Excitable Laser Processing Network Node in Hybrid Silicon: Analysis and Simulation

Excitable laser processing network node in hybrid silicon: analysis and simulation

Mitchell A. Nahmias,∗ Alexander N. Tait, Bhavin J. Shastri, Thomas Ferreira de Lima, and Paul R. Prucnal Electrical Engineering Dept., Princeton University, Princeton, NJ 08544, USA.

∗[email protected]

Abstract: The combination of ultrafast laser dynamics and dense on-chip multiwavelength networking could potentially address new domains of real-time signal processing that require both speed and complexity. We present a physically realistic optoelectronic simulation model of a circuit for dynamical laser neural networks and verify its behavior. We describe the physics, dynamics, and parasitics of one network node, which includes a bank of ﬁlters, a photodetector, and excitable laser. This unconventional circuit exhibits both cascadability and fan-in, critical properties for the large-scale networking of information processors based on laser excitability. In addition, it can be instantiated on a photonic integrated circuit platform and requires no off-chip optical I/O. Our proposed processing system could ﬁnd use in emerging applications, including cognitive radio and low-latency control. © 2015 Optical Society of America OCIS codes: (140.3538) Lasers, pulsed; (200.4700) Optical neural systems; (250.3140) Inte- grated optoelectronic circuits; (320.7085) Ultrafast information processing.

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#246538 Received 22 Jul 2015; revised 15 Sep 2015; accepted 22 Sep 2015; published 2 Oct 2015 © 2015 OSA 5 Oct 2015 | Vol. 23, No. 20 | DOI:10.1364/OE.23.026800 | OPTICS EXPRESS 26801 (ORNoC): Architecture and design methodology,” in “Design, Automation Test in Europe Conference Exhibition (DATE), 2011,” (2011), pp. 1–6. 41. D. G. Feitelson, Optical Computing: A Survey for Computer Scientists (MIT Press, USA, 1988).

1. Introduction Recently, there has been a surge of interest in the hybridization of photonic and electronic physics to achieve unique processing capabilities. In this context, there has been significant research in using laser dynamics for both information processing [1–3] and communication [4,5]. Multiplexing in optical networks have also been exploited for similar gains in performance [6, 7]. A more specific approach involves studying the dynamical property of excitability in lasers, in which discrete pulses are generated in response to perturbations that exceed a threshold [8–11]. Excitable lasers can process information in a way that resembles spiking in biological neuron models. In comparison, however, lasers can exhibit dynamics roughly eight orders of magnitude faster while being mathematically isomorphic to their biological counterparts [12]. In this manuscript, we simulate a unified, realistic model of a reconfigurable excitable laser processor, recently proposed in [13]. This model includes a detailed analysis and simulation of a processing-network node (PNN) as defined in a recent laser networking scheme [14]. Although parts of the signal pathway have been proposed [13–15], no single model has fully characterized the signal pathway of a PNN. The device uniquely allows for a large fan-in ( 10s to 100s ∼ per unit) without routing or packet switching, and utilizes the ultrafast dynamics of lasers for high bandwidth ( GHz) processing. We describe the photonic circuit techniques used in our ∼ approach and simulate the device based on experimentally measured parameters in a standard hybrid III-V/silicon platform [16]. The model allows us to explore critical properties such as energy consumption, cascadability, and signal bandwidth, and verifies that the PNN can be fabricated in a realistic device structure. Spiking neural networks (SNNs)—systems whose communication channels code information in events rather than bits—have received significant attention as an alternative to the von Neumann paradigm of computation. Hardware SNNs take advantage of distributed, sparse, and robust coding schemes to perform computations to minimize size, weight and power, and have been utilized both as biological network simulators and low-power data processors. Interconnects in neural network architectures require many-to-one fan-in and multicast communication, as opposed to von Neumann processors, where a number of point-to-point memory-processor communication bottlenecks arise [17]. In distributed systems, processing performance is more closely related to inter-node communication. Distributed electrical wires can introduce performance challenges in a multicast network due to RC and radiative physics, in addition to the typical bandwidth-distance-energy limits of point-to-point electronic links. Since biological-time SNNs typically operate far slower than underlying electronics, many large-scale systems employ a form of time-division multiplexing (TDM) or packet switching, notably, address event representation (AER). TDM and AER abstractions allow networks to have virtual interconnectivity densities that exceed the wire density by a factor related to the sacrificed bandwidth, which can be orders of magnitude [18]. While AER is effective for SNNs running at biological time scales, networks running at faster-than-biological time scales experience new challenges and limitations. The fastest large- scale SNN emulator is capable of an approximately 10 MHz synaptic simulation (104 speed-up over biological-time) [19]. In the GHz regime, the gap between this operation bandwidth and feasible real signal bandwidths shrinks, which creates a harsh tradeoff between operation bandwidth and virtual interconnectivity. Neuron emulators relying on electrical wires cannot, in the worst case, simultaneously cope with virtual interconnectivity and GHz signal bandwidths. This is a significant limitation for neuromorphic networks, which rely on high interconnectivity

w input 1 output WDM spectral output spikes spikes input pulses ﬁlter pulses

w2 …

wN dynamical laser system weights

Fig. 1. (a) A general depiction of a processor in a SNN. Input channels are weighted by wi, where they are summed together into a dynamical system (the neuron), which outputs its own set of spikes. (b) Concept of a photonic spike processor. Inputs are wavelength division multiplexed (WDM) channels on a single input channel. The spectral filter provides the weighted sum of inputs, which are summed together and drive a single excitable laser. The laser emits its a set of its own pulses into the network. densities for complex, parallelized operations. In contrast, it is well known that photonic technology possesses favorable properties for communication and can come to bear on problems in on-chip interconnect performance [20]. The research community has proposed a number of dynamical laser models [12, 21–26]. Unfortunately, many of these models suffer from network or interconnection limitations. Inter- connected systems require each node to robustly drive a number of other nodes—referred to as cascadability and fan-in (Fig. 1). If inputs are optically injected into the gain section, the pump wavelength must be shorter than that of the signal for population inversion to be achieved. As a result, network feedback requires wavelength conversion, a fairly expensive operation in terms of size, weight, and power. Alternatively, pumping the intensity directly requires a match with the cavity resonance, limiting inputs and outputs to the same wavelength. Optical fan-in with a single wavelength requires coherent optical summing to avoid beat noise, a practical impos- sibility in unsynchronized laser systems [27]. These complications would offset many of the advantages of using optical physics over more traditional RF electronic approaches. To address the issues of cascadability and fan-in for networks of ultrafast spiking laser neurons, a scalable networking architecture was recently proposed [28]. High interconnect density is achieved through the use of wavelength division multiplexing (WDM) signals fanning-in to a single photodetector, which directly drives an electronically-modulated laser neuron (see Fig. 1). In this setting, electronic physics act as a powerful complement to optics, which, as a result of the lack of charge and bosonic properties of photons, are unwieldy for cascadable computation on their own. O/E/O conversion is useful in many contexts, although it is conventionally associated with a high performance cost to demultiplex, digitize, and remodulate signals. In the neural networking context, the electronic link between a photodetector-laser pair can make a cascadable node without necessarily performing these costly functions (see Fig. 2). This work simulates a unified and realistic model, verifying that fabricating a PNN is within today’s technological capabilities in a hybrid silicon/III-V platform. This is an important stepping-stone towards larger scale simulations and the verification of these concepts in an experimental device. Application of the technology, once developed, would enable a new processing domains, including the manipulation of ultrafast physical phenomena, low-latency control, and—perhaps most pertinently—novel communication and classification schemes in the radio frequency (RF) domain. Examples of the latter include blind source separation, RF fingerprinting, spectral hole exploitation (i.e. non-interfering transmission and reception), and

#246538 Received 22 Jul 2015; revised 15 Sep 2015; accepted 22 Sep 2015; published 2 Oct 2015 © 2015 OSA 5 Oct 2015 | Vol. 23, No. 20 | DOI:10.1364/OE.23.026800 | OPTICS EXPRESS 26803 novel information encoding/retrieval through the direct modulation of carrier waveforms.

