A novel approach named off-spindle-axis (OSA) spiral grinding for fabricating aspheric microlens array (AMLA) mold inserts for precision glass molding (PGM) is presented. In OSA spiral grinding, three translational motions of the grinding wheel are synchronized with the rotation of the workpiece to form a local spiral wheel path for individual lens-lets. With this approach, the form accuracy of lens-lets can be compensated within sub-micrometer by means of the on-machine measurement. The determination of wheel path and form error compensation via on-machine measurement are systematically studied. A tungsten carbide mold insert with four convex aspheric lens-lets is fabricated to evaluate the grinding performance. PGM experiments are performed to produce glass AMLA using the ground insert. The experimental results indicate that both the ground and molded AMLA with homogeneous quality are achieved. The form accuracy and surface roughness of both the mold insert and the molded AMLA were less than 0.3 µm in PV and 10 nm in Sa, respectively.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Microlens arrays (MLA) are key components in both of imaging and non-imaging optical systems such as three-dimensional (3D) imaging, wavefront sensor, optical communication, laser beam collimation and 3D displays, for they demonstrate some excellent optical properties, including low aberration distortion, large field of view angles, infinite depth of field and high-performance homogenization [1–8]. With the increasing demand for optical products in industrial and consumer markets, more attention is being paid to the development of manufacturing microlens array. Various microlens array fabrication techniques have been developed, which can be classified based on material removal manner into non-mechanical and mechanical approaches. They include, but is not limited to, grayscale lithography , thermal reflow , microjet printing , melting photoresist  and direct laser writing , where non-mechanical techniques can provide more automation and versatility. While above-mentioned techniques are widely applied, some common problems shared by these techniques, such as the profile of microlens is not precisely controllable and the sag height of microlens is limited to the values of only tens of micrometers [14,15]. Moreover, the processing time is long and the manufacturing costs are considerable , which limit these techniques to the production of small quantities.
Mechanical machining techniques for fabricating MLAs are emerging into the industry as their potential for generating lens-lets with a truly aspheric profile or free-form structures and very high quality of optical surfaces [15,17–19]. In mechanical machining, ultra-precision diamond cutting and ultra-precision diamond grinding are two representative deterministic manufacturing methods for fabricating monolithic MLAs . There are a number of variants of ultra-precision diamond cutting, such as diamond micro milling (DMM), fast tool servo (FTS) diamond turning, slow slide servo (SSS) diamond turning and fly-cutting. With DMM, lens-let is machined individually by the high speed rotating diamond ball endmill. Owing to every lens-let shares the same machining process, it enables all the lens-lets to have highly consistent quality . However, due to the limited dynamic characteristic of the motion axes, the machining efficiency of DMM is quite low [14,22]. FTS and SSS diamond turning are widely regarded as the most effective methods for fabrication of non-rotationally symmetric surface like lens arrays . Differently from SSS achieve low-speed motion in the Z-axis by the slide, FTS usually generates the high frequency movement in the Z-axis by a specialized control system of a short travel system superimposed on the original axis-control of the machine. Consequently, FTS has higher machining efficiency than SSS. Multiple kinds of lens array like aspheric array , compound eyes structure , Alvarez lens array  etc. have been fabricated by FTS and SSS. Fly-cutting, that the diamond tool is mounted onto the work spindle and the part where the tool post is normally situated, is generally used to fabricate the micro groove array and micro pyramid arrays [24,25]. Zhu, et al [26–28], developed an end-fly-cutting-servo diamond machining method via a combination of end-fly-cutting and SSS to fabricate large-scale MLA and hierarchical micro-nanostructures. Integrating radial-fly-cutting with a new type of FTS that is erected on the Z-axis to realize fast servo of the workpiece in the vertical direction, Zhu, et al , introduced a new diamond cutting approach. With this new approach, hexagonal micro-sphere lens array, micro-freeform lens array, as well as a seamlessly stitched sinusoidal surface with quadrupled areas were successfully fabricated. More recently, some innovative diamond cutting processes, like diamond micro chiseling (DMC) , Guilloche machining technique  and virtual spindle based tool servo diamond turning  have been developed to generate discontinuously geometric structures. These techniques are able to manufacture 100%-fill-factor lens array with hexagonal individual apertures.
