Characterizing the chaotic dynamics of a semiconductor nanolaser subjected to FBG feedback

Pei Jiang; Pei Jiang; Pei Jiang; Pei Zhou; Pei Zhou; Pei Zhou; Nianqiang Li; Nianqiang Li; Nianqiang Li; Penghua Mu; Xiaofeng Li; Xiaofeng Li; Xiaofeng Li

doi:10.1364/OE.427541

1. Introduction

Semiconductor lasers can exhibit a variety of complex dynamics such as steady state, period, period doubling, quasi-periodicity and chaos when subjected to external perturbations [1–3]. Among these, optical chaos generation based on the simplest perturbation scheme, i.e., optical feedback was extensively studied due to its potential applications to secure communications [4,5], random number generation [6,7] and chaotic lidar/radar [8,9]. The generated chaos, however, shows certain periodicity related to the introduced external cavity for optical feedback, which has been termed time-delay signature (TDS) [10]. This periodicity flaw degrades the performance of chaos-based applications [11–13]. For example, it threatens the security of chaos secure communications due to the risk of information leakage of the key parameter, i.e., the time delay [14,15]. Therefore, it is of importance and interest to suppress TDS without complex structures. Li et al. numerically and experimentally demonstrated a TDS suppression and effective bandwidth enhancement (EBW) scheme using FBG feedback [16,17]. Then Wu et al. explored the effect of FBG feedback on VCSEL chaotic output. It is proved that TDS can be effectively suppressed when selecting proper bias currents and polarization angle [18–20]. In addition, a series of extensive studies have been carried out, such as introducing FBG feedback under optical injection [21], mutual injection [22,23] and adopting Gaussian apodized FBG [24] or chirped FBG [25,26] to replace conventional uniform FBG. These theoretical and experimental results show that quality of chaos can be improved effectively by dispersion-induced optical feedback [27,28]. Most of them were constructed with conventional semiconductor lasers, which, however, may cause problems with their large size. They might be not suitable for the increasingly prevalent photonic integration, which is desired for practical applications. In this sense, novel semiconductor nanolasers are of great interest due to their potential for incorporation in compact photonic integrated circuits [29,30]. Therefore, the current work is mainly intended to generate high-quality optical chaos by using nanolasers with broad parameters regions.

Nanolasers are important optical elements in the field of optical integration. A large number of experimental investigations have been explored on various nanolaser structures, including spaser [31], nanowire [32], and nano-pillar [33] lasers recent years, which aimed at continuous wave (CW) lasing for either optical [31] or electrical pumping [34]. These new-style lasers can exhibit enhanced dynamical performance because of two important parameters, i.e., the Purcell spontaneous emission enhancement factor F and enhanced spontaneous emission coupling factor β. Based on the introduction of the two phenomenological parameters in the rate equation [35,36], a series of theoretical analysis about the dynamics performance of nanolasers with the inclusion of certain external perturbation was implemented. For example, Sattar et al. systematically studied the dynamical properties of nanolasers with current modulation [37], conventional optical feedback [38], phase-conjugate feedback [39], as well as optical injection [40]. Han et al. further investigated mutual coupling induced nonlinear dynamics, where high-frequency modulation response and oscillations as well as rich dynamics including chaos were observed [41,42]. The above-mentioned evidence demonstrates that nanolasers allow to generate dynamics as rich as those expected in conventional semiconductor lasers. However, there are only a few studies dealt with the chaotic properties generated by nanolasers, which are crucial to the potential applications based on chaos. For example, Elsonbaty et al. studied the TDS suppression in a single semiconductor nanolaser with a hybrid all-optical and electro-optical feedback scheme [43]. Qu et al. demonstrated desired TDS suppression in a slave nanolaser subjected to double chaotic optical injection from two master nanolasers [44]. In our previous work, we proposed and demonstrated that optically injected nanolasers with the optical feedback and cascaded structure allowed for TDS suppression and communications [45]. The fundamental problem lies in their complex device structures. It is necessary to explore relatively simple structures to suppress TDS of nanolasers.

In the current contribution, we propose to achieve desired TDS suppression and bandwidth enhancement for nanolasers via FBG feedback. Different from the conventional mirror for localized feedback and linear response, FBG can provide distributed feedback and nonlinear response, which will increase dispersion-induced group delay. Time delay will be broadened and eventually suppressed by controlling frequency detuning to make the main peak of laser spectrum located in the position of sidelobes. It should also be mentioned that FBG is a filter device and thus there exists the filtering-induced optical loss. It is still expected that nanolasers subjected to FBG feedback will exhibit rich dynamic properties under different settings of the frequency detuning and feedback strength. Therefore, the current work clarifies the dynamics of nanolasers with FBG feedback and demonstrates high-quality chaos generated in a wide parameters range with a relatively simple structure, which brings more advantages for photonic integrated circuits. This paper is organized as follows. Section II introduces the model of nanolasers with FBG feedback, where the mathematical background and parameter definition are detailed. Section III provides the simulation results of dynamics and the corresponding chaotic characteristics. Finally, concluding remarks are presented in Section IV.

2. Theoretical model

The schematic diagram of a n anolaser subjected to either mirror feedback or FBG feedback is shown in Fig. 1. The only change in the configuration is the replacement of the mirror by a FBG and it will achieve different degrees of reflection under different wavelengths according to the reflectance spectrum. The corresponding dynamics of nanolasers in the presence of feedback from FBG are identified by using the modified form of single mode rate equations as follows [38]:

Fig. 1. Schematic of nanolasers subjected to (a) mirror feedback; (b) FBG feedback.

