Fig. 1. (a) Chaotic system of a multi-delay mutual coupling VCSEL ring network structure for chaotic signal generation: PC1, PC2, and PC3, polarization controllers; EDFA, erbium-doped optical fiber amplifier; VOA, variable optical attenuator; OC1 and OC2, optical circulators; FC, fiber coupler; LAC, higher-stability and low-noise diode controller; VCSEL1, VCSEL2, and VCSEL3, vertical-cavity surface-emitting lasers. (b) Conceptual diagram of the RC. (c) Photonic RC system based on VCSEL for chaos time series prediction: PC4, PC5, and PC6, polarization controllers; EDFA, erbium-doped optical fiber amplifier; VOA, variable optical attenuator; OC3, optical circulator; FC2, FC3, and FC4, fiber couplers; PD1 and PD2, photodetectors; OSC, oscilloscope; DL, delay line; TL, tunable laser; RF, RF amplifier; MZM, Mach–Zehnder modulator; VCSEL, vertical-cavity surface-emitting laser.

Fig. 2. Chaotic time series obtained in experiment. (a1)–(d1) Time-domain signal. (a2)–(d2) Power spectrum. (a3)–(d3) PE as a function of lag time. (a4)–(d4) ACF as a function of lag time.

Fig. 3. Time-domain chaotic signal (first column), power spectrum (second column), permutation entropy (third column), and autocorrelation curve function (fourth column) obtained by simulation.

Fig. 4. NMSE of the system as a function of the injection strength for (a) different MPEs and (b) approximate MPE.

Fig. 5. NMSE of the system as a function of the (a) injection strength kinj and (b) feedback strength kd. The black (red, blue, or green) line represents the result of 1- (2, 3, or 4) step-ahead prediction.

Fig. 6. (a) Optical spectrum of the free running VCSEL. (b) Optical spectrum of the VCSEL with the external optical injection. (c) Power-current curve of the VCSEL when the temperature is stabilized at 25°C.

Fig. 7. NMSE as a function of attenuation coefficient.

Fig. 8. Time domain prediction results for (a) 1-step-ahead task, (b) 2-steps-ahead task, (c) 3-steps-ahead task, and (d) 4-steps-ahead task. The red (blue) line is for the target (prediction) sequences.

Table 1. Parameter Values for Three VCSELs in the Numerical Simulation

Table 2. Parameter Values for the Photonic RC System in Our Numerical Simulation^a

Table 3. Impact of Dataset Size on the System