Open Access
Add to BibSonomy
Abstract
A novel time-delay signature (TDS) concealing electro-optic (EO) chaotic system with multiply feedback nonlinear loops is proposed and analyzed by numerical simulation. The proposed system employs mutual injection structure implemented by two asymmetric branches named as multiply feedback nonlinear loop which introduces an extra nonlinear factor to the system dynamic equation. The complexity of the chaos system is increased by introducing this multiply feedback nonlinear loop. The permutation entropy (PE) of the proposed system is improved to higher than 0.96 when feedback strength (β) equals 5. The proposed system can enter to chaos regime with a small β (β = 0.8). The TDS is concealed effectively due to the extra nonlinear factor introduced by multiply feedback nonlinear loop. Meanwhile, key-space of the proposed system is about 1012 times that of the classical EO system because more tunable time delay parameters are introduced. Furthermore, the performance of a secure communication system based on the proposed chaotic system is discussed, and the simulation results show that the system is sensitive to time delay parameters and robust to feedback strength, which proves the proposed system is suitable for secure communication.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recent years, owing to its special advantages of typically broadband, noise-like, and difficult to predict, optical chaotic source has been highlighted in the fields of optical secure communication [1,2]. It has been proved that the chaotic signal generated by the laser chaotic source can be transmitted directly in the existing optical communication system without any additional processing [3]. The optical chaotic source based on delay differential equation (DDE) attracted much attention in the past decades [4–6]. There are two main types of DDE. One is the Lang-Kobayashi rate equation mainly used to model the nonlinear behavior of semiconductor lasers subjected to external perturbation [6–10]. However, this kind of system demands a very high parameter matching degree, since the required synchronization is very sensitive to the physical parameters of the lasers between the transmitter and receiver [11]. Moreover, bandwidth enhancement is needed in this kind of system for realizing high speed processing. It is difficult to attain synchronization of bandwidth enhanced chaos [12]. The other type of DDE is the Ikeda equation, which is mainly used to model the dynamics of electro-optic (EO) delayed feedback system. Attribute to its high robustness and low cost, the system based on Ikeda equation has drawn wide attention [11,13,14].
Compared with low dimensional chaos like chua chaos, optical chaotic systems based on DDE can greatly increase the dimension and complexity of chaos [15]. But chaotic signals generated by the DDE system also introduce the time delay signature (TDS) caused by the periodic components. From the perspective of security communication, the feedback time delay parameter is a crucial secure key of the DDE system. It has been demonstrated that TDS can be extracted through many statistical analysis methods [16,17]. Once the eavesdroppers determines the value of delay parameter, they can easily reconstruct the chaotic system [18]. In this case, the communication system is not secure. Meanwhile, the key-space of traditional chaotic systems is relatively lower. Eavesdroppers can crack the system by brute force. Therefore, how to conceal TDS of the DDE system has become a severe question that needs to be settled urgently and the key-space of chaotic system needs to be extended. Up to now, a large amount of research has been reported on suppressing TDS [19–32]. Various types of schemes have been proposed, such as adjusting the parameters [19–23], mutual injection of dual feedback loop [24,25]. However, these methods include pseudo-random binary sequence (PRBS) generators [24,25] or digital signal processing (DSP) modules [19,21]. The clocks between the transmitter and receiver need to be synchronized and it is hard to realize. The system proposed in [26] introduces the optical microcavities. Due to the dispersion effect in the optical microcavity, different frequency components of the chaotic laser experience different delay time, which conceals the time correlation of original signal. However, the performance of the optical microcavity is easily affected by the external environment and it may be difficult to attain a good synchronization effect between the transmitter and receiver. The system proposed in [27] introduces an external noise source. And the experiment results show that the system can secretly transmit a 10 Gb/s on-off keying signal over 100km standard single-mode fiber. A chaos secure communication system based on VCSELs is proposed in [28]. Due to the variable-polarization optical feedback module, a large number of new frequencies are generated into the feedback loop, and the TDS is completely hidden when the feedback-gain is 3. In addition, [29] proposes that the time correlation of original chaotic signals can be reduced or even eliminated by nonlinear and non-reversible transformation. The system proposed in [30] employs mutual injection structure with a nonlinear function, and the TDS can be concealed due to the additional nonlinear effect introduced by the nonlinear function. But, to realize the system may need more than 4 laser diodes and a fairly high bias voltage (about 30V) to the modulator. In [31], an electrically coupled chaos system with three nonlinear feedback loops is demonstrated. Because of the nonlinear coupling of multiplex signals, the TDS of single output signal can be concealed. However the system has two output signals and the TDS can still be extracted by statistical analysis between the multiple outputs [32]. The system proposed in [32] converts three chaotic phase-modulated signals into three chaotic intensity-modulated signals by a 3 $\times$ 3 optical coupler, and the TDS can be suppressed while three chains have different center wavelength. The required feedback gain is very large (i.e. $\beta >6$) and that is difficult to achieve practically.
