A boundary diffraction wave (BDW) formulation is developed for predicting the collective radiation from large finite planar phased array antennas in a relatively efficient manner. This collective radiation is described in terms of a set of Floquet waves (FWs) arising from the array interior, together with the diffraction of these FWs by the finite array element truncation boundary; the latter is represented by a line integral of the incremental diffracted field over this boundary. In general, the BDW line integral must be evaluated numerically. The current distribution over the whole array is first obtained, for a given excitation, by full wave numerical methods. A traveling wave expansion is employed to represent the array distribution so as to facilitate the development of the BDW solution for arrays. Some numerical results based on the BDW are presented to demonstrate the utility and accuracy of the method for analyzing large planar arrays.
 A boundary diffraction wave (BDW) formulation is presented in this paper for predicting the collective radiation from large finite planar phased array antennas in a relatively efficient manner. The finite array element truncation boundary can be relatively arbitrary. The collective radiation based on the BDW formulation is described, in a physically appealing manner, in terms of a set of Floquet modal plane waves arising directly from the array interior, together with the diffraction of these Floquet waves (FWs) by the finite array element truncation boundary; the latter is represented by a line integral of the incremental diffracted field over this boundary. In general, the BDW line integral has to be evaluated numerically.
 The present BDW solution for finite planar arrays is developed via an extension of a technique presented much earlier by Rubinowicz [1962, 1965] for treating the diffraction by an aperture in a plane screen based on Huygens's principle. Miyamoto and Wolf  also arrived at a result similar to that obtained by Rubinowicz, but using an appropriate vector potential; the latter involved more complex mathematics. In some related work, Albani and Maci  extended Rubinowicz's approach to develop a line integral representation of the Physical Optics (PO) integral for the scattering from a perfect electric conducting (PEC) planar surface illuminated by elementary electric or magnetic dipoles. Some additional previous related work on the BDW line integral representation for the PO scattering from a PEC flat plate includes that due to Asvestas , Johansen and Breinbjerg , and Cui and Ando . Pelosi et al.  extended the procedure of Johansen and Breinbjerg  to treat the case of plane wave scattering by a finite planar impedance surface. In antenna applications, Mioc et al.  obtained a line integral representation for the field radiated by a modal current distribution in an open ended rectangular waveguide, while Infante and Maci  developed a line integral representation for the Kirchhoff-type aperture radiation from a parabolic reflector. Tiberio et al.  developed an incremental theory of diffraction (ITD) for the radiation/scattering by edged bodies, which arrives at a result similar to that obtained by Rubinowicz, but in a completely different way. A comprehensive review of the BDW related work is given by Albani and Maci .
 The BDW solution developed here can be applied to describe the collective radiation from a variety of practical, realistic finite planar arrays. For example, planar arrays of dipoles in air, or planar metallic slot arrays, or even more complex flush mounted radome-covered antenna arrays whose elements are slightly recessed in a cavity structure just below the skin line of a planar PEC boundary, etc., can be treated via the BDW method developed in this paper. The current distribution over the whole array for a given excitation is first obtained by full wave numerical methods, such as by a moment method (MoM) solution of the governing integral equation for the array element currents, or by a finite element boundary integral (FE-BI) method, etc. A traveling wave (TW) expansion [Janpugdee et al., 2005] is next employed to represent the array distribution so as to facilitate the development of the BDW solution for arrays. The TW basis set allows one to express a realistic complex array distribution in terms of a more compact sequence of much simpler distributions. The latter is crucial since the BDW formulation cannot be directly developed for realistic array distributions, which generally exhibit highly pronounced ripples especially near and at the array element truncation boundaries. In the case of the array types mentioned above, the collective radiation of each TW can be expressed in an exact fashion in terms of the BDW formulation; for other finite arrays such as planar patch arrays over an infinite grounded substrate, the BDW line integral is an approximate representation rather than exact, but is still quite accurate in this case as well. A TW expansion for a given array distribution can be obtained via its discrete Fourier transform (DFT), or by using any available parameter estimation methods. From numerical tests, it is found that the number of significant TW basis functions, for representing most practical phased array excitations, is equal to or less than 1% of the total number of array elements (or the number of array aperture field samples).
