On Absence and Existence of the Anomalous Localized Resonance without the Quasi-static Approximation

The paper considers the transmission problems for Helmholtz equation with bodies that have negative material parameters. Such material parameters are used to model metals on optical frequencies and so-called metamaterials. As the absorption of the materials in the model tends to zero the fields may blow up. When the speed of the blow up is suitable, this is called the Anomalous Localized Reconance (ALR). In this paper we study this phenomenon and formulate a new condition, the weak Anomalous Localized Reconance (w-ALR), where the speed of the blow up of fields may be slower. Using this concept, we can study the blow up of fields in the presence of negative material parameters without the commonly used quasi-static approximation. We give simple geometric conditions under which w-ALR or ALR may, or may not appear. In particular, we show that in a case of a curved layer of negative material with a strictly convex boundary neither ALR nor w-ALR appears with non-zero frequencies (i.e. in the dynamic range) in dimensions $d\ge 3$. In the case when the boundary of the negative material contains a flat subset we show that the w-ALR always happens with some point sources in dimensions $d\ge 2$. These results, together with the earlier results of Milton et al. ( [22, 23]) and Ammari et al. ([2]) show that for strictly convex bodies ALR may appear only for bodies so small that the quasi-static approximation is realistic. This gives limits for size of the objects for which invisibility cloaking methods based on ALR may be used.Comment: 30 pages, 7 figure

Tailoring Effective Media by Mie Resonances of Radially-Anisotropic Cylinders

This paper studies constructing advanced effective materials using arrays of circular radially-anisotropic (RA) cylinders. Homogenization of such cylinders is considered in an electrodynamic case based on Mie scattering theory. The homogenization procedure consists of two steps. First, we present an effectively isotropic model for individual cylinders, and second, we discuss the modeling of a lattice of RA cylinders. Radial anisotropy brings us extra parameters, which makes it possible to adjust the desired effective response for a fixed frequency. The analysis still remains simple enough, enabling a derivation of analytical design equations. The considered applications include generating artificial magnetism using all-dielectric cylinders, which is currently a very sought-after phenomenon in optical frequencies. We also study how negative refraction is achieved using magnetodielectric RA cylinders.Peer reviewe

Cloaking and magnifying using radial anisotropy

This paper studies the electrostatic responses of a polarly radially anisotropic cylinder and a spherically radially anisotropic sphere. For both geometries, the permittivity components differ from each other in the radial and tangential directions. We show that choosing the ratio between these components in a certain way, these rather simple structures can be used in cloaking dielectric inclusions with arbitrary permittivity and shape in the quasi-static limit. For an ideal cloak, the contrast between the permittivity components has to tend to infinity. However, only positive permittivity values are required and a notable cloaking effect can already be observed with relatively moderate permittivity contrasts. Furthermore, we show that the polarly anisotropic cylindrical shell has a complementary capability of magnifying the response of an inner cylinder.Peer reviewe

Electrostatic resonances of a negative-permittivity hemisphere

This article studies the electric response of an electrically small hemispherical object with negative permittivity by computing its polarizability which is determined by two orthogonal components, the axial one and the transverse one. A certain range of negative permittivity values is found where the mathematical determination of the polarizability becomes impossible due to an unlimited number of singularities. These singularities are due to surface plasmons, also referred to as electrostatic resonances, caused by the sharp edge of the hemisphere. It is also found that the planar surface of the hemisphere may support resonant surface modes. Furthermore, there exists a dipolar resonance determined by the overall geometry. In addition, it is shown that the resonances can be smoothened by introducing losses and, even more importantly, rounding the edge.Peer reviewe

Different homogenization methods based on scattering parameters of dielectric-composite slabs

The dispersion of the effective permittivity of a dielectric-composite slab is analyzed in a quasi-dynamic range using the simulated transmission and reflection data from the slab illuminated by an obliquely incident plane wave. Based on the retrieval results, the procedure for finding the dynamic trust region of the quasi-static Lord Rayleigh estimate for the effective permittivities of such composites is then developed. According to this process, the upper frequency limit of this trust region is numerically determined by an interpolation function. The proposed function of the inclusion area fraction p and relative permittivity ɛi is demonstrated as a good predictor within the ranges 0.1 ≤ p ≤ 0.5 and 10 ≤ ɛi ≤ 60. It is further shown that within the above ranges the effective wavelength inside the material should be at least 33 times the edge length of the unit cell, in order to ensure that the defined relative difference between the retrieved effective permittivity and the quasi-static estimate is not larger than 1%.Peer reviewe

On Absence and Existence of the Anomalous Localized Resonance without the Quasi-static Approximation

