Demanding performance required to antennas for new services and applications can be satisfied by recurring to arrays. A technical trend is in act for some years which sees array antennas more and more employed both on ground and in space applications.
In such situation the availability of modeling and design methods are obviously more and more important to reduce design risk and to reduce development time. A team composed of Italian universities and industry, focused by ESA, worked along the last years to develop methods and tools specifically suited to array modeling, with the aim to bridge the existing gap with respect to well assessed design procedures of other kind of radiators.

Different array configurations have been faced and specific methodologies for modeling have been developed, all of these sharing the features of accuracy (full-wave methods) and reduced computational cost:

- very large rectangular arrays, very effectively managed through the Truncated Full Wave - Floquet Wave, T(FW)2 algorithm developed by University of Siena; the overall array is simulated by taking into account both mutual coupling and truncation effects

- feed arrays of horns, managed by means of a hybrid GAM/MoM-SAF method developed by University of Florence and Politechnic of Turin; GAM (Generalized Admittance Matrix) to accurately model the internal propagation and the MoM-SAF to take into account the inter-element coupling

- arrays of patch antennas and metallic radiators (e.g. helices) by means of SIM-SFX method developed by Politechnic of Turin and Ingegneria Dei Sistemi; regular and irregular (i.e. not regular layout and also mixed radiators) configurations with 3D full dyadic Green Functions.

Furthermore, an innovative method for array synthesis have been developed by Politechnic of Turin, based on the Far Field Wavelet Expansion (FFWE) approach. The technique increases the efficiency of the synthesis process by working with an efficient representation of the radiated field that concentrates the information on the design task in fewer basis functions and therefore is able to converge faster.

All these methods and other interesting features to perform analyses and design have been integrated in the ADF-EMS (Antenna Design Framework - ElectroMagnetic Satellite) "Array Section", in which the user can found devoted tools for each working step, starting from array layout definition, excitation synthesis, preliminary modeling, detailed modeling, numerical optimization, tolerance analysis etc.

In this paper the main achievements of the before mentioned research activities are reported. Validations and real life applications up to date performed are demonstrated.

1 Introduction

It is now well known that periodic metamaterials can be used as superstrates to increase the directivity of a primary source. This paper depicts an efficient design method for a new type of antenna based on a slotted waveguide exciting such a periodic superstrate (Fig. 1). This method is based on the use of a MoM simulation code developed at UCL and the separate analysis of the guide and the metamaterial, i.e. separating the interior and exterior problems.

The exterior problem consists of a periodic metamaterial superstrate excited by a slot cut in a ground plane extending the top wall of the waveguide. Assuming that this ground plane is large enough, the problem can be seen as a periodic structure excited by magnetic currents M=-nXE, equivalent to fields in the slot, and flowing on an infinite ground plane. This kind of periodic structures illuminated by a single source can be analyzed very efficiently with our MoM simulation code and the Array Scanning Method (ASM) as described in [1]. By exciting only the TE₁₀ mode in the waveguide, one can assume that M_ext and H_ext, respectively the magnetic currents and H field in the slot for the exterior problem both have a sinusoidal distribution along the slot. Hence, one can find numerically the ratio of their amplitude Y_ext^HM.

The interior problem can then be seen as a closed waveguide (the slot is filled with metal) with two different excitations. The first excitation is the voltage at the feed probe and the second one comes from magnetic currents equivalent to fields in the slot. This structure can also be simulated, and using the superposition principle, we can find the relations binding I (amplitude of the current induced on the feed probe) with both V (amplitude of the voltage excitation at the probe level) and M_int (amplitude of magnetic currents along the slot).

When all the relations between fields and currents are known, the input impedance of the antenna, V/I at the proble level, can be determined by enforcing H_int=H_ext and M_int=-M_ext. With this method, the radiation pattern can be optimized in the exterior problem by tuning the properties of the superstrate. Then, under the single-mode approximation for fields in the slot, (1),(2) and (3) will allow the determination of the impedance at the waveguide feed level. This input impedance can be optimized by tuning the waveguide parameters in the interior problem.

