Asymptotic high-frequency Green's function for a large rectangular planar periodic phased array of dipoles with weakly tapered excitation in two dimensions

This paper deals with the derivation and physical interpretation of a uniform high-frequency representation of the Green's function for a large planar rectangular phased array of dipoles with weakly varying excitation. Thereby, our earlier published results, valid for equiamplitude excitation, and those for tapered illumination in one dimension are extended to tapering along both dimensions, including dipole amplitudes tending to zero at the edges. As previously, the field obtained by direct summation over the contributions from the individual radiators is restructured into a double spectral integral whose high-frequency asymptotic reduction yields a series of propagating and evanescent Floquet waves (FWs) together with corresponding FW-modulated diffracted fields, which arise from FW scattering at the array edges and vertexes. To accommodate the weak amplitude tapering, new generalized periodicity-modulated edge and vertex "slope" transition functions are introduced, accompanied by a systematic procedure for their numerical evaluation. Special attention is given to the analysis and physical interpretation of the complex vertex diffracted ray fields. A sample calculation is included to demonstrate the accuracy of the asymptotic algorithm. The resulting array Green's function forms the basic building block for the full-wave analysis of planar weakly amplitude-tapered phased array antennas, and for the description of electromagnetic radiation and scattering from weakly amplitude-tapered rectangular periodic structures

Abstract-This paper deals with the derivation and physical interpretation of a uniform high-frequency representation of the Green's function for a large planar rectangular phased array of dipoles with weakly varying excitation.Thereby, our earlier published results, valid for equiamplitude excitation, and those for tapered illumination in one dimension are extended to tapering along both dimensions, including dipole amplitudes tending to zero at the edges.As previously, the field obtained by direct summation over the contributions from the individual radiators is restructured into a double spectral integral whose high-frequency asymptotic reduction yields a series of propagating and evanescent Floquet waves (FWs) together with corresponding FW-modulated diffracted fields, which arise from FW scattering at the array edges and vertexes.To accommodate the weak amplitude tapering, new generalized periodicity-modulated edge and vertex "slope" transition functions are introduced, accompanied by a systematic procedure for their numerical evaluation.Special attention is given to the analysis and physical interpretation of the complex vertex diffracted ray fields.A sample calculation is included to demonstrate the accuracy of the asymptotic algorithm.The resulting array Green's function forms the basic building block for the full-wave analysis of planar weakly amplitude-tapered phased array antennas, and for the description of electromagnetic radiation and scattering from weakly amplitude-tapered rectangular periodic structures.

I. INTRODUCTION
I N A systematic sequence of previous studies [1]- [5], we have explored methods to reduce the often prohibitive numerical effort that accompanies an element-by-element full-wave analysis of large truncated plane periodic phased arrays.In our approach, the element-by-element array Green's function (AGF) formed by a planar periodic phased array of dipoles is restructured into an alternative "collective" formulation that represents the field radiated from the elementary Manuscript received October 6, 2003; revised July 19, 2004.The work of F. Mariottini, F. Capolino, and S. Maci was supported by the European Space Agency, The Netherlands.The work of L. B. Felsen was supported in part by Polytechnic University, Brooklyn, NY.
L. B. Felsen is with the Department of Aerospace and Mechanical Engineering and Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215 USA (e-mail: lfelsen@bu.edu).
Digital Object Identifier 10.1109/TAP.2004.841321dipoles in terms of the radiation from a superposition of continuous truncated Floquet wave (FW)-matched source distributions extending over the entire aperture of the array, with inclusion of other truncation-induced Floquet-modulated wave types.Since the FW series exhibits excellent convergence properties when the observation point is located far enough away from the array surface to render evanescent FWs and the corresponding diffracted fields negligible, this representation has been found to be more efficient than the direct summation of the spatial contributions from each element of the array, especially when each FW aperture distribution is treated asymptotically.The previous investigations have dealt with semi-infinite planar dipole arrays [1], [2] and right-angle sectoral planar phased arrays of dipoles [3], [4] with equiamplitude excitation, as well as strip arrays with one direction-tapered illumination [5].
Our new extension in this paper addresses planar rectangular arrays with slowly varying excitation profiles that are separable in the two orthogonal variables.The corresponding asymptotic treatment of each FW aperture distribution leads to locally amplitude-modulated truncated FWs, plus FW-excited diffracted contributions from the edges and vertexes of the array, which can be cast as previously in the format of a periodicity-adapted geometrical theory of diffraction (GTD).For the canonical finite planar phased array of dipoles with tapered excitation, the AGF is constructed by plane wave spectral decomposition in the two-dimensional complex wavenumber domain corresponding to the array-plane coordinates.This is followed by manipulations and contour deformations that prepare the integrands for subsequent efficient and physically incisive asymptotics parameterized by critical spectral points, i.e., saddle points and points at which the spectral amplitude function exhibits highly peaked characteristics very similar to those of poles (denoted later on as "quasi-poles").Different species of spectral quasi-poles define the various species of propagating and evanescent locally amplitude-modulated FWs.The other critical points in the double spectral integral define the asymptotic behavior of the edge and vertex diffracted rays; the confluence of these critical points in transition regions determines a variety of locally uniform new transition functions for truncated edge diffracted and vertex diffracted waves.
Arrays with tapered excitation have also been analyzed in [6], combining "global" FWs and diffracted fields developed in [1]- [3] with a numerical technique based on the discrete Fourier transform (DFT) that expands a general tapering in terms of its equiamplitude phased harmonics.Although the numerical DFT procedure can be applied to a wider class of taperings, our results here are in analytical closed form and thus can be rapidly computed for the considered class of applications [7]- [9].