This paper considers transmission problems for the Helmholtz equation with bodies that have negative material parameters. Such material parameters are used to model metals on optical frequencies and so-called metamaterials. As the absorption of the materials in the model tends to zero, the fields may blow up. When the speed of the blow up is suitable, this is called the anomalous localized resonance (ALR). In this paper we study this phenomenon and formulate a new condition, the weak anomalous resonance (w-AR), where the speed of the blow up of fields may be slower. Using this concept, we can study the blow up of fields in the presence of negative material parameters without the commonly used quasi-static approximation. We give simple geometric conditions under which w-AR or ALR may or may not appear. In particular, we show that in a case of a curved layer of negative material with a strictly convex boundary, neither ALR nor w-AR appears with nonzero frequencies (i.e., in the dynamic range) in dimensions d >= 3. In the case when the boundary of the negative material contains a flat subset, we show that w-AR always happens with some point sources in dimensions d >= 2.Peer reviewe

Electric Response of a Small Hemisphere

Työn tarkoituksena on selvittää homogeenisen, puolipallon muotoisen kappaleen sähköstaattinen vaste. Vasteen voimakkuutta voidaan kuvata laskemalla puolipallon polarisoituvuus. Polarisoituvuuden laskenta jaetaan kahteen erikoistapaukseen: aksiaaliseen ja transversaaliseen. Polarisoituvuutta ei pystytä selvittämään täysin analyyttisesti, vaan joudutaan muodostamaan ja ratkaisemaan suuri matriisiyhtälö. Polarisoituvuudet esitetään puolipallon suhteellisen permittiivisyyden funktiona. Positiivisilla permittiivisyyksillä saadaan tarkkoja tuloksia, jotka voidaan varmentaa laskemalla polarisoituvuus myös numeerisella valmisohjelmistolla. Polarisoituvuus lasketaan myös negatiivisilla permittiivisyyksillä, jolloin täytyy oletaan taajuus korkeaksi ja puolipallon mitat aallonpituuteen nähden pieniksi, jotta sähköstaattinen tarkastelu edelleen pätisi. Pienillä negatiivisilla permittiivisyyksillä puolipallon vaste on singulaarinen, eli havaitaan useita niin sanottuja staattisia resonansseja. Vastaavilla permittiivisyyden arvoilla tuloksia ei saada suppenemaan. Analyyttisesti voidaan osoittaa, että singulaarisuudet suhteellisen permitiivisyyden arvoilla −3 ≤ εr ≤ −1/3 aiheutuvat puolipallon terävistä nurkista. Ottamalla huomioon materiaalin kausaalisuus ja häviöt, singulaarisuudet katoavat, ja ratkaisu saadaan suppenemaan. Työ on luoteeltaan akateemista perustutkimusta, joka selvittää yhteyttä kappaleen geometrian, materiaalin ja sen sähköisten ominaisuuksien välillä.This thesis examines the electrostatic response of a homogeneous hemispherical object by computing its polarizability. Two special cases are investigated: one axial, the other transversal. Polarizability cannot be calculated purely analytically; hence a large matrix equation must be constructed and solved. The polarizability of a hemisphere is presented as a function of relative permittivity. With positive values for permittivity, the results are quite accurate and they are verified by computing the polarizability numerically. The thesis also considers negative values for permittivity. To obtain negative permittivities, the presumed frequency must be very high and the dimensions of the object very small in order for the static approach to be feasible. With small negative permittivities, the polarizability of the hemisphere is found to have several singularities, also known as static resonances, and the solution does not converge. It can be shown analytically that the resonances between relative permittivity values −3 ≤ εr ≤ −1/3 are due to the sharp corners of the hemisphere. By taking the causality and the losses of the material into consideration, the singularities vanish and the solution converges. This master's thesis is based on basic academic research observing the connection between the geometry, material and electrical characteristics of an object

Complex electromagnetic responses from simple geometries

The electromagnetic properties of a material arise from its intrinsic microstructure, which may often be very complex. However, materials are usually characterized more simply using macroscopic material parameters, electric permittivity and magnetic permeability. This thesis considers the principles of material modeling from the electromagnetics point of view. The analysis is mostly based on electrostatics. The aim of the thesis is to enhance the understanding of the interaction between matter and the electromagnetic fields, and further, the relation between matter and geometry. The contents of the thesis can be divided into three parts. The first part discusses the concepts of polarization and polarizability and considers the electric reponses of particles with different geometries. Polarizabilities of a three-dimensional hemisphere and a two-dimensional half-disk are solved. The second part studies negative material parameters. The emphasis lies on negative permittivity. Interfaces between permittivities of opposite signs are found supporting surface plasmons, or electrostatic resonances. The occurrence of these resonances is especially studied for a hemisphere and a half-disk. Moreover, it is showed that sharp edges with negative permittivity may support unphysically singular field modes, which in numerical simulations can result in non-convergent solutions. The most efficient way to overcome this problem in computational modeling is to slightly round all sharp corners. The third part focuses on homogenization of composite media. Effective material parameters modeling the response of a thin dielectric composite slab are retrieved. Computational homogenization techniques and their limitations are studied. The results indicate that for a successful homogenization, the unit cells of the slab must remain very small compared with the wavelength. Also, the boundary layers of the slab show higher effective permittivity than the corresponding bulk medium