1. Introduction

Recurrent neural networks (RNNs) are specialized neural networks designed to work with sequential data by maintaining memory of past inputs [1,2]. In contrast to traditional feedforward neural networks, RNNs achieved superior performance in uncovering temporal correlations within the sequential data [3]. Nonetheless, it is well known that training RNNs can be exceptionally challenging, primarily owing to the problems of vanishing and exploding gradients [4]. Furthermore, optimizing the objective function of an RNN model with numerous hyperparameters can prove to be extremely time-consuming [5,6]. Reservoir computing (RC) has been employed as a neural network architecture for sequential data processing because it effectively strikes a balance between training complexity and performance [7,8]. In RC system, input signals are mapped into higher-dimensional computational spaces using a nonlinear dynamical system called a reservoir. The reservoir comprises a network of thousands of nodes sparsely connected by fixed random weights. The key advantage of RC is that the training process is primarily focused on the linear readout layer, allowing RC to maintain good performance with low complexity [9,10].

The fundamental concept of RC is the necessity for relaxed constraints on the topology of the reservoir, a condition that can be observed in various physical systems [11,12]. Among these options, photonic RC holds the potential to serve as the perfect platform for hardware acceleration. This is attributed to its benefits, including an ultra-high operational bandwidth, minimal power consumption, and the ability to perform parallel computing through optical signal multiplexing [13–16]. There are outstanding demonstrations that can be classified into two separate categories: spatial reservoirs and time-delayed reservoirs [16]. The first approach closely resembles RNNs, where nodes are spatially distributed and can establish physical connections. Typical examples of spatial reservoirs include the network constructed with semiconductor optical amplifiers [17] and the network where a pixel of the spatial light modulator serves as a node [18]. Alternatively, time-delayed reservoirs can be efficiently implemented using a single non-linear dynamical system (node) subject to delayed feedback, and the spatial multiplexing of input in spatial RC with N nodes is substituted with time-multiplexing [19–21]. The reservoir comprises N sampled outputs of the non-linear node distributed along the optical delay loop, referred to as virtual nodes. Connections between these N virtual nodes are formed through delayed feedback when there is a misalignment between the delay and data injection times [13]. Nonlinear elements such as semiconductor lasers [22], VCSELs [23], electro-optic modulators [24], and photodetectors [25] have been integrated into an optoelectronic delayed feedback loop to create time-delayed reservoirs. These systems are constructed using readily available optical and electronic components, operating at gigahertz (GHz) speeds, and featuring thousands of virtual nodes [26]. The designed time-delayed reservoirs have achieved remarkable success in a range of benchmarking tests, including tasks like chaotic time series prediction, image and speech recognition, nonlinear channel equalization, and chromatic dispersion compensation in intensity modulation/direct detection (IM/DD) transmission systems [27–31]. However, to address the growing demands of increasingly complex computational tasks, it is anticipated that in the foreseeable future, the number of virtual nodes and the complexity of their interconnections will need to continually increase. This stems from the fundamental theoretical content of the delayed RC. Virtual nodes encapsulate a variety of input information features that are crucial for computational performance. As the number of virtual nodes grows, the encoded input information’s breadth and depth expand significantly, consequently boosting computational performance. Considering these factors, conventional splitting optoelectronic RC solutions seem impractical for extensive scaling, failing to meet the demands posed by architectures characterized by increasing spatial and temporal complexities [31–33]. This is the reason why various integrated optics-based RC schemes have been suggested, with some having been experimentally validated [34–36].

Silicon photonics provides an ideal platform for achieving extensive large-scale integration, making it possible to fabricate hundreds of individually addressable and reconfigurable optical elements within a small area of just a few square centimeters [37]. Most of the proposed RC schemes emphasize time-delayed reservoirs designed using passive integrated optical components, including optical dielectric waveguides [33,38,39], micro-ring resonators (MRRs) [40–43], and photonic crystal microcavities [44,45]. Within these RC systems, nonlinearity is commonly introduced through the use of square-law photodetectors or by leveraging the nonlinear response of a MRR [32,46]. The memory in these systems is determined by either the delay lines connecting the nodes [33] or by the photon lifetime within the microcavities [46]. In Ref. [33], to achieve sufficient memory, two spiral waveguides make up the 5.4 cm delay line. The device with centimeter-scale waveguides has an adverse effect on its integration and scalability. Using silicon-on-insulator (SOI) MRRs as nonlinear nodes, a 4 × 4 swirl reservoir topology has been theoretically established to perform a traditional nonlinear Boolean problem in Ref. [41]. An all-optical RC system, relying on a silicon MRR and time multiplexing, has been experimentally validated to operate at the GHz scale with thousands of nodes [32]. To ensure that the RC system possesses the necessary memory capacity (MC) for specific computational tasks, a recent proposal involves using a single silicon-based MRR in conjunction with an external optical feedback waveguide to construct the RC system [46]. The addition of an optical feedback waveguide allows for a large increase in the system’s linear MC. However, the length of the feedback waveguide in the optical feedback system is optimized at about 20 cm, which is far longer than the microring’s diameter. The long waveguide poses significant challenges in applications such as device fabrication, transmission loss, and temperature control. This is highly detrimental to the integration and scalability of photonic RC system.

