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Abstract
We propose and experimentally demonstrate the generation of dual-channels chaos with time delay signature (TDS) concealment by introducing a phase-modulated Sagnac loop in mutually coupled semiconductor lasers (MCSL). Furthermore, we demonstrate the utilization of the dual-channels chaos to solve multi-armed bandit (MAB) problem in reinforcement learning. The experimental results agree well with the numerical simulations. For the purpose of comparison, the MCSL with a conventional Sagnac loop is also considered. It is found that the TDS of dual-channels chaotic signals can be better concealed in our proposed system. Besides, the proposed system allows for a better decision making performance in MAB problem. Moreover, compared with the one-channel chaotic system, the proposed dual-channels chaotic system achieves ultrafast decision making in parallel, and thus, is highly valuable for further improving the security of communication systems and the performance of photonic intelligence.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since AlphaGo showed its amazing ability in Go, the field of artificial intelligence (AI) has been receiving more and more attention. With the development of photonic technology, the research on photonic intelligence has been carried out in depth. The photonic technologies are used to supplement the conventional computer algorithms [1–4], in order to overcome the limitations of conventional approaches such as von Neumann bottleneck [5]. Optical chaotic sources are promising candidates for diverse applications such as high speed random bit generators [6–8], and secure optical communication [9–11]. In recent years, more and more attractive chaotic sources have been reported, such as semiconductor ring lasers [12], vertical cavity surface emitting lasers [13–15], optoelectronic oscillator [16], optical fiber ring resonators [17], optomechanical oscillators [18], semiconductor lasers (SL) [19,20], etc. For these chaotic sources, the delayed optical feedback [21–23], optical injection [24–27] and other external disturbances [28–33] are commonly adopted due to their simple structures. However, by using autocorrelation function (ACF) and delayed mutual information to analyze chaotic signals, the time delay signature (TDS) of chaos can be identified successfully. Such TDS threatens the security of chaotic communication systems and reduces the randomness of chaotic signals. Therefore, the TDS concealment of chaotic sources is critical to the related applications.
At present, many methods have been proposed to suppress TDS in chaotic sources, such as SL with modulated opto-electronic feedback [22], SL with phase-modulated feedback [28–31], and SL subject to distributed feedback from a fiber Bragg grating [32,33]. Furthermore, the application of chaotic sources in reinforcement learning has been explored [34–39]. The multi-armed bandit (MAB) problem is one of the classical problems in reinforcement learning. There are a number of alternative slot machines, whose probability distribution is initially unknown. A player needs to decide which arm of K non-identical slot machines to select so as to maximize his reward. On the one hand, in order to identify the best slot machine, the environment must be sufficiently explored. On the other hand, experience made during learning must also be considered for action selection in order to minimize the costs of learning. Therefore, the trade-off between exploration and exploitation demands efficient exploration capabilities, maximizing the expected reward while minimizing the costs of exploration [34,35].
Remarkably, S. J. Kim et al. proposed a decision making system, which utilized optical energy transfer between quantum dots [36]. K. Morihiro et al. indicated that better exploration performance in reinforcement learning could be obtained by replacing the stochastic random generator with the deterministic chaotic generator based on the logistic map [38]. M. Naruse et al. combined the tug-of-war (TOW) algorithm with chaotic signals to solve the MAB problem. They suggested that the optimal decision making performance was achieved when the time lag that yielded the negative maximum of the autocorrelation was coincident with the sampling interval of the chaotic signals [34,35]. However, the dual-channels chaotic signals with TDS concealment to solve the MAB problem still remains largely unexplored, which may increase substantially the performance of decision making.