2. Methods As illustrated in Fig. 2, the PNN consists of three primary components: reconfigurable spectral filters, photodetectors, and an excitable laser. Although an inhibitory photodetector is shown in the schematic diagram, the behavior of the devices are similar, aside from a reversal of the ground and signal pads. In this analysis, we only examine the excitatory photodetector pathway for brevity and conciseness. In this scheme, WDM spike signals arrive along a dedicated waveguide. Weights are applied to each channel via a set of tunable spectral filters. Inputs from other laser neurons are weighted in the optical domain before reaching the photodetector. The photodetectors produce a photocurrent summing the total optical power. De- multiplexing many input channels is not necessary because the incoherent sum of all WDM channels is intentionally computed by the photodetector. The photodetectors receive optical pulses from a network and produce a current signal which modulates the laser carrier injection. The excitable laser performs nonlinear discrimination and regenerates the pulsed signal, analogous to the neural axon hillock. The photodetector front-end proposed here allows for significant signal fan-in, while tunable filters allow adjustments of the weights between neurons, allowing for network reconfigurability.

2.1. Technology Platform We describe an instantiation of the PNN in the hybrid/III-V platform. This platform includes III-V materials that are bonded to underlying passive silicon photonic interconnection networks. In a typical device, optical modes are hybridized between the silicon and III-V layers simultaneously [29]. Nanostructures necessary for waveguides, resonators, and gratings are fabricated strictly in silicon, while the III-V layers provide optical gain. This platform was chosen for its convenient ability to spatially organize many active elements together onto a single PIC. The analysis below uses realistic parameters derived from experimental papers, and is divided into three primary sections: (a) ﬁltering (i.e. neural weighting) of input WDM-pulsed signals, (b) summation and electrical conversion from photodetector to the adjacent laser, and (c) dynamics present within the laser cavity itself. A schematic of the full modeling structure is shown in Fig. 3.

2.2. Filter Bank In the proposed WDM network configuration, a large number of processing nodes share a single waveguide and selectively couple light in and out through a bank of adjacent add-drop filters. The pulses coupled from the broadcast waveguide are emitted from various lasers at different wavelengths, and have approximately equal amplitudes. The filter are tuned on and off resonance to control the weight or strength of connection between each element. The result is a series of pulses at different frequencies with an array of different amplitudes [30]. In a given broadcast ring with N nodes, N filters are associated with each node to adjust the strength of a total of N2 connections. In this analysis, we include only four in a given node for simplicity. We model the transmission function of a resonator drop filter with a Lorentzian function of the form: 1 Q T(δ) = 2 , where δ = (ω ω0) (1) 1 + δ ω0 − where δ is the linewidth-normalized frequency, Q is quality factor (Q 10,300), and ω is ≈ 0 the peak center frequency. Let δtun represent the tunability of each filter, and ∆δ the spacing between adjacent filters. We assume each tunable filter is composed of a single add-drop ring,