5 References

[1] X. Dardenne and C. Craeye, ``ASM Based Method for the Study of Periodic Metamaterials Excited by a Slotted Waveguide,'' Proc. of the 2005 IEEE Symposium on Antennas and Propagation, Washington DC

Planar structures include many different geometries, ranging from cavity-backed microstrip antennas, to frequency-selective surfaces and photonic band-gap materials, the common denominator being the presence of planar dielectric layers and of metallic surfaces.

From the point of view of mathematical and numerical modeling, one of the most successful approaches, providing a general framework for this type of structures, is the Integral Equation (IE) model combined with a discretization procedure like the Method of Moments (MoM). Essential to the IE-MoM approach is the knowledge of a pertinent set of Green’s function (GFs).

A GF representation of a multilayered structure, when it is bounded laterally with a given boundary condition (BC), can use an infinite series expansion. Each term of this series contains two main components. The first one is a modal function satisfying the lateral BCs and the second one is a spectral component that accounts for the BCs introduced between the piled layers in the longitudinal direction.

However, after application of the MoM, the calculation of the reaction terms shows two main difficulties that are usually solved by different approaches depending on the BCs applied. These are: a) the slow convergence of the series and b) the evaluation of overlapping integrals arising in each of its terms (integration between every pair of modal and basis functions).

In the first case, the Kummer’s transformation speeds-up the convergence of the infinite sums by acting only on the spectral component of each series term. For the second case, we propose a unified approach that greatly simplifies the evaluation of the overlapping integrals arising under any transverse BC (PEC/PMC, PBC) and with irrotational and constant-divergence basis functions (almost universally used for this type of problems).

First, the initial overlapping integrals are reduced from surface to contour integrals by expressing the modes in potential form and exploiting the characteristics of the basis functions. Then, assuming a polynomial variation of the basis function on the cells’ contour, all the possible overlapping cases (combinations of e,h-field and TE,TM modes) will be reduced to the evaluation of simply two line integrals. At this point, these integrals can still be solved analytically using the appropriate modal functions that satisfy the pertinent BCs of the problem.

This unified approach in the definition of Green’s functions and of the MoM overlapping integrals is, in addition, very efficient due to the analytical formulation of the integrated terms and the fast convergent series. The reduced complexity of the final solution also allows an easy check of the numerical stability and convergence of the algorithm as well as a considerable reduction of the computer code needed for its implementation.

As an example, analytical solutions for the aforementioned BCs on canonical shapes (i.e. rectangular and circular PEC/PMC and PBC on triangular lattice) together with RWG and rectangular rooftops will be presented and applied to the simulation of practical structures.

The thin circular loop antenna has been repeatedly addressed in the literature, and closed form expressions for the relevant current distribution have been well established [1]. However, these formulas involve complicated integrals of Lommel-Weber, Bessel and Modified Bessel functions, which render numerical computations cumbersome and inefficient, and thus their applicability is practically limited to loops smaller than 6 wavelengths in circumference [2]. On the other hand, numerical techniques may become unstable for very large loops, due to poor matrix conditioning, making convergence tests time-consuming and tedious. To alleviate these complications, a novel method was very recently devised, which accurately calculates the current and input conductance of very large loops, extending to hundreds of wavelengths in circumference [3]. The technique is based on the properties of circulant matrices, which are analytically invertible, regardless of their order. The algorithm, which yields semi-analytical expressions for the current distribution, is numerically robust and can be easily implemented, since it invokes almost exclusively elementary functions. However, for reasons of simplicity, [3] focuses on a delta source model only, and hence it does not present any data for the input susceptance, whose series expression can be proven to diverge, for increasingly finer discretization. The objective of this paper is the accurate computation of the input susceptance of arbitrarily large loops, using the basic mathematical procedures of [3], in conjunction with an alternative excitation simulation. Specifically, the magnetic frill generator has been proven to yield convergent values for the susceptance [4], and is therefore utilized in this work. However, the analysis in [4] is based on the classical current expressions of [1], and its usefulness is therefore limited to small loops. On the contrary, the technique proposed herein is much less elaborate, and much more efficient than [4], deriving semi-analytical expressions and yielding accurate susceptance values for loops as large as in the configurations examined in [3].