In this paper, we numerically and experimentally demonstrate the generation of dual-channels chaotic signals with TDS concealment in mutually coupled semiconductor lasers (MCSL) by introducing a phase-modulated Sagnac loop. Then, we present the utilization of the dual-channels chaotic signals to solve the MAB problem in parallel. For the purpose of comparison, the configuration of MCSL with a conventional Sagnac loop is also considered. This paper is structured as follows. In Section 2, the system model of MCSL with a Sagnac loop and the rate equation are introduced in detail. In Section 3, the numerical results of TDS concealment and decision making for the MAB problem are discussed. The effects of some controllable parameters, i.e., the coupling strength, frequency detuning, modulation index and coupling delays on the TDS properties are analyzed in detail. In addition, we extend the MCSL with a Sagnac loop to solve the MAB problem, and investigate the dependence of the TDS concealment and the decision making performance. Moreover, the decision making performances of dual-channels chaotic system and one-channel chaotic system are compared in detail. In Section 4, the experimental results of TDS concealment and decision making for the MAB problem are also presented, which agree well with the numerical simulations. Finally, concluding remarks are provided in Section 5.
2. System model
2.1 Experimental setup of a dual-channels chaotic system with a Saganc loop
The experimental setup of a dual-channels chaotic system with a phase-modulated Sagnac loop is shown in Fig. 1. Here, two distributed feedback (DFB) lasers without isolation are adopted as chaos sources. They are driven by laser diode controllers (ILX-Lightwave LDC-3724). The threshold of each DFB is about $11\textrm{mA}$. The wavelengths of two DFBs can be matched by controlling the laser diode controllers, and here, the wavelengths of free-running DFBs are set as $1552.998\textrm{nm}$ and $1552.998\textrm{nm}$, as shown in Figs. 1(a) and 1(b). In this setup, the output of DFB1 is divided into two parts through 10/90 fiber coupler FC2. The smaller part is sent to the measure module. The larger part is propagated through a fiber jumper ${t_1}$ to a 50/50 fiber coupler FC5. The output of DFB2 is connected in a similar fashion. After that, the outputs of the two lasers are injected into the Sagnac loop through a variable optical attenuator (VOA). The loop is formed by a 50/50 fiber coupler FC6 and an electro-optic phase modulator (PM, Covega Mach-10). The PM is driven by the PRBS, which is generated by the arbitrary waveform generator (AWG, Tektronix AWG7082C) and then amplified by the RF amplifier (Centellax OA3MVM3). Continuous wave light is split equally by the 50/50 fiber coupler FC6, with one half propagating in the clockwise (CW) direction and the other half propagating in the counterclockwise (CCW) direction. Since the velocity mismatch effect in the PM, the light modulated by the electrical signal that travels in the opposite direction has a small modulation index [40,41]. Therefore, only the incident light wave along the CW direction is effectively modulated, while the CCW light wave is not modulated. After combining the phase-modulated light and the unmodulated light at FC6, the light is sent back into DFB1 and DFB2 equally. The fiber jumpers with different lengths are utilized to adjust the coupling delay. The coupling strength is controlled with the VOA and monitored by an optical power meter (OPM). The optical spectra of the outputs of DFBs can be recorded by an optical spectrum analyzer (OSA, Ando AQ6317). The optical outputs of both DFBs are converted into electrical signals by a high-speed photodiode (PD, HP 11982A), and are recorded by a real-time oscilloscope (OSC, KEYSIGHT DSOV334A, 33GHz, 80GS/s). Throughout the work, a PMSL-MC represents the mutually coupled semiconductor lasers system with a phase-modulated Sagnac loop, while a CSL-MC stands for the mutually coupled semiconductor lasers system with a conventional Sagnac loop. Note, when the devices in the dashed box are removed, the configuration of CSL-MC is obtained. After acquiring the dual-channels chaotic signals, decision making for the MAB problem based on dual-channels chaotic signals is discussed later. The dual-channels chaotic signals are used to solve the MAB problem in parallel, and achieve ultrafast decision making.
