As artificial intelligence progresses rapidly, hardware neural networks, drawing inspiration from the human brain, are emerging as a potent remedy to the limitations imposed by the traditional von Neumann architecture. These networks have already achieved significant advancements in fields including pattern recognition [1], artificial vision [2], and cross-modal data processing [3]. While feed-forward neural networks merely transmit signals from input to output, this linear flow restricts their ability to handle dynamic spatial and temporal data. Recurrent neural networks (RNNs) have shown excellent performance in spatiotemporal tasks. However, RNNs suffer from slow training speeds and complications including issues such as gradient vanishing or exploding due to the complexity of backpropagation through time. To address these issues, the concept of reservoir computing (RC) has been introduced. RC simplifies network training by modifying only the output layer weights and employing straightforward algorithms such as linear regression for signal recognition. This innovation significantly lowers the cost and complexity of training networks, drawing the attention of researchers globally and marking a substantial step forward in the development of neural network technologies.

In the past two decades, RC has been implemented using different physical hardware platforms, which can be categorized into electronic and photonic architectures [4–6]. In particular, photonic-based technologies, including wavelength multiplexing and photonic integrated circuits, have facilitated high-speed and energy-efficient signal processing within the domains of communication and computation [7]. In general, the photonic RC is divided into two categories, i.e., spatiotemporal RC stemming from the free-space optical and diffractive optical element [8,9] and time-delayed RC consisting of the nonlinear node with the delay feedback loop [10–12]. The spatiotemporal RC exhibits ultra-fast parallel computing capabilities, but its scalability is significantly limited, which may result in unsatisfactory computing precision for complex task processing due to the limited node state. Therefore, a large number of optoelectronic elements are required to provide high dimensionality and nonlinearity, which in turn raises the cost and size of the RC system. For example, employing RC with free-space optics offers a promising method to enhance the node state size, though it comes at the cost of reduced system compactness [9]. Moreover, high dimensionality and nonlinearity can be obtained using a single nonlinear physical node with a time-delay feedback loop [13]. The time division multiplexing technology provides the possibility to achieve a sufficient node state by equally dividing the delay line. In 2012, Larger et al. proposed the first optoelectronic delay RC system based on the Mach–Zehnder modulator (MZM) and successfully performed the T146 digital speech recognition task with an error rate as low as 0.004 and a data processing rate of 47.9 kSa/s [14]. Since then, studies on the time-delay photonic RC have spread over a wide variety of novel optoelectric structures, including optical amplifiers [15], semiconductor mirrors [16], microring resonators with optical feedback [17,18], and semiconductor lasers with optical feedback [19–22] or optoelectronic feedback [23], to name just a few [24]. These architectures successfully improve the performance of signal processing in machine learning tasks, such as static image recognition [25], time-series prediction [26,27], reinforcement learning [28], and speech recognition [29].

The complex dynamics of delay systems have unlocked numerous opportunities for practical applications in machine learning. Recent studies have emphatically shown the significance of high-dimensional and nonlinear dynamics in RC. For example, Hou et al. demonstrated that the RC based on a semiconductor laser with double feedback can improve the prediction performance of the Santa Fe time-series task compared with the single feedback scenario [30]. Guo et al. revealed that memory capacity (MC) can be significantly increased by introducing double feedback in VCSEL-based RC [31]. Tavakoli et al. proved that an optoelectronic device equipped with multiple evenly spaced delays enhances the transformation of scalar inputs into higher-dimensional dynamics, thereby boosting its memory and predictive abilities for time-series inputs produced by low- and high-dimensional dynamical systems [32]. Cai et al. proposed a photonic RC using a semiconductor laser with randomly distributed optical feedback, where the randomly distributed optical feedback offers multiple external cavity modes, enhancing the nonlinearity of the reservoir laser [33]. Another example is the recently introduced deep time-delay RC by folding a deep neural network of arbitrary size into a single neuron with multiple time-delayed feedback loops [34]. The network’s connection weights can be adjusted by adjusting the feedback-modulation terms. Such a deep RC can perform more sophisticated tasks. These results indicate that the time-delayed dynamics is a critical factor that significantly affects the performance of time-delayed RC. However, understanding which mechanisms contribute to learning ability and leveraging the rich dynamic properties of time-delayed RC to effectively solve complex temporal computation tasks remain areas for further exploration.

