## 2 Optical Design of the LED Light Source System  
### The Optical Model Building of a Single LED  
In the design setup, we use Cy3 fluorescence molecules as a reference material for fluorescence absorption. To achieve the largest light flux passing through the excitation filters of the microscope, the spectrum of the selected LED must fit the absorption spectrum of the Cy3 fluorescent molecules as closely as possible. A Lumileds LED (LXML-PX02-0000) made by Philips company was chosen to build up our light source system. The absorption and emission spectrum of the Cy3 fluorescent molecules and the spectrum of the selected LED are shown in Figure 1.  
The luminous flux of the LED, which is measured by a 10-inch integrating sphere, is 143 lumens at 2.75 V (operating voltage) and 0.32 A (operating current). This luminous value will be used in the simulation to calculate the actual number of LEDs required. Based on the limitations of the fabrication and the measured luminous flux, we take 9 LEDs in a \(3\times 3\) array to construct the lighting system. The distance between each LED in the \(3\times 3\) array is 38.5 mm. The layout is shown in Figure 2.  
The Lumileds LED is a quasi-Lambertian LED. The emitting area of the LED is 1.06 mm \(\times\) 1.06 mm, and the shape of the packaging lens is dome-like, with a radius of about 2.8 mm. [PERSON] proposed a simple LED light source model construction in 2001 [43]. According to previous research by [PERSON] [44; 45; 46; 47], precision simulation results require constructing the structure of the LEDs to  
Figure 1: The absorption and emission spectrum of Cy3 fluorescent molecules, and the emission spectrum of the selected LED.  
Figure 2: The layout of the \(3\times 3\) LEDs array.  
the single LED. The optical model of the LED is shown in Figure 3a. Further, the rays emitted from the LED are traced, and the corresponding normalized intensity distribution is calculated by the ASAP optical analysis software (Breault Research Organization, Inc. Tucson, AZ 85706 USA). The precision of the model is verified by comparing the LED intensity distribution from the experimental result with that from the simulation result.  
The normalized intensity distribution comparison from the experimental result and that from the simulation result is shown in Figure 3b, in which the black and red dots respectively represent the experimental and simulation values. The accuracy of the optical model can also be verified by the normalized correlation coefficient (NCC) value that represents the similarity between the experimental result and the simulation result. It is obtained as follows,  
\[NCC=\frac{\sum\limits_{n}\left[A(\theta_{n})-\overline{A}\right]\left[B( \theta_{n})-\overline{B}\right]}{\sqrt{\sum\limits_{n}\left[A(\theta_{n})- \overline{A}\right]^{2}\left[B(\theta_{n})-\overline{B}\right]^{2}}}, \tag{1}\]  
where \(A(\theta_{n})\) and \(B(\theta_{n})\) are the experimental and simulation results, respectively, while \(\overline{A}\) and \(\overline{B}\) are the mean values. \(\theta_{n}\) is the divergent angle of the LED. After calculation, the NCC value is 99.5%, which means that this optical model is accurate enough to be used in subsequent optical designs.