Fig. 1. Experimental setup of a dual-channels chaotic system with a phase-modulated Sagnac loop. DFB1, DFB2: distributed feedback semiconductor lasers; FC: fiber coupler; VOA: variable optical attenuator; ${t_1},{t_2}$: fiber jumper with different length; PM: electro-optic phase modulator; PRBS generator: pseudo-random binary sequence generator; PC: polarization controller, LDC: laser diode controller, OI: optical isolator, OPM: optical power meter, OSA: optical spectrum analyzer; PD: photodiode; OSC: oscilloscope. Optical spectra are presented in the insets.
2.2 Rate equation model
The dynamics of the DFBs in the proposed scheme is described by the modified Lang-Kobayashi rate equations, by taking into account the MCSL with a phase-modulated Sagnac loop [28–31,40–42]. The rate equations of the slowly-varying complex electric field ${E_{1,2}}(t)$ and the corresponding carrier number ${N_{1,2}}(t)$ in the active region of two DFBs are written as:
3. Numerical results
In this section, we firstly present the numerical results of the TDS properties for the PMSL-MC system. For a direct comparison, the TDS properties for the CSL-MC system are also considered. The effects of coupling strength, frequency detuning, phase modulation index, as well as coupling delays on the TDS concealment are examined carefully. Furthermore, the dual-channels chaotic signals with TDS concealment generated in the proposed PMSL-MC system are used to solve the MAB problem in parallel.
The ACF is the most common method to identify the TDS. It is computed as following [29],
3.1 TDS concealment
The numerical results of time series, the corresponding ACFs, as well as the power spectra of the two DFBs in the CSL-MC system are shown in row 1 and row 2 of Fig. 2. Figures 2(a1) and 2(a2) show that chaos dynamics can be achieved in both DFBs in the CSL-MC system. The corresponding ACFs for the two DFBs are shown in Figs. 2(b1) and 2(b2), respectively. Obviously, there exist pronounced peaks. Furthermore, power spectra are shown in Figs. 2(c1) and 2(c2), which verify the chaotic dynamics. The results indicate that the TDS cannot be suppressed in the CSL-MC system. Row 3 and row 4 show the outputs of both DFBs in the PMSL-MC system. Obviously, chaotic dynamics are obtained in both DFBs (Figs. 2(a3) and 2(a4)), but the TDS are completely suppressed and are indistinguishable in the ACF traces (Figs. 2(b3) and 2(b4)). Moreover, the power spectra of both DFBs in proposed PMSL-MC system (Figs. 2(c3) and 2(c4)) are much flatter than those in CSL-MC system (Figs. 2(c1) and 2(c2)). Compared with the CSL-MC system, the PMSL-MC system exhibits great improvement of TDS concealment. This is because the phase modulation generates many new frequency components and introduces nonlinearity into the feedback light, which are beneficial to TDS concealment [30].
Fig. 2. The chaotic time series (column a), the corresponding ACFs (column b) and the power spectra (column c) for two lasers. The row 1 and row 2 are the results of DFB1 and DFB2 in the CSL-MC system respectively. The row 3 and row 4 are the results of DFB1 and DFB2 in the PMSL-MC system respectively.
Next, to quantify the TDS concealment of chaotic signals in both the CSL-MC system and the PMSL-MC system, we introduce ${\rho _\textrm{m}}$ as the most significant peak value in ACF trace. Note, a low value of ${\rho _\textrm{m}}$ indicates better TDS concealment [25]. For convenience, the ${\rho _\textrm{m}}$ of the two DFBs are denoted as ${\rho _{CSL - MC1}}$ and ${\rho _{CSL - MC2}}$ for the CSL-MC system, and are denoted as ${\rho _{PMSL - MC1}}$ and ${\rho _{PMSL - MC2}}$ for the PMSL-MC system. Figure 3 shows the values of ${\rho _\textrm{m}}$ when ${k_r}$ is varied. It can be seen that, the values of ${\rho _{CSL - MC1}}$ and ${\rho _{CSL - MC2}}$ are greater than ${\rho _{PMSL - MC1}}$ and ${\rho _{PMSL - MC2}}$ for all considered coupling strengths, indicating that the TDS cannot be concealed in the CSL-MC system. However, for the PMSL-MC system, the TDS values are always less than 0.2, which are much lower than those for the CSL-MC system under different coupling strength. That is to say, the TDS can be better suppressed in the PMSL-MC system.
