Introduction

During the past 25 years, there has been continued growth in the dynamics nonlinear delay system field and an ever-increasing appreciation of its applications among researchers. This interest was first motivated by Gibbs1 who demonstrated experimentally an optically bistable hybrid device that shows instabilities with periodic and chaotic dynamics, in good agreement with the theory predicted by the pioneering work of Ikeda2. The corresponding setups for this system are accurately modeled by Nonlinear Delay Differential Equations (NLDDEs)3. These systems are renowned for displaying a myriad of fascinating dynamical behaviors, including a sequence of intricate bifurcations4, and the concurrent presence of fast-scale chaos and slow-scale periodicity5.

Research has demonstrated that the type of delay in dynamic nonlinear systems can lead to a variety of distinct dynamical behaviors. Specifically, early studies on dynamic nonlinear delay systems have illuminated the many potential dynamical regimes associated with large time delay6, largely because the dynamics of the logistic map are well understood. These systems have been observed to exhibit intriguing phenomena such as square waves oscillation and multi-stability7. Additionally, when analyzed in the space–time representation, systems with large time delay have been shown to exhibit spatiotemporal behaviors such as coarsening8 and phase transitions9. More recently, a new type of dynamics, referred to as laminar chaos, has been identified through both theoretical analysis10,11 and experimental observation12 of nonlinear systems with time-varying delays. In principle, this newly identified dynamical behavior can be observed experimentally and may pave the way for new applications or enhancements to existing ones. For instance, it holds promise for advancing information processing technologies, including chaos communication13,14,15,16,17,18,19 and reservoir computing20,21,22,23,24,25,26,27,28,29. However, there remains a noticeable gap in comprehensive reports on the performance and dynamical behaviors of systems with small time delay. Further investigation into these systems is essential to fully understand their potential and limitations.

In this letter, we explore the impact of small time delay in a dynamic nonlinear system using a nonlinear time-delayed optoelectronic oscillator (OEO). The OEO is particularly suitable for this study due to its high quality-factor and its demonstrated ability to exhibit a wide range of dynamics, including periodic dynamics30,31,32, breathers33,34, and broadband chaos35. This versatility, combined with its capacity to generate ultra-stable and ultra-pure microwaves with low phase noise36,37,38,39,40 when employing fixed large time delay, makes the OEO a well-understood system. The oscillations in the OEO are primarily driven by nonlinearity and time-delayed feedback, making it an ideal candidate for investigating the effects of time delay. Our focus in this study is on the interplay of small time delay within the optoelectronic feedback loop. We discovered that bifurcation, chaos can occur even with small time delay, regardless of the applicability of the universal adiabatic approximation model35. Additionally, we found evidence of periodic modulations in the dynamical behaviors of OEO systems with small time delay, influenced by the offset phase.

Of greater significance, we demonstrate that the complex dynamic regime of OEO under small time delay holds potential for high-speed reservoir computing. This is largely due to its ‘fading memory capacity’ and the ability for high-dimensional state spatial mapping, which together yield a computing efficiency improvement of 1000 times compared to the large time delay counterpart. In addition, we introduce a novel approach for creating a tunable optoelectronic pulse generator. This is achieved by gradually modifying the time delay of the input electrical pulse into the small time delay OEO system.

Theoretical model

The schematic arrangement of the OEO system under investigation is shown in Fig. 1. It is realized from a Mach–Zehnder Modulator (MZM) which is seeded by a CW-laser at a wavelength of 1550 nm (193.1 THz). The modulated light is split by an X-Coupler with a cross-coupling coefficient specified to 0.1. Consequently, most portion of the average intensity of the modulated light can be adjusted by an optical attenuator (ATT), and then an adjustable time delay is applied to this signal by a passive optical element which can be performed experimentally by a thermalized fiber spool. Subsequently, the delayed light is monitored by a positive-intrinsic-negative photodiode detector (PIN), whose output current signal is amplified by a trans-impedance amplifier (TIA). Then a radio-frequency amplifier (AMP) converts the voltage proportional to the low-power RF signal from TIA to a higher-power signal and ultimately feeds back to the electrodes causing changes in the refractive index of MZM. Light signal coming from the other branch of the X-coupler is first amplified by the Erbium doped fiber amplifier (DEFA) and then recorded by the PIN. The radio frequency spectrum analyzer (RFSA) with the resolution bandwidth 10 GHz measures the power spectrum of the output signal versus its frequency, and the oscilloscope (OSC) measures the waveform of electrical signal. The dynamic evolutionary process of this delay system can be modeled by following integrodifferential delayed equation31,33:

