## 1. Introduction

[2] 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.

[3] 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* [1962] 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* [2002] 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* [1986], *Johansen and Breinbjerg* [1995], and *Cui and Ando* [2002]. *Pelosi et al.* [2000] extended the procedure of *Johansen and Breinbjerg* [1995] to treat the case of plane wave scattering by a finite planar impedance surface. In antenna applications, *Mioc et al.* [1999] obtained a line integral representation for the field radiated by a modal current distribution in an open ended rectangular waveguide, while *Infante and Maci* [2003] developed a line integral representation for the Kirchhoff-type aperture radiation from a parabolic reflector. *Tiberio et al.* [2004] 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* [2002].

[4] 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).

[5] 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.* [2000], *Janpugdee and Pathak* [2006] 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.

[6] 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.

[7] 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.

[8] 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.

[9] An *e*^{jωt} time convention for time harmonic sources and fields is assumed and suppressed throughout sections 2–4.