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Abstract
This work presents the nonlinear dynamics of a quantum cascade laser subject to optical injection. Within the stable locking regime, the optical power shows a hysteresis behavior as a function of the detuning frequency. Outside the stable locking regime, the laser mostly produces periodic oscillations. However, the laser pumped at a high pump current also generates spiking pulsations with uniform amplitude, which occur in the vicinity of the negative locking boundary.
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1. Introduction
Semiconductor lasers subject to optical injection usually produce rich dynamics, including stabilities, periodic oscillations, quasi-periodic oscillations, and chaotic oscillations [1]. The stable locking regime is bounded by the Hopf bifurcation at the positive detuning side and the saddle-node bifurcation at the negative detuning side. Within the locking regime, the lasing frequency and phase of the slave laser (SL) are synchronized with those of the master laser (ML). The optical injection improves a number of laser performances, such as raising the optical power [2], enhancing the modulation bandwidth while reducing the frequency chirp [3,4], as well as decreasing the relative intensity noise and the frequency noise [5,6]. Outside the locking regime, chaos has been widely studied, and has been found possible applications in chaotic Lidar [7], random bit generation [8], and secure communication systems [9]. In addition, periodic oscillations with one period (period-one oscillations, P1) have been demonstrated to be high-quality photonic microwave sources and hence have potential applications in 6G optical communication networks [10–12].
While most investigations of optical injection are based on near-infrared laser diodes, nonlinear dynamics of mid-infrared quantum cascade lasers (QCLs) excited by optical injection are attracting more and more interests in recent years. QCLs have an ultra-short carrier lifetime around 1.0 ps, which is smaller than the photon lifetime. Therefore, the nonlinear behaviors are expected to resemble those of class-A lasers rather than common class-B laser diodes [13]. The Hopf bifurcation and the saddle-node bifurcation of QCLs were found to be different to those of near-infrared laser diodes, especially in the weak injection regime [14–16]. Within the stable regime, optical injection enhances the modulation bandwidth [14,17], reduces both the intensity noise [18,19] and the frequency noise [20], which are similar to those of common laser diodes. In addition, optical injection has been used to synchronize QCLs with near-infrared optical frequency combs, which significantly reduced the optical linewidth of QCLs [21,22]. On the other hand, optical injection is also helpful to improve the performance of frequency combs of QCLs [23,24]. Outside the stable locking regime, our previous theoretical study showed that the QCLs mostly produced P1 oscillations, while quasi-periodic oscillations and chaotic oscillations were not observed without including optical noise [25,26]. The rare occurrence of complex nonlinear dynamics arises from the intrinsic stability against external perturbations, which is also extensively demonstrated in QCLs subject to optical feedback [27,28]. On the other hand, QCLs with optical feedback produce low frequency fluctuations and extreme pulses under specific operation conditions [29,30]. Interestingly, transmissions of chaos and giant pulses from the master QCL to the slave one were demonstrated recently [31,32]. This work experimentally investigates the nonlinear behavior of a QCL both within and out of the stable locking regime. It is found that the locking range for increasing the detuning frequency is broader than that for decreasing the detuning frequency. Within the locking regime, the optical power shows a clear hysteresis behavior with respect to the detuning frequency. Outside the locking regime, the QCL mostly produces periodic oscillations. However, the QCL also produces spiking pulsations in the vicinity of the negative locking boundary, when the pump current is operated at a high level.
2. Experimental setup
Figure 1 shows the experimental setup for the QCL subject to optical injection. A commercial single-mode distributed feedback QCL (Thorlabs) is used as the ML. The laser output is collimated by an aspherical lens with a focal length of 4.0 mm. The polarization of the laser is adjusted by a half-wave plate, which tunes the injection strength in combination with the polarization-dependent isolator (Thorlabs, I4500W4). Through a beam splitter (BS1), one branch is injected into the SL, which is also a distributed feedback QCL (Thorlabs). The other branch monitors the injection power of the ML using a power meter. The optical spectrum is measured by a high-resolution (0.08 /cm) Fourier transform infrared spectrometer (FTIR, Bruker Vertex 80). The optical signal is converted to the electrical one using a HgCdTe photodetector with a bandwidth of 560 MHz (PD, Vigo). The temporal waveform is recorded on a digital oscilloscope (OSC) with a bandwidth of 6.0 GHz, and the sampling rate is fixed at 20 GSample/s. The electrical spectrum is recorded on an electrical spectrum analyzer (ESA) with a bandwidth of 50 GHz, and the resolution bandwidth is fixed at 500 kHz. Both QCLs are pumped by continuous-wave current sources (Newport, LDC-3736). The operation temperature is maintained at 20 °C using thermoelectric coolers.
