Network Oriented Dyadic Time Domain Green's Function for a Sequentially Excited Infinite Planar Array of Dipoles in Free Space

During the past few years we have performed an effective wideband analysis for characterizing the electrodynamic behavior of phased array antennas, infinite periodic structures, frequency selective surfaces and related applications, with emphasis on gaining physical insight into the phenomenology of short-pulse radiation. The present contribution shows the current status of our network-oriented dyadic time domain Green's function (TD-GF) for a planar array of sequentially excited dipoles that constitutes a prototype study of sequentially short-pulsed radiation by infinite periodic arrays. The dispersive effects of the TD-Floquet waves (FW) and the consequences of a TE and TM field decomposition are discussed in details. In this network formulation, the equivalent TD transmission line (TL) voltages and currents excited by equivalent TD current generators are determined. An expression for the TM voltage TL-GF is provided for the first time in terms of incomplete Lipschitz-Hankel integrals and Bessel functions together with some remarks for its evaluation. As shown in previous publications for the scalar potentials, real-TD physical observables are determined by pairing TD Floquet waves with positive and negative indices. It is shown that the vectorial TD radiated field is reconstructed at any time with the superposition of a few TD-FWs.

thereby to better numerical convergence. Moreover, a network (transmission line) approach has conventionally been utilized in the analysis and design of such structures. These circumstances have motivated the present investigation of a TD network-oriented representation of the vector electromagnetic field radiated by an infinite periodic array of sequentially-pulsed electric dipoles, that constitutes a prototype study of sequentially short-pulsed radiation by infinite periodic arrays.
We have previously investigated canonical TD dipole-excited Green's functions (GF) for infinite [1] and truncated [2] periodic line arrays, and for infinite [3] and semi-infinite [4] periodic planar arrays. The radiated field there has been expressed and parameterized in terms of a few TD Floquet waves (FWs). A generalized frequency domain (FD)-TD transform for linear arrays of dipoles is in [5]. The resulting scalar TD-GF has already been used advantageously to construct a fast TD method of moments algorithm for wideband analysis of infinite periodic structures [6].
A principal feature of the network-oriented approach is that E (TM) and H (TE)-type TD-FW modes can be separated and treated individually. The field is expressed in terms of TD transmission line (TL) Green's functions that obey standard network theory. Therefore, possible infinite planar vertically inhomogeneous media may readily be incorporated into the formalism. Interesting causality issues accompany such E and H mode decompositions: it is found that individually, each E and H mode is noncausal. Causality on the total TD-FW vector mode field is recovered by summing the E and H mode contributions. In order to parameterize the TD-FW behavior, we begin with the solution in the FD, with subsequent inversion to the TD. The formal aspects of the analysis follow the traditional lines in [7], to which we refer frequently. Thus, the transverse th vector FW mode fields are expressed in terms of transverse FW-mode scalar eigenfunctions. Longitudinal field propagation is described by voltage and current TL-GFs, and ; in the TD, these TL-GFs can be evaluated in terms of Bessel functions and incomplete Lipschitz-Hankel integrals (see [8] for a summary of the main features for the scalar case combined with a preliminary analysis for the vectorial case). Numerical examples of radiation from infinite planar arrays of dipoles with short-pulse band-limited excitation demonstrate the accuracy of the TD-FW algorithm, and illustrate the rapid convergence of the (TD-FW)-based field representation since only a few FW terms are required for describing the off-surface field radiated by the planar array. Results for the case of an array simultaneously excited (nonphased) have already been used in a combined 0018-926X/$25.00 © 2007 IEEE Fig. 1. Geometry of an infinite planar periodic array of dipoles. d , d : interelement spacing along x and y, respectively. !=c = k: phase gradient of the excitation (i.e., the wavefront) along the direction 1 (see [3]). v = c=: phase speed along 1 .
(TD-FW)-FDTD algorithm, shown in [9], for the analysis of periodic arrays of complex scatterers.
The results of this paper lay the foundation to include layered media in the TD-FW analysis to widen the TD-FW applications already shown in [6] and [9].
The nonphased case (all array dipoles simultaneously excited) of the TD analysis shown in this paper is strictly related the TD analysis of wave propagation in closed waveguides (see, for instance, [10]).

