All-Optical Spiking Neuron Based on Passive Microresonator Jinlong Xiang , Axel Torchy, Xuhan Guo , and Yikai Su , Senior Member, IEEE

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 38, NO. 15, AUGUST 1, 2020 4019 All-Optical Spiking Neuron Based on Passive Microresonator Jinlong Xiang , Axel Torchy, Xuhan Guo , and Yikai Su , Senior Member, IEEE

Abstract—Neuromorphic photonics that aims to process and the neural network for a specific task and AI operating on store information simultaneously like human brains has emerged traditional computers consumes great power as well. In the as a promising alternative for next generation intelligent com- von Neumann computing architecture, the computing unit and puting systems. The implementation of hardware emulating the basic functionality of neurons and synapses is the fundamental the memory are physically separated from each other, thus the work in this field. However, previously proposed optical neurons machine instructions and data are serially transmitted between implemented with SOA-MZIs, modulators, lasers or phase change them. Consequently, the computational capacity is limited by materials are all dependent on active devices and pose challenges for the bandwidth and high energy consumption. Compared to the integration. Meanwhile, although the nonlinearity in nanocavities massively parallel processing of human brains, it’s undoubtedly has long been of interest, the previous theories are intended for specific situations, e.g., self-pulsation in microrings, and there is inefficient to train and simulate a neural network with software still a lack of systematic studies in the excitability behavior of on a von Neumann computer. the nanocavities including the silicon photonic crystal cavities. Neuromorphic computing takes inspiration from human Here, we report for the first time a universal coupled mode theory brains and aims to simultaneously process and store information model for all side-coupled passive microresonators. The excitabil- as fast as possible, which provides a promising alternative for ity of microresonators resulting from the subcritical Andronov- Hopf bifurcation is categorized as Class 2 neural excitability and next generation intelligent computing systems. It develops hard- can be used to mimic the Hodgkin-Huxley neuron model. We ware mimicking the basic functions of neurons and synapses, demonstrate that the microresonator-based neuron can exhibit the and then combines them into suitably scaled neural networks. typical characteristics of spiking neurons: excitability threshold, In the past few years, neuromorphic electronics, e.g., TrueNorth leaky integrating dynamics, refractory period, cascadability and at IBM [5], TPU at Google [6] and SpiNNaker at the University inhibitory spiking behavior, paving the way to realize all-optical spiking neural networks. of Manchester [7], have demonstrated impressive improvements in both energy efficiency and speed enhancement, allowing Index Terms—Coupled mode theory, microresonators, simultaneous operation of thousands of interconnected artificial nonlinearity, neuromorphic photonics, optical neurons. neurons. However, electronic architectures face fundamental limits as Moore’s law is slowing down [8]. Furthermore, the I. INTRODUCTION electronic communication links have intrinsic limitations on RTIFICIAL intelligence (AI) has attracted a lot of atten- speed, bandwidth and crosstalk. Distinct from those properties A tion lately because of its great potential to remarkably of electronics, silicon photonics naturally has the advantages of change almost every aspect of our daily life [1]. Applications low latency and low power consumption. Data is transmitted of AI such as face recognition [2], intelligent virtual assis- in silicon waveguides at the speed of light and the associated tant [3] and neural machine translation [4] have already been energy costs currently are just on the order of femtojoules per bit widely used and brought great convenience to us. However, [9]. Combined with the dense wavelength-division multiplexing it takes a considerable amount of computational time to train (WDM), mode division multiplexing (MDM) and polarization division multiplexing (PDM) technologies, the channel capacity Manuscript received December 24, 2019; revised March 7, 2020; accepted continues to increase. Consequently, neuromorphic photonic April 2, 2020. Date of publication April 8, 2020; date of current version systems could operate 6Ð8 orders-of-magnitude faster than neu- July 23, 2020. This work was supported in part by the National Key R&D romorphic electronics [10]. Program of China under Grant 2019YFB2203101, in part by the Natural Science Foundation of China under Grant 61805137 and Grant 61835008, in part by the Synapses and neurons are the basic building blocks of human Natural Science Foundation of Shanghai under Grant 19ZR1475400, in part by brains, which perform the linear weight-and-sum operation and Shanghai Sailing Program under Grant 18YF1411900, and in part by the Open the nonlinear activation operation. To obtain a complicated Project Program of Wuhan National Laboratory for Optoelectronics under Grant 2018WNLOKF012. (Corresponding author: Xuhan Guo.) brain-inspired optical chip, the hardware implementation of The authors are with the State Key Laboratory of Advanced the fundamental function units is essential. Recently, a number Optical Communication Systems and Networks Department of Elec- of different concepts for neuromorphic computing have been tronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]; [email protected]; proposed, including the reservoir computing [11]Ð[17], artificial [email protected]; [email protected]). neural networks (ANN) [18]Ð[25] and spiking neural networks Color versions of one or more of the figures in this article are available online (SNN) [26]Ð[44]. Utilizing the linear optics offered by the native at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2020.2986233 interference properties of light makes it possible to implement

