1. Introduction

Brain-inspired neuromorphic systems have attracted great interest for years, and are regarded as a promising candidate for breaking speed and power consumption bottlenecks in traditional Von Neumann architecture [1–3]. By using excitable nonlinear devices, human brain-like nervous system can be constructed on the physical layer, which can imitate the basic characteristics of biological neurons at an operation speed of several orders of magnitude higher. Photonic neuromorphic systems provide an efficient way for neuromorphic information processing, like coding, transmitting and processing, which have been explored by hardware implementations such as semiconductor optical amplifiers, fiber lasers, and vertical-cavity surface-emitting lasers with saturable absorber (VCSEL-SA) [4–11]. In the exploration of artificial intelligent development, memorizing capability is an indispensable building block, which enables high-speed reading, writing and buffering for manipulating data. The main motivation of using light as the information storage medium is attributed to its advantages in speed and energy. However, the boson property of photons requires extremely demanding conditions to store them in “optical capacitors” like electrons, which makes high-speed storage challenging. Generally, nonvolatile optical storage media is used to build an optical memory [2], which enables long-term storage of information. However, the storage performance is limited by material stability, access time and operation speed. Therefore, many approaches focus on the use of optical transmission in media to achieve delayed storage operation [12], e.g., slow light buffer, optical delayed line and feedback loop with active components. These schemes mainly adopt bistable state in amplitude, polarization or wavelength to realize the representation of logical states “0” and “1”, which makes optical storage upgrade to bit-level storage, and becomes the mainstream technology of high-speed optical caching. In spiking neuromorphic system, the delayed regeneration characteristics with low power consumption has been explored with the development of high-speed optical and optoelectronic technology [13–17]. In Ref. [13], dissipative optical localized states have been observed in silica optical fiber with delayed feedback, in which the excitable spiking existing in longitudinal localized structure can be incoherent triggered by the externally-injected perturbation. For its excellent phase topological properties, the proposed neuron-like operating mechanism is considered for all-optical storage and reshaping. In Ref. [14], properties of spiking encoding and storage have been experimentally studied in a coupled VCSEL-SA. Through injecting an optical pulse sequence into a single VCSEL-SA, intensity-coded spiking signals can be triggered and stored in the coupled VCSEL-SA. In Ref. [15], the regenerative memory property is achieved in a time-delayed neuromorphic photonic resonator, where a writing speed at the level of Mb/s is experimentally demonstrated. All-fiber laser scheme with similar excitable property has also been proposed [16]. In the implementation, the excitable nonlinear component is configured by using saturable-absorber Q-switched fibers, which supports homoclinic mechanism and microsecond-level spiking response. These above-mentioned photonic methods are of great significance in promoting the construction of neural morphological photonic regenerative memory for processing high bit rate information. However, the signal quality inevitably degrades after cycles due to the regenerative instability, as well as the complex spiking writing operation induced by the excitation sensitivity of the perturbation formats and the noise factors. In addition, these all-optical neuromorphic architectures are difficult to achieve on-chip integration [17], and require out-of-cavity optoelectronic devices for electrical access, which increases the system complexity for the current optoelectronic fusion signal processing.

Optoelectronic oscillators (OEOs), which combine the advantages of large photon bandwidth and high electronic robustness, have been deeply researched for its excellent performance in microwave signal generation with ultra-low phase noise in quasi-linear operation [18,19]. When the loop gain is greater than the loop loss, an OEO generates periodic oscillation through self-reproduction of the longitudinal modes from the noise, where the purity and the temporal characteristics of the generated microwave signal depends on the Q factor and the response characteristics of the loop filter. In essence, the self-regeneration of the single-tone oscillation reflects the temporal regenerative memory characteristic of the narrowband quasi-linear OEO. Under broadband nonlinear operating condition, OEOs can perform abundant nonlinear dynamic behaviors under the combined action of multiple time scales, nonlinearity and delayed feedback, such as periodic oscillation, breather and high-dimensional chaos [20–25]. Recently, many studies have shown that broadband OEOs exhibit excitable spiking regimes when the electro-optic intensity modulator in the loop is biased near its minimum transmission point (MITP) or maximum transmission point (MATP) [26–29]. The excitability mechanism is similar to that of the class II optical neurons. In this status, under noise [26,28] or externally-applied excitation [29], spiking response with a nanosecond or sub-nanosecond timescale can be triggered when the perturbation exceeds the explicitly-defined amplitude threshold. These characteristics indicate that the broadband nonlinear OEOs can operate at a speed of several orders of magnitude higher than that of biological neurons, which is potential to construct optoelectronic compatible high-speed spiking neuromorphic information processing network.

