Abstract-Terahertz quantum-cascade patch-antenna VECSEL with low power dissipation

Applied Physics Letters

Christopher A. Curwen, John L. Reno,  Benjamin S. Williams,

(a) Simulated peak reflectance for a ridge-based (Λ = 50 μm, w = 8.3 μm) and patch-based (Λ = 47 μm, w = 11.45 μm, L = 7.5 μm) metasurface design for a 10 μm thick 4.7 THz active region. Insets show field distribution for the two cases (note: relative strength of fields between the patch and ridge are not to scale).

https://aip.scitation.org/doi/abs/10.1063/5.0008867

We report a terahertz quantum-cascade vertical-external-cavity surface-emitting laser (QC-VECSEL) based upon a metasurface consisting of an array of gain-loaded resonant patch antennas. Compared with the typical ridge-based metasurfaces previously used for QC-VECSELs, the patch antenna surface can be designed with a much sparser fill factor of gain material, which allows for reduced heat dissipation and improved thermal performance. It also exhibits larger amplification thanks to enhanced interaction between the incident radiation and the QC-gain material. We demonstrate devices that produce several milliwatts of continuous-wave power in a single mode at ∼4.6 THz and dissipate less than 1 W of pump power. Use of different output couplers demonstrates the ability to optimize device performance for either high power or high operating temperature. Maximum demonstrated power is 6.7 mW at 4 K (0.67% wall-plug efficiency, WPE) and 0.8 mW at 77 K (0.06% WPE). Directive output beams are measured throughout with divergence angles of ∼5°.

Microfabrication was performed at the UCLA Nanoelectronics Research Facility, electron beam lithography was performed at the California NanoSystems Institute (CNSI) at UCLA, and wire bonding was performed at the UCLA Center for High Frequency Electronics. This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solution of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-NA-0003525. Partial funding was provided by the National Science Foundation (Nos. 1407711 and 1711892) and National Aeronautics and Space Administration (Nos. NNX16AC73G and 80NSSC19K0700).

Abstract-Broadband continuous single-mode tuning of a short-cavity quantum-cascade VECSEL

Christopher A. Curwen, John Reno,  Benjamin S. Williams,

Metasurface cavity design and electromagnetic simulations

https://www.nature.com/articles/s41566-019-0518-z

Changing the length of a laser cavity is a simple technique for continuously tuning the wavelength of a laser but is rarely used for broad fractional tuning, with a notable exception of the vertical-cavity surface-emitting laser (VCSEL). This is because, to avoid mode hopping, the cavity must be kept optically short to ensure a large free spectral range compared to the gain bandwidth of the amplifying material. Terahertz quantum-cascade lasers are ideal candidates for such a short cavity scheme as they demonstrate exceptional gain bandwidths (up to octave spanning)3 and can be integrated with broadband amplifying metasurfaces4. We present such a quantum-cascade metasurface-based vertical-external-cavity surface-emitting laser (VECSEL) that exhibits over 20% continuous fractional tuning of a single laser mode. Such tuning is possible because the metasurface has subwavelength thickness, which allows lasing on low-order Fabry–Pérot cavity modes. Good beam quality and high output power are simultaneously obtained.

Abstract-Robust Density Matrix Simulation of Terahertz Quantum Cascade Lasers

Benjamin A. Burnett, Andrew Pan, Chi On Chui, Benjamin S. Williams,

https://ieeexplore.ieee.org/document/8402230/

A common setback to electron transport models for quantum cascade laser active regions is the inability to freely simulate widely varying designs. One solution to this problem is to use a density matrix formalism with a generalized treatment of scattering, wherein the well-defined energy eigenbasis is used, and the relative simplicity of the density matrix can be taken advantage of for rapid simulations. We have developed such a model from first principles in the past, and now built a simulator for terahertz quantum cascade lasers that calculates a fully self-consistent solution to the coupled problem of bandstructure, lasing field strength, and space charge. This level of depth enables us to examine the model's performance across much of the design space and operating temperatures, for which we find generally good agreement. Areas for future improvement of the model are discussed, particularly the treatment of electron–electron scattering and continuum leakage. The model also enables us to make qualitative insights into the microscopic workings of the active regions, such as the nonequilibrium subband distributions and their response to the optical field, and the possibility for using two sequential optical transitions.