Although ultra-precision diamond cutting has been comprehensively studied and developed, it is limited to machining the MLAs whose substrate materials are plastics, crystals and non-ferrous metals, such as aluminum, brass, nickel and their alloys, which are most commonly being used as the material of mold tool inserts in injection molding process . However, compared to glass, plastic has poor deformation resistance, temperature resistance, scratch resistance and moisture resistance, high heat expansion coefficient, and low refractive index [33,34]. Additionally, ultraviolet light is either not transmitted at all or leads to solarizing of plastic . Optical glass represents the better alternative. For high volume production of glass optics precision glass molding (PGM) and hot embossing are the most common precision mass replicative techniques [34,36–38] These processes are characterized by high temperatures slightly above the transition temperature of the molded glass and high molding forces . Consequently, suitable materials of mold insert must possess high hardness, low heat expansion coefficient, high thermal stability and corrosion resistance . Furthermore, the mold insert surface should not react and adhere to the moldable glass. A few materials such as tungsten carbide (WC), silicon carbide (SiC) and glassy carbon are able to meet the harsh requirements [36,40]. Unfortunately, diamond cutting cannot be employed to machine these materials because of their extremely high hardness and brittleness, leading to excessive tool wear and poor quality of machined surfaces [41–43]. Ultra-precision diamond grinding is recognized as the most suitable technique to realize complex precision parts or structures on hard and brittle materials [44–46]. Xie, et al , fabricated micro pyramid arrays on a silicon surface using crossed grooving with a 60° V-tip of diamond grinding wheel. Guo, et al , applied ultrasonic vibration assisted grinding to generate micro V-groove array and micro pyramid array on binderless WC as well as SiC ceramic. The micro arrays with roughness of less than 50 nm and edge radius of less than 1 μm were achieved. Yamamoto, et al , ground a spherical concave microlens array, which has 19 lens-lets, on WC with a simultaneous 4-axes controlled grinding methods. Chen, et al , ground a microlens array mold insert consisting of 19 concave spherical lens-lets by C-axis fixed cross grinding. A double-sided glass MLA was obtained by precision glass molding press with the ground insert. The form accuracy and surface roughness of the glass MLA were less than 0.4 µm and 6 nm, respectively. However, the aforementioned grinding methods are unable to fabricate MLA that the lens-let has aspheric form. A new technique is needed that can make aspheric MLA mold insert has excellent quality for precision glass molding press or hot embossing.
In this study, an off-spindle-axis (OSA) spiral grinding technique is proposed to fabricate aspheric microlens array (AMLA). With this new technique, the form accuracy of lens-let could be improved by wheel path compensation based on the on-machine measured profile error. The principles and implementations of the proposed technique are elaborated. The wheel path generation for aspheric lens-let and profile error compensation approach based on on-machine measurement are detailed. The experiments on the ultra-precision grinding of a four-lens-let convex aspheric microlens array mold insert were done to validate the feasibility of OSA spiral grinding and evaluate the compensation efficiency and accuracy. PGM experiments were also conducted to fabricate glass AMLA using the ground insert. Both the ground and the molded AMLA achieved sub-micron form accuracy and nanometric surface roughness.
2. Off-spindle-axis spiral grinding
The contour grinding, such as cross grinding and parallel grinding , is generally employed to fabricate the symmetric aspheric surfaces. However, it fails to manufacture the AMLA because the axis of work spindle cannot be aligned to each symmetric axis of the lens-let in a single workpiece installation. The OSA spiral grinding provides a new perspective that constructs a spiral wheel path for individual lens-let by controlling the two translational motions, which are perpendicular to the axis of work spindle, while the work spindle rotation is controlled in the positioning mode. Meanwhile, the grinding wheel servos along the axis of work spindle according to the profile of lens-let. By repeating this procedure to every lens-let, discontinuous aspheric microlens array can be obtained sequentially.
