Dynamics of a Micro-VCSEL Operated in the Threshold Region Under Low-Level Optical Feedback

Dynamics of a micro-VCSEL operated in the threshold region under low-level optical feedback Tao Wang, Xianghu Wang, Zhilei Deng, Jiacheng Sun, Gian Piero Puccioni, Gaofeng Wang, and Gian Luca Lippi

Abstract—Semiconductor lasers are notoriously sensitive to quality, ease of integration and single-longitudinal mode emis- optical feedback, and their dynamics and coherence can be sion. In addition, they are less sensitive to optical feedback significantly modified through optical reinjection. We concentrate than their edge-emitting counterparts, rendering them more on the dynamical properties of a very small (i.e., microscale) Vertical Cavity Surface Emitting Laser (VCSEL) operated in attractive in many applications, in spite of their polariza- the low coherence region between the emission of (partially) tion sensitivity on which have focussed most of the optical coherent pulses and ending below the accepted macroscopic reinjection investigations (cf., e.g., [8], [9], [10], [11]). Our lasing threshold, with the double objective of: 1. studying the work focuses on the basic understanding of operation regimes feedback influence in a regime of very low energy consump- of very small devices which, in the future, could be used tion; 2. using the micro-VCSEL as a surrogate for nanolasers, where measurements can only be based on photon statistics. for transmissions and interconnects (e.g. datacenter applica- The experimental investigation is based on time traces and tions [12], [13]). Thus, we look at the dynamics of VCSELs radiofrequency spectra (common for macroscale devices) and due to non-polarization-selective feedback [14], [15], [16]. An correlation functions (required at the nanoscale). Comparison overview of the phenomenology observed in VCSELs with of these results confirms the ability of correlation functions to optical feedback can be found in [17]. satisfactorily characterize the action of feedback on the laser dynamics. Numerical predictions obtained from a previously VCSELs themselves are at the origin of laser miniatur- developed, fully stochastic modeling technique provide very close ization with the first design of a vertical semicounductor agreement with the experimental observations, thus supporting cavity [18], [19], which later branched along several inde- the possible extension of our observations to the nanoscale. pendent directions (e.g., photonic-crystal-based devices [20]). Index Terms—Semiconductor micro-VCSEL, optical feedback, The very low threshold and low power dissipation typical coherence, correlation functions, nonlinear dynamics. of nanolasers promise breakthroughs in a broad palette of applications [21], which, for our purposes, cover light sources for all-optical chips [22], data centers [23], [12], [13] and I.INTRODUCTIONANDOBJECTIVES quantum information [24], [25], [26], [27]. However, their Optical communications have been the prime mover behind extremely small photon numbers render an in-depth character- the investigation of optical feedback on the emission properties ization quite challenging. Thus, aside from a couple of older of semiconductor lasers, due to the extreme sensitivity of investigations [28], [29] (even at the single-photon level [30]), edge-emitting devices to even very low reinjection levels (e.g., only recently have concerted efforts surfaced, covering opto- from fiber ends [1]). The noise and coherence properties, as electronic feedback [31] or external light feedback both in well as the emitter’s dynamical stability, are thus modified, microcavities [32], [33], [34] and in photonic-crystal-based with detailed features depending on whether the reinjected Fano-lasers [35], [36]. light fraction carries phase information [2], [3] (coherence Aiming at future on chip and datacenter applications, which length Lc > Lf , Lf feedback length) or whether only the require extremely low power consumption, we concentrate reinjected photon fraction counts [4], [5] (Lc < Lf ). The large on the investigation of feedback on the emission at bias number of investigations dedicated to the study of feedback current levels in the region between the first light emission is reviewed in different papers, depending on the scope: for and the traditional threshold [37], where we have shown that instance, optical transmission systems [6] or understanding it is possible to obtain reliable pulse generation at lower energetic costs [38]. Our objective is threefold: understanding arXiv:1907.00145v1 [physics.optics] 29 Jun 2019 the dynamical features of semiconductor lasers with optical reinjection [7]. the dynamical influence of feedback on a small (mesoscale) Developed in the 1980’s and 1990’s, Vertical Cavity Surface laser at ultra-low bias (i.e., below the traditional thresh- Emitting Lasers (VCSELs) have become the most widespread old) where the deterministic dynamics mixes with stochastic coherent light sources thanks to their versatility, good beam behaviour [39]; testing the predictive capabilities of fully stochastic modeling [40]; and probing the second-order (zero- T. Wang, X. Wang, Z. Deng, J. Sun and G. Wang are with School of delay) autocorrelation as a suitable dynamical indicator. This Electronics and Information, Hangzhou Dianzi University, Hangzhou, 310018, last point aims at validating, on a micro-VCSEL [41], the China, e-mail: [email protected] G. P. Puccioni is with Istituto dei Sistemi Complessi, CNR, Via use of the only technique currently usable to characterize Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy, e-mail: gian- nanolasers, the target devices for datacom applications. [email protected] The need for photon statistics to investigate nanolaser G. L. Lippi is with Universite´ Coteˆ d’Azur, Institut de Physique de Nice (INPHYNI), UMR 7010 CNRS, 1361 Route des Lucioles, F-06560 Valbonne, behaviour (including their threshold properties) comes from France, e-mail: [email protected] the lack of available detectors with sufficient sensitivity and, 2 simultaneously, electrical bandwidth to obtain a full characterization of the laser output. The extremely low photon flux (typically at the sub-nW, or even pW level) renders interfer- ometry challenging for routine tests. Thus, photon statistics becomes the obvious choice, given that it possesses both the necessary time response and optical sensitivity. However, since it only provides statistical information, the identification of the emission features must be based on models, or – as is the case with our present proposal – on the comparison with other techniques. A discussion on the pertinency of microlaser investigations to learn about nanolaser behaviour and comparison between linear measurements and photon statistics can be found in [41].

