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The increasing demand for lightweight, miniaturized electronic devices has prompted the development of small, high-performance optical components for light-emitting diode (LED) illumination. As such, the Fresnel lens is widely used in applications due to its compact configuration. However, the vertical groove angle between the optical axis and the groove inner facets in a conventional Fresnel lens creates an inherent Fresnel loss, which degrades optical performance. Modified Fresnel lenses (MFLs) have been proposed in which the groove angles along the optical paths are carefully controlled; however, in practice, the optical performance of MFLs is inferior to the theoretical performance due to fabrication errors, as conventional design methods do not account for fabrication errors as part of the design process. In this study, the Fresnel loss and the loss area due to microscopic fabrication errors in the MFL were theoretically derived to determine optical performance. Based on this analysis, a design method for the MFL accounting for the fabrication errors was proposed. MFLs were fabricated using an ultraviolet imprinting process and an injection molding process, two representative processes with differing fabrication errors. The MFL fabrication error associated with each process was examined analytically and experimentally to investigate our methodology.
© 2015 Optical Society of America
1. Introduction
The growth in the demand for light sources in various fields (e.g., displays, automobiles, road lighting, and light bulbs), with an emphasis on eco-friendly technology, has prompted interest in low-power light sources. Light-emitting diodes (LEDs) are increasingly used as alternative lighting due to their low power consumption, long life, and shape variability compared with conventional fluorescent and incandescent lamp sources [1–3]. However, LEDs are limited by their Lambertian intensity distribution for lighting applications; thus, considerable research effort has focused on the addition of optical components to the front emitting area of LEDs [4–7]. Conventional spherical and aspherical lenses are capable of changing the light distribution of LEDs for specific applications. However, the use of these lenses reduces the light efficiency due to light absorption and scattering; additionally, these components increase the size of the optical system, which affects system miniaturization.
The Fresnel lens combines the attributes of conventional spherical and aspherical lenses by dividing the lens into a series of concentric annular grooves, each having its own refractive surface [8,9]. This lens has been widely applied in electronic devices requiring miniaturized, lightweight components due to the thin, compact nature of the lens and its minimal optical loss. Low material consumption and the short cycle time of Fresnel lenses during fabrication reduce manufacturing costs and increase production efficiency from an industry perspective [10,11].
Conventional Fresnel lenses are designed with a vertical groove angle between the optical axis and groove inner facets, as shown in Fig. 1(a). The vertical groove shape causes inherent Fresnel loss by refracting rays out of the target plane, which results in a decrease in optical efficiency. Kang et al. designed and fabricated a modified Fresnel lens (MFL) in which the Fresnel loss was reduced by controlling the groove angles on the optical paths, as shown in Fig. 1(b); improved optical performance was demonstrated with this design. However, the fabricated MFLs exhibit worse optical performance compared with simulations, due to fabrication errors, as shown in Fig. 1(c) [12,13]. This is because conventional design methods do not take fabrication errors into account during the design process.
Fig. 1 Schematic diagrams of the optical path by (a) a conventional Fresnel lens (groove angle: 0°), (b) the ideal modified Fresnel lens (MFL) (with a modified groove angle of θg,i), and (c) the fabricated MFL with rounded groove shapes [12,13].
These fabrication errors originate from the mold fabrication and replication processes. A highly durable micromold is fabricated by a direct machining process using a machining tool with a round peak shape; thus, it is impossible to fabricate a mold with completely sharp groove shapes using this tool. Additionally, the lens is replicated by filling a mold cavity with flowable resin and allowing the resin to solidify. During the polymerization process, shrinkage of the polymer occurs at the groove peaks, which prevents the solidified resin from completely replicating the mold shape [14–16]. For these reasons, the fabricated MFL has rounded groove peak shapes as opposed to the sharp groove peaks specified in the theoretical design, as shown in Fig. 1(c). Light rays passing through these rounded groove peaks are refracted in undesirable directions, which degrades the optical performance of the fabricated MFL compared with the predicted performance [17–19]. Thus, a design method is required to identify and accurately represent the fabrication errors associated with MFL design to predict its optical performance.
