Time domain Green's function for an infinite sequentially excited periodic planar array of dipoles

Periodic arrays of radiating or scattering elements play an important role in phased array antennas, frequency selective surfaces and related applications. To gain an understanding of the sparsely explored time domain (TD) behavior of such structures, we have initiated a systematic investigation of relevant canonical TD dipole-excited Green's functions (GF), which so far include those for infinite and truncated line periodic arrays, parameterized in terms of TD-Floquet waves (FW) and truncation-induced TD-FW-modulated tip diffractions. Such waves on semi-infinite and finite square arrays of dipoles have been investigated in the frequency domain, and shown to be useful in practical array applications. This article extends our TD studies to an infinite periodic sequentially pulsed planar array. Like the predecessor GFs, this canonical prototype is simple enough to admit closed form exact solutions, whose interpretation is discussed phenomenologically aided by asymptotic parameterization in terms of instantaneous frequencies. Preliminary numerical results demonstrate the efficiency of the TD-FW algorithms.


I. Introduction
Periodic arrays of radiating or scattering elements play an important role in phased array antennas, frequency selective surfaces and related applications.
To gain an understanding of the sparsely explored time domain (TD) behavior of such structures, we have initiated a systematic investigation of relevant canonical TD dipole-excited Green's functions (GF), which so far include those for infinite and truncated line periodic arrays [l], [2], parameterized in terms of TD-Floquet waves (FW) and truncation-induced TD-FW-modulated tip diffractions.Such waves on semi-infinite and finite square arrays of dipoles have been investigated in the frequency domain (FD) [3], [4], and shown to be useful in practical array applications [5].The present contribution extends our TD studies to an infinite periodic sequentially pulsed planar array.Like the predecessor GFs [!], [2], this canonical prototype is simple enough to admit closed form exact solutions, whose interpretation is discussed phenomenologically, aided by asymptotic parameterization in terms of instantaneous frequencies.Preliminary numerical results demonstrate the efficiency of the TD-FW algorithms.

