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.Laplace's equation is one of the most important partial differential equations of physics and engineering. It arises in many fields including electromagnetism, classical gravity, and fluid dynamics. It also has close links, through the Laplacian operator, with other important differential equations of physics, such as the wave equation and the diffusion equation. Analytical solutions of Laplace's equation, typically obtained via the method of separation of variables, are standard materials for physics textbooks [1]. The solution for a point source located outside a sphere plays a specially important role through its connection with the Green's function formalism [2]. We will focus on electrostatics in this article, but our results naturally extend to other applications of Laplace's equation.The standard electrostatics solution for a point source outside a dielectric sphere is relatively straightforward and obtained as a multipole expansion (infinite series) [2,3]. One important and often overlooked property of those series is that they can be very slowly convergent for sources close to the surface (often the most relevant situation), as shown explicitly in Ref. [4]. Moroz recently revisited this problem by focusing specifically on the decay rates (i.e. the self-field of a dipole in the quasi-static approximation) and used mathematical manipulations to express those series in a more convergent form [5]. Lindell also approached this problem from the point of view of image theory [6], but the resulting solutions involve integrals which must be computed numerically.In this work, we propose and demonstrate an alternative approach based on the use of spheroidal harmonics, which are the separable solutions of Laplace's equation in spheroidal coordinates [1,2]. This choice may appear counter-intuitive for a spherical object, but the point source breaks the spherical symmetry and we will show * eric.leru@vuw.ac.nz that the spheroidal harmonics are better suited to account for the singularities of the solution. With this original approach, we demonstrate dramatic improvements for the convergence of the solution series. We show that this idea is directly applicable to differ...
The T -matrix, often obtained with Waterman's extended boundary condition method (EBCM), is a widelyused tool for fast calculations of electromagnetic scattering by particles. Here we investigate the quasistatic or long-wavelength limit of this approach, where it reduces to an electrostatics problem. We first present a fully electrostatic version of the EBCM/T-matrix method (dubbed ES-EBCM). Explicit expressions are then given to quantitatively express the long-wavelength limit of the EBCM matrix elements in terms of those of the ES-EBCM formalism. From this connection we deduce a number of symmetry properties of the ES-EBCM matrices. We then investigate the matrix elements of the ES-EBCM formalism in the special case of prolate spheroids. Using the general electrostatic solution in spheroidal coordinates, we derive fully analytic expressions (in the form of finite sums) for all matrix elements. Those can be used for example for studies of the convergence of the T -matrix formalism. We illustrate this by discussing the validity of the Rayleigh hypothesis, where analytical expressions highlight clearly the link with analytical continuation of series.
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]
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