II.EXPERIMENTAL DETAILS AND OBSERVATIONS The experimental setup is shown in Fig. 1. The laser is a Fig. 1. Schematic illustration of the experimental design: LD, semiconductor VCSEL 980 (Thorlabs), designed for LAN data transmission laser diode (VCSEL); BS, beam splitter; M, dielectric mirror; PD, fast photodetector. The semiconductor micro-VCSEL is temperature stabilized at at 2.5 Gb/s [42], electrically supplied by a stabilized current 25◦C, powered by a commercial dc power supply (Thorlabs LDC200VCSEL) source (Thorlabs LDC200VCSEL), temperature-stabilized by with resolution 1µA and accuracy ±20µA. a home-made controller to better than 0.1◦C, with estimated β ≈ 10−4 [37]. The nominal threshold current declared by the Manufacturer [42] is typically ith ≈ 2.2mA, but can be as fied photodetector (Thorlabs PDA8GS, 9.5GHz bandwidth), high as ith,max = 3.0mA for some devices, while the maxi- to avoid backreflected contributions coming from the latter. mum operating current is imax = 10mA with corresponding The electrical signal from the photodetector is digitized by maximum laser output Pmax ≈ 1.85mW. The laser emits on a LeCroy Wave Master 8600A oscilloscope (6GHz analog a single polarization until a pump current value i ≈ 2.5mA bandwidth – acquiring 1 × 106 points in all measurements). (corresponding to the limit of the range we investigate) with a The data are stored in a computer through a GPIB interface rejection ratio of approximately 24 dB (spontaneous emission controlled in Python. The second-order autocorrelation func- is, of course, isotropic). Additional technical information can tion is numerically computed from the data trace acquired by be found in the Supplementary Information section of [37]. the linear detector, as in [37]. Comparison between the coherence properties of this de- Fig. 2 shows, in double-logarithmic scale, the average laser vice [37] and the manufacturer’s specifications suggest a output in the presence of the external cavity (squares) for consistency between the maximum specified threshold current different values of the pumping current, compared to the same value and the threshold for laser coherence, since the latter is response in the absence of feedback (circles). The error bars attained close to ith,max [37]. However lasing emission, in the are computed from the fluctuations in the measured signals. In form of partially coherent spikes, can already be obtained from the absence of feedback, emission from the laser in the form i ≈ 1.26mA [37], thus providing an interesting pumping range of irregular bursts has been observed for i = 1.26mA [37], below the coherence threshold where the influence of feedback thus we choose this bias point as reference (ith = 1.26mA on coherence buildup can be tested, and opening potential new hereafter) to normalize all currents to a common value. As −5 windows for applications of this self-pulsing regime. in macroscopic lasers (β / 10 ), the addition of feedback A dielectric, high-reflectivity mirror is aligned in front of introduces a reduction in the observed threshold (≈ 4%, com- the VCSEL at a distance Lec (2Lec = (70 ± 4) cm – i.e., an patible with the findings of [43]). The inset (Fig. 2) displays external cavity mode spacing ∆νec ≈ 0.23GHz) to provide a detail of the laser response in linear scale, confirming that the external cavity; inside the cavity we find a commercial, feedback lowers the emission threshold and proving that the AR-coated collimator (identified as “lens” in Fig. 1), a non- largest fluctuations occur at i ≈ 1.6ith. polarizing beamsplitter (BS) to send part of the output to a Radiofrequency (rf) power spectra characterize the observed detector, and a variable attenuator, used to fix the amount dynamics. Fig 3(a) shows the rf spectrum at i = ith: peaks of power reinjected into the laser by the external cavity: appear at the external cavity repetition frequency (∆νec ≈ throughout the experiment we used a reinjection coefficient 0.23 GHz) and its harmonics, where the third one, located ≈ 1.5%. This choice is made to match parasitic reinjection close to an intrinsic “resonance” in the spiking (cf. [37], values which would originate from residual and unwanted [44], [45] for its interpretation), presents a somewhat stronger backreflections. Since it is not a critical parameter (no dramatic amplitude and slightly narrower peak, but no additional feature changes appear as long as the reflectivity is kept low) and since (as in the free-running case [44]). the actual value would depend on the configuration (whether The onset of coherence is signalled by the appearance of a in free-space or integrated optics, etc.), we keep it constant broad resonance (Fig. 3b) – as in the free-running laser [46] throughout the experiment. – onto which a comb of external cavity modes is superim- A Faraday isolator (QIOPTIQ8450-103-600-4-FI-980-SC, posed. A partial loss of “long-term” memory is signalled by 40 dB isolation ratio) is placed in front of the fast, preampli- drowning of the first two harmonic components of ∆νec into 3

