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Abstract
We numerically study the synchronized chaos (SC) and spatiotemporal chaos (STC) in a three-element laterally-coupled laser array in the case of four waveguiding structures. The coupled rate equations are used to analyze the dynamics of the laser array, where spatiotemporal dynamic maps are generated to identify regions of SC, STC, and non-chaos in the parameter space of interest. First, we show that the key parameters of the laser array, i.e., the laser separation ratio, pump rate, linewidth enhancement factor, and frequency detuning play important roles in the array dynamics and synchronization properties. Then we show that the laser array composed of the purely real index guiding exhibits more obvious boundaries between SC and STC in wider parameter space with respect to these composed of either the positive index guiding with gain-indexing, the pure gain guiding, or the index antiguiding with gain-guiding. Finally, we show that the proposed laser array allows for two scenarios of parallel random bit generation (PRBG) by applying the same post-processing on chaos sources based on SC and STC dynamic states. Hence, our results provide a comprehensive study on the collective dynamics in the three-element laterally-coupled laser array and pave the way for PRBG based on laser arrays.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Chaotic semiconductor lasers (SLs) have attracted considerable interest due to their significant theoretic values and wide potential applications, such as chaos-based secure communication [1–3], chaotic radar/lidar [4,5], and random bit generation (RBG) [6,7]. As the representative of high-dimensional nonlinear systems, coupled SLs exhibit rich dynamical behaviors, which provide a good platform for different researches. In particular, synchronized chaos (SC) based on coupled SLs has been extensively investigated both theoretically and experimentally over the past decades, which has been employed for secure optical communications [2,8], secure key distribution [9], and analog of complex networks characteristics [10–12]. Thus, the understanding of the SC behaviors is essential for relevant applications and scientific researches.
There are a variety of spatial and temporal synchronization behaviors based on coupled SLs networks [13]. On the one hand, for time-delayed bidirectionally coupled lasers, the coupling delay time is non-negligible since the propagation time for the optical signal from one laser to the other is similar to or larger than the characteristic time scale of the lasers. Such coupling scenario supports time-delayed synchronization of identical intensities chaos between the two lasers [14]. However, isochronous synchronization can be achieved by adding the self-feedback [15], an externally driven laser [16], or a relay element to the laser [17,18]. Additionally, cluster synchronization of mutually-coupled SLs network with symmetric structure in complex topology has been numerically demonstrated in [19,20], where the lasers in the same clusters are synchronized with each other, while the lasers in different clusters are not synchronized. Various dynamic behaviors can also be observed in other coupled SLs systems, such as bubbling effects in star-type lasers network [21], interesting chaos synchronization regime under heterogeneous coupling delays [22], and rich synchronization properties in a ring SL network [23,24].
On the other hand, for phase-locked laser arrays, the coupling delay time is negligible and all the lasers have a common frequency and a constant phase difference relative to each other. These laser arrays have been employed as sources that can produce high output power in a spatially coherent beam. Phase locking can also be achieved with different coupling techniques, such as diffractive coupling with Talbot cavity [25,26], V-shape cavity [27], and self-Fourier cavity [28,29]. The motivation of these studies was to achieve diffraction-limited emission as close as possible to the array. In large arrays of diode lasers coupled via a decayed non-local coupling scheme, in-phase and anti-phase chaotic and periodic phase synchronization have been reported [30]. In addition to phase synchronization, various spatial and temporal synchronization can exist simultaneously in the laterally-coupled laser arrays, which are evanescently coupled between adjacent elements of the array [31–34]. A complex collective behavior, i.e., the so-called chimera state, has been found in large diode arrays, where synchronized clusters coexist with unsynchronized ones [35]. Theoretical and experimental investigations of SC have been reported in an array of three laterally-coupled solid-state lasers [36,37], where identical and synchronized chaotic signals can be realized within a range of coupling strength, while spatiotemporal chaos (STC) takes over beyond this range [36]. However, the authors only provided preliminary results for the existence of SC and STC determined by the coupling strength in the array of coupled lasers. In this paper, we extend prior analysis with an extensive parametric study of the synchronization properties, i.e., the internal parameters of the laser, the structure of the waveguide and the operating parameter, and apply the generated chaotic signals to the parallel random bit generation (PRBG) [38,39].
