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Abstract
Optical feedback exists in most laser configurations and strongly affects laser performances depending on the feedback strength, length, and phase. In this paper, we investigate the frequency comb behaviour of a semiconductor quantum cascade laser emitting around 4.2 THz with external optical feedback. A periodic evolution of the laser inter-mode beatnote from single-line to multiple-line structures is experimentally observed with a minor change of optical feedback length (phase) on the wavelength scale. The comb stability of the laser with feedback is also measured and compared with the same laser without feedback. Furthermore, our simulations reveal that the dynamical oscillations invoked by optical feedback are responsible for the measured multiple-line beatnotes. It is found that the characteristic feedback period is determined by the half wavelength of the laser, while the comb operation is maintained at most feedback length positions. Therefore, terahertz quantum cascade laser combs are robust against the minor position vibration of the feedback mirror in practice, owing to the much smaller feedback phase change than that of common near-infrared laser diodes.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical feedback is a widely existed phenomenon in all lasers. Depending on the types of lasers, the sensitivity to optical feedback that is related to the light reflected back into the laser cavity from an external target is different. With external optical feedback, the characteristics of a laser can be affected by varying the feedback strength, length, and phase, resulting in either narrowing or broadening of the mode linewidth, as well as affecting the frequency stability [1–4]. For example, it has been shown that under optical feedback, the mode stability of a mid-infrared quantum cascade laser (QCL) can be significantly improved and the laser linewidth has been reduced by more than 70 times compared to the case without feedback [5]. Concerning the linewidth reduction, the optical feedback technique is superior to other traditional methods including the voltage control [6], resistance control [7], optical delay techniques [8], etc. Up to now, most studies on optical feedback have been mainly carried out on the near-infrared and mid-infrared semiconductor laser platforms [9–13].
The electrical pumped and semiconductor-based terahertz QCL [14,15] is an ideal laser platform for optical feedback study due to its unique features, e.g., wide frequency coverage, high output power, and fast response [16–19]. There were only several reports on single-mode and multi-mode terahertz QCLs subject to optical feedback [20–25]. Particularly, M. Wienold, et al., proposed a self-mixing beatnote spectroscopy based on the optical feedback technique, which is able to assess the coherence of terahertz QCL frequency combs [21]. In [24], authors theoretically studied the laser optical feedback interference in a multi-mode terahertz QCL. It was found that the laser exhibited three different laser operation regimes, i.e., single-mode, multi-mode, and tunable-mode, depending on the optical feedback level and the laser gain bandwidth. The transient instabilities in a single-mode terahertz QCL under optical feedback were further revealed experimentally [25]. The investigation of optical feedback effects on the terahertz laser combs is of great importance for achieving highly stable frequency comb and dual-comb sources [16,26–29] and for implementing comb-based high resolution applications, e.g., fast spectroscopy, imaging, etc [30–32].
In this work, we investigate the frequency comb operation of a terahertz QCL emitting around 4.2 THz under external optical feedback. The laser output power, inter-mode beatnote and its stability are experimentally measured with monotonously changing the feedback length on the wavelength scale. A periodic change of the inter-mode beatnote from single-line to multiple-line structures is observed. In order to explain the observed phenomenon, a numerical simulation based on a two-mode rate equation model is performed to study the mode dynamics of the terahertz QCL comb. The simulation results agree well with the experimental observations. Furthermore, owing to the long wavelength of terahertz lasers, the frequency combs are naturally robust against the inevitable vibration of the feedback mirror position in practice.
2. Experimental setup and laser performance
Figure 1(a) shows the experimental setup that is employed to study optical feedback effects on a terahertz QCL. As shown in Fig. 1(a), the terahertz light emitted from the QCL is first collected using a parabolic mirror and part of the light is reflected back into the laser cavity using a gold mirror or a high resistivity silicon wafer that is placed in the parallel beam after the parabolic mirror. To evaluate the laser power under optical feedback, a silicon wafer is used for reflecting the terahertz light back for the feedback and simultaneously allowing the terahertz transmission for power measurement, as shown in the dashed rectangle of Fig. 1(a). For the frequency comb characterization, we replace the silicon wafer with a gold mirror with a reflectivity of 0.9 to increase optical feedback strength. The inter-mode beatnote signal that reflects the frequency comb behaviour is recorded at each $z$ position of the gold mirror using the laser self-detection [30,33]. In order to extract the inter-mode beatnote, the terahertz QCL is connected to the AC+DC port of a bias-T. The AC port of the bias-T is connected to a microwave amplifier with a gain of 30 dB and finally the inter-mode beatnote is registered on a spectrum analyzer (Rhode & Schwarz, FSW26). The gold mirror or silicon wafer is mounted on a motorized translation stage that can move precisely along the $z$-axis (with a step of 1 $\mu$m) to control optical feedback length or phase.
