## 2 Module Design and Simulation  
### Module Design  
In this section we present the theory of designing a uniform illumination system for single LEDs. We propose a novel uniform illumination system consisting of three parts: LED, collimator, and two perpendicular linear Fresnel lenses, as shown as in Figure 2. The collimator was used to collimate all light from the LED source. The collimated beam from the collimator lens was redistributed using double linear Fresnel lenses to provide uniform light distribution over the given target. Figure 1: The principle of our proposed uniform LED illumination system. Commercial collimators are cone-like lenses. We chose a collimator from Led-link Optics Inc. [8] for our experiment. The image of the commercial collimator lens is shown in Figure 3. Collimators are often designed for point sources and a fixed wavelength to create ideal collimated beams, while LED chips are usually \(1\times 1\) mm or larger. Therefore, the output beams from the collimator are slightly divergent. Details will be presented in the simulation part. The collimated beam needs to be distributed to create uniform illumination. We used the double linear Fresnel lenses as a redistribution module, the design method as shown in Figure 4. A single linear Fresnel lens can redistribute the collimated beam in one direction in the receiver area, as shown in Figure 4a. Although the incident beam does not have a uniform distribution, the Fresnel lens consists of many small grooves, where each groove is similar to a linear convex lens. When the collimated beam passes through the linear Fresnel lens, these grooves will split the incident beam (a divided beam can be considered to have a uniform distribution) and spread it over the receiver in one direction. When two linear Fresnel lenses are perpendicular to each other, they will direct the collimated beam to spread across the receiver in two directions. We applied the \"edge-ray principle\" and optical path length conservation to calculate the curves of Fresnel lens grooves. The calculation is shown in Figure 4b and consists of three steps. Step 1: Some initial conditions of light source (collimated light beam) such as left edge \(A_{L}B_{L}\), right edge \(A_{R}B_{R}\) of the segment, position of Fresnel lens, and receiver size (illumination space: RR ) are selected. Step 2: The ray to the left edge of \(A_{L}B_{L}\) passes through a linear Fresnel lens groove and refracts to the right side of the receiver (point R ). The ray to the right edge of \(A_{R}B_{R}\) passes through a groove and refracts to the left side of the receiver (point R). These two rays intersect at the focal line. Similarly,  
Figure 3: Illustration of collimator using in this research: (**a**) Principles of Total Internal Reflection (TIR) lens; (**b**) A commercial collimator lens. Figure 2: Structure of LED illumination system composed of an LED chip, collimator, and double linear Fresnel lenses. the \(A_{n}B_{n}\) rays between the \(A_{L}B_{L}\) and \(A_{R}B_{R}\) rays must also intersect at the focal line. Based on the law of optical path length (OPL) conservation, we have Equation (1):  
\[n_{1}|A_{L}B_{L}|+n_{0}|B_{L}F|=\ n_{1}|A_{R}B_{R}|+n_{0}|B_{R}F|=\ n_{1}|A_{n}B_ {n}|+n_{0}|B_{n}F| \tag{1}\]  
where \(n_{0}\), \(n_{1}\) are refractive indices of air and materials of Fresnel lens, respectively. Based on the above equation, all coordinates of the segment surface curves are calculated. Step 3: Repeat the calculation process, and all the other grooves of linear Fresnel lens are obtained by the same procedure. Because a linear Fresnel lens distributes light in only one direction, we placed two linear Fresnel lenses perpendicularly to propagate light in two directions over the lighting area, as shown in Figure 4c. So, a two-dimensional redistribution will create a square shape on the receiver. The focal line is important in this linear Fresnel lens. The LED light was focused on the focal line and was spread across the receiver RR'. Figure 5 shows the design process of the novel lens in Matlab.