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Abstract
Boolean chaos is widely used in physical systems for its digital-like behavior and complex dynamics. However, electronic logic devices limit the bandwidth of Boolean chaos and its development. Based on an autonomous optical Boolean network, a method of generating optical Boolean chaos with 14 GHz bandwidth is proposed, exploring the physical mechanism of the chaos generated by the system and analyzing the influences of external parameters on the dynamic characteristics of the system. The output status is mainly affected by the detection optical power, carrier recovery time of the semiconductor optical amplifier, and difference between the two self-feedback time delays.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Boolean networks are often used to study systems that exhibit switching behavior, such as those involved in cell cycle dynamics [1], neural interactions [2], and social networks [3]. However, all of these schemes require external clocks to synchronize state updates, and their state spaces are finite and discrete. Meanwhile, autonomous Boolean networks (ABNs) have continuous state spaces, fast time scales, and complex dynamics. ABNs are systems whose future behavior is determined by past time and transition states. Thus, they can be exploited in various applications, such as random number generation [4], radar [5,6], reservoir computing [7], and genetic circuits [8].
This study was focused on ABNs. The state transitions of outputs are typically called “events.” Ghil et al. introduced Boolean delay equations and ordinary differential equations to explain the continuous temporal evolution of ABNs [9–11]. On the premise that the logic gate in such a network can process input signals arbitrarily fast, they found that the dynamic characteristics of the system become complicated when the events of the entire system increase according to a power law. To the best of our knowledge, current systems that generate Boolean chaos are typically based on discrete electronic logic devices or field-programmable gate arrays, and the bandwidth of Boolean chaos is 1.4 GHz [12]. Owing to the bottleneck of the electronic signal processing speed, the bandwidth of Boolean chaos remains limited, which directly affects the application prospects of ABNs. Therefore, in this work, we considered the use of optical logic gates to construct optical ABNs (OABNs). Various groups have already proposed solutions for building optical logic gates [13–16], and semiconductor optical amplifier (SOA)-based optical logic gates seem to have the key advantages of high nonlinear coefficients and easy integration. However, they have only been used to process information in optical communication systems so far, but not to generate broadband chaos and complex signals.
This paper proposes an ABN based on optical logic gates, which consists of an optical XNOR logic gate with two self-feedback links. It provides an assessment of the theoretically expected performance and shows that the proposed scheme can generate optical Boolean chaotic signals with a bandwidth up to 14 GHz. Such high bandwidth Boolean chaotic signals will solve the low bandwidth problem of Boolean chaos, which can greatly expand the application field of ABNs. In addition, this report describes in detail the physical mechanism of optical Boolean chaos and discusses the effects of the feedback delay time, SOA carrier recovery time, and detection power.
2. Principle
The proposed network is composed of an optical XOR gate with an optical NOT gate combined with two feedback loops, as schematically shown in Fig. 1.
Fig. 1. Schematic of optical Boolean chaos. OXOR: optical exclusive OR gate; ONOT: optical NOT gate.
We chose to use an SOA-Mach–Zehnder interferometer (SOA-MZI) to realize the optical XOR gate, as shown in Fig. 2. The continuous-wave PIN1 is injected into the SOA-MZI as a detection signal and is divided into two equal parts with a 3 dB splitter. Two optical control signals, PE and PF, enter the upper and lower arms of the SOA-MZI in opposite directions. In each SOA, the two signals interact through cross-gain modulation (XGM) and cross-phase modulation (XPM).
Fig. 2. Schematic of optical Boolean chaos-based all-optical XOR, XNOR gate. SOA: semiconductor optical amplifier; 3 dB: 3 dB optical fiber coupler; τ1, τ2: optical delay lines.
As PE and PF are “H” or “L” (representing the optical power levels corresponding to the high and low logic states, respectively), the SOA-MZI is balanced and the output is “L.” However, when either PE or PF is “H,” the phase difference between the two arms of the SOA-MZI can generate “H” interference signals at the XOR output port. Through the above signal processing principle, this system can achieve XOR operation, and the output signal power can be expressed as
To obtain the desired XNOR operation, we decided to add another SOA after the XOR gate to realize the optical NOT gate. Through the XGM effect occurring in SOA3, the probe signal PIN2 experiences optical gains defined as GC, and its output power can be expressed as
The output of the XNOR gate is then sent back to SOA1 and SOA2 through two distinct feedback loops, introducing two distinct delays, τ1 and τ2, respectively. These delays correspond to the distances from the output of SOA3 to SOA1 and SOA2, respectively. Without loss of generality, τ1 can be taken to be smaller than τ2 and it can be assumed that the time delay difference Δτ is less than τ1. The working state of the system can be divided into the following time periods according to the delay time. Other delay situations can be analyzed analogously.