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II. COUPLED MODE THEORY OF MICRORESONATOR The nonlinearity of microresonators has long been of interest as a simple mechanism to manipulate light with light. The nonlinear phenomena have already been widely observed and carefully studied in different material platforms and various types of integrated resonators. All the self-pulsation, bistability and excitability behavior have been experimentally demonstrated in microrings [45]Ð[49], microdisks [50], [51] and two- Fig. 1. The passive microresonator couples with only one waveguide and the dimensional photonic crystal (PhC) cavities in InP platform [52], trigger pulse is applied in the opposite direction of the input pump light. [53]. The silicon two-dimensional PhC cavities can also exhibit bistability [54]Ð[56] and self-pulsation [57]Ð[60] behaviors. Besides, bistability has been observed in racetrack resonator [61] and silicon one-dimensional PhC cavity [62], [63]. While completely passive optical matrix multiplication units for neu- excitability behavior hasn’t been demonstrated in silicon PhC ral network applications in both free-space [21], [25] and the nanocavities. waveguide-based coherent circuitry [18]. Modulators combined The basic principle of nonlinear behaviors for Silicon-On- with photodetectors [23], [24], SOA-MZIs [19], [38] and laser- Insulator (SOI) microresonators is [48]: two-photon absorption cooled atoms with electro-magnetically induced transparency (TPA) effect first generates free carriers, which are then able to [25] are used to realize the nonlinear activation function in ANN. absorb light by free carrier absorption (FCA) effect. Besides, Meanwhile, lasers, e.g., micro-pillar lasers [36], two-section the presence of free carriers leads to a blueshift in the resonance lasers with saturable absorber regions [26], [27], [43], vertical- wavelength by free carrier dispersion (FCD) effect. Moreover, cavity surface-emitting lasers (VCSELs) [34], [35], [37], [40]Ð surface state absorption takes place everywhere at the silicon- [42], [44] and quantum dot (QD) lasers [39] are typically used to silica interface, and meanwhile some light is lost due to the implement the functionality of spiking neurons in SNN. Owing surface scattering and radiation loss. Therefore, the absorbed to its unique optical properties in different states, phase change optical energy is mainly lost in the form of heat. Owing to the material (PCM) has emerged as an attractive alternative to pro- thermo-optic effect, the heat induces a redshift in the resonance vide in hardware both the basic integrate-and-fire functionality wavelength. Typically, the free carriers relax at least one order of neurons and the plastic weighting operation of synapses of magnitude faster than the heat. Consequently, this difference [28]Ð[31]. Moreover, the broadcast-and-weight architecture [33] between the timescale of the fast free carrier dynamics and the has been proposed to unite well-defined neurons and synapses slow heating effects results in the self-pulsation, bistability and on the mainstream silicon photonics platform. However, the excitability phenomena in microresonators. optical neurons proposed so far mainly rely on active devices, Previously, different forms of CMT have been proposed to making it difficult to implement large-scale integrated neural analyze the nonlinearity in silicon microrings [46]Ð[48], mi- networks. Although the PCM-based spiking neuron itself is crodisks [50], [51] and two-dimensional silicon PhC cavities passive, extended lasers are needed to generate the output pulses [57], [59], [60], [64]. However, these CMT equations are in- and the output signals are amplified off-chip to form the feedback tended for specific situations, e.g., self-pulsation in microrings. links [31]. The lack of systematic studies in the nonlinear excitability In this paper, we report a new type of all-optical spiking behavior of nanocavities including the silicon PhC cavities is neuron implemented with the nonlinearity of passive microres- pressing. onators. As shown in Fig. 1, when triggered with a strong enough In this paper, we first propose, to the best of our knowledge, perturbation signal, the microresonator will emit a “negative” a universal CMT model for all passive side-coupled microres- output pulse, which can be used to emulate the spiking behavior onators. Considering the most common situation where a passive of neurons. To systematically study the interactions between dif- microresonator couples with only one waveguide, the nonlinear ferent nonlinear effects, we propose for the first time a universal dynamics of the microresonator can be described as follows [48], form of coupled mode theory (CMT) for all passive side-coupled [50], [55], [65], [66]: microresonators. The self-pulsation and excitability behaviors are successfully simulated to validate the presented model. We da 1 ± = j(ω − ω ) − a + κ S . point out that the excitability of passive microresonators can be ± ± ± ± (1) dt 2τ±,total categorized as Class 2 neural excitability and used to mimic the Hodgkin-Huxley neuron model. Then, we demonstrate that the microresonator-based neuron can exhibit the typical characteris- dN N Γ β c2 = − + FCA Si |a |4 +4|a |2|a |2 +|a |4 . tics of spiking neurons: excitability threshold, leaky integrating dt τ 2 2 + + − − fc 2ωVFCA ng dynamics, refractory period, cascadability and inhibitory spik- (2) ing behavior. Moreover, we analyze the impact of Q factor on the cascadability of the proposed neuron. Finally, we provide the guideline to design a microresonator-based neuron for scalable dΔT ΔT ΓthP = − + abs . (3) neural networks. dt τth ρSiCp,SiVcavity

