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
T HIS paper represents the third in a series of prototype studies [1], [2] of the time-domain (TD) behavior of sequentially excited periodic dipole array configurations, motivated by similar investigations in the frequency domain (FD) [3]- [6], which have already been applied effectively and efficiently to finite practical array antennas [7]- [10]. Referring to the more detailed introduction in [1] for background, we proceed in Section II to the formulation of the problem in terms of the frequency-domain (FD) and time-domain (TD) fields excited by the individual phased dipole radiators. Via Poisson summation, these discretely spaced individual FD and TD sources are reexpressed collectively in terms of equivalent global periodicity-induced continuous distributions which obey the Floquet wave (FW) dispersion relation and span the entire array surface. The FD-FW and TD-FW wavefields, i.e., the Floquet plane waves, radiated by these FW-modulated aperture distributions are developed in Sections III and IV, respectively, with emphasis in the TD on the new phenomenologies exhibited by the planar array, as well as on similarities with the previously investigated TD-infinite line dipole array [1]. As previously cited in [1], the TD-FW fields are found to be expressible in new exact closed forms which reduce to known results for special choices of the problem parameters. Numerical results in Section V furnish reference data which are used for comparison with considerably more efficient FW-generated fields. Conclusions are presented in Section VI.

A. Element-by-Element Formulation
The geometry of the planar array of dipoles oriented along the direction and excited by transient currents in free space is shown in Fig. 1. The period of the array is and in the and directions, respectively. The vector electric field is simply related to the directed magnetic scalar potential which shall be used throughout. A caret (^) tags time-dependent quantities; boldface symbols define vector quantities; , , and denote unit vectors along , , and , respectively. FD and TD quantities are related by the Fourier transform pair (1) 0018-926X/03$17.00 © 2003 IEEE coordinates. d , d : interelement spacing along x and x , respectively; ! = k: phase gradient of the excitation (i.e., the wavefront) along the direction i ; = =c: "slowness" (normalized wavenumber) along i ; v = c= = 1= : phase speed along i .
The phased array FD and TD dipole currents and , respectively, are given by (2) (3) where is the th dipole location, , denotes the ambient wavenumber, and denotes the ambient wave speed. Moreover, and , with (4) are the interelement phase gradients along and , respectively. Here, is chosen to match the form of the important nondimensional parameter introduced previously [1]. This yields (5) with now denoting the normalized (with respect to ) phase gradient, and the corresponding impressed phase speed, along the array in the direction (see Fig. 1). The FD phasing unit vector is rotated through the angle with respect the axis; this corresponds in the TD to sequentially pulsed dipole elements, with the element at turned on at time . Choosing the normalized form for along as in (5) systematizes subsequent notation and interpretation.
The respective regimes and characterize two distinct TD wave phenomenologies with phase speeds along larger or smaller than the ambient wave speed . Only the practically more important regime is examined in this paper.

B. Collective Formulation
To convert the individual element contributions in (2) into equivalent collective smooth aperture distributions, we use the Poisson sum formula in its most elementary form given by [11, pp. 117] . When applied sequentially to the double infinite series of phased -indexed FD and TD elements in (2), Poisson summation yields (6) The vector wavenumber (7)  (8) which combines the two Floquet-type dispersion relations with , has previously been employed in the FD studies of planar dipole arrays [3]- [5]. The subscript " " on denotes the vector component transverse to , and represents the -independent part of the vector dispersion relation in (7). Thus, in the frequency domain, Poisson summation converts the effect of the infinite periodic array of individual phased -indexeddipole radiators collectively into an infinite superposition of linearly smoothly phased -indexed equivalent planar distributions that furnish the initial conditions for propagating (i.e., radiating) PFW and evanescent (i.e., nonradiating) EFW Floquet-type waves. In the TD, the -indexed sequentially pulsed dipoles are converted collectively into smoothly phased, -indexed impulsive source distributions , which travel with phase speed in the direction, which is the direction of the wavefront shown in Fig. 1.

