Boolean logic gates implemented by a single photonic neuron based on a semiconductor Fano laser

Taiyi Chen; Taiyi Chen; Pei Zhou; Pei Zhou; Yu Huang; Yao Zeng; Shuiying Xiang; Nianqiang Li

doi:10.1364/OPTCON.461448

1. Introduction

With the emergence of increasing intelligent applications, a great amount of computing power is needed while bringing convenience to human beings with huge amounts of data. The traditional Von Neumann architecture, however, approaches essential efficiency bottlenecks, e.g., energy consumption, memory, and speed limitation [1]. Recently, neuromorphic computing systems as a typical Non-von Neumann structure have raised a lot of attention [2]. They aim to explore faster and more energy-efficient computing by imitating the way the brain works. More lately, the photonic realizations of neuromorphic computing systems have been explored due to their desired properties including high-speed, large bandwidth, high energy efficiency, and parallel processing ability compared with electronic counterparts [3–6]. In the brain, the information is encoded and processed in the spike trains, and the phenomenological response is called excitability. In the spike-based neuromorphic photonic computing, the so-called excitability has been investigated in various systems, such as the phase change materials (PCM) [6–8], single-photon avalanche diodes [9], resonant tunneling diodes [10,11], and different semiconductor lasers (SLs) [1,12–15]. In particular, SLs have attracted more and more attention for their rich dynamical characteristics. Most SL-based neuromorphic systems rely on the external injection from of a laser to generate excitability and the inherent relaxation oscillations (RO) may be clearly presented, which hinders the propagation of spiking dynamics in the photonic neural systems [16–19]. Due to the existence of RO, it takes more time for the system to recover to a stable state after the stimulus, which greatly slows down the operation speed. Additionally, the RO with large amplitudes may have a negative effect on the actual spiking response during the propagation, thus degrading the performance of the system. To solve this problem, an additional optical injection is added to suppress the unwanted relaxation oscillation [16]. Inevitably, the complexity of the system is increased rapidly, especially when complex tasks are performed.

In view of the above-mentioned shortcoming, we propose and demonstrate a model, in this Letter, to mimic biological neural dynamics based on a single semiconductor Fano laser (FL) under electrical modulation. An FL is well known for generating self-pulsing dynamics by replacing the active medium without extra optical injection [20]. Furthermore, one key property of an FL is that one or both laser mirrors are realized by a Fano interference between a waveguide (continuum) mode and the discrete resonance of a side-coupled nanocavity, which leads to the formation of Fano mirrors. In particular, it has been shown that the mirror can act as a saturable absorber [20] and may promote the generation of a self-sustained train of pulses at gigahertz frequencies with very low threshold powers. On the one hand, by controlling the Fano mirror, the spike can be adjusted to realize an all-optical nonlinear activation, which is suitable for neuromorphic photonic computing [21]. On the other hand, this distinct feature leads to the desired mitigation of relaxation oscillations and thus improves the stability of the laser with external perturbation [22]. In our previous work, a high-speed photonic reservoir computer was demonstrated based on the stable property in wide parameter space [23]. Additionally, as a specific type of microscopic lasers which are suitable for on-chip applications, the FL has an extremely small size (below 100 $\mathrm{\mu }{\textrm{m}^2}$) [20,24]; benefiting from the ultra-short photon lifetime in the FL [25], the information can be processed at a very high speed. Based on these advantages, an FL has great application potential in photonic neural networks. However, as such an FL as a photonic neuron has not been reported yet, herein, we demonstrate a photonic neuron based on a single FL under electrical modulation to mimic Class 1 and Class 3 neural dynamics [26], involving excitability, inhibition, threshold, and integrative properties. Moreover, by virtue of these properties, the optical Boolean operation is implemented by combining with the all-or-none law and timing encode method, which greatly reduces the complexity of the hardware implementation of neuromorphic photonic computing. In particular, several Boolean operations (AND, OR, and XOR) can be implemented in this simple system and are significantly faster than biological neurons on millisecond time scales and electrical spike methods on microsecond time scales, on account of inspiring by biological neurons and achieving by optical methods. These results may offer a great perspective for an FL as a candidate for neuromorphic photonic computing.

