Written by its inventors, this first tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces (BD-RISs) provides the readers with the basics and fundamental tools necessary to appreciate, understand, and contribute to this emerging and disruptive technology. Conventional (Diagonal) RISs (D-RISs) are characterized by a diagonal scattering matrix $\mathbf{\Theta}$ such that the wave manipulation flexibility of D-RIS is extremely limited. In contrast, BD-RIS refers to a novel and general framework for RIS where its scattering matrix is not limited to be diagonal (hence, the ``beyond-diagonal'' terminology) and consequently, all entries of $\mathbf{\Theta}$ can potentially help shaping waves for much higher manipulation flexibility. This physically means that BD-RIS can artificially engineer and reconfigure coupling across elements of the surface thanks to inter-element reconfigurable components which allow waves absorbed by one element to flow through other elements. Consequently, BD-RIS opens the door to more general and versatile intelligent surfaces that subsumes existing RIS architectures as special cases. In this tutorial, we share all the secret sauce to model, design, and optimize BD-RIS and make BD-RIS transformative in many different applications. Topics discussed include physics-consistent and multi-port network-aided modeling; transmitting, reflecting, hybrid, and multi-sector mode analysis; reciprocal and non-reciprocal architecture designs and optimal performance-complexity Pareto frontier of BD-RIS; signal processing, optimization, and channel estimation for BD-RIS; hardware impairments (discrete-value impedance and admittance, lossy interconnections and components, wideband effects, mutual coupling) of BD-RIS; benefits and applications of BD-RIS in communications, sensing, power transfer.

翻译：本文由发明者撰写，是关于超对角可重构智能表面（BD-RIS）的首篇教程，旨在为读者提供理解、欣赏并贡献于这一新兴颠覆性技术所需的基础知识与核心工具。传统的（对角型）可重构智能表面（D-RIS）以对角散射矩阵$\mathbf{\Theta}$为特征，其波束调控灵活性极为有限。相比之下，BD-RIS提出了一种新颖且通用的RIS框架，其散射矩阵不再局限于对角形式（故称“超对角”），因此$\mathbf{\Theta}$的所有元素均可用于调控电磁波，从而大幅提升调控自由度。其物理意义在于，BD-RIS通过单元间可重构元件，能够人为设计与重构表面单元间的耦合效应，使得被某一单元吸收的电磁波可经由其他单元传输。因此，BD-RIS开启了构建更通用、更灵活智能表面的可能性，并将现有RIS架构纳入其特例范畴。本教程将全面分享BD-RIS建模、设计与优化的核心技术，推动BD-RIS在众多应用领域实现变革性发展。涵盖主题包括：基于物理一致性与多端口网络的建模方法；透射、反射、混合及多扇区工作模式分析；互易与非互易架构设计及BD-RIS性能-复杂度帕累托前沿；面向BD-RIS的信号处理、优化与信道估计技术；BD-RIS硬件损伤（离散值阻抗与导纳、有损互连与元件、宽带效应、互耦）分析；以及BD-RIS在通信、感知与无线能量传输中的优势与应用前景。