In this article, we study the nonlinear dynamics of the three-element laser array coupled with the overlapping evanescent fields and restrict our main attention to the case of the purely real index guiding [40]. The influences of laser internal and external parameters on the dynamics of the laser array are analyzed. From our extensive simulations, the parameter regions of different spatiotemporal dynamics are obtained, which provides a guidance for the application of dynamic properties based on the proposed scheme. For completeness, we also consider three other representative laser waveguide structures [40], i.e., the positive index guiding with gain-indexing, the pure gain guiding and the index antiguiding with gain-guiding, and compare their results with those obtained from the purely real index guiding. Although SC dynamics would occur in the laser array regardless of the waveguide structure, the synchronization parameter region of the purely real index guiding is more consistent and beneficial to chaos-based applications. As an example, the results of SC and STC are applied to PRBG. After the same post-processing, the system could generate two highly correlated bit streams or three almost independent bit streams respectively.
2. Theoretical model
The schematic diagram of the three-element laser array is shown in Fig. 1, consisting of three identical laser waveguides, i.e., Guide A, B, and C. The width of each waveguide is ${2a}$, and the interval between the two waveguides is ${2d}$. The two adjacent lasers are mutually coupled by their overlapping evanescent field, as discussed in our previous work [40].
Fig. 1. Schematic of coupled slab waveguides, where ${{n}_{{1,2}}}$ represent the refractive indices in the waveguide cores and elsewhere, ${g}$ is the gain per unit length, and ${\alpha }$ is the background attenuation coefficient per unit length.
Based on the rate equations derived in two laterally-coupled lasers [40], the theoretical model of an array with i elements laterally-coupled lasers can be developed. Hence, the dynamic behaviors of the $i$th laser of the laser array are described by the equations
The complex coupling coefficient, ${\eta =\ |\eta |\textrm{exp}(i\theta )}$, describes the optical coupling between adjacent laser cavities, which is defined as
where ${{C}_{\eta }}$ and ${{C}_\theta }$ are calculated from numerical integration; ${{W}_{r}}$ and ${{W}_{i}}$ are the real and imaginary parts of the transverse propagation constant; ${d}/{a}$ is the laser separation ratio. It can be seen from Eq. (5) that the amplitude of the coupling coefficient decreases exponentially with the increase of ${d}/{a}$. We assume that the three laser cavities are approximately identical and the optical coupling is to be symmetrical.In the simulations, the parameter values are set as follows [40,41]:${\;\ \Gamma =\ 0}{.844}$, ${{g}_{{th}}}{ = 87}{.7\; \rm{c}}{{\rm m}^{{ - 1}}}$, ${{W}_{r}}{ = 1}{.26}$, ${{W}_{i}}{ = 0}$, ${{C}_{\eta }}{ = 83}{.6\; \rm{n}}{{\rm s}^{{ - 1}}}$, ${{C}_\theta }{ = 0}$, ${{a}_{{diff}}}{\ =\ 1\ \times 1}{{0}^{{ - 15}}}{\; \rm{c}}{{\rm m}^{2}}$, ${{\gamma }_{N}}{ = 1}{.0\; \rm{n}}{{\rm s}^{{ - 1}}}$, ${{\tau }_{p}}{ = 1}{.53\; {\textrm{ps}}}$, ${{\tau }_{N}}{ = 1\; {\textrm{ns}}}$, ${{N}_{0}}{\ =\ 1\ \times 1}{{0}^{{18}}}{\; \rm{c}}{{\rm m}^{{ - 3}}}$, ${n = }{{n}_{g}}{ = 3}{.4}$, ${\omega =}{\Omega _B}{ = 1}{.449\ \times }{10^{15}}{\; {\textrm{rad}}/\textrm{s}}$, and ${a\ =\ 4\ \mathrm{\mu} \mathrm{m}}$. These values are typical for laterally-coupled semiconductor lasers and they are fixed throughout the paper. We restrict attention to the case of equal pumping in each laser, so that ${P} \equiv {{P}_{A}}{ = }{{P}_{B}}{ = }{{P}_{C}}$. The pump rate is expressed in terms of the ratio of its threshold value ${P}/{{P}_{{th}}}$. The frequency detuning, defined as $\Delta {f = (}{{\Omega }_{A}}{ - }{{\Omega }_{B}}{)}/{2\pi =\ (}{{\Omega }_{C}}{ - }{{\Omega }_{B}}{)}/{2\pi }$, is considered in the current work due to the fact that this case might occur in practice caused unintentionally by minor manufacturing variations or design. The fourth-order Runge-Kutta method is utilized to solve Eqs. (1–3) with a fixed step of 1 ps. Each realization starts at a different initial condition, where the amplitudes and inverse populations are set to a uniform distribution on the interval [0, 0.1], and the phases are taken from a uniform distribution on the interval [-π, π].
