Open Access
Add to BibSonomy
Abstract
This work reports the nonlinear dynamics of a mid-infrared interband cascade laser (ICL) subject to optical injection. It is shown that the stable locking regime is asymmetric and broadens with increasing injection strength. Outside the locking regime, the ICL mostly produces period-one oscillations. However, three categories of periodic pulse oscillations are observed in the vicinity of the Hopf bifurcation and the saddle-node bifurcation. In particular, it is found that the ICL generates broadband chaos at a near-threshold pump current, and the chaos bandwidth is over 300 MHz.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Semiconductor lasers subject to optical injection usually exhibit rich nonlinear dynamics [1]. With optical injection, the light of a single-mode master laser (ML) uni-directionally injects into a slave laser (SL). The nonlinear dynamics are primarily dependent on the injection ratio and the detuning frequency. The injection ratio is defined as the power ratio of the ML to the SL, and the detuning frequency is the lasing frequency difference between both lasers. When the detuning frequency is small, the laser is operated in the stable locking regime, which is bounded by the Hopf bifurcation at the positive detuning side and the saddle-node bifurcation at the negative detuning side [1]. Within the locking regime, the electric field of the SL synchronizes with that of the ML and produces continuous wave. Meanwhile, a few dynamical characteristics of the SL are substantially improved. For instance, the modulation bandwidth is raised [2], the frequency chirp is reduced [3], both the relative intensity noise and the phase noise are suppressed [4,5]. Outside the locking regime, the laser produces various nonlinear pulse oscillations, including periodic oscillations, quasi-periodic oscillations, and chaotic oscillations [6]. Among these, chaotic oscillations have been extensively investigated, and have been applied in secure communication [7,8], chaos lidar [9,10], as well as physical random number generation [11,12]. In addition, period-one (P1) oscillations have been intensively studied, which serve as high-quality photonic microwave sources and can be used in radio-over-fiber communications [13–15].
Most investigations of nonlinear dynamics are based on near-infrared quantum well lasers (QWLs). Our recent work demonstrated that a mid-infrared quantum cascade laser (QCL) with optical injection produced both periodic oscillations and spiking pulsations [16,17]. In comparison with QCLs, the laser emission of interband cascade lasers (ICLs) is based on the interband transition of type-II quantum wells [18,19]. In addition, five to ten cascading gain stages are usually deployed in the active region of a single laser device [20]. The lasing wavelength of ICLs usually ranges from 3 µm up to more than 10 µm [21–23]. The carrier lifetime of ICLs is on the order of sub-nanosecond, which suggests that ICLs are similar to the common class-B semiconductor lasers [24,25]. It is well established that the linewidth broadening factor (also known as α factor) plays a crucial role in determining the nonlinear dynamics of semiconductor lasers. We have shown that the α factor of ICLs was in the range of 1.1 to 2.2 [26], which was smaller than that (3.0-5.0) of near-infrared semiconductor lasers [27]. Moreover, ICLs are usually strongly damped, leading to the absence of resonance peak in the modulation response [28–30]. Our previous work has shown that ICLs with the perturbation of optical feedback produced periodic oscillations, low frequency fluctuations, as well as broadband chaotic oscillations [31]. Furthermore, the onset of chaos (critical feedback level) of ICLs is higher than -10 dB, which is one to two orders of magnitude higher than that of near-infrared semiconductor lasers [32]. In particular, the chaos bandwidth of ICLs is over 450 MHz, which is one to two orders of magnitude broader than that of QCLs [33]. Recently, it was shown that ICLs with the perturbation of optoelectronic feedback produced periodic oscillations and burst oscillations, without the observation of chaotic oscillations [34]. This work aims to unveil and understand the nonlinear dynamics of an ICL with the perturbation of optical injection, especially those different to the common near-infrared QWLs. The measurement results show that the stable locking range generally broadens with increasing injection ratio, from 0.38 GHz at the injection ratio of -12.5 dB up to 2.66 GHz at 2.9 dB. Outside the locking regime, the laser mostly produces P1 oscillations. On the other hand, three types of periodic pulse oscillations are observed in the vicinity of the Hopf bifurcation and the saddle-node bifurcation. Interestingly, we find that the ICL produces broadband chaotic oscillations at a near-threshold pump current, and the chaos bandwidth is as high as 318 MHz. In addition, theoretical analysis based on a rate equation model with optical injection verifies the above experimental observations. We believe that this work is able to enrich the knowledge of nonlinear dynamics of semiconductor lasers. From the viewpoint of engineering applications, the P1 oscillations may be used in the mid-infrared modulation spectroscopy of gas molecules [35]. The chaotic oscillations are helpful for developing remote secure communication links and chaos lidar systems, taking advantage of the low-loss transmission windows of the atmosphere in the mid-infrared regime (3-5 µm and 8-12 µm) [36].