 A uniform geometrical theory of diffraction (UTD) solution for describing the collective array radiations developed by Capolino et al. [2000a, 2000b, 2000c], Çivi et al. , Janpugdee and Pathak  is expressed in closed form, and thus it is relatively more efficient than the BDW but it requires ray tracing. On the other hand, the BDW is more robust since it needs no ray tracing from the array boundary, and it does not require the calculation of any UTD transition function as does the UTD. However, the BDW line integral must in general be evaluated numerically. It can be shown that an asymptotic analytical evaluation of the BDW integral for large arrays reduces to the closed form UTD solution as expected. The BDW solution provides a physical picture for the array radiation mechanisms in terms of a diffracted wave emanating from the whole array boundary, in addition to direct field contributions arising from specific points in the array interior. The latter points move as a function of the observer at an external location. On the other hand, the UTD provides the same direct field contributions from the array interior, but the UTD diffracted ray fields emanate from specific diffraction (flash) points on the edges and corners of the array element truncation boundary which correspond to stationary phase and end points in the BDW line integral. In contrast, the conventional brute force array element-by-element field summation approach is not only highly inefficient for large arrays, but also lacks physical insight into the radiation mechanisms that is present in the BDW and also the UTD approach. Indeed, the present work is motivated by the need to predict the performance of large arrays which are gaining importance in high-resolution radar and other high-gain antenna applications.
 In practical applications, the large phased array antenna is usually situated on an even larger complex platform, e.g., a naval ship tower, etc. In such realistic problems, the collective BDW rays, which are launched from points on the large array aperture boundary as well as from a few interior points, can provide an efficient input to any existing ray-based UTD codes to predict the wave interactions of the array with the host platform via the UTD.
 It is noted that although the BDW formulation presented in this paper deals with finite planar arrays consisting of a periodic rectangular grid, the present method can be readily extended to treat finite planar arrays consisting of a skewed grid as well. Such an extension will be reported in the future.
 This paper is organized as follows. In Section 2, the BDW formulation for finite planar arrays of free standing electric and magnetic point currents is presented for the sake of simplicity in illustrating the concept. The procedure for the extension of the BDW solution in section 2 to treat some realistic array configurations is described in section 3. Typical numerical results based on the BDW for large planar arrays are presented in section 4 to demonstrate the utility and accuracy of the method. Conclusions are given in section 5.
 An ejωt time convention for time harmonic sources and fields is assumed and suppressed throughout sections 2–4.
2. BDW Formulation for Arrays of Point Currents
 Consider a finite, planar periodic array of (electric or magnetic) point currents located on a rectangular grid in the xy plane, at the positions described by
where dx and dy are the periodic element spacings in the and directions, respectively. The boundary of the array can be any polygonal shape. The rectangular array geometry is shown in Figure 1, and the polygonal array geometry is shown in Figure 2.
 It is assumed that every point current in the array has the same orientation represented by a unit vector . The current moment of the (n, m)th point current is defined by
Thus, for an array of electric point sources, the electric current density nm for the (n, m)th array element located at nm′ may be represented by nm = nmδ( − nm′), where nm is an electric current moment, is the position vector to an observer, and δ(.) is the Dirac delta function. Likewise, for an array of magnetic point sources, the magnetic current density nm = nmδ( − nm′), where nm is now a magnetic current moment of the (n, m)th element.
2.1. BDW Formulation
 The electric and magnetic fields radiated by the array are initially given by a summation of the fields produced by individual elements as
where denotes the discrete array aperture formed by the array element truncation boundary.
 For an array made up of electric point currents, the fields (nm, nm) are given by
where (; nm′) represents an appropriate vector potential function defined by (5); it is given by
and Rnm = ∣ − nm′∣. The quantity ɛ is the permittivity of the surrounding medium, which is assumed to be free space in the present case.
 On the other hand, an array made up of magnetic point currents in free space produces the fields (nm, nm) which are given by
 To facilitate the development of the BDW formulation, the array distribution Pnm is now represented by a TW basis set as
where i is a propagation constant and Ci is a complex amplitude, of the ith TW, respectively. These two parameters are to be determined numerically, from a given array current distribution, using a discrete Fourier transform (DFT) expansion [Brigham, 1974], or by any available parameter estimation methods, e.g., the CLEAN algorithm [Tsao and Steinberg, 1988], Prony's method [Van Blaricum and Mittra, 1978], etc. In the case of DFT, one notes that the expansion in (9) is exact if one includes all DFT terms which are equal to the total number of array elements; however, it is found in the cases of large arrays that only K terms are significant, where K is much less than the total number of array elements. The latter is true because the significant DFT spectrum is highly compact (narrow) in the spectral domain if the array support is defined over large (or wide) spatial domain; this is the well-known relationship that exists between a spatial quantity and its spectral transform or DFT.
