### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Formulation
- 3. Planar Array Synthesis
- 4. Numerical Results
- 5. Conclusions
- Appendix A:: Derivation of the Expansion Coefficients
- Appendix B:: Electromagnetic Fields Representation
- References
- Supporting Information

[1] An efficient uniform high-frequency representation of the radiation from a large irregularly contoured planar tapered phased array, which is accurate throughout the near, intermediate, and far field zones, is presented. The planar array is regarded as the superposition of a sequence of parallel line arrays, and each one of them is represented in terms of equivalent continuous tapered lines via the Poisson summation formula. The amplitude tapering along the line is asymptotically approximated by a physically based, observation-dependent linear combination of a small number (seven at most) of equiamplitude linearly phased excitations, which allows the reconstruction of the dominant contributions. After superimposing the asymptotic contributions of each line array, the total radiated field from the planar array is described in terms of rays emerging from the actual endpoints on the array rim and from a discrete sequence of points in linear loci on the surface of the array. As a consequence, high-frequency techniques may effectively be used to describe the interaction of these rays with the surrounding environment.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Formulation
- 3. Planar Array Synthesis
- 4. Numerical Results
- 5. Conclusions
- Appendix A:: Derivation of the Expansion Coefficients
- Appendix B:: Electromagnetic Fields Representation
- References
- Supporting Information

[2] The actual performance of any antenna is affected by the operating environment, which may cause pattern distortion, loss of signal due to blockage, multipath interference and interantenna coupling. Furthermore, on-site power density maps are needed to predict radiation hazards. As a consequence, in order to effectively design and install an array, an accurate and efficient modeling of its interaction with the surrounding environment is important. The relevant analysis requires an effective description of the radiated field in both the near and the far zones. However, the application of a full wave method frequently results in an overwhelming computational burden because of the very large electrical size of the antenna under investigation and because of the surrounding complex platform. This suggests to resort to high-frequency methods. In the present case, it is assumed that the current on the antenna is not significantly perturbed by the presence of the platform; then, the environment effects are introduced by means of asymptotic techniques, such as the uniform theory of diffraction (UTD) or the physical optics (PO). In this framework, the radiation from array antennas may be described in terms of rays emanating from individual array elements, or from small subarrays with well defined phase centers. However, this approach becomes highly inefficient for very large arrays because of the huge number of rays. A more convenient strategy consists in collectively representing the array radiation as the superimposition of continuous truncated Floquet wave (FW) distributions, defined over the entire aperture of the array [*Ishimaru et al.*, 1985; *Carin and Felsen*, 1992]. This approach has been proven to be very effective for the analysis of rectangular arrays [*Capolino et al.*, 2000a, 2000b, 2000c; *Maci et al.*, 2001]. However, for more irregularly contoured arrays, the number of ray contributions increases and an intrinsic ambiguity occurs in the definition of diffracting edges and vertices. In order to overcome this limitation, an alternative approach has been proposed by *Martini et al.* [2003]. It basically consists of reconstructing the field radiated by the planar array as the superimposition of the asymptotically dominant contributions from the constitutive parallel line arrays. These contributions are conical FWs truncated at their relevant shadow boundaries, spherical diffracted waves arising from the array tips and transition contributions across the FW shadow boundaries [*Capolino and Felsen*, 2002]. *Martini et al.* [2003] showed that this approach yields very accurate results when uniform amplitude line arrays are considered; however, the extension to the case of tapered arrays is not straightforward, since the strong localization of the dominant contributions leads to inaccuracies when the observation points approaches the far field region. A slowly varying tapering can be accommodated by weighting the conical wave and the two diffracted spherical waves by the value of the tapering function at the stationary phase point and at the endpoints, respectively. The accuracy can be improved by introducing in the asymptotic representation higher-order terms, related to the derivatives of the tapering function and to slope diffraction effects, as shown by *Mariottini et al.* [2005] for the strip array case. However, although the introduction of few terms is satisfactory for describing the field radiated in the vicinity of the array, this asymptotic approximation gradually fails as the observation point moves from the Fraunhofer to the Fresnel zone. There, the asymptotic ray description needs to be replaced by a more conventional FFT numerical representation of the radiated field.