$$\tau \frac{dx}{{dt}} + x + \theta^{ - 1} \mathop \smallint \limits_{{t_{0} }}^{t} x\left( s \right)ds = F\left( {x\left( {t - \tau_{D} } \right)} \right)$$
(1)
Figure 1
Figure 1
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Schematic arrangement of the optoelectronic oscillators.

The time constants are:

$${\uptau } = \frac{{\tau_{1} \tau_{2} }}{{\tau_{1} + \tau_{2} }}, \theta = \tau_{1} + \tau_{2}$$
(2)

where \(\tau_{1} = \frac{1}{{2\pi f_{L} }} = 3.185{ }\) μs, \(\tau_{2} = \frac{1}{{2\pi f_{H} }} = 1.32{ }\)  ns, \(\theta = 3.1863{ }\)  μs are used in our case. For generality, the feedback circuitry behaves as a second-order bandpass filter (BPF) with a 3-dB cutoff frequency \(f_{H} = 125\;\;{\text{ MHz}}\) of the first-order low-pass filter and low cutoff frequency \(f_{L} = 50\;\;{\text{ kHz}}\) (bandwidth \(\Delta_{f} = f_{H} - f_{L}\)). The derivative terms in Eq. (1) are physically due to the TIA that has an equivalent 4th-order low-pass Bessel filter effect. The integral terms are physically due to the response time of the feedback electronic circuit. The terms on the right side in Eq. (1) represent nonlinear feedback with MZM as a basic nonlinear unit with time delay \(\tau_{D}\). The normalized parameter also named bifurcation parameter \(\beta = \pi {\text{g}}AGP/2V_{{\pi_{rf} }}\) concerns the feedback strength which is proportional to the CW laser power P. Here, \(g = 5{ }\;{\text{V/mW}}\) is the conversion efficiency of PIN/TIA (the responsivity of the photodetector is 1 A/W, the trans-impedance gain of TIA is 5 kΩ), and G is the power gain of amplifier. The parameter A describes the overall attenuation of this feedback loop. \(x\left( t \right)\) is proportional to the input voltage applied on the RF electrodes of the MZM and \(V_{{\pi_{rf} }} = 4\;{\text{ V }}\) is the radio frequency half-wave voltage. \(\varphi = \pi V_{B} /2V_{{\pi_{DC} }} { }\) is a static offset phase in the interference condition, \(V_{{\pi_{DC} }} = 4\;{\text{ V }}\) is the dc half-wave voltage, the dc-voltage \(V_{B} { }\) is a bias voltage to tune the modulator at any point on the transmission curve.

Most of our understanding of the earlier efforts on OEO concentrate on the numerous possible dynamical regimes for the large time delay case, which involves the adiabatic approximation35, which rely on the adiabatic approximation \(\left( {\frac{\tau }{{\tau_{D} }} \ll 1} \right)\). This condition simplifies the system dynamics described by Eq. (1) to a discrete logistic map equation:

$$x_{n + 1} = \beta x_{n} \left( {1 - x_{n} } \right),$$
(3)

The Eq. (3) is particularly useful for exploring different dynamical behaviors. Under this condition, the system output exhibits irregular oscillations when the laser power operates near its threshold, often referred to as low-frequency fluctuation. Conversely, under small time delay conditions where \(\frac{\tau }{{\tau_{D} }}\sim 1\), the adiabatic approximation condition is not satisfied. As a result, Eq. (1) cannot be simplified to Eq. (3). Analyzing the system's dynamics under small time delay requires a different approach, focusing on non-adiabatic effects and how the system behaves when delay is closer to the system's intrinsic time constant τ.

