Abstract

Ring resonators are an interesting alternative cavity solution to the commonly used ridge-type waveguide for terahertz (THz) quantum cascade lasers. They either support a standing-wave pattern showing spatial hole burning if there are defects implemented or a traveling mode in a defect-free cavity. Here, we report on ring-shaped THz quantum cascade lasers emitting between 3.2 and 4.1 THz operating in four different emission regimes. The presence of defects in the cavities force the THz quantum cascade laser into a standing-wave pattern. The measurements show a complex behavior highlighting the effect of strong confinement and the optical nonlinearities leading to the generation of a harmonic state, as well as to a fundamental comb, exhibiting over 30 equidistant modes and covering a bandwidth of 622 GHz. The results are explained by numerical calculations based on the Maxwell–Bloch formalism, including the linewidth enhancement factor and reflection points. The compact geometry and high output power (4 mW detected) make these devices extremely appealing for on-chip frequency comb applications in the terahertz region.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Corrections

25 May 2021: A typographical correction was made to the author listing.

1. INTRODUCTION

The characteristic dynamics of light propagation in ring-shaped resonators have made them indispensable in the optical science. Due to the formation of whispering gallery modes (WGM), where the mode is guided by total internal reflection along the outer perimeter of the ring, confinement is high and losses are low. Several applications such as filters [1], resonant enhancement cavities [2], and passive elements for frequency comb formation [35] take advantage of the low scattering losses. In lasers, this leads to a decreased threshold current density, resulting in a lower required pumping power and therefore a decreased heat load of the active medium.

Particularly intriguing of ring resonators is their high potential for frequency comb generation on an integrated platform. Optical frequency combs [6] consist of phase-coherent equidistant emission lines, originally produced by mode-locked femtosecond lasers. Since their first realization, they have revolutionized time and frequency metrology [7,8]. There are several ways to realize optical frequency combs, such as intracavity phase modulation [9] or upconversion and downconversion from lower- and higher-frequency sources [10,11].

Frequency combs based on ring resonators serve as a compact alternative with the possibility of on-chip integration and have mostly been explored extensively in the near-infrared spectral region using passive cavities. Passive micro-ring resonators based on lithium niobate [3], indium phosphide [12], or silicon platforms [13,14] were used to produce broadband comb sources by exploiting nonlinear effects of the resonator medium. Their performance is mainly limited by scattering loss and thus quite sensitive to the surface quality.

Moving toward longer wavelengths scattering loss becomes more and more irrelevant. In the mid-infrared spectral region, frequency comb generators have been developed, consisting of quantum cascade or interband cascade semiconductor lasers [15,16], which are electrically driven devices based on semiconductor heterostructures. The unique properties of the gain of quantum cascade active regions have led to the observation of harmonic-state operation, indicated by a detected harmonic beat note [1719] and free-running comb formation in Fabry–Perot devices. Moreover, these devices can be forced to form a comb by injecting an RF signal at the round-trip frequency of the cavity. Comb sources in the terahertz (THz) region based on quantum cascade lasers (QCLs) so far were realized in ridge-type waveguides, utilizing four-wave mixing in the gain medium [15]. By using this waveguide geometry, either a dispersion compensation geometry [20] or an active region with a carefully designed gain curve [21,22] is necessary to achieve the actual comb formation. Recently, comb formation in the mid-IR in a ring-shaped QCL was reported [23,24], showing the high potential of ring resonators for realizing optical frequency combs. In the THz region, the developed ring resonators so far were mainly aimed to achieve surface-emission or single-mode operation [25,26]. However, ring-shaped cavities release their best properties exactly in this long-wavelength region to realize optical frequency combs. The effect of spatial hole burning (SHB) [27] with its negative side effects such as linewidth broadening [28], mode hopping, and wavelength chirping [29,30], and preventing the exploitation of the whole gain bandwidth [3134], is the sole mechanism in ridge-type waveguides causing multimode emission. In ring cavities, SHB can be switched on and off by implementing defects or not. Ring resonators with implemented defects exhibit a standing-wave pattern, formed by two counterpropagating waves if there is a well-defined reflection point such as a contacting pad or an air gap employed which breaks the circular symmetry. Contrary, defect-free ring cavities support a traveling unidirectional WGM, which is not affected by SHB.

In this work, we combine the beneficial optical characteristics of ring resonators and the advantage of long wavelengths in the THz spectral region. The long wavelength will decrease the sensitivity to scattering losses even further since high surface qualities at the few-micrometer level can be achieved easily. Supported by theoretical models [23], which predict comb formation by four-wave mixing in ring resonators in the THz region, we show self-starting comb operation in an ultrathin ring-shaped THz QCL. Our results are promising for implementation in spectroscopic applications such as dual-comb spectroscopy to overcome the current limitations of the THz technology for spectroscopic applications.

2. THZ QUANTUM CASCADE RING LASERS

In order to shed some light on the laser cavity dynamics behind the spontaneous frequency comb formation, we employ a physical model that is based on the Maxwell–Bloch formalism [23,3537]. The model additionally utilizes the concept of the so-called linewidth enhancement factor ($\alpha$ factor) that describes the change of the refractive index induced by the modulations of the optical gain [38]. It stems from the asymmetrical shape of the gain [39] and is known to induce substantial optical nonlinearities in fast gain media, such as QCLs, due to carrier lifetimes in the range of picoseconds. The $\alpha$ factor is implemented in the full Maxwell–Bloch system of equations similarly as in [23]. Other parameters of the model are listed in Table S1 (see Supplement 1). Spatially and time-dependent numerical simulations within the frames of the described model demonstrated spontaneous fundamental and harmonic frequency comb formation, both in laser cavities with and without defects, as will be shown later. To ensure that the laser has reached its stationary state, we simulate the evolution of the laser field for about 100,000 round trips.

The waveguide type of choice is the commonly used double metal waveguide (DMWG). This sort of waveguide where the active region is sandwiched between two metal layers has proven to be a reliable candidate to provide high optical mode confinement and reasonable low losses with typically a few ${\rm{c}}{{\rm{m}}^{- 1}}$ [40,41]. Due to the ring shape of the resonator, only WGM should be supported which leads to a further enhancement of the optical mode confinement. However, employing a DMWG leads to an edge-emitting device, resulting in an emission profile with a large vertical divergence angle. On the one hand, this emission behavior makes it more difficult to collect all the emitted light for detection. On the other hand, it is necessary to achieve multimode emission because surface-emitting ring devices always need a grating on top which leads to pre-defined emission modes [26]. To design the geometry of the resonator properly and to obtain the desired mode profile an eigenmode simulation using a commercial finite element solver is performed. In this simulation, undoped GaAs is used as the active region material. Since a DMWG is chosen, the surfaces of the two substrates are assumed as perfect electric conductors and only radiation losses are considered in the model. As expected, all computations yield WGMs [Fig. 1(b)] but depending on the simulation parameters the resulting mode profiles vary. The width of the ring is chosen to be 15 µm, determined by a compromise between the suppression of higher-order modes and outcoupling efficiency. The variation of the diameter of the ring leads to a changing mode spacing between adjacent modes since the mode spacing is inversely proportional to the optical length of the resonator. To achieve multimode emission and a small mode spacing of adjacent modes, we have chosen an outer ring diameter of 1.5 mm. The simulation for a ring with the above-mentioned parameters yields a spacing of 15.5 GHz between adjacent modes. Due to the small size of the ring resonator, it is possible to fabricate several devices on one single chip. Because THz QCLs are electrically driven devices it is necessary to provide electrical contacts on the top and bottom metal layer of the waveguide. To facilitate wire bonds to the device, two contacting pads (${{60}}\;{\rm{\unicode{x00B5}{\rm m}}} \times {{60}}\;{\rm{\unicode{x00B5}{\rm m}}}$) located at 6 and 12 o’clock are attached to the narrow ring on the inner perimeter. Since the contacting pads act as reflective points, a standing-wave geometry is obtained.

 figure: Fig. 1.

