## Appendix A Proof of Lemma 1
\] (A4)  
Note that in (A4), if \(\rho_{x}^{\prime}\
ot\in\left[\rho_{x,l}^{\min},\rho_{x,l}^{\max}\right]\) or \(\rho_{y}^{\prime}\
ot\in\left[\rho_{y,l}^{\min},\rho_{y,l}^{\max}\right]\), we can simply set \(S_{l}\left(\rho_{x,^{\prime}}^{\prime}\rho_{y}^{\prime}\right)\) to zero. Equation (A4) indicates that \(\frac{1}{\sqrt{N_{x}N_{y}}}\mathbf{b}\left(\rho_{x}^{\prime}\right)\otimes \mathbf{a}\left(\rho_{y}^{\prime}\right)\) is the eigenvector of \(\mathbf{C}_{l}^{\text{NLoS}}\), and the corresponding eigenvalue can be expressed as \(\frac{\beta_{l}}{K_{l}+1}S_{l}\left(\rho_{x,^{\prime}}^{\prime}\rho_{y}^{\prime}\right)\). For the commonly used PAS models, \(S_{l}\left(\rho_{x},\rho_{y}\right)\) achieves the maximum at \(\left\{\rho_{x},\rho_{y}\right\}=\left\{\rho_{x,l},\rho_{y,l}\right\}\)[15; 36], i.e., the virtual AoDs corresponding to sensors' physical AoDs in horizontal and vertical directions. Therefore, we can conclude  
\[ \begin{split}\mathbf{u}_{\max}\left(\mathbf{C}_{l}^{\text{NLoS}} \right)&=\frac{1}{\sqrt{N_{x}N_{y}}}\mathbf{b}\left(\rho_{x,l} \right)\otimes\mathbf{a}\left(\rho_{y,l}\right),\\ \lambda_{\max}\left(\mathbf{C}_{l}^{\text{NLoS}}\right)& =\frac{\beta_{l}}{K_{l}+1}S_{l}\left(\rho_{x,l},\rho_{y,l}\right).\end{split} \] (A5)  
Combining (A2) and (A5), in the large \(\left\{N_{x},N_{y}\right\}\) limit, we have  
\[ \begin{split}\mathbf{u}_{\max}\left\{\mathbf{C}_{l}\right\}& =\frac{1}{\sqrt{N_{x}N_{y}}}\mathbf{b}\left(\rho_{x,l}\right)\otimes\mathbf{a} \left(\rho_{y,l}\right),\\ \lambda_{\max}\left(\mathbf{C}_{l}\right)&=\frac{\beta_{ l}K_{l}}{K_{l}+1}N_{x}N_{y}+\frac{\beta_{l}}{K_{l}+1}S_{l}\left(\rho_{x,l},\rho_{y,l} \right).\end{split} \] (A6)With the channel model in (1) and (2), the harvest energy at sensor \(l\) can be written as  
\[ \begin{split}& P_{l}=\frac{\eta\tau}{1-\tau}p_{l}\mathbf{w}_{E,l}^{H} \mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{w}_{E,l}-\frac{E_{l}^{p}}{\left(1-\tau \right)T}\\ &=\frac{\eta\tau}{1-\tau}p_{l}\left\{\mathbf{w}_{E,l}^{H}\mathbf{ C}^{\text{LoS}}\mathbf{w}_{E,l}+\frac{\beta_{l}}{K_{l}+1}\middle|\int\limits_{ \begin{subarray}{c}\rho_{x}-\Delta\rho_{y}\\ \rho_{x}-\rho_{y}\end{subarray}}^{\rho_{x}+\Delta\rho_{y}}\int\limits_{\rho_{x }-\Delta\rho_{y}}^{\rho_{y}+\Delta\rho_{y}}r_{l}\left(\varphi,\theta\right) \mathbf{w}_{E,l}^{H}\mathbf{b}\left(\varphi,\theta\right)\otimes\mathbf{a} \left(\varphi,\theta\right)\text{d}\varphi\text{d}\theta\right|^{2}\right\}- \frac{E_{l}^{p}}{\left(1-\tau\right)T}\end{split}  \tag{10}\]  
Substituting (20) and (10) into (10), and using a similar procedure as that in (11), \(P_{l}\) (in the large \(\left\{N_{x},N_{y}\right\}\) limit) can be expressed asymptotically as  
\[P_{l}=\frac{\eta\tau}{1-\tau}p_{l}\left(\frac{\beta_{l}K_{l}}{K_{l}+1}N_{x}N_{ y}+\frac{\beta_{l}}{K_{l}+1}\middle|r_{l}^{\text{v}}\left(\rho_{x,l},\rho_{y,l} \right)\middle|^{2}\right)-\frac{E_{l}^{p}}{\left(1-\tau\right)T}, \tag{11}\]  
where \(r_{k}^{\text{v}}\left(\rho_{x,k},\rho_{y,k}\right)\) denotes the complex response gain with respect to the virtual AoDs \(\left\{\rho_{x},\rho_{y}\right\}\), which can be simply derived by using the formula of integration by substitution. Note that the term \(\frac{\beta_{l}}{K_{l}+1}\middle|r_{l}^{\text{v}}\left(\rho_{x,l},\rho_{y,l} \right)\middle|^{2}\) becomes negligible in the large \(\left\{N_{x},N_{y}\right\}\) limit since \(r_{l}^{\text{v}}\left(\rho_{x,l},\rho_{y,l}\right)\) is independent with \(\left\{N_{x},N_{y}\right\}\). This completes the proof.