Lorenz–Mie theory for 2D scattering and resonance calculations

Abstract

This PhD tutorial is concerned with a description of the two-dimensional generalized Lorenz-Mie theory (2D-GLMT), a well-established numerical method used to compute the interaction of light with arrays of cylindrical scatterers. This theory is based on the method of separation of variables and the application of an addition theorem for cylindrical functions. The purpose of this tutorial is to assemble the practical tools necessary to implement the 2D-GLMT method for the computation of scattering by passive scatterers or of resonances in optically active media. The first part contains a derivation of the vector and scalar Helmholtz equations for 2D geometries, starting from Maxwell's equations. Optically active media are included in 2D-GLMT using a recent stationary formulation of the Maxwell-Bloch equations called steady-state ab initio laser theory (SALT), which introduces new classes of solutions useful for resonance computations. Following these preliminaries, a detailed description of 2D-GLMT is presented. The emphasis is placed on the derivation of beam-shape coefficients for scattering computations, as well as the computation of resonant modes using a combination of 2D-GLMT and SALT. The final section contains several numerical examples illustrating the full potential of 2D-GLMT for scattering and resonance computations. These examples, drawn from the literature, include the design of integrated polarization filters and the computation of optical modes of photonic crystal cavities and random lasers. CONTENTS

Mie Scattering in the Macroscopic Response and the Photonic Bands of Metamaterials

“…The theory of multiple scattering from cylinders was pioneered in the 1950s by Twersky . In this section, we follow the study by Gagnon and Dubé to compute the solution of the multiple scattering problem of a finite array of dielectric cylinders. Consider first a single infinitely long dielectric cylinder of radius R and refractive index n , standing in vacuum with its axis aligned to the axis.…”

Structural entropy and spatial decay of quasimodes in Vogel spirals

“…Within this approach, the electromagnetic field is expanded in terms of Bessel-Fourier multipolar basis functions in the coordinate system centered at the origin of each individual cylinders. Using the Graf's addition theorem for Bessel functions and imposing the field continuity conditions at the boundary of each cylinder, we can obtain the polarization-dependent Mie coefficients by solving the linear system [49]:…”

“…where n indicates the n th -cylinder and is the multipolar order of the cylindrical Bessel J (x) and Hankel H (x) functions (x = k 0 r is the size parameter) used in the calculations [48,49].…”

“…Calculating the resonant electromagnetic quasimodes requires solving the homogeneous equation Tb = 0 [15,[48][49][50][51]. Specifically, we have to find the complex k values for which the relation det T( k) = 0 is satisfied [49].…”

“…Calculating the resonant electromagnetic quasimodes requires solving the homogeneous equation Tb = 0 [15,[48][49][50][51]. Specifically, we have to find the complex k values for which the relation det T( k) = 0 is satisfied [49]. The complex eigenvalue k has a direct physical interpretation, namely its real part is equal to the freespace wave number of the mode while its imaginary part describes the decay rate of the mode, which is also inversely proportional to its spectral width.…”