## 2 System Model and Analysis
### System Model
The mirror is first arranged via the clockwise rotation of the local \(z\)-axis with an angle \(\beta_{i,j}\) and the local negative \(x\)-axis with an angle \(\alpha_{i,j}\). The normal vector direction of the mirror after rotation is expressed as  
\[\mathbf{N_{i,j}}=\frac{\mathbf{R_{i,j}^{\ast}S+R_{i,j}D}}{\sqrt{2+2\mathbf{R_{ i,j}^{\ast}S^{T}R_{i,j}^{\ast}D}}}, \tag{3}\]  
where \(\mathbf{R_{i,j}^{\ast}S}=\frac{\mathbf{S-R_{i,j}}}{\|\mathbf{S-R_{i,j}}\|_{2}}\), \(\mathbf{R_{i,j}^{\ast}D}=\frac{\mathbf{D-R_{i,j}}}{\|\mathbf{D-R_{i,j}}\|_{2}}\), \(\|\cdot\|_{2}\) denote the \(\ell_{2}\)-norm, and \(\left(\cdot\right)^{T}\) denotes the transpose operator. Figure 1: Application scenario of the VLC system via mirror array-based IRS for parallel vehicles. Figure 2: Model of the VLC system via mirror array-based IRS for parallel vehicles. The relation between normal vector and rotation angle can be expressed as  
\[\hat{N_{ij}}=\left[ \begin{array}{c}\sin\left(\beta_{i,j}\right)\cos\left(\alpha _{i,j}\right)\\ \cos\left(\beta_{i,j}\right)\cos\left(\alpha_{i,j}\right)\\ \sin\left(\alpha_{i,j}\right)\end{array} \right]. \tag{4}\]  
In the actual scene, the distance of light transmission is much larger than the size of the light source, so it can be regarded as a point light source. The irradiance of the point light source after being reflected by the mirror \(R_{i,j}\) to the PD can be expressed as [31]  
\[E_{i,j}=\frac{\rho(m+1)P_{i}\cos^{m}\left(\theta_{R_{i,j}}^{\mathrm{S}}\right)} {2\pi\left(\left\|R_{i,j}D\right\|_{2}+\left\|R_{i,j}S\right\|_{2}\right)^{2} }\cos\left(\theta_{R_{i,j}}^{\mathrm{D}}\right), \tag{5}\]  
where \(E_{i,j}\) represents the irradiance at the detector center contributed by the mirror \(R_{i,j}\). \(m\) is the order of Lambertian emission [43] related to the half-power semiangle of LED \(\Phi_{1/2}\) which can be expressed as \(m=-ln2/ln(\cos\Phi_{1/2})\). According to Figure 2, \(\cos\left(\theta_{\mathrm{R_{i}}}^{\mathrm{S}}\right)=\mathbf{e_{2}^{T}R_{i,j}}\mathbf{S}=\mathbf{e_{2}^{T}\left(\mathbf{S}-\mathbf{R}_{i,j}\right)/ \left\|\mathbf{S}-\mathbf{R}_{i,j}\right\|_{2}}\), \(\cos\left(\theta_{\mathrm{R_{i}}j}^{\mathrm{D}}\right)=\mathbf{e_{2}^{T}R_{i,j}^{\mathrm{D}}D=\mathbf{e_{2}^{T}\left(\mathbf{D}-\mathbf{R}_{i,j}\right)/ \left\|\mathbf{D}-\mathbf{R}_{i,j}\right\|_{2}}\), and \(\mathbf{e_{2}^{T}}=\left[\mathbf{0,1,0}\right]\). According to the theory of VLC transmission [44], the direct current (DC) gain of the channel can be obtained as  
\[H_{i,j}^{IRS}=\frac{\varpi T_{s}\left(\theta_{\mathrm{R_{i}}j}^{\mathrm{D}} \right)A_{d}\rho(m+1)\cos^{m}\left(\theta_{\mathrm{R_{i}}j}^{\mathrm{S}} \right)}{2\pi\left(\left\|\mathbf{R_{i}j}\mathbf{D}\right\|_{2}+\left\| \mathbf{R_{i}j}\mathbf{S}\right\|_{2}\right)^{2}}\cos\left(\theta_{\mathrm{R _{i}}j}^{\mathrm{D}}\right)g\left(\theta_{\mathrm{R_{i}}j}^{\mathrm{D}}\right), \tag{6}\]  
\(g\left(\theta_{R_{i,j}}^{\mathrm{D}}\right)\) can be given as  
\[g(\theta)=\left\{ \begin{array}{ll}\frac{\mu^{2}}{sin^{2}\left(\Psi_{c} \right)}&0\leq\theta\leq\psi_{c}\\ 0&\theta>\psi_{c}\end{array} \right., \tag{7}\]  
The total DC gain can be obtained as  
\[H^{IRS}=\sum_{i=1}^{n_{\mathrm{R}}}\sum_{j=1}^{n_{\mathrm{I}}}H_{i,j}^{IRS}. \tag{8}\]