مرکز منطقه ای اطلاع رساني علوم و فناوري - On an optimum line source for maximum directivity

Abstract :

An exact solution is derived for the line source that maximizes a generalization $D_{\\alpha }$ of the directivity in any given direction $\\theta_{y}{0}$ for arbitrarily prescribed values of the aperture length $l$ , the edge behavior exponent $\\alpha$ , and the constraining superdirectivity ratio $\\gamma _{\\alpha }$ . The quantity $D_{\\alpha }$ represents exactly directivity for $\\alpha = 1$ and approximately for any other value of $\\alpha > -1$ . The solution is expressed in its natural form as a series of spheroidal functions $\\psi_{\\alpha n}(c,\\eta)$ of order $\\alpha$ . The most directive pattern space factor possible for an ( $N + 1$ )- term series expansion is shown to be $F(c \\cos \\theta_{y}) = \\sum \\min{n=0}\\max {N} a_{\\alpha n}\\psi_{\\alpha n}(c, \\cos \\theta_{y})$ and the resulting aperture distribution to be $A \\lgroup frac{2y}{l} rgroup = \\lgroup 1-[frac{2y}{l}]^{2} rgroup^{\\alpha } \\sum \\min{n=0}\\max {N} frac{a_{\\alpha n}}{\\nu_{\\alpha n}(c)} \\psi_{\\alpha n} \\lgroup c, frac{2y}{l} rgroup$ , where $a_{\\alpha 0}$ is an arbitrary (real) normalizing constant and the other $a_{\\alpha n}$ must satisfy the following set of $N$ linear equations: $\\sum \\min{n=1}\\max {N} x_{pn}(\\mu,D_{\\alpha })a_{\\alpha n} = -x_{p0}a_{\\alpha 0} , p = 1,2,3,...,N$ in which $X_{pn}(\\mu,D_{\\alpha }) = 2\\psi_{\\alpha n}(c, \\cos \\theta_{y0}) \\psi_{\\alpha p}(c, \\cos \\theta_{y0})(1 - \\cos^{2} \\theta_{y0})^{\\alpha } +(\\mu[\\gamma _{\\alpha n}(c) - \\gamma _{\\alpha }] - D_{\\alpha })\\Lambda _{\\alpha n}(c) \\delta _{pn}$ , and where the unknown directivity $D_{\\alpha }$ and Lagrange multiplier $\\mu$ , must be such that $D_{\\alpha }$ is the largest root of the determinantal equation $| {X_{pn}(\\mu,D_{\\alpha })} | = 0$ that satisfies the constraining condition $\\sum \\min{n=0}\\max {N} a_{\\alpha n}^{2}[\\gamma _{\\alpha n}(c) - \\gamma _{\\alpha }]\\Lambda _{\\alpha n}(c) = 0$ ; $c$ is $\\pi l/\\lambda$ , $\\nu_{\\alpha n}(c)$ , and $\\gamma _{\\alpha n}(c)$ are characteristic numbers of the spheroidal functions, and $\\Lambda _{\\alpha n}(c)$ is the normalization factor for the spheroidal functions on the interval (-1- ,1). By letting $N \\rightarrow \\infty$ the solution becomes exact. Numerical results obtained for $l = 2\\lambda$ and $5\\lambda$ , when $\\alpha = 0$ and $\\theta_{y0} = 90\\deg$ (broadside), reveal that for every unit increase in maximum directivity the value of $\\gamma _{0} - 1$ must be increased by a factor of about one hundred.