Fig. 1. Experimental setup and laser performances of a single plasmon terahertz QCL. (a) Experimental setup of the configuration under optical feedback. A gold mirror or silicon wafer is used to partially reflect the terahertz light emitted from the terahertz QCL back into the laser cavity. Optical feedback length or phase is controlled by precisely moving the gold mirror or silicon wafer along the $z$-axis that is mounted on a motorized translation stage. The laser output power under optical feedback is measured by collimating the terahertz light after the silicon wafer (with a thickness of 625 $\mu$m) to a terahertz power detector (Golay Cell) using a parabolic mirror (see dashed rectangle). The comb behaviour of the QCL is characterized by the self-detected inter-mode beatnote signals. (b) Light-current-voltage ($L-I-V$) characteristics of the terahertz QCL without optical feedback recorded at 20 K and 30 K in continuous wave mode. (c) Emission spectra of the terahertz QCL without feedback measured at currents of 660, 800, and 990 mA at 30 K. The ridge of the terahertz QCL is 6 mm long and 150 $\mu$m wide.
The terahertz QCL used in this work is based on a hybrid active region [34] and a single plasmon waveguide with a cavity length of 6 mm and a ridge width of 150 $\mu$m. The light-current-voltage ($L-I-V$) characteristics of the terahertz QCL measured in continuous wave (CW) mode are shown in Fig. 1(b). The dashed and solid curves denote the results recorded at 20 K and 30 K, respectively. The maximum power of the terahertz QCL is measured to be $\sim$2 mW at 20 K ($\sim$1.5 mW at 30 K). Figure 1(c) shows the emission spectra of the terahertz QCL measured at three different drive currents of 660, 800, and 990 mA at 30 K by employing a Fourier transform infrared spectrometer (FTIR) with a spectral resolution of 0.08 cm$^{-1}$. At a lower current of 660 mA around the laser threshould, the terahertz QCL emits a single frequency at 4.15 THz. When the current is increased to 800 or even 990 mA, the laser demonstrates a multi-mode emission due to the spatial hole burning effect and/or the Risken-Nummedal-Graham-Haken instability [35], and the frequency roughly spans from 4.13 to 4.27 THz.
3. Experimental results
For a laser under optical feedback, the light that is reflected back into the laser cavity mixes with the intra-cavity electric field and then generate a measurable self-mixing signal, for instance, the emission power, terminal voltage, and so on [5]. This self-mixing effect is remarkably universal and it has been demonstrated at numerous wavelengths for different laser structures [36–43].
Here, we first evaluate the laser output power under optical feedback. As shown by the dashed rectangle of Fig. 1(a), we use a silicon wafer to reflect the terahertz light back to the laser cavity and simultaneously measure the transmitted power under optical feedback. To facilitate an accurate power measurement, we operate the QCL in pulsed mode at a drive current of 660 mA with a repetition frequency of 10 kHz and a duty cycle of 50%. The power signal is measured using a room temperature Golay cell detector. Note that a slow modulation of 15 Hz is added to the QCL due to the slow response of the Golay cell detector. Figure 2(a) shows the transmission of the silicon wafer measured using an FTIR with a spectral resolution of 4 cm$^{-1}$. It can be found that the transmission of the silicon wafer is around 54%. The transmitted power is then measured as a function of the changed feedback length ($\Delta {L}_{\rm {ext}}$), as shown in Fig. 2(b). In this experiment, we move the silicon wafer along the $z$-axis with a step of 2 $\mu$m. The initial external cavity length is 19.5 cm which corresponds to $\Delta {L}_{\rm {ext}}$=0 in Fig. 2(b). The power values are translated from the numbers read from a lock-in amplifier by considering the calibration curve of the Golay cell detector. It can be clearly seen from Fig. 2(b) that the measured power values oscillates with a period of a half-wavelength of the laser ($\sim$36.29 $\mu$m). Like the conventional interferometry, the oscillation period shown in Fig. 2(b) relates to a target displacement of a half-wavelength, corresponding to a phase-shift of 2$\pi$. Similar phenomena were also observed in Ref. [24].
Fig. 2. Laser output power evaluation under optical feedback. (a) Transmission of the silicon wafer used in the experiment. (b) Laser output power as a function of the changed external cavity length ($\Delta {L}_{\rm {ext}}$). For the power measurement, the QCL is operated in pulsed mode at a drive current of 660 mA with a repetition frequency of 10 kHz and a duty cycle of 50%.
We further investigate the feedback effect on the frequency comb behaviour of the laser. To characterize the comb or non-comb operation, the inter-mode beatnote of the terahertz laser is recorded at each $z$ position. To enhance the reflected terahertz light power into the laser cavity, in this experiment we replace the silicon wafer with a gold mirror and the electrical beatnote signal is measured using the QCL itself as a detector. Two external feedback positions, i.e., $L_{\rm {ext1}}$=22 cm and $L_{\rm {ext2}}$=34 cm, are selected as shown in Figs. 3(a) and 3(b), respectively. At each position, we investigate the inter-mode beatnote signal by accurately tuning the feedback length with a step of 1 $\mu$m. The main results are shown in Figs. 3(c) and 3(d) for $L_{\rm {ext1}}$=22 cm and $L_{\rm {ext2}}$=34 cm, respectively. For both cases, we can see that the evolution of the beatnote signal shows a clear periodic feature as the feedback length (or the changed feedback length $\Delta {L}_{\rm {ext}}$) is monotonously increased. At each position, $L_{\rm {ext1}}$ or $L_{\rm {ext2}}$, we show two complete periods of the beatnote signal. Starting from a single-line beatnote signal (comb operation) at $\Delta {L}_{\rm {ext}}$=0, we can then observe dense multi-line beatnote signal and loose multi-line signal as $\Delta {L}_{\rm {ext}}$ is slightly increased. Finally, as $\Delta {L}_{\rm {ext}}$ is increased to 37 or 38 $\mu$m, the single-line beatnote signal is observed again. Similar as the measured power shown in Fig. 2(b), the evolution of the beatnote signal as a function of $\Delta {L}_{\rm {ext}}$ also shows a period of half-wavelength. To further prove the periodic change of the inter-mode beatnote signal with optical feedback length, we carried out another experiment employing two terahertz QCLs. The two lasers are optically coupled. One QCL is switched on and the other is switched off acting as a reflected mirror. The experimental setup and periodic change of the inter-mode beatnote signal with feedback length are shown in Fig. S1 (Supplement 1) and Visualization 1. To explain the observed periodic phenomenon shown in Figs. 3 and S1 (Supplement 1), a numerical simulation based on a two-mode rate equation model is carried out and the details are described in Section “Simulation and discussion".
