Images and plane waves : efficient field computation in electromagnetics and acoustics

In some simple or canonical problems, analytical solutions offer the most efficient way to compute the electromagnetic or acoustic fields. For arbitrary geometries, efficient numerical methods are needed. This thesis contains new or improved solutions of classical electrostatic problems and some further developments of a variant of the fast multipole method (FMM). In the first part, the electrostatic problems of pairs of both orthogonally intersecting and non-intersecting conducting spheres are solved using Kelvin's image theory. A new efficient method for evaluating the polarizability of two non-intersecting spheres is presented. Novel analytical solutions, and also computationally efficient approximative solutions, are obtained by applying Kelvin's inversion to the electrostatic image solution of a conducting wedge. Integral equation methods are popular for both electrodynamic and acoustic scattering problems. However, to be able to use very large number of unknowns, fast iterative methods, such as the fast multipole method, must be used. In the second part of this thesis, a new broadband variant of the multilevel fast multipole algorithm (MLFMA) is described and used for both acoustic and electromagnetic scattering problems. In particular, the implementation overcomes the low-frequency breakdown of the MLFMA using a combination of the spectral representation of the Green's function and Rokhlin's translation formula.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

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

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

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

Resonances in small scatterers with impedance boundary

With analytical (generalized Mie scattering) and numerical (integral-equation-based) considerations we show the existence of strong resonances in the scattering response of small spheres with lossless impedance boundary. With increasing size, these multipolar resonances are damped and shifted with respect to the magnitude of the surface impedance. The electric-type resonances are inductive and magnetic ones capacitive. Interestingly, these subwavelength resonances resemble plasmonic resonances in small negative-permittivity scatterers and dielectric resonances in small high-permittivity scatterers. The fundamental dipolar mode is also analyzed from the point of view of surface currents and the effect of the change of the shape into a non-spherical geometry

Positive and negative extinction of active particles

Using analytical Lorenz--Mie scattering formalism and numerical methods, we analyze the response of active particles to electromagnetic waves. The particles are composed of homogeneous, non-magnetic, and dielectrically isotropic medium. Spherical scatterers and sharp and rounded cubes are treated. The absorption cross-section of active particles is negative, thus showing gain in their electromagnetic response. Since the scattering cross-section is always positive, their extinction can be either positive, negative, or zero. We construct a five-class categorization of active and passive dielectric particles. We point out the enhanced backscattering phenomenon that active scatterers display, and also discuss extinction paradox and optical theorem. Finally, using COMSOL Multiphysics and an in-house Method-of-Moments code, the effects of the non-sphericity of active scatterers on their electromagnetic response are illustrated.Comment: preprint, 17 pages, 19 figure

Polarizability of a dielectric hemisphere

This article presents a method for solving the polarizability of a dielectric hemispherical object as a function of its relative electric permittivity. The polarizability of a hemisphere depends on the direction of the exciting electric field. Therefore, the polarizability can be written as a dyadic consisting of two components, the axial and the transversal polarizabilities, which can be solved separately. The solution is based on an analytical approach where the electrostatic potential function is written as a series expansion. However, no closed-form solution for the coefficients of the series is found, so they must be solved from a matrix equation. This method provides very high accuracy. However, it requires construction of large matrices which consumes both time and memory. Therefore, approximative expressions for the polarizabilities with absolute error less than 10−5 are also presented.Peer reviewe

Anomalous absorption, plasmonic resonances, and invisibility of radially anisotropic spheres

This article analyzes the response of a sphere with radially anisotropic permittivity dyadic (RA sphere), in both the electrostatic and full electrodynamic settings. Depending on the values and signs of the permittivity components, the quasistatic polarizability of the RA sphere exhibits several very different interesting properties, including invisibility, field concentration, resonant singularities, and emergent losses. Special attention is given to the anomalous losses that appear in the case of certain hyperbolic anisotropy values. We show that their validity can be justified in a limiting sense by puncturing the sphere at the origin and adding a small imaginary part into the permittivity components. A hyperbolic RA sphere with very small intrinsic losses can thus have significant effective losses making it an effective absorber. With different choices of permittivities, the RA sphere could also perform as a cloak or a sensor. The Mie scattering results by an RA sphere are used to justify the quasistatic calculations. It is shown that in the small parameter limit the absorption efficiency of an RA sphere is nonzero for certain lossless hyperbolic anisotropies. The absorption and scattering efficiencies agree with the quasistatic calculations fairly well for spheres with size parameters up to 1/3.Peer reviewe

Polarizability of conducting sphere-doublets using series of images.

The classical electrostatic problem of two nonintersecting conducting spheres in a uniform incident electric field is considered. Starting from the basic Kelvin’s image principle, the two spheres are replaced with equivalent series of image sources, from which the polarizability is calculated. Explicit expressions for the axial and transversal components of the polarizability dyadic are found by solving the recurrence equations. Efficient numerical evaluation of the different series is also discussed.Peer reviewe