Controllable non-uniformly distributed spiking cluster generation in broadband optoelectronic oscillator

Li Su; Li Su; Huan Tian; Huan Tian; Ziwei Xu; Ziwei Xu; Lingjie Zhang; Lingjie Zhang; Zhen Zeng; Zhen Zeng; Yaowen Zhang; Yaowen Zhang; Zhiyao Zhang; Zhiyao Zhang; Yali Zhang; Yali Zhang; Shangjian Zhang; Shangjian Zhang; Heping Li; Heping Li; Yong Liu; Yong Liu

doi:10.1364/OE.520246

1. Introduction

Spiking is a unique phenomenon in excitable systems, which has been observed in a variety of scientific fields, such as excitability in neurology [1,2], excitable pulsation in a laser system [3–5], and mixed-mode oscillation in chemistry [6–8]. The spiking response characteristic in these systems is similar to that of a biological neuron [9]. For example, there is a unique nonlinear threshold excitation effect that exhibits “all” or “none” response under external excitation. Additionally, the generated spiking waveform is asymmetric, which is with a refractory period after a brief discharge process [10,11]. There are abundant nonlinear dynamic behaviors in the spiking formation process, which not only attract extensive theoretical research, but also can be utilized in spiking neural networks (SNNs) [12–17].

Broadband optoelectronic oscillators (OEOs) are hybrid nonlinear systems composed of both optical and electrical links, which are widely used to generate various microwave signals [18–21]. In a broadband OEO cavity, the nonlinearity, the time-delay feedback and the filtering operations lead to complex dynamic behaviors, which can be described by using the modified-Ikeda equations [22]. In the past years, rich dynamic phenomena have been investigated in broadband OEOs such as limit-cycles [23], chaotic breathers [24–27] and hyperchaos [28–29]. Based on the above phenomena, a wide range of applications have also been explored, such as chaotic encryption communication [30], random number generation [31] and high-resolution radar ranging [32]. Spiking oscillation is also one of the dynamic behaviors in broadband OEOs, which can be well controlled by the externally-injected perturbation signals, such as random noise [33] and ultra-short pulse [34]. If a square-wave signal is injected into the broadband OEO cavity, switchable spiking and spiking cluster oscillation can be achieved under different gains or injection strengths [35]. Through temporally encoding the perturbation signal, a mapping between the spiking position and the waveform can be achieved, which can be used for spiking encoding [36], regenerative memory [37] and building up recurrent SNNs [38] with a higher information processing speed and a lower noise. However, in [35], the generated spiking pulses in a spiking cluster are uniformly distributed, which is not beneficial for achieving flexible spiking encoding and information processing.

In this paper, an approach to achieve non-uniformly distributed spiking cluster generation is proposed and demonstrated based on an externally-triggered broadband OEO. The peak voltage of each spiking pulse and the temporal interval between any neighboring spiking pulses in a spiking cluster are linearly proportional to the instantaneous amplitude of the externally-injected trigger signal, which can be used for spiking encoding and reservoir computing. Theoretical analysis, numerical simulation, and experiment are carried out to verify the feasibility of the proposed scheme, where the results are mutually consistent with each other.