In this paper, to achieve a substantial MC while keeping the device size compact, we present a silicon-based main cavity coupled in series with a linear-cavity array to construct a time-delayed RC system. In this designed reservoir, the single main cavity is used as a physical nonlinear node, and these series-coupled linear cavities further contribute to the formation of the delayed feedback loop. The system’s MC is greatly improved by means of an array of series-coupled linear-cavities. We numerically analyze and assess the proposed RC system’s performance on three classical tasks that require different compromises between nonlinearity and memory. When the task requires more memory than a single MRR can provide, the proposed system dramatically improves computation performance. The article is organized as follows: Section 2 describes the model of the main cavity coupled in series with a linear-cavity array. In Section 3, we describe its implementation in a time-delayed RC scheme. In Section 4, we conduct a quantitative analysis and evaluation of the performance of the proposed RC structure on tasks involving time-series prediction, including Narma 10, Mackey-Glass, and Santa Fe chaotic datasets. Finally, in Section 5, we perform an analysis of fabrication tolerances for the proposed RC structure.

2. Series-coupled microring resonators

Figure 1 illustrates the configuration of series-coupled microring resonators (SCMRRs) for building time-delayed RC system. As an all-pass filter structure, the configuration consists of a waveguide and series-coupled MRRs side coupled to the waveguide. Among these series-coupled MRRs, only the MRR directly connected to the waveguide displays nonlinear behavior, while the rest exhibit only linear behavior. The one with nonlinear behavior is termed the main cavity, while the others are referred to as the linear-cavity array in this paper. In this section, we provide a comprehensive model for the SCMRRs that serve as the foundational components in our reservoir architecture. The theoretical framework we employ is founded on the well-established coupled-mode theory (CMT) [47,48]. The model outlined in our research has already been introduced and effectively characterized a diverse set of experimentally observed dynamic behaviors in silicon-based MRRs [49,50].

 

Fig. 1. Schematic diagram of SCMRRs designed for constructing time-delayed reservoir computing system, which consists of a silicon-based main cavity coupled in series with a linear cavity array.

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Within the framework of CMT, the state variables for the main cavity consist of the following: ${U_1}$ represents the complex amplitude of the optical energy, $\Delta T$ is the mode-averaged temperature difference between the waveguide of the microring and its external surroundings, and $\Delta N$ is the free carrier density. Equations (1) to (3) provide the nonlinear dynamical equations governing the temporal evolution of the three state variables [46,49,51,52].

Equation (1) describes the evolution of the optical energy amplitude inside the main cavity over time, accounting for its coupling with the incident electric field amplitude ${E_{in}}(t)$ of the waveguide and the optical energy amplitude ${U_2}(t)$ of the linear cavity adjacent to the main cavity. Here, ${\omega _p}$ is the incident light angular frequency, ${\omega _1}(t)$ is the resonance angular frequency of the main cavity, $\gamma (t)$ is the total loss rate, and $\mu $, ${\mu _1}$ are the mutual energy coupling coefficients between the main cavity and the straight waveguide or the nearest linear cavity neighboring the main cavity, respectively. Equation (2) describes the incremental evolution of free carrier density ($\Delta N$) within the main cavity, originating from two-photon absorption (TPA) at a rate of ${G_{TPA}}$, and simultaneously undergoing recombination with a decay constant ${\tau _{FC}}$. Indeed, the three variables account for the physical phenomena occurring within the silicon-based nonlinear main cavity. TPA generates free carriers, leading to subsequent free carrier absorption (FCA) and free carrier dispersion (FCD). Both the generation and recombination of free carriers are affected by phonon emission within the silicon-based waveguide, resulting in a consequent alteration ($\Delta T$) in its mode-averaged temperature. Equation (3) is derived from Newton's law [53], with ${\tau _{TH}}$ representing the thermal decay time due to heat dissipation in the surrounding medium, ${P_{abs}}$ denoting the material’s absorption power responsible for heating, ${M_{ring}}$ as the mass of the main cavity, and ${c_{Si}}$ representing the heat capacity of the silicon-based waveguide.

The terms $\mu $ and ${\mu _1}$ correspond to the couplings of the main cavity are separately expressed as [48],

The refractive index of the main cavity is modified by changes in temperature and free carrier concentration. This, in turn, leads to alterations in the parameters ${\omega _1}(t)$ and $\gamma (t)$ within Eq. (1). The first term ${\omega _1}(t)$ is formulated as ${\omega _1}(t) = {\omega _1} + \delta {\omega _{nl}}(t)$, where ${\omega _1}$ represents the resonance frequency in the absence of nonlinearity and $\delta {\omega _{nl}}(t)$ accounts for the resonance frequency shift caused by nonlinear effects.

Within the configuration, the nonlinear impact denoted as $\delta {\omega _{nl}}(t)$ can be expressed as,

In Eq. (7), ${\gamma _l}$ is equal to ${\gamma _{i1}} + {\gamma _e}$, ${\gamma _{i1}}$ and ${\gamma _e}$ respectively represent the intrinsic loss rate of the main cavity and the extrinsic loss rates attributed to its coupling with the waveguide, while ${\eta _{FCA}}$ and ${\eta _{TPA}}$ denote the efficiencies of FCA and TPA, respectively.

