A Tutorial on Beyond-Diagonal Reconfigurable Intelligent Surfaces: Modeling, Architectures, System Design and Optimization, and Applications

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 Θ (commonly denoted as phase shift matrix in the literature). Since a very small percentage of the entries of that matrix, namely only the phases of its diagonal entries (in its passive form), are tunable, 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 Θ can potentially help shaping waves for much higher manipulation flexibility. In its passive form, Θ satisfies the unitary property ΘHΘ = I (for energy conservation in lossless ideal surfaces) and be either symmetric Θ = ΘTor asymmetric Θ ̸= ΘThence leading to reciprocal or non-reciprocal BD-RIS. Such scattering matrix properties correspondingly translate into novel passive (lossless) and reciprocal/non-reciprocal circuitry where each RIS element is not only connected to its own tunable impedance but also to other elements through reconfigurable components. 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. This offers an extra degree of freedom for reconfigurable surfaces that provides new opportunities and flexibility for manipulating, modulating, processing, and computing signals and waves in the analog domain. 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. We also point out challenges of BD-RIS which trigger directions that are promising for future research.