Circuit modeling of periodic structures

Abstract

In the present dissertation the main goal consists in the derivation of analytical circuit models for di_erent types of 1-D periodic structures by a method based on the integral equation. Thus, in Chapter 1 the techique to derive equivalent circuits is described in detail. It is applied for 1-D periodic structures, although for 2-D periodic structures it can be applied in a similar manner. Single-slit gratings and compound gratings are analyzed by using a slit-array formulation. For the derivation of the model we assume the grating to be sandwiched by two semi- in_nite dielectric susbtrates. Extensions to more complex environments are left for the following chapters. We will also present in the same chapter the strip-array formulation, which is adequate for slit gratings with large slit apertures. The obtaining of the circuits will be explained in detail, step by step, in order to see clearly the implicit physical insights. In Chapter 2 the investigation is focused on the study of the scattering response of a periodic single slit- and strip-grating under TE and TM nor- mal and oblique incidence. Their corresponding circuit models, derived in Chapter 1 assuming that the array is sandwiched by two semi-in_nite dielec- tric slabs, are now extended to account for multilayer systems. Additionally, we will also consider a pair of coupled gratings, which, under certain sym- metry conditions, can be studied from the perspective of a single grating by using an analysis based on even and odd excitations. It will be checked the excellent agreement shown by the circuit model in comparison with results provided by HFSS. In particular, the appearance of some kind of resonances such as Wood's anomaly or anomalous extraordinary transmission are well catched by the model. Finally, a discussion about the range of validity of the models is provided. In Chapter 3, the scattering response of the well-known mushroom struc- ture under TM normal and oblique incidence is analyzed in depth. The mushroom periodic structure is actually a periodic corrugated surface. Its corresponding equivalent circuit will be used not only to check the excellent performance and the reliability to reproduce complex resonant behaviors but also as an e_cient design tool. In order to corroborate this, an absorber is easily designed by _lling the corrugations with a lossy silicon dielectric. The model also incorporate modi_cations in order to account for possible ohmic losses in the metallic surfaces. At the end of the chapter a brief discussion about the performance of the model is carried out. Chapter 4 is devoted to the study of the scattering response of com- pound gratings under TM incidence. Periodic compound gratings contain more than one slit per period. The existence of two, three, or a more num- ber of slit apertures per each period and its mutual coupling introduces a new type of resonant: the so-called phase resonance. The appearance of pha- se resonance is accompanied with several phenomena whose study is quite interesting. The circuit model will provide an alternative explanation of pha- se resonance, and will allow us to undertand the associated complexity in a simple manner. The inclusion of ohmic and dielectric losses are incoporated in the model. Furthermore, it will also be checked that the model is capable to work accurately for frequencies close to the optic regime, by taking into account the properties of metals at these frequencies by the Drude model. This fact reveals that the Microwave Network Theory can be sucessfully extended to other frequency ranges under certain circumstances. Finally, Chapter 5 shows an exhaustive study about the scattering pro- perties of coupled slit gratings under TE and TM incidence. Departing from the model of a single slit grating, a _ topology is mathematically deduced to account for a pair of coupled gratings. These gratings can be geometrically di_erent and be misaligned each other, but their period must coincide. Sys- tems containing several gratings stacked will also be considered. From the circuit point of view, a stack of gratings is readily modelled by cascading their corresponding _ circuits. A brief discussion about the limits of validity of the model is also provided.