Matt Majic - Academia.edu (original) (raw)
Papers by Matt Majic
Physical Review Research
We introduce a new class of solutions to Laplace equation, dubbed logopoles, and use them to deri... more 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.
Applied Numerical Mathematics
Journal of Quantitative Spectroscopy and Radiative Transfer
Semi-analytic expressions for the static limit of the T -matrix for electromagnetic scattering ar... more 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.
Physical Review A
In electromagnetic scattering, the so-called T -matrix encompasses the optical response of a scat... more In electromagnetic scattering, the so-called T -matrix encompasses the optical response of a scatterer for any incident excitation and is most commonly defined using the basis of multipolar fields. It can therefore be viewed as a generalization of the concept of polarizability of the scatterer. We here calculate the series expansion of the T -matrix for a spheroidal particle in the small-size/longwavelength limit, up to third lowest order with respect to the size parameter,X, which we will define rigorously for a non-spherical particle. T is calculated from the standard extended boundary condition method with a linear system involving two infinite matrices P and Q, whose matrix elements are integrals on the particle surface. We show that the limiting form of the P -and Q-matrices, which is different in the special case of spheroid, ensures that this Taylor expansion can be obtained by considering only multipoles of order 3 or less (i.e. dipoles, quadrupoles, and octupoles). This allows us to obtain self-contained expressions for the Taylor expansion of T(X). The lowest order is O(X 3 ) and equivalent to the quasi-static limit or Rayleigh approximation. Expressions to order O(X 5 ) are obtained by Taylor expansion of the integrals in P and Q followed by matrix inversion. We then apply a radiative correction scheme, which makes the resulting expressions valid up to order O(X 6 ). Orientation-averaged extinction, scattering, and absorption cross-sections are then derived. All results are compared to the exact T -matrix predictions to confirm the validity of our expressions and assess their range of applicability. For a wavelength of 400 nm, the new approximation remains valid (within 1% error) up to particle dimensions of the order of 100 − 200 nm depending on the exact parameters (aspect ratio and material). These results provide a relatively simple and computationally-friendly alternative to the standard T -matrix method for spheroidal particles smaller than the wavelength, in a size range much larger than for the commonly-used Rayleigh approximation. arXiv:1810.06107v1 [physics.optics]
Physical Review E
We propose a powerful approach to solve Laplace's equation for point sources near a spherical obj... more 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.
Authorea
The electrostatics problem of a point charge next to a conducting plane is best solved by placing... more 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 consists of infinitely many rings on a disk with some complicated surface charge distribution. We attempt to describe the image as accurately as possible, and in doing so find simple accurate approximations for the potential, and derive an expression for the image charge density.
Physical Review Research
We introduce a new class of solutions to Laplace equation, dubbed logopoles, and use them to deri... more 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.
Applied Numerical Mathematics
Journal of Quantitative Spectroscopy and Radiative Transfer
Semi-analytic expressions for the static limit of the T -matrix for electromagnetic scattering ar... more 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.
Physical Review A
In electromagnetic scattering, the so-called T -matrix encompasses the optical response of a scat... more In electromagnetic scattering, the so-called T -matrix encompasses the optical response of a scatterer for any incident excitation and is most commonly defined using the basis of multipolar fields. It can therefore be viewed as a generalization of the concept of polarizability of the scatterer. We here calculate the series expansion of the T -matrix for a spheroidal particle in the small-size/longwavelength limit, up to third lowest order with respect to the size parameter,X, which we will define rigorously for a non-spherical particle. T is calculated from the standard extended boundary condition method with a linear system involving two infinite matrices P and Q, whose matrix elements are integrals on the particle surface. We show that the limiting form of the P -and Q-matrices, which is different in the special case of spheroid, ensures that this Taylor expansion can be obtained by considering only multipoles of order 3 or less (i.e. dipoles, quadrupoles, and octupoles). This allows us to obtain self-contained expressions for the Taylor expansion of T(X). The lowest order is O(X 3 ) and equivalent to the quasi-static limit or Rayleigh approximation. Expressions to order O(X 5 ) are obtained by Taylor expansion of the integrals in P and Q followed by matrix inversion. We then apply a radiative correction scheme, which makes the resulting expressions valid up to order O(X 6 ). Orientation-averaged extinction, scattering, and absorption cross-sections are then derived. All results are compared to the exact T -matrix predictions to confirm the validity of our expressions and assess their range of applicability. For a wavelength of 400 nm, the new approximation remains valid (within 1% error) up to particle dimensions of the order of 100 − 200 nm depending on the exact parameters (aspect ratio and material). These results provide a relatively simple and computationally-friendly alternative to the standard T -matrix method for spheroidal particles smaller than the wavelength, in a size range much larger than for the commonly-used Rayleigh approximation. arXiv:1810.06107v1 [physics.optics]
Physical Review E
We propose a powerful approach to solve Laplace's equation for point sources near a spherical obj... more 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.
Authorea
The electrostatics problem of a point charge next to a conducting plane is best solved by placing... more 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 consists of infinitely many rings on a disk with some complicated surface charge distribution. We attempt to describe the image as accurately as possible, and in doing so find simple accurate approximations for the potential, and derive an expression for the image charge density.