This thesis investigates wave propagation in waveguides with different cross-sectional geometries, filled with graded metamaterial structures. The growing interest in graded metamaterials in electromagnetic applications and models is motivated by their realism, mathematical simplicity, and versatility, compared to the conventional materials that are in use today. The majority of the research community resorts to the use of numerical implementations and solvers for obtaining a solution for the field distribution and propagation characteristics. However, these methods do not provide explicit physical insight into the connection between the steepness of the graded-index profiles and their respective field solutions. Thus, the motivation for the research in this thesis is to focus on analytically solving the wave equation for several graded-index profiles to gain physical insight into the field solutions and what phenomena may be predicted from these results.

The research in this thesis focuses on the graded impedance-matched RHM-LHM profile, which is a composite material involving an impedance-matched transition from a right-handed material to a left-handed material. The profile possesses a variable transition steepness of the relative material parameter values and dispersive characteristics. These variable properties act as degrees of freedom, thereby introducing a high degree of modeling flexibility and allowing for the study of wave propagation under more general conditions. Both non-periodic and periodic impedance-matched RHM-LHM transition profiles have been theoretically studied here using analytical functions. It is of interest to study these RHM-LHM composite materials, as the interaction between a regular material and a metamaterial may lead to newly discovered phenomena, novel analytical expressions of electromagnetic fields, and provide a deeper understanding of the underlying principles at the interface between such media.

The solution method is based on describing these graded metamaterial structures by their relative permittivity and permeability functions, where either one or both of these properties have a graded transition profile. The wave equation for each metamaterial structure is derived using Maxwell's equations and solved using the boundary conditions imposed by the waveguide geometry. The field distribution and propagation characteristics for a given electromagnetic mode are analytically expressed and visualized using the numerical software tool MATLAB. Furthermore, numerical results are obtained using the numerical software tool COMSOL Multiphysics, which are then compared with the analytical results to validate the analytical expressions derived from the wave equation.

Denna avhandling undersöker vågutbredning i vågledare med olika tvärsnitts-geometrier, som är fyllda med graderade metamaterialstrukturer. Det växande intresset för graderade metamaterial i elektromagnetiska tillämpningar motiveras av deras realism, matematiska enkelhet och mångsidighet, jämfört med de konventionella material som används idag. Majoriteten av forskningen inom området använder sig av numeriska implementeringar och lösare för att erhålla en lösning för fältfördelnings- och utbredningsegenskaperna. Dessa metoder ger dock inte explicit fysikalisk insikt i sambandet mellan de graderade materialprofilerna och deras respektive fältlösningar. Motivationen bakom denna forskning är därför att fokusera på att analytiskt lösa vågekvationen för realistiska graderade indexprofiler, för att få ökad fysikalisk insikt i fältlösningarna, samt vilka fenomen som kan förutsägas utifrån dessa resultat.

Forskningen fokuserar på den graderade impedansmatchade RHM-LHM-profilen, vilket är ett kompositmaterial som involverar en impedansmatchad övergång från ett högerhänt material till ett vänsterhänt material. Profilen har en variabel branthet i övergången av de relativa materialparametrarna och dispersiva egenskaper. Dessa variabla egenskaper fungerar som frihetsgrader, vilket introducerar en hög grad av modelleringsflexibilitet och möjliggör studier av vågutbredning och fältfördelningar under mer generella förhållanden. Både icke-periodiska och periodiska impedansmatchade RHM-LHM-övergångsprofiler har här teoretiskt studerats med hjälp av analytiska funktioner. Det är av intresse att studera RHM-LHM kompositmaterial eftersom interaktionen mellan ett konventionellt material och ett metamaterial kan leda till nyupptäckta fenomen, nya analytiska uttryck för de elektromagnetiska fälten, och bidra till en djupare förståelse av de underliggande principerna för ett gränssnitt mellan sådana medier.

Arbetet går ut på att beskriva dessa graderade metamaterialstrukturer utifrån deras relativa permittivitets- och permeabilitetsfunktioner, där antingen en eller båda dessa egenskaper har en graderad övergångsprofil. Vågekvationen för varje metamaterialstruktur härleds med hjälp av Maxwells ekvationer och löses med hjälp av de randvillkor som vågledargeometrin ställer. Fältfördelnings- och utbredningsegenskaperna för en given elektromagnetisk mod uttrycks analytiskt och visualiseras med hjälp av det numeriska programvaruverktyget MATLAB. Vidare erhålls numeriska resultat med hjälp av det numeriska programvaruverktyget COMSOL Multiphysics, vilka sedan jämförs med de analytiska resultaten för att validera de analytiska uttryck som härletts från vågekvationen.

We derive and investigate the longitudinal field solution for the TEM mode inside a waveguide with an arbitrarily shaped cross-section, filled with an impedance-matched metamaterial composite graded in the z-direction. The relative permittivity (ω, z) and permeability μ(ω, z) are assumed to vary according to hyperbolic tangent functions. We obtain exact analytical solutions to Maxwell's equations for both lossless and lossy media. We introduce an analytical ansatz for the longitudinal field solution and compare it to the derived longitudinal field solution. An excellent agreement between the results from the analytical solution and the assumed analytical ansatz is obtained. This confirms that the proposed analytical ansatz for the longitudinal field solution fulfills the governing wave equation.