II. STATEMENT OF THE PROBLEM
The canonical problem for the network formulation under study consists of the generic infinite periodic array of electric dipoles, shown in Fig. 1, oriented along the constant vector , with periodicities and along the and directions, respectively. The observation point is denoted by , with transverse coordinate . The th dipole source in the periodic array is located at , where , + and . Here, bold face symbols define vector quantities and , , denote unit vectors along , , , respectively. The FW-based modal FD and TD fields due to the array are related by the Fourier transform pair (1) in which can be either a scalar or a vector quantity; a caret denotes time-dependent quantities.
The phased FD and sequentially pulsed TD dipole array currents and , respectively, are given by . Moreover, denotes the ambient wavenumber, (with and representing the ambient permittivity and permeability) denotes the ambient wave speed, and and , with are the interelement phase gradients along and , respectively [3]. In the FD, the composite linear phasing on the array is along the direction , perpendicular to (see Fig. 1), and rotated through the angle with respect the axis. In the TD, this translates into interelement excitation delayed by and along and , respectively; i.e., the sequentially pulsed dipole elements located at are turned on at time . The important nondimensional single parameter (4) combines both phasings and [3]. In (4), is the impressed phase speed along . Here, we treat the case which implies excitation phase speeds (and corresponding projected phase speeds and ) larger than the ambient wavespeed [1], [3]. The nonphased case corresponds to simultaneous excitation of all the dipoles.

III. FD GREEN'S FUNCTION: DECOMPOSITION INTO TE AND TM MODES AND PREPARATION FOR FOURIER INVERSION
The FD electric and magnetic vector fields and at location , excited by the array of time-harmonic transverse dipole sources given in (2), oriented along , are represented a sum of FW modes Each th indexed FW is explicitly separated into its E and H parts The steps detailed in Appendix A lead to the following convenient form for the transverse components and of the modes and where the factor is given by (13) The present analysis is limited to transverse-to-dipoles, the simpler case with dipoles oriented along the axis can be treated analogously. Here, and are the E and H voltages at excited by a delta current generator at (see Fig. 2 with . The subscript " " on denotes the vector component transverse to , and represents the -independent part of the vector dispersion relation in (14). Floquet waves with transverse propagation constants or , with , are propagating or evanescent, respectively, along the -direction (we have assumed a lossless environment).
In (9)- (12) [as well as in (64)-(68)] we have introduced the FD admittance and voltage TL Green's functions (18) shown in Fig. 2, where the longitudinal Floquet wavenumber is (19) The quantity represents the current at produced by a voltage generator at , and it is here used in the derivation in Appendix A. The top Riemann -sheet in Fig. 3 is defined by , and or for or , respectively [7, p. 207], [1], [3]. The characteristic impedances for E and H modes are The -components of the fields can be determined from the transverse components.

IV. TD GREEN'S FUNCTION: FOURIER INVERSION OF THE FD MODAL EXPANSION
The total TD field is obtained as a superposition of the TD-FWs obtained by Fourier-inverting the FD modes in (5) and (6). Accordingly, each TD mode is decomposed into its E and H components Here, each th TD-FW is determined by Fourier inversion of (7) and (8) and is carried out by first analyzing the critical points in the complex -plane and then applying standard analytic Fourier transforms.

1) Physical Branch Points,
: The expressions for the modal admittance and voltage TL Green's functions in (18) contain the longitudinal wavenumber given in (19). Therefore they contain -dependent branch points at shown in Fig. 3 that cause vanish. They are physical because they are related to the usual radiation condition at [3]. Branch cuts are defined by imposing , and they separate the two Riemann sheets where the spectral wave components exponentially decay (the top one, where ) or grow (the bottom one) at .

2) Nonphysical Poles,
: Introducing the frequency shift , the vector in (14) is transformed into (27) and (28) The th E and H mode inversion integrals arising from (9) and (10) have poles at frequencies , where , i.e., for (29) with the sign denoting the pole above/below the real axis (see Fig. 3). Note that the poles move to for the nonphased case . These poles are nonphysical since their contribution violates causality for the individual E and H modes. However, when the th E and H mode contributions are superposed to obtain the total th TD-FW field, the pole singularities cancel out, and the sum in (7) is causal, having only branch point singularities above the -integration path in Fig. 3. The cancellation is demonstrated by first observing that (see Appendix B) Next, the two characteristic impedances in (22), evaluated at , become where we have used the fact that, by definition, the directed wavenumber in (19) for becomes . This implies that for (near the nonphysical poles), Therefore, (9) and (10) for , are such that , which proves the pole cancellation, and restores causality for the sum of the E and H modes in (7).

