Numerous physics textbooks for undergraduate and high-school students include optics with discussions on lenses and their functions as imaging devices, along with the lens formula[1, 2]. Despite the significance of lenses in optics education, it is not an easy task for students to create their lenses.

Traditionally, lenses are manufactured through glass polishing[3, 4]. This approach is still used in various applications and for mass production in the industry. For high-volume manufacturing, injection molding and precision glass molding methods are used in modern days[5,6]. However, these manufacturing methods require high-precision techniques and expensive facilities, such as those employed for machining nanoprecision molds. Thus, the polishing and precision molding methods are unsuitable for small-scale lens fabrication, especially for research and education purposes. Several studies have been conducted on semiconductor processes and soft molding methods, which are more appropriate for small-scale lens fabrication[7-11]. However, these approaches also demand high-precision techniques and expensive facilities. Recently, the precision achieved by 3D printers has significantly improved, with some 3D printers capable of fabricating optical components with nanometer precision[12]. However, these types of 3D printers are highly expensive for use in most laboratories and schools.

In this research, we propose an approach for fabricating lens arrays that are relatively easy to manufacture and possess good quality for small-scale production in the laboratory for research or optics education purposes. Furthermore, we will present the findings of the fabrication and characterization of the lenses through this approach. Our method includes vacuum forming, which was initially reported in[13] for smoothing the surface of a small prism array produced from a negative prototype fabricated using a 3D printer. The procedure in[13] entailed pouring a mixture of a polydimethylsiloxane (PDMS) base and agent into a negative prototype and hardening it to create a transparent silicone (PDMS) rubber prism. In this research, we employed a 3D printer and vacuum forming to create a lens prototype, then used PDMS to create a negative silicone mold. We poured thermal epoxy into the negative silicone mold and allowed it to harden to obtain the final hard and transparent epoxy lens. Making lenses can be more challenging than creating prisms because lenses have curved surfaces instead of planes, and the final performance necessitates high imaging capability rather than light deflection capability. However, lenses and lens arrays have broader applications and are more crucial in constructing optical systems than prisms[14-16].

Additionally, the epoxy employed in the lens fabrication process is characterized by greater hardness and a higher refractive index than silicone rubber, making it a popular material for producing optical components, including lenses for light-emitting diode (LED). The advantageous properties of epoxy may lead to broader applications of this fabrication approach in the future. Finally, we measured the focal lengths of the fabricated lens arrays with varying radii of curvature in the lens prototypes. We anticipate that the proposed fabrication approach will enable students to build their lenses and practice optical theory.

Figure 1 shows the process flow for creating the lens array, which was attempted in this study. First, as illustrated in Fig. 1(a), the 3D printer printed the first prototype of the lens array. The 3D printer employed in this experiment was a relatively inexpensive printer (ELEGOO MARS PRO). Ultraviolet (UV) images were obtained using a liquid crystal display to harden the UV resin layer by layer in increments of 0.05 mm. The vacuum forming method was used to smooth the surface of the first lens prototype. A JT-18 was the vacuum forming machine used, and the first lens prototype was coated with a melted dental plastic sheet (HMT GUOZIJIAN SPLINT) by vacuum action, as illustrated in Fig. 1(b). As demonstrated in Fig. 1(c), a mixture comprising PDMS base and curing agent was poured into the smoothed prototype to create a negative soft mold. The hardened PDMS mold was separated from the prototype, as illustrated in Fig. 1(d), and thermal epoxy was poured into it and covered with a flat lid, as demonstrated in Fig. 1(e). The flat lid was also made from PDMS using a flat plastic sheet as a prototype. This lid was necessary to create the flat side of the plano-convex lens array. The hardened epoxy lens was separated from the PDMS mold to obtain a final hard and transparent lens, as demonstrated in Fig. 1(f). The detailed process of making the PDMS mold and epoxy lenses followed the procedure used to produce microlenses[11]. To create a PDMS mold, we first mixed the base of Sylgard 184 with the curing agent in a ratio of 10:1 and poured the mixture into a container containing the prototype. The PDMS was cured in an oven at 120 °C for 1 h. The epoxy used to produce the lens contained an LED encapsulant, DC-5206, and DH-5260 (Daejoo.co.kr). The two parts were mixed in a ratio of 1:1 and poured into the PDMS mold. The epoxy was cured in an oven at 120 °C for 1 h. Figure 2 shows the intermediate and final products obtained from the lens-making process.

