l-th point in the k-th revolution of the spindle, the rotation angle of the spindle in the ow-xwywzw system can be expressed by [21, 35]
With fx being the feed per revolution along the owxw axis, the position of the spindle axis in the workpiece coordinate system can be determined by [35]
By means of the rotation transformation of vector, the point at the tool edge in the local coordinate of the workpiece ow-xwywzw yields
Assume the desired surface in the ow-xwywzw system can be described by zw = fw(xw,yw). The normal vector of the desired surface at the position corresponding to the projection point at the tool edge can be expressed by
How to determine the cutter contact point (CCP) can be equivalent to solving a position at which the cutting edge is tangential to the desired surface. With discrete calculation, the CCP can be approximately determined by finding the minimum value of the two directional vectors [21]
Assume that the initial distance between the ow-xwyw plane and the oT-xTyT plane is d, the required motion along the z-axis yields
In the ow-xwywzw system, the position of the CLP can be further determined by
Following through all the k and l by repeating the aforementioned steps, the toolpath for the basic array cell can be well generated.
4. Experiment setup
The cutting experiments were performed on a CNC ultra-precision lathe (Moore Nanotech 350FG, USA) with four-axis servo motions. A homemade fixture with mass balancing was especially designed to hold the diamond tool. The hardware configuration of the experiment setup is shown in Fig. 3, where the fixture with the diamond tool was clamped on the air-bearing spindle by the vacuum chuck, and the workpiece was installed on the slide, just following the translational servo motion along the z-axis of the machine tool. A commercial natural single crystal diamond tool with round edge (Contour Fine Tooling, UK) was used in the cutting. More details of the cutting conditions and the employed tool are shown in Table 1. After cutting, the surface was cleaned with alcohol to remove the attached chips. The Optical Surface Profiler (Zygo@ Nexview) and Olympus BX60 optical microscope were then employed to capture the topographies of the machined MLA with proper magnifications. To map a large area beyond the measurement range, a small group of images were stitched together by using the software system Mx of the Optical Surface Profiler.
Table 1. Cutting parameters and the tool geometry
A typical micro-aspheric lens array was fabricated for the demonstration of the EFCS system. The mathematical description of each aspheric lenslet can be expressed by [26]
where k is the conic constant determining the eccentricity of the conic surface by: k<−1: hyperboloid; k = −1: paraboloid; −1<k< 0: ellipsoid; k = 0: sphere; k>0: oblate ellipsoid. The shape parameter s determines whether the micro-aspheric lenslet is concave ( + 1) or convex (−1). R0 determines the radius of each lenslet, and C is the curvature parameter. ρ is the radial operator in the local Cylindrical coordinate system of the micro-aspheric lenslet.Details of the parameters for the MLA are summarized in Table 2, resulting in a part of the designed MLA in Fig. 4(a). Following the toolpath determination strategy proposed in Section 3, the optimal toolpath is obtained as illustrated in Fig. 4(b). The parameters were chosen as the ones in Table 1 except for the feedrate, which was set as 15μm/rev herein in order to have a much clearer view of the toolpath. As shown in Fig. 4(b), a smooth path on each lenslet is achieved, verifying the suitableness of the adopted azimuth sampling. Besides, the obtained toolpath appeared to be non-symmetric with respect to the y-axis, attributing to the inclination poses of the rake face plane with an angle of α. It indicates the effectiveness of the proposed toolpath determination strategy. In addition, due to the constant rotation radius, the sampling arc-length over the machined surface in each cutting circle is the same. Thus, evenly distributed control points for the motions of the diamond tool can be achieved without dependence on the relative positions between the diamond tool and the workpiece.
Table 2. Parameters of the micro-asphere array
5. Results and discussion
5.1 Characteristics of the machined structures
To demonstrate the capacity of large-scale fabrication, the MLA was fabricated over a length of 50mm. Since the experiments were conducted to validate the EFCS system for generating the MLA, only three regions in this large area were processed to reduce the machining time. The photography of the machined workpiece is illustrated in Fig. 5(a). Furthermore, three arbitrary regions in Zone A, Zone B and Zone C as marked in Fig. 5(a) were captured by Olympus BX60 optical microscope with an amplification of 20 × and shown in Figs. 5(b), 5(c) and 5(d), respectively. From the structures shown in Fig. 5, each lenslet in these regions shared almost the same features with no structure and position distortions. It suggested that a uniform quality of the array covering the relatively large area was well achieved.
Fig. 5 (a) Photography of the workpiece, and microscope diagram of the structures in (b) Zone A, (c) Zone B and (d) Zone C.
An arbitrary region was further captured and stitched by the Optical Surface Profiler with an amplification of 5 × , which has a lateral resolution of 0.824 μm. Figures 6(a) and 6(b) illustrate the three-dimensional (3D) structures containing 90 lenslets and the 2D cross-sectional profile passing through the centers, respectively. From the results shown in Figs. 6(a) and 6(b), homogeneous features of both the 3D structures and the profile are obtained, well indicating the uniformity of the machined MLA. Since distortion of the relative positions between the lenslets is a key factor forming the form error of the array, a square region containing 4 lenslets from Fig. 4(a) was further captured with an amplification of 20 × for the investigation of the structure distortion, the corresponding lateral resolution is 0.211 μm. The resulting structure is illustrated in Fig. 6(c), and the corresponding cross-hair profiles are presented in Fig. 6(d). The height of the lenslet is about 4.039μm, and the half-distance between two successive lenslets is about 125.316μm. The observed values have good agreements with the designed ones as shown in Table 2. More importantly, the shapes as well as the sizes of the two profiles along the cross-hair directions agree well with each other, showing high consistence and high accuracy of the obtained structures.
