Abstract

In this paper, a novel chaotic secure communication system based on vertical-cavity surface-emitting lasers (VCSEL) with a common phase-modulated electro-optic (CPMEO) feedback is proposed. The security of the CPMEO system is guaranteed by suppressing the time-delay signature (TDS) with a low-gain electro-optic (EO) feedback loop. Furthermore, the key space is enhanced through a unique secondary encryption method. The first-level encrypted keys are the TDS in the EO feedback loop, and the second-level keys are the physical parameters of the VCSEL under variable-polarization optical feedback. Numerical results show that, compared to the dual-optical feedback system, the TDS of the CPMEO system is suppressed 8 times to less than 0.05 such that they can be completely concealed when the EO gain is 3, and the bandwidth is doubled to over 22 GHz. The error-free 10 Gb/s secure optical transmission can be realized when the time-delay mismatch is controlled within 3 ps. It is shown that the proposed scheme can significantly improve the system performance in TDS concealment, as well as bandwidth and key space enhancement, which has great potential applications in secure dual-channel chaos communication.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Chaos synchronization was first verified by Pecora and Caroll in the circuit structure system and has been widely used in chaotic secure communication, signal processing, and life science research [13]. Compared with the circuit system, the laser chaotic system has many irreplaceable advantages, such as low attenuation, wide bandwidth, and complex dynamic behavior, which can be used to improve the security of current optical communication networks [4,5]. The generation methods of laser chaos in chaotic communication systems can be divided into two categories: all-optical (AO) and electro-optic (EO) sources. The AO source takes advantage of nonlinear characteristics inside the laser diode to generate chaotic signals, for example, optical injection (OI), optical feedback (OF), and photoelectric feedback [6]. The EO source uses a laser diode as the light source, and the chaotic signal is generated by the nonlinear characteristics of the external devices (such as Mach-Zehnder modulator) [7].

The chaotic output of the external-cavity semiconductor laser (ECSL) usually retains the time-delay signature (TDS), which is derived from the optical round trip between the laser and the external feedback mirror [8,9]. Therefore, utilizing some timing analysis methods, for instance, autocorrelation function (ACF), delayed mutual information (DMI), etc., the TDS can be obtained from the output signal of the laser, which seriously threatens the security of the chaotic secure communication system [10]. At the same time, the maximum data rate of chaotic optical transmission system is limited by the chaotic carrier bandwidth, which is closely related to the relaxation oscillation (RO) of the laser [11]. Owing to the energy of AO chaos is mainly concentrated near the RO frequency, the effective bandwidth is usually limited to a few gigahertz [12]. Therefore, it is significant to explore a secure chaotic communication method that can suppress the TDS of the laser chaotic system and increase the carrier bandwidth simultaneously.

In the past 20 years, vertical-cavity surface-emitting lasers (VCSEL) have received extensive attention in the fields of optical communication, optical interconnection and optical switch [1316]. Compared with the traditional edge-emitting laser (EEL), VCSELs have many desirable characteristics, such as low cost, low threshold current, single longitudinal mode operation, and ease of fabrication of 2D arrays [17]. Besides, due to weak material and cavity anisotropy, the output of VCSEL usually includes two orthogonal polarization components (i.e., x-polarization (XP) component and y-polarization (YP) component) [18], which is beneficial to realize dual-channel optical fiber communication. Therefore, VCSEL-based chaotic communication systems have broad application prospects in secure optical communication systems.

Recently, many VCSEL-based laser chaotic systems with different structures have been proposed to suppress the TDS. Liu et al. proposed a three-cascaded bidirectional VCSELs communication with concealment of TDS. However, the TDS are exposed through the cross-correlation between the master and slave lasers [19]. Zhong et al. experimentally studied the TDS of chaotic output in a 1550 nm VCSEL subject to fiber Bragg grating feedback [20]. Liu et al. proposed a novel double masking scheme based on two groups of mutually asynchronous VCSELs. This scheme can suppress the TDS to a certain extent, but it cannot completely hide it [21]. Zhang et al. proposed a method based on variable-polarization optical injection to achieve isochronous cluster synchronization with TDS suppression in VCSEL networks. However, the key space of this method is small and needs to be enhanced [22]. To sum up, in the VCSEL chaotic secure communication system, various issues need to be further studied, including TDS concealment, synchronization performance, bandwidth enhancement, and key space enhancement.

In this paper, we propose a novel dual-channel VCSEL chaotic communication system with secondary encryption based on common phase-modulated electro-optic (CPMEO) feedback. We theoretically and numerically analyze that the proposed system can achieve TDS concealment and bandwidth enhancement. Subsequently, the parameter spaces of chaos synchronization are numerically studied. Besides, the influence of laser internal parameters mismatch on the synchronized robustness is analyzed. Last but not least, communication quality and security performance are discussed in detail.

2. System model and rate equations

The schematic diagram of the dual-channel VCSEL chaotic communication system with secondary encryption based on CPMEO feedback is shown in Fig. 1. The output of the M-VCSEL is split into two parts by a beam splitter (BS), with one reflected back to the laser by the mirror $\rm M_1$, and the other by the mirror $\rm M_2$. By controlling the parameters of the dual-optical feedback (DOF), two linear polarization modes (XP-mode and YP-mode) with suppressed TDS can be output simultaneously. The rotating polarizer (RP) is used to achieve variable-polarization optical feedback (VPOF) [23]. By mixing XP and YP mode with a certain proportion and feeding it back into S-VCSELs, the TDS of the chaotic carrier (i.e. $x_2$) can be further suppressed.

 

Fig. 1. Schematic diagram of the proposed dual-channel chaos communication system with secondary encryption based on common phase-modulated electro-optic (CPMEO) feedback. DOF, dual optical feedback; VPOF, variable-polarization optical feedback; RP, rotating polarizer; M-VCSE, master VCSEL; S-VCSEL, slave VCSEL; D-VCSEL, driving VCSEL; M, mirror; OC, optical coupler; PM, phase modulator; PBS, polarization beam splitter; BS, beam splitter; FDL, fiber delay line; MZI, Mach-Zehnder interferometer; PD, photodetector; OI, optical isolator.

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A continuous wave (CW) laser light is fed into $\rm PM_1$. The output is transmitted through $\rm PM_2$, fiber delay line (FDL), and Mach-Zehnder interferometer (MZI) which converts phase fluctuations to intensity changes. Then the output is detected by a photodiode (PD) and amplified by a radio-frequency (RF) amplifier [24]. The $\rm PM_2$ and $\rm PM_4$ in the EO feedback loop of the transmitter and receiver are independently driven by the local S-VCSEL, and only the common driving signal generated by the DOF is transmitted through the public link. The VPOF module is used to reduce the correlation between the output signals of M-VCSEL (i.e. $x_1$) and $\rm S_1$-VCSEL (i.e. $x_2$), which can ensure that the messages cannot be correctly decrypted when the eavesdropper injects signal $x_1$ directly into EO-2. Therefore, the proposed CPMEO system has the security characteristic of secondary encryption with the help of WDM. The first-level encrypted keys are the TDs in the EO feedback loop, and the second-level keys are the TD and polarization angle of the VPOF module. The key space of the proposed CPMEO system has been significantly enhanced by using the unique secondary encryption method, thus can greatly reduce the risk of being deciphered.

Based on the spin-flip model, the rate equations for M-VCSEL and S-VCSEL are as follows [25]:

