Frequency Selective Surfaces (FSS's) are periodic arrays of patches and/or slots etched on a metal plate, having frequency and angular ??ltering properties. The FSS response to an excitation (for example a plane wave) is characterized in terms of its re ection and transmission coe??cient, and depends on the element type (slot or patch), the element shape (loop, cross, ...) and the array grid (rectangular or triangular). Cascading a number of FSS's to each other allows achieving larger bandwidths and steeper roll o??s. The angle dependence of the response can be reduced by sandwiching the FSS's between dielectric slabs. As a special application, the (multi-layer) selective structure is integrated with an array antenna: for example to reduce the antenna's radar cross section or to obtain wideband/multi-frequency antennas. The analysis and the design of a multi-layer FSS, possibly integrated with an antenna, requires the availability of a CAD package based on an e??cient analysis methodology. In this respect, microwave network techniques constitute an appropriate choice because of their modularity. Originally introduced for the analysis of waveguides, they have been extended to periodic structures. The basic idea is to derive a representation of layers and transitions between adjacent layers in terms of equivalent networks. The entire structure is then represented by the cascade of these networks. Di??erent choices of parameters to characterize the structure lead to di??erent types of equivalent network: for example, scattering parameters are used to derive the generalized scattering matrix. The adjective 'generalized' refers to the fact that the input and output ports correspond not only to the propagating modes but also to the evanescent ones. In the conventional approaches, the number of these ports is equal to the number of modes used to represent the electromagnetic ??eld at the transition. From this general equivalent network, a reduced form can be derived, which includes only the modes that actually contribute to the electromagnetic interaction between two successive transitions (accessible modes). If the structure consists of many layers, this reduced form is convenient to limit the calculation time, and it is necessary for some types of network representations to avoid the instability problem that arises when many networks are cascaded to each other. The distinguishing feature of the Integral Equation method for the derivation of Multimode Equivalent Networks (IEMEN), described in detail in this thesis, is that it resorts directly to an equivalent network representation in terms of the accessible modes only. The innovative idea is to retain as accessible, in the analysis of a certain transition, only those evanescent modes that arrive at the terminal planes with an attenuation factor that is smaller than a chosen maximum tolerance. The corresponding modal amplitudes are then identi??ed as the fundamental unknowns of the problem, in the sense that all the ??elds in the region between the terminal planes can be expressed as a linear combination of those quantities only. This results in the formulation of the scattering problem in terms of a single integral equation with reduced kernel, and corresponding reduced Green's function, and multiple forcing terms, one for each accessible mode. The formulation has been extended to e??ciently analyze structures containing thin layers, as for example bond ??lms. The IEMEN method has been implemented in a software tool and its analysis capabilities have been demonstrated for some representative examples presented in literature, and by comparing with analysis results obtained by means of commercial software. The tool is also a reliable and exible instrument for the design and it has been successfully used to solve a realistic design problem, consisting of an FSS to be integrated with a waveguide phased array. The FSS had to prevent interference between the array and a satellite communication antenna, located in its proximity. A dipole-based FSS was identi??ed as suitable geometry to meet the requirements. A classical design procedure has been adopted. It starts with a single-mode design, intuitive and Smith-chart based, to trace the relevant behavior of the structure and to perform a rough tuning of the FSS parameters. With respect to the transmission line corresponding to the main propagating Floquet mode, the FSS was characterized by a simple shunt equivalent admittance. The actual value of this admittance, as a function of the frequency, was obtained by means of the IEMEN approach. This initial design phase was followed by a re??nement phase, in which full-wave IEMENbased simulations, including all the relevant accessible modes, were used. It should be noted that, since the patch admittance is a slowly varying function of the frequency, it can be linearly approximated near the resonance. As a consequence, the ??rst design phase was very fast. This property holds also for the elements of the equivalent admittance matrix of a patch FSS, when a larger number of accessible modes is retained in the calculations, and it is a characterizing feature of the IEMEN method. In fact, all the fast frequence variations are accounted for at transmission-line level and not at equivalent-network level, because the modes that vary most rapidly with the frequency have been extracted from the IE kernel. The designed FSS has been manufactured and measured using di??erent setups. Subsequently, the properties of the reduced kernel integral equation have been investigated for an expansion of the unknown current by means of sub-domain basis functions. To comply with very stringent requirements set on the FSS, in terms of incidence angles and roll o??, dielectric slabs with high permittivity are required. As a consequence, the number of accessible modes to be included in the simulations increases and, for a sub-domain expansion of the unknown currents, the Method of Moments (MoM) matrix becomes ill conditioned. In particular, we have considered a simple two-dimensional geometry, consisting of an in??nite periodic array of metallic strips in free space under TM plane wave incidence. An asymptotic expression of the non-accessible Green's function has been derived where, besides the typical singularity, an oscillating term can be recognized. The amplitude of this term depends on the number of accessible modes extracted from the complete kernel and the period is the same as that of the Floquet waves with index equal to the index of the highest-order accessible mode. From a parametric study, it is observed that, if sub-domain functions are used to expand the unknowns, the MoM matrix condition number increases with the number of accessible modes and with the strip width. The intuitive explanation is that the natural modes of the array resonate with the Floquet modes. The study of the eigenvalue equation associated to the reduced IE operator has con??rmed this hypothesis. In fact, its eigenfunctions appear to be similar to combinations of the extracted Floquet waves. Consequently, the solution can be represented as a combination of Floquet modes. This con??nes the solution to a subspace of the solution space where the small unwanted eigenvalues are avoided. Thus, using a limited number of global basis functions can be seen as a way of regularizing an ill-conditioned problem. In particular, we have selected truncated Floquet waves (tfw), variations of those proposed in literature as entire domain basis functions for the analysis of large ??nite slot arrays. Furthermore, the particular formulation of the IEMEN approach, with a single IE and di??erent forcing terms, suggests the adoption, for each speci??c forcing term, of a di??erent compact set of tfw's, which can be used to solve the integral equation. This implies that a di??erent matrix has to be inverted for each accessible mode. Since only a few basis functions are needed to solve the problem corresponding to a given forcing term, the computational time is related to the calculation of the MoM matrix elements, rather than to the matrix inversion. Therefore, the e??ciency of the method of solution is not compromised. A number of test cases has been presented, demonstrating the advantage of using tfw's as basis functions instead of sub-domain functions, as well as a discussion on the nature of the eigenvalues of the reduced kernel IE.
|Qualification||Doctor of Philosophy|
|Award date||27 Jun 2005|
|Place of Publication||Eindhoven|
|Publication status||Published - 2005|