When the main cavity operates in a linear state, we can establish the conditions $\delta {\omega _{nl}}(t) = 0$ and $\gamma (t) = {\gamma _l}$. Consequently, the characteristic time scale of the main cavity in the absence of nonlinearity is determined by its photon lifetime ${\tau _{ph}} = \gamma _l^{ - 1}$. Since the nonlinear state is induced by TPA, two crucial timescale parameters, ${\tau _{FC}}$ and ${\tau _{TH}}$, are involved in the system's dynamic evolution. ${\tau _{FC}}$ is approximately two orders of magnitude smaller than ${\tau _{TH}}$. When the timescale of the input signal aligns with a specific timescale parameter, only the corresponding nonlinear effects have an impact on the dynamics. In the paper, the input signal is modulated to align with the timescale of ${\tau _{FC}}$, and we place particular emphasis on the nonlinear effect induced by the presence of free carriers in the main cavity.

Moreover, within the proposed SCMRRs framework, the temporal variations in the optical energy amplitude within the linear cavity array are derived from the following set of coupled differential equations:

The output electrical field signal ${E_{th}}$ through the waveguide can be expressed as,

The initial wavelength detuning between the laser wavelength and the main cavity resonance is defined as $\mathbf{\Delta }{\lambda _s} = {\lambda _p} - {\lambda _1}$. The resonance shift caused by nonlinear effects is denoted by $\mathbf{\Delta }{\lambda _1}(t) = {\lambda _1}(t) - {\lambda _1}$, where ${\lambda _1} = {{2\pi c} / {{\omega _1}}}$ and ${\lambda _1}(t) = {{2\pi c} / {{\omega _1}(t)}}$. The coupled differential Eqs. (1)-(9) are numerically solved using the Runge-Kutta method with an integration time step of 2ps, which is significantly smaller than the lowest timescale effects $\textrm{(}{\tau _{ph}} \approx 97\textrm{ps)}$ [46]. Before problem-solving, these equations are transformed into dimensionless form for convenience (see Supplement 1 for details). In this model, only one-way propagation is considered, and the values of the parameters are also provided in a detailed Table 1 in Supplement 1. In the remainder of this paper, to simplify the nomenclature, the configuration with varying numbers of series-coupled linear cavities is denoted using an abbreviated version. For instance, if the SCMRRs system has two linear cavities, it is referred to as SCMRRs-2.

3. Constructing time-delayed RC with SCMRRs

The configuration of SCMRRs is employed to build a time-delayed RC system. Figure 2 depicts the schematic of the proposed time-delayed RC system, which comprises an input layer, a reservoir, and an output layer [46]. In the input layer, the time-continuous input signals are first encoded as a sequence of bits, where ${x_i}$ represents the amplitude of the i-th bit and $\tau $ represents the bit period. Subsequently, a bit mask is applied by multiplying the bit stream with a set of random values $M(t)$. The mask set $M(t)$ has a size of N, and its elements follow a uniform distribution. $M(t)\textrm{ = }M(t + \tau )$ is periodic time series data with a period of $\tau $. The resulting signal is then modulated onto the intensity of the optical carrier, with the maximum laser input power denoted as ${P_M}$. Subsequently, the modulated optical signal enters the configuration of SCMRRs through the input port, propagating within all the MRRs via their coupling. Connected to the through port, a photodetector (PD) is used to convert the received optical signal into an electrical signal. During a single time period $\tau $, the electrical signal is synchronously sampled at the designated masking sampling interval $\theta $. By configuring the sampling interval to meet $\theta = {\tau / N}$, it ensures that N equidistant sampling points are evenly distributed in time by $\theta $. This results in the number of sampling points precisely matching the size of the mask set. The N equidistant points serve as N virtual nodes, playing roles similar to those of nodes in a conventional reservoir [19]. In the reservoir layer, the nonlinear features present in both the main cavity and the PD contribute to the nonlinear transformation of the input signal, while the series-coupled linear cavity array is designed to enhance its MC. As a result, the initial input signal bit undergoes a nonlinear transformation, transitioning from the physical system to a higher-dimensional space characterized by N virtual nodes. Ultimately, in the output layer, a predicted value ${o_i}$ corresponding to the input ${x_i}$ is obtained by a linear combination of the responses of the related virtual nodes, as follows,

