Chaos of interband cascade lasers in the mid-infrared regime

Chaos in nonlinear dynamical systems is featured with irregular appearance and with high sensitivity to initial conditions. Near-infrared semiconductor lasers subject to optical feedback from an external reflector are popular chaotic light sources, which have enabled multiple applications. Here, we report the fully-developed chaos in a mid-infrared interband cascade laser with external optical feedback. The chaos leads to significant electrical power enhancement over a frequency span of 500 MHz. In addition, the laser also exhibits periodic oscillations or low-frequency fluctuations before producing chaos, depending on the operation conditions. This work paves the way for extending chaos investigations from the near-infrared regime to the mid-infrared regime, which can stimulate potential applications in this spectral range.

Chaos is a common phenomenon in numerous nonlinear dynamical systems, with features of random appearance and high sensitivity to initial conditions 1 . Since Haken's prediction of chaos in laser systems in 1975 2 , chaotic oscillations have been observed in various types of lasers, including gas lasers [3][4][5] , solid-state lasers 6 , fiber lasers 7 , and semiconductor lasers 8 . Among these, semiconductor lasers are the most popular testbed for studying chaotic dynamics, owing to their compactness and the ease of control. Most semiconductor lasers belong to Class-B laser systems, where the carrier lifetime is much longer than the photon lifetime 8 . Consequently, the generation of chaos and other nonlinear dynamics requires some external perturbation, such as external optical or optoelectronic feedback, current or loss modulation, as well as optical injection 9 . However, chaos was also observed in a free-running vertical-cavity surface-emitting laser (i.e. without any external perturbation), and the physical origin was attributed to the nonlinear coupling between the two polarized modes in the vertical cavity 10 . In addition, it has been shown that quantum-dot micropillar lasers operated close to the quantum limit also exhibited chaos 11 , and the synchronization of these coupled lasers were reported as well 12 . The extensive and intensive investigations of chaos in semiconductor lasers have enabled the applications in chaotic secure communications 13 , in random number generations 14 , as well as in light detection and ranging (LIDAR) systems 15 . In recent years, chaos also advances the development of artificial intelligence in the field of optical reservoir computing 16,17 .
It is worthwhile to point out that chaos produced from the above semiconductor lasers are mostly operated in the nearinfrared regime, especially in the O-band (1260-1360 nm) and C-band (1530-1565 nm) communication windows of optical fibers.
Very recently, mid-infrared chaos is drawing more and more interests, because it has potential applications in long-reach freespace secure communications and in remote LIDAR systems, owing to the low transmission loss in the atmosphere (3-5 μm and 8-12 μm). In order to produce mid-infrared chaos, a few efforts have been made using quantum cascade lasers (QCLs) as the chaotic light source [18][19][20][21] . However, the QCLs only exhibited low frequency fluctuations (LFFs), while fully-developed chaos has been not observed 22,23 . This is because the carrier lifetime (around one picosecond) of QCLs is shorter than the photon lifetime, and hence resembles Class-A laser systems, which are much more stable than Class-A laser systems 24,25 . However, LFFs are of low dimensionality and the electrical bandwidth is reduced, which significantly limit the applications, and hence fully-developed chaos are preferred. In addition to QCLs, interband cascade lasers (ICLs) emit light in the mid-infrared regime as well 26,27 . Most ICLs are grown on GaSb substrates and the emission wavelengths are usually in range of 3-6 μm. On the other hand, InAs-based ICLs extend the lasing wavelength beyond 10 μm, which are nevertheless less mature than GaSb-based ones [28][29][30] . In addition, the power consumption of ICLs is more than one order of magnitude lower than the QCL counterpart 31 . On the other hand, The stimulated emission of ICLs relies on interband transition of carriers in type-II (mostly) quantum wells, and the carrier lifetime (sub-nanosecond) is longer than the photon lifetime 32,33 . Therefore, ICLs are classified into Class-B laser systems like common semiconductor lasers, and hence are more prone for chaos generation. Here, we report the first demonstration of fullydeveloped chaos in the mid-infrared regime using an ICL. The ICL is perturbed by optical feedback using an external reflection mirror. Beyond a critical feedback level, the ICL produces chaotic oscillations for both low and high pump currents. In addition, the ICL also generates periodic oscillations or LFFs through manipulating the operation conditions.

