Scaling Achievable Rates in SIM-aided MIMO Systems with Metasurface Layers: A Hybrid Optimization Framework

We investigate the achievable rate (AR) of a stacked intelligent metasurface (SIM)-aided holographic multiple-input multiple-output (HMIMO) system by jointly optimizing the SIM phase shifts and power allocation. Contrary to earlier studies suggesting that the AR decreases when the number of metasurface layers increases past a certain point for \emph{a fixed SIM thickness}, our findings demonstrate consistent increase. To achieve this, we introduce two problem formulations: one based on directly maximizing the AR (RMax) and the other focused on minimizing inter-stream interference (IMin). To solve the RMax problem, we apply Riemannian manifold optimization (RMO) and weighted minimum mean square error (WMMSE) methods to optimize the SIM phase shifts and power allocation alternately. For the IMin problem, we derive an efficient algorithm that iteratively updates each meta-atom's phase shift using a closed-form expression while keeping others fixed. Our key contribution is integrating these two approaches, where the IMin solution initializes the SIM phase shifts in the first algorithm. This hybrid strategy enhances AR performance across varying numbers of metasurface layers. Simulation results demonstrate that the proposed algorithms outperform existing benchmarks. Most importantly, we show that increasing the number of metasurface layers while keeping the SIM thickness fixed leads to significant AR improvements.