We first use the mutual information ${{M}_{{xy}}}$ to quantify the correlation between two variables ${x}$ and ${y}$ [20]. The entropy of ${x}$ is expressed as
The similar definition exists for variables ${y}$. In the same way, the joint entropy ${{H}_{{xy}}}$ can be defined in terms of the joint probability density ${P(}{{x}_{i}}{,}{{y}_{i}}{)}$ and written as
When the two variables ${x}$ and ${y}$ are highly correlated, one obtains ${{M}_{{xy}}}$ of large values. The value of ${{M}_{{xy}}}$ decreases as the two variables are relatively independent. Due to its computation efficiency, the cross-correlation function is used as a complementary measure to characterize the correlation between two variables [42].3. Results and discussion
We first discuss the perfectly symmetric situation where the internal parameters and operating conditions of the lasers are exactly the same. Later, a small frequency detuning between the lasers will be introduced. The mutual information and cross-correlation function are used to analyze the correlation between laser pairs. In order to distinguish the laser dynamics, the bifurcation diagram and the 0–1 test for chaos are employed. Note that the 0–1 test for chaos is a statistical method that can be used to identify the chaotic state. When the resulting value of the 0–1 test is close to 1, the output of the system can be regarded as chaos. More details can be found in Ref. [43]. In our simulations, the values of the mutual information, cross-correlation function, and 0–1 test in each state are calculated 5 times and then averaged. The interval of [160, 200] ns is used to calculate the correlation of the intensity time series and determine the chaotic state.
3.1 Spatiotemporal dynamics generation
The array exhibits rich dynamical behaviors, depending on different parameters. When the laser separation ratio ${d}/{a}$ is set at 0.5, the temporal evolutions of the three laser intensities are all chaotic and almost independent, as shown in Fig. 2(a1). Meanwhile, the output intensities of the three lasers are mainly concentrated at the lower values. In Figs. 2(a2-a3), the phase space trajectories are complex chaotic attractors in the ${{I}_{A}}{ - }{{I}_{C}}$ and ${{I}_{A}}{ - }{{I}_{B}}$ planes. This is the regime of STC, which is disordered in space and time. When ${d}/{a}$ is increased to 1.04, the coupling strength between lasers decreases. One can easily notice that the temporal evolutions of the output intensities in lasers A and C are almost identical, but quite different from that of laser B, as shown in Fig. 2(b1). This is the regime of SC. The phenomenon can also be seen in Figs. 2(b2) and 2(b3). In the ${{I}_{A}}{ - }{{I}_{C}}$ plane, the trajectory of SC is spatial order [Fig. 2(b2)]. However, the motion is a strange attractor in the ${{I}_{A}}{ - }{{I}_{B}}$ plane and the projection of ${{I}_{A}}$ and ${{I}_{B}}$ is simpler than the case of STC [Fig. 2(b3)]. These results coincide with previous studies that were limited to one case of mutual information versus the coupling strength [36]. In the SC case, lasers A and C are stably locked via the middle laser. When the laser spacing is further reduced, the nonlinearities in the array and the intensity of laser oscillations increase, leading to desynchronization of the outer lasers.