Fig. 1. Experimental setup. BS: Beam splitter; PD: photodetector; FTIR: Fourier transform infrared spectrometer; ESA: Electrical spectrum analyzer; OSC: Oscilloscope.
As shown in Fig. 2(a), the free-running ML exhibits a lasing threshold of Imth = 385 mA, and the maximum output power is around 50 mW. The free-running SL exhibits a lasing threshold of Isth = 425 mA, and the maximum output power is around 30 mW. Both lasers emit a single mode around 2182.5 /cm, when operated above the lasing threshold. Figure 2(b) shows that the lasing wavelength of the ML shifts from 2182.95 /cm at 410 mA down to 2180.88 /cm at 490 mA due to the Joule heating effect. The resulted frequency tunability of the ML is -775.1 MHz/mA, with an uncertainty of 6.3 MHz/mA. Then, the lasing peak frequency of the ML at each pump current is derived from the frequency tunability. In the optical injection setup, the detuning frequency (Δf) is defined as the lasing frequency difference between the ML and the SL, which is varied by changing the pump current of the ML. It is noted that the accuracy of the detuning frequency is limited by the resolution of the FTIR and the uncertainty of the frequency tunability. The injection ratio (Rinj) is defined as the power ratio of the ML to the free-running SL, which is varied by tuning the half-wave plate in Fig. 1. It is noted that the power of the free-running SL is measured by placing the power meter right after the collimator of the SL.
Fig. 2. (a) L-I curves of the free-running master laser and slave laser. (b) Optical spectra of the master laser for several pump currents.
3. Experimental results and discussion
3.1 Stable locking regime
When the SL is pumped at 1.02×Isth (435 mA), the lasing wavenumber is 2182.51 /cm. The ML exhibits the same lasing wavenumber when the pump current is operated at 1.11×Imth (428 mA), which corresponds to the situation of zero detuning frequency (Δf = 0 GHz) in Fig. 3(a). At zero detuning frequency, the maximum injection ratio is Rinj = 4.15 dB, which is achieved when the polarization of the light is well aligned with that of the optical isolator. When varying the detuning frequency from the negative side to the positive side through changing the pump current of the ML, the SL is stably locked from Δf=−0.85 GHz up to Δf=+1.42 GHz, leading to a full locking range of 2.27 GHz. Within the locking regime, the optical power of the locked laser increases almost linearly from 1.8 mW up to 8.9 mW. In contrast, the SL is locked to the ML from Δf=+0.10 GHz down to Δf=−0.89 GHz, when the detuning frequency is operated from the positive side to the negative one. Therefore, the locking range shrinks to 0.99 GHz. Meanwhile, the optical power decreases from 5.3 mW down to 1.8 mW. Consequently, the optical power of the injection-locked QCL exhibits a clear hysteresis phenomenon as a function of the detuning frequency. This hysteresis behavior is due to the fact that the optical injection reduces the gain and in turn raises the refractive index of the SL [11,33,34]. As a result, the lasing mode of the SL is shifted to the red (smaller frequency) side. Thus, the negative-detuning ML is more prone to capture and lock the SL than the positive-detuning one. It is noted that the injection ratio has little variation during the frequency detuning operation, because the change of the pump current is less than 3 mA within the stable locking range. On the other hand, the injection ratio is varied by rotating the half-wave plate in the experimental setup. Figure 3(b) shows the effect of the injection strength on the locking range. For the operation of increasing detuning frequency, the locking range rises from 0.66 GHz at Rinj=−8.3 dB to 2.27 GHz at Rinj = 4.15 dB. In comparison, the locking range becomes 0.15 GHz at Rinj=−8.3 dB and 0.99 GHz at Rinj = 4.15 dB, for the operation of decreasing detuning frequency. Consequently, for practical applications desiring a broad locking range, the frequency detuning should be operated from the negative side to the positive side rather than the opposite direction.
