Wideband and high-dimensional chaos generation using optically pumped spin-VCSELs

Yao Zeng; Yao Zeng; Yao Zeng; Pei Zhou; Pei Zhou; Pei Zhou; Yu Huang; Yu Huang; Penghua Mu; Nianqiang Li; Nianqiang Li

doi:10.1364/OE.477394

1. Introduction

Wideband chaos generation based on semiconductor lasers has attracted a great deal of attention in recent years for its potential in meeting the need of high-speed secure communication [1–4], achieving fast random bit generation [5–7], acquiring satisfactory resolution of chaotic lidar/radar [8–10], and realizing the accurate ranging to the multi targets [11]. However, limited to the relaxation oscillation frequency of semiconductor lasers, the resulting chaos bandwidth is of the order of magnitude of a few GHz and the power spectrum of chaos signals is not featureless. In addition, there exists an unwanted time delay signature (TDS) in most of chaos generation schemes. These two flaws may degrade the performance of the aforementioned applications [12]. To solve these problems, multifarious techniques have been proposed to generate the broadband chaotic signals with TDS suppression, such as optoelectronic hybrid feedback and parallel filtering [13], delay-interfered self-phase modulated feedback [14], frequency-detuned grating feedback [15], the combination of self-phase-modulated feedback and a microsphere resonator [16], asymmetric dual-path optical feedback [17], dual-chaotic optical injection [18], vertical-cavity surface-emitting lasers (VCSELs) with dual-path chaotic optical injections [19], VCSELs with optical feedback and injection [20], scattering feedback and optical injection [21], highly nonlinear configuration of inphase and quadrature-phase (IQ) modulator [22], electrical heterodyning [23], heterodyning two independent chaotic outputs from two external-cavity laser diodes with optical feedback [24], a ring network consisting of three semiconductor lasers coupled with heterogeneous delays [25], three-cascaded VCSELs [26,27]. Most of them indeed offer promising methods to achieve broadband chaos but at the sacrifice of the simplicity of the system. Therefore, it is of great importance to find alternative approaches for broadband chaos generation with novel lasers with no need of complex perturbation/control components.

As is known, a free-running VCSEL has the possibility to yield polarization chaos, which was first reported in 2013 by Virte et al. [28]. After that, much effort has been devoted to optimizing and synchronizing polarization chaos originally generated by solitary VCSELs [29,30]. Compared with the conventional VCSEL, a spin-polarized vertical-cavity surface-emitting laser (spin-VCSEL) is a specific example of spintronic devices that evinces various superiorities, including ultra-fast polarization dynamics, threshold reduction, and flexible spin control of the lasing output [31–33]. Benefiting from these merits, spin-VCSELs have multitudinous prospects for chaos-based applications, such as secure communication, secure key distribution, and lidar ranging [34–36], in which, however, the chaotic properties were not enhanced, leading to limited performance. For example, in our previous work, the maximal bit rate of message encryption was only 4 Gb/s [34], which is far from the reported records [2]. It is worth noting that the chaotic properties, i.e., the bandwidth, the spectral flatness (SF), and TDS performance, which play key roles in the aforementioned applications, have not been investigated yet in the reported spin-VCSEL systems.

On the other hand, the chaotic bandwidth is relatively low, i.e., only of the order of magnitude of a few GHz, which is not in favor of the chaos-based applications, e.g., limiting the bit rate of the generated RBG. Therefore, it is important to find alternative methods to generate high-quality chaos signals based on VCSELs. On the other hand, limited to the structure of a solitary spin-VCSEL, i.e., no delay loop, the chaotic dimension is relatively low, which may only support low-dimensional optical chaos-based applications. As is known, the optical injection configurations, such as mutual injection [37] and intensity-modulated optical injection [38–40], have been widely studied, because they are easy to implement and can support the generation of wideband chaos signals. However, similar to the structure of a solitary laser system, the chaotic dimension is relatively low for the structure of optical injection systems, where no delay loop is introduced. Fortunately, it is known that some specific feedback techniques either allow for broadband chaos [41,42] or enhance the correlation dimension (CD) of the generated chaos [43]. Therefore, it is valuable to investigate how to manipulate optical feedback in order to support higher chaotic dimension as well as enhance other chaotic features in spin-VCSELs. In addition to optical feedback, optical heterodyning was considered to be an alternative important technique for superior chaos generation in lasers with optical feedback [24]. By properly setting the feedback parameters and frequency detuning, the feature of the laser relaxation oscillation was eliminated and the dimensionality was enhanced. However, to the best of our knowledge, there is no work that has been undertaken to study the properties of chaotic signals obtained by heterodyning two independent chaos outputs generated from two free-running spin-VCSELs. Hence, it is also interesting to establish whether the optical heterodyne method can enhance the chaotic performance of free-running spin-VCSELs.

In this paper, we focus on optimizing the chaotic characteristics of free-running spin-VCSELs by either introducing optical feedback or heterodyning. We first focus on the influences of the feedback strength and pump power on the chaotic performance under the case of the spin-VCSEL with optical feedback in detail. Then, we turn to study the optical heterodyne configuration. Our results show that both schemes can obtain superior chaos signals with larger bandwidth and higher dimension/complexity compared to the chaos generated by the free-running spin-VCSEL under the same parameter conditions. More specifically, the bandwidth up to 30 GHz and above, no TDS of greater than 0.2, the flatness of 0.75 and above, and the high dimension/complexity can be simultaneously achieved over a large range of parameters in the proposed schemes. Furthermore, the optical feedback scheme outperforms the optical heterodyne scheme in terms of higher complexity/dimension of chaos signals, whereas the scheme of heterodyning two solitary spin-VCSELs only requires relatively low pump levels for achieving high-quality chaotic signals and may be more feasible for experimental implementation due to energy conservation.