In order to achieve the above purpose, a four-axis machine system is required. The configuration of OSA spiral grinding of a convex AMLA is illustrated in Fig. 1. It totally consists of three linear axes, i.e. X-, Y-, and Z-axis, and a positioned rotary axis, namely C-axis. The workpiece is fixed on a jig and then attached on the work spindle (C-axis) via the vacuum chuck. A round-off V-type grinding wheel is installed on the high speed grinding spindle, whose rotation axis is set parallel to the X-axis. The system O-xmymzm is fixed on the work spindle and defined as the global coordinate system of the machine system, where zm-axis is coincident with the work spindle axis, xm- and ym-axis are parallel to the X- and Y-axis, respectively.
The schematic view of the basic principle of OSA spiral grinding is shown in Fig. 2. Three coordinate systems, including O-xwywzw, A-xlylzl and A-xiyizi are defined as the workpiece coordinate system, lens-let coordinate system and intermediate coordinate system, respectively. O-xwywzw and A-xlylzl are attached to the workpiece center and the lens-let center, respectively. Both of them share the same axis directions and synchronously rotate around the work spindle. A-xiyizi is set at the same lens-let center, whose orientations are always kept consistent with that of O-xmymzm. During machining process, O-xmymzm remains static.
Figure 2(a) presents the initial machining stage of one of the lens-lets. At this stage, O-xwywzw and A-xlylzl are coincided with O-xmymzm and A-xiyizi, respectively. ρ is the distance between zw- and zl-axis, while α is the angle between OA and the xw-axis.
Figure 2(b) illustrates the motion decomposition of origin of the A-xiyizi coordinate system after the workpiece rotated with an angle of θj. Taking O-xmymzm as the reference frame, when the workpiece is rotating around the zm-axis, the motion of the A- xiyizi coordinate system can be decomposed into two simple harmonic oscillations along xm- and ym-axes, respectively. Such two oscillations have the same amplitude of ρ with π/2 phase difference.
It is noteworthy that when A- xiyizi is treated as the reference frame, the motion of A-xlylzl is regarded as rotation around the zi-axis. Furthermore, an equivalent contour grinding of the lens-let can be constructed in the reference system of A- xiyizi when the grinding wheel feeds along the xi-axis and servos along zi-axis, as shown in Fig. 2(c).
Based on this strategy, every lens-let is ground individually. By repeating these processes to the other lens-lets, the AMLA can be fabricated successfully.
2.2. Determination of wheel path for single aspheric lens-let
Since the whole AMLA can be machined by repeating the basic grinding process for a single lens-let, only the determination of the wheel path for a single lens-let is required.
As described in Section 2.1, the coordinate position of point A in the global coordinate system (O-xmymzm) after the workpiece rotated with an angle of θj can be obtained by
With f being the feed per revolution along the xi-axis for the cutter contact point (CCP), i.e. point P shown in Fig. 2(c), the position of CCP in the intermediate coordinate system can be expressed by
Let the cutter location point (CLP) fix on the center of the round-off grinding wheel. As shown in Fig. 2(c), the point C, i.e. the CLP, and the point P has the following position relationship in the intermediate coordinate system
By the principle of composition of motions, the position of the CLP in the O-xmymzm system can be further determined by
In real machining process, the motion components and are realized by the reverse direction movements of X- and Y-axial slides of the machine tool, and can be realized by the tool servo motion along the Z-axial slides of the machine tool. The work spindle generates the rotation angle θi.
2.3. Profile error compensation grinding with on-machine measurement
In practice, the ground part form deviates from the desired surface owing to many quasi-static systematic errors: wheel setting error, wheel dimensional and profile errors, error caused by wheel wear, etc. Conventionally, the form deviations were measured by off-machine measurement instruments and then compensated in the subsequent grinding cycle. Nevertheless, the off-machine measurement would bring the re-installation error into the next machining cycle. Apparently, utilizing the on-machine measurement unit to measure the ground surface can effectively eliminate the error resulted from the re-installing of the workpiece and substantially improve the machining efficiency.