In this paper, we propose a TDS concealing electro-optic chaotic system with multiply feedback nonlinear loops (MFNL). The proposed system can conceal the TDS effectively due to the extra nonlinear factor introduced by multiply feedback nonlinear loops. A larger key-space is achieved by introducing more tunable parameters which are nonlinearly coupled to each other. The chaotic signal generated from this system is highly complex, the probability distribution of this signal is closer to Gaussian distribution. The proposed system can conceal TDS with a smaller $\beta$ than the system in [32]. Furthermore, the performance of the secure communication system based on the proposed chaos source is also discussed. The communication system is very sensitive to delay parameters and robust to loop gain parameter which is susceptible to environmental change. In this way, the delay parameter of this system is suitable as the secret key and the system is reliable in the environment.
2. System construction and mathematical model
The traditional electro-optical chaotic system (classical EO) is illustrated in [11]. The Mach-Zehnder modulator (MZM) is seeded by a continuous-wave laser diode (LD) whose radio-frequency half-wave voltage is $V_\pi$. The modulated optical signal is detected by a photodiode (PD) after it goes through an optical fiber delay line. The generated electrical signal amplified by a radio-frequency amplifier (RF) drives the MZM. The dynamical model of the classical EO can be described by (1)
The left-hand of (1) represents a process of linear band-pass filter and the right-hand presents a non-linear, non-invertible transformation of delayed signal. From (1), it is clear to see that the numerical value of $x(t)$ is determined by the value of $x(t-T)$. In this case, some static analysis methods like autocorrelation function (ACF), and delayed mutual information (DMI) are usually adopted by eavesdroppers to extract TDS of the DDE system. The ACF and DMI are two crucial methods to evaluate the security of the DDE system. However, the classical EO system is demonstrated to be insecure, whose TDS can be identified by analysis of ACF and DMI [21]. The ACF and DMI are defined as follows [16,17]:
We can improve the complexity of system by increasing the gain coefficient of RF to reduce the temporal correlation [29]. The chaotic signal $x(t)$ is a non-periodic oscillating sequence with repeating intervals of finite growing behavior, and the external function $\cos (v)$ $(v=Gx(t))$ is a non-invertible function with many extrema.
With $G$ increasing, the $\cos (v)$ spans more lobes of the nonlinear transmission. Meanwhile, the chaos signal generated by the system become more complex. But, it may need a high $\beta$ to conceal the temporal correlation due to the directly proportional relationship between $\beta$ and $G$, and the value of $\beta$ may exceed the limit 5.1 [32]. In this case, we further analyze the situation where one chaotic signal is multiplied by the other chaotic signals. The schematic diagram is shown in Fig. 1. The $\beta =\pi Pg\eta G_1A/(2V_\pi )$. The gain coefficient of RF$_1$ and RF$_2$ are $G_1$ and $G_2$, respectively. The $\cos (x)$ represents the cosine transformation. The dynamical equation is described by (4):
Fig. 1. Conceptional diagram of nonlinear transformation with cosine. LD: laser diode, MZM:Mach-Zehnder modulator, PD:photodiode, T:optical fiber delay line.