Fig. 3. Schematic demonstrations of the feedback geometries for external feedback lengths of $L_{\rm {ext1}}$=22 cm (a) and $L_{\rm {ext2}}$=34 cm (b). (c) and (d) are inter-mode beatnote spectra for different changed external feedback lengths ($\Delta {L}_{\rm {ext}}$) measured around $L_{\rm {ext1}}$=22 cm and $L_{\rm {ext2}}$=34 cm, respectively. $\Delta {L}_{\rm {ext}}$ is equal to $L_{\rm {ext}}$-$L_{\rm {ext1}}$ for (c) and $L_{\rm {ext}}$-$L_{\rm {ext2}}$ for (d). The inter-mode beatnote are measured with a RBW of 50 kHz when the laser is operated at a drive current of 990 mA at 30 K.
From Fig. 3, it can be seen that the terahertz QCL under optical feedback can operate either in comb or non-comb regime depending on the feedback length or phase. In the following, we focus on the laser comb operation and investigate the linewidth of the inter-mode beatnote signal in the comb regime. In Figs. 4(a) and 4(b), we schematically show the evolution of the inter-mode beatnote in one period at two feedback length positions, i.e., $L_{\rm {ext1}}$=22 cm and $L_{\rm {ext2}}$=34 cm, respectively. The shaded areas marked by the “single inter-mode regime" are frequency comb regimes where single-line inter-mode beatnotes are observed. It is shown that for both cases, the comb operation can be obtained at most feedback length positions. Considering the period length of 38 $\mu$m (approximately a half-wavelength), the two numbers for the single-line regimes, i.e., 24 $\mu$m for $L_{\rm {ext1}}$ and 18 $\mu$m for $L_{\rm {ext2}}$ shown in Figs. 4(a) and 4(b), mean that 63% and 47% feedback length positions in one period are suitable for the frequency comb operation. The high percentages also indicate that the terahertz QCL frequency comb is robust against practical position vibration of the external feedback mirror. From this point of view, compared to near-infrared and mid-infrared lasers, it is easier for terahertz QCLs to maintain the comb operation when exposed to the same absolute feedback length variation because terahertz lasers experience a much smaller phase change. The typical inter-mode beatnote spectra recorded at the left-edge, center, and right-edge of the single-line regimes for both $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$ are shown in Fig. S2 (Supporting Information).
Fig. 4. Single-line regime and linewidth analysis of the terahertz QCL under optical feedback. (a) and (b) are schematic illustrations of the inter-mode beatnote evolution as the changed feedback length is increased at two feedback positions, i.e., $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$. The shaded areas in (a) and (b) depict the single-line regime or the frequency comb operation. In one period of $\sim$38 $\mu$m, the comb operation can be obtained at most positions, i.e., 24 $\mu$m and 18 $\mu$m at $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$, respectively. (c) Single-shot inter-mode beatnote in the single-line regime at $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$. For reference, the spectrum obtained without feedback is shown in the bottom panel. The data were recorded with a RBW of 500 Hz and a VBW of 500 Hz. (d) “Max-hold" inter-mode beatnote spectra in the single-line regime at $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$. the spectrum without feedback is shown in the bottom panel for reference. The time duration for the “max-hold" measurements is 30 seconds.
Figure 4(c) shows the measured single-shot inter-mode beatnote spectra at $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$ feedback length positions using a resolution bandwidth (RBW) of 500 Hz. It can be found that compared to the linewidth obtained for the laser without feedback (see bottom panel of Fig. 4(c), the linewidths measured for the laser with feedback are slightly larger, which indicate that in the current laser configuration external optical feedback slightly deteriorates the comb linewidth although the comb operation is still maintained. The “max-hold" measurements shown in Fig. 4(d) also illustrate that the laser without feedback has a better “max-hold" linewidth of 49 kHz. While the laser under optical feedback shows “max-hold" linewidths of hundreds of kilohertz. This is in contrast to the observation for a mid-infrared QCL frequency comb reported in Ref. [44] where optical feedback stabilizes the beatnote frequency. The slight deterioration in stability observed in Fig. 4 is likely due to the mechanical vibration of the experimental platform. On the other hand, we did not observe any chaotic oscillations, which commonly occur in the near-infrared laser diodes and significantly broadens the spectral linewidth [45]. Note that in the experiment it is difficult to compare and explain the linewidth differences at the two feedback length positions, i.e., $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$, due to the complex phase turbulence and imperfect optics during the move of the gold mirror.
