High-order accurate schemes for Maxwell's equations with nonlinear active media and material interfaces

TL;DR

This paper introduces a fourth-order accurate finite-difference time-domain scheme for Maxwell's equations with nonlinear media, capable of handling complex geometries and interfaces efficiently, suitable for advanced plasmonic applications.

Contribution

The paper presents a novel hierarchical modified equation approach for explicit, high-order Maxwell's scheme with curved interfaces and complex geometries, avoiding nonlinear iterations.

Findings

Achieves large CFL-one time-step stability.

Accurately models complex geometries with curvilinear and overset grids.

Validated for 2D and 3D nonlinear dispersive media applications.

Abstract

We describe a fourth-order accurate finite-difference time-domain scheme for solving dispersive Maxwell's equations with nonlinear multi-level carrier kinetics models. The scheme is based on an efficient single-step three time-level modified equation approach for Maxwell's equations in second-order form for the electric field coupled to ODEs for the polarization vectors and population densities of the atomic levels. The resulting scheme has a large CFL-one time-step. Curved interfaces between different materials are accurately treated with curvilinear grids and compatibility conditions. A novel hierarchical modified equation approach leads to an explicit scheme that does not require any nonlinear iterations. The hierarchical approach at interfaces leads to local updates at the interface with no coupling in the tangential directions. Complex geometry is treated with overset grids. Numerical stability is maintained using high-order upwind dissipation designed for Maxwell's equations in second-order form. The scheme is carefully verified for a number of two and three-dimensional problems. The resulting numerical model with generalized dispersion and arbitrary nonlinear multi-level system can be used for many plasmonic applications such as for ab initio time domain modeling of nonlinear engineered materials for nanolasing applications, where nano-patterned plasmonic dispersive arrays are used to enhance otherwise weak nonlinearity in the active media.