Nonlinear Dynamics of Interband Cascade Laser Subjected to Optical Feedback

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Article Nonlinear Dynamics of Interband Cascade Laser Subjected to Optical Feedback

Hong Han 1,*, Xumin Cheng 1, Zhiwei Jia 1 and K. Alan Shore 2

1 Key Laboratory of Advanced Transducers and Intelligent Control System, Ministry of Education, College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China; [email protected] (X.C.); [email protected] (Z.J.) 2 School of Electronic Engineering, Bangor University, Wales LL57 1UT, UK; [email protected] * Correspondence: [email protected]

Abstract: We present a theoretical study of the nonlinear dynamics of a long external cavity delayed optical feedback-induced interband cascade laser (ICL). Using the modiﬁed Lang–Kobayashi equations, we numerically investigate the effects of some key parameters on the ﬁrst Hopf bifurcation

point of ICL with optical feedback, such as the delay time (τf), pump current (I), linewidth enhancement factor (LEF), stage number (m) and feedback strength (f ext). It is found that compared with τf, I, LEF and m have a signiﬁcant effect on the stability of the ICL. Additionally, our results show that an ICL with few stage numbers subjected to external cavity optical feedback is more susceptible to exhibiting chaos. The chaos bandwidth dependences on m, I and f ext are investigated, and 8 GHz bandwidth mid-infrared chaos is observed.

Keywords: interband cascade laser; mid-infrared chaos; optical feedback; nonlinear dynamics

Citation: Han, H.; Cheng, X.; Jia, Z.; Shore, K.A. Nonlinear Dynamics of Interband Cascade Laser Subjected to 1. Introduction Optical Feedback. Photonics 2021, 8, As a mid-infrared semiconductor laser, the interband cascade laser (ICL) has made 366. https://doi.org/10.3390/ significant progress in the last two decades [1–6]. The RAND Corporation reports that photonics8090366 mid-infrared lasers in the 3–5 µm band of the atmospheric transmission window have good atmospheric transmission characteristics, lower transmission losses than other bands, Academic Editors: Ana Quirce and and are less susceptible to weather factors [7]. In addition, the mid-infrared band covers Martin Virte the absorption peaks of many atoms and molecules [8]. Therefore, ICL can be used in applications such as gas detection [9,10], clinical respiratory diagnosis [11] and free-space Received: 21 July 2021 optical communication [12]. Accepted: 25 August 2021 In contrast to the quantum cascade laser (QCL), the ICL is a bipolar device, with Published: 31 August 2021 the electronic transition of the ICL occurring between the conduction and the valence sub-bands [13]. Therefore, the carrier lifetime of the ICL is in the nanosecond order like Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in in more conventional semiconductor lasers. Furthermore, in recent experimental reports published maps and institutional affil- the linewidth enhancement factor of the ICL was found to be about 2.2, which is much iations. higher than that of QCL [14,15]. Both of these characteristics suggest that when subjected to external perturbation, the ICL will exhibit rich nonlinear dynamics. Wang et al.’s recent experiments confirm that with external optical feedback the ICL exhibits periodic oscillations and weak chaos [16]. 450 MHz low frequency oscillation (detector bandwidth limited) chaos was observed. However, the route to chaos and the identification of the Copyright: © 2021 by the authors. means for obtaining strong chaos are open for detailed study. A recent report shows that Licensee MDPI, Basel, Switzerland. based on a QCL with external optical feedback, a generated mid-infrared low frequency This article is an open access article distributed under the terms and chaotic oscillation was used to achieve 0.5 Mbit/s message private free-space optical conditions of the Creative Commons communication [17]. To realize a much higher-speed message secure transmission, strong Attribution (CC BY) license (https:// broadband chaos is essential. creativecommons.org/licenses/by/ In this paper, modified Lang–Kobayashi equations are used to investigate the dynam- 4.0/). ics of the ICL subject to external optical feedback. ICLs with short external cavities have

Photonics 2021, 8, 366. https://doi.org/10.3390/photonics8090366 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 366 2 of 13

Photonics 2021, 8, 366 2 of 12 In this paper, modified Lang–Kobayashi equations are used to investigate the dynamics of the ICL subject to external optical feedback. ICLs with short external cavities have been used to affect wavelength tuning [18]. In contrast, and with a view to perform- been used to affect wavelength tuning [18]. In contrast, and with a view to performing ing experiments with discrete devices [19–25], we focus on the case of optical feedback experiments with discrete devices [19–25], we focus on the case of optical feedback from fromlonger longer external external cavities, cavities, wherein wherein the external the external feedback feedback delay timedelay is time larger is thanlarger the than oscilla- the oscillationtion relaxation relaxation time of time the ICLof the [26 ICL]. The [26]. pump The current, pump current, feedback feedback strength, strength, stage number stage numberand linewidth and linewidth enhancement enhancement factor effects factor on effects the stability on the ofstability the ICL of with the ICL optical with feedback optical feedbackare analyzed, are analyzed, and the inﬂuence and the influence of the stage of the number, stage number, pump current pump and current feedback and feedback strength strengthon the bandwidth on the bandwidth of chaos of are chaos investigated. are investigated.

2.2. Theoretical Theoretical Model Model FigureFigure 11aa presentspresents an an ICL ICL structure structure having having three three stages. stages. Initially, Initially the, the electron electron transition transitionin the in ICLthe occursICL occurs between between the first the conductionfirst conduction sub-band sub-band and theand first the valencefirst valence sub-band, sub- bandas indicated, as indicated as Ee (blueas Ee potential(blue potential well) andwell) E hand(red E potentialh (red potential well) in well) the leftin the upper left cornerupper cornerin Figure in 1Figurea [ 27]. 1a After [27]. theAfter first the electron first electron transition, transition, the electron the electron reaches reaches the second the second stage stageconduction conduction sub-band sub- throughband through interband interb tunnelingand tunneli and thenng and repeats then the repeats electron the transition electron transitionin the second in the stage second and stage then inand the then third in stage.the third Figure stage.1b isFigure the schematic 1b is the diagramschematic of dia- an gramICL subjected of an ICL to subject externaled to mirror external feedback, mirror where feedback,τf is where the feedback τf is the time feedback delay. time delay.

FigureFigure 1. 1. ((aa)) 3 3-stage-stage number number structure structure o off interband interband cascade cascade laser( laser(ICL);ICL); ( (bb)) Schematic Schematic diagram diagram of of ICL ICL withwith optical feedback structure.