Fig. 3. (a) ${\rho _1}$, (b) ${\rho _2}$ in the CSL-MC system (${\rho _{\textrm{CSL - MC1}}}$, ${\rho _{\textrm{CSL - MC2}}}$) and the PMSL-MC system (${\rho _{\textrm{PMSL - MC1}}}$, ${\rho _{\textrm{PMSL - MC2}}}$), as a function of ${k_r}$.
Subsequently, to obtain the evolution of TDS patterns, the two-dimensional maps of ${\rho _\textrm{m}}$ values in the parameter space of coupling strength ${k_r}$ and frequency detuning $\Delta {f_{12}}$ are further presented in Fig. 4. A better TDS concealment region with ${\rho _\textrm{m}}\;<\;0.2$ is denoted as region A by the contour lines. As Figs. 4(a) and 4(b) illustrated, in the CSL-MC system, the region A is narrow and is greatly affected by the frequency detuning and coupling strength. However, the TDS concealment can be achieved in a much wider region in the PMSL-MC system, even when the coupling strength is large. These results suggest that, compared with the CSL-MC system, a better TDS concealment can be obtained in the PMSL-MC system. Hence, in the following, the PMSL-MC system is mainly considered.
Fig. 4. Two dimensional maps of ${\rho _m}$ in the parameter space of ${k_r}$ and $\Delta {f_{ 1 2}}$ for DFB1 (left column) and DFB2 (right column). The row 1 is for the CSL-MC system and the row 2 is for the PMSL-MC system.
Furthermore, to extensively investigate the properties of TDS concealment in the proposed scheme, the two dimensional maps of ${\rho _\textrm{m}}$ in the parameter space of ${k_{PM}}$ and ${k_r}$ are displayed in Fig. 5. The ${k_{PM}}$ is varied from 0 to 2 [28]. When the coupling strength is relative small, there is a wide range of region A. Moreover, the intermediate ${k_{PM}}$, especially ${k_{PM}} = 1$ corresponding to a better TDS concealment for all considered ${k_r}$. Besides, a lower ${k_r}$ leads to a wider region of A. Hence, in the proposed scheme, efficient TDS concealment can be achieved by using a proper modulation index.
Fig. 5. Two dimensional maps of ${\rho _m}$ in the parameter space of ${k_r}$ and ${k_{PM}}$ for (a) DFB1 and (b) DFB2.
Without loss of generality, we also consider other coupling delays, and present the values of ${\rho _\textrm{m}}$ in the parameter space of ${\tau _1}$ and ${\tau _2}$ in Fig. 6. It can be seen that, low values of ${\rho _\textrm{m}}$ can be obtained in the most space. However, there is a special line in the parameter space showing the clearly higher values of ${\rho _\textrm{m}}$, e.g., ${\tau _1} = {\tau _2}$, which is similar to the findings obtained in Ref. [25]. That is to say, the system with heterogeneous coupling delays has a better TDS concealment performance than the system with identical coupling delays. As a consequence, by introducing a phase-modulated Sagnac loop, dual-channels chaotic signals with TDS concealment can be simultaneously generated in a wide range of coupling delays.
Fig. 6. Two dimensional maps of ${\rho _m}$ in the parameter space of ${\tau _1}$ and ${\tau _2}$ for (a) DFB1 and (b) DFB2.