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(1)$$\begin{aligned} \frac{d I(t)}{d t}=& \Gamma\left[\frac{F \beta N(t)}{\tau_{\mathrm{n}}}+g_{\mathrm{n}}\left(N(t)-N_{0}\right)\right]-\frac{1}{\tau_{\mathrm{p}}} I(t)+\\ & 2 k_{f} \sqrt{I(t)} \operatorname{real}\left\{\left(e^{-i w \tau_{f}-i \varphi(t)} \sqrt{I\left(t-\tau_{\mathrm{d}}\right)} e^{i \varphi\left(t-\tau_{f}\right)} \otimes r(t) e^{i\left(\Delta \Omega+\Delta \Omega_{\mathrm{m}}\right) t}\right)\right\} \end{aligned}$$

(2)$$\begin{aligned} \frac{d \varphi(t)}{d t} &=\frac{\alpha}{2}\left(\Gamma g_{\mathrm{n}}\left(N(t)-N_{0}\right)-\frac{1}{\tau_{\mathrm{p}}}\right)+k_{f} \frac{1}{\sqrt{I(t)}} \\ & \operatorname{imag}\left\{\left(e^{-i w \tau_{f}-i \varphi(t)} \sqrt{I\left(t-\tau_{f}\right)} e^{i \varphi\left(t-\tau_{f}\right)} \otimes r(t) e^{i\left(\Delta \Omega+\Delta \Omega_{\mathrm{m}}\right) t}\right)\right\} \end{aligned}$$

(3)$$\frac{{{{d}}N(t)}}{{{\rm{d}}t}} = \frac{{{I_{{\rm{dc}}}}}}{{e{V_a}}} - \frac{{N(t)}}{{{\tau _{\rm{n}}}}}(F\beta + 1 - \beta ) - {g_{\rm{n}}}(N(t) - {N_0})I(t)$$

In the above rate equations, ${I(t)}$ is the photon density, ${\varphi{(t)}}$ is the phase, and ${{N(t)}}$ accounts for the carrier density. ${{F}}$ is the Purcell spontaneous emission enhancement factor, $\beta $ is the spontaneous emission coupling factor, $\Gamma $ is the confinement factor, ${{{V}}_{{a}}}$ is the volume of the active region, ${{{\tau }}_{\rm{p}}}\;$ is the photon lifetime, ${{{\tau }}_{\rm{n}}}$ is the carrier lifetime, ${g_n}$ is the differential gain, $\alpha $ is the linewidth enhancement factor, ${{e}}$ is the electron charge, ${{{N}}_{\rm{0}}}\;$ is the transparency carrier density, ${{w}}$ is the optical frequency, ${{{N}}_{{\rm{th}}}}$ is the threshold carrier density, and I_dc is the bias current. The last terms in Eqs. (1) and (2) represent FBG feedback. ${{{k}}_{{f}}}$ and ${{\;}}{\tau _{{f}}}$ are the feedback strength and feedback delay for the feedback path. $\delta_{\omega}= 2 \ast \pi \ast \delta_f$ is the angular frequency detuning of the Bragg frequency of the FBG away from the free-running frequency of the laser. And $\delta_{\omega_m} = 2 \ast \pi \ast \delta_{f_m}$ is the angular frequency offset of the free-running frequency of the laser away from the center of optical spectrum. r(t) represents the impulse response of the FBG and it equals to the inverse Fourier transform of the reflected frequency response [46]:

(4)$$r(\Omega ) = \frac{{\kappa \sinh (\sqrt {{\rm{|}}\kappa {{\rm{|}}^2} - {\delta ^2}} l)}}{{\delta \sinh (\sqrt {{\rm{|}}\kappa {{\rm{|}}^2} - {\delta ^2}} l) + i(\sqrt {{\rm{|}}\kappa {{\rm{|}}^2} - {\delta ^2}} )\cosh (\sqrt {{\rm{|}}\kappa {{\rm{|}}^2} - {\delta ^2}} l)}}$$

where |κ| is the magnitude of the coupling coefficient of the FBG, which can estimate the FBG bandwidth ${{{f}}_{{{BW}}}}$ (=c|κ|/πn_g). l is the length of the FBG, δ is the phase mismatch between the counter propagation modes that equals to n_gΩ/c (n_g is the refractive index, c is the speed of light in vacuum). The convolution processing of $\sqrt {I{{(t - }}{{{\tau }}_{{f}}}{{)}}} {{{e}}^{{{i\varphi(t - }}{{{\tau }}_{{f}}}{{)}}}}\otimes{{r(t)}}$ in domain achieves the corresponding product of ${\rm{FT}}\left\{ {\sqrt {I{{(t - }}{{{\tau }}_{{f}}}{{)}}} {{\rm{e}}^{{\rm{i\varphi(t - }}{{{\tau }}_{{f}}}{\rm{)}}}}} \right\}r(\Omega)$ in the frequency domain. ${{r(t)}}$ will become large and achieve optical feedback enhancement with the increase of FBG bandwidth. It should also be noted that the nanolaser at time t is affected by its emission at exactly ${{t - }}{{{\tau }}_{{f}}}$ for the mirror feedback and ${{r(t)}}$ can be replaced by δ(t). In contrast, it is affected by its emission at and before ${{t - }}{{{\tau }}_{{f}}}$ for FBG feedback.

In this study, we use the fourth-order Runge Kutta algorithm to solve Eqs. (1)–(3) with a step of 0.2 ps for a time span of 200 ns. During the calculation, the parameters are set as follows unless otherwise stated [38]:${{\;F = 14,\; = 0}}{\rm{.05,\; = 0}}{\rm{.645,\;\alpha = 5,\;}}{{{\tau }}_{\rm{n}}}{\rm{ = 1\;ns,\;}}{{{\tau }}_{\rm{p}}}{\rm{ = 0}}{\rm{.36\;ps,\;}}{{{V}}_{{a}}}{\rm{ = 3}}{\rm{.961}}{{\rm{0}}^{{\rm{ - 13}}}}{{\;c}}{{\rm{m}}^{\rm{3}}}$, ${{w}} = {\rm{1}}{\rm{.181}}{{\rm{0}}^{{\rm{15}}}}{{\;rad/s,\;}}{{{N}}_{\rm{0}}}{\rm{ = 1}}{\rm{.1\;1}}{{\rm{0}}^{{\rm{18}}}}\;{\rm{c}}{{\rm{m}}^{{\rm{ - 3}}}}{\rm{,\;}}{{{I}}_{{\rm{th}}}}{\rm{ = 1}}{\rm{.1\;mA,\;}}{{{I}}_{{\rm{dc}}}}{\rm{ = 2}}{{{I}}_{{\rm{th}}}}{\rm{,\;}}{{{\tau }}_{{f}}}{\rm{ = 0}}{\rm{.2\;ns}}$, ${\rm{|\kappa|= 600\;}}{{\rm{m}}^{{\rm{ - 1}}}}$, $\rm{n}_{g}=1.45$, $l = 0.02\;\rm{m}$. Throughout this paper, other parameters will be specified in the following sections.