UsingUsing appropriately appropriately modified modified Lang Lang–Kobayashi–Kobayashi equations, equations, the rate equations for the ICLICL with with optical optical feedback feedback are are as as follows follows [28,29]: [28,29]: dN()()() t IdN(t) NI t N t N(t) N(t) = − − − = η Γpυ g gS η ΓpυggS (1) dt q τsp τaug (1) dt q τspτ aug dS(t) 1 N(t) q = mΓpυgg −  S(t) + mβ + 2k S(t)S(t − τf )cosθ(t) (2) dS()() tdt 1τp Nτ tsp = mΓ pυgg  S()()()() t + mβ + 2k S t S t  τ f cosθ t (2) dt τp  τ sp s dϕ(t) αH 1 S(t − τf ) = mΓpυgg − − k sinθ(t) (3) dt 2  τp  S(t) dφ()t αH 1 S() t τ f where N(t), S(t )= and ϕ( mt)Γ respectivelypυ g g  represent  k the per sin stageθ()t carrier number, total photon(3) dt 2 τp  S() t number of all gain stages and phase of  the electric field. M is the number of gain stages, τsp is the spontaneous radiation lifetime, and τaug is the Auger recombination lifetime. Since where N(t), S(t) and φ(t) respectively represent the per stage carrier number, total photon the Auger recombination lifetime τaug in the ICL is smaller than the spontaneous radia- number of all gain stages and phase of the electric field. M is the number of gain stages, tion lifetime τsp, τaug must be considered in the current-carrying dynamics. In principle, τthesp is Auger the spontaneous coefficient radiation has a carrier lifetime density, and dependence,τaug is the Auger as recombination in ref. [30]. In thislifetime. work, Since we thefollow Auger [29 ]recombination in assuming a lifetime constant τaug value in the for ICL the is Auger smaller lifetime. than theη spontaneousis the current radiation injection sp aug lifetimeefficiency, τ ,Γ pτ is themust optical be considered confinement in the factor current per gain-carrying stage, dynamicsνg is the. groupIn principle velocity, the of Auger coefficient has a carrier density dependence, as in ref. [30]. In this work, we follow light, g is the material gain per stage which is given by g = a0[N(t) − Ntr]/A. τp is the [29] in assuming a constant value for the Auger lifetime. η is the currentp injection effi- photon lifetime, and k is the feedback coefficient which is given by k = 2Cl fext/τin, where ciency, Γp is the optical confinement factor per gain stage, νg is the group velocity of light, τin is the internal cavity roundtrip time, fext is the feedback strength which is defined as the power ratio between the feedback light and the laser output, and Cl is an external coupling

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√ coefficient. The external coupling coefficient can be expressed as Cl= (1 − R)/2 R, with R being the reflection coefficient of the laser front facet facing the external mirror. The steady-state solutions for the ICL operating above the threshold current are as follows [29]: 1 A N = + Ntr (4) m Γpvga0τp I − I S = mητ th (5) 0 p q q 1 A 1 1 Ith = + Ntr + (6) η m Γpvga0τp τsp τaug The descriptions of other symbols and their values used in the simulation are given in Table1, as taken from [14,27,29–32]. The integration time step in the simulation is 0.1 ps.

Table 1. ICL parameters used in the simulations.

Parameter Symbol Value Cavity length L 2 mm Cavity width W 4.4 µm 7 Group velocity of light vg 8.38 × 10 m/s Wavelength λ 3.7 µm Active area A 8.8 × 10−9 m2 Facet reflectivity R 0.32 Refractive index nr 3.58 Optical confinement factor Γp 0.04 Stage number m 3–20 Injection efficiency H 0.64 Photon lifetime τp 10.5 ps Spontaneous emission time τsp 15 ns Auger lifetime τaug 1.08 ns Threshold current Ith 19.8 mA (m = 5) Feedback strength fext 0~30% Time delay τf 1.5~5.0 ns −10 Differential gain a0 2.8 × 10 cm 7 Transparent carrier number Ntr 6.2 × 10 Spontaneous emission factor β 1 × 10−4 Linewidth enhancement factor αH 2.2

3. Numerical Results We calculate the carrier number and photon number for an increasing bias current, as shown in Figure2a,b, respectively. It is found that the number of stages m has little effect on the carrier number (in Figure2a) but that it inﬂuences the photon number (in Figure2b). For relatively large numbers of stages such as m = 10 (red dashed curve), the output power is much higher than in the case of m = 5 (black solid curve), as shown in Figure2b.

3.1. Route to Chaos Figure3 shows the output of the ICL with external optical feedback as the feedback strength increases, for the case of m = 5. The ICL output is stable when the feedback strength fext ranges from 0 to 0.019%. Without feedback, that is fext = 0, the relaxation oscillation frequency f R can be observed from the RF spectrum in Figure3(d-i) to be 1.02 GHz. This is in accordance with the value 1.035 GHz obtained by calculating the relaxation −1/2 oscillation frequency via the relation f R = (G0S0/τp) /(2π), where G0 = Γpvga0/A and S0 are found from Equation (5). As the feedback strength increases, the ICL enters into period-1 dynamics (ii), quasi-periodic dynamics (iii), weak chaos oscillations (iv), and then displays strong chaos (v). The frequency of period-1 oscillations is 1.03 GHz, as shown in Figure3(d-ii), which is a little larger than the relaxation oscillation frequency shown Photonics 2021, 8, 366 4 of 12

in Figure3(a-ii). As the feedback strength increases, a quasi-periodic oscillation is found, which is conﬁrmed by the RF spectrum and phase diagram; that is, more frequencies are induced in Figure3(d-iii) and more loops are found in the phase in Figure3(c-iii), respectively. An irregular laser intensity oscillation is observed where the ICL enters into weak chaos, as shown in Figure3(a-iv). Although the highest peak in the RF spectrum is still at 1.03 GHz, more frequency components appear, as presented in Figure3(d-iv). For a further increase in the feedback strength, more complex nonlinear dynamics are introduced, hence achieving strong mid-infrared optical chaos, as shown in Figure3(a-v–e-v). In the optical spectra of the chaos shown in Figure3(e-v), many external cavity modes with a frequency interval around 410 MHz are found. This can be conﬁrmed from the auto-correlation Photonics 2021, 8, 366 4 of 13 functions shown in Figure3(b-v), where the sidelobe peak is at 2.4 ns, corresponding to a cavity length of 36 cm.

10 m = 5 1.6 m = 5 9 (a) (b) ]

] m = 10 m = 10 7 7 8 1.4 10 10 [ [ 7 1.2 6 1.0 5 0.8 4 0.6 3 0.4 Carrier Carrier number 2 number Photon 0.2 1 0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 [ ] [ ] Pump current I/Ith Pump current I/Ith Figure 2. (a) Carrier number and ((b)) photon number vs. pump current. Black solid and red dashed curves representrepresent stage numbersnumbers m = 5 and m = 1 10,0, respectively.