2. Theoretical model

Based on the Taylor expansion at the steady-state solution $({{\mathrm{\omega }_\textrm{s}},{\textrm{N}_\textrm{s}}} )$ under oscillation conditions, the nonlinear dynamics of the FL can be easily deduced in combination with temporal coupled mode equations [20,27]:

(1)$$\begin{aligned} \frac{{d{A^ + }(t)}}{{dt}} &= \frac{1}{2}(1 - i\alpha )\Gamma {v_g}{g_N}(N(t) - {N_s}){A^ + }(t)\\ \textrm{ } &+ {\gamma _L}[\frac{{{A^ - }(t)}}{{{r_2}({\omega _s})}} - {A^ + }(t)] + {F_L}(t), \end{aligned}$$

(2)$$\begin{aligned} \frac{{d{A^ - }(t)}}{{dt}} &= ( - i{\delta _c} - {\gamma _T}){A^ - }(t) - P{\gamma _c}{A^ + }(t)\\ &\textrm{ + }\frac{1}{2}(1 - i\alpha ){\Gamma _C}{v_g}{g_N}({N_c}(t) - {N_0}){A^ - }(t), \end{aligned}$$

(3)$$\frac{{dN(t)}}{{dt}} = {R_P} - \frac{N}{{{\tau _s}}} - \Gamma {v_g}{g_N}(N(t) - {N_0})\frac{{{\sigma _s}{{|{{A^ + }(t)} |}^2}}}{{{V_c}}},$$

(4)$$\frac{{d{N_c}(t)}}{{dt}} = \frac{{{N_c}(t)}}{{{\tau _c}}} - {\Gamma _C}{v_g}{g_N}({N_c}(t) - {N_0})\frac{{\rho {{|{{A^ - }(t)} |}^2}}}{{{V_{NC}}}},$$

where ${A^ + }(t )$ and ${A^ - }(t )$ denote the right and left propagation envelope fields of the complex electric field in the laser cavity, respectively. $N(t )$ and ${N_c}(t )$ represent the averaged carrier densities in the laser L-cavity and nanocavity, respectively. $\mathrm{\alpha }$ is the nanocavity field confinement factor, $\mathrm{\Gamma }$ is the optical confinement factor, while ${\Gamma _C}$ is the nanocavity confinement factor. ${\gamma _L} = 1/{\tau _{in}} = {\nu _g}/({2L} )$ is the inverse of the roundtrip time in the L-cavity where L is the cavity length and ${\nu _g} = c/{n_g}$ is the group velocity with ${n_g}$ being the group refractive index of the photonic crystal (PhC). ${g_N}$, ${N_s}$, and ${N_0}$ separately denote the differential gain, the carrier density at steady state and the transparency. ${\delta _c} = {\omega _c} - \omega + \Delta {\omega _c}$ describes the difference between the laser frequency and the nanocavity resonance frequency, where $\Delta {\omega _c}$ is change of resonance frequency over time. Moreover, ${R_p} = K\ast J(t )/({e{V_c}} )$ is the pump rate where $J(t )$ is the input signal, K is the stimulus strength, and ${V_C}$ is the active region. ${\gamma _T} = {\gamma _c} + {\gamma _i} + {\gamma _p}$ is the total decay rate, which is related to the coupling rate of the waveguide ${\gamma _c}$, the intrinsic loss rate ${\gamma _i}$, and the coupling rate of the cross-port ${\gamma _p}$. The decay rates are related to the quality Q-factors: ${\gamma _x} = {\omega _0}/({2{Q_x}} )$ (x = T, c, i, and p) [27]. ${\tau _s}$ and ${\tau _c}$ are the effective carrier lifetimes of the waveguide and nanocavity, respectively. ${\sigma _s}$ and $\mathrm{\rho }$ separately represent the parameters related to the number of photons, which are given in Ref. [20]. The following parameters are used in the stimulation: $L = 5\; \mathrm{\mu}\textrm{m}$, $A = 0.21\; \mathrm{\mu}{\textrm{m}^2}$, ${\lambda _0} = 1554\; \textrm{nm}$, $\Gamma = 5$, ${\Gamma _C} = 0.3$, $\mathrm{\alpha } = 1$, ${Q_T} = 500$, ${Q_i} = 14300$, ${Q_p} = 10000$, $n = {n_g} = 3.5$, ${g_N} = 5 \times {10^{ - 20}}\; {\textrm{m}^2}$, ${N_0} = 1 \times {10^{24}}\; {\textrm{m}^{ - 3}}$, and ${\tau _s} = {\tau _c} = 0.5\; ns$. In addition, the nanocavity volume is ${V_{NC}} = 0.24\; \mu {m^3}$, and the cavity volume is $V = LA$. The last term in Eq. (1) is the complex Langevin noise source F_L(t).

