Fig. 1. Concept of QC and RC structures. (a−c) Design of the proposed QC, which has a similar operational criterion to convolutional coding in data processing, but meanwhile can extract features in the temporal dimension and provide memory capability. (d−f) Schematic diagram of different RC structures. The comparison of the nodes’ states between (d) Spatial RC, (e) TDRC and (f) proposed QRC verifies that the encoding data will provide the memory capability through QC. NL, nonlinear nodes.

Fig. 2. Schematic diagram of the TDRC and the QRC, as well as their physical implementation based on the VCSEL. (a) Schematic architectures of the TDRC and QRC based on the VCSEL. Experimental setup of (b) the TDRC and (c) the QRC. PC, polarization controller; Att, attenuator; MZM, Mach-Zehnder Modulator; DL, delay line; EDFA, erbium-doped fiber amplifier; OBPF, optical bandpass filter; PBS, polarization beam splitter; PD, photodetector; AWG, arbitrary waveform generator; OSA, optical spectrum analyzer; OSC, oscilloscope; Ch, channel. The blue (red) line represents the optical (electrical) connection. (d) Optical spectra of the VCSEL. The black line represents the optical spectra of the free-running VCSEL. The red line represents XM and YM separated by PBS and PC.

Fig. 3. (a) Bifurcation diagram with k_inj as the control parameter of the VCSEL. In all panels, the extrema (maxima and minima) of the intensity time series are shown as dots. (b−d) showcase the performance difference of the FFRC, TDRC and QRC, based on the benchmark tasks motioned before. With Q=6 and β=1.6 in (b); Q=9 and β=0.9 in (c); Q=39 and β=0.7 in (d). The injection power of the TDRC is set at 20 ns⁻¹. The detailed results of the chaotic time-series prediction are further demonstrated in (e), with k_inj=20 ns⁻¹ for FFRC; k_inj=20 ns⁻¹ and k_d=18 ns⁻¹ for TDRC; Q=6, β=1.6 and k_inj=20 ns⁻¹ for QRC. The target signal (red), prediction result (black), and error between them (blue) are shown.

Fig. 4. Two-dimensional maps of (a, d, g) NMSE, (b, e, h) SER, and (c, f, i) linear MC in the parameter space of k_inj and Δf. (a–c), (d–f) and (g–i) showcase the results of the FFRC, the TDRC and the QRC, respectively. A darker color indicates a smaller value, while the opposite means a larger value. With Q=6 and β=1.6 in (g); Q=9 and β=0.9 in (h); Q=39 and β=0.7 in (i). The feedback strength of the TDRC is set at 18 ns⁻¹, 12 ns⁻¹, 21 ns⁻¹ in (d, e, f), respectively. These results stem from the joint training of XM and YM.

Fig. 5. Analysis of the size of kernel Q and the step coefficient β. The (a) NMSE, (b) SER and (c) MC as a function of the kernel size Q, with k_inj=20 ns⁻¹ and β=0.8. The (d) NMSE, (e) SER and (f) MC as a function of step coefficient β, with k_inj=20 ns⁻¹ and Q=10. The original data and encoding data are injected in the XM and YM, respectively. Here, we use AF, which provides an additional nonlinear transformation for the output, to obtain the extended matrices [V_fx, V_fy]. The blue dashed lines illustrate the performance achieved by singularly injecting the original data into XM, and the red dashed lines represent the trained results of the optical injection terms with encoding signals.

Fig. 6. Two-dimensional maps of (a, d) NMSE, (b, e) SER, and (c, f) linear MC in the parameter space of Q and β. (a–c) and (d–f) showcase the results without and with AF, respectively. These results stem from the joint training of XM and YM. With k_inj=20 ns⁻¹ and Δf=0 GHz.

Fig. 7. The experimental performance comparison between the FFRC, the TDRC and the QRC on the (a) time-series prediction, (b) nonlinear channel equalization and (c) memory ability. With Q=6 and β=1.6 in (a); Q=9 and β=0.9 in (b); Q=39 and β=0.7 in (c). The performance of the QRC on nonlinear channel equalization is detailed in (d), while (e) depicts the three RCs’ memory details when k_inj is 1297.2 μW in the FFRC, the QRC and 1292.4 μW in the TDRC.