Lattice Boltzmann method for electromagnetic wave scattering

TL;DR

This work demonstrates that lattice Boltzmann methods can accurately solve electromagnetic scattering problems across one-, two-, and three-dimensional geometries, spanning Rayleigh to geometric-optics regimes. By solving Maxwell's equations on a D3Q7 lattice and employing an NTFF transformation, the authors validate LBM against analytical Mie theory for spheres and cylinders and against the Discretized Mie Formalism for hexagonal cylinders, including high-dielectric-contrast cases. The results show excellent agreement across canonical and noncanonical geometries, highlighting LBM's potential as a versatile, scalable alternative to FDTD, FEM, and DDA for complex scattering problems. The work also provides an open-source solver, emphasizes realistic extensions (TF/SF, absorbing boundaries, dispersive media), and positions LBM as a promising tool for photonics, atmospheric scattering, and related fields.

Abstract

In this paper, we propose the lattice Boltzmann method (LBM) as an alternative numerical approach for electromagnetic scattering. The method is systematically validated over a wide range of size parameters, thereby covering the Rayleigh, Mie, and geometric optics regimes, through comparison with established reference solutions. For circular cylinders, both perfect electrically conducting (PEC) and dielectric, LBM results are benchmarked against analytical Mie theory. For dielectric cylinders, comparisons are performed over a broad range of relative permittivities to assess accuracy across different material contrasts. Scattering from dielectric spheres is likewise compared with exact Mie solutions, showing excellent agreement. To assess performance for non-canonical geometries, we investigate a hexagonal dielectric cylinder and validate the results against the Discretized Mie-Formalism, demonstrating that LBM can accurately capture edge diffraction and sharp-facet effects. Overall, the study provides the first systematic benchmarking of LBM for electromagnetic scattering in one-, two-, and three-dimensional configurations, establishing it as a promising and versatile tool in computational electromagnetics.

Figures (12)

• Figure 1: Schematic of scattering problems studied: (a) planar dielectric interface, (b) circular cylinder, (c) hexagonal cylinder, and (d) sphere. Blue arrows indicate incident plane waves, red arrows indicate scattered waves (with varying lengths representing angular dependence), and green arrows denote transmitted fields inside the medium.
• Figure 2: Schematic of the near-to-far-field transformation. (a) The fields are first recorded on a fictitious boundary enclosing the scatterer. (b) These recorded near-field values are then converted to equivalent surface currents, which are analytically propagated to compute the scattered fields at a far-field observation point.
• Figure 3: Normalized amplitudes of reflected (black) and transmitted (red) electric fields relative to the incident field at a planar interface under normal incidence. (a) Dependence on dielectric constant $\varepsilon_r$ with $\mu_r=1$. (b) Dependence on relative permeability $\mu_r$ with $\varepsilon_r=1$. Solid lines represent analytical solutions, and markers denote LBM results.
• Figure 4: Real part of the total electric field for circular cylinders under plane-wave incidence. The top row shows perfect electrically conducting (PEC) cylinders, and the bottom row shows dielectric cylinders with $\varepsilon_r = 2$. Results are presented for three size-to-wavelength ratios: (a,d) $a/\lambda = 1/50$ (Rayleigh regime), (b,e) $a/\lambda = 1$ (Mie regime), and (c,f) $a/\lambda = 50$ (geometric optics regime). The dashed circle denotes the cylinder boundary. The plots illustrate the transition from weak Rayleigh scattering to resonance-dominated Mie scattering and eventually to shadowing and diffraction in the geometric optics limit.
• Figure 5: Comparison of the normalized scattering width ($\sigma_{2D}/\lambda$) of a perfect electrically conducting (PEC) circular cylinder across different scattering regimes. (a) Rayleigh regime with $a/\lambda = 1/50$ and $1/10$. (b) Mie regime with $a/\lambda = 1$. (c) Geometric optics regime with $a/\lambda = 10$ and $50$. Solid lines denote analytical Mie theory, while dashed lines represent LBM results. The close agreement across all regimes confirms the accuracy of LBM for PEC circular cylinders.