Time domain Green's function for a phased semi-infinite periodic line array of dipoles

We have begun a systematic investigation of the time domain (TD) behavior of Green's functions which are relevant for the characterization of truncated planar periodic arrays, with emphasis on the TD Floquet waves (FW) in the propagating and evanescent parameter regimes. Our principal aim is to understand the TD wave physics and phenomenologies on simple prototypes, invoking various methods that synthesize the solution from different perspectives. The first prototype, an infinite phased periodic line array of axial dipole radiators arranged along the axis of a cylindrical (/spl rho/,z) coordinate system has been studied in Felsen and Capolino (1998). This prototype is a basic building block for transverse truncated plane arrays formed by a finite number of infinite line sources, which has so far been studied only in the frequency domain (Capolino et al. 1998). Truncation aIong the axis of a phased line array of dipoles is explored in this paper for the frequency domain (FD) and TD. This modification of the infinite line dipole array leads to a new set of FD and TD truncation phenomena.


I. INTRODUCTION
We have begun a systematic investigation of the time domain (TD) behavior of Green's functions which are relevant for the characterization of truncated planar periodic arrays, with emphasis on the TD Floquet waves (FW) in the propagating and evanescent parameter regimes. Our principal aim is to understand the TD wave physics and phenomenologies on simple prototypes, invoking various methods that synthesize the solution from different perspectives. The first prototype, an infinite phased periodic line array of axial dipole radiators arranged along the z-axis of a cylindrical (p, z ) coordinate system ( Fig.1) has been studied in [l], and is reviewed in Sec. 111. This prototype is a basic building block for transversely truncated plane arrays formed by a finite number of infinite line sources, which has so far been studied only in the frequency domain (FD) [2]. Truncation along the axis of a phased line array of dipoles is explored in this paper for the FD and TD. This modification of the infinite line dipole array leads to a new set of FD and TI) truncation phenomena.

STATEMENT OF THE PROBLEM
The geometry of the truncated semi-infinite linear array of dipoles with period d , oriented along the t direction and excited by transient currents in free space, is shown in Fig.1. The E field is related to the z-directed magnetic scalar potential A which shall be used throughout. The FD (exp(jwt)) and TD currents J(w) and j ( t ) , respectively, due to the dipoles, are related via a Fourier transform pair, with the caret A denoting time-dependent functions. In the n-tagged element current amplitudes multiplying the delta function in (l), the FD portion wy,d accounts for an assumed (linear) phase difference between adjacent elements, with 7= denoting the interelement phase gradient normalized with respect to w. The TD portion identifies sequentially pulsed dipoles, with the element at z' = n d turned on at time t,, = -/,nd. In this paper, we only consider the radiating case v,@) = 7 ; l > c, where c is the ambient propagation wave speed.
The truncated Poisson summation formula plays an essential role in converting the collective radiation from the truncated periodic array of individual phased dipole radiators into an infinite superposition of truncated linearly smoothly phased equivalent line source radiators. For the series of spherical wave fields A,, due to the n-indexed individual dipole radiators, the formula is 5 An = -i-Ao + O0 A, , A', = hm A(u)~2*q"du, which converts the n-series fields into a q-series whose summands A, are the truncated Fourier transforms of the smoothed-out spherical wave function A ( v ) , sampled at integer multiples of 2n. Since n and therefore U in (2) are tied only to a spatial coordinate (see(l)), A,, and A ( v ) can be either functions of w or t . 0-7803-5639-X/99/$10.00 01999 EEE.

FD-AND TD-FLOQUET WAVES FOR THE INFINITE ARRAY
Results pertaining to the infinite prototype phased (sequentially excited) periodic line array of dipoles in [l] are now summarized. In this case the n-sums in (l), (2), and Fig.l+extend from -w to +ea, the truncation term Ao/2 in (2) is omitted, and the A, integral, now from v = -cm to f w , becomes the complete Fourier transform (FT). The "infinite version" of (2) is written ( 2~0 ) . Fiequency domain FW. AppLwg (2m) to the radiated spherical FD fields from each dipole [l] yields for A, at r = (p,z) a tabulated FT which is the FD-FW, smoothly phased line source field The FW, wavenumber along t, k,,(w), represents the Floquet-type dispersion relation. The square root defining the radial wavenumber k is defined so that Sm k, 5 0 on the top Riemann sheet, consistent with the ra%ation condition at p = 00. In (3), FWs with z-domain propagation constants lkr,l < Ikl characterize radially propagating FWs (PFW) while those with lkz,l > Ik/ characterize radially evanescent FWs (EFW). Each PFW contributes at r a ray asymptotic field lying on a ray cone with semiangle P,(w) = cos-'(k,,/k) (for both positive and negative frequencies)(see Fig.1 for q = 0). When Ikz,l > lkl, the  This implies that at the turn-on time to pertaining to the fixed observation point P, the wavefront amves from a direction coincident with that of the FD-FWo wave vector rCt w = (krO,kpO). At times t > t o (i.e. T > TO), the wavefront moves beyond P to the location shown in Fig. 2b, which is tagged on the z zi(t) = z -7 ; '

