### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Eigencurrent Description
- 3. General Aspects of Array Eigencurrents
- 4. Far Fields of Array Eigencurrents
- 5. Array-Eigenvalue Behavior
- 6. Conclusion
- Acknowledgments
- References

[1] To describe the behavior of a finite array, the current distribution on each of its elements is represented by a finite number of eigencurrents or modes. Subsequently, the eigencurrents of the array are expanded in terms of these element eigencurrents. For uniform linear arrays of loops and dipoles, the array-eigencurrent expansions and their associated eigenvalues are investigated. We focus on their parameter independent and diagonalizing features, and on their interpretation in terms of far-field characteristics and (standing) wave behavior.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Eigencurrent Description
- 3. General Aspects of Array Eigencurrents
- 4. Far Fields of Array Eigencurrents
- 5. Array-Eigenvalue Behavior
- 6. Conclusion
- Acknowledgments
- References

[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.

### 3. General Aspects of Array Eigencurrents

- Top of page
- Abstract
- 1. Introduction
- 2. Eigencurrent Description
- 3. General Aspects of Array Eigencurrents
- 4. Far Fields of Array Eigencurrents
- 5. Array-Eigenvalue Behavior
- 6. Conclusion
- Acknowledgments
- References

[18] In this section, we describe a number of features of array eigencurrents of linear array of loops and dipoles that we have found in our research and we illustrate these features by examples.

[19] Separating the array eigencurrents of linear arrays of dipoles and loops in groups, we obtain *N* groups of *Q* array eigencurrents. To order the array eigencurrents within each group, we adopt the strategy of *Bekers* [2004], where we index the corresponding array eigenvalues along the curve that they generate in the complex plane. The first array eigenvalue is the one of the array eigencurrent with the lowest phase variance in its expansion coefficients with respect to the template eigencurrent of the group. We choose this way of ordering, since it will reveal some striking properties of array eigencurrents.

[20] In each group, the expansions of the array eigencurrents are mainly described by the template eigencurrent of the group. Its expansion coefficients are at least a factor 10 larger than those of the other template eigencurrents. Moreover, its coefficients depend only weakly on the frequency, the spacing, and the template-element geometry, or more general, on the template eigencurrent of the group.

[21] As an example of the dependence on the template eigencurrent of the group, Figure 2 shows the coefficient distributions of array eigencurrent one in Groups 1, 2, and 3 of a linear array of 40 loops with spacing *d* = 0.5 *λ*. Since {*u*_{1}, *u*_{2}} and *u*_{3} are symmetric and antisymmetric with respect to the axis of the linear array, respectively, the array eigencurrents in the corresponding groups do not couple. For this reason, Figure 2 does not show the expansion coefficients for *u*_{1} and *u*_{2} in the first array eigencurrent of Group 3 and the expansion coefficients for *u*_{3} in the first array eigencurrents of Groups 1 and 2. We clearly observe in Figure 2 that, in each group, the coefficients for the template eigencurrent of the group describe the dominant behavior of the array eigencurrent. The coefficients for the other template eigencurrents may be regarded as a perturbation of order −20 dB. We observe also that, for all three groups, the corresponding template eigencurrent has the same coefficient distribution in the expansion of array eigencurrent one. For the other 39 array eigencurrents in the groups, we observed the same phenomenon. In addition, the perturbation reaches −10 dB for specific array eigencurrents.

[22] As an example of the dependence on the spacing, we consider a linear array of 40 loops with spacings 0.38 *λ*, 0.5 *λ*, and 0.58 *λ*. Figure 3a shows the coefficient magnitudes for array eigencurrent 33 in Group 1. As in the previous case, the coefficients for the template eigencurrent of the group, i.e., *u*_{1}, describe the dominant behavior of the array eigencurrent. The coefficients of the other template eigencurrents can be viewed as a perturbation of order −10 dB. Note that the presented coefficients correspond to *u*_{2}; the coefficients for *u*_{3} are zero. We also observe that, for the three spacings, the magnitudes of the coefficients are not identical, but they clearly show the same patterns, both for the template eigencurrent of the group, *u*_{1}, and for *u*_{2}. The same is valid for the phases. Figure 3b shows the phases for the coefficients of *u*_{1} after multiplication by the phase factors exp(*j*(*p* − 1)*π*) with *p* the element index [*Bekers et al.*, 2006]. The phase distributions for the three spacings are not identical, but a closer look reveals that they exhibit a similar standing wave pattern, which is particularly pronounced for *d* = 0.5 *λ*. Apparently, for all three spacings the template eigencurrent has the same coefficient distribution in the expansion of array eigencurrent 33. For the other 39 array eigencurrents, we observed the same phenomenon. Moreover, as in the previous case, the perturbation reaches up to −10 dB.