Figure 1. (Color online) Process flow involved in producing the lens array (a) producing the first lens prototype using a 3D printer (b) coating the prototype by vacuum forming (c) forming a negative PDMS mold (d) separating the PDMS mold from the lens prototype (e) pouring thermal epoxy into the PDMS mold (f) separating the epoxy lens from the PDMS mold.

Figure 2. (Color online) Intermediate and final products from lens-making processes (a) the first lens prototype coated using a vacuum forming machine (b) a lens prototype separated from the first prototype after vacuum forming (c) PDMS negative mold (d) a fabricated epoxy lens array after separation from the PDMS mold.

As demonstrated in Fig. 3, an image of the letter “A” was produced using the fabricated lenses to test if the lens array performed well as an imaging device. The lens array produced clear and inverted images of the character “A” when an upright letter “A” printed on paper was placed at a specific distance from the lens. This clear image indicates the excellent performance of the fabricated lens as an imaging lens. In other words, ray bundles starting from a point in “A” are refracted by the lens surface and converge to a small spot. The image spot size formed by an object point becomes large and blurs the image if there is an error in the spherical surface of the fabricated lens.

Figure 3. (Color online) Images of the letter “A” formed by a fabricated lens array: (a) an image produced by the lens array with the tapes on the side for fixing the lens on the supporting structure. (b) a magnified view of the image produced by a lens in the array with a marking dot on the surface.

The surface of the lens appeared very smooth to the naked eye and produced reflected images of the rectangular-shaped ceiling lamps, functioning as a convex mirror, as demonstrated in Fig. 3(b). This indicates that the roughness of the surface is significantly smaller than the visible wavelength, and thus the surface is optically smooth and produces very minimal scattering.

To examine the dependence between the shape of the first lens prototype and the final lens shape, different shapes of lens prototypes and two different plastic sheets were tested. It is crucial to assess the impact of prototype and processing on the final lens parameters, such as the focal length, since this knowledge can enable control and predictability over the final lens parameters. Figure 4 shows the lens prototypes used in the experiments, which have varying lens diameters and heights, resulting in different radii of curvature, except for lens c, which has the same parameters as lens b but with holes between the lenses. Holes were added to aid in the attachment of the plastic sheet to the prototype during the vacuum forming process. The holes in the prototype may offer more air paths to the vacuum holes located at the bottom of the sample in contact with the vacuum plate of the machine. The lens prototypes had radii of curvature of 2.69 mm, 3.63 mm, and 6.63 mm. First, lens prototypes a, b, and c have 5 × 5 lens arrays, while lens prototype d has a 2 × 2 lens array with a large lens diameter. According to the lens maker’s formula, a larger radius of curvature results in a larger focal length. Thus, if the final lens shape corresponds to that of the first lens prototype, we can regulate the focal length of the lens by adjusting the curvature of the lens prototype, which can be easily accomplished using a 3D printer. A lens array was selected as the prototype shape instead of a single lens to highlight the advantages of the lens-making method proposed in this research over the challenging conventional glass polishing method.

Figure 4. (Color online) Lens prototypes created by 3D printer: (a) r = 2.69 mm (h = 1.7 mm, d = 5.0 mm), (b) r = 3.63 mm (h = 1.0 mm, d = 5.0 mm), (c) r = 3.63 mm (h = 1.0 mm, d = 5.0 mm), and (d) r = 6.63 mm (h = 1.0 mm, d = 10.0 mm), where r, h, and d are the radius of curvature, lens height, and lens diameter, respectively. Prototype c has the same lens profile as that of prototype b but has holes between the lenses.

Furthermore, we modified the process by employing two different thicknesses of plastic sheets while coating the first prototype through vacuum forming. It is crucial to comprehend the extent to which the thickness of the plastic sheet impacts the final lens profile and focal length. The thicknesses tested were 0.6 mm and 1.0 mm, applied to prototypes a and b, respectively, creating two lenses from each prototype.