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where a± is the complex amplitude of the forward and backward The total absorbed power is given by: 2 propagation mode respectively and |a±| stands for the corre- P = P + P + P . (11) sponding mode energy in the microresonator. S± represents the abs abs,lin abs,T P A abs,F CA complex amplitude of the pump light and the perturbation signal 2πc injected in the opposite direction. ω± = is the frequency of λ± 1 ω = ω +Δω P = |a |2 + |a |2 . input light in the waveguide, r i denotes the shifted abs,lin τ + − (12) ω = 2πc lin resonance frequency of the microresonator, where r λr is the original resonance frequency and Δω is the total frequency i 1 1 2 2 shift caused by all nonlinear effects. κ± = is the cou- P = |a+| + |a−| . (13) τ±,in abs,F CA τ pling coefficient between the waveguide and the microresonator. FCA N is the concentration of free carriers in the microresonator ΔT β c2 and is the mode-averaged temperature difference with the P =Γ Si |a |4 +4|a |2|a |2 + |a |4 . abs,T P A TPA 2 + + − − surroundings. τth is the relaxation time for temperature and ng VTPA τfc is the effective free-carrier decay rate accounting for both (14) recombination and diffusion. βSi is the constant governing TPA, ρSi is the density of silicon, Cp,Si is the thermal capacity of III. NONLINEAR BEHAVIORS OF MICRORESONATORS silicon and V is the volume of the microresonator. n is the cavity g To validate the universal CMT-equations presented above, we group index, generally, the dispersion is neglected and n = n , g Si simulate the nonlinear self-pulsation and excitability behaviors n is the refractive index of bulk silicon. V and Γ denote the Si α α of microresonators taking the most common microrings and effective mode volume and confinement coefficient. nanobeams as examples, and the parameters used in the simula- The total loss for each cavity mode is: tion are provided in Table I. Since it takes a relatively long time 1 1 1 1 1 1 for the microring to reach its steady state from scratch, all the = + + + + . (4) τi,total τi,lin τi,v τi,in τi,T P A τi,,F CA simulation results for microrings begin with the steady state in demonstration. In simulation the pump light and the perturbation where 1 is the linear absorption loss, 1 represents the τi,lin τi,in/v signal have the same wavelength, and the backscattering is in-plane waveguide coupling loss and the vertical radiation loss neglected. As a consequence, there exists no interference effect respectively, given by 1 = ω ; 1 and 1 are τi,in/v Qin/v τi,T P A τi,F CA due to the coupling between the pump light and the trigger the loss due to TPA and FCA respectively. The mode-averaged light. Moreover, the dynamics of the microresonator will be TPA loss rates are: independent of the phase of the perturbation signal. The time 1 β c2 width of the trigger pulse is 10 ns and 1 ns for the microring and =Γ Si |a |2 +2|a |2 . TPA 2 ± ∓ (5) the nanobeam respectively. Besides, the maximum power of the τ±,T P A ng VTPA perturbation signal is set to be the corresponding pump power. The mode-average FCA loss rate is defined as: The optical energy in the microresonator generates both heat 1 c and free carriers, while the thermo-optic effect and the FCD = (σe + σh) N. (6) τi,F CA ng effect have an opposite influence on the resonance frequency shift. Due to the difference in the timescale between the slow σ is the absorption cross-section for electrons and holes. e/h relaxation process of heat and the fast free carrier generation The total detuning of the resonance frequency of the microres- and absorption dynamics, microresonators will exhibit self- onator Δωi can be expressed by: pulsation behavior, as shown in Fig. 2. The mode energy in the Δω Δn Δn Δn Δn i = − i = − i,Kerr + i,F CD + i,th . microcavity, the temperature difference between the microcavity ωr nSi nSi nSi nSi and the surroundings, the concentration of free carriers, and the (7) power of output light are all evolving periodically with time. The detuning due to the Kerr effect is: In fact, if the input power is below but very close to the Δn cn self-pulsation power, the microresonator can be excited to emit ±,Kerr = 2 |a |2 +2|a |2 . n n V ± ∓ (8) a “negative” output pulse when applied a strong enough pertur- Si g Kerr bation signal. As shown in Fig. 3, the microresonator initially where the effective mode volume for Kerr effects VKerr = rests at its steady state and the output power remains constant. VTPA and n2 is the Kerr coefficient. The detuning due to the When triggered with a strong enough perturbation signal, the FCD effect can be expressed by microresonator will be kicked out from its equilibrium. All Δn 1 dn 1 the mode-averaged cavity energy, the temperature difference i,F CD = − Si N = − (ζ + ζ ) N. i,e i,h (9) between the resonators and their surroundings and the concen- nSi nSi dN nSi tration of free carriers go through a sudden increase and then ζ Here i,e/h is the material parameter. Finally, the detuning slowly recover to their steady states, resulting in the appearance due to the thermal dispersion is: of a “negative” pulse in output power. Moreover, as long as Δn 1 dn the power of the trigger signal is above the threshold value, the i,th = − Si ΔT. (10) nSi nSi dT shape of the “negative” output pulse will keep the same. The