[23] To illustrate the dependence on the element geometry, we compare the array-eigencurrent expansions of the linear array of loops and dipoles with *d* = 0.38 *λ* and with 80 elements. Figure 4 shows the coefficients of array eigencurrent 73 in Group 1. Clearly, the coefficient magnitudes and phases of the template eigencurrent of Group 1 match very well. After multiplication by exp(*j*(*p* − 1)*π*) they reveal the same standing wave pattern as in the previous case. The coefficient magnitudes of the other template eigencurrents do not match and constitute perturbations of −14 dB and −20 dB. As in the previous two examples, the same findings apply to the other 79 eigencurrents in Group 1.

[24] Considering the patterns of the array eigencurrents, we observe that in each group of array eigencurrents, the coefficients of the corresponding template eigencurrent exhibit standing wave patterns. For the first ⌊*Q*/2⌋ eigencurrents in the group, these patterns are immediately clear. For the last ⌊*Q*/2⌋ array eigencurrents these patterns emerge after multiplication by exp(*j*(*p* − 1)*π*) as we have seen above. Moreover, after this multiplication, array eigencurrent *q* and array eigencurrent *Q* − *q* + 1 show the same patterns. As a specific example Figure 4b shows the unprocessed phases of the coefficients of template eigencurrent one in array eigencurrent eight. These are approximately the same as the processed phases for array eigencurrent 73. Moreover, although not shown, the corresponding magnitudes show the same pattern as the one of array eigencurrent 73 in Figure 4a. After processing the coefficients of array eigencurrent 73, both array eigencurrents show a sine pattern of four periods.

[25] The results presented above extend our results in the work of *Bekers et al.* [2007], where we presented results for frequency and element-geometry dependence for a single template eigencurrent. Moreover, we used the parameter independent features of the array eigencurrents to compute the array behavior by the Rayleigh-Ritz quotient applied to fixed coefficient distributions for the array eigencurrents. More examples of array-eigencurrent patterns are given by *Bekers* [2004].

[26] The orthogonal nature of the standing wave coefficient patterns (sampled cosines and sines), suggests that the operator is, up to a perturbation, diagonalized by the sampled aperture distributions, where the ‘sampling points’ are the template eigencurrents. In the work of *Addamo et al.* [2006] we found the same result for linear arrays of slots in a leaky coaxial cable. Whether our findings apply also to other types of template elements and arrays remains to be investigated. In this context, we note that the parameter dependence of the template eigencurrents can become an issue for more complex template elements, since the template eigencurrents do not necessarily scale with their geometry parameters as they do for dipoles and loops. In those cases, it may be convenient to separate the domain of the template element into several regions. Similar strategies are applied in, e.g., the Synthetic Functions Approach [*Matekovits et al.*, 2007] and the Characteristic Basis Function Method [*Prakash and Mittra*, 2003].

### 4. Far Fields of Array Eigencurrents

- Top of page
- Abstract
- 1. Introduction
- 2. Eigencurrent Description
- 3. General Aspects of Array Eigencurrents
- 4. Far Fields of Array Eigencurrents
- 5. Array-Eigenvalue Behavior
- 6. Conclusion
- Acknowledgments
- References

[27] Contrary to the coefficient distributions, the far fields of the eigencurrents depend strongly on the spacing and the element geometry. To illustrate the behavior of eigencurrent far fields, we introduce the spherical angles θ and ϕ in the classical way, i.e., θ is the angle with respect to the positive *z* axis and ϕ is the angle in the plane of the array with respect to the positive *x* axis. We consider the far fields in the *xz* plane.