Figure 5 shows the fabricated epoxy lens arrays. The lens arrays exhibited smooth surfaces, showing shiny reflections. Lens2 shows only a 5 × 3 array with two missing rows due to a lack of epoxy resin during the PDMS mold filling process. However, the remaining lenses from the Lens2 prototype were satisfactory for characterization. As previously stated, Lens1 and Lens2 were produced from prototype a with different thicknesses of plastic sheet used for vacuum forming, whereas Lens3 and Lens4 were produced from prototype b.

Figure 5. (Color online) Final epoxy lens arrays obtained from the first lens prototypes: Lens1 and Lens2 used prototype a with 0.6 mm and 1.0 mm thick plastic sheets, respectively; Lens3 and Lens4 used prototype b with 0.6 mm and 1.0 mm thick plastic sheets, respectively; Lens5 and Lens6 used prototypes c and d, respectively, with 1.0 mm thick plastic sheet.

To test the performance of the fabricated lens arrays, we imaged the character “A” printed on white paper from a certain distance away from the lens. Figure 6 shows the images obtained using the lens arrays. Figure 7 shows the magnified views of the images obtained from the central lenses of each type. Although some lenses far away from the center suffered from distortion and blurring due to aberration, the central lenses produced clear inverted images of “A,” indicating that all lens arrays functioned effectively as plano-convex lenses and possessed optically smooth spherical surfaces. It is challenging to achieve this imaging capability without an optically smooth spherical surface since the primary function of a refractive lens is to collect rays originating from an object point and focus them onto a small spot. Thus, our process of obtaining a smooth spherical shape through 3D printing and vacuum forming, then transferring this lens shape into a transparent and hard epoxy lens, proved successful.

Figure 6. (Color online) Images of letter “A” formed by the fabricated lens arrays.

Figure 7. (Color online) Magnified views of the images of the letter “A” formed by the fabricated lens arrays: (a) Lens1 (b) Lens2 (c) Lens3 (d) Lens4 (e) Lens5 (f) Lens6.

Finally, we characterized the fabricated lens by measuring their focal length and investigated the effect of the lens prototype and the thickness of the plastic sheet on the focal length. The focal length was determined using the Gaussian lens formula[1]. Although most commercial lenses are characterized by automated lens performance measurement systems using verified approaches[17, 18], the lenses fabricated in this research were characterized using the Gaussian lens formula. This formula is based on the paraxial theory and has some limitations, but it offers an approximate estimation of the focal lengths with a simple setup. In this research, we focus more on demonstrating the feasibility of the new fabrication technique in small-scale production for laboratory use and optics education rather than on obtaining precise lens data. Furthermore, this method provides an opportunity for optics education, allowing students to practice calculating the focal length using the textbook lens formula and lab-made lenses by measuring it through this method. For measurement, an object made of a rectangular grid pattern and a light source was imaged using a lens array fixed on a precision translation XYZ stage. Initially, the object pattern was imaged at infinity using a microscope objective lens and its accuracy was confirmed using an infinite conjugate camera. The resulting image, formed at infinity, was then projected again through the fabricated epoxy lens and observed using a universal serial bus (USB) microscope. Since a moving lens array with an XYZ stage can switch the focus of the USB microscope from the surface of the lens to the image formed by the lens, the distance between the lens surface and the image could be measured. Therefore, the focal length of the lens can be computed using the measured image distance from the lens based on the Gaussian lens formula. The measured focal lengths of the lens arrays, along with the other processing parameters, are summarized in Table 1.

Table 1 Measured focal lengths of the fabricated lens arrays with different processing parameters. The radius of curvature has two values: “Design,” indicating the value of the first lens prototype in the CAD design, and “Estimated,” which refers to the value of the final lens obtained from the measured focal length, assuming a refractive index of 1.55.