2. Operation principle and theoretical model

Figure 1 shows the schematic diagram of the broadband OEO to achieve externally-controllable non-uniformly distributed spiking cluster generation. The continuous-wave (CW) light from a laser diode (LD) is modulated in a dual-drive Mach-Zehnder modulator (DDMZM), where one radio-frequency (RF) port of the DDMZM is connected to a signal source to realize external triggering, and the other RF port is used to achieve signal feedback in the OEO cavity. The modulated light propagates through a spool of single-mode fiber (SMF) before entering a high-speed photodetector (PD). Then, the electrical signal from the PD is amplified by using a broadband low-noise amplifier (LNA), and is divided into two parts via an electrical splitter (ES). One part of the generated signal is fed back to the DDMZM to close the OEO loop, and the other part is recorded by using a high-speed real-time oscilloscope (OSC). Spiking pulse generation is achieved through nonlinear bistable effect in the OEO cavity, which is induced by the combined action of the high net gain, the nonlinear effect induced by the electro-optic and photoelectric conversion, and the linear effects including time-delay feedback and bandpass filtering [35]. In this scheme, the DDMZM is biased near its minimum transmission point (MITP) to enhance the nonlinear effect in the cavity. In addition, the LNA, which spans multiple frequency-doubled range, plays the role of a broadband passband filter (BPF). This broadband passband filtering effect is beneficial for manifesting the rich nonlinear effect in the OEO cavity. Hence, numerous modes can be excited in the OEO cavity, which is the kernel for spiking pulse generation. Additionally, a variable optical attenuator (VOA) is employed to keep the net gain of the OEO at a proper high level. In order to achieve a stable spiking cluster oscillation, the externally-injected signal is set to be with a repetition frequency equal to an integer multiple of the free spectral range (FSR) in the OEO cavity. Most importantly, the voltage of the externally-injected signal can be specially designed to adjust the spiking distribution in the high-voltage-level duration. It should be pointed out that the bistable effect provides a determined oscillation threshold for the externally-injected signal, where the noise in the OEO cavity cannot reach the bifurcation threshold. Hence, local stability is maintained.

Fig. 1. Schematic diagram of the broadband OEO to generate spiking cluster oscillation. (a) Experimental setup. LD: laser diode; DDMZM: dual-drive Mach-Zehnder modulator; VOA: variable optical attenuator; SMF: single-mode fiber; PD: photodetector; LNA: low-noise amplifier; ES: electrical splitter; OSC: oscilloscope. (b) Temporal waveform of the generated spiking cluster oscillation under triggering by using square-wave signals with different voltages. (c) Phase portrait of the generated spiking cluster (blue solid line) and the nullclines under different trigger signal voltages (dashed line). The black arrows represent the evolution direction of the phase trajectory. A0 and A-D are the characteristic points corresponding to those in (b).

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Mathematically, the dynamics process in the broadband OEO can be described by the following integral-differential delay equation (iDDE) [35]

(1)$$\left( {1 + \frac{\tau }{\theta }} \right){x}(t )+ \tau \frac{{d{x}(t )}}{{dt}} + \frac{1}{\theta }\int_{{t_0}}^t {{x}(t )dt} \textrm{ } = \textrm{ }\beta\, {\cos ^2}({{x}({t - T} )+ A(t )+ \varphi } )$$

where x(t)=πV(t)/2V_πRF and A(t)=πI(t)/2V_πRF are the dimensionless variables corresponding to the feedback signal voltage V(t) and the externally-applied trigger signal voltage I(t) at two RF ports of the DDMZM, respectively. φ=πV_bias/2V_πDC is the phase offset induced by the DC bias voltage V_bias. Thereinto, V_πDC and V_πRF are the DC half-wave voltage and the RF half-wave voltage of the DDMZM, respectively. T is the round-trip delay time in the OEO cavity. β is the gain coefficient of the OEO loop, which is determined by the LNA, the PD, the SMF, the VOA and the EC. θ=1/2πf_L and τ=1/2πf_H represent the characteristic times of the low cut-off frequency f_L and the high cut-off frequency f_H in the OEO cavity, respectively. Through introducing an integral variable y(t) and moving the fixed point to the origin, Eq. (1) can be rewritten as

(2)$$\begin{aligned} \frac{{d{x}(t )}}{{dt}} &={-} \left( {\frac{1}{\tau } + \frac{1}{\theta }} \right){x}(t )- \frac{1}{\tau }\textrm{y}(t )+ \frac{\beta }{\tau }[{{{\cos }^2}({{x}({t - T} )+ A(t )+ \varphi } )- {{\cos }^2}\varphi } ]\\ \frac{{dy(t )}}{{dt}} &= \frac{1}{\theta }x(t )\end{aligned}$$

Then, through setting the partial derivative to be 0 and eliminating the time delay term, the nullclines corresponding to Eq. (2) are obtained, which can be used to clearly describe the dynamic process of the spiking clusters in the phase space. The nullclines are given as

(3)$$\left\{ \begin{array}{l} \textrm{y}(\textrm{t} )={-} {x}(t )+ \beta [{{{\cos }^2}({{x}(t )+ A(\textrm{t} )+ \varphi } )- {{\cos }^2}\varphi } ]\\ x(\textrm{t} )= 0 \end{array} \right.$$

where an approximation 1+τ/θ≈1 holds for f_L≪f_H.

