1. INTRODUCTION

The nonlinear coupling between the optical modes of a cavity through the active medium of a laser have proven this system to be a very rich playground for nonlinear physics. Very soon after the discovery of the laser, Risken, Nummedal, and Graham and Haken (RNGH) predicted the appearance of a multimode instability in lasers with a fast gain medium in a ring cavity at large pumping levels from 8 to 14 times above laser threshold [1,2]. However, most laser systems where mode locking was initially achieved were based on an active medium with an upper-state lifetime that is long compared to the round-trip time of the cavity and therefore exhibited different dynamical properties. A bit more than ten years ago, it was realized that the quantum cascade laser (QCL) [3], being characterized by a very short upper-state lifetime, satisfied one key condition where such an RNGH instability could be observed [4]. Indeed, optical frequency comb generation was realized in QCLs in the mid-infrared [5] as well as in the terahertz (THz) range [6,7] in devices with a broad gain medium and a proper cavity dispersion compensation [6,8]. Moreover, these devices were shown to operate with fixed phases a linear frequency chirp characteristic of a mainly frequency-modulated output [9]. Recently, average powers of greater than 1 W and frequency coverage over ${100}\;{{\rm cm}^{ - 1}}$ have been demonstrated [10,11], making these devices indispensable tools for mid-infrared spectroscopy [12,13].

However, recent theoretical [14–17] and experimental studies [10] have shown the important role of the dynamical spatial hole effect that appears in the Fabry–Perot (FP) cavities used in QCLs in forcing the multimode operation and generating the frequency comb. Clearly, the strength of the spatial hole burning depends not only on the cavity structure but also on the electric field asymmetry within the cavity. Therefore, it would be of great interest to investigate the comb operation behavior from a ring cavity QCL [18–20], where the spatial hole burning is expected to be negligible in the absence of coupling between the clockwise (CW) and the counterclockwise (CCW) modes in the cavity [21]. As discussed in Supplement 1, Section 1, the difference between the FP and the ring configuration can also be understood when considering a Maxwell–Bloch model with the modal decomposition of the cavity field as used recently to model QCL-based frequency combs [14,15] and which mainly consists of a reduction of the self-saturation compared to the cross-saturation. Interestingly, the ring QCL is a system then very close to the original theoretical proposal studied by RNGH. Nevertheless, as compared to that proposal, the QCL ring exhibits additional complexities such as a finite dispersion originating from the waveguide and from the gain medium, the effect of electron transport, as well as a small but non-zero linewidth enhancement factor [22,23]. Recent works on ring QCLs have noted the similarity, under certain assumptions, of that system to a Gingsburg–Landau equation [24].

As shown schematically in Fig. 1(a), we report here a frequency comb generated directly from an electrically pumped ring QCL. In the frequency comb regime, the ring laser shows a narrow radio frequency (RF) beat note at the round-trip frequency over 85% of the current dynamic range, with a 3 dB linewidth below 900 Hz. As a further proof, a dual-comb experiment similar to the one reported in Ref. [25] has been carried out to reveal the frequency comb operation from the ring cavity QCL as well as to resolve the spectral phases.

 

Fig. 1. (a) Schematic of the ring QCL frequency comb. (b) Microscope image of a processed ring QCL with a width of 8 µm and a radius of 600 µm. (c) False-color scanning electron image showing the cross section of the same device. The purple area shows the facet of the active region. (d) Group velocity dispersions of plasmon-enhanced waveguide with different radii. The inset shows the 2D mode profile computed at ${1340}\;{{\rm cm}^{ - 1}}$ with a width of 8 µm and a radius of 600 µm. The ring-geometry-induced dispersion is evident from the maximum of the mode, which approaches the rim of the resonator compared with the straight waveguide. (e) Group velocity dispersion of a standard waveguide of the same dimensions, with the cladding consisting of 1500 nm n-doped InP (${2e16}\;{{\rm cm}^{ - 3}}$), 2000 nm ${ n}$-doped InP (${5e16}\;{{\rm cm}^{ - 3}}$), and 500 nm n-doped InP (${3e18}\;{{\rm cm}^{ - 3}}$). (f) Experimental and simulated dispersions of a FP device fabricated together with the ring QCL.