Fig. 2. (a) STC for ${d}/{a = 0}{.5}$. (b) SC for ${d}/{a = 1}{.04}$. (a1-b1) Time series for the intensities ${{I}_{A}}$, ${{I}_{B}}$ and ${{I}_{C}}$; (a2-b2) projection onto the ${{I}_{A}}{ - }{{I}_{C}}$ plane; (a3-b3) projection onto the ${{I}_{A}}{ - }{{I}_{B}}$ plane. ${{I}_{A}}$, ${{I}_{B}}$ and ${{I}_{C}}$ are normalized by ${1}{{0}^{{21}}}$. Other parameters are $\Delta {f = 0\;\textrm {GHz}}$, ${P}/{{P}_{{th}}}{ = 1}{.5}$, and ${{\alpha }_{H}}{ = 4}$.
In order to study the synchronization characteristics between laser pairs with the change of laser separation ratio, the mutual information between the lasers as a function of ${d}/{a}$ is plotted in Fig. 3(a). Since the correlations between lasers A and B, lasers B and C are quite similar, only the results of ${{M}_{{AB}}}$ and ${{M}_{{AC}}}$ are presented. This rule is also applied to subsequent correlation studies. Figures 3(b) and 3(c) show the evolutions of the extrema of ${{I}_{A}}$ and ${{I}_{B}}$, ${{I}_{A}}$ and ${{I}_{C}}$ as a function of ${d}/{a}$, respectively. For the case ${d}/{a\ < 1}{.02}$, both ${{M}_{{AB}}}$ and ${{M}_{{AC}}}$ are low and the laser outputs are chaotic. In this region, the laser outputs are not correlated in both space and time, which are STC dynamics. There is a region between ${d}/{a = 1}$ and 1.14 where ${{M}_{{AC}}}$ is high but ${{M}_{{AB}}}$ remains low [Fig. 3(a)], which indicates the correlation between lasers A and C is high, while that between lasers A and B is low. It can be seen in Figs. 3(b) and 3(c), the outputs of the three lasers in the region are chaotic. This is the domain of SC where the system is temporally chaotic but retains spatial symmetry. As ${d}/{a}$ is increased, both ${{M}_{{AB}}}$ and ${{M}_{{AC}}}$ are generally increased due to the fact that the dynamics of the lasers become simpler, which can be identified from the bifurcation diagrams shown in Figs. 3(b) and 3(c). Especially for the periodic region, one can obtain high values for ${{M}_{{AB}}}$ and ${{M}_{{AC}}}$. ${{M}_{{AC}}}$, however, is larger than ${{M}_{{AB}}}$ due to the symmetric position for lasers A and C.
Fig. 3. (a) Mutual information versus ${d}/{a}$. Red solid curve: ${{M}_{{AB}}}$ (mutual information of ${{I}_{A}}$ and ${{I}_{B}}$). Blue solid curve: ${{M}_{{AC}}}$ (mutual information of ${{I}_{A}}$ and ${{I}_{C}}$). (b) Bifurcation diagram for extrema of ${{I}_{A}}$ (dotted red) and ${{I}_{B}}$ (dotted blue). (c) Bifurcation diagram for extrema of ${{I}_{A}}$ (dotted red) and ${{I}_{B}}$ (dotted blue). Other parameters are $\Delta {f = 0\;\textrm {GHz}}$, ${P}/{{P}_{{th}}}{ = 1}{.5}$, and ${{\alpha }_{H}}{ = 4}$.
