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Abstract
We propose an approach to generate neuron-like spikes of vertical-cavity surface-emitting laser (VCSEL) by multi-frequency switching. A stable temporal spiking sequence has been realized both by numerical simulations and experiments with a pulse width of sub-nanosecond, which is 8 orders of magnitude faster than ones from biological neurons. Moreover, a controllable spiking coding scheme using multi-frequency switching is designed and a sequence with 20 symbols is generated at the speed of up to 1 Gbps by experiment. Furthermore, we investigate the factors related to time delay of spiking generation, including injection strength and frequency detuning. With proper manipulation of detuning frequency, the spiking generation delay can be controlled upto 60 ns, which is 6 times longer than the delay controlled by intensity. The multi-frequency switching provides another manipulation dimension for spiking generation and will be helpful to exploit the abundant spatial-temporal features of spiking neural network. We believe the proposed VCSEL-neuron, as a single physical device for generating spiking signals with variable time delay, will pave the way for future photonic spiking neural networks.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Neuromorphic photonics, deploying photonic physics to emulate powerful computational capability of human brains, have been intensively investigated in order to establish application specific architecture for accelerating hardcore algorithms [1,2]. The photonic platform has become a disruptive candidate for generating artificial neurons, synapses, and neural networks [3–14], due to the advantages of ultrafast speeds, wide bandwidth, low crosstalk, and large-scale integration capability [1]. As for the neurons, many types of lasers, such as semiconductor ring lasers [15], graphene excitable laser [5,16], semiconductor micro-disk lasers [6], opto-electronic oscillators [7], and vertical-cavity surface-emitting lasers (VCSELs) [4,6–8,13] have been adopted to emulate the dynamic response of biological neurons but at a much faster time scale. To emulate synapses, there are also various proposals of optical active components including the semiconductor optical amplifiers (SOAs) and vertical-cavity SOAs [9–11,17].
VCSEL, as a popular device widely used in the short-reach optical interconnects, has also been regarded as a promising chip for generating ultra-fast photonic neurons due to the merits of high-bandwidth, small footprint, scalability and low-cost [18,19]. In particular, VCSEL displays rich nonlinear dynamics under optical injection, including injection locking and spiking dynamics [20,21]. Based on these nonlinear dynamics, VCSELs can imitate the biological neuron in some essential dynamics through polarization switching, phase and amplitude modulated injection locking, such as tonic spiking, phasic spiking, and excitatory and inhibitory dynamics [4,8,13,15,18,21–23]. Moreover, functional processing tasks including pattern recognition [18], exclusive OR (XOR) classification task [24] and spike-timing-dependent plasticity (STDP)-based unsupervised spike pattern learning [23,25] have been demonstrated to show the applicability of the VCSEL based neuromorphic computing.
In the existing literature, the spiking dynamics are mostly realized by applying either positive or negative temporal perturbations on the injection amplitude or pulse occurrence period. Specifically, the VCSEL will cross the dynamic boundary to appear neuron-like behaviour when receiving an external stimulus on the injection light intensity, since the perturbation on the injection light breaks the stability of injection-locked VCSEL. However, the issues such as undesirable output spiking oscillation and unstable VCSEL operation, brought by large time-varying perturbation of injection amplitude should not be ignored.
In this paper, we propose an approach to generate spikes by multi-frequency switching based on an optical injection scheme. The frequency modulation with constant amplitude not only brings flexible spiking generation with tunable delay time, but also improves the stable operation of injection-locked VCSEL. The response characteristics of VCSEL are studied based on the spin flipping model (SFM), and the neuromorphic response of VCSEL is intensively studied by simulations and experiments. A stable temporal spiking sequence has been realized both by numerical simulations and experiments with a pulse width of sub-nanosecond. Moreover, a controllable spiking coding scheme using multi-frequency switching is designed and a sequence with 20 symbols is generated at the speed of up to 1 Gbps by experiment. The effect of injection light with different detuning frequencies on the time delay of spiking generation is also demonstrated, showing that spiking delay can be tuned upto 60 ns.
