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Abstract
To achieve an autonomously controlled reconfigurable microwave waveform generator, this study proposes and demonstrates a self-adjusting synthesis method based on a photonic delay reservoir computer with ring resonator. The proposed design exploits the ring resonator to configure the reservoir, facilitating a nonlinear transformation and providing delay space. A theoretical analysis is conducted to explain how this configuration addresses the challenges of microwave waveform generation. Considering the generalization performance of waveform generation, the simulations demonstrate the system's capability to produce six distinct representative waveforms, all exhibiting a highly impressive root mean square error (RMSE) of less than 1%. To further optimize the system’s flexibility and accuracy, we explore the application of various artificial intelligence algorithms at the reservoir computer's output layer. Furthermore, our investigation delves deeply into the complexities of system performance, specifically exploring the influence of reservoir neurons and micro-ring resonator parameters on calculation performance. We also delve into the scalability of reservoirs, considering both parallel and cascaded arrangements.
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1. Introduction
Microwave waveform generator (MWG) has garnered significant interest due to its pivotal role in boosting the signal processing functionalities [1,2]. Its versatile applications span numerous domains such as radar system, spread spectrum communications, imaging, sensing and metrology [3].
Microwave waveform generator is typically operated electronically, yet the frequency range of waveforms generated are usually constrained to lower spectrums, a limitation primarily due to electronic bandwidth bottlenecks. This restriction hinders the system's capacity for high-speed information processing. To counter the problem, an optical approach emerges as a compelling alternative. Optical systems are capable of exhibiting a wide bandwidth and can directly process optical signals, bypassing the need for optical-to-electrical and electrical-to-optical conversions, thus offering a more efficient solution.
Up to now, among the various optical methods available, the one based on frequency-to-time mapping [4,5] is prevailing and widely technology to generate microwave waveform. This method involves shaping the frequency spectrum in the frequency domain, and then obtain the microwave waveform by frequency-to-time mapping. A notable advantage of this method is its ability to generate microwave waveforms within a single time aperture. However, it necessitates the use of a high-frequency dispersion device. This presents a significant challenge, particularly in operating individual comb lines in scenarios with small comb spacing. Another major method is based on microwave photonic filtering [6,7]. The method accomplishes discrete optical spectral lines to be controlled by combining modulator and photonic filter. Its primary limitation is the tendency to generate a single, fixed waveform, owing to its limited flexibility. There was also scheme proposed to realize waveforms from the perspective of time domain envelope function calculation [8]. However, this scheme must change a different structure each time to produce a specific waveform.
Currently, an ideal MWG system [9] should encompass several key characteristics: (a) the ability to output reconfigurable microwave waveforms, demonstrating flexibility; (b) robustness against environmental variations, ensuring reliability; and (c) the capacity for automatic control of the output system, enhancing user-friendliness. While the integration of photonic computing into optical engineering can cater to the above problems with the advantages of artificial intelligence. Particularly, photonic delay reservoir computing (PDRC), which is derived from recurrent neural network (RNN), can be readily implemented in optical systems. This approach is characterized by a comparatively simple training process, especially when contrasted with other neural network architectures [10–12].
PDRC is suitable for temporal/sequential signal processing due to the RNN-based framework. Its main characteristic lies in the simplification of specific properties related to input weights, reservoir weights and output weights [13]. In the PDRC system, the weights of input layer and the weights of the reservoir layer don’t require training, only the readout weights of output layer undergo training with ordinary regression algorithms. Compared with other neural network structures [14], such simple and fast training structure can greatly reduce the computational cost of learning. The essence of the reservoir layer in PDRC is to nonlinearly transform input information into spatiotemporal patterns within a high-dimensional space and then are read out efficiently with an ordinary regression algorithm.
