Characteristic Mode Analysis of Crumpled Graphene Flakes and A New Green’s Functions Evaluation Method for Layered Media
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Graphene ﬂakes (GFs) in real composites are rarely perfectly ﬂat, and often exhibit complicated crumpled shapes. Therefore, the goal of this work was to quantify the electromagnetic scattering characteristics of individual crumpled GFs with shapes resembling those found in real composites. The extinction cross sections of tens of GFs, with different sizes and various levels of crumpleness, were calculated using multiple independent solvers. The results show that resonances in the extinction cross section spectrum decrease in amplitude as the GFs become more crumpled. Moreover, some crumpled GFs exhibited a broader resonance than that of perfectly ﬂat GFs. To explain these results, we used a characteristic mode analysis to decompose the graphene surface currents into a set of fundamental currents or modes. For perfectly ﬂat square GFs, the vertical and horizontal modes were found to overlap and resonate at the same frequencies. However, as the GFs became more crumpled, the horizontal/vertical symmetry broke down causing the corresponding modes to separate and resonate at different frequencies leading to an overall broader bandwidth. These results attest to the importance of modeling the exact shape of GFs to accurately characterize their electromagnetic response. In the second year of myPh.D.program, a slightly different path has been employed to develop a new method of Green’s functions evaluation that will eventually help with the electromagnetic scattering and propagation analysis of printed circuit board traces. With iii this goal in mind, modiﬁed Green’s function equations have been developed that were validated with existing numerical methods and commercially available electromagnetic wave solvers. The modiﬁed Green’s function method, otherwise called as analytical evaluation method, which replaces the tail region of Sommerfeld integrals with its equivalent closed form expressions. Further, a detailed study on the types of contours has been carried out to understand their merits and demerits. Finally, the best available contour method has been identiﬁed in conjunction with the analytical evaluation method. This combined technique has been applied to the Green’s functions of a single dielectric layer backed by a perfect electric conductor. This combination displayed good convergence even for the most difﬁcult case of zero vertical separation between the source and observation points in the space-domainsystem. Later, the modiﬁed Green’s functions were applied to the method of moments solution to obtain surface currents of PCB traces. In addition to full or semi-numerical studies of Green’s functions, the identiﬁcation of complex pole locations was also carried out. In this work, the Mittag-Lefﬂer (ML) expansion method has been applied to identify the initial locations of surface wave (SW) and leaky wave (LW) poles lying in the proper and improper sheet of Riemann surface (RS), respectively. To validate the accuracy and robustness of the proposed method, a detailed comparative analysis has been carried out between the ML expansion method and existing methods. The ML expansion method converts the transverse magnetic (TM) and transverse electric (TE) characteristic mode equations from transcendental to polynomial equations. The initial pole locations can be reﬁned using either conventional root-ﬁnding methods or Padè approximants. Numerical results for electrically thick, lossless and lossy substrates shows that the ML expansion has superior convergence properties. Further, a detailed iv study on numerical convergence of SW and LW poles to their ﬁnal roots is discussed. The ML expansion method is one of the less known methods that can efﬁciently identify and differentiate between the aforementioned type of poles. The transition region between LW and SW poles for electrically thick substrates (ETS) is minimally studied and efforts have been made in this work to understand and identify them as the frequency or dielectric constant changes. In addition, a convergence study of ML expansions has been carried out when applied to TE and TM modes of a single dielectric layered medium backed by a ground plane. Finally, the RF coupling experiments to PCB were debugged, analyzed and explained using CMA. The analysis has shown a match in coupling frequency between measurement and simulation and has shown hotspots on PCB indicating maximum and minimum regions for RF coupling. Further, the research was extended to PCB traces with matched loads and the CMA has shown signiﬁcant promise in understanding RF coupling.
Table of Contents
Introduction -- Theory and background -- Electromagnetic scattering analysis of crumpled graphene flakes -- Numerical and semi-analytical evaluation of Green's functions -- New pole extraction technique -- Method of moments -- CMA of PCB traces -- Conclusion and future work -- Appendix A. Coarse grained molecular dynamic graphene model -- Appendix B. List of required closed-form expressions -- Appendix C. Derivation of δ and Taylor series coefficients
Ph.D. (Doctor of Philosophy)