In this paper, we propose and demonstrate a novel conceptual model to achieve plastic neuromorphic regenerative memory of the input temporal encoding sequence in a broadband nonlinear OEO. Through biasing the electro-optic intensity modulator in the loop near its MITP, excitatory response in localized state is triggered by the injected on-off keying encoding sequence. Consequently, the injected data flow is converted into spiking encoding information. Due to the self-feedback of the OEO loop, the excited spiking sequence are periodic regenerated with a fixed temporal position, where the spiking shape and the trajectories of incoherently triggered spikes in the phase space are well-consistent. The pulse-to-spiking conversion and the regeneration process allow the neuromorphic OEO to reach high capacity and to realize high bit rate data buffering under large bandwidth and long delay settings, as well as improvement of anti-noise performance and energy efficiency in low-power spiking operation. The proposed optoelectronic regenerative memory is numerically and experimentally demonstrated, where the experimental results fit in with the numerical simulation results. Most importantly, high stability and robustness are maintained in the dynamic process. These results indicate that the spiking patterns are effectively built and regenerated in the broadband nonlinear OEO.

2. Theoretical model

The schematic diagram in Fig. 1(a) exhibits the simplified conceptual model of the proposed neuromorphic regenerative memory OEO, which includes a section of delayed single-mode fiber (SMF), a bandpass filter (BPF), electro-optic (E/O) conversion and optical-electric (O/E) conversion. Figure 1(b) shows the theoretical model of the proposed scheme, which is composed of an excitable unit and a memory unit. The excitable unit, including the nonlinear transfer function and the filtering term, is used to reshape and regenerate the externally-injected on-off keying temporal sequence. The memory unit is constructed by the timescale of the OEO cavity induced by the optical and electrical delay paths, which maps the temporal information to the one-dimensional spatial scale.

 

Fig. 1. Schematic diagram of the proposed regenerative memory OEO. (a) Conceptual model. E/O, electro-optic conversion; O/E, optical-electric conversion. (b) Theoretical model. The excitable unit is composed of the cosine-squared nonlinearity induced by E/O and O/E conversion. The loop filtering effect is introduced by the loop devices. The memory unit is the delayed timescale. (c) Iterative mapping of the oscillation process corresponding to the case that the bandwidth of the OEO loop is unlimited. V_π is the half-wave voltage of the electro-optic modulator. (d) System setup. LD: laser diode; PD: photodetector; VOA: variable optical attenuator; DDMZM: dual-drive Mach-Zehnder modulator; EA: electrical amplifier; EC: electrical coupler; SMF: single-mode fiber; DC: direct current; URZ: unipolar return to zero. (e) Phase portrait of the regenerated temporal waveform. The inset diagram depicts the temporal waveform of the generated spike, which corresponds to the phase trajectory. The purple arrow is the evolution direction of the dynamic field. The x and y are the dimensionless parameters of the temporal waveform and the temporal integral variable, respectively. The nullcline (i.e., the red dotted line) refers to the line with partial derivative of zero in the dynamic field (i.e., dx/dt = 0, dy/dt = 0). C, D, and E are the fixed point, the approximate threshold point and the ideal saturation point, respectively. The blue solid line is the phase trajectory.