For TDS quantification, delay mutual information (DMI) [17,25], autocorrelation function (ACF) [47,48] and permutation entropy (PE) [49] can be used. We mainly adopt the ACF, which has been widely used due to its high efficiency [45,50–52]. It quantifies the correlation between the time series and its delayed replica, which can be defined as follows [45]:

(5)$${\rm{C}}(\Delta t) = \frac{{\left\langle {\left[ {I(t + \Delta t) - \left\langle {I(t + \Delta t)} \right\rangle } \right]\left[ {I(t) - \left\langle {I(t)} \right\rangle } \right]} \right\rangle }}{{\sqrt {\left\langle {{{\left[ {I(t + \Delta t) - \left\langle {I(t + \Delta t)} \right\rangle } \right]}^2}} \right\rangle \left\langle {{{\left[ {I(t) - \left\langle {I(t)} \right\rangle } \right]}^2}} \right\rangle } }}$$

where <> denotes the average of the time series $I(t)$, ${\delta_{t}}$ is the time shift. Follow the literature [53], we define the peak size of ACF in time window of ${{t}} \in [{{{\tau }}_{\rm{d}}}{{ - j \times }}{{{\tau }}_{\rm{d}}}{\rm{,\;}}{{{\tau }}_{\rm{d}}}{{ + j \times }}{{{\tau }}_{\rm{d}}}{\rm{]\;ns}}$ with the value ${{j\; = \;20\%}}$ as the TDS.

3. Results and discussion

The main purpose of this paper is to focus on the evolution of the dynamics and chaotic characteristics of nanolasers subjected to FBG feedback. In the analysis, the effects of characteristic parameters related to FBG and nanolasers should be taken into account. It is interesting to explore the corresponding dynamics of nanolasers under filtered optical feedback and reveal the spectral characteristics.

Firstly, analysis of the spectrum of free running lasers is carried out. Compared with conventional semiconductor lasers, spontaneous emission cannot be ignored in nanolasers. The Purcell spontaneous emission enhancement F and the spontaneous $\beta $ are two important phenomenological factors, which determine the redshift of the laser wavelength. When the central wavelength of FBG cannot match with the wavelength of free running nanolasers, it is difficult to generate rich dynamics. So considering the change of wavelength drift and forming a mutual match by adjusting the operating temperature of the laser or the stress on FBG is necessary. Figure 2 shows the optical spectra for free running nanolasers under four different (F, ${{\;}}\beta $) settings. With the increase of F and ${{\;}}\beta $, the laser moves towards the negative frequency offset corresponding to the longer wavelength (pronounced redshift). The frequency offsets are estimated as −9, −45, −87, −163 GHz, respectively. Specifically, the change in wavelength is ∼1.29 nm from ${{F = 1}}$ and ${\rm{ = 0}}{\rm{.05}}$ to ${{F = 14}}$ and ${{\beta = 0}}{\rm{.1}}$, and the wavelength of 1591.7 nm with a frequency offset $\delta_{f_m} ={-} 87$ GHz at ${{F = 14}}$ and ${{\beta = 0}}{\rm{.05}}$ is used unless otherwise stated. The results in Fig. 2 also indicate that the wavelength matching is important for the current study, which may slightly complicate the practical utilization and operation of the proposed technique.

Fig. 2. Numerical results of the optical spectra for a free running nanolaser. (a)${{\;F = 1}}$ and ${{\beta = 0}}{{.05}}$; (b)${{\;F = 7}}$ and ${{\beta = 0}}{{.05}}$; (c)${{\;F = 14}}$ and ${{\beta = 0}}{{.05}}$; (d)${{\;F = 14}}$ and ${{\beta = 0}}{{.1}}$.

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Next, the dynamics of the nanolasers for different values of the FBG feedback strength are analyzed. The results for the cases of the frequency detuning of Δf=20 GHz, and three values of the feedback strength, i.e., k_f= 50, 60, 80 ns⁻¹, are shown in Fig. 3. When the feedback strength is weak, the output of nanolaser exhibits periodic oscillations. As can be seen in Fig. 3(a), period-one dynamics are presented for the considered feedback strength. As the feedback strength increases, it will enter quasi-periodic oscillations [Fig. 3(b)] and eventually yield chaos [Fig. 3(c)]. It should be emphasized that this resembles a quasi-periodic route-to-chaos. Such a dynamic path is commonly found in lasers with mirror feedback [54].

Fig. 3. Time series of photon densities (a) periodic signal at k_f=50 ns⁻¹; (b) quasi-periodicity at k_f = 60 ns⁻¹; (c) chaos at k_f = 80 ns⁻¹; Δf=20 GHz.

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In order to depict the dynamic evolution path clearly, bifurcation diagrams are obtained by plotting the maxima and minima of the photon density as a function of the feedback strength [55]. It can be seen in Figs. 4(a1) and 4(a2) that, as the feedback strength increases, the nanolaser with FBG feedback enters chaos through quasi-periodic route-to-chaos and then backs into a steady state or periodic state due to chaos crisis [56] when the frequency detuning is negative. However, for the considered cases of positive frequency detuning, there is no periodic state in the intermediate feedback strength after chaos occurs [Figs. 4(a3) and 4(a4)]. This implies that nanolasers with FBG feedback may exhibit asymmetric dynamics in the plane of the feedback strength versus the frequency detuning. Here we are only interested in the chaotic areas, so a chaos measurement is needed. Usually, the largest Lyapunov exponent (LLE) should be calculated since it is the benchmark for chaos identification, i.e., the system operates in a chaotic state when the LLE is greater than 0. However, the LLE calculation is relatively complex and extremely sensitive to noise. In this sense, we employ its effective alternative, i.e., the 0–1 test for chaos [57]. The test yields two distinct outputs, either close to 0, or close to 1. The former indicates a non-chaotic state, while the latter means a chaotic state. It is proved that the credible test results can be obtained by selecting an appropriate sampling rate and time series length. We set the sampling interval of $\omega_{\rm{s}}$=4 ps and the series length of N=500 by multiple parameter optimization and selection. The results for the 0–1 test for chaos are presented in Figs. 4(b1)–4(b4), which are compared to the bifurcation diagrams shown in Figs. 4(a1)–4(a4). It is clearly observed that the non-chaotic and chaotic windows depicted in the bifurcation diagrams are successfully identified by the 0–1 test for chaos. Such a simple measurement provides guidance for the analysis of two-dimensional dynamics distributions as shown below.