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2. DESIGN AND FABRICATION

The QCL active region is based on a strain-compensated InGaAs/AlInAs single-stack heterostructure grown by molecular beam epitaxy (MBE) [26]. To compensate for the positive dispersion of the InP cladding layer, a plasmon-enhanced waveguide has been designed following the method outlined in Ref. [27]. The width of the ridge forming the ring ($w$) was chosen to be 8 µm, determined by a compromise between the output coupling efficiency and the dispersion compensation. For comparison, the group velocity dispersion (GVD) of both a plasmon-enhanced waveguide and a normal waveguide was computed using a commercial two-dimensional finite element solver (Comsol 5.3). The corresponding GVD for ring radii ($R$) from 400 to 600 µm is shown in Fig. 1(d), where the inset of Fig. 1(d) shows a 2D mode profile computed at the designed wavenumber of ${1340}\;{{\rm cm}^{ - 1}}$ with a radius of 600 µm. At a wavenumber of ${1340}\;{{\rm cm}^{ - 1}}$, the computed GVD is $\sim{800}\;{{\rm fs}^2}/{\rm mm}$ for the plasmon-enhanced waveguide, which shows a reduction of $\sim{900}\;{{\rm fs}^2}/{\rm mm}$ compared with that of a standard waveguide [Fig. 1(e)] and thus demonstrates the effectiveness of the plasmon-enhanced waveguide. The computed waveguide loss of the mode is around ${2}\;{{\rm cm}^{ - 1}}$, which has taken into account both the bending radiation loss and the free-carrier absorption by using the Drude–Lorentz model for the material refractive index. The Comsol simulation has been proven to provide a good agreement with the experimental measurement of the QCL loss as pointed out in Ref. [28]. To test the validity of our model, the simulated waveguide dispersion is compared with experimental value obtained from a FP device using the same cladding structure as the ring waveguide (see Supplement 1, Section 2). As shown in Fig. 1(f), the experimental data shows a close-to-zero or slightly negative GVD. We attribute the small difference between the measurement and the simulation to uncertainties in the exact doping levels, cladding thickness, and waveguide width. The gain-induced dispersion is also measured, showing a dispersion of $ \lt {50}\;{{\rm fs}^2}/{\rm mm}$. Based on the simulation, ring structures with width of 8 µm and radius from 400 to 600 µm were processed following a buried-heterostructure process, where the laser ridges were enclosed by the Fe-doped InP for better heat extraction [29,30]. To further enhance the performance and result in continuous-wave operation, the devices were cleaved and epilayer-down mounted on AlN substrates attached to copper mounts for better heat extraction. For effective output power extraction, the ring structure was designed to be in close proximity to the cleaved edge as shown in Fig. 1(b). The output coupling efficiency is estimated to be around 0.2% (see Supplement 1, Section 3). The main mechanism for the radiation is the bending losses, which can be seen as a tunneling of the guided mode to the continuum induced by the waveguide curvature. The regrown InP on the sides and top of the laser ridge was very smooth and defect free despite the ring geometry (see the cross section of a finished device with $w={8}\;\unicode{x00B5}{\rm m}$ and ${R} = \;{600}\;\unicode{x00B5}{\rm m}$ in Fig. 1(c) and Supplement 1, Section 4). For this reason, we expect a very weak coupling between the CW and CCW modes of the ring laser.

3. RESULTS AND DISCUSSION

Spectral measurements with a resolution of $0.075\,\,{\rm cm}^{1}$ were carried out using a Fourier transform infrared spectrometer (FTIR). The light-current curve was measured using the lock-in scheme with a nitrogen-cooled mercury-cadmium-telluride detector (MCT) due to the weak evanescent wave out-coupling from the device. It should be noted that, for the current work, no special extractor (e.g., grating) was used for power extraction from the ring QCL. Thus, feedback and cavity perturbation effects were minimized compared with the FP combs, allowing us to effectively study the laser dynamics within the ring cavity. The light-current-voltage (LIV) characteristics and the spectra of the device with a width of 8 µm and a radius of 600 µm are presented in Fig. 2(a) at temperature of 255 K. The collected output power on the detector was about 90 µW at rollover. The laser spectrum starts with single-mode emission at the threshold (700 mA) and broadens at a current of 735 mA, thereafter remaining multimode until rollover current density, with a spectrum coverage of ${10}\;{{\rm cm}^{ - 1}}$ at 920 mA as shown in Fig. 2(b). Within the resolution of the FTIR, the spectra show a constant free spectral range (FSR) equal to $c/{n_\textrm{eff}}L$, where $c$, ${n_\textrm{eff}}$, and $L$ correspond to the speed of light in a vacuum, the mode refractive index, and the perimeter of the ring, respectively, demonstrating the travelling wave nature of the device. It is noted that the optical spectra can be nicely fitted by a ${{\rm sech}^2}$ envelope, which is an interesting indication of a possible operation with a bright pulse in the anomalous dispersion regime [31,32]. We attribute the pedestal in the FTIR spectra shown in Fig. 2(b), visible at a level of $ - {30}\;{\rm dB}$ to the main peaks, to the finite signal over noise and the slight nonlinearity of our MCT detector.

 

Fig. 2. (a) Light-current-voltage (LIV) curve in continuous-wave operation measured at a sink temperature of 255 K for a ring QCL with a width of 8 µm and a radius of 600 µm. The green area shows the single-mode regime, while the blue area shows the multimode regime. (b) Corresponding spectra as a function of current measured at a constant temperature of 255 K. Fits of spectra with ${{\rm sech}^2}$ envelope are shown in dash-dotted lines.

