An Expedient Approach to FDTD-based Modeling of Finite Periodic Structures

TL;DR

It is shown that this algorithm efficiently determines the size of a periodic structure necessary for fields to converge to the infinitely periodic case.

Abstract

This paper proposes an efficient FDTD technique for determining electromagnetic fields interacting with a finite-sized 2D and 3D periodic structures. The technique combines periodic boundary conditions---modelling fields away from the edges of the structure---with independent simulations of fields near the edges of the structure. It is shown that this algorithm efficiently determines the size of a periodic structure necessary for fields to converge to the infinitely-periodic case. Numerical validations of the technique illustrate the savings concomitant with the algorithm.

Figures (16)

• Figure 1: A period showing two-dimensional FDTD PBC updates (in the $z$ direction). The $E_x$ field at $z=d_z$ is used to update the corresponding electric field at $z=0$ (red arrow). The $H_y$ field at $z=\Delta z/2$ updates the $H_y$ node at $z=d_z+\Delta z/2$ (blue arrow).
• Figure 2: An illustration of a unit cell (right square) with PBCs (blue lines) on the left and right faces. A source in the unit cell produces fields (red) which travel rightwards within the dashed lines emanating from the source. The fields are then phase-shifted and translated to the left face, where they continue to propagate, as shown. These fields then appear as though they were generated by a phase-shifted image source in an adjacent unit cell (left square).
• Figure 3: An illustration of the ASM method removing unwanted image sources in a periodic structure (period $d$). The ASM integration order $M$ determines the location of the nearest parasitic image source. By selecting $M$ to be sufficiently large, fields (shown in red) generated by the image do not enter a region of interest $a$ units to either side of the source within runtime. By the end of the simulation, the wavefronts from unwanted sources travel a distance of at most $Md-a$.
• Figure 4: The fields in the inner region (between the vertical blue lines) of a finite structure can be estimated by fields from an infinite structure (top). The fields along the edges of the structure (outside the blue lines) can be calculated by simulating the edges of the finite structure (middle). The fields can be fused together to create a continuous field approximating those interacting with the finite structure (bottom).
• Figure 5: A flowchart describing the procedure of efficiently determining fields interacting with finite periodic structures and estimating the accuracy of the result. Note that the ASM step may be used to determine tangential fields along the boundaries of many differently-sized regions simultaneously. Subsequently, variously-sized edge-unit cell simulations can be run simultaneously as well.