Abstract-Metasurface terahertz laser with electronically-controlled polarization

 Daguan Chen,  Luyao Xu, Christopher A. Curwen,  Mohammad Memarian, John L. Reno, Tatsuo Itoh,  Benjamin S. Williams,

http://ieeexplore.ieee.org/document/8083356/

We report a terahertz metasurface quantum-cascade VECSEL laser without moving parts that can electronically switch between near-orthogonal linearly polarized output. It exhibits excellent beam pattern, single-mode operation, and power up to 93 mW at 77 K.

Abstract-Density matrix modeling of quantum cascade lasers without an artificially localized basis: A generalized scattering approach

Andrew Pan, Benjamin A. Burnett, Chi On Chui, and Benjamin S. Williams

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.085308

We derive a density matrix (DM) theory for quantum cascade lasers (QCLs) that describes the influence of scattering on coherences through a generalized scattering superoperator. The theory enables quantitative modeling of QCLs, including localization and tunneling effects, using the well-defined energy eigenstates rather than the ad hoc localized basis states required by most previous DM models. Our microscopic approach to scattering also eliminates the need for phenomenological transition or dephasing rates. We discuss the physical interpretation and numerical implementation of the theory, presenting sets of both energy-resolved and thermally averaged equations, which can be used for detailed or compact device modeling. We illustrate the theory's applications by simulating a high performance resonant-phonon terahertz (THz) QCL design, which cannot be easily or accurately modeled using conventional DM methods. We show that the theory's inclusion of coherences is crucial for describing localization and tunneling effects consistent with experiment.

• 
• 
• 
• 
• 
• 
• 

Abstract-Design strategy for terahertz quantum dot cascade lasers

Benjamin A. Burnett and Benjamin S. Williams

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-24-22-25471

The development of quantum dot cascade lasers has been proposed as a path to obtain terahertz semiconductor lasers that operate at room temperature. The expected benefit is due to the suppression of nonradiative electron-phonon scattering and reduced dephasing that accompanies discretization of the electronic energy spectrum. We present numerical modeling which predicts that simple scaling of conventional quantum well based designs to the quantum dot regime will likely fail due to electrical instability associated with high-field domain formation. A design strategy adapted for terahertz quantum dot cascade lasers is presented which avoids these problems. Counterintuitively, this involves the resonant depopulation of the laser’s upper state with the LO-phonon energy. The strategy is tested theoretically using a density matrix model of transport and gain, which predicts sufficient gain for lasing at stable operating points. Finally, the effect of quantum dot size inhomogeneity on the optical lineshape is explored, suggesting that the design concept is robust to a moderate amount of statistical variation.

© 2016 Optical Society of America

Full Article  |  PDF Article

Abstract-Origins of Terahertz Difference Frequency Susceptibility in Midinfrared Quantum Cascade Lasers

Benjamin A. Burnett and Benjamin S. Williams

https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.5.034013

We present a density-matrix-based transport model applicable to quantum cascade lasers which computes both linear and nonlinear optical properties coherently and nonperturbatively. The model is applied to a dual-active-region midinfrared quantum cascade laser which generates terahertz radiation at the difference frequency between two midinfrared pumps. A new mechanism for terahertz generation is identified as self-detection, ascribed to the beating of current flow following the intensity, associated with stimulated emission. This mechanism peaks at optical rectification but exhibits a bandwidth reaching significantly into the terahertz range, which is primarily limited by the subpicosecond intersubband lifetimes. A metric is derived to assess the strength of self-detection in candidate active regions through experiment alone, and suggestions are made for improvement of the performance at frequencies below 2 THz.

• 
• 
• 
• 
• 
• 
• 

Abstract-Terahertz quantum cascade VECSEL

Luyao Xu, Christopher A. Curwen, Philip W. C. Hon, Tatsuo Itoh, Benjamin S. Williams

Univ. of California, Los Angeles (United States)

