## 2 System Model
### Channel Model
#### 2.2.2 IRS-Aided MIMO Channel  
The geometric optics approximation is demonstrated to be highly accurate in this extremely near-field environment, as long as the size of the IRS unit significantly exceeds the wavelength of light. Numerical experiments' results indicate that the power density of the reflected beam in the near-infrared is highly concentrated, with energy focused along a straight line [36]. From geometric optics, it is understood that the light beam reflected by the optical IRS is equivalent to the beam emitted from an image source that is symmetrical to the IRS unit [28]. Based on the above analysis, under the condition of an extremely near field, there is almost no interference between the reflective elements of the optical IRS reflection channel. We can assume that a given IRS unit provides NLOS channel gain only when it is aligned with the LED/PD pair [25].  
Figure 2: Two-dimensional topological model of optical path based on IRS.  
For convenience, let \(\mathcal{M}\triangleq\{1,\cdots,M\}\), \(\mathcal{K}\triangleq\{1,\cdots,K\}\), \(\mathcal{N}\triangleq\{1,\cdots,N\}\). Define two binary matrices, \(\mathbf{R}\) and \(\mathbf{F}\), to represent the alignment between the IRS unit and the transmitting LED and receiving PD, respectively, in the following form:  
\[\mathbf{R}\triangleq [\boldsymbol{r}_{1},\cdots,\boldsymbol{r}_{M}]\in\{0,1\}^{N\times M }\,\ \boldsymbol{r}_{m}\in\mathbb{R}^{N\times 1}\,\ m\in\mathcal{M}, \tag{9}\] \[\mathbf{F}\triangleq [\boldsymbol{f}_{1},\cdots,\boldsymbol{f}_{K}]\in\{0,1\}^{N\times K }\,\ \boldsymbol{f}_{k}\in\mathbb{R}^{N\times 1}\,\ k\in\mathcal{K}. \tag{10}\]  
We represent the association between the \(n\)-th IRS unit and the corresponding LED/PD pair using the \(n\)-th row of the \(\mathbf{R}\) and \(\mathbf{F}\) matrix. Specifically, if \(r_{n,m}=f_{n,k}=1\) is satisfied, it means that the LED/PD pair consisting of the \(m\)-th LED and the \(k\)-th PD should be aligned with the \(n\)-th IRS unit. Thus, the IRS-assisted NLOS MIMO channel matrix should be written as follows:  
\[\mathbf{G}=\begin{bmatrix}(\boldsymbol{f}_{1}\odot\boldsymbol{r}_{1})^{T} \boldsymbol{h}_{1,1}&\cdots&(\boldsymbol{f}_{1}\odot\boldsymbol{r}_{M})^{T} \boldsymbol{h}_{1,M}\\ \vdots&\ddots&\vdots\\ (\boldsymbol{f}_{K}\odot\boldsymbol{r}_{1})^{T}\boldsymbol{h}_{K,1}&\cdots&( \boldsymbol{f}_{K}\odot\boldsymbol{r}_{M})^{T}\boldsymbol{h}_{K,M}\end{bmatrix}, \tag{11}\]  
where the channel gain \(\boldsymbol{h}_{k,m}\in\mathbb{R}^{N\times 1}\) between the \(m\)-th LED and the \(k\)-th PD is expressed as  
\[\boldsymbol{h}_{k,m}\triangleq[h_{k,1,m},\cdots,h_{k,N,m}]^{T}. \tag{12}\]  
Therefore, the channel vector \(\mathbf{g}_{k}\in\mathbb{R}^{M\times 1}\) of user \(k\) is expressed as  
\[\mathbf{g}_{k}=\Big{[}(\boldsymbol{f}_{k}\odot\boldsymbol{r}_{1})^{T} \boldsymbol{h}_{k,1}\,\ \cdots\,\ (\boldsymbol{f}_{k}\odot\boldsymbol{r}_{M})^{T} \boldsymbol{h}_{k,M}\Big{]}^{T}. \tag{13}\]  
Furthermore, we impose row sum constraints on \(\mathbf{R}\) and \(\mathbf{F}\) to ensure that each optical IRS unit can be assigned to at most one LED/PD pair [24; 25]:  
\[\sum_{m=1}^{M}r_{n,m}\leq 1,\ \sum_{k=1}^{K}f_{n,k}\leq 1,\ \forall n\in \mathcal{N}. \tag{14}\]