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Abstract
We propose a neuromorphic convolution system using a photonic integrated distributed feedback laser with a saturable absorber (DFB-SA) as a photonic spiking neuron. The experiments reveal that the DFB-SA laser can encode different stimulus intensities at different frequencies, similar to biological neurons. Based on this property, optical inputs are encoded into rectangular pulses of varying intensities and injected into the DFB-SA laser, enabling the convolution results to be represented by the firing rate of the photonic spiking neuron. Both experimental and numerical results show that the binary convolution is successfully achieved based on the rate-encoding properties of a single DFB-SA laser neuron. Furthermore, we numerically predict 4-channel quadratic convolution and accomplish MNIST handwritten digit classification using a spiking DFB-SA laser neuron model with rate coding. This work provides a novel approach for convolution computation, indicating the potential of integrating DFB-SA laser into future photonics spiking neural networks.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Convolutional neural networks (CNNs) have found widespread applications in various fields, including image classification [1–4], image segmentation [5], and face recognition [6,7]. Convolution computation, as a core step of CNNs, is indispensable in the field of artificial intelligence. In state-of-the-art artificial neural networks (ANNs), convolution computation involves a large number of multiply-accumulate operations. This implies that executing ANNs on traditional electronic platforms based on the von Neumann architecture, such as graphics processing units (GPUs), central processing units (CPUs), field programmable gate arrays (FPGAs) and application specific integrated circuits (ASICs), requires extremely high data throughput and energy consumption [8]. In the post-Moore's law era, the presence of the “memory wall” and “power wall” in the von Neumann architecture has severely constrained the computational efficiency of ANNs [9]. Therefore, there is an urgent need to develop a new platform offering high-speed and low-power consumption to overcome these limitations.
Photonics offers unique advantages such as high speed, high bandwidth, massive parallelism, and low power consumption, making it highly promising for hardware implementations of neural networks [10–14]. In recent years, many photonics-based CNNs have emerged to improve computational efficiency [15–21]. One example is an all-optical neuromorphic binary convolution system with a photonic spiking vertical-cavity surface-emitting laser neuron, which has already been experimentally demonstrated [15]. The system completed the multiplication operation through two cascaded Mach-Zehnder intensity modulators and characterized the convolution results by counting the output spikes of the photonic spiking neuron. In the experiments, it was successfully used for image edge detection. In addition, Xu et al. proposed and demonstrated a universal optical vector convolutional accelerator operating at more than ten trillion of operations per second [16]. This accelerator was based on simultaneously interleaving temporal, wavelength, and spatial dimensions, which was enabled by an integrated microcomb source. Dot products between weights and input data were computed using micro-ring resonators, and the final calculation results were represented by the intensities of the output neurons. Furthermore, Zou et al. proposed an integrated photonic tensor flow processor without digitally duplicating the input data [17]. The key functionalities of this processor were implemented by manipulating wavelengths and delay steps. Besides, Xu et al. proposed and experimentally demonstrated a simple energy-efficient photonic convolutional accelerator based on a monolithically integrated multi-wavelength DFB laser [18], and a real-time recognition task on the MNIST database was successfully conducted using this accelerator.
In this study, we propose for the first time to use a fabricated distributed feedback laser with a saturable absorber (DFB-SA) to implement rate-encoded convolution. Rate coding is a widely used encoding method in biological neural systems. For example, in auditory system, the firing rate of auditory nerve fibers increases with increasing sound intensity [22]. Rate coding enables the precise characterization of continuous and complex stimuli, fully utilizing limited neural resources, and exhibiting robustness and efficiency [23–25]. We designed and fabricated a photonic integrated DFB-SA laser [26], which exhibits rate coding characteristics similar to those of biological neurons. It generates spikes at different frequencies in response to input signals with varying intensities. Thus, we proposed to utilize it as a photonic spiking neuron to perform binary convolution calculations. Additionally, we developed a numerical model by modifying the well-known YAMADA model [27] to simulate quaternary convolution and accomplish MNIST handwritten digit classification.
The rest of the paper is organized as follows. In Section 2, we introduce the experimental setup, the rate equation model and some basic theories of the experiment. Section 3 describes the rate coding characteristics of the fabricated DFB-SA laser and provides the results of the binary convolution experiment. In Section 4, we utilize the rate equation model to reproduce the experimental results outlined in Section 3 and further simulate quaternary convolution calculations and complete the MNIST handwritten digit classification. Finally, Section 5 provides a summary of the conclusions.
