Dynamical model

The dynamical system underlying the behavior of our spike processing unit is a gain-absorber cavity model, describing single mode lasers with gain and SA sections. Despite its simplicity, it can exhibit a large range of dynamical behaviors48 and has been investigated in various contexts as the basis for an optical processor49. The system, in its simplest form, can be described using the following undimensionalized equations21,42:

where G(t) models the gain, Q(t) the absorption and I(t) the laser intensity. A is the gain bias current, B is the absorption level, γ G is the gain relaxation rate, γ Q is the absorber relaxation rate, γ I is the inverse photon lifetime and a is a differential absorption relative to the gain factor. We represent the spontaneous noise contribution to intensity via ϵf(G), for small ϵ and time-dependent input perturbations as θ(t). Although this model is typically used for single-mode lasers, it can also be applied to multi-mode lasers, provided that the modes all behave similarly and phase locking effects between modes are neglected (see Methods: fiber laser simulation).

When the dynamics of pulse generation are fast compared to the dynamics of the gain medium, one can compress the internal dynamics and obtain an instantaneous pulse-generation model42:

where the input θ(t) can include spike inputs of the form for spike firing times τ i , G thresh is the gain threshold and G reset ~ 0 is the gain at transparency.

This system is analogous to a leaky integrate-and-fire (LIF) neuron model, commonly employed in computational neuroscience for modeling biological neural networks. Although it is one of the simpler spike-based models, the LIF model is capable of universal computations50 and the transmission of information through spike timings51. The gain-absorber system has been predicted to exhibit cascadability, logic-level restoration and input-output isolation42, satisfying the basic criteria for optical computing18.

Excitable Laser Systems

Our demonstration of spike processing is based on a graphene fiber ring laser platform (Fig. 1). For comparison, we also perform numerical simulations of an analogous proposed integrated device (Fig. 2). Both devices along with their respective simulation models and parameters are described in detail in the Methods section. Although the accurate physical models differ for each device, the behavior of both models are fully encompassed in the dynamics of the simple, undimensionalized Yamada system (above). The fiber ring laser contains an erbium doped fiber amplifier (gain section) and liquid exfoliated graphene (absorber section), interacting with one another in a fiber ring (cavity). The ring laser pulses periodically if driven above a threshold, modulated by the passive saturation of graphene absorption. This behavior has been studied in the context of high power, wideband passively Q-switched lasers, for which graphene has many favorable properties52.

Figure 1 Graphene excitable fiber laser. The cavity consists of a chemically synthesized (see Methods: graphene sample preparation) graphene SA (GSA) sandwiched between two fiber connectors with a fiber adapter and a 75-cm long highly doped erbium-doped fiber (EDF) as the gain medium (see Methods: excitable fiber ring laser cavity). The EDF is pumped with a 980 nm laser diode via a 980/1550 nm wavelength-division multiplexer (WDM). An isolator (ISO) ensures unidirectional propagation. A polarization controller (PC) maintains a given polarization state, improving output pulse stability. The 20% port of an optical coupler provides the laser output at 1560 nm. To induce perturbations to the gain, 1480 nm excitatory pulses are incident on the system via a 1480/1550 nm WDM. These analog inputs—from other excitable lasers, for example—are directly modulated with an arbitrary waveform generator. Right: illustration of different possible phase-space dynamics associated with the laser intensity I and gain G of the system (see Text), as the various physical parameters (pump power, length of cavity, absorption) are varied. The desired excitable behavior, corresponding to the second phase-space schematic (red), is achieved when the parameter regimes drive the system toward a so-called homoclinic bifurcation1. Full size image