In this work, we propose photonic delay reservoir computing method to realize reconfigurable microwave waveform generator. Ring resonator is chosen as one nonlinear response (real) node to perform nonlinear transformation and the waveguide of ring introduces time delay. In presence of the time delay, one real nonlinear node is divided into multiple virtual nodes in a time-division multiplexing form. We analyze a ring resonator theoretically in the context of a reservoir layer and then demonstrate a PDRC system using a ring resonator as a reservoir layer to synthesis reconfigurable microwave waveform through simulations. To investigate the performance of proposed system, we show the generalization capabilities of system to generate six different microwave waveforms. And then, we calculate the memory capacity of the system and investigate the dependence on reservoir neurons, output layer regression algorithms and ring resonator parameters for the performance of output results. Finally, we discuss the scalability of reservoir based on ring resonator, including parallel and cascaded structure.
2. Ring resonator in delay photonic reservoir computing
Photonic delay reservoir computer (PDRC) is generally composed by three layers, which are input layer, reservoir layer and output layer [15] (Fig. 1). Input layer holds input data being fed into the system, and input data is written as defined in ${W_{in}}$ for tth timestep [16,17]. The reservoir layer of PDRC has N virtual nodes $x(t )$, which are implemented in a continuous medium space and wired together by weight ${W_{res}}$. N nodes equally divide the whole loop into N segments according to time division multiplexing, and the length of time each virtual node occupied is $\theta $. Input data is randomly connected with the N-dimensional reservoir layer. The random injection can provide motivation for the creation of a highly diverse reservoir response. Reservoir layer is generally comprised of photoelectric device that can provide nonlinear dynamics, and then reservoir computer can be an approximate chaotic system. Aiming at maximizing the diversity of reservoir responses, the internal connectivity ${W_{res}}$ of reservoir layer is random. Physical implementation of such connections is natural choice and highly practical. The neural state of reservoir layer [18] at the nth timestep can be determined recursively as nonlinear mapping:
where ${u_{bias}}$ is the vector of biases and ${f_{NL}}$ is the nonlinear function of the PDRC system. And then results Y(n) yielded by the output layer is defined as formula (2), given as a linear sum of all virtual nodes’ responses. where ${W_{out}}$ represents the N-dimensional vector of the output weights, which are trained by certain algorithm. In particular, this is the only part of the entire neural network that requires artificial control.PDRC architecture provides a viable hardware foundation for neuromorphic computing capability, which is contributable to its realization based on discrete nonlinear neurons, and implementations based on non-discrete continuous systems. Such configuration projects the input signals into a high-dimensional nonlinear dynamic system, which, in the absence of external stimuli, steadily return to equilibrium, indicating that computational system possesses memory characteristics. Considering the above key demand, we choose waveguide-ring resonator to complete the hardware implementation of the reservoir layer [19,20]. Ring resonator is a significant micro-optical device [21], and a schematic diagram of its structure is shown in Fig. 2(a). We describe the transmission of optical signals through the ring resonator to establish the scattering matrix (S-matrix) by employing the transfer matrix approach, as shown in Formula (3). It calculates by multiplying S-matrices of an X-coupler, ring waveguide and bus waveguide.
Fig. 2. (a) The concept map of micro-ring resonator, (b) micro-ring resonator to be reservoir layer.
In the transmission model, E represents the optical filed of optical signal and each S-matrix component $T_{n,m}^{\mu ,v}(f )$ is a complex-valued transfer function which describes coupling between input wave carried by the guided mode v at port m and the output wave carried by the guided mode $\mu $ at port l. In the model, it's assumed that each port is capable of supporting two types of guided modes: the fundamental TE-like mode whose amplitude is described by the $Ex$ -component of the optical signal, and the fundamental TM-like mode whose amplitude is described by the $Ey$ -component of the optical signal. In the ring resonator [22], each node is a segment divided by $\theta $ sampled by the detector as shown in Fig. 2(b). Optical signals circulating once in the ring waveguide convert an input signal into another under a specific transfer function, thereby implementing the functionality of ${W_{res}}$ during this process. ${W_{res}}$ matrix is a mapping of signal state at the beginning of ring loop to the signal state at the end of the ring loop, where the mapping covers physics of propagating the optical signal through the waveguide.