2. Experimental setup and results
Figure 1 shows the experimental setup for the ICL subject to optical injection. A commercial distributed feedback ICL (Nanoplus) is used as the ML, which emits a single mode around 3390 nm. The output light is collimated by an aspheric lens with a focal length of 4.0 mm. The unidirectional optical injection is warranted by a polarization-dependent isolator. Between the isolator and the collimating lens, a half-wave plate is employed to adjust the injection strength, through changing the direction of linear polarization. The light is divided into two branches via a beam splitter (BS1). One branch is injected into the SL, which is a distributed feedback ICL (Nanoplus) as well. The other branch monitors the ML’s injection power through using a power meter. The detuning frequency between both lasers is tuned by varying the pump current of the ML. The optical spectrum is measured by a Fourier transform infrared spectrometer (FTIR, Bruker) with a resolution of 0.08 /cm. The optical signal is converted to the electrical domain by a HgCdTe photodetector (PD, Vigo) with a detection bandwidth of 600 MHz. The temporal waveform is recorded on a digital oscilloscope (OSC, 59 GHz bandwidth), and the sampling rate is set at 20 GSample/s. Meanwhile, the electrical spectrum is measured by an electrical spectrum analyzer (ESA, 50 GHz bandwidth), and the resolution bandwidth is set at 500 kHz.
Fig. 1. Experimental setup. BS: beam splitter; PD: photodetector; FTIR: Fourier transform infrared spectrometer; OSC: oscilloscope; ESA: electrical spectrum analyzer. Both ICLs emit around 3390 nm.
In the experiment, both the free-running ML and the free-running SL are operated in the continuous-wave mode. The operation temperature of both lasers is controlled by the thermo-electric coolers. The temperatures of the ML and the SL are maintained at 15 °C and 22 °C, respectively. As shown in Fig. 2(a), the lasing threshold and the slope efficiency of the ML are Ithm = 18.0 mA and 0.26 mW/mA, respectively. By contrast, the lasing threshold and the slope efficiency of the SL are Iths = 20.6 mA and 0.22 mW/mA, respectively. The maximum output powers of both lasers are over 13 mW. In the experiment, the detuning frequency is finely adjusted through changing the pump current of the ML. Figure 2(b) shows that the optical wavelength of the ML red shifts from 3390.7 nm to 3393.6 nm when increasing the pump current from 40 mA to 80 mA. Consequently, the frequency tunability of the ML is about -1.9 GHz/mA. It is remarked that changing the pump current of the ML not only alters the detuning frequency but also the injection ratio. However, the maximum applied frequency tuning range in this work is less than 3.8 GHz, and hence the corresponding power change is less than 0.5 mW. To clarify the definition, the injection ratio in the measurement refers to the one at the detuning frequency of zero.
When the SL is pumped at 1.5×Iths(30 mA), the free-running output power is 2.2 mW and the lasing wavelength is 3392.4 nm. Figure 3(a) illustrates the injection-locking diagram formed by the injection ratio Rinj and the detuning frequency Δf. 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. At the Hopf bifurcation, the laser varies gradually from the state of weakly damped relaxation oscillation to the state of the undamped relaxation oscillation [6]. At the saddle-node bifurcation, the laser changes abruptly from the static state to the periodic state. The locking regime is asymmetric due to the non-zero α factor [26], which is similar to the case of common near-infrared semiconductor lasers [37]. On one hand, the stable locking regime broadens with increasing α factor. On the other hand, both the Hopf bifurcation and the saddle-node bifurcation shift towards the negative frequency detuning side. In case the α factor is zero, the Hopf bifurcation and the saddle-node bifurcation become symmetric with respect to the zero detuning frequency. Inside the stable locking regime, the injection-locked ICL exhibits a single lasing mode and produces continuous wave. For each injection ratio, the stable locking range is obtained from the difference of the detuning frequencies between the Hopf bifurcation and the saddle-node bifurcation. Figure 3(b) demonstrates that the locking range generally broadens with increasing injection ratio, which rises from 0.38 GHz at Rinj = -12.5 dB up to 2.66 GHz at Rinj = 2.9 dB. It is remarked that the actual locking range below Rinj = -7.5 dB is smaller than 0.38 GHz, but can not be tracked finely due to the limitation of the tuning step of the ML frequency (0.19 GHz).
