Stochastic induced dynamics in neuromorphic optoelectronic oscillators

Transcription

1 Opt Quant Electron DOI /s Stochastic induced dynamics in neuromorphic optoelectronic oscillators Bruno Romeira Ricardo Avó Julien Javaloyes Salvador Balle Charles N. Ironside José M. L. Figueiredo Received: 13 September 2013 / Accepted: 19 February 2014 Springer Science+Business Media New York 2014 Abstract We investigate the dynamics of optoelectronic oscillator (OEO) systems based on resonant tunneling diode photodetector (RTD-PD) and laser diode hybrid integrated circuits. We demonstrate that RTD-based OEOs can be noise-activated in either monostable or bistable operating conditions, providing a rich variety of signal s spiking, square pulses, bursting and behaviours stochastic and coherence resonances that are similar to that of biological systems such as neurons. The potential for fully monolithic integration of our OEO confers them a great potential in novel neuromorphic optoelectronic circuits for signal processing tasks including re-timing and re-shaping of pulsed signals exploiting either the monostable or the bistable operating conditions. Keywords Neuromorphic systems Optoelectronic oscillators Resonant tunneling diode Semiconductor laser Stochastic dynamics Excitability 1 Introduction Among the great diversity of nonlinear processes in electronics and optics, excitable nonlinear dynamical phenomena have recently received considerable attention in artificial photonic- B. Romeira (B) R. Avó J. M. L. Figueiredo Departamento de Física, Centro de Electrónica, Optoelectrónica e Telecomunicacões (CEOT), Universidade do Algarve, Campus de Gambelas, Faro, Portugal bmromeira@ualg.pt J. Javaloyes Dept. de Física, Univ. Illes Balears, C/Valldemossa km.7 5, Palma, Spain S. Balle Institut Mediterrani d Estudis Avançats (CSIC-UIB), C/Miquel Marqués, 21, Esporles, Spain C. N. Ironside School of Engineering, University of Glasgow, Glasgow G12 8QQ, UK

2 B. Romeira et al. neuron networks (Samardak et al. 2011). Excitability is a well-known nonlinear property (Lindner et al. 2004) that regulates several biological process like nervous impulse propagation (Hodgkin and Huxley 1952) or chemical reactions such as Belousov Zhabotinsky systems (Kuhnert et al. 1989). Excitable systems provide an all-or-none response to external stimuli: the system responds linearly and with small amplitude to stimuli that do not overcome a threshold, but otherwise it yields a large pulse (whose shape is independent of the stimulus) followed by a lapse, called the refractory time, during which the system does not respond to new stimuli. They can perform complex information processing tasks with promising applications in clock recovery and pulse reshaping functionalities in photonics (Barbay et al. 2011). However, the currently proposed systems are low speed and bulky making most of them not suitable for nowadays information processing needs. In this work, we investigate a neuromorphic high-speed optoelectronic oscillator (OEO) based on a double barrier quantum well (DBQW) resonant tunneling diode photo-detector (RTD-PD) driving a telecom laser diode (LD), Fig. 1a, b, c. The RTD-based OEO exhibits an asymmetric dc current voltage (I V) curve with a pronounced negative differential resistance (NDR) region (Slight et al. 2008), Fig. 1d. When dc biased in the NDR region, RTDs can operate as high-speed oscillators for THz generation (Suzuki et al. 2013), and recently RTD-based OEOs were proposed for high-purity phase locked oscillations employing either external injection locking (Figueiredo et al. 2008), or self-injection locking in a delay line feedback configuration (Romeira et al. 2013). When dc biased outside the NDR RTDs can display bistable behavior (Hartmann et al. 2011), and more recently we reported excitable behavior in RTD-based OEOs showing optical pulse generation with lapse time responses in the order of the nanosecond (Romeira et al. 2013). Here, we describe the noise activated dynamics observed in these RTD-OEOs operating in either monostable or bistable regimes. We observe spikes, square pulses, bursting, and stochastic and coherence resonances depending on the working conditions. The experimental results are well described employing a nonlinear Liénard oscillator model that describes the RTD stochastically activated by means of white Gaussian noise driving the LD, described by single mode rate equations (Romeira et al. 2013). The stochastically induced dynamics of this OEO can be exploited in a variety of applications including optoelectronic memories, or in artificial neural networks by coupling multiple OEO excitable units. 2 Experimental and theoretical description of the optoelectronic oscillator Our RTD-PDs consist of 10 nm wide AlAs/InGaAs/AlAs DBQW structures, grown on semiconducting InP substrates, embedded in a moderately doped InGaAlAs layers forming the core of a unipolar ridge optical waveguide, allowing the device to operate as a waveguide photodetector at λ 1,550 nm (Romeira et al. 2013). The RTD-PD is mounted in series with a LD (CST Global Ltd.) that operates at λ 1,550 nm with 6 ma threshold current, and 0.8 V threshold voltage. Figure 1a shows a photograph of a circuit prototype, and Fig. 1bpresents the equivalent electrical circuit. We measured the response of the OEO to white Gaussian noise signals injected using an Agilent 33250A arbitrary waveform generator (AWG), by recording the electrical and optical s employing a 2 GHz oscilloscope, Fig. 1c. The operating quiescent point of our circuit consists in the intersection of the nonlinear I V curve and a load line whose slope is defined by the inverse of the resistance of the circuit. AsshowninFig.1d, for low resistance the load line is almost vertical and there is only one intersection with the nonlinear characteristic function that allows for operation in either a rest

