Reconfigurable Intelligent Surface (RIS) is a breakthrough technology enabling the dynamic control of the propagation environment in wireless communications through programmable surfaces. To improve the flexibility of conventional diagonal RIS (D-RIS), beyond diagonal RIS (BD-RIS) has emerged as a family of more general RIS architectures. However, D-RIS and BD-RIS have been commonly explored neglecting mutual coupling effects, while the global optimization of RIS with mutual coupling, its performance limits, and scaling laws remain unexplored. This study addresses these gaps by deriving global optimal closed-form solutions for BD-RIS with mutual coupling to maximize the channel gain, specifically fully- and tree-connected RISs. Besides, we provide the expression of the maximum channel gain achievable in the presence of mutual coupling and its scaling law in closed form. By using the derived scaling laws, we analytically prove that mutual coupling increases the channel gain on average under Rayleigh fading channels. Our theoretical analysis, confirmed by numerical simulations, shows that both fully- and tree-connected RISs with mutual coupling achieve the same channel gain upper bound when optimized with the proposed global optimal solutions. Furthermore, we observe that a mutual coupling-unaware optimization of RIS can cause a channel gain degradation of up to 5 dB.

翻译：可重构智能表面（RIS）是一项突破性技术，通过可编程表面实现对无线通信传播环境的动态调控。为提升传统对角型RIS（D-RIS）的灵活性，超对角型RIS（BD-RIS）作为一类更通用的RIS架构应运而生。然而，现有对D-RIS和BD-RIS的研究普遍忽略了互耦效应，而考虑互耦的RIS全局优化、其性能极限及缩放规律仍属空白。本研究通过推导具有互耦效应的BD-RIS（特指全连接与树状连接RIS）最大化信道增益的全局最优闭式解，填补了上述空白。此外，我们给出了互耦存在时可实现的最大信道增益表达式及其闭式缩放规律。基于推导的缩放规律，我们通过理论分析证明：在瑞利衰落信道下，互耦平均而言能提升信道增益。理论分析（经数值仿真验证）表明：采用所提全局最优解优化后，具有互耦效应的全连接与树状连接RIS均能达到相同的信道增益上限。进一步研究发现，忽略互耦的RIS优化可能导致高达5 dB的信道增益损失。