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Abstract
A microlens array (MLA) is a fundamental optical element, which has been widely applied in the fields of imaging sensing, 3D display, and lighting source. However, it is still a challenge to design the MLAs simultaneously satisfying small size, wide field of view, and high image quality. Herein, a novel type of concave lens array on an aspheric convex substrate (CLAACs) is presented, which is composed of an aspheric substrate and a spherical concave subeye array. The facilely designed method of the CLAACs is described and its geometric model is also established by a numerical example. Furthermore, a fabrication method, which is directly machining the CLAACs on PMMA material, is proposed. To realize the ultra-precision machining of the lens, tool path planning is carried out before fabricating. The profile, surface quality, and imaging performance of the fabricated lens are then characterized to reveal its optical capabilities. The results show that the proposed method can realize the rapid design and fabrication of lenses flexibly and efficiently. The fabricated CLAACs exhibit excellent morphology uniformity, high imaging quality, and focusing performance. The study provides a feasible solution for the design and fabrication of such lens arrays with complex discontinuous surfaces.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microlens arrays (MLAs) is the essential components of optical systems. MLAs has been widely used in imaging sensing [1], 3D displays [2], illumination sources [3], and other devices [4–7] due to its good optical properties for diffusion, uniformity, focusing, and beam shaping of the incident light. The MLAs is usually designed as a convex lens to obtain a wide field of view. The convex lens is composed of a curved substrate and a convex subeye, which has the characteristics of thin thickness, compact volume, and easy integration. However, the arrangement of the microlens on the substrate has an important effect on the imaging quality and focusing effect of the lens. When the light rays come in obliquely, the lens array needs to adopt a spacing arrangement to ensure the focusing effect and imaging quality. But the filling factor and light energy utilization of the lens will decrease. Additionally, the optical crosstalk between subeyes has also an effect on the imaging quality of the lens [8,9]. Therefore, it is still a challenge to design MLAs with high imaging quality while maintaining a small volume and large field of view.
With the extensive application of MLAs in optics, imaging, and other fields, the development of various types of the microlens and its array is still one of the hot issues in current research. Microconcave lens arrays (MCLAs) has attracted extensive attention in recent years due to its wide field of view and high sensitivity [10–12]. The lens consists of a substrate and concave-subeye arrays. Compared with the convex lens, the subeye of the MCLAs is concave in the inner side of the substrate, which increases the filling factor and improves the acquisition efficiency of incident light. The lens also has good focusing even under off-axis illumination. Since the imaging between different subeyes is independent, the so-called “mosaic image” is formed. There is no crosstalk effect between the subeyes, and the imaging characteristics are significantly improved. So far, many scholars have investigated the MCLAs. For instance, Liu et al [13] proposed an amorphous silicon thin-film solar cell with a MCLAs coating, and the results showed that the cell has higher photovoltaic performance and its light reflectance was also significantly reduced compared with that of conventional solar cells. Meanwhile, the MCLAs is also widely used as the replication mold for the convex lens. Albero et al. [14] used isotropic wet etching and molding techniques to produce a concave mold, and then combined hot embossing and UV-molding to successfully replicate convex microlens. Additionally, some recent research results have shown the potential application of the MCLAs for optical imaging, e.g., compensation of imaging [15], improvement of imaging separability [16], optimization of integral imaging quality [17] can be performed using the MCLAs, as well as for sensors and laser systems [18–21], illumination uniformity [22] and imaging and sensing in the far-infrared [23,24].
Through the above literature research, it is known that the MCLAs can overcome the drawbacks of convex lens effectively. However, up to date, the previous research have mainly focused on: (1) Fabrication of plane concave lens arrays. (2) Development of replication mold for the convex lens. (3) Combined with the convex lens to realize achromatic aberrations, whereas the literature on the design and fabrication of the MCLAs on the curved substrate, especially the single-use of this lens as an optical device has not been reported. Furthermore, the development of injection molding [25], UV-cured printing [26], laser ablation [27], grayscale lithography [28], 3D printing method [29], ultra-precision diamond turning [30], and other processing techniques [31–35] have also laid the foundation for exploring and investigating new lenses.