Fig. 3. (a) Measured injection-locking diagram of the ICL with optical injection. (b) Stable locking range versus the injection ratio. The bias current of the SL is 1.5×Iths. The detuning frequency is tuned from the positive side to the negative side.
Outside the locking regime, the ICL produces pulse oscillations both above the Hopf bifurcation and below the saddle-node bifurcation. In Fig. 4(a), it is found that most of the unlocking regime produces P1 oscillations, which show a temporal waveform similar to the sinusoidal wave (see example of Rinj = -9.1 dB, Δf = + 0.42 GHz). In addition to the P1 oscillations, we observe three categories of periodic pulse oscillations as well. The temporal waveform of type-I oscillation (see example of Rinj = -9.1 dB, Δf = + 0.23 GHz) exhibits a sharp rising edge associated with a slow falling edge. The plot in Fig. 4(c) illustrates that type-I oscillation (dots) mainly occurs in the vicinity of the Hopf bifurcation. The temporal waveform of type-II oscillation (see example of Rinj = -9.1 dB, Δf = -0.54 GHz) is featured with a flat top. This kind of oscillation occurs near the saddle-node bifurcation in Fig. 4(c) (triangles), and the oscillations are observed at weak injection ratios of -9.1 dB and -7.5 dB. In comparison, the waveform of type-III oscillation (see example of Rinj = 1.5 dB, Δf = -1.49 GHz) exhibits a top with multiple ripples. Similar to type-II oscillation, type-III oscillation appears in the proximity of the saddle-node bifurcation in Fig. 4(c) (stars) as well. However, type-III oscillations are tracked at higher injection ratios of -6.0 dB, 0 dB, and 1.5 dB, respectively. Figure 4(b) analyzes the oscillation frequency of the pulse oscillations at the injection ratio of Rinj = -9.1 dB. It is shown that the oscillation frequency increases nonlinearly with the absolute detuning frequency, both below the saddle-node bifurcation and above the Hopf bifurcation (dashed lines). At the positive detuning side, the oscillation frequency rises from 0.10 GHz at Δf = + 0.23 GHz up to 1.85 GHz at Δf = + 1.75 GHz. In contrast, the oscillation frequency increases from 0.10 GHz at Δf = -0.54 GHz up to 1.71 GHz at Δf = -1.87 GHz, at the negative detuning side. The map in Fig. 4(c) demonstrates that the oscillation frequency indeed rises with increasing absolute detuning frequency for any injection ratio. For a fixed detuning frequency, the contour lines unveil that raising the injection strength generally reduces the oscillation frequency, both at the positive detuning side and at the negative detuning side. The variation of the oscillation frequency against the injection ratio is primarily determined by the interaction between the frequency pulling effect and the cavity resonance red-shift effect. In the frequency pulling effect, the electric field of the ML forces the lasing frequency of the SL away from its cavity resonance frequency, and towards the frequency of the ML [38]. In the cavity resonance red-shift effect, the optical injection reduces the gain of the laser medium, and thereby raises the refractive index [39]. As a result, the cavity resonance is shifted to the longer wavelength side. At the negative detuning side, both effects reduce the oscillation frequency in Fig. 4(c). At the positive detuning side, nevertheless, the frequency pulling effect reduces the oscillation frequency, while the cavity resonance red-shift effect raises the oscillation frequency [40]. Obviously, the frequency pulling effect in Fig. 4(c) is stronger than the red-shift effect, because the detuning frequency is small (less than 2.0 GHz). This is opposite to the case of near-infrared semiconductor lasers, where a large frequency detuning is usually employed so as to produce high-frequency P1 oscillations [40,41].