Fig. 1. (a) Sketch of the THz quantum cascade ring laser. Due to the narrow width of the waveguide, two contacting pads (${{60}} \times {{60}}\;{\rm{\unicode{x00B5}{\rm m}}}$ each) are connected to the ring at 12 and 6 o’clock for electrical contacting. (b) Eigenmode simulation (electric field in the $z$ direction) of the ring-shaped resonator using a commercial finite element solver (Comsol 5.5). The simulation yields WGMs. (c) SEM image of a ring resonator section with one contacting pad.

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The active medium of THz QCLs consists of a semiconductor quantum cascade heterostructure which is grown by molecular beam epitaxy. The QCL active region used in this experiment is based on a single-stack, three-well, longitudinal optical phonon depopulation design in the GaAs/AlGaAs material system. The heterostructure consists of 345 periods and has a total thickness of 15 µm. It has the same layer sequence as a temperature optimized structure reported in [42] but lower doping concentrations to allow continuous-wave operation. The structure is designed to emit in a frequency range between ${\sim}{3.3}$ and 4 THz, exploiting two intersubband transitions which are centered at 3.4 and 3.8 THz, respectively [43].

The ring resonators are fabricated by a standard process for DMWG. In detail, first the active region is Au–Au wafer bonded onto an n$+$- doped GaAs substrate. Afterwards, the active region substrate and the etch stop layer are removed by ${{\rm{H}}_2}{{\rm{O}}_2} + {{\rm{NH}}_4}{\rm{OH}}$ and HF etching. The device geometries are defined by a photolithography step applying a Ti/Au (10 nm/500 nm) metal contact acting as a self-aligned etch mask for the following reactive ion etching process. This results in an Au–Au double metal waveguide providing high mode confinement and low optical losses. No outcoupling elements (e.g.,  grating) are implemented in the ring to enhance the power extraction from the device.

The chip with the THz quantum cascade ring lasers is indium soldered to a copper plate, which is mounted onto the cold finger of a He flow cryostat. The emitted laser light from a part of the ring edge is collected by a parabolic mirror and guided into a Fourier transform infrared spectrometer, where spectra are recorded with a resolution of 2.25 GHz. In this way, only a fraction of the outcoupled light is collected for analysis in the spectrometer. To facilitate the electrical beat note signal measurement of the QCL, we install semi-rigid RF cables parallel to the DC bias lines. Previous works have shown that THz QCLs are feasible to generate a strong RF beat note signal at the round-trip frequency of the resonator [20,22]. The RF cable is connected to a signal analyzer via a DC-block element. The device is biased using a Keithley 2602.

Additionally, to the ring cavities, ridge waveguides are processed from the same active region to compare the characteristics of lasers with different resonator geometries. The ridge lasers exhibit an emission bandwidth of 447 GHz and 336 GHz for the upper- and lower-frequency gain region, respectively. The threshold current density is at ${{250}}\;{\rm{A}}/{{\rm{cm}}^2}$ and the device stops lasing at ${{390}}\;{\rm{A}}/{{\rm{cm}}^2}$ (see Supplement 1). In continuous-wave operation, the maximum operating temperature is 50 K.

3. EMISSION OF THZ RING QCL

Compared to the Fabry–Perot resonators fabricated as reference devices, the ring resonators exhibit the same electrical behavior. All devices show lasing with a threshold current density of ${{259}}\;{\rm{A/c}}{{\rm{m}}^2}$ in continuous-wave operation at a heat-sink temperature of 20 K and stop lasing at ${{390}}\;{\rm{A/c}}{{\rm{m}}^2}$. Moreover, the maximum operating temperature of the ring resonators is 80 K in continuous-wave operation. These results show that scattering loss does not limit the performance of the ring devices, and due to the absence of field discontinuities at the facets they are ideally suited for stable mode evolution.

In sharp contrast to our Fabry–Perot reference devices, the ring device exhibits four different emission regimes which can be localized along the light–current–voltage (LIV) curve [Fig. 2(a)]. At driving currents just above the threshold current ($\,J$th) of 200 mA, the laser operates in single mode [Fig. 2(b)]. The first lasing mode appears in the upper-frequency region around 3.8 THz. By increasing the driving current, the single-mode regime transits at ${1.07} \times J$th to a harmonic-state regime [Fig. 2(c)] in a current range between 214–256 mA. The transition is manifested by the kink in the IV- and the LI-curve, respectively. In this state, the device exhibits multimode operation with mode spacings of multiple times the fundamental spacing ${{\rm{f}}_{\rm{rep}}}$ of 15.5 GHz. Moreover, in the harmonic state, both transitions of the active region contribute to the emission showing modes in the upper- and the lower-frequency regions. Further increase of the driving current leads to a dense comb state, resulting in a mode spectrum with over 30 equidistant modes covering a bandwidth of 622 GHz [Fig. 2(d)]. Note that the dense comb is only present in the upper-frequency region which is due to the characteristics of the active region design at higher driving currents and higher temperatures [43]. The termination of the emission from the lower-frequency transition becomes evident in the intensity drop in the LI curve. The dense comb state collapses again to the single-mode regime at a driving current of 290 mA before the device stops to lase at 296 mA in constant current operation. Contrary, when driving the THz quantum cascade ring laser in constant voltage operation the device does not stop lasing after returning to the single-mode regime. It enters the negative differential resistance (NDR) region beyond the comb state, where a chaotic multimode emission is observed [Fig. 2(e)]. The modes are still equally spaced; however, some modes are missing (Fig. 3, 252 mA). It must be noted that each device exhibits the described formation of four emission regimes, including a stable comb formation. This shows that our ring devices are very robust and reliable frequency comb sources.

 figure: Fig. 2.

Fig. 2. Different emission regimes of the THz quantum cascade ring laser at a heat-sink temperature of 20 K. $Y$ axis in spectra in log-scale. (a) DC light–current–voltage measurement in the dynamic range of the ring laser. The device exhibits four different emission regimes, which are highlighted on the current–voltage curve. The device starts lasing single mode (b) at a threshold current of 200 mA (1). At (2) the emission transits to a harmonic state, showing multimode operation (c) with a mode separation by a multiple of the round-trip frequency of the cavity. The transition to the harmonic state is also manifested by the kink in the current–voltage and the current–light curves, respectively. (d) Further increase of the driving current leads to a dense comb state with several equidistant modes (3). Here, the transition results in an intensity drop on the current–light curve. The dense comb state collapses at a driving current above 290 mA to single-mode operation (4) and stops lasing at a driving current of 296 mA in constant current driving operation (5). (e) The chaotic multimode regime, located in the NDR region of the current–voltage curve, can be only accessed in constant voltage driving operation.

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 figure: Fig. 3.

Fig. 3. Emission mode evolution of the THz quantum cascade ring laser ($\,y$ axis in log-scale). The lasing of the device starts with single-mode emission. With increasing driving current, the emission spectrum jumps to a harmonic-state regime, consisting of multiple modes, which are separated by a multiple of the fundamental mode spacing ${{\rm{f}}_{\rm{rep}}}$ plus an offset (${0.32} \times {{\rm{f}}_{\rm{rep}}}$). At 254 mA, a 10th-order harmonic with the two remaining initial modes at 3.3 and 3.8 THz (marked orange) is observed. The spacing of these two modes to the next right mode is ${{6}} \times$ FSR. These two modes disappear with higher current leaving a sole 10th harmonic spacing. Further increase of the driving current leads to the dense comb regime, where sidebands proliferate around the main modes of the harmonic state, forming four mode groups which are visible in the spectrum at, e.g.,  288 mA centered at 3.6 THz, 3.78 THz, 3.9 THz, and 4.1 THz, respectively. This dense comb state collapses at a current of 290 mA to single-mode emission. When the device is driven in constant voltage driving operation a chaotic multimode regime in the NDR region is observed beyond the comb state instead (red curve).

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4. MODE EVOLUTION AND ELECTRICAL BEAT NOTE

In the following we will discuss the evolution of the different operating regimes.

A. Single Mode

At the lasing threshold, the device is lasing in the single-mode operation. This mode is centered between the four mode groups (Fig. 3, compare 208 mA and 288 mA) which arise at higher driving currents, and it disappears when the laser enters the dense comb regime.