The main cavity exhibits a quality factor of 5.91 × 10⁴, and self-pulsations may occur based on variations in free carrier concentration within its microring waveguide. The masking sampling interval is set to $\theta = 40\textrm{ps}$, and it is significantly smaller than three different timescales: the photon lifetime (here ${\tau _{ph}} \approx 97\textrm{ps}$), the thermal lifetime (here ${\tau _{TH}} \approx 83.3\textrm{ns}$) and the free carrier lifetime (here ${\tau _{FC}} \approx 3\textrm{ns}$) in the main cavity [46]. The main cavity nonlinearity is connected to the last two timescales. The masking interval is less than photon lifetime, representing the quickest characteristic time response of the main cavity associated with the photon lifetime. This is further utilized to maintain the system’s operation in a transient state. Hence, the current internal field in the main cavity does not fully dissipate when the next mask signal arrives, and this preservation of information from previous states facilitates the coupling of neighboring virtual nodes through inertia. The introduction of a series-coupled linear-cavity array into the system enables a greater amount of optical energy to be coupled into the linear cavity through its coupling with the main cavity. Consequently, this results in an extended retention time within the linear cavity, allowing for the preservation of information from earlier states. The fact that it enhances MC awaits further validation in Section 4. Due to the smaller free carrier lifetime, the bit period is configured at $\tau = 1\textrm{ns}$ to leverage the nonlinearity of free carriers, thereby achieving computation speeds in the GHz range due to its faster time response. Consequently, we obtain N = 25 virtual nodes based on the given expression $N = {\tau / \theta }$. For the subsequent simulations in the remainder of this paper, the number of virtual nodes is maintained at this value. The masked optical signal propagates through the main cavity and induces the generation of free carriers, causing a resonance shift by $\mathbf{\Delta }{\lambda _{FC}}$. Simultaneously, the resonance is also shifted by $\mathbf{\Delta }{\lambda _{TH}}$ due to thermal effects. However, because the signal speed ($\tau = 1\textrm{ns}$) is much faster than these variations (${\tau _{FC}}$, ${\tau _{TH}}$), they do not induce a nonlinear transformation of the signal but merely cause a resonance wavelength shift. When high optical energy enters the high-Q main cavity, it often exhibits self-pulsing dynamics, with thermal effects gaining significant. When the bit period $\tau $ is smaller than $1\textrm{ns}$, the number of virtual nodes is restricted, resulting in a decline in the computation performance of the reservoir. The larger number of virtual nodes can be obtained at $\tau > 1\textrm{ns}$, leading to a significant improvement in computation performance but at the expense of reduced computation speed. Thus, the bit period is selected as $\tau = 1\textrm{ns}$ to strike a balance between computation speed and performance.

 

Fig. 2. Schematic of the time-delayed RC with the SCMRRs configuration. The input information $x(t)$ is first masked by a random sequence $M(t)$. The masked electrical signal is then modulated onto the intensity of the optical carrier excited by the laser. At the through port, the optical signal is transformed into the corresponding electrical signal by the PD. The virtual nodes of the reservoir are created through time-multiplexing, and a predicted value ${o_i}$ is obtained by a linear weighted sum of the responses of the related virtual nodes. During the training process, the optimal output weights are obtained by minimizing the prediction error between the predicted value $o_i^{\prime}$ and its expected values $y_i^{\prime}$.

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4. Results

Using the numerical methods described earlier, three classical computation tasks named NARMA 10, Mackey-Glass, and Santa Fe are employed to evaluate the computation performance of the proposed RC system in Section 3. These tasks have diverse signal processing requirements. In the NARMA 10 task, the prediction of the next output value relies on analyzing a series of at least 10 consecutive outputs, including the current value and the preceding nine values [55], highlighting a substantial demand for memory. In contrast, the Mackey-Glass and Santa Fe tasks are benchmark one-step-ahead chaotic time-series prediction tasks, wherein the system must predict a future value (${x_{i + 1}}$) of the input series while processing the current value (${x_i}$) [56,57].

In photonic RC, the importance of nonlinear dynamics is undeniable, yet there is a potential for it to undermine MC [58]. To enhance computation performance, the RC system must find an equilibrium point between the nonlinear transformation of input information and the MC provided by the system. However, gauging the appropriate level of nonlinear transformation and the necessary MC for a specific task proves to be challenging. The following addresses the estimation of these two parameters. The assessment of the former can be indirectly conducted through the standard deviation of the resonance wavelength shift $\sigma \mathbf{(\Delta }{\lambda _0}(t))$ in the main cavity. A higher standard deviation signifies increased nonlinearity in the main cavity, and vice versa. The assessment of the latter is determined through the examination of linear MC, as presented in this context. The linear MC serves as a foundational benchmark in RC, designed to assess the echo state property. This involves training the reservoir to accurately reconstruct an input stream of values ranging from 0 to 0.5, drawn from an independent and uniform distribution, at a specified k timesteps later. It was originally provided in Ref. [59] and is written as,

Subsequently, we investigate the influence of the critical operational parameters within the SCMRRs-based system on the performance of the three selected tasks, including the maximum input laser power ${P_M}$, the initial wavelength detuning $\mathbf{\Delta }{\lambda _s}$, the ratio (Q₂/Q₁) of the linear cavity’s quality factor to the main cavity’s quality factor, and the total number M of the MRR in the SCMRRs system. The nonlinear dynamics within the main cavity are notably influenced by the first two parameters, with the system’s MC being substantially affected by both the quantity and quality factors of the linear cavities that are coupled in series. In the paper, the main cavity possesses a radius of 6.75$\mathrm{\mu}m$, while all the linear cavities exhibit a radius of 56$\mathrm{\mu}m$ and a higher quality factor. The maximum input laser power ${P_M}$ varies from 0.1 mW to 7 mW. The initial wavelength detuning $\mathbf{\Delta }{\lambda _s}$ is adjusted from -30pm to 30pm with a step size of 5pm to cover the wavelength range of the main cavity resonance, where its full width at half maximum satisfies that FWHM=$\lambda$₁/Q₁= 26pm. The ratio Q₂/Q₁ is adjusted from 5 to 60 by varying the linear cavity’s quality factor Q₂ while keeping the main cavity’s quality factor Q₁ fixed. The total number M of these MRRs is changed from 1 to 11 in the SCMRRs-based system. At M = 1, only the main cavity is present in the system. At M = 11, in addition to the main cavity, there are 10 series-coupled linear cavities in the system, and the configuration’s name is abbreviated as SCMRRs-10. During each assessment test, the previous 1000 data bits are first input into the system to remove any fluctuations induced by the inclusion of inputs. Then, 2000 input data bits are used for the training, and the next 1000 data bits are used for the test data. The same data bits were not shared between the training set and the test set. All the simulations employ the same random mask set, whose elements follow a uniform distribution between 0 and 1. The output layer of RC employs a linear classifier that utilizes ridge regression, with the ridge regression coefficient set to 10⁻⁴.