The ACF and DMI curves of $x(t)$ while $G_2=2,5,10$ are shown in Fig. 2(a) and Fig. 2(b). We set the $T=15$ ns and the $\beta =5$. It can be found that with $G_2$ increasing, the TDS on both the ACF curve and DMI curve is suppressed. When $G_2=10$, the TDS on the ACF curve can be concealed, but the TDS on the DMI curve can still be distinguished. In this way, a fairly large $G_2$ is needed to conceal TDS on both the ACF curve and DMI curve. In order to conceal the TDS more effectively, a novel system with a mutual injection structure (MFNL) is proposed, which is illustrated in Fig. 3. The whole system can be divided into two branches. One branch is named MZM branch and the optical signal in this branch is modulated by a MZM. The other branch is named PM branch and the optical signal in this branch is modulated by a phase modulator (PM). In MZM branch, the output signal of MZM is divided into two paths by a 50:50 coupler after amplification and delay. The signal in one path is transformed to a voltage signal by a PD and then subtracts a DC signal. The signal in the other path is also transformed to a voltage signal by a PD. Then the two paths of signal are combined by a multiplier, and the generated signal drives the PM after amplified by a RF. The process of PM branch is slightly different from MZM branch that the Mach-Zehnder interferometer (MZI) is employed to convert the phase fluctuation into intensity variation.
Fig. 3. Schematic of the proposed system. PM: phase modulator, E:Erbium-doped Optical Fiber Amplifier, MZI: Mach-Zehnder interferometer, DC: direct-current signal.
The power of LD is $2P$. The efficiency and gain of PD are considered which is $\eta$ and $g$, respectively. The gains of E$_1$ and E$_2$ are both $E$ and the gains of RF$_1$ and RF$_2$ are both $G$. The DC voltage is $\eta gEP/2$ ideally. In this situation, the signal after subtracter satisfies $(\eta gEP/2)\cdot [2\cos ^2 (\cdot )-\rm {VD}]$, which is the double angle formula when $\textrm {VD}=1$ ideally, and due to this process the nonlinear transformation function spans more lobes without increasing $G$. The bandwidth of the whole loop is determined by RF$_1$ and RF$_2$ whose low pass cut-off time are both $\theta$ and high pass cut-off time are both $\tau$. $\delta T_1$ and $\delta T_2$ are the differential delay of MZI$_1$ and MZI$_2$, respectively. $\phi _1=\pi f\delta T_1$ and $\phi _2=\pi f\delta T_2$ are the static offset phase of MZI$_1$ and MZI$_2$, respectively. The gain efficient $\beta _i =P^2 g^2 \eta ^2 E^2 GA\pi /(4V_\pi )$, ($i$=1, 2). The dynamic equation of the proposed system can be written as (5a) - (5f):
3. Numerical results and analyses
In this section, the numerical simulated results are presented. The corresponding parameters are $\tau =25$ ps, $\theta =5\mu$ s, $\delta T_1=600$ ps, $\delta T_2=400$ ps, $\phi _1=3\pi /8$, $\phi _2=\pi /4$, $T_1= 15$ ns, $T_2=25$ ns, $T_3=12$ ns, $T_4=12$ ns, $\beta _1=\beta _2=5$, $\textrm {VD}=1$.
3.1 Dynamical complexity analysis
The bifurcation diagram can be used to judge whether the system is in a chaotic state or not and reflect the dynamic behavior of the system. The bifurcation diagram of proposed system is shown in Fig. 4, which is similar to the classical EO system [11]. However, the proposed system could enter into the chaos regime with a lower feedback gain (i.e. $\beta _1> 0.8$) than the classical EO system. The qualitative explanation is that the steady state has a greater probability of losing stability due to multiple nonlinear transformations, thus causing a greater nonlinear disturbance than the classical EO system with the same feedback gain.