B. Harmonic State

The first evolving harmonic state is of the fourth order (Fig. 3, 216 mA), changing to the fifth order (Fig. 3, 246 mA), and finally transiting to the 10th order (Fig. 3, 254mA). At 254mA, a 10th-order harmonic with the two remaining initial modes at 3.3 and 3.8 THz (Fig. 3, 254 mA, marked orange) is observed. The harmonic spacing of these two modes to the next right mode is of the sixth order. These two modes disappear with higher current leaving a sole 10th-harmonic spacing. These laser bias dependent changes in the harmonic order are in agreement with the recent findings in [44]. However, the spacing of the modes in these states is not exactly 4, 5, 6, and 10 times the fundamental spacing ${{\rm{f}}_{\rm{rep}}}$ but is offset by ${0.32} \times {{\rm{f}}_{\rm{rep}}} = {{5}}\;{\rm{GHz}}$ (harmonic offset). By increasing the driving current, sidebands start to develop around the modes of the harmonic state forming four mode groups.

C. Comb State

Since these sidebands seem to be equidistant it gives reason to assume comb formation, which is verified by an RF beat note signal measurement [Fig. 4(a)]. Starting from a driving current of 256 mA, a stable and narrow beat note is detected, indicating an equidistant mode spacing. As the sidebands proliferate with the increase of the pumping current, additional beat note signals appear. The reason for this is that sidebands for every mode group evolve independently. At some point, the edge modes of two adjacent mode groups are separated by a frequency that differs from the integer multiple of the ${{\rm{f}}_{\rm{rep}}}$ and can be detected by the spectrum analyzer (upper limit 26.5 GHz). By a further increase of the driving current, the created sidebands of the mode groups overlap [Fig. 4(b)], producing a beat note signal at the sum and the difference of the fundamental mode spacing and the harmonic offset [Fig. 4(a)]. This harmonic offset varies at different driving currents which is manifested in the pulling of the two left mode groups and the rightest mode group toward the mode group which is centered around 3.9 THz.

 figure: Fig. 4.

Fig. 4. (a) RF signal of the THz quantum cascade ring laser at three different driving currents. The beat note signal at 15.563 GHz shows the intermodal spacing within the four mode groups. The inset shows a zoom-in of the free-running intermodal beat note at 15.563 GHz at a driving current of 270 mA indicating comb operation. The beat note signal at around 5 GHz arises due to the offset between the mode groups. Consequently, the sum- and difference-frequency signals appear around 10 and 20 GHz since the sidebands of the mode groups start to overlap with the rising driving current, shown in (b). The highlighted modes of two adjacent mode groups show the overlap of the rising modes with the increased driving current. The green- and purple-colored modes belong each to one mode group, respectively. Moreover, the mode groups are pulled toward each other with increasing pumping power, indicated by the decreasing offset beat note frequency, whereas the intermodal spacing stays constant.

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5. DISCUSSION

The investigated ring lasers show expected as well as truly unexpected behavior. As mentioned above, ring cavities should exhibit a decreased threshold current density due to the stronger confinement of the mode and reduced losses. However, the ring devices show slightly increased threshold current densities compared to the Fabry–Perot cavities. The reason for this is the profile of the supported mode in the ring cavity [Fig. 1(b)]. This mode is located on the edges of the waveguide resulting in a large evanescent field in free space. Consequently, the radiative losses are higher leading to a slightly increased threshold current density. Higher radiative losses are manifested by the value of the output signal as well. Even though only a fraction of the outcoupled light of the ring cavity is collected by the detector, the signal is more than doubled compared to the ridge cavity. Another difference between the two investigated waveguide geometries appears in the measured emission bandwidth. The emission bandwidth in the comb regime of the ring device spans 622 GHz, which is by a factor of 1.4 larger than the compared value for the ridge device. It seems that the locking mechanism of the comb formation leads to a progressive excitation of sidebands resulting in a full exploitation of the gain even in the weak gain regions located at the edges of the gain bandwidth.

Multimode operation in a ring cavity can be achieved via two distinct mechanisms. First, is the well-known spatial hole burning due to opposite propagating laser field components forming a standing-wave pattern, thus lowering the gain threshold for the side modes, and enabling multimode operation. However, it was found recently that a low-bias multimode instability can exist even in an ideal ring laser cavity, where the spatial hole burning effect is eliminated [23]. The instability is caused by the interplay of dispersive and nonlinear effects, which can lead to a phase turbulence regime and proliferation of the side modes after the initial single-mode emission. Both the dispersion and nonlinearity originate from the specific asymmetric gain shape, which is defined by the resonant transitions within the QCL structure. The presented model encompasses this with the linewidth enhancement factor. Employing the previously described numerical model, we analyze an ideal ring cavity that contains no defects for different values of the $\alpha$ factor. In the case $\alpha = {1.5}$, in Fig. 5(a) a frequency comb is formed with fundamental intermodal spacing. Increasing the value of $\alpha$ to 2.5 [Fig. 5(b)] leads to spontaneous second-harmonic comb generation. The spectra of both states form a single group of modes, which can be fitted very well with a ${\sec}{{\rm{h}}^2}$-type envelope in the case of the harmonic state. A possible intuitive picture behind the formation of a harmonic state over the fundamental one can be found from the time-domain analysis [35]. Optical nonlinearities are known to coherently couple the amplitude and phase of the laser light and thus determine the relation between the modal phases. In cases where the spectral bandwidth of a fundamental comb would be significantly different compared to the available gain bandwidth, a harmonic comb represents an energetically more favorable solution. The time trace of the laser intensity over two round trips can be seen in the bottom plots of Fig. 5. Both states exhibit high-contrast amplitude-modulated signal superimposed on a continuous background, which contradicts the common belief that amplitude variations of the laser intensity need to be washed out due to the fast gain dynamics. In Fig. 5(b) a spatial pattern known as Turing rolls is formed for $\alpha$ values that are in agreement with recent theoretical work [45]. Formation of coherent spatial patterns along with a ${\sec}{{\rm{h}}^2}$-type spectrum envelope indicates the potential of temporal soliton formation in these devices. Recently, it was demonstrated that the harmonic state can be stable and self-supported based on the frequency-domain analysis of the parametric gain [46].

 figure: Fig. 5.

Fig. 5. Numerical simulations of an ideal ring cavity QCL containing no defects using the linewidth enhancement factor of (a) $\alpha = {1.5}$ and (b) $\alpha = {2.5}$. The top plots depict the power spectra, and the bottom plots show the time trace of the laser intensity over two round trips. Numerical simulations have been run for 100,000 round trips to reach a stationary state forming (a) a fundamental frequency comb and (b) a second-order harmonic comb with a ${\sec}{{\rm{h}}^2}$ spectrum. Both time traces of the laser intensity show high-contrast amplitude modulations on a continuous background.

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 figure: Fig. 6.

Fig. 6. (a) Simulated spectra of a ring cavity QCL containing two opposing defects. For $\alpha = 0.4$ the laser has not reached the multimode instability and is operating in single mode. With the increase of the $\alpha$ factor, the laser transits from a fourth harmonic state to a dense state. Red crosses in the bottom plot indicate modes with 5% larger intermodal spacing than the remaining modes. This leads to an offset of 1.5 GHz between the two adjacent mode groups. The experimentally measured offset of 5 GHz indicates that the model is underestimating the nonlinearities in our devices. (b) Corresponding time traces. The single-mode emission exhibits a constant output intensity over the round trips, whereas the multimode emission shows a modulated output signal. The periodicity of the time traces at linewidth enhancement factor values of $\alpha =0.45 {-} 0.525$ originates from the equidistant spacing of the modes. However, in the bottom plot, the modes of the adjacent mode groups show a larger mode spacing, resulting in nonperiodic features in the time trace.