When the externally-injected square-wave signals are with different instantaneous voltages, the phase trajectory of the stable spiking cluster and the nullclines are shown in Fig. 1(c). The locations of the characteristic points A₀ and A-D in the time domain and in the phase space are marked in Fig. 1(b) and (c), respectively. Point A₀ corresponds to the fixed point when there is no trigger signal voltage, and the initial spiking pulse will start at point A₀. After the initial spiking pulse, the trajectory starts at point A and horizontally moves to point B, representing the rising edge of the pulse. Then, the trajectory moves to point C around the nullcline, and subsequently moves horizontally to point D, representing the falling edge of the pulse. Finally, the trajectory moves back to point A following the nullcline. This closed trajectory indicates that a stable dynamic evolution of the spiking cluster oscillation is established. When the externally-injected trigger signal voltage varies, the nullcline shifts and the locations of these four characteristic points change accordingly.

Assisted with the iDDE in Eq. (2) and the nullclines in Eq. (3), the distribution characteristics of the spiking cluster, such as the peak voltages and the intervals of the spiking pulses, can be clearly presented. For simplicity, the peak voltages represent the maximum peak voltages of the spiking pulses, which corresponds to the x value at point B in Fig. 1(c). Compared to the duration of the entire spiking pulse, the rising edge (point A to point B) is negligible. Hence, there is an approximation of y(A)≈y(B). It should be noted that point A is at the point of local minimum on the left side of the nullcline. Thus, the x value of point A can be obtained as

(4)$$x(A )= \frac{1}{2}\left( {\pi - \arcsin \left( { - \frac{1}{\beta }} \right)} \right) - A(t )- \varphi$$

Through substituting x(A) into Eq. (3), the y values of points A and B are calculated as

(5)$$\begin{aligned} y(A )\approx y(B )& =A(t )- \frac{1}{2}\left( {\pi - \arcsin \left( { - \frac{1}{\beta }} \right)} \right) + \varphi - \frac{1}{2}\sqrt {{\beta ^2} - 1} - \frac{\beta }{2}\cos ({2\varphi } )\\ & =A(t )+ {C_{\beta ,\varphi }} \end{aligned}$$

where C_β,φ is a variable determined only by β and φ as

(6)$${C_{\beta ,\varphi }} ={-} \frac{1}{2}\left( {\pi - \arcsin \left( { - \frac{1}{\beta }} \right)} \right) + \varphi - \frac{1}{2}\sqrt {{\beta ^2} - 1} - \frac{\beta }{2}\cos ({2\varphi } )$$

Therefore, when the parameters β and φ in the OEO are fixed, C_β,φ can be considered as a constant. The y values of points C and D can be calculated by using the same method as

(7)$$\begin{aligned}y(C )\approx y(D )& =- \frac{1}{2}\left( {2\pi + \arcsin \left( { - \frac{1}{\beta }} \right)} \right) + A(t )+ \varphi + \frac{1}{2}\sqrt {{\beta ^2} - 1} - \frac{\beta }{2}\cos ({2\varphi } )\\ & =A(t )+ {C_{\beta ,\varphi }} + {y_0} \end{aligned}$$

where y₀ refers to the difference between the maximum and the minimum values of y in a spiking pulse

(8)$${y_0} = y(C )- y(A )= \arcsin \left( {\frac{1}{\beta }} \right) + \sqrt {{\beta ^2} - 1} - \frac{1}{2}\pi$$