#246538 Received 22 Jul 2015; revised 15 Sep 2015; accepted 22 Sep 2015; published 2 Oct 2015 © 2015 OSA 5 Oct 2015 | Vol. 23, No. 20 | DOI:10.1364/OE.23.026800 | OPTICS EXPRESS 26804 2 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. X, MONTH DATE 2013 platform’s reliance on nonlinear fibers and other similar tech- nologies have made demonstrations bulky (on the order of meters), complex, and power-hungry (hundreds of watts). The platform is simply unscalable beyond2 a few neurons. Integrated IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. X, NO. X, MONTH DATE 2013 lasers, in contrast, are physically compact and are capable of using feedback rather than feedforwardplatform’s dynamics reliance on to nonlinear radically fibers and other similar tech- nologies have made demonstrations bulky (on the order of enhance nonlinearity. Feedback allowsmeters), for complex, the emergence and power-hungry of (hundreds of watts). The more complex behaviors, includingplatform bistability, is simply the unscalable formation beyond a few neurons. Integrated of attractors, and excitability. lasers, in contrast, are physically compact and are capable of using feedback rather than feedforward dynamics to radically Excitability is a dynamical systemenhance property nonlinearity. that Feedback underlies allows for the emergence of all-or-none responses. Its occurrencemore in complex a variety behaviors, of different including bistability, the formation lasing systems has received considerableof attractors, interest and excitability. [18], [19]. Excitability is a dynamical system property that underlies Excitability is also a critical propertyall-or-none of biological responses. Itsspiking occurrence in a variety of different neurons [20], [21]. More recently, severallasing systems excitable has received lasers considerablehave interest [18], [19]. Excitability is also a critical property of biological spiking (a) demonstrated biological-like spikingneurons features. [20], [21]. One More proposal recently, several excitable lasers have weighted !x 1(t)(t) (a) suggests using excitability in semiconductordemonstrated lasers biological-like [22], spiking [23] features. One proposal1 inputs! suggests using excitability in semiconductor lasers [22], [23]w1τ based on weakly broken Z2 symmetry close to a Takens- 1 1 based on weakly broken Z2 symmetry close to a Takens- summation! Integrate" Threshold" Bogdanov bifurcation, yet anotherBogdanov suggests bifurcation, using emergent yet another suggests using emergent xj(t) w x−(t)τ biological features from polarizationbiological switching features in from a vertical- polarization switchingxi(t)(t) in a vertical-jxij (ti j) I (t) j wτ s(t) cavity surface-emitting laser (VCSEL)cavity [24]. surface-emitting However, laser these (VCSEL) [24]. However,j i j these ∫ models have yet to demonstrate some key properties of spiking Σ output! Reset models have yet to demonstrate someneurons: key properties the ability to ofperform spiking computationsxi(t) without = Σn δ(t-τ informa-n) neurons: the ability to perform computationstion degradation, without clean-up informa- noise, or implement algorithms. synaptic! LIF neuron ! wNτ In this paper, we show for the first time that a photonict com-N N variable! model! tion degradation, clean-up noise, orputational implement primitive algorithms. based on an integrated, excitable laser with (b) x N(t)(t) In this paper, we show for the firstan time embedded that asaturable photonic absorber com- (SA) behaves analogouslyN to Fig. 2. (a) IllustrationBalanced and (b) functional description of a leaky integrate- putational primitive based on an integrated,a leaky integrate-and-fire excitable laser (LIF) with neuron. The LIF model is one and-fire neuron. Weighted(b) and delayed input signals are summed into the of the most ubiquitous models in computational neuroscience input current Iapp(Photodetectort), which travel to the! soma and perturb the internal state an embedded saturable absorber (SA)and behaves is the simplest analogously known model to for spike processing [25]. variable, the voltage V . Since V is hysteric, the soma performs integration Fig. 2. (a) Illustrationand and then applies(b) functional a threshold todescription make a spike of or no-spikea leaky decision. integrate- After a We also show that our laser neuron can be employed to a leaky integrate-and-fire (LIF) neuron. The LIF model is one and-fire neuron. Weightedspike is and released, delayed the voltage inputV is reset signals to a value areDCV summedreset bias. The! resulting into the spike carry out cortical algorithms throughinput several current smallI circuit(t),is which sent to travel other neurons to the in thesoma network. and perturb the internal state of the most ubiquitous models in computational neuroscience Spectral Filterapp demonstrations. Emulating this modelvariable, in a scalable the voltage device V . Since V is hysteric, the soma performs integration and is the simplest known model forrepresents spike the processing first step in [25]. building an ultrafastBank cognitive! Excitable Laser! and then applies a thresholdThe basic to make biological a spike structure or no-spike of a LIF neuron decision. is depicted After a in We also show that our laser neuroncomputing can platform. be employed to spike is released, the voltage V is reset to a value V . The resulting spike λ # excitatory ! Fig.inhibitory 2(a). It! consists of a dendriticreset treeGain that! collectsSA! and sums j carry out cortical algorithms through several small circuit is sent to other neuronsweight in the bank network.! II. LASER NEURON—THEORETICAL FweightOUNDATIONS bank! inputs from other neurons, a soma that acts as a low pass filter and integrates the signals over time, and an axon that demonstrations. Emulating this modelOur in device a isscalable based upon device a well-studied and paradigmatic carries an action potential, or spike,WDM when Waveguide the integrated! signal example of a hybrid computational primitive: the spiking represents the first step in building an ultrafast cognitive exceeds a threshold. Neurons are connected to each other via neuron. In this section, we briefly review the spiking neuron computing platform. The basic biologicalsynapses, structure or extracellular of a gaps,LIF acrossneuron which is chemicaldepicted signals in model and reveal the analogy between the LIF model and our Fig. 2(a). It consistsare transmitted.of a dendritic The axon, tree dendrite, that collects and synapse and all sums play an own. optical" electrical" λ1 … λi … λN" inputs from otherimportant neurons, role ina the soma weighting that and acts delaying as a of low spike signals.pass II. LASER NEURON—THEORETICAL FOUNDATIONS According to the standard LIF model, neurons are treated as A. Spiking Neuron Model filter and integrates the signals over time, and an axon that Our device is based upon a well-studied and paradigmaticFig. 2. (Top) A depiction ofelectrical an LIF devices.neuron The with membrane a synaptic potential variable,Vm(t) embedded, the voltage within a net- Studies of morphology andwork. physiology (Bottom)carries have an pinpointedA action schematic potential,difference of the across proposed or spike, their membrane,laser when neuron, the acts integrated as complete the primary withsignal internal filters, balanced example of a hybrid computationalthe LIF primitive: model as anthe effective spiking spiking model to describe a (activation) state variable. Ions that flow across the membrane photodetectors,exceeds anda threshold. an excitable Neurons laser. are In this connected model, only to each the excitatory other via photodetector neuron. In this section, we briefly reviewvariety of the different spiking biologically neuron observed phenomena [26]. experience a resistance R = Rm and capacitance C = Cm synapses, or extracellularassociated withgaps, the across membrane. which The soma chemical is effectively signals a first- model and reveal the analogy betweenFrom the theLIF standpoint model of computability and ourpathway and is complexity investigated. theory, LIF neurons are powerful computationalare primitives transmitted. that are Theorder axon, low-pass dendrite, filter, or a leaky and integrator, synapse with all the play integration an own. capable of simulating both Turing machines and traditional time constant ⌧m = RmCm that determines the exponential sigmoidal neural networks [27]. Signals areimportant ideally represented role in thedecay weighting rate of the and impulse delaying response of function. spike The signals. leakage by series of delta functions:and choose inputs andδ outputstunAccording= take4.4 the (0.66 form to the nm)current standard and through∆δ LIFR=m model,8drives.8 (1.3 the neurons membrane nm) to are achievevoltage treatedVm tolerable(t as) to levels of the x(t)=⌃n (t ⌧ ) for spike times ⌧ . Individual units 0, but an active membrane pumping current counteracts it A. Spiking Neuron Model j=1 jextinction ratio,electricalj crosstalk devices. and insertion The membrane loss [14]. potential A filterV bank(t), thewill voltage receive a series of pulses perform a small set of basic operations (delaying, weighting, and maintains a resting membranem voltage at a value of Studies of morphology and physiologyspatial summation, have temporal pinpointedfrom integration, other lasers,difference and thresholding) each across governed that theirVm(t by)= membrane, someVL. output acts power as the function primaryP internalout[ j](t) for neuron j at an the LIF model as an effective spikingare integrated model into to a describe singleassociated device a capable wavelength(activation) of implementingλ j state. variable.Fig. 2(b) Ions shows that the standardflow across LIF neuron the membrane model [27]. A a variety of processing tasks, including binary classification, neuron has: (1) N inputs which represent induced currents in There is alsoexperience a linewidth a resistance associatedR with= R them and output capacitance of each laser.C = SingleCm mode distributed variety of different biologically observedadaptive feedback, phenomena and temporal [26]. logic. input synapses xj(t) that are continuous time series consisting From the standpoint of computability and complexityfeedback theory, (DFB)associated lasers, withthe structures the membrane. assumed The here, soma tend is effectively to have narrow a first- linewidths. Nonethe- LIF neurons are powerful computational primitivesless, that since are theorder output low-pass is really filter, a time or aleaky dependent integrator, signal withPout[ thej](t) integration, the linewidth is fundamen- capable of simulating both Turing machines and traditionaltally broadenedtime by constant modulation,⌧m = andRm inCm addition,that determines can be further the exponential broadened by non-idealities sigmoidal neural networks [27]. Signals are ideally represented(such as carrier-induceddecay rate of chirp). the impulse Nonetheless, response we will function. first consider The leakage an ideal case in which by series of delta functions: inputs and outputs takebandwidth-limited the form current pulses through areR assumed.m drives This the allows membrane us to voltage approximateVm(t) theto output power spec- n x(t)=⌃ (t ⌧ ) for spike times ⌧ . Individualtrum units as Sout[0,i](λ but,t) = anPout active[ j](t)δ membrane(λ λi) for pumping each laser currenti in a givencounteracts loop where it δ(λ) is a Dirac j=1 j j − perform a small set of basic operations (delaying, weighting,delta functionand centered maintains at λi. Then a resting the broadcast membrane loop voltage signal B at(λ a,t) valuecontains of a sum of all output spatial summation, temporal integration, and thresholding)power that spectra,Vm i.e.(t)=VL. are integrated into a single device capable of implementing Fig. 2(b) shows the standard LIF neuron model [27]. A a variety of processing tasks, including binary classification, neuron has: (1) N inputsB(λ which,t) = ∑ representSout[i](λ induced,t) currents in (2) i adaptive feedback, and temporal logic. input synapses xj(t) that are continuous time series consisting

Each filter i in a given filter bank is associated with a wavelength λi corresponding to a laser i within the broadcast space. Although there will be some crosstalk due to the Lorentzian shape of the filter, we have minimized crosstalk by design (< 13 dB), so we can neglect its effect to first order. The power spectrum of the signal dropped from the broadcast waveguide for neuron j will take the form:

Sin[ j](λ,t) = ∑Ti j(λ)Sout[i](λ,t) (3) i

where Ti j(λ) is the transmission function of ring i for neuron j. Therefore, for a given photodetector spectral responsivity linear system Rλ , the current produced by the photodetector is as follows:

ip[ j](t) = ∑ Rλ Ti j(λ)Sout[i](λ,t) dλ (4) i Zλ Based on our assumptions, we can approximate the filter linewidth as significantly larger than the signal bandwidth.The 40 ps sech2 pulses used in this simulation have transform-limited bandwidth of ∆δ = .42 (.063 nm), which is much smaller than the bandwidth of the filters (as shown in Fig. 4(b)), allowing this approximation to be valid. We assume the photodetector response is approximately spectrally flat over the C-band [31], and neglect carrier diffusion limitations to frequency response, which are small compared to the RC limitations included in this model. This allows us to approximate the responsivity as a constant R P(t) = R P(t). λi { } PD Therefore, we can simplify the photodiode current to the following expression:

ip[ j](t) ∑RPDTi j(λi)Pout[i](t) (5) ≈ i A simulation of a filter bank with four filters is shown in Fig. 4. Pulses of different frequencies experience different weights—or amplitude shifts—based on the spectral position of each tunable ring. Based on the parameters used here, there can be a total of N = 34 filters per node within 45 nm gain band in hybrid III-V/Si. This number is synonymous with the upper bound on the fan-in per neuron. The networks themselves can be designed to have much larger numbers of neurons by using multi-loop strategies that can sidestep fan-in limitations [28]. In addition, the upper limit can be expanded: 62 channels have been demonstrated simultaneously in an experimental silicon photonics platform, a number that can be extended further using more advanced techniques [32]. Nonetheless, fan-in puts an upper limit on the complexity of the computation that a single unit can perform. The scalability of neural fan-in must be balanced with respect to spectral stability of the lasers, the linewidth of the pulses and the Q factor of the filters (this point is expanded in the Discussion section).

