Spike-based information encoding in vertical cavity surface emitting lasers for neuromorphic photonic systems

The ongoing growth of use-cases for artificial neural networks (ANNs) fuels the search for new, tailor-made ANN-optimized hardware. Neuromorphic (brain-like) computers are among the proposed highly promising solutions, with optical neuromorphic realizations recently receiving increasing research interest. Among these, photonic neuronal models based on vertical cavity surface emitting lasers (VCSELs) stand out due to their favourable properties, fast operation and mature technology. In this work, we experimentally demonstrate different strategies to encode information into ultrafast spiking events in a VCSEL-neuron. We evaluate how the strength of the input perturbations (stimuli) influences the spike activation time, allowing for spike latency input coding. Based on a study of refractory behaviour in the system, we demonstrate the capability of the VCSEL-neuron to perform reliable binary-to-spike information coding with spiking rates surpassing 1 GHz. We also report experimentally on neuro-inspired spike firing rate-coding with a VCSEL-neuron, where the strength of the input perturbation (stimulus) is continuously encoded into the spiking frequency (spike firing rate). With the prospects of neuromorphic photonic systems constantly growing, we believe the reported functionalities with the ultrafast spiking VCSEL-neurons provide a reliable, multifaceted approach for interfacing photonic neuromorphic platforms with existing computation and communication systems.


Introduction
The advent of artificial neural networks (ANNs) marked a significant milestone in computer science, allowing computers to perform new complex tasks that typically require human reasoning, such as object recognition in images [1,2] or natural language processing [3]. Nowadays, machine learning algorithms based on ANNs are quickly becoming ubiquitous, powering the services and systems we use in our daily lives [4][5][6]. However, conventional computers based on the von-Neumann architecture are not very well suited for ANNs due to their serial instruction processing and distinct memory and logic units. These facts, paired with the physical limits on further transistor size minimization (Dennard's scaling law [7]) and plateauing of Moore's law observed in recent years [8] have fuelled the search for new, 'beyond von-Neumann' computational architectures. The field of neuromorphic (brain-like) engineering aims at answering those needs by mimicking the functionality of biological brains and nervous systems in large-scale, densely interconnected networks of computing primitives (typically analog circuits or digital processors). These systems are characterised by propagating information using spikes: an asynchronous, sparse information encoding scheme observed in neurons that combines properties of both digital and analog signals. Whilst neuromorphic platforms such as Neurogrid [9] and Braindrop [10] from Stanford University, BrainScaleS [11] from Heidelberg University, Loihi [12] by Intel, SpiNNaker [13] from Manchester University and TrueNorth [14] by IBM have already demonstrated the viability and capabilities of brain-inspired hardware, there are certain limits imposed by their electronics-based character. Since one of the most fascinating characteristics of the brain is its remarkably low power consumption [15], neuromorphic systems are expected to enable highly energy-efficient functionality for future artificial intelligence applications. Given the fact that the processing performance in distributed systems significantly depends on the communication efficiency between the nodes, the reliance of electronic neuromorphic systems on metallic interconnects leads to performance challenges and introduces bottlenecks due to radiative and RC physics [16]. While digital many-core systems (such as SpiNNaker) offer high degree of tunability for each node, making them suitable for simulations of neuronal networks in brains [17], the capability of analog systems to directly exhibit the dynamical behaviour similar to that observed in neurons, without the need to perform resource-intensive numerical simulation of it [18], further raises the prospects of these systems for high-speed, energy-efficient neuro-inspired computation.
Optical and photonic systems are a prime example of analog models with high prospects for computation, thanks to their intrinsic nonlinearities and use of optical waveguides that carry signals passively, with lower latency, higher possible bandwidth [19] and low electromagnetic interference. These systems can also operate at ultrafast operation rates with high power efficiency [20] and with additional degrees of freedom such as wavelength and polarization. There are a number of different approaches aiming towards the realization of ANNs using optics and photonics, including matrix-vector multiplicators [21] or arrays of cascaded optical modulators [22]. Neuromorphic photonic platforms represent a subset of these approaches with various photonic devices being investigated, with a non-exhaustive list of examples including optical modulators [23,24], phase change materials [25][26][27], resonant tunnelling diode-laser diode coupled systems [28][29][30], quantum-dot lasers [31,32], silicon photonic components and circuits [33][34][35], micropillar lasers [36,37] and superconducting optoelectronic circuits [38,39].