Fig. 4. Bifurcation diagram of photon density and the corresponding 0–1 tests for chaos as a function of the feedback strength at (a1) and (b1) Δf=−30 GHz; (a2) and (b2) Δf=−20 GHz; (a3) and (b3) Δf=20 GHz; (a4) and (b4) Δf=30 GHz.

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Figure 5 shows the mapping of the dynamical state representing by the 0–1 test for chaos in the feedback parameter space, i.e., (k_f, Δf) under various settings of F and $\beta $. In these color maps cases, the pink color represents chaos, and the other states such as steady states, periodic oscillations, and quasi-periodicities are represented by dark blue. In all cases shown here, chaos can be obtained in nanolasers with FBG feedback, while F and ${{\;}}\beta $ play an important role in the laser dynamics. Firstly, uninterrupted pink region representing chaos can be seen for small values of F and $\beta $ [Figs. 5(a) and 5(b)]. On the contrary, two separated islands are acquired for relatively large values of F and $\beta $ [Figs. 5(c) and 5(d)], and chaos regions continue to shrink in size as F and $\beta $ increases. This is attributed to the stronger damping of the relaxation oscillation with the increasement of F and $\beta $. Secondly, the threshold of feedback strength required to render chaos will increase as the frequency detuning increases. This is expected since mismatching between the FBG frequency and the laser frequency leads to lower power reflection for actual feedback, which resembles some kind of the filter effect. Similar phenomena have been seen in the conventional semiconductor laser with frequency-detuned FBG feedback [17]. Thirdly, due to the red-shifting effect by the optical feedback, all maps are asymmetric, especially for large values of F and $\beta $. This confirms the asymmetric cases depicted in Fig. 4, where different bifurcation diagrams are obtained for equivalent positive and negative detuned frequencies.

Fig. 5. Two-dimensional maps of dynamics in the parameter space of the feedback strength and frequency detuning with (a)${{\;F = 1}}$ and ${{\beta = 0}}{{.05}}$; (b)${{\;F = 7}}$ and ${{\beta = 0}}{{.05}}$; (c)${{\;F = 14}}$ and ${{\beta = 0}}{{.05}}$; (d)${{\;F = 14}}$ and ${{\beta = 0}}{{.1}}$; |κ|=600 m⁻¹.

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Discussing the dynamics of nanolasers under different FBG bandwidths is also interesting. Dynamical maps in the (k_f, Δf) plane for four different FBG bandwidths are shown in Fig. 6. In Fig. 6(a) with ${\rm{|\kappa|}}$=100 m⁻¹, the laser remains non-chaos operation over a large region in the map and it requires high feedback strength to render chaos. Concretely, chaos is found in a small island around small detuned frequencies and strong feedback. As ${\rm{|\kappa|}}$ increases, the original chaos island expands in size, followed by the appearance of another island that is disconnected from the original one [Fig. 6(b)]. When ${\rm{|\kappa|}}$ continues to increase to large values, both chaos islands expand remarkably [Figs. 6(c) and 6(d)]. This phenomenon can be explained as follows. The coupling coefficient of ${\rm{|\kappa|}}$=100, 300, 600, and 900 m⁻¹ represents the FBG bandwidth of ${{{f}}_{{\rm{BW}}}}$=6.6, 20, 40 and 60 GHz, respectively. The narrow FBG bandwidth causes weak optical feedback and strong filtering effect, especially when there exists a certain frequency detuning and the main power component is concentrated in the sidelobe of the reflection spectrum. By contrast, the broad FBG bandwidth results in weak filtering effect and strong feedback, and thus abundant dynamics including chaos can be seen in large regions. It is worth noting that the laser dynamics will evolve towards mirror optical feedback with no frequency selectivity when the FBG bandwidth increase further.

Fig. 6. Two-dimensional maps of dynamics in the parameter space of the feedback strength and frequency detuning with (a) |κ|=100 m⁻¹; (b) |κ|=300 m⁻¹; (c) |κ|=600 m⁻¹; (d) |κ|=900 m⁻¹;${{\;F = 14}}$ and ${{\beta = 0}}{\rm{.05}}$.

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Until now, we have been focusing on the chaos generation in nanolasers with FBG feedback. In this subsection, we will pay attention to the chaotic characteristics, including TDS and EBW. This is important since these two measures determine the security and transmission rate in chaos-based communication and on the other hand, they degrade the randomness and bit rate in random-number generation. There is no doubt that chaos characteristics of nanolasers with FBG feedback are comprehensively determined by the relaxation oscillation time, FBG bandwidth, frequency detuning and feedback strength.

Figure 7 shows four typical examples of ACF for different feedback strengths under mirror feedback and FBG feedback. For a fair comparison, the parameters used of both feedback cases are set to the same. When the feedback strength is weak, chaos generated by both feedback structures contains obvious TDS, although the ACF peak around the feedback delay time in the mirror feedback is stronger than that in the FBG feedback. This is due to the linear effect with low dispersion in the mirror feedback. As the feedback strength increases, the ACF peak around the feedback time delay is almost kept the same for the mirror feedback, while the ACF peak is gradually suppressed for the FBG feedback. When the feedback strength increases to 120 ns⁻¹, the delay can be broadened or even completely eliminated under the distributed feedback. This is because such sufficient feedback power can perturb the laser to yield chaos without prominent external cavity features, i.e., reduced periodicity due to the external cavity. The comparison indicates that it is feasible to conceal the feedback delay in nanolasers with the FBG feedback. We have also checked the corresponding power spectra, similar features associated with the TDS suppression are acquired. For brevity, these results are not shown here.

Fig. 7. ACF computed from nanolasers chaotic intensity outputs with (a1)-(a4) mirror feedback; (b1)-(b4) FBG feedback; (a1) and (b1) k_f = 80 ns⁻¹; (a2) and (b2) k_f = 100 ns⁻¹; (a3) and (b3) k_f = 120 ns⁻¹; (a4) and (b4) k_f = 140 ns⁻¹; Δf=30 GHz.

