On an optimum line source for maximum directivity

An exact solution is derived for the line source that maximizes a generalizationDαof the directivity in any given directionθy{0}for arbitrarily prescribed values of the aperture lengthl, the edge behavior exponentα, and the constraining superdirectivity ratioγα. The quantityDαrepresents exactly directivity forα = 1and approximately for any other value ofα > -1. The solution is expressed in its natural form as a series of spheroidal functionsψα n(c,η)of orderα. The most directive pattern space factor possible for an (N + 1)- term series expansion is shown to beF(c cos θy) = ∑min{n=0}max{N} aα nψα n(c, cos θy)and the resulting aperture distribution to beA 2y/l = 1-[2y/l]² α ∑min{n=0}max{N} {aα n}{να n(c)} ψα n c, 2y/l , whereaα 0is an arbitrary (real) normalizing constant and the otheraα nmust satisfy the following set ofNlinear equations:∑min{n=1}max{N} xₚₙ(μ,Dα)aα n = -xₚ₀aα 0 , p = 1,2,3,...,Nin whichXₚₙ(μ,Dα) = 2ψα n(c, cos θy0) ψα p(c, cos θy0)(1 - cos² θy0)α +(μ[γα n(c) - γα] - Dα)Λα n(c) δₚₙ, and where the unknown directivityDαand Lagrange multiplierμ, must be such thatDαis the largest root of the determinantal equation| \{Xₚₙ(μ,Dα)\} | = 0that satisfies the constraining condition∑min{n=0}max{N} aα n²[γα n(c) - γα]Λα n(c) = 0;cisπ l/λ,να n(c), andγα n(c)are characteristic numbers of the spheroidal functions, andΛα n(c)is the normalization factor for the spheroidal functions on the interval (-1,1). By lettingN → solution becomes exact. Numerical results obtained forl = 25λ, whenα = 0andθy0 = 90(broadside), reveal that for every unit increase in maximum directivity the value ofγ₀ - 1must be increased by a factor of about one hundred.