3.1. RouteBy using to Chaos bifurcation diagrams, the dynamics of the ICL for increasing feedback strengthFigure can 3 beshows obtained, the output as shown of the inICL Figure with4 external(a-i) with opticalm = 5feedback and Figure as the4(b-i) feedback with mstrength= 10. Maximum increases, Lyapunov for the case exponents of m = [ 5.33 , The34] can ICL be output used to is determine stable when whether the feedback the ICL output with external optical feedback is chaotic (red dots) or not (blue dots), as shown strength fext ranges from 0 to 0.019%. Without feedback, that is fext = 0, the relaxation oscil- in Figure4(a-ii,b-ii). As shown in Figure4(a-i), the route to chaos for a 10-stage ICL with lation frequency fR can be observed from the RF spectrum in Figure 3(d-i) to be 1.02 GHz. externalThis is in optical accordance feedback with is the from value stable 1.035 (S) GHz to period-1 obtained (P1), by thencalculating quasi-periodic the relaxation (QP) and os- then chaos (C). This route is different from that of a 10-stage ICL where multiple-periodic cillation frequency via the relation fR = (G0S0/τp)−1/2/(2π), where G0 = Γpvga0/A and S0 are (MP) oscillations are observed. The number of stages m has an effect on the photon number found from Equation (5). As the feedback strength increases, the ICL enters into period-1 and phase, as presented in Equations (2) and (3), which results in different routes to chaos. dynamics (ii), quasi-periodic dynamics (iii), weak chaos oscillations (iv), and then displays 3.2.strong Hopf ch Bifurcationaos (v). The Analysis frequency of period-1 oscillations is 1.03 GHz, as shown in Figure 3(d-ii), which is a little larger than the relaxation oscillation frequency shown in Figure In this section, we explore the stability of ICLs with external optical feedback. We ﬁrst 3(a-ii). As the feedback strength increases, a quasi-periodic oscillation is found, which is compare two stage numbers, which are m = 5 and m = 10, to reveal the effects of the time confirmed by the RF spectrum and phase diagram; that is, more frequencies are induced delay, bias current and linewidth enhancement factor on the Hopf bifurcation points. Then, in Figure 3(d-iii) and more loops are found in the phase in Figure 3(c-iii), respectively. An we ascertain how the Hopf bifurcation changes for stage numbers m in the range of 3 to 20. irregular laser intensity oscillation is observed where the ICL enters into weak chaos, as The pump current is set a little above the threshold current, that is 1.1I . Figure5 shown in Figure 3(a-iv). Although the highest peak in the RF spectrum is still atth 1.03 GHz, shows that the external cavity delay has little effect on the Hopf bifurcation point. As the more frequency components appear, as presented in Figure 3(d-iv). For a further increase external cavity delay increases from 1.5 ns to 5.0 ns, the Hopf bifurcation point values, in the feedback strength, more complex nonlinear dynamics are introduced, hence achiev- that is the feedback power ratio where the ICL enters into P1, are around 0.02% to 0.33%. Asing shownstrong mid in Figure-infrared5, there optical is achaos, periodic as shown dependence in Figure of 3 the(a-v Hopf–e-v). bifurcation In the optical point spectra on theof the external chaos shown cavity in delay Figure time; 3(e for-v), a many 2.1 ns external delay time, cavity the modes value with of the a frequency Hopf bifurcation interval pointaround reaches 410 MHz a maximum are found. and This the secondcan be peakconfirmed is at 3.0 from ns. Asthe the auto delay-correlation time increases, functions the differencesshown in Figure between 3(b the-v), Hopfwhere bifurcation the sidelobe points peak reduce. is at 2.4 ns, corresponding to a cavity length of 36 cm.

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Figure 3. Output of ICL with external optical feedback asas feedbackfeedback strengthstrength increasesincreases withwithm m= = 5,5,I I= = 1.11.1IIthth, and τf = 2.4 2.4 ns ns;; fromfrom the the top top down down,, i: i: ffextext = =0.0%, 0.0%, ii: ii: fextf ext= 0.02%,= 0.02%, iii: iii:fext f=ext 0.14%,= 0.14%, iv: fext iv: =f ext0.20%,= 0.20%, v: fext v:= 0.84%;fext = 0.84%; columns columns (a–e) are (a– timee) are series time Photonics 2021, 8, 366 6 of 13 (TS),series autocorrelation (TS), autocorrelation curve functions curve functions (ACF), (ACF), phase phaseportrait portrait (PP), radio (PP),- radio-frequencyfrequency spectrum spectrum (RFS) (RFS)and optic andal optical spectra spectra (OS), respectively.(OS), respectively.

By using bifurcation diagrams, the dynamics of the ICL for increasing feedback strength can be obtained, as shown in Figure 4(a-i) with m = 5 and Figure 4(b-i) with m = 10. Maximum Lyapunov exponents [33,34] can be used to determine whether the ICL output with external optical feedback is chaotic (red dots) or not (blue dots), as shown in Figure 4(a-ii,b-ii). As shown in Figure 4(a-i), the route to chaos for a 10-stage ICL with external optical feedback is from stable (S) to period-1 (P1), then quasi-periodic (QP) and then chaos (C). This route is different from that of a 10-stage ICL where multiple-periodic (MP) oscillations are observed. The number of stages m has an effect on the photon number and phase, as presented in Equations (2) and (3), which results in different routes to chaos.

Figure 4. Bifurcations (i) and Maximum Lyapunov exponentsexponents (ii)(ii) ofof ICL output with external opticaloptical feedbackfeedback underunder II = 1 1.1.1Ith,, ττf f== 2.4 2.4 ns. ns. (a ()a )mm = =5, 5, (b (b) m) m = =10. 10.

Then, we ascertain how the Hopf bifurcation changes for stage numbers m in the range of 3 to 20. The pump current is set a little above the threshold current, that is 1.1Ith. Figure 5 shows that the external cavity delay has little effect on the Hopf bifurcation point. As the external cavity delay increases from 1.5 ns to 5.0 ns, the Hopf bifurcation point values, that is the feedback power ratio where the ICL enters into P1, are around 0.02% to 0.33%. As shown in Figure 5, there is a periodic dependence of the Hopf bifurcation point on the external cavity delay time; for a 2.1 ns delay time, the value of the Hopf bifurcation point reaches a maximum and the second peak is at 3.0 ns. As the delay time increases, the differences between the Hopf bifurcation points reduce.

1.0 0.9 m = 5 0.8 m = 10 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Feedback powerratio [%] 0.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 External cavity delay [ns]

Figure 5. Hopf bifurcation point vs. time delay τf for I = 1.1Ith.

To illustrate the pump current effects, the dependence of the Hopf bifurcation point on bias current is investigated. In Figure 6a,b, two external cavity delays are compared, viz τf = 2.0 ns and τf = 2.4 ns. As the pump current increases, the Hopf bifurcation point

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Figure 4. Bifurcations (i) and Maximum Lyapunov exponents (ii) of ICL output with external optical feedback under I = 1.1Ith, τf = 2.4 ns. (a) m = 5, (b) m = 10.

3.2. Hopf Bifurcation Analysis In this section, we explore the stability of ICLs with external optical feedback. We first compare two stage numbers, which are m = 5 and m = 10, to reveal the effects of the time delay, bias current and linewidth enhancement factor on the Hopf bifurcation points. Then, we ascertain how the Hopf bifurcation changes for stage numbers m in the range of 3 to 20. The pump current is set a little above the threshold current, that is 1.1Ith. Figure 5 shows that the external cavity delay has little effect on the Hopf bifurcation point. As the external cavity delay increases from 1.5 ns to 5.0 ns, the Hopf bifurcation point values, that is the feedback power ratio where the ICL enters into P1, are around 0.02% to 0.33%. As shown in Figure 5, there is a periodic dependence of the Hopf bifurcation point on the external cavity delay time; for a 2.1 ns delay time, the value of the Hopf bifurcation point Photonics 2021, 8, 366 6 of 12 reaches a maximum and the second peak is at 3.0 ns. As the delay time increases, the differences between the Hopf bifurcation points reduce.

