  In this paper, we introduce a physics-based 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 transformation. 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.