2.3. Electronic Junction The analysis of the electrical junction parasitics describes the signal as it is received from the photodetector and drives the adjacent excitable laser. It is converted from a series of pulses at different wavelengths to a single current signal (Fig. 4). Although current diffusion, saturation, and other nonidealities other can also play a role in the limiting the frequency response of the device, RC charging times tend to dominate [31]. We model the electronic junction as the lumped circuit (Fig. 3) using parasitic values that were determined in several experimental papers [29, 33, 34]. In this analysis, we only consider the bandwidth of a single photodetector driving a laser for simplicity, although multiple photodetectors may also be used for push-pull excitatory and inhibitory inputs, if desired. Multiple pulses of different wavelengths are incident on the photodetector, which provides the role of summation and converts signals to current pulses. A schematic of the junction is shown in Fig. 2, including a superimposed image of the parasitic circuit model. The purpose of the model is to ﬁnd the relationship between the photodetector current ip(t) and laser driving current ie(t): i (t) = F i (t) (6) e { p }

#246538 Received 22 Jul 2015; revised 15 Sep 2015; accepted 22 Sep 2015; published 2 Oct 2015 © 2015 OSA 5 Oct 2015 | Vol. 23, No. 20 | DOI:10.1364/OE.23.026800 | OPTICS EXPRESS 26806 ADFBCavity:ComparingFabry-P´erotandPhotonicCrystal Approaches

Mitchell A. Nahmias June 5, 2014

1 Introduction 2 One-Dimensional Model metal# I $ Cπ$=$.05pF$ contact# g A laser cavity requires some level of conﬁnement in Two,Sec/on$$ Designing and implementingPhotodetector$ a laser requires an energy source,again medium, and resonant feedback. all three dimensions. Typically, there are directions Excitable$Laser$ Rc(PD)$=$20Ω$ RL$=$10Ω$ Rc(L)$=$10Ω$ Resonant feedback, in its simplest form, involves re- transverse to lasing direction and those longitudinal Insula,ng#

(or along) the lasing direction. The transverse di- Layer# \$ flecting light back and forth between two mirrors. Equivalent)) III2V#Layer# This archetypal model is called a Fabry-Pérot cavity, rections are thoseCircuit)Model orthogonal to$ the propagation di- and this formalism has a+ surprisinglyVe$ amount of gen- rection,ip[j] and provide waveguiding (i.e. throughie[j total] 0$V$ erality. Laser cavities that can be understood within internal reflection). Shifts in the transverse profile dNph[j] this framework include solid stateRB$=$15Ω lasers,$ gas lasers, modulate the e↵ective index along the modulation = aRgag$(=$7.5Ωna[j] $n0a)Nph[j] Cp$=$1.5pF$ dt Si# distributed bragg reflector (DBR) lasers, and verti- direction. Because of the tight confinement along the N Cj$=$.16pF$ R $=$640MΩ C $=$.16pF ph[j] 2 transverse directions,Si we$ can approximate a three- d $ +sgs(ns[j] nSiO0s)N#ph[j] + VaBrna cal cavity surface emitting lasers (VCSELs). More 2 ⌧ph