The alteration in thermal and free carrier leads to changes in the refractive index of the ring resonator, resulting in nonlinear effects. This is illustrated in Formula (4), which depicts the temporal dynamics of the ring resonator [23] by following set of coupled differential equation:
3. System setup of PDRC based ring resonator to generate MWG
Generating microwave waveforms is a critical research endeavor in optical information processing. Given that any periodic waveform can be decomposed into a sum of multiple Fourier series, each characterized by appropriate amplification and phase weights, the generation of microwave waveforms can be effectively compounded by manipulating spectral lines of varying frequencies. Delay reservoir computing excels in facilitating the automatic optimization of temporal signal processing tasks under the stable structure and simple control. This section demonstrates the proposed system setup of photonic delay reservoir computing based micro-ring resonator, built for the purpose of generating microwave waveforms.
The schematic of the PDRC system to generate MWG is shown in Fig. 3. A Continuous Wave (CW) laser acts as a light source, supplying a 1500 nm carrier. The effective frequency information is provided by frequency comb sent from 5 GHz to 25 GHz at internals of 5 Hz as the basic frequency information of the waveform to be synthesized. And then the frequency information is directed to the Mach-Zehnder modulator to modulate the carrier. The above part makes up the input layer of the system. The scheme only needs to exploit the spectral lines of the input signal, eliminating the need for mask operations in the input layer, which is one of the advantages of the scheme.
Fig. 3. Scheme of delay line reservoir computing based ring resonator to generate OWG. (CW: continuous-wave laser, MZM: Mach-Zehnder modulator, FC: Frequency Comb, MR: Micro-ring Resonator, PD: Photodetector, OSC: oscilloscope).
In the reservoir, the micro-ring with high quality factor is able to offer enough free carriers [24] and exhibits self-pulsation dynamics to provide nonlinearity functionalities. The free carrier nonlinearity has a faster time response and provides for faster computing speed [21]. Meanwhile, the length of ring is 110um in the reservoir layer and the waveguide of ring resonator itself constitutes a feedback loop. Loaded with information input light waves is injected into the reservoir. In the feedback loop, the input signal captures the transient dynamics of micro-ring resonator in reservoir. When microwave photon signal carrying different frequency propagate in the resonator loop repeatedly, the intensification of the interaction between light and silicon has a dispersion effect. Hence the phase relationship between them may change, resulting in interference. In the output layer, the optical signal that passes through the reservoir layer is detected by a high-speed photodetector. Finally, the states of the virtual neurons are monitored on an oscilloscope, and subsequent postprocessing of the output is conducted.
3.1 Generalization performance of the system
We initially assess the generalization capability of this system in accomplishing microwave waveform generation. To quantitatively evaluate the performance of the system, we adopt normalized root mean square error (RMSE) as the evaluative metric, which is calculated as follows:
where ${y_i}$ represents the measured value of the point at which the output waveform is generated, ${\hat{y}_i}$ represents the idea value of the output waveform point, $A$ represents the peak of ideal waveform.Once the system structure is established, intelligent control of the regression weights in the output layer becomes crucial. Here, the neuron states are linearly combined to closely approximate the desired output. Ridge regression [16,25] is commonly opted to trained output weights ${W_{out}}$. Although ridge regression has been successfully applied in various contexts, it's important to note that other regression algorithms may surpass its performance in specific tasks. Gradient Boosting Regression (GBR) is a machine learning technique used for both regression and classification problems. It builds the model in a stage-wise fashion like other boosting methods do, but it generalizes them by allowing optimization of an arbitrary differentiable loss function. Comparing ridge regression, GBR is computationally more intensive, which offers greater flexibility and robustness against overfitting, it shines in handling non-linear datasets. Taking the above factors into account, we sample 4096 data points from the detected dataset to establish 32 virtual nodes. We then train the output layer weights using a GBR algorithm to synthesize a range of common waveforms.
Fig 4 illustrates the results for six commonly generated waveforms. As depicted in Fig. 4, the output waveforms closely match the ideal waveforms. The normalized RMSE values for the synthesized waveforms of the MWG are presented in Table 1. With the maximum RMSE remaining below 1%, this indicates means a high consistency between the OWG's output waveforms and the target waveforms. Therefore, our approach demonstrates strong generalization capabilities in microwave waveform generation.