To analyze the dynamics conveniently and intuitively, the cross-correlation function between different laser pairs and the 0–1 test are calculated. Because the dynamics of the three lasers are similar, only the results of lasers A and B are plotted. Figure 4 shows the absolute values of the cross-correlation coefficient of two laser pairs (${{C}_{{AB}}}$ and ${{C}_{{AC}}}$), the 0–1 test results of laser A (${{T}_{A}}$) and laser B (${{T}_{B}}$) as a function of ${d}/{a}$ with ${{\alpha }_{H}}$ of 2, 4, 6, and 8. As shown in Fig. 4(a) for ${{\alpha }_{H}}{ = 2}$, when ${d}/{a}$ is between 0 and 1, both ${{C}_{{AB}}}$ and ${{C}_{{AC}}}$ are much lower than 1, indicating that the lasers are asynchronous and asymmetry in space. Meanwhile, ${{T}_{A}}$ and ${{T}_{B}}$ are near to 1, signifying that the outputs of the laser array are chaotic and disordered in time. In this case, STC exists. For ${d}/{a\ > 1}$, although the correlations between lasers are high, the array output is non-chaotic since the 0–1 test values are close to 0. As ${{\alpha }_{H}}$ increases to ${{\alpha }_{H}}{ = 4}$, one can see that, in the interval between ${d}/{a = 1}$ and 1.14 in Fig. 4(b), lasers A and C are highly correlated with ${{C}_{{AC}}}$ close to 1, whereas they exhibit low correlation with laser B (${{C}_{{AB}}}\; $ is well below 1). Besides, by virtue of the 0–1 test results, the three lasers are confirmed to operate in chaotic regimes. This corresponds to the region of SC in Fig. 3(a). Furthermore, when ${{\alpha }_{H}}$ increases gradually, the region of STC gradually moves to a larger ${d}/{a}$, and the area of SC gradually decreases, as shown in Figs. 4(c) and 4(d). Note that, in the SC region, for a large value of ${{\alpha }_{H}}$, the three lasers become correlated, e.g., ${{C}_{{AC}}}{\; } \approx {\; 1}$ and ${{C}_{{AB}}}$ is slightly below 1 in Fig. 4(d).
Fig. 4. Evolutions of ${{C}_{{AB}}}$, ${{C}_{{AC}}}$, ${{T}_{A}}$, and ${{T}_{B}}$ versus ${d}/{a}$ with (a) ${{\alpha }_{H}}{ = 2}$, (b) ${{\alpha }_{H}}{ = 4}$, (c) ${{\alpha }_{H}}{ = 6}$, and (d) ${{\alpha }_{H}}{ = 8}$. ${{C}_{{AB}}}$: absolute value of the cross-correlation coefficient between ${{I}_{A}}$ and ${{I}_{B}}$; ${{C}_{{AC}}}$: absolute value of the cross-correlation coefficient between ${{I}_{A}}$ and ${{I}_{C}}$; ${{T}_{A}}$: the 0–1 test of ${{I}_{A}}$; ${{T}_{B}}$: the 0–1 test of ${{I}_{B}}$. Other parameters are $\Delta {f = 0\;\textrm {GHz}}$, ${P}/{{P}_{{th}}}{ = 1}{.5}$, and ${{\alpha }_{H}}{ = 4}$.
In addition to the effect of the laser separation ratio, we also analyze the influence of pump rate ${P}/{{P}_{{th}}}$ on the array dynamics. Figure 5 displays the evolutions of the correlation coefficients (${{C}_{{AB}}}$ and ${{C}_{{AC}}}$) and the 0–1 test of ${{I}_{A}}$ (${{T}_{A}}$) in the $({{P}/{{P}_{{th}}}{,d}/{a}} )$ plane for ${{\alpha }_{H}}{ = 4}$. Several important features can be identified from this figure. Firstly, there is a clear boundary for the correlation between lasers A and B, and between lasers A and C. One can obtain STC in the blue region in Figs. 5(a) and 5(b), where ${{C}_{{AB}}}$ and ${{C}_{{AC}}}$ are very low and the three lasers yield chaotic signals according to Fig. 5(c). Such a boundary is slightly dependent on the pump rate. Secondly, there is a region for synchronizing all the three lasers operating in non-chaotic regimes, i.e., the dark red region (the upper right corner of ${P}/{{P}_{{th}}}{ - d}/{a}$ plane) in Figs. 5(a) and 5(b). Thirdly, by comparing Figs. 5(a-c), one can find a narrow region (${{C}_{{AC}}}{\; } \approx {\; 1}\; $ and ${{T}_{A}}{\; } \approx {\; 1}$) in between the two regions mentioned above, which corresponds to the SC case. Last but not the least, Fig. 5(c) shows that there is a zone on the boundary between chaos and non-chaos (NC) where the value of the 0–1 test will fluctuate. This is due to the changes of steady-state, periodic, quasi-periodic, and chaotic states in this region. In order to better observe the parameter intervals of the highly synchronized and chaotic dynamics, the regions with cross-correlation coefficients of 0.95 and 0–1 test of 0.95 are marked with dashed lines. For better illustration, the region where ${{C}_{{AB}}}$ and ${{C}_{{AC}}}$ are less than 0.95 and ${{T}_{A}}$ is greater than 0.95 is defined as STC, the region where ${{C}_{{AC}}}$ is greater than 0.95 and ${{T}_{A}}$ is larger than 0.95 is defined as SC, and other regions are defined as NC. It should be noted that one can reasonably adjust this threshold which does not change the main conclusion below.
