Top-down extended meshing algorithm and its applications to Green’s tensor nano-optics calculations

We present a computational algorithm which speeds up Green’s tensor nano-optics calculations by means of optimizing the mesh that represents the system we want to investigate. The algorithm automates the process of creating a variable-size mesh that describes an arbitrary nanostructure. The total number of elements of this mesh is smaller than that of a regular mesh representing the same structure, and thus the Green’s tensor calculations can be performed faster. Precision, however, is kept at a similar level than for the regular mesh. Typically, the algorithm yields a mesh that speeds up Green’s tensor calculations by a factor of 4, while giving a maximum error in the field magnitude of about 5%. The speed-up factor makes it very suitable for otherwise lengthy calculations, and the error should be acceptable for most applications.

Figure 1

(Color online) Comparison between proper and extended mesh elements. A proper mesh element with no edge (top black square) can sit at the surface of the object. By introducing an edge, the proper mesh element (bottom black square) is forced to sit at a distance from the surface.

Figure 2

(Color online) Example of a mesh created by TEMA. A diameter 45 sphere with an asymmetric void is re-meshed with mesh sizes of 13 and below. The figure shows a cut through the center of the sphere. The mesh elements are color-coded according to their sizes.

Figure 3

Comparison between calculation times for the original, regular mesh, and the mesh created by TEMA. The two sets of calculation times are fitted to a quadratic function $t=A{N}^2$, where $N$ is the number of mesh elements in the original mesh.

Figure 4

(Color online) GT calculations of a “crescent moon”-like gold structure in vacuum. The wavelength is $\lambda =600\mathrm{nm}$, and the gold dielectric constant is $ϵ=-9.42+1.50i$. (a) Field enhancement, $2\mathrm{nm}$ below the surface, with the original mesh. (b) Same plot, but with the TEMA mesh. (c) Variable size mesh plot with no mesh edges. Notice the big difference between this plot and the previous two.

Figure 5

(Color online) The same sample as in Fig. 2, this time with variable edges so that larger mesh elements are placed in the inner part of the sample.