Fig. 4. (a) Temporal waveforms of different types of periodic oscillations. (b) Oscillation frequency versus the detuning frequency at the injection ratio of -9.1 dB. The dashed lines indicate the locking boundaries. (c) Oscillation frequency as functions of the injection ratio and the detuning frequency. The dots denote type-I oscillation, the triangles denote type-II oscillation, and the stars denote type-III oscillation. The detuning frequency is tuned from the positive side to the negative side.
It is known that near-infrared semiconductor lasers subject to optical injection usually produce chaotic oscillations in a small regime just above the Hopf bifurcation [42,43]. However, the ICL pumped at 1.5×Iths in Fig. 4 does not produce any chaotic oscillations. The absence of the chaotic oscillations can be attributed to the large damping factor [29] as well as the small α factor of ICLs [26]. Decreasing the pump current can reduce the damping factor, and thereby potentially stimulate the appearance of chaos [27]. Consequently, we change the pump current of the SL down to 1.02×Iths(21 mA). Then, the output power becomes 0.1 mW and the lasing wavelength becomes 3391.9 nm. For a fixed injection ratio of Rinj = -4.2 dB, Fig. 5(a) shows the temporal evolution of the nonlinear dynamics when increasing the detuning frequency from the negative frequency side to the positive frequency side. Below the saddle-node bifurcation, the ICL generates P1 oscillations at the detuning frequency of Δf = -0.41 GHz. Within the stable locking regime, the ICL produces continuous wave at Δf = -0.06 GHz. Once above the Hopf bifurcation, the ICL re-produces P1 oscillations at Δf = + 0.11 GHz. Finally, we observe chaotic oscillations at the detuning frequency of Δf = + 0.23 GHz. The P1 oscillations re-occur when increasing the detuning frequency to Δf = + 0.43 GHz. Figure 5(b) illustrates the corresponding electrical spectra of the nonlinear dynamics described in Fig. 5(a). It is shown that the oscillation frequencies of the P1 dynamics at Δf = -0.41 GHz, + 0.11 GHz, and +0.43 GHz are 394 MHz, 144 MHz, and 522 MHz, respectively. All the P1 oscillations exhibit a broad peak, which can be attributed to the poor stability of the lasing frequency, the poor injection coupling of both lasers, as well as the existence of phase noise [44]. The chaotic oscillations substantially raise the electrical spectrum within the bandwidth (600 MHz) of the photodetector. Below 400 MHz, the electrical power is at least 12 dB higher than the background noise level. The chaos bandwidth is commonly defined as the frequency span from DC to the cutoff frequency that accounting for 80% of the total power in the electrical spectrum [45]. Based on this definition, the corresponding chaos bandwidth is 318 MHz. However, we remark that the limited bandwidth of the photodetector is likely to underestimate this chaos bandwidth. This chaos bandwidth is smaller than those generated from ICLs with optical feedback, which is 465 MHz in [31] and is about 1 GHz in [46]. It is stressed that the chaos bandwidth is roughly determined by the resonance frequency of semiconductor lasers, and the maximum reported modulation bandwidth of ICL is around 5 GHz [28]. Figure 5(c) presents the bifurcation diagram as a function of the detuning frequency, which describes the extreme values (both local maxima and local minima) of the temporal waveform in Fig. 5(a). When increasing the detuning frequency from the negative side to the positive side, the bifurcation diagram clearly unveils that the nonlinear dynamics evolves from P1 oscillations below the saddle-node bifurcation (left dashed line), continuous wave within the stable regime, P1 oscillations above the Hopf bifurcation (right dashed line), through a small region of chaotic oscillations, and finally to P1 oscillations again. It is remarked that the nonlinear dynamics at this near-threshold current are similar to those at the high pump current in Fig. 4(c), except for the presence of chaos. Future work can plot a complete map of nonlinear dynamics for a detailed comparison with Fig. 4(c).
Fig. 5. Evolution of the nonlinear dynamics against the detuning frequency. The injection ratio is fixed at -4.2 dB. (a) Time series, (b) electrical spectra, and (c) bifurcation diagram. The dashed lines in (c) indicate the locking boundaries. The bias current of the SL is 1.02×Iths.
