## 4 Asymptotic Performance Analysis in a Multicast System  
In this section, we extend the system model to a multicast system and analyze the asymptotic optimality of the proposed scheme, leveraging the derived asymptotic results in the unicast system. In particular, we will demonstrate that achieving asymptotic optimality in the multicast system is readily attainable through a simple yet practical passive beam forming approach. Consider a SatCom system consisting of a single multibeam satellite, with \(K\) RISs, each capable of simultaneously serving \(M_{k}\) UEs. As the satellite is equipped with a reflective array antenna comprising \(K\) feeds, we examine \(K\) multicast groups (\(M_{k}\) UEs per group), with each multicast group \(k\) being served by RIS \(k\). On the basis of the multicast nature, the achievable rate of multicast group \(k\) is obtained as follows [30]:  
\[R_{\text{MC},k}=\min_{m\in\mathcal{M}_{k}}R_{k,m}, \tag{34}\]  
where  
\[R_{k,m}=\log\left(1+\rho_{k}\xi_{k}^{2}a_{k}|c_{k}|^{2}\Big{|}\sum_{n=1}^{N}f_ {k,n,m}^{*}e^{i\theta_{k,n}}\Big{|}^{2}\right). \tag{35}\]  
Here, \(\mathcal{M}_{k}\) is a set of UEs in multicast group \(k\) where \(|\mathcal{M}_{k}|=M_{k}\), and \(f_{k,n,m}\) is the fading channel between reflecting element \(n\) of RIS \(k\) and UE \(m\). As discussed in (19), optimal reflection performance can be achieved by aligning all phase shift values of RIS with the channel phases from RIS to UE. However, in the multicast system, it is impossible to design the optimal passive beamforming to simultaneously match the phases of channels for multiple UEs. Based on (34), we can observe that maximizing the achievable rate in the multicast system is synonymous with maximizing the individual rate of each UE. Hence, we introduce a practical passive beamforming approach for the multicast system, wherein the reflecting elements of the RIS are evenly divided and allocated to each UE within a multicast group. In particular, \(N\) reflecting elements of each RIS are divided into \(M_{k}\) reflecting groups, and the reflection phases of reflecting group \(m\) are obtained as follows:  
\[\bar{\theta}_{k,n}=-\angle\bar{f}_{k,n,m}^{*}, \tag{36}\]  
for \(n\in\mathcal{N}_{k,m}\). Here, \(\bar{f}_{k,n,m}\) is the LoS component of \(f_{k,n,m}\) and \(\mathcal{N}_{k,m}\) is a set of reflecting elements in reflecting group \(m\) of RIS \(k\), where \(\sum_{m=1}^{M_{k}}\lvert\mathcal{N}_{k,m}\rvert=N,\forall k\). For notational simplicity, we assume that \(N\) mod \(M_{k}=0,\forall k\). In order to achieve the SNR order of \(\mathcal{O}\big{(}N^{2}\big{)}\) for a large \(N\), \(N\) reflecting elements are equally divided into \(M_{k}\) reflecting groups (i.e., \(\frac{N}{M_{k}}\) elements per each reflecting group) and, similarly in (26), we have:  
\[\bar{R}_{k,m} \approx\log\left(\rho_{k}\xi_{k}^{2}a_{k}|c_{k}|^{2}\bigg{|}\bar{ \beta}_{k,m}\left(\frac{N}{M_{k}}+\sum_{n\in\mathcal{N}_{k,m}}\bar{f}_{k,n,m}^ {*}e^{i\theta_{k,n}}\right)\bigg{|}^{2}\right)\] \[\approx\log\left(N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_{k}|^{2}\bar{ \beta}_{k,m}^{2}/M_{k}^{2}\right), \tag{37}\]  
where \(\bar{\beta}_{k,m}=\sqrt{\xi_{k,m}\kappa_{k,m}/(\kappa_{k,m}+1)}\) and \(\kappa_{k,m}\) and \(\zeta_{k,m}\) are, respectively, the Rician factor and the path loss between RIS \(k\) and UE \(m\). From (34) and (37), we can observe that \(\bar{R}_{\text{MC},k}\) is determined exclusively depending on \(\bar{\beta}_{k,m}\). Define the minimum \(\bar{\beta}_{k,m}\) for all \(m\in\mathcal{M}_{k}\) as \(\bar{\beta}_{k,m^{\prime}}\), then we have:  
\[\bar{R}_{\text{MC},k}=\log\left(N^{2}\rho_{k}\xi_{k}^{2}a_{k}|c_{k}|^{2} \bar{\beta}_{k,m^{\prime}}^{2}/M_{k}^{2}\right). \tag{38}\]  
Next, we derive a theoretical ideal performance for the multicast system under consideration, with the goal of capturing the asymptotic optimality of the multicast system. In this regard, we define an ideal RIS capable of reflecting signals to multiple UEs simultaneously with perfect phase alignment for all UEs.