arXiv subjects

Matt Majic

Publications and source records attributed to Matt Majic.

11 recordsLinked to original sources

arXiv2021

Images of point charges in conducting ellipses and prolate spheroids

This is an investigation into the exact forms of images of point charges in 2D conducting ellipses and 3D prolate spheroids. For an ellipse with exterior point source, we compare two previous expressions for the analytic continuation inside the ellipse down to the focal line, the location of the image charge. For an interior source, we discover a sequence of image charges lying outside. For a point source close the surface of a spheroid, series solutions for the potential diverge in a region that encompasses the singularities of the continuation of the reflected potential inside the spheroid. We uncover an image system for a point charge on the axis of a prolate spheroid that extends from the focal segment somewhat further up the axis. For an off axis charge we find a new approximate image charge. For a point charge at the center of the spheroid, the image singularity is found to lie on an infinitely wide, flat sheet with a hole around the spheroid. For a point charge located anywhere inside the spheroid, we find two new approximate image charge solutions, a point charge in real space which applies when source lies near the surface, or a point charge in complex space which projects as a ring in the real space which applies when the source lies near the rotation axis. When the point charge lies exactly on a focal point, a multipole of order 1/2 is an ideal image approximation.

physics.class-ph↗

arXiv2021

Mean path length inside non-scattering refractive objects

It has recently been argued that the geometric-optics mean path length of rays inside a refractive object under Lambertian illumination is independent of the scattering strength of the medium [Savo et al., Science 358, 765 (2017)]. We here show that it is in fact different in the case of zero-scattering. We uncover and explain the role of trapped ray trajectories in creating this unexpected discontinuity from zero- to low-scattering. This allows us to derive new analytic results for the zero-scattering mean path length of simple refractive shapes. This work provides a fresh perspective on the study of path length inside refractive objects, with possible applications in for example the study of scattering by large particles or the design of optical systems.

physics.optics↗

arXiv2021

Definition and properties of logopoles of all degrees and orders

Logopoles are a recently proposed class of solutions to Laplace's equation with intriguing links to both solid spheroidal and solid spherical harmonics. They share the same finite line singularity with the former and provide a generalization of the latter as multipoles of negative order. In [Phys. Rev. Res. 1, 033213 (2019)], we introduced and discussed the properties and applications of these new functions in the special case of axi-symmetric problems (with azimuthal index $m=0$). This allowed us to focus on the physical properties without the added mathematical complications. Here we expand these concepts to the general case $m\neq 0$. The chosen definitions are motivated to conserve some of the most interesting properties of the $m=0$ case. This requires the inclusion of Legendre functions of the second kind with degree $-m\leq n<m$ (in addition to the usual $n\geq |m|$) and we show that these are also related to the exterior spheroidal harmonics. We show that Logopoles can also be defined for $n\le m$, and discuss in particular logopoles of degree $n=-m$, which correspond to the potential of line segments of uniform polarization density.

math-ph↗

arXiv2020

The image of a point charge in an infinite conducting cylinder

The electrostatics problem of a point charge next to a conducting plane is best solved by placing an image charge placed on the opposite side. For a charge between two parallel planes this can be solved with image charges outside the planes at evenly spaced intervals moving out to infinity. What is the corresponding image of a point charge is when placed on the axis of a cylinder?. The potential of a point charge in a cylinder is well known and may expressed in many forms involving integrals or series of Bessel functions, but none of which elude to an image. In fact the image is complex (in both senses), consisting of infinitely many rings on a disk with some surface charge distribution. This manuscript attempts to describe the image as accurately as possible, and in doing so finds simple accurate approximations for the potential.

physics.class-ph↗

arXiv2019

Relationships between solid spherical and toroidal harmonics

We derive new relationships expressing solid spherical harmonics as series of toroidal harmonics and vice versa. The expansions include regular and irregular spherical harmonics, ring and axial toroidal harmonics of even and odd parity about the plane of the torus. The expansion coefficients are given in terms of a recurrence relation. As an example application we apply one of the expansions to express the potential of a charged conducting torus on a basis of spherical harmonics.

math-ph↗

arXiv2019

A new class of solutions to Laplace equation: Regularized multipoles of negative orders