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The evolution of the ACF peak value around feedback delay and EBW with the feedback strength is offered in Fig. 8. The EBW can be defined as the frequency range of 80% total power covers in the signal power spectrum [58]. Compared with the mirror feedback, FBG feedback requires a higher feedback strength to render chaos which is due to the filtering effect that weakens part of the spectral component and suppresses the coherent collapse. Thus, we set different minimum feedback strengths to assure nanolasers work in chaos states in both feedback cases. As for the ACF property shown in Fig. 8(a), it can be found that the TDS value gradually decreases to a minimum and then increases again for both feedback cases. These features are consistent with those in the conventional semiconductor lasers [10,53] and those for previous results in nanolasers [44,45]. However, the FBG feedback requires a higher feedback strength to reach the valley which is due to the filtering effect, e.g., k_f is roughly larger than 100 ns⁻¹. It is interesting to find that much lower ACF peak values are obtained for the FBG feedback compared to the mirror feedback, consistent with the findings in conventional semiconductor lasers [16]. We then turn to the chaos bandwidth and find that the chaos EBW induced by FBG monotonically increases from k_f= 100 ns⁻¹, while in the mirror case, it increases first, reaches to its peak at an intermediate feedback strength, and finally decreases gradually [Fig. 8(b)]. This leads to crossover between the EBW curves of FBG and mirror feedback. Thus, the EBW in the FBG feedback can be greater than that in the mirror feedback when the feedback strength increases to 230 ns⁻¹ and above, which might be due to the dispersion which excites oscillations at higher frequencies. Besides, two frequency detuning values of Δf=20 and 30 GHz are compared and similar trends are found. This indicates that low TDS and high EBW can be achieved in nanolasers with FBG feedback for relatively flexible parameter settings.

Fig. 8. (a) ACF and (b) EBW as a function of the feedback strength for nanolasers with mirror feedback and FBG feedback.

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To understand the effect of dispersion on TDS suppression, we turn to analyze the optical spectrum with the change of frequency detuning, which is obtained by the inverse Fourier transform of the complex amplitude $\sqrt {{{I(t)}}} {{{e}}^{{{i\varphi (t)}}}}$. For the purpose of comparison, both mirror feedback and two cases of frequency detuning in FBG feedback are studied. The results are shown in Fig. 9, where the redshift occurs and the main peak moves towards the negative frequency detuning in both feedback configurations. The red line represents the reflectance spectrum and the blue line stands for the group delay, which is obtained from the phase of the FBG frequency response and reflects the dispersion effect. Without loss of generality, the feedback strength k_f is fixed at 120 ns⁻¹, which only needs to ensure that sufficient optical feedback can drive the laser into chaos.

Fig. 9. Optical spectrum for chaotic laser emission in nanolasers with mirror feedback and FBG feedback. (a) Mirror feedback; (b) FBG feedback, Δf=30 GHz; (c) FBG feedback, Δf=−30 GHz.

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When the FBG is replaced by mirror feedback, the group delay spectrum is close to zero and the reflection has no frequency selectivity. The optical spectrum for mirror feedback is illustrated in Fig. 9(a). As can be seen, at the position of negative frequency detuning, there are many obvious small peaks with an interval of about 5 GHz, corresponding to the reciprocal of the feedback delay ${{{\tau }}_{\rm{f}}}$. These peaks will reveal the external cavity modes (ECMs) and result in profound TDS in the ACF computed from the laser intensity. Therefore, it is necessary to provide distributed optical feedback to broaden and even conceal the delay. In Fig. 9(b), FBG is adjusted to positive detuning at Δf = 30 GHz with the main peak located in the sidelobe position of the reflection spectrum. The low-frequency sidelobe with high dispersion is exactly at the original position that contains some ECM structures. Consequently, it leads to the reconstruction of the spectral structure and obscures the information of the feedback delay [see Fig. 7(b3)]. However, such ECM structures will appear again in Fig. 9(c) when the frequency detuning is negative, e.g., when the center of reflectance spectrum locates at Δf=−30 GHz. The mainlobe with low group delay is exactly at the position that contains most of the emitted power as expected. Therefore, it is important to achieve TDS concealment by choosing a proper frequency detuning. A clear comparison with previous works for conventional semiconductor lasers indicates that FBG feedback indeed improves the TDS property regardless of the laser type. In addition, the corresponding spectral structure will change and affect the EBW after filtering. On the one hand, compared with the mirror feedback in Fig. 9(a), the power under the mainlobe is kept the same, while the power under the sidelobes will be weakened. On the other hand, compared with the conventional semiconductor laser [17], a nanolser with FBG feedback exhibits a more flat spectrum, which means that it allows for high bandwidth and TDS suppression in wider parameter regions.

The aforementioned results only qualitatively describe the effects of the frequency detuning, which prove that the delay information contained in the spectral component can be weakened moderately after the variation of the sidelobe group delay. Here, we also introduce ACF and DMI to quantitatively analyze the delay due to its high accuracy. DMI is defined as the sum of ${\rm{p}}({{{I,}}{{{I}}_{{\tau }}}} ){\rm{log}}[{{\rm{p}}({{{I,}}{{{I}}_{{\tau }}}} ){\rm{/p}}({{I}} ){{p}}({{{{I}}_{{\tau }}}} )} ]$, where ${\rm{p}}({{I}} )$, ${{\;\rm{p}}}({{{{I}}_{{\tau }}}} )$ is the probability density function of I, ${{{I}}_{{\tau }}}$ and ${\rm{p}}({{{I,}}{{{I}}_{{\tau }}}} )$ is the joint probability density function of $({{{I,}}{{{I}}_{{\tau }}}} )$ [17,25].