Here, an air bearing linear variable differential transformer (LVDT) probing on-machine measurement system is used to measure the profile error of the ground aspheric lens-let. Figure 3 schematically illustrates the principle of the measurement. The probe moves along the ideal measuring path under control of the X- and Z-axis, while the LVDT sensor captures the position deviation of the probe. However, the measured error data cannot be directly applied to characterize the profile error of the ground aspheric lens-let due to the effect of the probe size. For a specific point D (xD v, 0, zD v) on the ideal probe centre path, the sensor detects a deviation of δD z in the zi direction. The corresponding actual position of the probe centre is point F (xF v, 0, zF v), the relationship between point D and F can be expressed as
To obtain the position of the actual contact point M (xM i, 0, zM i) between the probe and the actual ground surface when the probe centre is located at point F, the continuous actual probe centre path has to be generated to determine the normal vector at nF at point F. The cubic B-Spline fitting method  is applied to construct the actual lens-let profile based on the discrete point sets of F. Before performing the fitting process, high-frequency components of the discrete point set of F need to be removed. Here, the robust second order Gaussian regression filter , which is insensitive against outlier and can simultaneously process profiles with form components, is adopted. The normal vector nF is determined by computing the normal direction at point F using the fitted actual lens-let profile.
The continuous actual ground aspheric profile is reconstructed based on the point sets of M with the cubic B-Spline fitting method. For any point P(xP i, 0, zP i) on the designed aspheric profile, the corresponding point P’(xP’ i, 0, zP’ i) is the intersection of the continuous actual ground aspheric profile and the normal of point P. δP n, which is the signed distance between point P and P’, is defined as the normal residual error of point P. δP n is positive while P’ is above the designed lens-let profile and is negative while P’ is below the designed lens-let profile.
To improve the form accuracy of the ground aspheric lens-let, the new wheel path for the next grinding needs to be generated based on the previous wheel path and the measured profile error. The relationship between them is demonstrated in Fig. 4. For a theoretical CCP P, moving the grinding wheel at point P to point P” with a distance δP n, the error can be eliminated. Therefore, the following compensated wheel path can be expressed as
3. Experimental setups
3.1. Ultra-precision grinding of AMLA mold inserts
Grinding experiments were conducted to verify the feasibility of OSA spiral grinding and the efficiency of the compensation method proposed on a five-axis ultra-precision machine tool, as shown in Fig. 5. The workpiece was clamped on the work spindle via a homemade fixture. The work spindle can maintain an angular position to less than 1 arcsec in a modulated mode. The high speed grinding spindle was installed on the B-axis rotary table. Three linear axes (X-, Y- and Z-axis) translate linearly with positioning resolution of 1 nm. The rough grinding was performed to take the bulk removal of material and obtain a preformed microlens array using a resin bond diamond grinding wheel of grit size #325. In fine grinding, a resin bond diamond grinding wheel of grit size #2400 was utilized to achieve the required profile accuracy and finish. Prior to grinding, wheels were trued on-machine with a rotary tantalum truer to enhance the geometric accuracy of the arc nose. The detailed conditions of grinding experiments are summarized in Table 1.
The Nanotech Workpiece Error Correction System (WECS) installed on the B-axis rotary table near the grinding spindle was used to measure the profile error of the ground lens-lets. In this system, an air bearing Linear Variable Differential Transformer (LVDT) probe is driven under the ideal measuring path across the radial direction of a symmetric surface in contact mode, while LVDT sensor captures the displacement of the probe along Z-axis. The measured deviation is recorded by the data acquisition subsystem and then used to compensate for the wheel path. The accuracy and the resolution of the LVDT along the Z-axis are 10 nm and 1 nm, respectively. A ruby probe with radius of 0.2 mm was used in measurements.