2. Principle of frequency-switched photonic spiking neurons
For the existing literature [6,15,26], the spiking dynamics are mostly realized by applying either positive or negative temporal perturbations on the injection amplitude or pulse occurrence period. As the steady-state of injection locking is destructed due to the fluctuation of external laser, the VCSEL will cross the dynamic boundary to appear neuron-like behaviors. However, large time-varying perturbation of injection amplitude will make the VCSEL suffer from undesirable output spiking oscillation. Alternatively, when the VCSEL initially works at the boundary of the injection-locking state in dynamic space, the slight perturbation on frequency detuning will make the VCSEL shift from the injection-locking state to multi-cycle dynamic region, presenting a series of spikes during the perturbation. On the contrary, if we apply the opposite scheme of excitatory spiking generation, we will naturally be able to observe the inhibitory spiking response, in which the VCSEL is back to calm when the frequency detuning drops into the locking range. For clarity, we define the steady-state frequency detuning $\Delta {f_{x0}}$ and the spiking-state frequency detuning $\Delta {f_{x1}}$, respectively.
We consider a typical spin-flip model of VCSEL with rate equation given by [6,27]:
Table 1. VCSEL Parameters
With these parameters, the parallel polarization (Y-p) is the main mode for the free-running case. As stated in [21], there exist stable regions for different injection powers and detuning frequencies for the injection-locked state, while on the boundaries of these dynamic regions, the VCSEL status is unstable, presenting various neuron-like behaviours. By changing the injection conditions, the VCSEL will perform different dynamic features.
We consider an orthogonal injection (X polarization direction) with constant amplitude and a time-varying frequency detuning, denoted by ${E_{\rm inj}}e^{i\Delta {\omega }(t)}$. By taking frequency detuning as a controlled variable, bifurcation diagrams are illustrated in Fig. 1, with frequency detuning ranging from −15 GHz to 10 GHz. As shown in each figure, for a relatively small ${\Delta {f_x}}$, the orthogonal polarization (X-p, blue dots) becomes the main mode while the parallel polarization (Y-p, orange dots) is suppressed, indicating an injection-locked state, in which the polarization switching occurs. With the increase of frequency detuning, both parallel (Y-p) and orthogonal (X-p) polarization coexist and the status of VCSEL becomes complex, in which the chaotic dynamics can be observed. The target status, referred as multi-cycle region, can be obtained at the boundary between the locked state and the chaotic state with around 500-MHz range, where the operating status of VCSEL will appear neuron-like behaviour. For even larger frequency detuning, the parallel (Y-p) mode comes to the dominated mode again. The unlocked region [19] can be observed since the frequency detuning is out of the locking range. Consequently, the detuning frequency switching will break the stability of the injection locking state, which provides another manipulation dimension for spiking generation. As an alternative, we can apply the temporal perturbation on frequency detuning instead of injection intensity. Moreover, from the analysis of bifurcation diagrams in Fig. 1, we can see that the X-p mode oscillates violently on the right boundary with positive detuning. While on the left boundary with negative detuning, the X-p mode is relatively stable and can generate periodic spikes. Hence, by switching $\Delta {f_x}$ between the injection-locked state and the boundary state, we can obtain the reproducible spikes. In addition, from the Fig. 1(a) to (d), we can see that the locking range increases with the injection intensity, which is consistent with the results described in [20].
Fig. 1. Bifurcation diagrams of orthogonal polarization (blue) and parallel polarization (orange). ${E_{\rm injx}}$ are (a) 0.1, (b) 0.15, (c) 0.2, (d) 0.25, respectively, with $\mu =2$, ${E_{\rm injy}}=0$, ${\Delta {f_y}=0}$.
3. Experiments and results
3.1 Experimental setup
In order to study the behaviors of VCSEL under detuning frequency, we establish a testbed to build a VCSEL neuron relying on the optical injection scheme. As shown in Fig. 2, a tunable laser (TSL, Santec TSL-710) is used as the master laser, which can be tuned at wavelength and output power respectively. A single-mode 1550 nm VCSEL is utilized as the slave laser. The VCSEL is packaged in transistor-outline form with a typical output power of around −8 dBm at a bias current of 8 mA. The threshold current of VCSEL is about 2 mA. The optical spectra of free-running VCSEL for diverse current values are shown in Fig. 3. Almost four modes can be seen in the free-run situation with the side mode suppression ratio (SMSR) of around 40 dB. As we finely tune the bias current of VCSEL, the operating wavelength shifts at 0.635 nm/mA. We refer to the main lasing mode as the Y-polarized mode (Y-p mode), and the subsidiary mode as the X-polarized mode (X-p mode).
