On an optimum line source for maximum directivity

An exact solution is derived for the line source that maximizes a generalizationD_{\alpha}of the directivity in any given direction\theta_{y}{0}for arbitrarily prescribed values of the aperture lengthl, the edge behavior exponent\alpha, and the constraining superdirectivity ratio\gamma_{\alpha}. The quantityD_{\alpha}represents exactly directivity for\alpha = 1and 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 beF(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 beA \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, wherea_{\alpha 0}is an arbitrary (real) normalizing constant and the othera_{\alpha n}must satisfy the following set ofNlinear equations:\sum\min{n=1}\max{N} x_{pn}(\mu,D_{\alpha})a_{\alpha n} = -x_{p0}a_{\alpha 0} , p = 1,2,3,...,Nin whichX_{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 directivityD_{\alpha}and Lagrange multiplier\mu, must be such thatD_{\alpha}is the largest root of the determinantal equation| \{X_{pn}(\mu,D_{\alpha})\} | = 0that 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;cis\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 lettingN \rightarrow \inftythe solution becomes exact. Numerical results obtained forl = 2\lambdaand5\lambda, when\alpha = 0and\theta_{y0} = 90\deg(broadside), reveal that for every unit increase in maximum directivity the value of\gamma_{0} - 1must be increased by a factor of about one hundred.