3. Result and discussion

In the FL, the spiking dynamics similar to the Class 1 neural excitability can be found. For continuous stimulus strength, the frequency of output spikes varies with the current is shown in Fig. 1(a), where the frequency-input curve is so smooth that spike trains of an arbitrary frequency can be stimulated [Fig. 1(b)] or suppressed [Fig. 1(c)] by applying an external stimulus (raising or dropped rectangular stimulus) of different strengths. Figure 1(d) shows that the dynamics in response to a raising rectangular stimulus out of the Class 1 neural dynamic region has only one spike in the duration, which resembles Class 3 neural excitability and is known as phasic spiking. Interestingly, the unwanted relaxation oscillation, which affects the speed, accuracy and stability of the computation in other systems [17–19], disappears at the end of the external stimulus in the proposed system.

Fig. 1. (a) The frequency-input curve of the spiking dynamic in the FL. (b) The excitatory dynamics and (c) the inhibitory dynamics of the FL subject to external stimulus. (d) Phasic spiking response in FL subject to stimulus.

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To begin with, we demonstrate the excitation and threshold properties of the FL photonic neuron in a simple case. The duration of the stimulus is fixed at $\mathrm{\Delta t\;\ } = {\; }2$ ns. Figure 2(a) shows the three different levels (K = 0, K = 1.5, and K = 3) of the input for photonic neuron based on an FL, the response spiking dynamics of the FL neuron is shown in Fig. 2(b). Here, the output is calculated as ${P_t} = 2{\varepsilon _0}nc|{A^ + }(t )- {A^ - }(t ){|^2}$ with ${\varepsilon _0}$ being the free space permittivity. As can be seen from Fig. 2(b), there is no spike response when no stimulus (K = 0) or a small stimulus (K = 1.5) is applied, whereas a phasic spike with a large amplitude is generated in the FL neuron under a large stimulus (K = 3). Next, the logic OR and AND operations based on the properties are explored, as shown in Fig. 2(c). For the sake of simplicity, the duration of each rectangular stimulus is set as $\mathrm{\Delta t\;\ } = {\; }1.5$ ns to encode the input bit. Bit “0” represents the strength level ${K^{(0 )}}\; \equiv \; 0$ while bit “1” represents a small level strength ${K^{(1 )}}\; \equiv \; 1.5$, which corresponds to a raising rectangular stimulus similar to Fig. 2(a). Specifically, the input term can be written as ${K_1}{J_1}(t )+ {K_2}{J_2}(t )$. Then an addition term (${K_a}\; = \; 1.5$) is applied to realize the real-time switching between OR and AND operations. The spiking dynamics conforming to the all-or-none criterion is one of the most prominent properties of biological neurons, denoting the response result of the Boolean operation. During the rectangular stimulus, bit “0” represents no-spike response while bit “1” corresponds to a phasic spike.

Fig. 2. (a) Input of the FL neuron with different stimulus. (b) The responses of the FL under the input of (a). (c) Schematic diagrams of an FL neuron for realizing logic OR and AND operations.