Complex temporal signals often exhibit variable time scales and high spectral richness, which can be efficiently processed by the brain. For example, the brain can easily recognize sound signals across different time scales, as illustrated in Fig. 1(a). In this paper, we propose and demonstrate a novel photonic RC system based on a semiconductor laser with photonic-filter feedback, which enhances the reservoir’s ability to process input signals across different time scales. The photonic-filter feedback is obtained simply through an optical ring cavity, formed by linking two ports on an optical coupler. Such a ring cavity structure is known as an infinite impulse response single-source microwave photonic filter or fiber ring resonator [35], which can also be used to generate wideband chaotic signals and suppress the time-delay signature of chaos [36–38]. Our proposed photonic-filter feedback scheme offers at least two benefits to the RC. First, it enables the enhancement or suppression of specific frequency components in the feedback loop through spectrum adjustment, resulting in RC with selective memory retention. This filtering enhances time-series processing by allowing greater focus on relevant signal components while mitigating the impact of noise and irrelevant information. Second, fiber ring feedback can be viewed as introducing multiple delay loops, which extend the system’s temporal response range, enabling the reservoir to distinguish input signals across different time scales more effectively. We compare the performance of the RC system with conventional single feedback and our photonic-filter feedback through experimental demonstrations and simulations using Mackey–Glass chaotic time series and nonlinear channel equalization (NCE) tasks. The main contents of the remaining sections of this paper are summarized as follows. Section 2 describes the experiment and simulation schemes for our proposed RC system. The results for chaotic time prediction and signal classification are illustrated in Section 3. The conclusion is presented in Section 4.

Organisms utilize feedback mechanisms to collect and store diverse information in the brain [see Fig. 1(a)]. However, this memory gradually deteriorates over time. Yet, consistent or repetitive feedback serves as a potent strategy to boost memory retention, a process underpinned by neuroplasticity. Similarly, employing feedback mechanisms in artificial neural networks, e.g., RNN, effectively captures the temporal correlation in time series. Building on this principle, the implementation of a multidelay feedback scheme can further enhance the MC of neural networks, enabling them to capture input information across various time scales [39]. For a clearer visualization, a sketch of the time evolution of neural state y(t) is presented in Fig. 1(b). For the single feedback scenario [see Fig. 1(b2)], the neural node state yi(t) can be time-continuously sampled. In contrast, Fig. 1(b3) illustrates the multidelay feedback scenario, where neural node states yi(t) exhibit staggered sampling intervals due to different time delays (Δt). This staggered structure allows the system to capture richer temporal information, effectively increasing the MC of the network by processing input signals over various time scales. We have thus proposed a multidelay RC system designed to optimize these memory retention capabilities, as shown in Fig. 1(c). The RC, inspired by neural processes, consists of three distinct layers: the input, the reservoir, and the output layer. In the input layer, the initial signal x(t) undergoes a sample-hold process with a sampling interval of T, and then it is combined with a random mask sequence to produce the input stream S(t) for the reservoir. Within the reservoir layer, the feedback delay line is partitioned to uniformly distribute the virtual nodes, and the distance between successive virtual nodes is represented by θ. Consequently, the number of virtual nodes M=τ/θ. In scenarios with multiple feedback delays (τ1<τ2<τ3<…<τi), typically only τ1 is utilized for extracting the virtual nodes [31,32,40]. In the output layer, the optimal weight values are determined through the application of the linear least-squares technique.