We further perform the phase noise, amplitude Allan deviation, and frequency Allan deviation measurements for the inter-mode beatnotes as shown in Figs. 5(a), 5(b), and 5(c), respectively. Similar to what we observed in Fig. 4, we find that the laser without feedback shows the best stability in terms of phase noise, amplitude and frequency Allan deviations. The phase noise below 10$^{6}$ Hz in Fig. 5(a) is slightly reduced by external optical feedback. However, the phase noise above 10$^{7}$ Hz is raised by more than 5 dB. Figures 5(b) and 5(c) show that both the amplitude Allan deviation and the frequency Allan deviation with the longer feedback length ($L_{\rm {ext2}}$) are larger than those with the shorter one ($L_{\rm {ext1}}$). It can be because the feedback strengths at the longer feedback length decrease due to the reduced feedback coupling ratio. Consequently, the laser becomes more sensitive to the mechanical vibration of the platform and hence the feedback phase [12]. Although at both $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$ feedback length positions the frequency comb operation is observed under optical feedback, the laser that is configured with a closer feedback mirror ($L_{\rm {ext1}}$) shows a superior stability performance.
Fig. 5. Stability evaluations of the terahertz QCL at $L_{\rm {ext1}}$ and $L_{\rm {ext2}}$. (a), (b), and (c) are measured phase noise (RBW: 10%, tolerance: 10%), amplitude Allan deviation, and frequency Allan deviation of the inter-mode beatnote signals. For reference, the black curves measured from the QCL without feedback are also plotted.
4. Simulation and discussion
In order to understand the experimental observations, we theoretically investigate the effects of optical feedback on terahertz QCLs, using a rate equation approach. The rate equation model takes into account three electronic levels, i.e., upper, lower and bottom levels as shown in Fig. 6. Under a current pumping, carriers in the injector region are firstly injected into the upper laser level. Then, the carriers transit either to the lower laser level with a lifetime of $\tau _{32}$ or to the bottom level with a lifetime of $\tau _{31}$. The stimulated emission occurs between the upper and lower levels. The carriers in the lower laser level relax to the bottom level with a lifetime of $\tau _{21}$, and finally tunnel into the next stage with a time of $\tau _{\rm {out}}$. Note that in order to simplify the simulation of the laser comb, we only consider two coupled longitudinal modes with a free spectral range (FSR) of 6.15 GHz that is the same as we experimentally observed in Fig. 3.
The rate equations describing the carrier dynamics of the terahertz QCL can be written as [24,46–48],
The minor change of the feedback mirror position in the experiment can be translated into the variation of the initial feedback phase in our rate equation model. Therefore, in the simulation, we can alternatively study optical feedback phase effects on the dynamics of the terahertz QCL. Based on the experiment, in the simulation the external cavity length is fixed as $L_{\rm {ext}}$=22 cm which leads to an external cavity frequency of 0.68 GHz; the feedback ratio is set as $r_{\rm {ext}}=- 16$ dB; and the pump current is $I=1.57I_{\rm {th}}$. Figures 7(a) and 7(b) show the calculated electrical and optical spectra, respectively, of the laser for various feedback phase conditions. The results obtained without optical feedback are also shown in the top panels for reference. It can be seen that without feedback, the electrical spectrum in Fig. 7(a) exhibits one peak at 6.15 GHz, which corresponds to inter-mode beatnote frequency of the two longitudinal modes shown in Fig. 7(b). When the QCL is subject to optical feedback with the initial phases $\phi$ between 0 and −0.5$\pi$, both the electrical and optical spectra are almost identical to those of the laser without feedback. Interestingly, for the feedback phases of $\phi _0=-0.7\pi$, $-0.74\pi$, and $-0.76\pi$, the electrical spectra in Fig. 7(a) exhibit multiple side peaks around the main peak, which are also observed in the experiment (see Fig. 3). The existence of the side peaks is due to the beating of the side modes in the optical spectra shown in Fig. 7(b). As the feedback phase is further increased to -$\pi$ or even larger, we see the electrical and optical spectra go back to the situations of $\phi _0=0$ and then repeat the behaviour as we observed for the phases between 0 and -$\pi$. It can be seen that our simulation can reveal the same periodic feature observed in the experiment and the phase period of $\pi$ exactly corresponds to the half-wavelength of 35.5 $\mu$m (see Fig. 3).
Fig. 7. Simulated electrical (a) and optical spectra (b) of the terahertz QCL with optical feedback for feedback phases varying from 0 to -$\pi$. The two top panels show the results of the laser without feedback for reference. In (b), the x-axis is shifted by 4.2 THz for a clear comparison.