The rapid development of optical technology has put forward more stringent requirements for optical elements, such as volume, imaging quality, and field of view. In our previous research work [36], it was successfully fabricated spherical lens array on a planar substrate using molding technique, but the selection of lens materials was limited and the molding process was prone to defects such as fragmentation, bubbles, and underfilling. The processing and precision of the microstructure mold are still difficult to be guaranteed. The lens is also limited by the plane substrate in terms of field of view and volume.
On the basis of these advancements, we present a new type of concave lens array on aspheric curved surface for use as the element of optical systems. This new design can obtain high material removal rate, weight light and volume small of lens in composed of a spherical concave subeye array and aspheric substrate manner. No auxiliary equipment is required because the lens is designed based on an aspheric substrate, and the wide field of view of the lens is effectively guaranteed. The proposed lens is different from the spherical lens array [36] in two ways. The first difference is that the proposed design uses an aspheric substrate and subeyes to control the lens imaging effect actively, whereas the substrate of spherical lens array is plane and passively changes the image quality by molding depth. The second difference is that the proposed lens the lens adopts direct machining on PMMA, which avoids the limitations of molding technology in terms of material, molding process and molding accuracy, and enables flexible manchining of curved lens arrays.
The remainder of this paper is arranged as follows: Section 2 introduces the design method and presents the geometric model of the proposed lens. Section 3 performs the tool path planning and describes the machining principles of the proposed lens. Section 4 conducts the machining process and characterizes the surface quality and optical properties of the fabricated lens. Finally, the conclusions are summarized in Section 5.
2. Mechanical design of the CLAACs
Generally, shape and materials of the lens play an important role in its optical properties. Compared with materials, the optical performance of the lens can be significantly improved by changing its structural parameters [37–39]. The optical surface curvature and subeye arrangement will affect its focal length and imaging quality. Therefore, it is necessary to study the design method of the proposed lens to choose the range of lens design parameters reasonably, such as radius of curvature of substrate surface, radius, depth and phase angle of the subeye.
The concepts of MCLAs and aspheric convex substrate are synthesized to complement each other, resulting in a novel lens structure with good optical properties. In the following, considering the lens machining, the envelope approach is used to design the new lens in this paper. In Section 2.1, the characteristics of the new lens is described. In Section 2.2, the design method of the lens is illustrated. In Section 2.3, a numerical example is given to perform the geometric modeling of the lens.
2.1 Traditional and new concept of a lens array
The convex lens is one of the most widely applied optical elements, and its structure is shown in Fig. 1. The new type of lens that is composed of a spherical concave subeye array and aspheric substrate in this paper, called concave lens arrays with an aspheric convex substrate (CLAACs), as shown in Fig. 2. Compared with convex lens, the subeye of the proposed lens is concave, together with the curved substrate structure, which incorporates the huge inherent advantages of size, weight, and volume. Compared with the MCLAs, the substrate of the proposed lens is aspheric, which makes its structure more compact and the field of view is also further larger. Furthermore, since the large curvature difference between the subeye and substrate, the aspheric substrate first images the target object (AB) to produce an enlarged virtual image (A′B′). This virtual image is then secondarily imaged (A″B″) by the concave subeye, which leads to a further improvement in the resolution of the lens, as shown in Fig. 2(b).
Meanwhile, studies have shown that the best way to obtain a high-quality and distortion-free curved microlens array is to directly fabricate the microlens array on a curved substrate [31,35]. Therefore, we attempt to use diamond turning with a swinging tool to directly fabricate the whole lens on PMMA material in Section 4.
2.2 Design method
In order to improve the efficiency and accuracy of lens design, the free-form surface combination theory [22] is used to subdivide the CLAACs into two parts: aspherical convex substrate and spherical concave subeye array. The aspheric convex substrate is designed first, and then the spherical concave subeye array is also performed and compounded onto the substrate. Finally, the mathematical model of the CLAACs can be obtained.
- (a) Aspherical convex substrate design. Figure 3 shows a schematic diagram of the aspherical convex substrate profile. Assuming that a moving circle of radius r move along the substrate spiral, Pp (x0, z0) is the center of the circle, Pc (x1, z1) is the contact point, θ is the initial rotation angle, α is the initial angle of the moving circle, ω·t is the angle of rotation of the moving circle, and the line m is the tangent line of the curve L at the point Pc, then we have the point Pc (x1, z1) on the moving circle can be expressed as the straight line m is the tangent line at the point Pp, then the slope of the line PpPc is combining Eqs. (1)–(3), the coordinates of the center of the moving circle Pp can be obtained, and the aspherical convex substrate formed by the moving circle envelope can be also obtained.
