Measurements and Modeling of a Monolithically Integrated Self-Spiking Two-Section Laser in InP

The self-spiking behavior of an integrated saturable absorber and gain section laser fabricated in an InP technology platform is analyzed. The gain, absorber and intensity dynamics are first inspected using the normalized Yamada model. This model shows excitable behavior as well as the relative refractory period, both of which are also present in biological neurons. Measurements of a two-section laser show irregular spike generation on the millisecond timescale, with a saturable absorber voltage controlled spike density. From our simulations, and from the quasi-random character and millisecond timescale at which these pulses occur, we conclude the laser is triggered by noise, an important characteristic in the operation of biological neurons. Simulations of the laser around the excitability threshold using a newly proposed model with an optical noise term show qualitatively similar self-spiking behavior as measured.


I. INTRODUCTION
A FTER two periods of setbacks in the '70s and '80s, research into artificial intelligence (AI) is experiencing a steady growth since the beginning of the 21 st century. The reason for this revival was the improvement in computational power in microprocessors. In recent years, research into AI is experiencing an exponential growth, mainly driven by novel algorithms, neural network modeling, big data and the predictions of the end of Moore's Law [1].
Currently, neural network models are implemented in software frameworks such as Tensorflow to build large deep neural networks (DNNs). Such networks consist of artificial neurons arranged in multiple layers, each applying a filtering operation on input data. Since neurons are densely interconnected, calculations in these networks are performed in a parallel manner. These models are often trained and put into practice using conventional electronic hardware such as CPUs and GPUs. These processors are build following the Von-Neumann architecture, where data processing and memory units are physically separated. The CPU fetches instructions from its memory unit on every clock cycle, processes the input data, and then generates the output. Thus, data is processed sequentially. This serial way of processing limits the speed at which parallel computations can be done.
Another limiting factor in electronics is power dissipation, mainly driven by electrical interconnects. The skin effect, dielectric losses, and wiring density all add to power dissipation [2], [3]. This energy consumption of electronic hardware for AI calculations is an important factor to consider. Electronic neuromorphic processors such as IBM's neurosynaptic core and Neurogrid are optimized from an architectural perspective, but still face speed and interconnection challenges due to the fundamental bandwidth fan-in tradeoff [3]. The limited bandwidth of electronic solutions also restricts the operational speed. For low-speed application such as image recognition in the kHz regime, the operational speed of electronics is sufficient, however, tasks such as manipulating the radio spectrum for hypersonic aircraft must be performed in the GHz regime [1], and fiber non-linearity compensation, model predictive control, and high-energy particle classifications need low-latency solutions [4].
The most energy efficient neural network is the human brain, where data is encoded using short electrical signals or spikes. These spikes are digital in amplitude but analogue in the time domain, and are generated by individual neurons and transmitted through axons [3]. In [5], an extensive comparison between the human brain and digital computing in terms of energy efficiency is given. The combined analogue and digital data representation scheme allows for a less strict error tolerance compared to digital computers. In digital computers, the error probability is around 10 −24 , whereas for the human brain, this value is estimated to be around 0.65. This means that the human brain with its spiking neural network operates in an incredibly noisy environment. However, the human brain is optimized such that it operates at an energy efficiency almost 60 times greater than digital computers, at only 20 Watts.
The principles of a biological spiking neural network (SNN) can be transferred to integrated photonics due to the spiking capabilities of integrated lasers [6], [7], [8]. Integrated photonics benefits from high switching speed, high communication bandwidth, low crosstalk [9], [10], and temporal characteristics governed by ultra-fast carrier dynamics [11]. This means that an optical neuron might operate orders of magnitude faster than its biological counterpart and can be the nonlinear building block in an all-optical spiking neural network on a photonic integrated circuit (PIC). There are several challenges in realizing an all-optical SNN. Next to research into the non-linear neuron [12], [13] also low-energy synaptic weighting and scalable neuromorphic architectures that allow for efficient training are being investigated [14]. Synaptic weighting can be performed using a single photonic ring resonator [15] and can be scaled up by increasing the number of resonators to form a weight bank [16] and leveraging the benefits of wavelength division multiplexing (WDM) [10]. Another approach is by using phase changing materials (PCM) to manipulate the transmission of waveguides [17], [18].
In this work, we are addressing the excitable nonlinearity [19], [20] within an SNN and in specific how lasers can be used as a self-spiking source and analyze its characteristics. Given the fact that the human brain operates in an incredibly noisy environment, it is essential to identify the effects of optical noise on its photonic counterpart. The influence of noise on a self-pulsating two-section laser has been studied before in terms of timing statistics, jitter and amplitude variations [19], [21]. This shows that noise is an important aspect in the dynamics of a two-section laser. We report on an integrated saturable absorber and gain laser that shows quasirandom self-spiking in the presence of noise. We adapted the well-known Yamada model [3], [15], [22] to investigate the effects of optical noise on the laser's self-spiking behavior in simulations. To the best of our knowledge, this is the first time the self-spiking behavior of such a laser structure is investigated in the presence of optical noise in simulations and measurements.