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The contacting pads depicted in Fig. 1(a) represent reflection points for the laser field which breaks the rotational symmetry of the cavity. In the numerical simulations, they are modeled as 60 µm wide defects with a reflectivity strength of 5%. The results of these simulations are shown in Fig. 6. The newly introduced defects give rise to spatial hole burning, which lowers the threshold for the side mode appearance. Consequently, the multimode instability is reached for much smaller $\alpha$ values compared to the defect-free ring. One can see that for $\alpha = {0.45}$ the laser is in a fourth-harmonic comb state, and for smaller values of $\alpha$, the laser operates in single mode. The order of the harmonic is susceptible to many parameters among which are the driving current, defects, and of course, the linewidth enhancement factor (see Supplement 1). By increasing the value of the $\alpha$ factor to 0.475, the first weak side modes appear neighboring the strong modes of the harmonic state. With a further increase of $\alpha$, the laser gradually transitions to a dense spectrum state. This behavior qualitatively matches the spectral evolution in Fig. 3 since the value of the $\alpha$ factor is known to increase linearly with increasing driving current. Additionally, the spectrum consists of several mode groups that are formed around the original modes of the harmonic state. This is in sharp contrast to the single group of modes in Fig. 5, indicating the importance of the cavity defects on the laser field evolution.

Furthermore, at the bottom plot of Fig. 6, red crosses mark the modes whose intermodal spacing differs from the remaining modes and is around 5% larger. This difference would produce an additional beating, thus explaining the RF signal consisting of multiple beat notes in Fig. 4.

The measured formations of the four emission regimes in the THz ring QCL are accurately described by our implemented numerical model based on the Maxwell–Bloch formalism, which contributes to the ongoing understanding of the dynamics in ring devices including the interaction of modes.

6. CONCLUSION

In conclusion, we demonstrate the realization of ultrathin ring-shaped THz QCLs, which exhibit four different emission regimes (single-mode, harmonic state, dense comb, and chaotic multimode) in continuous-wave operation. Comb formation is measured in each device, showing the robustness of our ring lasers. The comb regime shows a spectrum with 30 equidistant modes with a spacing around 15.56 GHz and a correspondingly strong and narrow beat note signal. Moreover, a numerical model is used to simulate the mode evolution in the ring cavities. The calculated results are in very good agreement with the measured data and describe the spectral behavior of the devices accurately. For future experiments, the measurement of the THz emission time-dependent by electro-optic sampling would be very interesting to study the temporal behavior of the signal and to confirm the amplitude modulation. To further explore the frequency comb, higher output powers are necessary. This can be achieved by employing collecting features all around the ring resonator to collect all emitted light or by coupling to a ridge waveguide which is added close to the ring cavity. Coupling to a ridge waveguide enables the investigation of coupling phenomena such as the physical behavior in the vicinity of an exceptional point [47].

Funding

Österreichische Forschungsförderungsgesellschaft (FFG 849614); European Research Council (853014); Austrian Science Fund (DiPQCL P30709-N27, DK Solids4Fun W1243); Air Force Office of Scientific Research (FA9550-17-1-0340).

Acknowledgment

The authors acknowledge financial support from the Austrian Science Fund FWF. N. O. and B. S. acknowledge the funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme. A. M. A acknowledges funding through COMTERA, a program under the auspices of ERA.NET RUS PLUS, and AFOSR.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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13. I. Demirtzioglou, C. Lacava, K. R. H. Bottrill, D. J. Thomson, G. T. Reed, D. J. Richardson, and P. Petropoulos, “Frequency comb generation in a silicon ring resonator modulator,” Opt. Express 26, 790–796 (2018). [CrossRef]  

14. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007). [CrossRef]  

15. A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012). [CrossRef]  

16. B. Schwarz, J. Hillbrand, M. Beiser, A. M. Andrews, G. Strasser, H. Detz, A. Schade, R. Weih, and S. Höfling, “Monolithic frequency comb platform based on interband cascade lasers and detectors,” Optica 6, 890–895 (2019). [CrossRef]  

17. T. S. Mansuripur, C. Vernet, P. Chevalier, G. Aoust, B. Schwarz, F. Xie, C. Caneau, K. Lascola, C. Zah, D. P. Caffey, T. Day, L. J. Missaggia, M. K. Connors, C. A. Wang, A. Belyanin, and F. Capasso, “Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator,” Phys. Rev. A 94, 063807 (2016). [CrossRef]  

18. M. Piccardo, D. Kazakov, N. A. Rubin, P. Chevalier, Y. Wang, F. Xie, K. Lascola, A. Belyanin, and F. Capasso, “Time-dependent population inversion gratings in laser frequency combs,” Optica 5, 475–478 (2018). [CrossRef]  

19. F. Wang, V. Pistore, M. Riesch, H. Nong, P.-B. Vigneron, R. Colombelli, O. Parillaud, J. Mangeney, J. Tignon, C. Jirauschek, and S. S. Dhillon, “Ultrafast response of harmonic modelocked THz lasers,” Light Sci. Appl. 9, 51 (2020). [CrossRef]  

20. D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014). [CrossRef]  

21. A. Forrer, M. Franckié, D. Stark, T. Olariu, M. Beck, J. Faist, and G. Scalari, “Photon-driven broadband emission and frequency comb RF injection locking in THz quantum cascade lasers,” ACS Photonics 7, 784–791 (2020). [CrossRef]  

22. M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nat. Photonics 9, 42–47 (2015). [CrossRef]  

23. M. Piccardo, B. Schwarz, D. Kazakov, M. Beiser, N. Opačak, Y. Wang, S. Jha, J. Hillbrand, M. Tamagnone, W. T. Chen, A. Y. Zhu, L. L. Columbo, A. Belyanin, and F. Capasso, “Frequency combs induced by phase turbulence,” Nature 582, 360–364 (2020). [CrossRef]  

24. B. Meng, M. Singleton, M. Shahmohammadi, F. Kapsalidis, R. Wang, M. Beck, and J. Faist, “Mid-infrared frequency comb from a ring quantum cascade laser,” Optica 7, 162–167 (2020). [CrossRef]  

25. L. Mahler, M. I. Amanti, C. Walther, A. Tredicucci, F. Beltram, J. Faist, H. E. Beere, and D. A. Ritchie, “Distributed feedback ring resonators for vertically emitting terahertz quantum cascade lasers,” Opt. Express 17, 13031–13039 (2009). [CrossRef]  

26. E. Mujagić, C. Deutsch, H. Detz, P. Klang, M. Nobile, A. M. Andrews, W. Schrenk, K. Unterrainer, and G. Strasser, “Vertically emitting terahertz quantum cascade ring lasers,” Appl. Phys. Lett. 95, 011120 (2009). [CrossRef]  

27. A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008). [CrossRef]  

28. M. Aoki, K. Uomi, T. Tsuchiya, S. Sasaki, M. Okai, and N. Chinone, “Quantum size effect on longitudinal spatial hole burning in MQW lambda/4-shifted DFB lasers,” IEEE J. Quantum Electron. 27, 1782–1789 (1991). [CrossRef]  

29. M. G. Davis and R. F. O’Dowd, “A transfer matrix method based large-signal dynamic model for multielectrode DFB lasers,” IEEE J. Quantum Electron. 30, 2458–2466 (1994). [CrossRef]  

30. T. Yamanaka, S. Seki, and K. Yokoyama, “Numerical analysis of static wavelength shift for DFB lasers with longitudinal mode spatial hole burning,” IEEE Photonics Technol. Lett. 3, 610–612 (1991). [CrossRef]  

31. J. Kröll, J. Darmo, K. Unterrainer, S. S. Dhillon, C. Sirtori, X. Marcadet, and M. Calligaro, “Longitudinal spatial hole burning in terahertz quantum cascade lasers,” Appl. Phys. Lett. 91, 161108 (2007). [CrossRef]  

32. K. Kikuchi, M. Kakui, C.-E. Zah, and T.-P. Lee, “Observation of highly nondegenerate four-wave mixing in 1.5 µm traveling-wave semiconductor optical amplifiers and estimation of nonlinear gain coefficient,” IEEE J. Quantum Electron. 28, 151–156 (1992). [CrossRef]  