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TABLE I PARAMETERS FOR MICRORING AND NANOBEAM [48], [55], [65], [66]

Fig. 2. Schematics of the self-pulsation behavior of microresonators. (a) For microrings: δλ = −20 pm and Pin = 5 mW; (b) For nanobeams: δλ = −12 pm and Pin = 5.6 μW.

detuned, which results in Class 2 neural excitability. We believe the proposed neuron can mimic the Hodgkin-Huxley neuron model as the Andronov-Hopf bifurcations have been scrutinized numerically and analytically in this model [67]Ð[72]. Therefore, simulation results agree quite well with previous studies [48], the microresonator-based neuron should be categorized as Class [49], which guarantees the correctness of the proposed universal 2 neuron. CMT-equations. The spiking neuron has several typical characteristics: (i) excitability threshold: The spiking neuron will only emit an output pulse when the power of the perturbation signal exceeds IV. OPTICAL SPIKING NEURON BASED ON PASSIVE a certain threshold. Otherwise, the neuron will not respond MICRORESONATOR and remain silent. (ii) leaky integrating dynamics: The spiking Although the microresonator emits a “negative” pulse when neuron can be excited with several sufﬁciently closely-spaced excited with a perturbation signal, it can still be used to mimic subthreshold pulses by integrating their combined perturbation the basic functions of a spiking neuron. In SNN, information (iii) refractory period: After an excitation, the neuron needs to is encoded and transmitted in the form of spikes. It doesn’t relax to its steady-state before it can be triggered once again, matter what the shape of the spike is, and whether the spike and the recovery time needed is called the refractory period. is “positive” or “negative” also makes no difference to the (iv) cascadability: The output spike of the previous neuron operation of the whole SNN. Here, we explicitly point out that should be strong enough to excite the next neuron, which is the passive microresonator can serve as an all-optical spiking the foundation of forming a multi-layer SNN. (v) inhibitory neuron. According to the phase-plane analysis [48], the ex- spiking behavior: The arrival of a stimulus can stop the spik- citability of passive microrings can be explained by the sub- ing activity, which plays an important role in the well-known critical Andronov-Hopf bifurcation when the input light is blue spike-timing-dependent plasticity (STDP) learning rules [73].