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In the extreme direct-current (DC) bias scenario (i.e., the electro-optic intensity modulator is biased far from its linear transmission point), the externally-injected intensity perturbations, such as optical or electrical pulses, exhibits “all” or “none” response in each localized interval, which correspond to the high level and the low level of the injected signal in the time domain, respectively. Hence, the response of the OEO cavity is synchronously triggered by the injected waveform. The OEO model with unlimited bandwidth can be mathematically described by using iterative mapping as

In practice, the bandpass filtering effect of the OEO loop can be achieved by the frequency limitation of the electrical devices in the electrical path, where the frequency response function can be modeled as a 2^nd-order BPF as

Figure 1(d) exhibits the experimental setup of the proposed neuromorphic regenerative memory OEO, where the red line and the blue line represent the optical path and the electrical path, respectively. The system is composed of a laser diode (LD), a dual-drive Mach-Zehnder modulator (DDMZM), a photodetector (PD) and a variable optical attenuator (VOA). The DDMZM and the PD constitute a nonlinear node as an excitable element to achieve spiking response. The EA is used as a bandwidth-limited gain unit, which determines the frequency response of the OEO loop. In addition, the SMF provides long-delayed time for larger capacity, and the VOA is used to tune the loop gain. According to Eq. (2), the dynamic evolution and the final state are determined when there is an externally-applied localized perturbation as shown in Fig. 1(e), where the low cut-off frequency, the high cut-off frequency, the phase offset and the gain coefficient are set to be 31.83 MHz, 3.18 GHz, 0.55π and 1.4611, respectively. The spiking threshold equation can be approximately deduced by using the nullcline equations in Eq. (4) as

The points C (0,0), D (0.49,0) and E (1.41,0) in Fig. 1(e) correspond to the three solutions of Eq. (5). Thereinto, the point C is the fixed point. The point D is approximately the amplitude threshold. The point E is the saturation value. It should be pointed out that, due to the limited bandwidth in the actual OEO loop, the generated spike is with a slower rising edge, which leads to the deviation between the actual amplitude and the ideal saturation value, i.e., the point E in Fig. 1(e). If the perturbation exceeds the amplitude threshold, i.e., the point D in Fig. 1(e), the phase trajectory evolves for a circle in the dynamic field and returns to the fixed point (0,0), which corresponds to a short spiking pulse in the time domain as shown in the inset of Fig. 1(e). In this process, the existing form of the information ranges from binary-coded voltage to spiking-coded pulse sequence. In addition, the dynamic field determines the robust and anti-noise excitation process, which binds the spiking dynamics to completely coherent localized states, and allows the excitation of data flow in the time domain for spiking coding and information storage.

The necessary condition for excitation is that the DDMZM should be biased near its extreme DC bias points (e.g., positive-polarity excitation at its MITP, or negative-polarity excitation at its MATP) [29], where the voltage net gain is positively correlated with the injection strength of the square pulse. Typically, an OEO is a dissipative physical system, in which the precondition for mode establishment should satisfy the gain condition (i.e., the net gain of the OEO loop should be larger than 0 dB). In addition, the system gain should be set to be below the Hopf bifurcation threshold to keep the global stability of the OEO system. By using net voltage gain function and considering the Hopf bifurcation threshold in Ref. [29], the gain coefficient β range should meet the following inequality.