II. STATEMENT OF THE PROBLEM
Consider a large rectangular periodic array of linearly phased dipoles located in the plane (Fig. 1).The array dimensions are and along and , respectively; the interelement spatial periods along the and directions are given by and ; and the interelement phase gradients of the excitation by and , respectively.An dependence is implied and suppressed.All dipoles are oriented along the unit vector (boldface symbols denote vector quantities, and boldface symbols with a caret denote unit vectors).Superimposed upon that background is a -dependent amplitude-separable tapering function , sampled at the dipole locations (1) with denoting the dipole current amplitude, and denoting the location of -th dipole.Without compromising practical utility, we assume real and positive in the domain , and zero elsewhere; here, and .The electromagnetic vector field at any observation point can be derived from the magnetic vector potential (2) where , which is synthesized by summing over the individual dipole radiations.Our goal is the efficient and phenomenologically insightful evaluation of the high-frequency near and far fields radiated by this complex physical configuration.

A. Spectral Domain Analysis: Floquet Waves
Our analysis is carried out in the spectral wavenumber domain.Accordingly, we employ the spectral Fourier representation of the scalar free space Green's function with ; this Fourier transform is given by [10], where and the upper and lower signs apply to and , respectively.Because of symmetry, from here on, we shall deal with only.On the top Riemann sheet of the complex -plane, for real , we define for and for .The location of branch points and branch cuts with respect to the real-axis integration path in the plane is found by introducing small losses , which are eventually removed [1], [3], with respect to the wavelength.The inset shows the local coordinate system associated with a vertex.Note that is related to r by r = + z for i = 1; 2, respectively.[4].Substituting the spectral Fourier representation of the scalar free-space Green's function into (2) and interchanging the sequence of the -summation and the spectral integration operations, leads to (see Section II for notation and definitions) The -sum in ( 4) is manipulated via the truncated Poisson sum formula [4], [11] into (5) where (6) are the Fourier-transformed tapering functions.The FW wavenumbers, which define the FW dispersion, are given by (7) with .Since the tapering function has a wavenumber bandwidth narrow with respect to the spectral period, the aliasing of the spectra is small (outside the visible range, the oscillations are exponentially damped to zero).