In order to understand the nonlinear dynamics, we can analyse the time characteristics of the laser for various feedback phase conditions. Figure 8 shows the calculated time traces for different feedback phases. For $\phi _0=0$, both Mode 1 and Mode 2 produce continuous wave. While, the combined electric field exhibits a sinusoidal wave with a frequency of 6.15 GHz, and hence results in the single peak in the electrical spectrum shown in Fig. 7(a). For the feedback phase of $\phi _0=-0.7\pi$, both modes exhibit sine-wave like oscillations, which is known as period-one oscillations in the aspect of nonlinear dynamics [57]. The oscillation frequency of both modes is 0.43 GHz, which is fundamentally determined by the external cavity frequency (0.68 GHz). The deviation of the oscillation frequency from the external cavity frequency is due to the modulation of optical feedback phase and feedback strength [58]. In contrast, the combined electric field produces quasi-periodic oscillations, due to the phase difference of the two periodic oscillations as well as the difference of the lasing frequency. This quasi-periodic oscillation results in the appearance of the sidebands both in the electrical spectrum and in the optical spectrum in Fig. 7. When the feedback phase is tuned to $\phi _0=-0.74\pi$, the dynamics is similar to the case of $\phi _0=-0.7\pi$. However, the oscillation frequency of both modes is changed to 0.41 GHz. For the feedback phase of $\phi _0=-0.76\pi$, the oscillations of both modes become quasi-periodic, resulting in the complex oscillation of the combined electric field. Consequently, both the electrical spectrum and the optical spectrum exhibit many side modes.
Fig. 8. Simulated time series of the photon number for the terahertz QCL with various optical feedback phases. From top to bottom, the feedback phases are set in sequence as $\phi _0=0$, −0.7$\pi$, −0.74$\pi$ and −0.76$\pi$. The red and blue curves represent the photon number of Mode 1 and Mode 2, respectively. The gray curves denote the photon number of the combined electric field of the two longitudinal modes. In the bottom row, the right panel shows the zoom-in of the traces marked by a pink rectangular box in the left panel.
Although the simulations in Figs. 7 and 8 are based on two longitudinal modes, the conclusion can be generalized to the frequency comb with many equally spaced modes. It is expected that the combined electric field of multiple modes is more complicated than that of only two modes. This is also the reason that we observed much smaller line spacing in Figs. 3(c) and 3(d) ($\sim$20 MHz) for a frequency comb. However, the fundamental mechanism remains the dynamics of each longitudinal mode.
Regarding the external feedback length, in this experiment we demonstrate the effect for relatively short length of tens microns. Note that the variation of the dynamics in Figs. 3 and 7 is due to the laser sensitivity to the feedback phase. Based on the basic theory of optical feedback, the external cavity modes exist for both short-cavity regime (delay time ${\tau }_{\rm {ext}}$ is much shorter than the inverse of resonance frequency 1/$f_{\rm {r}}$, here $f_{\rm {r}}$ is equal to the fundamental inter-mode beatnote frequency of the comb) and long-cavity regime (${\tau }_{\rm {ext}}\gg$1/$f_{\rm {r}}$) [59]. Furthermore, the dynamics is sensitive to the feedback phase for both regimes. Therefore, we believe the dynamics discussed in the article also occur for feedback length of a few meters or even longer. However, the spacing of the beatnote lines is expected to decrease with increasing feedback length, due to the reduced external-cavity frequency.
As we noted in the introduction, in Ref. [21] the authors developed a new method (beatnote spectroscopy) to assess the coherence of the comb based on self-mixing of the signal reflected from an FTIR. As it was claimed, weak optical feedback induced a shift of the beat-note frequency, depending on the feedback strength and feedback phase. However, optical feedback triggered nonlinear effects including the generation of multiple beatnotes, which was not discussed in details since it went beyond the scope of that paper. In contrast, in our work, we focus on the nonlinear dynamics of the QCL frequency comb induced by optical feedback. Particularly, we unveil the effect of feedback phase (through minor change of external mirror) on the nonlinear dynamics. In Ref. [21], the feedback mirror inside the FTIR is continuously displaced so as to obtain the beatnote interferogram, based on which only one power spectrum is derived. In contrast, we never record the beatnote interferogram. Instead, we record one electrical spectrum for each position of the external mirror using the spectrum analyzer, so as to identify the nonlinear dynamics. That is, we obtain multiple electrical spectra for multiple mirror positions. Besides, the mirror position is changed one by one rather than continuously.
It is worth noting that different from the near-infrared and mid-infrared semiconductor lasers that are extremely sensitive to the change of external optical feedback length, terahertz QCLs can maintain the frequency comb operation in most feedback conditions (feedback lengths or phases), which can be seen from both the experiment (Fig. 3) and simulation (Fig. 7). This is because for the terahertz lasers, the wavelength is much longer and the oscillation period is a half wavelength. For a given change of optical feedback length or phase, the terahertz frequency comb operation won’t be destroyed and still demonstrate a relatively high stability. Our work verifies that terahertz QCL frequency combs are naturally less sensitive to the position vibration of the external mirror compared to other laser combs emitting in shorter wavelengths. Note that if we consider the ratio of ${\Delta }L_{\rm {ext}}$/$\lambda$ with $\lambda$ being the laser wavelength, the effect in terahertz QCLs and near-infrared lasers may be similar. However, in a practical experiment, the absolute mechanical sensitivity is important. For example, for a near-infrared laser, when the feedback mirror changes the position by 1 $\mu$m, the laser comb operation can be destroyed. However, for a terahertz QCL, the comb operation can be maintained even when the mirror position is changed by tens of microns (see Figs. 3 and 4).