Among the most promising neuromorphic photonics technologies are vertical cavity surface emitting lasers (VCSELs). Thanks to their viability for large-scale, high throughput fabrication, these compact devices are inexpensive and readily commercially available. They can operate at standard telecom wavelengths, have low lasing thresholds and exhibit rich nonlinear dynamics [40]. For example, when subject to different types of polarised optical injection, VCSELs can show bistability, polarization switching [41,42] and a wide range of complex nonlinear responses, ranging from periodical dynamics to chaos and excitable responses [43,44]. Therefore, the use of these devices as artificial photonic spiking neurons has been proposed in recent years [45,46]. Specifically, controlled excitation [47], inhibition [48] and propagation [49] of neuron-like spiking regimes have been demonstrated in VCSELs at operation rates up to seven orders of magnitude higher than the timescales of biological neurons. These functionalities can be used to demonstrate toy models of optical [50] and auditory [51] pathways or for supervised pattern recognition [52,53].
While the mystery of biological neural coding has not yet been fully uncovered, multiple different ways of stimuli representation have been observed in neural codes [54]. In general, information communicated between neurons can be encoded using either rate-coding of spike trains (number of spikes in a given time window) or precise spike timing (temporal coding), with spike timing precision (at the ms timescales) being crucial for information encoding and processing in biological neural systems [55]. Hence, an implementation that delivers reliable, digital-to-spike conversion at high data rates is essential to facilitate brain-inspired information encoding and bridge future neuromorphic computing platforms with conventional digital computers and telecommunication networks [56]. Previous numerical studies have reported a wide range of spike encoding capabilities in neuromorphic photonic systems. Both precise spike timing and rate coding regimes were observed in a single section quantum-dot InAs/GaAs laser model [63]. In the VCSEL-SA based models, sound azimuth information was encoded in the interval between two spikes [51] and rate-coded trains were elicited from a single laser [64] as well as propagated between two mutually coupled devices [65].
In this work, we demonstrate the viability of photonic spiking VCSEL-neuron for high-speed, spike-based information encoding in the optical domain. The information injected to the neuron is encoded either using a binary-to-spike encoding scheme for digital signals, or biologically inspired intensity-to-firing rate encoding scheme for continuous signals. Both absolute and relative refractoriness in the neuron are evaluated and used to estimate the maximum operation frequency of the neuron in the binary-to-spike regime. The dependence of spike activation time as a function of input perturbation strength is also investigated, with prospects for latency-based information encoding. Moreover, these experiments are realized using a commercially available VCSEL operating at a standard telecom wavelength of 1310 nm, which makes our approach fully compatible with current optical telecommunication networks.

Methods
Simplified schematics of our proposed VCSEL-based photonic spiking neuron and the two optical approaches for high-speed temporal spike encoding and intensity-to-firing rate encoding are provided in figures 1(a) and (b), respectively. All the experiments carried out in this work have been realized using the experimental setup shown in figure 2(a). The optical spiking neuron is implemented using an off-the-shelf fibre-coupled 1310 nm VCSEL (RayCan) mounted in a thermoelectrically cooled laser diode mount. The laser is temperature stabilized (Thorlabs TED200C) at the temperature of 298 K and biased with a current  well above the device's lasing threshold (I = 5 mA, I thr = 1.65 mA). The spectrum of the laser, shown for I = 5 mA at 298 K in figure 2(b), exhibits two peaks that correspond to the two orthogonal polarizations of the fundamental transverse mode of the device. These are referred to as parallel (the dominant peak, denoted λ y ) and orthogonal (the secondary peak, denoted λ x ) polarized modes.