In this study, the Fresnel loss due to microscopic fabrication errors of the MFL was theoretically derived, and a design method that takes the fabrication errors into account was proposed. The effect of the inherent fabrication errors on optical performance was investigated analytically and experimentally. To achieve this, the Fresnel loss area of the MFL was theoretically defined. Optical simulations were conducted to change the groove peak radius (GPR) representing the extent of the fabrication errors at the groove peak of the MFL. To verify the proposed design method experimentally, MFLs were fabricated using ultraviolet (UV) imprinting and injection molding processes, which are two representative processes presenting different fabrication errors. The GPR and the optical performance of the fabricated MFLs were measured and compared with simulation results to validate the proposed design methodology.
2. Design of the modified Fresnel lens and optical simulation including the fabrication errors
To verify how the fabrication errors of the MFL influence Fresnel loss, a MFL for uniform illuminance distribution at the target plane was designed. The optical properties of the MFL were analyzed using the design method with application of the associated fabrication errors. An illumination control algorithm that controls the energy distribution of LEDs having a Lambertian distribution was developed for effective lens design. The energy relationship was defined by dividing the target plane into multiple areas to be illuminated; the same light intensity was used in each area to create uniform illuminance distribution over the entire target plane. Thus, the proposed process used the inverse relationship between the light source and target plane to optimize the lens profile [13].
Figure 2(a) shows a schematic diagram of the light ray path from the light source to the target plane through the MFL. A ray emitted by the light source is refracted at angle θout1,i by the flat bottom surface of the MFL, as expressed by
where θs,i is the angle of the i-th ray emitted by the light source and nl is the refractive index of the MFL. When the i-th ray reaches point (xt,i, h) on the target plane, passing through point (xl,i, zi) on the grooves of the MFL, the angle of the i-th ray is refracted by the MFL grooves by θout2,i, as defined byFig. 2 Schematic diagrams of (a) the light ray path from the light source to the target plane through the MFL, (b) Fresnel loss due to the fabrication errors of the MFL grooves, and (c) enlarged ideal and fabricated groove profiles showing the fabrication errors in one groove and the ray path due to the grooves.
When fabrication errors are present at the groove peaks of the MFL, Fresnel loss occurs, as shown in Fig. 2(b). In these areas, the light rays are refracted out of the target plane; thus, the luminous flux efficiency decreases and differences develop between the simulated and measured light distributions. The wider the Fresnel loss area, the greater the difference between the simulated and measured optical properties, as evidenced by the reduction in optical performance. Figure 2(c) shows a schematic diagram of ideal and fabricated groove profiles; fabrication errors are shown for one groove, along with the ray path. The angle of the groove surfaces on the MFL where the i-th ray passes, θg,i, is defined by
for an angle of incidence of θout1,i and a refracted angle by the MFL groove of θout2,i. The height of the point on the MFL groove surface where the i-th ray passes is given byThis expression is valid excluding the Fresnel loss area where the fabrication errors between the designed and fabricated MFL occur. The Fresnel loss area, Lk, extends from xl,k1 to xl,k2 in the k-th groove of the MFL. In this study, the GPR is defined as the radius of the imaginary circle corresponding to the round groove peak shape of the MFL; this allows the fabrication errors to be expressed numerically. Lk is defined as
where r is the GPR at the groove peak of the MFL, θr,k is the groove angle of the k-th inner groove facets, and θg,k2 is the groove angle at xl,k2 of the k-th outer groove facets. Therefore, the total Fresnel loss area of a MFL having n number of grooves, Ltotal, is expressed asThe illuminance uniformity, which is used as an index to evaluate the optical performance of the designed MFL, is given for the target plane by
where Eavg is the average illuminance of the entire target area, Ei is the illuminance at the i-th target area, and n is the number of divisions at the whole target area [20].Based on the above expressions, the MFL was designed and optical simulations were carried out. As shown in Fig. 3, the MFL was positioned 1 mm from the light source, and the target plane (130 × 130 mm) was placed 100 mm from the light source. A white light source was formed by combining a blue LED with a yellow phosphor, which provided a Lambertian distribution. The target illuminance distribution was uniform at the target plane. The final MFL design had 46 microgrooves, a diameter of 8.85 mm, and a maximum groove height of 406 μm.