[3] Different procedures, based on a Fourier transform rigorous decomposition of the tapering function into a set of uniform amplitude, phased excitations, have been proposed for both planar [*Nepa et al.*, 1999] and line [*Cicchetti et al.*, 2003] arrays. Although these representations exhibit a high flexibility, the number of uniform excitations required to accurately represent the radiated field depends on the array size and tapering. This partially nullifies the benefits deriving from the use of the Floquet wave concept.

[4] An alternative solution to provide a uniform description from the near to the far region, limiting at the same time the number of ray contributions, is presented in this paper. While previous approaches rely on a mathematical transform to precisely represent the current distribution in terms of exponential functions, the expansion employed in this work is directly defined to provide an accurate estimate of the radiation integral. To this end, the asymptotically dominant characteristics of the integrand function are conveniently reconstructed by approximating the tapering along the line array through a proper superposition of a few equiamplitude, linearly phased excitations, whose number is small (seven at most) and completely independent of the array size and tapering. In particular, these excitations are linearly combined to match the actual tapering function and its derivative at the endpoints and at the stationary phase point of the spatial integral along the line. Also, an additional constraint is imposed on the approximated tapering function in order to match the area subtended to the line array. This adaptive asymptotic procedure is able to uniformly cover the region ranging from the near to the far field zone. Once the field from each tapered array has been obtained, the planar array radiation is calculated by superimposing the line arrays dominant contributions. This leads to a limited number of conical and spherical rays emanating from points located on the array surface and on the actual contour of the array, respectively.

[5] It is worth pointing out that the present approach can also be employed to describe the scattering from a periodic arrangement of elements illuminated by a plane wave. Furthermore, it can be applied to speed up the computation of the field radiated by a current distribution calculated with a numerical method. Finally, by exploiting the concepts presented by *Civi et al.* [2000], *Neto et al.* [2000a, 2000b], and *Cucini et al.* [2003], the proposed field representation can be used as a basic constituent for an efficient full wave analysis of large phased-array antennas.

[6] The problem is formulated in section 2. The high-frequency representation of the radiation from a linear tapered phased array is derived in section 3. In section 4, it is shown how these results can be employed to build up the radiation from the planar array. Finally, some representative numerical results are presented in section 5, and conclusions are drawn in section 5.

### 2. Problem Formulation

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Formulation
- 3. Planar Array Synthesis
- 4. Numerical Results
- 5. Conclusions
- Appendix A:: Derivation of the Expansion Coefficients
- Appendix B:: Electromagnetic Fields Representation
- References
- Supporting Information

[7] The geometry for an arbitrarily contoured planar phased array of parallel elements in free space is depicted in Figure 1a. For the sake of simplicity, two perpendicular periodicity directions have been assumed and a rectangular reference system (*x*_{0}, *y*_{0}, *z*_{0}) is defined with its *x*_{0} and *y*_{0} axes parallel to them. However, it should be noted that this procedure is also directly generalizable to arrays with non orthogonal grids. No restrictions are imposed on the *x*_{0} dependence of the excitation, while along the *y*_{0} direction the amplitude tapering is assumed to be smooth and the phase predominantly linear. For current distributions provided by a numerical analysis, a smooth tapering can be obtained by filtering out rapid variations, which does not provide significant contributions to the field radiated away from the array plane and edges.

[8] In the following, an array of elementary electric dipoles will be considered for the sake of simplicity; however, the procedure is easily applicable to arrays with different elements by weighting each ray contribution by the relevant element pattern [*Capolino et al.*, 2001].

[9] The radiated electric field by the planar array is obtained as the superposition of the radiated fields by linearly phased, tapered finite line arrays that are parallel to the *y*_{0} axis (Figure 1). Therefore the basic constituent of the procedure is the formulation and asymptotic evaluation of the radiated field by a tapered phased line array.

