## 1. Introduction

[2] Modeling the behavior of a large planar patch array, either in free space or on top of a layered substrate, is a classical problem in electromagnetic theory. The main challenge is to find a compromise between physical understanding, accuracy and computational efficiency. Almost all approaches rely on the radiation integral, which relates currents, either induced or equivalent, in identical subdomains of a planar aperture to the radiated field far away from the antenna. The subdomains or cells are organized according to a Bravais lattice, which is generated from two primitive vectors. The structure of the Bravais lattice determines the directions in which the radiation lobes are observed, while their relative strength is also determined by the behavior of the field inside the cell.

[3] The traditional engineering approach, as described in textbooks such as *Hansen* [1966], *Amitay et al.* [1972], and *Mailloux* [2005], relies on an approximate representation of the current in each cell, in which elementary distributions are weighted by appropriately chosen complex amplitudes. When the array is infinite and the excitation is phased, Bloch functions are obtained in which the elementary currents are identical and the amplitude is an exponent whose wave vector depends on the phase factor in the excitation, and on the structure of the Bravais lattice [*Stark*, 1966; *Wasylkiwskyj and Kahn*, 1970]. The radiated field is then obtained as the product of so-called element and array patterns. For finite arrays, a slow variation of the current distributions and the resulting element patterns is neglected, while the amplitudes per cell are optimized. This leads to a variety of analytically or numerically determined sets of excitation coefficients with corresponding array patterns, also called tapers, which depend on the specific problem at hand.

[4] Computational techniques for determining the electromagnetic behavior have a similar history. The traditional approach is to approximate a finite array by an infinite one, so that the computational analysis can be restricted to evaluating the radiating current distribution in a single cell. The current in this so-called template element or elementary cell, typically the one nearest to the origin, is then repeated across an infinite aperture, multiplied by two phase factors that correspond to the primitive vectors that span the Bravais lattice. The current in the template element is then determined by solving an electric or magnetic field integral equation by the method of moments. The choice of basis functions, in particular between global or local ones, again depends on the specific problem at hand.

[5] The infinite-array approach breaks down when the array becomes resonant. The occurrence of a resonance can still be predicted from the infinite-array formulation, by searching for the so-called blind scan angles for which electromagnetic energy propagates in the transverse direction from cell to cell, so that the excitation coefficients represent a sampled traveling wave. The radiation pattern of the entire array, however, depends on the distribution of the excitation coefficients, which now show a standing wave behavior. This explains, for example, the success of the descriptions in the work of *Skivervik and Mosig* [1993], where current distributions computed for an infinite array were combined with a standing wave taper.

[6] In recent years, computational approaches for modeling the electromagnetic behavior of finite arrays were gradually refined. Typically, a number of current distributions are computed for a single antenna element in a more simple environment for which the requirements on computer time and memory are strongly reduced. A Gram-Schmidt orthogonalization or a singular-value decomposition then is used to extract a basis of linearly independent distributions. Each basis function is a local solution to Maxwell's equations with appropriate boundary conditions. Subsequently, the interaction between the cells is handled by expressing the current distribution across the entire array in terms of the basis functions obtained in this manner. This may be envisaged as the computational counterpart of the engineering approach outlined above, but now with numerically computed current distributions and excitation coefficients.

[7] Several ideas have been proposed to generate the set of current distributions. In the work of *Craeye et al.* [2004], they are generated from a computation for finite-by-infinite arrays. In the expansion wave concept of *Vandenbosch and Demuynck* [1998] a single antenna element is excited by plane waves at various directions of incidence. In the synthetic functions and computed basis function approaches [*Matekovits et al.*, 2007; *Prakash and Mittra*, 2003], they are found by positioning elementary dipole and voltage sources near the antenna element. In both these approaches, the choice of the set of excitations is ad hoc, without a guarantee of completeness of the resulting collection of current distributions. In the latter three, the extraction of the basis from this collection requires additional computational effort.

[8] In the work of *Bekers* [2004], the problem of selecting a set of excitations is avoided by considering the eigenfunctions, or eigencurrents, of the impedance operator that relates the current on the elements to the excitation field. The eigencurrents are first determined for a single element. Next, the eigencurrents for the entire array are expanded in terms of the element eigencurrents. This results in a hierarchy that is similar to the ones described above. The difference is that at both levels, exact solutions to the relevant integral equations are found. Once the eigencurrents have been found, it is straightforward to relate the radiated field to a given excitation, which offers new possibilities for pattern synthesis. Moreover, by identifying suitably chosen subarrays, our two-level approach can be extended to a multilevel one.

[9] The idea of considering antenna arrays as eigenvalue problems is not new. Floquet's theorem is based on the observation that, for a periodic geometry with a phased excitation, the electromagnetic field and associated current distributions are eigenfunctions of translation operators corresponding to the two primitive vectors. The resulting eigenfunctions are the well known Bloch functions. In an array of rectangular waveguides, the field is expressed in terms of eigenfunctions of the Helmholtz operator Δ + *k*^{2} for the electric or magnetic field, with Dirichlet boundary conditions and the corresponding modes are cosine and sine functions that scale with the length and width of the aperture [*Marcuvitz*, 1951]. At a finite distance from the aperture, only a finite number of modes will contribute to the electromagnetic field. These modes, referred to as accessible modes [*Rozzi and Mecklenbrauker*, 1975; *Conciauro et al.*, 2000], represent either propagating waves or evanescent waves that have a significant contribution to the field at the specified distance. The propagation constants and the cutoff frequencies of the modes follow from their eigenvalues, which only depend on the length and the width of the aperture.

[10] In the work of *Bekers* [2004], the attention was focused on the formulation of the approach, and its computational efficiency. The test geometry was a linear array of straight of circular thin strips. The surprising conclusion was that only a few eigencurrents per element are needed to achieve an acceptable description of the global array response. In the works of *Bekers et al.* [2005, 2006], the relation between the eigencurrents and resonances was investigated. The results confirm the classical analysis of *Baum* [1976], that a resonance corresponds to a small value of an eigenvalue.

[11] In the present paper, we continue the development of *Bekers* [2004] and *Bekers et al.* [2006]. The logical next step is to analyze the relation between the eigencurrents and the radiated field. We consider the same uniform linear arrays of dipoles and loops. First, we investigate the spatial variation of the expansion coefficients, and their dependence on parameters as element geometry, spacing, and frequency. Second, we investigate the far fields of the array eigencurrents, and we relate them to specific radiation behavior of the array. Finally, we discuss the interpretation of the array eigencurrents and their eigenvalues in relation to the conventional Floquet modes and propagation constants.