In grinding experiments, a microlens array mold insert with four convex aspheric lens-lets was fabricated. Binderless tungsten carbide was used as the workpiece material. Parameters of microlens array shown in Table 2 were used in the experiments. Four lens-lets are arranged in the square pattern as the 3D model shown in Fig. 6(a). Figure 6(b) illustrates the profile and slope feature of single lens-let. The sag is approximately 0.12 mm and the max slope angle is about 15°. Following the tool path determination strategy presented in Section 2.2, the tool path for rough grinding, which has a larger spiral pitch of CCP than fine grinding to show a much clear view of the wheel path, of #1 lens-let is generated as illustrated in Fig. 6(c). The machining parameters were adopted as the ones shown in Table 1. The relationships between the rotation angle of C-axis and the xm-, ym-, and zm-component of the wheel path is illustrated in Fig. 6(d). As the C-axis rotates, the x-axial slide tends to rise with oscillation; the y-axial slide oscillates around the Om; and the z-axial slide descends following the profile of the lens-let. As expressed by Eq. (5), the oscillations in x- and y-axial directions are rigidly harmonic motions with π/2 phase difference. The frequency of each oscillation is exactly the same as that of the rotation speed of the work spindle, while the amplitude the oscillations is equal to the distance between the lens-let axis and the work spindle axis. The rising trend of x-axial slide was contributed by the wheel feed in the direction of xi-axis.
3.2. PGM of AMLA
To fabricate the glass AMLA, a precision glass mold press machine (GMP-311V) was used. The glass selected for molding experiment was a moldable glass (D-ZK3, CDGM), whose transition temperature is 511 °C. The glass blank was optically polished disk with a diameter of 10 mm and thickness of 5 mm. The glass AMLA was molded under the following conditions: (i) heating of molds and glass gobs to a molding temperature of 552 °C, (ii) forming at 552 °C with a lower mold velocity of 0.2 mm/s in the upward direction, (iii) initial slow cooling of the molded lens at a cooling rate of 0.5 °C/s, (iv) rapid cooling of the lens at a cooling rate of 1 °C/s to a release temperature of 140 °C, (v) the molded lens was released and subsequently cooled to room temperature (~20 °C). Nitrogen gas was used first to purge oxygen and then in the second and final stages to control the cooling rate. The molding load was maintained at a constant magnitude of a 0.3 kN force during the initial forming to minimize the residual stresses in the lenses from the deformation and temperature changes. To improve replication accuracy, a second pressing at 0.2 kN was applied at the beginning of the fast cooling period.
4. Results and discussion
The photographs of the ground AMLA mold insert and the molded glass AMLA are illustrated in Figs. 7(a) and 7(b), respectively. It can be seen that mirror surfaces were obtained for both of them.
4.1. Characteristics of the AMLA
For characterizing the structure of ground and molded AMLAs, both of them were captured and stitched by laser scanning confocal microscope (OLS4000) with an amplification of 20 × , which has a lateral resolution of 0.5 μm. Figures 8(a) and 8(b) show the 3D structures of the ground AMLA mold insert and the corresponding cross-hair profiles, respectively. From the results illustrated in Figs. 8(a) and 8(b), homogeneous features of both the 3D structures and the profiles are obtained, well indicating the uniformity of the ground AMLA. Since distortion of the relative positions between the lens-lets is a critical factor forming the form error of the array, the distance between two lens-lets were measured, which are 2.5 mm and 2.5005 mm in horizontal and vertical direction, respectively. The observed values have good agreements with the designed ones as shown in Table 2. More importantly, the shapes along with the sizes of the two profiles along the cross-hair directions align well with each other, demonstrating high consistence and high accuracy of the obtained structures. The 3D structures of the molded AMLA and the corresponding cross-hair profiles shown in Figs. 8(c) and 8(d), respectively, also exhibit the same characterizations as the ground AMLA mold insert, except for the reverse shape.
4.2. Form accuracy
Figure 9 illustrates the evolution of profile error of the #1 aspheric lens-let for the grinding processes without and with profile error compensation. The first finish grinding was done without any compensation process, which obtained a profile error of 522 nm in peak-to-valley (PV) value measured by the WECS, as shown in Fig. 9(a). The profile error curve has a “Λ” shape, indicating that an inward tool setting error in X-axis was generated in the tool setting stage. Conventionally, the tool setting error should be analyzed and then adjusted to eliminate the form error with obvious shape characteristic. In order to prove the validity of the compensation strategy, the proposed compensation procedures were carried out directly. After the first compensation grinding, the profile error was decreased to 207 nm as shown in Fig. 9(b). To further improve the accuracy, the compensation procedures were repeated, which improved the profile accuracy to 135 nm in PV as shown in Fig. 9(c). To evaluate the accuracy of the results of WECS, the profile error of the first ground lens-let was also measured using the Form Talysurf profilometer, which has a stylus with 2 μm tip radius. Figure 9(d) shows the profile error measured by the profilometer, giving a PV value of 231 nm. Compared the profile error shown in Figs. 9(c) and 9(d), both profile errors have the same fluctuation pattern and same level PV value, demonstrating the consistency of the measurements. Also worth noting is that the PV value obtained by the WECS is slightly smaller than that measured by the profilometer, which is caused by the size effects of the probe.