The photonic neuron operation relies on coherent injection locking of the VCSEL to a signal from an external tunable laser source (TL; Santec TSL210V). The information input to the system is realized by modulating the intensity of TL's light using a Mach-Zehnder modulator (MZM), which is controlled by RF signals from an arbitrary waveform generator (AWG) (Keysight, M8190A) operating at 12 GSa s −1 . These RF signals were amplified using a 10 dBm amplifier (Mini-Circuits, ZX60-14012l-S+) before entering the MZM. First, the TL signal is passed through an optical isolator to ensure unidirectional coupling between the lasers. Following that, the power is adjusted using a variable optical attenuator and the signal is polarization matched to the MZM using a polarization controller. The modulator bias point is set to achieve maximal average power while preserving full modulation amplitude (modulations are typically rectangular-shaped drops, see figure 1(b)). The modulated signal is polarization matched to the orthogonally polarised mode of the VCSEL, before its injection into the device through a 50:50 coupler and an optical circulator. The other branch of this coupler is used to record the average injection power P inj using a power meter (PM). The TL wavelength is matched to the selected peak in the laser spectra with a certain frequency detuning ∆ f . The detuning and injection power are the main operational parameters defining the distance of the unlocking boundary (threshold) from the locked (steady) state. In this steady (non-spiking) state, the VCSEL-neuron is fully locked to the TL. By bringing the system close to the unlocking boundary and modulating the amplitude of the injection signal, the VCSEL-neuron responds with well controlled, very fast (sub-nanosecond) optical spiking responses [47]. In general, higher detuning and lower injection power bring the threshold closer to the steady state. With a high injection power, locking can be achieved more easily, but large drops may be needed to activate spikes. Larger drops also facilitate faster continuous spiking (rate coding). The detuning and power values used through this work were found experimentally based on values used in previous works (see [66]). The laser output passes again through the circulator into a 50:50 coupler for analysis in an optical spectrum analyser and a real-time oscilloscope (RT OSC). In the oscilloscope line, the optical signal is converted to the electrical domain using a high-speed fibre-coupled amplified photodetector (PD; Thorlabs PDA8GS). The signal being injected into the VCSEL-neuron is recorded using the PD and RT OSC from the PM branch. The setup is fully realized using commercial fibre optic components at standard telecom wavelengths. There is a clear transition between absolute (no spiking observed) and relative refractory period (reduced spiking probability) followed by recovered state (spiking probability very close to unity).
The modulation schemes are depicted in figure 1(b). For the investigation of the refractory behaviour and the binary-to-spike format conversion in VCSEL-neurons, the modulation entails short rectangular-shaped intensity drops, where each drop serves to elicit one single spike during its duration. The amplitude of the drops can be variable or fixed (binary modulation). Only drops with sufficient amplitude cross the spiking threshold (depicted as red dashed line I T in figure 1(b)) and elicit the spikes. In the rate-coding demonstration of the VCSEL-neuron, an arbitrary continuous signal is used for intensity modulation. The information is encoded into the signal's intensity, such that 'stronger' stimuli correspond to larger drops in injection power. The modulation signal must cross a certain threshold level to trigger continuous spiking responses.

Results and discussion
Multiple injection modulation schemes were used to demonstrate the different properties and responses of the VCSEL-neuron investigated in this work. In figures 3-6, upper (red) time-traces denote the intensity of the modulated signal from the TL injected into the VCSEL-neuron, whilst lower (blue) time-traces correspond to the response of the VCSEL.