Results

Hopf and Neimark-Sacker bifurcations in time-domain

The system output states are illustrated generically via a bifurcation diagram, which plots the amplitude of the input after modulation as a function of feedback strength \(\beta\). Consequently, as the feedback strength \(\beta\) is progressively increased, Hopf bifurcations and Neimark-Sacker bifurcations (also referred to as secondary Hopf bifurcations or torus bifurcations41,42) sequentially emerge. The bifurcation diagram typically exhibits a wide range of rich dynamic regimes, labeled from A to H, for both large and small time delays (5 µs and 5 ns), as shown in Fig. 2a and b. These periodicity windows are more abundant compared to those of a discrete-time nonlinear delay oscillator, as discussed in Ref.35. Furthermore, there is agreement between the bifurcation diagrams obtained through numerical theory using Eq. (1) for both large and small time delays (illustrated in the insets of Fig. 2a and b) and the simulations conducted using commercial VPI software, which models the behavior of various components (shown in the main plots of Fig. 2).

Figure 2
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(a, b) Bifurcation diagram of large/small time delays; (a1, b1) Lyapunov exponent as a function of feedback strength \(\beta\) of large/small time delays, and typical dynamics regimes include: (a2, b2) steady state; (a3, b3) periodic oscillation; (a4, b4) laminar chaos (a5, b5) turbulent chaos regime;

Figure 2a1, b1 displays the Lyapunov exponent43 as a function of feedback strength \(\beta\), Chaotic behavior is characterized by a strictly positive Lyapunov exponent, whereas a negative or null Lyapunov exponent indicates non-chaotic behavior. The signs of the Lyapunov exponents seem to correspond to the dynamic states indicated by the bifurcation diagram. The open circles in Lyapunov exponent at Fig. 2a1, b1 mark the laser power corresponding to the four scenarios illustrated in (a2–a5) and (b2–-b5). For large time delay, the basic stationary steady-state solution in dynamic regime A is stable, independent of the varied control parameters of AMP gain (10 dB, 15 dB, 20 dB) and \(\tau_{D}\) (5 ns, 50 ns, 10 µs, 15 µs) as displayed in Fig. 2a2. As soon as the input intensity is sufficiently high and up to a critical point, the classical square-wave solution of the period \(2\tau_{D}\) appears after the Hopf-bifurcation of a stationary solution. It represents that the system operates as a square-wave generator and converts the CW light beam into an arbitrary periodic square-wave at regime C by adjusting time delay (5 μs,10 μs,15 μs cases plotted in Fig. 2a3). With further increasing feedback strength \(\beta\), this nice \(2\tau_{D}\)-period square-wave undergoes a sequence of Neimark-Sacker bifurcations and shows the period-doubling phenomenon in regime D (Fig. 2a4), this phenomenon can be regarded as a laminar chaos. The study of the progressive transformation of laminar chaos is discussed in supplementary material Figs. S1 and S2. Then a high value of the input power could yield a turbulent chaos regime E which have already been shown the potential application for chaotic optical communications with the waveform transition to a more complex pattern as illustrated in Fig. 2a5. Noticeably, for small time delay \(\tau /\tau_{D} = 3.787\) with \(\tau_{D}\) = 5 ns at low power branches, we found that their change of stability and transition to these bifurcations are very similar to the larger ones. The fundamental bifurcation diagram with a small time delay emerges with the same trend as the one with a large time delay, as plotted in Fig. 2b. When the incident power is low, the noticeable similarity in behavior between large and small time delay within the chaotic regime E is evident in the periodicity windows of the bifurcation diagram. Specifically, for time series analysis, one can distinguish many similar regimes for large and small delay, ranging from A to E. For instance, regimes such as (a2, b2) represent steady states, and (a5, b5) correspond to turbulent chaos regimes. However, it is important to note that the square-wave oscillation observed in Fig. 2a3 changes to a quasi-sinusoidal oscillation in Fig. 2b3 in time traces. This distinction highlights the nuanced differences in the system's behavior under varying time delays, despite the overall similarity in the bifurcation trends. Theoretically, the adiabatic approximation is not valid and does not apply to small time delay. This observation indicates that Hopf and Neimark-Sacker bifurcations are always possible, regardless of how small the time delay is. Furthermore, we discussed the changes in the time series with varying time delay values in Supplementary Material Figs. S3 and S4.