$$\begin{aligned} \frac{{dE_{x,y}^M}}{{dt}} = &{k^M}(1 + i{\alpha ^M})({{{N}}^M}E_{x,y}^M + i{n^M}E_{y,x}^M - E_{x,y}^M) - (\gamma _a^M + i\gamma _p^M)E_{x,y}^M\\ &+ {\sigma _{1,2}}E_{x,y}^M(t - {\tau _{f1,2}})\exp ( - i{\omega ^M}{\tau _f}_{1,2}) + \sqrt {{\beta _{sp}}} \zeta _{x,y}^M \end{aligned}$$
$$\begin{aligned} \frac{{dE_{x,y}^{{S_i}}}}{{dt}} = &{k^{{S_i}}}(1 + i{\alpha ^{{S_i}}})({{{N}}^{{S_i}}}E_{x,y}^{{S_i}} + i{n^{{S_i}}}E_{y,x}^{{S_i}} - E_{x,y}^{{S_i}}) - (\gamma _a^{{S_i}} + i\gamma _p^{{S_i}})E_{x,y}^{{S_i}}\\ &+ {\sigma _3}E_{x,y}^{{S_i}}(t - {\tau _{f3}})co{s^2}({\theta _p})exp( - i{\omega ^{{S_i}}}{\tau _{f3}}) + {\sigma _3}E_{y,x}^{{S_i}}(t - {\tau _{f3}})cos({\theta _p})sin({\theta _p})exp( - i{\omega ^{{S_i}}}{\tau _{f3}})\\ &+ \eta E_{x,y}^M(t - {\tau _{in_M}})exp( - i{\omega ^{_M}}{\tau _{in_M}} + i\Delta \omega t) + \sqrt {{\beta _{sp}}} \zeta _{x,y}^{S_i} \end{aligned}$$
$$\frac{{d{N^{M,{S_i}}}}}{{dt}} = - \gamma _N^{M,{S_i}}{{[}}{{{N}}^{M,{S_i}}}(1 + {\left| {E_x^{M,{S_i}}} \right|^2} + {\left| {E_y^{M,{S_i}}} \right|^2}) + i{n^{M,{S_i}}}(E_y^{M,{S_i}}E_x^{M,{S_i}^{{*}}} - E_x^{M,{S_i}}E_y^{M,{S_i}^{{*}}}) - \mu ]$$
$$\frac{{{{d}}{{{n}}^{M,{S_i}}}}}{{dt}} = - \gamma _s^{M,{S_i}}{n^{M,{S_i}}} - \gamma _N^{M,{S_i}}[i{N^{M,{S_i}}}(E_y^{M,{S_i}}E_x^{M,{S_i}^{{*}}} - E_x^{M,{S_i}}E_y^{M,{S_i}^{{*}}}) + {n^{M,{S_i}}}({\left| {E_x^{M,{S_i}}} \right|^2} + {\left| {E_y^{M,{S_i}}} \right|^2})]$$
where ${{{E}}_{x,y}}$ represents the slowly varied complex field amplitudes of XP and YP. $N$ is the total carrier inversion between conduction and valence bands, and $n$ is the difference in concentration between carriers with spin-up and spin-down. The superscript $M$, $S_i$ stand for M-VCSEL, $\rm S_1$-VCSEL and $\rm S_2$-VCSEL. ${\omega _{M,S,D}} = 2\pi c/{\lambda _{M,S,D}}$ is the central frequency of M-VCSEL, S-VCSEL and D-VCSEL with central wavelength $\lambda _M$ = 1550 nm, $\lambda _D$ = 1553.3 nm. $\Delta \omega = 2\pi \Delta f$, where $\Delta f = {f_M} - {f_s}$ represents the frequency detuning between M-VCSEL and S-VCSEL. $\tau _{fi}$ and $\sigma _i$ are the feedback delay and the feedback strength. ${\tau _{in_M}}$ and $\eta$ are the injection delay and the injection strength from M-VCSEL to S-VCSEL. The last terms in Eqs. (1)–(2) are the spontaneous emission noise, where ${\beta _{sp}}$ is the spontaneous emission factor, and $\zeta$ is the Gaussian white noise terms of zero mean value and unitary variance. For simplicity, we have neglected noise effects (i.e. $\sqrt {{\beta _{sp}}} \zeta _{x,y}^{M,S}=0$) and assumed the internal parameters of VCSELs are identical [26]. Other parameters and the selected values are described in Table 1 [22].

Tables Icon

Table 1. The parameters for the CPMEO system

Next, the generation process of the EO feedback chaos signal driven by VPOF source will be analyzed. The modified Ikeda-based CPMEO feedback equation driven by the AO source can be written as:

$$\begin{aligned} {{X}}_{x,y}^{s,r}(t) &+ \frac{1}{{{\varepsilon ^{s,r}}}}\int_{{t_0}}^t {{{X}}_{x,y}^{s,r}(\xi)d\xi} + {\tau_h ^{s,r}}\frac{d}{{dt}}{{X}}_{x,y}^{s,r}(t) = \\ &{\beta^{s,r}} \cos ^2 \Big[\Delta {({{X}}_{x,y}^{s,r})_{{T^{s,r}}}} + \Delta {\big(({\varphi _{AO}})_{x,y}^{s,r}\big)_{{T^{s,r}}}} + \Delta {({m_{x,y}})_{{T^{s,r}}}} + \varphi _0 \Big]{} \end{aligned}$$
$${\phi _{{{AO}}}}(t) = \frac{{\rho \pi |{E_{AO}}(t - {\tau _{in_S}}){|^2}}}{{{V_\pi }}}$$
where ${ X}(t) = \frac {{\pi {V}(t)}}{{2{V_{\pi }}}}$, ${V}(t)$ is the driving voltage of PM, ${V_{\pi }}$ is the half-wave voltage of the PM. The EO feedback loop gain is $\beta = \frac {{\pi g {P_0}G{A_t} }}{{2{V_\pi }}}$, where $A_t$ is the overall attenuation of feedback loop, $P_0$ is the power of D-VCSEL, $g$ and $G$ are the gain coefficients of PD and RF amplifier, respectively. $\Delta {({{F}})_{{T}}} = {{F}}(t - {T}) - {{F}}(t - {T} - \delta {T})$, where ${ F}$ represents ${ X(t)}$, message ${m(t)}$, and ${\varphi _{AO} (t)}$. $T$ and $\delta T$ are the time delays caused by the FDL and the interference delay of MZI. The phase modulation index ${\rho }/{{{V_\pi }}}$ is set to 1 for calculation convenience. The subscripts $x$, $y$ stand for XP and YP of chaos signal, and the superscripts $s$, $r$ refer to the transmitter and receiver. $\tau _{in_S}$ is the flight time from S-VCSEL to $\rm PM_2$. It can be seen that a large number of new frequency components are generated in the phase terms of the Ikeda-based CPMEO equation due to the AO source.

3. Calculation results and discussion

The fourth-order Runge-Kutta algorithm with an integration step of 1 ps is used to numerically solve the Eqs. (1)–(5) of the proposed CPMEO chaotic system. The major parameters in the system are $\tau _{f1}=\tau _{f3}=\tau _{in_M}=\tau _{in_S}=3 \rm ~ns$, $\tau _{f2}=4~\rm ns$, $\sigma _1=30 \rm ~ns^{-1}$, $\sigma _2=40 \rm ~ns^{-1}$, $\sigma _3=5 \rm ~ns^{-1}$, $\eta =100 \rm ~ns^{-1}$, $T=15 \rm ~ns$, and $\delta T=400 \rm ~ps$ [22,27].

Due to the weak anisotropy of the active region and the cavity, two orthogonal polarization modes can be output under the appropriate parameters of the VCSEL [28]. In this paper, the single polarization switching method is adopted, in which $\gamma _a>0$ and $\gamma _p$ is large enough [29]. Figure 2 shows the P-I curve of the two polarization mode outputs in a free-running VCSEL. It can be clearly seen that when the normalized injection current $\mu$ is 1.5, only the YP component starts to oscillate. Both XP and YP signals start to oscillate when $\mu$ reaches 2.1. The output intensity of XP and YP is exactly the same when $\mu$ is 2.7. Since two polarization components with similar output strength are more conducive to achieving dual-channel chaotic communication, the normalized bias current of the chaotic system is set to 2.7.

 

Fig. 2. P-I curve for free-running VCSEL.

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3.1 Dynamic complexity analysis

Free-running VCSELs can exhibit rich dynamic characteristics with appropriate parameter selection. In order to explore their applications in the secure communication, it is necessary to perform numerical simulations to generate a wide-band and highly-complex chaotic carrier.

In this section, we will analyze the complexity of the chaotic carrier generated from the DOF (i.e. $x_1$), DOF+VPOF (i.e. $x_2$) and CPMEO (i.e. $x_3$) modules in the proposed dual-channel VCSEL system. Figures 3(a1)–3(a3) and Figs. 3(b1)–3(b3) respectively show the chaotic time traces and the corresponding RF power spectrum output from the three modules when $\beta$ = 3. The outputs of the three modules in chaotic dynamics are displayed in Figs. 3(a1)–3(a3).

 

Fig. 3. Time series (first row), RF spectrum (second row), the probability density function with the best Gaussian fit (third row) and the permutation entropy (fourth row) of $x_1 {(t)}$ (first column), $x_2 {(t)}$ (second column) and $x_3 {(t)}$ (third column) for XP output and YP output. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

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A flatter power spectrum can make the energy distribution more uniform, which can effectively increase the bandwidth of the chaotic carrier. As shown in Figs. 3(b1)–3(b3), compared with DOF and DOF + VPOF, the spectrum of the chaotic carrier output from the CPMEO system is significantly flatter and wider. We adopt the effective bandwidth as the span between the direct current (DC) and the frequency where 80$\%$ of energy is contained in the power spectrum [30]. The effective bandwidth of S-VCSEL is expanded up to 11 GHz and is 1.5 times of M-VCSEL, as shown in Figs. 3(b1)–3(b2). The expansion of the bandwidth benefits from the optical-locking injection. After the CPMEO module, the effective bandwidth of XP and YP is doubled to 22.31 GHz and 22.40 GHz. The significant increase in bandwidth is due to the redistribution of spectrum energy [31]. Using the chaotic source output from the VPOF to drive $\rm PM_2$, a large number of new frequency components will be generated in the EO feedback loop. Thus, the complex chaotic carriers with high nonlinearity can be obtained. In the proposed CPMEO system, the dominance of the laser relaxation oscillation is eliminated by the strong nonlinearity of the EO feedback loop.

Next, the randomness of chaotic sequences will be discussed. If the probability density function (PDF) of the chaotic carrier is closer to the Gaussian distribution, its time-domain waveform will be more similar to a random driving source (i.e. noise) [32]. The Gaussian fit of the data is difined as $R$:

$${{R}} = 1 - \frac{\sum\limits_{i = 1}^n {{({y_i} - \mathop{y_i}\limits^{\wedge})}^{2}}}{\sum\limits_{i = 1}^n {{({y_i} - \mathop{y_i}\limits^{-})}^{2}}}$$
where ${y}$ is the original data, ${{\hat y}}$ is the Gaussian fitted data, ${{\bar y}}$ is the mean of the original data, and $R \in [0,1]$. The Gaussian model fits the data better when $R$ is close to 1. Figures 3(c1)–3(c3) present that for general DOF, the time series distribution has an obvious asymmetry, and the fitting coefficients of the two polarization modes are $R_{\textrm{DOF}} - \textrm{x} = 0.7586$ and $R_{\textrm{DOF}} - \textrm{y} = 0.7517$. After the VPOF module, the fitting coefficients have been further improved to $R_{\textrm{VPOF}} - \textrm{x} = 0.8548$ and $R_{\textrm{VPOF}} - \textrm{y} = 0.8591$. Finally, the distribution of the chaotic time series output from EO module has better symmetry and is closer to the Gaussian distribution. The fitting coefficient reaches $R_{\textrm{CPMEO}} - \textrm{x} = 0.9942$ and $R_{\textrm{CPMEO}} - \textrm{y} = 0.9972$. This quantitatively proves that the CPMEO feedback significantly improves the random characteristics of VCSEL chaotic carriers.