[1] T. T. Wu, "Theory of thin circular loop antenna", J. Math. Phys., vol. 3, pp. 1301-1304, Nov.-Dec. 1962. [2] L. -W. Li, C. -P. Lim and M. -S. Leong, "Method of moments analysis of electrically large circular loop antennas: non-uniform currents", IEE Proc. -Microw. Antennas Propag., vol. 146, no. 6, pp. 416-420, Dec. 1999. [3] H. T. Anastassiu, "Fast, simple and accurate computation of the currents on an arbitrarily large circular loop antenna", IEEE Trans. Antennas and Propagation, AP-54, no. 3, pp. 860-866, Mar. 2006. [4] G. Zhou and G. S. Smith, "An accurate theoretical model for the thin-wire circular half-loop antenna", IEEE Trans. Antennas and Propagation, AP-39, no. 8, pp. 1167-1177, Aug. 1991.

Current standardized procedures for measurements of the Specific Absorption Rate (SAR) of mobile phones and radio base station antennas include a volumetric scan of the electric field strength induced in a head or body phantom. Assessment of multi-band and whole-body SAR requires repeated volumetric scanning over a large part of the phantom and is time-consuming. In order to reduce the total evaluation time, different methods have been proposed to estimate the SAR from measurement data based on sparse volumetric scanning and surface scanning. These methods rely on data fitting with underlying assumptions about the spatial distribution of the fields. In order not to be biased by previous or current antenna design, and to be able to use currently available assessment systems, a model-independent dual-plane-scan method is investigated based on amplitude measurements of the electric field components. The amplitude of the electric field components are measured in two planes close to the phantom surface, and the phase is recovered using an iterative process. The plane wave spectrum of the resulting complex electric field components is then used to propagate the field into the phantom. The measurement time is typically reduced by a factor 5 and in some cases even more. Furthermore, the plane wave spectrum is utilized for fast calculation of the mass-averaged local SAR values. A numerical tolerance study, using single and multi-peak fields with relevant errors superposed, is performed to demonstrate the robustness of the method. The resulting errors in the estimated SAR values are below 1\% for realistic positioning errors and signal to noise ratio. Comparisons with measurements in a flat phantom are also made. Moreover, the underlying algorithm can be applied to curved surfaces.

The paper will investigate the transparency of various multi-layered frequency-selective surfaces (FSSs) with relatively basic element shapes and compare simulation results to microwave measurements. Simulation results are carried out with two in-house software tools for FSSs analysis which are very different in terms of the applications they are intended for:

The second tool follows a rigorous periodic method of moments (PMM) approach. In contrast to Ben Munk’s work [1], it implements a mixed current formulation, allowing arrays with magnetic and electric currents to exist simultaneously, and current distributions are represented by rectangular and triangular rooftop functions. The FSS geometry is generated with a layout editor. In future, active components will be included in the FSS layout. The tool is intended as building block for rigorous simulation of complete quasi-optical systems.

The paper will compare the two simulation techniques for multi-layered FSSs with simple element types. The effect of finite metal thickness will be presented and simulations will be compared to microwave measurements.

For example, a free-standing FSS consisting of two identical arrays of horizontal, rectangular slots was investigated. (Parameters: slot length =2.5mm, slot width=0.25mm, unit cell spacing = 2.8mm in both dimensions, distance between arrays = 2mm). Figure 1 shows the simulation results of both methods for various angles of incidence for TE polarisation, assuming zero metal thickness.