Fig. 4. Schematic of the normalized strength as a function of the power of one single perturbation pulse for δλ = −12 pm, Pin =4.4 μW, Ttr =1ns.

In the case of the subcritical Andronov-Hopf bifurcation, there doesn’t exist a well-defined threshold manifold, and a continuous transition between subthreshold oscillations and excitability usually exists [75]. We define the normalized strength of the “negative” output pulses as: S =1− Pmin , where Pconstant P P Fig. 3. Schematics of the excitability behavior of microresonators. Under min and constant represent the minimum of the output power proper pump conditions, the microresonator can be excited with a strong after the perturbation pulses and the constant output power at enough perturbation signal. (a) For microrings: δλ = −20 pm, Pin =2mW, the steady state respectively. To determine the threshold of our and Ptr =0.4mW; (b) For nanobeams: δλ = −13 pm, Pin =5μW,and Ptr =2μW. proposed neuron under specific pump conditions, we fix the duration Ttr and change the power Ptr of the perturbation pulse, and we define the threshold value at the point where the slope dS/dPtr is the maximum. For δλ = −12 pm, Pin =4.4 μW, Next, we provide a detailed exposition of these characteristics Ttr =1ns, Fig. 4 presents how the normalized strength changes for the nanobeam-based spiking neuron. The microring-based with the power of one single perturbation pulse. The normalized neuron can exhibit the same behaviors as nanobeams, and the strength first increases slowly when Ptr is below 1.25 μW, then simulation results are available in [74]. it experiences a sharp jump when Ptr increases from 1.25 μW to 1.4 μW. After that, the normalized strength increases very slowly again and remains almost constant when P is above A. Excitability Threshold tr 1.5 μW. Ptr =1.36 μW is defined as the threshold where the In ANN, the nonlinear unit is a necessary part for learning slope of the curve is the maximum at this point. With one single tough tasks. Without the nonlinear activation function, the output trigger pulse, perturbation above the threshold will result in of ANN will only be the simple linear combination of the input a remarkable stronger strength increase than the subthreshold information. Consequently, the ANN turns out to be the primitive perturbation and the spiking shape will keep almost constant. perceptron machine. It’s the nonlinear unit that makes it possible However, no functional excitability will be induced in the low- for ANNs to approximate any functions, thus realizing com- Ptr region of one single trigger, as the shape of the “negative” plicated tasks such as image classification, speech recognition output pulse is dependent on the perturbation power and the and so on. The sigmoid function and tanh function are usually normalized strength isn’t strong enough. adopted in the ANN. In contrast, the information in SNN is Fig. 5 presents the distribution of the excitability threshold encoded in the timing or rate information of spikes, which are of nanobeams over different input power and resonance shift. assumed to be all-or-none binary events. The spikes are used It’s worth mentioning that the microresonator will exhibit self- for interneuron communication and carry information through pulsation or stay at the steady state at the upper right gray region. the sheer fact of their appearance, whereas their amplitude and Moreover, it can be excited at the lower left gray region, but width are neglected. Consequently, the condition on which the the trigger power needs to be higher than the pump power, spiking neuron can be triggered to emit an output spike, or more which is not discussed in this paper. According to Fig. 5, as the exactly, the excitability threshold behavior, directly determines wavelength shift increases, the excitability threshold power of the performance of the SNN. microresonators increase gradually. On the contrary, it becomes