3 Stochastic induced dynamics (a) input (c) 500 µm ridge waveguide Collector windows Emitter windows Collector windows silica input in microstrip line Resonant Tunneling Diode (RTD) Photo-detector (PD) Laser Diode (LD) RTD-PD V, I input LD N, S out PD (b) + - (d) R L RTD-PD in LD V out I-V of monostable circuit A I-V of bistable circuit B Load Line A Load Line B I C II III F(V) input Bias tee AWG Channel 2 Channel 1 Oscilloscope Fig. 1 a Photograph of the hybrid optoelectronic integrated circuit. Inset are shown the RTD and LD dies. b Corresponding lumped electrical circuit showing the equivalent inductance L, resistance R, and RTD-PD capacitance C and I V nonlinear F(V ). c Schematic block diagram of the RTD in series with the LD forming the optoelectronic oscillator with quadruple electronic and optical inputs/s. d Experimental I V curves of the monostable (solid) and bistable (dashed) circuits state regime, regions I and III, or a self-oscillatory regime, region II, depending on the bias conditions, while under high resistance conditions several (three) intersections may occur. The OEO is described by single mode laser rate equations describing LD s photon and carrier number density (S, N) dynamics coupled to a Liénard system which accounts for the current and voltage across the RTD-PD-LD (I, V ) (Romeira et al. 2013): V = 1 μ [I f (V ) χξ(t)], İ = μ [ V 0 γ I V ] (1) Ṅ = 1 [ I N N δ ] {1 ɛs}s, Ṡ = 1 [ ] N δ {1 ɛs}s S + β N τ n I th 1 δ τ p 1 δ Here, time has been scaled by the LC resonance frequency ω 0 = ( LC) 1, f (V ) (Romeira et al. 2013) describes the I V curve, and μ = V 0 /I 0 C/L and γ = R(I0 /V 0 ) describe the equivalent circuit parameters (inductance, L, capacitance C, and resistance R, Fig. 1b); V 0 = V dc V th,wherev dc is the applied bias voltage and V th is the LD s voltage drop at threshold. Finally, we introduce the stochastic effects in our model employing an effective delta-correlated Gaussian white noise ξ(t) of zero mean and unit variance, so χ stands for the total noise strength. 3 Results and discussion When the RTD-PD-LD operates in monostable conditions and dc biased within the NDR, region II (Fig. 1d), it exhibits voltage-controlled self-sustained current oscillations driving the laser (Romeira et al. 2013). When dc biased in regions I and III (Fig. 1d), the OEO responds

