On this page, we show how to set up bulk electric and magnetic properties to have a negative index medium over a range of wavelengths.
Material setup
FDTD includes a magnetic and electric Lorentz medium, described at Permittivity models. We can use this material model to create a bulk negative index medium where both real(ε) and real(μ) are negative at the same wavelength.
The media have a relative permittivity and permeability given by
$$ \begin{array}{l}{\varepsilon(\omega)=\varepsilon_{\text {bare }}(\omega)+\chi_{e}+\frac{\Delta \varepsilon \omega_{e}^{2}}{\omega_{e}^{2}-2 i \delta_{e} \omega-\omega^{2}}} \\ {\mu(\omega)=1+\chi_{m}+\frac{\Delta \mu \omega_{m}^{2}}{\omega_{m}^{2}-2 i \delta_{m} \omega-\omega^{2}}}\end{array} $$
where the subscript e and m refer to the electric and magnetic properties respectively and ω is the angular frequency. In this example, we choose the following properties:
- We use no base material. This has the effect that εbase = 1. In other words, the base material is actually the vacuum.
- \(\Delta \varepsilon\) = \(\Delta \mu\) = 1
- ωe = ωm = 2e15
- δe = δm = 1e13
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Note: model coefficients In real materials (and real metamaterials), the permittivity and permeability are obviously different. However, for the purposes of simplicity in this example, we have used the same model coefficients for both eps and mu. |
We can open the file negative_index.fsp which includes this material and can then use the script file magnetic_electric_lorentz.lsf to plot the resulting properties. The relative permittivity (which in this case is equal to the relative permeability) is shown below.
The real and imaginary parts of the permittivity.
We see that from 700 nm to 800 nm, the permittivity (and permeability) are negative while the imaginary part is less than 0.1. We further expect that near 760 nm, the real part of the refractive index will be close to -1.
Simulation setup
We setup a simulation with a beam at 45 degrees incidence on a 2 micron slab of our negative index medium in a background of air. In addition, we have added a mesh override region over the slab, and set the equivalent index for the mesh to 2. The reason is that the magnetic electric Lorentz medium is included with FDTD but is implemented as a plugin. At mesh time the software will base the target mesh size for this material on the background material (which defaults to the vacuum if none is selected). Therefore we may want to use a mesh override region to force a smaller mesh size if we know that the material will need it. In this case, for safety, we override the mesh with an equivalent index of 2, which is more than sufficient for the electric and magnetic properties over the wavelength range of 700 to 800 nm.
Additionally, since the beam is incident at 45 degrees, which is relatively steep, we increased the minimum number of layers of PML from 12 to 24 in the Advanced Options of the FDTD simulation region.
Note, the PML default settings are modified to overcome possible diverging simulation, which need some knowledge on PML.
Results
After running the simulation, we can visualize the E field over the entire structure from the monitor called 'profile', as shown below.
We can select the parameter lamba and the drag the slider on the lower right to look at the profiles at different wavelength. The profile at approximately 761 nm has the lowest reflection and highest transmission, as we would expect from the curve permittivity data. We can see this result, which clearly shows the unusual refractive properties of a bulk negative index medium.
See also