The experimental setup of our proposed RC system is shown in Fig. 1(d). A tunable laser (TL, Newkey Photonics, Inc., maximum power 16 dBm) is used as a drive laser, and its output is injected into the reservoir laser through polarization controllers (PC1 and PC2), a variable optical attenuator (VOA), an MZM (bandwidth 10 GHz, Vπ=1.8), a 2×1 fiber coupler (FC, 70% port), and an optical circulator (CIR). The masked input information S(t) generated by an arbitrary waveform generator (AWG, 7001A) and amplified by an RF amplifier is modulated into the MZM. A standard binary mask signal featuring two random discrete levels {1, −1} is utilized as the input mask sequence. The distributed feedback (DFB) laser is employed as a reservoir laser, which is driven by a high-stability and low-noise diode controller. The pump current is fixed at 7.77 mA (threshold current Ith=7.83mA), and the temperature is set at 25°C, which causes a lasing wavelength of 1551.173 nm, as shown in Fig. 2(c). The DFB laser output is split into two paths: the first path is reintroduced into the DFB laser through a 2×1 FC1 (30% port), a VOA, a PC3, the photonics filter, and a 1×2 FC2 (50% port); the second path is directed through the 1×2 FC2 (50% port) to a photodetector (PD, HP11982A, bandwidth 15 GHz) for photoelectric conversion and oscilloscope (OSC, LeCroy WaveMaster 820Zi-B, bandwidth 20 GHz, sampling rate 80 GSa/s) for collecting information from the reservoir. The photonic-filter feedback is constructed by connecting ports 2 and 4 of FC3.

The external injection power Pinj is measured before the injection beam enters FC1. Frequency detuning is defined as the difference between the injection optical frequency and the free-running optical frequency of the DFB laser. The feedback power Pf is measured at port 3 of FC3; however, losses at FC1 and CIR are accounted for in the feedback strength calculation. The coupling coefficient for the ring cavity is determined by the proportion of power that is transferred from port 1 to port 3 in FC3. For example, FC3 is a 10:90 fiber coupler, and 90% optical power is coupled to the ring cavity constructed by connecting ports 2 and 4 of FC3; thus the coupling coefficient is r=0.1. Besides, when ports 2 and 4 are disconnected, the coupling coefficient r is defined as 1, i.e., conventional single feedback. Under this condition, the feedback delay τ is determined by measuring the external cavity modes induced by single optical feedback.

To evaluate the performance of our proposed RC system, we adopt two benchmark tasks: Mackey–Glass chaotic time-series prediction and NCE. The objective of this task is to forecast the next step in a chaotic time series, which has been employed in machine learning. For this purpose, we use the initial 4000 data points from the Mackey–Glass chaotic time series in the delay-based RC system, with the first 3000 points allocated for training and the remaining 1000 points set aside for testing. The performance of the RC predictions is assessed using the normalized mean square error (NMSE), which is defined as follows [41]: NMSE=⟨||y(z)−y¯(z)||2⟩⟨||y¯(z)−⟨y¯(z)⟩||2⟩,(1)where ⟨·⟩ and ∥·∥ stand for the average and the norm over time, respectively. y¯(z) and y(z) separately denote the target value and the predicted value, where z is the discrete time index. Usually, when the NMSE≤0.1, the prediction performance of the RC system can be considered to be good enough for the prediction task of chaotic time series. Regarding the NCE task, it holds substantial practical importance in the telecommunications sector. The original signal d(i) is transmitted through a communication channel that introduces nonlinear distortion as follows [42]: $$\begin{gathered} q(i)=0.08d(i+2)−0.12d(i+1)+d(i)+0.18(i−1) \\ −0.1d(i−2)+0.091d(i−3)−0.05d(i−4) \\ +0.04d(i−5)+0.03d(i−6)+0.01d(i−7), \end{gathered}$$(2)u(i)=q(i)+0.026q(i)2−0.11q(i)3+v(i),(3)where v(i) is the Gaussian white noise with a tunable signal-to-noise ratio (SNR) between 10 and 32 dB, q(i) is the linear channel output, and u(i) is the noisy nonlinear channel output. For RC, the term u(i) is used to confirm d(i) in this task. The classification performance of the RC is evaluated by the signal error rate (SER), which is defined as the ratio of the number of false signals to the total number of signals [42].

