RESEARCH ARTICLE

Time-delay signature characteristics of the chaotic output from an optoelectronic oscillator by introducing an optical feedback

  • Xixuan LIU ,
  • Xi TANG ,
  • Zhengmao WU ,
  • Guangqiong XIA
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  • School of Physical Science and Technology, Southwest University, Chongqing 400715, China

Received date: 10 Aug 2019

Accepted date: 23 Sep 2019

Published date: 15 Dec 2020

Copyright

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature

Abstract

In this work, via autocorrelation function (ACF) and permutation entropy (PE) methods, we numerically investigate the time-delay signature (TDS) characteristics of the chaotic signal output from an optoelectronic oscillator (OEO) after introducing an extra optical feedback loop. The results demonstrate that, for such a chaotic system, both the optoelectronic feedback with a delay time of T1 and the optical feedback with a delay time of T2 contribute to the TDS of generated chaos. The TDS of the chaotic signal should be evaluated within a large time window including T1 and T2 by the strongest peak in the ACF curve of the chaotic signal, and the strongest peak may locate at near T1 or T2. Through mapping the evolution of the TDS in the parameter space of the optical feedback strength and time, certain optimized parameter regions for achieving a chaotic signal with a relatively weak TDS can be determined.

Cite this article

Xixuan LIU , Xi TANG , Zhengmao WU , Guangqiong XIA . Time-delay signature characteristics of the chaotic output from an optoelectronic oscillator by introducing an optical feedback[J]. Frontiers of Optoelectronics, 2020 , 13(4) : 402 -408 . DOI: 10.1007/s12200-019-0960-z

Introduction

Chaotic dynamics of optical systems have attracted considerable attention due to their promising applications such as secure optical communication system [13], high speed random number generation (RNG) [46], optical time domain reflectometer [7,8] and chaotic radar [911], etc. Semiconductor lasers are the most popular sources to achieve optical chaos, and the generation ways of optical chaos based on semiconductor lasers can be divided into two categories: one is based on the nonlinear effect of lasers through introducing one or more external perturbations such as optical feedback [1215], optical injection [16,17] and optoelectronic feedback [18,19], which can be described by Lang-Kobayashi rate equations [20]; the other is based on the nonlinear effect of an external passive nonlinear device, where an optoelectronic oscillator (OEO) [2124] is a typical device and can be described by Ikeda’s delay differential equations [25]. For traditional OEO-based chaotic systems, the chaos can be generated by that a continuous wave (CW) light output from a laser goes through an optoelectronic feedback loop, in which a passively nonlinear optical device is contained. It is worth noting that the laser is treated as a linear light source operating at a stable state [26]. Usually, time-delay signature (TDS) inevitably exists in the chaotic output of a traditional OEO due to optoelectronic feedback loop. Generally, taking a chaotic signal with obvious TDS as a chaotic carrier, the security of the chaotic secure communication will be reduced [27]. Taking a chaotic signal with obvious TDS for high speed RNG, the randomness property of RNG will be degraded due to the recurrence features induced by the TDS [28]. As a result, some effective schemes have been proposed in succession to reduce (or suppress) the TDS of the chaotic output from OEO-based chaotic systems. For example, instead of a CW by a chaotic signal output from an optical feedback laser, the TDS in the OEO-based chaotic system is successfully hidden [29]. By adopting multiple electro-optic nonlinear loops, the TDS can also be effectively suppressed [30]. Based on a two-dimensional coupled optoelectronic delay feedback system with variable parameters, an obvious reduction of the TDS can be accomplished [31].
Recently, a simple and easily implemented scheme is proposed to reduce the TDS in an OEO-based chaotic system [32]. For this scheme, an extra optical feedback loop with a delay time of T2 is introduced into an OEO-based chaotic system. The TDS of the chaotic output is analyzed in a vicinity of the optoelectronic feedback time T1, and the results demonstrate that the TDS can be effectively suppressed. However, as is well known, an extra optical feedback loop will lead to another TDS located near T2, and the TDS should be inspected simultaneously at the vicinities of T1 and T2. As a result, in this work, within a large time window including T1 and T2, the TDS of the chaotic signal from an OEO after introducing an extra optical feedback loop is analyzed, and some optimized parameters for generating a chaotic signal with reduced TDS have been specified.