We introduce a new class of solutions to Laplace equation, dubbed logopoles, and use them to derive a new relation between solutions in prolate spheroidal and spherical coordinates. The main novelty is that it involves spherical harmonics of the second kind, which have rarely been considered in physical problems because they are singular on the entire z axis. Logopoles, in contrast, have a finite line singularity like solid spheroidal harmonics, but are also closely related to solid spherical harmonics and can be viewed as an extension of the standard multipole ladder toward the negative multipolar orders. These new solutions may prove a fruitful alternative to either spherical or spheroidal harmonics in physical problems.

math-ph↗

arXiv2019

Electrostatic T-matrix for a torus on bases of toroidal and spherical harmonics

Semi-analytic expressions for the static limit of the T-matrix for electromagnetic scattering are derived for a circular torus, expressed in bases of both toroidal and spherical harmonics. The scattering problem for an arbitrary static excitation is solved using toroidal harmonics and the extended boundary condition method to obtain analytic expressions for auxiliary Q and P-matrices, from which the T-matrix is given by their division. By applying the basis transformations between toroidal and spherical harmonics, the quasi-static limit of the T-matrix block for electric multipole coupling is obtained. For the toroidal geometry there are two similar T-matrices on a spherical basis, for computing the scattered field both near the origin and in the far field. Static limits of the optical cross-sections are computed, and analytic expressions for the limit of a thin ring are derived.

physics.comp-ph↗

arXiv2019

Image theory for a sphere with negative permittivity

An image system for a point charge outside a dielectric sphere is presented for all complex values of relative permittivity $ε=ε'+iε''$. The standard image integral solution of a point charge outside a dielectric sphere involving an image point charge plus a line source is shown to diverge for $ε'<-1$, and a correction is proposed for this case, involving image multipoles of infinite magnitude that regularise the divergent line integral. The number of these multipoles depends on the position of $ε$ relative to the resonant values $ε=-1-1/n$ for positive integer $n$. The internal potential and dipole sources are also considered.

physics.comp-ph↗

arXiv2018

Quasistatic limit of the electric-magnetic coupling blocks of the $T$-matrix for spheroids

The $T$-matrix formally describes the solution of any electromagnetic scattering problem by a given particle in a given medium at a given wavelength. As such it is commonly used in a number of contexts, for example to predict the orientation-averaged optical properties of non-spherical particles. The $T$-matrix for electromagnetic scattering can be divided into four blocks corresponding physically to coupling between either magnetic or electric multipolar fields. Analytic expressions were recently derived for the electrostatic limit of the electric-electric $T$-matrix block $\mathbf T^{22}$, of prolate spheroids. In such an electrostatic approximation, all the other blocks were zero. We here analyse the long-wavelength limit for the other blocks ($\mathbf T^{21}$, $\mathbf T^{12}$, $\mathbf T^{11}$) corresponding to electric-magnetic, magnetic-electric, and magnetic-magnetic coupling respectively. Analytic expressions (finite sums) are obtained in the case of spheroidal particles by expressing the fields with solutions to Laplace's equation, expanding the fields in terms of spheroidal harmonics and applying the boundary conditions. Similar expressions are also presented for the auxiliary matrices in the extended boundary condition method, often used in conjunction with the $T$-matrix formalism.

physics.class-ph↗

arXiv2018

Exact gravitational potential of a homogeneous torus in toroidal coordinates and a surface integral approach to Poisson's equation

New exact solutions are derived for the gravitational potential inside and outside a homogeneous torus as rapidly converging series of toroidal harmonics. The approach consists of splitting the inter- nal potential into a known solution to Poisson's equation plus some solution to Laplace's equation. The full solutions are then obtained using two equivalent methods, applying differential boundary conditions at the surface, or evaluating a surface integral derived from Green's third identity. This surface integral may not have been published before and is general to all geometries and volume density distributions, reducing the problem for the gravitational potential of any object from a volume to a surface integral.

physics.class-ph↗

arXiv2016

Spheroidal harmonic expansions for the solution of Laplace's equation for a point source near a sphere

We propose a powerful approach to solve Laplace's equation for point sources near a spherical object. The central new idea is to use prolate spheroidal solid harmonics, which are separable solutions of Laplace's equation in spheroidal coordinates, instead of the more natural spherical solid harmonics. We motivate this choice and show that the resulting series expansions converge much faster. This improvement is discussed in terms of the singularity of the solution and its analytic continuation. The benefits of this approach are illustrated for a specific example: the calculation of modified decay rates of light emitters close to nanostructures in the long-wavelength approximation. We expect the general approach to be applicable with similar benefits to a variety of other contexts, from other geometries to other equations of mathematical physics.

physics.class-ph↗