Dynamical behaviors in high-dimensional phase-space

In order to further demonstrate that small time delay can also exhibit limit-cycle oscillations, quasi-periodic oscillations, and chaos, similar to the large time delay, we have introduced the concept of phase space into the field of nonlinear time-delayed optoelectronic oscillators. The reconstructed phase space yields well-separated trajectories for the system dynamics of the OEO system, which provides a new perspective on the mechanism of the transition process. In such a high-dimensional state space, \(F\left[ {x\left( t \right)} \right]\) and \(F[\left( {x\left( {t - \tau_{D} } \right)} \right]\) in Eq. (1) are along the x and y axes, respectively. Figure 3a–c and the insets illustrate a series of intricate dynamic behaviors for small (5 ns) and large (5 μs) time delays, depicted by phase-space trajectory distributions in various dynamic regimes. The blue and red dots denote the measured data points with varying bifurcation parameter β. Figure 3d–f display the corresponding RF power spectrum up to 0.1 GHz for the 5 ns time delay corresponding to (a–c). Clearly, as shown in Fig. 3a and its inset, the phase-space orbit is a stable single-limit cycle when the control parameter β is 1.287. The orbit between large and small delays on the phase planes exhibits similar evolutionary trends, thereby confirming the similarity between small and large delays. This regime is usually optimal for a high-purity signal generator. Additionally, in Fig. 3d, we can see that the energy spectrum for the small delay has a fundamental frequency with a single peak. Furthermore, the presence of Neimark-Sacker bifurcations is triggered at the bifurcation value β = 2.446 and is evidenced by the frequency characteristics and phase-space properties, as shown in Fig. 3b. The appearance of a three-limit cycle with the torus oscillating near a stable periodic orbit, along with the power spectra in Fig. 3e displaying three energy peaks with side-bands, indicates the emergence of a series of incommensurate frequencies. It is interesting to stress that the evolutionary trends of trajectory, as shown in Fig. 3b for small delay, still keep pace with the large delay ones in the inset of Fig. 3b. Subsequently, as shown in Fig. 3c and its inset, the phase-space trajectories for both small and large delays exhibit similar chaotic behavior for the value of β equal to 3.089. Unlike the power spectrum of periodic signals, the spectrum in chaotic regimes, as shown in Fig. 3f, lacks a clear dominant frequency or harmonic components. The energy distribution is relatively uniform without any prominent frequency components, indicating the chaotic nature of the system.

Figure 3
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(a–c) Compared the projections between large and small delay on (\(F\left[ {x\left( t \right)} \right],F[\left( {x\left( {t - \tau_{D} } \right)} \right]\)) phase planes where \(\beta = 1.287,{ }2.446,{ }3.089,{ }4.377\), the \(V_{B}\) is fixed at 2 V under an RF gain constant G = 4 dB (d–f) shows corresponding power spectra obtained at small delay for different dynamic regimes respectively.

Periodic modulations for dynamical behaviors

Moreover, the OEO is based on the time-delayed resonators which exhibit lower loss than their electronic counterpart significantly, it is meaningful to discuss the nonlinear interactions of the system. Different from previous experiments focusing on the standard bias set to the half-transmission point30,31 or a maximum of the transmission curve35 of MZM at large time delay only, various static offset phases have been explored in our case at the different initial dynamic regime A-G for small time delay. Figure 4a shows the system function (blue dot line) \(x\left( t \right){ }\) and its slope distribution (red dot line), which was obtained by applying a bias voltage \(V_{B}\) to the radio frequency input of the MZM and then increasing \(V_{B}\) from 0 to 8 V (relative modulation phase \(\varphi = \pi V_{B} /2V_{{\pi_{DC} }}\) is range from [0, π]) to bias it at any point on the transmission curve. The occurrence of transition between numerous states of each dynamic regime is periodically modulated under different biases of MZM. To avoid redundancy, the transient process of other periods is not shown since it sustains the complete same behavior as shown in Fig. 4b. The transient phenomena shown in Fig. 4b are in well agreement with the numerical theory of the periodic bifurcation diagram (see Fig. S5 in the supplementary material), also make further discussion about the bifurcation diagram at regime C and time traces at different bias. We note that the common feature among these orbits is that the amplitude tendency (envelope) under various increasing \(V_{B}\) is consistent with the slope \(dx\left( t \right)/dt\) change of the nonlinear function. The envelope amplitude becomes larger and larger as the \(V_{B}\) increases from 0 to 2 V and reaches to the maximum value for \(V_{B}\) = 2 V, i.e. φ = π/4 at maximum slope regime. Subsequently, the amplitude began to gradually decrease until the slope minimum point. A further increased \(V_{B}\) leads to \(dx\left( t \right)/dt < 0\) which limits the nonlinearity of system until the minimum slope regimes at \(V_{B} = 6 V\), i.e. \(\varphi = 3\pi /4\), where the phase-space trajectory is fast decayed to a dot and the system is ineffective at any operating points and dynamic regimes. The transition from steady-state behavior to a single-limit cycle (periodic solution regime B), single-limit cycle to a three-limit cycle (regime C), and chaos to a two-limit cycle (regime F and G) are identified and shaded in Fig. 4b. Here we have only discussed the small time delay at \(\tau_{D} = 5 ns\), but we believe that there should also exist the same rich dynamics with other any time delays.