B. Spectral Plane Inspection and Critical Points: High-Frequency Parameterizations
The integrand of (3) consists of two contributions that characterize the asymptotic behavior of the integral: the modulating spectral array factor (SAF) , and the spectral Green's function (GF) .Henceforth, we assume (legitimately for realistic tapering functions on large arrays) that varies slowly with respect to the wavelength (see Fig. 1).For this type of weak variation, and since is positive in the domain , its spectrum in (6) is localized around .Since the function is synthesized by superposition of spectrum contributions translated to with or for or , respectively, this function appears concentrated around all the spectral points.For example, considering a 30 30 element dipole array with , i.e., , Fig. 2 .On the basis of this decomposition, the behavior of the exact solution is parameterized by the critical points in the spectral double integral, which also govern the strategies for the asymptotic approximations.The critical points in (3) are the following.

i)
. These points, where the SAF in Fig. 2(a) exhibits peaks, describe the same phenomenology and localization property as the spectral poles for the semi-infinite array [2].Accordingly, we shall refer to these points as quasi-poles.
ii) .This is a first-order two-dimensional stationary phase point (SPP) of the individual-element spectral GF in Fig. 2(c), which satisfies for , with nonvanishing Hessian determinant, .This SPP is related via to the spherical coordinate angles and which locate the observation point with respect to the and axes, respectively (see Fig. 1).
The critical points in i) and ii) play a different role in the asymptotic evaluation of the integral in (3) with respect to their dependence on the observer location.For observation points close to the array surface and far from the array edges and vertexes, the oscillations of the individual element spec-tral GF are rather slow; therefore, the periodic, highly peaked spectral function acts like a passband periodic filter for that GF.As shall be demonstrated in the next section, the outcome of this sampling is the FW summation corresponding to the infinite array, with adiabatic modulation due to the tapering function.When the observation point is in the far zone of the array (i.e., at distance greater than 2 ), the oscillations around the stationary phase point are so rapid as to localize the contributions from the periodic function to its values at .Consequently, the far field is proportional to .The recognition that this expression is the ordinary "array factor" associated with the pattern multiplication law [12] has motivated our designation of as the spectral array factor.In the intermediate zone (Fresnel zone) where the distance of the observation point from the array and the size of the array may be comparable, both types of critical points and their interaction have to be taken into account in the asymptotic evaluation of the integral.In this intermediate regime, Fig. 2(b) depicts the SAF in one periodicity cell, and Fig. 2(c) displays the corresponding phase modulation associated with the GF.The two plots exhibit comparable variation, and it is not evident how to identify a spectral filtering that quantifies the influence of one factor on the other.We therefore refine the asymptotics and introduce additional critical points to account for transitions between the various wave types.iii) and .These points satisfy the one-variable stationary phase conditions and respectively, with .The asymptotic wavefields corresponding to these two saddle points tag diffracted fields from edge 2 (located at ) and edge 1 (at ), respectively.For a rectangular array, diffraction from the other two edges can be found similarly by including the appropriate phase reference in the second term on the right-hand side of (5).
The asymptotic contributions pertaining to the three types of critical points are examined next.Though the critical points are the same as those in [3], the localization process is conducted differently here by introducing the quasi-poles and a spatial-spectral expansion [see (17) and (36)] of the spectral integrand in (3) for the end-point evaluations (edge-and vertex-diffracted waves).

IV. UNIFORM HIGH-FREQUENCY SOLUTIONS
Because of the slowly varying assumption for the tapering function , adiabatic methods can be applied, based on perturbation about const.The case of equiamplitude excitation, which has been treated in [3] and [4], is briefly summarized here for convenience.

A. Equiamplitude Excitation
1) Finite array: Since , the series in ( 4) can be evaluated in closed form: ; here, exhibits singularities at that are cancelled by the zeros of the numerator in (see Fig. 2).Because the functions are recognized as the analytic continuation of the spectral array factor in the complex -plane, the SAF exhibits peaks at the values , i.e., at the FW wavenumbers in (7).
2) Infinite array: When the number of array elements in both dimensions increases, the peaks of the SAF become correspondingly enhanced and, in the limit of an infinite number of dipoles, tend to two separate sequences of Dirac delta functions (8) with . This reduces the integral in (3) to the periodicity-modulated FW series for the infinite AGF (9) where .
3) Planar sectoral array: When the upper limit in the series ( 4) is extended to infinity one obtains (assuming small losses), . For this case, the asymptotic evaluation of (3) can be performed as in [3] via sequential deformation of the original double integration contour into the complex -plane local steepest descent path (SDP) along the 45 line through the saddle point of the phase in the integrand, with extraction of the residues at intercepted poles.These residues reconstruct the FW series of the infinite AGF with the proper truncation functions.

