## Abstract

A Fresnel lens is an optical component that can be used to create systems more compact, cost-effective, and lightweight than those using conventional continuous surface optics. However, Fresnel lenses can usually cause a loss of flux efficiency and non-uniform distribution of illuminance due to secondary refraction by surface discontinuities, especially along the groove facet. We therefore proposed to modify a groove angle in the Fresnel lens and analyzed interrelation between the groove angle and multiple optical performances, such as flux efficiency and the uniformity of illuminance and color. The groove angle was optimized to maximize the uniformity and efficiency in the target viewing angle considering various weights of merit functions. Specifically, in our study, when the uniformity of illuminance had a little more weight than the flux efficiency (ratio of 0.6:0.4), final optimum groove angles of 24.7°, 29.4°, and 31.3° were obtained at target viewing angles of 20°, 30°, and 40°, respectively. We also fabricated a modified Fresnel lens with a groove angle of 29.4° using UV-imprinting. The real optical performance of the fabricated Fresnel lens was then compared to that of a spherical lens.

©2009 Optical Society of America

## 1. Introduction

Light-emitting diodes (LEDs) have many favorable characteristics, such as high flux efficiency and reliability, low power consumption, long life, and better environmental effects compared to traditional light sources. Moreover, LEDs have been increasingly applied in practical applications, such as electric devices, traffic and automotive lighting, and general lighting due to their flux efficiency and cost-effectiveness [1,2]. However, LEDs also have fundamental limits in many illumination applications because LED sources are ideal Lambertian emitters, whereby the intensity distribution I_{θ} is a cosine function of the emission angle. The flux efficiency of the original source with viewing angle θ_{1} is therefore lower within the specific target area than that of the original source with an optical lens of viewing angle θ_{2}, because θ_{1} is greater than θ_{2}. Here, flux efficiency is defined as the ratio of the incident flux within the target area, centered on the viewing angle θ_{2}, to the total flux emitted by the LED. Viewing angle is defined as the angle at which the luminous intensity is reduced to half its value at 0°, as shown in Fig. 1
. Optical components, however, can be used to improve the flux efficiency at typical target-field distances > 10 A_{LED}
^{1/2}, where A_{LED} is the LED emitting area. The flux efficiency in the target area is easily improved by simple optical components with continuous surface such as spherical lenses because the intensity distribution of the original source becomes a power law of the cosine function. However, it causes the illuminance distribution E_{θ} to decrease more rapidly away from the optical axis. Color separation also occurs because of the different thickness of the phosphor coating layer around the blue LED chip, as shown in Fig. 2
. That is, the rays emitted from the side surfaces of the blue chip could be yellowish in color, because these rays pass through more phosphor material than rays emitted from the top surface. When an optical lens is used, the main yellowish rays are refracted in a much more regular direction. Thus, ordinary optical components cannot meet many other optical requirements, such as the distribution of uniform illuminance and color in the target area [3,4].

The improvement in the optical power of the lens, via traditional optical components, such as a spherical lens with a continuous refractive surface, is limited by constraints on the geometrical dimensions, namely between the radius of curvature and the thickness, with the lens diameter fixed. Therefore, Fresnel lenses have become a viable solution in design of optical components due to their advantages of geometric design freedom and low thickness. In a Fresnel design, the continuous refractive surface of a traditional optics is divided and collapsed onto a bottom plane. Each discontinuous refractive surface connects to a groove facet that is generally parallel to the optical axis. Figure 3 shows the design concept of a Fresnel lens. The optical path difference of chief rays due to the groove angle, which is the angle between the groove facet and the optical axis, is also shown. When the groove angle is 0°, as shown in Fig. 3(a), some rays transmitted through each discontinuous refractive surface deviate because of secondary refraction by the groove facet, which causes great part of optical loss at outer region in the target area. Therefore, we noted that the angle of this groove facet could affect optical efficiency and uniformity, and thus proposed a modified Fresnel lens with a constant groove angle to compensate illuminance of outer region owing to reduction of secondary refraction as shown in Fig. 3(b) [5]. We also verified our design concept, and optimized the groove angle for various target viewing angles via iterative ray tracing to maximize flux efficiency and the uniformity of the illumination and color distributions. Practically, we fabricated the modified Fresnel lens and evaluated those optical performances.

