On the Impact of Phase Errors in Phase-Dependent Amplitudes of Near-Field RISs

This paper investigates mutual coupling between phase-dependent amplitudes (PDAs) and designed phase shifts within pixels of near-field (NF) reconfigurable intelligent surfaces (RISs) in the presence of phase errors (PEs). In contrast to existing research that treats phase shifts with errors (PSEs) and the PDAs separately, we introduce a remaining power (RP) metric to quantify the proportion of power preserved in the signals reflected by the RIS, and we prove its asymptotic convergence to theoretical values by leveraging extended Glivenko-Cantelli theorem. Then, the RP of signals passing through RIS pixels is jointly examined under combined phase and amplitude uncertainties. In addition, we propose four pixel reflection models to capture practical conditions, and we derive approximate polynomial upper bounds for the RP with error terms by applying Taylor expansion. Furthermore, based on Friis transmission formula and projected aperture, we propose a general NF channel model that incorporates the coupling between the PSEs and the PDAs. By using Cauchy-Bunyakovsky-Schwarz inequality and Riemann sums, we derive a closed-form upper bound on spectral efficiency, and the bound becomes tighter as the pixel area decreases. We reveal that as the RIS phase shifts approach the ends of their range, the RP under independent and identically distributed PEs is smaller than that under fully correlated PEs, whereas this relationship reverses when the phase shifts are near the middle of their range. Neglecting the PEs in the PDAs leads to an overestimation of the RIS performance gain, explaining the discrepancies between theoretical and measured results.