This inequality is deduced under the ideal condition that the low cut-off frequency is infinite proximity to DC. However, due to the bandpass filtering effect in Eq. (2), the OEO cavity shows frequency selection mechanism. When an on-off keying binary coded signal is injected, the low-frequency components cannot obtain enough net gain and are lost. Therefore, only the high-frequency components satisfy the oscillating condition. In this status, the jump edge of each binary code is preferentially excited as a stable short spiking under the balance of gain and loss, where the phase trajectories fit in with the dynamic field. Hence, the injected on-off keying encoding sequence is reshaped into a spiking sequence, and is stored through self-excitation induced by the optoelectronic feedback. In fact, the low cut-off frequency acts as differential function for oscillating waveforms, which introduces an annealing time (i.e., the waveform component with negative voltage) as shown in the inset of Fig. 1(e). The waveform components with positive and negative voltages are distributed in the different time domain, which may result in the destroy of global stability if the gain coefficient is set to be too large. In addition, the loss of the oscillation amplitude induced by the BPF should be considered. These factors decrease the upper limit gain and increase the lower limit gain in Eq. (6). Furthermore, in order to enhance the neuromorphic regenerative memory performance, the low cut-off frequency, the 3-dB bandwidth of the BPF and the cavity length of the OEO should be set to be large enough to achieve a high operation speed and a large storage capacity.

3. Simulation results

In the simulation, the BPF is set to be with a high cut-off frequency of f_H= 3.18 GHz and a low cut-off frequency of f_L= 31.83 MHz, which correspond to the characteristic time of τ=1/2πf_H= 0.05 ns and θ=1/2πf_L= 5 ns, respectively. The gain factor is set to be β=1.4611. The phase offset induced by the DC bias voltage of the DDMZM is set to be ϕ=0.55π. The loop delay and the half-wave voltage of the DDMZM are set to be 2.145 µs and 2.2 V, respectively. For coherent injection, the injected amplitude is set to be 0.2 V, where the period of the temporal sequence is equal to the loop delay. In this status, stable oscillation modes are built from the same temporal sequence, which avoids the inter-symbol crosstalk of adjacent periods, and provides an effective way for studying the changes of dynamic characteristics under the injection state and the regeneration state, respectively. For incoherently random injection, the injected amplitude is set to be 0.9 V, and the duration of the injected sequence is equal to the loop delay. These parameter settings guarantee the excitation condition for stable spiking oscillation in the OEO cavity. Then, the numerical simulation is implemented by using pulse tracing method [31]. Thereinto, the cosine-squared nonlinearity induced by intensity modulation and beat frequency is calculated in the time domain and the bandpass filtering equation is calculated in the frequency domain. In the simulation, the stability of the spiking oscillation is demonstrated by the pulse-to-pulse consistency, which can be described by the dynamic trajectory induced by the modified Ikeda-like equations.

Figure 2 shows the simulation results of the regenerated spiking evolution under a single pulse injection, where the injected square-wave pulse is set to be with an amplitude of 0.9 V. Thereinto, the injected pulse widths in Fig. 2(a)-(b), (c)-(d) and (e)-(f) are set to be 1 ns, 2 ns and 10 ns, respectively. It can be seen from the temporal evolution in Fig. 2(a)-(f) that the injected square-wave sequence is converted into spiking oscillation, and is continuously regenerated due to the loop feedback after the excitation source is “off”. With the increasement of the loop cycles, the injected pulse is compressed into a localized narrow pulse with a full width at half maximum (FWHM) of 0.57 ns, and the generated narrow pulse gradually reaches a stable state. Under external injection with different pulse widths, the pulse evolution performs an identical dynamic path, and the final spiking steady states are consistent. These results indicate that the proposed regenerative memory OEO allows reshaping and storage of encoded information at different rates in a robust way. Figure 3 exhibits the phase portrait of the spiking oscillation, which corresponds to the evolution process in Fig. 2. With the increasement of cycles, the phase trajectories of the injected square-wave pulse are compressed and converges to a smaller closed track as shown in Fig. 3(a)-(c). Finally, they reach saturation and remain stable. Correspondingly, in the time domain, narrow spiking pulses determined by the phase space manifold is formed. Meanwhile, it can be seen from the phase trajectories in Fig. 3(a)-(c) that the injected pulses with different pulse widths evolve with an identical dynamic path, indicating the robustness of the spiking evolution process.

 

Fig. 2. Simulation results of the regenerated spiking evolution under square-wave pulse injection with different pulse widths. (a)-(b), (c)-(d) and (e)-(f) are the simulation results under external injection with a pulse duration of 1 ns, 2 ns and10 ns, respectively.