Fig. 6 Characteristics of the machined MLA, (a) the 3-D structure of a large area, (b) the 2D profile of the cross-section, (c) an extracted square area, and (d) the corresponding profiles along the cross-hair directions.
To evaluate the form accuracy of the machined MLA, an arbitrary lenslet from Fig. 6(c) was extracted and illustrated in Fig. 7(a). Since the lenslet is rotationally symmetrical, the cross-hair profiles were extracted to have a comparison with the desired profiles determined by Eq. (12), resulting in the profiles as well as the matching errors in Figs. 7(b) and 7(c). As shown in Figs. 7(b) and 7(c), the desired and obtained profiles agree well with each other in both directions, and the resulted form errors are less than ± 100nm.
Fig. 7 Details of the machined lenslet, (a) the 3D structure, (b) and (c), the profiles and the machining errors along the cross-hair directions.
5.2 Characteristics of surface micro-topography
By removing the aspheric surface from the lenslet shown in Fig. 7(a), the resulted surface micro-topography is further illustrated in Fig. 8. From the analysis modular of the software, the obtained roughness value was about Sa = 16nm. In the side-feeding direction, the slight fluctuations may correspond to the residual tool marks. Obviously, the surface micro-topography is predominated by the undesired fluctuations along the cutting direction. It might be induced by the relative vibrations between the diamond tool and the workpiece, which is a common phenomenon in both EFCS and FTS/STS diamond cutting [16, 33].
In MLA cutting, each lenslet serves as an impulse excitation source. In other words, the toolpath for the MLA can be regarded as a signal with a set of discontinuous impulses. It is much easier for the mechatronic system to be vibrated when forced by these impulse commands as well as the induced external impulse excitations (i.e. impulse cutting forces). This is currently the main challenge for further improvement of surface quality. To avoid this phenomenon, there are two possible solutions: a) modify the toolpath to be continuous; b) improve the dynamics performances of the servo system. Further research into this issue will be conducted in the very near future.
5.3 Efficiency of the EFCS system
In the present study, only three different zones, each with an area of 2.5 × 7.5mm2, were fabricated to demonstrate the basic principle of the EFCS system. The number of control points for the toolpath is 1.8 × 106, and the volume of data file is about 44MB. The overall cutting time is about 11.5 hours. It is to be noted that the current volume of data file is sufficient for processing the workpiece with arbitrary dimensions by repeating the basic cutting process. Considering that the required volume of the data file for processing a micro-structured surface with a diameter of 150mm is about 3GB [32], the proposed EFCS contributes significantly to the reduction of the volume of data file.
In FTS/STS assisted diamond turning, the constraints for spindle speed are two folds, one is the working bandwidth of the servo system and the other one is the data transfer rate of the control system. In EFCS system, a much higher spindle speed can be achieved due to the reasons that a) significant reduction of the number of toolpath control points per revolution is achieved; b) the requirement for maximum tracking bandwidth is decreased with respect to a given spindle speed. Both the two benefits are derived from the employment of much shorter rotation radius when comparing with the aperture of the workpiece. It suggests that although only a part of a cutting cycle is employed in the EFCS system, it will not decrease the cutting efficiency due to the relatively very high spindle speed. For instance, the cutting time for such an area (Φ10mm) in STS is estimated to be about 20 hours by adopting the same feedrate and proper sampling numbers.
On the other hand, if the geometry of the desired surface is fixed, the required sampling number per revolution and the working bandwidth are unchangeable in FTS/STS. Thus, there will be “cask effects” in determining the maximum spindle speed, and it is even impossible to reconcile the two key factors. However, in the EFCS system, the rotation radius can be flexibly adjusted with respect to different surface structures, and an optimal radius can be found to avoid the “cask effects”. Besides, since the cutting requires approximately constant working frequency at any cutting revolution, a full usage of the bandwidth of the servo system can always be achieved during the cutting, resulting in much higher efficiency of the EFCS system.
6. Conclusions
A novel end-fly-cutting-servo (EFCS) assisted single point diamond cutting method is adopted and demonstrated for large-scale fabrication of the micro-lens array (MLA) with uniform quality in the present study. It mainly combines the concepts of fast/slow tool servo (FTS/STS) and end-face fly-cutting, transferring the cutting from the Cylindrical coordinate system to the Cartesian coordinate system.
The diamond tool is attached on the spindle and rotates with a constant radius in the EFCS system, thereby endowing the cutting system with consistent cutting velocity, consistent arc-length sampling, and consistent tracking bandwidth requirement in any cutting revolution. The intrinsic consistence of the cutting process leads to the homogeneous quality of the machined surface.
With consideration of the rigorous periodicity of the common MLA, it is subdivided into several basic lens array cells. Shifting the efforts on only the basic array cell can significantly reduce the computational costs as well as the volume of data file of the toolpath. Besides, by repeating the cutting for the basic array cell, the whole MLA possessing arbitrary size can be uniformly achieved with high quality.
The irreconcilable factors that determine the spindle speed in FTS/STS, namely the tracking bandwidth and the data transfer rate of the servo system, can be optimized in the EFCS system to maximize the cutting efficiency by adjusting the rotation radius. Thus, a higher cutting efficiency can be achieved by adopting much higher spindle speed, especially for large-scale fabrication of the MLA.
Finally, an aspheric MLA is experimentally fabricated over a relatively large area to demonstrate the effectiveness of the EFCS system. Homogeneous machining quality is obtained. The form error is less than ± 100nm, and the surface roughness inside the lenslet is about Sa = 16nm. The relative vibration between the diamond tool and the workpiece is the main issue affecting surface quality.
Acknowledgments
The work described in this paper was jointly supported by the Research Committee of The Hong Kong Polytechnic University (RTJZ) and the National Natural Science Foundation of China (51275434).
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