Proc. SPIE 9734, Vertical External Cavity Surface Emitting Lasers (VECSELs) VI, 97340G (March 10, 2016); doi:10.1117/12.2213230

http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=2503446

Vertical-external-cavity surface-emitting lasers (VECSELs) have been successfully used in the visible and near-infrared to achieve high output power with excellent Gaussian beam quality. However, the concept of VECSEL has been impossible to implement for quantum-cascade (QC) lasers due to the "intersubband selection rule". We have recently demonstrated the first VECSEL in the terahertz range. The enabling component for the QC-VECSEL is an amplifying metasurface reflector composed of a sparse array of metallic sub-cavities, which allows the normally incident radiation to interact with the electrically pumped QC gain medium. In this work, we presented multiple design variations based on the first demonstrated THz QC-VECSEL, regarding the lasing frequencies, the output coupler and the intra-cavity aperture. Our work on THz QC-VECSEL initiates a new approach towards achieving scalable output power in combination with a diffraction-limited beam pattern for THz QC-lasers. The design variations presented in this work further demonstrate the practicality and potential of VECSEL approach to make ideal terahertz QC-laser sources.
 © (2016) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.

Active terahertz waveguides based on transmission-line metamaterials

Benjamin S. Williams, Amir Ali Tavallaee, Philip Hon and Tatsuo Itoh The demonstration of a 1D left-handed metamaterial waveguide for terahertz quantum-cascade lasers opens the door to new techniques for beam steering and shaping. 22 August 2013, SPIE Newsroom. DOI: 10.1117/2.1201308.005081 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.2Where a conventional transmission line has series inductance LR and shunt capacitance CR, a metamaterial line is modeled by adding series capacitance CL and shunt inductance LL (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 CL and enable left-handed propagation. Cx, Lx: 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. Benjamin S. Williams, Amir Ali Tavallaee, Philip Hon, Tatsuo Itoh University of California at Los Angeles Los Angeles, CA References: 1. B. S. Williams, Terahertz quantum-cascade lasers, Nat. Photon. 1, p. 517-525, 2007. 2. A. Lai, C. Caloz, T. Itoh, Composite right/left-handed transmission line metamaterials,IEEE Microw. Mag. 5, p. 34-50, 2004. 3. A. A. Tavallaee, P. W. C. Hon, Q.-S. Chen, T. Itoh, B. S. Williams, Active terahertz quantum-cascade composite right/left handed metamaterial, Appl. Phys. Lett. 102, p. 021103, 2013.

Abstract-Active terahertz quantum-cascade composite right/left-handed metamaterial

http://apl.aip.org/resource/1/applab/v102/i2/p021103_s1?isAuthorized=no

Amir A. Tavallaee^1,2, Philip W. C. Hon¹, Qi-Sheng Chen³, Tatsuo Itoh¹, and Benjamin S. Williams^1,2

¹Electrical Engineering Department, University of California, Los Angeles, California 90095, USA
²California NanoSystems Institute, University of California, Los Angeles, California 90095, USA
³Northrop Grumman Aerospace Systems, Redondo Beach, California 90278, USA 

View Map

(Received 31 October 2012; accepted 19 December 2012; published online 14 January 2013)

•   Alerts 
•   Tools 
•   Share  

• Abstract
• References (22)
• Article Objects (4)
• Related Content

We report the demonstration of a composite right/left-handed (CRLH) metamaterial waveguide for terahertz quantum-cascade (QC) lasers. By incorporating gap capacitors (∼ 250 nm) in the top metallization of a metal-metal waveguide operating in a higher order lateral mode, we have realized a CRLH transmission line that supports traveling modes with negative effective phase indices (i.e., left-handed or backward-wave propagation). The CRLH metamaterial waveguide is employed as an active leaky-wave antenna for a terahertz QC-laser. Directional single-lobed beams launched in the backwards direction at angles of −4° and −63° were experimentally observed at excitation frequencies 2.59 and 2.48 THz, respectively.

Benjamin Williams at UCLA presents talk" Next generation terahertz sources: quantum cascade lasers and active metamaterials"

Benjamin Williams

UCLA, Electrical Engineering Department

what
when	Sep 19, 2012 from 01:00 PM to 02:30 PM
where	Engr. IV Bldg., Shannon Room 54-134
contact name	Prof. Ben Williams

Abstract

The terahertz frequency range (roughly 0.3-10 THz, or wavelengths of 30-1000 μm) remains one of the least developed regions of electromagnetic spectrum. Compared to the neighboring microwave/mm-wave or infrared spectral ranges it remains challenging to manipulate, to detect, and particularly to generate THz radiation. However, the location of the terahertz range offers unique opportunities for hybrid devices, as photonic techniques for achieving gain (i.e. stimulated emission and lasing) can be combined with lower-frequency circuit and antenna techniques for waveguiding and radiation control. In the past decade, the terahertz quantum cascade (QC) laser has emerged as a promising and flexible source of THz continuous-wave radiation with milliwatt power in the 2-5 THz range. However challenges remain in issues of output power, beam quality, operating temperature, and frequency tunability.

