Green's function for a planar phased sectoral array of dipoles: uniform high-frequency solution

A truncated Floquet wave (TFW) formulation has been presented previously for the potential of a right-angle sectorial planar phased array of directed dipoles. In this paper, we perform an asymptotic evaluation of the spectral wavenumber integrals, uniformly valid in the vicinity of the vertex and of the shadow boundary of any FW or edge-diffracted wave species.


I. INTRODUCTION
In the first part of this paper [l], a truncated Floquet wave (TFW formulation has directed dipoles Fig 1 1 (because repeated reference will be made to equations, figures, etc., in [l\, suck references are preceded by 1.). In (l.ll), we have shown a decomposition of the array potential A ( i ) i that can be more explicitly rearranged
The (TFWI-(edge-diffracted wave) compensation mechanijm away from the vertex is formalized by the uniform asymptotic evaluation of A,!' in (3 which exhibits FW-modulated edge-diffracted waves with respect to the array ge along the Eaxis. The uniform asymptotiu is performed by the Van der Waerden (Vdd) method as in [1.2], leading to been presented for the potential of a right-angle sectoral planar p h ased array of iias follows,

U. UNIFORM VERTEX DIFFRACTION
Near the vertex, the 2,-ed e and z2-edge FW-shadow boundary transitions interact with the vertex-induced S#Cs centered on the zl-axis and I -axis respectively, due to the truncation of the corresponding edge diffracted fieids (Pigs. 1, 1.3). The confluence of these four SB transitions near the vertex defines the asymptotics pertainin to vertex diffraction, which is obtained by the VdW method applied to the doubfe integral A"(?). Within the VdW method, the asymptotic evaluation of integrals characterized by specific arrangements of critical parameters (saddle points and singularities) is addressed by mapping the given integrand (both phase and amplitude) onto the simplest canonical integrand that accommodates the relevant c o n f y t i o n of critical points. The reduction to the canonical form is accomplished by se ectively adding and subtracting "regularizing" portions of the integrand, which can involve an arbitrary number of poles; for simplicity here, we develop expressions only for regularization of the p,q) pole which is closest to the saddle point (SP   Table, which we now explain. Starting in Table A with the spectral amplitude in the integrand of (1.1), we refer to the discussion in Sec.l.111 on parameterizing the uniform sectoral array asympto~ics in terns of the interaction, via the vertex, of the uniform semi-infinite array (SIA) solutions pertaining to each edge. These phenomenologies are identified in the first column, beginning sequentially with edge 1. The. second column identifies the relevant spectral amplitude terms. Note that D, integrations, respectively (see (1.2),(1.3)). The third column in Table A gives tlie VdW regularization which isolates, via the Wi&-jd, (kZ1 ;k,,, )]-I, Wz,p+-jd&z -ki2,g)]-1 functions, the effect of the (p,q) poles under consideralion. Whde the pole extraction is direct for edge 1, the corresponding treatment of edge 2 is more involved because it is preconditioned by the presence of edge 1. Altogether, the regularization of S(k,,,k,,) leads to the nine individual terms in the 2nd row of Table A, which are rearranged in the second column of Table B into four groups S. (k,,,kSz), i=0,..,3. Each group addresses uniform transition through a critical siatial domain listed in the first column. Note that these regularizing decompositions are ezact for the ropagating FW spectrum (evanescent effects are neglected here) and they provi& the formal structure for subsequent uniform asymptotics. The lowest-order locally uniform asymptotic evaluation is performed next.

i * r ' kr -(AFz1-k~J)2 + ~~( k z i -S i , ) ( k z a -L ) + ~P , z -L ) ' ) (4)
with A=kr/[2%n24z) B=kr/(2~sin2q5,), H=kpcosP,cosal~(~sin2~2sjn'~~ (see Fig. 1.1). Thls quadratic form is the lowest order approximation to the exact phase over a limited r e p n Q centered at the SP in the (kJl, k,) domain, which is "suffici&ly large to uniformly accommodate the poles in the various specbral terms Si(ktl, kx2) in Table B. Since the A," integral is the only one that has two poles (one m each variable), its asymptotic evaluation is carried out first. The other integrals can be asymptotically evaluated by reduction of A: . First, we chan5e variables to (=a(kZ1 -6r1,), q=o(kr2 -6 J2,) to transform the quadratic phase in '

(4) into rs' . r'n kr-(('+h(q+qa), with t u = H / ( A B ) " 2 = c o t~l w t~z =~~l~~~
Next, the regular slowly varying amplitude function (k )-I in the i n k and (see Table B for

IV. NUMERICAL RESULTS
Numerical tests have been performed on a "large" square array of dipoles in ordes to validate the asymptotic solution in (7). The electric ficld has leen derived from the potential by using a dyadic spectral form in the integrand of (1.1), which introduces a simple additional factor into (7). An element-by-element summation over the contribution from each dipole serves as a reference. The 10 x 10 element test array has equi-amplitude dipoles oriented along c i d 1 with interclement phasing 71=7a=0 and period d,+=f.YA, A=, %/ %.
In a spherical coordinate system (r,e,d) with origin at the center of the array and polar axis perpendicular to the array plane, the array radiates a broadside ( 0 4 ) main beam but, due to the large interelement distances [Z] F. Capolino, S. Maci "Simplified, clod-form expressions for computing the generdiaed Frsnel integral and lheir application (0 vertex diffraction" Microm. and 9 1 . Tech. Le::en, Vol. 9, N.l, pp. 32-37, May 1995.