A system with higher complexity can defense parameter estimation attacks more effective. It is necessary to discuss the dynamical complexity of the proposed system. We use permutation entropy (PE) to evaluate the complexity of system. As suggest in [31], we select the ordinal pattern length $L=6$ and the embedding delay $D=2$. The length of time series used to calculate PE is $10^{5}$, which is large enough to allow for a reasonable distribution of possible ordinal patterns ($L!$) [31]. To better reflect the performance of the proposed system, the mutual injection system (MIS) is introduced as a reference substance which is the system without nonlinear transformation function in [30]. The PE curves of $x_1(t)$ as a function of $\beta _1$ of MIS and proposed system when $\beta _2=5$ are shown in Fig. 5. It is observed that the output signal $x_1(t)$ becomes more complex as $\beta _1$ increases. The proposed system can attain a highly chaotic state ($\textrm {PE}>0.9$) with lower $\beta _1$ (i.e. $\beta _1>1.1$).
Fig. 5. PE$(x_1)$ as a function of $\beta _1$ of the proposed system and MIS for the ordinal pattern length $L=6$ and the embedding delay $D=2$.
The randomness of the signal can be reflected by the probability distribution density graph. The probability distribution of MIS and the proposed system with $\beta _i=5$ ($i=1,2$) are shown in Fig. 6(a) and Fig. 6(b). It is observed that the proposed system is closer to Gaussian distribution, which means the output signal of proposed system has better randomness.
3.2 Security analysis
The security of the system mainly depends on the confidentiality of TDS. The degree of confidentiality of TDS is evaluated through statistical analysis including ACF and DMI. The ACF curve and DMI curve of MIS and the proposed system when $\beta _i=5$ ($i=1,2$) are shown in Fig. 7(a) and Fig. 7(b). It is clear to see, the MIS can conceal the TDS at $T_1$ and $T_2$ respectively, but the TDS at $T_1$+$T_2$ is exposed. This is due to the temporal correlation of signals in such mutual injection system depending on the sum of the time-delays accumulated in the different branches during the propagation. Instead, due to introducing the multiply feedback loops which enhances nonlinear degree of the system, the TDS of the proposed system can be concealed both on the ACF curve and DMI curve. To further analyze the performance in TDS concealing of the system, the ACF and DMI curve of $x_1(t)$ while $\beta _i$ = 2, 2.7, 3.3 and 3.8 are analyzed in Fig. 8(a) and (b) respectively. To make the result more precise, the $T_4$ is set to 7ns, whose value is different from $T_3$. As is shown in Fig. 8(a), when $\beta _i=2$ the peaks of TDS at $\delta T_2$, $T_4$, $T_1+T_2$, and $T_1+T_2+T_4$ are both exposed on the ACF curve. When $\beta _i$ grows to 2.7, the absolute value of peaks at $\delta T_2$ and $T_4$ get smaller but the peaks still can be distinguished. The peaks at $T_1+T_2$ and $T_1+T_2+T_4$ are concealed. When $\beta _i=3.3$, the peaks at $T_4$ are further suppressed and the other peaks are concealed. When $\beta _i$ attains 3.8, all the peaks are concealed. It can be see in Fig. 8(b), when $\beta _i=2$ the peaks of TDS at $T_3$, $T_4$, $T_1+T_2$, $T_1+T_2+\delta T_1$, $T_1+T_2+T_3$, $T_1+T_2+T_3+\delta T_2$, $T_1+T_2+T_4$, $T_1+T_2+T_4+\delta T_1$, $T_1+T_2+T_3+T_4$ and $T_1+T_2+T_3+T_4+\delta T_2$ are both exposed on the DMI curve. When $\beta _i$ grows to 2.7, all the peaks get smaller, but still can be distinguished. When $\beta _i$ = 3.3, the peaks at $T_1+T_2+T_3$ and $T_1+T_2+T_3+\delta T_2$ still can be distinguished and other peaks are concealed. When $\beta _i$= 3.8, all the peaks on the DMI curve are concealed. Thus, the security of the proposed system can be guaranteed when $\beta _i>=3.8$.