Permutation entropy (PE) is used to quantify the unpredictability of time series. Due to the simplicity and high robustness of the algorithm, PE is widely used in nonlinear system analysis and sequence complexity calculation [33]. We embed a scalar time series $\{y(i) , i = 1 ,2 ,\ldots ,n\}$ to a $m$-dimensional space $Y_i =[y(i),y(i+L), \ldots , y(i + (m-1)L)]$, where $m$ is the embedding dimension and $L$ is the embedding delay time. Here $Y_i$ is in increasing order. For a permutation with number $j$, let $f (j)$ denote its occurrence frequency. The probability of relative occurrence frequency can be defined as $p({j}) = {f(j) }/{{[n - (m - 1)L]}}$. Therefore, the PE is defined as [34]:

$$PE(m) = - \sum {p({j})} \log p({j})/\log (m!)$$
The system will generate highly complex chaotic signals when PE is larger than 0.9 [27]. As suggested in [24], we choose $L$ = 2 and $m$ = 6. Figures 3(d1)–3(d2) show that when feedback strength is 50 $ns^{-1}$, the PE of the chaotic carrier output from VPOF is 0.4188 in XP and 0.4812 in YP, which is 0.2 higher than the DOF. Figure 3(d3) demonstrates that the PE of the chaotic carrier output from the EO module has been significantly improved when $\beta =3$. We notice that the difference is that the PE is 0.8023 when the feedback strength is 0. The main reason is that the low-complexity chaotic carrier can be generated by an independent EO feedback loop (i.e. without the AO driving source), but the TDS can be easily obtained [27]. Both polarization modes in the CPMEO system can output high-complexity chaotic carriers (i.e. PE $>$ 0.9) when feedback strength is larger than 20 $ns^{-1}$. More importantly, the PE in XP mode is 0.9782 and 0.9804 in YP mode when feedback strength is 50 $ns^{-1}$.

3.2 Concealment of chaotic time-delay signature

For chaotic secure communication systems, suppressing the TDS of chaotic carriers is the key to ensureing system security. The autocorrelation function (ACF) and delayed mutual information (DMI) of the chaotic time series are used to analyze the TDS of external cavity feedback. The ACF and DMI are defined as [35]:

$$\textrm{ACF}(s) = \frac{{\langle [x(t + s) - \langle x(t)\rangle ] [x(t) - \langle x(t)\rangle ]\rangle }}{{\sqrt {\langle {{(x(t) - \langle x(t)\rangle )}^2}\rangle \langle {{(x(t + s) - \langle x(t)\rangle )}^2}\rangle } }}$$
$$\textrm{DMI}(s) = \sum_{}^{} {p[x(t),x(t + s)]log\frac{{p[x(t),x(t + s)]}}{{p[x(t)] p[x(t + s)]}}}$$
where $s$ stands for time-shift, $\left \langle \cdot \right \rangle$ means time average, $p[x(t)]$ and $p[x(t), x(t + s)]$ are the probability of marginal and joint distribution, respectively.

Figure 4 demonstrates the ACF and DMI of the chaotic carriers output by the DOF, VPOF and EO modules in the proposed CPMEO feedback system when $\beta = 3$. The first row of Fig. 4 is the XP and YP of the chaotic carriers output by the DOF module (i.e. $x_1$), indicating that both ACF and DMI have obvious peaks at the delays ${\tau _{f1}} = 3~\rm ns$, ${\tau _{f2}} = 4~\rm ns$ and the difference ${\tau _{f2}} - {\tau _{f1}} = 1~\rm ns$. After the chaotic carrier passes through the VPOF module (i.e. $x_2$) (second row), the TDS can be clearly seen although they are suppressed to a certain extent. Figures 4(e)–4(f) show the ACF and DMI of the chaotic carrier output from the EO module (i.e. $x_3$) for both polarization modes. The system has no obvious peak at all the feedback delays (i.e. ${\tau _{f1}} = 3~\rm ns$, ${\tau _{f2}} = 4~\rm ns$, $T=15~\rm ns$, $T + \delta T = 15.4 \;\textrm{ns}$) of XP and YP, which means that the TDS are effectively hidden.

 

Fig. 4. The ACF(x) (first column) and DMI(x) (second column) of chaotic carrier output from DOF (first row) , VPOF (second row) and EO modules (third row). The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

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Next, we focus on the effect of the EO feedback gain $\beta$ on TDS suppression. Figure 5 shows the peak size at the TDs as a function of $\beta$ in ACF($x_3$) and DMI($x_3$). If the extreme values of XP and YP are in the background range (i.e. the area between the two red lines), the eavesdropper cannot extract the TDS. The background $\textrm{Q}_{ACF}$ and $\textrm{Q}_{DMI}$ in ACF(x) and DMI(x) are defined as:

$${Q_F}(\beta) = [{\underline {{P}} _F}(\beta),{\overline {P} _F}(\beta)]$$
$${\underline {{P}} _F}(\beta) = mean\{ F({x_\beta})\} - SD\{ F({x_\beta})\}$$
$${\overline P _F}(\beta) = mean\{ F({x_\beta})\} + SD\{ F({x_\beta})\}$$
where $SD$ is the standard deviation, and $F$ represents the function of ACF and DMI. Figure 5 displays that when $\beta$ is small, the system has a significant peak at 3 ns, and the peaks at other TDs are relatively small. In order to perfectly hide all TD peaks in the background $\textrm{Q}_{ACF}$, the minimum $\beta$ required in the XP and YP are 3 and 2.5, respectively. Through a similar analysis method, the threshold of $\beta$ in DMI is 1.5. Therefore, all TD peaks are located in the background $\textrm{Q}_{ACF}$ and $\textrm{Q}_{DMI}$ when $\beta \geq 3$, which satisfies the condition that the bifurcate parameter $\beta$ in the electro-optic feedback loop needs to be less than 5.1 [36]. Therefore, the TDS concealment in the proposed CPMEO system can be easily implemented in practice, and the eavesdropper cannot decipher the system through the existing timing analysis methods.

 

Fig. 5. Value of the peaks of $x_3$ in ACF(x) (first row) and DMI(x) (second row) for increasing the feedback gain $\beta$ at ${\tau _1} = 3\;\textrm{ns}$, ${\tau _2} = 4\;\textrm{ns}$, ${{T = 15~\rm {ns}}}$ and ${{T + }}\delta {T = 15.4}\,\textrm{ns}$ when $\Delta f= 20~\rm GHz$; The red lines correspond to the background $\textrm{Q}_{\textrm{ACF}}$ or $\textrm{Q}_{\textrm{DMI}}$.

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These phenomena in both the TDS concealment and complexity performance could be explained by the following reasons. 1) The VPOF module would generate a large number of new frequencies into the EO feedback loop. Therefore, the AO driving source significantly improves the nonlinearity of the EO chaotic carrier, which reduces the time-domain correlation of the chaotic carrier. 2) The more key space dimensions covered in a chaotic system, the more complex chaotic carriers will be output. The CPMEO system has 4 key space dimensions: the feedback TD and polarization angle in the VPOF, and the time delay $T$ caused by the FDL and the interference delay $\delta {T}$ of MZI in the EO feedback loop.

3.3 Chaos synchronization

The performance of chaos synchronization in the system directly affects the realizability of laser chaotic communication [37]. In the VCSEL chaotic communication based on CPMEO feedback, the injection locking synchronization method is adopted. By selecting the appropriate system parameters, $\rm S_1$-VCSEL and $\rm S_2$-VCSEL driven by M-VCSEL can generate the same chaotic carriers (i.e. $x_2=x_5$) to further drive the EO feedback loop for information encryption and decryption. The cross-correlation (CC) is used to evaluate the synchronization quality [38]:

$$\textrm{CC} = \frac{{\langle [x(t) - \langle x(t)\rangle ] [y(t) - \langle y(t)\rangle ]\rangle }}{{\sqrt {\langle {{[x(t) - \langle x(t)\rangle ]}^2}\rangle \langle {{[y(t) - \langle y(t)\rangle ]}^2}\rangle } }}$$
where $x(t)$ is the time trace of the transmitter, $\textrm{y}(t)$ is the time trace of the receiver. When CC is higher than 0.95, the system can achieve the high-quality chaos synchronization [5].

3.3.1 Conditions for chaos synchronization

The parameter conditions for achieving chaos synchronization between transmitter (i.e. $x_3$) and receiver (i.e. $x_6$) in the proposed CPMEO feedback chaotic system is explored. The maps of the CC coefficients evolution is firstly calculated in the parameter space of feedback strength $\sigma _1$ and $\sigma _2$ of DOF. As shown in Figs. 6(a)–6(b), the regions with high CC values are symmetrically distributed along the main diagonal. Accordingly, it is easier for the system to achieve chaos synchronization when both feedback strengths are set to a larger value.