Fig. 2. Main panel: log-log laser response with (squares, blue online) and without feedback (circles, red online). Inset: detail in linear scale.

Fig. 4. Temporal intensity data traces at (a)1.00ith, (b)1.10ith, and (c)2.00ith. The insets show corresponding sample traces in the free-running laser.

that the laser output is gaining in overall phase coherence by being capable of maintaining and partially imprinting the (relatively) long timescale features of the external cavity onto the laser output, while at the onset of coherent output [45] memory was very limited (panel b). Notice that the low- frequency part of the spectrum, ν < 0.25 GHz, remains practically unchanged between panels (b) and (c) – unlike the free-running laser [46] –, suggesting that the low-frequency part of the spectrum be a mixture of “slow” intrinsic dynamics and of LFFs, where the latter gain in relevance with bias (presumably due to growing coherence). Inspection of the temporal traces provides additional information. At low pump rate (panel (a), Fig. 4) the temporal output is rendered more regular by feedback (main panel) which encourages photon emission through re-injection (compare to the irregular bursts of the free-running regime – inset). A slightly negative residual bias in the detector’s Fig. 3. Radiofrequency spectra for (a) ipump = 1.00ith, (b) 1.10ith, and (c)2.00ith. The very narrow lines appearing in all spectra are artefacts due output produces, at times, “negative” values for output power to spurious rf components presents in the environment. The three dashed (rather than drops to zero). Larger fluctuations (i.e., involving vertical lines, across the three panels, mark the position of the 4th, 8th, and 12th harmonic component of the external cavity’s roundtrip frequency. sharper drops in photon emission) are instead observed at i = 1.1ith (Fig. 4(b)), but no strong differences are identifiable when comparing to the feedback-less case. Power drops (LFF) the low-frequency spectral enhancement (0 ≤ ν / 0.25 GHz), become more noticeable at larger pump values (Fig. 4c) compatible with weak LFFs. Finally, for well-established laser with dynamics clearly different from the free-running regime operation (Fig. 3c) a stronger broad peak appears centered at (inset). ν ≈ 2 GHz, signature of free-running laser operation [46], The tools employed so far for the characterization of the upon which the same external cavity frequency comb is super- experimental observations are those typical of macroscopic posed. It is interesting to notice, however, that the frequency lasers. We now consider indicators which become indispens- th th harmonics of ∆νec are extended: hints of the 12 and 13 able for micro- and nanoscale devices [28], [32], [34], but component are visible here, while in panel (b) one could barely have only occasionally been used for macroscopic lasers [47]. distinguish the 9th component. Also on the low-frequency side Photon counting is the only experimentally available tech- there is an improvement in the visibility of the harmonics, with nique with sufficient bandwidth and sensitivity for the very even an identifiable trace of the 1st component. This suggests low photon fluence values of nanolasers. It is based on the 4 measurement of the second-order correlation function g(2)(τ): hI(t + τ)I(t)i g(2)(τ) = , (1) hI(t + τ)ihI(t)i where I is the measured instantaneous laser intensity, τ is the time delay, and h·i denotes statistical averaging. If the emitted light is coherent, g(2)(τ) = 1 independently of the τ value, while thermal radiation is characterized by g(2)(τ = 0) = 2, with a gradual decay towards 1 as the emitted field acquires a degree of coherence. Since our setup measures the dynamics of the intensity, I, with a temporal resolution ∆to = 0.1ns, we can directly compute the correlation, eq. (1), from the time series [37], [45]. Notice that the filtering introduced by the limited electronic bandpass reduces the absolute values of the correlations [37], [48], while maintaining its functional shape. Fig. 5 compares the zero-delay (τ = 0) autocorrelation (2) function obtained from data in the absence of feedback [37] Fig. 5. Measured second order autocorrelation (g (τ = 0)) as a function of bias: free-running laser (dashed, black, line); fed-back laser (solid, red, line). (dashed, black line) and in the presence of feedback (solid, Inset: detail of the autocorrelation in the presence of feedback for low pump red line). Below ith (cf. inset for details) feedback induces values. strong changes in