Fig. 4. Comparison of output waveforms (black solid line) of the MWG and the ideal ones (orange region), from left to right: triangle, square, positive sawtooth, oblique sawtooth1, oblique sawtooth2, negative sawtooth.
Table 1. Normalized RMSE of output waveforms
3.2 Memory capacity of PDRC
The efficacy of reservoir computing relies on its ability to effectively facilitate nonlinear transformations and leverage the memory capacities inherent in dynamic systems. The aspect of nonlinear transformation was elaborated upon in the section 2. This section analyzes the linear memory capability of the PDRC. Similar to connections in the brain that travel at the limited speeds [26], reservoir forms temporarily delayed interconnected loops. The current output is influenced by data from diverse initial time intervals, which means the system exhibits short-term memory capabilities. To measure the linear memory capability, we adopt the linear memory capacity (MC) task [27,28] to evaluate the proposed system.
In previous research, computations were conducted by feeding random bit sequences into the system, with values sourced from a uniform distribution. However, this computing method, when employed in nonlinear systems, may yield inconsistent results. Diverse input data can trigger various nonlinear transformations within the system. Our system is designed for waveform synthesis task; therefore, it adopts the computation method described in the following formula.
3.3 Dependence on reservoir neurons and output layer regression algorithm
The quantity of neurons affects both the complexity and the computational power of the network connections within the reservoir computer, as Hebbian says, “neurons that fire together, wire together” [29]. We investigate dependence of the reservoir computing performance on the number of virtual nodes in the PDRC. As shown in Fig. 6, it shows that the normalized RMSE variations as the number of virtual nodes is changed. Blue, green and red dashed curves represent the RMSEs of the triangle, square and sawtooth waveforms, respectively. The overarching pattern depicted in the graph indicates a decline in error rates for all three synthesized waveforms as the number of nodes in the system increases. An increase in the number of virtual nodes signifies a more intricate structure in the reservoir layer and a higher count of readout weights from the output layer engaged in training. For a node count less than 16, all three waveforms exhibit high error rates, with the square waveform being particularly affected. This is primarily because synthesizing square waves that closely resemble the ideal waveform is challenging; it demands a greater variety of frequency components for satisfactory results. Once the count of virtual nodes reaches 32, the system demonstrates adept control over the error rates of the output waveforms.
Then, we investigate the dependence of RMSE on the different regression algorithms applied to the weights of the output layer. Three regression algorithms, random forest, gradient boosting regression, and ridge regression respectively, are evaluated to ascertain their impact on the quality of the waveform produced by the PDRC system, using varying numbers of nodes. To ensure a fair comparison, the focus of this part work is on generating triangular waves of identical frequency, with the normalized RMSE results displayed in Fig. 7. This figure illustrates that the RMSE for all three methods decreases as the number of nodes increases, aligning with the observations noted in the previous part.
Fig. 7. RMSE of the PDRC with different regression algorithm. (The green, blue, and red columns represent random forest, gradient boosting algorithm and ridge regression towards triangle waveform.)
When the neuron count in the PDRC system reaches 64, it performs exceptionally well across all scenarios. However, with the number of virtual nodes falling below 64, the performance of the ridge regression method shows a significant discrepancy, struggling to generate high-quality waveforms. In contrast, the use of random forests remains effective, consistently yielding better results even with a low node count. As the node count increases, random forests are outperformed by the gradient boosting regression algorithm, which exhibits greater stability. A notable limitation of the gradient boosting regression algorithm is its difficulty in closely replicating the ideal waveform at extremely low node counts. As the number of virtual nodes in the PDRC increases slightly, it becomes easier to accurately train a perfect triangular waveform. Additionally, the number of sampling points in the training set also plays a crucial role in determining how closely the output waveform matches the desired pattern.
3.4 Dependence on micro-ring resonator parameters
In the reservoir layer, the parameters of the micro-ring resonator, which facilitate nonlinear dynamics, exert a notable influence on the entire dynamic system. This section is dedicated to exploring the impact of the Ring's parameters on the output results. To characterize the changes before and after, Pearson correlation coefficient is introduced [30], which is defined as the covariance of the two variables divided by the product of their standard deviations, as shown in formula (7). The Pearson correlation coefficient is a measure of the degree of correlation between two variables. It is a value between 1 and -1, where 1 means that the variables are completely positively correlated, 0 means that there is no linear correlation, and -1 means that the variables are completely negatively correlated.