0–τ1: During this period, when the continuous wave PIN1 passes through the upper and lower arms of the SOA-MZI as the detection signal, owing to the existence of a feedback delay, feedback signals PE and PF remain in the initial no-signal state, as shown by the dotted line in Fig. 2. The carrier densities of SOA1 and SOA2 are not affected by the feedback signal. The detection signals of the upper and lower arms experience the same gain and phase modulation and output high-logic-state PA and PB signals. The two signals are interfered at a 3 dB coupler to output the XOR signal PC. PC and the probe signal PIN2 pass through SOA3 in the reverse direction. PIN2 is affected by the XPM effect in SOA3 and outputs signal PD in a high logic state, which is equivalent to the XNOR signals of PE and PF.
τ1–τ2: The high logic output of the system at time 0 arrives at SOA1 as PE, which consumes numerous carriers of SOA1. Because of the low carrier concentration in the SOA, the continuous wave passing through it cannot achieve a high gain, and the output signal PA is a low-logic-state signal. Moreover, during this time, the system output is not fed back to SOA2, and PF maintains the initial no-signal state, so the output signal PB of SOA2 is in a high logic state. The phase difference between PA and PB is π, PC is in a high logic state, and the XNOR output after SOA3 is in a low logic state.
The evolution of the subsequent time period can be obtained through the same analysis. However, depending on the values of τ1 and τ2, the system evolution can be increasingly complex. As discussed below, even chaotic Boolean dynamics can be achieved. In particular, signals PE and PA and signals PF and PB are complementary, and there is a time difference of Δτ between PE and PF.
3. Simulation model and typical behavior
The mathematical description using a Boolean delay equation [9] showed that an ABN generates chaotic oscillation, producing aperiodic dynamic characteristics except when the initial output is zero. Moreover, when the initial value is zero, a stable fixed point will occur, and the solution of the Boolean equation will be in a low logic state.
Cavalcante et al. [17] numerically simulated a system composed of an XOR gate with two-way delayed feedback. When the two delay times were not equal, one initial transition at time t = 0 caused the system to oscillate. The number of conversions per unit time increased significantly, which is consistent with the reports in [9] and [10].
To verify the feasibility of realizing aperiodic dynamics in OABNs, we constructed a feedback system composed of optical XNOR gates. Using this configuration, the entire system can start to oscillate without transformation in the initial state. We built this structure in the simulation software VPItransmissionMaker. Table 1 lists the main parameters of the SOA and other parameters as the standard parameters in VPI [18].
Table 1. SOA Default Simulation Parameter Values
Notably, the carrier density and gain of the SOA are closely related to the bias current. To operate the system, the bias current of the SOA should meet the following conditions. First, when the two arms of the SOA-MZI have the same logic state signal input, a phase difference of π exists between the upper and lower arms. Second, the high logic state signal output of SOA3 can consume the carriers of SOA1 and SOA2.
3.1 Static state
The output state of the entire system is mainly determined by the difference between the two feedback links. When the feedback times of the two channels are equal, the time difference is 0. In this case, the system always outputs a high-logic-state signal. Initially, because no feedback signals are sent to the two arms of the SOA-MZI, PIN1 obtains the same gain and phase shift through SOA1 and SOA2, respectively. The XOR output of the low-logic-state signal and PIN2 pass through SOA3 in the reverse direction, at which time the system outputs a high logic state. When the high-logic-state signal feeds back to the two SOAs, the carriers in them are consumed to the same extent. Thus, PIN1 produces the same gain and phase change, and the XOR outputs a low-logic-state signal. In this case, the system has the same output as in the high logic state. The process loops repeatedly, with the system remaining in a static state.
3.2 Periodic signal
When the lengths of the two delay lines are commensurate, that is, when the shortest delay is a simple fraction of the longest one (e.g., 1/2, 1/3; see Section 3.3 for further details), a periodic signal is observable.