Fig. 5. Maps of the evolutions of the cross-correlation coefficient (a) ${{C}_{{AB}}}$, (b) ${{C}_{{AC}}}$ and the 0–1 test (c) ${{T}_{A}}$ in the $({{P}/{{P}_{{th}}}{,d}/{a}} )$ parameter space. The dashed contour curves in Figs. 4(a-b) correspond to the cross-correlation coefficient of 0.95, while the dashed contour curves in Fig. 4(c) correspond to the ${{T}_{A}}$ of 0.95. Other parameters are $\Delta {f = 0\;\textrm {GHz}}$ and ${{\alpha }_{H}}{ = 4}$.
Figure 6 presents the above-defined dynamical regions of the laser array in the (${P}/{{P}_{{th}}}$, ${d}/{a}$) plane with ${{\alpha }_{H}}$ of 2, 4, 6, and 8. We consider a wide range of pump rates from ${1}{.1}{{P}_{{th}}}$ to ${2}{.5}{{P}_{{th}}}$ for the lasers. In general, the size of the STC region expands gradually with the increase of ${{\alpha }_{H}}$, while the size of the SC region increases first and then decreases. For smaller ${{\alpha }_{H}}$, SC only exists when ${P}/{{P}_{{th}}}$ is relatively large. Based on the simulation results, to achieve the parallel chaotic signal output in the laser array, the manufacturing and operating parameters of the laser need to be adjusted. When ${{\alpha }_{H}}$ is equal to 4, 6, and 8, it can be seen from Figs. 6(b-d) that there are scattered SC regions near the boundary of SC and NC regions. This corresponds to the phenomenon of the boundary between 0 and 1 in the test, as shown in Fig. 5(c). We present a region surrounded by the white dotted lines in the figure where the spatiotemporal dynamic states, i.e., SC and STC, can switch flexibly with the change of the pump rate as the value of ${d}/{a}$ is fixed. This corresponds to the scenario of a designed three laterally-coupled laser array. After conducting the same post-processing unit for the three chaotic signals generated by the laser array, two types of PRBG can be generated in this proposed laser array. One is the generation of two nearly identical sequences of RBG, corresponding to SC dynamics, and the other is the generation of three sequences of RBG with low correlation, corresponding to STC dynamics. The details will be further explained below.
Fig. 6. The spatiotemporal dynamics of the laser array in the (${P}/{{P}_{{th}}}$, ${d}/{a}$) parameter space for (a) ${{\alpha }_{H}}{ = 2}$, (b) ${{\alpha }_{H}}{ = 4}$, (c) ${{\alpha }_{H}}{ = 6}$, and (d) ${{\alpha }_{H}}{ = 8}$ with $\Delta {f = 0\;\textrm {GHz}}$. SC: synchronized chaos; STC: spatiotemporal chaos; NC: non-chaos.