II. THEORETICAL FRAMEWORK
The laser structure under investigation is a two-section laser comprising a saturable absorber and gain section ( Figure 1). By including a loss mechanism in an optical cavity, the laser becomes bistable [23], [24], [25], [26] and as a result, the laser can exhibit various dynamical properties such as Q-switching, mode-locking, or a combination of both [27]. For example, a minor change of a few percent of the saturable absorber and gain length ratios and carrier lifetimes may already affect the stability, modulation depth and pulse width of a mode-locked laser [28]. In addition, also the time-bandwidth product and side mode suppression ratio depend on the saturable absorber and gain length ratio, as demonstrated in the InP technology platform [29]. Another dynamical property of these types of lasers is excitability. This refers to the state where the system, initially at rest, can be brought with an external trigger into a different state (such as a pulsating state) before it returns to its rest state. Similar dynamics are observed in biological neurons, where a stimulus must be of sufficient strength to trigger a neuron to fire [3]. This means that from an external source, either deterministic or random noise, the laser can  generate an optical spike or pulse. The intracavity loss element can be active such as an acoustic-or electro-optic modulator, or passive, such as a saturable absorber. In the last case, the laser dynamics are governed by the different carrier lifetimes in the gain and absorber region [25], which are on the order of hundreds and tens of picoseconds, respectively.
An optical pulse is formed if the absorber is bleached, i.e. if the losses are reduced significantly. This can be done by changing the absorption by means of carrier injection, or by increasing the intensity in the cavity by an optical trigger. The dynamical properties of such a laser can be described by the Yamada model, which has been extensively studied [19], [20], [25]. In this model, the gain, loss and intensity dynamics can be described by three coupled differential equations for the gain G, loss Q and optical intensity I : The parameter description is given in Table I. The time scale at which the solutions for G, Q, and I are calculated using this model is in the unit of the photon lifetime τ p . Figure 2 shows the response of G, Q and I (marked by squares, triangles and circles, respectively) under a deterministic optical pulse injection at t = 50 τ p . Under steady-state conditions (i.e. t < 50 τ p ), only low-intensity spontaneous emission contributes to Eq. (3). Since I is low, Eq. (1) and (2) , respectively, so the equations are dynamically decoupled and the system is at rest. At t = 50 τ p , an external optical pulse θ(t) is added to I , and the intensity starts to increase. Consequently, the losses rapidly decrease, and a very sharp increase in the optical intensity is formed. This depletes the gain and results in a reduction of intensity, forming a pulse. After the pulse formation, the gain and loss recover following their respective recovery rates to their steady-state conditions.
The model is used first to investigate the strength that is needed for the optical pulses to achieve an excitable response. Figure 3 shows the simulated pulse intensity amplitude for five values of the absorption B, a fixed value of 4.3 for gain bias A and optical pulses intensities ranging from 0 to 0.10. For B > 3.42, a step in the optical intensity is observed. This means that for an injected optical pulse strength of 0.006 a response pulse is formed. For a gain bias current A = 4.3 and level of absorption B = 3.42, the threshold for an excitable response is almost six times lower compared to the case where A = 4.3 and B = 3.72, see Figure 3. Similar results can be achieved by tuning A and setting B to a specific fixed value. This means that depending on the biasing conditions, a response pulse can be formed only after a single small optical trigger is injected into the cavity.
A special case occurs when the value for B is lowered even further, e.g. 3.32, a response is observed regardless the injected optical pulse strength. This means that the laser is able to form a pulse without any optical pulse injected, e.g. it is selfpulsating. This is a different operation regime compared to the excitable regime. Another special case is when multiple subthreshold pulses close in time are injected into the model. Due to the recovery rates, a short train of subthreshold pulses applied within the recovery period is also able to saturable the absorber and result in an excitable response.
Another aspect of the pulse formation is the time delay between the injected pulse and its response. Figure 4 shows the response amplitude after a first and second optical pulse injection. The time delay between the two pulses is varied from 0 to 140 τ p . The solid line shows the response after the first pulse is injected. This is a flat line, since the system  is initially at a steady state. Note that for a second injected pulse, there is a dead time of about t = 60 τ p . During this window, the system has not recovered yet, i.e. the gain is not strong enough and losses are too high to form a pulse. After this window, a second pulse starts to form, although at a lower amplitude. If the time delay between the two pulses is increased further, both responses approach the same amplitude. The period during which a second pulse is formed, although at a lower amplitude, is defined as the relative refractory period, which is a similar property biological neurons show. This is an important factor, since it sets a lower limit on the time a second pulse can be formed.

