Revisiting the Fraunhofer and Fresnel Boundaries for Phased Array Antennas

This paper presents the characterization of near-field propagation regions for phased array antennas, with a particular focus on the propagation boundaries defined by Fraunhofer and Fresnel distances. These distances, which serve as critical boundaries for understanding signal propagation behavior, have been extensively studied and characterized in the literature for single-element antennas. However, the direct application of these results to phased arrays, a common practice in the field, is argued to be invalid and non-exact. This work calls for a deeper understanding of near-field propagation to accurately characterize such boundaries around phased array antennas. More specifically, for a single-element antenna, the Fraunhofer distance is $d^{\mathrm{F}}=2D^2 \sin^2(\theta)/\lambda$ where $D$ represents the largest dimension of the antenna, $\lambda$ is the wavelength and $\theta$ denotes the observation angle. We show that for phased arrays, $d^{\mathrm{F}}$ experiences a fourfold increase (i.e., $d^{\mathrm{F}}=8D^2 \sin^2(\theta)/\lambda$) provided that $|\theta-\frac{\pi}{2}|>\theta^F$ (which holds for most practical scenarios), where $\theta^F$ is a small angle whose value depends on the number of array elements, and for the case $|\theta-\frac{\pi}{2}|\leq\theta^F$, we have $d^{\mathrm{F}}\in[2D^2/\lambda,8D^2\cos^2(\theta^F)/\lambda]$, where the precise value is obtained according to some square polynomial function $\widetilde{F}(\theta)$. Besides, we also prove that the Fresnel distance for phased array antennas is given by $d^{\mathrm{N}}=1.75 \sqrt{{D^3}/{\lambda}}$ which is $\sqrt{8}$ times greater than the corresponding distance for a conventional single-element antenna with the same dimension.