2. Theoretical model

For optical feedback and optical heterodyning, the rate equations of optically pumped spin-VCSELs can be modeled as follows [24,44–46], where the spontaneous emission noise is not considered:

(1)$$\begin{aligned} \frac{{dA_{x,y}^{1,2}}}{{dt}} &= [{\kappa ({{N^{1,2}} - 1} )\mp {\gamma_a}} ]A_{x,y}^{1,2} \pm {n^{1,2}}\kappa A_{y,x}^{1,2} \times [{\sin ({\varphi_{x,y}^{1,2} - \varphi_{y,x}^{1,2}} )- \alpha \cos ({\varphi_{x,y}^{1,2} - \varphi_{y,x}^{1,2}} )} ]\\ &+ {k_f}A_{x,y}^{1,2}({t - \tau } )\cos [2\pi {f_0}\tau + \varphi _{x,y}^{1,2} - \varphi _{x,y}^{1,2}(t - \tau )], \end{aligned}$$

(2)$$\begin{aligned} \frac{{d\varphi _{x,y}^{1,2}}}{{dt}} &= \alpha \kappa ({{N^{1,2}} - 1} )\mp {\gamma _P} \pm {n^{1,2}}\kappa \frac{{A_{y,x}^{1,2}}}{{A_{x,y}^{1,2}}} \times [{\cos ({\varphi_{x,y}^{1,2} - \varphi_{y,x}^{1,2}} )+ \alpha \sin ({\varphi_{x,y}^{1,2} - \varphi_{y,x}^{1,2}} )} ]\\ &- {k_f}\frac{{A_{x,y}^{1,2}(t - \tau )}}{{A_{x,y}^{1,2}}}\sin [2\pi {f_0}\tau + \varphi _{x,y}^{1,2} - \varphi _{x,y}^{1,2}(t - \tau )], \end{aligned}$$

(3)$$\frac{{d{N^{1,2}}}}{{dt}} ={-} \gamma {N^{1,2}}({1 + {{|{A_x^{1,2}} |}^2} + {{|{A_y^{1,2}} |}^2}} )+ \gamma {\eta _{1,2}} - i\gamma {n^{1,2}}A_x^{1,2}A_y^{1,2}[{{e^{i({\varphi_y^{1,2} - \varphi_x^{1,2}} )}} - {e^{i({\varphi_x^{1,2} - \varphi_y^{1,2}} )}}} ],$$

(4)$$\frac{{d{n^{1,2}}}}{{dt}} = \gamma P{\eta _{1,2}} - {\gamma _s}{n^{1,2}} - \gamma {n^{1,2}}({{{|{A_x^{1,2}} |}^2} + {{|{A_y^{1,2}} |}^2}} )- i\gamma {N^{1,2}}A_x^{1,2}A_y^{1,2}[{{e^{i({\varphi_y^{1,2} - \varphi_x^{1,2}} )}} - {e^{i({\varphi_x^{1,2} - \varphi_y^{1,2}} )}}} ],$$

where the superscripts and subscripts 1 and 2 represent spin-VCSEL1 and spin-VCSEL2, respectively. ${A_{x,y}}$ and ${\varphi _{x,y}}$ stand for the amplitudes and phases of x LP and y LP, respectively. N is the total carrier inversion between the upper conduction and the lower valence bands, and n is the difference between the population inversions of the spin-up and spin-down radiation channels. The linear dichroism ${\gamma _a}$ and the linear birefringence ${\gamma _p}$ are employed to model the amplitude anisotropy and the phase anisotropy, respectively. Other parameters of this model are defined as follows: $\kappa $ is the optical field decay rate, $\alpha $ is the linewidth enhancement factor, $\gamma $ is the carrier decay rate, ${\gamma _s}$ is the spin-flip relaxation rate, $\eta = {\eta _ + } + {\eta _ - }$ represents the total normalized pump power of the optical pumping, and $P = ({{\eta_ + } - {\eta_ - }} )/({{\eta_ + } + {\eta_ - }} )$ is the pump polarization ellipticity, where ${\eta _ + }$ $({{\eta_ - }} )$ is the right (left) circularly polarized pumping component describing the polarized light pumping. The third terms in Eqs. (1) and (2) account for the optical feedback, where ${k_f}$ corresponds to the feedback strength, $\tau $ represents the feedback delay time, and ${f_0}$ stands for the center frequency of the spin-VCSEL. Moreover, the heterodyne signal can be expressed as:

(5)$${I_H} = 2{A^1}{A^2}\sin [{2\pi \Delta \upsilon t + ({{\varphi^2} - {\varphi^1}} )} ]$$

where ${A^1}$ and ${A^2}$ are the amplitudes of free-running spin-VCSEL1 and spin-VCSEL2, respectively. ${\varphi ^1}$ and ${\varphi ^2}$ denote the phases of spin-VCSEL1 and spin-VCSEL2, respectively. $\mathrm{\Delta }v$ represents the frequency detuning of two solitary spin-VCSELs. Note here that the last terms in Eqs. (1) and (2) do not exist for the case of optical heterodyning.

In our analysis, the fourth-order Runge-Kutta algorithm is utilized to solve Eqs. (1–4) with a fixed step of 1 ps and the following parameters used for the spin-VCSELs are kept constant [30,47–50]: $\kappa = 600\; \textrm{n}{\textrm{s}^{ - 1}}$, $\alpha = 3$, ${\gamma _a} ={-} 0.7\; \textrm{n}{\textrm{s}^{ - 1}}$, ${\gamma _p} = 25\; \textrm{n}{\textrm{s}^{ - 1}}$, $\gamma = 1\; \textrm{n}{\textrm{s}^{ - 1}}$, and ${f_0}$ = 193.55 THz. Additionally, we choose the feedback delay time of $\tau = 3\; \textrm{ns}$ by following our previous work [47], since it has no obvious influence on the results reported below. For brevity, in the case of optical feedback, we only present the results for x LP of spin-VCSEL1/spin-VCSEL2, since the output results for both polarizations are almost identical. In the scenario of optical heterodyning, we only consider the results for heterodyning two chaotic outputs from the x LPs of spin-VCSEL1 and spin-VCSEL2, owing to the fact that the heterodyne results for four cases, i.e., heterodyning x LP/y LP output of spin-VCSEL1 and x LP/y LP output of spin-VCSEL2, are nearly the same.