Statement of the Problem
The geometry of the planar array of dipoles oriented along the Jo direction and excited by transient currents in free space is shown in Figla.The period of the array is dl and dz in the z1 and zz directions, respectively.The E field component is simply related to the Jo-directed magnetic scalar potential A which shall be used throughout.A caret A tags timedependent quantities; bold face symbols define vector quantities; izl, iz2 and i, denote unit vectors along zl, 22, and z, respectively FD and T4quantities are related by the Fourier transform pair A(w) = IFm A(t)e-jwtdt, A(t) = & J_", A(w)eJwtdw.The phased array FD and TD dipole currents J ( w ) and j ( t ) , respectively, are given by with 7 = 71 i,,+yz i,,, xm, = mdl i,,+ndz i,,, and 6(x'xm,) = 6(ximdl)b(zLndz).In the m, n-dependent element current amplitudes multiplying the delta function in (1) the FD portions wyl& and wyzdz account for an assumed (linear) phase difference between adjacent elements in the xi and xz directions, respectively.Combined in the vector 7, y1 and yz denote interelement phase gradients normalized with respect to w.The TD portion identifies sequentially pulsed dipole elements, with the element at x = x, , turned on at time t,, = The wavenumber k,,p,(w) = kz1,,izl + kzl,,iz2 is given by the two Floquet-type dispersion relations with p , q = 0, fl, f2, ....The vector apq = al,,iZl + ( Y Z , ~~~~, represents the part of kt,pq = w-y + apq that does not depend on w , will be extensively used throughout the formulation.Thus, in the frequency domain, Poisson summation converts the effect of the infinite periodic array of individual phased m, windexed dipole radiations collectively into an infinite superposition of linearly smoothly phased p , q-indexed equivalent planar aperture distributions that furnish the initial conditions for propagating (PFW) and evanescent (EFW) Floquet-type waves.In the TD, the m, n-indexed sequentially pulsed dipoles are converted collectively into smoothly phased, p , q-indexed impulsive source distributions b(t -7 .x') which travel with phase speed y-' (y = 1 7 1 = d-) in the 7 direction.
On the right side of (2), this yields the collective FW-phased plane waves Here, k,"," = kt,pq + kZ& denotes the total FW,, propagation vector, and k,,,(w) = (IC2 -kZl2, -k&,-'/', where IC = w / c , with k the ambient wavenumber and c the ambient wave speed.The square root function is defined so that S'mk,,, 5 0 in the top Riemann sheet, consistent with the radiation condition at p = CO.In (4), Floquet waves with transversedomain propagation constants kt,p, < k or kt,pq > k, with kt,pq = (ki1,, + kzz,p)-1/2, characterize PFW or EFW, respectively, in the z-direction.Owing to the exponential attenuation of EFW, along z, the EFW portion of E&, A,"," converges rapidly away from the array plane and a few terms may suffice for an adequate approximation of the total radiated field.
The same operations applied to the right hand side of ( 2 ) , or direct FD inversion of (4), yields the TD-FW The integrand in (5) contributes only for those real (z;,z!J-values which satisfy T + 7 .(xx') + e-'R(x') = 0 , r = t -7 .x.For the radiating case (y < e-'), this condition defines time dependent "equal delay" ellipses with major axis along the phasing direction 7 (see Fig. la).For the nonradiating case (y < e-l), with Bey, 2 0 and %my, 5 0, the ellipses are replaced by single branch hyperbolas.The integral in ( 5) is evaluated using first the change of coordinates (x'x) '7 = u l y , ( x ' -x ) .( i + x ~) = uzy.For the radiatingcase, the resulting ul-inner integral has been reduced in [ l ] for a line array of dipoles, and the u2 integral is then evaluated via the formula J ! l e-j'" cos [ b m ] / m d u = Jo(d-), leading to the exact expression with b, , = 7 .a,,, TO = yzz, and U ( T ) = 0 or 1 for T < 0 or r > 0, respectively.
The phenomenology and interpretation of ( 6) is directly analogous to that for the line dipole array [l], where we explain the complex-valued TD-field in (6) and define-a p , q-paired "physically observable" TD-FW, yielding the real field A : ; : + AT;; = 2 Re A ; : " .Note that all the zkp, f q contributions arrive simultaneously at a stationary observer.Asymptotic Inversion.
This phase contributes to the inverse Fourier integral through the a:ymptotic local frequencies wpg(r, t ) which satisfy the saddle point condition $$(U) = 0, and parameterize the TD-FW wave dynamics.The solutions are real in the causal domain t > to = 7 .x + r , ( T > TO).As in [l], [2], the pqth TD-FW obtained via FD inversion asymptotics is parameterized by these instantaneous frequencies, and its form agrees with the large argument (for Jo) asymptotic approximation of (6).
VI. Band Limited Pulse Excitation When each dipole in (1) radiates a practically useful band-limited (BL) pulse G(t -7 .xmn), the corresponding band-limited TD-FW 2gwsBL for p , q # 0 can be evaluated by including the pulse spectrum G(w) in the impulsive inversion integral.For wideband (short duration) pulses, G(w) can be considered slowly varying with respect to the phase &U) [7], and can therefore be approximated by its value at the saddle point frequencies wpq,,(t), i = 1,2.The asymptotic BL-TD field A F y is found by multiplying the ordinary asymptotic AFqy by G(wpq,j).
For p = q =, O, the FD-FW is not inverteble by w-asymptotics and is calculated convolving G ( t ) with the TD-FW Akw(t).

Fig. 1 .
Fig. 1. a) Infinite periodic planar array of electric dipoles.Problem-matched coordinates: ( u I , ~)with SI in the direction of the phasing 7.For t > to =(&st arrival time), contributions arrive at the observer simultaneously from time dependent expanding "equal delay" ellipses.b) Radiated field; parameters in Sec.VI.