B. TD Transmission Line Green's Functions
A TD version of the TL Green's functions, schematized in Fig. 2, is obtained by Fourier-inverting the FD-TL Green's functions in (18) (33) in which is given in (19). The integrand in (33) has branch points at , as shown in Fig. 3, and the indentation of the integration path is chosen in accord with the radiation condition at (causality) for any ; therefore the integration path from to is shifted below the branch cuts (see Fig. 3), where or for or in accord with the radiation condition specified in the text after (21) (see also [7, p. 35] where, to ensure the existence of the Fourier transform pair in (1), the variable and therefore the contour of integration in (33) is shifted slightly below the real -axis into ). The dashed area in Fig. 3 represents the side of the branch cut where (see also [5]). Note that, because of the -dependency in (22), the evaluation of is analogous to that of .

1) The Nondispersive Case
: For the nondispersive case , , and the TD voltage and current are evaluated exactly as defines the turn-on time at which the impulsive wavefront reaches the observer at , and is the free space impedance.
2) The Dispersive Case, or : Dispersive, higher order -mode voltages and currents are evaluated as in Appendix C (see also [9] for the nonphased case, and [11]- [13] for similar dispersive problems), yielding the closed form expressions Here, and are the zero-th and first order Bessel functions of the first kind, and is the zeroth-order incomplete Lipschitz-Hankel integral of the first kind (see [14] and references therein for more details). In (36)-(39) The evaluation of is rather complex and it is carried out in this paper for the first time. The main steps of its derivation are shown in Appendix C. Note that all the th TD-TL GF have the same turn-on time , defined by the unit step function .

C. Fourier Inversion of the E-H-Decomposed th FD-FWs
The TD electric and magnetic fields in (25) and (26) where we have used the shifted frequency defined just before (27). After decomposition of (46) into three integrals with distinct dependency, we obtain the expression (48) with TD coefficients where is a function of as shown in (28), and the common factor arises from the mentioned shift of frequency. Note that is a noncausal function since it has been evaluated by applying Cauchy's theorem with residue contributions from the nonphysical poles at in (29), both above and below the integration path which is located slightly below the real axis (Fig. 3). The remaining dyad for the electric field in (42) Note that in (42) and (43) the TD-TL Green's functions defined in (36)-(38) are causal since the signal arrives at at the turn-on time ( ) (see [3] for a physical interpretation of and reference systems in terms of moving coordinates.) The E,H-decomposed TD field expressions in (42) and (43) are noncausal since they are determined by a time convolution between casual TD-TL Green's functions and noncausal dyads that are time-spread around . However, as demonstrated in Section IV-A, causality is recovered by summing the E and H constituents in (25) and (26). This can easily be seen directly in the TD for the magnetic field in free space where the TL Green's functions for the E and H cases coincide: the sum of the E and H components is equivalent to the sum of the TD dyads , which is not spread around Although obtained by conventional Fourier inversion from the frequency domain, the results in (42),(43) are complex for or . Indeed, in this case and, from (19), and thus the expressions (36)-(38) are complex. The phenomenology is analogous to that observed previously for the TD potential radiated by a linear [1] and planar [3] array of dipoles, and is thus addressed as in [1], [3] by , pairing to obtain the "physically observable" real TD-FW field. Noting from (19) that and , while , it follows that in (42) the following properties hold: and , where the asterisk denotes complex conjugation. The property follows from noticing that in (37) is real and from . Therefore, the total electric field can be represented as a sum of "physically observable" real TD-FWs as The terms in the series on the right-hand side of (56) can also be rearranged so as to include only positive (and zero) indexes. Analogous conclusions can be drawn for the magnetic field.