## 2. Designing an initial lens shape

The first step in designing a optical lens for a given target viewing angle is finding the initial design values of the refractive surface. These determine the optical power of the lens. For this step, we applied 2 × 2 ray matrices using derivatives between the direction and position of the ray vector emitted from the center of the LED source [6]. The initial radius of curvature of a normal spherical lens can be calculated using the geometric variables as shown in Fig. 4(a) .

_{in}is a viewing angle, which is decided by the intensity distribution of the LED. Finally, θ

_{out}is a target viewing angle for each illumination application, and n

_{1}, n

_{L}, and n

_{2}are the refractive indices of each medium. If θ

_{out}is determined, the radii of curvature with respect to t are derived from Eq. (1). t is defined as the distance that an incident ray emitted from the LED source transmits between R

_{1}and R

_{2}, in the direction of the optical axis Z, as shown in Fig. 4(a). The total thickness T of the lens is approximately equal to t for paraxial rays. We were therefore able to draw the initial configuration of the Fresnel lens using the pair values R and T, as shown in Fig. 4(b).

The basic construction of the Fresnel lens in this study is such that a continuous refractive surface is divided into a number of equal heights, N. The profile of each discontinuous refractive surface Z_{N}(y), and positions y_{n} and y_{n}’, where the value of Z_{N}(y) is zero and T/N, respectively, are

After the initial configuration of the discontinuous refractive surfaces was designed using Eqs. (2)–(4), each discontinuous refractive surface connects to modified groove facets and the base substrate of the Fresnel lens, allowing the selection of h. However, because the geometric variables of spherical and Fresnel lenses are based on on-axis ray tracing, the off-axis rays emitted from the LED are not considered. Hence, ray-tracing simulations for given practical conditions are also necessary to calculate the optical power of the lenses [7,8]. We selected s, n_{1}, n_{L}, n_{2}, θ_{in}, θ_{out}, and θ_{groove} as the fixed variables, choosing values of 1 mm, 1, 1.5, 1, 60°, 30°, and 0°, respectively. Using these values in Eqs. (1)–(4), we performed a three-dimensional (3D) ray-tracing simulation of the entire optical system. We then corrected h and N in accordance with the target viewing angle. The groove height of the Fresnel lens T/N may be altered due to fabrication issues. In this paper, the values of N are 17, 15, and 10 for the respective target viewing angles 20°, 30°, and 40°. Table 1
shows the corrected specifications of the Fresnel lens according to the target viewing angle, and Fig. 5
shows simulated intensity distributions in each case. The angles corresponding to half of the normalized intensity at 0° matched the target viewing angles.

Figure 6 shows the simulated illumination distribution for different target areas of each viewing angle. The radius of the target area on the detecting plane is z•tan θ, where z is 100 mm from the LED source to the detecting plane. As previously stated, the illumination curve follows a power cosine law. Thus, the smaller the target viewing angle, the more rapidly the illumination distribution decreased on the same target plane.

## 3. Designing an optimal Fresnel lens

The final purpose of this study was to design an optimal Fresnel lens, so that high flux efficiency and a uniform distribution of illuminance and color could be achieved at the target plane for any target viewing angle. These characteristics were derived as functions of the groove angle, which was a variable in the optimal design process, as shown in Fig. 7
. The aims were to maximize the flux efficiency F_{1}(θ), which is a ratio of the incident flux within the target viewing angle Φ_{θ} to the total input flux Φ_{input} emitted by the LED.; to minimize the normalized deviation of illuminance F_{2}(θ), which is a ratio of the standard deviation of illuminance to the average illuminance L_{avg}.; and to minimize the normalized deviation of color F_{3}(θ), which is a ratio of the standard deviation of color to the average color [9–11]. Here, the functions F1, F2, and F3 are

_{groove}is the groove angle of the Fresnel lens, n is the total number of bins in the mesh of the detecting area, L

