Pulsed radiation by a phased semi-infinite periodic planar array of dipoles

To gain an understanding of the sparsely explored time domain (TD) behavior of periodic arrays of radiating or scattering elements (phased array antennas, frequency selective surfaces and related applications), 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, as well as for infinite planar periodic arrays. Such Green's functions have been parameterized in terms of TD-Floquet waves (FW) of cylindrical and planar type, and truncation-induced TD-FW-modulated diffractions. Such waves on semi-infinite and finite square arrays of dipoles have been investigated in the frequency domain (FD), and shown to be useful in practical array applications. The present contribution extends our TD studies to an infinite periodic sequentially pulsed semi-infinite planar array. The phenomenology associated with truncated TD-FW and truncation-induced diffraction is explained in terms of instantaneous frequencies aided by asymptotic parameterization. Preliminary numerical results demonstrate the efficiency of the TD-FW algorithms.


I. Introduction
To gain an understanding of the sparsely explored time domain (TD) behavior of periodic arrays of radiating or scattering elements (phased array antennas, frequency selective surfaces and related applications), we have initiated a systematic investigation of relevant canonical TD dipoleexcited Green's functions (GF), which so far include those for infinite and truncated line periodic arrays [l], [2], as well as for infinite planar periodic arrays [3].Such Green's functions have been parameterized in terms of TD-Floquet waves (FW) of cylindrical [l] and planar type [3], and truncation-induced TD-FW-modulated diffractions [Z].
Such waves on semi-infinite and finite square arrays of dipoles have been investigated in the frequency domain (FD) [4], and shown to be useful in practical array applications [5].The present contribution extends our T D studies to an infinite periodic sequentially pulsed semi-infinite planar array.The phenomenology associated with truncated TD-FW and truncation-induced diffraction is explained in terms of instantaneous frequencies aided by asymptotic parameterization.Preliminary numerical results demonstrate the efficiency of the TD-FW algorithms.