1.0 0.9 m = 5 0.8 m = 10 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Feedback powerratio [%] 0.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 External cavity delay [ns]

Figure 5. Hopf bifurcation point vs. time delay τf forfor II == 1 1.1.1IIthth. .

Photonics 2021, 8, 366 To illustrateillustrate thethe pumppump currentcurrent effects,effects, thethe dependencedependence ofof thethe HopfHopf bifurcationbifurcation7 pointpointof 13 on bias current is investigated.investigated. In Figure 66aa,b,,b, twotwo externalexternal cavitycavity delaysdelays areare compared,compared, viz τff = 2.0 ns and τf = 2.4 ns. As the pump current increases,increases, the HopfHopf bifurcationbifurcation pointpoint gradually increases for both m = 5 (squares) and m = 10 (circles) cases, as well as for gradually increases for both m = 5 (squares) and m = 10 (circles) cases, as well as for τf = τf = 2.0 ns and τf = 2.4 ns. Compared with τf = 2.4 ns, at τf = 2.0 ns there is a need for a 2.0 ns and τf = 2.4 ns. Compared with τf = 2.4 ns, at τf = 2.0 ns there is a need for a larger larger feedback power ratio to enable the ICL to enter an unstable state, which is around feedback power ratio to enable the ICL to enter an unstable state, which is around 1.3 1.3 times that of τf = 2.4 ns for a pump current of 3Ith. Since these two delays show times that of τf = 2.4 ns for a pump current of 3Ith. Since these two delays show similar similar trends for the Hopf bifurcation point versus bias current, we focus our attention on trends for the Hopf bifurcation point versus bias current, we focus our attention on τf = τf = 2.4 ns in the following results. 2.4 ns in the following results.

6 5 m = 5 (a) m = 5 (b) 5 m = 10 4 m = 10 4 3 3 2 2 1 1 Feedback power ratio[%] power Feedback 0 Feedbackpower[%] ratio 0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Pump current [I/Ith] Pump current [I/Ith]

FigureFigure 6. 6. HopfHopf bifurcation bifurcation point point vs. vs. bias bias current. current. (a ()a τ) f τ=f 2.=0 2.0 ns, ns, (b ()b τ)f τ= f2.=4 2.4ns.ns.

ItIt is is well well-appreciated-appreciated that that the the linewidth enhancement factor,factor,α αHH,, playsplays a a crucial crucial role role in insemiconductor semiconductor laser laser nonlinear nonlinear dynamics. dynamics. For For common common quantum quantum well well laser laser diodes, diodes,αH isαH in isthe in rangethe range of 2.0 of to2.0 5.0 to [ 355.0,36 [3].5,3 A6 recent]. A recent report report shows shows that thethat below-threshold the below-threshold linewidth lin- ewidthenhancement enhancement factor factor of ICL of is ICL in the is in range the range of 1.1–1.4 of 1.1 [14–1]..4 Here,[14]. Here we calculate, we calculate the Hopf the Hopfbifurcation bifurcation point point versus versus linewidth linewidth enhancement enhancement factor, factor which, which ranges ranges from 1from to 5, 1 as to shown 5, as shownin Figure in Figure7. As the 7. linewidthAs the linewidth enhancement enhancement factor increases factor increases from 1 to 2,from the 1 Hopf to 2, bifurcation the Hopf bifurcationpoint value point reduces value rapidly reduces and rapidly then and tends then to betends stable to be as stableαH increases as αH increases further. further. Thus, as Thusexpected,, as expected, a small aα Hsmallimparts αH imparts the ICL the with ICL considerable with considerable dynamic dynamic stability. stability. For ICLs, the number of stages, m, is usually less than 20. Figure8 shows the Hopf bifurcation point value versus stage number with I = 1.5I , τ = 2.4 ns. It is seen that an 0.6 th f increased stage number gives risem = 5 to an exponential increasing Hopf bifurcation point value,0.5 which indicates that ICLs m with = 10 a large number of stages are more stable. 0.4

0.3

0.2

0.1

Feedback [%] ratio power 0.0

1 2 3 4 5 Linewidth enhancement factor [a ] H

Figure 7. Hopf bifurcation point vs. linewidth enhancement factor under I = 1.1Ith, τf = 2.4 ns.

For ICLs, the number of stages, m, is usually less than 20. Figure 8 shows the Hopf bifurcation point value versus stage number with I = 1.5Ith, τf = 2.4 ns. It is seen that an increased stage number gives rise to an exponential increasing Hopf bifurcation point value, which indicates that ICLs with a large number of stages are more stable.

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gradually increases for both m = 5 (squares) and m = 10 (circles) cases, as well as for τf = 2.0 ns and τf = 2.4 ns. Compared with τf = 2.4 ns, at τf = 2.0 ns there is a need for a larger feedback power ratio to enable the ICL to enter an unstable state, which is around 1.3 times that of τf = 2.4 ns for a pump current of 3Ith. Since these two delays show similar trends for the Hopf bifurcation point versus bias current, we focus our attention on τf = 2.4 ns in the following results.

6 5 m = 5 (a) m = 5 (b) 5 m = 10 4 m = 10 4 3 3 2 2 1 1 Feedback power ratio[%] power Feedback 0 Feedbackpower[%] ratio 0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Pump current [I/Ith] Pump current [I/Ith]

Figure 6. Hopf bifurcation point vs. bias current. (a) τf = 2.0 ns, (b) τf = 2.4 ns.

It is well-appreciated that the linewidth enhancement factor, αH, plays a crucial role in semiconductor laser nonlinear dynamics. For common quantum well laser diodes, αH is in the range of 2.0 to 5.0 [35,36]. A recent report shows that the below-threshold linewidth enhancement factor of ICL is in the range of 1.1–1.4 [14]. Here, we calculate the Hopf bifurcation point versus linewidth enhancement factor, which ranges from 1 to 5, as shown in Figure 7. As the linewidth enhancement factor increases from 1 to 2, the Hopf Photonics 2021, 8, 366 7 of 12 bifurcation point value reduces rapidly and then tends to be stable as αH increases further. Thus, as expected, a small αH imparts the ICL with considerable dynamic stability.

0.6 m = 5 0.5 m = 10

0.4

0.3

0.2

0.1

Feedback [%] ratio power 0.0

1 2 3 4 5 Photonics 2021, 8, 366 Linewidth enhancement factor [a ] 8 of 13 Photonics 2021, 8, 366 H 8 of 13

Figure 7.7. Hopf bifurcation point vs. linewidth enhancement factor under II = 1 1.1.1Ith,, ττf f== 2.4 2.4 ns. ns.

1.0For ICLs, the number of stages, m, is usually less than 20. Figure 8 shows the Hopf 1.0 bifurcation point value versus stage number with I = 1.5Ith, τf = 2.4 ns. It is seen that an increased0.8 stage number gives rise to an exponential increasing Hopf bifurcation point 0.8 value, which indicates that ICLs with a large number of stages are more stable. 0.6 0.6

0.4 0.4

0.2 0.2 Feedback powerFeedbackratio [%]

Feedback powerFeedbackratio [%] 0.0 0.0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Stage number Stage number Figure 8. Hopf bifurcation point value vs. stage number m with I = 1.5Ith, τf = 2.4 ns. Figure 8. Hopf bifurcation point value vs. stage number m with II = 1 1.5.5Ithth,, ττf f== 2.4 2.4 ns. ns.