Furthermore, we study the influence of frequency detuning $\Delta {f}$ on the array dynamics. The detuning is set that the outer lasers share a common frequency and there is a detuning with the inner laser. Here we take the case of ${P}/{{P}_{{th}}}{ = 2}{.5}$ as an example, since SC dynamics is more easily generated for this high pump rate. Figure 7 presents the dynamics of the laser array in the $({\Delta {f,\; d}/{a}} )$ plane with ${{\alpha }_{H}}$ of 2, 4, 6, and 8. It is known that frequency detuning plays an important role in the synchronization properties of coupled SLs. For the case of two delay- coupled lasers, detuning leads to a leader-laggard phenomenon, with the high-frequency laser dominating the dynamics. For zero detuning, the two lasers spontaneously switch leader and laggard roles [14]. In addition, for the case of three delay-coupled lasers, the two outer lasers exhibit isochronous synchronization and advance the dynamics of the central laser. The dominant sign changes only for very high positive detuning of the central laser. When the frequencies of the outer lasers are the same, the zero-lag synchronization between outer lasers is always maintained in the considered range of the detuning, and the correlation between the outer lasers and inner laser will decrease with the increase of detuning [18]. In the proposed scheme here, we consider instantaneous coupling and it can be seen from Fig. 7 that SC and STC still exist although there is a large frequency mismatch between the adjacent lasers. As shown in Fig. 7(a), the proportion of the SC region under positive detuning is greater than that under negative detuning, which indicates that the frequency of the outer lasers higher than that of the inner laser is beneficial to the SC. Similar behaviors exist when ${{\alpha }_{H}}$ is equal to 4, 6, and 8, as shown in Figs. 7(b)-(d). We find that under different combinations of ${P}/{{P}_{{th}}}$ and ${{\alpha }_{H}}$, there is such a proportion rule: when ${P}/{{P}_{{th}}}$ is fixed, the region of SC and STC will gradually expand in size with the increase of ${{\alpha }_{H}}$.
Fig. 7. The spatiotemporal dynamics of the laser array in the ($\Delta {f}$, ${d}/{a}$) parameter space for (a) ${{\alpha }_{H}}{ = 2}$, (b) ${{\alpha }_{H}}{ = 4}$, (c) ${{\alpha }_{H}}{ = 6}$, and (d) ${{\alpha }_{H}}{ = 8}$ with ${P}/{{P}_{{th}}}{ = 2}{.5}$. SC: synchronization chaos; STC: spatiotemporal chaos; NC: non-chaos.
The studies so far have been focused on the synchronization properties of the laser array composed of the purely real index guiding. It is known that waveguide structures affect the cavity loss and the gain of lasers, and thus the output of lasers in the arrays [26]. Therefore, we consider the influence of waveguide structures based on the cases analyzed in [40]. Here, the spatiotemporal dynamics results of the three-element laterally-coupled laser array composed of other three waveguide structures, i.e., the positive index guiding with gain-indexing [ Figs. 8(a1-d1)], the pure gain guiding [Figs. 8(a2-d2)], and the index antiguiding with gain-guiding [Figs. 8(a3-d3)], are presented in the $({\Delta {f,d}/{a}} )$ plane with ${{\alpha }_{H}}$ of 2, 4, 6, and 8, where ${P}/{{P}_{{th}}}{ = 2}{.5}$. The three waveguides show some properties similar to those observed in the purely real index waveguide in Fig. 7. More specifically, there exist the three regions of SC, STC, and NC independent of the waveguiding structure. Moreover, the linewidth enhancement factor indeed plays a role in the evolution of these three regions. However, these regions are dispersive and interrupting with each other. It is also worth noting that the SC regions can be hardly found in the case of the positive index guiding with gain-indexing [Figs. 8(a1-d1)]. The comparison among the results for the four waveguide structures shows that the three-element laterally-coupled laser array composed of purely indexing guiding performs the best SC properties, where the SC and STC regions in the two dimensional parametric plane are more intensive. This may provide a guidance for the future fabrication of waveguide laser arrays based on the presented research.
Fig. 8. The spatiotemporal dynamics of the laser array composed of (a1-d1) positive index with gain-guiding, (a2-d2) pure gain guiding, and (a3-d3) index antiguiding with gain-guiding in the ($\Delta {f}$, ${d}/{a}$) parameter space for (a1-a3) ${{\alpha }_{H}}{ = 2}$, (b1-b3) ${{\alpha }_{H}}{ = 4}$, (c1-c3) ${{\alpha }_{H}}{ = 6}$, and (d1-d3) ${{\alpha }_{H}}{ = 8}$ with ${P}/{{P}_{{th}}}{ = 2}{.5}$. SC: synchronized chaos; STC: spatiotemporal chaos; NC: non-chaos.