III. FABRICATED DEVICE
The gain and saturable absorber integrated laser was fabricated in a commercially available active-passive multiproject wafer (MPW) InP integration platform [30]. In this relatively mature platform, photonic integrated circuits can be designed using predefined building blocks such as gain sections, phase shifters, distributed Bragg reflectors (DBR), multimode interference reflectors (MIR), electrical isolation sections and 2.0 μm and shallow and 1.5 μm wide deeplyetched waveguides. In Figure 5, the cross-section of the two types of waveguides are shown. An InGaAsP multi quantum well (MQW) layer is used for active regions, surrounded by InP for passive components and waveguides. To probe the active regions electrically, layers of InGaAs and gold are deposited on top of the InP layer.
Using these building blocks, linear cavity lasers with an 80 μm saturable absorber, 500 μm gain section, a 500 μm phase shifter, a 350 μm DBR optimized for reflections at  1550 nm and an MIR with estimated reflectivities of 0.87 and 0.40, respectively, were fabricated. The active regions are separated by 30 μm electrical isolations sections and are connected using shallow and deeply etched waveguides. The motivation for the chosen dimensions was to fabricate a laser that has the smallest form factor possible on this platform, and yet would be able to show excitability. This means that all individual components should be as small as possible. The gain section length was based on a Fabry-Pérot laser cavity comprising a 420 μm gain section and two multimode interference reflectors previously fabricated on the same platform [30]. To account for the photon absorption by the saturable absorber, a larger gain section and higher reflection DBR were used. The DBR length of 350 μm was chosen to ensure single mode operation and a relatively high reflectivity of approximately 0.8 [31]. The electrical isolation sections of 30 μm are chosen to yield high enough isolation between sections (∼M). The outputs of the laser are coupled to the edge of the chip, where light can be collected using lensed fibers. An isolator prevents back reflections from the measurement equipment to the chip. Figure 6 shows a micrograph of one of the fabricated devices. Clearly visible are both mirrors, the folded waveguides going to the edges of the chip, the saturable absorber and gain sections, and a phase shifter.
IV. MEASUREMENTS Figure 7 shows a schematic overview of the DUT and the measurement setup. The chip temperature was stabilized on a copper chuck using a Peltier element and ILX Lightwave 3900 TEC to a temperature of 20 • C. Using a lensed fiber, light was collected at the MIR output of the chip, amplified by a Keopsys EDFA before it was fed into a high speed photodiode and 6GHz LeCroy Wavemaster 8600A real time oscilloscope to collect time traces. A power meter is used to monitor the optical output and to align the lensed fiber optimally to the edge coupler. The ILX Lightwave 3900 was also used to pump current into the gain section, a Keithley  2401 source meter unit (SMU) was used to apply a voltage and to measure the current through the saturable absorber.
The integrated laser showed single mode operation well above the lasing threshold, as shown in Figure 8. At the lasing threshold, the laser was operating in multimode regime.
To validate that our devices indeed show bistable operation, the L-I characteristics and saturable absorber I-V characteristics are measured first. In the L-I characteristics, the bistability manifests itself in a hysteresis when a fixed bias voltage or current is applied at the saturable absorber, while the current through the gain section is swept from low to high and vice versa [24]. Figure 9 shows a sudden onset in the optical output power in both gain current sweep directions. In this experiment, the current was swept from 0 to 100 mA in steps of 1 mA with a scan dwell time of 0.1 sec. The moment at which this sharp increase happens, depends on the direction the current is swept, and thus a hysteresis is observed. Although the maximum optical output power only reaches 400 μW, this was sufficient to observe the hysteresis and verify our device shows bistable operation.
The bistability is also present when the saturable absorber I-V characteristics are measured [24]. In Figure 10, the voltage  at the saturable absorber was swept from 0.0 to 1.0 V in steps of 0.01 V, while the current through the saturable absorber was monitored. This procedure was then repeated for 10 different gain pump currents. At a bias of 0 V, the absorption is relatively strong and stimulated emission in the cavity is suppressed. Increasing the positive junction voltage applied at the saturable absorber decreases the absorption, which increases stimulated emission. As a result, the photon density in the cavity increases and a larger negative photo current is generated. This is observed between approximately 0.5 V and 0.8 V for gain currents between 50 and 70 mA. In this case, the saturable absorber is bleached and the laser is turned on. When the voltage is further increased to a value above 0.8 V, the typical forward bias diode characteristic becomes dominant.
Next, the gain section is biased slightly below the hysteresis in Figure 9 at 50.11 mA. From simulations, it was found that at this biasing point excitable behavior is to be expected. Over 50 ms (1M samples), the optical output is monitored for different saturable absorber voltages. At a saturable absorber voltage of approximately 0.720 mV, the laser starts to generate pulses without any optical injection. By increasing the voltage only slightly, the absorption decreases and the density of pulses rises. This is clearly visible in Figure 11 and Figure 12 for voltages of 0.727 V and 0.730 V, respectively. Note that the pulses do not occur at a fixed repetition rate, which would be expected for a self-pulsating laser.
Note that the pulses do not occur at a fixed repetition rate, which would be expected for a self-pulsating laser. Given the quasi-random character of the pulses and the fact that pulses occur on the millisecond timescale, we conclude that the pulses are not generated by self-pulsating mechanisms and relaxation oscillations, but that these pulses are likely triggered by noise.