Fig. 7. ACF (a) and DMI (b) curves of chaos generated by the proposed system and MIS with $\beta _i=5$.
Fig. 8. ACF (a) and DMI (b) curves of chaos generated by the proposed system with $\beta _i=2$, 2.7, 3.3 and 3.8.
However, considering the actual situation, the output of DC power may not be accurate. And the value of VD is probably not equal to 1. So it is necessary to analyze the TDS suppressing effect in non-ideal case. The ACF and DMI curves of proposed system are shown in Fig. 9(a) and Fig. 9(b), while VD changes from 0 to 2 with 0.2 step size, equaling to error ratio varied from $-100\%$ to 100$\%$. As is shown in Fig. 9, the detuning VD hardly influences the security of proposed system. When the value of VD is between 0 and 2, the TDS can hardly be extracted from the ACF curve. And only when $\left |\rm {VD}-1\right |$ $\ge 1$ the peak at $T_1$+$T_2$ can be distinguished on the DMI curve.
Fig. 9. ACF(a) and DMI(b) curves of chaos generated by the proposed system with VD changed from 0 to 2 with 0.2 step size.
4. Chaotic synchronization communication
Due to the advantage of TDS concealing, the proposed chaotic system can be applied to secure communication. The frame of the chaotic synchronization communication system is depicted in Fig. 10. The system of transmitter side is the same as the proposed system. At the receiver side, an open loop system composed like transmitter is utilized for decryption. The dynamic equations of Fig. 10 can be given by (6a) - (6h) and (7a) - (7h) (the variables of dynamic equation of receiver side is distinguished by adding $'$):
Transmitter side:
Receiver side:
It is clear to see, when the proposed system is used for communication, the added message will disturb the generated chaotic signal which may influence the chaotic dynamic of the system. In fact, the modulation depth of message is lower than 5$\%$ in the actual chaotic communication process. So according to the previous analysis, it can be determined that the added message will not threaten the security of the proposed system.
Chaotic secure communication is based on synchronization. If the physical parameters of transmitter side and receiver side mismatch, the synchronization quality will be influenced. So it is crucial to quantitatively analyze the effect of parameter mismatch on synchronization performance. In this case, the cross-correlation (CC) function is introduced. The CC is defined as follow:
Fig. 11. The correlation coefficient of $x(t)$ and $x^{'} (t)$ when parameters are mismatched between the transmitter and the receiver. (a) the mismatch of time delay $\Delta T_1$, $\Delta T_2$, $\Delta T_3$ and $\Delta T_4$. (b) the detuning of feedback gain $\beta _i^{'}$ when $\beta _i=5$.(c) the detuning of $\theta ^{'}$ when $\theta =5\mu$ s.(d) the detuning of $\tau ^{'}$ when $\tau =25$ ps. (e) the detuning of $\textrm {VD}^{'}$ when VD = 1.
5. Conclusion
In summary, we propose a TDS concealing electro-optic chaotic system with multiply feedback nonlinear loops. The simulated results show that the permutation entropy of the output signal of proposed chaos source can attain 0.96 when $\beta _i=5$ $(i=1,2)$. Furthermore, the TDS can be completely concealed, which ensures the security of the proposed system. The key-space of the proposed system is about $10^{12}$ times larger than the classical EO system due to more tunable parameters are nonlinearly coupled. All the analysis on security and complexity of the proposed system above was done when $\beta _i\leq 5$ $(i=1,2)$, which is attainable to implement. Moreover, the synchronization system and the chaotic communication scheme are discussed. The time-delay parameters are sensitive for synchronization, so the time-delay parameters can serve as a secret key. In addition, due to the excellent Gaussian like distribution characteristics of the proposed system’s output signal, the chaos source can also be applied in many other areas, such as random number generation.