[28] The array eigencurrent 1 in Figure 2 can be viewed in several ways, e.g., as a standing wave of half a period on the aperture or as a cosine taper. The eigencurrent far field is a sum pattern with side lobes of −22 dB for Group 1 and 3 and a difference pattern for Group 2. The pattern differences are straightforwardly explained by the differences in the far fields of the template eigencurrents.

[29] The far-field patterns for the array eigencurrents in Figure 3 are shown in Figure 5. In this case, the pattern differences are entirely due to the spacing differences since we consider only array eigencurrents of Group 1 with corresponding template eigencurrent 1. For *d* = 0.5 *λ* and *d* = 0.58 *λ*, the patterns have clearly distinguishable main lobes at ±55° and at ±44.7°. The lobes correspond to linear phase tapers along the array with total phase reversals ((*Q* − 1) *kd*sinθ) of 100.4 rad and 100.0 rad. These reversals are approximately the same as the total phase reversal of the array eigencurrent (100.5 rad). Note that array eigencurrent 33 has 32 phase reversals along the array; only seven reversals are shown in Figure 3b because of the multiplication by exp(*j*(*p* − 1)*π*). These considerations suggest that an array eigencurrent will be specifically excited by a linear phase taper if its total phase reversal is the same as that of the taper. For *d* = 0.38 *λ*, the array eigencurrent radiates much less power into the far field; the array operates as a (reactive) end-fire array. Moreover, the array eigencurrent has a larger total phase reversal than the maximum total phase reversal of a linear phase taper (93.1 rad). Therefore, the excitation of the array eigencurrent will not be significant unless its array eigenvalue is close to zero. In that case, any phase taper can excite it, as we discussed in the work of *Bekers et al.* [2006]. In that case, the eigencurrent is a homogeneous solution of the operator equation *w* = −*v*^{ex}. Finally, for more examples of far-field patterns, we refer to *Bekers* [2004].

### 5. Array-Eigenvalue Behavior

- Top of page
- Abstract
- 1. Introduction
- 2. Eigencurrent Description
- 3. General Aspects of Array Eigencurrents
- 4. Far Fields of Array Eigencurrents
- 5. Array-Eigenvalue Behavior
- 6. Conclusion
- Acknowledgments
- References

[30] In this section, we consider the behavior of the array eigenvalues. We focus in particular on the linear arrays of 40 loops with the three spacings. Figure 6a shows the normalized array eigenvalues of Group 1. We clearly observe the strong spacing dependence of of these values. For example, the array eigenvalue 33 moves from the origin as the spacing increases. As a first atempt to interpret the eigenvalues, we compare the array eigenvalues of the finite linear array with those of the corresponding infinite array, where we sum the mutual-coupling contributions of the first 100 neighbors on both sides of a reference element. For the infinite array, the discrete spectra of the groups {*ν*_{nq}}_{q = 1}^{Q} turn into continous spectra *ν*_{n}(Ψ) with corresponding group eigencurrents *u*_{n}(Ψ). Here Ψ is the phase of the Fourier factor *e*^{jpΨ} (*p* = −100, …, 100) in the infinite-array series. Figure 6b shows that the continuous infinite-array spectrum of Group 1 follows the discrete spectrum of the finite array, except near the array eigenvalues 31 and 32. In this respect we note that at Ψ = ±*kd* + 2*πm*, the infinite-array series does not converge. The position Ψ = *kd* = 0.76 *π* is indicated on the curve. At this value, the first unit-cell mode of the infinite array changes from propagation into evanescent or vice versa. This observation suggests that the eigenvalues 30–40 correspond to evanescent modes on the finite array and the eigenvalues 1–29 correspond to propagating modes. The transition at eigenvalue 30 corresponds to a surface wave phenomenon. In the work of *Bekers* [2004] we have considered similar divergences, but for the case that the first grating lobe is in the plane of the array.