3.3.3.3. Bandwidth of Chaos AA broadband RFRF spectrum isis oneone of the significant signiﬁcant characteristics ofof chaos.chaos. The ba band-nd- widthwidth ofof chaoschaos determinesdetermines thethe rangerange resolutionresolution ofof chaoticchaotic lidarlidar [[37],37], thethe bitbit raterate ofof randomrandom sequencesequence generationgeneration [[38]38] andand the transmission rate of optical chaos communicationscommunications [3 [399].]. WeWe useuse aa conventionalconventional deﬁnitiondefinition ofof bandwidthbandwidth ofof chaoticchaotic signalssignals asas thethe frequencyfrequency spanspan betweenbetween thethe DCDC andand thethe frequencyfrequency wherewhere 80% 80% of of the the energy energy is is contained contained [ 40[40],], and and investi- inves- gatetigate the the bandwidth bandwidth of of mid-infrared mid-infrared chaos. chaos. Similar Similar to to regularregular quantumquantum wellwell laserlaser diodes,diodes, thethe bandwidthbandwidth ofof chaoschaos fromfrom ICLICL withwith externalexternal opticaloptical feedbackfeedback increasesincreases asas thethe feedbackfeedback powerpower ratioratio increases,increases, asas shownshown inin FigureFigure9 9..

2.4 2.4 m == 55 m = 10 2.2 m = 10 2.2

2.0 2.0

1.8 1.8

1.6

Bandwidth [GHz] Bandwidth 1.6 Bandwidth [GHz] Bandwidth 1.4 1.4 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Feedback power ratio [%] Feedback power ratio [%] FigureFigure 9.9. BandwidthBandwidth ofof chaoschaos vs.vs. feedbackfeedback powerpower ratioratio underunder II == 1.11.1IIth,, ττf = 2.4 ns, mm == 55 (black(black Figure 9. Bandwidth of chaos vs. feedback power ratio under I = 1.1Ithth, τf = 2.4 ns, m = 5 (black squares)squares) andand mm == 1010 (red circles).circles).

Increasing the pump current and hence the relaxation oscillation frequency is expected to enhance the bandwidth of chaos,, and this is confirmed in Figure 10. Here, we notice that mid-infrared chaos from ICL with different stage numbers has the same increasing tendency. Once the pump current increases to 2Ith, a 6 GHz chaos bandwidth is creasing tendency. Once the pump current increases to 2Ith, a 6 GHz chaos bandwidth is obtained when the stage number is 10, and a 8 GHz chaos bandwidth can be achieved with 3Ith, as shown in Figure 10. with 3Ith, as shown in Figure 10.

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Increasing the pump current and hence the relaxation oscillation frequency is expected to enhance the bandwidth of chaos, and this is conﬁrmed in Figure 10. Here, we notice that mid-infrared chaos from ICL with different stage numbers has the same increasing Photonics 2021, 8, 366 tendency. Once the pump current increases to 2Ith, a 6 GHz chaos bandwidth is obtained9 of 13 Photonics 2021, 8, 366 9 of 13 when the stage number is 10, and a 8 GHz chaos bandwidth can be achieved with 3Ith, as shown in Figure 10. 9 9 m = 5 8 m m = = 5 10 8 7 m = 10 7 6 6 5 5 4 Bandwidth [GHz] Bandwidth 4 Bandwidth [GHz] Bandwidth 3

3 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 1.0 1.2 1.4Pump1.6 1.8 current2.0 2.2 2.4[I/I2.6th] 2.8 3.0 Pump current [I/I ] Figure 10. Bandwidth of chaos vs. thbias current when fext = 28% and τf = 2.4 ns with m = 5 (black squares), m = 10 (red circles). Figure 10. Bandwidth of chaos vs.vs. bias currentcurrent whenwhen ffextext = 2 28%8% and τff == 2.4 2.4 ns with m = 5 (black squares), m = 1 100 (red circles). Figure 11 respectively presents the time series, auto-correlation, phase diagram, RF spectrumFigure and 11 respectivelyoptical spectr presentspresental of mids the-infrared timetime series,series, chaos auto-correlation,autofor m-correlation, = 5 (in Figure phasephase 11a) diagram,diagram, and m = 1 RFRF0 spectrum(in Figure andand 11b). opticaloptical This spectralindicatesspectral of ofthat mid-infrared mid for-infrared relatively chaos chaos high for fornumberm m= 5= (in5 of (in Figure stages Figure 11 the a)11a) bandwidth and andm = m 10 = (inof10 (inFigurechaos Figure 11fromb). 11b). ICLs This This indicatesis further indicates thatenhanced that for relatively for, as relatively shown high in high numberFigure number 11 of(a stages-iv of,b- ivstages the). The bandwidth the enhanced bandwidth of band- chaos of chaosfromwidth ICLs from of the is ICLs furtherchaos is further is enhanced, due toenhanced the asrelaxation shown, as shown in oscillation Figure in Figure 11 frequency(a-iv,b-iv). 11(a-iv ,ofb The- ivthe) enhanced. TheICL enhancedincreasing bandwidth band- with widthofthe the number chaosof the chaos isof duestages. is to due the to relaxation the relaxation oscillation oscillation frequency frequency of the of ICL the increasingICL increasing with w theith thenumber number of stages. of stages.

Figure 11. Output chaos of ICL with external optical feedback as stage number increases with I = 1.9I , τ = 2.4 ns and Figure 11. Output chaos of ICL with external optical feedback as stage number increases with I = 1.9Ith, τthf = 2.4f ns and fext = fext28%.= 28%. (a) m (a =) m5, =(b 5,) m (b =) 1m0.= Columns 10. Columns (i-v) are (i–v) TS, are ACF, TS, ACF,PP, RFS, PP, RFS,OS, respectively. OS, respectively. Figure 11. Output chaos of ICL with external optical feedback as stage number increases with I = 1.9Ith, τf = 2.4 ns and fext = 28%. (a) m = 5, (b) m = 10. ColumnsTheThe (i-v) stagestage are TS, numbernumber ACF, PP, effecteffect RFS, isis OS, presentedpresented respectively. in Figure 12,12, where mm rangesranges from from 3 3 to to 20 20 and and thethe pump pump currents currents are are 1.1 1.1IthIth (black(black squares)squares) andand 2Ith (red(red circles). circles). Although Although the the increase increase in in The stage number effect is presented in Figure 12, where m ranges from 3 to 20 and bandwidthbandwidth forfor thethe relativelyrelatively highhigh pumppump current,current, 2Ith,, is is faster faster than than that that of of the the relatively relatively the pump currents are 1.1Ith (black squares) and 2Ith (red circles). Although the increase in lowlow pump pump current, current, 1.11.1IIthth, thethe tendency of the stage number number effects effects is is the the same, same, that that is is the the bandwidthbandwidth offorof chaos chaosthe relatively increasesincreases high asas thethe pump stagestage current, numbernumber 2 increases.Ith, is faster This than tendency tendency that of theis is verified veriﬁed relatively in in lowFiguresFigure pumps9 –911– 11.current,. 1.1Ith, the tendency of the stage number effects is the same, that is the bandwidth of chaos increases as the stage number increases. This tendency is verified in Figures 9–11.