Fig. 3. (a) Optical power of the injection-locked QCL as a function of the detuning frequency. The injection ratio is 4.15 dB, and the pump current is 435 mA. (b) Stable locking range as a function of the injection ratio. Increasing or decreasing the detuning frequency results in different locking range.
3.2 Periodic oscillations
Outside the stable-locking regime, we investigate the nonlinear dynamics through decreasing the detuning frequency, since its stable regime is narrow as described in Fig. 3. It is found that the QCL with optical injection mostly produces P1 oscillations, which is in agreement with our previous theoretical study in [25,26]. The time series in Fig. 4(a) shows that the QCL produces typical P1 oscillations for detuning frequencies both out of the Hopf bifurcation (Δf=+0.62, +0.18 GHz) and out of the saddle-node bifurcation (Δf=−0.93, −1.07 GHz). The corresponding electrical spectra in Fig. 4(b) present that the P1 oscillation frequency varies with the detuning frequency. The oscillation peak is noisy and the electrical linewidth is on the order of tens of megahertz. This broad electrical linewidth is determined by the optical linewidth of both QCLs (< 25 MHz), the lasing frequency stability, as well as the phase coherence between the two lasers [35]. However, the electrical linewidth can be narrowed through employing optoelectronic feedback, external modulation, or dual-loop optical feedback, in case narrow linewidth is demanded [10,36]. Figure 4(c) illustrates the map of the electrical spectra as functions of the detuning frequency and the Fourier frequency. Within the locking regime (from +0.10 GHz to −0.89 GHz), the electrical spectra do not exhibit any sharp oscillation peak. Outside the locking regime, the P1 oscillation frequency increases when the frequency of the ML is detuned away from the SL. Interestingly, the variation of the P1 oscillation frequency versus the detuning frequency in Fig. 4(d) is asymmetric. Generally, the oscillation frequency is smaller than the detuning frequency at the negative detuning side, while larger than the latter at the positive detuning side. Again, this is due to the mechanism that the optical injection raises the refractive index and hence red shifts the lasing mode of the SL [33].
Fig. 4. Period-one oscillations. (a) Time traces; (b) electrical spectra; (c) map of electrical spectra; (d) oscillation frequency versus detuning frequency. The dynamics is tracked by decreasing the detuning frequency. The injection ratio is 4.15 dB, and the pump current is 435 mA.
When the pump current of the SL is increased to 1.08×Isth (460 mA), the lasing wavenumber becomes 2181.87 /cm. The ML exhibits the same wavenumber at the pump current of 1.18×Imth (453.3 mA). When the waveplate is well aligned with the polarization of the ML, the maximum injection ratio becomes Rinj = 0.55 dB. The SL at this pump current also exhibits stable locking regime and P1 oscillations as those described in Fig. 3 and Fig. 4. In addition, the QCL produces periodic pulse oscillations in the vicinity of the saddle-node bifurcation with Δfsn=−0.39 GHz, which are observed only when increasing the detuning frequency. The Hopf bifurcation at this pump current occurs at the detuning frequency of Δfhp=+1.0 GHz. The time series in Fig. 5(a) shows that the pulse durations vary with the detuning frequency. At the detuning frequency of Δf=−0.94 GHz, several ripples appear in each oscillation pulse. The electrical spectra in Fig. 5(b) show that the fundamental frequencies for all the three detunings are around 17.9 MHz, corresponding to an oscillation period of 55.9 ns. The electrical spectra show multiple harmonic peaks due to the pulse distortion from the sinusoidal waveform. In addition, the noise levels of the pulse oscillations are significantly higher than the noise level of continuous-wave emission (Δf=+0.01 GHz), which suggests that the oscillation of the pulses is unstable.
Fig. 5. Pulse oscillations. (a) Time traces and (b) electrical spectra for several detuning frequencies. The dynamics is tracked by increasing the detuning frequency. The injection ratio is 0.55 dB, and the pump current is 460 mA.