33. W. Fang, A. Hsu, S. L. Chuang, T. Tanbun-Ek, and A. M. Sergent, “Measurement and modeling of distributed-feedback lasers with spatial hole burning,” IEEE J. Sel. Top. Quantum Electron. 3, 547–554 (1997). [CrossRef]  

34. J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” IEEE J. Quantum Electron. 34, 879–886 (1998). [CrossRef]  

35. N. Opačak and B. Schwarz, “Theory of frequency-modulated combs in lasers with spatial hole burning, dispersion, and Kerr nonlinearity,” Phys. Rev. Lett. 123, 243902 (2019). [CrossRef]  

36. J. R. Freeman, J. Maysonnave, S. Khanna, E. H. Linfield, A. G. Davies, S. S. Dhillon, and J. Tignon, “Laser-seeding dynamics with few-cycle pulses: Maxwell-Bloch finite-difference time-domain simulations of terahertz quantum cascade lasers,” Phys. Rev. A 87, 063817 (2013). [CrossRef]  

37. C. Jirauschek, M. Riesch, and P. Tzenov, “Optoelectronic device simulations based on macroscopic Maxwell–Bloch equations,” Adv. Theory Simul. 2, 1900018 (2019). [CrossRef]  

38. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–an overview,” IEEE J. Quantum Electron. 23, 9–29 (1987). [CrossRef]  

39. N. Opačak, S. D. Cin, J. Hillbrand, and B. Schwarz, “Frequency comb generation by Bloch gain induced giant Kerr nonlinearity,” arXiv:2102.08167 (2021).

40. M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16, 3242–3248 (2008). [CrossRef]  

41. M. Martl, J. Darmo, C. Deutsch, M. Brandstetter, A. M. Andrews, P. Klang, G. Strasser, and K. Unterrainer, “Gain and losses in THz quantum cascade laser with metal-metal waveguide,” Opt. Express 19, 733–738 (2011). [CrossRef]  

42. M. A. Kainz, S. Schönhuber, A. M. Andrews, H. Detz, B. Limbacher, G. Strasser, and K. Unterrainer, “Barrier height tuning of terahertz quantum cascade lasers for high-temperature operation,” ACS Photonics 5, 4687–4693 (2018). [CrossRef]  

43. M. A. Kainz, S. Schönhuber, B. Limbacher, A. M. Andrews, H. Detz, G. Strasser, G. Bastard, and K. Unterrainer, “Color switching of a terahertz quantum cascade laser,” Appl. Phys. Lett. 114, 191104 (2019). [CrossRef]  

44. A. Forrer, Y. Wang, M. Beck, A. Belyanin, J. Faist, and G. Scalari, “Self-starting harmonic comb emission in THz quantum cascade lasers,” Appl. Phys. Lett. 118, 131112 (2021). [CrossRef]  

45. L. Columbo, M. Piccardo, F. Prati, L. A. Lugiato, M. Brambilla, A. Gatti, C. Silvestri, M. Gioannini, N. Opacak, B. Schwarz, and F. Capasso, “Unifying frequency combs in active and passive cavities: temporal solitons in externally-driven ring lasers,” Phys. Rev. Lett. 126, 173903 (2021). [CrossRef]  

46. Y. Wang and A. Belyanin, “Harmonic frequency combs in quantum cascade lasers: time-domain and frequency-domain theory,” Phys. Rev. A 102, 013519 (2020). [CrossRef]  

47. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014). [CrossRef]  

References

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    [Crossref]
  24. B. Meng, M. Singleton, M. Shahmohammadi, F. Kapsalidis, R. Wang, M. Beck, and J. Faist, “Mid-infrared frequency comb from a ring quantum cascade laser,” Optica 7, 162–167 (2020).
    [Crossref]
  25. L. Mahler, M. I. Amanti, C. Walther, A. Tredicucci, F. Beltram, J. Faist, H. E. Beere, and D. A. Ritchie, “Distributed feedback ring resonators for vertically emitting terahertz quantum cascade lasers,” Opt. Express 17, 13031–13039 (2009).
    [Crossref]
  26. E. Mujagić, C. Deutsch, H. Detz, P. Klang, M. Nobile, A. M. Andrews, W. Schrenk, K. Unterrainer, and G. Strasser, “Vertically emitting terahertz quantum cascade ring lasers,” Appl. Phys. Lett. 95, 011120 (2009).
    [Crossref]
  27. A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
    [Crossref]
  28. M. Aoki, K. Uomi, T. Tsuchiya, S. Sasaki, M. Okai, and N. Chinone, “Quantum size effect on longitudinal spatial hole burning in MQW lambda/4-shifted DFB lasers,” IEEE J. Quantum Electron. 27, 1782–1789 (1991).
    [Crossref]
  29. M. G. Davis and R. F. O’Dowd, “A transfer matrix method based large-signal dynamic model for multielectrode DFB lasers,” IEEE J. Quantum Electron. 30, 2458–2466 (1994).
    [Crossref]
  30. T. Yamanaka, S. Seki, and K. Yokoyama, “Numerical analysis of static wavelength shift for DFB lasers with longitudinal mode spatial hole burning,” IEEE Photonics Technol. Lett. 3, 610–612 (1991).
    [Crossref]
  31. J. Kröll, J. Darmo, K. Unterrainer, S. S. Dhillon, C. Sirtori, X. Marcadet, and M. Calligaro, “Longitudinal spatial hole burning in terahertz quantum cascade lasers,” Appl. Phys. Lett. 91, 161108 (2007).
    [Crossref]
  32. K. Kikuchi, M. Kakui, C.-E. Zah, and T.-P. Lee, “Observation of highly nondegenerate four-wave mixing in 1.5 µm traveling-wave semiconductor optical amplifiers and estimation of nonlinear gain coefficient,” IEEE J. Quantum Electron. 28, 151–156 (1992).
    [Crossref]
  33. W. Fang, A. Hsu, S. L. Chuang, T. Tanbun-Ek, and A. M. Sergent, “Measurement and modeling of distributed-feedback lasers with spatial hole burning,” IEEE J. Sel. Top. Quantum Electron. 3, 547–554 (1997).
    [Crossref]
  34. J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” IEEE J. Quantum Electron. 34, 879–886 (1998).
    [Crossref]
  35. N. Opačak and B. Schwarz, “Theory of frequency-modulated combs in lasers with spatial hole burning, dispersion, and Kerr nonlinearity,” Phys. Rev. Lett. 123, 243902 (2019).
    [Crossref]
  36. J. R. Freeman, J. Maysonnave, S. Khanna, E. H. Linfield, A. G. Davies, S. S. Dhillon, and J. Tignon, “Laser-seeding dynamics with few-cycle pulses: Maxwell-Bloch finite-difference time-domain simulations of terahertz quantum cascade lasers,” Phys. Rev. A 87, 063817 (2013).
    [Crossref]
  37. C. Jirauschek, M. Riesch, and P. Tzenov, “Optoelectronic device simulations based on macroscopic Maxwell–Bloch equations,” Adv. Theory Simul. 2, 1900018 (2019).
    [Crossref]
  38. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–an overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
    [Crossref]
  39. N. Opačak, S. D. Cin, J. Hillbrand, and B. Schwarz, “Frequency comb generation by Bloch gain induced giant Kerr nonlinearity,” arXiv:2102.08167 (2021).
  40. M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16, 3242–3248 (2008).
    [Crossref]
  41. M. Martl, J. Darmo, C. Deutsch, M. Brandstetter, A. M. Andrews, P. Klang, G. Strasser, and K. Unterrainer, “Gain and losses in THz quantum cascade laser with metal-metal waveguide,” Opt. Express 19, 733–738 (2011).
    [Crossref]
  42. M. A. Kainz, S. Schönhuber, A. M. Andrews, H. Detz, B. Limbacher, G. Strasser, and K. Unterrainer, “Barrier height tuning of terahertz quantum cascade lasers for high-temperature operation,” ACS Photonics 5, 4687–4693 (2018).
    [Crossref]
  43. M. A. Kainz, S. Schönhuber, B. Limbacher, A. M. Andrews, H. Detz, G. Strasser, G. Bastard, and K. Unterrainer, “Color switching of a terahertz quantum cascade laser,” Appl. Phys. Lett. 114, 191104 (2019).
    [Crossref]
  44. A. Forrer, Y. Wang, M. Beck, A. Belyanin, J. Faist, and G. Scalari, “Self-starting harmonic comb emission in THz quantum cascade lasers,” Appl. Phys. Lett. 118, 131112 (2021).
    [Crossref]
  45. L. Columbo, M. Piccardo, F. Prati, L. A. Lugiato, M. Brambilla, A. Gatti, C. Silvestri, M. Gioannini, N. Opacak, B. Schwarz, and F. Capasso, “Unifying frequency combs in active and passive cavities: temporal solitons in externally-driven ring lasers,” Phys. Rev. Lett. 126, 173903 (2021).
    [Crossref]
  46. Y. Wang and A. Belyanin, “Harmonic frequency combs in quantum cascade lasers: time-domain and frequency-domain theory,” Phys. Rev. A 102, 013519 (2020).
    [Crossref]
  47. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
    [Crossref]

