# Spectra Prediction for WLEDs with High TLCI
## 2 Materials and Methods
### Experimental Methods
Another issue that needs to be addressed is that the peak wavelength of the emission spectrum of the phosphor changes with increasing temperature, i.e., spectral shift. Figure 4 shows the red shift of the emission spectrum of the green phosphor used in this paper. If we assume that the normalized spectral shape remains the same under different driving currents, the relationship between the shift value and driving current can be established by fitting the peak wavelength:  
\[\Delta\lambda=\lambda_{P,I}-\lambda_{P,0}=f(I), \tag{3}\]  
Figure 3: Relative power spectrum of LED modules. Figure 2: LED ceramic substrate package. (**a**) The size of the LED package in dimensions of mm. (**b**) The LED package without phosphors. (**c**) Schematic diagram of LED package, where d represents the thickness of the phosphor. (**d**) Diagram of the experimental setup. and the shifted emission spectrum is \(S/(\lambda)=S(\lambda+\Delta\lambda)\). In this study, we assume that the peak wavelength shift is linearly related to match the driving current, \(\Delta\lambda=f(I)=kI+b\); \(k\) and \(b\) are coefficients. If the absorption of green light by the red phosphor is ignored, the final emission spectrum can be estimated using the weighted average method:  
\[S_{mix}(\lambda)=S_{B}^{\prime}(\lambda)\cdot(1-W_{G}-W_{R})+S_{G}^{\prime}( \lambda)\cdot Q_{G}\cdot W_{G}+S_{R}^{\prime}(\lambda)\cdot Q_{R}\cdot W_{R}, \tag{4}\]  
where \(S_{mix}(\lambda)\) is the mixed spectrum. \(Q_{G}\) and \(Q_{R}\) are the conversion efficiencies of green and red phosphors, respectively. \(W_{G}\) and \(W_{R}\) are the proportion of blue light absorbed by green and red phosphors, respectively. \(S_{B}^{\prime}(\lambda)\), \(S_{G}^{\prime}(\lambda)\), and \(S_{R}^{\prime}(\lambda)\) are the emission spectra of the chip, green phosphor, and red phosphor, respectively. Equation (4) assumes that green and red phosphors have the same particle size distribution, are uniformly mixed, and have constant conversion efficiencies [23]. In Equation (4), both green and red phosphors are excited by the blue light from the LED chip. However, there may be an overlap between the red phosphor excitation spectrum and the green phosphor emission spectrum. It can be seen from Figure 5 that the excitation spectrum of the red phosphor and the emission spectrum of the green phosphor used in this study overlaps in the range of 400 nm to 600 nm. This indicates that the red phosphor absorbs a considerable amount of the light emitted by the green phosphor. The reabsorbed light is converted to red light by the red phosphor subsequently. The final emission spectrum would be written as follows:  
\[ \begin{array}{c}S_{mix}(\lambda)=S_{B}^{\prime}(\lambda)\cdot(1-W_{G}-W_{R} )+S_{G}^{\prime}(\lambda)\cdot Q_{G}\cdot W_{G}\cdot(1-W_{G-R})\\ \hskip 14.226378 pt+S_{R}^{\prime}(\lambda)\cdot(Q_{R}\cdot W_{R}+Q_{G-R}\cdot W _{G-R}),\end{array}  \tag{5}\]  
where \(S_{mix}(\lambda)\) is the mixed spectrum, and \(Q_{G}\) and \(Q_{R}\) are the conversion efficiency of green and red phosphor, respectively. \(W_{G}\) and \(W_{R}\) are the proportion of blue light absorbed by green and red phosphors, respectively. \(S_{B}^{\prime}(\lambda)\), \(S_{G}^{\prime}(\lambda)\), and \(S_{R}^{\prime}(\lambda)\) are the emission spectra of the chip, green phosphor, and red phosphor, respectively. \(W_{G-R}\) is the proportion of reabsorbed green light by red phosphor, which is related to the concentration ratio of green phosphor and red phosphor. \(Q_{G-R}\) is the re-emission conversion efficiency of red phosphor. It can be seen from Equation (5) that \(W_{G-R}=0\) represents that no green light is absorbed by the red phosphor. A higher \(W_{G-R}\) means more green light is absorbed by the red phosphor and converted to red light. The actual \(W_{G-R}\) value depends on the absorption characteristics of the red phosphor, the mixing ratio of the phosphor, the volume of the phosphor paste, and the packaging structure [23].