## 2 System Model and Analysis  
### System Model  
The considering scenario is that VLC via mirror array-based IRS for the parallel vehicles in adjacent lanes. The right headlamp (LED light source) of the right vehicle is used as the transmitter, and the PD is installed in the middle of the two headlamps of the left vehicle. The mirror array-based IRS is installed on the street light pole, and the height of the center point is consistent with the headlamps. Figure 1 shows the application scenario of the VLC system via mirror array-based IRS for parallel vehicles. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Definitions** \\ \hline \(x_{s}\) & X-coordinate of the transmitter \(S\) as measured from the upper left corner of the IRS \\ \(y_{s}\) & Y-coordinate of the transmitter \(S\) as measured from the IRS along the road \\ \(z_{s}\) & Z-coordinate of the transmitter \(S\) as measured from the upper left corner of the IRS \\ \(x_{d}\) & X-coordinate of the receiver \(D\) as measured from the upper left corner of the IRS \\ \(y_{d}\) & Y-coordinate of the receiver \(D\) as measured from the IRS along the road \\ \(h_{d}\) & Z-coordinate of \(D\) as measured from the transmitter \(S\) \\ \(w_{m}\) & Width of each mirror \\ \(h_{m}\) & Height of each mirror \\ \(\Delta w_{m}\) & Edge-to-edge inter-mirror separation distances along the \(x\)-axis \\ \(\Delta h_{m}\) & Edge-to-edge inter-mirror separation distances along the \(z\)-axis \\ \(n_{k}\) & The number of mirrors of each column in the IRS \\ \(n_{l}\) & The number of mirrors of each row in the IRS \\ \(\rho\) & Mirror reflection efficiency \\ \(P_{i}\) & Transmitted power \\ \(m\) & Order of Lambertian emission \\ \(\Phi_{1/2}\) & Half-power semiangle of an LED \\ \(\theta_{R_{i}/i}^{\mathrm{S}}\) & Irradiance angle of the LED from the transmitter \(S\) to mirror \(R_{i,j}\) \\ \(\theta_{R_{i}/i}^{\mathrm{D}}\) & Incidence angle of the PD from mirror \(R_{i,j}\) to the receiver \(D\) \\ \(\omega\) & Current-to-light conversion efficiency \\ \(A_{d}\) & Physical area of the PD \\ \(T_{s}(\cdot)\) & Optical filter gain \\ \(g(\theta)\) & Optical concentrator gain \\ \(\mu\) & Refractive index \\ \(\Psi_{c}\) & FOV of the PD \\ \(\zeta\) & Efficiency of the transmit power amplifier \\ \(I_{DC}\) & DC-offset \\ \(A\) & Amplitude constraint of the signal \\ \(\epsilon\) & The variance of the signal \\ \(\eta\) & Responsivity of the PD \\ \(N\) & Total number of mirrors in the IRS \\ \(B\) & VLC system modulation bandwidth \\ \(P_{max}\) & The maximum power threshold \\ \(BER_{t}\) & The maximum acceptable BER \\ \(N_{max}\) & The maximum number \\ \(\mathbb{E}(\cdot)\) & Expectation operator \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mathematical notations and definitions. The model diagram of this scenario is given in Figure 2 for the convenience of analysis. For the mirror array-based IRS, the rotation angle of each mirror in the IRS can be controlled independently without interfering with others. One of the mirrors is analyzed as an example. We define a Cartesian coordinate system whose origin is at the center of the mirror \(R_{i,j}\) (\(1\leq i\leq n_{k}\), \(1\leq j\leq n_{l}\)). The position vector of the transmitter \(\mathbf{S}\) can be expressed as  
\[\mathbf{S}=\left[ \begin{array}{c}-\left(x_{s}+\frac{y_{m}}{2}+(j-1)(w_{m}+ \Delta w_{m})\right)\\ y_{s}\\ -\left(z_{s}+\frac{h_{m}}{2}+(i-1)(h_{m}+\Delta h_{m})\right)\\ \end{array} \right], \tag{1}\]  
The position vector of the PD can be expressed as  
\[\mathbf{D}=\left[ \begin{array}{c}-\left(x_{d}+\frac{y_{m}}{2}+(j-1)(w_{m}+ \Delta w_{m})\right)\\ y_{d}\\ h_{d}-\left(z_{s}+\frac{h_{m}}{2}+(i-1)(h_{m}+\Delta h_{m})\right)\\ \end{array} \right], \tag{2}\]  
To ensure that the reflected light reaches the receiver, each mirror must be rotated according to the position of the transmitter and receiver to obtain the appropriate angle.