Figures 10(a1)−10(a3) shows the corresponding time series which act as noise-like oscillations. The chaotic characteristic can be analyzed by ACF, DMI and power spectrum. When chaos is generated via conventional optical feedback, TDS is obvious and there are multiple peaks located at around the delay time and its harmonics, as shown in ACF [Fig. 10(b1)] and DMI [Fig. 10(c1)]. In Fig. 10(d1), there exist peaks equally spaced with the interval of 5 GHz, which represents the ECM structure. However, such delay signature cannot be easily extracted from ACF [Fig. 10(b2)] and DMI [Fig. 10(c2)] when adopting FBG feedback with a frequency detuning of Δf=30 GHz. The power spectrum shown in Fig. 10(d2) becomes much smoother since less peaks can be observed. Here, we also offer the case of Δf = −30 GHz. Though the TDS cannot be eliminated, the peak value around the delay in ACF [Fig. 10(b3)] and DMI [Fig. 10(c3)] slightly decreases and the peak is broadened as expected. Besides, the attenuated ECM structure in the power spectrum is shown in Fig. 10(d3). In short, it is further proved that the frequency detuning is an important parameter to achieve chaotic TDS concealment of nanolasers with FBG feedback, which is in accordance with the case of the conventional laser the same feedback scheme [17]. In addition, the relaxation oscillation frequency of a nanolaser is much higher than a conventional semiconductor laser. As a result, nanolasers have the potential for large bandwidth and can be used as a better source of broadband chaos.

Fig. 10. Time series (a1)-(a3) and their corresponding ACF (b1)-(b3), DMI (c1)-(c3), as well as frequency spectra for chaotic laser emission. The laser is under (a) mirror feedback; (b) FBG feedback, Δf=30 GHz; (c) FBG feedback, Δf=−30 GHz.

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The above analysis indicates how to achieve TDS concealment and desirable EBW of nanolasers under specific settings of the feedback strength and frequency detuning. It is interesting to evaluate the TDS and EBW properties in a wide range of key parameters. More concretely, the influence of the feedback strength, frequency detuning, ${{F}}$, ${{\;}}\beta $ and ${\rm{|\kappa|}}$ on the TDS and EBW performance will be studied. Figure 11 shows the variation of TDS and EBW in the parameter space of the feedback strength and frequency detuning with different ${\rm{|\kappa|}}$. It should be noted that larger ${\rm{|\kappa|}}$ leads to wider FBG bandwidths, as shown in Figs. 11(c1)–11(c4). In Figs. 11(a1)–11(a4), the red color represents obvious TDS, corresponding to the stable state or periodic state while the dark blue represents the case of the suppressed TDS. A careful comparison shows that TDS can be almost completely eliminated in chaotic states when ${\rm{|\kappa|}}$ is relatively small; for example, see the small dark blue region in Figs. 11(a1) and 11(a2). The zone of TDS suppression, i.e., low TDS, will expand gradually with FBG bandwidth increases. However, too large FBG bandwidth goes against TDS concealment. This is because the mainlobe of FBG covers most of the optical spectrum, and in this case, the group delay and dispersion beneficial for TDS suppression is negligible. Even so, under a proper setting of the FBG feedback, nanolasers can still achieve TDS suppression over a wide range of parameters, which is due to its characteristics of flat spectrum. Figures 11(b1)-(b4) show the evolution of EBW. It can be seen that the EBW increases when the FBG bandwidth becomes larger and allows more powers to perturb the nanolaser; see dark red regions. The zone of high EBW also expands gradually with the increment of the FBG bandwidth. Besides, for slightly larger FBG bandwidths, higher EBW can be found in the negative frequency detuning than that in the positive frequency detuning. It is due to red shift induced by the feedback which causes more powers to be concentrated on longer wavelengths.

Fig. 11. Two-dimensional maps of ACF (a1)-(a4) and EBW (b1)-(b4) in the parameter space of the feedback strength and frequency detuning, reflection spectra (c1)-(c4) with (a1)-(c1) |κ|=100 m⁻¹; (a2)-(c2) |κ|=300 m⁻¹; (a3)-(c3) |κ|=600 m⁻¹; (a4)-(c4) |κ|=900 m⁻¹.

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On top of all that, it is necessary to discuss the influence of F and $\beta $ on TDS concealment with FBG feedback, though it has been proved that larger F and $\beta $ will cause stronger damping of the relaxation oscillation and lead to more obvious TDS under optical feedback, single injection and dual injection [43–45]. As can be seen in Figs. 12(a1)–12(a4), the chaotic region shrinks in size greatly and the TDS concealment becomes worse as F and $\beta $ increases. This means similar conclusions can be drawn for FBG feedback and, for example, the TDS can be hardly concealed when F=14 and $\beta {\rm{ = 0}}{\rm{.1}}$, since no dark bule region can be seen. The corresponding EBW maps are presented in Figs. 12(b1)–12(b4). By combining the findings in Figs. 11 and 12, one can clearly see that it is possible to achieve both TDS suppression and EBW enhancement by appropriate selections of the key parameters considered above.

Fig. 12. Two-dimensional maps of ACF (a1)-(a4) and EBW (b1)-(b4) in the parameter space of the feedback strength and frequency detuning with (a1) and (b1)${{\;F = 1}}$ and ${{\beta = 0}}{\rm{.05}}$; (a2) and (b2)${{\;F = 7}}$ and ${{\beta = 0}}{\rm{.05}}$; (a3) and (b3)${{\;F = 14}}$ and ${{\beta = 0}}{\rm{.05}}$; (a4) and (b4)${{\;F = 14}}$ and ${{\beta = 0}}{\rm{.1}}$.

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Finally, we display the evolution maps of chaotic characteristics under FBG feedback in the parameter space of the delay time and feedback strength with Δf = 30 GHz, as shown in Fig. 13. The results show the variation of the feedback delay time takes no effect on TDS suppression and EBW enhancement. In other words, whatever the feedback delay time is, TDS disappears gradually and then appears again with the increase of the feedback strength, as shown in Fig. 13(a1). The EBW is enhanced gradually as the feedback strength increases regardless of the values of the feedback delay, as shown in Fig. 13(b1). Moreover, the results for the grating length of l=0.02 m and 0.04 m are compared in Fig. 13. It is interesting to find that maps shown in Figs. 13(a1) and 13(b1), i.e., corresponding to l=0.02 m, and Figs. 13(a2) and 13(b2) i.e., corresponding to l=0.04 m, are nearly the same. From the physical viewpoint, we understand that, though different grating lengths will affect the numbers of sidelobes from reflection spectrum, the dispersion which is determined by the FBG bandwidth remains unchanged. It represents that stable and desirable chaos with low TDS and high EBW can be feasibly generated without strict restrictions on the length of the feedback path and FBG gratings in practical applications.