System model and theory

Fig.1 Schematic diagram of a chaotic signal generation based on an OEO after introducing an extra optical feedback. LD: laser diode; FC: fiber coupler; PC: polarization controller; MZM: Mach-Zehnder modulator; DL: delay line; PD: photodetector; RFA: radio frequency amplifier; VA: variable attenuator

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Figure 1 is a schematic diagram for a chaotic signal generation based on an OEO after introducing an extra optical feedback. A laser diode is used to generate a CW light, which is sent to a Mach-Zehnder modulator (MZM) with a radio frequency (RF) half-wave voltage of VπRF after passing through a polarization controller 1 to match the polarization of the CW light with that of the MZM. The MZM output is divided into two parts by an optical fiber coupler 2. One passes through a delay line 1 and is converted into an electrical signal by a photodetector (PD). The electrical signal is further amplified by a radio frequency amplifier (RFA), and the amplifier also plays a role of a band-pass filter with a low cut-off frequency of fL and a high cut-off frequency of fH. Finally, the amplified electrical signal V(t) depended on the conversion efficiency of PD and the magnification of RFA, is sent into the MZM and constitutes an optoelectronic feedback loop with a delay time of T1. The other part is injected into an optical feedback loop composed by a delay line 2, a variable attenuator, a polarization controller 2, a fiber coupler 1 and a polarization controller 1, and the delay time of the feedback loop is T2. The variable attenuator is employed to control the optical feedback strength K, and the polarization controller 2 is utilized to match the polarization of the optical feedback with that of MZM for maximizing the coupling efficiency.
Based on the Ikeda rate equations [25]. and taking an extra optical feedback into account, for convenient for comparing with the results obtained in Ref. [32], the dynamical equations of the chaotic system mentioned above can be described by [32]
dx(t)dt= 1 tH{ x(t)+ 1tLy(t) Pin(t T 1)×Q ×cos2 [x (t T1)+φ]},
Pin(t)=P0 +K×Pin(t T2)× cos2[ x(t T 2)+φ],
dy(t) dt=x(t),
where x(t) = pV(t)/(2VpRF), tH = 1/(2pfH), tL = 1/(2pfL), and Q = pgAG/(2VpRF) (A represents the total attenuation of the optoelectronic feedback loop, g and G are the conversion efficiency of PD and the gain of RFA, respectively). P0 is the laser output power, and Pin(t) is the coupled input power of MZM. P0Q represents the strength of the nonlinear function, which plays an important role in the chaotic behavior with high complexity.
Some methods such as autocorrelation function (ACF), delay mutual information method [33], permutation entropy (PE) [34,35], sample entropy [36] and Kolmogorov-Sinai entropy [37] can be used to qualitatively evaluate the TDS, where each approach possesses its unique virtues and respective limitations. In this work, we adopt both ACF and PE methods. For a delay-differential system, ACF is defined as follow:
ACF(Δt)= [x(t) x(t)] [x(t+Δ t) x(t)] (x(t) x(t) )2 (x(t+Δ t) x(t))2,
where Dt indicates the time shift,<·>denotes the time average, and x(t) stands for chaotic time series. We set Dt ∈ [0 ns, 40 ns] and the time series length is taken as 10 ms during calculating ACF.
The PE method, which is based on information theory, has some unique advantages such as simplicity, extremely fast calculation, robustness to noise and easily estimation for any type of time series [34,35,37], and can be defined as follows: an arbitrary time series X = {x(t), t = 1,2,…, T} is first reconstructed into a set of D-dimensional vectors after choosing an appropriate embedding dimension D and embedding delay time t. Then we study all D! permutation p of order D. For each p, the relative frequency is determined as
p(π) = {t| tTD,Xhas typeπ}TD+1
where # notes number. Hence, PE is defined as
h(P )= Σ p(π )logp( π),
where P = {p(p)} stands for probability distribution. The normalized PE can be further expressed as
H( P)=h(P)hmax =Σp(π) logp(π)log(D! ) ,
where the value of H(P) ranges from 0 to 1. H(P) = 0 means that the time series is predictable and regular while H(P) = 1 corresponds to a completely stochastic process. After considering the suggestion in Ref. [34], D is fixed at 6 and the time series length is set as 10 ms for the calculation of PE.

Results and discussion

Fig.2 (a) Time series, (b) power spectra, (c) zoom-in power spectra, (d) ACF curves and (e) PE curves of the chaotic signal from an OEO after introducing an extra optical feedback with T1 = 30 ns and T2 = 25 ns, where the first, second, third and fourth rows represent an optical feedback strength K of 0, 0.2, 0.5 and 0.8, respectively

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Equations (1)–(3) can be numerically solved by adopting fourth-order Runge-Kutta algorithm with a step of 5 ps via MATLAB software. During the calculations, the parameters are chosen as [32]: tH= 25 ps, tL= 5 μs, φ= −π/4. Unless otherwise specified, Q and P0 take 1000 W-1 and 5 mW, respectively, i.e., P0Q= 5, which can be attained in practical systems [38,39].
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.
Fig.3 (a) Calculated map of ACF curves for T1 = 30 ns and T2 = 25 ns with K varied from 0 to 1 and the expanded versions of the ACF curve at the vicinities of (b) T2 and (c) T1, respectively