Finally, the x value at point B can be obtained by solving Eq. (3) in conjunction with Eq. (5) as

(9)$${x}(B )+ A(t )+ {C_{\beta ,\varphi }} = \beta [{{{\cos }^2}({{x}(B )+ A(t )+ \varphi } )- {{\cos }^2}\varphi } ]$$

Considering x(B)+A(t) as an overall solution, it is a fixed value when the parameters β and φ are constant. Therefore, it can be deduced that there is a negative linear relationship between the peak voltages of the spiking pulses and the external trigger intensity. Correspondingly, the envelope of the spiking cluster will have a shape opposite to the externally-applied trigger signal.

The spiking interval T_w is evaluated as the time interval between the peak voltages of two adjacent spiking pulses. The duration of the rising and falling edges of the spiking pulse is negligible compared to the entire spiking interval T_w. According to Eq. (2), T_w can be given by

(10)$${T_w} = \int\limits_{t(B )}^{t(C )} {dt} + \int\limits_{t(D )}^{t(A )} {dt = } \int\limits_{y(B )}^{y(C )} {\frac{\theta }{x}dy} + \int\limits_{y(D )}^{y(A )} {\frac{\theta }{x}dy}$$

where A, B, C, and D correspond to the points shown in Fig. 1. In order to accomplish this integral in Eq. (10), Eq. (3) needs to be approximated as a linear equation, since the original equation is unable to derive the inverse function. Thus, the left and the right sides of the nullcline can be described as

(11)$$2y(t )={-} 2x(t )+ {a_i}({x(t )+ A(t )} )+ {b_i}$$

where a_i and b_i are the coefficients of the linear equation, and i = 1,2 correspond to the left side and the right side, respectively. a_i and b_i can be obtained by fitting a linear polynomial to the nullcline in the numerical simulation. Eventually, the analytical spike interval T_w can be derived as

(12)$$\begin{aligned} {T_w} &= \int\limits_{y(D )}^{y(A )} {\frac{{\theta ({{a_1} - 2} )}}{{2y - {a_1}A(t )- {b_1}}}dy} + \int\limits_{y(B )}^{y(C )} {\frac{{\theta ({{a_2} - 2} )}}{{2y - {a_2}A(t )- {b_2}}}dy} \\& = \theta \left[ {\frac{{{a_1} - 2}}{2}\ln \left( {1 + \frac{{2{y_0}}}{{2A(t )- {a_1}A(t )- {b_1} + 2{C_{\beta ,\varphi }}}}} \right) + \frac{{{a_2} - 2}}{2}\ln \left( {1 + \frac{{2{y_0}}}{{2A(t )- {a_2}A(t )- {b_2} + 2{C_{\beta ,\varphi }}}}} \right)} \right] \end{aligned}$$

It should be noted that the coefficients a_i and b_i are only related to the parameters β and φ. Therefore, the same analytical model can be used to study the OEO system with fixed open-loop gain and DC bias voltage but with different externally-injected trigger signals. It can be seen from Eq. (12) that the spiking interval is proportional to the characteristic time of the low cut-off frequency, i.e., θ. Thus, it is possible to achieve spiking interval modulation through changing the low cut-off frequency. Nevertheless, the cut-off frequency of a certain OEO cavity is hard to control. Moreover, changing the cut-off frequency results in a change of the gain threshold, which would fail to obtain spiking pulses. Therefore, through varying the voltage of the externally-injected trigger signal, the spiking interval can be regulated more effortlessly, which exhibits a negative correlation with the trigger intensity. It is worth noting that the voltage of the externally-injected triggering signal should be set in a certain range, which is determined by the oscillation threshold and the linear fitting range of Eq. (12). If the externally-injected voltage is below the oscillation threshold, stable spiking oscillation will not be established. On the contrary, if the externally-injected voltage is above the linear fitting range, the negative linear relationship between the intervals of the spiking pulses and the voltage of the externally-injected triggering signal is no longer in force.