2. Experimental setup and theoretical model
We describe here the experimental setup and theoretical model of the neuromorphic convolutional system based on a photonic spiking DFB-SA laser neuron. We set an input feature map and a kernel as the two inputs of the convolution system. The value of one element in the input feature map and kernel is limited to 0 or 1.
2.1 Experimental setup
Figure 1(a) shows the experimental setup used for binary convolution calculations. A tunable laser (TL) generates an optical carrier that is injected into the Mach-Zehnder Modulator (MZM, Fujitsu FTM7928FB). We utilize two MZMs (MZM1 and MZM2, Fujitsu FTM7928FB) in cascade to implement multiplication operations. The electronic signals on the two MZMs are provided by two independent channels (channel 1 and channel 2) of a field-programmable gate array (FPGA, Xilinx xczu48dr-ffvg1517-2-e) with high-speed digital-to-analog converters. Polarization controllers (PC1, PC2 and PC3) are placed here to match the polarization state. A variable optical attenuator (VOA) is used to adjust the power of the modulated optical signal. The modulated optical signal is injected into the spiking DFB-SA laser neuron via a three-port optical circulator (CIRC). The output of the DFB-SA laser is detected by a photodetector (Agilent HP11982A) and then sent to a real-time oscilloscope (SCOPE, Keysight DSOZ592A). Here, the summation is realized offline, but it can be realized in experiment similar to Ref. [28]. For instance, we can employ two cascaded spike frequency detectors (SFD1 and SFD2) to realize the summation, as denoted by the dashed box, and is further provided in Fig. 2 (a). The optical spectrum is observed by using an optical spectrum analyzer (OSA, Advantest Q8384). A laser diode controller (LDC, ILX Lightwave LDC3724B) provides precise temperature control and low-noise bias currents to the active region, while a voltage source (VS) supplies reverse bias voltages to the saturable absorber region. During the experiment, the temperature of the DFB-SA laser is fixed at 298.02 K, and the reverse bias voltage of the saturable absorber region and the bias current of the active region are maintained at 0.4 V and 98 mA, respectively.
Fig. 1. (a) Experimental setup of binary convolution operation. TL, tunable laser; PC1, PC2 and PC3, polarization controllers; FPGA, field-programmable gate array; RF PA, radio frequency power amplifier; MZM, Mach-Zehnder modulator; VOA, variable optical attenuator; OC, optical couplers; PD, photodetectors; SCOPE, oscilloscope; CIRC, circulator; VS, voltage source; LDC, laser diode controller; OSA, optical spectrum analyzer; SFD1 and SFD2, spike frequency detector. (b) Optical spectrum of the free-running DFB-SA laser. (c) Optical spectrum of DFB-SA laser subject to constant optical injection. (d) Microscopic top view of the photonic integrated DFB-SA laser. (e) The appearance of the packaged module.
Fig. 2. (a) The schematic diagram of the SFD. DC1 and DC2, direct current source; AND, logic AND gate. (b1) An exemplary input for the SFD. (b2) The example signals following the logic AND gate. (b3) An exemplary output from the SFD.
The free-running optical spectrum and the spectrum after constant optical signal injection of the DFB-SA laser are presented in Figs. 1(b) and 1(c), respectively. The wavelength of the optical injected signal in our experiments is 1548.684 nm. The fabricated DFB-SA laser chip and the packaged module are presented in Figs. 1(d) and 1(e).
2.2 Spike frequency detector
To detect the spike frequency and deploy the convolution system onto the hardware of large-scale CNNs, we propose and simulate a SFD [28]. The schematic diagram of the SFD is shown in Fig. 2(a). The output of the DFB-SA laser is converted into electrical signals by a photodetector (Fig. 2(b1)) and is subsequently fed into the SFD. Two direct current sources provide constant threshold signals (DC1) and logic high-level signals (DC2). These signals, in conjunction with a subtractor and a logic AND gate, reshape the spikes within the spike train into square waves of equal amplitude (Fig. 2(b2)). The signal is then integrated over the encoding duration of one number, with the integration value rising faster as the spike frequency increases (Fig. 2(b3)). This indicates that the SFD can convert discrete spikes into continuous analog signals. Subsequently, these continuous analog signals can be seamlessly injected into other layers of complex CNNs, thereby expanding the applicability of the convolution system within larger and more intricate neural networks.