_{k}is the illuminance of each bin, u

_{avg}and v

_{avg}are average color coordinate for the total bins, u

_{k}and v

_{k}are the color coordinate of each bin. The position of the detection plane was changed so that the areas of bins were equal, regardless of the target viewing angle. In this simulation, considering a trade-off between resolution and accuracy in statistical ray-tracing process, the detecting area was divided into 25 bins. To reduce the optimization time, the groove angle was changed in intervals of 5°, and multiple merit functions, Eqs. (5)–(7), were calculated from simulation data as shown in Fig. 8 . For each target viewing angle, the flux efficiency F

_{1}(θ) and the normalized illuminance deviation F

_{2}(θ) had only one optimum point inside the groove angle range, whereas no optimum point was found for the normalized color deviation F

_{3}(θ). That is, for each target viewing angle, there was no relationship between F

_{3}(θ) and the groove angle. However, it should be noted that the uniformity of color could be improved by using Fresnel lenses since the values of the normalized deviation of color were markedly reduced compared to conventional spherical lens as result of Fig. 10 and 11 . Therefore, the notations for F

_{1}(θ) and F

_{2}(θ) were emphasized so that each term functionally depended on the groove angle values. We used these two functions objectively for optimization. As we had two independent objectives, we fitted the data using a third-degree polynomial regression function based on the least-squares method. Each function was normalized between 0 and 1, as in Eqs. (8) and (9).

From a practical point of view, the weight values should be reflected in the merit function MF to show the relative importance of each objective function in Eq. (10). Optimum draft angles were calculated with various sets of weight values using the differential of MF. When the uniformity of illuminance had a little more weight than the flux efficiency (0.6:0.4 ratio), final optimum groove angles of 24.7°, 29.4°, and 31.3° were obtained at target viewing angles of 20°, 30°, and 40°, respectively, as shown in Fig. 9 . Finally, a modified Fresnel lens with an optimum groove angle of 29.4° was chosen for the final 3D geometrical specification. It was compared to a spherical lens and a Fresnel lens of groove angle 0° to verify the performance of the optimum groove angle. In particular, we conducted separate simulations for each emitting surface of the LED chip, taking into account the spectral distribution variation by phosphor coating layer. We verified that the color deviation resulting from ray separation by a spherical lens with a viewing angle of 30° was markedly reduced by using Fresnel lenses, because the distribution of rays emitted from side surfaces of chip was scattered more moderately compared with spherical lens as shown in Fig. 10. Finally, we also reconfirmed quantitatively that the optimized Fresnel lens produced a more uniform distribution for the normalized illuminance and color deviations (which were 0.218 and 0.007, compared with 0.334 and 0.01761 for the spherical lens) for a target viewing angle of 30° as shown in Fig. 11. However, for a Fresnel lens with a groove angle of 0°, the color distribution was left unchanged, while the illuminance distribution was improved, as results in Fig. 8.

## 4. Results and discussion

We used an ultraviolet (UV) imprinting process to fabricate a modified Fresnel lens with optimum groove angle. UV imprinting consists of the chain of subprocesses shown in Fig. 12(a) . The mold, which has the inverted shape of the original circular groove facets, was fabricated by diamond turning. An UV-curable polymer was selected as the replication material. During UV exposure, a relatively low pressure was applied to ensure replication quality. The height of each groove increased with the compression pressure. The pressure applied during photopolymerization reduced material shrinkage and unfilled gaps [12]. Therefore, complete filling of each groove cavity by an appropriate molding process was expected to prevent the illumination system from losing optical efficiency.

Figure 12(b) shows the surface profiles of each groove of the fabricated lens. The optimized groove angle was measured to be 29.5°. Figure 13 shows the fabricated modified Fresnel lens with optimum design and accompanying LED module. To measure the illuminance and color distribution quality, a diffuser plate (120 × 120 mm) was placed 100 mm above the LED source. The diffuser plate was also divided into 25 detecting regions to calculate the normalized deviation of illuminance and color. For a target viewing angle of 30°, the measured values of normalized deviations of illuminance and color were 0.331 and 0.0214 for spherical lenses, and 0.214 and 0.0079 for the fabricated Fresnel lenses, respectively, as shown in Fig. 14(b) . It was noted that the measured and simulated results coincided closely. Therefore, we could assure that optimizing the groove angle improved multiple optical properties of the system.