 

Fig. 6. Two-dimensional maps of the max cross-correlation function for XP (the first column) and YP (the second column) output between EO-1 (i.e. $x_3$) and EO-2 (i.e. $x_6$) under: (a,b) different feedback strength $\sigma _1$ and feedback strength $\sigma _2$ of DOF (c,d) different frequency detuning $\triangle f$ and injection strength $\eta$ (e,f) different feedback strength $\sigma _3$ and polarization angle $\theta _p$ of VPOF when $\beta$ = 3.

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Figures 6(c)–6(d) show the evolution of the maximum CC between the XP output and the YP output of the chaotic carriers in the parameter space of injection strength ${\eta }$ and frequency detuning $\Delta {{f}}$. The results show that the higher the injection strength is, the easier the system could achieve the high-quality chaos synchronization. This could be attributed to the fact that the initial value and noise interference of the laser are different, and these factors will greatly influence the quality of chaos synchronization at a small injection strength. Furthermore, compared with the negative frequency detuning, the area with the maximum CC higher than 0.95 is significantly increased when the frequency detuning is positive.

In order to further explore the effect of system parameters on the synchronization performance, Figs. 6(c)–6(d) show the evolution of the maximum CC under different polarization angles $\theta _p$ and the feedback strengths ${\sigma _3}$ of VPOF. When the feedback strength is less than $10 \;\textrm{ns}^{ - 1}$, the system can maintain high quality synchronization. Unlike the results in Figs. 6(a)–6(b), as the feedback strength of closed-loop VPOF-2 gradually increases, the synchronization quality of the system gradually deteriorates. The difference of the two trends can be explained as follows: compared with an open-loop structure, the closed-loop synchronization has better symmetry, security and synchronization quality, but the parameter conditions for achieving chaos synchronization are more stringent [39]. In addition, the areas of high-quality chaos synchronization in XP or YP are the largest when the polarization angle is ${40^ \circ }$.

Finally, according to the areas where $\rm CC>0.95$ is simultaneously satisfied in the above three cases, we choose a set of system parameters, namely $\sigma _1=30 \rm ~ns^{-1}$, $\sigma _2=40 \rm ~ns^{-1}$, $\sigma _3=5 \rm ~ns^{-1}$, $\eta =100 \rm ~ns^{-1}$, $\Delta f= 20~\rm GHz$, and $\theta _p={40^ \circ }$, to simulate the chaos synchronization between transmitter and receiver. Figure 7 shows that high quality chaos synchronization (i.e. $\rm CC=0.996$) is achieved, and the synchronization error between the chaotic signal of transmitter and receiver gradually stabilizes to 0 after $\rm 15~ns$.

 

Fig. 7. (a) The cross-correlation coefficients and (b) the synchronization error between the chaotic carriers generated by EO-1 (i.e. $x_3$) and EO-2 (i.e. $x_6$). The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

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3.3.2 Effects of internal parameter mismatch on synchronization robustness

The above discussion is based on the fact that the parameters of all components are completely consistent. Since it is almost impossible to have identical lasers, analyzing the influence of laser internal parameters mismatch on chaos synchronization robustness is crucial. The parameters of $\rm S_1$-VCSEL are set to fixed values, and the parameters of $\rm S_2$-VCSEL are changed with the mismatch rate. The parameter mismatch rate is defined as:

$$\Delta \rho = \frac{{{\rho^{r}} - {\rho^{s}}}}{{{\rho^{s}}}}$$
where $\rho$ represents the internal parameters of the VCSEL, the superscripts $s$ and $r$ represent the transmitter and receiver, respectively.

Figure 8 reveals the maximum CC of XP and YP mode under the mismatch of different internal parameters. Compared with other parameters, the CC of the linewidth enhancement factor $\alpha$, the field decay rate $k$, and the carrier decay rate ${\gamma _{\textrm{N}}}$ are lower under the same parameter detuning rate. This means that their changes have a greater impact on the chaos synchronization between the lasers. Moreover, the CC always remains above 0.95 when the internal parameters detuning of the laser are between −20% and 20%. Thus the internal parameters of the laser can be selected from a wide range, which indicates the strong robustness to chaos synchronization.

 

Fig. 8. Maximum of cross-correlation for (a) XP and (b) YP with the internal mismatched parameters. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

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3.4 Chaos communication

Two orthogonal polarization modes XP and YP output from the VCSEL are respectively used as chaotic carriers to realize dual-channel chaotic secure communication, which can increase the transmission capacity of information. In this paper, we use the phase chaotic modulation method to load the signal on the chaotic carrier (as shown in Fig. 1), and the scheme is transparent to the modulation format of the signal. Taking the BPSK signal as an example, the performance of chaotic secure communication is simulated. A pseudorandom code sequence with bit rate of 10 Gbit/s is used as the original message. Figure 9 displays the time sequence of messages after encryption and decryption for both XP and YP channels. The encrypted messages can be perfectly hidden in the chaotic carrier, and can be successfully decoded when $\beta$ mismatch is 1%.

 

Fig. 9. (a) Original, (b) encrypted, and (c) recovered messages of XP and YP channel, considering 1% mismatch in $\beta$. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

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The bit error rate (BER) performance of the proposed CPMEO feedback communication system under different TD and feedback gain $\beta$ detuning is evaluated. From Fig. 10(a), with a slight detuning of $T$, the BER first increases rapidly and then reaches a flat top. As shown in Fig. 10(b), the BER rises slowly with $\beta$ detuning, and the overall trend is similar to $T$. In order to ensure that the BER is less than $3.8 \times {10^{-3}}$, which is the decision forward error correction (FEC) threshold [2], the TD mismatch must be controlled within 3 ps, and the $\beta$ mismatch need to be controlled within 12%. In general, if the feedback time delay of the system is not accurately known, eavesdroppers will not be able to intercept the messages.

 

Fig. 10. Variation of BER versus the mismatch of (a) time delay $T$ (b) feedback gain $\beta$ of EO feedback loop. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

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Subsequently, the security of the proposed VCSEL secure communication system based on CPMEO feedback under three typical attack modes will be discussed. 1) The first attack method is to directly extract messages from the chaotic signals transmitted on the fiber channel. The message is well masked in the chaotic carrier and cannot be directly extracted, as shown in Fig. 9. 2) The second attack method is to directly inject the chaos carrier $x_1$ into the electro-optic feedback loop EO-2 to intercept the message (assuming that the eavesdropper knows the device structure and key parameters of the EO module). We use the VPOF module to reduce the correlation between $x_1$ and $x_5$, which can greatly reduce the possibility of messages being illegally decrypted. From Fig. 11(a), we can see that the CC value of $x_1$ and $x_5$ in the XP and YP modes can be reduced to 0.44 and 0.3275, respectively. As Fig. 12 illustrated, the message cannot be decrypted correctly in this case. 3) The third eavesdropping method is to perform the cross-correlation calculation on chaotic signals transmitted in the public link (i.e. $x_1$ and $x_4$) to extract the TDS. Figure 11(b) depicts the CC between $x_1$ and $x_4$ around zero time-shift is 0.002. At the same time, there are no obvious peaks at the feedback TDs (i.e. 3 ns, 4 ns, 15 ns, 15.4 ns), which greatly guarantees the security of the system.

 

Fig. 11. The cross-correlation coefficients between (a) the output of M-VCSEL (i.e. $x_1$) and $\rm S_2$-VCSEL (i.e. $x_5$) (b) the output of M-VCSEL (i.e. $x_1$) and EO-1 (i.e. $x_4$). The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

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Fig. 12. (a) Original and (b) recovered message of XP and YP channel when the chaotic carrier $x_1$ is directly injected into electro-optic feedback loop EO-2. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

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Finally, our work is compared with other researches on VCSEL chaos communication in Table 2. We label the system in [21] as System 1 and the system in [19] as System 2 to distinguish them. The parameters of each system are set to the same as those in [21] for comparison, and the time step is set as 1 ps. The TDS of System 1 and System 2 in ACF are about 0.33 and 0.25, as shown in the first column of Table 2. In DMI, there is a TDS peak of about 0.17 at $\tau _f$ in System 1. More importantly, the CPMEO system has no obvious TDS peak in ACF and DMI, and is basically stable at a position close to 0. The effective bandwidth of the System 2 and the CPMEO system are both around 21 GHz. However, in System 2, the maximum CC between S-VCSEL and R-VCSEL at the time delay is 0.2262 in XP and 0.3604 in YP. Namely, the TDS will be exposed through CC calculation, which will pose a certain threat to security. Overall, the proposed CPMEO system can achieve secure communication with wide bandwidth under different parameter conditions.

Tables Icon

Table 2. Comparison of the three VCSEL systems.

4. Conclusion

In this paper, the TDS concealment, wide bandwidth and key space enhancement can be achieved simultaneously in a novel dual-channel VCSEL chaos secure communication system based on CPMEO feedback. It is demonstrated that compared with the conventional all-optical VCSEL system, the time series distribution of the CPMEO feedback system is more symmetrical and close to Gaussian distribution, and the PE increases significantly. The TDS of the CPMEO system is suppressed 8 times to less than 0.05 to be completely hidden, and the bandwidth is doubled to lager than 22 GHz. The key space is significantly improved by using the VPOF module to reduce the time-domain correlation between the DOF output and the EO input chaotic carrier. High-quality chaos synchronization ($\rm CC=0.996$) between the transmitter and receiver is achieved by properly setting the system parameters. Moreover, the system BER performance with bit rate of 10 Gbit /s is simulated and analyzed. The demodulation of messages between legitimate users can be guaranteed when the TD and $\beta$ mismatch are controlled within 3 ps and 12%, respectively. In conclusion, the proposed scheme makes it a reality to achieve a secure dual-channel chaotic communication with high bit-rate and long-diatance.