3. Results and discussion

3.1 Chaotic characteristics of the free-running spin-VCSEL

Figures 1(a1) and 1(b1) show the typical two-dimensional bifurcation diagrams of a solitary spin-VCSEL in the parameter space $({\eta ,P} )$, where ${\gamma _s}$ = 100 $\textrm{n}{\textrm{s}^{ - 1}}$ and ${\gamma _s}$ = 20 $\textrm{n}{\textrm{s}^{ - 1}}$, respectively. Here dynamical states are conveniently defined by calculating the extrema of the time series. The continuous-wave (CW), period-one (P1), period-two (P2), and complicated dynamics including chaos are identified as a constant intensity, two intensity extrema, four intensity extrema, and even more extrema, respectively. As shown in Figs. 1(a1) and 1(b1), we utilize white, blue, and cyan to represent the CW, P1, and P2, respectively. We use other colors to represent complicated dynamics including chaos where the number of extrema exceeds 4 and gradually increases using green to yellow, red, and dark red. Here, we choose the 0–1 test for chaos to confirm the chaotic states and the non-chaotic states. We would like to emphasize that the output of the test close to 0 indicates a non-chaotic state, whereas the output of the test close to 1 means a chaotic state, and more details about the 0–1 test for chaos were described in the previous report [51–54]. The results of the 0–1 test for chaos calculated in the same parameter space are shown in Figs. 1(a2) and 1(b2), where the chaotic states corresponding to the value close to 1 are marked in dark red and the non-chaotic states corresponding to the value close to 0 are marked in white and blue. The results indicate that the complicated dynamic region obtained from our bifurcation analysis almost coincides with that identified by the 0-1 test for chaos. Apparently, as ${\gamma _s}$ is decreased, the complicated dynamic region becomes larger, which coincides with the previous reports about VCSELs [30,44]. Considering that the chaotic dynamics generated from the free-running spin-VCSEL with ${\gamma _s}$ = 20 $\textrm{n}{\textrm{s}^{ - 1}}$ can exist within a broader region in the $({\eta ,P} )$ plane, we further discuss the characteristics of chaos generated by the spin-VCSEL with ${\gamma _s}$ = 20 $\textrm{n}{\textrm{s}^{ - 1}}$. An example of the typical chaotic intensity time series and power spectrum is shown in Figs. 1(c1) and 1(c2), in which a broad, flat, and featureless power spectrum can be observed.

Fig. 1. Two-dimensional (a1-b1) bifurcation diagrams and (a2-b2) maps of the 0-1 test for chaos of a solitary spin-VCSEL in the $({\eta ,P} )$ plane, where (a1-a2) ${\gamma _s} = 100\; \textrm{n}{\textrm{s}^{ - 1}}$ and (b1-b2) ${\gamma _s} = 20\; \textrm{n}{\textrm{s}^{ - 1}}$. (c1) Time series and (c2) power spectrum of chaotic states for ${\gamma _s} = 20\; \textrm{n}{\textrm{s}^{ - 1}}$, $\eta = 6.8$, and $P ={-} 0.27$.

Download Full Size | PDF

The characteristics of the chaos reported in Fig. 1 call for more quantitative analyses. Therefore, in the following, we further analyze the evolution of bandwidth and SF of chaos generated in the free-running spin-VCSEL. Here, the definition of the chaotic bandwidth used in our simulation is the effective bandwidth which sums up only those discrete spectral segments accounting for 80% of the total power [55]. To demonstrate whether the frequencies are widely distributed, we define SF as the ratio of the arithmetic mean to the geometric mean of the 80% highest energy portion of the spectrum [56,57]:

(6)$$\textrm{SF} = \frac{{\prod _{n = 1}^{{N_s}}\textrm{PSD}{{({{f_n}} )}^{1/{N_s}}}}}{{\frac{1}{{{N_s}}}\sum {_{n = 1}^{{N_s}}} \textrm{PSD}({{f_n}} )}}$$

where PSD$({{f_n}} )$ represents the power spectral density at the nth discrete frequency ${f_n}$ in a given bandwidth and ${N_s}$ is the total number of samples. SF varies from 0 to 1, and a value of 1 corresponds to a perfectly flat spectrum, on the contrary, a value of 0 is associated with a pure spectral tone. We first focus on the dependence of the bandwidth of the generated chaos on the pump power $\eta $ and polarization ellipticity P, as shown in Figs. 2(a) and 2(b), respectively. As we can see, the chaotic bandwidth can be improved by increasing the values of $\eta $ or P, and a larger slope is observed in the case of $\eta $ [see Fig. 2(a)]. Likewise, we display the evolution of SF as a function of $\eta $ or P, as shown in Figs. 2(c) and 2(d). One can see from Fig. 2(c) that the flatness of the power spectrum slightly increases with the enhancement of $\eta $. Interestingly, as the value of P is enlarged, the SF increases obviously at first and then trends to be an almost stationary value for $P$ > −0.4. Synthesizing the results of Fig. 2, we find that the free-running spin-VCSEL can achieve both bandwidth and flatness enhancement simultaneously through increasing the pump power $\eta $ or/and the pump polarization ellipticity P. Under the selected value of $\eta $ and P, the chaos signal with a bandwidth greater than 30 GHz and the flatness of 0.75 or more can be obtained.

Fig. 2. (a) The bandwidth (BW) and (c) flatness of chaos as functions of the pump power $\eta $, with $P ={-} 0.27$. (b) The bandwidth and (d) flatness of chaos as functions of the pump polarization ellipticity P, with $\eta = 6$.

Download Full Size | PDF

3.2 Enhanced characteristics of the spin-VCSEL with optical feedback

Now, Although the addition of spin degrees of freedom greatly enriches the chaotic regime of spin-VCSELs, the chaos dimension is lower than hyperchaos [34]. Thus, to improve the properties of chaos generated by a free-running spin-VCSEL, we take into account the effect of optical feedback [43]. Figure 3 shows the power spectra, autocorrelation functions (ACFs), and CD curves for the chaos signals. To quantitatively evaluate the TDS of obtained chaos signals, we adopt the ACF which is described as [13,14,19,58]:

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

where $I(t )$ denotes the intensity time series, $\Delta t$ is the time shift, and $\langle{\cdot}\rangle$ stands for the time average. Here, the measurement limit of the ACF related to the data size is 300000. For estimating CD of the chaotic outputs, we use the Grassberger-Procaccia (GP) algorithm [24,43,59,60], which calculates the correlation integral C(r), i.e., the probability of point pairs with Euclidean distance not greater than r in the phase space of delayed embedding, and estimates CD by the slope of convergence of the logarithmic plot of C(r) versus r. To begin with, we show a typical example of the chaos signal generated from the free-running spin-VCSEL. The effective bandwidth and CD of chaos are calculated to be 32.29 GHz and 5.71, respectively. As expected, we find the TDS value in the ACF is close to 0. Afterwards, we consider three cases of chaos signals generated from the spin-VCSEL with optical feedback, as shown in Figs. 3(b)–3(d), where the feedback strengths are set as ${k_f}$ = 10 $\textrm{n}{\textrm{s}^{ - 1}}$, ${k_f}$ = 20 $\textrm{n}{\textrm{s}^{ - 1}}$, and ${k_f}$ = 30 $\textrm{n}{\textrm{s}^{ - 1}}$, respectively. It is seen that the bandwidth and CD are enhanced compared with the case of ${k_f}$ = 0. Nevertheless, no obvious TDS can be identified from ACF plots until ${k_f}$ is increased to 30 $\textrm{n}{\textrm{s}^{ - 1}}$, where a pronounced peak located around the position of the feedback delay time $\tau $ appears in the ACF [see Fig. 3(d2)]. These results indicate that by properly adjusting the feedback strength, the bandwidth and CD of chaos can be improved, meanwhile there is no obvious TDS in the spin-VCSELs with optical feedback.