In this talk, I will discuss our efforts to address these challenges for THz sources using novel material and electromagnetic approaches. Using the THz QC-laser gain material as a foundation, we have introduced the concept of active composite right/left handed (CRLH) THz metamaterial waveguides. By loading sub-wavelength transmission-line laser waveguides with additional inductive and capacitive circuit elements, we can engineer the waveguide dispersion to enable new functionality, such as laser antennas for beam-steering and shaping and efficient out-coupling of THz radiation. We have demonstrated proof-of-concept devices including passive CRLH metamaterial waveguides, and active QC-laser metamaterial leaky-wave antennas. I will also discuss the prospects for novel THz laser devices based upon metamaterial waveguides, including widely tunable THz sources, and integrated THz laser phased arrays.

Biography

Benjamin Williams is an Assistant Professor of Electrical Engineering at the University of California, Los Angeles. He received the Ph.D. degree from the Massachusetts Institute of Technology, Cambridge, Massachusetts in 2003 in Electrical Engineering and Computer Science, and was a Postdoctoral Associate at the Research Laboratory of Electronics at MIT from 2003-2006. His research interests include quantum cascade lasers, intersubband and intersublevel devices in semiconductor nanostructures, and terahertz metamaterials and sub-wavelength plasmonics. He is the recipient of the DARPA Young Faculty Award (2008) and the NSF CAREER award (2012).

Abstract: Terahertz composite right-left handed transmission-line metamaterial waveguides

http://apl.aip.org/resource/1/applab/v100/i7/p071101_s1?isAuthorized=no

We report terahertz metamaterial waveguides based on the concept of composite right/left-handed transmission-lines. The waveguides are implemented in a metal-insulator-metal geometry fabricated with spin-coated Benzocyclobutene and contact photolithography. Angle-resolved reflection spectroscopy shows strong resonant absorption features corresponding to both right-handed and left-handed (backward wave) propagating modes within the leaky-wave bandwidth. Tuning of the waveguide dispersion is achieved by varying the effective lumped element series capacitance. The experimental results are in good agreement with full-wave finite element method simulations as well as an intuitive transmission-line circuit model.

© 2012 American Institute of Physics

Metamaterial concepts applied to terahertz-laser waveguides

http://spie.org/x47667.xml?highlight=x2404&ArticleID=x47667

Benjamin S. Williams, Amir Ali Tavallaee, Philip Hon and Tatsuo Itoh

A terahertz quantum-cascade laser with a transmission-line metamaterial coupler antenna offers an intermediate step toward a zero-index laser.

11 April 2011, SPIE Newsroom. DOI: 10.1117/2.1201103.003597

Researchers have devoted considerable recent effort to development of ‘electromagnetic metamaterials.’ Such materials can be engineered to exhibit customizable or conventionally unobtainable electromagnetic properties, including propagation with near-zero or even negative refractive index (i.e., backward wave propagation). This is typically done by incorporating lumped inductive or capacitive elements (or, at optical frequencies, plasmonic or dielectric elements) on length scales that are sufficiently smaller than the wavelength so that the medium appears homogeneous. Electromagnetic metamaterials are currently being used to investigate and implement novel antenna concepts, subwavelength resonators and waveguides, superlenses that beat the diffraction limit, and even electromagnetic cloaking.

A particular challenge is coping with the absorption that accompanies the various metallic inclusions. One approach offsets these losses by incorporating a source of gain into the metamaterial structure. For example, an active photonic material can provide gain through stimulated emission of photons.¹ The terahertz frequency range is particularly well suited for investigation of active photonic metamaterials, since metal is still a relatively good conductor, inductor-capacitor circuit elements can be fabricated using contact photolithography, and photonic gain is available through intersubband transitions in terahertz quantum-cascade (QC)-laser material.