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9 9 I = 1.1I 8 I = 1.1I th 8 th I = 2Ith 7 I = 2Ith 7 6 6 f = 30% 5 f ext = 30% 5 ext 4 4

3 fext = 3.9% fext = 3.9% Bandwidth [GHz] 3 Bandwidth [GHz] 2 2 1 1 2 4 6 8 10 12 14 16 18 20 2 4 6 8Stage10 number12 14 16 18 20 Stage number

Figure 12. Bandwidth of chaos vs. stage number I = 1.1Ith (black squares) and I = 2Ith (red circles) Figure 12.12. BandwidthBandwidth ofof chaoschaos vs.vs. stagestage numbernumber II == 1.11.1IIthth (black(black squares)squares) andand II == 22IIthth (red circles)circles) with τf = 2.4 ns. with τff = 2.4 2.4 ns. ns. By using Maximum Lyapunov exponents, we distinguish chaos from QP, and calcu- By using MaximumMaximum LyapunovLyapunov exponents, exponents, we we distinguish distinguish chaos chaos from from QP, QP, and and calculate calculate the 80% energy power bandwidth versus feedback power ratio and pump current, as thelate 80% the 80% energy energy power power bandwidth bandwidth versus versus feedback feedback power power ratio and ratio pump and current,pump current as shown, as shown in Figure 13. By using dashed curves, we distinguish QP and C, while the white inshown Figure in 13Figure. By using13. By dashed using dashed curves, curves, we distinguish we distinguish QP and QP C, whileand C, the while white the region white region represents the stable(S) output. It is found, in Figure 13a, that S is observed for a representsregion represents the stable(S) the stable output.(S) output It is found,. It is found in Figure, in Figure13a, that 13a, S isthat observed S is observed for a small for a small number of stages m = 5 when the bias current is below 1.2Ith and the feedback power numbersmall number of stages of stagesm = 5m when = 5 when the the bias bias current current is belowis below 1.2 1.2IthIthand and the the feedback feedback powerpower ratio is in the range of 4% to 30%. Stable output is also found in the top left corner of Figure ratio is is in in the the range range of of 4% 4% to to30% 30%.. Stable Stable output output is also is alsofound found in the in top the left top corner left corner of Figure of 13a. For a large stage number, that is m = 10, S is observed when the pump current is small, Figure13a. For 13 aa. largeFor astage large number, stage number, that is thatm = 1 is0,m S =is 10,observed S is observed when the when pump the pumpcurrent current is small, is small,below below 1.2Ith, 1.2andI the, and feedback the feedback power power ratio is ratio around is around 28%, 28%,as indicated as indicated in the in bottom the bottom right below 1.2Ith, andth the feedback power ratio is around 28%, as indicated in the bottom right corner of Figure 13b. S is also observed for m = 10 when the bias current is higher than rightcorner corner of Figure of Figure 13b. S13 isb. also S is observed also observed for mfor = 10m when= 10 when the bias the current bias current is higher is higher than than2.16I 2.16th andI theand feedback the feedback power power ratio is ratio less is than less 4% than, as 4%, seen as in seen Figure in Figure 13b. These 13b. Theseresults 2.16Ith andth the feedback power ratio is less than 4%, as seen in Figure 13b. These results indicate that an ICL with a relatively high bias current will exhibit S, QP and then enter resultsindicate indicate that an thatICL anwith ICL a withrelatively a relatively high bias high current bias current will exhibit will exhibit S, QP S,and QP then and enter then into C as the feedback power ratio increases. As the stage number increases from 5 to 10 enterinto C into as the C asfeedback the feedback power power ratio increases. ratio increases. As the Asstage the number stage number increases increases from 5 fromto 10 and the ICL is subject to a relatively high bias current, both S and QP regions are extended, 5and to 10the and ICL theis subject ICL is to subject a relatively to a relatively high bias high current, bias both current, S and both QP Sregions and QP are regions extended are, extended,as shown towards as shown the towards left of Figure the left 13a of Figure,b. This 13 confirmsa,b. This that conﬁrms for relatively that for few relatively stage num- few as shown towards the left of Figure 13a,b. This confirms that for relatively few stage num- stagebers an numbers ICL with an external ICL with optical external feedback optical is feedback amenable is amenableto exhibiting to exhibiting chaos. The chaos. results The for bers an ICL with external optical feedback is amenable to exhibiting chaos. The results for resultsboth cases for bothconfirm cases that conﬁrm in order that to inobtain order broadband to obtain broadbandmid-infrared mid-infrared chaos, one needs chaos, a one rel- both cases confirm that in order to obtain broadband mid-infrared chaos, one needs a rel- needsatively a high relatively pump high current pump as current well as as a large well asfeedback a large feedbackpower ratio. power Furthermore ratio. Furthermore,, a 8 GHz atively high pump current as well as a large feedback power ratio. Furthermore, a 8 GHz ab 8andwidth GHz bandwidth of mid-infrared of mid-infrared chaos can chaos be canobtained be obtained for m = for 10m, as= 10,shown as shown in the in top the right top bandwidth of mid-infrared chaos can be obtained for m = 10, as shown in the top right rightcorner corner of Figure of Figure 13b. 13b. corner of Figure 13b.

(a) (a) Figure 13. Cont.

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(b)

FigureFigure 13.13. BandwidthBandwidth vs. vs. feedback feedback power power ratio ratio and and pump pump current, current, with with τf = τ2.4f = ns. 2.4 (a ns.) m = (a 5), m(b)= m 5, = (1b0.) m = 10.

4.4. DiscussionDiscussion AA mid-infraredmid-infrared chaotic chaotic laser laser will will be be a promising a promising source source for forimplementing implementing secure secure free- free-spacespace optical optical communication. communication. In this In thispaper, paper, we weshow show how how to obtain to obtain broadband broadband mid mid--in- infraredfrared chaos chaos using using an an ICL ICL with with long long-cavity-cavity optical optical feedback. feedback. The The analysis analysis of the of thelaser laser sta- stabilitybility is effected is effected by byidentifying identifying the the Hopf Hopf bifurcation bifurcation point. point. From From such such considerations considerations we wefind ﬁnd that that ICLs ICLs with with a relatively a relatively small small number number of stages of stages are aremore more unstable unstable and andthus thus sus- susceptibleceptible to toexhibiting exhibiting chaos. chaos. The The chaos chaos bandwidth bandwidth of of the the ICL ICL is is related to thethe relaxationrelaxation oscillationoscillation frequency,frequency, which which increases increases linearly linearly with with the the cube cube root root of of the the stage stage number number andand laserlaser pumppump current.current. Therefore,Therefore, toto obtainobtain broadbandbroadband chaoschaos fromfrom anan ICLICL therethere isis aa needneed forfor aa highhigh pumppump current,current, largelarge numbernumber ofof stagesstages asas wellwell asas strongstrong opticaloptical feedbackfeedback strength.strength. OurOur calculationscalculations showshow thatthat 88 GHzGHz bandwidthbandwidth mid-infraredmid-infrared chaoschaos cancan bebe generatedgeneratedusing using aa 10-stage10-stage ICLICL biasedbiased atat threethree timestimes thethe thresholdthreshold current.current. WithWith thethe availabilityavailability ofof suchsuch broadbandbroadband mid-infraredmid-infrared chaos, chaos, it it is is expected expected that that of-order of-order Gbit/s Gbit/s securesecure free-spacefree-space opticaloptical communication is feasible. communication is feasible.