Fig. 13. Two-dimensional maps of ACF (a1) and (a2) and EBW (b1) and (b2) in the parameter space of the delay time and feedback strength with (a1) and (b1) l=0.02 m; (a2) and (b2) l=0.04 m.

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4. Conclusion

In summary, numerical simulations have been conducted to analyze the dynamic behaviors and chaotic characteristics of nanolasers under FBG feedback via modified rate equations. Attention has been given on how to produce high-quality chaos which exhibits suppressed TDS and high EBW through optimizing the operation parameters. Numerical results show the phenomenon of obvious wavelength redshift with the increase of F, β. It is important to find such a specific emission wavelength then match it with the FBG central wavelength for generating rich dynamics via filtering. And in that case, large regions of chaos can be achieved when increasing FBG bandwidth and decreasing F, β. Above all, generated chaos subjected to FBG feedback, which is characterized by high bandwidth and suppressed TDS can be obtained within a wide parameter range space of (k_f, Δf). It is due to the increment of dispersion-induced group delay that reconstructs the external cavity and obscures the time delay information. Therefore, our findings are beneficial to achieving chaos-based applications by using the novel, compact nanolasers without complex structures.

Funding

National Natural Science Foundation of China (62004135, 62001317); Natural Science Research of Jiangsu Higher Education Institutions of China (20KJA416001, 20KJB510011); Natural Science Foundation of Jiangsu Province (BK20200855); Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P.R. China (IPOC2020A012); State Key Laboratory of Advanced Optical Communication Systems Networks, China (2021GZKF003); Project of Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing University of Aeronautics and Astronautics), Ministry of Education (RIMP2020001); Startup Funding of Soochow University (Q415900119).

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. M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015). [CrossRef]

2. M. C. Soriano, J. G. Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013). [CrossRef]

3. S. Y. Xiang, Z. X. Ren, Y. H. Zhang, X. X. Guo, G. Q. Han, and Y. Hao, “Computing primitive of fully-VCSELs-based all-optical spiking neural network for supervised learning and pattern classification,” IEEE Trans. Neural Netw. Learning Syst. 1–12 (2020).

4. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef]

5. N. Jiang, W. Pan, L. Yan, B. Luo, W. Zhang, S. Xiang, L. Yang, and D. Zheng, “Chaos synchronization and communication in mutually coupled semiconductor lasers driven by a third laser,” J. Lightwave Technol. 28(13), 1978–1986 (2010). [CrossRef]

6. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]

7. N. Li, B. Kim, V. N. Chizhevsky, A. Locquet, M. Bloch, D. S. Citrin, and W. Pan, “Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser,” Opt. Express 22(6), 6634–6646 (2014). [CrossRef]

8. F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40(6), 815–820 (2004). [CrossRef]

9. F. Y. Lin and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004). [CrossRef]

10. D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32(20), 2960–2962 (2007). [CrossRef]

11. M. Cheng, L. Deng, H. Li, and D. Liu, “Enhanced secure strategy for electro-optic chaotic systems with delayed dynamics by using fractional Fourier transformation,” Opt. Express 22(5), 5241–5251 (2014). [CrossRef]

12. Y. Hong, P. S. Spencer, and K. A. Shore, “Wideband chaos with time delay concealment in vertical-cavity surface-emitting lasers with optical feedback and injection,” IEEE J. Quantum Electron. 50(4), 236–242 (2014). [CrossRef]

13. R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011). [CrossRef]

14. Q. Zhao, Y. Wang, and A. Wang, “Eavesdropping in chaotic optical communication using the feedback length of an external-cavity laser as a key,” Appl. Opt. 48(18), 3515–3520 (2009). [CrossRef]

15. Y. N. Han, S. Y. Xiang, Y. Wang, Y. T. Ma, B. Wang, A. J. Wen, and Y. Hao, “Generation of multi-channel chaotic signals with time delay signature concealment and ultrafast photonic decision making based on a globally-coupled semiconductor laser network,” Photonics Res. 8(11), 1792–1799 (2020). [CrossRef]

16. S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor Laser,” IEEE Photonics J. 4(5), 1930–1935 (2012). [CrossRef]

17. S. S. Li and S. C. Chan, “Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback,” IEEE J. Sel. Top. Quantum Electron. 21(6), 541–552 (2015). [CrossRef]

18. Z. Q. Zhong, S. S. Li, S. C. Chan, G. Q. Xia, and Z. M. Wu, “Polarization-resolved time-delay signatures of chaos induced by FBG-feedback in VCSEL,” Opt. Express 23(12), 15459–15468 (2015). [CrossRef]

19. Z. Q. Zhong, Z. M. Wu, and G. Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550 nm VCSEL subject to FBG feedback,” Photonics Res. 5(1), 6–10 (2017). [CrossRef]

20. Y. Li, Z. M. Wu, Z. Q. Zhong, X. J. Yang, S. Mao, and G. Q. Xia, “Time-delay signature of chaos in 1550 nm VCSELs with variable-polarization FBG feedback,” Opt. Express 22(16), 19610–19620 (2014). [CrossRef]

21. J. Song, Z. Q. Zhong, L. X. Wei, Z. M. Wu, and G. Q. Xia, “Experimental investigations on nonlinear dynamics of a semiconductor laser subject to optical injection and fiber Bragg grating feedback,” Opt. Commun. 354, 213–217 (2015). [CrossRef]

22. W. Cai, S. Xiang, X. Cao, A. Wen, and Y. Hao, “Experimental investigation of the time-delay signature of chaotic output and dual-channel physical random bit generation in 1550 nm mutually coupled VCSELs with common FBG filtered feedback,” Appl. Opt. 59(15), 4583–4588 (2020). [CrossRef]

23. Y. Wang, S. Xiang, B. Wang, X. Cao, A. Wen, and Y. Hao, “Time-delay signature concealment and physical random bits generation in mutually coupled semiconductor lasers with FBG filtered injection,” Opt. Express 27(6), 8446–8455 (2019). [CrossRef]

24. J. Qi, Z. Q. Zhong, G. N. Wang, G. Q. Xia, and Z. M. Wu, “Characteristics of chaotic output from a Gaussian apodized ﬁber Bragg grating external-cavity semiconductor laser,” Acta. Phys. Sin. 66(24), 102–109 (2017). [CrossRef]