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Fig. 5. Schematic of the distribution for the excitability threshold of nanobeams over different pump power Pin and wavelength shift δλ.

lower when pumped with higher input power. In addition, the excitability threshold is more sensitive to the wavelength shift than the pump power. Fig. 6. Schematics of the leaky integrating dynamics of nanobeams. For δλ = −12 pm, Pin =4.4 μW, Ptr =1μW,andTtr =1ns,twosame perturbation pulses with the time interval of 1nscan excite the nanobeam while B. Leaky Integrating Dynamics one single pulse just introduces slight attenuation to the output power. Similar to the leaky integrate-and-fire (LIF) neuron, the nanobeam-based neuron can also be excited by integrating several closely-spaced subthreshold perturbation pulses. To demonstrate this, we keep δλ = −12 pm, Pin =4.4 μW, Ptr =1μW and Ttr =1ns. As shown in Fig. 6, when a perturbation pulse is sent to the nanobeam at t =20ns, small variations are intro- duced to the mode energy, the temperature shift and the concentration of free carriers in the nanobeam. Consequently, the output power experiences slight attenuation and then recovers slowly to the steady state. At t =40ns, two identical perturbation pulses with an interspike interval of 1nsare applied to the nanobeam. As opposed to the case of single pulse, the output power attenuates to almost 0 μW and a distinct “negative” spiking dip output pulse is generated. The nanobeam-based neuron can also be excited by two subthreshold perturbation pulses, which is the direct evidence of the integrating behavior. Moreover, during the time interval from t =41nsto t =42ns, the mode energy, the temperature variation and the concentration of free carriers decrease slowly, and the output spiking dip experiences a weak Fig. 7. Response of the nanobeam to two identical perturbation pulses (nor- recovery approximately from 2 μW to 2.2 μW, which means the malized strength) as a function of the interspike interval for δλ = −12 pm, integrating process is leaky. We also notice that the two same Pin =4.4 μW, Ptr =1μW,andTtr =1ns. perturbation pulses have different impacts on the nanobeam, which can be attributed to the resonance wavelength shift caused by the first perturbation pulse. For example, the first perturbation Ttr =1ns, the response of the nanobeam to two identical per- pulse leads to the output power attenuates from about 3 μW to turbation pulses is depicted in Fig. 7. The normalized strength 2 μW, while the second one causes the output power attenuates remains almost constant when the interspike interval is below from about 2.2 μW to 0 μW. 1nsand decreases very slowly when the interval increases from To further study the timing information during this process, 1nsto 1.45 ns. Actually, almost constant output spiking pulses we change the interspike interval between the two identical can be observed in this region. Then the normalized strength subthreshold perturbation pulses and record the normalized decreases sharply from about 0.9 to 0.6 when the interval strength. For δλ = −12 pm, Pin =4.4 μW, Ptr =1μW and increases from 1.45 ns to 1.7ns. After that, the normalized

Fig. 9. Schematic of the cascadability behavior of microresonators (N: nanobeam). For δλ = −18.5pm, Pin =11μW, Ptr =0.75 μW, the perturbation signal excites the ﬁrst nanobeam and its negative output pulse triggers the second nanobeam.

nanobeams (∼ 20 ns) is approximately one eighth of that for microrings (∼ 160 ns).