Fig. 3. Power spectra (first column), ACF traces (second column), and CD curves (third column) when the feedback strength (a) ${k_f} = 0\; \textrm{n}{\textrm{s}^{ - 1}}$, (b) ${k_f} = 10\; \textrm{n}{\textrm{s}^{ - 1}}$, (c) ${k_f} = 20\; \textrm{n}{\textrm{s}^{ - 1}}$, and (d) ${k_f} = 30\; \textrm{n}{\textrm{s}^{ - 1}}$, with $\eta = 6.8$, $P ={-} 0.27$.

Download Full Size | PDF

Figure 4 presents the influence of the pump power on the bandwidth, TDS, and CD of chaos generated in the spin-VCSELs with optical feedback, where ${k_f}$ = 30 $\textrm{n}{\textrm{s}^{ - 1}}$. Similar to Fig. 3, increasing the pump power can also improve the chaotic bandwidth and dimension. Especially, in Figs. 4(a2)–4(d2), as the pump power is increased, the TDS values in ACF traces gradually decrease, which implies that the enhancement of the pump power can suppress the TDS. Moreover, it is expected that, as the pump power is further increased, e.g., $\eta = 10.8$, the chaotic bandwidth and dimension are further improved and meanwhile the TDS value in ACF trace further decreases. Therefore, for spin-VCSELs subject to optical feedback, the pump power can be used to enhance the chaotic performance. More specifically, by reasonably increasing the pump power, one can obtain broadband chaos with desired TDS suppression and CD performance.

Fig. 4. Power spectra (first column), ACF traces (second column), and CD curves (third column) when the pump power (a) $\eta = 2.8$, (b) $\eta = 4.8$, (c) $\eta = 5.8$, and (d) $\eta = 8.8$, with $P ={-} 0.27$, ${k_f} = 30\; \textrm{n}{\textrm{s}^{ - 1}}$.

Download Full Size | PDF

Next, we further investigate the evolution of the chaos bandwidth, the SF, the TDS values (the ACF maximum in the time window [2.7, 3.3]ns), and the CD when the feedback strength ${k_f}$ is used as a control parameter for four values of the pump power $\eta $. The evolution of the bandwidth is shown in Fig. 5(a), where in all cases of $\eta $, the chaotic bandwidth is increased as ${k_f}$ is enhanced. In addition, for a given ${k_f}$, the bandwidth monotonically increases as the value of $\eta $ is increased, which indicates that the bandwidth strongly depends on $\eta $. As shown in Fig. 5(b), the flatness of the chaotic signal is insensitive to the variation of ${k_f}$ for low ${k_f}$. However, it gradually decreases with the increase of ${k_f}$ when the feedback power is strong enough. It should be noted that under a strong pump power, e.g., $\eta $ = 8.8, the effect of the feedback strength on the SF is drastically weakened. From Fig. 5(c), one can see that, for all cases of the pump power, before the value of ${k_f}$ reaches a certain threshold (the corresponding TDS is close to 0.2, which can be considered as a low level and may have negligible effect on chaos-based applications [25,61,62].), the residual TDS is considered to be low and has negligible effect on chaos-based applications. Once the feedback strength exceeds the threshold, the value of TDS significantly increases. Furthermore, we also find that such a threshold can be increased by enhancing the pump power $\eta $. As for the CD evolution shown in Fig. 5(d), almost the same trend can be seen, i.e., CD is enhanced with increasing ${k_f}$ and a higher dimension is associated with a larger $\eta $ for a selected ${k_f}$. Hence, from these results, we conclude that, through properly tailoring the feedback strength and the pump power, the performance of chaos generated from spin-VCSELs with optical feedback can be enhanced, e.g., in the range of 0 $\textrm{n}{\textrm{s}^{ - 1}}$ < ${k_f}$ < 20 $\textrm{n}{\textrm{s}^{ - 1}}$ for all cases of $\eta $, the chaos signals with broader bandwidth, satisfactory SF, no obvious TDS, and higher dimension can be obtained.

Fig. 5. Evolution of (a) the chaos bandwidth (BW), (b) the spectral flatness, (c) the TDS value around $\tau $, and the CD as a function of ${k_f}$ for four values of $\eta $.

Download Full Size | PDF

To obtain a global view of the effects of the feedback strength and the pump power for the chaotic properties, we plot two-dimensional maps of the bandwidth, SF, TDS, and dimension/complexity in the $({{k_f},\; \eta } )$ plane. Note that, although the CD is the standard measure to quantify the dimension property, it is not easy to draw two-dimensional maps of the CD. Therefore, we turn to permutation entropy (PE) estimation of the chaotic output, which can provide valuable information for the complexity [56,58,63], due to its computation efficiency and accuracy. Herein, the embedding dimension is D = 5 and the embedding delay is ${\tau _e}$ = $\tau $/${\mathrm{\Omega }_s}$, where the sampling period is ${\mathrm{\Omega }_s}$ = 10 ps. In Fig. 6(a), we can find that the chaos bandwidth can be improved by either increasing the feedback strength or the pump power. Interestingly, the comparison among Figs. 6(b)–6(d) shows that the regions for larger flatness and better TDS suppression coincide with that for complexity enhancement, where the feedback strength trends to a smaller value and the pump power trends to a larger value. Such chaos signal can be identified as strong chaos [64], which is desirable for many chaos-based applications. Specifically, for a spin-VCSEL subject to optical feedback, chaotic signals with better performance, i.e., the bandwidth of > 30 GHz, no TDS of greater than 0.2, the flatness of ∼0.75 (the flatness value of 0.75 can be a threshold value to distinguish whether the spectrum of a chaotic state is flat or not [56,57]), and the complexity of > 0.97, can be achieved in a wide range of the parameter space.

Fig. 6. Mapping diagrams of the (a) bandwidth (BW), (b) SF, (c) TDS value, and (d) PE in the parameter space of $({{k_f},\eta } )$.