Metamaterials have generally been studied by means of the time-harmonic approach, where the wave propagation is described in terms of the spatial coordinates. However, recently studies of temporal metamaterials, where the electromagnetic properties change in time, have been initiated by metamaterials researchers. We therefore use the space-harmonic method to study a 'temporal step' metamaterial. By gradually switching the material parameters in time between two values, the medium can be described using the hyperbolic tangent mathematical function. Exact analytical solutions for the fields are obtained and analyzed. The space-harmonic analysis of temporally graded metamaterials is dual to the time-harmonic analysis of spatially graded metamaterials, that have been extensively studied by the present authors.

Metamaterials are mostly studied using the time-harmonic approach, where the wave propagation is spatially described. Recently, studies of media having electromagnetic properties that change in time have been given the attention of the scientific community, aiming to describe wave-matter interaction in both space and time. In the present paper, we use the space-harmonic method for the general description of wave propagation in time. Such a method can be used to effectively describe a "temporal multilayered" (or temporal multi-stepped) metamaterial by alternating the effective material parameters of the medium in time between two values. We obtain the exact analytical solution for the fields in two examples of temporally periodic metamaterials. Numerical simulations for the impedance-matched scenario are also carried out, showing an excellent agreement when compared to the exact analytical field. Our results also demonstrate the duality between the descriptions of the temporally periodic metamaterial and its spatial counterpart, spatially periodic metamaterials.

Over the last decades, metamaterials have mostly been studied using the time-harmonic approach, with wave propagation described in terms of the spatial coordinates. The metamaterials community has only recently initiated studies of temporal metamaterials, with electromagnetic properties that change in time. In this work, we propose the space-harmonic method to study a "temporal multistepped"metamaterial. By alternating the material parameters in time, the medium can be described using periodic mathematical functions. The exact analytical solutions for the fields in temporally periodic metamaterials are then obtained. The space-harmonic analysis of temporally periodic metamaterials is dual to the time-harmonic analysis of spatially periodic metamaterials, previously studied by one of the present authors.

Over the last decades, metamaterials have mostly been studied using the time-harmonic approach, with wave propagation described in terms of the spatial coordinates. The metamaterials community has only recently initiated studies of temporal metamaterials, with electromagnetic properties that change in time. In this work, we propose the space-harmonic method to study a "temporal multistepped" metamaterial. By alternating the material parameters in time, the medium can be described using periodic mathematical functions. The exact analytical solutions for the fields in temporally periodic metamaterials are then obtained. The space-harmonic analysis of temporally periodic metamaterials is dual to the time-harmonic analysis of spatially periodic metamaterials, previously studied by one of the present authors.

In this paper, we study transverse electric (TE) wave propagation inside a hollow circular waveguide filled with a lossy graded multilayered dielectric composite. The dielectric composite grading and the wave propagation are directed along the z-direction. The z-dependent permittivity of the dielectric composite is modeled using a periodic sinusoidal function. The exact analytical solutions to Maxwell's equations are obtained, and the field solutions and wave behavior confirm the expected properties of a lossy graded multilayered dielectric medium inside a hollow circular waveguide. Thereafter, through a numerical study performed using the commercial software COMSOL Multiphysics, we show that the analytical and numerical results are in perfect agreement. The analytical model applies to any combination of the material parameters relevant to the graded multilayered dielectric medium. The significance of the proposed method is that it can be utilized for analytically studying wave propagation and wave phenomena in a variety of media with characteristics including, but not limited to, periodicity, grading, negative refraction, and spatial- and frequency dependence. The validity is not restricted to any given frequency regime, therefore, allowing the proposed method to be useful for different types of applications, such as super-resolution imaging, electromagnetic cloaking, sub-wavelength focusing, and microwave absorbers.

We study TE-wave propagation in a circular waveguide with a graded transition from a lossy right-handed material (RHM) filling the left-hand half of the waveguide to the impedance-matched lossy left-handed material (LHM) filling the right-hand half of the waveguide. We obtain exact analytical solutions to Maxwell's equations and perform corresponding numerical simulations in COMSOL. An excellent agreement is obtained between the numerical simulations and analytical results. The presented method can model smooth realistic material transitions, where the interface width is an additional degree of freedom in the design of practical RHM-LHM interfaces.

A theoretical study of TE-wave propagation in a hollow circular waveguide with a graded dielectric medium is presented. The medium is modeled using a squared cosine function that oscillates between two unique complex dielectric media to simulate a periodic multilayer structure. The proposed model is applicable to arbitrary complex permittivities of the two media, including negative values for chiral metamaterials. The exact analytical result for the longitudinal magnetic field component H-z is obtained, while the transverse electric and magnetic field components could be determined using the appropriate Maxwell's equations.

We present a parametric analysis of the absorption cross-section of small ellipsoidal composite structures (particles) with gold nanoparticles (GNP) embedded in lossy human tissue. The optimal permittivity of the ellipsoidal particle that maximizes the absorption at any given frequency, derived in our previous work with the surrounding tissue as a saline water, is used. We present new results for realistic tissue material parameters and for different GNP-host media and surrounding media. Present results are useful to assess the feasibility of the proposed radiotherapeutic hyperthermia-based methods to treat cancer, based on electrophoretic heating of gold nanoparticle suspensions using microwave radiation, as well as to improve the general understanding of plasmonic resonances in lossy media.