[10] The geometry for this problem is depicted in Figure 1b. The array consists of *N* elementary -oriented electric dipoles, with interelement period *d*; the excitation *I*_{n} of each element is a time harmonic function given by *I*_{n} = *f*_{n}*e*^{−jkd(n−0.5)cosγ}, *n* = 1, …, *N*, where *k**d* cos γ is the interelement phasing and *f*_{n} is a complex coefficient accounting for non uniform amplitude and phase excitation. Local cylindrical (ρ, ϕ, *z*) and spherical (*R*, β, ϕ) reference systems are introduced, with their *z* axis along the line array and their origin half a period shifted beyond the first array element. The magnetic vector potential radiated by the array at *P* ≡ (ρ, ϕ, *z*) is obtained by an element-by-element summation over the individual dipole contributions:

where *R*_{n} = is the distance from the observation point to the *n*th dipole. By applying the Poisson summation formula [*Civi et al.*, 1999] the summation in (1) is rigorously rephrased as

where *k*_{zp} = *k* cos γ + are the corresponding infinite line array FW wave numbers and *f*(*z*′) is a continuous complex function which varies slowly with respect to the wavelength, and satisfies the condition *f*[(*n* − 0.5)*d*] = *f*_{n}. The function *f* may be either given in analytic form or reconstructed by polynomial interpolation of its sampled values. Notice that the Poisson summation formula has been applied to give an equivalent continuous line whose tips are half a period shifted toward the external side with respect to the array elements. This eliminates the residual spatial contribution from the final representation and locates the diffracted contributions at two points symmetrically disposed with respect to the array elements.

[11] The summation in (2) can be interpreted as the superimposition of a series of truncated FW contributions, consisting of the radiation from finite continuous current lines, characterized by the same tapering *f* and different phase gradients *k*_{zp} = *k* cos γ_{p}. The FWs are propagating for ∣*k*_{zp}∣ < *k* and evanescent otherwise. For propagating FWs and real *f*, the integral in (2) exhibits a stationary phase point at *z*′ = *z*_{s} = *z* − ρ cot γ_{p}. For *f* complex, *z*_{s} still provides a good estimate of the stationary phase point position, provided that the phase of *f* is a slowly varying function. This results in the condition ε′(*z*_{s}) ≪ *R*″(*z*_{s}), where ε(*z*′) is the phase of *f*(*z*′). On the other hand, for evanescent FWs there is not a stationary phase point along the line, and the spatial integral is dominated by the end points *z*′ = 0, *z*′ = *L*, where *L* = *Nd* indicates the line length. Upon these considerations, the contribution of each continuous line is asymptotically evaluated as described in the following subsections.

#### 2.1. Tapered Phased Current Line

[12] In order to uniformly asymptotically evaluate the integral in (2), the tapering function *f*(·) is approximated by a linear combination of exponentials as follows

with

[13] The expansion coefficients *C*_{i}, _{i} are designed to match those characteristics of the tapering function which asymptotically mostly affect the radiation. In particular, these are the value and derivative of *f*(·) at the two endpoints and at the stationary phase point plus the area subtended by *f*(·). Hence the following conditions are imposed

The conditions (5c) imply a dependence of the equivalent current on the position of the observation point; however, they are only required for propagating FWs and when the stationary phase point lies within the finite line; otherwise, these conditions are neglected, therefore reducing the number of terms in the summation. In any case, the equations (5) lead to a square linear system, whose solution provides a closed form expression for the expansion coefficients, as shown in Appendix A.

[14] Figure 2 shows, for different positions of the observation point, the comparison between the approximated and the actual current distributions along a uniformly phased current line fed with a Taylor amplitude taper ( = 5, *SLL* = −20 dB); the position of the stationary phase point *z*_{s}, which moves with the observer, is indicated by a circle. It is apparent that the two curves differ significantly only in some portion of the line, but they coincide at the critical points and the area subtended by the two curves is always the same. These conditions are sufficient to guarantee a good agreement between the values of the relevant radiation integrals both in the near and in the far zone. (See also Animation 1.) In Figure 3, the real part of the integrand function is plotted for various observation distances. The first curve, relevant to a near field observation, clearly illustrates how in this case the main contributions to the radiation come from a region around the stationary phase point, and from the end points. The integrand oscillates so rapidly over the remaining domain that the relevant integral provides a negligible contribution to the overall result. However, the radiation phenomenon becomes less and less localized for increasing observation distances. Indeed, as the observer moves away from the array, the oscillatory behavior of the integrand function gradually slows down and the value of the radiation integral tends to depend on the whole current distribution along the line. At those aspects, the integrand function shape closely resembles that of the tapering function and the accuracy of the approximation is guaranteed by the condition on the subtended area. (See also Animation 2.)