Results
Experimental setup and laser emission spectra. The ICL under study has an active region of 7 cascading stages of InAs/GaInSb type-II quantum wells, and it is fabricated into a Fabry-Perot laser with a narrow ridge (see Methods). In order to produce chaos, the ICL is perturbed by external optical feedback, which is provided by a gold mirror as illustrated in Fig. 1a. The gold mirror is placed 36 cm away from the laser sample, which results in an external cavity frequency of 417 MHz. Because the light of ICL is transverse electric polarized, the feedback strength can be adjusted by rotating the polarizer. The feedback ratio is defined as the ratio of the mirror reflected power to the laser output power, which is monitored by the power meter. For different feedback ratios, the optical spectra, the time traces, and the electrical spectra are all recorded in the experimental setup (see Methods).
The free-running ICL in Fig. 1b shows four visible longitudinal modes around 3392 nm, at a low pump current (1.09×Ith) slightly above the lasing threshold (Ith =78 mA). When applying optical feedback to the ICL, the number of longitudinal mode increases. This is because the optical feedback reduces the lasing threshold, and hence more modes surpasses the threshold (see Methods) 8 .
Although chaos can broaden the spectral linewidth of the longitudinal modes, it is hard to be identified from the measured spectra due to the limited resolution (0.1 nm) of the optical spectrum analyzer. Both the time trace and the corresponding phase portrait in Fig. 2b prove that the dynamics is period-one oscillation, which shows a single period in the time trace and one cycle in the phase portrait. The physical origin of period-one oscillation is that the relaxation oscillation is undamped by optical feedback via the Hopf bifurcation [34][35][36][37] . Period-one oscillations have been widely investigated in near-infrared semiconductor lasers, which are high-quality photonic microwave sources for applications in radioover-fiber communications. When the feedback strength exceeds the critical feedback level, which quantifies the onset of fullydeveloped chaos, the ICL begins to produce chaotic oscillations. The critical feedback level of the ICL is measured to be around - Further raising the feedback level reduces the Lyapunov exponent down to 1.6 /ns at the feedback ratio of -4.2 dB. The reduction of the Lyapunov exponent suggests that the laser system may evolve into LFFs or steady state for feedback ratios beyond -4.0 dB 44 . It is worthwhile to mention that these Lyapunov exponents are more than three orders of magnitude larger than those of LFFs in QCLs 45 . Consequently, the fully-developed chaos in the ICL is much more sensitive to the initial conditions, and much less predictable than the low-dimensional LFFs in QCLs. Frequency domain characteristics. In Fig. 4a, the free-running ICL shows a smooth electrical spectrum except the low-frequency part below 100 MHz, where the noisy spikes are mainly due to the technical noise sources (current source noise, thermal noise, and mechanical noise) as well as the mode partition noise 46,47 . The spectrum does not show any relaxation resonance peak, suggesting that the ICL is strongly damped. This observation is consistent with the measured intensity modulation responses 48-50 , where no resonance peak occurs either. Therefore, ICLs resemble quantum dot lasers, the relaxation oscillations of which are usually over-damped 51,52 . When optical feedback is applied to the ICL with a feedback ratio of -12.7 dB, a small peak appears around 168 MHz, which determines the oscillation period of the corresponding time trace in Fig. 2a. The peak frequency is much smaller than the external cavity frequency of 417 MHz, and therefore the peak must be due to the weakly damped relaxation oscillations. It is known that the oscillation frequency of semiconductor lasers with optical feedback varies around the resonance frequency of the free-running laser 8 . Therefore, the resonance frequency of the ICL is deduced to be around 168 MHz, which is smaller than the common GHz-level resonance of near-infrared semiconductor lasers. This is because the tested ICL is not designed for high-speed operations, such as the long cavity length as well as the long carrier transport time across the thick separate confinement layer, which limit the relaxation oscillation frequency 46 . However, ICLs can reach GHz-level resonance through proper optimizations [48][49][50] . Increasing the feedback level to -9.4 dB, the ICL exhibits a typical period-one oscillation at 155 MHz, and the oscillation peak amplitude is about 40 dB higher than the background noise level. At feedback ratios of -7.9 dB and -4.2 dB, the ICL produces chaotic oscillations and the electrical power levels are substantially raised in a broad frequency range, up to the bandwidth limit (450 MHz) of the photodetector. The map in Fig. 4b displays the evolution of the electrical power distribution with feedback strengths. It is shown that the frequency of the weak relaxation oscillation slightly increases with increasing feedback level from -16 to -10 dB. At the onset of period-one oscillation (around -10 dB), the oscillation frequency abruptly shifts to a slightly smaller value. Beyond the critical feedback level of -8 dB, the ICL exhibits chaotic oscillations within a wide feedback strength window, up to the feedback limit (-4.2 dB) of the experimental configuration. In order to quantify the bandwidth of the chaotic signals, we employ the definition that the frequency span from DC to the frequency where 80% of the total power is contained within the power spectrum 53 . Using this definition, Fig. 4c demonstrates that the chaos bandwidth (dots) firstly declines and then rises with the increasing feedback ratio. The maximum chaotic bandwidth is 269 MHz, which is reached right above the critical feedback level. The chaos bandwidth of the ICL is much higher than the LFF bandwidth (less than 50 MHz) in QCLs 18,21 , which is very beneficial for high-speed applications. In addition, Fig. 4c proves that the chaos significantly raises the intensity noise (stars) of the ICL by more than 15 dB once beyond the critical feedback level. Besides, the noise level climbs up with increasing feedback strength.
Chaos at a high pump current. In the above analysis, the ICL is operated near the lasing threshold. However, we find that the ICL produces chaos when it is operated well above threshold as well. When it is pumped at 1.35×Ith, the bifurcation diagram in Fig. 5a and the electric power distribution map in Fig. 5b demonstrate that the ICL does not show any periodic oscillations, which is different to the near-threshold case. Instead, the ICL produces LFFs before transferring to the regime of fully-developed chaos.
Consequently, the ICL undergoes the LFF route to chaos 22,23,54 . For feedback levels in range of -16 to -14 dB (example of -14.3 dB) in Fig. 5d, the multimode hopping slightly raises the noise level at frequencies below 200 MHz, which leads to the weak fluctuations in the corresponding time trace in Fig. 5c. In the feedback range of -14 to -8 dB (example of -11.9 dB), the ICL produces LFFs, which are confirmed by the statistical analysis (see Supplementary Note Two). The LFFs in Fig. 5c show irregular power jump-ups with gradual power increase and following drastic power decrease. This is in contrast to typical LFFs observed in common semiconductor lasers, which exhibits random power dropouts with sudden power decrease followed by gradual power recovery 22,23 . However, LFFs with power jump-ups have been indeed observed in semiconductor lasers biased well above threshold 55 . In addition, we also observe coexistence of power jump-ups and dropouts in the experiment (not shown). The LFF leads to the power enhancement of the electrical spectra in Figs. 5b, d, and the bandwidth broadens with rising feedback level.
Besides, a dominate peak appears around the resonance frequency of the ICL, and the peak frequency increases with the feedback level as well. Further increasing the feedback level to the range of -8 to -6 dB (example of -7.1 dB), both the LFF and the chaos coexists in the time trace of Fig. 5c, and the bandwidth of the power spectrum in Fig. 5d further broadens. The dynamics is finally evolved into fully-developed chaos when the feedback level surpasses -6 dB, and an exampled time trace at -4.2 dB is displayed in Fig. 5c. As illustrated in Figs. 5b, d, the chaos bandwidth is higher than that of LFFs. Detailed comparisons of the chaos and the LFFs are discussed in Supplementary Note Two.