4 B. Romeira et al. (a) (b) Fig. 2 Simulated noise induced resonances in the RTD-PD-LD excitable system in the a electrical (V)and b optical (S) domains as a function of noise strength to external perturbations by generating spikes in both electrical and optical domains when the perturbation exceeds a given threshold (Romeira et al. 2013). In this excitable regime, the system is highly sensitive to noise emitting fixed-form spikes or bursts of spikes depending on the dc bias point, and showing coherence resonance (CR) (Lindner et al. 2004), i. e., a maximum of temporal regularity in the for a finite noise level. Figure 2 shows a numerically example of the occurrence of spiking and CR behavior in the RTD-PD-LD due to noise perturbation, and selecting the bias operation in region I, i.e., in an equilibrium situation in the positive differential resistance. For a very weak noise input level, χ = 0.12, the RTD-PD-LD maintains its steady (rest) state. For a given noise threshold perturbation, however, the RTD-PD-LD can leave the rest state. When χ = 0.13 (weak noise) we observe the spiking in both electrical and laser s (V and S, respectively). The peaks appear random, and the interval between excitations varies substantially. For a moderate noise, χ = 0.18, the spiking is rather regular, which implies that the interspike intervals do not differ much, providing an example of CR behavior. The pulse triggering mechanism is explained as follows. The RTD-PD-LD can be considered as a slow fast excitable system in which the fast variable is the voltage and the slow one the current. During the excitable orbit, the two fast stages correspond to sudden peak and drop of the voltage during which the current does not changes appreciably. This two fast stages are interleaved by two slow stages in which the current evolves along the attracting nullcline as definedby the f (V ) function while the voltage follows adiabatically (for more details see Romeira et al. 2013). The excitable regimes predicted in the numerical simulations were verified experimentally. By adding to the dc bias in region I a stochastic voltage generated by a Gaussian noise source with a cut-off frequency of 80 MHz, the RTD-PD-LD circuit A passes from emitting occasional spikes [see Fig. 3a, plots (i) and (ii)] to emission of an almost periodic pulse train [see Fig. 3a, plot (iii) and (iv)] as the noise level increases. The upward electrical pulses have a full width half maximum (FWHM) of 13 ns, while the LD light exhibits downward pulses of FWHM of 200 ns. The LD emits pulses in response to current modulation and its response is determined by carrier and photon lifetimes, yielding optical pulses longer than the current spikes. The OEO has potential to achieve sub-ns response (Romeira et al. 2013), limited by the parasitics, e.g. the gold wires inductance, and by the laser relaxation oscillation in a monolithic version. Further increasing the noise level above 225 mv destroys the periodicity of the signal, yet the pulse shapes remain. Further analysis of the corresponding

5 Stochastic induced dynamics (a) (b) Fig. 3 a Experimental time traces of electrically noise induced neuron-like phenomena in both the electrical (V) and the optical (a.u.) domains: (i) (ii) random pulsing firing (V dc = 2.85 V and V noise = 180 mv); (iii iv) coherence resonance behavior (V dc = 2.85 V and V noise = 225 mv); (v) (vi) multi-pulsing bursting behavior (V dc = 3.36 V and V noise = 250 mv). b Corresponding histograms of the ISI statistics of the laser : (i) random pulsing firing; (ii) coherence resonance behavior; (iii) multi-pulsing bursting behavior Fig. 4 Experimental time traces of electrically noise induced neuron-like phenomena measured in the laser for a circuit dc biased in a bistable condition, V dc = 2 V. Random fired squared pulses at (i) V noise = 125 mv, and (ii) V noise = 150 mv; (iii) periodic sequence of square pulses, V noise = 250 mv inter-spike interval (ISI) series data, Fig 3b, reveals the pulse statistics showing a typical exponential behavior of a Kramer s escape process (Lindner et al. 2004), displaced by the refractory time of the excitable orbits. Figure 3b (ii) shows the ISI of coherence resonance. Similar behavior is observed when the RTD-PD-LD is biased in region III, but now emitting downward upward electrical pulses, Fig. 3a, plot (v), and upward optical pulses that appear in bursts [see Fig. 3a, plot (vi)] due to the asymmetry of the I V curve, which has smaller slope on the valley side. Note that, since the operating point is now at the low current value in region III, which defines the starting point of the excitable orbit, the system experiences a current increase which explains the upward optical pulses. The analysis of the inter-spike interval [Fig. 3b (iii)] reveals that the bursts manifest themselves in the histograms as a peaked structure at n/t l,wheret l = 350 ns corresponds to the excitable refractory time. This means that in this bursting regime, if we have an excitable orbit at T 1, it is likely to be followed by another one at T 2 = T 1 + T l. We see also the appearance of the statistics with its normal tail, that is very similar to the Fig. 3b (i), as a result of the excitable response. When the RTD-PD-LD is loaded with high resistance, circuit B, it shows a region of bistability around 2 V, Fig. 1d. This bistability is evidenced as noise-triggered hopping between the two stable states in regions I and III. When dc biased in such a condition, switching between these two states leads to some irregular waveforms composed of two plateaux whose duration is governed by the Kramer escape time in both basins of attraction, Fig. 4, plots (i) and (ii).