- (b) Spherical concave subeye array design. Figure 4 shows the spatial position of the subeye of lens. $O^{\prime}$ is the center of the circle of the subeye, and $O^{\prime\prime}$ is the center of the circle of the moving circle. In $\triangle O{O_1}O^{\prime}$, we have$$\begin{array}{c} {\left|{\overrightarrow {OO^{\prime}} } \right|= r^{\prime}\sin \varphi ^{\prime}}\\ {\left|{\overrightarrow {OO^{\prime\prime}} } \right|= \sqrt {{r^{\prime 2}} - {{({r^{\prime}\sin \varphi^{\prime}\sin \alpha } )}^2}} } \end{array}$$where $\angle OO^{\prime\prime}{O_D} = \varphi ^{\prime}, \cdot \left|{\overrightarrow {O{O_1}} } \right|= r^{\prime}\cos \varphi ^{\prime},\left|{\overrightarrow {{O_1}O^{\prime}} } \right|= r^{\prime}\sin \varphi ^{\prime}$.
Similarly, in $\;\mathrm{Rt}\;\mathrm{\Delta}O^{\prime\prime}{O_1}O^{\prime}$, the $\mathrm{\mid }\overrightarrow {{O_1}O^{\prime\prime}}$ and $\left|{\overrightarrow {O^{\prime}O^{\prime}} } \right|$ have following relationship:
Figure 4 shows a cross section view of the lens. In $\mathrm{\Delta }OO^{\prime\prime}{P_p}$, we have
So, the $\left|{\overrightarrow {O^{\prime\prime}{P_p}} } \right|$ have2.3 Geometric model
In this section, a numerical example of lens surface generation is conducted to demonstrate the effectiveness of the novel design method. Based on the above design method, the generating procedure is programmed using MATLAB software. Then the points on the CLAACs surface are calculated and imported into CAD software to generate the CLAACs.
Taking the lens in a short-focus vision system as an example, the lens array with a focal length of 6.388 mm is designed using the above design method. The lens substrate is parabolic, the spacing between two adjacent rings of lenses a is 0.7 mm, and the lens material is PMMA [23]. The remaining design parameters and geometric models of the CLAACs are shown in Table 1, Table 2 and Fig. 5, respectively.
Table 1. Design parameters of the CLAACs.
Table 2. Number of subeye for CLAACs.
3. Fabrication method of the CLAACs
In this section, we attempt to machine the MCLAs by ultraprecision cutting, namely, diamond turning, which can produce higher geometric freedom and lower surface roughness than grayscale lithography. However, the machining of aspheric lenses by ultra-precision diamond turning technology is still a challenge. In the cutting of non-continuous surfaces, the tool and the workpiece are prone to problems such as overcutting or edge biting, which will affect the surface quality of the lens [40]. Meanwhile, much of the previous research on the diamond turning of PMMA was performed on flat or spherical surfaces, while there is little research on the diamond turning of PMMA for aspherical concave lens arrays. Therefore, it is necessary to plan the tool path before machining lens arrays by ultra-precision diamond turning technology.
Motivated by this, we propose a fabrication method for direct machining of the whole lens on PMMA material. This method is divided into two steps: (1) Based on the single-point diamond turning technology, one is to propose directly fabricating the proposed lens on PMMA by introducing a rotational motion (B-axis) to swing the diamond cutting tool and adjust the actual tool clearance angle in real-time, which can prevent tool interference effectively. (2) The optimal tool path determination strategy has been developed with consideration of geometries and installation poses of the diamond tool.
3.1 Machining principle
The CLAACs has non-linear characteristics. In the machining process, not only the linear motion of X-axis, the angle control of the main axis (C-axis), but also the rotation of the B axis and the micro feed of the sharp tool axis should be considered. In this paper, a B-axis is added to the single-point diamond fast tool servo machine, which makes the C-axis, Z-axis, B-axis, and FTS-axis linkage, realizing the synchronous rotation and radial feed motion of diamond turning tools. Finally, the fine machining of the proposed lens is completed.