[15] The insertion of (3) in (2), when *z*_{s} ∈ [0, *L*], leads to

where *k* cos γ_{pi} = *k*_{zp} + Φ_{i}.

[16] The summation in (6) can be interpreted as the superimposition of contributions from uniform finite current lines characterized by different phase gradients. Although it resembles a truncated Fourier representation, it has been obtained in a completely different manner. As a consequence, the number of uniform excitations required is always very small (seven at most), and completely independent of the actual length and tapering of the line array.

[17] The uniform asymptotic evaluation of the single line contribution is presented in the following subsection.

#### 2.2. Uniform Current Line

[18] The radiation of a current line of length *L* with phase gradient *k* cos γ_{pi} can be obtained as the difference between the radiation of two spatially shifted semi-infinite lines. For a semi-infinite line with a vertex at *z* = 0, the magnetic vector potential is given by

Introducing in the above expression the spectral representation of the free space Green's function [*Felsen and Marcuvitz*, 1994], interchanging the order of the integrals, and analytically performing the inner integral, leads to

where *k*_{ρ} = and the branch of the root is chosen so that *m*{*k*_{ρ}} < 0 for *k*^{2} < *k*_{z}^{2} and *k*_{ρ} > 0 for *k*^{2} > *k*_{z}^{2} on the top Riemann sheet of the complex *k*_{z} plane. The approximation of the Hankel function for large argument can be introduced under the hypothesis *k*_{ρ}ρ ≫ 1 (a distance of λ between the observer and the array axis is sufficient to have a very good approximation), thus obtaining

Using spherical coordinates in both spectral (*k*_{z} = *k* cos θ) and spatial domains (*z* = *R* cos β) leads to

with *G*(θ) = .

[19] This spectral integral representation is finally uniformly asymptotically evaluated for observation points far from the line (*kR* ≫ 1), using a Van der Waerden procedure as in work by *Martini et al.* [2003].

[20] The final result for the finite line may be interpreted as the sum of a truncated conical wave emerging from the inner part of the line (only for propagating modes) and two endpoint diffracted spherical waves arising from the tips, along with the relevant compensation terms. Specifically,

where

is a single indexed FW truncated at conical shadow boundaries and

is the summation of a spherical wave contribution arising from the line *l*th tip and the relevant compensation term. In (12) and (13), *U*(·) is the Heaviside unit step function, β_{SB} is the shadow boundary (SB) angle, (*R*_{l}, β_{l}, ϕ_{l}) are the coordinates of the observation point in a spherical system centered at the line *l*th tip (see Figure 1b), *F*(·) is the UTD transition function and δ_{pil}^{2} = 2*kR*_{l} sin^{2}. The asymptotic representation is accurate and well behaved at most observation aspects. Inaccuracies may only arise when the observer approaches the array axis and, simultaneously, the conical wave front collapses (γ_{pi} and β_{l} 0), owing to the approximation of the Hankel function for large argument.

[21] The electromagnetic fields can be represented through integral expressions whose arguments only differ from that in (10) for a smooth and slowly varying term. As a consequence, the derivation of the relevant asymptotic representations is quite similar to the one presented above for the magnetic vector potential. The explicit expressions are reported in Appendix B, for the sake of completeness.

[22] Finally, it is worth noting that the results in (11)–(13) can also be obtained by applying a stationary phase uniform asymptotic evaluation to the space domain integral in (7) [*Felsen and Carin*, 1994]. This allows one to quite straightforwardly generalize the previous procedure to weakly aperiodic arrays by introducing the concept of local periodicity. In practice, this implies that the radiated field can be represented as the superimposition of locally adapted FW contributions, which are obtained from (11) and (13) by replacing the global periodicity *d* with the local periodicity at the footprint. Such an extension has relevance, for instance, for reflect array applications. Similarly, by letting the distance ρ from the array axis be a slowly varying function of the coordinate *z*, the formulation can be extended to accommodate arrays whose elements do not all lie on the same line, but are slightly displaced from a central axis, as those realized with slotted waveguides.