III. FLOQUET WAVES: FREQUENCY DOMAIN
To obtain for the potential fields radiated by the linearly phased dipole array element currents at an equivalent sum of FW potentials radiated by the smoothly phased FW-modulated aperture distributions, we multiply the FD portion of (6) by the FD element Green's function (11) and perform the integration , for 1 and 2, to generate on the left-hand side (LHS) of (6). Here, denotes the position vector. On the right-hand side (RHS) of (6), this yields the collective FW-phased plane waves (it is the inverse of the transform shown in [12, p. 481]) (12) Here, denotes the total propagation vector, and (13) is the wavenumber along . The square root function in (13) is defined so that on the top Riemann sheet, consistent with the radiation condition at . Furthermore, or for or 0, respectively, in order to satisfy the radiation condition for positive and negative real frequencies. In (12), Floquet waves with transverse propagation constants or , where , characterize PFW or EFW, respectively, along . Note that by phase matching along , , each PFW contributes at the observation point a ray asymptotic field originating at a point on the -plane. The ray emanating from the point lie on a ray with angular displacement from the -axis, and azimuthal displacement from the axis (see Fig. 1), for positive or negative frequencies. For or , and give rise to two different propagation angles. When approaches , the polar angle tends to . Beyond that limit, when , the polar angle becomes complex and the field becomes evanescent along , with , i.e., , defining the th FW cutoff condition. Owing to the exponential attenuation of along , the EFW portion of converges rapidly away from the array plane and a few terms may suffice for an adequate approximation of the total radiated field.

IV. FLOQUET WAVES: TIME DOMAIN
Three distinct approaches are analyzed, each describing different aspects of TD-FWs. The first two lead to exact expressions for TD-FWs, while the third leads to an asymptotic description of the same phenomena.

A. Fourier Inversion From the FD
The TD Floquet Wave is obtained through Fourier inversion from the frequency domain (15) (16) with given in (12). The wavenumber is rewritten as (17) in which we used the frequency shift (18) and the definitions  , respectively, in accord with the definitions for in (13). The positive-negative transition at occurs between the two branch points. The indentation of the integration path in (24) 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. 2), where or for or in accord with the radiation condition specified in the text after (13) (see also [12, p. 35] where, to ensure the existence of the Fourier pair in (1), the variable and therefore the contour of integration in (24) is shifted slightly below the real axis into ). Since for , the integrand in (24) decays exponentially in , the integration contour can be closed by addition of the noncontributing portion ; because no singularities are located within the contour, the integral vanishes by Cauchy's theorem. For , the integration contour can be closed by the noncontributing portion , and is therefore deformable into . Using the relation (demonstrated in Appendix A) in which and are the zeroth-order Hankel functions of the first and second kind, respectively, and combining , leads directly to the closed form exact expression (26) with or 0 for or , respectively. Although obtained by conventional Fourier inversion from the frequency domain, the result in (26) is complex for or since and, from (19), in this case. The phenomenology is directly analogous to that observed previously for the line dipole array [1], and is addressed as in [1] by , pairing to obtain the "physically observable" real TD-FW field. Noting from (8) and (19) that and , it follows that the "physically observable" real TD-FW field is given by For the mode, one has , with and ; i.e., the argument of the Bessel function in (26) vanishes. Since , we have which agrees exactly with the real field radiated by an impulsively excited smooth infinite plane source with phasing specified by .