Figure 2 Proposed integrated graphene excitable laser. (a) Cutaway architecture of the hybrid InGaAsP-graphene-silicon evanescent laser (not to scale) showing a terraced view of the center. The device comprises a III-V epitaxial structure with multiple quantum well (MQW) region bonded to a low-loss silicon rib waveguide that rests on a silicon-on-insulator (SOI) substrate with a sandwiched heterostructure of two monolayer graphene sheets and an hexagonal boron nitride (hBN) spacer. Structure is very similar to70 with added graphene and boron nitride sections between the layer stacks. Note that, unlike the fiber laser of Fig. 1, the gain section of this structure is electrically pumped. The cavity and waveguide are formed by the presence of a half-wavelength grating in the silicon. Silicon gratings provide feedback for the lasing cavity. The full cavity structure includes III-V layers bonded to silicon and a quarter-shifted wavelength grating (quarter shift not shown). The laser emits light along the waveguide structure into a passive silicon network. (b) Cross-sectional profile of the excitable laser with an overlaid electric field (E-field) intensity profile. The optical mode of the laser lies predominately in the silicon waveguide with a small portion of the mode overlapping the QWs of the III-V structure for optical gain and the 2D materials heterostructure for absorption. The silicon waveguide has a width, height and rib etch depth of 1.5 μm, 500 nm and 300 nm, respectively. The calculated overlap of the optical mode with the silicon waveguides is 0.558 while there is a 0.00046 overlap in the graphene sheets and 0.043 overlap in the QWs. The E-field intensities are calculated at a wavelength of 1.5 μm. Graphene’s thickness of 0.335 nm and absorption coefficient of 301,655 cm−1 are used for the simulations (see Methods: integrated-device simulation). Full size image

The integrated device contains electrically pumped quantum wells (gain section), two sheets of graphene (absorber section) and a distributed feedback-grating (section). In this design, we consider a hybrid silicon III-V laser platform in which the graphene layers are sandwiched in between the silicon and III-V layers. The hybrid III-V platform is highly scalable and amenable to both passive and active photonic integration53. The integrated device is capable of exhibiting the same behaviors as the fiber prototype, but on a much faster time scale and with lower pulse energies. Supplementary Figure S1 compares the pulse repetition rate and pulse widths as a function of input power between the integrated device and fiber laser. In both cases, the rate of output pulses depends monotonically on the amount of power being consumed. This has many behavioral similarities with the behavior of rate neurons, which code information through spike frequency modulation54. Although both lasers consume similar amounts of power, the integrated device pulses are ~106 times faster. This corresponds to a ~106 times decrease in the energy consumed per pulse. The devices (and their respective simulation models) are described in more detail in the Methods section.

Excitability

We demonstrate that both the fiber ring laser and the integrated device are excitable and capable of performing spike processing tasks. Excitability is defined by three main criteria: (i) an unperturbed system rests at a stable equilibrium; (ii) a perturbation above the excitability threshold triggers a large excursion from this equilibrium; and (iii) the system then settles back to the attractor in what is called the refractory period, after which the system can be excited again55.

Figure 3a–c demonstrates excitability within the fiber ring laser. In this system, an excitatory pulse increases the carrier concentration within the gain region by an amount proportional to its energy (integrated power) through gain enhancement. Beyond some threshold excitation energy, the absorber is saturated, resulting in the release of a pulse. This is followed by a relative refractory period during which the arrival of a second excitatory pulse is unable to cause the laser to fire as the gain recovers. The system is also capable of emitting spike doublets or triplets (see Fig. 3d) in which the inter-spike timing encodes information about the pulse width and amplitude, a useful encoding scheme for selective activation56.

Figure 3 Excitable dynamics of the graphene fiber laser. Note the blue and red curves correspond to input and output pulses, respectively. (a–c) Excitatory activity (temporal integration of nearby pulses) can push the gain above the threshold, releasing spikes. Depending on the input signal, the system can lead to (a) repeatable pulse generation, or suppressed response due to the presence of either (b) sub-threshold input energies (integrated power ) or (c) a refractory period during which the gain recovers to its resting value and the laser is unable to pulse (regardless of excitation strength). (d) Typical bursting behavior i.e. emission of doublets (two spikes) and triplets (three spikes) when a strong input drives the system over the threshold to fire repetitively. (e,f ) Restorative properties (repeatable pulse reshaping) of spike processing; inputs with either the (e) same or (f ) different energies. Test conditions: (a) three 4 μs input pulses separated by 3 μs followed by a 10 μs pulse after 52 μs delay and two 5.5 μs pulses separated by 4.5 μs after 50 μs delay; (b) three 4 μs input pulses separated by 3 μs followed by a 4.5 μs pulse after 52 μs delay and two 5.5 μs pulses separated by 4.5 μs after 50 μs delay; (c) five 10 μs input pulses separated by 10, 50, 20 and 25 μs delays, respectively; (d) three input pulses with widths 25, 40 and 80 μs separated by 55 μs; in (a–d) the 980 nm pump is biased at 60.4 mW. (e) three input pulses with energies 252, 264 and 256 nJ separated by 85 and 65 μs, result in outputs with pulses widths (i) 7.4, (ii) 6.1 and (iii) 6.9 μs, respectively; 980 nm pump is biased at 61 mW. (f ) three input pulses with energies of 265 nJ separated by 50 and 65 μs, result in outputs with pulse widths (i) 6.3, (ii) 6 and (iii) 6.5 μs; 980 nm pump is biased at 67.6 mA. Full size image

