Reconfigurable intelligent surfaces (RISs) provide an interface between the electromagnetic world of the wireless propagation environment and the digital world of information science. Simple yet sufficiently accurate path loss models for RISs are an important basis for theoretical analysis and optimization of RIS-assisted wireless communication systems. In this paper, we refine our previously proposed free-space path loss model for RISs to make it simpler, more applicable, and easier to use. In the proposed path loss model, the impact of the radiation patterns of the antennas and unit cells of the RIS is formulated in terms of an angle-dependent loss factor. The refined model gives more accurate estimates of the path loss of RISs comprised of unit cells with a deep sub-wavelength size. The free-space path loss model of the sub-channel provided by a single unit cell is also explicitly provided. In addition, two fabricated RISs, which are designed to operate in the millimeter-wave (mmWave) band, are utilized to carry out a measurement campaign in order to characterize and validate the proposed path loss model for RIS-assisted wireless communications. The measurement results corroborate the proposed analytical model. The proposed refined path loss model for RISs reveals that the reflecting capability of a single unit cell is proportional to its physical aperture and to an angle-dependent factor. In particular, the far-field beamforming gain provided by an RIS is mainly determined by the total area of the surface and by the angles of incidence and reflection.
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