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Fig. 3. Simulated phase portrait of the spiking oscillation under square-wave pulse injection with an injection pulse width of 1 ns, 2 ns, and 10 ns, respectively. (a), (b), and (c) correspond to the simulation results in Fig. 2(a)-(b), (c)-(d), and (e)-(f), respectively.

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Due to the impulse response induced by the bandpass filtering effect of the loop filter, the regenerated spiking has a slowly-changing duration with a timescale of about 9.38 ns, which is related to the cut-off frequency of the filtering parameters, and corresponds to the “annealing time” in the neuron activity. Figure 4(a)-(b) show the temporal evolution of the regenerated spiking pulses under dual square-wave pulse injection with a temporal interval of 2 ns and 9 ns, respectively. When the temporal interval of the injected dual pulses is far smaller than the annealing time, one of the pulses is attenuated by the loss modulation induced by the previous pulse as shown in Fig. 4(a). On the contrary, when the second pulse is injected far away from the previous pulse, it can get enough gain and keep spiking oscillation as shown in Fig. 4(b). Hence, the spiking width and the annealing time become the main limitation factors of the spiking encoding rate in the proposed neuromorphic regenerative memory OEO system.

 

Fig. 4. Simulation results of the regenerated spiking evolution under dual square-wave pulse injection with different pulse intervals. (a) and (b) show the results under external injection with a pulse interval of 2 ns and 9 ns, respectively. The annealing time refers to the duration of the negative peak part of the spiking pulse rising to zero voltage, which is with a timescale of 9.38 ns in the simulation.

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The regeneration characteristics are demonstrated by comparing the dynamic characteristics of spiking encoding sequence under coherent injection state and regeneration state. The perturbation source is selected as a unipolar return-to-zero encoding sequence with a pattern of “1011011101”, which is compatible with the response characteristics of the OEO system. Figure 5 shows the simulation results of the built spiking oscillation under the injection state and the regeneration state. It can be seen from Fig. 5(a) that each rising edge of the injected pulse corresponds to a short spike, indicating that the injected signal is successfully converted into a spiking coding sequence. The excited spikes are with an identical FWHM of 1.26 ns and a repetition period of 2.145 µs (equal to the loop delay), indicating that they are strictly synchronized with the injected waveform. Figure 5(b) shows the integral characteristic of the temporal waveform. The integral characteristic curve (i.e., the blue line) is with a similar slowly-varying envelope as that of the injected waveform, which reflects the effect of high-pass differential operation introduced by the BPF on spiking reshaping. The spiking components in the integral envelope also indicates that the spatiotemporal dynamics between the regenerated spiking patterns and the injection information are separated and independent. Figure 5(c) shows the phase portrait of the temporal waveform x and its integral variable y, which corresponds to the local waveform in Fig. 5(d) and the ultra-wideband spiking power spectrum in Fig. 5(e). The unique phase trajectory includes the robust coupling dynamics of the injected waveform and the regeneration regime, indicating that the dispersed localized states in the time domain are with an identical dynamic path.

 

Fig. 5. Simulation results of the spiking encoding signal generation. (a), (b), (c), (d) and (e) are the temporal waveform, the integral curve, the phase portrait, the localized waveform and the spectrum under the injection state, respectively. (f), (g), (h), (i) and (j) are the temporal waveform, the integral curve, the phase portrait, the localized waveform and the spectrum under the regeneration state, respectively. In (a), (b), (f) and (g), the different background colors represent different delay periods. The blue dotted line in (a) and (f) represents the temporal waveform of the injected signal, and the blue solid line in (b) and (g) represents the temporal integral waveform y. The blue arrow represents the evolution direction of the spiking dynamics. P is the fixed point, and A is the quasi-stable point under injection excitation.