2.3 Multiplication theory based on cascaded MZMs
We used two cascaded MZMs to implement multiplication in the experiments. This process can be described by the transfer function of cascaded MZMs.
The derivation of the transfer function for a single MZM is as follows:
The subscripts $in$ and $out$ correspond to the input and output of the MZM, respectively. $E$ represents the optical signal, while $I$ represents the optical intensity. ${V_{b1}}$ and ${V_{b2}}$ are the bias voltages on the two arms of the MZM. ${V_{RF1}}$ and ${V_{RF2}}$ are the radio-frequency signals carrying information. ${V_\pi }$ represents the half-wave voltage. In our experiment, we utilize a single-arm driven MZM, where both ${V_{b2}}$ and ${V_{RF2}}$ are set to 0. To simplify this analysis, we set ${V_{b1}}$ equal to ${V_\pi }$, enabling us to transform Eq. (3) into Eq. (4):
Here, we denote the constant $\pi /2{V_\pi }$ as $\alpha $ to simplify the description. Equation (4) shows that within the range $\mathrm{0\ < }{\textrm{V}_{\textrm{RF}}}\mathrm{\ < }{\textrm{V}_\pi }$, the intensity of the modulated signal is approximately proportional to the ${\textrm{V}_{\textrm{RF}}}$, which includes data information. Consequently, the data value can be mapped to ${\textrm{V}_{\textrm{RF}}}$ in this range and result in an increase in the intensity of the modulated signal generated at the MZM output as the data loaded onto it increases.
The transfer function of two cascaded MZMs is as follows:
From Eq. (5), it is evident that by designing the radio-frequency signal appropriately, we can ensure that an increase in the product of the data and kernel leads to an increase in the intensity of the corresponding beam after two cascaded MZMs. This enables the two cascaded MZMs to successfully carry out the multiplication operation.
2.4 Modified Yamada model
We modified the well-known YAMADA model to model the spiking DFB-SA laser neuron with external injection in the numerical simulation [27]. The equations are as follows:
The subscripts 1 and 2 correspond to the active region and the saturable absorber region, respectively. $N$ denotes the carrier density, $S$ represents the number of photons in the laser, ${S_{ext}}$ refers to the number of externally injected photons, ${\beta _{sp}}$ is the spontaneous emission coefficient, and $I$ indicates the bias current of the active region. For the sake of simplicity, carrier diffusion between the active and the saturable absorber region is not considered. The significance and values of the remaining parameters used in numerical simulations are presented in Table 1.
Table 1. DFB-SA laser parameters
2.5 Mechanisms of AM to FM transformation in the DFB-SA laser
Based on the modified Yamada model described in section 2.4, the behavior of the DFB-SA laser injected with an external optical signal is simulated with various intensities. Figures 3(a) and 3(b) depict the photon number of externally injected signals and the output photon number of the DFB-SA laser, respectively. Figures 3(c) and 3(d) display carrier density in both the active region and the saturable absorber region. As the photon number of externally injected light increases, the carrier density changes in both the active region and the saturable absorber region accelerate, leading to a corresponding increase in the rate of change of the output photon number. Therefore, as the external input signal becomes stronger, the frequency of the DFB-SA laser's output signal changes more rapidly. This modulation of the input signal amplitude is converted into frequency modulation in the output signal of the DFB-SA laser.
Fig. 3. (a) External photon injection number of the DFB-SA laser. (b) Output photon number of the DFB-SA laser. (c) Carrier density in active region. (d) Carrier density in saturable absorber region.
3. Experimental results
In this section, we first describe the rate coding characteristic of the spiking DFB-SA laser neuron experimentally. Then we present the experimental results of a binary convolution calculation.
3.1 Experimental results on rate coding of the spiking DFB-SA laser neuron
Figure 4(a) displays the response of the spiking DFB-SA laser neuron to varying stimulus intensities (S). In our experiments, we encoded the stimulus as a positive perturbation in the form of a rectangular pulse whose amplitude increases as the stimulus intensity rises. We used four different stimuli of intensities and defined the stimulus intensity as $({{V_S} - {V_{CW}}} )/{V_{CW}}$. The stimulus intensities for S1-S4 were 2.0612, 2.6211, 2.9124, and 3.5011, respectively, with a duration of 26.67 ns. The corresponding number of spikes in the spike train were 15, 17, 18, and 19. This indicates that the firing rate of the neuron increases with the stimulus intensity.