The results shown in Fig. 8(a) are Pearson correlation coefficient (pcc) value, obtained by sampling the length of micro-ring waveguide at intervals of 10um, benchmarked with a micro-ring length of 110um as a baseline waveform. It is observable that the coefficient remains close to 1 across a broad range of conditions. A minor deviation from the baseline waveform is observed when the micro-ring length extends to 270um. More pronounced deviations from the benchmark of 1 occur at lengths of 40, 80, 120, 160, 200, 240, and 280um. This result underscores the noticeable impact of micro-ring length on the waveform's output, highlighting the sensitivity of the system's performance to variations in micro-ring dimensions. In areas of the image where deviations from pcc = 1 are most significant, specifically where notches are found, the lengths of the micro-ring exactly correspond to multiples of 40um. This unexpected observation may suggest that within this system, the nonlinear transformation capabilities arising from every 40-um increment in micro-ring length have a detrimental effect on the output quality of the entire reservoir system. A probable explanation for the phenomenon depicted in this figure could be the existence of a numerical correlation between the delay times produced at every 40-um interval within the micro-ring and the common multiples of response times throughout the entire feedback loop of the reservoir. And each additional 40-um for the micro-ring waveguide causes the effective information to almost complete a full circuit around the reservoir, failing to enhance high-dimensional mapping capabilities, not enhancing the system's effective computational capabilities.
Fig. 8. Pearson correlation coefficient with different (a) length of ring waveguide and (b) ring phase shift and (c) Q-factor.
Figure 8(b) shows pcc value as a function of the ring additional phase shift. To calculate correlation coefficient value using Eq. (7), no phase shift of the ring is calculated as a fixed variable. In this case, if the phase shift degree alteration is minimal, the correlation coefficient values remain near 1, signifying that the output result is largely unaltered. However, when the ring's phase shift exceeds 60°, particularly when it reaches 80°, the output waveform will be severely deformed. This shows that the system can tolerate a certain degree of tolerance in the phase of the micro-ring waveguide during practical application without affecting the final output.
Figure 8(c) shows pcc value as a function of the quality factor of the micro-ring resonator, where the values are measured using a quality factor of $Q = 20000$ as a baseline. In this study, we examine the quality factor of Q ranging from 5000 to 50000. The results, as depicted in the figure, indicate significant variations in output with changes in the quality factor. In particular, at lower quality factors, the output waveform substantially deviates from the baseline. Beyond a quality factor of 20000, the output consistency shows minor fluctuations. This phenomenon can be partially attributed to the findings in Ref [23], which suggest that micro-rings with a high quality factor ($Q > 2 - 3 \times {10^4}$) have self-pulsation behavior and can generate enough free carrier to provide dynamics.
4. Discussion
In the previous chapter, we discussed the system parameters, neuronal division, optimization algorithm of the delay line reservoir computing based on micro-ring, resulting in generalization and higher performance of microwave waveform generation. In this chapter, we will provide a conceptual possibility for extending the reservoir computer to improve the performance of processing high-speed information [31].
Firstly, we show that the extended reservoir layer is configured by two same parallel micro-ring resonators as shown in Fig. 9. The input layer, serving as the optical interface to external information sources, remains unchanged. It acquires data for processing and channels this data to the subsequent layer in the neural network [32]. The input layer uniformly injects the same input signals into each reservoir, each comprising a ring resonator, to analyze the dynamic properties of each reservoir's output. Every virtual node on each resonator in the reservoir layer contributes to processing the output signal, significantly increasing the number of neuronal nodes involved. In the next step, all neuronal states are linearly weighted and summed to generate the final output. This enhanced reservoir computing structure improves the performance of information processing tasks by increasing the number of parallel reservoirs.
Then, we show another extended reservoir layer [33] is configured by two same cascaded micro-ring resonators as shown in Fig. 10. The input layer still functions as an external signal injected into the reservoir. The input signal is initially fed into the first reservoir layer, where all virtual nodes undergo a linearly weighted summation to produce the layer's output. This output is then used as the input for the cascaded reservoir of the next layer. In this subsequent layer, the state of the virtual nodes is collected, and their weighted summation forms the output of that layer. This procedure is repeated, with the output of each reservoir layer serving as the input for the next, until the result from the final reservoir becomes the output of the entire system. This reservoir employs a cascading structure where each layer possesses the capabilities of a traditional reservoir, significantly enhancing the system's information processing ability and expanding the scope of its applications.
5. Conclusion
In conclusion, we proposed a photonic scheme for reconfigurable microwave waveform generator using photonic delay reservoir computing with micro-ring resonator. Different from the ideas of frequency-to-time mapping and microwave photonic filtering, this method takes the view of the incorporation of optical computing with intelligent control into waveform generator engineering. We demonstrated the generalization performance to synthesis microwave waveforms under autonomous control and showed that the normalized RMSE of six distinct representative generated microwave waveforms were below 1%. And we gave the memory capacity of the system to characterize the part performance of reservoir computing. We concluded by quantitatively analyzing the impact of reservoir neurons, the output layer regression algorithm, and micro-ring resonator parameters on the system's performance. To further improve the computing capability of PDRC, the reservoir can be expanded in a parallel and cascading manner. This paper provided a new access to automatically control flexible microwave waveform generation.
Funding
National Natural Science Foundation of China (62221001, 62235003).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. Z. Jiang, D. E. Leaird, and A. M. Weiner, “Line-by-line pulse shaping control for optical arbitrary waveform generation,” Opt. Express 13(25), 10431–10439 (2005). [CrossRef]
2. S. Liao, Y. Ding, and J. Dong, “Photonic arbitrary waveform generator based on Taylor synthesis method,” Opt. Express 24(21), 24390 (2016). [CrossRef]
3. Y. Xie, L. Zhuang, and A. J. Lowery, “Picosecond optical pulse processing using a terahertz-bandwidth reconfigurable photonic integrated circuit,” Nanophotonics 7(5), 837–852 (2018). [CrossRef]
4. R. Cao, G. Wang, and M. Li, “Photonic generation of a microwave waveform with an ultra-long temporal duration using a frequency-shifting dispersive loop,” Opt. Express 30(4), 4737–4747 (2022). [CrossRef]
5. H. Y. Jiang, L. S. Yan, and Y. F. Sun, “Photonic arbitrary waveform generation based on crossed frequency to time mapping,” Opt. Express 21(5), 6488–6496 (2013). [CrossRef]
6. C. Wang, T. Ning, and J. Li, “Photonic generation of frequency-quadrupled triangular waveform based on a DP-QPSK modulator with tunable modulation index,” Opt. Laser Technol. 137, 106818 (2021). [CrossRef]
7. X. Zhang, R. Wang, W. Jiang, et al., “Generation of Broadband Reconfigurable LFM Waveforms Via Heterodyne-Beating Synchronized Lasers,” J. Lightwave Technol. 40(13), 4110–4118 (2022). [CrossRef]
8. X. Chen, Y. Jiang, and Q. Yu, “All-optical microwave waveform transformation based on photonic temporal processors,” Opt. Express 30(7), 10428–10442 (2022). [CrossRef]
9. B. Fischer, M. Chemnitz, B. MacLellan, et al., “Autonomous on-chip interferometry for reconfigurable optical waveform generation,” Optica 8(10), 1268–1276 (2021). [CrossRef]