3. Simulation model and results
In order to verify the experimental observations, we theoretically study the nonlinear dynamics of ICLs with optical injection, through the framework of rate equations [50]. The dynamics of the carrier number N of the SL, the photon number S of the SL, and the phase difference ϕ between the SL and the ML are described by:
Table 1 lists the detailed ICL parameters used for the simulation. The ICL under study exhibits a lasing threshold of Ith = 23 mA and is biased at 1.5×Ith in the following simulation. The simulation shows that the stable locking diagram for αH = 2.2 in Fig. 6(b) is more asymmetric than that for αH = 1.1 in Fig. 6(a). In addition, both the Hopf bifurcation and the saddle-node bifurcation in the former case shift to the negative detuning side. Outside the stable locking regime, the ICL primarily produces P1 oscillations for both values of α factor. This simulation result agrees well with the experimental one in Fig. 4 and Fig. 5. Besides, the chaotic oscillations only occur in a small region, which is consistent with the experimental observation in Fig. 5 as well. As expectation, the chaos region for αH = 2.2 in Fig. 6(b) is larger than the one for αH = 1.1 in Fig. 6(a). That is, a large value of α factor is helpful for the generation of chaos in semiconductor lasers [1]. Interestingly, the simulation in Fig. 6 shows that the ICL with optical injection can generate period-two oscillations and quasi-periodic oscillations as well. However, both oscillations are not observed in the experiment, which is likely due to the existence of noise and the instability of the experimental setup.
Fig. 6. Simulated maps of the nonlinear dynamics of the ICL with optical injection. (a) αH = 1.1, (b) αH = 2.2. The bias current in simulation is 1.5×Ith.
Table 1. ICL parameters used in the simulation
Based on the analysis of the experimental results (Fig. 4 and Fig. 5) and the simulation results (Fig. 6), it is found that the regimes of chaos and complex dynamics of ICLs are generally smaller than those of common QWLs. This suggests the ICLs are more “stable” against the perturbation of optical injection than QWLs. This conclusion is in agreement with our previous observation of ICLs with the perturbation of optical feedback [31,52]. The high stability of ICLs against external perturbations arises from the small α factor [26] as well as the large damping factor [29].
4. Conclusion
In conclusion, we have unveiled the nonlinear dynamics of an ICL subject to the optical injection both in experiment and in theory. The measurement shows that the stable locking regime is asymmetric and broadens with increasing injection strength. Outside the stable locking regime, the ICL mainly produces P1 oscillations, while three types of periodic pulse oscillations occur in the vicinity of the saddle-node bifurcation and the Hopf bifurcation. In addition, broadband chaotic oscillations are observed at a near-threshold pump current, which exhibit a bandwidth as high as 318 MHz. This mid-infrared chaos bandwidth is one to two orders of magnitude broader than those of mid-infrared QCLs, but is smaller than those of near-infrared QWLs (GHz to tens of GHz). The simulation analysis obtained from the rate equation model confirms that the ICL indeed primarily produces P1 oscillations, while generates chaotic oscillations in a small region. Future work will investigate the detailed characteristics of the broadband chaos as well as its applications in chaos lidar and secure communication.
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 are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, 3rd ed. (Springer, 2017).
2. T. B. Simpson, J. M. Liu, and A. Gavrielides, “Bandwidth enhancement and broadband noise reduction in injection-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 7(7), 709–711 (1995). [CrossRef]
3. C. Lin and F. Mengel, “Reduction of frequency chirping and dynamic linewidth in high-speed directly modulated semiconductor lasers by injection locking,” Electron. Lett. 20(25-26), 1073–1075 (1984). [CrossRef]
4. N. Schunk and K. Petermann, “Noise analysis of injection-locked semiconductor injection lasers,” IEEE J. Quantum Electron. 22(5), 642–650 (1986). [CrossRef]