[31] The match between the two spectra of the finite and the infinite array suggests that each finite-array eigenvalue corresponds to a specific scan angle with associated Floquet wave and propagation constant. While the infinite array excites only one eigencurrent per group, the finite array excites Q array eigencurrents per group. Therefore, for the case presented in Figure 6 the infinite array will, in simulation, not reveal the finite-array resonance due to the 33rd array eigenvalue being close to zero. However, considering the match between the two spectra, we may be able to determine from the infinite array whether the corresponding finite-array suffers from a resonance phenomenon. Such a consideration may be combined with numerical search techniques as those described by *Li and Scharstein* [2006] and *Fischer et al.* [2005].

[32] More insight into the correspondences between array eigencurrents and Floquet waves may be obtained by considering an aperture in the *xy* plane that is infinitely long in the *y* direction and has opening [−*L*, *L*] in the *x* direction. The *y* component of the electric field distribution is then described by the Helmholtz equation in two dimensions. The eigenfunctions of the Helmholtz operator are then described by

where *ν* is the eigenvalue. Note that we have inserted a minus sign, since this sign is also present in our definition of or, more precisely, in the Helmholtz operator in (3). By separation of variables, *E*(*x*, *z*) = *X*(*x*)*Z*(*z*), we obtain

where the square roots need to be defined in correspondence with boundary condtions and propagation characteristics. These functions are the Floquet modes of this problem. In the literature, they are in general derived as homogeneous solutions, i.e., *ν* = 0. Instead, we have computed eigencurrents with nonzero eigenvalues. Only in case a resonance occurs, one of the array eigenvalues lies close to zero. If we take *ν* = 0 in (5) and we apply Dirichlet boundary conditions in *x*, we obtain the standing waves of the aperture, sin(*mπx*/*L*) and cos((2*m* − 1)*x*/2*L*), which are sampled at the antenna elements. For other values of *ν*, (5) may also provide insight into the propagation characteristics, which in our opinion merits further research.

### 6. Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Eigencurrent Description
- 3. General Aspects of Array Eigencurrents
- 4. Far Fields of Array Eigencurrents
- 5. Array-Eigenvalue Behavior
- 6. Conclusion
- Acknowledgments
- References

[33] For uniform linear arrays of dipoles and loops, we have demonstrated that the corresponding impedance operator is, up to a perturbation, diagonalized by the sampled aperture distributions, where the ‘sampling points’ are represented by a finite number of element eigencurrents, or template eigencurrents, on each of the elements. To achieve this, we expand the array eigencurrents in terms of the template eigencurrents. Then, the array eigencurrents are naturally separated into groups, where each group correspond to one specific template eigencurrent. The array eigencurrents in a group are mainly described by the corresponding template eigencurrent and their expansion coefficients with respect to this eigencurrent depend only weakly on parameters as the spacing, the frequency, and the element geometry, or more general, the template eigencurrent. In that group, the coefficients for the other template eigencurrents are of lower order.

[34] In each group of array eigencurrents the coefficients for the template eigencurrent of the group exhibit the same sequence of standing wave patterns. These patterns are linked to specific radiation behavior of the array, which can be observed from their far-field pattern of the array eigencurrents and by investigating the array eigenvalues. A comparison between the eigenvalues of a finite array and the corresponding infinite array reveals that the discrete spectrum of the finite array follows closely the continuous spectrum of the infinite array except when the grating lobe or a surface wave occurs. In this way, array eigencurrents and array eigenvalues can be linked to Floquet waves and their propagation constants. Further research on this remarkable correspondence is recommended.

[35] The eigenfunction approach provides a mathematically rigorous procedure for obtaining the response of a large, finite antenna array in terms of local expansion functions and global expansion coefficients. Moreover, the approach is internally consistent, since both the basis functions and the expansion coefficients are found from the eigenfunction problem for the impedance operator in an electric or magnetic field integral equation. The extension to a multilevel approach, in which subarrays are analyzed at an intermediate level, is therefore straightforward. The array eigenfunctions are characteristic of the induced surface current and the associated radiated far field. Therefore, they are the natural ‘ports’ for pattern synthesis.