### 3. Planar Array Synthesis

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Formulation
- 3. Planar Array Synthesis
- 4. Numerical Results
- 5. Conclusions
- Appendix A:: Derivation of the Expansion Coefficients
- Appendix B:: Electromagnetic Fields Representation
- References
- Supporting Information

[23] The radiation from the planar array is reconstructed by superimposing the contributions (11) corresponding to the various uniform lines. Its final global form is represented as the sum of diffracted fields arising from the actual rim plus conical waves from points located along hyperbolic lines on the array surface (Figure 4). The efficiency of the approach relies on the fact that the infinite summation in (6) is very rapidly convergent since, in most practical instances, only one or two conical FWs for each line array and linear phasing are propagating, while the remaining are evanescent, thus providing only diffracted contributions. Since the difference between the phase gradients corresponding to the same FW (index *p*) decreases for increasing lengths *Nd*, it can be inferred that for the cases of interest (large arrays) the nature of the mode (propagating or evanescent) usually does not change with index *i*. As far as diffraction terms are concerned, for a given phase gradient Φ_{i}, all the infinite FW contributions arising from the same tip can be rigorously collected in a single term by exploiting the following relationship

Finally, the compensation terms are required only in proximity of the SBs.

[24] The asymptotic expressions derived are accurate when the observation point is both in the far field region of each array element and at least a few wavelengths from the array boundaries and from the linear arrays axis. The final representation is simple, physically appealing and significantly more efficient than the element-by-element summation also for linear arrays consisting of a few tens of elements. Therefore it is particularly well suited to be used as a first step in a PO procedure to model the interaction of a large array with the surrounding environment; in these cases, indeed, typically the scattering obstacles are outside the main beam region, and therefore are illuminated only by the diffracted fields. Furthermore, by interfacing the proposed procedure with a GTD solver, a significant speeding up of the ray tracing process in complex platform can be obtained.

### 4. Numerical Results

- Top of page
- Abstract
- 1. Introduction
- 2. Problem Formulation
- 3. Planar Array Synthesis
- 4. Numerical Results
- 5. Conclusions
- Appendix A:: Derivation of the Expansion Coefficients
- Appendix B:: Electromagnetic Fields Representation
- References
- Supporting Information

[25] In order to assess its accuracy and feasibility, first the present approach has been applied to evaluate the free space radiation from an irregularly shaped, tapered planar phased array consisting of 2160 elementary magnetic dipoles. The array, whose geometry is depicted in Figure 5a, lies on the *xy* plane of a cartesian reference system and the radiated field is observed in the plane ϕ = 0°.

[26] In the first example, the dipoles are excited with a Taylor current distribution with SLL = 20 dB and = 3, and phased so as to obtain broadside radiation. Figure 5b shows a remarkable agreement between the numerical results from the present asymptotic solution and the reference solution, obtained through an element-by-element summation, both in the near and in the far zone of the array.

[27] As a further example, the interaction of the same array with a nearby metallic obstacle has been analyzed. The geometry for the problem is depicted in Figure 6a; the array is fed with a Gaussian amplitude tapering with σ = 14λ and phased for a bearing angle θ_{0} = 10°, ϕ_{0} = 0°. The present procedure has been employed to obtain both the unperturbed incident electric field from the array at the observation point and the unperturbed magnetic field on the plate. Then, the scattering from the plate has been computed through a discrete Fourier transform. In Figure 6b the results provided by this procedure are successfully compared with the results provided by a physical optics simulation performed using a commercial software (FEKO). The saving in computation time is in this case significant (more than a factor of 13), being the obstacle outside the main beam region, and hence illuminated only by the diffracted field of the array. An even more remarkable gain in computation times is obtained when the rays have to be tracked through a complex platform; indeed, with the proposed approach, the number of ray contributions to be accounted for is given by the number of endpoints of the line arrays (108 for the array considered), instead of by the number of dipoles (2160).