Fig. 2. Experimental setup of a VCSEL neuron using multi-frequency switching. TSL: tunable laser. PC: polarization controller. SSB: single-sideband modulator. AWG: arbitrary waveform generator. PD: photodetector. DSO: digital storage oscilloscope.
The complete optical injection architecture is illustrated in Fig. 2. The tunable laser emission is passed through a polarisation controller (PC) before being modulated by a single side-band (SSB) modulator, which is controlled by a 64 GSa/s arbitrary waveform generator (AWG, Keysight M8195A). The SSB is used to apply the continuous-phase frequency modulation onto the light to generate frequency shifting signals, presenting the frequency-coded stimuli pulses. Before the modulated light with controlled polarization is injected into the working VCSEL, a circulator is inserted in-between to prevent light reflection back to the master laser and route the injection-locked light for transmitting. On the receiving side, the signals are converted into electrical signals by a photodetector (PD, Finisar XPDV21x0) with 50-GHz bandwidth, and then captured by a real-time digital storage oscilloscope (DSO, Keysight DSOZ592A) with an 80 GSa/s sampling rate. Therefore, the injection light with frequency modulation will behave in the form of a series of frequency-coded signals to push VCSEL work in the steady state (’0’) or spiking state (’1’). The dynamic regions and spiking behaviors of VCSEL are analyzed in detail through numerical simulations and experiments. Specifically, the spiking behaviors of VCSEL under two polarization modes are described in terms of injection intensity, injection frequency detuning and injection timing of master laser. Moreover, the effect of injection light with different detuning frequencies on the spiking responses is investigated in order to tune the pulse width of the abrupt spike and the time delay of spiking generation.
3.2 Spiking generation based on multi-frequency switching
Based on the VCSEL neuron model set up in Section 2, we can obtain the continuous excitatory spiking and inhibitory spiking response, which are similar to the neuromorphic behaviors in biological systems. Specifically, $\Delta {f_{x0}}=-5.27$ GHz and $\Delta {f_{x1}}=-9.27$ GHz are selected from Fig. 1(c) to emulate the spiking behaviors through the numerical simulation. In the experiment, we use the SSB to apply the frequency modulation onto the light to generate frequency-shifting signals. We finely tuned the frequency detuning between the tunable laser and the X-p mode of VCSEL until the locking state occurs, where the X-p mode becomes the main mode and the Y-p mode is suppressed. After that, we apply the frequency modulation onto the injection light, with a frequency scan from 9 GHz to 11 GHz, corresponding to the state shifting from the steady to the spiking.
Figure 4 demonstrates the generated spike under frequency switching. The dotted line represents the numerical result, and the solid line represents the experimental result, respectively. The temporal pulses present a similar trend both in simulation and experiment, verifying the feasibility of the method. Therefore, the experimental spikes generated by frequency switching have the characteristics similar to biological neurons, but have the width and speed of sub-nanosecond, which are 8 orders of magnitude faster than biological neurons.
In Fig. 5, a time series of frequency coding and reproducible spikes are presented, using the proposed coding scheme with multi-frequency switching. The numerical results are shown in Fig. 5(a), where we present an arbitrary binary coding of ’ 01101 01101 10101 11100 ’. As explained before, the symbol ’0’ denotes $\Delta {f_{x0}}=-5.27$ GHz, representing the steady state, and ’1’ denotes $\Delta {f_{x1}}=-9.27$ GHz, representing the spiking state. Other parameters we choose are ${E_{\rm injx}}=0.2$, $\mu =2$, ${E_{\rm injy}}=0$, corresponding to Fig. 1(c). Under a continuous injection light with $\Delta {f_{x1}}=-9.27$ GHz, the VCSEL will generate the reproducible spikes with a pulse width estimated to be 1 ns. When the rising edge of the code ’1’ arrives, the VCSEL generates a spike with a peak intensity of 9. When the successive codes ’1’ arrive, suggesting the continuous injection disturbance, the same number of spikes are generated by VCSEL. When the falling edge of the code ’0’ arrives, the spike response is immediately suppressed, without intensive oscillation or abrupt pulses.
