İki boyutlu yalıtkan ve metal nano boyutlu yapıların momentler yöntemi temelli sayısal bir teknik ile optik karakteristiklerinin elde edilmesi ve ayrık karmaşık imaj yöntemi için hata kriterinin ve otomatik derece seçiminin geliştirilmesi

The main goal in this study is to develop an efficient and accurate numerical techniquethat could help to simulate and analyze two-dimensional nano-structures in multilayeredenvironment in the optical regime. Since the Method of Moments (MoM) hasbeen one of the most widely-used numerical techniques for electromagnetic problemsin layered open environment, it has been employed in this study as the main numericaltool that the development is based on. As the MoM solves mixed-potential integralequations in electromagnetic and optic problems, it translates the integral equationsinto a set of algebraic equations by expanding the unknown functions under the integralsign into a series of known functions with unknown coefficients, referred toas basis functions. It should be remembered that converting governing differentialequations into integral equations requires the knowledge of Green's functions, whichis the response of the medium of interest to a point source. In this endeavor, thereare couple of computational difficulties that need to be overcome, which are mainlyin the computations of Green's functions in multilayered media via Discrete ComplexImage Method (DCIM), and in the computations of the entries of resulting algebraicequations, i.e., MoM matrix entries. In this thesis, primarily these difficulties havebeen addressed properly and their efficient and robust remedies have been proposed,demonstrated and verified.Before providing some more details on the contributions in this thesis, it shouldbe remembered that the performance of the MoM for the problems in layered mediais strongly dependent on the accurate and efficient evaluation of the spatial-domainGreen's functions in layered media. Although the DCIM has eliminated some of themajor bottlenecks in the efficient computation of Green's function in the spatial domain,there still exist two hurdles that render the DCIM not robust and reliable tobe employed in the development of CAD tools for EM and optics simulations, whichare namely the lack of a spatial-domain error metric assuring the accuracy of the endresult and an automatic order selection algorithm finding the optimum number ofcomplex images. Therefore, in the first part of the dissertation, covering the first twochapters, these two remaining issues in the core of the original and improved DCIMshave been discussed, and theoretically sound, easy-to-implement and computationallyefficient novel remedies have been proposed, developed and demonstrated. Asa result of these improvements, the DCIM is now ready to be used, not only in thecalculations of Green's functions but also in the evaluations of Fourier, Laplace andHankel transformations, accurately and reliably in commercial softwares.As it was stated in the opening sentence, the ultimate goal is to develop an efficient,accurate and robust algorithm for the analysis of 2D metal and dielectricnano-structures in multilayered media. In the last part of the dissertation, a novelMoM-based approach has been proposed and developed for the analysis of such structuresin free space. The main power of the developed method is that it provides aclosed-form expression for a generic matrix entry of the MoM, and, without referringto any numerical integration algorithms, all matrix entries can be obtained by a simpleevaluation of the closed-form expression at the location of the point of interest.Although this method can be extended to multilayered environment in a straightforwardmanner by considering each term in the closed-form approximation of theGreen's function as the Green's function in free space with complex source location,this part has been left for the future study because of its numerical inefficiency.