IV. FD-AND TD-FW F O R THE SEMI-INFINITE ARRAY
With the infinite array results in mind, we procede to the semi-infinite case. Frequency domain truncated FW. Applying (2) t s t h e FD radiated fields A,,?, w ) leads to the truncated integral representation A,(r, U ) which is asymp totically evaluated via the following steps. First, the spherical wave function A(r, v, w ) is expressed as a wavenumber spectral integral over axially phased cylindrical waves. Interchanging the order of integration, the v-integral is calculated exactly as a wavenumber spectral pole. The remaining spectral integral is deformed via the saddle point method into the steepest descent path (SDP) and where the FW A Y arises from the residue of the pole encountered in the deformation, and the p t h spherical diffracted fields A,d due to the truncation are obtained from a uniform saddle point evaluation along the SDP. U is the Heaviside unit function; 4" is the shadow boundary of the truncated FW, which, for propagating FWs, coincides with the FW, propagation angle p, = cos-'(k,,/k) (see  (7) is the same as that for the infinite array, except that its domain of existence is the region a d < a,"". The disconthuity of the PFW, at the shadow boundary p d = a,"" is compensated by the diffracted field At.
Time domain truncated FW. The TD-FW field 2, is merely a truncated version of the infinite array field in (5) since it is generated by the v 2 0 contributions in the v-integral of (2) instead of the -m < v < 00 range in (5). + Accordingly, z g,(r,t) = 5 A:rW(r,t)U(zj(t)), i=i where the fundamental TD-FW 2;rw(r, t) and zi(t) are the same as in (5),(6) and the U function expresses the truncation at the v = 0 end point. The space-time phenomenology has already been explained in connection with (5), except that due to the truncation, at later times (Fig.Zc) than those in Figs.2a,b, the observer receives only the contribution arriving from zi(t). The (dashed) contribution from G(t) is off the array and is therefore not excited. Since z i ( t ) = 0 corresponds to Bi(t) = p d , the FW existence condition U(z:(t)) = U(pi(t)p d ) , i = 1,2, can be parameterized in terms of instantaneous propagation angles.

4
Asymptotic inuersion. The TD field $,(r, t ) can also be found through Fourier inversion of the high-frequency FD field in (7). Due to their nondispersive behavior, the q = 0 FW and diffracted field in (7) can be inverted exactly, while the dispersive case IpI # 0 is evaluated asymptotically. The FW field AfW(r, w ) exp(jwt) has a high frequency asymptotic phase $(CO) = kz,z+kp,p-wt (see (3),(4)). The uniform diffracted field A,d(r, w ) exp(jwt) in (7) has a more complicated phase which simpliies, however, close to the SB at P d = Pq where it can be shown to have the same phase as the FW [2]. This common phase 4 contributes to the inverse FT through the two asymptotic instantaneous frequencies w,,<(r, t), i = 1,2, which satisfy the saddle point condition $$(U) = 0 and are real in the radiating domain T > TO (t > to). Thus the asymptotics yields A, N ~~= , [~[~" U(pi(t)pa) + A;,+], in which A:fw is the same as in (5) and the TD diffracted field A$ is a spherical contribution centered at the array truncation. The instantaneous shadow boundaries @f(t) = cos-l(kzq/k),,(t) (see (7)) and the FW propagation angle &,i = psg are found to be independent of q and equal to the above pi(t). Thus, all ,$:fir, t) fields at a given instant t reach an observer at r from the same direction Pi(t). The asymptotic diffracted field $$ provides the required continuity across the instantaneous FW shadow boundaries and is negligible elsewhere (see also (8)). When each ditole in (1) radiates a band-limited pulse &(t -yznd), the band-limited TD field AqBL is found by multiplying each i = l, 2 constituent by the pulse spectrum G(w) evaluated at the instantaneous frequency w,+. Thus, only those TD-FW, fields with wp,i in the &(t) signal bandwidth contribute to the actual field. In Fig. 3, this TD asymp totic solution is compared with a reference solution obtained by an element by element summation over the pulsed radiation from all dipoles, for an array of 10 elements with interelement phasing 7= = 0.2/c (Po = 78"). Both left and right truncation effects have been considered, treating the actual array as the difference between two semi-infinite arrays. In Fig.3, we have chosen a Rayleigh puke

&(t) = sie[j/(j + ~~t / 4 )~] ,
with w~ = 4nc/d, d being the interelement spacing. The field is plotted vs. normalized time t/T, with T = d/c. The observer is located at two points z = 0 (a), and z = -5d (b) with the same radial distance p = 10d. In both cases, the included asymptotic terms are I Q ( 5 3, thereby demonstrating the good convergence property of the TD-FW field representation.