Fig. 5. A Coding scheme using multi-frequency switching. (a) Numerical results, (b) experimental results.
Similar results can also be observed in the experiment, as illustrated in Fig. 5(b), confirming the feasibility of the coding scheme. To demonstrate the coding scheme, we rely on an SSB to generate the frequency shifting signals, presenting the frequency-coded stimuli pulses. For the steady state ’0’, $\Delta {f_{x0}}=9$ GHz, and for spiking state ’1’, $\Delta {f_{x1}}=10$ GHz. Therefore, the injection stimuli pulses are frequency coded by ’ 00101 10111 ’ (’ 01110 00110 ’), with the pulse width of 1 ns. We reproduce the signal of ’ 00101 10111 ’ (’ 01110 00110 ’) for 10 times and select the time scale of 20 ns to show the spiking response in the experiment results. As plotted in Fig. 5(b), the spiking response signals are in good agreement with the injection stimuli pulses, with a reproducible spiking response of up to 1 Gbps. Therefore, the proposed approach to generate spikes by multi-frequency switching is successfully demonstrated both in simulation and experiment, with the spiking response of up to 1 Gbps.
3.3 Time delay of spiking generation
A time delay is observed between the perturbation arrival and the pulse firing, which can be controlled by both the intensity and frequency detuning of the injection light. However, only the effect of intensity on the spiking delay is studied in previous researches. For instance, the temporal map in previous work [28] shows the effect of the perturbation strength on the time delay of spiking generating, indicating a 1 ns delay at most with the injection ratio varying from 1 dB to 10 dB. In this paper, the effect of the detuning frequency on the time delay is investigated and we discover that under different injection frequency detuning, the VCSEL will output spikes with designed time delay.
For comparison, the factors related to delay time, including injection strength and frequency detuning, are investigated independently. By taking injection strength as a controlled variable, the response delay is illustrated in Fig. 6(a), with the injection strength ${E_{\rm injx}}$ ranging from 0.48 to 0.57. The frequency detuning is fixed to −5 GHz. As shown in Fig. 6(a), the response delay decreases with the decrease of the injection strength, which means that the further the working state is from the locked region, the faster the spikes are generated. The time delay can achieve 10 ns at most when ${E_{\rm injx}}$ is around $0.57$. Meanwhile, we analyze the relationship between the number of generated spikes and injection strength and define the period of spikes observed during the perturbation as the ${T_{\rm spikes}}$. As illustrated in Fig. 6(b), the perturbation on injection strength at the time of 4 ns will make the VCSEL shift from the injection-locking state to multi-cycle dynamic region, presenting a series of spikes during the perturbation. The duration window is denoted as ${T_{\rm w}}=4$ ns. When the injection strengths are set to 0.57, 0.56, 0.55, respectively, the responses can be observed in Fig. 6(b), with the spike numbers of 0, 1, 2, correspondingly. The ${T_{\rm spikes}}$ declines with the decrease of the injection strength, similar to the trend of response delay. In order to find the relationship between ${T_{\rm spikes}}$ and response delay, we add the orange dots which represent the value of ${T_{\rm spikes}}$ under different injection strengths, which is obtained as illustrated in Fig. 6(b). For instance, in the bottom subfigure of Fig. 6(b), when the injection strength is 0.55, the duration of stimuli (perturbation) is 4 ns and the number of spikes is 2. Thus, the value of $T_{\rm spikes}$ is 2 ns, corresponding to the point $(0.55, 2)$ in Fig. 6(a). However, when the injection strength increases (decreases) slightly, the number of spikes observed during the ${T_{\rm w}}$ remains the same, indicating the same $T_{\rm spikes}$. Specifically, when the injection strength is set to 0.552 and 0.548, the number of spikes remains to 2 during the 4 ns disturbance duration, indicating the ${T_{\rm spikes}} = 2$ ns, corresponding to the point (0.552, 2) and (0.548, 2) in Fig. 6(a) respectively. In general, the same trend can be obtained between ${T_{\rm spikes}}$ and response delay, indicating that we can use ${T_{\rm spikes}}$ to approximate response delay.
Fig. 6. The relationship between spikes firing and injection strength. (a) Response delay and ${T_{\rm spikes}}$ under different injection strengths.(b)Temporal spiking response under different injection strengths.
