###### Abstract  
This paper proposes a lens design method for effectively collimating the light emitting from a light-emitting diode (LED). This collimating lens contains two aspherical lens surfaces which can be mathematically characterized using a few designing parameters, and hence is called an analytic collimating lens. An optical ray-tracing algorithm has been developed for these analytic collimating lenses to analyze their optical performance and to optimize their designs. Six high-power and commercially available ultraviolet (UV) LEDs are chosen as examples for demonstrating the optimal collimating lens design. For each UV-LED, the corresponding optical collimating lens is determined by inputting the ray data file provided by the manufacture over a finite-size emitting area. The divergent angles of the six UV-LEDs have been successfully collimated to a narrow range in between 1.56\({}^{\circ}\) to 2.84\({}^{\circ}\) from their original radiation angle around 46\({}^{\circ}\) to 120\({}^{\circ}\). Furthermore, the proposed analytical collimating lenses are suitable for mass-production using standard mold injection methods, and hence possess great potentials for industry applications of LEDs.  
UV-LED; analytic collimating lens; ray tracing; quantitative evaluation; photolithography +
Footnote †: journal: Applied Sciences  
surfaces were also proposed [15; 16] for LED's light collimating, but the collimation ability is limited near the optical axis and hence the efficiency is low.  
There are many methods to manufacture axisymmetric aspherical lenses, [PERSON] [17] using wafer-level etching to fabricate high numerical aperture silicon collimating lens in parallel. [PERSON] [18] made use of surface tension self-assembly to construct refractive collimating microlens arrays. Another more important approach to produce optical lenses is glass molding technology [19], [PERSON] improved quality of lenses according to performance evaluation of experiments and analysis. For the application of UV-LEDs in photolithography, collimation of UV light is critical for it directly affects the dimensional accuracy of developed photoresist microstructures and the verticality of structural sidewalls [20; 21]. Another important application of UV-LEDs is for the imaging homogenization system [22; 23], in which fly's eye homogenizers are typically used. Such an imaging homogenization system requires high collimation of incident light [24] to reduce optical crosstalk on the detector plane and improve energy efficiency.  
At present, most of the research works related to lens design investigate the problem of optical illumination instead of optical collimation. However, if we consider a light source of the finite luminous area with divergence angle in the case of optical collimation, the collimating ability of the lens is restricted. Therefore, this research extends the previously proposed analytic collimating lens [25] based on the hypothesis of a point source with an infinite light-emitting area, and then explores an extended UV-LED light source with a finite light-emitting area. Refer to Appendix A for the design principle and derivation process of the analytic collimating lens.