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Here, all four patterns (11, 10, 01, and 00) for two-bit Boolean operations are considered. Eight cases for testing the AND operation are shown in Fig. 3. Figures 3(a) and 3(b) show the two-bit sequences, which are used as input 1 and input 2, respectively. The response results of the FL neuron (which are also the arithmetic results of the two-bit Boolean operation) are shown in Fig. 3(c). It is verified that one spike is generated only in the case of “11”, while no spike is obtained in other cases. In other words, the AND logical operation is implemented by the proposed system. However, for the cases of the OR operation, the addition term is applied to switch the AND operation to the OR operation, so that the input term can be replaced by ${K_1}{J_1}(t )+ {K_2}{J_2}(t )+ {K_a}{J_a}(t )$. The corresponding test results of OR are shown in Fig. 4, where no spike is generated for the case of “00” and a phasic spike is fired for other three patterns. This indicates that the OR operation is also performed successfully in the photonic neuron based on the FL.

Fig. 3. (a) The input 1 and (b) the input 2 of two-bit operations. (c) The response result of the AND.

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Fig. 4. (a) The input 1 and (b) the input 2 of two-bit operations. (c) The response result of the OR

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Subsequently, we also consider the influence of some key parameters, i.e., $\mathrm{\Delta }t$ and K, on the successful results of the Boolean operations. We assume that the strength and the duration of the stimulus of the bit “1” in the AND operation are the same as in the OR operation, and the switching between them is implemented only by the addition term ($0 < {K_a} \le {K^{(1 )}}$). In Fig. 5, we illustrate the response results of the logic gates in the parameter space ($\mathrm{\Delta }t,\; K$). Here, the duration of the rectangular stimulus ($\mathrm{\Delta }t$) for a bit “1” range is set from 0 ns to 2 ns. We focus on the region where the AND operation can be implemented successfully by varying the input strength of bit “1” (${K^{(1 )}}$). For the response result with the varying of K, we can find three distinct regions. In Region A (blue), the AND operation is unattainable due to a weak input strength of bit “1” (${K^{(1 )}}$). In this region, there is no spike response in any of the four patterns including “11”, which is termed as the non-logical region. In the Region B (green), once ${K^{(1 )}}$ exceeds the threshold $K_{th}^{(1 )}$, the AND operation is easily available, i.e., one spike occurs only in the pattern “11”. Further, by attaching an addition term ${K_a}$, the OR operation can be also obtained, i.e., one spike occurs in the patterns “11”, “10”, “01”. That is, the AND can switch to the OR when the patterns “10” and “01” emit a spike in the condition:

(5)$$K_{th}^{(1)} + K_{th}^{(1)} \le {K^{(0)}} + {K^{(1)}} + {K_a}.$$

where the strength of bit “0” ${K^{(0 )}}\; \equiv \; 0$, As ${K^{(1 )}}$ further increases to $2K_{th}^{(1 )}$, it is obvious from the Eq. (5) that the spike can be observed in the patterns “10” and “01”. Namely, only OR operation is available in the Region C(yellow). Here, both Regions B and C are defined as the logical regions.

Fig. 5. Two-dimensional map of the response results of Boolean operations in the parameter space ($\mathrm{\Delta t},{\; \textrm{K}}$)

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Furthermore, the spike dynamics based on an FL proposed here has the integrated property similar to the biological neuron dynamics and the ability to integrate the weighted sum over time. In other words, the spikes appear earlier with the increase of stimulus intensity, which has been widely discussed in the previous report [1]. The duration of the rectangle stimulus in the following four cases is kept constant to investigate the influence of the input strength (K) on the time when the spike emits, as shown in Fig. 6(a). For clarity, the input rectangle stimulus (in pink dotted lines) and the response results (in blue solid lines) are plotted in the same coordinate region, the starting time of the rectangle stimulus and the spike time are defined as ${t_{in}}$ and ${t_p}$, respectively. Here, the stimulus strength (K) of the stimulus is set as 2, 3, 4, and 7, and the duration of the stimulus is fixed at $\mathrm{\Delta t\;\ } = {\; }2$ ns. As shown in Fig. 6(a1), a spike is generated near the end of the rectangle stimulus (${t_p} - {t_{in}} \approx \; 2\; $ ns) for the strength K = 2, which means that the stimulus strength is close to the threshold at $\mathrm{\Delta t\;\ } = {\; }2$ ns. However, a typical phasic spike appears as the strength is further increased. For the cases of K = 3 and K = 4, as can be seen in the Figs. 6(a2) and 6(a3), a phasic spike without the unwanted relaxation oscillation at the end of the rectangle signal is generated, which is attributed to the Fano interference suppressing the relaxation oscillation in the laser [23]. The latency time of the spike, i.e., the interval between initiation time of stimulus and spike time ${t_p} - {t_{in}}$, is about 1.3 ns and 0.8 ns, respectively. Obviously, the latency time is decreasing rapidly at first. Figure 6(a4) shows that the stimulus strength is further increased to 7. Here, a phasic spike with a large amplitude is still generated but the latency is decreased slightly compared to the large increase of the stimulus strength. That is, as the spike time approaches the initial time of the stimulus, more energy is required, in other words, a large K is needed.