Funding
National Natural Science Foundation of China (61831003, 62021005).
Disclosures
The authors declare no conflicts of interest.
References
1. X. Jiang, M. Cheng, C. Luo, F. Luo, L. Deng, S. Fu, C. Ke, M. Zhang, M. Tang, P. Shum, and D. Liu, “Reproducible optical noise-like signal generation subjected by digital sequences,” Opt. Express 25(23), 29189–29198 (2017). [CrossRef]
2. N. Jiang, A. Zhao, C. Xue, J. Tang, and K. Qiu, “Physical secure optical communication based on private chaotic spectral phase encryption/decryption,” Opt. Lett. 44(7), 1536–1539 (2019). [CrossRef]
3. Y. Qu, S. Xiang, Y. Wang, L. Lin, A. J. Wen, and Y. Hao, “Concealment of time delay signature of chaotic semiconductor nanolasers with double chaotic optical injections,” IEEE J. Quantum Electron. 55(2), 1–7 (2019). [CrossRef]
4. P. Colet and R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19(24), 2056–2058 (1994). [CrossRef]
5. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef]
6. V. Annovazzi-Lodi, G. Aromataris, and M. Benedetti, “Multi-user private transmission with chaotic lasers,” IEEE J. Quantum Electron. 48(8), 1095–1101 (2012). [CrossRef]
7. C. R. Mirasso, P. Colet, and P. García-Fernández, “Synchronization of chaotic semiconductor lasers: Application to encoded communications,” IEEE Photonics Technol. Lett. 8(2), 299–301 (1996). [CrossRef]
8. H. D. Abarbanel, M. B. Kennel, L. Illing, S. Tang, H. Chen, and J. Liu, “Synchronization and communication using semiconductor lasers with optoelectronic feedback,” IEEE J. Quantum Electron. 37(10), 1301–1311 (2001). [CrossRef]
9. N. Li, W. Pan, L. Yan, B. Luo, and X. Zou, “Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers,” Commun. Nonlinear Sci. Numer. Simul. 19(6), 1874–1883 (2014). [CrossRef]
10. T. Deng, G. Xia, and Z. Wu, “Broadband chaos synchronization and communication based on mutually coupled vcsels subject to a bandwidth-enhanced chaotic signal injection,” Nonlinear Dyn. 76(1), 399–407 (2014). [CrossRef]
11. R. Lavrov, M. Peil, M. Jacquot, L. Larger, V. Udaltsov, and J. Dudley, “Electro-optic delay oscillator with nonlocal nonlinearity: Optical phase dynamics, chaos, and synchronization,” Phys. Rev. E 80(2), 026207 (2009). [CrossRef]
12. H. Someya, I. Oowada, H. Okumura, T. Kida, and A. Uchida, “Synchronization of bandwidth-enhanced chaos in semiconductor lasers with optical feedback and injection,” Opt. Express 17(22), 19536–19543 (2009). [CrossRef]
13. J.-P. Goedgebuer, L. Larger, H. Porte, and F. Delorme, “Chaos in wavelength with a feedback tunable laser diode,” Phys. Rev. E 57(3), 2795–2798 (1998). [CrossRef]
14. L. Larger, J.-P. Goedgebuer, and V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C. R. Phys. 5(6), 669–681 (2004). [CrossRef]
15. M. Le Berre, E. Ressayre, A. Tallet, H. Gibbs, D. Kaplan, and M. Rose, “Conjecture on the dimensions of chaotic attractors of delayed-feedback dynamical systems,” Phys. Rev. A 35(9), 4020–4022 (1987). [CrossRef]
16. V. S. Udaltsov, J.-P. Goedgebuer, L. Larger, J.-B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003). [CrossRef]