Next, we analyze the MC of our proposed RC system, which quantifies measures of how many past steps the system can recall. The memory function m(j) stands for the correlation between the current output oj(n) and input y(n−j) of j time steps before, which is defined as [43] m(j)=Cov2[y(n−j),oj(n)]σ2[y(n)]σ2[oj(n)].(4)

Here, a pseudo-random number y(n) between −1 and 1 is used as the input signal. We select the first 3000 points for training and the next 1000 points for verification. Cov represents the covariance, and σ2 is the variance. The MC can be described as the sum of m(j) as follows [26]: MC=∑j=1∞m(j).(5)

In addition, the photonic RC could be simulated using the modified Lang–Kobayashi model to account for the DFB laser with photonic-filter feedback. The rate equations can be written as follows [44,45]: $$\begin{gathered} dEr(t)dt=1+iα2{GN[Nr(t)−N0]1+ε|Er(t)|2−1τp}Er(t)+F(t) \\ +kinjεr(t)+ξr(t), \end{gathered}$$(6)dN(t)dt=J−Nτe−GN[N(t)−N0]1+ε|Er(t)|2|E|2,(7)F(t)=ηrEr(t−τ)e−iωrτ+η∑nnum=1∞rnnum−1(1−r)2⁢Er(t−τ−nnumτ0)e−i[ωr(τ+nnumτ0+φ)],(8)where Er(t) and N(t) represent the complex amplitude and average carrier number density of a slowly changing electric field of the reservoir laser, respectively. α is the linewidth enhancement factor, GN is the gain coefficient, N0 is the carrier density at transparency, ε is the saturation coefficient, τp and τe are the photon and carrier lifetimes, and kinj is the injection strength from the drive to reservoir laser.

The optical feedback term F(t) is described in Eq. (8), where the photonic filter can be regarded as an infinite impulse response [35]. Herein, η is the feedback strength of the reservoir laser, nnum is the number of feedback loops, ωd,r are the optical angular frequency of the drive and reservoir lasers, and Δω is the angular frequency detuning [Δω=2πΔf=2π(fd−fr)], where fd and fr represent the center frequency of the drive and reservoir laser, respectively. φ is the feedback phase in the external cavity, J is the injection current density of the reservoir lasers, τ is the feedback delay time of the reservoir laser, and τ0 is the delay time for one ring cavity recirculation. In the same experimental setting, the coupling coefficient is represented by r. For comparison, a single feedback scheme is investigated as well. A single feedback system is achieved when the coupling coefficient r is set to 1, corresponding to the experimental condition where ports 2 and 4 are disconnected. Therefore, the feedback term can be defined as Fsingle(t)=ηEr(t−τ)e−iωτ for the single feedback scheme.

The amplitude of the electric field of the drive laser is modulated by using MZM. S(t) is described as follows: εr(t)=|ε0|2{1+ei[S(t)+Φ0]}ei2πΔft,(9)where |ε0| represents the injected field amplitude, and Φ0 denotes the bias voltage of MZM. In addition, S(t)=x(t)×mask(t)×γ, where x(t) is the feature value of the input information, mask(t) is the mask matrix, and γ is the scaling factor. The system of equations is resolved numerically using a fourth-order Runge–Kutta algorithm with 2 ps per step. The simulation parameters are shown in Table 1. Besides, spontaneous emission is simulated by adding uncorrelated white Gaussian noise, ξr(t), to the amplitude of the electric field for the reservoir laser. The output SNR of the reservoir laser is maintained at 23 dB. This noise level is achieved by introducing white Gaussian noise during the drive signal injection without any input modulation [44].Table 1.

Parameter Values Used in Simulations [44]

In this work, we mainly want to demonstrate the benefits of photonic-filter feedback for time-delay RC. For comparison, the reservoir laser with single feedback is also shown below. At first, we increased the feedback power Pf causing the DFB laser to enter chaotic dynamics, which allowed us to calculate the feedback delay time τ. In this experiment, the feedback delay time τ is fixed at 100 ns. Moreover, we evaluate the dimension/complexity of the chaos signal generated from the cases of single feedback and photonic-filter feedback. From Figs. 2(a) and 2(b), one can find that the case of photonic-filter feedback demonstrates a higher chaotic dimension. This is advantageous for RC as it requires a high-dimensional space to effectively map the input signal. Next, the external injection light is introduced to stabilize the DFB laser output because the RC operates in the steady-state region. The optical spectrum of the TL and the free-running DFB laser is shown in Fig. 2(c). The frequency detuning Δf between TL and DFB laser is −3GHz. When the injection power Pinj is fixed at 0.8 mW and the feedback power Pf is set at 0.001 mW, the injection locking phenomenon can be observed, and the corresponding optical spectra are plotted in Fig. 2(d).