4.1 NARMA 10 benchmark test

The NARMA 10 task is a discrete-time 10th nonlinear autoregressive moving average system [55]. The output of the NARMA 10 system is described as follows:

Figure 3 illustrates the performance of the NARMA 10 benchmark task for the proposed SCMRRs-based system. Figure 3 (a) and (b) show the variations of NMSE and MC versus the ratio (Q₂/Q₁) of their quality factors and the total number M of the MRRs, respectively. Overall, when the MC is larger, the corresponding NMSE tends to be smaller, and vice versa. However, as the number of MRRs and the ratio of quality factors change, their MC or NMSE does not exhibit continuous and gradual changes but rather shows certain localized fluctuations. Compared with the model in Ref. [46], the proposed model is more complex. For the RC system based on the single MRR with optical feedback, the changes in the feedback waveguide length or phase variation were continuous, and their influence on computation performance was continuously gradual. In the proposed model, there is insufficient continuity observed in the variations of the number of MRRs and the ratio of quality factors. Therefore, when both parameters change, the system may exhibit non-linear instability, leading to localized oscillatory effects on the computation performance. Nevertheless, the overall variation pattern still adheres clearly to the conclusion that NMSE is smaller when MC is larger. As the number of linear MRRs or the quality factor ratio increases, the temporal span for photon information to couple these linear MRRs and subsequently re-couple into the main cavity extends. This prolonged duration enables the system to preserve bit information from a more distant past while processing current bit information, ultimately resulting in a larger MC. The minimum error NMSE_min= 0.156 is found at MC_max= 15.03, with a quality factor ratio of Q₂/Q₁= 10 and a total number of these MRRs M = 11. Specifically, the main cavity’s resonance wavelength is 1549.66 nm in the absence of nonlinearity, while the linear MRR’s resonance wavelength is 1549.92 nm. As the number of linear microcavities increases from 1 to 10, the system can preserve bit information from an even earlier past while processing current bit information. However, as the number of linear microcavities continues to increase, the energy entering these coupled microcavities saturates and no longer undergoes further changes, resulting in a plateau in MC. When the ratio of quality factors begins to increase, a greater amount of energy is coupled into these MRRs, and its presence within these MRRs persists for a longer duration, thereby leading to a larger MC. As the ratio of quality factors continues to exceed 10, despite the longer duration of optical energy coupled into the microcavities, the power coupled into these linear MRRs decreases significantly. Consequently, when the quality factor surpasses 10, there may actually be a decrease in MC.

 

Fig. 3. Performance of the NARMA 10 benchmark task for the proposed SCMRRs-based RC system. (a) NMSE and (b) MC versus the quality factor’s ratio Q₂/Q₁ and the total number M of MRRs. (c) MC (memory function $M{C_k}$, with ${l_{max}}\textrm{ = }45$) when the designed RC system has a varying number of MRRs. (d) The calculated weight values for the task to remember the previous input value x_i-1 based on the RC system with a single main cavity (red curve) and the proposed SCMRRs-based system that result in the lowest NMSE (black curve).

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Figure 3(c) shows the comparison of memory function $M{C_k}$ between the single MRR-based RC system and the proposed SCMRRs-based RC system. The single MRR-based RC system without optical feedback (M = 1) possesses a constrained MC stemming from the inertia between the reactions of the last virtual nodes to the preceding input value x_i-1 and the responses of the initial virtual nodes to the current input value x_i. The single MRR-based RC system can only remember the preceding input value x_i-1 in relation to the current input x_i (Fig. 3(c), red curve). Figure 3(d) displays the calculated readout weights for the task to remember the preceding input value x_i-1 (red curve). Considering the reservoir’s response to the current input x_i in the training step, the calculated results suggest that the weight values of the initial virtual nodes predominantly contribute to the computation, indicating a very limited MC. On the other hand, within the RC system based on SCMRRs, these series-coupled linear cavities function as linear analog shift registers. As illustrated in Fig. 3(c), the system’s MC is notably improved when the number of linear cavities increases from 2 to 10. At the beginning, the main cavity receives its initial excitation from the optical signals injected through the input waveguide. When the frequency of the optical signal closely aligns with the resonant frequency of the series-coupled linear cavities, a portion of the optical signal gradually transfers from the main cavity to these linear cavities. These signals undergo multiple round-trips within the linear cavities before eventually coupling back into the main cavity. As optical signals continuously couple into these series-coupled linear cavities with a high quality factor, the proposed RC system based on SCMRRs achieves the function of an extended linear memory. As depicted in Fig. 3(d) (black curve), nearly all virtual nodes participate in the task computation, signifying a notable enhancement in MC compared to the single MRR-based RC without optical feedback.