Download Full Size | PDF

3.3 Enhanced characteristics of spin-VCSELs through optical heterodyning

In this section, we present a heterodyne scheme based on two free-running spin-VCSELs to boost the quality of chaotic signals. In this method, it is compulsory to explore the influence of frequency detuning $\mathrm{\Delta }v$ between spin-VCSEL1 and spin-VCSEL2 on heterodyne spectra. Some typical results are shown in Fig. 7, for three values of the frequency detuning with the fixed ${\eta _1}$ of 3 and ${\eta _2}$ of 3.5. Note that both lasers work in chaotic regimes for all cases as shown in Figs. 7(a1-c1). Figures 7(a1-a3) show the case of the detuning of 40 GHz of laser center frequencies. It can be seen that the optical spectra of two spin-VCSELs are primarily separated in Fig. 7(a2), which results in an upswept heterodyne spectrum shown in Fig. 7(a3). This is due to the power enhancement at the detuning originating from the beat between center frequencies of the lasers. As the frequency detuning is decreased to 20 GHz, one can see that the overlap of optical spectra of two lasers is enhanced [see Fig. 7(b2)] and a flatter spectrum [see Fig. 7(b3)], where the components of low frequencies and high frequencies are almost equally distributed. Further reducing the frequency detuning to 0 GHz, the optical spectra of two spin-VCSELs are almost overlapped with each other, resulting in a declivitous spectrum shown in Fig. 7(c3). This is similar to the spectrum of a free-running spin-VCSEL operating in the chaotic regime, where the power around the relaxation oscillation frequency (∼ a few GHz) is high. These results are similar to previous findings for heterodyning two conventional external-cavity semiconductor lasers [24]. One can observe from Figs. 7(a3-c3) that the chaotic bandwidth can be weakened from 40.45 GHz to 23.07 GHz with the decrease of the frequency detuning and Fig. 7(b3) shows a flatter microwave spectrum compared with Figs. 7(a3) and 7(c3). This implies that a proper setting of the detuning is important for promoting the spectrum feature.

Fig. 7. (a1-c1) Time series of two initial chaos signals, (a2-c2) optical spectra and (a3-c3) microwave spectra of heterodyne signals obtained at different frequency detuning of two spin-VCSELs. (a1-a3) $\mathrm{\Delta }v = 40\; \textrm{GHz}$, (b1-b3) $\mathrm{\Delta }v = 20\; \textrm{GHz}$, and (c1-c3) $\mathrm{\Delta }v = 0\; \textrm{GHz}$, with ${\eta _1} = 3$, ${\eta _2} = 3.5$, and $P ={-} 0.2$.

Download Full Size | PDF

Figure 8 gives the trend of chaotic bandwidth and flatness as a function of the frequency detuning for several cases of the pump power ${\eta _1}$ and ${\eta _2}$. It can be seen that the effects of increasing ${\eta _1}$ or/and ${\eta _2}$ on the bandwidth and flatness of heterodyne signals are similar. Therefore, in the following analyses, we focus on one of three scenarios shown in Fig. 8. From Fig. 8(a2), we can find that, as the frequency detuning increases, the bandwidth of the heterodyne signal first increases, then reaches its maximum, and finally decreases for all cases of ${\eta _2}$; for a given $\mathrm{\Delta }v$, the larger the pump power is, the larger the bandwidth is. Likewise, it can be seen in Fig. 8(b2) that the flatness of heterodyne signals is slightly increased at first (e.g., $\mathrm{\Delta }v$ < 20 GHz) before achieving its maximum and then dramatically decreased as the value of $\mathrm{\Delta }v$ increases for all cases. In addition, we notice that the effect of $\mathrm{\Delta }v$ on the flatness is weakened for the case of larger pump powers in the special parameter region, e.g., $\mathrm{\Delta }v$ > 20 GHz. These results show that the bandwidth and flatness of heterodyne signals can be simultaneously enhanced when the frequency detuning is properly selected, which confirms the finding in Fig. 7.

Fig. 8. Evolution of (a) the bandwidth (BW) and (b) the spectral flatness of the heterodyne signal as a function of $\mathrm{\Delta }v$ for four values of (a1-b1) ${\eta _1}$, with ${\eta _2} = 3.5$ and $P ={-} 0.2$, (a2-b2) ${\eta _2}$, with ${\eta _1} = 3$ and $P ={-} 0.2$, and (a3-b3) ${\eta _1}$, ${\eta _2}$, with $P ={-} 0.2$.

Download Full Size | PDF

Further, we consider the influence of the pump power on the heterodyne signals with a selected frequency detuning of 30 GHz. The first column of Fig. 9 shows the time series of two chaotic outputs from the free-running spin-VCSEL1 and spin-VCSEL2, the second column of Fig. 9 shows the optical spectra of heterodyne signals, and the corresponding microwave spectra are shown in the third column of Fig. 9. We can see from Figs. 9(a2-c2) that, as the pump power of spin-VCSEL2 is increased, the optical spectra of spin-VCSEL2 are greatly broadened, which leads to more overlap between the optical spectra of the two lasers; the low-frequency components of the heterodyne signals are enhanced. In addition, owing to the reduction of the power of center frequencies of spin-VCSEL2 with increasing ${\eta _2}$, there are no obvious peaks in the microwave spectra, which originate from the beat between the center frequencies of the two spin-VCSELs. Therefore, one can find in Figs. 9(a3-c3) that the bandwidth and flatness of heterodyne signals can be enhanced as the pump power is stronger.

Fig. 9. (a1-c1) Time series of two initial chaos signals, (a2-c2) optical spectra and (a3-c3) microwave spectra of heterodyne signals obtained at the different pump power of spin-VCSEL2. (a1-a3) ${\eta _2} = 3.5$, (b1-b3) ${\eta _2} = 5.5$, and (c1-c3) ${\eta _2} = 7.5$, with $\mathrm{\Delta }v = 30\; \textrm{GHz}$ and ${\eta _1} = 3$ and $P ={-} 0.2$.

Download Full Size | PDF

Figure 10 shows the influence of the pump power on the bandwidth and flatness of heterodyne signals for four cases of frequency detuning $\mathrm{\Delta }v$. It can be seen that, in all cases, the bandwidth of the heterodyne signal monotonously increases as the pump power is gradually boosted. Nevertheless, there is a great difference in the variation of flatness for four cases. In the scenarios of smaller values of the frequency detuning, e.g., $\mathrm{\Delta }v$ = 10 GHz and $\mathrm{\Delta }v$ = 20 GHz, the flatness depends weakly on the pump power, namely, the flatness is basically unchanged as the pump power consistently increases. One can also see, however, that ${\eta _2}$ has a significant influence on the SF for larger values of the frequency detuning, e.g., $\mathrm{\Delta }v$ = 30 GHz and $\mathrm{\Delta }v$ = 40 GHz. Specifically, the SF is increased at first and then trends to a stationary value. Such a trend becomes more obvious when a larger value of the frequency detuning is selected. One can conclude from these results that the increase of pump power has little effect on the improvement of the flatness for the case of smaller frequency detuning where the flatness has a relatively large value [see Fig. 8(b)]. On the contrary, the flatness can be obviously boosted before reaching its saturation point with the increase of the pump power for the case of larger frequency detuning.