 By using the TW expansion of (9) in (3), the total fields (, ) can now be expressed as the superposition of the fields (i, i) that are produced by each ith TW current (of normalized amplitude), namely,
 By using the result developed later in Section 2.2, the electric and magnetic fields (i, i) in (11) can be expressed within the BDW formulation by a superposition of the FWs and the BDW line integral as
The FW and BDW rays in the near zone, which are launched from the rectangular array aperture, are shown in Figure 3 for a near zone observer at P.
 The terms (i,pqFW, i,pqFW) are the (p, q)th mode of the electric and magnetic FWs produced by the ith TW current. The FWs are in the form of a modal sum of plane waves. For the array of electric point currents, they are given by
where i,pq± is the Floquet wave number given by
with i,pq and ki,pqz given by
and βi,pq = ∣i,pq∣. The negative (−j) branch of the square root in (18) must be chosen for βi,pq2 > k2 to satisfy the radiation condition. Let i,pq± be the unit vectors in the propagation directions of the FWs; they are given by i,pq± = i,pq±/k. The quantity Z in (15) is the characteristic impedance of free space.
 On the other hand, for the array of magnetic point currents, the fields (i,pqFW, i,pqFW) are given by
The FWs represent the field produced by an infinite periodic array. Because of the array element boundary truncation of the finite array, each FW exists only within a confined region of space, and it vanishes at its shadow boundary defined by a unit step function Ui,pqFW, which becomes unity if the observation point is in the lit region of the (p, q)th FW, and is zero otherwise. For a given frequency, generally only one or just the first few FW modes, for which βi,pq2 < k2, propagate away from the array aperture; whereas the remaining infinite number of FW modes corresponding to βi,pq2 > k2 become evanescent away from the array face.
 The terms (i,pqBDW, i,pqBDW) in (12) and (13) represent the BDWs, which are expressed in terms of a line integral along the array aperture boundary Γ as
where (i,pqBDW, i,pqBDW) are the incremental BDW contributions. They are given by
 The vector ′ represents the point along the integration contour Γ. The terms (i,pqFW, i,pqFW) are the electric and magnetic FW fields; they are given by (14)–(15) for the array of electric point currents, and by (19)–(20) for the array of magnetic point currents. The subscript “greater than” and “less than” symbols on the vector ′ indicate the points at ′ just above and just below the aperture plane, respectively. R0 is the distance from the integration point ′ to the observation point ; it is defined by
The dyadics are the diffraction coefficients for the incremental BDWs. The superscript “plus” and “minus” indicate their correspondence to the FWs at boundary points just above and below the aperture plane, respectively. The BDW diffraction coefficients are given by
where is a unit tangential vector to the aperture boundary Γ at the integration point ′, in the positive sense (i.e., right-hand rule) around the surface normal in the direction. 0 is the unit vector defined by 0 = 0/R0.
2.2. Detailed Derivation of the BDW Formulation
 Consider the expressions of the fields (i, i) produced by the ith TW current, which are given by (11). By applying the partial Poisson's sum formula [Papoulis, 1968; Çivi et al., 1999], the element-by-element summation in (11) can be converted into a (Floquet) modal spectrum summation as
where ds′ denotes the integral over a continuous aperture formed by a boundary of the actual array. It is noted that the mathematical boundary of the aperture A here is defined at half a period away from the actual array element truncation boundary, as shown by the shaded area in Figure 4; this half period extension is to avoid the presence of any explicit end point correction terms which otherwise must be included in the partial Poisson sum formula [Çivi et al., 1999]. The (p, q) are the indices of the Floquet modal spectrum. The vector ′ represents the integration point in the aperture A, which is the continuation of the discrete nm′ into the corresponding continuous space (defined within A) via the Poisson's summation. The functions 0 (′) and 0 (′) are the continuation of nm and nm in (11), respectively, according to the continuation of nm′ → ′, where
Therefore, for the array of electric point currents, 0 (′) and (′) are given by (4) and (5), respectively, with nm′ being replaced by ′. On the other hand, for the array of magnetic point currents, 0 (′) and 0 (′) are given by (7) and (8), respectively, with nm′ being substituted once again by ′.