Funding

National Natural Science Foundation of China (61831003); National Key Research and Development Program of China (2018YFB1800802).

Disclosures

The authors declare no conflicts of interest.

References

1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990). [CrossRef]  

2. J. Ke, L. Yi, Z. Yang, Y. Yang, Q. Zhuge, Y. Chen, and W. Hu, “32 Gb/s chaotic optical communications by deep-learning-based chaos synchronization,” Opt. Lett. 44(23), 5776–5779 (2019). [CrossRef]  

3. X. Guo, S. Xiang, Y. Zhang, L. Lin, A. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020). [CrossRef]  

4. Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated sagnac loop,” Opt. Express 28(2), 1665–1678 (2020). [CrossRef]  

5. 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]  

6. P. Zhou, Q. Fang, and N. Li, “Phased-array assisted time-delay signature suppression in the optical chaos generated by an external-cavity semiconductor laser,” Opt. Lett. 45(2), 399–402 (2020). [CrossRef]  

7. X. Gao, “Enhancing Ikeda time delay system by breaking the symmetry of sine nonlinearity,” Complexity 2019, 1–14 (2019). [CrossRef]  

8. 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]  

9. M. Li, X. Zhang, Y. Hong, Y. Zhang, Y. Shi, and X. Chen, “Confidentiality-enhanced chaotic optical communication system with variable RF amplifier gain,” Opt. Express 27(18), 25953–25963 (2019). [CrossRef]  

10. 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]  

11. Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, “Bandwidth-enhanced chaos synchronization in strongly injection-locked semiconductor lasers with optical feedback,” Opt. Lett. 28(5), 319–321 (2003). [CrossRef]  

12. A. Zhao, N. Jiang, Y. Wang, S. Liu, B. Li, and K. Qiu, “Correlated random bit generation based on common-signal-induced synchronization of wideband complex physical entropy sources,” Opt. Lett. 44(24), 5957–5960 (2019). [CrossRef]  

13. X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020). [CrossRef]  

14. P. Mu, W. Pan, and N. Li, “Analysis and characterization of chaos generated by free-running and optically injected VCSELs,” Opt. Express 26(12), 15642–15655 (2018). [CrossRef]  

15. A. Elsonbaty, S. F. Hegazy, and S. S. Obayya, “Simultaneous concealment of time delay signature in chaotic nanolaser with hybrid feedback,” Opt. Laser Eng. 107, 342–351 (2018). [CrossRef]  

16. D. Zhong, G. Xu, W. Luo, and Z. Xiao, “Real-time multi-target ranging based on chaotic polarization laser radars in the drive-response VCSELs,” Opt. Express 25(18), 21684–21704 (2017). [CrossRef]  

17. F. Denis-le Coarer, A. Quirce, Á. Valle, L. Pesquera, M. Sciamanna, H. Thienpont, and K. Panajotov, “Polarization dynamics induced by parallel optical injection in a single-mode VCSEL,” Opt. Lett. 42(11), 2130–2133 (2017). [CrossRef]  

18. S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017). [CrossRef]  

19. Y.-Z. Liu, Y.-Y. Xie, and Y.-C. Ye, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017). [CrossRef]  

20. Z.-Q. Zhong, Z.-M. Wu, and G.-Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550 nm VCSEL subject to FBG feedback,” Photonics Res. 5(1), 6–10 (2017). [CrossRef]  

21. B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018). [CrossRef]  

22. L. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, and M. Xu, “Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression,” Opt. Express 27(23), 33369–33377 (2019). [CrossRef]  

23. S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013). [CrossRef]  

24. 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]  

25. L. Zhang, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Common-injection-induced isolated desynchronization in delay-coupled VCSELs networks with variable-polarization optical feedback,” Opt. Lett. 44(15), 3845–3848 (2019). [CrossRef]  

26. S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012). [CrossRef]  

27. W. Cheng, Y. Ji, H. Wang, and B. Lin, “Security-enhanced electro-optic feedback phase chaotic system based on nonlinear coupling of two delayed interfering branches,” IEEE Photonics J. 10(4), 1–15 (2018). [CrossRef]  

28. N. Li, H. Susanto, B. Cemlyn, I. Henning, and M. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017). [CrossRef]  

29. C. Masoller and M. Torre, “Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(4), 483–489 (2005). [CrossRef]  

30. S.-S. Li, X.-Z. Li, and S.-C. Chan, “Chaotic time-delay signature suppression with bandwidth broadening by fiber propagation,” Opt. Lett. 43(19), 4751–4754 (2018). [CrossRef]  

31. A. Wang, Y. Yang, B. Wang, B. Zhang, L. Li, and Y. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013). [CrossRef]  

32. N. Jiang, A. Zhao, S. Liu, C. Xue, and K. Qiu, “Chaos synchronization and communication in closed-loop semiconductor lasers subject to common chaotic phase-modulated feedback,” Opt. Express 26(25), 32404–32416 (2018). [CrossRef]  

33. 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]  

34. Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004). [CrossRef]  

35. Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated sagnac loop,” Opt. Express 28(2), 1665–1678 (2020). [CrossRef]  

36. L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Philos. Trans. R. Soc., A 371(1999), 20120464 (2013). [CrossRef]  

37. X. Li, W. Pan, D. Ma, and B. Luo, “Chaos synchronization of unidirectionally injected vertical-cavity surface-emitting lasers with global and mode-selective coupling,” Opt. Express 14(8), 3138–3151 (2006). [CrossRef]  

38. A. Zhao, N. Jiang, C. Chang, Y. Wang, S. Liu, and K. Qiu, “Generation and synchronization of wideband chaos in semiconductor lasers subject to constant-amplitude self-phase-modulated optical injection,” Opt. Express 28(9), 13292–13298 (2020). [CrossRef]  

39. M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photonics Technol. Lett. 21(7), 426–428 (2009). [CrossRef]  

References

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  • |

  1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
    [Crossref]
  2. J. Ke, L. Yi, Z. Yang, Y. Yang, Q. Zhuge, Y. Chen, and W. Hu, “32 Gb/s chaotic optical communications by deep-learning-based chaos synchronization,” Opt. Lett. 44(23), 5776–5779 (2019).
    [Crossref]
  3. X. Guo, S. Xiang, Y. Zhang, L. Lin, A. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
    [Crossref]
  4. Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated sagnac loop,” Opt. Express 28(2), 1665–1678 (2020).
    [Crossref]
  5. 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]
  6. P. Zhou, Q. Fang, and N. Li, “Phased-array assisted time-delay signature suppression in the optical chaos generated by an external-cavity semiconductor laser,” Opt. Lett. 45(2), 399–402 (2020).
    [Crossref]
  7. X. Gao, “Enhancing Ikeda time delay system by breaking the symmetry of sine nonlinearity,” Complexity 2019, 1–14 (2019).
    [Crossref]
  8. 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]
  9. M. Li, X. Zhang, Y. Hong, Y. Zhang, Y. Shi, and X. Chen, “Confidentiality-enhanced chaotic optical communication system with variable RF amplifier gain,” Opt. Express 27(18), 25953–25963 (2019).
    [Crossref]
  10. 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]
  11. Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, “Bandwidth-enhanced chaos synchronization in strongly injection-locked semiconductor lasers with optical feedback,” Opt. Lett. 28(5), 319–321 (2003).
    [Crossref]
  12. A. Zhao, N. Jiang, Y. Wang, S. Liu, B. Li, and K. Qiu, “Correlated random bit generation based on common-signal-induced synchronization of wideband complex physical entropy sources,” Opt. Lett. 44(24), 5957–5960 (2019).
    [Crossref]
  13. X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
    [Crossref]
  14. P. Mu, W. Pan, and N. Li, “Analysis and characterization of chaos generated by free-running and optically injected VCSELs,” Opt. Express 26(12), 15642–15655 (2018).
    [Crossref]
  15. A. Elsonbaty, S. F. Hegazy, and S. S. Obayya, “Simultaneous concealment of time delay signature in chaotic nanolaser with hybrid feedback,” Opt. Laser Eng. 107, 342–351 (2018).
    [Crossref]
  16. D. Zhong, G. Xu, W. Luo, and Z. Xiao, “Real-time multi-target ranging based on chaotic polarization laser radars in the drive-response VCSELs,” Opt. Express 25(18), 21684–21704 (2017).
    [Crossref]
  17. F. Denis-le Coarer, A. Quirce, Á. Valle, L. Pesquera, M. Sciamanna, H. Thienpont, and K. Panajotov, “Polarization dynamics induced by parallel optical injection in a single-mode VCSEL,” Opt. Lett. 42(11), 2130–2133 (2017).
    [Crossref]
  18. S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
    [Crossref]
  19. Y.-Z. Liu, Y.-Y. Xie, and Y.-C. Ye, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
    [Crossref]
  20. Z.-Q. Zhong, Z.-M. Wu, and G.-Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550 nm VCSEL subject to FBG feedback,” Photonics Res. 5(1), 6–10 (2017).
    [Crossref]
  21. B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
    [Crossref]
  22. L. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, and M. Xu, “Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression,” Opt. Express 27(23), 33369–33377 (2019).
    [Crossref]
  23. S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
    [Crossref]
  24. 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]
  25. L. Zhang, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Common-injection-induced isolated desynchronization in delay-coupled VCSELs networks with variable-polarization optical feedback,” Opt. Lett. 44(15), 3845–3848 (2019).
    [Crossref]
  26. S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012).
    [Crossref]
  27. W. Cheng, Y. Ji, H. Wang, and B. Lin, “Security-enhanced electro-optic feedback phase chaotic system based on nonlinear coupling of two delayed interfering branches,” IEEE Photonics J. 10(4), 1–15 (2018).
    [Crossref]
  28. N. Li, H. Susanto, B. Cemlyn, I. Henning, and M. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
    [Crossref]
  29. C. Masoller and M. Torre, “Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(4), 483–489 (2005).
    [Crossref]
  30. S.-S. Li, X.-Z. Li, and S.-C. Chan, “Chaotic time-delay signature suppression with bandwidth broadening by fiber propagation,” Opt. Lett. 43(19), 4751–4754 (2018).
    [Crossref]
  31. A. Wang, Y. Yang, B. Wang, B. Zhang, L. Li, and Y. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
    [Crossref]
  32. N. Jiang, A. Zhao, S. Liu, C. Xue, and K. Qiu, “Chaos synchronization and communication in closed-loop semiconductor lasers subject to common chaotic phase-modulated feedback,” Opt. Express 26(25), 32404–32416 (2018).
    [Crossref]
  33. 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]
  34. Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004).
    [Crossref]
  35. Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated sagnac loop,” Opt. Express 28(2), 1665–1678 (2020).
    [Crossref]
  36. L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Philos. Trans. R. Soc., A 371(1999), 20120464 (2013).
    [Crossref]
  37. X. Li, W. Pan, D. Ma, and B. Luo, “Chaos synchronization of unidirectionally injected vertical-cavity surface-emitting lasers with global and mode-selective coupling,” Opt. Express 14(8), 3138–3151 (2006).
    [Crossref]
  38. A. Zhao, N. Jiang, C. Chang, Y. Wang, S. Liu, and K. Qiu, “Generation and synchronization of wideband chaos in semiconductor lasers subject to constant-amplitude self-phase-modulated optical injection,” Opt. Express 28(9), 13292–13298 (2020).
    [Crossref]
  39. M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photonics Technol. Lett. 21(7), 426–428 (2009).
    [Crossref]