2021 (2)

A. Forrer, Y. Wang, M. Beck, A. Belyanin, J. Faist, and G. Scalari, “Self-starting harmonic comb emission in THz quantum cascade lasers,” Appl. Phys. Lett. 118, 131112 (2021).
[Crossref]

L. Columbo, M. Piccardo, F. Prati, L. A. Lugiato, M. Brambilla, A. Gatti, C. Silvestri, M. Gioannini, N. Opacak, B. Schwarz, and F. Capasso, “Unifying frequency combs in active and passive cavities: temporal solitons in externally-driven ring lasers,” Phys. Rev. Lett. 126, 173903 (2021).
[Crossref]

2020 (5)

Y. Wang and A. Belyanin, “Harmonic frequency combs in quantum cascade lasers: time-domain and frequency-domain theory,” Phys. Rev. A 102, 013519 (2020).
[Crossref]

F. Wang, V. Pistore, M. Riesch, H. Nong, P.-B. Vigneron, R. Colombelli, O. Parillaud, J. Mangeney, J. Tignon, C. Jirauschek, and S. S. Dhillon, “Ultrafast response of harmonic modelocked THz lasers,” Light Sci. Appl. 9, 51 (2020).
[Crossref]

A. Forrer, M. Franckié, D. Stark, T. Olariu, M. Beck, J. Faist, and G. Scalari, “Photon-driven broadband emission and frequency comb RF injection locking in THz quantum cascade lasers,” ACS Photonics 7, 784–791 (2020).
[Crossref]

M. Piccardo, B. Schwarz, D. Kazakov, M. Beiser, N. Opačak, Y. Wang, S. Jha, J. Hillbrand, M. Tamagnone, W. T. Chen, A. Y. Zhu, L. L. Columbo, A. Belyanin, and F. Capasso, “Frequency combs induced by phase turbulence,” Nature 582, 360–364 (2020).
[Crossref]

B. Meng, M. Singleton, M. Shahmohammadi, F. Kapsalidis, R. Wang, M. Beck, and J. Faist, “Mid-infrared frequency comb from a ring quantum cascade laser,” Optica 7, 162–167 (2020).
[Crossref]

2019 (5)

N. Opačak and B. Schwarz, “Theory of frequency-modulated combs in lasers with spatial hole burning, dispersion, and Kerr nonlinearity,” Phys. Rev. Lett. 123, 243902 (2019).
[Crossref]

M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. M. Kahn, and M. Lončar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
[Crossref]

B. Schwarz, J. Hillbrand, M. Beiser, A. M. Andrews, G. Strasser, H. Detz, A. Schade, R. Weih, and S. Höfling, “Monolithic frequency comb platform based on interband cascade lasers and detectors,” Optica 6, 890–895 (2019).
[Crossref]

C. Jirauschek, M. Riesch, and P. Tzenov, “Optoelectronic device simulations based on macroscopic Maxwell–Bloch equations,” Adv. Theory Simul. 2, 1900018 (2019).
[Crossref]

M. A. Kainz, S. Schönhuber, B. Limbacher, A. M. Andrews, H. Detz, G. Strasser, G. Bastard, and K. Unterrainer, “Color switching of a terahertz quantum cascade laser,” Appl. Phys. Lett. 114, 191104 (2019).
[Crossref]

2018 (3)

2016 (1)

T. S. Mansuripur, C. Vernet, P. Chevalier, G. Aoust, B. Schwarz, F. Xie, C. Caneau, K. Lascola, C. Zah, D. P. Caffey, T. Day, L. J. Missaggia, M. K. Connors, C. A. Wang, A. Belyanin, and F. Capasso, “Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator,” Phys. Rev. A 94, 063807 (2016).
[Crossref]

2015 (1)

M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nat. Photonics 9, 42–47 (2015).
[Crossref]

2014 (2)

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
[Crossref]

2013 (1)

J. R. Freeman, J. Maysonnave, S. Khanna, E. H. Linfield, A. G. Davies, S. S. Dhillon, and J. Tignon, “Laser-seeding dynamics with few-cycle pulses: Maxwell-Bloch finite-difference time-domain simulations of terahertz quantum cascade lasers,” Phys. Rev. A 87, 063817 (2013).
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2012 (2)

N. Dupuis, C. R. Doerr, L. Zhang, L. Chen, N. J. Sauer, P. Dong, L. L. Buhl, and D. Ahn, “InP-based comb generator for optical OFDM,” J. Lightwave Technol. 30, 466–472 (2012).
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A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
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2011 (2)

J. C. Pearson, B. J. Drouin, A. Maestrini, I. Mehdi, J. Ward, R. H. Lin, S. Yu, J. J. Gill, B. Thomas, C. Lee, G. Chattopadhyay, E. Schlecht, F. W. Maiwald, P. F. Goldsmith, and P. Siegel, “Demonstration of a room temperature 2.48–2.75 THz coherent spectroscopy source,” Rev. Sci. Instrum. 82, 093105 (2011).
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M. Martl, J. Darmo, C. Deutsch, M. Brandstetter, A. M. Andrews, P. Klang, G. Strasser, and K. Unterrainer, “Gain and losses in THz quantum cascade laser with metal-metal waveguide,” Opt. Express 19, 733–738 (2011).
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2010 (3)

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010).
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L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
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P. Dong, N.-N. Feng, D. Feng, W. Qian, H. Liang, D. C. Lee, B. J. Luff, T. Banwell, A. Agarwal, P. Toliver, R. Menendez, T. K. Woodward, and M. Asghari, “GHz-bandwidth optical filters based on high-order silicon ring resonators,” Opt. Express 18, 23784–23789 (2010).
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2009 (3)

J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3, 99–102 (2009).
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L. Mahler, M. I. Amanti, C. Walther, A. Tredicucci, F. Beltram, J. Faist, H. E. Beere, and D. A. Ritchie, “Distributed feedback ring resonators for vertically emitting terahertz quantum cascade lasers,” Opt. Express 17, 13031–13039 (2009).
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E. Mujagić, C. Deutsch, H. Detz, P. Klang, M. Nobile, A. M. Andrews, W. Schrenk, K. Unterrainer, and G. Strasser, “Vertically emitting terahertz quantum cascade ring lasers,” Appl. Phys. Lett. 95, 011120 (2009).
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2008 (2)

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
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M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16, 3242–3248 (2008).
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2007 (3)

J. Kröll, J. Darmo, K. Unterrainer, S. S. Dhillon, C. Sirtori, X. Marcadet, and M. Calligaro, “Longitudinal spatial hole burning in terahertz quantum cascade lasers,” Appl. Phys. Lett. 91, 161108 (2007).
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K.-L. Yeh, M. C. Hoffmann, J. Hebling, and K. A. Nelson, “Generation of 10 μJ ultrashort terahertz pulses by optical rectification,” Appl. Phys. Lett. 90, 171121 (2007).
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P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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2002 (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
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1999 (1)

T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, “Absolute optical frequency measurement of the cesium D-1 line with a mode-locked laser,” Phys. Rev. Lett. 82, 3568–3571 (1999).
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1998 (1)