VI. TD FIELD PRODUCED BY BAND-LIMITED SHORT-PULSE DIPOLE EXCITATION
In the previous Sections the dipoles were excited by impulsive currents. In the more realistic case of short-pulse current excitation with band-limited (BL) spectrum , the Dirac delta function in the TD expressions in (2) is replaced by . Therefore the fields radiated by the array, and , is given by the convolution of the waveform with the impulsive fields in (23) and (24). Accordingly, each BL-TD-FW is decomposed into its E and H components given by the time convolution For numerical purposes however it may be convenient to evaluate the TD fields as follows. First, performing and storing the time-convolution between the TD dyads and the BL exciting signal: . Next, the resulting equivalent "dyadic waveform excitation" is time-convolved with the TD-TL Green's functions and . This may be convenient since both and have a short time duration (the dyad is almost like an impulse in the case .) Furthermore, once the convolution is evaluated and stored, it can be re-used for TD field evaluations at various heights from the array plane. The number of -Floquet waves sufficient to reconstruct the total field is proportional to the bandwidth of the exciting waveform . Indeed, one should include all -Floquet waves whose -cutoff frequencies with , 2, are within the bandwidth of , as shown in [3, Sec. IV.C and V]

VII. ALTERNATIVE REPRESENTATION
The representation for the th FW excited by an array of impulsive dipole currents described in the previous sections is given by (42) where the causal TD-TL Green's functions are time-convoluted with the noncausal dyadic expressions . We have already seen that though the E and H FWs are individually noncausal their sum restores causality. However, it is possible to have causal dyadic expressions for individual E and H components if one chooses the path shown in Fig. 4 for the inverse Fourier transform of the th electric field in (7). Indeed, the poles shown in Figs. 3 and 4 are not present in (7) and therefore one can choose either the path in Fig. 3 (as in this paper) or the one in Fig. 4. If one chooses the Fourier inversion path as in Fig. 4, the decomposition into E and H modes leads to a field representation similar to that in (42) with the only difference that now the dyads have causal coefficients Though simpler in appearance, one can easily recognize that due to the hyperbolic functions these coefficients grow exponentially in almost all practical cases. Analytically, the correct numbers are restored by the sum of the E and H modes but round-off errors may appear in the calculations.

VIII. A NOTE ON THE EVALUATION OF THE FUNCTION IN (39)
The complex exponent in (39) may give rise to extremely large numbers of the order of that may cause overflow or round-off errors when either high -order FWs are needed (wide band excitation waveforms) or for observation points far from the array plane (large ). The arrangement and grouping shown in (37) with (39) limit this effect since the exponential growth is canceled by the term . This is shown by using the asymptotic expansion of in [15,Eq. (3)] (see also [14]), valid for , that when inserted in (39) leads to (63) where the exponentially growing terms are canceled analytically. The above formula is also used to avoid the numerical evaluation of the function when resulting in a fast evaluation when only a few terms are retained.
Furthermore, one can show that the function in (39) has pure real values for any , and thus we suggest to take its real part a priori.

IX. ILLUSTRATIVE EXAMPLES
To check the accuracy of the TD-FW-based network formalism for the pulse-excited planar phased array of dipoles, we have implemented the electric field solution (23), with (25), (36) and (37) convolved with a BL excitation as in (58). The solution is compared here with a reference solution obtained via element-by-element summation over the pulsed radiations from all dipoles [16]. It consists of the natural superposition of different "spatial" contributions, and it is considered an independent solution compared to the spectral TD-FW sum. Agreement of both methods implies that they have been both implemented correctly. The various spherical TD contributions reach the observer at the delayed times , with , and . The element-by-element -sum has been truncated when contributions from far-away elements are negligible, i.e., retaining elements with . Further elements would contribute only to the tail of the TD signal, not shown in the time scale of the following figures. The selected BL excitation is a normalized Rayleigh pulse (i.e., ) [17], with FD spectrum and central radian frequency , which corresponds to a central wavelength that determines the time duration of the pulse (see [3,Fig. 6] where the same pulse has been denoted by the symbol ). In [8] only a result for the magnetic field, which involves the TL-GFs , was shown. Here, we present results for the electric field that have never been shown previously because they are computed by using the more involved TL-GF in (37) whose analytic expression has been shown here for the first time. Consider an array of dipoles oriented along , with periods . In Fig. 5 the exciting waveform is chosen such that so that the the average length of the pulse is eight times the period of the array (see [3] for more details). Results for both nonphased ( ) and phased ( along the direction ) cases are displayed (more appropriately we should say that the dipoles are sequentially excited with a straight wavefront moving along the direction as in Fig. 1.) The electric field is observed at the location , versus time . The exciting TD waveform is centered at , and the signal arrival time to the observer is determined by using (45) and (35) in (42) (for the geometry considered ). It is remarkable that only the dominant TD-FW with is necessary to represent the radiated field at any time . The agreement with the element-by-element reference solution is excellent. As a further check on the numerics, both the TD-FW expansion and the element-by-element reference solution yield a negligible component. The total electric field is obtained by numerically summing its E and H-mode constituents; for the phased case , it has been verified that this sum cancels the higher order small noncausal components, and renders the total signal causal.
In Fig. 6 the exciting waveform is chosen such that so that the average length of the pulse is twice the period of the array. All other parameters are as for the case in Fig. 5. Due to the larger bandwidth of the exciting waveform this time a slightly larger number of TD-FWs (those with ) is needed, though still showing that a limited number of TD-FW constituents is sufficient to reconstruct the field at any time .
It has been verified that in all cases encountered in these numerical examples, the parameters needed to compute the incomplete Lipschitz-Hankel integrals in (39) are such that and thus the asymoptotic form in (63) has been used.