2020 (6)

2019 (7)

A. Zhao, N. Jiang, Y. Wang, S. Liu, B. Li, and K. Qiu, “Correlated random bit generation based on common-signal-induced synchronization of wideband complex physical entropy sources,” Opt. Lett. 44(24), 5957–5960 (2019).
[Crossref]

L. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, and M. Xu, “Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression,” Opt. Express 27(23), 33369–33377 (2019).
[Crossref]

L. Zhang, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Common-injection-induced isolated desynchronization in delay-coupled VCSELs networks with variable-polarization optical feedback,” Opt. Lett. 44(15), 3845–3848 (2019).
[Crossref]

X. Gao, “Enhancing Ikeda time delay system by breaking the symmetry of sine nonlinearity,” Complexity 2019, 1–14 (2019).
[Crossref]

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]

M. Li, X. Zhang, Y. Hong, Y. Zhang, Y. Shi, and X. Chen, “Confidentiality-enhanced chaotic optical communication system with variable RF amplifier gain,” Opt. Express 27(18), 25953–25963 (2019).
[Crossref]

J. Ke, L. Yi, Z. Yang, Y. Yang, Q. Zhuge, Y. Chen, and W. Hu, “32 Gb/s chaotic optical communications by deep-learning-based chaos synchronization,” Opt. Lett. 44(23), 5776–5779 (2019).
[Crossref]

2018 (8)

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]

P. Mu, W. Pan, and N. Li, “Analysis and characterization of chaos generated by free-running and optically injected VCSELs,” Opt. Express 26(12), 15642–15655 (2018).
[Crossref]

A. Elsonbaty, S. F. Hegazy, and S. S. Obayya, “Simultaneous concealment of time delay signature in chaotic nanolaser with hybrid feedback,” Opt. Laser Eng. 107, 342–351 (2018).
[Crossref]

W. Cheng, Y. Ji, H. Wang, and B. Lin, “Security-enhanced electro-optic feedback phase chaotic system based on nonlinear coupling of two delayed interfering branches,” IEEE Photonics J. 10(4), 1–15 (2018).
[Crossref]

B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
[Crossref]

S.-S. Li, X.-Z. Li, and S.-C. Chan, “Chaotic time-delay signature suppression with bandwidth broadening by fiber propagation,” Opt. Lett. 43(19), 4751–4754 (2018).
[Crossref]

N. Jiang, A. Zhao, S. Liu, C. Xue, and K. Qiu, “Chaos synchronization and communication in closed-loop semiconductor lasers subject to common chaotic phase-modulated feedback,” Opt. Express 26(25), 32404–32416 (2018).
[Crossref]

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]

2017 (7)

N. Li, H. Susanto, B. Cemlyn, I. Henning, and M. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

D. Zhong, G. Xu, W. Luo, and Z. Xiao, “Real-time multi-target ranging based on chaotic polarization laser radars in the drive-response VCSELs,” Opt. Express 25(18), 21684–21704 (2017).
[Crossref]

F. Denis-le Coarer, A. Quirce, Á. Valle, L. Pesquera, M. Sciamanna, H. Thienpont, and K. Panajotov, “Polarization dynamics induced by parallel optical injection in a single-mode VCSEL,” Opt. Lett. 42(11), 2130–2133 (2017).
[Crossref]

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Y.-Z. Liu, Y.-Y. Xie, and Y.-C. Ye, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

Z.-Q. Zhong, Z.-M. Wu, and G.-Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550 nm VCSEL subject to FBG feedback,” Photonics Res. 5(1), 6–10 (2017).
[Crossref]

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]

2015 (1)

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]

2013 (3)

S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

A. Wang, Y. Yang, B. Wang, B. Zhang, L. Li, and Y. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
[Crossref]

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Philos. Trans. R. Soc., A 371(1999), 20120464 (2013).
[Crossref]

2012 (1)

S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012).
[Crossref]

2009 (1)

M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photonics Technol. Lett. 21(7), 426–428 (2009).
[Crossref]

2006 (1)

2005 (1)

C. Masoller and M. Torre, “Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(4), 483–489 (2005).
[Crossref]

2004 (1)

Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004).
[Crossref]

2003 (1)

1990 (1)

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
[Crossref]

Adams, M.

N. Li, H. Susanto, B. Cemlyn, I. Henning, and M. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

Cao, Y.

Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004).
[Crossref]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
[Crossref]

Cemlyn, B.

N. Li, H. Susanto, B. Cemlyn, I. Henning, and M. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

Chan, S.-C.

Chang, C.

Chen, X.

Chen, Y.

Cheng, M.

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]

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]

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]

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]

Cheng, W.

W. Cheng, Y. Ji, H. Wang, and B. Lin, “Security-enhanced electro-optic feedback phase chaotic system based on nonlinear coupling of two delayed interfering branches,” IEEE Photonics J. 10(4), 1–15 (2018).
[Crossref]

Colet, P.

M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photonics Technol. Lett. 21(7), 426–428 (2009).
[Crossref]

Deng, L.

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]

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]

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]

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]

Denis-le Coarer, F.

Du, Y.-X.

B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
[Crossref]

Elsonbaty, A.

A. Elsonbaty, S. F. Hegazy, and S. S. Obayya, “Simultaneous concealment of time delay signature in chaotic nanolaser with hybrid feedback,” Opt. Laser Eng. 107, 342–351 (2018).
[Crossref]

Fang, Q.

Fu, S.

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]

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]

Fu, Y.

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]

Gao, J.

Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004).
[Crossref]

Gao, X.

X. Gao, “Enhancing Ikeda time delay system by breaking the symmetry of sine nonlinearity,” Complexity 2019, 1–14 (2019).
[Crossref]

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]

Guo, X.

Guo, X. X.

X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Hao, Y.

X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

X. Guo, S. Xiang, Y. Zhang, L. Lin, A. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated sagnac loop,” Opt. Express 28(2), 1665–1678 (2020).
[Crossref]

Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated sagnac loop,” Opt. Express 28(2), 1665–1678 (2020).
[Crossref]

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Hegazy, S. F.

A. Elsonbaty, S. F. Hegazy, and S. S. Obayya, “Simultaneous concealment of time delay signature in chaotic nanolaser with hybrid feedback,” Opt. Laser Eng. 107, 342–351 (2018).
[Crossref]

Henning, I.

N. Li, H. Susanto, B. Cemlyn, I. Henning, and M. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

Hively, L. M.

Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004).
[Crossref]

Hong, Y.

M. Li, X. Zhang, Y. Hong, Y. Zhang, Y. Shi, and X. Chen, “Confidentiality-enhanced chaotic optical communication system with variable RF amplifier gain,” Opt. Express 27(18), 25953–25963 (2019).
[Crossref]

S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

Hu, H.

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]

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]

Hu, W.

Huang, L.

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]

Huang, Y.

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]

Ji, Y.

W. Cheng, Y. Ji, H. Wang, and B. Lin, “Security-enhanced electro-optic feedback phase chaotic system based on nonlinear coupling of two delayed interfering branches,” IEEE Photonics J. 10(4), 1–15 (2018).
[Crossref]

Jiang, N.

Jiang, X.