Finally, it should be pointed out that the spectrum of the generated spiking cluster sequence exhibits a comb-like shape, where the comb tooth interval is inversely proportional to the loop delay. For the generated uniformly distributed spiking cluster, there is another larger spectral periodicity, which is determined by the temporal repetition period between neighboring spiking pulses in a spiking cluster duration [35]. For the generated non-uniformly distributed spiking cluster sequence, the large spectral periodicity cannot be observed any more since the periodic characteristic in a spiking cluster duration is broken. In addition, most of the energy is located in the low-frequency range, since the broadband OEO is operating at large timescales. Through increasing the low cut-off frequency and the loop bandwidth, the spiking interval is in a smaller time scale. On this condition, the frequency range is broadened.

3. Simulation results

The iDDE in Eq. (2) is numerically solved by using a standard constant step size Runge-Kutta method of the fourth order to simulate this nonlinear OEO system. In the simulation, the loop delay is set to be T = 1.018 µs (corresponding to a spool of SMF with a length of 200 m). The low cut-off frequency and the high cut-off frequency of the bandpass filter are set to be f_L= 14.89 MHz and f_H= 8.03 GHz, respectively. The DDMZM is with a DC half-wave voltage of V_πDC=4 V and an RF half-wave voltage of V_πRF=4 V. The phase offset induced by the DC bias voltage is set to be φ=0.55π, which makes the DDMZM work near its MITP. The gain coefficient is set to be β=1.93. The coefficients a₁, a₂, b₁, and b₂ in Eq. (12) can be obtained by linear fitting. Under the condition of β=1.93 and φ=0.55π, the linearization coefficients can be calculated as a₁=-0.67, a₂= 1.17, b₁=-0.05 and b₂= 2.15.

In order to demonstrate the effect of externally-injected trigger signal voltage on the distribution characteristics of the spiking cluster, two square-wave signals with high voltage levels of 0.64 V and 1.02 V, respectively, are applied to the DDMZM, where both signals are with a period of 1.018 µs and a duty cycle of 50%. Figure 2(a) and (d) show the temporal waveforms under stable spiking cluster oscillation. It is worth noting that each spiking cluster has a higher initial spiking pulse which is excited by the short rising edges of the trigger waveform. In this work, only the characteristics of the spiking pulses under continuous excitation are considered. It can be seen from the detailed views in Fig. 2(b) and (e) that the peak voltages of the spiking clusters are 3.60 V and 3.22 V, where the voltage difference is equal to the difference between the trigger voltages. In addition, the spiking intervals under the two trigger voltages are 18.33 ns and 15.36 ns, respectively. The spiking interval decreases as the trigger voltage increases, which agrees with the theoretical analysis. Figure 2(c) and (f) represent the phase trajectories of the spiking cluster oscillation under different trigger signal voltages, which fit in with the theoretical analysis.

Fig. 2. Simulation results of stable spiking cluster oscillation under square-wave signal injection. (a) and (d) are the temporal waveforms of the generated spiking clusters (blue line) and the externally-injected square-wave signals (red line). (b) and (e) are the temporal waveform details of the spiking cluster oscillation (blue line) and the externally-injected square-wave signals (red line). (c) and (f) are the phase portraits of the generated spiking clusters (blue solid line) and the nullclines under external triggering (red dashed line). The black arrows represent the evolution direction of the phase trajectory. A₀ and A-D are the characteristic points.