3.2 Decision making for the MAB problem
In this section, we demonstrate the utilization of the dual-channels chaotic signals to solve the MAB problem in parallel. In order to distinguish K non-identical slot machines, we code them in an $M$-bit binary with ${D_i}(i = 1,..,M)$ being 0 or 1. It is obvious that $K = {2^M}$. As shown in Fig. 7, for example, when $K = 4 (M = 2)$, the slot machines are numbered by ${D_1}{D_2} = \{ 00,01,10,11\} $ and the reward probability of each slot machine i is ${P_i}(i = 1,..,N)$. Here, dual-channels chaotic signals are used to determine the slot machine with the highest probability of reward in parallel. The identity of the slot machine to be selected is determined bit by bit from the most significant bit (MSB) to the least significant bit. For each bit, the decision is made based on a comparison between the level of the sampled signal and the threshold value. For clarity, we take the 4-armed bandit problem as an example. First, we sample dual-channels chaotic signals at $t = {t_1}$ and obtain two sampled signals ${s_1}$ and ${s_2}$. Then, ${s_1}$ is compared with a threshold value $T{V_1}$. If ${s_1} \le T{V_1}$, the MSB of the slot machine is determined to be 0, which is denoted as ${D_1} = 0$. Otherwise, the MSB is determined to be ${D_1} = 1$. Here, we suppose ${s_1} \le T{V_1}$, then ${D_1} = 0$. Based on the determination of ${D_1}$, ${s_2}$ is compared with another threshold value denoted as $T{V_{2,0}}$. The first number in suffix means the 2nd-MSB of the slot machine, while the second number is related to the determination of MSB. Note, the $m\textrm{ - th}$ bit relates to ${2^{m - 1}}$ kinds of threshold values. In other words, if there are K slot machines, there are a total of $K - 1$ threshold values. Similar to the decision rule of ${D_1}$, if ${s_2} \le T{V_{2,0}}$, the 2nd-MSB of the slot machine is determined to be 0, which is denoted as ${D_2} = 0$. Otherwise, the 2nd-MSB is determined to be ${D_2} = 1$. In practice, the levels of threshold values are limited to a finite value. More precisely, the actual threshold value is denoted by [35,37],
In Eq. (6), the first (second) term only exists in the case when the selected slot machine yields (not yields) a reward at cycle $t$. a is a memory parameter, $\Omega $ and $\Delta $ are increment parameters which are determined based on the history of rewards. In this paper, the initial value of $\lfloor{TV(t)} \rfloor $ is 0, a is 0.99, $\Omega $ and $\Delta $ are given as follows [34–37],
The chaotic signals of this scheme are generated by the system in Fig. 1, the time series and their spectra are presented in Fig. 2. After normalization, the ranges of two chaotic signals are varied from $- 0.5$ to $0.5$. The constant k is set as 0.05 and L is set as 10, so that the $T(t)$ spans from $- 0.5$ to $0.5$. The sampling interval is chosen as 100ps [34,35]. The slot machines are played 400 cycles consecutively, and such play is repeated 5000 times. The reward probability of each slot machine i is $\{ {P_1},{P_2},{P_3},{P_4}\} = \{ 0.8,0.2,0.2,0.2\} $. The correct decision rate (CDR) is defined as the ratio of the number of correct selection (best slot machine selection) to the number of total selection.
The CDR as a function of cycle for the CSL-MC system and for the PMSL-MC system are displayed in Fig. 8(a). It can be seen that the CDR of PMSL-MC system converges more rapidly than that of CSL-MC system. Here, we define the cycle at which the CDR reaches 0.9 as convergence cycle (CC). For instance, the CC of PMSL-MC system is 91 and the CC of CSL-MC system is 117. Without loss of generality, CC as a function of ${k_r}$ is further presented for both systems in Fig. 8(b). It can be seen that, the values of CC in the PMSL-MC system are much smaller than those in the CSL-MC system for all ${k_r}$. Interestingly, the trend is closely related to Fig. 3. The ${\rho _m}$ varies with the change of ${k_r}$. For a lower ${\rho _m}$, the value of CC is much smaller, which means the CDR can rapidly converge. With the increase of ${\rho _m}$, the value of CC is gradually increased. The lower ${\rho _m}$ corresponds to the higher complexity of laser chaos, which is an important factor affecting the convergence rate of decision making [34,44]. With the increasing of ${k_r}$, the complexity of chaos is decreased, and thus, CC is increased. That is to say, a rapid convergence of CDR is favored by a smaller ${\rho _m}$, and the CDR performance of the PMSL-MC system is better than that of the CSL-MC system.