B. Weakly Amplitude-Modulated FW Contributions and Shadow Boundary Planes
We begin by assuming that the observer is located far from the array edges in terms of wavelength.Therefore the dominant contributions to the field arise from the peaks associated with the quasi-poles.Generalizing the one-dimensional taper analysis in [5], by inserting ( 5) into (3), the contributions from the critical points at are found by expanding the exponent of the integrand in (3) in Taylor series in a neighborhood of (10) with Next, approximating , and retaining only the dominant asymptotic term of the remainder, one finds plus higher order terms of type .Here, , with the branches chosen as for (see Section III-A), is real for propagating FW.Inserting (10) into (3) [and approximating the slowly varying portion of the integrand by its value at ], leads to (11) The two spectral integrals are calculated directly, using the definition in (6), yielding In ( 11) and ( 12), is the th FW for equiamplitude excitation [see ( 9)], which is multiplied in ( 12) by the tapering function evaluated at the footprint of the th FW.Note that the two vertex-truncation terms in (5), and , for both , are not included here because they provide additional contributions from edge and vertex diffractions, which are calculated separately using (10).
The stationary phase evaluation of the Kirchhoff spatial radiation integral associated with each th equivalent FW-matched aperture distribution would provide the same result as in (12) since is the corresponding space domain stationary phase point.We note that since for , the tapering function automatically truncates the FW domain of existence at the shadow boundary (SB) planes defined by the conditions and , which correspond to and , respectively.In the angular space domain, these two conditions become and , where is the transverse-to-observation angle (see Figs. 1 and 3) and (13) Here, and define the shadow boundary planes (SBPs) associated with the two edges that intersect at the vertex ; in Fig. 3, these SBPs are displayed for two edges of the rectangular array.

C. FW-Induced Amplitude-Modulated Diffracted Fields and Shadow Boundary Cones
As noted in Section III-B, the critical points ii) give rise to edge-diffracted field contributions from the four edges of the Fig. 3. (p; q)th FW shadow boundaries for rectangular array.The (p; q)th propagating FW exists inside the four (p; q)th shadow boundaries (SBs), originating at the point (z ; z ) = (z 0yk =k ; z 0yk =k ).
The shadow boundary planes relevant to edge 1 and 2 are defined by = , and = in (13), respectively.rectangular array.The procedure to obtain these contributions is based on that presented in [5] and is summarized in Appendix I.
Denoting by and the derivative of and with respect to their arguments, the diffracted field from edge 1 is given by (14) where the amplitude tapering function in the direction is evaluated at the footprint of the th edge-diffracted field along edge .Since for and , the domain of existence of the th edge-diffracted field is bounded automatically.The truncation at corresponds to , which becomes in the angular space domain , where is the conical angle (see Fig. 1).Equation characterizes the shadow boundary cone (SBC ) which, centered at the vertex (Fig. 4), confines the diffracted field.The SBC , centered at the vertex (Fig. 4), has the same aperture as the diffraction cone [analogous considerations apply to the -indexed independent -edge diffracted rays, whose domain of existence is confined by ], where is the Heaviside unit step functions.In (14), denotes the vector wavenumber of the th diffracted field, and lies on the surface of a diffraction cone centered at on the array edge, forming an angle with the -axis.Moreover, in (14) is the standard uniform theory of diffraction (UTD) transition function [13] (15) (16) is the slope UTD transition function (see Appendix I and [14]).It can be shown that when the nondimensional parameter is large, and both tend to unity.The diffraction cone of the propagating z -edge (z -edge) diffracted ray originates at the q-dependent (p-dependent) point z on edge 1 (z on edge 2).The SBC that truncates the domain of existence of the qth edge diffracted field has the same aperture angle as the diffraction cone and is centered at the vertex.The diffracted rays from edges 1 and 2 are associated with the wavevectors k = ẑ k + ẑ k cos + ŷk sin and k = ẑ k + ẑ k cos + ŷk sin , respectively.These rays are truncated at the SBCs = and = with respect to the axes along edges 1 and 2, respectively.These two SBCs intersect along the intersection line of the two FW-SBs, which coincides with the direction of the propagation vector k = ẑ k + ẑ k + ŷk of the pq-FW.
The vertex diffracted rays propagate with spherical spreading factor along the wavevector k = rk.
The dominant asymptotic term [the first term in (14)] is the same as that for equiamplitude case (see [2]), except for multiplication by the tapering function evaluated at on the edge.The second contribution is of higher asymptotic order since , whereas .This agrees with the description derived previously for the single tapered edge in [5], as is to be expected in view of the assumptions stated at the beginning of Section IV-B.
The edge diffracted fields from the other edges of the rectangular array have analogous expressions easily deducible from (14).