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Next, three types of externally-injected trigger waveforms are applied to the DDMZM. Figure 3 shows the simulation results of stable spiking cluster oscillation under different trigger signal injections. Thereinto, Fig. 3(d)-(f) exhibits the waveform details corresponding to Fig. 3(a)-(c). As shown in Fig. 3(a), a trapezoidal waveform with a period of 1.018 µs, a duty cycle of 60% and a high voltage level from 1 V to 0.5 V is applied to the DDMZM. It can be seen from Fig. 3(d) that the peak voltage of the spiking pulse increases linearly as the trigger voltage decreases. Meanwhile, from a qualitative perspective, the spiking interval increases as the trigger voltage decreases. As shown in Fig. 3(b), a parabola waveform with a period of 1.018 µs, a duty cycle of 50% and a high voltage level from 0.75 V to 0 V is applied to the DDMZM. Similarly, both the peak voltage and the interval of the spiking pulse nonlinearly increase as the trigger voltage decreases. In addition, a half-sinusoidal waveform with a period of 1.018 µs, a duty cycle of 50% and a peak voltage of 1.5 V is injected into the OEO cavity as shown in Fig. 3(c). As can be seen in Fig. 3(f), the peak voltage and the interval of the spiking pulse are at minimum values when the trigger signal voltage reaches the maximum. Therefore, the peak voltages of the spiking pulses in a cluster are modulated by the voltage of the externally-injected trigger waveforms according to a negative linear mapping, which is manifested by the similarity of the envelope shapes of the spiking clusters to the externally-injected trigger waveforms. This is consistent with the theoretical analysis in Eq. (9).

Fig. 3. Simulation results of stable spiking cluster oscillation under different external trigger signal injection. (a)-(c) are the temporal waveforms of the generated spiking clusters (blue line) and the externally-injected trigger signals (trapezoidal waveform, parabola waveform, and half-sinusoidal waveform, respectively. Red line). (d)-(f) are the temporal waveform details of the spiking cluster oscillation (blue line) and the externally-injected trigger signals (red line).

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Figure 4(a)-(c) illustrate the dimensionless temporal waveforms and their corresponding integral curves of the spiking clusters under triggering by using the above-mentioned waveforms. It can be seen that the envelope shapes of the integral curves are similar to the trigger waveforms. Based on the values of x and y, the associated phase portraits of the spiking clusters are obtained as shown in Fig. 4(d)-(f). It is worth noting that the red dashed line and the yellow dashed line represent the nullclines under the maximum trigger signal voltage and the minimum trigger signal voltage, respectively. In a single spiking cluster duration, the phase trajectories of the spiking pulses are shifted following the movement of the nullcline under various trigger voltages.

Fig. 4. Simulation results of the dimensionless variables x and y. (a)-(c) are the dimensionless temporal waveforms (blue line) and the corresponding integral curves (vermilion line) of the spiking cluster. (d)-(f) are the associated phase portraits of the spiking cluster (blue solid line) and the nullclines under the maximum external trigger (yellow dashed line) and the minimum external trigger (red dashed line). The black arrows represent the evolution direction of phase trajectory. A₀, A₁-D₁ and A₂-D₂ are the characteristic points.

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Finally, the spiking interval in a single cluster duration is quantitatively investigated. Figure 5 shows the simulation results of the spiking intervals under triggering by using the same externally-injected signals as those in Fig. 2 to Fig. 4, where the peak voltage of the square wave signal is set to be 0.75 V. The blue dotted line represents the simulated spiking intervals, and the purple solid line represents the theoretical values of the spiking intervals. The simulated spiking intervals are obtained by taking the time interval between two adjacent peak voltage points of spiking pulses, and the theoretical values of the spiking intervals are calculated by using Eq. (12). It can be seen that the theoretical values are consistent with the simulation results in Fig. 5. The maximum deviation (about 0.1 ns) appears in the lower right of the curve in Fig. 5(d). This is due to the high voltage of the externally-injected trigger signal, which exceeds the range of linearizing the nullcline and leads to a larger calculation error in Eq. (12). In addition, it is worth noting that the spiking interval has a negative correlation with the external trigger intensity. Thus, the spiking intervals can be modulated by changing the external trigger voltage.

Fig. 5. Simulated spiking intervals in a cluster duration under triggering of (a) a square waveform, (b) a trapezoidal waveform, (c) a parabola waveform, and (d) a half-sinusoidal waveform. The blue dotted line and the purple solid line represent the simulated spiking intervals and the theoretical values of the spiking intervals, respectively.