 For an array made up of electric point currents, the fields (i, i) in (26)–(27) can be expressed as
with R = ∣ − ′∣, and i,pq is defined by (17). Each term in the summation of (29) and (30) can be interpreted as the (electric or magnetic) field produced by a continuous, unit amplitude TW electric surface current located in the aperture A, which has a propagation constant i,pq, and is polarized along . The propagation constant i,pq can be considered as the Floquet harmonics of the TW propagation constant i as seen from (17).
 By following a procedure similar to that described by Rubinowicz  and Albani and Maci , one now applies the equivalence theorem for the electromagnetic field to a volume V bounded by a closed surface S, which encloses the TW surface current, as shown in Figure 5. In particular, the surface S consists of two semi-infinite lateral surfaces, S1 and S2, which are generated by the translation of the boundary contour of the aperture A in the direction i,pq± into the region above and below the aperture plane (xy plane), respectively, and the two end caps, C1 and C2, which recede to infinity. The lateral surfaces, S1 and S2, are chosen in these particular directions to facilitate the development of the BDW formulation, as will be seen in the derivation which follows.
 On the basis of the equivalence theorem, one has the relation
where A is the electric field produced by the TW surface current residing in the aperture A, S is the electric field produced by the equivalent electric and magnetic currents on the surface S with S = S1 + S2 + C1 + C2, and ∞ is the electric field produced by the same TW surface current if it was of infinite planar extent. Note that V1,2 is bounded by the closed surface A + S1,2 + C1,2.
 The electric field A is given by
which is essentially the same as each term in the summation of (29).
 One can show that the electric and magnetic fields (∞, ∞) are the plane waves given by
where i,pq± is given by (16), and i,pq± = i,pq±/k. It is noted that the fields (∞, ∞) are indeed the same as the (p, q)th modal FWs given in (14)–(15), except for a factor of 1/(dxdy).
 The electric field S is given by
where the contributions from the two end caps at infinity, C1 and C2, are negligible, and
where is a unit normal vector to the surface S pointing into the volume V, and the fields (∞, ∞) are given by (34)–(35).
where U() is the Heaviside unit step function defined by
The surface integral on S1 (and S2) in (36) can be expressed as a twofold integral consisting of a line integral along the semi-infinite strip (as shown in Figure 6) and a contour integral along the boundary Γ of the aperture A, namely,
where l′ is a variable of integration along the boundary Γ, and ξ′ is a variable of integration along the semi-infinite strip. The unit vector and the contour integral along the boundary Γ are defined in the positive sense (i.e., right-hand rule) around the surface normal in the direction.
 By following the same procedure as Rubinowicz  and Albani and Maci , one can evaluate the semi-infinite integral in an exact fashion along the dark strip (see Figure 6) yielding the contribution from the end point on the boundary Γ, whereas the contribution from another end point at infinity is negligible. This closed form evaluation of the integral over the dark strip is possible because of special properties of the integrand along the strip, as described by Rubinowicz  and Albani and Maci . The remaining contour integration along the boundary Γ is to be evaluated numerically. Finally, one obtains
 For the case where (∞, ∞) are evanescent modes, i.e., βi,pq > k, the corresponding propagation vector i,pq± becomes complex. One can obtain the BDW solution for evanescent modes from the solution developed for the propagating modes by an analytical continuation of the propagation vector i,pq± to complex values, as well as the continuation of other associated parameters into the complex space. It is noted that, for evanescent modes, the associated lateral surfaces S1 and S2 are located in complex space, therefore the unit step function U() = 0 for any observation point in real space. Thus only the boundary integral term, S, contributes to A for evanescent modes.
Also, by substituting A from (46) for the term within the braces in (29), one obtains the expression given for i in (12), where
and Ui,pqFW = U().