D. Cascadability Behavior To provide the required learning capacity for complicated artificial intelligence applications such as computer vision and medical diagnosis, a multi-layer neural network is the inevitable choice, which means the cascadability behavior of optical neurons is quite necessary. In order to simulate the cascadability Fig. 8. Schematics of the refractory period behavior of nanobeams. For δλ = −13 pm, Pin =5μW, Ptr =2μW, (a) the nanobeam keeps silent to behavior of the proposed optical neuron, two microresonators the second trigger pulse when the time interval is 12 ns, (b) but it can be excited with the same conditions are cascaded. The perturbation signal once again when the time interval increases to 16 ns. is only applied to the first microresonator, and the output signal of the first neuron directly serves as the input signal of the second strength decreases slowly again and remains about 0.3 when neuron. The simulation results are presented in Fig. 9. For δλ = the interval exceeds 2.5ns. −18.5pm, Pin =11μW, Ptr =0.75 μW the perturbation signal excites the first nanobeam and its “negative” output pulse C. Refractory Period triggers the second nanobeam. All the physical parameters of the second microresonator including the mode-averaged energy After an excitation, the spiking neuron will be insensitive to in the cavity, the temperature difference with the surroundings the second trigger signal until recovering to its steady state. and the free carrier concentration evolve in the same way as When scaled to a multi-layer SNN, the refractory period will the first microresonator after a certain latency. Consequently, directly determine the operation speed of the whole network. there exist two “negative” pulses at the output of the second The microresonator-based optical neuron has this property microresonator. The first “negative” pulse corresponds to the as well. As shown in Fig. 8, if the interval between the two output of the first neuron, and the second pulse is the triggered trigger pulses is shorter than the refractory period, the mi- output of the second neuron, which is the direct evidence for croresonator will not respond to the second perturbation signal. δλ = −13 P =5μW P =2μW the cascadability behavior of the microresonator-based spiking For pm, in , tr when the time neuron. interval is 12 ns, the nanobeam stays silent to the second trigger pulse. However, when the time interval increases to 16 ns, E. Inhibitory Spiking Behavior the nanobeam emits an output spike once again, which is the direct evidence of the refractory period behavior. The effective Both the spiking excitation and inhibition are integral parts relaxation time of free carrier τfc and the relaxation time of of the STDP learning rules. Recently, some innovative schemes temperature τth are two key factors deeply influencing the have been proposed to experimentally demonstrate the excita- excitability behavior of microresonators. The former determines tory and inhibitory spiking dynamics simultaneously in neu- the shape of the “negative” pulse, while the latter determines romorphic photonic systems [40]Ð[43]. We point out that the how fast the microresonator can be excited. Beneficial from proposed neuron can also emulate the inhibitory spiking be- the shorter thermal relaxation time, the refractory period for havior. We keep δλ = −12 pm, Pin =4.4 μW, Ttr =1ns.

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TABLE II PERFORMANCE COMPARISON OF OPTICAL NEURON

can be pumped and excited with serval milliwatts. Owing to the ultra-high Q/V ratio, the light-matter interactions in the nanobeam is much stronger than that in the microring. Con- sequently, the nanobeam-based neuron can work under serval microwatts, three orders-of-magnitude lower than the microring. Moreover, since the refractory period is mainly limited by the relaxation time of heat, the nanobeam-based neuron can operate more than eight times faster than the microring-based neuron. We emphasize that the nanobeam-based neuron can operate at the speed of GHz, six orders-of-magnitude faster than the timescale of biological neural networks: kHz. Table II depicts the performance comparison between our proposed optical neuron with previous work. The microresonator-based neuron can process information passively, which is extremely difﬁcult for other types of neuron to achieve. Moreover, they can be easily fabricated in photonic integrated circuits together with other optical components in standard CMOS process, thus making large-scale SNNs available. Combined with the PCM-implemented synapse in [30], an all-optical passive spiking neural network can be obtained. Furthermore, we investigate the impact of Q factor on the distribution of the cascadability behavior of the nanobeam-based Fig. 10. Inhibitory spiking dynamics of the proposed neuron. From the left to neuron. Since ultra-high Q of 3.6 × 105 on SiO2 claddings and the right, the excitatory pattern is lagging the inhibitory pattern by 3ns, 1ns, 7.2 × 105 and 0nsfor the ﬁrst three cases, and it is leading by 1.4ns, 3nsfor the last on air claddings have been experimentally demon- two cases. strated [76], we keep the Qv/Qin ratio constant and increase the Q value from original 114400 to 171600 and 228800 suc- cessively. The simulation results are depicted in Fig. 11. As Unlike the previous setup, the power of the perturbation light is the value of Q increases from left to right, relatively larger set to be 0.35 μW at the steady state. Under these conditions, wavelength detuning and higher pump power are needed for a single excitatory perturbation pattern with the high level of the nanobeam-based neuron to demonstrate the cascadability 1 μW can excite the nanobeam. To demonstrate the inhibitory behavior. It can be attributed to the high temperature sensitivity spiking dynamics, inhibitory perturbation patterns with the low of the resonance frequency resulting from the ultra-high Q/V level of 0 μW are sent to the nanobeam. As shown in Fig. 10, as ratio. the inhibitory pattern moves toward the excitatory pattern, the Finally, we give the general guideline to design a practical “negative” output spike is gradually suppressed to almost null; as microresonator-based neuron: (i) Choose the type of microres- the inhibitory pattern moves far away from the excitatory pattern, onator. Compared with microrings, the nanobeam-based neuron the “negative” output spike recovers to the normal state again. can work with faster speed and much lower power. However, Therefore, the proposed neuron can emulate both the excitatory owing to the ultra-high Q/V ratio, nanobeams can be more sen- and inhibitory spiking dynamics. sitive to the temperature variation of the surroundings, affecting the stability of the whole system. Moreover, the condition for the cascadability of nanobeams is more critical; (ii) Design the V. D ISSCUSSION quality factor Qv and Qin of the microresonator. Generally, a We analyze the performance of the microresonator-based higher Q can enhance the light-matter interaction, thus making neuron: the power consumption and the operation speed are the microresonator exhibit the expected nonlinear behaviors. determined by the pump power and the refractory period re- While as discussed above, a lower Q is preferred to more easily spectively. As discussed above, the microring-based neuron achieve the cascadability with lower threshold; (iii) Choose