the autocorrelation as shown not only by its “oscillations” in the mean value but also by the large error bars. This regime (i < ith) corresponds to the feedback- modulating the revival peak and originating from the intrinsic induced extension of the emission below threshold, where laser resonance (spectral component at ≈ 1 GHz in Fig. 3b); (2) the photons reinjected into the laser by the external cavity and 2. the revival’s width (two full oscillations in g (τ)) act as an additional source of “spontaneous emission noise”, which corresponds to the spectral width of the laser resonance whence the strongly fluctuating correlations. The autocorrela- (≈ 2 GHz in Fig. 3b) – broader than in the self-spiking regime. tion is (slightly) lower in the presence of feedback than in its The same holds at the larger pump value (Fig. 6c), where the (2) absence, in agreement with the temporal traces (Fig. 4a) which “faster” oscillation in g (τ) comes from the larger central show a less irregular signal with optical reinjection. A more frequency value in Fig. 3c, with very similar spectral width. important observation, instead, is that coherence is degraded The comparison offered by our experiment, capable of by feedback for i > 1.04ith. Although this behaviour requires providing at the same time direct (through temporal traces further investigation, it is plausible to relate its occurrence and easy access to rf power spectra) and statistical information to the fact that even when the laser emits a (noisy) cw field (through autocorrelations) enables us to compare the perfor- (i ' 1.12ith [45]), its coherence remains low, as proven by mance of the two techniques and confirm that correlations the lack of convergence of g(2)(0) towards 1 (reached only for can be used to interpret experimental results obtained from i > 3mA [37]). It appears that in this less-coherent regime much smaller lasers (e.g., [34]), where only statistical mea- optical reinjection reduces the, already low, coherence rather surements are possible. Although experimental proof for our than enhancing it, as observable in larger devices and well claim is currently missing, as we are validating the comparison above threshold. between dynamical and statistical techniques for the first time, The delayed (τ 6= 0) autocorrelation, eq. (1), provides it is plausible that the concurrence of the information should additional information on the laser dynamics for the three hold irrespective of the laser size (i.e., cavity β), or degree selected pump values already shown in Figs. 3 and 4. Fig. 6a of coherence of the emitted radiation. Indeed, while at larger shows a coherence revival at τ ≈ 4.3ns with a second order values of β noise plays a more substantial role, correlation multiple (the third is barely recognizable) which corresponds functions, spectra and time traces carry the corresponding to the roundtrip time in the external cavity. In the finer details, information, even if its relevance may be at times more one can recognize the signature of an intrinsic periodicity with difficult to identify in one of the chosen indicators. In addition, δt ≈ 0.6ns stemming from pulse repetition [37], [44]. How- fully Stochastic Simulations [40] have been shown to well- ever, since no precise spectral feature is recognizable in the reproduce experimental observations [37] and can span the self-spiking regime in the absence of feedback [37], [44], this whole range of meso- and nanolaser scales [41] (cf. Sup- component would be expected to be very small. Alternately, plementary Material for additional information). Below, we the additional ripples superposed on the correlation signal test this numerical technique to compare its predictions to our (Fig. 6a) may also originate from the details of the spiking observations in the limit of incoherent feedback. dynamics which, for the moment, remain unclear. In addition to the limitations already found in matching A drastic change in the laser output characteristics is visible experimental observations in “large” VCSELs [49] to Lang- in the autocorrelation at i = 1.1ith (panel b) where the trace Kobayashi models [50], theoretical support for experiments is much smoother and, in addition to the signature of the is difficult to obtain for small lasers, where the traditional external cavity (as in panel a), one recognizes in the coherence rate-equations-based models (or even Maxwell-Bloch ones) revival two features: 1. an oscillation with period Tlas ≈ 1ns, cannot be used in conjunction with noise sources. Although 5