Refractoriness of the VCSEL-neuron
To evaluate the refractory period of the VCSEL-neuron, multiple pairs of rectangular-shaped intensity perturbations (0.249 ns wide), with varying temporal separation between them (figure 3), were used as the input modulated signals, The separation ∆t, measured between the two input pulse rising edges, was gradually increased from 0.332 ns to 1.494 ns, in steps of 83 ps. The first recorded perturbation arrives when the VCSEL-neuron is in a locked (resting) state and always triggers a spiking response at the device's output. When the second perturbation arrives too soon after the first one (figure 3(a)), the system is still inside the so-called absolute refractory period (∆t < τ abs ) and cannot elicit another spiking response. However, when stimuli separation time reaches the edge of the absolute refractory period (∼0.83 ns, figure 3(b)), occurrences of second spikes can be observed, with probability of second spike increasing as ∆t is increased further from τ abs . The absolute refractory period is highlighted as a red shaded area in figures 3(a)-(c). Simultaneously, it was observed that the amplitude of the second spike was slightly lower for spikes very close to τ abs , further increasing with larger ∆t. The relative number of observed second spiking events for the same dataset is depicted in figure 3(d) as a function of trigger pulse separation. The blue dash-dotted line shows a sigmoid curve fit to the calculated spike probability. For ∆t above 1.08 ns, figure 3(d) shows the probability of second spiking event reaching near certainty. After this time period (denoted as relative refractory period τ rel , highlighted in orange shading in figures 3(a)-(c)), the dynamical system is sufficiently recovered and capable of properly responding with another spike. Hence, to ensure highly reliable activation of spikes with precise timing, the VCSEL-neuron should be operated with minimal inter-spike temporal separation equal or larger to the relative refractory period.

Spike latency in VCSEL-neurons for stimulus strength encoding
The perturbation intensity of the injected signals (stimuli) also influences the spiking response in the VCSEL-neuron. To analyse this phenomenon, we devised an experiment where we inject stimuli into the VCSEL-neuron with different controlled intensities and analyse the temporal features of the triggered spiking responses. To do this, the available amplitude range of the digital-to-analog converter (DAC) on the AWG was divided into ten equidistant levels. These were used to create a modulation waveform of ten rectangular-shaped drops of varying intensity, each with a temporal length of 0.5 ns and separated by 2 ns. Figure 4(a) shows the response of the VCSEL-neuron to gradually increasing perturbations (with 20%-100% of DAC amplitude range). The bottom plot in figure 4(a) shows (in blue) five superimposed time-series captured at the output of the VCSEL-neuron in response to five identical input perturbation patterns (shown in red in figure 4(a)). These patterns were injected into the device in consecutive measurements. All the injected perturbations successfully pushed the VCSEL-neuron through its spiking threshold, triggering the firing of a single spike. For the low-amplitude (leftmost) perturbations shown in figure 4(a), the spiking response exhibits both a pronounced onset delay and jittering. More information regarding the responses across multiple consecutive measurements can be seen in the temporal maps in figure 4(b) for two selected perturbation depths (20% and 90%). These maps show the spikes generated at the VCSEL-neuron's output when an identical input perturbation enters the laser 150 times. Green colour represents high output signal amplitude (spike crests) while blue represents low-amplitude (background, no activity). The maps further confirm that both the low and high amplitude input signals resulted in spiking responses in all the cycles. The spikes triggered by the low-amplitude perturbation exhibit increased temporal jittering. This effect vanishes when perturbations with larger amplitude are used. Figure 4(c) shows the average spike activation delay for each of the stimuli strengths, calculated from 150 measurement cycles. The last spike was used as a timing reference between measurement cycles, hence its delay is fixed to zero. For very low-amplitude stimuli, there is a significant increase in mean spike onset delay as well as the standard deviation (due to the jittering). The high R 2 -fit suggests that for linearly increasing perturbation strength, the mean spike activation time in the VCSEL-neuron decays exponentially and later stabilizes near a certain fixed value. It is worth noting that all recorded spiking responses in this experiment had the same amplitude. The results shown on figure 4(b) suggest that the larger drop amplitudes result in improved control over the spike activation process, both in terms of stable onset delay and reduced temporal jittering. Furthermore, these observations hint towards the capability of the VCSEL-neuron to encode stimuli information into spike latency coding, an encoding approach observed, among others, in retinal ganglion cells [57]. This approach will be further investigated in following studies.

