Active terahertz waveguides based on transmission-line metamaterials

Electromagnetic metamaterials are artificial structures that can be engineered to exhibit customizable or conventionally unobtainable electromagnetic properties, such as propagation with near-zero or even negative refractive index. In a material with a negative index, the flow of energy is opposite to the movement of the wavefronts, an effect known as backward-wave or left-handed propagation (so named because the electric field, magnetic field, and wavevector form a left-handed triple). At IR and optical frequencies, left-handed materials can be made by incorporating plasmonic structures into a dielectric. Provided the size and periodicity of the structures is sufficiently small compared to the wavelength, waves propagate as if the medium were uniform with new values for the refractive index (or other bulk properties). Current research in this area investigates electromagnetic metamaterials for novel antenna concepts, sub-wavelength resonators and waveguides, superlenses that beat the diffraction limit, and even cloaking from electromagnetic radiation.

Our research group has been working on methods to apply metamaterial concepts to the terahertz (THz) frequency range, where the wavelength is approximately a hundred times longer than in the visible. The novelty in our work is the combination of metamaterial-inspired waveguides with a THz quantum-cascade laser-gain medium. In this way, stimulated emission of THz photons from intraband transitions in the gallium-arsenide-based medium compensates for losses and allows active devices.¹

To design and describe the metamaterial waveguide, we adopt the transmission-line formalism, where negative and zero-index propagation are modeled by the introduction of additional lumped element capacitance and inductance into the series and shunt branches of the transmission line.² Where a conventional transmission line has series inductance L_R and shunt capacitance C_R, a metamaterial line is modeled by adding series capacitance C_L and shunt inductance L_L (the subscripts L and R stand for left-handed and right-handed propagation, respectively). We can adapt this scheme to THz quantum-cascade devices, which are fabricated into a metal-dielectric-metal waveguide. Figure 1 shows the calculated dispersion relation for a typical design with left-handed propagation below about 2.6THz and right-handed propagation above 2.6THz: a composite right-/left-handed (CRLH) metamaterial waveguide. At 2.6THz, the dispersion relation crosses between the two types of propagation, without a stopband, while maintaining non-zero group velocity. Such a condition is referred to as balanced and results from proper engineering of the effective capacitance and inductance on the transmission line.

Figure 1. Calculated dispersion relation for a balanced terahertz (THz) metamaterial waveguide exhibiting left-handed (LH) and right-handed (RH) propagation. GaAs: Gallium arsenide. AlGaAs: Aluminum gallium arsenide. p: Unit cell size.

The key advance in this recent work is the inclusion of 200nm-size gaps in the top metallization of the waveguide: see Figure 2. These gaps play the role of a series capacitance in the transmission-line model for the waveguide, and are the key feature that enables left-handed propagation. We demonstrated the existence of left-handed propagation indirectly by using a section of the CRLH metamaterial waveguide as a leaky-wave coupling antenna for a THz quantum-cascade laser. The laser feeds the antenna with the THz signal, which is then radiated into the far-field at an angle that depends on the THz frequency and its location on the dispersion diagram. While propagation in the right-handed region will result in a beam angled in the forward direction, propagation in the left-handed region generates a beam angled in the backward direction (off normal). Propagation with a zero effective index (β=0) gives a beam directed in the surface normal direction. Therefore, by measuring the far-field beam pattern and the radiation frequency, we can reconstruct the dispersion relation: see Figure 1. We recently observed a backward-directed beam for the first time, demonstrating the existence of left-handed propagation.³

Figure 2. Image of a composite right-/left-handed metamaterial waveguide implemented in a THz quantum-cascade (QC) metal-metal waveguide. The dielectric of this ‘transmission line’ is made up of active THz QC gain material grown in GaAs/AlGaAs quantum wells. The inset image shows a close-up of 200nm gaps in the metallization that create the series capacitance C_L and enable left-handed propagation. C_x, L_x: Capacitors, inductors (where x denotes R or L). Cu: Copper. Cr: Chromium. Au: Gold.

Beyond this proof of principle, we now have access to a wide array of microwave circuit, antenna, and metamaterial design techniques that can be applied to THz lasers. For example, such a metamaterial antenna could be used to steer a beam between the forward and backward directions (depending on the exact frequency). Or, if we can develop dynamic control of the circuit elements, tunable resonators and phase shifters become possible. Our future work focuses on using these design techniques to create a new class of lasers with flexible and dynamic control of spectral and radiation properties, including beam shaping and steering, wavelength tuning, and polarization state.