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When the injection source is turned off, the spiking oscillation is still maintained as can be seen in Fig. 5(f). In Fig. 5(f), two adjacent periods in 60^th and 61^th cycle after turning off the injection source are exhibited. The temporal position of the spiking sequence corresponds to that of the previously injected binary coding sequence. Due to the fact that the generated spiking waveform is with a relatively slow falling edge, the superposition of the injected waveform and the regenerated waveform leads to that a larger spiking pulse duration is with its amplitude exceeding the threshold in the injection state. As a result, the pulse width of the spiking in the injection state as shown in Fig. 5(d) is larger than that in the regeneration state as shown in Fig. 5(i). Therefore, in Fig. 5(g), the integral characteristic is changed to absolutely regenerated spiking regime, where the injection characteristics are lost. As the injection state is switched to the regeneration state, the phase trajectory of x and y collapses to a smaller range, as shown in Fig. 5(h), which corresponds to a narrower pulse in Fig. 5(i). As a result, the spectrum of the regenerated waveform is broadened as shown in Fig. 5(j). Therefore, the injected sequence is converted into spiking regime, and stored due to the self-excitation induced by the feedback loop for cycles. Compared with the results in Fig. 2, the pulse width in Fig. 5(i) is larger, which is attributed to that the spiking evolution in 60^th and 61^th cycle has not reached the stable state. In fact, the pulse width of the regenerated spiking will gradually approach 570 ps in Fig. 2 as the cycle increases. It should be pointed out that the OEO under incoherent random injection is with an identical dynamic path. The only difference lies in that the strength should be set with a larger amplitude to reach the threshold condition, and the duration of the injected sequence should be set no longer than the loop delay to avoid inter-symbol crosstalk. In addition, through biasing the DDMZM closer to its linear transmission point, the power consumption of the OEO system can be reduced due to the decline of the saturable amplitude. Nevertheless, the regenerative memory characteristic may be broken down if the DDMZM is biased too close to the its linear transmission point.

4. Experiment results

A proof-of-concept experiment is carried out based on the architecture in Fig. 1(d). The half-voltage, the voltage corresponding to the MITP, and the DC bias voltage of the DDMZM (EOSPACE AE-DD-0VPP-40-PFA-SFA) are 2.2 V, 2.9 V and 3.35 V, respectively. The OEO cavity is measured with a loop delay of 2.1411 µs, a 3-dB high cut-off frequency of 364.1 MHz and a low cut-off frequency of 38.3 MHz, respectively. In addition, the LNA (GT-HLNA-0022 G) with a small-signal gain of 28 dB is applied in the loop. The loop gain is tuned by adding a VOA in the optical path. The binary unipolar return to zero encoding sequence with a pattern of “1011011101” is generated by using an arbitrary waveform generator (RIGOL DG5352) with an operation bandwidth of 250 MHz. Each code is with a duration of 209.1 ns and a rising edge of 1.44 ns, which is injected into the upper branch of the DDMZM. The injected amplitude is set to be 0.5 V. In addition, the loop delay is concisely tuned to be equal to the period of the injected signal by adding a variable optical delay line in the OEO loop. It should be pointed out that the encoding sequence is injected with negative polarity for the π-phase change induced by the common ground of the DDMZM electrode.

Figure 6 shows the experimental results of the built spiking oscillation under the injection state and the regeneration state, where the perturbation source is a unipolar return-to-zero encoding sequence with a pattern of “1011011101”. The results in Fig. 6 are well-consistent with the simulation results in Fig. 5. It can be seen from Fig. 6(a)-(b) and Fig. 6(f)-(g) that the unipolar return-to-zero encoding spiking sequence is regeneratively reconstructed through injection excitation, and is repeated among cycles. The dynamic trajectories of the injected waveform disappear when the injection source is “off”. The existing form of encoding sequence changes to another state, and maintains a stable memory state. Therefore, the regenerated spiking temporal sequence has absolutely different dynamic characteristics compared with the initial state, which is called regeneration state. The phase portraits, the local waveform, and the spectrum of the signal under injection state and regeneration state are shown in Fig. 6(c)-(e) and Fig. 6(h)-(j), respectively. The unique phase trajectories and the ultra-wideband spectrum demonstrate the robustness of the regenerative memory in the broadband nonlinear OEO cavity. It should be pointed out the bandpass filtering effect has effectively limited the spiking response range of the input pulse sequence, which leads to the localized space-time structure. Hence, the spiking unit are relatively independent of each other. The experimental results verify the feasibility of achieving high-speed cache memory in a broadband OEO.

