# Double-Sided Freeform Lens for Light Collimation of Light Emitting Diodes
## 3 Lens Design and Simulation with UV-LED
The calculated \(S_{1}\) and \(S_{2}\) surfaces obtained from Equations (6a,b) and (12) are shown in Figure 4a,b, respectively (note that the refraction index (\(n\)) was taken as 1.51 in every case and the collecting angle (\(\theta_{B}\)) was set as 60\({}^{\circ}\)). The numerical labels in Figure 4a,b represent the Fresnel transmittance values at the corresponding locations on the boundary surface. It is seen that for the \(S_{1}\) surface, most of the optical transmittance values were around 95% for radiation angles \(\theta\) = 0\({}^{\circ}\)-60\({}^{\circ}\) irrespective of the value of \(\xi\). However, for the \(S_{2}\) surface, the optical transmittance at higher radiation angles reduced rapidly (and even reached zero) with increasing \(\xi\)  
Figure 3: Fresnel optical transmittance as function of incident angle for: (**a**) Air to poly(methyl methacrylate) (PMMA) interface, and (**b**) PMMA to air interface. (Note that the refraction index of the chosen PMMA material is 1.51). Figure 2: Angular distribution of radiation intensity of NCSU276A ultra violet light emitting diode (UV-LED): (**a**) Rectangular plot and (**b**) polar plot. due to the effects of internal total reflection. In other words, the flux energy received at larger radiation angles of the LED was completely rejected by the lens. Based on the angular distribution of the radiation intensity of the chosen NCSU276A UV-LED (see Figure 2a,b) and the Fresnel losses computed at surfaces \(S_{1}\) and \(S_{2}\) (see Figure 4a,b), one can roughly estimate the intensity distribution of the collimated LED beam on a projected plane perpendicular to the \(z\)-axis. Figure 5 presents the corresponding results for the \(\xi\) values considered above. In the present study, the lens profile design calculated for \(\xi=0.3\) was chosen for illustration purposes since,  
Figure 4: Designed lens surface profiles for different values of \(\xi\): (**a**) Surface \(S_{1}\) and (**b**) surface \(S_{2}\). (Note that the numerical labels show the Fresnel transmittance percentages at the corresponding locations). compared to larger values of \(\xi\), a value of \(\xi\) = 3 generally resulted in lower Fresnel losses and limits the lens aperture size and projected area size thereby assuring a higher average projected light intensity. Moreover, a smaller value of \(\xi\) (e.g., \(\xi\) = 0.1) resulted in a positive slope of surface \(S_{2}\) at point \(B\) and hence complicates the computer numerical control (CNC) machining of the steel mold and makes the injection molding of the lens almost impossible. The optical performance of the double freeform lens with \(\xi\) = 0.3 was evaluated by means of Zemax simulations. The simulations commenced by determining the collimating performance of the lens given the radiation pattern of the NCSU276A UV-LED shown in Figure 2. The simulation results showed that the light emerging from the lens was perfectly collimated with a zero divergent angle. In other words, the accuracy of the proposed collimating lens design under the assumption of a point source was confirmed. The angular distribution of the collimated LED light was then re-simulated using the true finite-sized model of the NCSU276A LED provided by the manufacturer. In performing the simulations, the LED illuminating chip size was set as \(1.0\times 1.0\) mm\({}^{2}\), while the distance between the LED and lens surface (i.e., \(r_{o}\) in Figure 1) was set as 10.65 mm based on a practical consideration of the injection-molded lens size and the LED intensity level after collimating. Based on the light source model of the NCSU276A, one million rays were used in the Zemax simulation process together with design parameters of \(\xi\) = 0.3, \(\theta_{B}\) = 60\({}^{\circ}\), and \(r_{o}\) = 10.65 mm. The angular distribution of the light intensity was statistically analyzed at a detector plane with a size of \(40\times 40\) mm\({}^{2}\) located at z = +40 mm.