10. T. Bu, S. Kumar, M. Jin, et al., “Efficient optical reservoir computing for parallel data processing,” Opt. Lett. 47(15), 3784–3787 (2022). [CrossRef]
11. T. Hülser, “Role of delay-times in delay-based photonic reservoir computing,” Opt. Mater. Express 12(3), 1214–1231 (2022). [CrossRef]
12. F. Duport, A. Smerieri, and A. Akrout, “Virtualization of a Photonic Reservoir Computer,” J. Lightwave Technol. 34(9), 2085–2091 (2016). [CrossRef]
13. G. Tanaka, T. Yamane, and J. B. Héroux, “Recent advances in physical reservoir computing: A review,” Neural Netw. 115, 100–123 (2019). [CrossRef]
14. J. Torrejon, M. Riou, and F. A. Araujo, “Neuromorphic computing with nanoscale spintronic oscillators,” Nature 547(7664), 428–431 (2017). [CrossRef]
15. G. Van der Sande, D. Brunner, and M. C. Soriano, “Advances in photonic reservoir computing,” Nanophotonics 6(3), 561–576 (2017). [CrossRef]
16. U. Paudel, M. Luengo-Kovac, and J. Pilawa, “Classification of time-domain waveforms using a speckle-based optical reservoir computer,” Opt. Express 28(2), 1225–1237 (2020). [CrossRef]
17. F. Duport, A. Smerieri, and A. Akrout, “Fully analogue photonic reservoir computer,” Sci. Rep. 6(1), 22381 (2016). [CrossRef]
18. E. Picco, P. Antonik, and S. Massar, “High speed human action recognition using a photonic reservoir computer,” Neural Netw. 165, 662–675 (2023). [CrossRef]
19. M. Borghi, D. Bazzanella, and M. Mancinelli, “On the modeling of thermal and free carrier nonlinearities in silicon-on-insulator microring resonators,” Opt. Express 29(3), 4363–4377 (2021). [CrossRef]
20. M. Borghi, S. Biasi, and L. Pavesi, “Reservoir computing based on a silicon microring and time multiplexing for binary and analog operations,” Sci. Rep. 11(1), 15642 (2021). [CrossRef]
21. G. Donati, C. R. Mirasso, and M. Mancinelli, “Microring resonators with external optical feedback for time delay reservoir computing,” Opt. Express 30(1), 522–537 (2022). [CrossRef]
22. J. Kullig, D. Grom, and S. Klembt, “Higher-order exceptional points in waveguide-coupled microcavities: perturbation induced frequency splitting and mode patterns,” Photonics Res. 11(10), A54 (2023). [CrossRef]
23. T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14(2), 817–831 (2006). [CrossRef]
24. T. Van Vaerenbergh, M. Fiers, and P. Mechet, “Cascadable excitability in microrings,” Opt. Express 20(18), 20292–20308 (2012). [CrossRef]
25. S. Sackesyn, C. Ma, and J. Dambre, “Experimental realization of integrated photonic reservoir computing for nonlinear fiber distortion compensation,” Opt. Express 29(20), 30991 (2021). [CrossRef]
26. B. J. Shastri, A. N. Tait, T. Ferreira de Lima, et al., “Photonics for artificial intelligence and neuromorphic computing,” Nat. Photonics 15, 102–114 (2021). [CrossRef]
27. S. Boshgazi, A. Jabbari, and K. Mehrany, “Virtual reservoir computer using an optical resonator,” Opt. Mater. Express 12(3), 1140–1153 (2022). [CrossRef]
28. R. M. Nguimdo, P. Antonik, and N. Marsal, “Impact of optical coherence on the performance of large-scale spatiotemporal photonic reservoir computing systems,” Opt. Express 28(19), 27989 (2020). [CrossRef]
29. D. O. Hebb and W. A Kandel, The Organization of Behaviour (Wiley, 1949).
30. K. Kanno, A. Amalina Haya, and A. Uchida, “Reservoir computing based on an external-cavity semiconductor laser with optical feedback modulation,” Opt. Express 30(19), 34218–34238 (2022). [CrossRef]
31. J. Feldmann, N. Youngblood, and C. D. Wright, “All-optical spiking neurosynaptic networks with self-learning capabilities,” Nature 569(7755), 208–214 (2019). [CrossRef]
32. H. Hasegawa, K. Kanno, and A. Uchida, “Parallel and deep reservoir computing using semiconductor lasers with optical feedback,” Nanophotonics 12(5), 869–881 (2023). [CrossRef]
33. D. Zhong, K. Zhao, and Z. Xu, “Deep optical reservoir computing and chaotic synchronization predictions based on the cascade coupled optically pumped spin-VCSELs,” Opt. Express 30(20), 36209–36233 (2022). [CrossRef]