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]

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]

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]

Ke, C.

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]

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]

Ke, J.

Larger, L.

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Philos. Trans. R. Soc., A 371(1999), 20120464 (2013).
[Crossref]

Li, B.

Li, J. F.

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Li, L.

Li, M.

Li, N.

Li, N. Q.

S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012).
[Crossref]

Li, S.-S.

Li, X.

Li, X.-Z.

Lin, B.

W. Cheng, Y. Ji, H. Wang, and B. Lin, “Security-enhanced electro-optic feedback phase chaotic system based on nonlinear coupling of two delayed interfering branches,” IEEE Photonics J. 10(4), 1–15 (2018).
[Crossref]

Lin, L.

X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

X. Guo, S. Xiang, Y. Zhang, L. Lin, A. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

Liu, B.-C.

B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
[Crossref]

Liu, D.

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]

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]

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]

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]

Liu, L.

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]

Liu, S.

Liu, Y.

B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
[Crossref]

Liu, Y.-Z.

B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
[Crossref]

Y.-Z. Liu, Y.-Y. Xie, and Y.-C. Ye, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

Luo, B.

Luo, C.

Luo, W.

Ma, D.

Ma, Y.

Masoller, C.

C. Masoller and M. Torre, “Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(4), 483–489 (2005).
[Crossref]

Mirasso, C. R.

M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photonics Technol. Lett. 21(7), 426–428 (2009).
[Crossref]

Mu, P.

P. Mu, W. Pan, and N. Li, “Analysis and characterization of chaos generated by free-running and optically injected VCSELs,” Opt. Express 26(12), 15642–15655 (2018).
[Crossref]

S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012).
[Crossref]

Obayya, S. S.

A. Elsonbaty, S. F. Hegazy, and S. S. Obayya, “Simultaneous concealment of time delay signature in chaotic nanolaser with hybrid feedback,” Opt. Laser Eng. 107, 342–351 (2018).
[Crossref]

Ohtsubo, J.

Ohyagi, K.

Pan, W.

Panajotov, K.

Pecora, L. M.

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
[Crossref]

Pesquera, L.

Pierce, I.

S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

Priyadarshi, S.

S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

Protopopescu, V. A.

Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004).
[Crossref]

Qiu, K.

Quirce, A.

Sciamanna, M.

Shi, Y.

Shore, K. A.

S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

Shum, P.

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]

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]

Song, Z.

Soriano, M. C.

M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photonics Technol. Lett. 21(7), 426–428 (2009).
[Crossref]

Susanto, H.

N. Li, H. Susanto, B. Cemlyn, I. Henning, and M. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

Takiguchi, Y.

Tang, M.

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]

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]

Thienpont, H.

Torre, M.

C. Masoller and M. Torre, “Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(4), 483–489 (2005).
[Crossref]

Tung, W.-w.

Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004).
[Crossref]

Valle, Á.

Wang, A.

Wang, B.

Wang, H.

W. Cheng, Y. Ji, H. Wang, and B. Lin, “Security-enhanced electro-optic feedback phase chaotic system based on nonlinear coupling of two delayed interfering branches,” IEEE Photonics J. 10(4), 1–15 (2018).
[Crossref]

Wang, Y.

Wen, A.

Wen, A. J.

X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Wu, Z.-M.

Z.-Q. Zhong, Z.-M. Wu, and G.-Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550 nm VCSEL subject to FBG feedback,” Photonics Res. 5(1), 6–10 (2017).
[Crossref]

Xia, G.-Q.

Z.-Q. Zhong, Z.-M. Wu, and G.-Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550 nm VCSEL subject to FBG feedback,” Photonics Res. 5(1), 6–10 (2017).
[Crossref]

Xiang, S.

Xiang, S. Y.

X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012).
[Crossref]

Xiao, Z.

Xie, F.

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]

Xie, Y.-Y.

B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
[Crossref]

Y.-Z. Liu, Y.-Y. Xie, and Y.-C. Ye, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

Xu, G.

Xu, M.

Xue, C.

Yan, L.

Yan, L. S.

S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012).
[Crossref]

Yang, Y.

Yang, Z.

Ye, Y.-C.

Y.-Z. Liu, Y.-Y. Xie, and Y.-C. Ye, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

Yi, L.

Yu, Q.

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]

Zhang, B.

Zhang, H.

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Zhang, L.

Zhang, M.

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]

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]

Zhang, X.

Zhang, Y.

X. Guo, S. Xiang, Y. Zhang, L. Lin, A. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

M. Li, X. Zhang, Y. Hong, Y. Zhang, Y. Shi, and X. Chen, “Confidentiality-enhanced chaotic optical communication system with variable RF amplifier gain,” Opt. Express 27(18), 25953–25963 (2019).
[Crossref]

Zhang, Y. H.

X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

Zhao, A.

Zheng, J.

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]

Zheng, W.-J.

B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
[Crossref]

Zhong, D.

Zhong, Z.-Q.

Z.-Q. Zhong, Z.-M. Wu, and G.-Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550 nm VCSEL subject to FBG feedback,” Photonics Res. 5(1), 6–10 (2017).
[Crossref]

Zhou, P.

Zhu, X.

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]

Zhuge, Q.

Zou, X.

Zou, X. H.

S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012).
[Crossref]

Complexity (1)

X. Gao, “Enhancing Ikeda time delay system by breaking the symmetry of sine nonlinearity,” Complexity 2019, 1–14 (2019).
[Crossref]

IEEE J. Quantum Electron. (1)

C. Masoller and M. Torre, “Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(4), 483–489 (2005).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (4)

S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

X. Guo, S. Xiang, Y. Zhang, L. Lin, A. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

X. X. Guo, S. Y. Xiang, Y. H. Zhang, L. Lin, A. J. Wen, and Y. Hao, “Polarization multiplexing reservoir computing based on a VCSEL with polarized optical feedback,” IEEE J. Sel. Top. Quantum Electron. 26(1), 1–9 (2020).
[Crossref]

S. Y. Xiang, H. Zhang, X. X. Guo, J. F. Li, A. J. Wen, W. Pan, and Y. Hao, “Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

IEEE Photonics J. (4)

Y.-Z. Liu, Y.-Y. Xie, and Y.-C. Ye, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017).
[Crossref]

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]

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]

W. Cheng, Y. Ji, H. Wang, and B. Lin, “Security-enhanced electro-optic feedback phase chaotic system based on nonlinear coupling of two delayed interfering branches,” IEEE Photonics J. 10(4), 1–15 (2018).
[Crossref]

IEEE Photonics Technol. Lett. (2)

S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, “Randomness-enhanced chaotic source with dual-path injection from a single master laser,” IEEE Photonics Technol. Lett. 24(19), 1753–1756 (2012).
[Crossref]

M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open- and closed-loop receivers in all-optical chaos-based communications,” IEEE Photonics Technol. Lett. 21(7), 426–428 (2009).
[Crossref]

J. Lightwave Technol. (1)

Nonlinear Dyn. (1)

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]

Opt. Express (10)

P. Mu, W. Pan, and N. Li, “Analysis and characterization of chaos generated by free-running and optically injected VCSELs,” Opt. Express 26(12), 15642–15655 (2018).
[Crossref]

D. Zhong, G. Xu, W. Luo, and Z. Xiao, “Real-time multi-target ranging based on chaotic polarization laser radars in the drive-response VCSELs,” Opt. Express 25(18), 21684–21704 (2017).
[Crossref]

Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated sagnac loop,” Opt. Express 28(2), 1665–1678 (2020).
[Crossref]

Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated sagnac loop,” Opt. Express 28(2), 1665–1678 (2020).
[Crossref]

A. Wang, Y. Yang, B. Wang, B. Zhang, L. Li, and Y. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
[Crossref]

N. Jiang, A. Zhao, S. Liu, C. Xue, and K. Qiu, “Chaos synchronization and communication in closed-loop semiconductor lasers subject to common chaotic phase-modulated feedback,” Opt. Express 26(25), 32404–32416 (2018).
[Crossref]

L. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, and M. Xu, “Isochronous cluster synchronization in delay-coupled VCSEL networks subjected to variable-polarization optical injection with time delay signature suppression,” Opt. Express 27(23), 33369–33377 (2019).
[Crossref]

M. Li, X. Zhang, Y. Hong, Y. Zhang, Y. Shi, and X. Chen, “Confidentiality-enhanced chaotic optical communication system with variable RF amplifier gain,” Opt. Express 27(18), 25953–25963 (2019).
[Crossref]

X. Li, W. Pan, D. Ma, and B. Luo, “Chaos synchronization of unidirectionally injected vertical-cavity surface-emitting lasers with global and mode-selective coupling,” Opt. Express 14(8), 3138–3151 (2006).
[Crossref]

A. Zhao, N. Jiang, C. Chang, Y. Wang, S. Liu, and K. Qiu, “Generation and synchronization of wideband chaos in semiconductor lasers subject to constant-amplitude self-phase-modulated optical injection,” Opt. Express 28(9), 13292–13298 (2020).
[Crossref]

Opt. Laser Eng. (1)

A. Elsonbaty, S. F. Hegazy, and S. S. Obayya, “Simultaneous concealment of time delay signature in chaotic nanolaser with hybrid feedback,” Opt. Laser Eng. 107, 342–351 (2018).
[Crossref]

Opt. Laser Technol. (1)

