ENZ Waveguides

Introduction

Waveguides are used to transmit electromagnetic energy from one location to another. This can be used to transmit either power or information. In some sense, every electromagnetic conduit is a waveguide including power cords, striplines, RF coaxial cables, and optical fibers. However, we only concern ourselves with the waveguiding nature when the wavelength becomes short enough that we have to be careful about reflections as cross-section and length dimensions become on the order of and larger than the wavelength. In other words, we happily fold power cords behind our desks in giant balls and depend on Kirchoff to electrically untangle our mess, but we must be careful not to bend or kink an optical fiber in the slightest.

In these works, we have explored how we can use metamaterials (materials with artificially created permittivities and permeabilities) to give optical fibers and RF waveguides the resiliency of the power cord. Our method is to create waveguides with abnormally large phase velocities and therefore large propagating wavelengths. It is well known that while energy can not travel faster than the speed of light, the phase velocity may be much larger. In this way two points on the waveguide can be physically far but electrically near and any abruption in between them will appear electrically small in comparison with the wavelength. We can then employ Kirchoff at frequencies and distances where his laws are generally considered useless.

Most metamaterial work involves the creation of vast 3D arrays of small resonant structures that have separations of under one-tenth of the free space wavelength. The material is then "homogenized" to yield an effective permittivity and permeability. This is done in an effort to create an isotropic material that behaves the same for a wave traveling in any direction. In contrast, in these works, the wave energy is confined to travel either forward or backward along the waveguide and isotropy at the small scale is not necessary to capture the wave behavior. Instead, we take advantage of the waveguide structure itself and employ different levels of homogenization.

TE10

Cross-section geometry for TE10 mode at cut-off.

In our PRL 2008, JAP 2009, PRB 2009 works, we built a section of waveguide operating under the TE10 mode. The width was designed so that at the operating frequency this mode was precisely at cutoff so that the k-vector along the length of the waveguide was zero. The mode can be imagined as superposition of two plane waves propagating in the +y and -y directions. Since the k-vector along the length near zero, the waveguide index also near zero, giving the waveguide epsilon-near-zero (ENZ) properties

The thin ENZ region is able to transmit energy and operates independent of length.

Normally one would expect that a waveguide at cutoff would not transmit energy, but we found that by reducing the height of the waveguide in the x-direction, we could better match to normal waveguides on either side. The energy is transferred in a mechanism similar to electron tunneling. Since the wave vector is zero along the waveguide, the length of this ENZ section is irrelevant and the phase at the input and exit faces are always identical.

Additionally, since all abruptions are small compared to the wavelength, we can apply twists and turns in it that we would not expect a typical waveguide to survive.

ENZ waveguide channeling energy with infinite phase velocity through a tortuous channel spelling ENZ.

The electric field is magnified inside the ENZ region and this can be used to enhance nonlinear effects.

D-Dot

Cross-section geometry for TE10 mode at cut-off.

In our PRL 2012 works, we made use of a different waveguide geometry. Here we employ a hybrid mode with primarily longitudinal electric field, leading to a longitudinal displacement current (or D-Dot current). Again, this waveguide is sized so that it has a propagation constant which is zero along the waveguide direction at the frequency of interest. The central region is filled with dielectric which can be used to guide the wave along arbitrary paths.

This geometry is interesting because it has many of the same qualities of a capacitor and a coaxial cable, all at frequencies and size scales where neither would seem applicable. Similar to our previous work, the distance between the two conducting plates can be used to define an effective permittivity. Within the dielectric region, this effective permittivity is positive while in the air regions, this effective permittivity is negative (i.e. plasmonic). Since the effective displacement current is proportional to the effective permittivity and the electric field is along the +z-direction everywhere, there is a forward displacement current in the center and a "return" displacement current on either side, similar to a coaxial cable. Since the energy travels with a primarily longitudinal current across dielectric space, it is similar to a capacitor.

The D-Dot wire concept is particularly interesting at optical frequencies where we would not need to create artificial plasmonic materials as they exist naturally. The DDot wire could potentially be used as a new type of circuit connector individual devices electrically close, even if they are many free-space wavelengths apart.