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In this paper, we introduce a physics-based analytical characterization ofthe free-space path-loss of a wireless link in the presence of a reconfigurableintelligent surface. The proposed approach is based on the vectorgeneralization of Green's theorem. The obtained path-loss model can be appliedto two-dimensional homogenized metasurfaces, which are made of sub-wavelengthscattering elements and that operate either in reflection or transmission mode.The path-loss is formulated in terms of a computable integral that depends onthe transmission distances, the polarization of the radio waves, the size ofthe surface, and the desired surface transformation. Closed-form expressionsare obtained in two asymptotic regimes that are representative of far-field andnear-field deployments. Based on the proposed approach, the impact of severaldesign parameters and operating regimes is unveiled.

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