The machining device for the CLAACs is shown in Fig. 6(a). By linkage of the Z-axis and B-axis, the aspheric trajectory motion with large curvature variation is realized, and the machining of the CLAACs is also achieved with the fast tool axis, as shown in Fig. 6(b). In addition, other free-form microstructures with large curvature variations can be machined by synergistically controlling the variation of the Z-axis and B-axis. Therefore, using the proposed method, a wide range of lens geometries and arrays can be fabricated.
3.2. Tool path planning
The tool path directly affects the lens surface quality and performance. If the tool path does incompatible with the surface characteristics of the lens, overcutting or edge biting will occur easily [41]. Therefore, planning the tool path reasonably is the key to fabricating the lens array with high precision.
The solution of the 3D coordinates of the tool location points is a key problem in tool path planning. However, since the CLAACs is different from the conventional free-form surface, it cannot be described by a single determined equation. The tool path planning is more complicated. Herein, the CLAACs is divided into two parts: the substrate surface and the lens array.
The tool path planning is carried out separately for the above two feature surfaces, and then the planned tool path are combined to generate an NC program for machining the lens. And the principle can also be applied to lens arrays with other shapes.
Specifically, the tool path planning and overall procedure for fabricating the CLAACs is shown schematically in Fig. 7, and the steps are as follows:
- Step 1: According to the design requirements of the optical element, the main design parameters of the lens are obtained, such as lens diameter, the number, and depth of the subeyes, etc.
- Step 2: Based on the form accuracy and surface roughness of the lens, determine the machining process parameters, such as tool radius, spindle speed, radial feed speed, etc.
- Step 3: The 3D model of the lens is built, and the tool path planning is performed for this lens to get the spatial coordinates of the tool position point.
- Step 4: Convert the tool point coordinates into a CNC program to machine the lens on a fast tool servo machine and evaluate its surface quality and optical properties.
As mentioned above, the essential to achieving ultra-precision machining of the CLAACs is to find the spatial position of each tool point. The tool path of the CLAACs is obtained by taking the lens contour as the base and shifting the tool radius by an equal distance, as shown in Fig. 8 (a). The blue curve is the tool path of the substrate profile and the green curve is the tool path of the subeye profile. Using the lens design method in Section 2, the radius of the moving circle is set as the tool radius, so that the tool position points of each part on the lens surface can be obtained directly.
However, there is a specific area where the substrate surface and the subeye intersect. The tool tip does not contact the substrate surface and subeye, leaving residuals in the transition area, which will affects the lens machining accuracy. The angle range of the specific area is θB. Therefore, in order to ensure obtaining a complete surface of the fabricated lens, a tool path optimization of the specific areas must be performed.
First, the angular value of θB is obtained using the geometric relationship. On this basis, the intersection point Pco of the substrate surface and the subeye is taken as the center of the circle, and the arc of the tool tip is taken as the radius to make an arc, so that the tool path in the area is smoothly transitioned, thus the optimized tool path is obtained, as shown in Fig. 8(b). The value of θB can be calculated as follows.
In $\mathrm{\Delta }OO^{\prime\prime}{P_{CO}}$, the $\angle OO^{\prime\prime}{P_{CO}}$ can be obtained by
Similarly, in $\mathrm{\Delta }OO^{\prime\prime}{P_D}$, the length of $\left|{\overrightarrow {O{P_D}} } \right|$ can be obtained by
In $\triangle O{P_D}{P_{CO}},\angle OO^{\prime\prime}{P_{CO}}, \cdot \left|{\overrightarrow {O{P_D}} } \right|$ and θB are related as follows
To verify the correctness of the tool path planning and optimization method, MATLAB software is used to simulate the machining of the CLAACs in Section 2.3. Figure 9(a) shows the comparison results of the original tool path and theoretical profile of the lens. The blue curve indicates the theoretical profile of the lens, and the red curve indicates the original tool path. It can be found that there is an error in the transition area between the tool path and the theoretical contour of the lens, i.e., there is a margin at the subeye edge. Figure 9(b) shows the comparison results of the optimized tool path and the theoretical profile of the lens. It can be revealed that the optimized tool path is match well with the theoretical profile of the lens. The results show that the tool path planning method for machining the CLAACs is feasible. Therefore, by deliberately choosing tool geometry, tool path, and cutting parameters, complicated CLAACs structures with different shapes and dimensional sizes can be obtained.