D. FW-Induced Amplitude Modulated Vertex Diffraction
The critical points iii) in Section III-B parameterize the truncation-induced amplitude modulated vertex-diffracted field effects.These describe truly new phenomena that were not encountered in [3]- [5].For instance, near the vertex at , the -edge and -edge FW-shadow boundary transitions interact with the vertex-induced SBCs centered on the -axes and -axes, respectively, due to the truncation of the corresponding edge diffracted fields.The confluence of these four SB transitions near the vertex defines the asymptotics pertaining to vertex diffraction, which is implemented by the following steps.First, in ( 5) is conveniently expanded in such a way as to highlight the behavior of at the truncation as (17) where is defined in Section IV-A (finite array) and is its derivative (the derivation of ( 17) is the same as that shown for in (33)-(36) of Appendix I).Note that, due to the local approximation (17), pole singularities not present Fig. 5. Topology of the k complex planes.The original integration paths of (3) are along the real axis with clockwise indentation around the poles.As described in Section IV-D and in [3], the original integration paths are deformed locally along the 45 -line SDPs. in (3) are now introduced.In the following, physical residue contributions associated with these poles are not accounted for here since they represent the physical FW and edge-diffracted fields already discussed in Section IV-C and -B.The presence of the pole singularities allows us to describe the diffraction mechanism by (poles)-(saddle point) interaction implemented through operations on the already developed canonical integral in [3].From now on, the topology of the spectral plane is analogous to that treated in [3] for the uniform sectoral AGF.
Insertion of the asymptotic expansion (17) of into (3) permits the diffracted field from vertex to be expressed as a sum of four terms (18) with (19) where are the local SDPs in the plane .The superscript denotes the th derivative; for ; for ; for ; and for .The integration paths in (19) are along the local-SDP for each variable , as shown in Fig. 5, intercepting the real axis at the saddle point (SP) .The double integral (19) for the case was evaluated in [3], with details given of the path deformations in both variables.The regularization process in that case involves the Van der Waerden (VdW) method, which uses selective addition and subtraction of pole singularities.We have also suggested in [3] the use of the Pauli-Clemmow method [15] because it incorporates the relevant phenomenologies in a simpler format than the VdW method.In particular the Pauli-Clemmow method involves selective multiplication and division by the regularizing functions .In this paper we extend the Pauli-Clemmow method to accommodate "slope diffraction effects."For simplicity, we develop expressions only for regularization of the pole which is closest to the SP.For the vertex problem, the critical parameters are tied to the first-order SP, and to the -and -poles in (3), i.e., that minimize the spectral distances and .For details see [3] and  [16].The resulting expressions for each -indexed integral in ( 19) are (20) where the locally uniform canonical functions , are defined as (21) The remaining parameters are defined as and , with angles , and defined in Figs. 1 and 3, and SB angles defined after (14) and in Fig. 4 (see [3] for details).Results obtained from (20) are comparable to those obtained applying the VdW procedure in a region "not too close" to the vertex of the array.Again, as in [3], approaching the vertex, the diffraction coefficient due to the VdW is superior; however, when the Pauli-Clemmow evaluation is augmented with the "slope" diffraction term, (20) and the globally regularized version of the VdW perform comparably.For the case of (nontapered amplitude), the treatment of the vicinity of the vertex has been carried out in [3] and it has been shown that the function , denoted in [3] simply by , is based on a modification of a less explicit four-parameter function introduced in [17], and thereafter used in [18].The new remaining vertex transition functions are defined by mapping the relevant configurations of critical points in the corresponding original integrals (19) onto the simplest canonical integrands.The resulting functions are normalized in such a way as to tend toward unity for large values of (see Table I).Thus, "far" from the SBCs and the vertex (or, alternatively, for high enough frequencies), the (nonuniform asymptotic) vertex-diffracted field is represented entirely in terms of four spherical wave contributions."Near" the SBCs (i.e., near the vertex), the locally uniform transition functions differ from unity and can be expressed as combinations of Fresnel integrals and "ordinary" functions as shown in Appendix II and Section IV-A.