When τ1 = 1 ns and τ2 = 1.5 ns, the system outputs a periodic signal, as shown in Fig. 3. In this case, the system output can be analyzed according to the following time periods. At 0–1 ns, the feedback signal has not reached SOA1 and SOA2 and the system outputs a high-logic-level signal. At 1–1.5 ns, only the system output at time 0 reaches SOA1 as a control signal, the two detection signals undergo different gains and phase modulations, and the system outputs low-logic-level signals. At 1.5–2 ns, the system output at 0.5 ns reaches SOA1 and the output at 0 ns reaches SOA2, so the system outputs high-logic-level signals. Finally, at 2–3.5 ns, the control signal of SOA1 experiences a low–high–low state change, that of SOA2 experiences a high–low–high state change, and the duration of each level is a delay time difference (0.5 ns). At this point, one signal operation cycle ends. Owing to the phase shift mutation of the transmission signal and the different durations of the rising and falling edges, the system will output flaw and burr signals.
Fig. 3. Periodic oscillations produced by the all-optical autonomous Boolean network. (a) XOR output waveform; (b) XNOR output waveform; (c) frequency spectrum. Delay lines 1 and 2 are 1 and 1.5 ns, respectively.
3.3 Chaotic signal output
When the lengths of the two delay lines are incommensurate, the XNOR output as the control signal will arrive at SOA1 and SOA2 at different times. The previous analysis shows that the system XNOR output at moment 0 is a high-logic-level signal.
Further, the entire system can generate a Boolean chaotic signal when the two delays are incommensurate. Because of the change in the time delay difference, as the system continues to run, it will interfere with narrow pulses. In addition, there are rising and falling edges in the signal logic state transformation, and the duration of these signals corresponds to the fast consumption time and slow recovery time of the SOA carrier. Burr and flaw signals are also generated when the two signals interfere. When these narrow pulse signals are fed back to SOA1 and SOA2, the carrier will not be consumed completely, and an incomplete signal will be output. The above process repeats, and the system generates Boolean chaotic signals. When the shortest delay time and delay time difference are 1 and 0.15 ns, respectively, the carrier recovery times of commercial SOAs are on the order of tens to hundreds of picoseconds. To determine the influence of the delay time difference on the system output, we used delay times as small as 1 ns. The development of the time series in Fig. 4(a) shows that the whole system outputs complex, unrepeated signals with a Boolean transformation behavior. Notably, the Burr and flaw signals account for a small proportion of the output time sequence, which will not affect the phase discrimination or logic results. In the frequency domain shown in Fig. 4(b), the spectrum extends from DC to 14 GHz (10 dB bandwidth) [12].
Fig. 4. Chaotic oscillations generated by an ABN. (a) Output Boolean chaotic waveform; (b) measured power spectrum.
To confirm the chaotic nature of the dynamics, we calculated the permutation entropy and correlation dimension of the output data. Permutation entropy is used as an indicator of the complexity of time series [19,20]. A more regular time series has a smaller arrangement entropy, while a more complex time series has a greater arrangement entropy. We obtained the normalized permutation entropy of the output sequence using the method proposed by Bandt and Pompe [20], as shown in Fig. 5(a). In this example, the number of points in the time series is N = 150,000, and we used an ordinal pattern length D of 4. This method ensures that N ≫ D! and maintains a practical computation time [21]. The abscissa represents the embedding time, and the ordinate represents the permutation entropy. The figure shows that the permutation entropy increases and tends to be flat near 1. Because there is no entropy drop anywhere in the graph, the signal is considered to be completely random.
Fig. 5. Results of permutation entropy and maximum Lyapunov exponent analysis of the Boolean chaotic data set shown in Fig. 4. (a) Permutation entropy; (b) maximum Lyapunov exponent.
Another indicator used to measure the divergence of chaotic signal trajectory indices is the maximum Lyapunov exponent [10,12]. When this value is greater than 0, the signal is in a chaotic state. We quantized a 40 μs chaotic sequence into a 0/1 sequence and calculated its Boolean distance [10], where T = 10 ns and δ = 0.01 is a fixed parameter. The solid blue line in Fig. 5(b) represents the time evolution of ln < d(s)>, where <> denotes an average function. It displays the slope of the time evolution curve when < ln(d(s))> reaches a constant value after a period of time. We found that λmax = 0.25 ns−1(±0.01 ns−1), which demonstrates that the network is chaotic.
