## 2 Theoretical Model  
### Optical Model  
In the optical model without a homogenizer, the irradiance distributions of concentrated light were analyzed near the focal plane of the Fresnel lens. The physical structure of the concentrated photovoltaic cell is shown in Figure 1a. The irradiance is concentrated by a Fresnel lens, and the multijunction cell is placed directly under the Fresnel lens. The parameters of the Fresnel lens are as follows: material: PMMA (Poly (methyl methacrylate); aperture area: 15,125 mm\({}^{2}\); thickness: 2 mm; focal distance estimated at 500 nm: 110 mm; and ring width: 0.5 mm. In this model, the position where the refracted light less than 500 nm intersects the optical axis will be at less than 110 mm. At the same time, due to the non-ideality of focusing, the intensity and spectral distribution of the concentrated illumination on the design focal plane are non-uniform.  
The geometric relationships and the fundamental laws of refraction have been implemented in a programming environment to calculate the reflection position and the light power on the cell surface. The refracted light will go through three processes from the Fresnel lens to the surface of cells: (1) the incidence on the front surface of the lens; (2) the propagation through the lens; and (3) the exit from the rear surface of the lens. Figure 1b shows a 2D cross-section geometric relationships sketch of an incident light and its refracted light for a certain ring of the Fresnel lens. In the sketch, f represents the focal distance; d is the distance between the incident light and the lens center; t represents the distance between the refracted light and the front surface of the lens which equals half of the ring height if the light incidents from the middle of the ring; \(\alpha\) is the incident angle and also equals angle of the slope of the facet; and \(\Omega\) is the angle between the refracted light and the optical axis. According to the geometric relationships, \(\Omega\) at 500 nm (\(\Omega_{500}\)) is estimated as follows:  
\[\Omega_{500}=tan^{-1}\bigg{(}\frac{d}{f-t}\bigg{)} \tag{1}\]  
Once this angle is estimated, \(\alpha\) at 500 nm (\(\alpha_{500}\)) is obtained by using the following relationship:  
\[\alpha_{500}=tan^{-1}\bigg{(}\frac{sin\Omega_{500}}{n_{500}-cos\Omega_{500}} \bigg{)} \tag{2}\]  
Figure 1: (**a**) The structure of the concentrating photovoltaic cell. (**b**) The geometrical relationships between the incident light and the refracted light.  
where \(n_{500}\) is the refractive index at 500 nm, its value being 1.49 [28], the \(\Omega\) angle as a function of the wavelength can be estimated from the well-known Snell's law as:  
\[\Omega(\lambda)=sin^{-1}(n(\lambda)\cdot sin\ \alpha_{500})-\alpha_{500} \tag{3}\]  
Based on the expressions above, the geometry slope of the refracted light functions as the wavelength and can be approximated. Ray tracing can be then performed for each ray height and wavelength. According to the spectral response of the triple junction cell, the spectral range is 300-1700 nm. The calculation takes a wavelength every 20 nm in this spectral range. The energy of this wavelength is obtained by integrating the wavelength in the positive and negative 10 nm range according to the solar power spectral density curve. A parameter (optical efficiency) is introduced here and it is defined as follows [29]:  
\[\mu=\frac{\mathrm{G}}{\mathrm{I}}\, \tag{4}\]  
where G represents the irradiance actually irradiated on the cell surface; and I represents the incident irradiance with the area of Fresnel lens. The incident spectral irradiance inputted in the optical model is AM1.5D and the DNI is taken as a constant value of 1000 W/m\({}^{2}\).