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Fig. 3. (a) Experimental setup for studying the mode switching in the ring laser. The dashed line indicates the optical axis. AWG, arbitrary waveform generator. (b) Beam pattern from the ring laser after collimation. The inset shows the relative positon between the collimating lens and the laser. (c) Upper: waveform of a typical driving pulse. In the pulse, the ${T_L}={T_H}={0.1}\;{\rm s}$, and ${T_f}$ is varied for different rising times to ensure constant thermal dissipation for each pulse. Lower: optical intensites for both the counterclockwise (CCW) and clockwise (CW) modes during the pulse. (d) The counts for both CCW and CW modes as a function of rise time ${T_r}$.

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As shown schematically in Fig. 3(a), due to the nature of the device losses, we can identify the nature, CW or CCW, from the position of the emission on the ring. Figure 3(b) shows the typical beam pattern after collimation, with the inset indicating the relative position between the lens and the device. To study the switching between the modes, a driving scheme as shown in the upper part of Fig. 3(c) was used. In this scheme, a train of fixed-length current pulses, parameterized by the rise time ${T_r}$, are superimposed onto a DC bias of 692 mA and delivered to the laser. The joule dissipation per period was kept the same for all waveforms used by adjusting the fall time ${T_f}$. For this measurement, we used a total pulse length of 1 s, with the two additional parameters ${T_H}$ and ${T_L}$ kept constant at 0.2 s apiece, and swept ${T_r}$ from 20 to 580 ms. For each rise time, we record the detected intensity on an oscilloscope 25 times.

Two typical waveforms were observed to occur as shown in Fig. 3(c), bottom: one with clear lasing (the blue trace) and another with a complete absence of signal after a short time. The number of lasing instances was counted for each rise time. We then repositioned the lens to the opposite side of the ring and repeated the experiment.

 

Fig. 4. (a) Intermode beat-note spectra as a function of driver current at 255 K with a span of 75 MHz, resolution bandwidth of 100 Hz, and 10 averages (b) Beat-note linewidth at $ - {3}\;{\rm dB}$ and peak frequency as a function of bias current at 255 K.

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Counts for each rise time at both positions are plotted in Fig. 3(d), and what is observed to emerge is a complementary pair of curves from which we can identify the dark traces in (${c}$) as corresponding to lasing in the opposite direction, not collected by the lens. At the extema, we find the device to almost exclusively lase in CW for short rise times and CCW for slower ramps. In between, the laser chooses one mode or another probabilistically. We note that extending the high-time ${T_H}$ did not reveal any further dynamics once the CW or CCW state had been selected. The switching between the two modes with varying rising time indicates a thermal origin of the effect; operation of ring combs on predominantly CW or CCW modes and switching between those was already observed in extended cavity ring QCLs [21].

In the data shown in Fig. 2 as well as in the remainder of the paper, the device was driven such that it operated in a CCW mode (see Supplement 1, Section 5).

To verify the frequency comb operation from this ring QCL, the intermode beat notes were measured electrically, possible owing to the ultrafast dynamics in QCL active regions [33]. In this approach, the current driver and the QCL were connected to the DC port and the AC + DC port of a bias tee, respectively. The generated RF signal due to intracavity intensity modulation was detected with a spectrum analyser for RF characterization. Shown in Fig. 4(a) are beat-note spectra recorded for different currents at 255 K. To capture the narrow linewidth of the beat note, the span, the bandwidth, and the spectral averaging were set to be 200 kHz, 100 Hz, and single shot, respectively. Notably, the ring QCL shows a sharp beat note in most of the multimode regime (from 735 to 990 mA) and a linewidth (3 dB) below 900 Hz as shown in Fig. 4(b). The beat-note frequency shows a current tuning coefficient of $\sim{180}\;{\rm kHz/mA}$ [see Fig. 4(a)] and temperature tuning coefficient of 1.9 MHz/K (see Supplement 1, Section 6), as the repetition rate ${f_\textrm{rep}}$ is effective refractive index dependent. Compared with the FP frequency comb, no high-phase-noise regime was observed [8], which thus demonstrates the stability of comb operation in terms of the bias current. The single low-noise beat note strongly indicates that all the modes are phase locked and equidistant [34]. However, a narrow beat note at the round-trip frequency of the ring laser is a necessary but not a sufficient condition to prove the frequency comb operation [35].