## 5. Conclusions

We noted that general Fresnel lenses with small groove angles within 5° lose flux efficiency and illuminance uniformity due to vignetting by their groove facets. We therefore designed a modified Fresnel lens to minimize the optical loss and maximize the uniformity. The initial optical power of the Fresnel lens was derived from 2 × 2 ray matrices and verified by an iterative ray-tracing simulation. Other optical effects of the dominant design variable, the groove angle of the Fresnel lens, were also verified analytically and experimentally. The optimum groove angle for viewing angles of 20°, 30°, and 40° was derived from the merit function considering the weights of each optical function. As a result, the flux efficiency and uniform illuminance distribution could be maximized over the target area compared to the other lenses. In addition, the color deviation could be reduced due to moderate mixing between the blue and yellow rays emitted from the LED by each refractive surface of the Fresnel lens. Finally, the modified Fresnel lens optimized with a groove angle of 29.4° for a viewing angle of 30° was fabricated by UV imprinting, and its superior optical properties were verified experimentally. The optimization of fabrication processes to reduce the replication error of each groove of a Fresnel lens is the subject of ongoing research.

## Acknowledgments

This research was supported by a grant from Center for Nanoscale Mechatronics and Manufacturing (M102KN010005-08K1401-00510), one of the 21st Century Frontier Research Programs, which are supported by Ministry of Science and Technology, Korea.

## References and links

**1. **Y. Uchida and T. Taguchi, “Lighting theory and luminous characteristics of white light-emitting diodes,” Opt. Eng. **44**(12), 124003–1 (2005). [CrossRef]

**2. **S. Bierhuizen, M. Krames, G. Harbers, and G. Weijers, “Performance and trends of high power light emitting diodes,” Proc. SPIE **6669**, 66690B (2007). [CrossRef]

**3. **P. Manninen, J. Hovila, P. K¨arh¨a, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. **18**(1), 223–229 (2007). [CrossRef]

**4. **I. Moreno, M. Avendaño-Alejo, and R. I. Tzonchev, “Designing light-emitting diode arrays for uniform near-field irradiance,” Appl. Opt. **45**(10), 2265–2272 (2006). [CrossRef] [PubMed]

**5. **O. E. Miller, J. H. Mcleod, and W. T. Sherwood, “Thin Sheet Plastic Fresnel Lenses of High Aperture,” J. Opt. Soc. Am. **41**(11), 807–815 (1951). [CrossRef]

**6. **L. Frank, Pedrotti, S.J, Leno M. Pedrotti, Leno S. Pedrotti, *Introduction to Optics 3nd Edition* (Pearson, ST., San Francisco, 2007)

**7. **C.-C. Sun, T.-X. Lee, S.-H. Ma, Y.-L. Lee, and S.-M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. **31**(14), 2193–2195 (2006). [CrossRef] [PubMed]

**8. **Á. Borbély and S. G. Johnson, “Performance of phosphor-coated light-emitting diode optics in ray-trace simulations,” Opt. Eng. **44**(11), 111308 (2005). [CrossRef]

**9. **G. Wyszecki, and W. S. Stiles, *Color Science 2nd Edition* (Wiley, New York, 1982).

**10. **F. Fournier and J. Rolland, “Optimization of freeform lightpipes for light-emitting-diode projectors,” Appl. Opt. **47**(7), 957–966 (2008). [CrossRef] [PubMed]

**11. **Y. Zhen, Z. Jiaa, and W. Zhang, “The Optimal Design of TIR Lens for Improving LED Illumination Uniformity and Efficiency,” Proc. SPIE **6834**, 68342K (2007). [CrossRef]

**12. **S. M. Kim, H. Kim, and S. Kang, “Development of an ultraviolet imprinting process for integrating a microlens array onto an image sensor,” Opt. Lett. **31**(18), 2710–2712 (2006). [CrossRef] [PubMed]