 The expression of i in (13) can be derived by the same procedure. Furthermore, the BDW solution for the electric and magnetic fields produced by an array made up of magnetic point currents can be developed by following the same procedure presented above, or alternatively by employing the duality theorem.
 In some cases, such as for a finite planar array of microstrip patches on an infinite grounded substrate, the integral over the semi-infinite dark strips in Figure 6 can no longer be evaluated in an exact fashion because of the complexity of the integrand which contains the associated Green's function for a grounded substrate; but it can nevertheless be evaluated asymptotically in an accurate manner via its end point contribution. The remaining contour integration along the boundary Γ is then evaluated numerically as before. The resulting BDW formulation for this latter case is thus not exact, but is an asymptotic high-frequency approximation.
3. Applications to Some Realistic Finite Planar Arrays
 The BDW solution for a finite planar array of point currents presented in Section 2 can be applied to represent the collective radiation from some practical, realistic finite planar arrays as described below.
3.1. Slot Arrays
 For an array of slots in a planar metallic platform, one can employ the equivalence theorem to replace a slot array by an array of equivalent tangential magnetic currents on an infinite PEC surface, which now covers the slot apertures as well. These equivalent magnetic currents are directly proportional to the tangential electric fields inside the slot apertures, which must be found by using any numerical method of solution as mentioned above. By using image theory, one can find the fields in the region external to the PEC platform as the fields radiated by an array of equivalent magnetic currents in free space, but with the amplitude of magnetic currents multiplied by a factor of two. For an array of short and thin slots, one can assume that the field distributions in every slot have an identical shape (analogous to the dominant waveguide mode distribution for waveguide fed slots), but each with different amplitudes that must be determined by numerical methods. The amplitudes of equivalent magnetic currents can then be represented by a set of TWs. The electric and magnetic fields produced by the array are given by (10), with (i, i) given by (12)–(13). The electric FW, i,pqFW, is now given by
where F(.) is the Fourier transform of the equivalent magnetic current in the slot defined by
The function f(.) represents the equivalent magnetic current distribution in the slot. The superscript ″ denotes the local coordinate system centered on the slot, and Λ denotes the support of the equivalent magnetic current distribution in each slot. The magnetic FW, i,pqFW, is given by (20). The BDWs (i,pqBDW, i,pqBDW) are now given by
where F(.) is again the Fourier transform of the equivalent magnetic current in the slot defined by (49), and (i,pqBDW, i,pqBDW) are given by (22)–(23).
 For an array of extended slots, the field distribution in each slot generally can be more accurately represented by a set of basis functions. Typically, the number of basis functions per slot is much less than the total number of array elements for large arrays. A set of the associated basis functions on every slot will form its own mathematical array. Each mathematical array associated with each set of basis can be treated separately by the solution presented above for an array of short and thin slots; the total fields produced by the actual array can be found via superposition.
3.2. Arrays of Dipoles
 For an array of dipoles in air, one can replace the dipoles by equivalent electric currents. For an array of short and thin dipoles, one can assume that the electric current distribution on every dipole have an identical shape of a piecewise sinusoidal function, but each with different amplitudes that must be determined by numerical methods. The amplitudes of electric currents can then be represented by a set of TWs. The electric and magnetic fields produced by the array are given by (10), with (i, i) given by (12)–(13). The magnetic FW, i,pqFW, is now given by
where F(.) is the Fourier transform of the equivalent electric current on the dipole defined by (49). The electric FW, i,pqFW, is given by (15), and the BDWs (i,pqBDW, i,pqBDW) are given by (50).
 For an array of dipoles with extended lengths, the equivalent current distribution on every dipole generally can be more accurately represented by a set of basis functions. A set of the associated basis functions on every dipole will form its own mathematical array. Each mathematical array associated with each set of basis can be treated separately as before, and then the total fields produced by the actual array can be found by superposition.