Since pulse generation results from internal dynamics in excitable systems, such systems exhibit important restorative properties. Different input perturbations often result in the same output, an important criteria for cascadability. Figure 3e,f illustrate the response of the device as a result of a variety of input pulses. The excitable system responds in a stereotyped and repeatable way; all emitted pulses having identical pulse profiles. Outputs trigger asynchronously from input pulses, preserving analog timing information.

Figure 4 show some key behaviors associated with excitability. Figure 4a provides information about the refractory period for both the fiber laser and integrated device, which sets an upper bound on the pulse rate for a given unit. Similarly, Fig. 4b shows the output pulse width as a function of an input pulse for both the integrated and fiber lasers. The fiber experiment is corroborated with matching simulation results (see Methods: fiber laser simulation). Although the pulse profile stays the same, its amplitude may change depending on the value of the perturbation. The integrated device exhibits the same behavior on a much faster time scale, recovering in nanoseconds with pulse widths in picoseconds, a factor of about ~103 and ~106 faster, respectively than the fiber prototype. The width of the pulses—which puts a lower bound on the temporal resolution of the information encoded between spikes—is bounded by both the SA recovery time and the round trip cavity time. Although graphene’s incredibly fast response time (~2 ps) makes it effectively instantaneous in the fiber lasers, our simulations suggest that graphene can shorten the pulse widths in an integrated device. The relative refractory period on the other hand is bounded by the speed of either the gain or the SA, although the gain recovery times tends to be larger. Comparing to the recently demonstrated semiconductor micropillar laser49, the integrated device’s refractory period is slightly slower (2 ns vs. 0.5 ns) while the pulse widths are 100 times smaller (2 ps vs. 200 ps). Also, comparing to the resonant tunneling photo-detector driving a laser diode31, the proposed integrated laser’s refractory period is 250 times faster (2 ns vs. 500 ns) with pulse widths around 105 shorter (2 ps vs. 200 ns).

Figure 4 Second-order properties of excitability. (a) Response of the excitable laser (output pulse energy) to a second input pulse as a function of the interspike interval between two identical (first and second) excitatory pulses after the first pulse has triggered an excitable response. Both the fiber based excitable laser experimental results and the integrated excitable laser simulations exhibit absolute refractory periods (where the second pulse produces no output) and relative refractory periods (where the output response is reduced from its resting value). However, the later operates ~103 faster (ns compared to μs) with ~103 lower output pulse energies (pJ compared to nJ). Insets show the transient dynamics of the integrated excitable laser, i.e. the intensity I(t) and recovery of gain carriers G(t) as a result of input signal θ(t), before (i) and after (ii) its refractory period. (b) Excitable laser’s output behavior in response to a single input pulse with different energies. The integrated excitable laser simulations also follows a similar relationship profile but with output pulse widths ~106 smaller (ps compared to μs). (c) Response of the excitable laser when implemented a coincidence detector: the excitable laser is biased such that it will not fire unless two excitatory pulses are temporally close together. Output response is strongly dependent on the temporal correlation of two inputs. Average input power is kept constant with changing pulse interval. Insets show the simulated pulse dynamics for the integrated laser for pulses that are (i) closer vs (ii) farther apart. Full size image

Temporal pulse correlation is an important processing function that emerges from excitability. An integrating excitable system is able to sum together multiple inputs if they are close enough to one another in time. This allows for the detection of pulse clusters, or potentially, coincidence detection of pulses across channels through the use of incoherent optical summing47. Coincidence detection underlies a number of processing tasks, including associative memory57 and a form of temporal learning called spike timing-dependent plasticity (STDP)58,59. Temporal pulse correlation in the fiber laser experiment and simulation and integrated laser simulation are shown in Fig. 4c. Reducing the time interval between input pulses (i.e. simultaneous arrival) results in an output pulse. Although the fiber laser can function at kHz speeds, the internal dynamics of the integrated device allow it to function much faster, putting it in the GHz regime.