Author Contributions: Conceptualization, H.H. and K.A.S.; software, X.C.; formal analysis and Author Contributions: Conceptualization, H.H. and K.A.S.; software, X.C.; formal analysis and val- validation, H.H. and K.A.S.; investigation, K.A.S.; resources, H.H.; data curation, X.C.; writing— idation, H.H. and K.A.S.; investigation, K.A.S.; resources, H.H.; data curation, X.C.; writing—origi- original draft preparation, H.H.; writing—review and editing, K.A.S.; visualization, X.C.; supervision, nal draft preparation, H.H.; writing—review and editing, K.A.S.; visualization, X.C.; supervision, H.H. and Z.J.; project administration, H.H.; funding acquisition, H.H. and Z.J. All authors have read H.H. and Z.J.; project administration, H.H.; funding acquisition, H.H. and Z.J. All authors have read andand agreedagreed toto thethe publishedpublished versionversion ofof thethe manuscript.manuscript. Funding: Funding: ThisThis researchresearch waswas fundedfunded byby NationalNational NaturalNatural ScienceScience FoundationFoundation ofof ChinaChinagrant grant numbernumber 6180516861805168 andand 61741512,61741512, ResearchResearch ProjectProject SupportedSupported byby ShanxiShanxi ScholarshipScholarship CouncilCouncil ofof ChinaChina grantgrant numbernumber 2021032,2021032, ShanxiShanxi “1331“1331 Project”Project” KeyKey InnovativeInnovative Team; ProgramProgram forfor TopTop YoungYoung andand Middle-Middle- agedaged Innovative Innovative Talents Talents of of Shanxi, Shanxi, andand Key Key Research Research and and Development Development Project Project grant grant number number 201903D121124.201903D121124. InstitutionalInstitutional ReviewReview BoardBoard Statement:Statement:Not Not applicable.applicable. InformedInformed ConsentConsent Statement:Statement:Not Not applicable.applicable. DataData AvailabilityAvailability Statement:Statement:The The datadata presentedpresented inin thisthis studystudy areare availableavailable onon requestrequest fromfrom thethe correspondingcorresponding author.author. ConﬂictsConflicts of of Interest: Interest:The The authors authors declare declare no conﬂictno conflict of interest. of interest. The fundersThe funders had no had role no in therole design in the ofdesign the study; of the in study; the collection, in the collection, analyses, analyses, or interpretation or interpretation of data; inof thedata; writing in the ofwriting the manuscript, of the manu- or inscript, the decision or in the to decision publish to the publish results. the results.

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Abbreviations The following abbreviations are used in this manuscript: ICL interband cascade laser LEF linewidth enhancement factor QCL quantum cascade laser TS time series ACF autocorrelation curves functions PP phase portrait RFS radio-frequency spectrum OS optical spectral S stable P1 period-1 QP quasi-periodic MP multiple-periodic C chaos

References 1. Lin, C.-H.; Yang, R.Q.; Zhang, D.; Murry, S.; Pei, S.; Allerman, A.; Kurtz, S. Type-II interband quantum cascade laser at 3.8 µm. Electron. Lett. Abbrevi. 1997, 33, 598–599. [CrossRef] 2. Yang, R.Q.; Bradshaw, J.L.; Bruno, J.D. Room temperature type-II interband cascade. Appl. Phys. Lett. 2002, 81, 397–399. [CrossRef] 3. Yang, R.Q.; Hill, C.J.; Yang, B.H.; Wong, C.M.; Muller, R.E.; Echternach, P.M. Continuous-wave operation of distributed feedback interband cascade lasers. Appl. Phys. Lett. 2004, 84, 3699–3701. [CrossRef] 4. Kim, M.; Canedy, C.L.; Bewley, W.W.; Kim, C.S.; Lindle, J.R.; Abell, J.; Vurgaftman, I.; Meyer, J.R. Interband cascade laser emitting at λ = 3.75 µm in continuous wave above room temperature. Appl. Phys. Lett. 2008, 92, 191110. [CrossRef] 5. Bagheri, M.; Frez, C.; Sterczewski, L.; Gruidin, I.; Fradet, M.; Vurgaftman, I.; Canedy, C.L.; Bewley, W.W.; Merritt, C.D.; Kim, C.S.; et al. Passively mode-locked interband cascade optical frequency combs. Sci. Rep. 2018, 8, 3322. [CrossRef][PubMed] 6. Yang, H.; Yang, R.Q.; Gong, J.; He, J.-J. Mid-Infrared Widely Tunable Single-Mode Interband Cascade Lasers Based on V-Coupled Cavities. Opt. Lett. 2020, 45, 2700–2702. [CrossRef][PubMed] 7. Chen, C.C. Attenuation of Electromagnetic Radiation by Haze, Fog, Clouds, and Rain; Rand Corp.: Santa Monica, CA, USA, 1975; pp. 1–39. 8. Vurgaftman, I.; Weih, R.; Kamp, M.; Meyer, J.R.; Canedy, C.L.; Kim, C.S.; Kim, M.; Bewley, W.W.; Merritt, C.D.; Abell, J.; et al. Interband cascade lasers. J. Phys. D Appl. Phys. 2005, 48, 123001. [CrossRef] 9. Horstjann, M.; Bakhirkin, Y.; Kosterev, A.; Curl, R.; Tittel, F.; Wong, C.; Hill, C.; Yang, R. Formaldehyde sensor using interband cascade laser based quartz-enhanced photoacoustic spectroscopy. Appl. Phys. B 2004, 79, 799–803. [CrossRef] 10. Wysocki, G.; Bakhirkin, Y.; So, S.; Tittel, F.K.; Hill, C.J.; Yang, R.Q.; Fraser, M.P. Dual interband cascade laser based trace-gas sensor for environmental monitoring. Appl. Opt. 2007, 46, 8202–8209. [CrossRef] 11. Risby, T.H.; Tittel, F.K. Current status of midinfrared quantum and interband cascade lasers for clinical breath analysis. Opt. Eng. 2010, 49, 111123. 12. Soibel, A.; Wright, M.W.; Farr, W.H.; Keo, S.A.; Hill, C.J.; Yang, R.Q.; Liu, H.C. Midinfrared Interband Cascade Laser for Free Space Optical Communication. IEEE Photonics Technol. Lett. 2010, 22, 121–123. [CrossRef] 13. Yang, R.Q.; Li, L.; Jiang, Y.C. Interband cascade laser: From original concept to actual device. Prog. Phys. 2014, 34, 169–190. 14. Deng, Y.; Zhao, B.B.; Wang, C. Linewidth broadening factor of an interband cascade laser. Appl. Phys. Lett. 2019, 115, 181101. [CrossRef] 15. Green, R.P.; Xu, J.-H.; Mahler, L.; Tredicucci, A.; Beltram, F.; Giuliani, G.; Beere, H.E.; Ritchie, D. Linewidth enhancement factor of terahertz quantum cascade lasers. Appl. Phys. Lett. 2008, 92, 071106. [CrossRef] 16. Deng, Y.; Fan, Z.F.; Wang, C. Optical feedback induced nonlinear dynamics in an interband cascade laser. Proc. SPIE 2021, 11680, 116800J. 17. Spitz, O.; Herdt, A.; Wu, J.; Maisons, G.; Carras, M.; Wong, C.-W.; Elsäßer, W.; Grillot, F. Private communication with quantum cascade laser photonic chaos. Nat. Commun. 2021, 12, 3327. [CrossRef] 18. Caffey, D.; Day, T.; Kim, C.S.; Kim, M.; Vurgaftman, I.; Bewley, W.W.; Lindle, J.R.; Canedy, C.L.; Abell, J.; Meyer, J.R. Performance characteristics of a continuous wave compact widely tunable external cavity interband cascade lasers. Opt. Express 2010, 18, 15691. [CrossRef] 19. Wang, A.; Li, P.; Zhang, J.; Zhang, J.; Li, L.; Wang, Y. 4.5 Gbps high-speed real-time physical random bit generator. Opt. Express 2013, 21, 20452–20462. [CrossRef] 20. Wang, A.; Yang, Y.; Wang, B.; Zhang, B.; Li, L.; Wang, Y. Generation of wideband chaos with suppressed time-delay signature by delayed self-interference. Opt. Express 2013, 21, 8701–8710. [CrossRef][PubMed] Photonics 2021, 8, 366 12 of 12