X. CONCLUSIONS
A network-oriented dyadic Green's function has been investigated here for a planar infinite periodic array of sequentially band limited pulse-excited dipoles. Accordingly, the E (TM) and H (TE) mode contributions can be separated and treated individually in a systematic fashion. The thus reduced modal fields are expressed in terms of transmission line (TL) Green's functions that behave according to standard network theory. Therefore, possible infinite-planar transversely homogeneous layers with longitudinal inhomogeneities can be readily incorporated within the formalism (see [18] and [19] for similar problems.) It has been found that individually, each TD-FW E and H mode is noncausal and can be obtained in closed form in terms of a convolution between characteristic noncausal dyadic functions and the causal TL Green's functions. Causality of the total mode field is recovered in the E and H mode sum. The total radiated field can be constructed at any location and at any time within our numerical experiments by retaining only a few TD-FWs. It should also be noted that "physically observable" th TD-FWs are synthesized by , superposition. The attention in this paper has been devoted to a famous network formalism that has been used extensively in the FD [7, Ch. 2 and 3], and we thus felt that an investigation in the TD of the same formalism was necessary to better understand and model short-pulse propagation of the dispersive relation peculiar of radiation and scattering in spatially periodic problems.
The TD field network representation for the case of a truncated semi-infinite planar array of dipoles is not in closed form, and can be treated asymptotically by generalizing the steps in [4] for E and H modes.

APPENDIX A FD MANIPULATION OF H AND E MODES
The electric and magnetic field expressions in (9)-(12) are obtained as follows. First, the fields are expressed in terms of E-mode ( ) and H-mode ( ) scalar Hertz potentials and , respectively, as in [7, p. 444]. The Hertz potentials are related to scalar functions and as shown in [7, p. 445]. From a transmission line analysis along the axis, with eigenvalues evaluated in the cross section transverse to , one should represent the scalar potentials and in terms of a -modal expansion (see [7, pp. 196-198, and p. 446]). Thus, and are expressed in terms of scalar transverse eigenfunctions , in which the transverse wavenumber, given in (14), corresponds to the th FW, as can be inferred from the plane wave case treated in [7, p. 251]. One should note that for the infinite array in free space, the transverse wavenumber in [7, p. 251] is the same for E and H modes.
For each FW (each term in the -summation) we use the equivalence , , and , and the scalar Hertz potentials and are rewritten as where the factor is given in (13). These expressions are inserted into the field representations [7, p. 444]. After performing the differentiations on the summands, and the use of the equivalence in the E part, the transverse parts of the electric fields in (7)  Note that , which leads to the desired expression for the th FD-FW in (9) and (10). Analogous treatment is done to the transverse component of the magnetic field in (8) leading to (11) and (12) where . The first term produces a delta Dirac while the second term is evaluated as in [20, p. 1027], leading to (38). An analogous argument is used for in (33). First the shift of frequency leads to (72) The first integral is evaluated in a procedure similar to the one in (71) (see the Tables in [21, p. 248]) while the second is exactly the one evaluated for the potential Green's function in [3,Eq. (24)]. This leads to (36). The evaluation of in (33) carried out in the following is rather more involved. After the same shift of frequency used for the two previous cases, the expression of is rewritten as This special integral has already been evaluated in [11,Eq. (11)] and [13, Eq. (7)] under a more general condition, and here readapted as (75) where (76) with arguments and given in (40)  He is also working as a consultant for CommProve, a wireless telecommunications company. His scientific interests include theoretical and numerical studies of propagation and radiation in quasi crystals, periodic structures, leaky wave antennas. He has a strong interest in computer science and its innovative applications on the applied electromagnetic research field.