3.3 Spiking pulsations
When further increase the pump current of SL to 1.13×Isth (480 mA), the lasing wavenumber becomes 2181.33 /cm. Operating the ML at 1.23×Imth (473.9 mA) leads to the zero frequency detuning condition. When the waveplate is well aligned with the polarization of the ML, the maximum injection ratio becomes Rinj= −0.24 dB. At this high pump current, the QCL generates stable regime and periodic oscillations as well. In addition, the QCL produces aperiodic spiking pulsations in the vicinity of the saddle-node bifurcation with Δfsn=+0.14 GHz, which are observed only when increasing the detuning frequency. The Hopf bifurcation at this pump current occurs at the detuning frequency of Δfhp=+1.0 GHz. As shown in Fig. 6(a), reducing the detuning frequency from the saddle-node bifurcation substantially raises the occurrence probability of the spikes. It is stressed that the time spacing between adjacent spikes is aperiodic. In addition, each group of spikes may contain one individual spike (such as Δf=+0.05, −0.03, −0.11 GHz), two successive spikes (such as Δf= −0.03, −0.11 GHz), and three or more successive spikes (Δf= −0.11, −0.19 GHz). Interestingly, the amplitudes of the spikes for all the detuning frequencies are highly uniform. Figure 6(b) shows that the electrical power levels of the spiking pulsations are higher than that of the free-running QCL, and the power level goes up with the increasing occurrence probability of spikes in Fig. 6(a). The raised power level of the electrical spectra unveils the aperiodic nature of the spiking pulsations. Spiking dynamics have been widely observed in near-infrared laser diodes with either optical injection or optical feedback, which is also known as the excitability phenomenon [37–39]. However, the amplitude of spikes in near-infrared lasers is much less uniform than that in QCLs, and the reason is yet unknown. The underlying physical mechanism of spikes is attributed to the noise triggered excitable pulses in the vicinity of the saddle-node bifurcation on a limit cycle [37]. The stable limit cycle undergoes a homoclinic bifurcation, leading to a pair of stable and unstable states connected by long and short heteroclines. With the perturbation of noise, the laser jumps from the stable state to the vicinity of the unstable one, resulting in the generation of spiking pulsations. Precise control of the occurrence of spikes or excitable pulses has been found to be useful for applications in spiking neural networks in recent years [40–42].
Fig. 6. Spiking pulsations. (a) Time traces and (b) electrical spectra for several detuning frequencies. The dynamics is tracked by increasing the detuning frequency. The injection ratio is −0.24 dB, and the pump current is 480 mA.
3.4 Discussion
Figures 3–6 have shed some light on the nonlinear dynamics of QCLs with optical injection. We remark that while P1 oscillations appeared for both increasing and decreasing the detuning frequency, pulse oscillations and spiking oscillations were observed only when increasing the detuning frequency. The reason is likely due to the frequency pulling effect, which requires theoretical confirmation in future work. Secondly, the maximum operation current of the SL is limited to 1.13×Isth in this work, due to the power saturation effect shown in Fig. 2(a), the frequency matching and the injection strength requirements of the ML. Investigation of nonlinear dynamics far above the lasing threshold demands a QCL with weak saturation effect as well as a widely tunable external cavity ML instead of a distributed feedback laser. External cavity MLs with independent control of injection power and detuning frequency can enable systematical mapping of the locking boundaries and nonlinear dynamics at various pump currents. In addition, the narrow bandwidth of the photodetector used in this work prevents the exploration of high-frequency nonlinear dynamics. While the cutoff bandwidth of most commercial HgCdTe photodetectors is limited up to be around 1.0 GHz, future investigation of high-frequency dynamics demands advanced mid-infrared photodetectors with broad bandwidth, like quantum well infrared photodetectors [43].
4. Conclusion
In conclusion, this work unveiled the nonlinear dynamics of a QCL subject to optical injection both inside and outside the stable locking regime. Inside the locking regime, the optical power shows a hysteresis behavior when decreasing and increasing the detuning frequency between the ML and SL. The stable locking range for the operation of raising the detuning frequency is broader than the opposite operation. Outside the locking regime, the QCL mainly produces periodic oscillations both for positive and for negative detunings. However, for the QCL pumped at a high current, aperiodic spiking pulsations appear in the vicinity of the saddle-node bifurcation, which exhibit highly uniform amplitudes. Chaotic oscillations are not observed in the QCL with optical injection, which is unlike QCLs with optical feedback. Future work will investigate detailed characteristics and mechanisms of spiking pulsations as well as its control methods.
Funding
Natural Science Foundation of Shanghai (20ZR1436500).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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