Keywords

1 Introduction

Biological systems possess a wealth of complexity and diversity in its architecture, that sometimes exhibit advanced capability in performing highly complex cognitive tasks. Recent development of artificial intelligence has successfully introduced such biological architectures into their algorithms. One of the most successful examples should be Deep Neural Network (DNN), consisting of multiply connected layers composed of nonlinear elements (neurons), where information propagates unidirectionally from layer to layer. In spite of the fact that DNN profits by its extremely high accuracy in many recognition tasks [1–3], high learning cost is still critical bottle neck, and fundamentally its function is oriented to static recognition. Recurrent Neural Network (RNN), another bio-inspired architecture, consists of connected nonlinear elements but has recurrent loop of information propagation. Owing to its recurrent feedback loop, RNN has internal memory that allows recognition of dynamic (sequential) information. Despite that advantage, however, RNN also suffers from learning cost as same as DNN.

Reservoir Computing (RC) is an emerging computation algorithm, which can overcome drawback of high learning cost that dangles about Neural Network (NN) [4, 5]. RC consists of mainly two parts; RNN with random connections, called reservoir, and linear classifier called output layer. The feature of RC is that randomly given weight values between nodes in reservoir are left untrained, and only weights between reservoir and output layer are updated during training process. That trick allows RC’s learning cost to be low compared with other NNs. Provided that the number of reservoir’s nodes is large enough, input data is nonlinearly projected into high dimensional space with plenty of random basis, therefore can be linearly classified at output layer, being analogous with kernel method of machine learning. Leveraging its low learning cost and high capability in classification of sequential data, RC has been applied to several classification tasks such as spoken digit recognition [6], hand-written digit recognition [7], phoneme recognition [8], etc.

RC has another great advantage in its ease of implementation as hardware. Due to serial processing and Von-Neumann bottleneck, software implementation of bio-inspired algorithms has been facing critical issue of energy-consumption. While a human brain consumes only 10 W for daily cognitive tasks, today’s high-performance computer is anticipated to consume the order of 10 kW in 2020 for the same tasks even on the ideal extrapolation of Moore’s Law. In contrast, hardware implementation of bio-inspired algorithms can overcome that issue thanks to its parallel processing and distributed information representation [9]. Owing to small amount of tunable weights, RC is suitable for this purpose. Several hardware implementations of RC have been reported so far by Mackey-Glass electronic circuit [10], soft silicon material [11], connected Semiconductor Optical Amplifier (SOA) [12], laser with time-delayed feedback loop [13], etc. What is remarkable moreover in hardware implementation is that a reservoir is not necessarily a RNN but can be any dynamical physical system, only provided that the system possesses rich nonlinearity. If one can implement reservoir as nonlinear physics with short time constant, processing in reservoir is performed as physical phenomena, therefore high-speed and low-power processing will be achieved.

Among several types of hardware implementations of RC, photonic implementation will be in particular of importance, because photon-electron nonlinear interaction have extremely short time constant around ps to ns, and moreover, it is highly compatible with existing interconnect technology, Silicon Photonics (SiPh) that is highly compatible with matured CMOS process [14]. Multi-integration of electric circuit, SiPh circuit, and photonic RC circuit on the identical silicon chip will tremendously expand the computing capability for cognitive tasks at edge devices, that are indispensable for IoT and cognitive era.

In this paper, we propose a photonic RC consisting of a semiconductor laser with a tunable external mirror, which is suitable for SiPh owing to its simple and compact configuration. The reservoir is tested by a simple task of waveform classification with varying system’s parameter, carrier injection rate to the laser and mirror reflectivity. The best operation window as a reservoir is scrutinized.

2 Methods

2.1 Chaotic Dynamics of Laser

Three-variable systems like Lorentz model are well-known to exhibit chaotic behavior under some conditions met. It was 1975 when distinctive analogy between Maxwell-Bloch equations that describe light-matter interaction of laser and Lorenz equations that describe fluid convection was pointed out by Haken. Despite that analogy, in most of lasers including semiconductor lasers, damping of polarization (γ p ) is much faster than other two parameters; population inversion (γ N ) and electric field (κ), then γ p is adiabatically eliminated, and the lasers operate as stable two-variable systems. Although a semiconductor laser itself oscillates with stable state, external perturbation introduces additional freedom to this two-variable system, and induces chaotic dynamics.

External feedback to a semiconductor laser is one of the most popular methods to bring on chaotic state [15]. Under critical amount of external feedback, intrinsic laser mode with relaxation oscillation frequency f RO and external frequency f EC compete each other and construct complex temporal pattern. The key of RC is to make the system to be highly nonlinear but stay verging on chaotic state, so-called “edge of stability”. This delicate state is realized by tuning several laser parameters.