Fig. 1. Transmission of rectangular slot FSS with EC modelling and with PMM
1. Munk, B.A. Frequency Selective Surfaces: Theory and Design, John Wiley & Sons, 2000.

The solution of electromagnetic problems in periodic structures by means of numerical full-wave methods requires the efficient and accurate computation of periodic Green's functions. The application of the Floquet-Bloch theorem reduces the infinite computational domain to a single unit cell, but leads to the evaluation of very slowly converging series.

Different techniques have been developed in the past to accelerate the convergence of the relevant series; we limit here our attention to the free space Green's functions. Poisson's formula and Kummer's decomposition represent the most commonly adopted approaches. Numerical techniques like the Shanks' transformations or the rho algorithm may also be necessary to further accelerate the result. Improved convergence can be obtained by applying the Ewald's transformation, which splits the Green's function into two contributions; the computational burden between them is optimized by carefully choosing a splitting parameter. An implementation of the Ewald's transformation has been proposed in the literature for 2-D lattices and for an array of line sources with 1-D periodicity. Furthermore, an integral representation has been proposed by Veysoglu to convert free-space 2-D Green's function with 1-D periodicity to an integral representation whose integrand exhibits gaussian decay.

Though a real phase shift among adjacent cells is usually considered in the evaluation of the periodic Green's functions, the possible extension of the different acceleration techniques to the case of general complex phase shift is necessary when modal analysis is performed. Such extension is not trivial: in fact, when the free-space periodic Green's function is used to study complex modes (i.e., modes with complex propagation constant) some of the involved series can diverge. This behavior is a consequence of the fact that complex modes are not required to verify the usual boundary conditions at infinity, even if they can be excited by finite sources and can describe a physical field in limited regions of space.

In this work we treat in detail the acceleration of the free space Green's function in 3-D structures periodic along one direction in the general case of complex phase shifts. We extend the application of the Ewald's method by performing an optimization of the relevant splitting parameter. The two resulting series have both a gaussian convergence, but they require the computation of a double series and of the complementary error function of complex argument. In addition, we obtain a new integral representation of the Veysoglu-kind of the Green's function, which transforms the original series in an integral. Finally, we present a spectral Kummer-Poisson's decomposition, which grants an exponential convergence. An extensive comparison among these techniques is performed and the most efficient way to compute the Green's function is found to be the latter one. Every method presented allows for an easy extraction of the spatial singularity, needed in applications in the spatial domain; the integral representation and the Kummer's decomposition holds for general complex phase shifts, while the Ewald's method is possible only with real phase shifts.

The accuracy problem of magnetic-field integral equation (MFIE) and combined-field integral equation (CFIE) with Rao-Wilton-Glisson (RWG) basis functions applied on very large scattering problems is reported. Our recent studies on the inaccuracy of the MFIE for moderate-size problems have shown that the source of the error is the RWG functions and the accuracy can be improved by decomposing the RWG functions into first-order-complete linear-linear (LL) basis functions. (O. Ergul and L. Gurel, "Improving the accuracy of the magnetic-field integral equation with the linear-linear basis functions," Radio Science, to appear.) On the other hand, it was thought that the accuracy problem is limited to small geometries, especially those including sharp edges and corners. In this work, we show that MFIE and CFIE with the RWG functions are significantly inaccurate even for very large and smooth geometries, and the accuracy can again be improved with the LL functions.

Table 1 lists the values of the forward-scattered electric field from a sphere of radius 6 lambda illuminated by a plane wave. The scattering problem is solved by multilevel fast multipole algorithm (MLFMA) employing MFIE and CFIE. The numerical integrals on the basis and testing functions are evaluated with at most 1 % error and the far-field interactions are calculated with three digits of accuracy. However, Table 1 shows that the relative error compared to the Mie-series solution is above 2 % for the discretization with 132,003 RWG functions corresponding to a triangulation with lambda/10 mesh size. This inaccuracy is unacceptable considering all the efforts for controlling errors in an MLFMA implementation. We observe that the relative error decreases below 1 % by employing 528,786 RWG functions. On the other hand, only 65,724 LL functions corresponding to triangulation with lambda/5 mesh size give more accurate results in spite of coarser modeling of the curvatures. Comparing the number of unknowns required for the same level of accuracy, it is observed that the LL functions are much more efficient compared to the RWG functions for the MFIE and CFIE. In the presentation, we will provide more examples on the significance of employing the LL functions in the MFIE and CFIE.