modern designs, however, can still provide feedback dimensional laser as a one-dimensional e↵ective index dna[j] Nph[j] na[j] JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. X, NO. X, FEBRUARY 2014 profile along the longitudinal direction.1 [include fig- = aga(na[j] n40a) JOURNALbut fall outside OF of LIGHTWAVE the domain of Fabry-Pérot TECHNOLOGY, theory. VOL. X, NO. X, FEBRUARY 2014 dt Si# Va ⌧a 4 These include photonic crystal and nano-beam lasers, ure which shows the collapse] Ia + i (t) + e[j] which are better understood under the framework of Let us designate the x, y directions as transverse eVa photonic crystal theory. and the z direction as longitudinal. Suppose we are dns[j] Nph ns[j] Is = sgs(ns[j] n0s[j]) + Photodetector) dt output) Vs ⌧s eVs Here, we investigate a laser design that falls on dealing with a single transverseRate)Equa4on)Model) slice of our DFB laser. the boundary between the FabrySpectral)Response) Pérot and pho- In that case, we want to solve for the profiles E~ (x, y) copies of a single output signal to multiple receiver neurons optical link" electrical link" control wire" ⌘c hc optical link" electrical link" control wire" JOURNAL OF LIGHTWAVEcopies TECHNOLOGY, of a single VOL. X,output NO. X, signal FEBRUARY~ to multiple 2014 Nph[j receiver] Nph neurons[j] Pout[j](t) Nph[j](t) 4 tonic crystal methods: a distributedRPD =0 feedback.81A/W (DFB) and H(x, y) givend a refractive index profile nxy(x, y). ⇡ ⌧p laser. The entire cavity is periodically structured This amounts to solvingn thea[j eigenproblem:] = +Vf " ...na[j] +V" (Figure 1). Each input is modulated independently by(Figure a con- 1). EachP" i ( inputt)= isR P modulated(t) independentlydt 0 1 0 byCurrent a1 con-By" solving forP" a given mode in 2D (which can be Current" as a di↵raction grating, withp[j] the gain and reflectioni in[i] ns[j] ns[j] done by i.e. matching boundary conditions), we ar- distributed throughout the structure.i Of particular Pin[j] @= Ti(i)APout[i] @ PumpA "rive at an e↵ective index neff and a Pump" stant multiplier (a.k.a. weight), which can be positive, negative,stant multiplier (a.k.a.X weight), which can bei positive, negative, interest is a quarter-shifted DFB grating, which in- X copies of a single outputcludes an signal quarter wavelength to multiple o↵set (or defect receiver state) neurons optical 3link Fabry-Pérot" Output Technique"electrical link" control wire" Output" λ" ip[j] = R Pin[j] λ" or zero. After weighting, all inputs to the neuron are summed,or zero.within the After periodic structure. weighting,Spectral)Filtering) This cavity all is well- inputs to the neuron{ } are summed,Laser Laser known for it’s extremely good single-mode proper- PD" Cross-" +V" PD" Cross-" (Figure 1). Each input is modulated independentlyP" byP" a con-1 : Pout[1] P" 4 Photonic CrystalP" Technique P" Current" before modulating a nonlinear dynamical element: here,before a laserties, and modulating is a popular choice for a networks nonlinear involving dynamical element: here,Neuron a laser (λj)" Coupler" Neuron (λj)" Coupler" stant multiplier (a.k.a. weight),wavelength division which multiplexing can (WDM) be positive, [cite]. A negative,input(s)) 2 : Pout[2] 5 Scattering Pump" quarter-shifted DFB is unique in that it can analyzed λ" (λj)" λ" λ" (λj)" neuron device. The configuration of the system is determinedneuron device. The configurationλ" of the system3 : Pout[3] is determined Output" in two ways: as a Fabry-Pérot structure and a pho- λ" or zero. After weighting,tonicall crystal inputs with a defect to state. the In this neuron report, we are summed, ... -V" Laser -V" by its weight matrix, where element wij signifies the influenceby its weight matrix,Pin[i](t)= whereTi (i) Pout element[i](t) wij signifies the influence PD" Cross-" will compare the analysis in the two methods and the 1This is a significant approximation, with the mainP problem" P" before modulating a nonlinear dynamical element: here, a laser Broadcast Waveguide" Neuron (λj)Broadcast" Waveguide" strength of neuron i on neuron j. A processor canstrength exhibitresults thatof follow. neuron i on neuronthat it cannotj. predictA processor scattering losses. can exhibit Coupler" neuron device. The configuration ofExcitatory theFig. system 3. Cross Weight section is " determined ofInhibitory the deviceand Weight full modeling " structure for theλ" hybrid silicon/III-VExcitatoryλ" Weight " Inhibitory Weight " (λj)" a large variety of behaviors through reconfigurationa of large the variety of behaviors1 through reconfiguration of the IN# THROUGH# IN# laser neuron (inhibitory photodetector not included). A series of pulsesTHROUGH# along n different -V" Bank {λ ; i≠j}" Bank {λ ; i≠j}" Bank {λi ; i≠j}" Bank {λi ; i≠j}" by its weight matrix, where element wijwavelengthsignifiesi channels theλ influence...λ are spectrallyi filtered (i.e. weighted). This results in an exci- weight matrix, although this weight tuning should typicallyweight matrix, although this weight1 N tuning should typically Broadcast Waveguide" strength of neuron i on neuron j. Atatory processor photodetector can current exhibit response, which propagates into the equivalent circuit depicted happen on timescales much slower than spiking dynamics.happen The on timescalesabove. much The interaction slower between than the spiking photons in thedynamics. cavity, gain and The SA sectionsFig. are 3. modeled A processing-network node (PNN) is coupled to a broadcast a large variety of behaviorsFig. through 3. A reconfiguration processing-network of nodethe (PNN)IN# Excitatory is coupled Weight to a " broadcastInhibitory Weight " THROUGH# problem of neural networking contains prominent one-to-manyproblem ofwaveguide. neural networkingusing The rate front-end equations. contains The consists resulting prominent output of two power banks along one-to-manyBank wavelength of {λ continuously ; i≠j}λ" j becomeswaveguide. tunableBank the input {λ to The; i≠j}" front-end consists of two banks of continuously tunable other neurons in a given network. i i weight matrix, although(multicast) thismicroring weight and many-to-one droptuning filters should that (fan-in) partially typically components. drop WDM channels In the that case are present.microring Two drop filters that partially drop WDM channels that are present. Two (multicast) and many-to-one (fan-in) components. In the case waveguide integrated photodetectors (PDs) convert the optical signal to an happen on timescales much slowerwaveguide than integrated spiking dynamics. photodetectors The (PDs)Fig. convert 3. the A opticalprocessing-network signal to an node (PNN) is coupled to a broadcast of spiking networks, communication signals are pulses:of binary spiking networks, communication signals are pulses: binary electronic current and perform2 the summation operation of all weighted inputs. electronicwhere F currentis the linear and perform transfer function the summation of the modeledwaveguide. operation circuit. The of all front-end weighted consists inputs. of two banks of continuously tunable problem of neural networkingin amplitude contains and asynchronous prominent one-to-many in time. Spike timing is the A short wire subtracts these photocurrents and modulates current injection in amplitude and asynchronous in time. Spike timing is the A shortA simulationwire subtracts of multiple these input photocurrents pulses is shownmicroring and in modulates Fig drop 5. The filters laser current thatis pumped partiallyinjection with drop a stable WDM channels that are present. Two (multicast) and many-to-one (fan-in) components. In the case into an excitable laser neuron, which performs threshold detection and pulse predominantinto informaticcurrent an excitable source laserI parameter,p, while neuron, the PD which so is reverse temporal performs biasedwaveguide thresholdmultiplexing with a integrated large detection voltage photodetectors (> and5 V) pulse to offset (PDs) the convert the optical signal to an predominant informatic parameter, so temporal multiplexing formation in an optical cavity. The output of the laser is coupled back into of spiking networks, communicationformationinfluence signals in of anIp optical. Responsivityare pulses: cavity. is assumedbinary The output to beelectronic of 0.81 theA/W laser current[16]. is The coupled and junction perform back is kept the into fairly summation short operation of all weighted inputs. (TDM) and switch-based routing techniques are not(TDM) viable and switch-based( 100s of µm) to avoid routing transmission techniques line effects. are not viable the broadcast waveguide and sent to other PNNs. Insets represent example in amplitude and asynchronousthe broadcastin∼ time. waveguide Spike timing and sent is tothe otherA PNNs. short wireInsets subtracts represent these example photocurrents and modulates current injection strategies for networkingWhile the general physical expression pulses. for the complex The impedance goal of of our the linkspectrograms is more involved, of the in broadcast and drop waveguides. The broadcast waveguide strategies for networking physical pulses. The goal of our spectrograms of the broadcast and drop waveguides.into an excitable The broadcast laser neuron, waveguide which performs threshold detection and pulse predominant informatic parameter,practice, so temporalthe dominant parasiticsmultiplexing are the capacitance of the metal wire C hasand the 6 WDM contact channels. resis- Three of these channels are shown partially dropped network designhas 6 WDM will channels. be to support Three of athese large channelsformation number are shown in of an par- optical partiallyπ cavity. dropped The output of the laser is coupled back into network design will be to support a large number of par- tances RPD, RL. The metal wire capacitance Cπ is the easiest to adjust lithographically,into the excitatory either PD, and two other channels are shown partially dropped (TDM) and switch-based routinginto the techniques excitatory PD, are and not two viable other channelsthe broadcast are shown waveguide partially and dropped sent to other PNNs. Insets represent example allel, asynchronous,by changing and the heightreconfigurable of the oxide layer connections or the area occupied between by the metalinto bridge the inhibitory to change PD. The channel subsets that are dropped are determined allel, asynchronous, and reconfigurable connections between into the inhibitory PD. The channel subsetsspectrograms that are dropped of the are broadcast determined and drop waveguides. The broadcast waveguide strategies for networkinga distributed physical groupthe characteristics pulses. of photonic The of the goal junction. processing of our primitives that is by the tuning state of each filter (driving circuitry not shown). a distributed group of photonic processingnetwork primitives design will that be is to supportby theA tuning common a large state first ofnumber order each model filter of for (driving synapticpar- dynamicscircuitryhas 6 WDM is not an integratorshown). channels. with Three decay of [35]: these channels are shown partially dropped compatible with the approach of spikes represented as physicalcompatible with the approach of spikes representedinto the as excitatory physical PD, and two other channels are shown partially dropped allel, asynchronous, and reconfigurable connections betweends s optical pulses. = into+ theI(t) inhibitory PD. The channel subsets that are dropped are determined optical pulses. a distributed group of photonic processing primitives that isdt −byτ the tuning state of each filter (driving circuitry not shown). set of available input signals. Reconfiguration of the filters’ compatible with the approachset of spikeswhere of availableI( representedt) is the input, inputτ asis signals. the physical time constant Reconfiguration and s(t) is the synaptic of the variable. filters’ A neuron will A. Broadcast-and-weighttypically sum signals from multiple synapses si(t) and receive them asdrop inputs. states, In biological corresponding to weight adaptation or learning, A. Broadcast-and-weight optical pulses. drop states, corresponding to weight adaptation or learning, systems, synaptic variables typically represent neurotransmitter concentrations.intentionally In our case, occurs it on timescales much slower than spike Wavelengthintentionallyrepresents division the multiplexed RC occurs charged on signal. channelization timescales This behavior takes much of place the slower independently spec- than of temporal spike integra- Wavelength division multiplexed channelization of the spec- set of available inputsignalling. signals. A Reconfiguration reconfigurable of filter the could, filters’ for example, be im- trum is onesignalling. way to A efficiently reconfigurable use the filter full could, capacity for example, of a be im- trum is one way to efficiently useA. the Broadcast-and-weight full capacity of a drop states, correspondingplemented to by weight a microring adaptation resonator or learning, whose resonance is tuned waveguide,plementedwhich can by have a microring useable transmission resonator whose windows resonance up is tuned waveguide, which can have useable transmissionWavelength windows division multiplexed up channelization of the spec- intentionally occursthermally on timescales or electronically. much slowerIn thana group spike of N nodes with N to 50nm widethermally (>1THz or bandwidth) electronically. [53].InIn a fiber group communica- of N nodes with N to 50nm wide (>1THz bandwidth)trum [53]. In is fiber one waycommunica- to efficiently use the full capacity of a signalling. A reconfigurablewavelengths, filter each could, node for needs example, a dedicated be im- weighting filter for tion networks,wavelengths, a WDM each protocol node called needs abroadcast-and-select dedicated weighting filter for tion networks, a WDM protocol calledwaveguide, broadcast-and-selectwhich can have useable transmission windows up plemented by a microringall (N resonator1) possible whose inputs resonance plus one is filter tuned at its own wavelength can createall many(N potential1) possible connections inputs plus between onethermally filter nodes: at its or own theelectronically. wavelength In a group of N nodes with N can create many potential connectionsto 50nm between wide (> nodes:1THz the bandwidth) [53]. In fiber communica- to add its output to the broadcast medium. The total number of active connectionto add its is output selected, to the not broadcast by altering medium. the intervening The total number of 2 active connection is selected, not bytion altering networks, the intervening a WDM protocol called broadcast-and-select wavelengths, each nodefilters needs in the a dedicatedsystem would weighting thus scale filter quadratically for with N . medium,filters but rather in the by system tuning would a filter thus at scaleall the(N quadratically receiver1) possible to with inputsN 2 plus. one filter at its own wavelength medium, but rather by tuning acan filter create at the many receiver potentialthe desired to connections wavelength between [54]. We nodes: present the a similar protocol A filter design example is given in Section III-D. the desired wavelength [54]. We presentactive connection a similar protocol is selected,A not filter by design altering example the intervening is given into Section add its output III-D. to the broadcast medium. The total number of for a spike processing network and call itfilters “broadcast-and- in the system would thus scale quadratically with N 2. for a spike processing network andmedium, call it but “broadcast-and- rather by tuning a filter at the receiver to weight.” It differs by allowing multiple inputsA tofilter be design selected example is given in Section III-D. weight.” It differs by allowing multiplethe desired inputsto wavelength be selected [54]. We present a similar protocol B. Processing-network node simultaneouslyB. Processing-network and with intermediate node strengths between 0% simultaneously and with intermediatefor strengths a spike processing betweenand 0% 100%. network and call it “broadcast-and- In a biological neural network, the complicated structure weight.” It differs by allowing multiple inputs to be selected and 100%. In this scheme,In a biologicala group of nodesneural shares network, a commonB. the Processing-network complicated medium structureof physical node wires (i.e. axons) connecting neurons largely In this scheme, a group of nodes sharessimultaneously a common and mediumin with which intermediateof the physical output strengths of wires every between (i.e. node axons) 0% is assigned connecting a unique neuronsdetermines largely the network interconnectivity pattern, so the role in which the output of every nodeand is 100%. assigned atransmission unique determines wavelength the and network made interconnectivity available toIn every a biological pattern, other so neuralof the neurons role network, is predominantly the complicated computational structure (weighted addition, transmission wavelength and made availableIn this scheme, to every anode group other (Figure of nodesof neurons 2). shares Each is node a predominantly common has a tunable medium computational spectralof physical filter (weighted bank wires at addition,integration, (i.e. axons) thresholding). connecting neurons The contrasting largely all-to-all nature of node (Figure 2). Each node has a tunablein which spectral the filter output bankits front-end. of at everyintegration, By node tuning is thresholding). assigned continuously a unique The between contrastingdetermines0%-100% all-to-all the drop network natureoptical interconnectivity of broadcast saddles pattern, the photonic so the role neuron primitive units its front-end. By tuning continuouslytransmission between 0%-100% wavelengthstates, drop and eachoptical made filter available broadcast drops a portion to saddles every of the other its photoniccorrespondingof neurons neuron wave- is primitivepredominantlywith units additional computational responsibilities (weighted of addition, network control (routing, states, each filter drops a portion ofnode its corresponding(Figure 2). Eachlength wave- node channel, haswith a tunable additional thereby spectral applying responsibilities filter a bank coefficient at ofintegration, of network transmission control thresholding). (routing,wavelength The contrastingconversion, all-to-all WDM signal nature generation, of etc). length channel, thereby applying a coefficientits front-end. of By transmission tuninganalogous continuouslywavelength to a neural between conversion, weight.0%-100% The WDM filters drop signal ofoptical a given generation, broadcast receiver etc). saddlesThe the proposed photonic design neuron of a primitive processing-network units node (PNN) analogous to a neural weight. Thestates, filters ofeach a given filter drops receiveroperate a portion in parallel,The of proposed its correspondingallowing design it to of wave- receive a processing-networkwith multiple additional inputs node responsibilitiescan (PNN) perform allof networkof these necessary control (routing, functions, achieving com- operate in parallel, allowing it tolength receive channel, multiple therebysimultaneously. inputs applyingcan a perform coefficient An interconnectivity all of transmissionthese necessary pattern iswavelength functions, determined conversion, achieving by pactness com- WDM by signal flattening generation, the dual etc). roles of processing and net- simultaneously. An interconnectivityanalogous pattern is to determined a neuralthe weight. local by statespactness The of filters filters by of flattening anda given not the receiver a state dual of roles theThe transmission of proposed processing design andworking net- of a into processing-network a single set of node devices. (PNN) It attains rich com- the local states of filters and not aoperate state of in the parallel, transmissionmedium allowing betweenworking it to nodes. receive into Routing a multiple single in set this inputs of network devices.can is perform transparent, It attains all of rich theseputational com- necessary capabilities functions, by achieving leveraging com- analog physics offered medium between nodes. Routing in thissimultaneously. network is transparent, Anmassively interconnectivityputational parallel, pattern and capabilities switchless, is determined by making leveraging by itpactness ideal analog to supportby physics flatteningby offered optoelectronics.the dual roles of Overall, processing the and PNN net- is an unconventional the local states of filters and not a state of the transmission massively parallel, and switchless, making it ideal to supportasynchronousby optoelectronics. signals of a neural Overall, character. the PNNworking is an into unconventional a singlerepurposing set of devices. of conventional It attains optoelectronic rich com- devices, thereby medium between nodes. Routing in this network is transparent, putational capabilities by leveraging analog physics offered asynchronous signals of a neural character. The abilityrepurposing to control of each conventional connection, optoelectronic each weight, devices, in- appearing thereby as a strikingly simple circuit with potential to massively parallel, and switchless, making it ideal to support by optoelectronics. Overall, the PNN is an unconventional The ability to control each connection, each weight,dependently in- appearing is critical as for a strikingly creating differentiation simple circuit amongst with potentialgeneralize to to existing – and prospective – photonic platforms. asynchronous signals of a neural character. repurposing of conventional optoelectronic devices, thereby dependently is critical for creating differentiation amongstthe processinggeneralize elements. to existing A great – and variety prospective of possible – photonic weight platforms.One possible implementation of a PNN is depicted in Figure 3, The ability to control each connection, each weight, in- appearing as a strikingly simple circuit with potential to the processing elements. A great variety of possibleprofiles weight allowsOne possiblea group of implementation functionally similar of a PNN units is to depicted compute in Figurewhile 3, the dual purpose of its devices are summarized in dependently is critical for creating differentiation amongst generalize to existing – and prospective – photonic platforms. profiles allows a group of functionally similar units to computea tremendouswhile variety the dual of functions purpose despite of its sharing devices a arecommon summarizedTable I. in the processing elements. A great variety of possible weight One possible implementation of a PNN is depicted in Figure 3, a tremendous variety of functions despite sharing a common Table I. profiles allows a group of functionally similar units to compute while the dual purpose of its devices are summarized in a tremendous variety of functions despite sharing a common Table I. Table 1. Hybrid/III-V Laser Parameters series Param. series Description series Value 11 3 Vg gain section volume 1.68 10− cm × 12 3 Vα SA section volume 3.36 10 cm × − Γ QW conf. factor 0.056 n transparency carrier density 1.75 1018 cm 3 0 × − vg group velocity c/3.49 1 g0 QW gain coefficient 966 cm− τg gain carrier lifetime 1.1 ns τα SA carrier lifetime 100 ps