3.2 Parallel random bits generation
In this section, we apply the results of SC and STC to PRBG. For the designed laser arrays, the distance between the lasers is fixed. The spatiotemporal behavior of chaotic state can be changed by adjusting the operating current/frequency. Since we focus on the functionality of the proposed laser array for two different types of RBG, the post-processing method is not so important for illustration here. To simplify the processing, we first compute the high-order finite differences (HFD) of the chaotic signals proposed in our previous work [44] to post-process the chaotic signals and extract the same number of least significant bits (LSBs). Therefore, we can obtain three binary bit streams ${{b}_{A}}{(t)}$, ${{b}_{B}}{(t)}$, and ${{b}_{C}}{(t)}$, each with the bit rate reaching hundreds of Gb/s to the order of Tb/s.
To clarify the correlation between the generated bit sequences, the bit error ratio (BER) is utilized. According to previous reports [45,46], if two bit sequences are completely independent (identical), their BER will be 0.5 (0). With loss of generality, we only consider the purely real index guiding and take the situation of ${d}/{a = 0}{.88}$ and ${{\alpha }_{H}}{ = 4}$ for example. Figure 9 shows the sequence correlation results after calculating 35th-order HFD from the chaotic sources and selecting 25 LSBs. The length of each sequence is 1Gbit, and all the bit sequences can pass the standard NIST tests if the lasers operate in the SC and STC regimes. Figure 9 shows the variation of BER ${{b}_{{AB}}}{(t)}$, ${{b}_{{AC}}}{(t)}$, and ${{b}_{{BC}}}{(t)}$ with the change of pump rate ${P}/{{P}_{{th}}}$ for $\Delta {f = 0\;\textrm {GHz}}$. As shown in Fig. 9, the values of ${{b}_{{AB}}}{(t)}$ and ${{b}_{{BC}}}{(t)}$ almost remain constant (BER ≈ 0.5) with the increase of ${P}/{{P}_{{th}}}$, while the value of ${{b}_{{AC}}}{(t)}$ decreases abruptly when the value of ${P}/{{P}_{{th}}}$ exceeds 1.7 and tends to saturate (BER ≈ 0) finally. This is because of the shift of laser dynamics from STC to SC [see Fig. 6(b)]. The results show that when the array output is STC, the correlation between two arbitrary bit streams is very low, while SC corresponds to two high correlation bit streams. In a certain range of laser separation ratio, two types of PRBG can be switched flexibly by changing the operating current.
Fig. 9. Bit error ratio of ${(}{{b}_{A}}{,\; }{{b}_{B}}{)}$, ${(}{{b}_{A}}{,\; }{{b}_{C}}{)}$ and ${(}{{b}_{B}}{,\; }{{b}_{C}}{)}$ versus ${P}/{{P}_{{th}}}$ with $\Delta {f = 0\;\textrm {GHz}}$. Other parameters are ${d}/{a = 0}{.88}$ and ${{\alpha }_{H}}{ = 4}$.
4. Conclusion
In summary, we have numerically investigated the SC and STC behaviors in a three-element laterally-coupled laser array, by mainly focusing on the waveguiding structure of the purely real index guiding. The coupled rate equations have been utilized to model the system. The cross-correlation function and the 0–1test have been used to visualize the spatiotemporal dynamic variation in the parameter space, where SC, STC, and NC regions of interest have been revealed in some detail. Parameters such as the laser separation ratio, the pump rate, the linewidth enhancement factor, and the frequency detuning have been shown to have a strong effect on the synchronization properties in the laser array. We have also studied the influence of waveguide structures, and the results highlight the importance of waveguide design on spatiotemporal dynamic states. Furthermore, an application of PRBG in such a coupled laser array system has been implemented. Through the same post-processing, i.e., by computing the HFD of the chaotic intensities time series and extracting LSBs, either two highly correlated bit streams or three almost independent bit streams based on the SC and STC dynamics can be generated. The results suggest a new insight into the research of PRBG in laser arrays.
Funding
National Natural Science Foundation of China (62004135, 62001317, 62171305); 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.
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