On the Path-Loss of Reconfigurable Intelligent Surfaces: An Approach Based on Green’s Theorem Applied to Vector Fields

In this paper, we introduce a physics-consistent analytical characterization of the free-space path-loss of a wireless link in the presence of a reconfigurable intelligent surface. The proposed approach is based on the vector generalization of Green's theorem. The obtained path-loss model can be applied to two-dimensional homogenized metasurfaces, which are made of sub-wavelength scattering elements and that operate either in reflection or transmission mode. The path-loss is formulated in terms of a computable integral that depends on the transmission distances, the polarization of the radio waves, the size of the surface, and the desired surface transformations. Closed-form expressions are obtained in two asymptotic regimes that are representative of far-field and near-field deployments. Based on the proposed approach, the impact of several design parameters and operating regimes is unveiled.