6 B. Romeira et al. Increasing the noise level allows to reduce the residence times in both states, and leads to an almost regular square waveform, see Fig. 4 (iii). This is the basis for stochastic resonance, whereby a signal of small amplitude can be amplified and regenerated in bistable systems by adding an optimum amount of noise. 4Conclusion We have investigated a novel optoelectronic oscillator consisting of a resonant tunneling diode-photodetector driving a laser diode that can be noise-activated, providing a rich variety of neuron-like electrical optical signal s. In monostable operating conditions, excitable single spiking, bursting, and coherence resonance were successfully achieved. When operating in bistable conditions, stochastic resonance phenomena was demonstrated by adding an optimum amount of noise. Since the N-shape I V of this optoelectronic circuit is maintained almost from DC up to GHz frequencies, RTD-based OEOs can be designed for either low or fast speed response for neural emulation applications, or ultra-fast data processing. Acknowledgments J.J. acknowledges financial support from the Ramon y Cajal fellowship. B.R. thanks FCT Portugal for a Postdoctoral Fellowship (Grant SFRH/BPD/84466/2012). J.J. and S.B. acknowledge financial support from project RANGER (TEC C03-01) and from the Direcció General de Recerca del Govern de les Illes Balears and the FEDER funds. References Barbay, S., Kuszelewicz, R., Yacomotti, A.M.: Excitability in a semiconductor laser with saturable absorber. Opt. Lett. 36(23), (2011) Figueiredo, J.M.L., Romeira, B., Slight, T.J., Wang, L., Wasige, E., Ironside, C.N.: Self-oscillation and period adding from resonant tunnelling diode-laser diode circuit. Electron. Lett. 44(14), (2008) Hartmann, F., Gammaitoni, L., Höfling, S., Forchel, A., Worschech, L.: -induced stochastic resonance in a nanoscale resonant-tunneling diode. Appl. Phys. Lett. 98(24), (2011) Hodgkin, A.L., Huxley, A.F.: A quantitive description of membrane current and its application to conduction end exitation in nerve. J. Physiol. 117(4), (1952) Kuhnert, L., Agladze, K.I., Krinsky, V.I.: Image processing using light-sensitive chemical waves. Nature 337, (1989) Lindner, B., Garcia-Ojalvo, J., Neiman, A., Schimansky-Geier, L.: Effects of noise in excitable systems. Phys. Rep. 392(6), (2004) Romeira, B., Javaloyes, J., Ironside, C.N., Figueiredo, J.M.L., Balle, S., Piro, O.: Excitability and optical pulse generation in semiconductor lasers driven by resonant tunneling diode photo-detectors. Opt. Express 21(18), (2013) Romeira, B., Javaloyes, J., Figueiredo, J.M.L., Ironside, C.N., Cantu, H., Kelly, A.E.: Delayed feedback dynamics of Liénard-type resonant tunneling-photo-detector optoelectronic oscillators. IEEE J. Quantum Electron. 49(1), (2013) Samardak, A.S., Nogaret, A., Janson, N.B., Balanov, A., Farrer, I., Ritchie, D.A.: Spiking computation and stochastic amplification in a neuron-like semiconductor microstructure. J. Appl. Phys. 109(10), (2011) Slight, T.J., Romeira, B., Wang, L., Figueiredo, J.M.L., Wasige, E., Ironside, C.N.: A Liénard oscillator resonant tunnelling diode-laser diode hybrid integrated circuit: model and experiment. IEEE J. Quantum Electron. 44(12), (2008) Suzuki, S., Shiraishi, M., Shibayama, H., Asada, M.: High-power operation of terahertz oscillators with resonant tunneling diodes using impedance-matched antennas and array configuration. IEEE J. Sel. Top. Quantum Electron. 19(1), (2013)