A modified recursive transfer matrix algorithm for radiation and scattering computation of a multilayer sphere

TL;DR

This work addresses the numerical instability encountered when computing scattering and radiation for multilayered spheres within the Lorentz-Mie framework. It introduces a modified Recursive Transfer Matrix Algorithm (mRTMA) that leverages Debye potentials and logarithmic derivatives to avoid overflow and singularities, incorporating a rescaled coefficient and a hybrid recursion to cover thin shells and strongly absorbing media. The method provides stable, scalable computation across many layers and supports outgoing-wave scenarios, with validated accuracy against Mie theory and existing recursive approaches. The resulting approach broadens the practical applicability of multilayer sphere analyses in optics, remote sensing, and thermal radiation to highly layered and absorptive systems.

Abstract

We discuss the electromagnetic scattering and radiation problems of multilayered spheres, reviewing the history of the Lorentz-Mie theory and the numerical stability issues encountered in handling multilayered spheres. By combining recursive methods with the transfer matrix method, we propose a modified transfer matrix algorithm designed for the stable and efficient calculation of electromagnetic scattering coefficients of multilayered spheres. The new algorithm simplifies the recursive formulas by introducing Debye potentials and logarithmic derivatives, effectively avoiding numerical overflow issues associated with Bessel functions under large complex variables. Moreover, by adopting a hybrid recursive strategy, this algorithm can resolve the singularity problem associated with logarithmic derivatives in previous algorithms. Numerical test results demonstrate that this algorithm offers superior stability and applicability when dealing with complex cases such as thin shells and strongly absorbing media.

Figures (5)

• Figure 1: Schematic of an $n$-layer sphere embedded in an infinite dielectric medium. The outer radius, permittivity, and permeability of each layer are $r_i$, $\epsilon_i$, and $\mu_i$, respectively, with $i = 1, 2, ..., n$. The surrounding medium has vacuum permittivity $\epsilon_{n+1} = \epsilon_0$.
• Figure 2: (a) Plot of the extinction efficiency $Q_{ext}$ for the coated sphere as a function of the outermost layer's size parameter. (b) Loglog plot of $Q_{ext}$ - 2 of the coated sphere as a function of the outermost layer's size parameter. The plot compares the results with those obtained using RTMA. The refractive indices of the core and shell are $m_1=1.33+0.0\mathrm{i}$ and $m_2=1.33+\mathrm{i}$, respectively.
• Figure 3: (a) The extinction efficiency $Q_{\rm ext}$ of the multilayer sphere as a function of the number of layers. The refractive index of each layer is randomly generated. (b) Absolute difference between $Q_{\rm ext}$ and $Q_{\rm ext}^{\rm Mie}$, where $Q_{\rm ext}^{\rm Mie}$ is the extinction efficiency for a homogeneous sphere, computed using the Mie algorithm mishchenko02. The refractive index is fixed at $1.33 + \mathrm{i}$. The size parameter of the sphere in (a) and (b) is $4\pi$.
• Figure 4: The emitted energy of a SiC+gold coated sphere at a temperature of 300K, as a function of the outer radius R, is normalized with respect to the Stefan-Boltzmann result. Different curves correspond to different ratios of inner to outer layer radii. For SiC and gold, the optical constants' data are from laor93 and johnson72 respectively.
• Figure 5: Performance validation. Prog. 1: Our implementation of the mRTMA. Prog. 2: Our implementation of Yang's recursive algorithm. Prog. 3: Scattnlay's implementation of Yang's recursive algorithm. Size parameter of the sphere is $4\pi$.