On the Maximum End-Fire Directivity of Compact Antenna Arrays Based on Electrical Dipoles and Huygens Sources

Abstract

This article investigates the superdirectivity limits of end-fire linear arrays based on closely spaced radiating elements. First, directivity upper bounds have been derived analytically in the case of Huygens-source- and electrical-dipole-based arrays using spherical wave expansion (SWE). The fundamental bounds are derived when the interelement spacing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> tends to 0 and as a function of the number of the elements ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> ) composing the array. Furthermore, the complex excitation coefficients associated with end-fire arrays of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> infinitesimal Huygens sources and electrical dipoles are synthesized as a function of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> to achieve maximum directivity. For this purpose, synthesis procedures based on SWE and array theory are used. When <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> tends to 0, the numerical results are in excellent agreement with the proposed upper bounds. The maximum superdirectivity approaches a value of P <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{2}+$ </tex-math></inline-formula> 2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P^{2} + P - 1/2$ </tex-math></inline-formula> , respectively, in the case of Huygens sources and electrical dipoles. A numerical method to estimate the antenna gain, when the arrays are optimized in terms of directivity, is also provided. The theoretical results are then successfully validated through electromagnetic simulations in the case of half-wavelength-dipole-based arrays as a function of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> . Three prototypes are designed and experimentally characterized as well to demonstrate the proposed analysis.