A Novel Transformation Optics-Based FDTD Algorithm for Fast Electromagnetic Analysis of Small Structures in a Large Scope

Abstract

In this paper, we present a novel Transformation Optics based Finite-difference time-domain (TO-FDTD) algorithm with Square Transformed Region, which is suitable for fast electromagnetic analysis of small structures in a large computational domain. A small structure can be solved with coarse grids in a transformed region, since it is enlarged by transformation optics. No fine grid is applied for the small structure so that computational efficiency can be improved greatly. In addition, the TO-FDTD algorithm can also avoid time-space field interpolations and the late-time instability of the subgridding algorithm. To eliminate the staircasing errors introduced by curved boundaries of its transformed region, we propose the square transformed region instead of the circular one. We parameterize the square boundary in polar coordinates so that it is easy to be implemented. Through coordinate transformation, new anisotropic permittivity and permeability can be obtained in the transformed region. Then we develop a stable anisotropic FDTD algorithm to solve the transformed Maxwell's Equations. Numerical results on diffraction and scattering of small structures show that our proposed TO-FDTD algorithm has high computational efficiency and accuracy.INDEX TERMS Finite-difference time-domain (FDTD), staircasing errors, transformation optics.

Faster and More Accurate Time Domain Electromagnetic Simulation Using Space Transformation

“…The goal of transformational optics is to create two equivalent optical systems, with two different geometries and different material parameters which are related by a specified diffeomorphism. This allows us to alter the geometry of our initial problem to make it easier to be solved either numerically [14,[25][26][27][28] or analytically [17][18][19][20][21]. For instance, consider the nonuniform structure which is illustrated in Fig.…”

“…It is clear that the above dispersion formulas are just the Drude model for a lossless medium. Although such a material will make the simulation stable, it limits the bandwidth of the simulation [14,[25][26][27]. This constraint should not be considered as an obstacle when simulating many potential SERS devices as the excitation in Raman spectroscopy is done using a monochrome laser with a very short bandwidth.…”

“…For many optical applications, FDTD is preferred since it directly allows us to investigate the nonlinear behavior of materials and obtain their frequency responses by performing a single simulation. However, when the problem contains sub-wavelength geometrical details [14] across a relatively large computational domain it is very difficult to obtain accurate results by FDTD. Sub-gridding FDTD algorithms are one of the most popular approaches that can help to overcome this problem [15,16].…”

“…In [14,[25][26][27][28] the simulation geometry was transformed so that more meshing cells were placed on a small object by enlarging the area around it. Ref [14] showed that transformation optics FDTD is superior in terms of stability and accuracy in comparison to sub-gridding FDTD. However, all these methods still suffer from discretization errors as the cubic grid of FDTD points cannot easily follow curved boundaries.…”

Novel Time-Domain Electromagnetic Simulation Using Triangular Meshes by Applying Space Curvature

“…It is however possible to ensure that the solutions of the transformed equations map one to one onto the solutions of the real physical system provided that in addition to transforming the spatial grid the material parameters are also changed appropriately [26]- [30]. This approach takes concepts from transformation optics [31] and uses them to solve Maxwell's Equations over an arbitrary domain and such methods are known as TO-FDTD.…”