Propagation in and scattering from a matched metamaterial having a zero index of refraction

Planar metamaterials that exhibit a zero index of refraction have been realized experimentally by several research groups. Their existence stimulated the present investigation, which details the properties of a passive, dispersive metamaterial that is matched to free space and has an index of refraction equal to zero. Thus, unlike previous zero-index investigations, both the permittivity and permeability are zero here at a specified frequency. One-, two-, and three-dimensional source problems are treated analytically. The one- and two-dimensional source problem results are confirmed numerically with finite difference time domain (FDTD) simulations. The FDTD simulator is also used to treat the corresponding one- and two-dimensional scattering problems. It is shown that in both the source and scattering configurations the electromagnetic fields in a matched zero-index medium take on a static character in space, yet remain dynamic in time, in such a manner that the underlying physics remains associated with propagating fields. Zero phase variation at various points in the zero-index medium is demonstrated once steady-state conditions are obtained. These behaviors are used to illustrate why a zero-index metamaterial, such as a zero-index electromagnetic band-gap structured medium, significantly narrows the far-field pattern associated with an antenna located within it. They are also used to show how a matched zero-index slab could be used to transform curved wave fronts into planar ones.

Figure 1

Comparison of the electric field generated at $t=7500\Delta t$ by an infinite current sheet driven by a $30\mathrm{GHz}$ source in a matched zero-index Drude medium slab and in free space.

Figure 2

Electric field generated by the infinite current sheet driven by a $30\mathrm{GHz}$ source in a matched zero-index Drude medium slab. (a) Early times; (b) late times.

Figure 3

Comparison of the electric field time histories generated by an infinite current sheet driven by a $30\mathrm{GHz}$ source in a matched zero-index Drude medium slab at two spatially separated points shows that the phase difference between them is zero.

Figure 4

Comparison at $t=13500\Delta t$ of the electric field distribution resulting from a $30\mathrm{GHz}$ plane wave propagating through a matched zero-index Drude medium slab and propagating in free space.

Figure 5

Electric field distribution at late times resulting from a $30\mathrm{GHz}$ plane wave propagating through a matched zero-index Drude medium slab.

Figure 6

Comparison of the electric field time histories at two spatially separated points in a matched zero-index Drude medium slab that result from a $30\mathrm{GHz}$ plane wave propagating through it shows that the phase difference between them is zero.

Figure 7

Field components generated at late times by an infinite line current driven at $30\mathrm{GHz}$ in a zero-index matched Drude medium cylinder that is surrounded by free space. (a) Electric field $E_z$; (b) magnetic field component $H_x$.

Figure 8

The electric field distribution produced by an infinite line current driven at $30\mathrm{GHz}$ and centered in a matched zero-index Drude medium cylinder that is surrounded by free space. (a) $t=0$, (b) $t=240\Delta t$, and (c) $t=1800\Delta t$.

Figure 9

The electric field distribution produced by a line source driven at $30\mathrm{GHz}$ and centered in a rectangular matched zero-index Drude medium. (a) $t=0$, (b) $t=200\Delta t$, (c) $t=800\Delta t$, and (d) $t=2200\Delta t$.

Figure 10

The electric field intensity distribution produced by a line source driven at $30\mathrm{GHz}$ external to a rectangular matched zero-index Drude slab. (a) $t=0$, (b) $t=500\Delta t$, (c) $t=1000\Delta t$, and (d) $t=8000\Delta t$.

Figure 11

The FDTD predicted electric field values along the $y$-axis of the simulation space for an external line source radiating at $30\mathrm{GHz}$ in the presence of a rectangular matched zero-index Drude medium slab. A spatially constant, time varying electric field intensity is generated within the slab. (a) Electric field along the $y$-axis; (b) zoom-in.

Figure 12

The FDTD predicted electric field values at $t=8000\Delta t$ along the transverse centerline of the slab and a transverse line exterior to the slab, 100 cells from the exit face of the slab.