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4. Experiment results

An experiment is carried out to verify the theoretical analysis and the simulation results. In the optical path, a home-made LD is used to generate the CW light with a center wavelength of 1559.904 nm and an output power of 15.55 dBm. The employed DDMZM (Sumitomo T.DKH1.5-10PD-ADC) is with a DC half-wave voltage of 2.9 V, of which the MITP voltage is measured to be 3.49 V. The DC bias voltage is set to be 3.1 V, ensuring that the DDMZM works near its MITP. A section of SMF with a length of 200 m is applied to provide a loop delay of 1.018 µs. A high-speed PD (HP 11982A) is employed to achieve photoelectric conversion in the OEO cavity. In the electrical path, an LNA (GT-HLNA-0022G) with an operation frequency range from 34.25 MHz to 22 GHz and a small-signal gain of 28 dB is used to amplify the electrical signal as well as to achieve bandpass filtering. A 3-dB electrical power divider (GTPD-COMB50G) is used to provide the feedback signal in the OEO cavity and output the generated spiking cluster. An arbitrary waveform generator (AWG, RIGOL DG5352) with a maximum output frequency of 350 MHz and a sampling rate of 1 GSa/s is used to generate the externally-injected trigger signals. The temporal waveforms of the generated spiking clusters are recorded by using a high-speed real-time oscilloscope (Tektronix DPO75002SX) with a sampling rate of 50 GSa/s.

Firstly, stable spiking cluster oscillation under square-wave signal injection with different voltages is observed. The injected square-wave signals are set to be with a period of 1.018 µs, a duration of 50% and high voltage levels of 0.40 V and 0.60 V, respectively. Figure 6 shows the experimental results. It can be seen from Fig. 6(b) and (e) that the peak voltages of the generated spiking clusters are 3.09 V and 2.85 V, where the peak voltage difference is almost equal to the voltage difference between the two trigger signals. In addition, the spiking intervals under triggering by the two square-wave signals are measured to be 8.61 ns and 7.13 ns, respectively, indicating that the spiking interval decreases as the trigger voltage increases. The phase portraits of the generated spiking clusters are also obtained, as shown in Fig. 6(c) and (f), which are consistent with the simulation results in Fig. 2(c) and (f), respectively.

Fig. 6. Experimental results of stable spiking cluster oscillation under square-wave signal injection. (a) and (d) are the temporal waveforms of the generated spiking clusters (blue line) and the externally-injected square-wave signals (red line). (b) and (e) are the temporal waveform details of the spiking cluster oscillation (blue line) and the externally-injected square-wave signals (red line). (c) and (f) are the phase portraits of the generated spiking clusters (blue solid line) and the nullclines under external triggering (red dashed line). The black arrows represent the evolution direction of the phase trajectory. A₀ and A-D are the characteristic points.

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Next, the spiking pulse distribution in a single cluster duration under external injection by using different types of trigger waveforms is evaluated. Figure 7 shows the experimental results of stable spiking cluster oscillation under triggering by using a trapezoidal waveform, a parabola waveform and a half-sinusoidal waveform. It can be seen that both the peak-voltage envelop of the cluster and the interval of the spiking pulse increase as the trigger voltage decreases. However, the negative linearity relationship between the externally-injected trigger signal voltage and the peak voltage of the spiking pulse is not as obvious as those in the numerical simulation, which is attributed to the noise introduced by the high-speed oscilloscope and the active devices in the OEO cavity.

Fig. 7. Experimental results of stable spiking cluster oscillation under different external trigger signal injection. (a)-(c) are the temporal waveforms of the generated spiking clusters (blue line) and the externally-injected trigger signals (trapezoidal waveform, parabola waveform, and half-sinusoidal waveform, respectively. Red line). (d)-(f) are the temporal waveform details of the spiking cluster oscillation (blue line) and the externally-injected trigger signals (red line).