Figure 4
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(a) Nonlinear transmission function of the MZM and slopes distribution of the nonlinear function, (b) transients between various states at different initial dynamic regimes at different \(V_{B}\) from 0 to 8 V.

Dynamical regime B for high-speed reservoir computing

The dynamics properties render OEO a potential solution for the physical implementation of reservoir computing (RC), which is a highly efficient computing system for processing time-dependent data. An opto-electronic RC architecture is depicted in Fig. 5a. This reservoir is formed by segmenting the OEO’s time delay loop into N intervals and applying time multiplexing. The input signal is sampled and maintained for a time delay \(\tau_{D}\), which corresponds to the delay in the feedback loop. At any instance, the input signal is multiplied by a mask, creating a temporal input stream. This stream is combined with the state \(x_{{t_{i} }}\) of the reservoir and then directed to the nonlinear node \(F\left( x \right)\). The transient dynamic response of the reservoir is extracted by an output layer, which consists of linear weighted sums of the reservoir node states to perform a training procedure on a computer for each input task. Reservoir computing should satisfy several key properties24: the first is to transform nonlinearly the input signal into a high- dimensional state space, and the second is the short-term memory. For memory capacity, we set the system in a typical dynamics regime B of small delay \(\tau_{D} = 5 \;{\text{ns}}\) and input two different kinds of electrical signals (sine-wave signals at frequency of 12 GHz and square-wave signals at period of T = 1 μs) into the loop. It is observed a quite interesting and unreferenced phenomenon, namely, the emergence of a hybrid regime where dynamics regime B is superimposed onto the input signal periodicity, as shown in Fig. 5b1 and b2. These indicate that time-delayed recurrent loops have a potential capacity as memory carriers. In addition, we input a sequence of 6-bit binary-coded signal (i.e., 101010) in typical dynamic regime C of large and small time-delayed cases then record the time traces. It appears that input data streams are modulated in the amplitude of periodic square-wave oscillation and sinusoidal-wave oscillation with a slowly varying envelope. Remarkably, these two types of oscillations are damped just after their birth in Fig. 5b3 and b4, and the attenuation rate is more rapid for the small time delay case than for the large one. The system gradually forgets previous inputs as new inputs come in, and this regenerative memory output means a clear system of “fading memory”. Actually, dynamics regime B has the strongest memory ability than other regimes (not shown here), and it is also the optimal regime for reservoir computing to be discussed next.

Figure 5
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(a) Schematic diagram of the reservoir computing, (b) memory carriers in the delayed-feedback system, (c, d) signal classification task for both large and small-time scales at different dynamic regimes.

We evaluate the performance of the OEO RC with small time delay on three different tasks. For a benchmark classification task of differentiating between rectangular and triangular waveforms. Further details on this task are given in the Supplementary Material S5. We chose 1500 values of input signal \(u\left( n \right)\) for training and each value of the time continuous input stream was held during one time delay interval by sample and hold procedure, then we obtained a continuous function \(u\left( t \right)\):

$$u\left( t \right) = u\left( n \right), n\tau^{\prime} \le t < \left( {n + 1} \right)\tau^{\prime}$$
(4)