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Axel Torchy followed an intensive mathematics and physics training in Lycée Yikai Su (Senior Member, IEEE) received the B.S. degree from the Hefei du Parc (Lyon, France) before he joined Télécom Paris, an engineering school University of Technology, China, in 1991, the M.S. degree from the Beijing specializing in computer science. He majored in embedded systems and mobile University of Aeronautics and Astronautics (BUAA), China, in 1994, and the networks. In September 2019, he joined Shanghai Jiao Tong University as a dual- Ph.D. degree in EE from Northwestern University, Evanston, IL, USA in 2001. degree student, then he joined the State Key Laboratory of Advanced Optical He worked at Crawford Hill Laboratory of Bell Laboratories before he joined Communication Systems and Networks to work on the realization of photonic the Shanghai Jiao Tong University, Shanghai, China, as a Full Professor in 2004. synapses and neurons and the implementation of related algorithms. His research areas cover silicon photonic devices for information transmission and switching. He has over 400 publications in international journals and conferences, with more than 3700 citations (scopus search). He holds six US patents and ∼50 chinese patents. Prof. Su served as an associate editor of APL Xuhan Guo received the B.S. degree from the Huazhong University of Science Photonics (2016Ð2019) and Photonics Research (2013Ð2018), a Topical Editor and Technology, Wuhan, China in 2009 and the Ph.D. degree from the University of optics letters (2008Ð2014), a Guest Editor of IEEE JSTQE (2008/2011), and of Cambridge, Cambridge, UK, in 2014. He has worked as a Research Associate a Feature Editor of Applied Optics (2008). He is the Chair of IEEE Photonics in University of Cambridge until 2017, then he moved to Shanghai Jiao Tong Society Shanghai chapter, the General co-chair of ACP 2012, and the TPC University, China, where he is currently an Associate Professor of photonics. co-chair of ACP 2011 and APCC 2009. He also served as a TPC member He is the author or co-author of more than 30 publications. His current research of a large number of international conferences including CLEO (2016Ð2018), interests include hybrid silicon lasers, silicon photonics devices and circuit, and ECOC (2013Ð2017), OFC (2011Ð2013), OECC 2008, CLEO-PR 2007, and PON. He also serves as a TPC member of a number of international conferences LEOS (2005Ð2007). Prof. Su is a senior member of IEEE and a member of including ECOC (2018Ð), CLEO Paciﬁc Rim (2018Ð), IEEE GLOBECOM OSA. (2016Ð2017), ECIO (2010), the secretary (2018Ð) of IEEE Photonics Society Shanghai chapter. Dr. Guo is a member of IEEE and a member of OSA.

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