5. F. Mogensen, H. Olesen, and G. Jacobsen, “FM noise suppression and linewidth reduction in an injection-locked semiconductor laser,” Electron. Lett. 21(16), 696–697 (1985). [CrossRef]
6. S. Wieczorek, B. Krauskopf, T. B. Simpson, et al., “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rep. 416(1-2), 1–128 (2005). [CrossRef]
7. A. Argyris, D. Syvridis, L. Larger, et al., “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef]
8. Y. Xie, Z. Yang, M. Shi, et al., “100 Gb/s coherent chaotic optical communication over 800 km fiber transmission via advanced digital signal processing,” Adv. Photon. Nexus 3(01), 016003 (2023). [CrossRef]
9. F. Y. Lin and J. M. Liu, “Chaotic lidar,” IEEE J. Select. Topics Quantum Electron. 10(5), 991–997 (2004). [CrossRef]
10. R. Chen, H. Shu, B. Shen, et al., “Breaking the temporal and frequency congestion of LiDAR by parallel chaos,” Nat. Photonics 17(4), 306–314 (2023). [CrossRef]
11. A. Uchida, K. Amano, M. Inoue, et al., “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]
12. B. Shen, H. Shu, W. Xie, et al., “Harnessing microcomb-based parallel chaos for random number generation and optical decision making,” Nat. Commun. 14(1), 4590 (2023). [CrossRef]
13. S. K. Hwang, J. M. Liu, and J. K. White, “Characteristics of period-one oscillations in semiconductor lasers subject to optical injection,” IEEE J. Select. Topics Quantum Electron. 10(5), 974–981 (2004). [CrossRef]
14. S. C. Chan, S. K. Hwang, and J. M. Liu, “Radio-over-fiber AM-to-FM upconversion using an optically injected semiconductor laser,” Opt. Lett. 31(15), 2254–2256 (2006). [CrossRef]
15. X. Q. Qi and J. M. Liu, “Photonic microwave applications of the dynamics of semiconductor lasers,” IEEE J. Select. Topics Quantum Electron. 17(5), 1198–1211 (2011). [CrossRef]
16. Y. B. Peng, B. B. Zhao, and C. Wang, “Nonlinear dynamics of a quantum cascade laser with optical injection,” Opt. Express 30(15), 27593–27601 (2022). [CrossRef]
17. Y. Peng, S. Liu, V. Kovanis, et al., “Uniform spike trains in optically injected quantum cascade oscillators,” Chaos 33(12), 123127 (2023). [CrossRef]
18. R. Q. Yang, “Infrared laser based on intersubband transitions in quantum wells,” Superlattices Microstruct. 17(1), 77–83 (1995). [CrossRef]
19. C. H. Lin, R. Q. Yang, D. Zhang, et al., “Type-II interband quantum cascade laser at 3.8 µm,” Electron. Lett. 33(7), 598–599 (1997). [CrossRef]
20. J. R. Meyer, W. W. Bewley, C. L. Canedy, et al., “The interband cascade lasers,” Photonics 7(3), 75 (2020). [CrossRef]
21. D. A. Díaz-Thomas, O. Stepanenko, M. Bahriz, et al., “Interband cascade lasers with AlGaAsSb cladding layers emitting at 3.3 µm,” Opt. Express 27(22), 31425–31434 (2019). [CrossRef]
22. W. Huang, S. Hu, J. Tu, et al., “InAs-based interband cascade laser operating at 5.17 µm in continuous wave above room temperature,” IEEE Photon. Technol. Lett. 36(2), 91–94 (2024). [CrossRef]
23. J. A. Massengale, Y. Shen, R. Q. Yang, et al., “Low threshold, long wavelength interband cascade lasers with high voltage efficiencies,” IEEE J. Quantum Electron. 59(6), 1–7 (2023). [CrossRef]
24. I. Vurgaftman, W. W. Bewley, C. L. Canedy, et al., “Mid-IR type-II interband cascade lasers,” IEEE J. Select. Topics Quantum Electron. 17(5), 1435–1444 (2011). [CrossRef]
25. I. Vurgaftman, W. W. Bewley, C. L. Canedy, et al., “Interband cascade lasers with low threshold powers and high output powers,” IEEE J. Select. Topics Quantum Electron. 19(4), 1200210 (2013). [CrossRef]
26. Y. Deng, B. B. Zhao, and C. Wang, “Linewidth broadening factor of an interband cascade laser,” Appl. Phys. Lett. 115(18), 181101 (2019). [CrossRef]
27. L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).
28. B. Schwarz, J. Hillbrand, M. Beiser, et al., “Monolithic frequency comb platform based on interband cascade lasers and detectors,” Optica 6(7), 890–895 (2019). [CrossRef]
29. Z. F. Fan, Y. Deng, C. Ning, et al., “Differential gain and gain compression of an overdamped interband cascade laser,” Appl. Phys. Lett. 119(8), 081101 (2021). [CrossRef]
30. S. Zaminga, P. Didier, H. Kim, et al., “Free-space gigabit data transmission with a directly modulated interband cascade laser epitaxially grown on silicon,” in 2023 IEEE Photonics Conference (IPC) (2023).
