Revisiting the Fraunhofer and Fresnel Boundaries for Phased Array Antennas

Abstract

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 $d^{\mathrm{F}}=2D^2 \sin^2(θ)/λ$ where $D$ represents the largest dimension of the antenna, $λ$ is the wavelength and $θ$ denotes the observation angle. We show that for phased arrays, $d^{\mathrm{F}}$ experiences a fourfold increase (i.e., $d^{\mathrm{F}}=8D^2 \sin^2(θ)/λ$) provided that $|θ-\fracπ{2}|>θ^F$ (which holds for most practical scenarios), where $θ^F$ is a small angle whose value depends on the number of array elements, and for the case $|θ-\fracπ{2}|\leqθ^F$, we have $d^{\mathrm{F}}\in[2D^2/λ,8D^2\cos^2(θ^F)/λ]$, where the precise value is obtained according to some square polynomial function $\widetilde{F}(θ)$. Besides, we also prove that the Fresnel distance for phased array antennas is given by $d^{\mathrm{N}}=1.75 \sqrt{{D^3}/λ}$ which is $\sqrt{8}$ times greater than the corresponding distance for a conventional single-element antenna with the same dimension.

Figures (5)

• Figure 1: Franhofer and Fresnel characterized regions for a phased array antenna (dashed lines), versus a center-fed single-element antenna having the same dimension (dotted lines).
• Figure 2: Antenna with diameter $D$. (a): UE located at an arbitrary distance. (b): UE located at Fraunhofer distance.
• Figure 3: $N$-element ULA antenna with diameter $D$ for two scenarios regarding the position of the transmitter.
• Figure 4: Fraunhofer distance per wavelength ($d^{\mathrm{F}}/\lambda$) versus observation angle $\theta$ for $N$-element ULA with half-wavelength inter-element spacing (i.e., $D=\frac{(N-1) \lambda}{2}$). The dashed lines relate the Fraunhofer distance $d^{\mathrm{F0}}$ for the case of a single-element antenna whose diameter is the same as the corresponding antenna array diameter where $d^{\mathrm{F0}}=2D^2\sin^2(\theta)/\lambda$.
• Figure 5: Fresnel distance function per wavelength ($d^{\mathrm{N}}/\lambda$) versus observation angle $\theta$ for $N$-element ULA with half-wavelength inter-element spacing. The dashed lines relate the Fresnel distance $d^{\mathrm{N0}}$ for the case of a single-element antenna having an equal diameter to that of the corresponding array antenna.

Theorems & Definitions (5)

• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof