1. Introduction

In today's society, we are in the midst of a data explosion era, with an urgent demand for new computing technologies that are ultra-fast, high-bandwidth, large-capacity, high-density, low-power, and cost-effective [1–3]. Neuromorphic computing, inspired by the information processing methods of the human brain, uses neurons and synapses as basic units to simulate the structure and function of biological neural systems [4,5]. It aims to construct a new computing paradigm, the ‘artificial super brain,’ by emulating various aspects of the biological brain [6,7]. With the implementation of neuromorphic computing projects in various countries around the world, a new technological frontier in the field of neuromorphic computing is rapidly emerging. This development will lead humanity into the ‘Brain Era” [7–9].

Reservoir Computing (RC), as a simple and efficient neuromorphic computing framework resembling the human brain's cortical circuits, has gained significant attention [10–12]. A typical RC system consists of three components: the input layer, the reservoir layer, and the output layer [13,14]. In the input layer, the input signals after a preprocessing procedure are then fed into the reservoir layer via fixed weight connections [11]. The reservoir layer includes a network of randomly interconnected nodes, enabling input signals to be nonlinear projected into high-dimensional state space, achieving linear separability [15,16]. In the output layer, based on the state of nodes in the reservoir network, recognition can be achieved through simple linear weighting and simple linear regression algorithms [11,14,15]. Compared with traditional recurrent neural networks that face challenges such as low computational efficiency, complex training algorithms, and susceptibility to local optima, RC has advantages such as fast learning and low training costs, making it highly promising in various applications such as pattern recognition, time series prediction, wireless communication, and stock price prediction [11,17].

Especially, the hardware implementation of RC holds the potential to develop the next generation of machine learning hardware devices and chips [13,14]. Compared to software-based implementations, hardware-based RC can reduce the power consumption of machine learning hardware and significantly enhance data processing speed [17,18]. Particularly, a simplified hardware RC implementation method has been proposed by Appeltant et al. in 2011, which simplifies large-scale nodes in traditional RC systems to a single nonlinear node with delayed feedback signals (regarded as time delay RC), greatly reducing the difficulty of hardware implementation [19]. Recently, several physical devices and materials have been employed for the hardware implementation of RC systems, such as field-programmable gate arrays [20–22], photonics [23,24], dynamic memristors [25,26], atomic switch devices [27], spintronic oscillators [28], et al. Among them, the photonics time delay RC system has received widespread attention in recent years, benefiting from inherent advantages of photonics such as ultra-high speed, large bandwidth, and multi-dimensional capabilities [19,29,30]. For example, in 2013, Brunner et al first proposed a time delay RC system based on semiconductor laser and verified its feasibility through experiments. Parallel recognition of digital types and digital speakers for speech digital recognition tasks could be achieved in the propose, with an information processing rate of 1Gbps.Besides, it was found that the best performance was achieved near the laser threshold current, with word error rate could be achieved at 0.014% [31]. In 2019, Vatin et al. proposed a numerical model of a time delay RC system based on a vertical cavity surface emitting laser (VCSEL). It was found that compared to the case where there is only one polarization mode in VCSEL, the RC system has higher computational performance, larger memory capacity, and better classification ability when two polarization modes coexist in VCSEL [32]. In the same year, Vatin et al. further experimentally implemented the time delay RC system based on the VCSEL [33]. In 2022, Yang et al numerically verified a RC system based on the spin VCSEL. It indicated that, the proposed RC realized both single task processing and parallel tasks processing because of the feasible tunability and multiplexing of the left and right circularly polarized modes of the spin VCSEL [34]. In 2023, Jiang et al proposed a time delay RC system utilizing a reflective semiconductor optical amplifier as the reservoir layer. It was found that, compared to the RC system based on traditional semiconductor optical amplifier, the proposed RC system with reflective semiconductor optical amplifier achieved a wider consistency interval and superior robustness [35]. In 2024, Qi Qu, et al introduced and demonstrated a system to generate microwave waveforms, in which employed a delay feedback reservoir computing with SOA. Here, the SOA is used to provide rich nonlinearity functionalities [46]. Note that, most works of the photonics RC system have been limited to implementations of the reservoir layer. However, linear matrix computing units (the input layer) and nonlinear computing units (the reservoir layer) cannot be simultaneously implemented in the optical domain, thereby limiting the information processing speed of photonics RC system.

