First, for
T1 = 30 ns and
T2 = 25 ns, the output characteristics of the OEO subject to optical feedback with different feedback strength
K are investigated. Figure 2 displays the time series (a), power spectra (b), zoom-in power spectra (c), ACF curves (d) and PE curves (e) of the chaotic output for
K = 0 (first row), 0.2 (second row), 0.5 (third row) and 0.8 (fourth row), respectively. As shown in Figs. 2(a1)−2(a4), the evolutions of the time series are complex, from which it is hard to directly observe a TDS. However, the TDS can be extracted from the power spectra, ACF and PE curves. For
K = 0 (first row), there are some equidistant peaks emerging in the power spectrum, and the frequency interval is about 33.3 MHz which is equal to a reciprocal of the optoelectronic feedback delay time
T1. Correspondingly, an obvious peak (0.62) located at
T1 can be observed in Fig. 2(d1), and meanwhile some obvious downward peaks located at
T1 and the subharmonics of
T1 can be found in Fig. 2(e1). As demonstrated in Ref. [
35], when an embedding delay matches harmonics and subharmonics of the feedback time, PE will arrive at its minima and emergences a downward peak. For an embedding dimension
D, there are
D-2 subharmonics peaks. Since we set
D = 6 and
T1 = 30 ns, there are four subharmonic peaks located at 15, 10, 7.5, and 6 ns, respectively. Under this case, TDS is only determined by the optoelectronic feedback. For
K = 0.2 (second row), the frequency intervals between two adjacent peaks in power spectrum are not the same, and a new peak originating from optical feedback appears at the optical feedback time
T2 in the ACF curve. Meantime, some extra sharp downward peaks can be observed in PE curve at
T2 and its integer fractions. Under this circumstance, TDS should be quantified by the strongest peak (0.48) in ACF curve or the deepest downward peak in PE curve. Obviously, the TDS is still determined by optoelectronic feedback, which is weaker than that for
K = 0. The reason for the weakening of TDS may be due to the complexity improvement originating from the introduction of optical feedback [
32]. For
K = 0.5 (third row), the power spectrum becomes much smoother and no significant peaks appears, then the TDS cannot be identified from the power spectrum due to the joint action of optical feedback and optoelectronic feedback. However, from Figs. 2(d3) and 2(e3), it can be seen that the amplitude of the peak (downward peak) locating at
T1 is similar with that locating at
T2 in ACF curve (PE curve). As a result, the optical feedback devotes the same degree TDS as that for the optoelectronic feedback, and the maximum of ACF is about 0.26. In particular, for
K = 0.8, the TDS resulted by optoelectronic feedback is greatly reduced, but the TDS is still very obvious (as shown in Figs. 2(d4) and 2(e4)) due to the recurrence feature induced by the strong optical feedback strength. Meanwhile, some peaks with an equal frequency interval emerge in the power spectrum, and the frequency interval is equal to a reciprocal of
T2. Under this case, the TDS is mainly originated from the optical feedback. Therefore, for an OEO subject to an extra optical feedback, the TDS of the generated chaotic signal should be inspected not only at the vicinity of
T1 but also at the vicinity of
T2.