Fig. 8. (a) The CDR as a function of cycle for the CSL-MC system and for the PMSL-MC system. (b) The CC as a function of ${k_r}$ for the CSL-MC system and the PMSL-MC system.
In order to show the evolution of the threshold value more intuitively, the threshold values of some representative cases as functions of cycle are discussed in Fig. 9. Here, the threshold values adjustments of the 1-st, 2500-th, and 5000-th trials are presented. As shown in Fig. 9(a), the $T{V_1}$ gradually increases as the number of cycles increases, and finally fluctuates around the upper limit of 0.5. Therefore, the chaotic signal ${s_1}$ will more likely be lower than the $T{V_1}$, so that ${D_1}$ is more likely to be determined as 0. The evolutions of $T{V_{2,0}}$ are similar to $T{V_1}$, as seen in Fig. 9(b). The chaotic signal ${s_2}$ will more likely be lower than $T{V_{2,0}}$, making ${D_2}$ more likely to be determined as 0. However, as shown in Fig. 9(c), it can be seen that the variation of $T{V_{2,1}}$ is small and only fluctuates in previous cycles.
Fig. 9. Threshold value as a function of cycle for the 1-st, 2500-th, and 5000-th decision making process for (a) $T{V_1}$ (b) $T{V_{2,0}}$ (c) $T{V_{2,1}}$ in the PMSL-MC system.
Furthermore, in order to discuss the ultrafast decision making of the proposed system, we compare the CDRs of dual-channels chaotic signals and one-channel chaotic signal in the PMSL-MC system as presented in Fig. 10. For the dual-channels scheme, two chaotic signals are sampled simultaneously at time ${t_1}$, and the sampling signals ${s_1}$ and ${s_2}$ are obtained. They are used to determine the MSB and the 2nd-MSB of the slot machine, respectively. However, for the conventional one-channel scheme, the MSB of the slot machine is determined by sampling the chaotic signal at time ${t_1}$, and the 2nd-MSB of the slot machine is determined by sampling the chaotic signal at time ${t_2}$. As shown in Fig. 10(a), it is clearly that the dual-channel chaotic system converges more rapidly than the one-channel chaotic system. We further present CC as a function of ${k_r}$ for dual-channels and one-channel chaotic signals in the PMSL-MC system in Fig. 10(b). Obviously, compared with one-channel chaotic signal, the dual-channels chaotic signals have a faster convergence rate for all ${k_r}$. That is to say, the proposed system with dual-channels chaotic signals, which permits parallel decision making, converges more rapidly.
Fig. 10. (a) The CDR as a function of the number of cycles for dual-channels and one-channel in the PMSL-MC system. (b) The CC as a function of ${k_r}$ for dual-channels and one-channel in the PMSL-MC system.
In reality, the external environment is uncertain. As shown in Fig. 11, we suppose that slot machines are played 800 cycles consecutively, and the reward probabilities of four slot machines are switched at the 400-th cycle to emulate the changing environment. At the beginning, the reward probabilities of four slot machines are $\{ {P_1},{P_2},{P_3},{P_4}\} = \{ 0.8,0.2,0.2,0.2\} $. After 400 cycles, the reward probabilities are switched to $\{ {P_1},{P_2},{P_3},{P_4}\} = \{ 0.2,0.2,0.2,0.8\} $. It can be seen in Fig. 11 that the CDR of the PMSL-MC system converges faster than that of the CSL-MC system. At the 400-th cycle, the CDRs of both systems are reduced to the lowest value. With the further increase of cycles, the CDRs of both systems can increase rapidly. Moreover, the PMSL-MC system recovers faster than the CSL-MC system. This indicates that the PMSL-MC system can provide a faster decision making rate and a better environmental adaptability.