Fig. 7. Numerical time traces computed for normalized pump values P : Pth (a) 1.0; (b) 1.1, and (c) 2.0.

Fig. 6. Time-delayed second-order autocorrelation for: ipump = 1.00 ith (a), 1.10 ith (b) and 2.00 ith (c). through the output mirror, with a time delay equal to the experimentally measured one, and entering the cavity with sufficiently far from threshold predictions of stochastic rate a Poissonian probability law, in agreement with the whole equations hold for macroscopic lasers, at the meso- (and conceptual framework of the Stochastic Simulator [40] (cf. nano-) scale there appear intrinsic violations of the necessary Supplementary Material for additional information). conditions for a proper Langevin noise description [51], [52], Fig. 7 shows a sample of the numerically predicted temporal due to memory effects in the carriers typical of Class B laser dynamics in the presence of feedback for pump values lasers [53], [54]. We therefore resort to using a fully stochastic corresponding to the experimental ones. Comparison to the technique [40], based on recurrence relations which make use experimental measurements (Fig. 4) shows a good qualitative of Einstein’s semiclassical field theory [55] (for more infor- agreement and comparable features: an irregular spiking be- mation cf. Supplementary Material available online). Since haviour is found in Fig. 7a, while Fig. 7b and Fig. 7c show a the latter does not consider field phases, it is necessary to noisier output. check the field’s coherence in our experiment. Transfer of A good qualitative agreement also exists when compar- these findings to other systems will entirely depend on the ing the numerically predicted rf spectra (Fig. 8) with the coherence length of the emission regime (low, for instance, in experimentally measured ones: in the self-spiking regime nanolasers in the threshold region) compared to the external (panel a) there is no apparent background and the four peaks cavity length. corresponding to the external cavity (multiples) are closely In the self-spiking regime (i / 1.1ith), the emission reproduced; at coherence onset (panel b) the intrinsic rf laser resembles (filtered) Amplified Spontaneous Emission, thus resonance appears with the superposed external cavity comb, one expects emission linewidths of the order of a nanometer while for established oscillation (but not full coherence! – (experimentally confirmed) and thus coherence lengths entirely cf. Fig. 5) the broad rf laser resonance moves between 2 negligible compared to the external feedback length 2Lec. In and 3 GHz with the superposed frequency comb (Fig. 8c). the partially coherent regime (1.1ith / i / 2ith), the spectral The main discrepancy appears in the low-frequency part of width is of the order of hundreds of GHz (in the middle of the the spectrum, which decreases instead of increasing (Fig. 3). bias range), leading to coherence lengths Lc ≈ K mm 2Lec This is probably due to the fact that the model does not keep (K = O(1)). Thus, we can consider the feedback contribution long-term memory and cannot properly reproduce the LFFs, in as being constituted (mostly) of incoherent photons (as in [5], spite of a qualitative resemblance between the temporal traces but with an entirely different physical origin) reinjected into (Figs. 4 and 7, panels (c)). the cavity and simply added to the intracavity photon number. As a last step, we compare the predicted correlations, This process is simulated by considering a stochastic trans- eq. (1), to the experimental ones in Fig. 9. Aside from mission of a given fraction (1.5%, as in the experimental a somewhat smoother signal, devoid of the technical noise estimates) of the backreflected photon number coming back which affects the measurements (especially in the self-pulsing 6