We divided every time delay interval \(\tau^{\prime}\) into N = 50 segments with duration \(\theta = \frac{{\tau_{D} }}{N} = 20{ }ps\) for each one. Here N is the number of virtual nodes. The input mask \(m_{i}\) was chosen from the discrete set [0, 1] randomly. Subsequently, we carried out the multiplication \(m_{i} u\left( n \right)\) which breaks the symmetry of each virtual node to enrich system dynamics. The time evolution equation can be expressed by the following discretized state equations25,29:

$$x_{i} \left( n \right) = \left\{ {\begin{array}{*{20}c} {\sin (\alpha x_{i - k} \left( {n - 1} \right) + \beta m_{i} u\left( n \right) + \varphi ) k \le i < N} \\ {\sin (\alpha x_{N + 1 - k} \left( {n - 2} \right) + \beta m_{i} u\left( n \right) + \varphi ) 0 \le i < k} \\ \end{array} } \right.$$
(5)

Here \(\tau^{\prime}\) is typically set to \(\tau^{\prime} - \tau_{D} = k\theta\) that makes the system yield rich dynamics by coupling the discretized variables \(x_{i} \left( n \right)\) to each other in the unsynchronized regime and we chose \(k = 1, \alpha = 0.2\). Then we obtained the reservoir response state which was collected from the photodetector during the training period. The aim is to minimize the mean square error \(W_{out} W_{res} - y_{t\arg et}^{2}\) to find the optimal weight values \(W_{out}\) that can be determined by taking Tikhonov regularization \(W_{out} = y_{t\arg et} W_{res}^{T} (W_{res} W_{res}^{T} + \lambda I)^{ - 1}\). λ = 1e−7 represents the regularization coefficient and I represent an identity matrix. Thereafter, the approximation of the target function \(\hat{y}\left( n \right) = W_{out} W_{res}\) can be calculated through this offline learning approach. It is important to note that the output refresh rate \(1/\tau_{D}\) equal approximately to \(0.2\) MHz for the large time delay and 0.2 GHz for the small time delay can be adopted as the processing speed of RC, which thus reveals a computing-efficiency improvement of 1000 times at the small time delay compared to the large time delay. Figure 5c and d show the final \(\hat{y}\left( n \right)\) of the system at both large and small time delays, from which regime B is identified as an optimal operating point because of its strongest memory ability for the signal classification task. However, state space in the oscillatory regimes C and D exhibits fluctuation which cannot be mapped onto a target value. In addition, high separation properties at chaotic regime E make the system sensitive to small input variations, and therefore it is difficult to achieve the linear algorithm at this classification task. Apart from the benchmark classification task, our small time delay RC architecture demonstrates its classification capabilities in a communication system and its ability to predict future values in a temporal sequence through tasks such as channel equalization25 (trained over a sequence of 3000 inputs and tested over another sequence of 6000 inputs) and Mackey–Glass time series prediction44,45 (using sequences of 5000 symbols for training and predicting the next 1000 symbols). Further details on these two tasks are given in the supplementary information S6 and S7. Figure 6 illustrate the performance for different dynamic regimes. In Fig. 6a, the symbol error rate (SER) for nonlinear channel equalization at small time scales is shown. We achieved an error rate of 9.7e−5 at 28 dB and 1.0e−4 at 32 dB of signal-to-noise ratio in dynamic regime B, which is lower than in other dynamic regimes. Figure 6b presents the prediction output of the trained small time delay OEO (green) overlaid with the correct continuation (blue) for Mackey–Glass time series. The associated normalized root mean square error (NRMSE) for Mackey–Glass time series prediction is depicted in Fig. 6c, with the lowest value being 0.027 in dynamic regime B, which is also lower than in other dynamic regimes. These results highlight the superior performance of the system in dynamic regime B, indicating better prediction accuracy and lower error rates compared to other regimes. All the tasks we tested exhibited excellent performance under small time delay, while ensuring a high rate of 0.2 GHz. Finally, we summarized representative works on optoelectronic RC in Table 1. To the best of our knowledge, the proposed RC with the small time delay OEO topology has the highest computing efficiency of 0.2 GHz signal rate (Table 2).

Figure 6
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(a) Results for nonlinear channel equalization at \(\tau_{D} = 5\;\;{\text{ns}},\) \(\varphi = \pi /4,\) G = 4 dB. (b) Prediction output of the trained small time delay OEO and correct continuation for Mackey–Glass time series in dynamic regime B. (c) NRMSE for Mackey–Glass time series prediction in each dynamic regime at \(\tau_{D} = 5\;\;{\text{ns}},\) \(\varphi = { }\pi /4,\) G = 4 dB.