Figures 10(a) and 10(b) compare the four-lens-let profile errors of the ground and molded AMLA measured by the Form Talysurf, respectively. As can be seen, the lens-lets of ground AMLA mold insert shows good consistency in shape of profile errors, because of the identical processing method and compensation strategy. The profile errors of molded glass AMLA lens-lets present a completely opposite outline in comparison to the mold on account of a precision glass molding process. The details of profile errors of both mold insert and molded glass elements matches well, which illustrates the fact that the PGM is executed adequately. The specific measured data, including peak-valley value and root-mean-square value of profile errors, are shown in Table 3. Results show that the magnitude of profile errors of mold insert and molded glass AMLA is same while the value of the latter is slightly smaller. This phenomenon can be attributed to the shrinkage of the glass during the cooling stage of precision glass molding and would not cause any negative effect to the performance of molded glass element in this case.
4.3. Surface micro-topography and roughness
Scattering losses and blurred images would occur if the surface quality is inadequate. Accordingly, the ground and molded AMLA quality in terms of surface roughness were investigated by the 3D Optical Surface Profiler (Nexview) with 50 × objective lens. Figure 11 illustrates the surface topography at different regions of the #1 lens-let for ground and molded AMLA. It can be seen that the ground lens-let possesses high uniformity in the nanometric surface roughness in central region (Sa 7.045 nm) and the off-central region (Sa 7.296 nm), as shown in Fig. 11(a). The same pattern has been replicated onto the molded glass component as shown in Fig. 11(b). It is noted that two periodic fluctuations with different spatial frequencies emerge in the micro-topography. The high frequency spiral pattern texture is corresponding to the residual tool marks, which is a common phenomenon occurred in the contour machining process. However, the predominant low frequency ring pattern fluctuations might be induced by the improper wheel dressing conditions. As the amplitudes of these undesired fluctuations are about 50 nm, they can be eliminated by a subsequent polishing process. Surface roughness (Sa) of all the lens-lets of mold inserts and molded glass are listed in Table 4. It is indicated that a homogeneous surface quality has been achieved via the fabrication strategy according to the data that the average Sa is about 7.5 nm with a sub-nano standard deviation (STD). Since the grinding wheel leading in from the center of lens-let, the STD in the central region is larger than that in the fringe region, which agrees to the machining process.
In conclusion, a novel off-spindle-axis spiral grinding is developed for the fabrication of molds used in the manufacturing of aspheric microlens arrays by precision glass molding. The core idea of the method is to construct an off-spindle-axis spiral wheel path via two translational motions that are perpendicular to the work spindle axis while the work spindle is controlled in the positioning mode. With this method, the form accuracy of lens-let can be improved by the on-machine profile measurement. The strategy of wheel path generation and profile error compensation based on on-machine measurement were detailed. The proposed methods were verified experimentally by grinding a four-lens-let convex AMLA mold insert. The experimental results indicated that the AMLA mold insert can be fabricated with high homogeneous quality. The obtained form accuracy and surface roughness were less than 0.25 μm and 10 nm in Sa, respectively. Precision glass molding was also conducted to replicate the glass AMLA. The molded glass AMLA possess almost identical features as the mold insert, except the inverse shape. The proposed technique is ideal for production of mold for molding press glass AMLA.
National Key Research and Development Program (2016YFB1102200); National Natural Science Foundation of China (61635008, 51320105009); ‘111’ project by the State Administration of Foreign Experts Affairs and the Ministry of Education of the People’s Republic of China (B07014).
The authors would like to express their sincere thanks to Mr. Zhen Li and Mr. Chao Xiao for the preparation of experiments.
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