Time-delay signatures hidden in the phase of chaotic output of mutually delay-coupled vertical-cavity surface-emitting lasers

Time-delay signature (TDS) extracted from the phase time series (named as P-TDS) generated by a mutually delay-coupled vertical-cavity surface-emitting lasers (MDC-VCSELs) chaotic system is investigated systematically. By adopting self-correlation function (SF) and permutation entropy (PE) methods, the influences of coupling strength and frequency detuning on P-TDS are analyzed, and also compared with the case that TDS is extracted from the intensity of field (named as I-TDS). Through mapping the evolutions of P-TDS and I-TDS in the parameter space of coupling strength and frequency detuning, an optimal range can be determined for simultaneously generating four-channel chaotic signals with both weak P-TDS and I-TDS.