A new 3D Fast Physical Optics method is presented for computing backscattered fields at high frequencies of PEC parabolic surfaces. This method can be easily extended to more complex surfaces. In virtue of Cauchy theorem a suitable surface deformation is used to convert a double highly oscillatory integral into a double slow varying integral, easily calculable by means of Gaussian quadrature.

High frequency scattering produced by arbitrary shaped bodies is conventionally analyzed by using different asymptotic methods such as PO. In this article a new 3D Fast Physical Optics (3DFPO) method is introduced to solve electrically large scattering problems. The basic approach presented here consists of an efficient and accurate evaluation of the PO integral, which becomes very time consuming at high frequencies. The efficient evaluation of highly oscillatory integrals is a problem with a great number of applications within different fields of science such as: quantum cosmology, nuclear physics, acoustic scattering or, in the matter at hand, electromagnetic scattering.

This paper shows a new method of integration by means of complex variable techniques. In virtue of the Cauchy theorem, the path of integration is modified, so that the double highly oscillatory integral is converted into a double integral extended on a bidimensional manifold with a slow varying behaviour, easily calculable by means of bidimenisonal Gaussian Quadrature.

The above mentioned method is applied to calculate the scattered field produced by one parabolic surface. The resolution used was 30 points per degree. The following figure (fig 2) shows the scattered field at D=96 Lambda. The total computation time is 9 min 10 seconds with numerical brute force integration and 10 seconds with the integration method above mentioned. We can see the agreement between the numerical brute force integration (green) and the method above mentioned (blue). The scattered field at D=3200000 Lambda is shown in fig 2. The total computation time is 10 seconds.

The dependency of the solution time with the frequency in both numerical brute force integration and 3DFPO method is shown in fig 3. We observe that the computation time does not increase with frequency.

Fig 1 Fig 2 Fig 3

Substrate integrated circuits [1] are an effective alternative to metallic waveguide structures. Their realization relies on the consolidated PCB technology inheriting its advantage and in particular the simplicity of the realization process and reduced costs. The basic device is the substrate integrated waveguide (SIW). This structure is built realizing, in a conventional board of laminate, a waveguiding channel with an array of metallic via holes. SIWs inherit the advantage of metallic waveguides like high power capacity and low dispersion and combine them with the possibility of integration typical of microstrip structures. Recently many SIW based devices like filters and antennas have been presented [2]. The analysis of SI structures has been carried out with several methods [3],[4]. In [5] the EFIE integral equation has been solved with a filamentary unknown current flowing on the metallic posts. In this paper a full wave analysis of substrate integrated structures is proposed. In particular the cylindrical wave expansion of the magnetic dyadic Green’s function for the parallel plate waveguide is firstly considered. Then the scattering from the metallic posts is included to obtain the Green’s function of the post walled structures. In the present analysis only magnetic current sources have been considered. Radiating slots have been also taken into account considering the equivalent magnetic current distributions and imposing the continuity of the field along the slot. The resulting integral equation has been solved with the Galerkin method of moments. As a preliminary result the analysis of a SIW structure already presented in literature has been reproduced. In fig.1 is shown a filter obtained inserting posts in a SIW with characteristics reported in [2]. In fig.2 are shown the simulated S11 and S12. It is observed that they compare favorably with the measurements published in [2]. More results will be shown in the presentation.

_{Fig.1 SIW filters as in [2]}

_{Fig.2 Return losses: full wave simulation(solid line), measurements in[2](*), HFSS(--)}