## 2 Methods
### Mesopic Luminance and Color Appearance  
To calculate mesopic luminance, the S/P-ratio of a light source is essential. It is the scotopic lumens divided by photopic lumens as Equation (5). \[\mathrm{S/P}=\frac{L_{s}}{L_{p}}=\frac{\int\limits_{380}^{780}k_{s}V^{\prime} (\lambda)S(\lambda)d\lambda}{\int\limits_{380}^{780}k_{p}V(\lambda)S(\lambda)d\lambda} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Components** & **FWHM (nm)** & **Wavelength Shifts (nm)** & **Density Ratios** & **Records** \\ \hline
**MS1** & Blue, green, red & (25, 36, 13) & B: 450–480 & 0.1–1.0 & 5273 \\
**MS2** & Blue, green, red & (22, 73, 88) & G: 500–550 & 0.1–1.0 & 45,382 \\
**MS3** & Blue, green, orange & (25, 70, 70) & R: 605–645 & 0.1–1.0 & 39,076 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Spectral parameters for test. where \(k_{s}\) and \(k_{p}\) are 1699 and 683 lm/W, respectively, which are the peak values of luminous efficiency in photopic and scotopic ranges, and \(S(\lambda)\) is the SPD of a light source. The coefficient and the mesopic luminance \(L_{mes}\) can be obtained by Equations (6)-(8), which is an iterative method. \(L_{p}\) is the photopic luminance, \(L_{s}\) is scotopic luminance, and \(V^{\prime}(\lambda_{0})\) represents the scotopic luminous efficiency at 555 nm, where the value is 683/1699, \(n\) is the number of iterative steps, \(a\) and \(b\) are parameters which are 0.7670 and 0.3334, respectively. \[m_{0}=0.5, \tag{6}\]  
\[L_{mes,n}=\frac{m_{(n-1)}L_{p}+(1-m_{(n-1)})L_{s}V^{\prime}(\lambda_{0})}{m_{(n -1)}+(1-m_{(n-1)})V^{\prime}(\lambda_{0})}, \tag{7}\]  
\[m_{n}=a+b\log_{10}(L_{mes,n}),\ for\ 0\leq m_{n}\leq 1 \tag{8}\]  
Equation (7) can be rewritten as Equation (9), showing that S/P-ratio and the photopic luminance are the primary factors and positive correlation with \(L_{mes}\) for calculating the mesopic luminance. To reach the maximum value of mesopic luminance, one aim of the study is to optimize SPD with the highest S/P-ratio in the same \(L_{p}\). \[L_{mes,n}=\frac{m_{(n-1)}+(1-m_{(n-1)})(S/P)V^{\prime}(\lambda_{0})}{m_{(n-1) }+(1-m_{(n-1)})V^{\prime}(\lambda_{0})}L_{p} \tag{9}\]  
The mesopic appearance model describes that the opponent process converts the cone responses from \(L\), \(M\), and \(S\) to red/green, yellow/blue opponent-color, and luminance channels denoted by \(L\)-\(2M\), \(L\)+\(M\)-\(S\), and \(L\)+\(M\), respectively. The model considered the characteristic of the nonlinear shift in spectral luminous efficiency, reduction of saturation at low illuminance levels, and the variations of hue and chroma loci under different illuminance levels from 1000 to 0.01 lux. The experimental results showed that chroma decreases continuously with the decrease of the illuminance level until 0.01 lux. The mesopic color appearance model puts a rod intrusion varied with different illuminance levels to each channel of the two-stage color-vision model. The outputs could be acquired with Equations (10) and (12). \[A(E)=\alpha(E)K_{w}((L_{p}+M_{p})/(L_{p}+M_{p})_{w})+\beta(E)K^{\prime}{}_{w} (Y^{\prime}/Y^{\prime}{}_{w})^{\gamma} \tag{10}\]  
\[r/g(E)=l(E)(L_{p}-2M_{p})+a(E)Y^{\prime} \tag{11}\]  
\[b/y(E)=m(E)(L_{p}+M_{p}-S_{p})+b(E)Y^{\prime} \tag{12}\]  
where \(A(E)\), \(r/g(E)\), and \(b/y(E)\), respectively, represent the outputs of the luminance, red-green, and blue-yellow channels of an illuminance \(E\), and \(L_{p}\), \(M_{p}\), and \(S_{p}\) represent cone responses in photopic condition. \(Y^{\prime}\) is the scotopic luminance where the rod signals are involved. \(K_{w}\) and \(K^{\prime}{}_{w}\) are the maximum responses of luminance in photopic and scotopic conditions in which the values are 100 and 78.4, respectively. \(\gamma\) is the parameter of the nonlinear relationship between photopic and scotopic luminance channels attained by the least-squares fitting to the experimental data. \(\alpha(E)\), \(\beta(E)\), \(l(E)\), \(a(E)\), \(m(E)\), and \(b(E)\) are the different functions of weighting coefficients under illuminance \(E\).