For an RC system, the property of nonlinearity is a primary factor that facilitates the separation of different input categories and thus ensures the ability to precisely process complex tasks. We first perform the Mackey–Glass chaotic time-series prediction task to evaluate the performance of our proposed RC system. Some key parameters are set as follows: the number of virtual nodes M=500, the feedback power Pf=0.01mW, and the coupling coefficient r=0.1. Figures 3(a1)–3(a3) display the experimental input signal S(t) and response output of the reservoir laser with single feedback and photonic-filter feedback. The significant differences between the input and response output reveal the nonlinear transformation. For quantitative analysis, we calculate the correlation coefficient to investigate the response properties of the single feedback and photonic-filter feedback. Figures 3(b1) and 3(b2) plot the correlation diagram between input and response output generated from single feedback and photonic-filter feedback scenarios, respectively. We can find that the photonic-filter feedback shows a lower correlation coefficient compared with single feedback. In Figs. 3(c1) and 3(c3), we show the temporal waveforms of the origin (red), prediction result (blue), and prediction error. For the single feedback scenario, the NMSE is 0.0389, which decreases to 0.0078 in the case of photonic-filter feedback. Further, the NCE task is also performed to examine our proposed RC system. Likewise, the experimental input signal S(t) and response output of the reservoir laser are depicted in Figs. 3(d1)–3(d3), respectively. The nonlinear mapping leads to an obvious difference between the input and response output. In this task, a similar phenomenon can be observed again, where the correlation coefficient between the input and response output of the reservoir laser with photonic-filter feedback is smaller than that of single feedback. Furthermore, the prediction results and errors for two cases of single feedback and photonic-filter feedback are drawn in Fig. 3(f). The SER values for the case of photonic-filter feedback decrease from 0.015 to 0.007 compared with the case of single feedback. These results unveil that the photonic-filter feedback scheme can significantly enhance the RC prediction performance.

For a more comprehensive analysis, we investigate the effect of feedback power on our proposed RC performance. In Fig. 4(a1), the different coupling coefficients are shown. From this figure, one can see that the value of NMSE increases with the increase of the coupling coefficient. When the coupling coefficient is 0.5, the prediction error on the Mackey–Glass time series task is almost equivalent to the single feedback scenario, indicating that photonic-filter feedback does not improve the RC performance. At the same time, the results of the NCE task are shown in Fig. 4(b1). Similarly, an optimal performance (minimum SER value) can be found under the case of r=0.1. However, when the coupling coefficient increases from 0.2 to 0.5, it is difficult to observe an improvement in RC performance compared to single feedback. Furthermore, we consider the effect of node number on RC performance, as shown in Figs. 4(a2) and 4(b2). It can be seen that when the node number is increased, the RC performance is significantly improved for both the Mackey–Glass time series and the NCE task.

Finally, we evaluate the MC of our proposed RC system under different coupling coefficients. The MC represents the number of previous steps that the system can remember. From Fig. 4(c1), we can see that when the coupling coefficient r=0.1, the MC values are raised compared with the single feedback. This is mainly due to the increased relevance of the third step. This is attributed to the enhancement of correlation function m(j) in the third step, as shown in Fig. 4(c2). When the coupling coefficient increases from 0.2 to 0.5, the MC values are slightly increased. We speculate that there are two main aspects of RC performance improvement. On the one hand, photonic filters can be approximately regarded as forming multiple feedback loops, enhancing the MC of the system and thus leading to improved prediction performance. On the other hand, multifeedback increases the dimension of the system and enriches the system dynamics, thus improving the performance of RC to some extent.

In this section, we numerically investigate the effect of the photonic-filter feedback scheme on the time-delay RC system. In agreement with the experiment, the Mackey–Glass time series and NCE task are employed to evaluate our proposed RC performance. To ensure computing efficiency in numerical simulations, the feedback delay number and feedback time are set as nnum=100 and τ=1ns (which is much larger than the relaxation oscillation period). In the multidelay dynamic system, the feedback delay interval of each delay should be carefully adjusted [31,32], which directly affects the dynamical evolution, e.g., the edge of chaos. Moreover, in some cases, resonance phenomena may occur when the delay interval spacing approaches the clock cycle, potentially leading to a degradation in RC performance. In our simulation, we observe that when the ring cavity delay time, τ0, approaches an integral multiple of τ, there is a significant decrease in RC performance for both Mackey–Glass chaotic time-series prediction and NCE classification tasks. Since the optimal delay interval spacing depends on the specific task, here we fix the ring cavity delay time τ0 at 2.1 ns.