Fig. 10. Evolution of (a) the bandwidth (BW) and (b) the spectral flatness of the heterodyne signal as a function of ${\eta _2}$ for four values of $\mathrm{\Delta }v$, with ${\eta _1} = 3$ and $P ={-} 0.2$.

Download Full Size | PDF

For comparison purposes, the power spectra, ACF, and CD curves for two initial chaos signals generated by the free-running spin-VCSEL1 and spin-VCSEL2 and the resultant signal by heterodyning these two initial chaos signals are shown in Fig. 11. Here, we set the frequency detuning, pump power of spin-VCSEL1, and the pump power of spin-VCSEL2 as 20 GHz, 3, and 3.5, respectively. It can be clearly seen in Figs. 11(a1) and 11(b1) that most of energies of the power spectra of the initial chaos of spin-VCSEL1 and spin-VCSEL2 concentrate on the frequencies around the relaxation oscillation and degrade quickly as the frequency increases, resulting limited chaos bandwidths. However, for the heterodyne signal shown in Fig. 11(c1), the power spectrum is significantly broadened and much flatter compared with these of the two initial chaos outputs. Meanwhile, we can observe that the laser relaxation oscillation is obviously eliminated and thus the corresponding bandwidth can be enhanced. In addition, Figs. 11(a2-c2) indicate that no peaks associated with the delay time are identified in the ACF traces owing to no external delay loop introduced in our proposed optical heterodyne scheme. To evaluate the complexity/dimensionality of the initial chaos and optical heterodyne signals, we also conduct the CD analyses in Figs. 11(a3-c3). The estimated CD values of the two initial chaos outputs are about 5.18 and 5.64, while the CD value of the heterodyne signal is as high as 7.56, which indicates that the complexity of the spin-VCSEL output can be improved through optical heterodyning. Synthesizing the results of Fig. 11, we find that with proper selection of the frequency detuning, the optical heterodyne scheme can achieve the indistinguishable TDS in ACF trace, as well as simultaneous bandwidth and complexity enhancement under a moderate pump power.

Fig. 11. Power spectra (first column), ACF traces (second column), and CD curves (third column) of (a) spin-VCSEL1, (b) spin-VCSEL2, and (c) heterodyne signal, with $\mathrm{\Delta }v = 20\; \textrm{GHz}$, ${\eta _1} = 3$, ${\eta _2} = 3.5$, and $P ={-} 0.2$.

Download Full Size | PDF

To gain more details on the characteristics of the heterodyne chaotic signals, Fig. 12 displays the two-dimension mapping diagrams of the bandwidth, flatness, and PE in the plane of frequency detuning and pump power of spin-VCSEL2. For PE calculation, the embedding dimension is D = 5, the embedding delay is ${\tau _e}$ = 1, and the sampling period is ${\mathrm{\Omega }_s}$ = 10 ps [63]. One can see that increasing the pump power results in a larger bandwidth, which is similar to Fig. 6. However, it should be noted that the TDS value is always low in the parameter plane considered since there is no delay loop for the heterodyne scheme, which is different from Fig. 6. From these figures, it is clearly observed that under the condition of tunable pump powers, strong chaos can be achieved in a wider range of the frequency detuning, e.g., 20 GHz < $\mathrm{\Delta }\nu $ < 40 GHz. Based on the above analyses, we can find that both optical feedback and optical heterodyne schemes can enhance chaotic characteristics and achieve strong chaos in large parameter space compared with a solitary spin-VCSEL. Remarkably, through employing optical heterodyne technique, with properly selecting the frequency detuning between the two spin-VCSELs, strong chaos can be obtained in relatively lower pump powers compared to the optical feedback configuration, where the bandwidth depends on the pump power and the TDS needs to be suppressed. It is well known that the larger pump power may bring about a series of problems, such as energy consumption and heat accumulation. However, comparison between Figs. 6(d) and 12(d) shows that the optical feedback scheme is beneficial to boosting the complexity/CD of chaos signals, which is favorable for some chaos-based applications requiring complex and high-dimensional chaos [65,66].

Fig. 12. Mapping diagrams of the (a) bandwidth (BW), (b) SF, and (c) PE in the parameter space of $({\mathrm{\Delta }v,\; {\eta_2}} )$, where ${\eta _1} = 3$ and $P ={-} 0.2$.

Download Full Size | PDF

4. Conclusion

In conclusion, we have numerically demonstrated that the spin-VCSEL can provide plentiful chaotic dynamic behaviors by controlling the spin degree of freedom. Herein, we first analyzed the properties of chaos generated from a free-running spin-VCSEL, including the bandwidth and flatness. To further improve the chaotic properties, we proposed two schemes, i.e., a spin-VCSEL with optical feedback and optical heterodyning two independent chaos outputs from two solitary spin-VCSELs. By analyzing key characteristics of chaos signals, such as the bandwidth, SF, TDS, and dimension/complexity, we found that strong chaos regimes, i.e., chaos signals with large bandwidth, no TDS of greater than 0.2, the flatness of 0.75 and above, and the high dimension/complexity, were maintained in broad parameter space for both schemes. Remarkably, for the optical feedback scheme, the strong chaos signal relies on the feedback strength and/or pump power, while the strong chaos depends on the frequency detuning for the optical heterodyne scheme. In particular, compared with the optical feedback scheme, the strong chaos can be obtained at a smaller pump power for the optical heterodyne scheme, which is beneficial for experimental operations. Whereas the complexity/dimension of chaos generated from the optical feedback scheme outperforms that of the optical heterodyne scheme. Overall, both configurations have prosperous potential in many fields, such as fast random bit generation, high-speed secure chaos communication, and high-resolution chaotic lidar/radar.

Funding

National Natural Science Foundation of China (62004135, 62001317, 62171305, 62111530301); Natural Science Research Project of Jiangsu Higher Education Institutions of China (20KJA416001, 20KJB510011); Natural Science Foundation of Jiangsu Province (BK20200855).