Fig. 9. Numerical time-delayed second-order autocorrelation for normalized Fig. 8. Numerical rf spectra for normalized pump values P : (a) 1.0; (b) P Pth pump values P : (a) 1.0; (b), 1.1, and (c),2.0. 1.1, and (c) 2.0. th regime) the agreement is excellent! All features analyzed in field’s phase progressively gains relevance in the gradual Fig. 6 are found here, thus validating the interpretations that transition towards coherent laser emission. Nonetheless, the we have previously offered. A careful look at Fig. 8c shows a scheme’s validity in the fully incoherent regime promises to faster drop in the recurrences of the autocorrelation function, provide insights into the feedback dynamics at the smallest while the slightly negative overall slope is currently attributed laser scales [45]. to numerical problems. ACKNOWLEDGMENT III.CONCLUSIONS The authors are grateful to the Region´ PACA and BBright An experiment performed in a micro-VCSEL, pumped in for support, and to B. Garbin, F. Gustave and M. Marconi the transition region between incoherent and coherent emis- for assistance and discussions. Technical support from J.-C. sion with reinjection levels (from a “long” external cavity) Bery (mechanics) and from J.-C. Bernard and A. Dusaucy compatible with parasitic reflections shows a lowering of the (electronics) is gratefully acknowledged. T. W. thanks the minimum bias value to obtain light emission accompanied by a scientific research starting fund (KYS045618036) and national reduction of the photon bursts typical of the free-running laser. natural science foundation of China (61804036). G.L. L. The influence of the optical reinjection remains moderate until acknowledges discussions with T. Ackemann, M. Giudici, J. the actual coherence threshold (g(2)(0) = 1) is approached. Mørk and S. Reitzenstein. The authors are grateful to two Comparison between traditional dynamical indicators and anonymous Referees for constructive criticism and valuable correlation functions shows that the latter are capable of advice. characterizing the influence even of weak feedback, thus enabling them as a tool for the analysis of optical reinjection in REFERENCES nanolasers. The possibility of basing the analysis of the laser [1] C. H. Henry and R. Kazarinov, “Instability of semiconductor lasers due dynamics on the sole second-order correlations represents a to optical feedback from distant reflectors,” IEEE J. Quantum Electron., vol. QE-22, p. 294, 1986. substantial step in furthering the investigation of the temporal [2] C. Serrat, S. Prins, and R. Vilaseca, “Dynamics and coherence of a behaviour of micro- and nanolasers even in the absence of multimode semiconductor laser with optical feedback in an intermediate- sequential information. Finally, we have proven that predic- length external-cavity regime,” Phys. Rev. A, vol. 68, pp. 053804(1-7), 2003. tions obtained from fully stochastic numerical simulations, [3] J. Ohtsubo, “Dynamics of semiconductor lasers with optical feedback,” in the incoherent feedback regime, are in good qualitative Semiconductor Lasers, pp. 113-182, 2017. agreement with the observations, except for the reproduction [4] J.S. Cohen, F. Wittgrefe, Maarten D. Hoogerland, and J.P. Woerdman, “Optical Spectra of a Semiconductor Laser with Incoherent Optical of low frequency fluctuations. This is probably due to reliance Feedback,” IEEE Journal of Quantum Electronics, vol. 26, pp. 982-990, of the numerical scheme on the photon number, while the 1990. 7