 

Fig. 6. Experimental results of the spiking encoding signal generation. (a), (b), (c), (d) and (e) are the temporal waveform, the integral curve, the phase portrait, the localized waveform and the spectrum under the injection state, respectively. (f), (g), (h), (i) and (j)are the temporal waveform, the integral curve, the phase portrait, the localized waveform and the spectrum under the regeneration state, respectively. In (a), (b), (f) and (g), the different background colors represent different delay periods. The blue solid line in (a) and (f) represents the temporal waveform of the injected signal, and the blue solid line in (b) and (g) represents the temporal integral waveform y. The blue arrow represents the evolution direction of the spiking dynamics. P is the fixed point, and A is the quasi-stable point under injection excitation.

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Figure 7 exhibits the temporal evolution of spiking encoding sequence in multiple round trips. Thereinto, Fig. 7(a)-(b) show the simulation results of the regenerated spiking oscillation under incoherent random injection. With the increasement of cycles, the injected codes are gradually narrowed into a short spiking sequence, and keep circulating steadily through periodic feedback. From the linear evolution path, the spiking amplitude and the shape remain steady after several cycles, verifying the robustness and the reliability of the proposed regenerative optoelectronic memory oscillation. Figure 7(c)-(d) show the experimental recorded results of the continuously regenerated spiking oscillation. The duration of the sampled temporal waveform is 200 µs, and is recorded after ten minutes. In fact, the experiment includes more complex influencing factors, such as dispersion, optical fiber transmission nonlinearity and environmental change. However, the experimental results indicate that stable regenerated spiking sequence oscillation is built for million cycles, verifying the feasibility of long-term caching in the proposed regenerative memory OEO.

 

Fig. 7. Temporal evolution of the regenerative memory spiking encoding temporal sequence. (a) and (b) are the simulation results of 100 cycles in the time domain under incoherent random injection (i.e., the injection duration is less than the loop delay). The injected unipolar return-to-zero encoding sequence is with an amplitude of 0.9 V, a duration of 2.145 µs and a bit rate of 4.662 Mb/s. (c) and (d) are the experimental results of 200 µs duration under stable regenerated spiking encoding sequence oscillation. The injected unipolar return-to-zero encoding sequence is with an amplitude of 0.5 V, a duration of 2.1411 µs and a bit rate of 4.7825 Mb/s.

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The encoding error analysis is shown in Fig. 8. The spiking interval of the adjacent regenerated spiking pulses are measured among cycles to demonstrate the time departure and the encoding quality of the OEO cavity. With reference to the spiking interval “0110” in the injected sequence of “1011011101”, autocorrelation test is carried out for the regeneration spiking interval of 92 delay cycles. Figure 8(a) shows the autocorrelation result of the truncated time series in the first period, where the spiking interval is measured to be 198.36 ns. The time interval comparison of multiple cycles is shown in Fig. 8(b), where no time departure is exhibited. The measurement results verify the stability of the regeneration regime. However, due to the bandwidth limitation of the arbitrary waveform generator used in the experiment, each injected square-wave waveform is with a rising edge of 1.44 ns (a slowly-changing edge of 7.8 ns), which leads to inevitable temporal encoding error. In addition, the time of switch keying may have a great impact on the results. As shown in Fig. 8(d), temporal encoding errors are measured between each spiking code, where the theoretical error covers a ∼5% random range. Through using a wideband signal generator, the encoding error can be effectively reduced.