The permutation entropy and maximum Lyapunov exponent estimation provide consistent results, indicating that the simulated time series indeed corresponds to chaotic behavior.
4. Physical mechanism
Cavalcante et al. identified the memory effect as an important dynamical feature of systems that generate chaotic dynamics [17,22]. They termed the memory effect “degradation” and described the effect mathematically with a degradation function. The degradation effect is present when an input transition occurs, and the gate has not yet switched from the previous transition propagation. In the scheme considered in this work, the main degradation effect can be expected to originate from the non-zero recovery time of the SOAs, which is directly linked to the carrier dynamics.
We used an optical NOT gate composed of a single SOA to simulate this degradation effect. In Fig. 6(a), the horizontal axis is the time axis, the ordinate represents the width of the input Gaussian pulse, and the vertical coordinate represents the power of the output signal. The curves in this figure represent the output of Gaussian pulses with widths of 10–100 ps passing through the SOA. It is evident that as the pulse width gradually decreases, the duration of the low logic state gradually decreases as well. This state even disappears when the pulse width is less than 10 ps. In fact, when the pulse width is greater than the consumption time of the SOA carrier, the carriers in the SOA are quickly consumed, and the carrier concentration recovers slowly after reaching a low level when the pulse goes through the SOA. The change in carrier concentration causes the output signal to maintain a low logic state for a period significantly longer than the pulse duration. This change is clearly visible for an input pulse width of 100 ps. In contrast, when the input pulse width is less than the carrier consumption time of the SOA, the carriers in the SOA start to recover before they are consumed at the lowest concentration, and a pulse is output without a low logic state, as shown in the 10 ps pulse width case.
Fig. 6. Sketches of the degradation effect. (a) Optical behavior causing the degradation effect; (b) tp/tp0 versus TG, showing regions with the degradation effect.
To characterize the influence of the degradation effect on the output pulse width more precisely, we computed the results shown in Fig. 6(b) by adjusting the pulse width of the input pulse in detail, where tp0 is the transmission time without the degradation effect, tp is the signal transmission time with the degradation effect, and TG represents the full width at half maximum of the transmission Gaussian signal. We took the value of tp/tp0 as the standard to evaluate the degradation effect. A reduction in tp/tp0 was observed with decreasing TG, producing solutions such as that shown in Fig. 6(b). Thus, when the pulse width is less than the SOA carrier consumption time, an incomplete pulse will be output owing to the degradation effect.
We performed a simulation to determine the effects of the carrier recovery time and input intensity of PIN2 on the output signal dynamics, producing the results shown in Fig. 7. When the delay of one channel was 1 ns, we fixed the power of PIN1 and delay time difference at 10 mW and 0.15 ns, respectively. The horizontal and vertical coordinates in the figure represent the SOA carrier recovery time and detection signal intensity, respectively. The different colors represent different types of adjacent event intervals in the output temporal waveforms. The red region represents a chaotic state signal, the green region represents a periodic state signal, and the blue region indicates that the system continues to output a high logic state after operation.
When the power of PIN2 is too small, the signal amplification is not sufficient after the SOA. The system output is fed back to the SOA-MZI, and the output pulse energy of the XOR interference is small. Owing to the degradation effect, the system outputs a constant high logic signal after some time. As the optical power of the detection signal increases, the extinction ratio of the XOR interference output signal begins to increase, and the system can output periodic signals. However, the burr and depression energies in the periodic signal are not sufficient to generate Boolean chaos, and Boolean chaos can be produced by slightly increasing the detection signal power. In the case of the system outputs periodic signals, increasing the detection signal power will consume SOA carriers. As a result, the carrier recovery time of SOA becomes shorter, and the extinction ratio of the output signal begins to decrease. Since the rising and falling edges of the signal become shorter, the pulse width of the interference output signal will be narrowed, and then the degradation effect will produce Boolean chaos. However, increasing the detection optical power consumes numerous carriers in the SOA. The extinction ratio of the output signal begins to deteriorate, thereby starting to produce periodic and constant high logic signals.