Our group has proposed an approach to develop planar metamaterial waveguides that are suitable for integration with QC-laser material.² It is adapted from the transmission-line formalism where negative and zero-index propagation can be modeled by introduction of additional lumped-element capacitance and inductance into the series and shunt branches of the transmission line (see Figure 1).³ Figure 2 shows the calculated dispersion relation for a typical balanced design. This can be readily applied to terahertz QC devices, which are typically fabricated into a double-metal waveguide that is similar in form to microstrip transmission line. Gallium arsenide (GaAs)/aluminum GaAs multiple quantum wells comprise the dielectric of the transmission line and provide amplification through stimulated emission of terahertz photons.⁴

 

Figure 1. (a) Schematic of a candidate quantum-cascade (QC)-laser double-metal metamaterial waveguide, where gold contacts and ground plane are indicated in yellow. (b) 1D transmission-line metamaterial obtained by incorporating both shunt and series inductors and capacitors. THz: Terahertz. GaAs: Gallium arsenide. AlGaAs: Aluminum GaAs. C_x, L_x: Capacitors, inductors (where x denotes R or L). ℏω_LO: Stimulated emission energy.

 

Figure 2. Typical calculated dispersion relation for a balanced terahertz metamaterial waveguide exhibiting backward (at frequencies f < f₀) and forward wave propagation at (f > f₀), as well as zero-index propagation at f₀.

One device proposed by our group is a metamaterial ‘zero-index laser.’ This laser cavity is designed to oscillate in a mode with a zero phase index at frequency f₀ and, as such, exhibits a uniform mode in the longitudinal direction. Figure 3 shows the associated calculated electric-field pattern. This is very different from a conventional laser cavity in which the laser mode exhibits a sinusoidal standing-wave pattern and, hence, interacts nonuniformly with the gain medium. For this reason, a zero-index laser may be useful to suppress spatial-hole burning, a common phenomenon in lasers that can cause undesirable multimode oscillation. Other applications include traveling-wave metamaterial antennas that radiate with high efficiency and directivity in the forward and backward directions (depending on the exact frequency). The ability to engineer the beam and radiative coupling efficiencies would benefit terahertz QC-lasers in particular, since double-metal waveguides have notoriously poor beam patterns and coupling efficiencies.

 

Figure 3. Full-wave simulation of electric-field vectors and intensity in zero-index laser cavity oscillating at frequency f₀. Note the uniformity of the electric field in the longitudinal direction.

Since the amount of gain provided by the QC-laser material is limited, the success of this approach depends on designs that minimize losses. Our full-wave electromagnetic simulations indicate that while the ohmic and radiative losses are larger than in conventional terahertz QC lasers, they are not insurmountable. As an intermediate step toward a zero-index laser, we have demonstrated that a section of metamaterial waveguide can be used as a coupling antenna when fed by the signal from an adjacent, conventional terahertz QC-laser. The metamaterial section is active—i.e., gain is available to provide amplification to the signal injected from the master oscillator—and a directional beam (in one direction) is obtained in the forward direction.

Our next step is to demonstrate backward-wave operation from these antennas and optimize the designs to reduce losses so that a zero-index laser can be realized. However, in general, use of transmission-line metamaterial concepts is useful for designing laser waveguides and resonators with engineered phase characteristics. These techniques have the potential to allow design of lasers with flexible control of spectral and radiation properties, including beam shaping and steering, wavelength tuning, and polarization control.

---

Benjamin S. Williams, Amir Ali Tavallaee, Philip Hon, Tatsuo Itoh

University of California at Los Angeles

Los Angeles, CA

---

References:

1. T. A. Klar, A. V. Kildishev, V. P. Drachev, V. M. Shalaev, Negative-index metamaterials: going optical, IEEE J. Sel. Top. Quant. Electron. 12, pp. 1106-1115, 2006.

2. A. A. Tavallaee, P. Hon, K. Mehta, T. Itoh, B. S. Williams, Zero-index terahertz quantum-cascade metamaterial lasers, IEEE J. Quant. Electron. 46, pp. 1091-1098, 2010.

3. A. Lai, C. Caloz, T. Itoh, Composite right/left-handed transmission line metamaterials, IEEE Microw. Mag., pp. 34-50, 2004.

4. B. S. Williams, Terahertz quantum cascade lasers, Nat. Photon. 1, pp. 517-524, 2007.