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Since PE curves exhibit similar properties with ACF curves, we adopt the ACF method in the following discussion. Figure 3(a) shows the maps of ACF curves for T1 = 30 ns and T2 = 25 ns with K varied from 0 to 1, and Figs. 3(b) and 3(c) is the correspondingly expanded version at the vicinity of T2 and T1, respectively. Obviously, with the increase of K, the TDS at T2 gradually enhances while the TDS at T1 gradually weakens. In other words, the dominant role for contributing TDS undergoes a switch from optoelectronic feedback to optical feedback during increasing K. Under the other parameters given above, there exists an optimal K (named as Koptimal), under which the ACF peak within a large time window including T1 and T2 is the weakest.
Fig.4 (a) Dependence of s on the optical feedback strength K under P0 = 5 mW; (b) dependence of Koptimal on the laser power P0 and corresponding smin under fixed P0 and Koptimal

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Next, to find the Koptimal value, we calculate the maximum value (s) of ACF within a large time window including T1 and T2 under different K, which is shown in Fig. 4(a). Under P0 = 5 mW, with the increase of K, s first decreases, after passing through a minimum of 0.26, and then increases. Koptimal is about 0.55 and the minimum of s (named as smin) is 0.26 for P0 = 5 mW. For P0 taking different values between 4 to 10 mW (corresponding to different P0Q values between 4 to 10), the calculated results are presented in Fig. 4(b). It can be observed that both Koptimal and smin show monotonously downward trends with the increase of P0, and smin arrives at about 0.04 for P0 = 10 mW. As a result, a relatively large P0 is helpful for effectively reducing TDS.
Fig.5 Calculated map of s in the parameter space of T2 and K under T1 = 30 ns

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Since the above results are obtained under a fixed T2 = 25 ns, we further analyze the evolution of the TDS under different T2. Figure 5 displays a calculated map of s in the parameter space of T2 and K under T1 = 30 ns. From this diagram, it can be seen that the TDS cannot be efficiently reduced for too small or too large K. As pointed out in Ref. [13], for a chaotic system with two feedback loops, the TDS of the chaotic output is formed due to a joint action of the two feedback loops, and meanwhile the intrinsic characteristic time of the oscillator takes an important role for the TDS reduction/suppression. For a too small (or too large) K, the TDS is mainly depended on the optoelectronic feedback (or optical feedback) loop, and the joint action of the optical feedback and optoelectronic feedback is less effective. On the contrary, as shown in this diagram, for K ∈ [0.4, 0.7], the TDS can be effectively reduced except that T2 is located at the vicinity of 30 ns or 15 ns.
The result in Fig. 5 shows that the value of T2 seriously affects the TDS of the chaotic output. Finally, we calculate s in the parameter space of T1 and T2 under K = 0.55 in Fig. 6. Obviously, for the relationship of T2 and T1 is close to an integral multiple, the TDS of the chaotic signal is enhanced. Especially, under the case of T2 = T1, the TDS distinctly strengthens, which is easy to understand as follows. Under this case, the location of TDS originating from optical feedback is the same as that from optoelectronic feedback, and then the introduction of an extra optical feedback will further enhance the chaotic TDS. Therefore, the optical feedback time T2 should be deviated from T1 and its integer fractions in practice.
Fig.6 Calculated map of s in the parameter space of T1 and T2 under K = 0.55

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Conclusions

In summary, the TDS of the chaotic signal output from an OEO subject to an extra optical feedback is numerically investigated via ACF and PE methods. The simulated results show that, with the increase of the optical feedback strength, the TDS at T1 originating from the optoelectronic feedback is weakened, but an extra TDS emerging at the vicinity of the optical feedback time T2 is strengthened. Therefore, the TDS of this chaotic system should be evaluated by the joint action of both the optoelectronic and optical feedback loops. Through adopting the strongest peak value (s) of ACF curve within a large time window including T1 and T2 to characterize the TDS of the chaotic signal, the TDS evolution in the parameter space of K and T2 has been calculated. The results show that, under a laser power of P0 = 5 mW, the TDS can be effectively reduced for an optical feedback strength of K ∈ [0.4, 0.7] and T2 deviated from T1 and its integer fractions. Additionally, a larger P0 is helpful for achieving a chaotic signal with lower TDS.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61575163, 61775184, 11704316, and 61875167).
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