The proposed SCMRRs-based RC system is assessed in comparison to various MRR-based RC systems. These include the single MRR-based RC system and the RC system based on the MRR with external optical feedback. For the sake of fairness, these comparisons are conducted under identical environmental conditions, including the structural and material parameters of the main cavity. Table 1 presents the NMSE comparison between the proposed RC system and several MRR-based RC systems for the NARMA 10 task. In the task, both memory and nonlinearity play a role in the computation of the task. Within these photonic RC systems, nonlinearity arises from both the nonlinearity of the main cavity and the PD. Given the low input laser power (${P_M} = 0.1\textrm{mW}$), the main cavity operates in a linear state, making PD nonlinearity the predominant factor [46]. Therefore, a large MC is crucial for improving the computation performance of this task. Both the proposed RC system based on SCMRRs-10 and the MRR-based RC system with optical feedback exhibit larger MC, leading to lower NMSE compared to other systems. The NMSE for SCMRRs-10 is 0.156, which is less than the NMSE of 0.187 for the MRR-based RC system with optical feedback. Moreover, the MRR-based RC system is designed to incorporate a feedback waveguide exceeding 20 cm to achieve a larger MC. This introduces some challenges, including device fabrication, transmission loss, temperature control, etc [33]. The proposed SCMRRs-based system, employing optical microcavity-based resonators, features an overall size more than 350 times shorter than that of the feedback waveguide in the MRR-based RC system with optical feedback. Due to multiple cascaded MRRs, the fabrication complexity of this proposed structure may be greater than that of the single MRR-based system with optical feedback. Nevertheless, recent scientific advancements in the preparation of series-coupled MRRs [60–62] have made it feasible to fabricate the SCMRRs using existing fabrication techniques.

Table 1. The NMSE comparison of the proposed SCMRRs-based RC system and several MRR-based RC systems for the NARMA 10 task

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4.2 Mackey-Glass benchmark test

The Mackey-Glass time series serves as a standard benchmark for chaotic time series prediction tasks [56]. The series is defined by the following differential equation:

Figure 4 displays the performance of the Mackey-Glass benchmark task for the SCMRRs based RC system. We initially obtain the optimal laser energy input value of ${P_M} = 5\textrm{mW}$ and the optimal wavelength detuning value of $\mathbf{\Delta }{\lambda _s} ={-} 20\textrm{pm}$. Under these conditions, the lowest NMSE of the predicted value occurs at the quality factor’s ratio Q₂/Q₁= 40 and the total number of these MRRs M = 7, as shown in Fig. 4(a) (black circle). This corresponds to the smaller value of $\sigma (\mathbf{\Delta }{\lambda _0}(t))$ shown in Fig. 4(b). The highest NMSE value is observed at Q₂/Q₁= 5 and M = 3, as denoted by the red circle in Fig. 4(a). This corresponds to the larger value of $\sigma (\mathbf{\Delta }{\lambda _0}(t))$ depicted in Fig. 4(b). On the whole, system’s nonlinearity plays a key role in the Mackey-Glass benchmark test, and this aspect is quite distinct from the NARMA 10 task. The system’s nonlinear dynamics are associated with the power level within the main cavity, evaluated through the standard deviation of its resonance shift $\sigma (\mathbf{\Delta }{\lambda _0}(t))$ . For one-step-ahead forecasting, the Mackey-Glass task does not require too much MC. Although the single MRR-based RC system without optical feedback cannot provide a large MC, it still achieved good performance (NMSE = 0.0137). At ${P_M} = 5\textrm{mW}$, the nonlinearity of the main cavity contributes to the system’s nonlinearity. The linear cavity array functions as the memory provider, relying on the coupling between these cavities. When the resonance wavelength of the main cavity is close to that of these linear cavities, a strong coupling occurs, enabling the system to provide a larger MC. However, when the resonance wavelengths of the main cavity and these linear cavities are far apart, almost no coupling occurs, and the MC provided by the system is equivalent to that of a single main cavity. In the system, the nonlinear response of the main cavity significantly influences its resonance wavelength, thereby affecting the coupling between the main cavity and linear cavities and ultimately impacting the system’s MC. However, another aspect involves high power in the main cavity, possessing a large value of $\sigma (\mathbf{\Delta }{\lambda _0}(t))$ and high system’s nonlinearity. In this case, the SCMRRs-based RC system is in a seriously detuned state, and its NMSE performance is severely degraded. Therefore, there exists a trade-off between the system nonlinearity and the MC. Figure 4(c) shows the resonance shift $\mathbf{\Delta }{\lambda _0}(t)$ versus time when operating the computation. At Q₂/Q₁= 5 and M = 3, the shift of the main cavity’s resonance wavelength over time is depicted by the red curve in Fig. 4(c). The multiple-peaked bursts appear in the curve. In this time period, the system generates too much detuning between the resonance wavelength and the input wavelength. The self-pulsation phenomenon takes place along with a thermal warming-up step and then a thermal cool-down step [49–52]. In this case, as shown in Fig. 4(d) (path 1), the light signals mainly propagate through the main cavity during these time intervals, and they are not coupled into these series-coupled linear cavities. Consequently, the proposed system has higher nonlinearity, but loses a lot of MC. In this way, the proposed system eventually produces a relatively large NMSE. At Q₂/Q₁= 40 and M = 7, the time-dependent shift in the resonance wavelength of the main cavity is represented by the black curve in Fig. 4(c). The value of $\mathbf{\Delta }{\lambda _0}(t)$ is changed slightly with a small detuning with respect to the input wavelength. The self-pulsation phenomenon has almost not occurred, and the light signal is coupled into series-coupled linear cavities during the entire time evolution (Fig. 4(d), path 2).

