## 2 Optoelectronic Modeling Methods
### Optical Modeling
The time-averaged Poynting vector \(\overrightarrow{S}(\overrightarrow{r},\lambda,t_{b})\), indicating the amount of the optical power flow, is given by  
\[\overrightarrow{S}(\overrightarrow{r},\lambda,t_{b})=\frac{1}{2}\text{Re} \bigg{\{}\overrightarrow{E}(\overrightarrow{r},\lambda,t_{b})\times \overrightarrow{H}(\overrightarrow{r},\lambda,t_{b})\bigg{\}}, \tag{2}\]  
where \(\overrightarrow{H}(\overrightarrow{r},\lambda,t_{b})\) is the complex conjugate of the magnetic field amplitude. The amount of optical absorption is described by the optical power dissipation, which is given by  
\[Q(\overrightarrow{r},\lambda,t_{b})=-\
abla\cdot\overrightarrow{S}( \overrightarrow{r},\lambda,t_{b}). \tag{3}\]  
The absorptivity can then be obtained by taking the integration over the optical power dissipation within the entire region of the CIGS solar cell in reference to the wavelength-dependent incident power from the uppermost boundary, which corresponds to the solar irradiance of AM 1.5 sunlight. Thus, the absorptivity is written as  
\[A(\lambda,t_{b})=\frac{1}{S(\lambda)}\iint Q(\overrightarrow{r},\lambda,t_{b })d\overrightarrow{r}. \tag{4}\]  
According to the ETAM, the effect of the incoherent layer can be modeled by averaging the coherent simulation results over a set of equispaced phase thicknesses [18; 19]. In the coherent simulations, the set of equispaced phase thicknesses of the EPL used in the cover glass layer is matched with the equally spaced optical phase changes ranging between 0 and \(2\pi\), which provide different optical calculation results of the CIGS solar cell. However, if the cover glass is appropriately modeled as optically incoherent, all calculation results with respect to the set of equispaced phase thicknesses should be equal to the ETAM-applied incoherent calculation result irrespective of these equispaced phase thicknesses. Thus, coherent simulation results over the set of equispaced phase thicknesses of the EPL can be interpreted as the variation of the calculation inaccuracy when an incoherent cover glass layer is inappropriately modeled as optically coherent. Then, the ETAM-applied optical power dissipation and absorptivity, which include the effect of the incoherent cover glass layer, can then be expressed as  
\[Q^{ETAM}(\overrightarrow{r},\lambda)=\frac{1}{N}\underset{b=1}{\overset{N}{ \sum}}Q(\overrightarrow{r},\lambda,t_{b}), \tag{5}\]  
\[A^{ETAM}(\lambda)=\frac{1}{S(\lambda)}\iint Q^{ETAM}(\overrightarrow{r}, \lambda)d\overrightarrow{r}, \tag{6}\]  
where \(N\) indicates the number of equispaced phase thicknesses. Finally, the optical power dissipation \(Q(\overrightarrow{r},\lambda,t_{b})\) is related with the generation rate of the electron-hole pair, which will be used as one of the input parameters in the continuity equation of the electrical modeling. The electron-hole-pair generation rate \(G(\overrightarrow{r},t_{b})\), depending on the thickness of the EPL, is given by  
\[G(\overrightarrow{r},t_{b})=\int S(\lambda)\frac{Q(\overrightarrow{r}, \lambda,t_{b})}{hc/\lambda}d\lambda, \tag{7}\]where \(h\) is the Plank constant and c is the speed of light in a free space. When the optically incoherent characteristics of the thick cover glass were considered, the ETAM-applied generation rate of the electron-hole pair \(G^{ETAM}(\overrightarrow{r})\) is written as  
\[G^{ETAM}(\overrightarrow{r})=\int S(\lambda)\frac{Q^{ETAM}(\overrightarrow{r },\lambda)}{\hbar c/\lambda}d\lambda. \tag{8}\]