 

Fig. 5. (a) Schematic of the multiheterodyne experiment for frequency comb characterization. OBP, optical band-pass filter; MCT, mercury-cadmium-telluride detector. (b) Spectrum for the reference laser at 549 mA and the ring laser at 942 mA, both at 255 K, recorded immediately after the dual-comb acquisition. (c) Multiheterodyne spectrum through coherent averaging for an integration time of 40 ms. Multiple multiheteorodyne beat-note lines are observed, in an excellent agreement with the optical spectrum of the ring laser. (d) Recovered phases of the ring laser by adding the measured reference phase to the multiheterodyne phase. The phase offset and slope have been removed for visual clarity. Each shade of the red dots represents one multiheterodyne measurement, and the blue ellipses represent the 5%–95% confidence intervals based on the uncertainty in the SWIFTS measurement. (e) Reconstructed temporal intensity of the ring laser output. The blue bands correspond to the 5%–95% confidence intervals including the error in both the SWIFTS and multiheterodyne phase.

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We performed a dual-comb experiment on the free-running devices, which directly reveals the operating characteristics of both lasers used in the experiment [25]. The schematic of the experimental setup is shown in Fig. 5(a). For a stronger signal, we used a FP frequency comb as the reference laser with a free spectral range (${{\rm FSR}_\textrm{FP}}$) of approximately half of the one of the ring QCLs (${{\rm FSR}_\textrm{ring}}$). An optical band-pass filter was used to select the modes of interest only and to meanwhile increase the signal-to-noise ratio (SNR). Typical emission spectra for both lasers are shown in Fig. 5(b). In the experiment, both lasers were directed to a high-bandwidth (1 GHz) mid-infrared MCT detector (Vigo PV-4TE-10.6). The generated heterodyne signal was acquired by a high-bandwidth oscilloscope (analogue bandwidth of 1 GHz and sampling rate of 2.5 GS/s). During the experiment, in order to locate the multiheterodyne signal within the bandwidth of the detector, both the temperatures and currents of both lasers have to be carefully tuned. The corresponding beat-note frequencies are $\sim{23.80}\;{\rm GHz}$ and $\sim{11.94}\;{\rm GHz}$ for the ring and reference lasers, respectively, during experiment, leading to a repetition frequency difference $\Delta {f_\textrm{rep}}={2}{f_{F - P}} - \;{f_\textrm{ring}} = \;\sim{80}\;{\rm MHz}$.

By performing a sequential post-correction of the ${\Delta }{f_\textrm{rep}}$ and the difference in carrier envelope offset frequencies ${ \Delta }{f_\textrm{ceo}}$ [36,37], we are able to substantially enhance the SNR of our multiheterodyne spectra from the raw measurement through coherent averaging (see Supplement 1, Sections 7 and 8). Multiple multiheterodyne signal peaks are observed between 100 and 1000 MHz with a constant spacing of $\sim{80}\;{\rm MHz}$, and, post-correction, Fourier-limited linewidths of $\sim{25}\;{\rm Hz}$. These are shown in Fig. 5(c). Within experimental error, the shape of the multiheterodyne spectrum has been verified to correspond to the product of the two amplitudes of the overlapping modes as measured in situ by FTIR. The narrow beat-note spectra at round-trip frequency, a constant spacing between the multiheterodyne beat notes, and the good agreement between RF and optical domains confirm the frequency comb from the ring QCL.

The spectral phases could be retrieved from the experiment [25] using a separate shifted wave interference Fourier-transform spectroscopy (SWIFTS) measurement of the spectral phases for the reference FP devices [6,9] (see Supplement 1, Sections 7 and 8), and they are displayed in Fig. 5(d). The phases are stable within the experimental accuracy and, in contrast to the results for FP devices [9], do not exhibit a parabolic profile. Consequently, the reconstructed intensity profile obtained from the amplitudes and phases displays a significant intensity modulation.

There is a striking similarity between the spectrum and intensity profile of the output with the one predicted for the RNGH instability. Nevertheless, the fact that the multimode operation is initiated very close to lasing threshold, in contradiction to the predictions from the theory from RNGH, points towards additional elements contributing to lowering of the threshold of the multimode instability. As mentioned earlier, the latter may include a non-zero linewidth enhancement factor, dispersion of the cavity, and ultrafast electron transport. We hope this work will stimulate additional theoretical analysis in this direction.

4. CONCLUSION

In conclusion, we have demonstrated a mid-infrared frequency comb from a novel ring cavity QCL. Narrow beat notes ($ \lt {900}\;{\rm Hz}$) are shown in 85% of the dynamic range, which may be attributed to the travelling wave nature of the ring cavity, contrary to the standing waves in standard FP QCL frequency combs. The frequency comb operation was verified by the multiheterodyne experiment, with the RF domain beat-note lines in good agreement with the optical spectrum of the ring QCL measured in situ. Higher output power from the frequency comb is required to efficiently take advantage of the pulse regime. This can be achieved by using new cavity designs, e.g., adding a bus waveguide to the ring cavity to fully explore the ultra-short pulse and even soliton regimes [32] of such devices.