Discussion
We have reported the fully-developed chaos generation from a mid-infrared ICL subject to external optical feedback. The ICL produces chaos when the feedback strength surpasses the critical feedback level around -8 to -6 dB. The chaos bandwidth is more than 200 MHz, whereas the electrical power spectra are significantly raised over a frequency span of 500 MHz. Before producing chaos, the ICL exhibits periodic oscillations when it is operated close to the threshold, while exhibits LFFs when operated well above threshold.
In comparison with common quantum well lasers, the ICL is more stable against optical feedback and its critical feedback level is more than 20 dB higher than the former ones 56,57 . On the other hand, the critical feedback level of the tested ICL is comparable to those of quantum dot lasers 58,59 . The critical feedback level of semiconductor lasers are primarily determined by the linewidth broadening factor (α-factor) and the damping factor 60 . Therefore, the strong resistance to optical feedback of the ICL can be attributed to its small α-factor (2.2) as well as its large damping factor 48,61 , which are worthy of more physical investigations. On the other hand, the chaos bandwidth of the ICL is smaller than the common GHz bandwidth of near-infrared lasers 62 . Because the chaotic bandwidth is limited by the relaxation resonance frequency of the ICL, it can be improved by highspeed design and the bandwidth is expected to reach the GHz level in future work.
In comparison with its QCL counterpart, the ICL owns a few advantages for the production of chaos. Firstly, the ICL is proved to show fully-developed chaos in this work, whereas QCLs can only generate low-dimensional LFFs 18,21 . Secondly, the chaos bandwidth of the ICL is much broader than the LFF bandwidth of QCLs, which is very favorable for high-speed applications.
Thirdly, the generation of LFFs in QCLs requires critical operation conditions, and the LFFs only exist in a narrow parameter window 18,21 . In contrast, chaos in ICL appears for a broad range of feedback strengths, and for both low and high pump currents.
Consequently, we strongly believe that this work can stimulate extensive investigations of chaos in the mid-infrared regime. Its potential applications in long-reach secure free-space communications and in remote chaotic LIDAR systems may emerge in the future.

Methods
ICL sample. The ICL sample was grown on a GaSb substrate by solid source molecular beam epitaxy. The laser consists of 7 cascading gain stages, which are formed by W-shape InAs/GaInSb type-II quantum wells. The laser has a ridge waveguide with a ridge width of 9.0 μm and a cavity length of 1.5 mm. Both laser facets are as-cleaved without any coatings. At an operation temperature of 20 °C, the lasing threshold is Ith=78 mA. When the ICL is subject to optical feedback with a feedback ratio of -4.2 dB, the threshold is reduced down to 73 mA. The linewidth broadening factor of the ICL operated above threshold was measured to be around 2.2 61 .
Experimental setup. The ICL sample is mounted on a heat sink and its temperature is maintained at 20 °C by using a thermos-electric controller. The pump current is supplied by a low-noise battery current source (LDX-3620B). The laser output is collimated by an aspherical lens with a focal length of 4.0 mm. The light is split into two paths by a beam splitter (BS1). One path provides the optical feedback through a gold mirror, which is placed 36 cm away from the laser sample.
The feedback strength is adjusted by rotating the polarizer, and the feedback power is monitored by a power meter. Data availability. All data supporting this study are available from the corresponding author upon request.