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

2. J. X. Ke, L. L. Yi, G. Q. Xia, and W. S. Hu, “Chaotic optical communications over 100-km fiber transmission at 30-Gb/s bit rate,” Opt. Lett. 43(6), 1323–1326 (2018). [CrossRef]

3. L. Jiang, J. C. Feng, L. S. Yan, A. L. Yi, S. S. Li, H. Yang, Y. X. Dong, L. S. Wang, A. B. Wang, Y. C. Wang, W. Pan, and B. Luo, “Chaotic optical communications at 56 Gbit/s over 100-km fiber transmission based on a chaos generation model driven by long short-term memory networks,” Opt. Lett. 47(10), 2382–2385 (2022). [CrossRef]

4. N. Q. Li, W. Pan, L. S. Yan, B. Luo, and X. H. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014). [CrossRef]

5. 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]

6. X. Tang, Z. M. Wu, J. G. Wu, T. Deng, J. J. Chen, L. Fan, Z. Q. Zhong, and G. Q. Xia, “Tbits/s physical random bit generation based on mutually coupled semiconductor laser chaotic entropy source,” Opt. Express 23(26), 33130–33141 (2015). [CrossRef]

7. Y. Guo, Q. Cai, P. Li, R. N. Zhang, B. J. Xu, K. A. Shore, and Y. C. Wang, “Ultrafast and real-time physical random bit extraction with all-optical quantization,” Adv. Photonics 4(03), 035001 (2022). [CrossRef]

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

9. W. Z. Feng, N. Jiang, Y. Q. Zhang, J. Y. Jin, A. K. Zhao, S. Q. Liu, and K. Qiu, “Pulsed-chaos MIMO radar based on a single flat-spectrum and Delta-like autocorrelation optical chaos source,” Opt. Express 30(4), 4782–4792 (2022). [CrossRef]

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

11. D. Z. Zhong, W. A. Deng, K. K. Zhao, Y. L. Hu, P. Hou, and J. B. Zhang, “Detections of the position-vectors of the multi targets located in a circular space based on an asymmetric coupling semiconductor lasers network,” Opt. Express 30(21), 37603–37618 (2022). [CrossRef]

12. Y. H. Hong, X. F. Chen, P. S. Spencer, and K. A. Shore, “Enhanced flat broadband optical chaos using low-cost VCSEL and fiber ring resonator,” IEEE J. Quantum Electron. 51(3), 1–6 (2015). [CrossRef]

13. A. K. Zhao, N. Jiang, Y. Q. Zhang, J. F. Peng, S. Q. Liu, K. Qiu, M. L. Deng, and Q. W. Zhang, “Semiconductor laser-based multi-channel wideband chaos generation using optoelectronic hybrid feedback and parallel filtering,” J. Lightwave Technol. 40(3), 751–761 (2022). [CrossRef]

14. A. K. Zhao, N. Jiang, S. Q. Liu, C. P. Xue, J. M. Tang, and K. Qiu, “Wideband complex-enhanced chaos generation using a semiconductor laser subject to delay-interfered self-phase-modulated feedback,” Opt. Express 27(9), 12336–12348 (2019). [CrossRef]

15. 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]

16. N. Jiang, A. Zhao, S. Liu, C. Xue, B. Wang, and K. Qiu, “Generation of broadband chaos with perfect time delay signature suppression by using self-phase-modulated feedback and a microsphere resonator,” Opt. Lett. 43(21), 5359–5362 (2018). [CrossRef]

17. Q. Yang, L. J. Qiao, X. J. Wei, B. X. Zhang, M. M. Chai, J. Z. Zhang, and M. J. Zhang, “Flat broadband chaos generation using a semiconductor laser subject to asymmetric dual-path optical feedback,” J. Lightwave Technol. 39(19), 6246–6252 (2021). [CrossRef]

18. S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Q. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012). [CrossRef]

19. J. J. Chen, C. X. Li, X. L. Mu, L. F. Li, and Y. N. Duan, “Time-delay signature suppression of polarization-resolved wideband chaos in VCSELs with dual-path chaotic optical injections,” Appl. Opt. 59(24), 7217–7224 (2020). [CrossRef]

20. Y. H. 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]

21. J. Z. Zhang, M. W. Li, A. B. Wang, M. J. Zhang, Y. N. 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]

22. H. W. Luo, M. F. Cheng, C. M. Huang, B. L. Ye, W. D. Shao, L. Deng, Q. Yang, M. M. Zhang, and D. M. Liu, “Experimental demonstration of a broadband optoelectronic chaos system based on highly nonlinear configuration of IQ modulator,” Opt. Lett. 46(18), 4654–4657 (2021). [CrossRef]

23. C. H. Cheng, Y. C. Chen, and F. Y. Lin, “Chaos time delay signature suppression and bandwidth enhancement by electrical heterodyning,” Opt. Express 23(3), 2308–2319 (2015). [CrossRef]

24. A. B. Wang, B. J. Wang, L. Li, Y. C. Wang, and K. Alan Shore, “Optical heterodyne generation of high-dimensional and broadband white chaos,” IEEE J. Sel. Top. Quantum Electron. 21(6), 531–540 (2015). [CrossRef]

25. S. Y. Xiang, A. J. Wen, W. Pan, L. Lin, H. X. Zhang, H. Zhang, X. X. Guo, and J. F. Li, “Suppression of chaos time delay signature in a ring network consisting of three semiconductor lasers coupled with heterogeneous delays,” J. Lightwave Technol. 34(18), 4221–4227 (2016). [CrossRef]

26. H. J. Liu, N. Q. Li, and Q. C. Zhao, “Photonic generation of polarization-resolved wideband chaos with time-delay concealment in three-cascaded vertical-cavity surface-emitting lasers,” Appl. Opt. 54(14), 4380–4387 (2015). [CrossRef]

27. Y. H. Hong, A. Quirce, B. J. Wang, S. K. Ji, K. Panajotov, and P. S. Spencer, “Concealment of chaos time-delay signature in three-cascaded vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 52(8), 1 (2016). [CrossRef]

28. M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7(1), 60–65 (2013). [CrossRef]

29. M. Virte, M. Sciamanna, and K. Panajotov, “Synchronization of polarization chaos from a free-running VCSEL,” Opt. Lett. 41(19), 4492–4495 (2016). [CrossRef]

30. P. H. Mu, W. Pan, and N. Q. Li, “Analysis and characterization of chaos generated by free-running and optically injected VCSELs,” Opt. Express 26(12), 15642–15655 (2018). [CrossRef]

31. M. Lindemann, G. Xu, T. Pusch, R. Michalzik, M. R. Hofmann, I. Zutic, and N. C. Gerhardt, “Ultrafast spin-lasers,” Nature 568(7751), 212–215 (2019). [CrossRef]

32. K. Harkhoe, G. Verschaffelt, and G. Van der Sande, “Neuro-inspired computing with spin-VCSELs,” Appl. Sci. 11(9), 4232 (2021). [CrossRef]

33. P. Georgiou, C. Tselios, G. Mourkioti, C. Skokos, and D. Alexandropoulos, “Effect of excited state lasing on the chaotic dynamics of spin QD-VCSELs,” Nonlinear Dyn. 106(4), 3637–3646 (2021). [CrossRef]

34. N. Q. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-VCSELs,” Opt. Lett. 42(17), 3494–3497 (2017). [CrossRef]

35. Y. Huang, P. Zhou, and N. Q. Li, “High-speed secure key distribution based on chaos synchronization in optically pumped QD spin-polarized VCSELs,” Opt. Express 29(13), 19675–19689 (2021). [CrossRef]