In this paper, we propose a photonic implementation of an RC system based on a VCSEL with two MZMs (VMM-RC). In contrast to previous approaches, the input and reservoir layers have been achieved in optical domain, allowing for higher processing speed and lower power consumption while maintaining comparable performance. In this work, the application of the system in Santa-Fe chaotic time series prediction and handwritten digits recognition have been demonstrated. Here, Two-level mask, Six-level mask and chaos mask signal are considered in both simulation and experiment with the system dealing with the Santa-Fe chaotic time series prediction. Furthermore, for handwritten digits recognition Two-level mask, Six-level mask, chaos mask signal and random mask signal are employed. The minimum NMSE can reach 0.0020 (0.0456) for Santa-Fe chaotic time series prediction in simulation (experiment) while the minimum WER can be 0.0677 for handwritten digits recognition. The main contents of the remaining sections of this paper are summarized as follows. In Section 2, the methods are presented in detail, including the VMM-RC set up and methods for generating input signal u(t) for different tasks. Additionally, section 2.1 is used for experimental scheme while section 2.2 for simulation mode. The results and discussion are provided in Section 3. The conclusion is presented in Section 4.

2. Methods

2.1 Experimental scheme

While the process of information processing in the human brain involves hundreds of millions of nerve cells and hundreds of billions of synaptic connections, we can abstract it as a sensory organ receiving information (INPUT). The brain interprets, categorizes, and integrates information before making decisions (OUTPUT). Thus, the human brain could be considered as a directed connection, Random neural networks with memory decay and complex spatiotemporal dynamics. RC, is a machine learning framework inspired by human brain, consisting of the three components: input layer, reservoir layer, and output layer. As shown in Fig. 1 (a), the reservoir layer has functions like the human brain. In addition, the input layer generates electrical signals containing input and mask information, and then uses two MZMs to modulate the input and mask signals separately on the optical carrier to achieve the input layer in the optical domain. After converting the optical signal into an electrical signal through the photodetector (PD) of the output layer, the results are collected using an oscilloscope (OSC) and calculated by a computer. In Fig. 1(b), the experimental set up of the VMM-RC system is presented which is built using a single VCSEL and two MZMs. The experimental devices could be summarized in the following steps. Firstly, a commercial 1550 nm VCESL (SEOUL VIOSYS) without isolation is regard as a nonlinear node. The VCSEL is driven by a low-noise laser diode controller (ILX-Lightwave, LAC-3724C). In addition, the external optical light generated by the tunable laser (TL) with isolation is directed into MZM₁, then modulated with the input signal u(t) and the output of MZM₁ is sent into MZM₂ which modulated with the mask signal mask(t). This configuration realizes the input weight in the optical domain, where the mask signal is also referred to as the input weight. Two polarization controllers (PCs) are employed in the generation process of the injection optical signal. PC₁(PC₂) are used to adjust to align with the modulation axis of the MZM₁(MZM₂). Thirdly, the input signal u(t) and mask signal mask(t) are generated by a Field Programmable Gate Array (FPGA, ZYNQ ultrascale + rfsoc zcu216) and amplified by RF amplifiers 1 and 2, respectively. Besides, multi-channel DAC internally at the speed of 6.4GSa/s is integrated in the FPGA employed. The injection optical signal modulated with u(t) and mask(t) is divided into two parts using a 20%:80% FC, with 20% of the ports utilized for observing the optical injection signal. These port pass through photodetectors (PD₁, Agilent/HP11982a) for photoelectric conversion and are acquired by an oscilloscope (OSC, Keysight DSOV334A). The output of the remaining port passes through an erbium-doped optical fiber amplifier (EDFA) and a variable optical attenuator (VOA), which could adjust the power of injection optical signal. CIR is employed to inject the signal into VCSEL. Furthermore, 10% of the output of the VCSEL is feedback to the VCSEL by CIR after passing through the delay line (DL) and PC₃. PC₃ is employed to adjust the polarization of the feedback signal. The 50%:50% FC is employed to combines the injection optical signal with feedback signal. Another 90% of the output of VCSEL pass through PD₂ for photoelectric conversion, and finally acquire information by the OSC.

 

Fig. 1. (a) The VMM-RC scheme compares with human brain. (b) The VMM-RC system experimental scheme schematic. c1(c2) shows how to obtain the input signal u(t) in Santa-Fe chaos time series prediction (Handwritten digit recognition) task.

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For Santa-Fe chaos time series prediction, the Santa-Fe chaos time series can be directly employed as u(t) by the CH1 of FPGA as shown in Fig. 1(c1). However, for Handwritten digits recognition, pre-processing steps, as shown in Fig. 1 (c2), are necessary to generate u(t). Firstly, the database, comprising 70,000 images of size 28 × 28, is processed through feature extraction, resulting in 70,000 vectors sized 192 × 1. In addition, the vectors are subjected to 2D Mask Preprocessing (with dimensions 192×N, where N represents the number of virtual nodes in the reservoir). This preprocessing step involves multiplying the vectors by a 2D mask consisting of randomly assigned -1 and 1 values, transforming the input information from two dimensions to one dimension [45]. Importantly, each point of u(t) needs to be repeated ten times, ensuring that the input weight operates effectively at every point of the input information.

It is worth noting that the VCSEL with self-feedback light is utilized as reservoir. The number of virtual nodes is N = 100. The interval between the adjacent virtual nodes is θ≈780ps. The sampled period of input signal (equal to the feedback time delay) is τ=N×θ (the information processing rate is R = 1/τ). The input weight is realized in optical domain and transient responses from VCSEL are extracted for post-processing. In the post-processing phase, the virtual nodes states extracted from VMM-RC system are put into matrix X, which can be multiplied by the output weight to obtain the prediction or recognition results.

2.2 Simulation mode

Therefore, the simulation model for VMM-RC is established by using the well-known spin-flip model to analyze the nonlinear dynamics of VCSEL, which could be described as following rate equation [36]:

In addition, for Santa-Fe chaotic time series, we consider different mask signals, namely the Two-level mask signal, Six-level mask signal, and Chaos mask signal shown in Fig. 2, to analyse their influence on the performance of the VMM-RC system. Two-level mask signal is consisted of random -1 and 1 values, while the Six-level mask signal consist of ±1, ± 0.6 and ±0.3 stochastically. Therefore, the Chaos mask signal was obtained from the Mackey-Glass chaotic system at a sampling rate of 20 GHz [43]. As we progress from Two-level mask signal to Six-level mask signal and then to chaos mask signal, the complexity of mask(t) gradually increases.

 

Fig. 2. The different mask signals consider in simulation and experiment. Two-level masks signal (blue line), six-level masks signal (red line) and chaos masks signal (green line) are displayed in a, b, and c, respectively

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Table 1. Parameter Values for the VMM-RC System in our Numerical Simulation^a

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3. Result and discussion

3.1 Simulation results

Here, we verify the performance of VMM-RC system by the Santa-Fe chaos time series prediction and handwritten digits recognition.

On the one hand, Santa-Fe chaotic time series were recorded from a far-infrared laser operating in a chaotic state in experiments [41]. The Santa-Fe dataset contains 9000 sample points, and 3000 points are employed for training while 1000 points are used to test the system [42]. In this task, the value of the next moment in the sequence is predicted by the known sequence. The Santa-Fe chaotic time series prediction task detected the nonlinearity and memory ability of RC. NMSE can be used to evaluate the performance of the system [43]:

For Santa-Fe chaotic time series prediction task, the influence of feedback power K_d and virtual number N are examined in our work. Firstly, the NMSE values of the VMM-RC system as functions of K_d or N are revealed in Fig. 3(a) or Fig. 3(b). It is evident that for Two-level mask, Six-level mask or Chaos mask signals used in the system, the NMSE values remain stable at 0.004, 0.0023 or 0.0020, respectively, when N = 100, 2ns^-1< K_d< 25ns^-1 and K_inj= 10ns^-1. Subsequently, for Two-level mask (Six-level mask or chaos mask) signal, it can be observed that the NMSE values of the VMM-RC system based on single VCSEL is decreasing when N < 100 (N < 120 or N < 100). Then, the NMSE values fluctuate at 0.0058 (0.0035 or 0.00202) when 100 < N < 200 (120 < N < 200 or 100 < N < 200). It is indicated that larger N and more complex mask(t) are desired for better performance achieved in the VMM-RC system for Santa-Fe chaotic time series prediction.