B.-C. Liu, Y.-Y. Xie, Y.-Z. Liu, Y. Wang, Y.-X. Du, W.-J. Zheng, and Y. Liu, “A novel double masking scheme for enhancing security of optical chaotic communication based on two groups of mutually asynchronous VCSELs,” Opt. Laser Technol. 107, 122–130 (2018).
[Crossref]

Opt. Lett. (7)

L. Zhang, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Common-injection-induced isolated desynchronization in delay-coupled VCSELs networks with variable-polarization optical feedback,” Opt. Lett. 44(15), 3845–3848 (2019).
[Crossref]

S.-S. Li, X.-Z. Li, and S.-C. Chan, “Chaotic time-delay signature suppression with bandwidth broadening by fiber propagation,” Opt. Lett. 43(19), 4751–4754 (2018).
[Crossref]

Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, “Bandwidth-enhanced chaos synchronization in strongly injection-locked semiconductor lasers with optical feedback,” Opt. Lett. 28(5), 319–321 (2003).
[Crossref]

A. Zhao, N. Jiang, Y. Wang, S. Liu, B. Li, and K. Qiu, “Correlated random bit generation based on common-signal-induced synchronization of wideband complex physical entropy sources,” Opt. Lett. 44(24), 5957–5960 (2019).
[Crossref]

F. Denis-le Coarer, A. Quirce, Á. Valle, L. Pesquera, M. Sciamanna, H. Thienpont, and K. Panajotov, “Polarization dynamics induced by parallel optical injection in a single-mode VCSEL,” Opt. Lett. 42(11), 2130–2133 (2017).
[Crossref]

J. Ke, L. Yi, Z. Yang, Y. Yang, Q. Zhuge, Y. Chen, and W. Hu, “32 Gb/s chaotic optical communications by deep-learning-based chaos synchronization,” Opt. Lett. 44(23), 5776–5779 (2019).
[Crossref]

P. Zhou, Q. Fang, and N. Li, “Phased-array assisted time-delay signature suppression in the optical chaos generated by an external-cavity semiconductor laser,” Opt. Lett. 45(2), 399–402 (2020).
[Crossref]

Philos. Trans. R. Soc., A (1)

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Philos. Trans. R. Soc., A 371(1999), 20120464 (2013).
[Crossref]

Photonics Res. (2)

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]

Z.-Q. Zhong, Z.-M. Wu, and G.-Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550 nm VCSEL subject to FBG feedback,” Photonics Res. 5(1), 6–10 (2017).
[Crossref]

Phys. Rev. A (1)

N. Li, H. Susanto, B. Cemlyn, I. Henning, and M. Adams, “Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers,” Phys. Rev. A 96(1), 013840 (2017).
[Crossref]

Phys. Rev. E (1)

Y. Cao, W.-w. Tung, J. Gao, V. A. Protopopescu, and L. M. Hively, “Detecting dynamical changes in time series using the permutation entropy,” Phys. Rev. E 70(4), 046217 (2004).
[Crossref]

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[Crossref]

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Figures (12)

Fig. 1.
Fig. 1. Schematic diagram of the proposed dual-channel chaos communication system with secondary encryption based on common phase-modulated electro-optic (CPMEO) feedback. DOF, dual optical feedback; VPOF, variable-polarization optical feedback; RP, rotating polarizer; M-VCSE, master VCSEL; S-VCSEL, slave VCSEL; D-VCSEL, driving VCSEL; M, mirror; OC, optical coupler; PM, phase modulator; PBS, polarization beam splitter; BS, beam splitter; FDL, fiber delay line; MZI, Mach-Zehnder interferometer; PD, photodetector; OI, optical isolator.
Fig. 2.
Fig. 2. P-I curve for free-running VCSEL.
Fig. 3.
Fig. 3. Time series (first row), RF spectrum (second row), the probability density function with the best Gaussian fit (third row) and the permutation entropy (fourth row) of $x_1 {(t)}$ (first column), $x_2 {(t)}$ (second column) and $x_3 {(t)}$ (third column) for XP output and YP output. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.
Fig. 4.
Fig. 4. The ACF(x) (first column) and DMI(x) (second column) of chaotic carrier output from DOF (first row) , VPOF (second row) and EO modules (third row). The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.
Fig. 5.
Fig. 5. Value of the peaks of $x_3$ in ACF(x) (first row) and DMI(x) (second row) for increasing the feedback gain $\beta$ at ${\tau _1} = 3\;\textrm{ns}$, ${\tau _2} = 4\;\textrm{ns}$, ${{T = 15~\rm {ns}}}$ and ${{T + }}\delta {T = 15.4}\,\textrm{ns}$ when $\Delta f= 20~\rm GHz$; The red lines correspond to the background $\textrm{Q}_{\textrm{ACF}}$ or $\textrm{Q}_{\textrm{DMI}}$.
Fig. 6.
Fig. 6. Two-dimensional maps of the max cross-correlation function for XP (the first column) and YP (the second column) output between EO-1 (i.e. $x_3$) and EO-2 (i.e. $x_6$) under: (a,b) different feedback strength $\sigma _1$ and feedback strength $\sigma _2$ of DOF (c,d) different frequency detuning $\triangle f$ and injection strength $\eta$ (e,f) different feedback strength $\sigma _3$ and polarization angle $\theta _p$ of VPOF when $\beta$ = 3.
Fig. 7.
Fig. 7. (a) The cross-correlation coefficients and (b) the synchronization error between the chaotic carriers generated by EO-1 (i.e. $x_3$) and EO-2 (i.e. $x_6$). The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.
Fig. 8.
Fig. 8. Maximum of cross-correlation for (a) XP and (b) YP with the internal mismatched parameters. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.
Fig. 9.
Fig. 9. (a) Original, (b) encrypted, and (c) recovered messages of XP and YP channel, considering 1% mismatch in $\beta$. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.
Fig. 10.
Fig. 10. Variation of BER versus the mismatch of (a) time delay $T$ (b) feedback gain $\beta$ of EO feedback loop. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.
Fig. 11.
Fig. 11. The cross-correlation coefficients between (a) the output of M-VCSEL (i.e. $x_1$) and $\rm S_2$-VCSEL (i.e. $x_5$) (b) the output of M-VCSEL (i.e. $x_1$) and EO-1 (i.e. $x_4$). The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.
Fig. 12.
Fig. 12. (a) Original and (b) recovered message of XP and YP channel when the chaotic carrier $x_1$ is directly injected into electro-optic feedback loop EO-2. The corresponding parameters are $\beta$ = 3 and $\Delta f= 20~\rm GHz$.

Tables (2)

Tables Icon

Table 1. The parameters for the CPMEO system

Tables Icon

Table 2. Comparison of the three VCSEL systems.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

d E x , y M d t = k M ( 1 + i α M ) ( N M E x , y M + i n M E y , x M E x , y M ) ( γ a M + i γ p M ) E x , y M + σ 1 , 2 E x , y M ( t τ f 1 , 2 ) exp ( i ω M τ f 1 , 2 ) + β s p ζ x , y M
d E x , y S i d t = k S i ( 1 + i α S i ) ( N S i E x , y S i + i n S i E y , x S i E x , y S i ) ( γ a S i + i γ p S i ) E x , y S i + σ 3 E x , y S i ( t τ f 3 ) c o s 2 ( θ p ) e x p ( i ω S i τ f 3 ) + σ 3 E y , x S i ( t τ f 3 ) c o s ( θ p ) s i n ( θ p ) e x p ( i ω S i τ f 3 ) + η E x , y M ( t τ i n M ) e x p ( i ω M τ i n M + i Δ ω t ) + β s p ζ x , y S i
d N M , S i d t = γ N M , S i [ N M , S i ( 1 + | E x M , S i | 2 + | E y M , S i | 2 ) + i n M , S i ( E y M , S i E x M , S i E x M , S i E y M , S i ) μ ]
d n M , S i d t = γ s M , S i n M , S i γ N M , S i [ i N M , S i ( E y M , S i E x M , S i E x M , S i E y M , S i ) + n M , S i ( | E x M , S i | 2 + | E y M , S i | 2 ) ]
X x , y s , r ( t ) + 1 ε s , r t 0 t X x , y s , r ( ξ ) d ξ + τ h s , r d d t X x , y s , r ( t ) = β s , r cos 2 [ Δ ( X x , y s , r ) T s , r + Δ ( ( φ A O ) x , y s , r ) T s , r + Δ ( m x , y ) T s , r + φ 0 ]
ϕ A O ( t ) = ρ π | E A O ( t τ i n S ) | 2 V π
R = 1 i = 1 n ( y i y i ) 2 i = 1 n ( y i y i ) 2
P E ( m ) = p ( j ) log p ( j ) / log ( m ! )
ACF ( s ) = [ x ( t + s ) x ( t ) ] [ x ( t ) x ( t ) ] ( x ( t ) x ( t ) ) 2 ( x ( t + s ) x ( t ) ) 2
DMI ( s ) = p [ x ( t ) , x ( t + s ) ] l o g p [ x ( t ) , x ( t + s ) ] p [ x ( t ) ] p [ x ( t + s ) ]
Q F ( β ) = [ P _ F ( β ) , P ¯ F ( β ) ]
P _ F ( β ) = m e a n { F ( x β ) } S D { F ( x β ) }
P ¯ F ( β ) = m e a n { F ( x β ) } + S D { F ( x β ) }
CC = [ x ( t ) x ( t ) ] [ y ( t ) y ( t ) ] [ x ( t ) x ( t ) ] 2 [ y ( t ) y ( t ) ] 2
Δ ρ = ρ r ρ s ρ s

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