Lens number	Lens prototype	Radius of curvature (mm)		Thickness of the plastic sheet used for vacuum forming (mm)	Focal length (mm)
		Design (first prototype)	Estimated (final lens)
1	a	2.69	2.4	0.6	4.0
2	a	2.69	2.9	1.0	5.0
3	b	3.63	4.5	0.6	8.0
4	b	3.63	5.2	1.0	9.3
5	c	3.63	3.7	1.0	6.5
6	d	6.63	6.0	1.0	10.7

As illustrated in Table 1, there is a relationship between the radius of curvature of the initial prototype and the final focal length, with a larger radius of curvature resulting in a longer focal length. Furthermore, the thickness of the plastic sheet also affects the radius of curvature and focal length, depending on the manufacturing process and prototype shape. Regarding Lens1 and Lens2, using a thicker plastic sheet results in a larger radius of curvature. This pattern also holds for Lens3 and Lens4. However, Lens5 has a significantly smaller radius of curvature than Lens4, despite having the same thickness of the plastic sheet and prototype radius of curvature as Lens4. This can be attributed to the holes in prototype c, as they likely help in attaching the plastic sheet to the prototype during the vacuum forming process, as intended. The radii of curvature of the final lens can be determined using measured focal lengths and ray tracing software. The experiment and simulation used an aperture diameter of 1.45 mm. The ray tracing software determined the radius of curvature by positioning the image plane at the measured focal length and minimizing the spot size for a given aperture size. The minimum spot size corresponds to the best focus achieved in the experiment. For the simulation, we assumed that the epoxy had a refractive index of approximately 1.55. The estimated radii of curvature for the lenses were about 2.4 mm, 2.9 mm, 4.5 mm, 5.2 mm, 3.7 mm, and 6.0 mm, respectively, as listed in Table 1. To validate the accuracy of our estimation, we used a Mahr contour measuring machine (XC10) to measure the radii of curvature of the fabricated lenses. The measured radii of curvature for Lens1 and Lens6 were 2.40 mm and 5.98 mm, respectively. The measured data values were in good agreement with the estimated values obtained from the measured focal lengths, within the measurement error. These findings show that the proposed approach of lens-making based on 3D printing and vacuum forming can be employed to fabricate lens arrays with varying focal lengths by modifying the shape of the prototype with a 3D printer.

In this research, we proposed and experimentally demonstrated an approach for fabricating lens arrays using a 3D printer, vacuum forming, and PDMS soft molding methods. Initially, the 3D printer was used to create the first prototype of the lens arrays with different lens diameters and heights. Different parameters of the lens shape resulted in different radii of curvature and focal lengths. The vacuum forming process was then applied to smooth the rough surface of the first prototype by coating it with a melted plastic sheet. The smoothed plastic prototype was separated from the first prototype and employed to produce a negative PDMS mold. Subsequently, thermal epoxy was poured into the PDMS mold and hardened to form a transparent and hard epoxy lens array.

The fabricated lens arrays performed well as imaging lenses, generating clear inverted images of the letter “A” printed on paper, which indicates that the fabricated lens had an optically smooth spherical surface with insignificant scattering from the surface. Furthermore, the image produced by reflection from the lens surface shows that the surface is optically smooth.

The Gaussian lens formula was used to determine the focal lengths of the fabricated lens arrays. The difference in the radius of curvature of the first prototype affected the focal length, with a larger radius of curvature resulting in a greater focal length for the final lens. This tendency can be used to adjust the focal length of the final lens by modifying the curvature of the initial prototype, which can be easily achieved using a 3D printer. Furthermore, we observed that the thickness of the plastic sheets (0.6 mm and 1.0 mm) and the holes between lenses influenced the radius of curvature and focal lengths of the final lens arrays. Additionally, we used ray tracing software to estimate the radii of curvature of the final lenses and discovered that the estimated values closely matched those obtained directly using a contour measuring machine.

In summary, the proposed approach for creating lenses using 3D printers, vacuum forming, and PDMS molds is suitable for fabricating a small amount of intricate optical components such as lens arrays, which are challenging to fabricate using conventional glass polishing techniques. Since the proposed fabrication approach does not necessitate costly or complex facilities such as nano-3D printers and polishing machines, it can be conducted in schools or small laboratories by students. This study’s findings can be applied to assist students in their optics education.

The authors wish to express their sincere gratitude to Jae-dong Lee, a researcher at the Daegu Mechatronics Institute, for measuring the radius of curvature of the fabricated lenses.