 

Fig. 3. The NMSE values of the VMM-RC system as functions of K_d (N) are revealed in a (b). The yellow line (blue line or red line) for Two-level mask (Six-level mask or Chaos mask) signal.

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On the other hand, for handwritten digits recognition, as shown in Fig. 1(c2), we utilized the mixed National Institute of Standards and Technology (MNIST) database to evaluate the performance of the VMM-RC system. The MNIST database contains 70,000 handwritten digital images, with 60,000 images in the training subset and another 10,000 images in the test subset [44]. All images in the MNIST database are standardized to a size of 28 × 28 pixels, and the pixel values have been normalized to [0,1]. Besides, a convolution neural network is employed to extract features from each image in the database, resulting in a 192 × 1 feature vector. Moreover, the 2D-Mask processing also is applied to generate u(t), effectively reducing the amount of input information. To evaluate the image recognition performance, the WER is defined as in Eq. (7), where T_error is the number of incorrectly recognized samples and T_total is the total number of images [45].

In Fig. 4, we considered Four different mask signals: Two-level mask signal (yellow line), Six-level mask signal (blue line), Chaos mask signal (red line), and Random mask signal (green line) for handwritten digits recognition with the VMM-RC system. The random mask signal is composed of values between -1 and 1. On the one hand, the WER values of the system as a function of the optical injection power K_inj are shown in Fig. 4(a). For Two-level mask signal (Six-level mask signal, Chaos mask signal or Random mask signal) with K_d= 10ns^-1 and N = 100, the WER decrease when K_inj< 10ns^-1 (K_inj< 10ns^-1, K_inj< 7ns^-1 or K_inj< 5ns^-1). Subsequently, the WER values fluctuate around 0.1864 (0.1652, 0.0863 or 0.0677) when 10ns^-1< K_inj< 38ns^-1 (10ns^-1< K_inj< 38ns^-1, 10ns^-1< K_inj< 38ns^-1, 7ns^-1< K_inj< 38ns^-1 or 5ns-1< Kinj < 38ns-1). On the other hand, the WER values of the system as a function of the K_d are revealed in Fig. 4(b) when K_inj= 30ns^-1 and N = 100. Within the monitored range of K_d, the WER values of system remain stable at 0.1796 (0.1644, 0.07835 or 0.0683) when Two-level mask signal (Six-level mask signal, Chaos mask signal or Random mask signal) is employed. The average and standard deviation are computed based on simulating the system under each condition 5 times. As observed in Fig. 4, when Two-level mask signal or Six-level mask signal is employed in the VMM-RC system, the similar performance could be achieved, with slightly smaller deviations in the case of Six-level mask signal. Therefore, opting for more complex mask signal, such as Chaos mask signal and Random mask signal, significantly improves the system’s performance. Furthermore, the system exhibits greater stability with the utilization of more complex mask(t) for the handwritten digits recognition task, leading to lower standard deviation.

 

Fig. 4. Two-level mask signal (yellow line), Six-level mask signal (blue line), Chaos mask signal (red line) and Random mask signal (green line) employ as mast(t) are considered for handwritten digits recognition with the VMM-RC system. (a) The WER of the system as a function of K_inj. (b) The WER of the system as a function of K_d.

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3.2 Experimental results

In the experiment, Santa-Fe time series prediction is implemented using Two-level mask signal, Six-level mask signal, and Chaos mask signal in the VMM-RC system. Figure 5 illustrates the signal evolution of the VMM-RC system during the implementation of the input weight matrix in the optical domain. Different input weights correspond to different mask signals, such as Two-level mask signal (a₁-a₃), Six-level mask signal (b₁-b₃), and Chaos mask signal (c₁-c₃). When only MZM₁ operates, the modulation result of u(t) is obtained, as indicated by the red line in Fig. 5. The blue line represents the modulation of mask(t) into the injection optical signal when only MZM₂ operates. Combining MZM₁ and MZM₂ enables the multiplication of mask(t) and u(t) in the optical domain, depicted by the green line in Fig. 5.