17. V. Udaltsov, L. Larger, J. Goedgebuer, A. Locquet, and D. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay-differential equations,” J. Opt. Technol. 72(5), 373–377 (2005). [CrossRef]
18. M. Prokhorov, V. Ponomarenko, A. Karavaev, and B. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Phys. D 203(3-4), 209–223 (2005). [CrossRef]
19. L. Liu, S. Miao, M. Cheng, and X. Gao, “A new switching parameter varying optoelectronic delayed feedback model with computer simulation,” Sci. Rep. 6(1), 22295 (2016). [CrossRef]
20. D. Rontani, M. Sciamanna, A. Locquet, and D. Citrin, “Multiplexed encryption using chaotic systems with multiple stochastic-delayed feedbacks,” Phys. Rev. E 80(6), 066209 (2009). [CrossRef]
21. X. Gao, F. Xie, and H. Hu, “Enhancing the security of electro-optic delayed chaotic system with intermittent time-delay modulation and digital chaos,” Opt. Commun. 352, 77–83 (2015). [CrossRef]
22. H. Hu, W. Su, L. Liu, and Z. Yu, “Electro-optic intensity chaotic system with varying parameters,” Phys. Lett. A 378(3), 184–190 (2014). [CrossRef]
23. H. Hu, S. Shi, and F. Xie, “Electro-optic intensity chaotic system with an extra optical feedback,” Opt. Commun. 402, 140–146 (2017). [CrossRef]
24. M. Cheng, C. Luo, X. Jiang, L. Deng, M. Zhang, C. Ke, S. Fu, M. Tang, P. Shum, and D. Liu, “An electrooptic chaotic system based on a hybrid feedback loop,” J. Lightwave Technol. 36(19), 4259–4266 (2018). [CrossRef]
25. Z. Zhao, M. Cheng, C. Luo, L. Deng, M. Zhang, S. Fu, M. Tang, P. Shum, and D. Liu, “Semiconductor-laser-based hybrid chaos source and its application in secure key distribution,” Opt. Lett. 44(10), 2605–2608 (2019). [CrossRef]
26. Q. Li, D. Chen, Q. Bao, R. Zeng, and M. Hu, “Numerical investigations of synchronization and communication based on an electro-optic phase chaos system with concealment of time delay,” Appl. Opt. 58(7), 1715–1722 (2019). [CrossRef]
27. Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019). [CrossRef]
28. H. Wang, T. Lu, and Y. Ji, “Key space enhancement of a chaos secure communication based on vcsels with a common phase-modulated electro-optic feedback,” Opt. Express 28(16), 23961–23977 (2020). [CrossRef]
29. J. J. Suárez-Vargas, B. A. Márquez, and J. A. González, “Highly complex optical signal generation using electro-optical systems with non-linear, non-invertible transmission functions,” Appl. Phys. Lett. 101(7), 071115 (2012). [CrossRef]
30. Y. Huang, H. Hu, F. Xie, and J. Zheng, “An innovative electro-optical chaotic system using electrical mutual injection with nonlinear transmission function,” IEEE Photonics J. 10(1), 1–12 (2018). [CrossRef]
31. X. Gao, M. Cheng, L. Deng, L. Liu, H. Hu, and D. Liu, “A novel chaotic system with suppressed time-delay signature based on multiple electro-optic nonlinear loops,” Nonlinear Dyn. 82(1-2), 611–617 (2015). [CrossRef]
32. X. Zhu, M. Cheng, L. Deng, X. Jiang, C. Ke, M. Zhang, S. Fu, M. Tang, P. Shum, and D. Liu, “An optically coupled electro-optic chaos system with suppressed time-delay signature,” IEEE Photonics J. 9(3), 1–9 (2017). [CrossRef]
33. L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Philos. Trans. R. Soc., A 371(1999), 20120464 (2013). [CrossRef]