Fig. 3 Schematic diagrams of the optical system for MFL design, and the illuminance distribution with and without the MFL.
To verify the effect of the Fresnel loss area on optical performance, the GPR of the designed MFL was varied from 0 to 20 μm in the optical simulations, and the relative flux efficiency and illuminance uniformity were determined. In the simulations, the revised GPR was applied to all of the MFL grooves, and the optical performance by the GPR was numerically analyzed. Our simulation results showed that when the GPR increased from 0 to 20 μm, the relative flux efficiency decreased from 91.5% to 66.9%, and the illuminance uniformity decreased from 98.4% to 92.9%, which differed significantly from the targeted optical performance. From these simulation results, we verified that a greater GPR resulted in a smaller relative flux efficiency and illuminance uniformity.
3. Fabrication of the modified Fresnel lens based on simulation results
The designed MFL was fabricated to verify the effect of the fabrication errors on the optical performance experimentally using two representative processes: UV imprinting and injection molding, each involving its own fabrication error [21,22]. A mold having the reverse MFL profile is required for the replication process [23,24]. In this study, a mold having micro MFL grooves was fabricated using a direct machining process and an ultra-precision machining system (NANOFORM-200, Precitech Inc., USA). The diamond tool used in the machining process had a tip angle of 30° and a tip radius of 5 μm.
First, the MFL was fabricated using the UV imprinting process with the fabricated mold. A UV photopolymer was heated to 40 °C to improve transcribability and to remove the air bubbles. The mold was irradiated with UV light (wavelength: 365 nm; dose: 100 mW cm−2) under a compression pressure of 90 kPa between the substrate and the mold to improve the transcribability of the photopolymer. Figure 4(a) shows an image of the UV-imprinted MFL with a maximum groove height of 400 μm.
Fig. 4 Images of the MFL fabricated using (a) an ultraviolet (UV) imprinting process and (b) an injection molding process.
The MFL was also fabricated using an injection molding process. Cyclic olefin copolymer (COC) was used as the thermoplastic material for the replication process due to its high transmittance, low birefringence, and low water absorption rate [25]. Figure 4(b) shows an image of the injection-molded MFL with a maximum groove height of 395 μm with an injection pressure of 1620 kgf cm−2, and a mold temperature of 110 °C.
The profiles of the fabricated MFLs were measured using a Form Talysurf PGI 840 system (Taylor Hobson Ltd., Leicester, UK); these profiles were then compared with the designed profile to identify the fabrication error. Figure 5 shows the measured images of the MFLs fabricated by UV imprinting and injection molding; fabrication errors in the form of rounded peak shapes were evident in the imprinted and molded MFLs. The measured maximum GPR of the UV-imprinted MFL was 6 μm for a groove height of 400 μm. The injection-molded MFL exhibited a maximum GPR of 12 μm at a groove height of 395 μm.
Fig. 5 Designed and measured profile and cross-sectional images of (a) a UV-imprinted MFL with a groove peak radius (GPR) of 6 μm and (b) an injection-molded MFL with a GPR of 12 μm.
4. Measurement and analysis of simulated and fabricated modified Fresnel lenses
To analyze the optical performance of the fabricated MFLs, we compared the relative flux efficiency and illuminance uniformity of an ideal MFL with the fabricated MFLs. The relative flux efficiency was calculated by measuring the total flux emitted from the lens using an integrating sphere. The illuminance was measured using a diffuser plate positioned 100 mm away from the LED source; illuminance uniformity was calculated by dividing the diffuser plate (130 × 130 mm) into 25 areas for analysis purposes.