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” IEEE J. Quantum Electron. 34, 879–886 (1998).
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1997 (1)

W. Fang, A. Hsu, S. L. Chuang, T. Tanbun-Ek, and A. M. Sergent, “Measurement and modeling of distributed-feedback lasers with spatial hole burning,” IEEE J. Sel. Top. Quantum Electron. 3, 547–554 (1997).
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1994 (1)

M. G. Davis and R. F. O’Dowd, “A transfer matrix method based large-signal dynamic model for multielectrode DFB lasers,” IEEE J. Quantum Electron. 30, 2458–2466 (1994).
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1993 (2)

M. Kourogi, K. Nakagawa, and M. Ohtsu, “Wide-span optical frequency comb generator for accurate optical frequency difference measurement,” IEEE J. Quantum Electron. 29, 2693–2701 (1993).
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K. Fiedler, S. Schiller, R. Paschotta, P. Kürz, and J. Mlynek, “Highly efficient frequency doubling with a doubly resonant monolithic total-internal-reflection ring resonator,” Opt. Lett. 18, 1786–1788 (1993).
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1992 (1)

K. Kikuchi, M. Kakui, C.-E. Zah, and T.-P. Lee, “Observation of highly nondegenerate four-wave mixing in 1.5 µm traveling-wave semiconductor optical amplifiers and estimation of nonlinear gain coefficient,” IEEE J. Quantum Electron. 28, 151–156 (1992).
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1991 (2)

T. Yamanaka, S. Seki, and K. Yokoyama, “Numerical analysis of static wavelength shift for DFB lasers with longitudinal mode spatial hole burning,” IEEE Photonics Technol. Lett. 3, 610–612 (1991).
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M. Aoki, K. Uomi, T. Tsuchiya, S. Sasaki, M. Okai, and N. Chinone, “Quantum size effect on longitudinal spatial hole burning in MQW lambda/4-shifted DFB lasers,” IEEE J. Quantum Electron. 27, 1782–1789 (1991).
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1987 (1)

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–an overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
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M. A. Kainz, S. Schönhuber, B. Limbacher, A. M. Andrews, H. Detz, G. Strasser, G. Bastard, and K. Unterrainer, “Color switching of a terahertz quantum cascade laser,” Appl. Phys. Lett. 114, 191104 (2019).
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M. A. Kainz, S. Schönhuber, A. M. Andrews, H. Detz, B. Limbacher, G. Strasser, and K. Unterrainer, “Barrier height tuning of terahertz quantum cascade lasers for high-temperature operation,” ACS Photonics 5, 4687–4693 (2018).
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Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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Banwell, T.

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A. Forrer, Y. Wang, M. Beck, A. Belyanin, J. Faist, and G. Scalari, “Self-starting harmonic comb emission in THz quantum cascade lasers,” Appl. Phys. Lett. 118, 131112 (2021).
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A. Forrer, Y. Wang, M. Beck, A. Belyanin, J. Faist, and G. Scalari, “Self-starting harmonic comb emission in THz quantum cascade lasers,” Appl. Phys. Lett. 118, 131112 (2021).
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Y. Wang and A. Belyanin, “Harmonic frequency combs in quantum cascade lasers: time-domain and frequency-domain theory,” Phys. Rev. A 102, 013519 (2020).
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M. Piccardo, D. Kazakov, N. A. Rubin, P. Chevalier, Y. Wang, F. Xie, K. Lascola, A. Belyanin, and F. Capasso, “Time-dependent population inversion gratings in laser frequency combs,” Optica 5, 475–478 (2018).
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T. S. Mansuripur, C. Vernet, P. Chevalier, G. Aoust, B. Schwarz, F. Xie, C. Caneau, K. Lascola, C. Zah, D. P. Caffey, T. Day, L. J. Missaggia, M. K. Connors, C. A. Wang, A. Belyanin, and F. Capasso, “Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator,” Phys. Rev. A 94, 063807 (2016).
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L. Columbo, M. Piccardo, F. Prati, L. A. Lugiato, M. Brambilla, A. Gatti, C. Silvestri, M. Gioannini, N. Opacak, B. Schwarz, and F. Capasso, “Unifying frequency combs in active and passive cavities: temporal solitons in externally-driven ring lasers,” Phys. Rev. Lett. 126, 173903 (2021).
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M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5, 4034 (2014).
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M. Martl, J. Darmo, C. Deutsch, M. Brandstetter, A. M. Andrews, P. Klang, G. Strasser, and K. Unterrainer, “Gain and losses in THz quantum cascade laser with metal-metal waveguide,” Opt. Express 19, 733–738 (2011).
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Burghoff, D.

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
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M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. M. Kahn, and M. Lončar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
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M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers–an overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
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T. S. Mansuripur, C. Vernet, P. Chevalier, G. Aoust, B. Schwarz, F. Xie, C. Caneau, K. Lascola, C. Zah, D. P. Caffey, T. Day, L. J. Missaggia, M. K. Connors, C. A. Wang, A. Belyanin, and F. Capasso, “Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator,” Phys. Rev. A 94, 063807 (2016).
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J. Kröll, J. Darmo, K. Unterrainer, S. S. Dhillon, C. Sirtori, X. Marcadet, and M. Calligaro, “Longitudinal spatial hole burning in terahertz quantum cascade lasers,” Appl. Phys. Lett. 91, 161108 (2007).
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T. S. Mansuripur, C. Vernet, P. Chevalier, G. Aoust, B. Schwarz, F. Xie, C. Caneau, K. Lascola, C. Zah, D. P. Caffey, T. Day, L. J. Missaggia, M. K. Connors, C. A. Wang, A. Belyanin, and F. Capasso, “Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator,” Phys. Rev. A 94, 063807 (2016).
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L. Columbo, M. Piccardo, F. Prati, L. A. Lugiato, M. Brambilla, A. Gatti, C. Silvestri, M. Gioannini, N. Opacak, B. Schwarz, and F. Capasso, “Unifying frequency combs in active and passive cavities: temporal solitons in externally-driven ring lasers,” Phys. Rev. Lett. 126, 173903 (2021).
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M. Piccardo, B. Schwarz, D. Kazakov, M. Beiser, N. Opačak, Y. Wang, S. Jha, J. Hillbrand, M. Tamagnone, W. T. Chen, A. Y. Zhu, L. L. Columbo, A. Belyanin, and F. Capasso, “Frequency combs induced by phase turbulence,” Nature 582, 360–364 (2020).
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M. Piccardo, D. Kazakov, N. A. Rubin, P. Chevalier, Y. Wang, F. Xie, K. Lascola, A. Belyanin, and F. Capasso, “Time-dependent population inversion gratings in laser frequency combs,” Optica 5, 475–478 (2018).
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T. S. Mansuripur, C. Vernet, P. Chevalier, G. Aoust, B. Schwarz, F. Xie, C. Caneau, K. Lascola, C. Zah, D. P. Caffey, T. Day, L. J. Missaggia, M. K. Connors, C. A. Wang, A. Belyanin, and F. Capasso, “Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator,” Phys. Rev. A 94, 063807 (2016).
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M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16, 3242–3248 (2008).
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D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
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J. C. Pearson, B. J. Drouin, A. Maestrini, I. Mehdi, J. Ward, R. H. Lin, S. Yu, J. J. Gill, B. Thomas, C. Lee, G. Chattopadhyay, E. Schlecht, F. W. Maiwald, P. F. Goldsmith, and P. Siegel, “Demonstration of a room temperature 2.48–2.75 THz coherent spectroscopy source,” Rev. Sci. Instrum. 82, 093105 (2011).
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Chen, W. T.