Fig. 11. The CDR as a function of the number of cycles for the CSL-MC system and the PMSL-MC system in a changing environment.
4. Experimental results
4.1 TDS concealment
In experiments, the PM with an insertion loss of $\textrm{4dB}$ is used in the system. The half-wave voltage of the PM is $\textrm{7V}$, and the peak amplitude of the PM driving signal is $\textrm{5V}$, which corresponds to a peak phase shift of $0.7\pi $. The PM is driven by the $\textrm{8GS/s}$ PRBS, which is generated by the AWG and then amplified by the RF amplifier. The injection optical power ${P_{inj}}$ is measured at the OPM. The experimental results of injection optical power-dependent behavior of ${\rho _m}$ for the CSL-MC system and the PMSL-MC system are presented in Fig. 12. Here, the bias currents are set as $30.1\textrm{mA}$ and $31\textrm{mA}$ for two DFBs, and the corresponding output powers are $2.915\textrm{mW}$ and $2.913\textrm{mW}$. The coupling delays are calculated as ${\tau _1} = 144.2\textrm{ns}$ and ${\tau _2} = 149.6\textrm{ns}$. As shown in Fig. 12, the TDS values in the PMSL-MC system are always much lower than those in the CSL-MC system at different ${P_{inj}}$. Due to the limitation of the intrinsic parameter values of different DFB, DFB2 performs slightly better TDS concealment. Comparisons with the results in Fig. 3 indicate that the experimental results are well in line with the simulation results.
Fig. 12. Experimental (a) ${\rho _1}$, (b) ${\rho _2}$ in the CSL-MC system (${\rho _{\textrm{CSL - MC1}}}$, ${\rho _{\textrm{CSL - MC2}}}$) and the PMSL-MC system (${\rho _{\textrm{PMSL - MC1}}}$, ${\rho _{\textrm{PMSL - MC2}}}$), as a function of ${P_{inj}}$.
4.2 Decision making for the MAB problem
We experimentally demonstrate the decision making performance in the PMSL-MC system and in the CSL-MC system. The chaotic signals for decision making are acquired experimentally from the proposed systems. As shown in Fig. 13(a), the CDR of PMSL-MC system converges more rapidly than that of CSL-MC system. Furthermore, Fig. 13(b) shows the values of CC in the PMSL-MC system are much smaller than those in the CSL-MC system for all ${P_{inj}}$. These experimental results agree well with our numerical results shown in Fig. 8.
Fig. 13. Experimental (a) The CDR as a function of cycle for the CSL-MC system and for the PMSL-MC system with ${P_{inj}} = 98.15\mu \textrm{W}$. Experimental (b) The CC as a function of ${P_{inj}}$ for the CSL-MC system and the PMSL-MC system.
5. Conclusion
In conclusion, we propose to generate dual-channels chaos with TDS concealment by introducing a phase-modulated Sagnac loop, and the dual-channels chaos are used to solve MAB problem in parallel for the first time. Compared to the CSL-MC system, the concealment of TDS in the PMSL-MC system can be achieved simultaneously in both DFBs with a wide range of coupling strength, frequency detuning, modulation index and coupling delays. Besides, a lower TDS is favorable for a rapid convergence of decision making. The experimental results are well in line with the numerical simulation. Moreover, to the best of our knowledge, the parallel decision making based on dual-channels chaos is realized in our proposed system for the first time, which converges more rapidly than one-channel chaos. Furthermore, the proposed system is scalable in a larger MAB problem. We consider that this work has potential value for secure optical communication system and future photonic intelligence.
Funding
National Natural Science Foundation of China (61974177, 61674119); National Postdoctoral Program for Innovative Talents (BX201600118); China Postdoctoral Science Foundation (2017M613072); Shaanxi Province Postdoctoral Science Foundation.
Disclosures
The authors declare no conflicts of interest.
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