Information encoding with precise spike timing in a VCSEL-neuron
By injecting controlled intensity perturbations into the VCSEL-neuron with: (a) temporal separation higher than the system's relative refractory period, and (b) sufficiently high amplitude beyond the excitation threshold, spike trains with precise timing can be obtained with high reliability and low temporal jitter. We utilize this capability of the VCSEL-neuron to demonstrate its function for information encoding of digital binary signals into spike train representation in the optical domain. Figure 5(a) plots the time-series captured at the output of the VCSEL-neuron under the injection of a binary ASCII representation of the word 'IoP' , an acronym of the Institute of Photonics at the University of Strathclyde. Figure 5(a) shows an overlay of five consecutive responses from the VCSEL-neuron to the injection of the 'IoP' pattern, clearly revealing that the binary digital information is encoded into a spike train generated by the VCSEL-neuron. The binary representation is shown as series of ones and zeros at the top of the figure. The injection signal is modulated using unipolar, return-to-zero (RZ) serial encoding scheme. Since spikes in the VCSEL-neuron are elicited by short optical power drops, the Mach-Zehnder modulator is biased in such an operating point that the polarity of the modulation RZ waveform is inverted, with zero level now set at higher power level. In accordance with results shown in previous section, any injection intensity drop that overcomes the threshold (see figure 1(b)) causes spiking. In order to obtain reliable, low latency spikes for each bit, full range of the AWG DAC was used, resulting in approx. 18% power drops in the injection signal (K p = 0.18 , see [67] for more details). The temporal length of one bit is 0.916 ns, with 54.5% duty cycle. This corresponds to a bit rate of 1.09 Gbps. For a '1 ′ input bit, a single spike is excited at the VCSEL-neuron's output, whilst for a '0 ′ input bit, the VCSEL-neuron remains quiescent as it is in its injection locked (steady) state. In agreement with [58], it was observed that the duration of the incoming perturbations (input bits in this case) does not have an effect on the response as long as it remains shorter than the absolute refractory period of the device. The temporal stability map shown in figure 5(b) depicts in a single plot 150 consecutive responses (cycles) of the VCSEL-neuron to the same input 'IoP' digital pattern, clearly illustrating the reproducibility of the proposed digital-to-spiking encoding mechanism. Good spike timing consistency is further confirmed by the means of standard deviation of activation times, shown in figure 5(c). By operating at 0.916 ns bit −1 , the VCSEL-neuron facilitates binary-to-spike conversion rate of 1.09 Gbps, over four times higher than the values 0.27 Gbps theoretically predicted on other photonic neuronal models [59]. It is worth noting that the phrase length (24 bits) was selected for proof-of-concept due to its easy visualization. In principle, there are no limits imposed by the VCSEL-neuron in terms of encoded phrase length.

Information encoding with spike firing rate-coding
The functionality of the VCSEL-neuron is not only limited to digital input operation. Figure 6 demonstrates the capability of the VCSEL-neuron to encode the intensity of varying amplitude stimuli into its spike firing frequency. In biological neurons, this behaviour is referred to as Class 1 neural excitability and it can be observed for example in cortical excitatory pyramidal neurons [60]. Unlike classical analog-to-digital converters, which convert continuous-time signals into discrete-time, binary-coded forms, the VCSEL-neuron converts the signal magnitude into a spike firing rate, effectively realizing analog-to-spike rate-coding directly in the optical domain. Figures 6(a) and (b) plot results measured at the VCSEL-neuron's output under the injection of a sawtooth-shaped input signal with a repetition frequency of f = 25 MHz. As the intensity of the injected signal decreases, it crosses the activation threshold of the VCSEL-neuron, resulting in the firing of spikes at its output. These are initially triggered at a low spike firing rate, which rapidly grows as the input intensity decreases. A similar behaviour can be observed under the injection of a sinusoidal input signal with a frequency of f = 25 MHz, as seen in figures 6(c) and (d). In the second case, it is clearly observable how the spike firing rate periodically increases and decreases in response to the input intensity-modulated signal. To the best of our knowledge, this is the first demonstration of spike rate-coding as a function of input stimuli intensity in a photonic neuron. The temporal maps and histograms in figures 6(b) and (d) merge in a single plot the response of the VCSEL-neuron to 150 consecutive input triangle and sinusoidal input signals. For better clarity, a single response (or cycle) is highlighted in colour shaded areas in figures 6(a) and (c). For both sawtooth (periodically monotonous) and sinusoidal (non-monotonous) cases of injected analogic signals, consistent behaviour can be observed in the output patterns. Simultaneously, a clear relationship between the amplitude of the input signal and the observed spiking frequency at the output of the VCSEL-neuron is seen. As the modulation strength increases in time, the density (frequency) of spikes increases across all the recorded cycles. Unlike in the precise spike timing case (see figure 5), there is now higher randomness in the timing of the first spike between the cycles. This can be attributed to small noise variations in system conditions between the subsequent measurements, which cause the laser to cross the spiking boundary at slightly different time among different cycles. However, since the information in the rate-coding approach is represented in the frequency and not the spike time itself, we believe this behaviour is not detrimental to the reported functionality. Once the system crosses the spiking regime boundary, the timing of subsequent spikes among the cycles exhibits good level of consistency. It is also worth noting that in situations where the second spike is elicited within the relative refractory period (like continuous spiking at high frequencies), a slight decrease in the output spike amplitude can be observed. Since this amplitude suppression effect is also observed in biological neuronal models, we believe this is not detrimental to the computational capability of a system that would rely on rate-coded VCSEL-neurons. The effect of reduced spike amplitude during excitation in the relative refractory period was reported in the classical Hodgkin & Huxley dynamical model of squid axons [68]. Based on this effect, these neuronal models exhibit spike amplitude attenuation in their elicited spike trains during high-rate continuous spiking [69]. Figure 6(e) shows the measured average inter-spike temporal separation for the case of the sawtooth wave modulation, measured across 135 cycle repetitions. The average is calculated for the first 18 spikes in each cycle, with an exponential line of best fit shown as dashed red line. For this case of approximately linearly increasing modulation strength, the inter-spike temporal spacing follows an exponential decay, until it reaches minimal values close to the absolute refractory period of the system. The exponential relation between observed spiking frequency and function of stimuli strength is in agreement with behaviour observed in Class 1 neurons [61] modelled using the Wilson-Cowan model [62], further confirming the similarity between the dynamical behaviour of our VCSEL-neuron and biological neurons.

Conclusions
We demonstrate experimentally the information encoding capabilities of a spiking photonic neuron running at ultrafast rates, based on an off-the-shelf VCSEL operating at telecom wavelengths. First, we experimentally investigate the refractory behaviour of the VCSEL-neuron and observe both absolute and relative refractory periods of ∼ 0.8ns and ∼ 1.1ns respectively. We also demonstrate and evaluate how the amplitude of the input stimulus influences the spike onset timing, permitting the use of spike latency to encode the intensity information of incoming stimuli. Based on the refractory period findings, we demonstrate experimentally the capability of the VCSEL-neuron to perform all-optical spike information encoding. Two different ways of signal encoding into spikes are introduced. First, we demonstrate precise spike timing encoding, where we utilize standard RZ serial encoding scheme for conversion of a 24-bit binary sequence into an optical spike train at rates over 1 Gbps. This bit sequence was selected as a representation of the three-letter word 'IoP' (acronym of Strathclyde's Institute of Photonics) in ASCII, but the same VCSEL-neuron is also capable of encoding arbitrarily long bit sequences. Secondly, we demonstrate biologically inspired rate-coding for periodically monotonous and non-monotonous input signals, where the strength (amplitude) of input optical stimuli is translated into the firing rate of the VCSEL-neuron. Both the temporal and rate-coding of spike trains have been observed in biological neurons, and as such, a device enabling reliable encoding using both of these information representation schemes at very high speeds (GHz rates) will be crucial for interfacing future photonic neuromorphic systems with both digital and analog inputs from devices such as sensors as well as additional (optical) processing and communication systems.