Table 1 Comparison of representative works on Reservoir Computing.
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Table 2 Comparison of representative works on optoelectronic pulse generators.
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Dynamical regime C for optoelectronic pulse generator

The multiple time delay complex dynamics of OEO might also offer attractive high-performance features at the generation of microwave oscillations or optical pulse for another engineering applications like radar sources. Different from the former time-lens soliton-assisted compression of the ultralow jitter optical pulses48,49,50,51,52,53,54, we propose a simpler, more stable, and tunable optical pulse source using single optoelectronic oscillators as shown in Fig. 7a. By gradually changing the time delay of the input electrical pulse \(\tau_{P}\) in the typical dynamics regime C, instead of utilizing an additional phase modulator for pulse compression, ultralow jitter optical pulses are generated, as depicted in Fig. 7c and d, when a single electrical pulse is injected into the loop. The variation of time delay in input electrical pulses from 1 to 50 ns and from 1 to 50 ps is illustrated in Fig. 7b. When the time delay \(\tau_{P}\) and the amplitude of the input single electrical pulse are small, it results in a perturbation decay of the generated short and regularly spaced optical pulses in the optical domain. Increasing \(\tau_{P}\) (from 1 to 50 ns in the case of large time delay and from 1 to 50 ps in the case of small delay) and input pulse amplitude leads to \(\tau_{D}\)-periodic transient pulsing behavior, where the pulse width can be expanded while the total period \(\tau_{D}\) remains constant in a round-trip through the loop for \(\tau_{P}\) ≤ \(\tau\). Here, \(\tau\) represents the response times of the system components. The expansion of pulses for different \(\tau_{P}\) is depicted in Fig. 7e, f within the specific range of \(\tau\) ≤ \(\tau_{P}\) ≤ \(\tau_{D}\). The oscillations gradually transition from sinusoidal to square waves and exhibit a period close to one optical time delay \(\tau_{D}\). Moreover, a further increase in the input time delay \(\tau_{P}\) could monotonically alter the pseudo-duty cycle of the output waveform, leading to irregular square wave oscillations in the system output. For instance, \(\tau_{P}\) = 1 us and 2 us are illustrated in Fig. 7e and f, with the duty ratio being 0.2 and 0.4 at one optical time delay \(\tau_{D}\), respectively. This indicates that different plateau lengths of periodic square-wave oscillations are adjustable for both the optical domain and the electrical domain. Additionally, we have summarized and compared the characteristics of previous optoelectronic pulse generators. Our solution stands out for its novelty in terms of an adjustable output pulse duty cycle.

Figure 7
Figure 7
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(a) Adjustable optoelectronic pulse generator. (b) Time delayed input single electrical pulse of both ns and ps time scale. (c, d) Generated ultralow jitter optical pulses of optoelectronic pulse generator at different time delayed input single electrical pulse from 1 to 50 ns and 1 to 50 ps. (e) and (f) Tunable pseudo duty cycle at fixed periodic oscillations.

Conclusion

In this paper, we analyzed the complex dynamic behavior by changing different bifurcation parameters at different dynamic regimes of an optoelectronic time delayed oscillator. It is found that Hopf and Neimark-Sacker bifurcation can appear at small time delay, the changes of bifurcation depend on the feedback laser power, and system dynamic behavior is periodically controllable through the system’s nonlinear function. High-dimensional state space reveals the time evolution between various periodicity and full chaos and has important implications for understanding the stability of general time delay systems. Moreover, we have demonstrated that the typical dynamics regime B of the OEO system is a potentially effective regime for high-speed reservoir computing. Additionally, in typical dynamics regime C, self-sustained periodic oscillation pulsing and different pseudo-duty cycles are also possible while the time delay of the input single electrical pulse (\(\tau_{P}\)), the response time of the system components (\(\tau\)), and the optical time delay (\(\tau_{D}\)) are within the specific range for ultra-high spectral purity level generation. Furthermore, it needs to be emphasized that regime E, under our small time delay condition, also holds promising applications for fast chaos-based communication.