Lang-Kobayashi equation is well-known equations that describe laser system with time-delayed feedback [16, 17]. Electric field E(t) with slowly varying amplitude in a single mode laser and carrier density N(t) are coupled in the form of rate equations,

$$ \frac{{d\varvec{E}_{1} (t)}}{dt} = \frac{1 + i\alpha }{2}\left[ {\frac{{G_{N} \left( {N_{1} \left( t \right) - N_{0} } \right)}}{{1 + \varepsilon \left| {\varvec{E}_{1} \left( t \right)} \right|^{2} }} - \frac{1}{{\tau_{p} }}} \right]\varvec{E}_{1} \left( t \right) + \frac{\kappa }{{\tau_{in} }}\varvec{E}_{2} \left( {t - \tau } \right)\exp \left( { - i\omega t} \right) , $$
(1)
$$ \frac{{dN_{1} (t)}}{dt} = J_{1} - \frac{{N_{1} \left( t \right)}}{{\tau_{s} }} - \left[ {\frac{{G_{N} \left( {N_{1} \left( t \right) - N_{0} } \right)}}{{1 + \varepsilon \left| {\varvec{E}_{1} \left( t \right)} \right|^{2} }} - \frac{1}{{\tau_{p} }}} \right]\left| {\varvec{E}_{1} \left( t \right)} \right|^{2} , $$
(2)

where physical variables J 1, N 1, E 1, E 2, represent carrier injection rate, carrier density, electric field in a laser cavity, electric field reflected from an external mirror respectively. κ represents coupling coefficient between internal mode and external mode, given by κ = (1 − r 21 )r 2/r 1, where r 1 and r 2 are laser facet reflectivity and external mirror reflectivity. Other optical parameters are given in Table 1 [16].

Table 1. Optical parameters used in the calculation.
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We numerically solve those equations to investigate temporal dynamics of the laser-feedback system with varying laser’s parameters. And then, we connect a linear classifier to the system and test its capability as a reservoir to nonlinearly transform input signal to allow efficient classification.

2.2 Computational Set-up

We assume the reservoir to consist of a semiconductor laser and an external mirror whose reflectivity is tunable (see Fig. 1). The laser oscillates under the constant carrier injection to the gain medium, being perturbed by external feedback light. Input information to the system is supplied as electric signal pattern of other carrier injection that modulates the mirror reflectivity. The input signal pattern is transported to the laser as modulated optical field pattern, and interacts with intrinsic laser field in the cavity and electrons. Optical output power from the laser is periodically sampled, being regarded as output information from the reservoir. By closely chaotic dynamics in the laser, signal pattern is nonlinearly transformed. Here, each sampling point on temporal axis represents “node” of the reservoir, being connected to output nodes. Sampled optical intensities are weighted-summed and fed into output nodes consisting of nonlinear activation functions such as sigmoid functions. The weights connecting temporal “nodes” and output nodes are updated during training.

Fig. 1.
figure 1

Conceptual diagram of photonic reservoir. Input signal is supplied to the reservoir as electric signal to modulate DBR mirror reflectivity. Output power from the laser is converted to digital signal, and sampled at FPGA/CPU, then weighted summed.

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Considering integration with SiPh, we assume a reservoir consisting of available photonic devices familiar with today’s SiPh. For an external mirror, we can leverage Distributed Bragg Reflection (DBR) mirror, which has periodic structure in wavelength scale. Integrating pn junction, electric current is supplied to the DBR to change carrier density inside the structure, causing to modulate the reflectivity. By carefully designing DBR mirror’s periodicity to make it’s photonic band-edge around wavelength of incident light, the reflectivity can be modulated from 0.0 to 1.0 by small amount of electric current. Optical power is transferred by Si waveguide (SiWG), and detected by Photo Detector (PD), and converted to digital signal by Analogue Digital Converter (ADC), and then its sampling and weighted-summation are performed at CPU/FPGA, that is located outside of the SiPh chip. Compared with looping feedback system, the proposed system has advantage of spatial compactness owing to in-line device configuration free from optical coupler for output monitoring.

To simulate dynamics of the proposed system, we performed calculation by numerically solving Eqs. (1) and (2) by 4th order of Runge-Kutta algorithm.

2.3 Recognition Task

For testing performance of the photonic reservoir, we set a simple task to classify sine/triangular wave forms. Each waveform is supplied to the system independently as a modulation pattern of mirror reflectivity. Weight values of output layer are trained to exhibit 01 and 10 for sine and triangular waveforms respectively. For varying sample’s patterns and making the task more complex, frequency ω n of n th waveform sample is given by \( \omega_{n} = \omega_{0} + \alpha_{n} \omega_{0} \), where \( \alpha_{n} \) is a random value satisffying \( \alpha_{n} \in [0, A) \). Higher A expands sample’s variation and makes the task more difficult. In this paper we set parameters A = 0.3 for the first test. The variations of waveforms are shown in Fig. 2.

