Characterization of Continuous Electromagnetic Manifolds via Calculus of Variations

We present a novel calculus of variations (CoV)-based framework for the characterizing of, and beamforming over, continuous electromagnetic manifolds of arbitrary multiple-input multiple-output (MIMO) array geometries. Building upon the discrete moment-matrix formulation of the state-of-the-art (SotA), the proposed framework simultaneously overcomes three of its fundamental limitations: (i) the point-source approximation error incurred by the near-field radiation operator; (ii) the confinement of the beamforming space to the N-dimensional subspace dictated by the hardware port count; and (iii) the generalization to arbitrary array geometries. To this end, each mesh element is modeled as a two-dimensional planar patch whose spatially averaged Green's function is evaluated via Gauss-Legendre (GL) quadrature, yielding a strictly more accurate near-field representation at negligible additional cost, while a continuous feeding function w(p) in L^2(S_T), introduced as the infinite-dimensional limit of the N-port network, lifts the optimization onto a hardware-decoupled current subspace of dimension K >> N. As an application example, we employ the proposed CoV-based framework to derive closed-form optimal beamformers for both unconstrained field-strength maximization, and a near-field pattern synthesis under a power density (PD) and region constraints, establishing their exact analogy to the discrete and generalized matched filters. Full-wave MATLAB Antenna Toolbox validation confirms consistent near-field accuracy gains over the SotA baseline for both linear and planar geometries at comparable computational cost.