21. Wang, L.; Wang, D.; Gao, H.; Guo, Y.; Wang, Y.; Hong, Y.; Shore, K.A.; Wang, A. Real-time 2.5-Gb/s correlated random bit generation using synchronized chaos induced by a common laser with dispersive feedback. IEEE J. Quantum Electron. 2020, 56, 1–8. [CrossRef] 22. Zhang, T.; Jia, Z.; Wang, A.; Hong, Y.; Wang, L.; Guo, Y.; Wang, Y. Experimental observation of dynamic-state switching in VCSELs with optical feedback. IEEE Photonics Technol. Lett. 2021, 33, 335–338. [CrossRef] 23. Jumpertz, L.; Schires, K.; Carras, M.; Sciamanna, M.; Grillot, F. Chaotic light at mid-infrared wavelength. Light Sci. Appl. 2016, 5, e16088. [CrossRef] 24. Spitz, O.; Wu, J.; Carras, M.; Wong, C.-W.; Grillot, F. Low-frequency fluctuations of a mid-infrared quantum cascade laser operating at cryogenic temperatures. Laser Phys. Lett. 2018, 15, 116201. [CrossRef] 25. Jumpertz, L.; Carras, M.; Grillot, F. Regimes of external optical feedback in 5.6 µm distributed feedback mid-infrared quantum cascade lasers. Appl. Phys. Lett. 2014, 105, 131112. [CrossRef] 26. Panajotov, K.; Sciamanna, M.; Arteaga, M.A.; Thienpont, H. Optical feedback in vertical-cavity surface-emitting lasers. IEEE J. Sel. Top. Quantum Electron. 2013, 19, 1700312. [CrossRef] 27. Yang, R.Q. Mid-infrared interband cascade lasers based on type-II heterostructures. Microelectron. J. 1999, 30, 1043–1056. [CrossRef] 28. Lang, R.; Kobayashi, K. External optical feedback effects on semiconductor injection laser properties. IEEE J. Quantum Electron. 1980, 16, 347–355. [CrossRef] 29. Deng, Y.; Wang, C. Rate Equation modeling of interband cascade lasers on modulation and noise dynamics. IEEE J. Quantum Electron. 2020, 56, 2300109. [CrossRef] 30. Bewley, W.W.; Lindle, J.R.; Kim, C.S.; Kim, M.; Canedy, C.L.; Vurgaftman, I.; Meyer, J.R. Lifetimes and Auger coefficients in type-II W interband cascade lasers. Appl. Phys. Lett. 2008, 93, 041118. [CrossRef] 31. Vurgaftman, I.; Canedy, C.L.; Kim, C.S.; Kim, M.; Bewley, W.W.; Lindle, J.R.; Abell, J.; Meyer, J.R. Mid-infrared interband cascade lasers operating at ambient temperatures. New J. Phys. 2009, 11, 125015. [CrossRef] 32. Vurgaftman, I.; Canedy, C.L.; Kim, C.S.; Kim, M.; Bewley, W.W.; Lindle, J.R.; Abell, J.; Meyer, J.R. Mid-IR type-II interband cascade lasers. IEEE J. Sel. Top. Quantum Electron. 2011, 17, 1435–1444. [CrossRef] 33. Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from a time-series. Phys. D Nonlinear Phenomena. 1985, 16, 285–317. [CrossRef] 34. Wolf, A. Wolf Lyapunov Exponent Estimation from A Time Series. MATLAB Central File Exchange. Available online: https: //www.mathworks.com/matlabcentral/fileexchange/48084-wolf-lyapunov-exponent-estimation-from-a-time-series (accessed on 5 January 2021). 35. Wang, D.; Wang, L.; Li, P.; Zhao, T.; Jia, Z.; Gao, Z.; Guo, Y.; Wang, Y.; Wang, A. Bias Current of Semiconductor Laser-An Unsafe Key for Secure Chaos Communication. Photonics 2019, 6, 59. [CrossRef] 36. Huang, Y.; Zhou, P.; Li, N.Q. Broad tunable photonic microwave generation in an optically pumped spin-VCSEL with optical feedback stabilization. Opt. Lett. 2021, 46, 3147–3150. [CrossRef] 37. Wang, B.; Wang, Y.; Kong, L.; Wang, A. Multi-target real-time ranging with chaotic laser radar. Chin. Opt. Lett. 2008, 6, 868–870. [CrossRef] 38. Li, P.; Guo, Y.; Guo, Y.; Fan, Y.; Guo, X.; Liu, X.; Shore, K.A.; Dubrova, E.; Xu, B.; Wang, Y.; et al. Self-balanced real-time photonic scheme for ultrafast random number generation. APL Photonics 2018, 3, 061301. [CrossRef] 39. Wang, L.; Mao, X.; Wang, A.; Wang, Y.; Gao, Z.; Li, S.-S.; Yan, L. Scheme of coherent optical chaos communication. Opt. Lett. 2020, 45, 4762. [CrossRef] 40. Lin, F.Y.; Liu, J.M. Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback. Opt. Commun. 2003, 221, 173–180. [CrossRef]