A. Formal Solution
To synthesize the total high-frequency asymptotic solution for the rectangular array potential, referring to Sections IV-A-D, we obtain (22) Note that the domains of existence of the various wave species are automatically embedded in the definitions of inside the expressions of the FWs in ( 12)and of the diffracted fields in (14).The domains of existence of the FWs and diffracted fields are given in terms of and the SB planes and cones in the text in connection with ( 12) and ( 14), respectively (see Figs. 3 and 4).

B. Transition Functions
Since the numerical evaluation of in ( 21) can be performed efficiently as in [16], [19], and [20] in terms of standard general-izedFresnelintegrals [20],itisconvenienttonumericallyevaluate in( 21)intermsofthefunction andtheUTDtransition functions and .In Appendix II, the transition functions in (21) have been expressed exactly in terms of the function and as follows: and , where and are defined in ( 15) and ( 16), respectively.
For simplicity of notation, we discuss the asymptotic order estimates and field behavior associated with vertex .For the other vertexes the same rules apply in their respective local coordinates.The vertex-diffracted field [associated with ] incorporates the transition from a vertex-dominatedsphericalwave toanedge-dominatedcylindricalwave andit compensatesfor the discontinuities (across the SBCs) of the edgediffracted rays at and (see Fig. 4); the respective approximate parameters vanish there.The asymptotically dominant terms in the compensation mechanism at are those involving and , with the latter behaving like for significantly distinct from .Analogous considerations apply for .At the simultaneous intersection of the conical SBC and the truncated FW planar SB ,both and vanish(see[3,Fig.3]),andthecorresponding functiontransformsthevertex-induced fieldlocally into a plane wave to match the FW.The transitional behavior of the other vertex diffracted transition functions is summarized in Table I.The functions tend toward unity for large values of and .Large values of both and imply significant spectral domain plane separation between the poles and the SP, and define the nonuniform ray regime.When only one of the parameters or is large, only one of the two poles is asymptotically far away from the SP, and yields the canonical UTD transition function.The limit for can be obtained directly from ( 21) on approximating the denominator in the integrand by its value at [i.e., ] and recognizing the remaining single-pole integral as the ordinary one-parameter UTD transition function .Concerning the functions and in (21), the large parameter (or range is characterized by a double-pole integral that can be expressed in terms of the UTD slope transition function (see Table I).The vertex-diffracted contribution containing accounts for the transition from a vertex-centered spherical wave to an edge-centered cylindrical wave and it compensates for the discontinuities across the relevant SBCs (see [3]).The other vertex diffracted terms not present in [3], are of higher asymptotic order and compensate for the discontinuities across the SBCs of the diffracted wave associated with the derivative of the tapering functions at the edges.Note that when the excitation function tends to zero at the vertex from both directions and , only the contribution remains.The various compensation mechanisms are summarized in Table II.The spreading factors and complex amplitudes are shown for each field contribution associated with vertex , namely, , and .As shown in the last column, diffracted field species reduce to other wave species at the planar and conical SBs.

VI. TOTAL VECTOR ELECTRIC FIELD SYNTHESIS
The electric vector fields are obtained from the vector potential via and .When the differential operators are applied to the spectral representation (3), interchanging the order of integration and differentiation yields (noting that in the spectral domain) (25) where denotes the unit dyadic, is the free space impedance and the notation in implies a dependence on .The magnetic field is treated formally in similar fashion, leading to the replacement of by the magnetic dyadic .The asymptotic evaluation of ( 25) is carried out in the same manner as for the potential in (3), except that we now take into account the extra term .The integral in (25) is dominated asymptotically by the same critical spectral points as in Section III-B, whence the evaluation procedure is the same as in Sections II-IV.
The polarization dyadic may be assumed to be slowly varying so that (25) can be evaluated by the procedure used in Section IV, where the extra term is approximated by its value at the critical spectral points (26) with (27) (28) (29) in which the wavevectors , and are defined in Fig. 4. Similar expressions hold for the other field contributions, which can be expressed conveniently in their ray-fixed reference systems.The reader is referred to [1], [2], and [3] for more details about the asymptotic pertaining to the vector fields.