31. Y. Deng, Z. F. Fan, B. B. Zhao, et al., “Mid-infrared hyperchaos of interband cascade lasers,” Light: Sci. Appl. 11(1), 7 (2022). [CrossRef]
32. J. Helms and K. Petermann, “A simple analytic expression for the stable operation range of laser diodes with optical feedback,” IEEE J. Quantum Electron. 26(5), 833–836 (1990). [CrossRef]
33. L. Jumpertz, K. Schires, M. Carras, et al., “Chaotic light at mid-infrared wavelength,” Light: Sci. Appl.5(6), e16088 (2016). [CrossRef]
34. J. Liu, Z. Wu, M. Zhao, et al., “Low-frequency regular pulse and intermittent oscillation in a mid-infrared interband cascade laser with optoelectronic feedback,” Opt. Express 31(18), 29012–29018 (2023). [CrossRef]
35. B. Zhao, Y. Peng, X. Wang, et al., “Modulation characteristics of period-one oscillations in quantum cascade lasers,” Appl. Sci. 11(24), 11730 (2021). [CrossRef]
36. O. Spitz, A. Herdt, J. Wu, et al., “Private communication with quantum cascade laser photonic chaos,” Nat. Commun. 12(1), 3327 (2021). [CrossRef]
37. S. Wieczorek, B. Krauskopf, and D. Lenstra, “A unifying view of bifurcations in a semiconductor laser subject to optical injection,” Opt. Commun. 172(1-6), 279–295 (1999). [CrossRef]
38. W. A. van der Graaf, A. M. Levine, and D. Lenstra, “Diode lasers locked to noisy injection,” IEEE J. Quantum Electron. 33(3), 434–442 (1997). [CrossRef]
39. C. Wang, K. Schires, M. Osiński, et al., “Thermally insensitive determination of the linewidth broadening factor in nanostructured semiconductor lasers using optical injection locking,” Sci. Rep. 6(1), 27825 (2016). [CrossRef]
40. S. C. Chan, S. K. Hwang, and J. M. Liu, “Period-one oscillation for photonic microwave transmission using an optically injected semiconductor laser,” Opt. Express 15(22), 14921–14935 (2007). [CrossRef]
41. C. Wang, R. Raghunathan, K. Schires, et al., “Optically injected InAs/GaAs quantum dot laser for tunable photonic microwave generation,” Opt. Lett. 41(6), 1153–1156 (2016). [CrossRef]
42. T. B. Simpson, J. M. Liu, K. F. Huang, et al., “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclass. Opt. 9(5), 765–784 (1997). [CrossRef]
43. S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun. 183(1-4), 195–205 (2000). [CrossRef]
44. M. AlMulla and J. M. Liu, “Linewidth characteristics of period-one dynamics induced by optically injected semiconductor lasers,” Opt. Express 28(10), 14677–14693 (2020). [CrossRef]
45. F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221(1-3), 173–180 (2003). [CrossRef]
46. F. Grillot, O. Spitz, S. Zhao, et al., “Dynamic and nonlinear properties of mid-infrared interband quantum cascade lasers,” Proc. SPIE 12430, 15 (2023). [CrossRef]
47. Y. Deng, B. B. Zhao, X. G. Wang, et al., “Narrow linewidth characteristics of interband cascade lasers,” Appl. Phys. Lett. 116(20), 201101 (2020). [CrossRef]
48. W. W. Bewley, J. R. Lindle, C. S. Kim, et al., “Lifetimes and Auger coefficients in type-II W interband cascade lasers,” Appl. Phys. Lett. 93(4), 041118 (2008). [CrossRef]
49. R. Q. Yang, “Mid-infrared interband cascade lasers based on type-II heterostructures,” Microelectron. J. 30(10), 1043–1056 (1999). [CrossRef]
50. Y. Deng and C. Wang, “Rate equation modeling of interband cascade lasers on modulation and noise dynamics,” IEEE J. Quantum Electron. 56(2), 1–9 (2020). [CrossRef]
51. R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982). [CrossRef]
52. X. Y. Li, Z. F. Fan, Y. Deng, et al., “30-kHz linewidth interband cascade laser with optical feedback,” Appl. Phys. Lett. 120(17), 171109 (2022). [CrossRef]