Fig. 2.
figure 2

Waveform variations of input data. Sine wave with lowest (a) and highest (c) frequency and triangular wave with lowest (b) and highest frequency (d).

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3 Results

3.1 Dynamics of Reservoir

Fist we investigate fundamental dynamics of the reservoir without injecting input signal patterns, meaning, mirror reflectivity is fixed. We scrutinize temporal dynamics of output optical intensity under different laser parameters; normalized carrier injection rate J/J 0 and reflectivity r ex of the external mirror. It should be noticed that J 0 is laser oscillation threshold without external feedback, but actual threshold slightly decreases under feedback. External cavity length L is fixed to 1 cm.

In order to measure the complexity of laser dynamics, we investigate number of temporal extremal values of output power, N ex , for temporal range from 30 to 50 ns, that is enough after passing transient state required to start laser oscillation. N ex is 2D-plotted as a function of J/J 0 and reflectivity r ex in Fig. 3(a). It is shown that N ex is zero under low carrier injection and mirror reflectivity, meaning the laser oscillates in stable state, but increases with J/J 0 and r ex, and creates variety of temporal patterns. Most representative four patterns, stable oscillation, periodic oscillation, pulse package, and chaos are shown (see Fig. 3(b–e)). With increase of J/J 0 and r ex , those four states appear in seemingly random manner, but statistically pulse package and chaos states occur more frequently. We investigate reservoir’s performance as a function of J/J 0 under different r 0, central mirror reflectivity.

Fig. 3.
figure 3

Number of temporal extremals N ex is color plotted as a function of mirror reflectivity and carrier injection to the laser (a). Typical four temporal dynamics are shown; stable state (b), periodic state (c), pule packages (d), and chaotic state (e), each corresponding to b–e in (a).

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3.2 Performance of Classification

To test performance of classification, Eq. (1) and (2) are numerically solved with modulating external mirror reflectivity by \( r_{ex} = r_{0} + r_{mod} sin(\omega_{n} t \)) or \( r_{0} + r_{mod} triangular(\omega_{n} t \)). We train the system with 400 training samples (200 sine and 200 triangular waveforms), with different J/J 0 ranging from 0.5 to 2.0 under different r 0, central mirror reflectivity. r mod is fixed to 0.2. Output intensity is periodically sampled in 30 ns < T < 50 ns. Sampled 1000 data is fed into output nodes consisting of sigmoid functions and their connection weights are trained. Simple ordinary least square algorithm is used for updating the weights. After training, error rate is measured by 100 test samples independent from training samples.

The error rates as a function of J/J 0 with different r 0 are shown in Fig. 4. Remarkable is that the reservoir exhibits best classification performance when J/J 0 is close to 1 in each r 0. This is the range, according to Fig. 3, where the system transits from periodic to pulse package state by sweeping of r ex . Meanwhile, if the system is fixed around stable state or reaches chaotic state, its classification performance is substantially degraded. That feature will be qualitatively explained by complexity of laser dynamics. Higher carrier injection rate or feedback amount enhance nonlinearity of photon-electron interaction, therefore input signals are transformed in more complex manner, making waveform classification easier because slight but essential differences of waveforms are enhanced. If nonlinearity of the system is too high, inessential trivial differences are also enhanced by “sensitivity to initial condition”, and classification becomes even more difficult.

Fig. 4.
figure 4

Error rate of waveform classification task as a function of J/J 0 at different central mirror reflectivity r 0.

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Another point to notice is that with increase of mirror reflectivity, “operation window” of reservoir, where error rate is low, broadens and carrier injection rate to achieve that window decreases. From the point of view of low-energy consumption and stability of operation, it will be desired to work with external mirror with high reflectivity.

4 Conclusion

We proposed a photonic RC system consisting of a semiconductor laser with a tunable external mirror, possessing advantage in compactness of integration owing to its in-line device configuration. Complex interaction between internal laser field and reflected laser field realizes highly nonlinear projection of input sequential data to output. Testing by waveform classification task, it is confirmed that the reservoir exhibits best classification performance when laser is tuned to effective oscillation threshold, that is around “edge of stability”.

In this paper, we selected J/J 0 and r 0 for reservoir’s tuning parameters, but it is also possible to tune laser’s parameters more directly, for example Q-factor of laser cavity. It is crucially important as future works in this area to identify the ideal tuning parameters and its operation window as a reservoir, from the viewpoint of practical flexibility after integration.