 

Fig. 4. Performance of the Mackey-Glass benchmark task for the proposed SCMRRs-based RC system. (a) NMSE and (b) $\sigma \textrm{(}\Delta {\lambda _\textrm{0}}\textrm{(}t\textrm{))}$ versus the quality factor’s ratio Q₂/Q₁ and the total number M of these MRRs. (c) In the testing phase of the task, the temporal evolution of the resonance shift and the bit error have many extreme values: the black curve corresponds to the lowest NMSE (the black circle in (a)) and the red curve corresponds to the largest NMSE (red circle in (a)). (d) Dynamical evolution of the SCMRRs based system exhibiting the self-pulsation phenomenon: light is not coupled into (path 1, upper) these linear cavities or enters (path 2, lower) these cavities.

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Table 2 shows the NMSE comparison of the proposed SCMRRs-based RC system and several MRR-based RC systems for the Mackey-Glass task. The SCMRRs-6-based system achieves a minimum NMSE of 0.0083, significantly lower than the NMSE value of 0.0137 obtained by the single MRR without optical feedback. The SCMRRs-6’s NMSE is larger than the result of the MRR with optical feedback (NMSE = 0.0014) [46]. The results confirm that the Mackey-Glass task requires consideration of both nonlinear transformations and MC for optimal NMSE performance. Because of the nonlinearity of the main cavity and the MC offered by these linear cavities, the proposed system with a smaller size achieves a suboptimal NMSE value compared to the single MRR-based system with optical feedback.

Table 2. The NMSE comparison of the proposed SCMRRs-based RC system and several MRR-based RC systems for the Mackey-Glass task

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4.3 Santa Fe benchmark test

The Santa Fe laser time series involves one-step-ahead prediction on data acquired by sampling the intensity of a far-infrared laser in a chaotic state [57]. In the task, the goal is to predict only the future step, and the MC provided by the single MRR (without optical feedback) is enough [46]. Additionally, this task also requires moderate system nonlinearity. Compared with the NARMA 10 and the Mackey-Glass timeseries task, the available Santa Fe dataset contains experimental noise in its values, this noise introduces additional challenges in accurately predicting the future step.

Figure 5 displays the calculated performance of the Santa Fe task in the SCMRRs-based RC system. As shown in Fig. 5(a), the minimum NMSE of 0.018 (black circle) is found at the maximum input laser power ${P_M} = 1\textrm{mW}$ and the initial wavelength detuning $\mathbf{\Delta }{\lambda _s} = 10\textrm{pm}$ for Q₂/Q₁= 30 and M = 7. Figure 5(b) shows the value of $\sigma \mathbf{(\Delta }{\lambda _0}(t))$ versus the maximum input laser power P_M and $\mathbf{\Delta }{\lambda _s}$ for the SCMRRs-based RC system. Almost all computational results show NMSE values close to those obtained from the single-MRR-based system without optical feedback. At ${P_M} = 1\textrm{mW}$ and $\mathbf{\Delta }{\lambda _s} = 10\textrm{pm}$, as indicated by the minimum NMSE in Fig. 5(a), $\sigma \mathbf{(\Delta }{\lambda _0}(t))$ exhibits a moderate value. This observation confirms that the system necessitates a moderate level of nonlinearity to achieve a low NMSE. At ${P_M} = 1\textrm{mW}$ and $\mathbf{\Delta }{\lambda _s} = 10\textrm{pm}$, Fig. 5(c) discusses the influence of both the quality factor’s ratio Q₂/Q₁ and the total number M of these linear cavities on the computation performance at ${P_M} = 1\textrm{mW}$ and $\mathbf{\Delta }{\lambda _s} = 10\textrm{pm}$.The SCMRRs-based RC system still achieves the minimum NMSE value at Q₂/Q₁= 30 and M = 7. Figure 5(d) discusses the impact of both the quality factor’s ratio Q₂/Q₁ and the total number M of MRRs on the standard deviation of the resonance wavelength shift $\sigma \mathbf{(\Delta }{\lambda _0}(t))$ in the main cavity. It is noted that $\sigma \mathbf{(\Delta }{\lambda _0}(t))$ at the lowest NMSE also corresponds to an intermediate value.

 

Fig. 5. Performance of the Santa Fe task for the proposed SCMRRs-based RC system. (a) NMSE and (b) $\sigma \mathbf{(\Delta }{\lambda _0}(t))$ versus the maximum input laser power ${P_M}$ and the total number M of these MRRs. (c) NMSE and (d) $\sigma \mathbf{(\Delta }{\lambda _0}(t))$ versus the quality factor’s ratio Q₂/Q₁ and the total number M of these MRRs.

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Table 3 shows the NMSE comparison of the proposed SCMRRs-based RC system and several MRR-based RC systems for the Santa Fe task. The SCMRRs-6-based system achieves a minimum NMSE of 0.018, which is lower than the NMSE of 0.038 obtained by the single MRR-based system without optical feedback. The NMSE of the SCMRRs-6-based system is a little lower than the result of the single MRR-based system with optical feedback (NMSE = 0.020) [46]. These results clearly demonstrate that the Santa Fe task only requires a moderate level of nonlinear transformation for achieving optimal performance. The task doesn’t demand a significant amount of MC, and the MC provided by the single MRR without optical feedback is enough. Therefore, several MRR-based systems achieve almost the same optimal performance by employing their respective modest nonlinearities under different conditions.

