Since XFdtd^® includes frequency-dependent dielectric and magnetic materials, it is capable of making three-dimensional calculations for double negative materials, also called negative index materials and meta-materials. Learn how XFdtd can be applied to these new materials.

There is currently considerable interest in the electromagnetic behavior of materials that have both permittivity and permeability with negative real parts. These materials have several names including Negative Index Materials (NIM) and Double Negative Materials (DNG). This interest is based in large part on the availability of meta-materials with this characteristic. XFdtd, even though a time domain solver, has the capability to make calculations for these materials. This example illustrates this capability and presents some interesting results for these unusual materials.

While XFdtd is intended for full 3D calculations, these figures were made for a 2D geometry, which is efficient for illustrating the behavior of these materials. The geometry is shown in Figure 1. The green rectangle is the location of a 2D slab of material. The simple horn antenna is fed by a waveguide with a 30 GHz voltage source. The antenna is tilted at a 20 degree angle with the slab normal. The voltage waveform is ramped in amplitude over the first cycle. The electric field is polarized perpendicular to the plane of the figures so the magnetic field is in the plane of incidence.

There are four different materials considered. First the slab is removed and the antenna radiates in free space. Next a dielectric slab with relative mu = epsilon = 4 is considered. This slab has the same impedance as free space so should have no reflections at its interfaces. Finally, a double negative material, as described in “Wave propagation in media having negative permittivity and permeability” by Richard W. Ziolkowski and Ehud Heyman, in Physical Review E, Volume 64, 056625, composes the slab. At 30 GHz the dielectric/magnetic materials in the slab have relative complex permittivity/permeability of –0.99891 +j3.91E-2. This is for engineering exp(+j t) time convention. Thus the materials have a small loss which must be true for physically realizable passive materials. For both the dielectric and magnetic materials with negative real permittivity/permeability, calculations are made using the recursive convolution FDTD implementation for Drude materials which is implemented in XFdtd. The cell size is 0.1 mm and the time step is 0.0963 ps.

Consider the sequence of Figure 2 , Figure 3, and Figure 4. These are for the four different slab materials being considered. All are after 9900 time steps and are for the same false color dB scale. Except for the free space result the slab material is present in all calculations, but the display of the slab material is turned off so that the interior fields can be clearly seen. The slab boundaries are marked on the figures. The free space calculation shows the fields that exist with no slab present. The slab with epsilon=mu=4 has the same impedance as free space, so there is no reflection at the slab boundaries. However, the velocity of propagation in the slab is ¼ that of free space so the wave fronts are ¼ of the free space separation. Beyond the slab the fields are essentially those of the free space calculation, except for a phase difference, since this slab does not attenuate the field energy and also since there must be a phase match on both slab interfaces with free space. Note that at both slab surfaces the wavefronts are continuous. That is, each wave peak and null can be traced across the slab boundaries with free space.

Finally consider Figure 4. For this slab both the dielectric and magnetic materials which compose it have the double negative relative permittivity/ permeability given above. This results in a material with an impedance very close to that of free space, so that there is little reflection at the surfaces of the slab. Once the wavefronts pass through the slab and back into free space the wavefronts realign and are very similar to those for both the free space and epsilon = mu =4 material, showing proper satisfaction of the phase matching conditions at both slab surfaces. The amplitude is reduced after the waves pass through the double negative slab due to attenuation.

Since both permittivity and permeability have negative real parts, the double negative material has a negative phase velocity so that the wavefronts propagate in the opposite direction than is the case for a “normal” material. Figure 5 shows a time step movie of the fields after the calculation has reached steady state. Within the slab the wavefronts clearly appear to be propagating toward the source, while in both free space regions outside the slab the wave fronts propagate away from the source.

Does this indicate lack of causality in the XFdtd result? Let’s look at some earlier time results which show the initial interactions of the fields from the source with the slab. Time domain snapshots when the fields from the source first interact with the double negative slab are shown in sequence of Figure 6, Figure 7, Figure 8, and Figure 9. In order to show the low level signals at early time, the dB scale and reference level are changed, saturating the colors for the higher field values. This allows observation of the gradual bending of the wavefronts as the reverse propagation direction in the slab is developed. But note that the leading edge of the electric field passing through the slab maintains a spherical wave front.

More information on using XFdtd to investigate the properties of these materials can be obtained by viewing the presentation in the XFdtd Showcase.