36. D. Z. Zhong, N. Zeng, H. Yang, Z. Xu, Y. L. Hu, and K. K. Zhao, “Precise ranging for the multi-region by using multi-beam chaotic polarization components in the multiple parallel optically pumped spin-VCSELs with optical injection,” Opt. Express 29(5), 7809–7824 (2021). [CrossRef]

37. L. J. Qiao, T. S. Lv, Y. Xu, M. J. Zhang, J. Z. Zhang, T. Wang, R. K. Zhou, Q. Wang, and H. C. Xu, “Generation of flat wideband chaos based on mutual injection of semiconductor lasers,” Opt. Lett. 44(22), 5394–5397 (2019). [CrossRef]

38. C. H. Tseng and S. K. Hwang, “Broadband chaotic microwave generation through destabilization of period-one nonlinear dynamics in semiconductor lasers for radar applications,” Opt. Lett. 45(13), 3777–3780 (2020). [CrossRef]

39. Y. Zeng, P. Zhou, Y. Huang, and N. Q. Li, “Optical chaos generated in semiconductor lasers with intensity-modulated optical injection:a numerical study,” Appl. Opt. 60(26), 7963–7972 (2021). [CrossRef]

40. C. H. Tseng, R. Funabashi, K. Kanno, A. Uchida, C. C. Wei, and S. K. Hwang, “High-entropy chaos generation using semiconductor lasers subject to intensity-modulated optical injection for certified physical random number generation,” Opt. Lett. 46(14), 3384–3387 (2021). [CrossRef]

41. G. Bouchez, C. H. Uy, B. Macias, D. Wolfersberger, and M. Sciamanna, “Wideband chaos from a laser diode with phase-conjugate feedback,” Opt. Lett. 44(4), 975–978 (2019). [CrossRef]

42. C. J. Chen, Z. W. Jia, Y. X. Lv, P. Li, B. J. Xu, and Y. C. Wang, “Broadband laser chaos generation using a quantum cascade laser with optical feedback,” Opt. Lett. 46(19), 5039–5042 (2021). [CrossRef]

43. J. Y. Ruan and S. C. Chan, “Chaotic dimension enhancement by optical injection into a semiconductor laser under feedback,” Opt. Lett. 47(4), 858–861 (2022). [CrossRef]

44. N. Q. Li, H. Susanto, B. R. Cemlyn, 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]

45. Y. Huang, P. Zhou, and N. Q. Li, “Broad tunable photonic microwave generation in an optically pumped spin-VCSEL with optical feedback stabilization,” Opt. Lett. 46(13), 3147–3150 (2021). [CrossRef]

46. A. Uchida, Optical Communication with Chaotic Lasers, Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

47. Y. Zeng, P. Zhou, Y. Huang, and N. Q. Li, “Extreme events in optically pumped spin-VCSELs,” Opt. Lett. 47(1), 142–145 (2022). [CrossRef]

48. M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87(1), 013834 (2013). [CrossRef]

49. M. S. Torre, H. Susanto, N. Q. Li, K. Schires, M. F. Salvide, I. D. Henning, M. J. Adams, and A. Hurtado, “High frequency continuous birefringence-induced oscillations in spin-polarized vertical-cavity surface-emitting lasers,” Opt. Lett. 42(8), 1628–1631 (2017). [CrossRef]

50. M. Virte, M. Sciamanna, E. Mercier, and K. Panajotov, “Bistability of time-periodic polarization dynamics in a free-running VCSEL,” Opt. Express 22(6), 6772–6777 (2014). [CrossRef]

51. G. A. Gottwald and I. Melbourne, “On the implementation of the 0–1 test for chaos,” SIAM J. Appl. Dyn. Syst. 8(1), 129–145 (2009). [CrossRef]

52. N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Mapping bifurcation structure and parameter dependence in quantum dot spin-VCSELs,” Opt. Express 26(11), 14636–14649 (2018). [CrossRef]

53. P. Jiang, P. Zhou, N. Q. Li, P. H. Mu, and X. F. Li, “Characterizing the chaotic dynamics of a semiconductor nanolaser subjected to FBG feedback,” Opt. Express 29(12), 17815–17830 (2021). [CrossRef]

54. N. Q. Li, H. Susanto, B. R. Cemlyn, I. D. Henning, and M. J. Adams, “Nonlinear dynamics of solitary and optically injected two-element laser arrays with four different waveguide structures: a numerical study,” Opt. Express 26(4), 4751–4765 (2018). [CrossRef]

55. 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]

56. D. Rontani, E. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016). [CrossRef]

57. G. Bouchez, T. Malica, D. Wolfersberger, and M. Sciamanna, “Optimized properties of chaos from a laser diode,” Phys. Rev. E 103(4), 042207 (2021). [CrossRef]

58. B. C. Liu, Y. Y. Xie, T. T. Song, Y. C. Ye, X. Jiang, J. X. Chai, Q. F. Tang, and M. Y. Feng, “Dynamic behaviors and bandwidth-enhanced time-delay signature suppression of mutually coupled spin-polarized vertical-cavity surface-emitting lasers,” Opt. Laser Technol. 149, 107811 (2022). [CrossRef]

59. P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50(5), 346–349 (1983). [CrossRef]

60. J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, “Automated correlation dimension analysis of optically injected solid state lasers,” Opt. Express 17(9), 7592–7608 (2009). [CrossRef]

61. Y. A. 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]

62. Y. T. Ma, S. Y. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated Sagnac loop,” Opt. Express 28(2), 1665–1678 (2020). [CrossRef]

63. M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011). [CrossRef]

64. S. Heiligenthal, T. Jungling, O. D’Huys, D. A. Arroyo-Almanza, M. C. Soriano, I. Fischer, I. Kanter, and W. Kinzel, “Strong and weak chaos in networks of semiconductor lasers with time-delayed couplings,” Phys. Rev. E 88(1), 012902 (2013). [CrossRef]

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

66. K. Kanno, A. Uchida, and M. Bunsen, “Complexity and bandwidth enhancement in unidirectionally coupled semiconductor lasers with time-delayed optical feedback,” Phys. Rev. E 93(3), 032206 (2016). [CrossRef]

Wideband and high-dimensional chaos generation using optically pumped spin-VCSELs

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

3.1 Chaotic characteristics of the free-running spin-VCSEL

3.2 Enhanced characteristics of the spin-VCSEL with optical feedback

3.3 Enhanced characteristics of spin-VCSELs through optical heterodyning

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Equations (7)

Optics Express