25. D. M. Wang, L. S. Wang, T. Zhao, H. Gao, Y. C. Wang, X. F. Chen, and A. B. Wang, “Time delay signature elimination of chaos in a semiconductor laser by dispersive feedback from a chirped FBG,” Opt. Express 25(10), 10911–10924 (2017). [CrossRef]

26. D. M. Wang, L. S. Wang, Y. Y. Guo, Y. C. Wang, and A. B. Wang, “Key space enhancement of optical chaos secure communication:chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019). [CrossRef]

27. J. Ke, L. Yi, T. Hou, Y. Hu, G. Q. Xia, and W. Hu, “Time delay concealment in feedback chaotic systems with dispersion in loop,” IEEE Photonics J. 9(2), 1–8 (2017). [CrossRef]

28. J. Zhang, M. Li, A. Wang, M. Zhang, Y. Ji, and Y. C. Wang, “Time-delay-signature-suppressed broadband chaos generated by scattering feedback and optical injection,” Appl. Opt. 57(22), 6314–6317 (2018). [CrossRef]

29. S. I. Azzam, A. V. Kildishev, R. M. Ma, C. Z. Ning, R. F. Oulton, V. M. Shalaev, M. I. Stockman, J. L. Xu, and X. Zhang, “Ten years of spasers and plasmonic nanolasers,” Light Sci. Appl. 9(1), 90 (2020). [CrossRef]

30. R. M. Ma and R. F. Oulton, “Applications of nanolasers,” Nat. Nanotechnol. 14(1), 12–22 (2019). [CrossRef]

31. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009). [CrossRef]

32. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef]

33. M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. V. Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009). [CrossRef]

34. K. Ding, M. T. Hill, Z. C. Liu, L. J. Yin, P. J. V. Veldhoven, and C. Z. Ning, “Record performance of electrical injection sub-wavelength metallic-cavity semiconductor lasers at room temperature,” Opt. Express 21(4), 4728–4733 (2013). [CrossRef]

35. E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express 17(10), 7790–7799 (2009). [CrossRef]

36. K. A. Shore, “Modulation bandwidth of metal-clad semiconductor nanolasers with cavity-enhanced spontaneous emission,” Electron. Lett. 46(25), 1688–1689 (2010). [CrossRef]

37. Z. A. Sattar and K. A. Shore, “Analysis of the direct modulation response of nano-lasers,” J. Lightwave Technol. 33(14), 1 (2015). [CrossRef]

38. Z. A. Sattar and K. A. Shore, “External optical feedback effects in semiconductor nanolasers,” IEEE J. Sel. Top. Quantum Electron. 21(6), 500–505 (2015). [CrossRef]

39. Z. A. Sattar and K. A. Shore, “Phase conjugate feedback effects in nano-lasers,” IEEE J. Quantum Electron. 52(4), 1–8 (2016). [CrossRef]

40. Z. A. Sattar, N. A. Kamel, and K. A. Shore, “Optical injection effects in nanolasers,” IEEE J. Quantum Electron. 52(2), 1–8 (2016). [CrossRef]

41. H. Han and K. A. Shore, “Dynamics and stability of mutually coupled nano-lasers,” IEEE J. Quantum Electron. 52(11), 1–6 (2016). [CrossRef]

42. H. Han and K. A. Shore, “Analysis of high-frequency oscillations in mutually-coupled nano-lasers,” Opt. Express 26(8), 10013–10022 (2018). [CrossRef]

43. A. Elsonbaty, S. F. Hegazy, and S. S. A. Obayya, “Simultaneous concealment of time delay signature in chaotic nanolaser with hybrid feedback,” Opt. Lasers Eng. 107, 342–351 (2018). [CrossRef]

44. Y. Qu, S. Xiang, Y. Wang, L. Lin, A. J. Wen, and Y. Hao, “Concealment of time delay signature of chaotic semiconductor nanolasers with double chaotic optical injections,” IEEE J. Quantum Electron. 55(2), 1–7 (2019). [CrossRef]

45. P. Jiang, P. Zhou, N. Li, P. Mu, and X Li, “Optically injected nanolasers for time-delay signature suppression and communications,” Opt. Express 28(18), 26421–26435 (2020). [CrossRef]

46. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]

47. P. Zhou, Q. Fang, and N. Li, “Phased-array assisted time-delay signature suppression in the optical chaos generated by an external-cavity semiconductor laser,” Opt. Lett. 45(2), 399–402 (2020). [CrossRef]

48. R. Zhang, P. Zhou, Y. Yang, Q. Fang, P. Mu, and N. Li, “Enhancing time-delay suppression in a semiconductor laser with chaotic optical injection via parameter mismatch,” Opt. Express 28(5), 7197–7206 (2020). [CrossRef]

49. C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002). [CrossRef]

50. N. Li, R. M. Nguimdo, A. Locquet, and D. S. Citrin, “Enhancing optical feedback-induced chaotic dynamics in semiconductor ring lasers via optical injection,” Nonlinear Dyn. 92(2), 315–324 (2018). [CrossRef]

51. N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48(10), 1339–1345 (2012). [CrossRef]

52. N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600109 (2013). [CrossRef]

53. N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015). [CrossRef]

54. A. Uchida, Optical Communication with Chaotic Lasers: Applications of Nonlinear Dynamics and Synchronization (Wiley, 2012), Chap. 4.

55. N. Li, H. Susanto, B. R. Cemly, I. D. Henning, and M. J. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017). [CrossRef]

56. M. Virte, A. K. D. Bosco, D. Wolfersberger, and M. Sciamanna, “Chaos crisis and bistability of self-pulsing dynamics in a laser diode with phase-conjugate feedback,” Phys. Rev. A 84(4), 043836 (2011). [CrossRef]

57. G. A. Gottwald and I. Melbourne, “On the implementation of the 129–145 Test for chaos,” SIAM J. Appl. Dyn. Syst. 8(1), 129–145 (2009). [CrossRef]

58. F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48(8), 1010–1014 (2012). [CrossRef]

Characterizing the chaotic dynamics of a semiconductor nanolaser subjected to FBG feedback

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Equations (5)

Optics Express