τph photon lifetime 2 ps nsp spontaneous noise factor 2 Ig gain pumping current 21 mA ηi injection efficiency 0.6 ηc differential quantum efficiency 0.26 λ lasing wavelength 1550 nm tion in the soma. The synaptic time constant thus has a correspondence with this circuit’s RC time constant. As shown in Fig. 4, the electronic junction does not significantly degrade the bandwidth of input pulses with FWHMs on the order of ps. In fact, the RC time constant can be as ∼ low as 30 ps using standard lithographic techniques [15]. This shows that optical injection is not necessary to maintain the information content in high bandwidth pulses, if the electrical junction is short. Nonetheless, a slower synaptic time constant—which is useful for various processing applications—is still possible by engineering a larger capacitance across the wire junction.

2.4. Laser Neuron 2.4.1. Dynamics The dynamical system underlying the behavior of our processing model is a gain-absorber cavity, describing single mode lasers with gain and saturable absorber (SA) sections. Despite its simplicity, it can exhibit a large range of possible behaviors [9], and has been investigated in various contexts as the basis for an optical processor [22]. The system in its simplest form can be described using the following undimensionalized equations:

G˙(t) = γ [A G(t) G(t)I(t)] + θ(t) (7a) G − − Q˙(t) = γ [B Q(t) aQ(t)I(t)] (7b) Q − − I˙(t) = γ [G(t) Q(t) 1]I(t) + ε f (G) (7c) I − − 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, γG is the gain relaxation rate, γQ is the absorber relaxation rate, γI is the inverse photon lifetime , and a is a differential absorption relative to the gain factor. We represent spontaneous noise via ε f (G), for small ε, and time-dependent input perturbations as θ(t).

#246538 Received 22 Jul 2015; revised 15 Sep 2015; accepted 22 Sep 2015; published 2 Oct 2015 © 2015 OSA 5 Oct 2015 | Vol. 23, No. 20 | DOI:10.1364/OE.23.026800 | OPTICS EXPRESS 26808 Under a desired parameter regime, the internal dynamics can be compressed so that pulse generation is instantaneous. The behavior simpliﬁes to [12]:

dG(t) = γ (G(t) A) + θ(t); (8a) dt − G − if G(t) > Gthresh then (8b) release a pulse, and set G(t) G → reset where the input θ(t) can include spike inputs of the form θ(t) = ∑ δ (t τ ) for spike ﬁring i i − i times τ , G is the gain threshold, and G 0 represents the gain at transparency. This i thresh reset ∼ 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 computations [36], and is capable of coding information in the timing between spikes [37].

2.4.2. Device We consider a two-section distributed feedback (DFB) excitable laser in the hybrid platform. We use a DFB cavity to guarantee a single longitudinal lasing mode, defined through the lithographic definition of lasing wavelength via silicon-on-insulator grating pitch. The quarter wavelength-shifted defect is placed asymmetrically in the cavity to direct the light in one direction. An ion implantation region electrically isolates the semiconductor absorber and gain sections, and a separate implant provides a smaller lifetime in the absorber section. Small modulation currents that exceed the Q-switch threshold will trigger large pulse discharges (see Fig. 5(a)). We define the effective gain and absorption factors for each section as:

vgg0 g˜(ng[ j]) = Γ (ng[ j] n0) n0 − vgg0 α˜ (nα[ j]) = Γ (n0 nα[ j]) n0 − where Γ denotes the conﬁnement factor of the QW sections with the transverse mode, vg is the waveguide group velocity, g0 is the gain coefﬁcient of the material, n0 is the transparency density and ng,nα denote the carrier concentrations in the gain and absorber sections, respectively. For laser j, we can describe the internal cavity dynamics using the following system of equations:

dNph[ j] 1 n = g˜(n ) α˜ (n ) N + g˜(n ) s (10a) dt g[ j] − α[ j] − τ ph[ j] g[ j] V ph g dng[ j] Ig[ j] + ie[ j](t) ng[ j] Nph[ j] = g˜(ng[ j]) (10b) dt eVg − τg − Vg dnα[ j] nα[ j] Nph[ j] = + α˜ (nα[ j]) (10c) dt − τα Vα where Nph(t) is the total number of photons in the cavity, n(t) is the number of carriers, n0 is the transparency carrier density, V is the cavity volume, Γ is the conﬁnement factor, τ is the carrier lifetime, τph is the photon lifetime, ns the spontaneous emission factor, and ie(t) represents the electrical modulation in the gain provided by the photodetector system, separate from the constant current bias Ig. Subscripts g and α identify the active and absorber regions. Parameter values are shown in Table 1 and based on those found in [31, 34]. The absorber is

#246538 Received 22 Jul 2015; revised 15 Sep 2015; accepted 22 Sep 2015; published 2 Oct 2015 © 2015 OSA 5 Oct 2015 | Vol. 23, No. 20 | DOI:10.1364/OE.23.026800 | OPTICS EXPRESS 26809 assumed to have a faster lifetime, which helps improve the dynamics and consistency of the input signals [13]. The equations can be simpliﬁed to the undimensionalized set of equations through a set of variable substitutions [12,38]. The results are shown in Fig. 5. The output power of laser j can be computed via:

ηc hc Pout[ j](t) = Nph[ j](t) (11) τph λ j for output coupling efﬁciency ηc, photon lifetime τph, Planck’s constant h, speed of light c, wavelength λ j, and photon number Nph(t). The resulting single-mode power spectrum of the laser takes the form S (λ,t) = P (t)δ(λ λ ), as mentioned in section 2.2. The signal out[ j] out[ j] − i Sout[ j] is coupled into the broadcast loop, becoming the input of other PNNs.

2.5. Non-Idealities Although this analysis has considered the physics of the signal pathway to first order, there are many second order non-idealities that will require further study and consideration. Discussed here is a brief overview of the major sources that could lead to decreased performance. First, let us consider fan-in limitations. In addition to the fundamental transform-limited bandwidth δλB, there is also temperature-induced linewidth fluctuations δλT and carrier-induced laser chirp δλC. If the filters have a linewidth of δλF , we must space filters by at least

(∆λ)2 (δλ )2 + (δλ )2 + (δλ )2 + (δλ )2 (12) ≈ B T C F to avoid significant interference. Closely spaced, overlapping filters can unintentionally allow the same signal to interfere with itself, resulting in coherent crosstalk. This manifests as phase differences δφ between each path becoming visible as amplitude noise δP with an SNR that depends on the magnitude difference between the filtered signals. Therefore, for a given a channel spacing ∆λ and gain bandwidth G (typically around 50 λ ∼ nm), the maximum channel fan-in capacity Cf of the system can be defined as: G C λ (13) f ≈ δλ This puts a fairly hard limit on the fan-in of neurons. In the very best case, (i.e. bandwidth- limited channels with 10 ps pulses in a 50 nm gain bandwidth), we can expect a maximum of 200 channels. Although it is a hard limit, this number is still very high compared to what ∼ is possible in electronics, and still unmatchable at such enormous signal bandwidths (i.e. 40 ∼ GHz). In addition, networks using clever organizational strategies can include many more PNNs than the fan-in limit, as mentioned earlier in the manuscript. We must also consider noise sources that can compromise the SNR of the signal as it travels through the system, which has not been studied here. There are many sources of noise to consider including amplified spontaneous emission (ASE) noise, thermal noise, and shot noise for both the optical and electrical signal. Since the laser pump current Ip is the largest signal in the pathway, we can expect that its shot noise contribution σs will dominate as input power is reduced. Nonetheless, all noise sources can play a contribution in reducing the signal SNR as it travels through the pathway and obfuscates the lasers ability to make a discrete decision based on its input. The effect of noise on this pathway remains to be studied, although it will not be significantly higher than what is already seen in directly modulated laser diodes and photodetectors.