M. Piccardo, B. Schwarz, D. Kazakov, M. Beiser, N. Opačak, Y. Wang, S. Jha, J. Hillbrand, M. Tamagnone, W. T. Chen, A. Y. Zhu, L. L. Columbo, A. Belyanin, and F. Capasso, “Frequency combs induced by phase turbulence,” Nature 582, 360–364 (2020).
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M. Piccardo, D. Kazakov, N. A. Rubin, P. Chevalier, Y. Wang, F. Xie, K. Lascola, A. Belyanin, and F. Capasso, “Time-dependent population inversion gratings in laser frequency combs,” Optica 5, 475–478 (2018).
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M. Aoki, K. Uomi, T. Tsuchiya, S. Sasaki, M. Okai, and N. Chinone, “Quantum size effect on longitudinal spatial hole burning in MQW lambda/4-shifted DFB lasers,” IEEE J. Quantum Electron. 27, 1782–1789 (1991).
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W. Fang, A. Hsu, S. L. Chuang, T. Tanbun-Ek, and A. M. Sergent, “Measurement and modeling of distributed-feedback lasers with spatial hole burning,” IEEE J. Sel. Top. Quantum Electron. 3, 547–554 (1997).
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M. Piccardo, B. Schwarz, D. Kazakov, M. Beiser, N. Opačak, Y. Wang, S. Jha, J. Hillbrand, M. Tamagnone, W. T. Chen, A. Y. Zhu, L. L. Columbo, A. Belyanin, and F. Capasso, “Frequency combs induced by phase turbulence,” Nature 582, 360–364 (2020).
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T. S. Mansuripur, C. Vernet, P. Chevalier, G. Aoust, B. Schwarz, F. Xie, C. Caneau, K. Lascola, C. Zah, D. P. Caffey, T. Day, L. J. Missaggia, M. K. Connors, C. A. Wang, A. Belyanin, and F. Capasso, “Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator,” Phys. Rev. A 94, 063807 (2016).
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M. Martl, J. Darmo, C. Deutsch, M. Brandstetter, A. M. Andrews, P. Klang, G. Strasser, and K. Unterrainer, “Gain and losses in THz quantum cascade laser with metal-metal waveguide,” Opt. Express 19, 733–738 (2011).
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J. R. Freeman, J. Maysonnave, S. Khanna, E. H. Linfield, A. G. Davies, S. S. Dhillon, and J. Tignon, “Laser-seeding dynamics with few-cycle pulses: Maxwell-Bloch finite-difference time-domain simulations of terahertz quantum cascade lasers,” Phys. Rev. A 87, 063817 (2013).
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M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16, 3242–3248 (2008).
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M. G. Davis and R. F. O’Dowd, “A transfer matrix method based large-signal dynamic model for multielectrode DFB lasers,” IEEE J. Quantum Electron. 30, 2458–2466 (1994).
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T. S. Mansuripur, C. Vernet, P. Chevalier, G. Aoust, B. Schwarz, F. Xie, C. Caneau, K. Lascola, C. Zah, D. P. Caffey, T. Day, L. J. Missaggia, M. K. Connors, C. A. Wang, A. Belyanin, and F. Capasso, “Single-mode instability in standing-wave lasers: the quantum cascade laser as a self-pumped parametric oscillator,” Phys. Rev. A 94, 063807 (2016).
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Supplementary Material (1)

NameDescription
» Supplement 1       Simulation parameters, reference ridge data, different harmonics spectra

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. (a) Sketch of the THz quantum cascade ring laser. Due to the narrow width of the waveguide, two contacting pads ( ${{60}} \times {{60}}\;{\rm{\unicode{x00B5}{\rm m}}}$ each) are connected to the ring at 12 and 6 o’clock for electrical contacting. (b) Eigenmode simulation (electric field in the $z$ direction) of the ring-shaped resonator using a commercial finite element solver (Comsol 5.5). The simulation yields WGMs. (c) SEM image of a ring resonator section with one contacting pad.
Fig. 2.
Fig. 2. Different emission regimes of the THz quantum cascade ring laser at a heat-sink temperature of 20 K. $Y$ axis in spectra in log-scale. (a) DC light–current–voltage measurement in the dynamic range of the ring laser. The device exhibits four different emission regimes, which are highlighted on the current–voltage curve. The device starts lasing single mode (b) at a threshold current of 200 mA (1). At (2) the emission transits to a harmonic state, showing multimode operation (c) with a mode separation by a multiple of the round-trip frequency of the cavity. The transition to the harmonic state is also manifested by the kink in the current–voltage and the current–light curves, respectively. (d) Further increase of the driving current leads to a dense comb state with several equidistant modes (3). Here, the transition results in an intensity drop on the current–light curve. The dense comb state collapses at a driving current above 290 mA to single-mode operation (4) and stops lasing at a driving current of 296 mA in constant current driving operation (5). (e) The chaotic multimode regime, located in the NDR region of the current–voltage curve, can be only accessed in constant voltage driving operation.
Fig. 3.
Fig. 3. Emission mode evolution of the THz quantum cascade ring laser ( $\,y$ axis in log-scale). The lasing of the device starts with single-mode emission. With increasing driving current, the emission spectrum jumps to a harmonic-state regime, consisting of multiple modes, which are separated by a multiple of the fundamental mode spacing ${{\rm{f}}_{\rm{rep}}}$ plus an offset ( ${0.32} \times {{\rm{f}}_{\rm{rep}}}$ ). At 254 mA, a 10th-order harmonic with the two remaining initial modes at 3.3 and 3.8 THz (marked orange) is observed. The spacing of these two modes to the next right mode is ${{6}} \times$ FSR. These two modes disappear with higher current leaving a sole 10th harmonic spacing. Further increase of the driving current leads to the dense comb regime, where sidebands proliferate around the main modes of the harmonic state, forming four mode groups which are visible in the spectrum at, e.g.,  288 mA centered at 3.6 THz, 3.78 THz, 3.9 THz, and 4.1 THz, respectively. This dense comb state collapses at a current of 290 mA to single-mode emission. When the device is driven in constant voltage driving operation a chaotic multimode regime in the NDR region is observed beyond the comb state instead (red curve).
Fig. 4.
Fig. 4. (a) RF signal of the THz quantum cascade ring laser at three different driving currents. The beat note signal at 15.563 GHz shows the intermodal spacing within the four mode groups. The inset shows a zoom-in of the free-running intermodal beat note at 15.563 GHz at a driving current of 270 mA indicating comb operation. The beat note signal at around 5 GHz arises due to the offset between the mode groups. Consequently, the sum- and difference-frequency signals appear around 10 and 20 GHz since the sidebands of the mode groups start to overlap with the rising driving current, shown in (b). The highlighted modes of two adjacent mode groups show the overlap of the rising modes with the increased driving current. The green- and purple-colored modes belong each to one mode group, respectively. Moreover, the mode groups are pulled toward each other with increasing pumping power, indicated by the decreasing offset beat note frequency, whereas the intermodal spacing stays constant.
Fig. 5.
Fig. 5. Numerical simulations of an ideal ring cavity QCL containing no defects using the linewidth enhancement factor of (a)  $\alpha = {1.5}$ and (b)  $\alpha = {2.5}$ . The top plots depict the power spectra, and the bottom plots show the time trace of the laser intensity over two round trips. Numerical simulations have been run for 100,000 round trips to reach a stationary state forming (a) a fundamental frequency comb and (b) a second-order harmonic comb with a ${\sec}{{\rm{h}}^2}$ spectrum. Both time traces of the laser intensity show high-contrast amplitude modulations on a continuous background.
Fig. 6.
Fig. 6. (a) Simulated spectra of a ring cavity QCL containing two opposing defects. For $\alpha = 0.4$ the laser has not reached the multimode instability and is operating in single mode. With the increase of the $\alpha$ factor, the laser transits from a fourth harmonic state to a dense state. Red crosses in the bottom plot indicate modes with 5% larger intermodal spacing than the remaining modes. This leads to an offset of 1.5 GHz between the two adjacent mode groups. The experimentally measured offset of 5 GHz indicates that the model is underestimating the nonlinearities in our devices. (b) Corresponding time traces. The single-mode emission exhibits a constant output intensity over the round trips, whereas the multimode emission shows a modulated output signal. The periodicity of the time traces at linewidth enhancement factor values of $\alpha =0.45 {-} 0.525$ originates from the equidistant spacing of the modes. However, in the bottom plot, the modes of the adjacent mode groups show a larger mode spacing, resulting in nonperiodic features in the time trace.