 

Fig. 5. The signal evolution of the VMM-RC system during the implementation of input layer in the optical domain. Two-level mask signal (a₁-a₃), Six-level mask signal (b₁-b₃) and Chaos mask signal (c₁-c₃). The blue (red or green) line for mask signal mask(t) modulation results (input signal u(t) or the result of u(t) multiplied by mask(t))

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The optical spectrum of the free running VCSEL is delayed in Fig. 6(a). Here, the temperature is fixed at 25°C, and the bias current is 1.6 mA. As can be seen, there are two peaks observed, which are defined as X-PC mode on the left (the wavelength is 1554.41 nm) and the Y-PC mode on the right (the wavelength is 1554.65 nm). Correspondingly, the X-PC mode is the dominant mode and the Y-PC mode of the VCSEL is the suppression mode. In addition, the power-current (PI) curve for VCSEL is shown in Fig. 6(b) and the threshold of VCSEL used is approximately 1.5 mA. When the bias current adds up to 5 mA, the optical power of VCSEL output could reach 195.5$\mathrm{\mu}\textrm{W}$. Moreover, the optical spectra of VCSEL with optical injection are displayed in Fig. 6(c). Just as revealed, the optical injection signal is injected into Y-PC mode, so that the Y-PC mode becomes the domain mode and X-PC mode is suppressed after carrying the Santa-Fe chaotic time series. Finally, the NMSE values as a function of frequency detuning for Two-level mask employed as mask(t) are further presented in Fig. 6(d). It could be found that the prediction performance of the VMM-RC system based on the single VCSEL and two MZMs is better when the frequency detuning $\mathrm{\Delta f} \in ({ - 12\textrm{GHz},20\textrm{GHz}} )$, and the performance deteriorates when the frequency detuning is less than -12 GHz or greater than 20 GHz.

 

Fig. 6. (a) the optical spectrum of the free running VCSEL. (b) the PI curve of the VCSEL when the temperature is stabilized at 25°C. (c) the optical spectrum of the VCSEL with the external optical injection. (d) the MMSE values obtained from the VMM-RC system with Two-level mask used as mask(t) at the power of optical injection =176µW, the feedback power K_d =10 µW and N = 100.

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Figure 7 illustrates the NMSE values as a function of the power of the optical injection signal for different mask signals, including Two-level mask (yellow line), Six-level mask (blue line), and Chaos mask (red line). Regardless of the mask signals employed in the VMM-RC system, the NMSE values decrease as the power of optical injection increase. This trend indicates that the VMM-RC system achieves better performance with higher optical injection power. More specifically, the NMSE values could fluctuate at 0.0502 (0.0463 or 0.0456) with Two-level mask (Six-level mask or Chaos mask) signal. Within more complex mask(t) used, the system would achieve better and more performance under lower optical injection power which is agree with the simulation results.

 

Fig. 7. The NMSE values of the VMM-RC system as a function of the optical injection power. The yellow (blue or red) line stands for the system with Two-level mask signal (Six-level mask signal or chaos mask signal) employed.

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4. Conclusion

In this paper, a photonic VMM-RC system based on single VCSEL and two MZMs in series is proposed. The performance in Santa-Fe chaotic time series prediction and handwritten digits recognition is investigated. The VMM-RC Scheme achieves both the input and the reservoir layers in optical domain, demonstrating higher processing speed and lower power consumption compared to previous approaches. Additionally, the influence of different mask signals mask(t), such as Two-level mask, Six-level mask, and chaos mask signal, employed in system are thoroughly researched. It is observed that employing more complex mask signals leads to better system performance. The minimum NMSE can reach 0.0020 (0.0456) for Santa-Fe chaotic time series prediction in simulation (experiment) while the minimum WER can 0.0677 for handwritten digits recognition numerically. The VMM-RC system proposed is helpful for further enhancing the development of photonic RC, which break through the long- standing limitation of photonic RC system on reservoir implementation. It enables the simultaneous implementation of linear matrix computing units (the input layer) and nonlinear computing units (the reservoir layer) in the optical domain, significantly enhancing the information processing speed of photonic RC systems.