VII. NUMERICAL RESULTS
Numerical tests have been performed on a "large" square array of dipoles in order to validate the high frequency formulation in (26).An element-by-element summation over the radiated contributions from each dipole serves as a reference.From a variety of near-field scans carried out for different array parameters and dipole orientations, we have selected only some of the most challenging examples because of space limitations.The quality of the analytic-numerical comparisons in the examples is typical of what we have found throughout.The 30 30 element test array of dipoles oriented along with identical periods but with different excitations, is shown in the insets of Fig. 6.Only one propagating FW is excited due to the small array period.Consequently, the total propagating contributions are one truncated FW, four truncated edge-diffracted rays, and four vertex-diffracted rays.The electric field component observed at a distance in the diagonal scan plane depicted in the insets, is shown in Fig. 6, for three different coordinate-separable excitations.In Fig. 6(a), the interelement phasings are (broadside radiation, i.e., main beam at ) with excitation function given by a Taylor tapering [27] (SLL dB; the Taylor weights from the edge to the center in both directions are for with the remaining weights constructed by symmetry).In Fig. 6(b), the interelement phasings are , yielding a radiated main beam at with respect to the normal to the array plane, and at with respect to both the and axes with tapered excitation functions , where .In Fig. 6(c), the interelement phasing is , yielding a radiated main beam at with respect to the normal to the array plane, and at with respect to both the and axes.The tapering function here is .In all three cases, solid curves denote the uniform asymptotic expression for the total radiated field in (26), while dashed curves denote the radiated field in (26) without the vertex contributions.Clearly, the vertex-diffracted waves compensate for the disappearence of the edge-diffracted waves at their SBCs, rendering the total radiated field (solid curves) continuous.The agreement between the asymptotic and numerical reference solutions, which coincide almost everywhere on the scale of the drawing, has been found quite satisfactory even for scan radii smaller than 12 .Note that the tapered excitation vanishes at the four array edges, i.e., .This implies that the four edge-diffracted fields and the four vertex-diffracted fields , involve only the

slope-diffracted terms
, thereby providing a good test case for the usually subdominant additional edge-and vertex-"slope diffracted" contributions.In particular, also in this case, the vertex slope-diffracted fields compensate via the transition function (last row of Table II) for the disappearance of the edge slope-diffracted fields.

VIII. CONCLUSION
In this paper, we have analyzed and validated the high-frequency diffraction phenomena pertaining to large rectangular arrays of dipoles with different linear phasings along the two principal coordinates, and with weakly tapered different excitation profiles along these coordinates that may tend to zero at the array edges.The results have been expressed in terms of our previously formulated asymptotic uniformized, periodicity-adapted, FW-modulated GTD for the canonical dipole AGF generated by an infinite plane rectangular sector, which has been generalized here to accommodate the above-mentioned amplitude tapering.The uniform AGF assumes its most intricate form in the vicinity of the vertex, with transition and compensation mechanisms that have been quantified analytically, interpreted phenomenologically, and computed efficiently by our asymptotic algorithm, with accuracy validated through a preliminary set of numerical simulations.The high-frequency results from this analysis can be applied directly to the prediction of the radiation pattern distortion and the interantenna coupling that occurs when the actual array is placed in an electromagnetically complex environment like that of a large array in the antenna farm of a space platform.Such predictions are conventionally obtained by computation-intensive tracing of ray fields from each individual element or a subgroup of elements of the array through the complex environment [7].Alternatively, global ray tracing based on the direct application of this formulation permits characterization of the entire array aperture radiation in terms of a few rays whose number is independent of the number of array elements, thereby drastically reducing the computational effort.The main effect of the array truncation on the array currents is a global modulation of these currents with a very large period [28], [29].Although this modulation may perturb the input impedance of the elements, it does not affect significantly the field from the Fresnel zone to the far zone.In other words, the spatial perturbation is filtered by the free-space Green's function (except possibly at grazing aspects).To correctly evaluate this current deformation near the edges of the array, the same FW parametrization of the active array Green's function can be used in a full-wave analysis of the actual rectangular array elements, as demonstrated for short-wire dipoles in [28]- [30] and for open-ended waveguide in an infinite ground plane in [8] and [9].This paper terminates our long series of investigations of the FW-modulated UTD pertaining to large planar rectangular phased dipole arrays in free space, escalating sequentially from single-edge geometries to two-edge, single-vertex infinite sectors, culminating in the rectangular array.Simultaneously, we have dealt with dipole excitation profiles ranging from equiamplitude through one-dimensional weak tapering to the two-dimensional tapering here.We regard to these studies, in their totality, to have furnished the foundation for the phenomenologically incisive, numerically efficient full-wave treatment of a variety of practical weakly amplitude-tapered phased array antennas (see [12]).While we feel confident that our results so far may be extended to accommodate particular array requirements not covered by weakly tapered or other assumptions, we have initiated the application of these free space algorithms to the important case of dipole arrays printed on, or in, multilayer substrate slabs (see [22]- [24]).However, still ongoing in free space is the separate series of time-domain (TD) canonical studies of sequentially pulsed dipole arrays, which have led to a new FW-modulated TD-UTD [25], [26].