Figures 5(a1) and 5(a2) plot the dynamic evolution of the reservoir laser with both feedback schemes as a function of the feedback strength, where the input signal S(t) is 0. The employment of photonic-filter feedback results in the unstable bifurcation point shifting to a lower feedback strength, indicating a corresponding shift in the optimal working point of the RC. Figure 5(b1) illustrates the prediction error for the Mackey–Glass chaotic time series as a function of the feedback strength for the single feedback and photonic-filter feedback scenarios. Some key parameters are set as follows: M=100, r=0.1, γ=1, and φ=−0.16π. From Fig. 5(b1), it is evident that as the feedback strength increases and approaches the edge of instability, the NMSE value for the photonic-filter feedback scheme remains lower than that for single feedback. A similar trend can also be observed in the NCE task, as shown in Fig. 5(c1), where γ=0.2 and φ=0. To gain more details on the effect of the key parameter on the RC performance, we plot two-dimensional maps of the NMSE and SER values in the (kf,r) plane in Figs. 5(b2) and 5(c2), respectively. From these figures, we can see that good RC performance can be obtained with a small coupling coefficient and moderate feedback strength (the edge of instability).

In Fig. 5(d1), m(j) is shown when step j is changed for the single feedback and photonic-filter feedback, where the m(j) reduces monotonically as step j is increased. A larger m(j) is achieved for the photonic-filter feedback scheme. m(j) is used to calculate MC as shown in Fig. 5(d2). Compared with the case of single feedback, the MC can be improved by nearly three times in the photonic-filter feedback. Moreover, from two-dimensional maps [see Fig. 5(d3)], we can see that a larger MC value can be obtained with a smaller coupling coefficient, which is consistent with our experimental results.

The benefits of multidelay feedback structures have been successfully demonstrated in various fields including optical chaos generation, photonic microwave, and neural networks [32,40,45–49]. Especially in the field of neural networks, the introduction of the additional delay increased the complexity of system dynamics. This feature can significantly enhance RC performance, e.g., higher MC, which benefits multistep prediction or autonomous prediction [26,50]. Nevertheless, physical implementations of multidelay architectures present great challenges in control and implementation. In this work, we propose and experimentally demonstrate a multidelay RC system with excellent performance using a simple 2×2 FC. Owing to the rapid advancements in photonic integration technology, our proposed system is readily integrated on a chip [43]. Moreover, here we also consider a potential application based on our proposed RC system. In 2019, Penkovsky et al. proposed a deep convolutional neural network based on cascaded time-delayed RC, where the radio frequency filter acts as convolution kernels to extract local features [51]. In our system, the photonic filter is regarded as an infinite impulse response filter, potentially serving as an optical convolution kernel in a physical deep convolutional neural network. Further work will explore the deep convolutional neural network based on photonic-filter feedback RC architectures.

In conclusion, we have both numerically and experimentally demonstrated a novel time-delayed RC based on a DFB laser with photonic-filter feedback and injection. On the one hand, the photonic-filter feedback can adjust the spectrum of the signal in the feedback loop, enhancing or suppressing specific frequency components. On the other hand, the photonic-filter feedback is equivalent to optical feedback from multiple external cavities with different lengths, which enhances the dimensionality/complexity of reservoir dynamics. These characteristics enhance the RC system’s ability to respond to input signals across different time scales. The experimental results show that with a small coupling coefficient, our proposed RC system has superior performance in time-series prediction and signal classification compared to the conventional single feedback scheme. Moreover, benefiting from the rich neuron dynamics generated by the photonic-filter feedback scheme, the MC of RC can also be boosted. Our numerical results are in good agreement with the experimental observations, which indicate that the photonic-filter feedback can significantly improve the time-delayed RC performance due to the multiple delay feedback enriching the neural dynamics.