Extremely Large Beyond-Diagonal RIS: Low-Rank Modal Optimization for Near-Field Communications

Beyond-diagonal reconfigurable intelligent surfaces (BD-RIS) achieve their best performance when fully connected, at the price of an optimization and hardware burden that grows quadratically, and per iteration cubically, with the number of elements. Extremely large surfaces make this burden prohibitive, while their sheer aperture places both the base station and the users in the radiative near field, where far-field design tools break down. This paper introduces the extremely large BD-RIS (XL-BD-RIS) concept and shows that near-field geometry is precisely what makes fully connected performance affordable at scale. Modeling the cascade with the free-space Green function, we prove that the aperture fields live in a low-dimensional subspace spanned by the spherical-wave responses of the terminal positions, and we design a compact unitary modal matrix on this subspace, built from localization information alone, that provably attains the fully connected optimum with a number of reconfigurable entries set by the geometry and independent of the panel size. A weighted-MMSE Riemannian algorithm optimizes the beamformers and the modal matrix with monotone convergence at panel-size-independent cost. Numerical results show that a $24\times24$-element panel reaches the fully connected optimum with about two hundred entries instead of three hundred thousand. A mismatched DFT beamspace pays a sixty-fold entry penalty rooted in the beam spread of spherical wavefronts, while the classical block-wise architecture delivers strictly lower rates at any matched entry budget.