B. Spatial Synthesis of TD-FWs Via Poisson Summation
Since the FD-series in (6), when applied to , has summands composed of two -dependent functions (see text after (11)), the TD involves a convolution which yields (29). Alternatively, first, one finds that . When this function is time-convolved with the TD portion on the left-hand side of (6), i.e., , followed by , one obtains the field (29) excited by the impulsive th dipole current in (2) which represents a spherical impulsive wavefront radiated by the dipole at at the delayed time . The same operations applied to the right-hand side of (6), or direct FD inversion of (12), yields the TD-FW (30) The argument of the delta function in (30) identifies the two-dimensional (2-D)integral as a Radon slant-stack projection transform [13] (normalized to the unit cell area ). The integrand in (30) contributes only for those real -values which satisfy (31) To understand the implications of this condition, we change coordinates to (32) with (see Fig. 1), which orients the coordinate along the direction of propagation of the traveling impulse excitation (see Figs. 3 and 4). Therefore, the integral in (30) becomes (33) and (31) is written as (34) Fig. 3. Phenomenology matched coordinate system (u ; u ), rotated with respect to (x ; x ), whose transformation is given in (32). u points in the direction i i i = cos i i i + sin i i i of the propagating wavefront. The first signal arrival at the observation point (x; z) originates at the earlier point Successively, for t > t , contributions arrive at the observer simultaneously from points whose locus is a distinct "equal delay" ellipse (see also, Fig. 4).
with and in accord with in (13) (since ; see (7) and (13)). In the -plane, (35) defines -independent "equal delay" curves in the plane, whose shape depends on the parameters , and . We now explore the behavior of the solutions for the FW for various -parameter ranges. Equation (35) describes an ellipse with foci  , with  and  , 2, center  and  axis ratio , as shown in Fig. 4. At the turn on time , the foci coincide and the ellipse reduces to a point at . At later time instants, the ellipse becomes larger, with the focus moving along the direction and the focus moving toward the origin .For the nonphased case , the ellipse degenerates into a circle with center fixed at . When approaching cutoff , the axis ratio tends to infinity, and the two foci as well as the launch point move to . (For , which corresponds to evanescent in the FD, the equal delay contours becomehyperbolas(see(35)),ofwhichonlytheright-handbranch is relevant. Whether TD radiation is now possible under special phase-matched conditionsremains to be exploredfurther.) The -integral in (33) with . The two real solutions of (37) for coincide at time which, at the observer, corresponds to the causal (wavefront) arrival time of a signal due to a smoothly phased infinite line current along , located at , with launch point in the plane at . In the moving coordinate system along the excitation wavefront, represents the signal arrival delay that the moving observer encounters with respect to the exciting current impulse located at (Fig. 3). For , these solutions separate according to (37) and move toward and (Fig. 4) in which is defined in (22). This result, obtained by applying the Poisson summation formula directly to the TD element-by-element field representation, is coincident with that in (26) obtained from the direct Fourier inversion of the FD-FW. The remarks after (26), concerning the "physically observable" TD-FW, apply here as well.

C. Asymptotic Inversion From the FD 1) Local Frequencies and Wavenumbers:
The behavior of the high-frequency asymptotic evaluation of the FD inversion integral in (1) (41) provides additional insight and parameterizes the TD-FW dispersion process. The manipulations here are 2-D generalizations of those carried out in [1] for the line dipole array, and the principal steps are given below. Referring to the last expression for in (12), accounts for the slowly varying amplitude terms in the integrand. The phase is given by (42) with and defined in (7) and (13) (16) and (23), respectively. Positive and negative frequencies are denoted by and , respectively. This expression agrees with that obtained via the operation performed directly on the time-dependent phase in (26), after replacing the Bessel function by its large argument asymptotic approximation.
The two instantaneous frequencies of the in (43) at a given point and a given instant ( in the moving reference system; see (31)) are real in the causal domain , increase with mode indexes but decrease with time , and approach their observer-independent cutoff frequency when , (defined by ) (see Fig. 7 (44)); thus, the wavenumbers , with , as in the text after (14).
2) The Nondimensional Estimator: In order to assess the accuracy of the asymptotics in (50), we use the nondimensional estimator defined as [15] (54) which combines the various critical problem parameters and variables. We have noted here that . The range of validity of the asymptotic solution is expressed thereby through the condition , with the limits given by . As a function of , this eliminates the near-wavefront regime and the late-time regime , for both of which . However, the validity of the asymptotic result is extended to when the dipoles are excited by a band-limited waveform (see Section V-B).  on the array plane; these points all lie on the -instantaneous "equal delay" ellipse defined in (31), as shown in Fig. 5. We first recall the definition of in (11), and thus (see Fig. 5) which, when inserted together with (56) into (31), leads to (57) This identity is verified from (45)-(47) and (43), and it represents the equation of the "equal delay" ellipse in terms of instantaneous wavenumbers. In summary, at each time , all TD-FWs propagate toward the observer along -dependent cones, from directions , , with the same group velocity . These TD-FW emerge earlier from points located on the "equal delay" ellipse at time .