5. Conclusions
In summary, we have investigated the frequency comb behavior of a terahertz QCL emitting around 4.2 THz with external optical feedback. Experimental results demonstrated that the measured power and inter-mode beatnote showed an oscillation behaviour characterized by a period of half-wavelength, when the optical feedback length is changed. For the present laser feedback configuration, we observed that the comb operation can be obtained at most feedback length positions (63%). Although the comb operation was obtained under optical feedback, the comb linewidth and stability were deteriorated compared to the laser without feedback. To explain the observed phenomenon, a numerical simulation based on a two-mode terahertz QCL rate equation model was carried out to investigate the mode dynamics of the laser under optical feedback. The simulation can fully reproduce the periodic feature of the inter-mode beatnote with varied feedback lengths. The multi-line inter-mode beatnotes originate from complex oscillations of the combined electric field at proper feedback length positions or phases. Furthermore, from both experiment and simulation, we revealed that the terahertz QCL demonstrated frequency comb behaviour in a large range of optical feedback length positions or phases, which indicated that the terahertz QCL was robust against the inevitable position vibration of the feedback mirror in practice, owing to the small feedback phase change.
Funding
National Natural Science Foundation of China (62235019, 61875220, 62035005 61927813, 61991430, 61804095, 62022084); Chinese Academy of Sciences (YJKYYQ20200032, ZDBS-LY-JSC009); Science and Technology Commission of Shanghai Municipality (20XD1424700, 21ZR1474600).
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.
References
1. E. Patzak, H. Olesen, A. Sugimura, S. Saito, and T. Mukai, “Spectral linewidth reduction in semiconductor lasers by an external cavity with weak optical feedback,” Electron. Lett. 19(22), 938–940 (1983). [CrossRef]
2. G. Agrawal, “Line narrowing in a single-mode injection laser due to external optical feedback,” IEEE J. Quantum Electron. 20(5), 468–471 (1984). [CrossRef]
3. H. Olesen, J. Osmundsen, and B. Tromborg, “Nonlinear dynamics and spectral behavior for an external cavity laser,” IEEE J. Quantum Electron. 22(6), 762–773 (1986). [CrossRef]
4. N. Schunk and K. Petermann, “Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback,” IEEE J. Quantum Electron. 24(7), 1242–1247 (1988). [CrossRef]
5. B. Zhao, X. Wang, and C. Wang, “Strong optical feedback stabilized quantum cascade laser,” ACS Photonics 7(5), 1255–1261 (2020). [CrossRef]
6. L. Tombez, S. Schilt, D. Hofstetter, and T. Südmeyer, “Active linewidth-narrowing of a mid-infrared quantum cascade laser without optical reference,” Opt. Lett. 38(23), 5079–5082 (2013). [CrossRef]
7. I. Sergachev, R. Maulini, A. Bismuto, S. Blaser, T. Gresch, Y. Bidaux, A. Müller, S. Schilt, and T. Südmeyer, “All-electrical frequency noise reduction and linewidth narrowing in quantum cascade lasers,” Opt. Lett. 39(22), 6411–6414 (2014). [CrossRef]
8. A. Shehzad, P. Brochard, R. Matthey, T. Südmeyer, and S. Schilt, “10 kHz linewidth mid-infrared quantum cascade laser by stabilization to an optical delay line,” Opt. Lett. 44(14), 3470–3473 (2019). [CrossRef]
9. S. Borri, G. Insero, G. Santambrogio, D. Mazzotti, F. Cappelli, I. Galli, G. Galzerano, M. Marangoni, P. Laporta, V. Di Sarno, S. Luigi, P. Maddaloni, and P. D. Natale, “High-precision molecular spectroscopy in the mid-infrared using quantum cascade lasers,” Appl. Phys. B 125(1), 18 (2019). [CrossRef]
10. Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012). [CrossRef]
11. F. Mezzapesa, L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. Vitiello, H. Beere, D. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21(11), 13748 (2013). [CrossRef]
12. X.-G. Wang, B.-B. Zhao, F. Grillot, and C. Wang, “Spectral linewidth reduction of quantum cascade lasers by strong optical feedback,” J. Appl. Phys. 127(7), 073104 (2020). [CrossRef]
13. L. Jumpertz, F. Michel, R. Pawlus, W. Elsässer, K. Schires, M. Carras, and F. Grillot, “Measurements of the linewidth enhancement factor of mid-infrared quantum cascade lasers by different optical feedback techniques,” AIP Adv. 6(1), 015212 (2016). [CrossRef]
14. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264(5158), 553–556 (1994). [CrossRef]
15. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002). [CrossRef]
16. M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nat. Photonics 9(1), 42–47 (2015). [CrossRef]
17. H. Li, P. Laffaille, D. Gacemi, M. Apfel, C. Sirtori, J. Leonardon, G. Santarelli, M. Rösch, G. Scalari, M. Beck, J. Faist, W. Hänsel, R. Holzwarth, and S. Barbieri, “Dynamics of ultra-broadband terahertz quantum cascade lasers for comb operation,” Opt. Express 23(26), 33270–33294 (2015). [CrossRef]
18. R. Paiella, R. Martini, F. Capasso, C. Gmachl, H. Y. Hwang, D. L. Sivco, J. N. Baillargeon, A. Y. Cho, E. A. Whittaker, and H. Liu, “High-frequency modulation without the relaxation oscillation resonance in quantum cascade lasers,” Appl. Phys. Lett. 79(16), 2526–2528 (2001). [CrossRef]
19. L. H. Li, L. Chen, J. R. Freeman, M. Salih, P. Dean, A. G. Davies, and E. H. Linfield, “Multi-watt high-power THz frequency quantum cascade lasers,” Electron. Lett. 53(12), 799–800 (2017). [CrossRef]
20. L. Columbo and M. Brambilla, “Multimode regimes in quantum cascade lasers with optical feedback,” Opt. Express 22(9), 10105–10118 (2014). [CrossRef]
21. M. Wienold, B. Röben, L. Schrottke, and H. T. Grahn, “Evidence for frequency comb emission from a Fabry-Pérot terahertz quantum-cascade laser,” Opt. Express 22(25), 30410–30424 (2014). [CrossRef]
22. G. Agnew, A. Grier, T. Taimre, Y. L. Lim, K. Bertling, Z. Ikonić, A. Valavanis, P. Dean, J. Cooper, S. P. Khanna, M. Lachab, E. H. Linfield, A. G. Davies, P. Harrion, D. Indjin, and A. D. Rakić, “Model for a pulsed terahertz quantum cascade laser under optical feedback,” Opt. Express 24(18), 20554–20570 (2016). [CrossRef]
23. H.-W. Hübers, H. Richter, R. Eichholz, M. Wienold, K. Biermann, L. Schrottke, and H. T. Grahn, “Heterodyne spectroscopy of frequency instabilities in terahertz quantum-cascade lasers induced by optical feedback,” IEEE J. Sel. Top. Quantum Electron. 23(4), 1–6 (2017). [CrossRef]
24. X. Qi, G. Agnew, T. Taimre, S. Han, Y. L. Lim, K. Bertling, A. Demić, P. Dean, D. Indjin, and A. D. Rakić, “Laser feedback interferometry in multi-mode terahertz quantum cascade lasers,” Opt. Express 28(10), 14246–14262 (2020). [CrossRef]
25. X. Qi, K. Bertling, T. Taimre, G. Agnew, Y. L. Lim, T. Gillespie, A. Robinson, M. Brünig, A. Demić, P. Dean, L. H. Li, E. H. Linfield, A. G. Davies, D. Indjin, and A. D. Rakić, “Observation of optical feedback dynamics in single-mode terahertz quantum cascade lasers: Transient instabilities,” Phys. Rev. A 103(3), 033504 (2021). [CrossRef]
26. 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(6), 462–467 (2014). [CrossRef]
27. Y. Yang, D. Burghoff, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz multiheterodyne spectroscopy using laser frequency combs,” Optica 3(5), 499–502 (2016). [CrossRef]
28. Z. Li, W. Wan, K. Zhou, X. Liao, S. Yang, Z. Fu, J. Cao, and H. Li, “On-chip dual-comb source based on terahertz quantum cascade lasers under microwave double injection,” Phys. Rev. Appl. 12(4), 044068 (2019). [CrossRef]
29. Y. Zhao, Z. Li, K. Zhou, X. Liao, W. Guan, W. Wan, S. Yang, J. Cao, D. Xu, S. Barbieri, and H. Li, “Active stabilization of terahertz semiconductor dual-comb laser sources employing a phase locking technique,” Laser Photonics Rev. 15(4), 2000498 (2021). [CrossRef]
30. H. Li, Z. Li, W. Wan, K. Zhou, X. Liao, S. Yang, C. Wang, J. C. Cao, and H. Zeng, “Toward compact and real-time terahertz dual-comb spectroscopy employing a self-detection scheme,” ACS Photonics 7(1), 49–56 (2020). [CrossRef]
31. L. A. Sterczewski, J. Westberg, Y. Yang, D. Burghoff, J. Reno, Q. Hu, and G. Wysocki, “Terahertz spectroscopy of gas mixtures with dual quantum cascade laser frequency combs,” ACS Photonics 7(5), 1082–1087 (2020). [CrossRef]
32. L. A. Sterczewski, J. Westberg, Y. Yang, D. Burghoff, J. Reno, Q. Hu, and G. Wysocki, “Terahertz hyperspectral imaging with dual chip-scale combs,” Optica 6(6), 766–771 (2019). [CrossRef]
33. P. Gellie, S. Barbieri, J.-F. Lampin, P. Filloux, C. Manquest, C. Sirtori, I. Sagnes, S. P. Khanna, E. H. Linfield, A. G. Davies, H. Beere, and D. Ritchie, “Injection-locking of terahertz quantum cascade lasers up to 35 GHz using RF amplitude modulation,” Opt. Express 18(20), 20799–20816 (2010). [CrossRef]
34. W. Wan, H. Li, T. Zhou, and J. Cao, “Homogeneous spectral spanning of terahertz semiconductor lasers with radio frequency modulation,” Sci. Rep. 7(1), 44109 (2017). [CrossRef]
