Gravina, Giovanni (2022) Basis of surface modes for the scattering from penetrable objects. [Tesi di dottorato]

Preview

Text gravina_giovanni_34.pdf Download (43MB) | Preview

Collection description

Accurate and efficient solutions of integral formulations heavily depends on the choice of basis functions. Two broad categories of basis functions may be identified: sub-domain functions, which are non-zero only over a portion of the object, or entire-domain functions, which extend over the entire domain of one or few individual objects. Although the former approach may have a wider applicability, and is arguably more robust when dealing with objects of irregular shape and sharp corners, the latter is very appealing when multiple scattering problems are considered, where the electromagnetic system under investigation is a collection of mutually-coupled identical object. Representative examples of sub-domain functions are those involved in the finite element method and discontinuous Galerkin method: for instance the Rao-Wilton-Glisson rooftop functions, loop-star functions, loop-tree functions, Trintinalia-Ling functions and Buffa-Christiansen functions. Dually, classic examples of entire-domain basis functions are the vector spherical wave functions, vector spheroidal wave functions. A first strategy to generate analytic entire domain basis function is by exploiting coordinate systems where the Helmholtz equation is separable. In particular, it has been observed that this equation can be solved by separation of variables in eleven coordinate systems. A different but effective strategy to generate entire domain basis function even in irregular domains is by introducing a convenient auxiliary eigenvalue problem. A possible example is represented by the characteristic modes. They do not depend on the particular excitation conditions and they are effective to describe the electromagnetic scattering from collections of identical objects of given material at a fixed operating frequency. Nevertheless, characteristic modes do depend on the frequency, and their interesting properties are lost if they are used as a basis at a frequency different from the one at which they are computed. Thus, they may not be the best choice when multiple frequencies are involved since they have to be recalculated at each frequency. The goal of this thesis is to look for different basis sets that can simplify the numerical solution of electromagnetic scattering problems from a given object at multiple frequencies and give physical insights. In particular, we follow two approaches. In Chapter 4 we introduce longitudinal and transverse static surface modes and use them to solve the surface integral equation governing the full-wave electromagnetic scattering from penetrable objects. The longitudinal modes are the eigenmodes with zero surface curl of the electrostatic surface integral operator, which gives the scalar potential as a function of the surface charge density. The transverse modes are the eigenmodes with zero surface divergence of the magnetostatic surface integral operator, which returns the vector potential as a function of the surface current distribution. These modes are orthogonal. The static modes only depends on the shape of the object, thus, the same static basis can be used regardless of the frequency of operation and of the material constituting the object. We expand the unknown surface currents of the Poggio-Miller-Chang-Harrington-Wu-Tsai surface integral equations in terms of the static surface modes and solve them using the Galerkin-projection scheme. The introduced expansion allows the regularization of the singular integral operators and yields a drastic reduction of the number of unknowns compared to a discretization based on standard sub-domain basis functions. In Chapter 5 we examine the electromagnetic modes and the resonances of homogeneous, finite size, two-dimensional bodies in the frequency domain by a rigorous full wave approach based on an integro-differential formulation of the electromagnetic scattering problem. Using a modal expansion for the current density that disentangles the geometric and material properties of the body the integro-differential equation for the induced surface (free or polarization) current density field is solved. The current modes and the corresponding resonant values of the surface conductivity (eigen-conductivities) are evaluated by solving a linear eigenvalue problem with a non-Hermitian operator. They are inherent properties of the body geometry and do not depend on the body material. The material only determines the coefficients of the expansion and hence the frequencies at which their amplitudes are maximum (resonance frequencies). The eigen-conductivities and the current modes are studied in detail as the frequency, the shape and the size of the body vary. Open and closed surfaces are considered. The presence of vortex current modes, in addition to the source-sink current modes (no whirling modes), which characterize plasmonic oscillations, is shown. Important topological features of the current modes, such as the number of sources and sinks, the number of vortexes, the direction of the vortexes are preserved as the size of the body and the frequency vary. Unlike the source-sink current modes, in open surfaces the vortex current modes can be resonantly excited only in materials with positive imaginary part of the surface conductivity. Eventually, as examples, the scattering by two-dimensional bodies with either positive or negative imaginary part of the surface conductivity is analyzed and the contributions of the different modes are examined.

Downloads

Actions (login required)

View Item