 

Fig. 8. Experimental results of the generated spiking encoding sequence. (a) The autocorrelation curve of the temporal waveform. The operation data “0110” from the injected sequence “1011011101” includes two adjacent spiking pulses, and covers a temporal length of 0.88 µs. (b) The autocorrelation time offset within 92 cycles in (a). (c) The rising edge of the injected encoding sequence. (d) The error analysis of the encoding spiking sequence.

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5. Discussion

The proposed neuromorphic regenerative memory OEO is operating in its selective mechanisms for voltage level and waveform reshaping induced by multiple time scales. Therefore, its pulsating principle is different from that of mode-locking systems and conventional nonlinear excitable devices [16]. In the excitation process, the regenerative memory occurs in a low Q-factor condition, where stable spiking oscillation is established among several cycles in localized structure. Specifically, the cosine-squared nonlinear transfer function defines the perturbation threshold in the extremely-biased setup [29], which is similar to the excitation characteristics of the biological neurons. Therefore, the extremely-biased OEO is considered as a neuromorphic nonlinear system, which performs “all” or “none” spiking response under external perturbation. This property provides an effective way for spiking excitation and operation in full-physical layer. Moreover, through increasing the loss by injecting inhibitory temporal waveform with negative voltage or increasing the excitation threshold by tuning the bias voltage, the regenerated spiking encoding sequence can be effectively wiped out selectively.

In the proposed scheme, the localized regime is achieved by the differential-integral effect of the OEO response function, which determines the operation speed and energy efficiency of spiking response. Under incoherent injection, considering the step response of the high-pass filtering and the threshold-excitation characteristic (i.e., only the waveform components higher than the voltage threshold can obtain enough gain), the response spiking width of the injected square-wave signal can be approximately calculated as Δτ=θln(x₁*/x₂*). Consequently, if the 3-dB bandwidth is unlimited, the response spiking width can be infinitely narrowed when the saturation point approaches the threshold point x₁*. However, the slow timescales of the phase portrait also lead to a long “annealing time” in the response asymmetric spiking waveform as shown in Fig. 5(d) and Fig. 5(h), which limits the operation speed of the regenerative memory OEO. Although the writing speed is only demonstrated at a level of Mb/s in the experiment, it can be further improved through optimizing the filter design, i.e., enhancing the bandpass filtering effect and the operation bandwidth of the BPF. For example, by applying broadband microwave photonic filter [26], picosecond-level high-speed spiking response can be achieved. Moreover, the long-delayed SMF split as thousands of storage units provides a larger capacity for storing binary data, which provides optical/electrical regenerative memory compatibility. In addition, due to the threshold excitation effect and the anti-noise performance introduced by the waveform reshaping function of the OEO, the storage capacity can be further improved by employing an optical amplifier to enlarge the loop delay. Since the spiking operation has many unique advantages in information processing, spiking neuromorphic system has attracted broad research interests in recent years. For example, in [32], the hardware implementation of integrated photonic spiking neuron chip paves the way for complex task processing in the future. In the current requirements of high-speed switching, neuromorphic computing and high data throughput, the regenerative memory oscillation in neuromorphic OEO yielded with a small access time exhibits potential applications for high-speed information processing.

6. Conclusion

In summary, we have proposed and demonstrated a neuromorphic regenerative memory scheme based on a time-delayed broadband nonlinear OEO. Through setting the broadband OEO cavity in excitable state, the injected binary temporal sequence is converted into a spiking sequence, and is regenerated through periodic self-feedback in the OEO loop. The response timescale is limited to nanosecond-level, and the regenerated spiking is with a high stability in the evolution among cycles. These characteristics are beneficial for achieving high-speed spiking neuromorphic information processing. The proposed scheme allows spiking operation in localized structure, which provides a potential way for spiking encoding, pulse reshaping and optical storage in future neuromorphic system.