APPENDIX I TAPERED EDGE DIFFRACTED FIELDS
Here we summarize the procedure for deriving the FW modulated diffracted field at edge 1 (i.e., that at ): all other edges may be treated similarly.The relevant critical point is the SPP [see ii) in Section III-B] which satisfies .Inserting ( 5) into (3), the edge contributions due to the critical point at are found by first expanding the exponent of the integrand in Taylor series around , i.e., , where and and then evaluating 1 in a neighborhood of .The pertinent asymptotic contribution may be derived from (3) as , where which, after exact evaluation of the inverse Fourier transform of the integral in , yields Note that a) depends on the spectral variable and b) the two vertex-truncation terms and in (5) are not included in (30), (31) since they will be incorporated subsequently into vertex-diffracted fields to obtain an asymptotic expansion of that highlights the behavior of at the truncations and .The inverse Fourier transform in ( 6) is approximated asymptotically, using two sequential integrations by parts (we consider only the end-point ) (32) and the -integral in the third term of (41) can be evaluated in closed form, while the second term reconstructs so that (42) The last equality in (42) involves use of (37) in the first and third terms on the right-hand side.After rearrangement, we obtain (23).The expression for is obtained from symmetry in (41).To obtain the canonical form of the transition function in (21), we first perform -differentiation of (or -differentiation of ), using (21) (43) Insertion of (42) into (43), and term by term differentiation thereafter, yields (44) Use of the identities and in (44), followed by simple manipulations, leads to (24).

Fig. 1 .
Fig. 1.Ĵ -directed parallel dipoles, weighted by an amplitude taper and linear phasing, with d and d denoting the interelement spatial periods along z and z , respectively.The tapering function f (z )f (z ) is slowly varying
(a) and (b) displays the SAF in (4) which appears in the integrand of (3), whereas Fig. 2(c) shows the rapid phase variation, with stationary phase at the center, of the individual element spectral Green's function

Fig. 4 .
Fig.4.Ray description of the field radiated by the rectangular array of dipoles.The diffraction cone of the propagating z -edge (z -edge) diffracted ray originates at the q-dependent (p-dependent) point z on edge 1 (z on edge 2).The SBC that truncates the domain of existence of the qth edge diffracted field has the same aperture angle as the diffraction cone and is
Asymptotic High-Frequency Green's Function for a Large Rectangular Planar Periodic Phased Array of Dipoles With Weakly Tapered Excitation in Two Dimensions F. Mariottini, F. Capolino, Senior Member, IEEE, S. Maci, Fellow, IEEE, and L. B. Felsen, Life Fellow, IEEE

TABLE I ASYMPTOTIC
BEHAVIOR OF THE TRANSITION FUNCTIONS T (a; b; w) FOR LARGE VALUES OF a AND/OR b.THE PARAMETERS a (b) ARE LARGE WHEN THE OBSERVER IS "FAR" FROM SBC (SBC ), OR WHEN THE OPERATING FREQUENCY IS HIGH ENOUGH