Optical WDM Weight Bank Filter Electrical Wire ... LASER Broadcast Broadcast Waveguide *Waveguide (b) (d)

input signal spectrum

filter transmission

Fig. 4. Simulation of the response of the processing network node at each step. (a) Input WDM spike signals coming from the broadcast waveguide with FWHM = 40 ps. Trace color indicates the carrier wavelength of each pulse. (b) Transmission of spectrum of the weight bank filter (dotted lines) and input signal power spectra (solid curves). Input signals are assumed to be at the transform limit for 10 ps pulses. (c) WDM signals after transmission through the filter bank. Pulses on the same wavelength channel acquire the same weight and are then detected. (d) Electronic current signal, ie(t), that modulates the laser neuron after traversing the parasitic circuit model in Fig. 3. Pulses are low pass filtered to FWHM = 56 ps after traveling through device parasitics.

3. Discussion The model developed in this paper—including the ﬁlter bank, electrical junction, and laser neuron—enables the exploration of both analog signal properties and the effect of physical design decisions on the behavior of networked PNNs. An example simulation of the complete model with WDM optical-in and optical-out is shown in Figs. 4 and 5. The front-end weight bank and photodetector in Figs. 4(a)–4(c) act to generate an electronic representation of the weighted sum of WDM inputs carrying FWHM=40 ps pulses. This signal that modulates the laser neuron is shown in Fig. 4(d), after traversing the parasitic circuit model from Fig. 3. While some pulse spreading is visible (FWHM=56 ps), pulse amplitude and timing information is clearly maintained when realistic parasitic values are used. Figure 5 illustrates the internal dy-

60 40 50 20 20 0 6.1 6.2 6.3

Input Current (mA) 0 0 0 2 4 6 8 10

(b) Time (ns) Output Optical Power (mW) # 18

) 10 6 -3 gain region 4

2 transparency carrier density

saturable absorber (SA)

carrier density (cm 0 0 2 4 6 8 10 Time (ns)

Fig. 5. Simulated internal dynamics of the laser neuron in response to the modulation signal in Fig 4.d. (a) input electrical modulation ie(t) (black line) causes the release of a large optical pulse near 6ns (red line). Inset shows the magniﬁed output spike, which is approximately a sech2 pulse with FWHM=16.5 ps. (b) Simulated carrier densities in both the gain (blue line) and saturable absorption (purple line) sections of the laser. The dashed line represents the transparency carrier density, n0. namics of the excitable laser neuron in response to this modulation signal. Small perturbations to the gain population produce no optical output, but a larger input causes saturation of the SA and release of a short pulse (FWHM=16.5 ps), even narrower than the original input pulses. This is a testament to the system’s regenerative properties, which can continue to propagate pulses without their eventual degradation [12]. This simulation indicates that the PNN design meets a number of important scalability crite- ria: the dynamical processing model is both cascadable and restorative, the signal pathway can accept large fan-in and perform summation, elements are compatible with a high-volume multiplexing scheme, and the processing nodes are constructed from standard elements that can be implemented on PICs. Prior work on spiking laser neurons has largely focused on single laser dynamical phenomena, while suggesting the possibly of all-optical processing networks. No realistic approach for many-to-one interconnection with optically injected lasers has been proposed, and the well-known challenges of all-optical modulation (wavelength conversion, phase noise accumulation, two-beam beat interference) remain a question in these systems. The electronic mediation in the PNN enables a cascadable node based on a spiking laser neuron, but does not bring the typical downsides of O/E/O conversion in digital communication systems because transduction is separated from receiving (i.e. sampling, quantization, retiming). The energy and cost of electronic conversion in optical systems results largely from high-speed clocked transistor circuitry [39] and the need to demultiplex WDM channels before conversion [40]. In our system, electronic transduction does not regenerate or receive signals, but instead exploits the electronic physics for intermediate analog processing. The conversion between optical and electronic domains curtails the propagation of optical phase noise and the need for direct wavelength conversion, eliminating two major barriers facing scalable optical

#246538 Received 22 Jul 2015; revised 15 Sep 2015; accepted 22 Sep 2015; published 2 Oct 2015 © 2015 OSA 5 Oct 2015 | Vol. 23, No. 20 | DOI:10.1364/OE.23.026800 | OPTICS EXPRESS 26812 computing [41]. At the same time, the temporary conversion to the electrical domain does not signiﬁcantly degrade the signal, as this work has simulated. Every device in the primary signal pathway performs both physical and computational roles, resulting in a robust, ultrafast, and efﬁcient signal pathway. Utilization complementary physics along the signal pathway is not unlike this pathway in biological neurons, in which electrical action potentials are converted to chemical signals called neurotransmitters upon reaching a synapse (the junction between neurons). Chemical signaling is a relatively short-range process, but it introduces much more functionality compared to direct electrical modulation (e.g. both excitation and inhibition, a variety of synaptic timeconstants, and the ability to adapt the connection strength). The post-synaptic neuron body passively ac- cumulates the effect of many synapses, and its internal spiking dynamics generate a single channel of electrical action potentials, which is suitable for long-distance transmission. This process is roughly paralleled by O/E/O conversion in the PNN: the balanced photodetector and analog electrical link enable cascadable summation of multiwavelength inputs, from which an excitable laser generates a new spiking optical signal. This functionality does not impact bandwidth performance because the electrical link can be very short-range, similar to how the neurotransmitter link is very short in the biological pathway.

4. Conclusion We have investigated a device that can act as a node in an ultrafast photonic neural network, using a computational model that fully exploits the temporal resolution of optical signals. The architecture is physically enabled by three primary discoveries. First, a laser with gain and saturable absorption sections behave analogously to a biological neuron, but with dynamics and temporal resolutions on the order of nanoseconds and picoseconds, respectively. Secondly, a WDM broadcast-and-weight networking approach allows a large number of signals to be superimposed on a single waveguide where connection strengths are easily configured by tunable filter banks. Finally, a short receiver-less electronic link can carry a WDM fan-in signal without significant distortion or loss. By using the full model constructed here that takes into account many physical effects, we show that these optical subcircuits constitute a complete PNN capable of participating in a cascadable, scalability photonic neural network. Although the PNN promises several unique properties advantageous for processing, there still remain aspects that require exploration as the number of elements increase—these include the reliability and configurability of WDM filter-based weighting, the effect of noise accumulation, the effect of transform-limited pulses and chirp on the interconnection network density, and perhaps most importantly, power consumption and the effect of temperature on a large number of active elements clustered together on a PIC. Nonetheless, this represents an important direction in the consideration of scalability for photonic neural networks. Further developments could pave the way for processing systems that could perform real-time, complex processing of ultrafast (GHz) signals in real-time, which has the potential to revolutionize radio cognition or the control of ultrafast physical phenomena.

Funding Information The authors acknowledge the support of the National Science Foundation Graduate Research Fellowship Program (NSF GRFP) and the Banting Postdoctoral Fellowship administered through the Natural Sciences and Engineering Research Council of Canada.