4. Machining experiment and performance evaluation of the CLAACs
In this section, a machining experiment is carried out and surface quality and optical properties of the fabricated CLAACs are also measured and analyzed. Surface quality mainly includes profile and surface roughness. Optical properties mainly include focal length, imaging quality, and light focusing.
4.1. Experimental setup
To validate the effectiveness of the proposed machining method, a machining experiment is first carried out, and then the optical properties of the fabricated lens are measured. Figure 10 illustrates a single-point diamond ultra-precision fast tool servo-turning machine, which consists of an air-bearing spindle, B-axis, tool holder, and X, Y, and Z coordinate axis.
An air-bearing spindle is mounted on the X-axis, and the direction of the spindle axis is perpendicular to the X-axis. The Z-axis is equipped with a rotatable B-axis (model HDS-AB-B-NFS250), a tool holder with adjustable tool position is mounted on the B-axis table, a fast tool servo drive is mounted on the tool holder independent of the machine tool, the tool is fixed on the fast tool servo drive, the workpiece is fixed on a special fixture, the special fixture is fixed on the spindle by vacuum suction cup.
Lens arrays can be fabricated on free-form surfaces or on steep surfaces by X-axis, Z-axis and B-axis linkage. Two-step process is adopted to fabricate the CLAACs on PMMA. As the first step, the aspherical convex substrate is fabricated using single-point diamond turning technology. The spindle speed is 2500 r/min, the feed rate is 1 mm/min. Subsequently, a spherical concave subeye array is also fabricated by linkage of the Z-axis and B-axis. In addition, the tool front angle and back angle are 0° and 10°, respectively, and the radius of the tool tip arc is 0.05 mm. the radius of the workpiece is 16 mm. The other machining parameters are also provided in Table 3.
Table 3. Finishing machining parameters of the CLAACs.
The ZYGO 3D Optical Profiler (NewView 9000 Series 3D) is used to measure geometric properties such as surface roughness and profile of the lens, and the focal length test set (SSFC-IV) is used to measure the focal length of the lens. The curved microlens array imaging system is used to measure the imaging quality and focusing effect of the lens.
4.2. Surface profile
Tool wear or incorrect tool position will affect the surface profile of the lens, resulting in the optical properties. In order to further evaluate the form accuracy of the machined lens, the whole lens and cross-sectional profiles of three subeyes are meansured, and the results are shown in Fig. 11.
Fig. 11. Geometric characteristics and three-dimensional morphology of lens, (a) machined lens, (b) and (c), the 3D morphology and profiles, (d)∼(f) comparison of actual and theoretical profiles of subeyes in different positions, (g) and (i), 3D morphology of the corresponding subeyes.
It can be seen that the fabricated CLAACs has a complete surface and a clear cross-sectional profile, which demonstrates the correctness of the tool path planning, as shown in Figs. 11(a) and (c). Figures 11(d)-(i) illustrates the measured profiles and 3D morphology of the 2nd to 4th ring subeyes of the CLAACs and compared with that of the theoretical profiles. The measured profile of the lens is smooth and match well with that of the calculated profile. The absolute error is less than 3%. Overall, it is demonstrated that the proposed method can fabricate high-quality the CLAACs, which can meet the basic requirements of micro-optics applications.
4.3. Surface roughness
The optical properties of microlens are important factors affecting the performance of optical devices [42]. Due to the surface roughness of the lens being too high, the lens will produce diffuse reflection and its image quality will be also reduced. Therefore, it is very important to detect and evaluate the surface roughness of the fabricated lens.
Figure 12 shows the surface morphology of subeye on each ring, and its measured mean roughness are also listed in Table 4. On the same cross-sectional profile, the roughness of the subeye on the first ring is significantly better than that of the other rings. This may be due to the fact that the tool cutting line speed of the outer ring is larger at the same rotational speed, and the faster the frequency of the fast tool entering and leaving the lens from the base surface, the more frequent the impact generated by the fast tool, causing the surface roughness of the lens to be larger. But in general, the lens subeye still has a good optical surface smoothness.