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Figure 8(a)-(c) exhibit the dimensionless temporal waveforms and the corresponding integral curves of the spiking clusters. It can be seen that the envelope shapes of the integral curves are similar to the trigger waveforms, which match with the simulation results in Fig. 4(a)-(c). Based on the values of x, y in Fig. 8(a)-(c), the phase portraits of the spiking clusters in Fig. 8(d)-(f) can be obtained. The phase trajectories are identical to those in Fig. 4(d)-(f). However, point D₂ and the trajectories from point B₁ to point C₁ in Fig. 8(d)-(f) deviate from the nullclines, which is mainly attributed to the non-ideal bandpass filtering characteristic and the noise in the experiment. In addition, the phase portrait in Fig. 8(f) is cluttered, which is attributed to the range of y -value variations being so large that aliasing occurs.

Fig. 8. Experimental results of the dimensionless variables x and y. (a)-(d) are the dimensionless temporal waveforms (blue line) and the corresponding integral curves (vermilion line) of the generated spiking clusters. (e)-(h) are the associated phase portraits of the spiking cluster (blue solid line) and the nullclines under maximum external triggering (yellow dashed line) and minimum external triggering (red dashed line). The black arrows represent the evolution direction of the phase trajectory. A₀, A₁-D₁ and A₂-D₂ are the characteristic points.

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Finally, the spiking interval under different externally-injected trigger waveforms is specially investigated. Figure 9 shows the experimental results of the spiking intervals under triggering by using the same externally-injected signals as those in Fig. 6 to Fig. 8. The experimental results are obtained by taking the time interval between two adjacent peak voltage points of the spiking pulses in a cluster duration as shown in Fig. (5). The theoretical values are obtained by substituting the experimental parameters into Eq. (12). It can be seen that there are some deviations between the experimental results and the theoretical values. The maximum deviations in Fig. 9(a)-(d) are 0.96 ns, 1.58 ns, 0.62 ns, and 1.03 ns, respectively. These deviations are mainly attributed to the ripples in the frequency response curve of the OEO cavity, which leads to the deviation of the evolutionary trajectory for the spiking cluster from the nullcline in the ideal case. In addition, the noise in the OEO cavity is equivalent to changing the externally-injected trigger signal voltage, which likewise causes a change in the spiking interval. It is worth noting that both the simulation and the experimental results indicate that the spiking interval has a negative correlation with the external trigger intensity. Therefore, it is possible to tune the spiking interval or the pulse width by changing the externally-injected trigger signal voltage in an appropriate range, i.e., 0 V-1.5 V.

Fig. 9. Spiking intervals in a cluster duration under triggering of (a) a square waveform, (b) a trapezoidal waveform, (c) a parabola waveform, and (d) a half-sinusoidal waveform. The blue dotted line and the purple solid line represent the actual spiking intervals and the theoretical values of the spiking intervals, respectively.

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5. Conclusion

In summary, we have proposed and demonstrated an approach to achieve controllable non-uniformly distributed spiking cluster generation based on an externally-triggered broadband optoelectronic oscillator. The dynamic and the distribution characteristics, i.e., peak-voltage envelop and interval of spiking pulses in a cluster duration, are theoretically analyzed in the stable spiking cluster oscillation under different trigger signal voltages, where the mapping relationships between the distribution characteristics and the externally-applied trigger waveforms are revealed. Both the peak-voltage envelop of the cluster and the interval of the spiking pulses increase as the trigger signal voltage decreases, which is numerically and experimentally verified by investigating stable spiking cluster oscillation under different trigger waveforms, i.e., square waveforms, trapezoidal waveforms, parabola waveforms and half-sinusoidal waveforms. This scheme has the capability of controlling spiking cluster distribution by easily tuning the trigger signal voltage, which can be applied in spiking neural networks, neuromorphic calculation, and spiking encoding.

Funding

National Natural Science Foundation of China (62301120, 61927821); Fundamental Research Funds for the Central Universities (ZYGX2020ZB012).

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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Controllable non-uniformly distributed spiking cluster generation in broadband optoelectronic oscillator

Abstract

1. Introduction

2. Operation principle and theoretical model

3. Simulation results

4. Experiment results

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (12)

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