Fig. 4. (a) Input stimuli of varying intensities (S1 to S4) and the corresponding output spike trains of the spiking DFB-SA laser neuron are presented, with the spike counts in each spike train labeled. (b) Firing rate of the spiking DFB-SA laser neuron under stimuli of different intensities during 200 consecutive cycles.
We injected the input signal shown in Fig. 4(a) into the spiking DFB-SA laser neuron for 200 consecutive cycles, and the results are shown in Fig. 4(b). The firing rate is defined as the reciprocal of the mean inter-spike-interval in the spike train under a given stimulus. Over these 200 cycles, the average firing rate for S1-S4 were 0.5522 GHz, 0.6193 GHz, 0.6474 GHz, and 0.6951 GHz, respectively, with standard errors of $\textrm{2}\textrm{.2312} \times \textrm{1}{\textrm{0}^{\textrm{ - 4}}}$, $\textrm{1}\textrm{.6117} \times \textrm{1}{\textrm{0}^{\textrm{ - 4}}}$, $\textrm{1}\textrm{.6067} \times \textrm{1}{\textrm{0}^{\textrm{ - 4}}}$, and $\textrm{1}\textrm{.3478} \times \textrm{1}{\textrm{0}^{\textrm{ - 4}}}$. The results demonstrate that the firing rate of the photonic spiking neuron is consistent and stable when subjected to the same intensity of stimulus. Additionally, the firing rate is distinguishable when subjected to varying stimulus intensities. Therefore, the spiking DFB-SA laser neuron can encode external stimuli in terms of spike frequency. The experimental setup in this part was similar to Fig. 1(a), and we simply removed PC2 and one of the MZMs.
3.2 Experimental results on binary convolution calculation
To conduct convolution operations, we randomly generated two binary matrices of sizes $\mathrm{5\ \times 5}$ and $\mathrm{2\ \times 2}$, respectively (Fig. 5(a)). The $\mathrm{5\ \times 5}$ matrix acted as the input feature map, and the $\mathrm{2\ \times 2}$ matrix was used as the convolutional kernel. The convolution was performed with a stride of 1 and no padding, resulting in a $\mathrm{4\ \times 4}$ output matrix (feature map). We take an example of two successive elements in the feature map (indicated by the red boxes in Fig. 5(a)) to illustrate the calculation steps. Each result was obtained by summing up the element-wise product of the sub-matrix (receptive field) and the kernel. Data of these matrices were flattened by row and input to MZMs (Fig. 5(b)). A value of “1” was encoded as a high-level signal that could modulate the light from the TL to produce power raises, while a value of “0” was encoded as a low-level signal that produced no intensity modulation to the light from the TL. Here, we denote the intensities of radio-frequency signals encoding numbers 0 and 1 as ${\textrm{k}_\textrm{0}}$ and ${\textrm{k}_\textrm{1}}$, respectively, with ${\textrm{k}_\textrm{0}}$ being lower than ${\textrm{k}_\textrm{1}}$. According to the theory presented in Section 2.3, $\textrm{si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{0}}\textrm{)}$ is less than $\textrm{si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{1}}\textrm{)}$. Additionally, the intensities of the modulated optical signals after two cascaded MZMs corresponding to $\mathrm{1\ \times 1}$, $\mathrm{0\ \times 1}$, $\mathrm{1\ \times 0}$, and $\mathrm{0\ \times 0}$ are directly proportional to $\textrm{si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{1}}\textrm{)si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{1}}\textrm{)}$, $\textrm{si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{0}}\textrm{)si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{1}}\textrm{)}$, $\textrm{si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{1}}\textrm{)si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{0}}\textrm{)}$, and $\textrm{si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{0}}\textrm{)si}{\textrm{n}^\textrm{2}}\mathrm{(\alpha }{\textrm{k}_\textrm{0}}\textrm{)}$, respectively. Consequently, these four types of dot products yield three different intensities (S1, S2 and S3), as depicted in Fig. 5(c). Notably, these three intensities are associated exclusively with two product results: 0 and 1. Moreover, the product result 0 corresponds to two lower intensities (S2 and S3), while the product result 1 corresponds to the highest intensity (S1). According to the experimental result in Section 3.1, we know that higher stimulus intensity will lead to higher firing rate of the spiking DFB-SA laser neuron. So, the product result 1 will lead to the highest firing rate of the photonic spiking neuron (Fig. 5(d)). Therefore, counting the number of highest frequency spike trains within the time window for each convolution step provides the value of that feature map element. In this experiment, the encoding duration of each number was set to 5.1 ns. To align the modulated optical signals on MZM1 and MZM2, a time delay was applied to the electronic kernel signal. This delay length was equal to the time required for the optical signal to propagate between MZM1 and MZM2. The alignment process was performed directly by the FPGA.