D. The Total Physically Observable Radiated Field
The total "physically observable" field radiated by the array is expressed as a sum of , paired TD-FWs, (58) where the th TD-FW is given by (26) or (40). The terms in the series on the RHS of (58) can also be rearranged so as to include only positive (and zero) , indexes.

V. BAND-LIMITED PULSE EXCITATION
We now analyze the effects of physically realizable band-limited (BL) pulsed dipole excitation on the field radiated by the planar array. The pulse excitation function is represented as with spectrum . Accordingly, the factor multiplying in (2) becomes for the FD dipole currents and for the TD dipole currents.
The total BL response of the planar array is then obtained by convolving the total TD impulse response in (58) with the BL signal , yielding The BL Floquet-modulated signal due to the planar array can be calculated either by convolution with the exact TD-FW or by inversion of FD asymptotics.

A. Convolution With the Exact FW
Here, the exact FW field in (26) or (40) is used in (60). Again, the , pairing defines "physisically observable" BL-TD-FW, yielding the real field , that also demonstrates (59) for BL excitation.

B. Band-Limited Asymptotics
Avoiding the convolution in (60), the th BL field can be calculated as the inverse Fourier transform of . Therefore, for or , using the high-frequency asymptotics in Section IV-C, can be evaluated approximately by including the pulse spectrum in the inversion integral (41). For these short pulses, can be considered slowly varying with respect to the phase in the integrand of (41) [2], [16], and can therefore be approximated by its value at the saddle point frequencies , , 2. Thus, near the wavefronts and . For , which is not amenable to -domain saddle point asymptotics (see Section IV-C), the pulsed response is calculated by the convolution in (60). Although the impulsively excited asymptotic wavefields in Section IV-C are valid only for early times close to (behind) the wavefronts, convolution with a waveform having a band-limited spectrum may enlarge the range of validity to later observation times behind the wavefront. For or , the relevant fields are those with in the signal bandwidth.