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[51] G.L. Lippi, J. Mørk, and G.P. Puccioni, “Numerical solutions Gaofeng Wang (S’93-M’95-SM’01) received the Ph.D. degree in electrical to the Laser Rate Equations with noise: technical issues, imple- engineering from the University of Wisconsin-Milwaukee, Milwaukee, WI, mentation and pitfalls,” Proc. SPIE, vol. 10672, Nanophotonics USA, in 1993, and the Ph.D. degree in scientiﬁc computing from Stanford VII, 106722B (1-14), 2018. Freely available for personal use at: University, Stanford, CA, USA, in 2001. From 1993 to 1996, he was a https://sites.google.com/site/gianlucalippi. Scientist with Tanner Research Inc., Pasadena, CA. From 1996 to 2001, he was [52] G.L. Lippi, J. Mørk, and G.P. Puccioni, “Analytical vs. Numerical a Principal Research and Development Engineer with Synopsys Inc., Mountain Langevin Description of Noise in Small Lasers”, arXiv:1903.08859, View, CA. In 1999, he served as a Consultant with Bell Laboratories, Murray 2019. Hill, NJ, USA. From 2001 to 2003, he was the Chief Technology Ofﬁcer [53] F.T. Arecchi, G.L. Lippi, G.P. Puccioni, and J.R. Tredicce, “Determinis- (CTO) of Intpax, Inc., San Jose, CA. From 2004 to 2010, he was the CTO of tic chaos in laser with injected signal,” Opt. Commun., vol. 51, 308-314, Siargo Inc., Santo Clara, CA. From 2004 to 2013, he was a Professor and the 1985. Head in the CJ Huang Information Technology Research Institute with Wuhan [54] J.R. Tredicce, F.T. Arecchi, G.L. Lippi, and G.P. Puccioni, “Instabilities University, Wuhan, China. From 2010 to 2013, He was the Chief Scientist in lasers with an injected signal,” J. Opt. Soc. Am. B, vol. 2, 173-183, with Lorentz Solution, Inc., Santa Clara, CA. He is currently a Distinguished 1985. Professor with Hangzhou Dianzi University, Hangzhou. He has authored over [55] A. Einstein,“Zur Quantentheorie der Strahlung,” Physikalische Zeitschr., 210 journal articles and holds 30 patents. His current research interests include vol. 18, 121-128, 1917. integrated circuit and microelectromechanical system design and simulation, computational electromagnetics, electronic design automation, and wavelet applications in engineering.

Tao Wang received his Ph.D. degree in physics from the Universite´ de Nice- Sophia Antipolis, France, in 2016. From 2013 to 2016, he worked in the Institut Non Lineaire´ de Nice (now the Institut de Physique de Nice) as a PhD student. Since 2016, he was an post-doctoral fellow in the Institut National de la Recherche Scientiﬁque, Canada. He currently is an associate professor in the School of Electronics and Information, Hangzhou Dianzi University, Hangzhou, China. His research interests include micro/nano scale laser dynamics, optical sensors based on lasers, nonlinear optics, light-matter interactions, and optical materials.

Xianghu Wang is studying in Hangzhou Dianzi University, Hangzhou, China. His research interest includes semiconductor laser physics, theoretical Gian Luca Lippi (Laurea in Fisica, University of Florence, Italy, 1984; modelling and understanding of dynamics in micro-/nanoscale lasers Ph.D. Bryn Mawr College, USA, 1990; Habil. Dir. Rech. Universite´ de Nice-Sophia Antipolis, France, 1998) is currently Distinguished Professor at the Physics Department of the Universite´ Coteˆ d’Azur and member of the Institut de Physique de Nice (formerly Institut Non Lineaire´ de Nice) since 1994. Between 1990 and 1993 he was active as Post-Doctoral Fellow at the Institut fur¨ Angewandte Physik (Westfalische¨ Wilhelms-Universitat¨ Munster,¨ Germany) ﬁrst as Alexander-von-Humboldt Fellow, then with DFG support. His research covers laser dynamics, nonlinear dynamics in optical Zhilei Deng is studying in Hangzhou Dianzi University, Hangzhou, China. systems, laser-matter interactions, particle trapping, and physical properties His research interest focuses on theoretical design and modelling of small of nanolasers, mainly from an experimental point of view, but including size VCSELs and optical sensors based on the laser devices. modelling aspects. He is co-author and author of over 60 papers in refereed journals, 24 conference proceedings and nearly 120 contributions to confer- ences (22 invited presentations). He has been director of a French Doctoral School in Sciences and subsequently Director of the European Doctorate EDEMOM. Former International Consultant for the project “Laser Trapped Mirror Proposal” (MSMT, NASA), past referee for INTAS (EU) and EPSRC (UK) programmes, referee for all the main physics journals, he is currently Jiacheng Sun is studying in Hangzhou Dianzi University, Hangzhou, China. member of the Strategic Council of REA. His research interest mainly focuses on light-matter interaction in low dimen- sional structures, especially exploring novel optical properties of nanolasers.

Gian Piero Puccioni earned his degree in Physics at the Universita` degli Studi di Firenze (Italy) and had a Post-doctoral Fellowship at the Univerity of Toronto (Canada). Between 1987 and 2004 he was a researcher at the National Institute of Optics in Florence where he also held the course in Numerical Computations of the Specialization School in Optics and was the Head of the Computer Center of the Institute. In 2004 he moved to the Institute of Complex Systems (CNR) in Firenze. He published several articles both experimental and theoretical on chaotic behavior in lasers, and more recently on mathematical models for nanolasers.