Fig. 5. An example of the calculation steps of two successive elements in the feature map. (a) The binary convolution operation verified by this experiment. (b) Upper subfigure: optical signal modulated by MZM1 (input feature map). Lower subfigure: optical signal modulated by MZM2 (kernel). (c) Inputs of the spiking DFB-SA laser neuron. S1: high-intensity stimulus; S2: middle-intensity stimulus; S3: low-intensity stimulus. (d) Outputs of the DFB-SA laser.
To obtain the complete feature map, the signals on MZM1 were encoded by the receptive field in its sliding order (see upper subfigure in Fig. 6(a)), while the signals on MZM2 were encoded by repeating the process 16 times for the kernel (see lower subfigure in Fig. 6(a)). Figures 6(b) and 6(c) present the modulated signals after two cascaded MZMs (data after multiplication) and the outputs of the spiking DFB-SA laser neuron. The statistical results of 50 consecutive experiments are presented in Fig. 6(d). The positive error bar signifies the maximum firing rate observed at that time across all the experiments, whereas the negative error bar indicates the minimum firing rate. The red dots indicate a product result of 1, while the dots with other two colors indicate a product result of 0, and they are separated by red and green shadows, respectively. It is evident that the spike frequency associated with a product result of 1 is clearly distinguishable from the other two spike frequencies with a product result of 0.
Fig. 6. (a) Upper subfigure: optical signal modulated by MZM1 (input feature map). Lower subfigure: optical signal modulated by MZM2 (kernel). (b) Inputs of the spiking DFB-SA laser neuron. S1: high-intensity stimulus; S2: middle-intensity stimulus; S3: low-intensity stimulus. (c) Outputs of the DSB-SA laser. (d) Statistical results of the DFB-SA laser output spike train frequency in 50 consecutive cycle experiments. The positive error bar represents the maximum firing rate observed at that time across all the experiments, whereas the negative error bar represents the minimum firing rate. The black dashed line is used to indicate the time window for calculating a single element in the feature map.Fig. 7. (a) Output of the SFD1. (b) Output of the SFD2.
Here, two cascaded SFDs are used to perform summation during the convolution step. SFD1 integrates over the duration of a single-number encoding period, while SFD2 integrates over the duration of a convolution step. The signal after SFD1 is depicted in Fig. 7(a), where higher-frequency spikes result in higher integration values. The signal after SFD2, shown in Fig. 7(b), accurately represents the convolution results. Based on these results, it can be concluded that using rate coding of the spiking DFB-SA laser neuron for convolution calculation produces stable and accurate results.
4. Numerical simulation
In this section, we firstly validated the experimental results of Section 3 using the modified Yamada model. Additionally, we expanded the device depicted in Fig. 1(a) to (a) 4-channel system, incorporating optical phase shifters into each channel, and converted the input data from binary to quaternary to achieve quaternary convolution computations. We also complete the MNIST handwritten digit classification. All of these tasks are based on the rate coding characteristic of the spiking DFB-SA laser neurons.
4.1 Validation of the binary convolution
In order to verify the feasibility of the numerical model, we initially simulated the binary convolution experiment in Section 3 using the aforementioned model. The simulation results are depicted in Fig. 8. Similar to the phenomenon observed in Section 3, stronger stimulus resulted in a higher firing rate of the photonic spiking neuron model. By comparing Fig. 6(c) and Fig. 8(b), it can be found that the trend of the firing rate of the photonic spiking neuron is similar in the experiment and simulation, indicating that the simulation results agree well with the experimental results. This reveals that the numerical model presented in Section 2.4 can accurately describe the phenomena observed in our experiment.
Fig. 8. Numerical simulation results corresponding to the binary convolution experiment in Section 3. (a) Inputs of the DFB-SA laser in the simulation, corresponding to Fig. 6(b). S1: high-intensity stimulus; S2: middle-intensity stimulus; S3: low-intensity stimulus. (b) Outputs of the DFB-SA laser in the simulation, corresponding to Fig. 6(c).