V. NOISE MODELING
To model optical noise, a white-noise fluctuating δ-correlated function (Langevin force) F I (t) is added to Eq. (3) in the normalized Yamada model as follows: The equations for gain and absorption remain as in Eq. (1) and (2). In (4), the Langevin force F I (t) satisfies [32]: where D I I is the diffusion coefficient for the intensify. In each integrated step, a random number is drawn from a Gaussian probability distribution with zero mean and standard deviation 2R S I t, where R s is the average of spontaneous emission rate and the fixed the integration timestep t. The extended model with optical noise is used to simulate the optical output of a laser biased in the excitability regime. For weak absorption, in this case B ≤ 3.680, noise is able to trigger the laser to create an irregular train of pulses. This is shown in Figure 13 and Figure 14 for values of B = 3.6802 and 3.6794, respectively.
From these figures is clear that by decreasing the absorption only slightly (B = 0.0008), the pulse density increases significantly. Similar results are obtained by changing the value for the gain A. By eliminating the noise, i.e. setting R s in both cases to zero, no pulses are generated. From  these simulations it is clear that an external optical noise is able to trigger an excitable laser to generate an irregular train of pulses, similar to the measurements. Changing B in simulations, which is similar to changing the absorber voltage, also changes the pulse density.
For a total of 50 simulated time traces with the parameter settings as in Figure 14, the timing distribution and its fit are shown in Figure 15. The timing follows a positively skewed distribution with a lower bound related to the relative refractory period dead-time. From simulations of multiple values of B, it is clear that the mean value decreases if B is decreased, i.e. if the absorption becomes less, the time difference between consecutive pulses stabilizes.

VI. DISCUSSION
Integrated lasers with a saturable absorber and gain section share similar dynamical properties as biological neurons. For this reason, the proposed integrated laser might be a suitable candidate for an optical neuron. The biological neuron is highly optimized to operate in a noisy environment, therefore we have analyzed the effects of optical noise on an integrated two-section laser that can produce quasi-random spikes. The normalized Yamada model has been used before to study excitability and describe the gain, absorber and intensity dynamics of optical neurons. With the extension described here, it is possible to study the effects of optical noise on the excitable laser. The qualitative agreement between the measurements and the data generated by the model indicates that noise may influence the spike generation when operated in the excitable regime. This is an important aspect to consider, especially when the integrated laser is used as an optical neuron.

VII. CONCLUSION
In summary, we have measured and simulated a selfspiking saturable absorber and gain section laser. Integrated lasers show similar dynamical properties as biological neurons such as excitability and the relative refractory period, and might therefore be a suitable candidate to be used as an integrated optical neuron in an SNN. The laser dynamics were simulated first using the well-known Yamada model. A twosection laser was fabricated in a commercially available activepassive multi-project wafer (MPW) InP integration platform. The fabricated laser shows a bistability and hysteresis in the I-V and L-I characteristics, and shows a quasi-random selfspiking output on the millisecond timescale. An addition to the well-known Yamada model was presented to include the effects of optical noise on an excitable two-section laser, which is also an important aspect in biological neurons. This model shows qualitative agreement between absorption controlled simulated and measured spike densities.