By taking frequency detuning as a controlled variable, the response delay is illustrated in Fig. 7(a), with the frequency detuning ranging from −8 GHz to −10 GHz. When the detuning frequency varies from −9 GHz to −10 GHz, there is a time difference of 0.3 ns at the moment of pulse firing. As the frequency detuning drops to −8.5 GHz, the time delay increases rapidly, and the pulse generation time delay can be as long as 50 ns. The injection strength is fixed to 0.2. As plotted in Fig. 7(a), the response delay decreases with the decrease of the frequency detuning, similar to the situation in Fig. 6(a). The experimental results plotted in Fig. 7(b) are in good agreement with the numerical results in Fig. 7(a). As the frequency detuning varies from 9.6 GHz to 10.6 GHz, the VCSEL will output spikes with different time delay up to 60 ns. The trends of the curves in the experiment and simulation are consistent with each other. The experimental results illustrated in Fig. 8 indicate the different temporal spiking responses with the corresponding frequency detuning states one-to-one. In specific, Fig. 8(a) shows the temporal spiking response at frequency detuning of 9.75 GHz, 9.9 GHz, and 10.6 GHz from top to bottom respectively. When the VCSEL works in the multi-cycle region, switching the frequency detuning at the same time will result in different output sequences. In our experiment, the optical power of the signal injected to the VCSEL is fixed to −5 dBm. And the VCSEL will work in a multi-cycle region with delicate controlling the frequency detuning and polarization, generating a series of periodic spike pulse signals within the disturbance duration window ${T_{\rm w}}$. As shown in Fig. 8, when we keep the $\Delta {f_{x1}}$ fixed to 9.75 GHz, there is only 1 spike within the 10 ns time scale. Whereas, when $\Delta {f_{x1}}=9.9$ GHz, the number of spikes changed to 4, indicating the ${T_{\rm spikes}}=2.5$ ns. As $\Delta {f_{x1}}= 10.6$ GHz, there are 17 spikes within the 10 ns time scale, with ${T_{\rm spikes}}= 0.59$ ns. As we mentioned before, we can use ${T_{\rm spikes}}$ to approximate the value of response delay, thus every temporal spiking sequence illustrated in Fig. 8 corresponds to the particular point on the curve in Fig. 7(b).
Fig. 7. The relationship between spikes firing and frequency detuning. (a) Simulation results, (b) experiment results.
Fig. 8. Temporal spiking response under different frequency detuning.(a) Spiking sequence, (b) the frequency detuning corresponding to the spike sequence.
The results indicate that with proper manipulation of detuning frequency, the spiking generation delay can be controlled within 60 ns. Based on the multi-frequency switching method, the frequency of neuron-like spikes is controllable with the proper manipulation of detuning frequency, which opens the possibility for spiking frequency coding in photonic neurons with VCSELs, and is interesting and valuable for the ultrafast photonic neuromorphic systems and brain-inspired photonic information processing. On the other hand, we believe the proposed VCSEL-neuron will pave the way for future photonic spiking neural networks, especially for reducing the size of the chip dramatically since we can use a single physical device to generate a spike signal with variable time delay, without the need for extra delay lines.
4. Conclusion
In this paper, we propose an approach to generate spikes by multi-frequency switching based on an optical injection scheme. The frequency modulation with constant amplitude not only brings flexible spiking generation with tunable delay time, but also improves the stable operation of injection-locked VCSEL. A frequency modulation is realized both by numerical simulation and experiment to obtain a stable spiking response of up to 1 Gbps. The multi-frequency switching provides another manipulation dimension for spiking generation and will be helpful to exploit the abundant spatial-temporal features of spiking neural network. Furthermore, the factors related to the time delay of spiking generation, including injection strength and frequency detuning, are investigated separately. Both the numerical and experimental results indicate that with proper manipulation of detuning frequency, the spiking generation delay can be controlled within 60 ns, while the delay can be controlled within 10 ns by intensity. This shows that compared to the intensity dimension, the frequency dimension can provide a larger delay control range. We believe the proposed VCSEL-neuron will pave the way for future photonic spiking neural networks.
Funding
National Natural Science Foundation of China (62175146).
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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