3.3. Complex Array Recessed in a Planar Metallic Platform
 For an array of complex antenna elements slightly recessed in a cavity structure just below the skin line of a planar metallic platform and covered by flush mounted radome, one can employ a standard equivalence theorem to replace the array radome aperture (formed at the outer radome surface) by a tangential equivalent magnetic surface current density on a planar PEC surface which also covers the array radome aperture. The equivalent magnetic surface current density is defined by the tangential electric field in that aperture. This equivalent magnetic surface current generates the same fields outside the array aperture region as produced by the antenna array in the actual problem. The array radome aperture field distribution must be found first by numerical methods. The numerically determined equivalent magnetic current in the array aperture is then quantized or sampled into a finite set of elemental magnetic currents, which form a mathematical array on this PEC surface. The equivalent magnetic current distribution must be sampled slightly over the Nyquist rate in order to avoid the error caused by aliasing. The polarizations of each quantized magnetic surface current are generally not identical. One can express each quantized magnetic current in terms of a sum of two components, namely, the and components. Mathematical arrays formed by each vector component of quantized magnetic currents can be treated separately, and then the total fields produced by the actual array can be found by superposition. By using image theory, one can remove the PEC surface, and have a mathematical array of quantized magnetic currents with double the amplitude and located in free space. The BDW solution for an array of magnetic point currents presented in Section 2 can be applied directly to treat this mathematical array.
4. Numerical Results
 To demonstrate the utility and accuracy of the BDW solution, numerical examples of the radiation from a rectangular periodic array of 501 × 501 magnetic point currents are illustrated in this section. Each point current is oriented in the direction, where = cos(2π/9) + sin(2π/9). The interelement spacing in the and directions is dx = dy = 0.2 λ, where λ is the wavelength in free space. Thus the whole array aperture size is 100 λ × 100 λ. The array distribution is assumed to be a hypothetical distribution with Gaussian tapered amplitude as shown in Figure 7, and linear phase that produces a main beam in the direction (θ0 = 30°, ϕ0 = 0°). In practice, this example array could be a mathematical array of sampled equivalent currents over the physical aperture of a complex flush mounted radome-covered antenna array whose elements are slightly recessed within a cavity structure in a planar metallic platform, with the aperture size of 1 m × 1 m, and the operating frequency of 30 GHz (Ka band). The aperture field distribution must be found by numerical methods, such as FE-BI, etc. The aperture distribution is then uniformly sampled at every 0.2 λ in the two orthogonal ( and ) directions; this sampling rate is over the Nyquist rate.
 The TW expansion is employed to represent the Gaussian tapered array distribution in the present example. The TWs are computed by using the DFT, and only 55 dominant TWs, which are about 0.02% of the total number of array elements, are used for the numerical results shown below. The fields produced by each TW are computed in terms of the FWs and the corresponding BDWs. The present solution, denoted by TW-BDW, is compared with the reference solution which is obtained by the conventional element-by-element field summation approach.
Figures 8a–8b show the comparison of the electric fields radiated by the array at a radial distance r = 100 λ in the xz plane (ϕ = 0°), which are in the radiative near-field (or Fresnel) region of the array. The TW-BDW solution (dashed line) agrees well with the reference solution (solid line). Figures 9a–9b show the comparison of the electric fields produced by the array at a radial distance r = 100 λ in the yz plane (ϕ = 90°) using the TW-BDW and the reference solutions.
Figures 10a–10b show the comparison of the electric far fields radiated by the array, observed in the xz-plane (ϕ = 0°), which are obtained by the TW-BDW and the reference solutions. The numerical results show that the TW-BDW solution agrees extremely well with the reference solution in each case.
 A BDW solution for representing the collective fields radiated by large finite planar phased arrays is presented in this paper. A relatively arbitrary realistic array current distribution is expressed in terms of a sequence of simple distributions via a TW expansion. The field produced by each TW is represented in terms of the FWs arising from the array interior, and the BDWs which are expressed as a line integral of the incremental diffracted field over the array element truncation boundary. It is found that, for most practical phased array excitations, the number of significant TW basis functions is equal to or less than 1% of the total number of array elements. The BDW for each TW is exact, but it requires the numerical evaluation of the BDW line integral. Thus the BDW solution is less efficient than the closed form UTD ray solution for arrays; however, the BDW is more robust since it needs no ray tracing from the array boundary, and it does not require the calculation of any UTD transition function as does the UTD. Both the BDW and the UTD are more efficient than the conventional element-by-element field summation approach. Numerical results for the near and far fields of an array of electric point currents based on the TW-BDW solution are seen to compare extremely well with the reference solution.
 The authors thank Matteo Albani for a discussion on the BDW. The authors also thank the reviewers for carefully reviewing the manuscript and pointing out some additional references.