Figures 6(a)–6(c) show images of the simulated illuminance distribution by an ideal MFL and the MFLs accounting for fabrication errors of the UV imprinting process and injection molding process. The relative flux efficiencies for an ideal MFL and the fabricated MFLs, with fabrication errors applied, were 91.5%, 84.5%, and 76% respectively, and the illuminance uniformity values were 98.4%, 97%, and 95.2%, respectively. Figures 6(d) and 6(e) show images of the measured illuminance distribution for the MFLs fabricated using a UV imprinting process and an injection molding process; the relative flux efficiencies were-80% and 69%, respectively, and the illuminance uniformity values were 96.1% and 95%, respectively. Figure 6(f) shows the simulated and measured relative illuminance distribution at the target plane, accounting for the fabrication errors; a degradation in the optical performance was observed as the GPR increased, providing confirmation of the design methodology.
Fig. 6 Simulation results of the illuminance distribution for (a) an ideal MFL, and a lens applying (b) the UV-imprinted MFL profile (groove peak radius: 6 μm) and (c) the injection-molded MFL profile (groove peak radius: 12 μm). Measurement results of the illuminance distribution for a MFL fabricated by (d) UV imprinting (groove peak radius: 6 μm) and (e) injection molding (groove peak radius: 12 μm). (f) Comparison of relative illuminance for individual lenses.
Figure 7 shows the simulated and measured relative flux efficiency and the illuminance uniformity by changing the GPR of the MFL. In the simulations, the GPR was varied based on the optical system shown in Fig. 3, and the optical properties were analyzed. As shown in Fig. 7, the greater the GPR, the smaller the relative flux efficiency and illuminance uniformity, thus confirming the effect of the GPR on the optical performance. A comparison of simulation and experimental results verified the feasibility of the design method and the ability to predict the real optical performance analytically by applying the appropriate fabrication error.
Fig. 7 Simulation and measurement results of (a) the relative flux efficiency and (b) the illuminance uniformity as a function of the GPR.
5. Conclusions
The Fresnel loss due to fabrication errors at the groove peak of a MFL for LED illumination was theoretically derived, and a design method that takes fabrication errors into account was proposed. The effect of the inherent fabrication errors on the optical performance was investigated analytically and experimentally. First, the MFL was designed and the Fresnel loss area was theoretically defined. Optical simulations were then conducted to determine the optical performance of the designed MFL. The fabrication errors were then analyzed with respect to the GPR, which varied from 0 to 20 μm. Simulation results showed that when the GPR was 0 μm, the relative flux efficiency was 91.5% and the illuminance uniformity was 98.4%. However, as the GPR increased from 0 to 6 μm and 12 μm, the relative flux efficiency decreased to 84.5% and 76%, respectively, and the illuminance uniformity decreased to 97% and 95.2%, respectively. Thus, the greater the GPR, the smaller the relative flux efficiency and illuminance uniformity, and the greater the difference from the targeted optical performance. To verify the present design method experimentally, MFLs were fabricated using a UV imprinting process and an injection molding process, which are two representative processes with different fabrication errors. The MFLs fabricated by UV imprinting and injection molding demonstrated GPR values of 6 μm and 12 μm, respectively, measured relative flux efficiencies of 80% and 69%, respectively, and measured illuminance uniformity values of 96.1% and 95%, respectively.
Using the proposed approach, the degradation in optical performance of a MFL by fabrication errors was determined analytically, and the feasibility of the design method was verified experimentally for two MFL fabrication processes: UV imprinting and injection molding. This approach can be extended to conventional Fresnel lenses and patterned optical components with grating structures. Additional studies are currently underway to design and realize the optical performance of various optical systems that include Fresnel lenses or patterned optical components.
Acknowledgments
This research was supported by the R&D program for Industrial Core Technology through the Korea Evaluation Institute of Industrial Technology supported by the Ministry of Trade, Industry & Energy in Korea (Grant No. 10040225).
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