35. 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(5), 053804 (2008). [CrossRef]
36. P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation nd: Yag microchip laser,” Electron. Lett. 33(6), 491–492 (1997). [CrossRef]
37. G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A: Pure Appl. Opt. 4(6), S283–S294 (2002). [CrossRef]
38. D. Han, M. Wang, and J. Zhou, “Self-mixing speckle in an erbium-doped fiber ring laser and its application to velocity sensing,” IEEE Photonics Technol. Lett. 19(18), 1398–1400 (2007). [CrossRef]
39. X. Dai, M. Wang, Y. Zhao, and J. Zhou, “Self-mixing interference in fiber ring laser and its application for vibration measurement,” Opt. Express 17(19), 16543–16548 (2009). [CrossRef]
40. Y. L. Lim, M. Nikolic, K. Bertling, R. Kliese, and A. D. Rakić, “Self-mixing imaging sensor using a monolithic vcsel array with parallel readout,” Opt. Express 17(7), 5517–5525 (2009). [CrossRef]
41. S. Donati, “Developing self-mixing interferometry for instrumentation and measurements,” Laser Photonics Rev. 6(3), 393–417 (2012). [CrossRef]
42. A. Valavanis, P. Dean, Y. L. Lim, R. Alhathlool, M. Nikolic, R. Kliese, S. P. Khanna, D. Indjin, S. J. Wilson, A. D. Rakic, E. H. Linfield, and G. Davies, “Self-mixing interferometry with terahertz quantum cascade lasers,” IEEE Sens. J. 13(1), 37–43 (2013). [CrossRef]
43. T. Taimre, M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić, “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photonics 7(3), 570–631 (2015). [CrossRef]
44. C. C. Teng, J. Westberg, and G. Wysocki, “Optical-feedback-stabilized quantum cascade laser frequency combs,” Conference on Lasers and Electro-Optics p. STu4N.3 (2019).
45. M. Tlidi, K. Panajotov, M. Ferré, and M. G. Clerc, “Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in kerr optical frequency comb and in all fiber cavities,” Chaos 27(11), 114312 (2017). [CrossRef]
46. T. Gensty and W. Elsäßer, “Semiclassical model for the relative intensity noise of intersubband quantum cascade lasers,” Opt. Commun. 256(1-3), 171–183 (2005). [CrossRef]
47. Y. Petitjean, F. Destic, J.-C. Mollier, and C. Sirtori, “Dynamic modeling of terahertz quantum cascade lasers,” IEEE J. Sel. Top. Quantum Electron. 17(1), 22–29 (2011). [CrossRef]
48. X. Wang, F. Grillot, and C. Wang, “Rate equation modeling of the frequency noise and the intrinsic spectral linewidth in quantum cascade lasers,” Opt. Express 26(3), 2325–2334 (2018). [CrossRef]
49. D. Sukow, T. Heil, I. Fischer, A. Gavrielides, A. Hohl-AbiChedid, and W. Elsäßer, “Picosecond intensity statistics of semiconductor lasers operating in the low-frequency fluctuation regime,” Phys. Rev. A 60(1), 667–673 (1999). [CrossRef]
50. T. Carr, D. Pieroux, and P. Mandel, “Theory of a multimode semiconductor laser with optical feedback,” Phys. Rev. A 63(3), 033817 (2001). [CrossRef]
51. I. Koryukin and P. Mandel, “Dynamics of semiconductor lasers with optical feedback: Comparison of multimode models in the low-frequency fluctuation regime,” Phys. Rev. A 70(5), 053819 (2004). [CrossRef]
52. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
53. C. Otto, B. Globisch, K. Luedge, E. Schoell, and T. Erneux, “Complex dynamics of semiconductor quantum dot lasers subject to delayed optical feedback,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 22(10), 1250246 (2012). [CrossRef]
54. K. Zhou, H. Li, W. Wan, Z. Li, X. Liao, and J. Cao, “Ridge width effect on comb operation in terahertz quantum cascade lasers,” Appl. Phys. Lett. 114(19), 191106 (2019). [CrossRef]
55. P. Tzenov, D. Burghoff, Q. Hu, and C. Jirauschek, “Time domain modeling of terahertz quantum cascade lasers for frequency comb generation,” Opt. Express 24(20), 23232–23247 (2016). [CrossRef]
56. A. Hamadou, S. Lamari, and J.-L. Thobel, “Dynamic modeling of a midinfrared quantum cascade laser,” J. Appl. Phys. 105(9), 093116 (2009). [CrossRef]
57. X. Wang, B. Zhao, Y. Deng, V. Kovanis, and C. Wang, “Nonlinear dynamics of a quantum cascade laser with tilted optical feedback,” Phys. Rev. A 103(2), 023528 (2021). [CrossRef]
58. S.-Y. Ye and J. Ohtsubo, “Experimental investigation of stability enhancement in semiconductor lasers with optical feedback,” Opt. Rev. 5(5), 280–284 (1998). [CrossRef]
59. J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos (Springer, 2013).