C. Illustrative Examples
To check the accuracy of the TD-FW-based BL Green's function algorithm for the impulse-excited planar phased dipole array, we have implemented two numerical examples (see Figs. 8 and 10). The TD asymptotic solution (59), with (61), is compared there with a reference solution obtained via element-by-element summation over the pulsed radiation from all dipoles, i.e., The -series has been truncated when contributions from the far elements are negligible, i.e., when . The chosen BL excitation is a normalized Rayleigh pulse (i.e., ) [17], with FD spectrum and central radian frequency , shown in Fig. 6. To explain the results in Figs. 8 and 10, we shall utilize plots of the TD-FW instantaneous frequency dispersions shown in Fig. 7. The relevant spectral range of that contributes significantly to the total radiated field at the observer can be assessed from Fig. 7 (14)), and . At turn-on , all instantaneous frequencies with or . It is also seen that for , defined in (44). The index , 2 tags negative/positive -frequencies, respectively. The RHS of  . Fig. 8 shows the field radiated by the array with parameters , and , observed at , . The central radian frequency is chosen as , with central wavelength ; this implies from Fig. 7 that the for only those TD-FWs with lie in the region where is nonvanishing, and therefore furnish the dominant contributions. Indeed, in Fig. 8, excellent agreement with the reference solution has been obtained by retaining only the asymptotic terms , thereby demonstrating good convergence of the TD-FW field representation. Since the median wavelength is larger than the interelement spacings , the main feature of the pulse shape in Fig. 8, is contributed by the integrated excitation waveform of Fig. 6 and represents the , which is evaluated by the convolution in (60) [see text after (63)]. The tail after the wavefront is due to the higher order FWs with , which oscillate at their distinct local instantaneous frequencies , , 2, and thereby form the noted interference pattern.
The quality of the asymptotic results in Fig. 8 up to (and beyond) is assessed by the behavior of the nondimensional estimators in (54), as shown in Fig. 9 for . The estimator for is not included since it is not amenable to saddle point asymptotics as noted in Section IV-C. In the plotted range, for all except near turn-on where . Near , the local instantaneous frequencies tend to infinity, but due to the bandlimited excitation frequency spectrum in Fig. 6, TD-FWs with or are not excited there. Fig. 10 shows plots for an infinite planar array under the same conditions as in Fig. 8, except that the central radian frequency is now . This changes the relevant spectral range of to The quality of the asymptotics for the pqth TD-FW is assessed by how well each satisfies the condition E 1, with respect to an arbitrarily set reference level. , as can be seen from Fig. 7. Accordingly, it is noted that excellent agreement with the reference solution has been obtained also in this case by retaining the relevant asymptotic terms , thereby again demonstrating good convergence of the TD-FW field representation. At these shorter wavelengths, features of individual element arrivals become more pronounced but are well synthesized by a correspondingly larger number of TD-FWs.

VI. CONCLUSION
In this paper we have extended previous studies of periodicity-induced impulsive Green's functions for phased arrays of dipoles from the line dipole array (infinite [1] and truncated [2]) to an infinite planar array. From the detailed analyses in [1] and [2], we have gained substantial insight into relevant techniques for quantifying and interpreting TD periodicity-induced global phenomena in terms of TD-FW wavefields. The new features introduced by the assembly of an infinite periodic array in terms of phased line dipole arrays in Section IV-B therefore bear strong notational resemblance to the FD planar array sector geometry in [5], and strong phenomenological resemblance to that of the TD infinite line dipole array in [1]. To highlight these analogies, we have used phrasings similar to those in [1] and [2] for similar concepts and methodologies. As in [1], the present prototype problem is sufficiently simple to yield the exact closed form TD solutions in (28) for Floquet-type dispersive wave phenomena, which are dispersive TD-FW radiating plane waves for ; the nonradiating case will be presented separately. The most interesting and new finding here is the excitation mechanism (for ) of the TD-FWs along "equal delay" ellipses in the array plane (see (35) and Figs. 3 and 4), and the appealing physical interpretations that follow from it (see Fig. 5). The next prototype studies will be of the TD-GFs for a semiinfinite planar [18], and thereafter for a plane-sectoral, phased dipole array. This will furnish the tools for analyzing actual finite planar arrays under short pulse conditions. The practical utility of FW-based dipole GFs for finite planar phased arrays has already been demonstrated in the frequency domain [3]- [10], and application of its TD counterpart will be guided by these FD studies.

APPENDIX A DETAILS PERTAINING TO (25)
We perform the change of variable in the integral in (25). Along the paths and in Fig. 2 . After recalling that , (67) has the two -indexed solu-tions in (43). To sort out the correct solution for , we substitute (43) into the original (66) [recalling (13)] to obtain (68) Real values of in (43) are obtained only for . Since, , the sign of depends on through the sign of the second term inside the parentheses in (43), i.e., . Since , we have . Thus, both the LHS and RHS of (68) have the same sign for and opposite sign for , for , 2. This means that for both and are real solutions of (66), while neither is a solution for negative .