With the delay time and probe power fixed, we observed the system output state by adjusting the recovery time of the SOA carrier. The SOA carrier recovery time was initially set to 100 ps. In this case, the pulse width of the XOR operation output was too narrow. When this kind of signal passes through the SOA, the signal power generated by the XGM effect is insufficient, and the system outputs a constant high-logic-state signal after running for some time. As the SOA carrier recovery time decreases, the signal power of the narrow pulse width pulse generated by the XGM effect gradually increases, so the system can generate periodic signals. Then, to reduce the recovery time of the SOA carrier, the system can output an optical Boolean chaotic signal. However, if the recovery time of the SOA carrier is continuously reduced, then the pulse width of the interference pulse in the system will become larger than the fast consumption time of the SOA carrier. These pulses do not meet the conditions of the degradation effect, and the system will start to generate periodic or constant high-logic-state signals.
According to the previous analysis results, the output of the system is closely related to the delay difference when the other external conditions are fixed. To confirm this point, we fixed the delay time of one channel at 1 ns and analyzed the influence of the delay difference on the durations of adjacent events in the output sequence, as shown in Fig. 8. Other delay cases have similar rules, which can be analyzed analogously.
First, regarding the change in the delay difference, the system displays either periodic or chaotic dynamics, because a change in the delay difference will lead to a change in the XOR pulse width. As the delay difference changes, the system will interfere with the narrow pulse signals. In addition, Burr and flaw signals are also generated when the rising and falling edges of the logic signal interfere. When a narrow pulse passes through the SOA, the system will produce a degradation effect and then output a chaotic Boolean signal. When outputting other pulse width signals, the system outputs periodic signals. Second, the graph is basically symmetrical between 0.3 and 0.7 ns, and the axis of symmetry is half of the time difference. With one delay time fixed and the other adjusted, the system always interferes with pulses with widths equal to the delay difference or the difference between the delay difference and fixed delay. This characteristic is the fundamental cause of the symmetrical distribution of the system output signal bifurcation. Third, when the delay difference is too small, the system will output a pulse with a very small width at the beginning of operation, which will cause a degradation effect. Although chaotic signals can be generated, the duration of adjacent event intervals in the output signal is small.
By studying the effects of external parameters on the system output, the following rules can be identified. When the SOA carrier recovery time is fixed, the system can be made to output a periodic signal of any frequency by adjusting the delay time difference and detection signal power.
5. Conclusion and discussion
Photonic integrated chips constitute a future development trend. In this study, we focused on the integration of optical Boolean chaos. The on-chip integration of an optical Boolean chaotic system can be realized by combining optical logic gate integration technology and optical delay line integration technology. The integration technology for optical logic gates composed of SOA-MZIs is already very mature. The integrated adjustable optical delay line can be realized by combining the ring resonator and the switchable delay line [23]. By adjusting the delay difference between the two channels, the system outputs optical Boolean chaotic signal. Furthermore, the slow SOA carrier recovery time limits the maximum speed of an optical logic gate constructed with an SOA as the main device. If the main device is replaced with an SOA with a faster carrier recovery time or another high-speed optical logic gate, then the system can produce higher bandwidth Boolean chaos.
This paper proposed a method of generating optical Boolean chaos based on optical logic gates. Our simulation results show that this structure can generate not only periodic, controllable square wave signals, but also optical Boolean chaotic signals with bandwidths of up to 14 GHz. This method solves the problem of the low bandwidth of Boolean chaos generated by electronic logic devices. By calculating the permutation entropy and correlation dimension of the system output sequence, it was proven that Boolean chaos rather than noise is produced. This study also demonstrates that the degradation effect is the fundamental cause of Boolean chaos in such a system; that is, when the input pulse width is too small, the logic device cannot fully respond and outputs an abnormal pulse. In addition, we analyzed in detail the effects of the delay time difference between the two channels, detected signal power, and SOA carrier recovery time on the system output.
Funding
National Natural Science Foundation of China (61671316, 61705160, 61731014); Shanxi Scholarship Council of China; Natural Science Foundation of Shanxi Province (201801D121145); Guangdong Provincial Key Laboratory of Photonics Information Technology; Research Foundation-Flanders (FWO) (postdoctoral fellowship of MV) (G0E7719N).
Disclosures
The authors declare no conflicts of interest.
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