4.2 Numerical results on quaternary convolution calculation
We achieved the quaternary convolution here. The experimental scheme is shown in Fig. 9(a). Optical phase shifters are integrated to assist in summing the intensities of the optical waveforms at the same wavelength from all four channels. The quaternary convolution used in the simulation is shown in Fig. 9(b). Figure 9(c) presents the quaternary convolution calculation steps of an element in the feature map as an example. Unlike the single-channel experiment in Section 3, the four multiplication operations were assigned to four parallel channels and the summation of the element-wise products was performed by adding the intensities of the optical signals from all of the 4 channels. Therefore, only one SFD is needed to obtain the convolution results. The simulation results are presented in Fig. 10, nine different intensities of stimuli were injected into the DFB-SA laser, resulting in nine different firing rates of the spiking DFB-SA laser neuron (represented by the differently colored transparent bands in Fig. 10(d)). The integrated signal after the SFD is depicted in Fig. 10(c). Figures 10 demonstrate that the numerical simulation results of the quaternary convolution are correct, indicating the potential of the spiking DFB-SA laser neuron to perform multi-radix calculations.
Fig. 9. (a) The schematic diagram of the 4-channel convolution system used in the simulation. TL, tunable laser; PC, polarization controllers; FPGA, field-programmable gate array; RF PA, radio frequency power amplifier; MZM, Mach-Zehnder modulator; VOA, variable optical attenuator; OC, optical couplers; PD, photodetectors; SCOPE, oscilloscope; PM, power meter; CIRC, circulator; VS, voltage source; LDC, laser diode controller; OSA, optical spectrum analyzer; SFD, spike frequency detector. (b) The quaternary convolution principle used in the simulation. (c) An example of the calculation steps of an element in the feature map.
Fig. 10. (a) Input of the DFB-SA laser in the quaternary convolution simulation. (b) Output of the DFB-SA laser. (c) Output of the SFD. (d) Statistics of the spike frequency for each spike train. The red numbers above the images represent the convolution results.
4.3 Numerical results on MNIST handwritten digit classification
In this section, we implemented MNIST handwritten digit classification. The architecture of the CNN used in the simulation is depicted in Fig. 11(a). We simulated the first convolution layer using the DFB-SA laser. The experimental setup is similar to that shown in Fig. 9(a). The first convolutional layer is designed to accept single-channel input images and generate four-channel outputs. These input images are binary representations, and their convolution with four distinct kernels (Fig. 11(b)) results in the four-channel outputs (Fig. 11(c)). Figure 11(d) represents the raw output of the DFB-SA laser and the integrated signal after the SFD processing. The SFD transforms discrete spikes into continuous analog signals, which are then injected into the subsequent network structure. The final classification accuracy reaches 95.85% on the test set of the MNIST dataset (Fig. 11(e)), which is consistent with the results obtained on a 64-bit digital computer.
Fig. 11. (a) Schematic of the CNN used in MNIST handwritten digit classification. (b) Four kernels used in the first convolution layer during the simulation. (c) Output of the four channels in the first convolution layer. The pixel values represent the firing rates. (d) The output of the DFB-SA laser and the integrated signal output from the SFD, corresponding to the region marker by red box in (c). (e) Confusion matrix in the simulation.
5. Conclusion
In this work, we first explored the relationship between the firing rate of the photonic spiking neuron based on the DFB-SA laser and the intensity of the stimulus through experiments. It is found that the firing rate of the DFB-SA laser increases with higher received stimulus intensity. Taking advantage of this property, we achieved binary convolution based on rate coding and reproduced the experimental results using the rate equation model. Additionally, we numerically simulated quaternary convolution and complete MNIST handwritten digit classification successfully. The results show that the DFB-SA laser has great potential in optical computing as a photonic neuron combined with the rate coding method. Furthermore, for our DFB-SA laser neuron, the minimum optical power level that excites it is $\textrm{128}{.7\; }\mu W$. When the injected optical power is $\textrm{328}{.5\; }\mu W$, the firing rate of our DFB-SA laser neuron can reach 1.44 GHz. This means that the DFB-SA laser neuron can perform data representation and computation tasks with extremely low power consumption. Therefore, further research on incorporating DFB-SA lasers into spiking neural networks is worth exploring in the future.
Funding
National Key Research and Development Program of China (2021YFB2801900, 2021YFB2801901, 2021YFB2801902, 2021YFB2801903, 2021YFB2801904); National Outstanding Youth Science Fund Project of National Natural Science Foundation of China (62022062); National Natural Science Foundation of China (61974177); Fundamental Research Funds for the Central Universities (QTZX23041).
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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