Statement of the P r o b l e m
The geometry of the semi-infinite planar array of dipoles oriented along the Jo direction and excited by transient currents in free space is shown in Fig. 1a.The period of the array is dz and d, in the 2 and z directions, respectively.The E field component is simply related to the Jo-directed magnetic scalar potential A which shall be used throughout.A caret ~ tags timedependent quantities; bold face symbols define vector quantities; i,, i, and i, denote unit vectors along x, y, and z , respectively.FD and T D quantities are related by the Fourier transform pair A(w) = J -m m A(t)e-Jwtdt, A(t) = & II",A(w)eJwtdw.The phased array FD and TD dipole currents J ( w ) and j ( t ) , respectively, are given by In the m, n-dependent element current amplitudes multiplying the delta function in (1) the FD portions wqzx'/c and wqzz'/c account for an assumed (linear) phase difference between adjacent elements in the z and z directions, respectively, and qz/c and qz/c denote interelement phase gradients normalized with respect to w.The T D portion identifies sequentially pulsed dipole elements, with the element at (x', z') = (nd,, md,) turned on at time t, , , = (qZndz + qzmd,)/c.111.FD and TD Floquet Waves for the Infinite A r r a y A):Frequency D o m a i n FW.Applying the infinite Poisson summation formula to the doubly-infinite sum over the radiation by each dipole we obtain the total field expressed as Atot(r,w) = E&, A,"," where theFW Arqw(r,w) = e-jk.F.W"/(2jd,d2ky~c,,,).
Here, qqw = k,,i, + kypqiy + kzgir denotes the total FW,, propagation vector, and The inversion of the FD-FW is written as Arqw(r,t) = F(w) exp(-j$(w))dw in which F ( w ) accounts for the slowly varying amplitude terms and $(w and kz, functions of w, accounts for all the w phase terms 7 : the exponent.For p = q = 0, the phase is linearly dependent on w and the inverse Fourier transform is evaluated as with up, = c(q.a,+ qzaq)/(lq'), W,, = [G,", + a,",c2/(1q 2 ) ] ' / 2 , are real in the causal domain t > to = (7.To parameterize the truncated FW phenomenologies in terms of instantaneous frequencies and wavenumbers, we access the time domain through Fourier inversion of Atot(r,w) in (5) which has been solution to early observation times near the wavefronts.Each of the AFqw(r,w) and its corresponding diffracted field A$r,w) has a particular arrival time, near which the FD asymptotics will be most accurate.Again we need to distinguish between the dispersive q # 0 and the nondispersive q = 0 cases.The asymptotic inversion procedure is analogous to that in [Z], but differs in detail.We shall only list the results.The "quasi nondispersive" q = 0 (a, = 0) term is decomposed based on the relations kPo(w) = w / ~( l -9 2 ) ' / ~ and B ( w ) = l / Z + j C,"=-,[w/c(1-q:)1/2(cos $-cos$oo)-ap]-'.Therefore, Fourier inversion for the terms in (6) corresponding to the p = 0 contributions can be done in closed form, obtained by high-frequency asymptotics This restricts the validity of the truncated TD with rd = m p / c and t d = 9xz/c + 7 d .All the other pterms (with q = 0) yield truncated TD-FW via moving shadow boundaries $ p : ( ~~, , ~( t ) )  and the diffracted fields can be approximated in the neighborhood of $ B ( ~~o , , ( t ) ) as in [2] for the truncated line array.For q # 0, TD inversion from the high-frequency result in (6) is based on the stationary (saddle) points w,, defined by (d$d/dw)I,: = 0, [l] of the composite phase q d ( w ) = k,,p + k,,aw t , which are real in the diffracted-field causd domainr' > Td (t > t d ) .The two solutiodfh (8) identify the local instantaneous frequencies of oscillation of the q-th diffracted wave at a given point r and a given instant 7'.The corresponding instantaneous wavenumbers kq,i(t) = w;,j(t)/c, kzq,i(t) E %w:,i(t)/C + a,, and kpq,i(t) = (k&(t) -kZ,,i(t))-'/2 are all real for t > td, and kpq,j(t) -+ 0 f o r t -+ CO.Standard asymptotic8 leads to i = 1,2.It can be shown that wqd,cutoff P wt(b + 00) 5 ug:rf E wPq,i(t + XI), arid thus, since t d 2 to, at a certain time t = t&? 2 t d , the y-th diffracted and pq-th FW local instantaneous frequencies are equal, i.e., ut($:) = wpq(t::).Furthermore, it can be shown that cos q5p,(wpq(ti:)) = cos q5, which means that at t = t : : the py-th moving SB intercepts the stationary observer at r; there the q-th TD diffracted field has a transitional behavior that compensates for the truncation of the py-th TD-FW, and restores total field continuity.
V. Band Limited Pulse Excitation When each dipole in (1) radiates a practically useful band-limited (BL) pulse G[t-(qrz+qzz)/c], the corresponding band-limited TD-FW xFqwiBL forp or y # 0 and TD-diffracted field A$BL for y # 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 phases G(w) and @(U), and can therefore be approximated by its value at the saddle point frequencies wp,,,i(t), and w,&(t) for FWs and diffracted fields, respectively.The asymptotic BL-TD fields and are found by multiplying the ordinary asymptotic A ; $ and A$ by G(w,,,+) and G(w&J, respectively.The FD-FWoo and y = 0 diffracted field are not inverteble by w-asymptotics and are calculated by convolving 6(t) with the TD-FW 2tow(t) and x f ( t ) in (7), respectively.Preliminary numerical experiments have been carried out to test the accuracy of the asymptotic solutions for 2FFBL, and to compare the results with a reference solution obtained by an element-by-element summation over the pulsed BL radiation from all dipoles, i.
respectively, is the Heaviside step function; and to = (qzz + qzz + m y ) / .is the turn on time[3].For p # 0 or q # 0, the phase G(w) contributes to the inverse Fourier integral through the asymptotic local frequencies wpq(r, t ) which satisfy the saddle paint condition ( d $ / d ~) ~~~ = 0, and parameterize the TD-FW wave dynamics.The solutions[3]