This example shows how to use FDTD to find the bandstructure of an ideal all dielectric metamaterial that exhibits a zero effective index at optical frequencies.
Bandstructure: The following figure shows the simulation setup in the FDTD Solution's file dielectric_zim_band.fsp. It consists of a 2D stack of infinitely long silicon rods embedded in a Si02 cladding both of which are assigned their respective non-dispersive index given in Valentine . Due to the axial invariance of the ZIM, only a 2D FDTD simulation domain is required to obtain the bandstructure. Furthermore, to efficiently simulate the ZIM only a single unit cell is included in which the boundaries are assigned Bloch periodic boundary conditions with wave vectors kx and ky specified across them.
The dipole cloud is used to excite the ZIM. It consists of a 10 randomly distributed electric dipole sources which are oriented along the rod's axis to only excite the ZIM's TM modes. The frequency range of the source is set to 0 to 250THz.
A set of randomly distributed time monitors to measure the electric fields, a bandstructure calculation which apodizes the measured time signal to filter out its beginning and end, and a FFT, we are able to find the resonances of the ZIM at a particular value of kx and/or ky. By sweeping over all kx and ky values, we can reconstruct the entire dispersion diagram (or bandstructure) of the ZIM. This is done in the optimizations and sweeps toolbox which contains a sweep of kx and ky over the entire Brillouin zone (i.e., from the Gamma - X, X - M, and M - Gamma).
Results and Analysis
The script file dielectric_zim_band.lsf is used to run all the sweeps and generate the complete dispersion diagram of the ZIM shown in the above figure. An interesting feature of the dispersion diagram is the frequency in which the two transverse bands (TM2 and TM4) with linear dispersion intersects a quasi-longitudinal band (TM3). It is at this triple degeneracy frequency in which the metamaterial exhibits a zero effective index in which both permittivity and permeability are zero. This can be verified through a retrieval of the bulk effective medium properties of the metamaterial using a field averaging of the Bloch modes (as detailed in Valentine ). Since the dipole cloud will excite all the resonant modes of the ZIM, the field profiles will consist of a linear combination of each modes. In order to extract a single mode's field profile (such as the TM4 mode's profile which is used in the field averaging), the source(s) need to be placed strategically inside the simulation to excite only a single mode and not others. More details on setting up the simulation can be found in the Bloch mode profile page.
The triple degeneracy response in the dispersion diagram is attributed to exciting the dielectric rod's 1st and 2nd Mie resonances (the electric monopole and magnetic dipole). To verify this in the simulation, the script finished by turning off the dipole cloud and turning on a single dipole source located at the center of the simulation. The simulation is run twice with the orientation and dipole type changed between each simulation in order to excite each resonance individually. Afterwords, the script generates the electric and magnetic field plots at the triple degeneracy frequency which are shown below.
1. J. Valentine et al., "Realization of an all-dielectric zero-index optical metamaterial", Nature Photonics, vol. 7, (2013).
Metamaterial simulation methodology, Square 2D lattice bandstructure, Bloch mode profile