Fig. 6. The response of an FL under constant duration ($\Delta \textrm{t} = 2$ ns) of stimulus with (a1) K = 2, (a2) K = 3, (a3) K = 4, and (a4) K = 7. (b) XOR output for time encoding. (c) The change of $\textrm{K}_{\textrm{th}}^{(1 )}$ with different duration of bit “1”.

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Meanwhile, precise timing can be used to encode information in biological neurons, which is one of the greatest advantages for the spiking neural network [1]. The time information contained in the spike dynamics as shown in Fig. 6(a) can also be used to represent the results of the Boolean operation. The XOR operation is shown here as an example. For simplicity, bit “0” still represents the strength level ${K^{(0 )}}\; \equiv \; $0, the strength bit “1” is set as ${K^{(1 )}} = 2$, and the duration of each bit is fixed at $\Delta \textrm{t}\; = 2$ ns. Time information in the spike of the FL output is used to denote the response results of the XOR operation. During the rectangular stimulus, the response of the XOR operation will be treated as bit “0” when the following conditions are fulfilled:

(6)$${t_p} - {t_{in}} \in [0,\frac{1}{2}\Delta t]\textrm{ }or\textrm{ }no\textrm{ spike}.$$

otherwise, the response denotes bit “1” when the following condition is satisfied:

(7)$${t_p} - {t_{in}} \in (\frac{1}{2}\Delta t,\Delta t].$$

several cases including all the four patterns for testing the XOR operation are shown in Fig. 6(b). To show the results clearly, black dotted lines are used to divide the rectangular stimulus at the central timing. It is obvious that a spike is generated before the black dotted line in the duration of a bit for the pattern “11” and no spike is fired for the pattern “00”, which fits the conditions in Eq. (6), whilst a spike appearing near the end of the stimulus satisfies Eq. (7). Namely, the XOR operation is successfully implemented in the proposed system by time encoding. Certainly, a lower bit width can be used to increase the speed of the operations, but a larger $K_{th}^{(1 )}$ follows as shown in Fig. 6(c), which corresponds to the conclusion in Fig. 6(a).

4. Conclusion

In conclusion, we have proposed and numerically demonstrated a high-speed photonic neuron model based on an FL under electrical modulation, which can mimic the Class 1 and Class 3 neuronal dynamics. The properties of the dynamics were mainly involved with the excitability, inhibition, threshold, and integration in the photonic neuron. Interestingly, benefiting from the Fano interference, which forms the unique Fano mirrors, the unwanted relaxation oscillation found at the end of the stimulus in other systems disappeared. The result directly promotes to the improvement of the computing efficiency and speed. Furthermore, based on these properties with the all-or-none law and timing encode, several Boolean operations (AND, OR, and XOR) were testified successfully with a fast time scale. These results pave the way for the implementation of integrated and low-energy neuromorphic photonic spiking processing and computing systems.

Funding

National Key Research and Development Program of China (2021YFB2801900, 2021YFB2801901, 2021YFB2801902, 2021YFB2801904); National Natural Science Foundation of China (62004135, 62001317, 62171305); Natural Science Research Project of Jiangsu Higher Education Institutions (20KJA416001, 20KJB510011); Natural Science Foundation of Jiangsu Province (BK20200855).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Boolean logic gates implemented by a single photonic neuron based on a semiconductor Fano laser

Abstract

1. Introduction

2. Theoretical model

3. Result and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (7)

Optics Continuum