Temporal pattern recognition

We demonstrate a simple pattern recognition circuit using several interconnected graphene fiber lasers. Pattern recognition of spatiotemporal phenomena is critical in the real-time processing of analog data. In the context of biological neural systems, networks of spiking neurons convert analog data into a spikes and recognize spatiotemporal bit patterns60. Spatiotemporal patterns play an important role in both visual61 and audio62 functionalities and underly the formation of polychronized groups in the context of learning63.

As shown in Fig. 5a, we construct a simple two-unit pattern recognition circuit by cascading two excitable graphene (sic.) lasers with a delay τ between them. In our case the objective is to distinguish (i.e. recognize) a specific input pattern: a pair of pulses separated by a time interval Δt ≈ τ, equal to the delay between the excitable lasers. Coincidence detection provides the discriminatory power for classification.

Figure 5 Temporal pattern recognition. (a) Simple circuit with two cascaded graphene excitable lasers. (b) Measured output pulse peak power and pulse duration as a function of the time interval between the two input pulses. (c) Measured input and output waveforms at specific instances: (i) Δt − τ = −45 μs, (ii) Δt ≈ τ = 135 μs and (iii) Δt − τ = 35 μs. The output pulse energy is largest when Δt ≈ τ showing the system only reacts to a specific spatiotemporal input pattern. Full size image

A pulse doublet travels to both lasers, created using a modulator and arbitrary waveform generator. The output from the first laser travels to the second through a long single mode fiber (~km), which acts as a delay element. The second laser is biased with a larger threshold such that it will not fire unless two excitatory pulses—the original input and output from the first laser—arrive at the same time (Δt ≈ τ = 135 μs) (see Fig. 5b). Synchronous arrival of these two spikes causes the release of a pulse. Experimental time traces for the inputs and outputs are shown in Fig. 5c. A well-formed output pulse appears only for the desired two-pulse pattern.

One can reduce the occurrence of pulses that are at a non-normalized amplitude (i.e. Figs 5b and 3c) through a sharper threshold function. The sharpness of the transition of these curves depends on where the system resides in the parameter space. Biasing the pump (980 nm) closer to the laser threshold decreases the excitable threshold49 and can make these transition sharper. This ratio can be optimized for application-specific purposes. In this case, for example, the first laser acts as a nonlinear stage to simply regenerate input pulses because it is biased close to the threshold. The second laser, on the other hand, requires two coincident pulses to reach its threshold. It therefore plays the role of a pattern classifier.

Between the laser stages, a photodetector (PD), rather than direct optical input, modulates the laser driver (LD) (allowing wavelength conversion from 1560 to 1480 nm) (see Methods: excitable fiber ring laser cavity). This PD-driven architecture (see outlined dashed box in Fig. 5a) has been explored in an integrated context43, as a potential route to scalable on-chip networking47. The dynamics introduced by the PD are analogous to synaptic dynamics governing the concentration of neurotransmitters in between signaling biological neurons64. This simple circuit demonstrates several important features necessary for robust optical processing: well isolated input/output ports allow for the construction of feedforward networks and the spatio-temporal recognition of spikes allows the system to classify patterns. More complex recognition and decoding would be possible as the system is scaled.

Stable recurrent circuit

We also demonstrate a self-recurrent graphene laser that can sustain a pulse traveling around the loop ad infinitum, providing a proof-of-principle demonstration of cascadability and pulse regeneration. Recurrently-connected dynamical networks which evolve toward a stable pattern over time (i.e. attractor networks) can exhibit hysteresis and play a critical role in memory formation and recall65. Equivalently, since a single unit with a self-referent connection can be mapped to an infinite chain of lasers, this system can be viewed as a demonstration of stability in arbitrarily many layered feed-forward networks.

Figure 6a illustrates an excitable graphene laser with a self-referent connection. The output is fed back to the input via single-mode fiber which acts as a delay element (100 μs). An electronic weight W controls the modulation depth of the PD providing an all-or-none response depending on whether it is above or below a given threshold. Figure 6b depicts the system’s ability to demonstrate bistability when feedback is present. It is capable of settling to an attractor in which a single pulse travels around the loop indefinitely. This circuit represents a test of the network’s ability to handle recursive feedback and the stability of the pulse is a sign that the system is cascadable.