Table 3. The NMSE comparison of the proposed SCMRRs-based RC system and several MRR-based RC systems for the Santa Fe task

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5. Fabrication tolerance analysis

When examining these cascaded MRRs in SCMRRs from design and fabrication standpoints, it is crucial to investigate the impact of fabrication size errors on the computation performance of the proposed RC system. However, this analysis is conducted to confirm the theoretical feasibility of achieving a large MC using the proposed RC system based on cascaded MRRs, without delving into a very detailed and comprehensive analysis of device design and fabrication. The main fabrication challenge of these cascaded MRRs-based structure lies in the mismatch of resonance wavelengths among these linear cavities [60–62], preventing effective mutual coupling and consequently significantly hampering their ability to effectively increase MC. Therefore, this discussion focuses on the impact of errors in device fabrication dimensions on the consistency of resonance wavelengths in these linear cavities.

In our analysis of fabrication tolerance, we introduced a variable indicated by Δr to represent the radius error of the linear MRR. Then, we proceeded to investigate its influence on the computation performance of the proposed RC system. We chose ten sets of random values within a certain range as the radius errors (Δr) of ten linear cavities to create the SCMRRs-10-based RC system. For the sake of simplicity, only the NARMA 10 task was selected to evaluate the proposed RC system. Figure 6(a) and (b) show MC and (b) NMSE versus the index of a random choice within the range, respectively. The range spans from ±500pm, ± 1 nm, ± 10 nm, to ±50 nm. When the Δr value is within a small range (±500pm), resulting in a resonance wavelength fluctuation range of about ±10.3pm, the resonance wavelengths of ten linear cavities show relatively good consistency. In this case, the MC does not decrease significantly from the ideal scenario, and accordingly, the NMSE does not increase substantially. Conversely, when the Δr value fluctuates within a larger range (±50 nm), resulting in a resonance wavelength fluctuation range of about ±1.03 nm, optical signals can hardly effectively couple into the linear cavities. This leads to a significant reduction in MC. The NMSE also increases significantly, and its results are similar to those of the single-MRR-based system without optical feedback. The change in radius error also affects the coupling between these linear MRRs; refer to Supplement 1 for a detailed analysis of the error calculation process.

 

Fig. 6. The linear cavity’s radius error Δr is assigned some random values within various ranges of ±500pm, ± 1 nm, ± 10 nm, and ±50 nm. Specifically, within each range, we selected separately ten sets of random values as the radius errors of ten linear cavities to create the SCMRRs-10-based RC system. Then, the system’s performance is assessed using the NARMA 10 task. (a) MC and (b) NMSE versus the index of a random choice within each range, where ${P_M} = 0.1mW, \Delta {\lambda _s} ={-} 30pm$, and ${{{Q_2}} / {{Q_1} = 10}}$.

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While the proposed SCMRRs-based RC system places stringent demands on the consistency of resonance wavelengths in linear cavities, existing fabrication techniques are fully capable of meeting these requirements. In Ref. [60,61], a silicon-based adiabatic elliptic microring (AEM) structure has been designed and fabricated, successfully producing a high-order filter comprising 10 series-coupled AEMs. Without requiring individual thermal tuning and meticulous calibration for each AEM, all 10 AEMs have maintained exceptional consistency in the resonance wavelengths, even within a waveguide width range with a tolerance of ±40 nm. In Ref. [62], researchers have also employed a three-dimensional MRR-based optical switch element on a multi-layer Si₃N₄-on-SOI platform to achieve high-performance 8 × 8 optical switch fabrics. Here, the resonance wavelengths of all MRRs were nearly perfectly aligned, underscoring the outstanding fabrication tolerance of these cascaded MRRs.

6. Conclusion

In this paper, we conducted a numerical investigation of a SCMRRs-based system as a versatile computation platform in time-delayed RC. Compared with the single MRR-based system with an ultra-long optical feedback waveguide [46], our proposed system shares nearly the same MC while reducing the device size by 350 times. Moreover, the feedback waveguide in Ref. [46] reaches a length of over twenty centimeters, making both the fabrication and waveguide loss handling quite challenging. In contrast, the resonant coupling structure we employ is more feasible under current preparation technologies. To evaluate its computation performance, we have computed three typical tasks that have different MC requirements. For the NARMA 10 task, the proposed SCMRRs-10-based system possesses greater MC than the system based on the MRR with optical feedback, and the former achieve better performance than the latter. For the Mackey-Glass prediction task, because the proposed system fulfills the system’s requirements for nonlinearity and MC, it shares almost the same lowest prediction error with the system based on the MRR with optical feedback. For the Santa Fe task, the one-step-ahead prediction does not need large MC, and the proposed system achieves a little better performance than the system based on the MRR with optical feedback [46]. The proposed SCMRRs-based system shares almost the same computation performances as the system based on the MRR with optical feedback, but with a significantly smaller footprint. With existing fabrication techniques, this proposed RC system paves the way for scalable integrated photonic RC systems.