Table 4. Mean roughness of subeye on each ring.
Generally, the surface roughness of optical elements is required to have a microscale or nanoscale [43,44]. From the above results, the actual profile of the fabricated lens is very close to that of theoretical profile, and its surface roughness has also achieved the nanoscale. Thus, the fabricated lenses fulfill the optical functional surface requirements, and it also indicates that this technology can produce precision microlens arrays on an aspheric substrate.
4.4. Focal length
Focal length is one of the most important ways to check the quality of the lens structure and machining quality. In this paper, the test method in Ref. [36] is used for measured the focal length of the CLAACs to evaluate its optical performance. The refractive index, n, of PMMA material is 1.49, and the theoretical focal length of subeye is 6.388 mm.
Ten subeyes are randomly selected on the surface of the CLAACs for measuring the focal length, and the results are shown in Table 5. It can be found that the arithmetic means deviation of the lens is 6.879 mm, the mean standard deviation is 0.1349 mm, and the relative standard deviation is 1.96%. The relative error between the measured and theoretical focal length of the lens is 7.6%, which demonstrates that the experimental results agreed well with the theoretical calculation.
Table 5. Measured focal length of the CLAACs (mm).
4.5. Imaging quality
The imaging quality of the lens can be affected by poor surface finish or aberration between the subeyes. To investigate the characteristics of the fabricated CLAACs, a simple optical system is utilized to analyze the imaging quality of the laser beam. The schematic illustration of the optical system is shown in Fig. 13. The device mainly consisting of a He-Ne laser, beam expander, mask, sample table, and CCD. The light is emitted from the laser, passes through a skeletonized letter “E” mask, and forms an image on the CCD that is refracted by the CLAACs. The results are shown in Fig. 14.
As shown in Fig. 14, a clear image of “E” with no deformation is obtained for every subeye, which demonstrates that the fabricated CLAACs exhibit a good optical imaging performance. Note that the boundary between each group of the center of the lens is clear than that in the edges. The main reason is that the subeyes of the CLAACs are not in the same plane, and the formed image is a three-dimensional figure. Conversely, the formed image in the CCD is a two-dimensional projection of the lens on a plane. The edge subeyes of the lens form an angle with the digital amplifier, which causes the image of the outer subeyes to look blurry. The experimental results verify that the CLAACs has good imaging function and high spatial resolution.
4.6. Light focusing
The light focusing properties of the CLAACs are also tested and the results are displayed in Fig. 15. The grayscale intensities of the focal spots are almost the same on the same focal plane formed by subeye on the same distance from the center. It can be also observed that each subeye of the CLAACs can form a sharp focusing point on the focal plane due to the uniformity and excellent profile of subeye on the fabricated CLAACs. The Gaussian shape of the laser beam induced relatively strong light intensity at the center area when compared to the edge of the area. It is noted that the angle of incident light is increased for subeye from center to edge of CLAACs, which results in the decrease of the peak intensities of focal spots. This is mainly related to the angle between the incident light and the lens. Therefore, the presence of light intensity distribution further confirmed the lens effect of CLAACs.
From the above results, the proposed lens has good dimensional consistency, imaging and focusing ability, and the effectiveness of the design method and process technique of the lens are verified.
5. Conclusions
This paper presented a novel type of concave lens array on an aspheric convex substrate (CLAACs) to tackle the lens problems, such as volume, field of view, and image quality. The design principles of the CLAACs have been described and its 3D modeling was also carried out. Subsequently, a method for direct machining the proposed lens on PMMA with a swinging tool was proposed. And the tool path of this processing method was planned before fabrication of the lens, followed by testing of the fabricated lens to reveal its optical properties. The results show that the proposed design method can conveniently design the CLAACs. A high-accuracy and high-quality of the CLAACs have been successfully fabricated in experiments. The fabricated CLAACs has excellent optical properties, such as high imaging quality and focusing performance. The CLAACs also show advantages in thickness, volume, and field of view. We believe that the CLAACs may contribute to the fabrication of optical systems and has potential applications for imaging sensing, 3D display, and lighting sources.
Funding
National Natural Science Foundation of China (51875491, 51975501).
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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